Status and Future Developments in the Study of Transport Properties

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Series C: Mathematical and Physical Sciences - Vol. 361

Status and Future Developments in the Study of Transport Properties edited by

W. A. Wakeham Department of Chemical Engineering and Chemical Technology, Imperial College, London, UK

A. S. Dickinson Department of Physics, University of Newcastle, Newcastle upon Tyne, U.K.

F. R. W. McCourt Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada

and

V. Vesovic Department of Chemical Engineering and Chemical Technology, Imperial College, London, UK

Springer-Science+Business Media, B.V.

Proceedings of the NATO Advanced Research Workshop on Status and Future Developments in the Study of Transport Properties Porto Carras, Halkidiki, Greece May 29-31, 1991

Library of Congress Cataloging-in-Publication Data

ISBN 978-90-481-4125-8 ISBN 978-94-017-3076-1 (eBook) DOI 10.1007/978-94-017-3076-1

Printed on acid-free paper

All Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo­copying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Table of Contents

Preface ......................................................................... vii

Overview on Intermolecular Potentials A. van der Avoird ............................................................. 1

Traditional 'Transport Properties W. A. Wakeham and V. Vesovic .............................................. 29

Classical Path Methods for Lineshape Cross Sections J. M. Hutson ........... " .................................................... 57

Crossed Beam Studies M. Faubel .................................................................... 73

Status of Kinetic Theory F. R. W. McCourt .......................................................... 117

Overview on Experimental Data from Senftleben-Beenakker Effects and Depolarized Rayleigh Scattering

L. J. F. Hermans ............................................................ 155

Elastic and Inelastic Cross-Sections from Laser Studies of Small Molecules A. J. McCaffery ............................................................. 175

Atomic Ion/Molecular Systems L. A. Viehland .............................................................. 189

Classical and Semi-classical 'Treatment of Energy Transfer in Small Molecules G. D. Billing ................................................................ 205

Generalized Cross-Sections for Senftleben-Beenakker Effects and Laser Studies of Molecules

W.-K. Liu ................................................................... 217

From Line-Broadening to Van der Waals Molecules: Complementary Ways to Probe the Anisotropic Interaction

Ph. Brechignac ............................................................. 237

Calculation of Pressure Broadened Spectral Line Shapes Including Collisional 'Transfer of Intensity

S. Green .................................................................... 257

Concluding Remarks J. J. M. Beenakker .......................................................... 285

List of Participants ............................................................ 289

Index .......................................................................... 293

PREFACE

This volume contains the fourteen papers presented at the NATO-sponsored Ad­vanced Research Workshop on the 'Status and Future Developments in the Study of Transport Properties' held in Porto Carras, Halkidiki, Greece from May 29 to May 31, 1991. The Workshop was organised to provide a forum for the discussion among prac­titioners of the state-of-the-art in the treatment of the macroscopic, non-equilibrium properties of gases. The macroscopic quantities considered all arise as a result of the pairwise interactions of molecules in states perturbed from an equilibrium, Maxwellian distribution.

The non-equilibrium properties of gases have been studied in detail for well over a century following the formulation of the Boltzmann equation in 1872. Since then the range of phenomena amenable to experimental study has expanded greatly from the properties characteristic of a bulk, non-uniform gas, such as the viscosity and thermal conductivity, to the study of differential scattering cross-sections in molecular beams at thermal energies, to studies of spectral-line widths of individual molecules and of Van der Waals complexes and even further. The common thread linking all of these studies is found in the corresponding theory which relates them all to the potential energy function describing the interaction of pairs of molecules. Thus, accompanying the experimental development there has been a corresponding improvement in the theoretical formulation of the quantities characterising the various phenomena.

In 1973 a seminar was held at Brown University, Providence, USA to mark the centenary of the publication of the Boltzmann equation. At that time, one of the principal topics of discussion was the use of the available macroscopic information to elucidate the forces between atomic systems, most especially between the noble gas atoms helium, neon, argon, krypton and xenon. The theory of the behaviour of such systems was well understood but the means of measuring accurately the traditional transport properties such as viscosity and thermal conductivity, of studying molecular­beam scattering and making spectroscopic measurements on Van der Waals' dimers were still new. However, just about ten years later, it was possible to assert that the forces of interaction among the noble gases were essentially known, although careful work continues today to obtain the most refined atomic potentials.

Naturally, since that time, there has been a steadily increasing effort devoted to achieving the same success for polyatomic molecular systems. However, the topic has proved both richer and more complicated than might have been thought originally. Because polyatomic molecules possess internal energy in a variety of modes which can be exchanged upon collision and because intermolecular interaction is not spherically symmetric, a whole new set of macroscopic phenomena exist. Their effects are varied in character and can be studied by diverse techniques encompassing observations of Senftleben-Beenakker effects on the viscosity and thermal conductivity of a gas in a magnetic field, spectroscopic measurements on Van der Waals' complexes, laser spectroscopy of interacting small molecules, drift-tube mobility studies of atomic­ion/molecule systems, state-to-state scattering cross-sections in molecular beams and spectral line broadening. A consequence of this variety of phenomena has been a

vii

viii

divergence of direction among the workers in the field, often into different disciplines, in an attempt to circumvent the barriers to progress and, in particular, to develop new experiments to probe new phenomena. Thus, while the intermolecular pair potential has remained the common goal, it has often been, apparently, a secondary motivation.

The same complicating characteristics of polyatomic molecules make the calcu­lation of the results of their encounters very much more difficult than for atomic systems. On approximately the same timescale, developments in high-speed comput­ers and in their appplication to the treatment of molecular collision processes have rendered possible what had hitherto been impossible. That is, it is now nearly possible to evaluate routinely, for assumed intermolecular potentials, many of the quantities that characterise the observable phenomena. This Advanced Research Workshop was therefore conceived by the organizing committee to be an ideal opportunity to bring together experts from the various disparate fields to pool their experiences and ideas with a view to re-emphasising the commonality of their endeavours. Indeed it was thought that it might even be possible to agree upon a set of representative systems to be studied by all techniques. To this end each of the papers presented was intended to provide the stimulus for lively debate - and so it proved.

As the concluding remarks of Professor J.J.M. Beenakker make clear, the work­shop confirmed that we are now at a turning point in the field of molecular collision processes when much of the experimentation is complete and we must await the in­terpretation of the results. It seems, though, that each experimental technique has its own optimum molecular systems so that there is less complementary information available than one would wish. Nevertheless, as the papers contained in this vol­ume demonstrate, the immediate outlook for an understanding of the interaction of relatively small molecular systems is quite positive.

The Advanced Research Workshop therefore accomplished its aims, and all of the participants are indebted to NATO for the financial support that made it possible. The success of the event was the result of the combined efforts of a number of groups of people. Firstly, there are the speakers, who provided thought-provoking oral pre­sentations at the meeting and so promptly delivered written versions of their material. Secondly, there are the other participants who contributed to stimulating and valu­able discussions. Finally, the management and staff of the Village Inn, Porto Carras also deserve the thanks of the participants for their warm, efficient hospitality.

Special thanks are due to the members of the organising committee: Alan Dickin­son, Fred McCourt, and particularly Marc Assael and Velisa Vesovic who made it all possible.

Imperial College London September 1991

William A. Wakeham

OVERVIEW ON INTERMOLECULAR POTENTIALS

A. VAN DER AVOIRD Institute of Theoretical Chemistry University of Nijmegen 6525 ED Nijmegen The Netherlands

ABSTRACT. The various types of contributions to the interaction between two molecules and their representations are discussed. In particular the employ­ment of spherical harmonic and atom-atom (or site-site) expansions, as well as other analytic representations, such as the generalized Lennard-Jones form, is described. A brief description of spin-dependent potentials, which arise when open-shell molecules interact, is also given. The question of additivity of inter­molecular potentials is addressed, and the importance of many-body interac­tions, especially for liquid and solid-state properties, is considered. An overview is presented of recent ab initio calculations of interaction potentials for simple molecular systems, such as N2 , O2 , CO, H2 , Ar-H20, and Ar-NH3 . Finally, the status of intermolecular potentials is presented, and illustrated by means of their applications to the calculation of second virial coefficients, Van der Waals spectra, and the properties of molecular solids, including lattice dynamics. It is concluded that for the near future the best multi-dimensional intermolecular potentials will likely be obtained by combining the results of ab initio calcu­lations with the fitting of a limited number of variable parameters to accurate experimental data.

1. Introduction

As will be substantiated in this workshop, the knowledge of intermolecular potentials opens the way to (the calculation of) many observable properties, for microscopic as well as macroscopic systems. In the first category are thermodynamic stability, the spectra of Van der Waals molecules [1-4], and molecular beam scattering cross sections [5-7], elastic or inelastic state-to-state, total or differential. In the second category are various bulk gas and condensed matter properties. Measured gas phase properties [8,9] which depend directly on the intermolecular potential are virial coef­ficients, viscosity and diffusion coefficients, thermal conductivity, sound absorption, pressure broadening of spectral lines, nuclear magnetic relaxation and depolarized Rayleigh scattering. Additional information is obtained from the effects of electric and magnetic fields on the transport properties (Senftleben-Beenakker effects). In the condensed phases one may calculate (by liquid state theory) or simulate (by Monte Carlo or Molecular Dynamics methods) the behaviour of liquids [10]' or study the

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W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 1-28. ©1992 Kluwer Academic Publishers.

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stability and lattice vibrations of molecular solids [11]. On the other hand, all the measured data may be used, and have actually been used in several examples, to construct or improve (semi- ) empirical intermolecular potentials.

Several reviews on intermolecular potentials have appeared during the past five years [2,3,12-16]' and hence I shall simply outline the most important points. In­teractions between molecules are usually divided into long-range interactions and short-range interactions. At long range, i.e. when the charge clouds of the interacting molecules do not overlap, the interaction energy can be obtained formally by standard Rayleigh-Schrodinger perturbation theory. The perturbation, which is the intermolec­ular interaction operator, can be expanded as a multipole series in powers of R- 1 ,

where R is the distance between the centers-of-mass of the molecules. The first-order energy is the electrostatic multipole-multipole interaction energy. The second-order energy contains the induction (multipole-induced multipole) energy and the (non­classical) dispersion energy. For molecular ions the electrostatic and induction in­teractions are strongly dominant. For polar, e.g. hydrogen bonded, molecules the electrostatic interactions are still the most important contribution, while the induc­tion and dispersion energies are comparable. For apolar molecules, i.e. molecules with small dipole moments, the dispersion energy becomes the most important (attractive) long range interaction. The long range interactions are completely determined by the permanent multipole moments and the, static as well as frequency-dependent, multi­pole polarizabilities of the monomers.

Since the molecular charge clouds have exponential tails, there is always some overlap between them. The effects of this overlap are twofold. Penetration causes the exact electrostatic interaction between continuous, overlapping charge clouds to deviate from its representation by a multipole series. This is correctly included in the Rayleigh-Schrodinger perturbation theory if one avoids the expansion of the electro­static interaction operator [14]. Not included in the standard perturbation theory are the exchange effects, which arise from the antisymmetrization of the overall electronic wave functions, as required by the Pauli postulate. Both penetration and exchange effects modify the interaction energy in all orders of perturbation theory.

Most of the current work on intermolecular interaction potentials is concerned with closed-shell molecules, but it is worth noting that interactions between open-shell molecules are especially interesting. As a direct consequence of the relation between the spin and the permutation symmetry of electronic wave functions [17], different cou­plings between the non-zero spin states of interacting open-shell monomers will lead to different exchange interactions. In other words interacting open-shell molecules possess a manifold of intermolecular potential energy surfaces, one surface for each total spin state. The splitting between these surfaces is caused by exchange interac­tions. Some of these potential surfaces may correspond to chemical bonding, in the same way that it occurs between open-shell atoms. A very weak bond of this type seems to be present [18,19] in the singlet state of (NOh. In the (02h dimer, on the other hand, the singlet, triplet and quintet state all show a net, although different, exchange repulsion [20] between the triplet O2 molecules. This case provides a very interesting system in which the Van der Waals interaction potential is spin-dependent.

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2. Representation of intermolecular potentials

For most applications it is practical to write the intermolecular potential in analytic form. This allows easy calculation for many different distances and orientations of the molecules, as required in Monte Carlo simulations, for example. Specific forms will be convenient for scattering, liquid state or lattice dynamics calculations. Moreover, if the potential is to be improved through such studies, this form must contain a limited number of variable parameters. In practice two basic forms can be recognized, a spherical expansion and an (isotropic) atom-atom or site-site potential.

2.1. THE SPHERICAL EXPANSION

In this expansion the orientational dependence (anisotropy) of the intermolecu­lar potential between two arbitrary, non-linear, molecules A and B is explicitly

expressed in (symmetric top) free-rotor functions, D<t:kA (nA )* for molecule A

and D<t:kB (UB)* for molecule B [21J. The Euler angles n A = (GA' ,BA,"'A) and U B = (GB,,BB,'YB) describe the orientations of the molecules with respect to an arbitrary space-fixed (SF) coordinate frame. The direction of the vector R, which connects the center-of-mass of molecule A to that of molecule B, is given with respect to the SF frame by the polar angles (e, cf> ) • The intermolecular potential can be expressed as [1]

(1)

in terms of the complete orthogonal set of angular functions

(2)

The functions c~)(e, cf» are spherical harmonics in the Racah normalization and the expression in large round brackets is a 3-j symboI[21J. The expansion coeffi­cients Vf:t:L(R, qA, qB) depend upon both the distance R between the molecules and the internal molecular coordinates, symbolized by qA and qB. Any intermolecu­lar pair potential can be expressed to any desired accuracy when a sufficient number of these expansion coefficients is given. For molecules with well-defined equilibrium structures (sometimes called nearly-rigid molecules) it is convenient to make a Taylor expansion of the expansion coefficients about the equilibrium values of the internal coordinates qA and qB. In practice this has been done only for very simple sys­tems [22]. Most often the rigid-molecule approximation, in which qA and qB are

4

replaced by their equilibrium values, or a Born-Oppenheimer separation between the internal and external molecular coordinates, in which qA and qB are replaced by their vibrationally-averaged values [23] is assumed.

All long-range contributions to the interaction potential defined in the intro­duction depend upon the distance R as power series in R-1 . When the multipole

operators QV;:) and QV;:) are defined as tensors (including the summation over all particles i with charges Zi) with spherical components

(3)

then the first order electrostatic interactions automatically adopt the form of Eq. (1), in which the expansion coefficients are

Further, the second order induction contribution can also be expressed in the form of Eq. (1) with, for the polarization of A,

(5) and an analogous expression for the polarization of B. Finally, the dispersion contri­bution is given by the generalized Casimir-Polder formula

(6)

in which the frequency-dependent coupled polarizabilities are defined by

(7)

5

(I I')L and an analogous expression for aK~ B B(W). The symbol (l,m;I',m'IL,K) denotes a Clebsch-Gordan coefficient, and the purely algebraic coefficients ( are given [24] by

LALBL _(_ )IA+I;" [(2lA + 2lB + 1)!(21~ + 21~ + I)!] 1/2 (IAI:"IBI~ - 1 (2IA)!(2l~)!(2IB)!(21~)!

X [(2LA + 1)(2LB + 1)(2L + 1)]1/2 (lA + IB' 0; l~ + l~, OIL, 0) (8)

l' A

l' R

l~ + l~ with the expression between curly braces being a Wigner 9-j symbol [21]. The short­range overlap (penetration and exchange) contributions to the expansion coefficients Vf:t.BL (R, qA, qB) depend exponentially on R.

In specific cases and with a specific choice of the coordinate frame the general form of the spherical expansion can be considerably simplified. If we choose a body­fixed (BF) frame with the z-axis along the vector R, we may employ the result

0;;)(0,0) = 8M,0' For linear molecules the free-rotor functions are simply spherical harmonics [21],

(L) * _ (L) _ 47r (L) [ ]1/2

DMO(a, (3,"f) -OM ((3,a)- 2L+1 YM ((3,a) , (9)

while, if either A or B is an S-state atom, we have 060 ) ((3, a) = 1. Thus one obtains, for instance, the well known Legendre expansion for an atom-diatom potential

V(R,qA,(3A) = LVLA(R,qA) hA (cos (3A) (10) LA

or, more generally, an expansion in spherical harmonics for an atom-molecule potential

V(R, qA, (3A, "fA) = L VLAKA (R, qA) o)fAA) ((3A, "fA)' (11) LAKA

2.2. ATOM-ATOM OR SITE-SITE POTENTIALS

A widespread approximation of the intermolecular potential between two molecules A and B is to write it as a sum of atom-atom potentials, which are assumed to be isotropic, i.e. to depend only upon the distance Rab between the atoms a in molecule A and the atoms b in molecule B

VAB = L L vab(Rab). (12) at=A bt=B

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The atom-atom potential functions are most commonly written as

vab(Rab ) = aab exp( -babRab) - CabR-;;b6 + QaQbR-;;b1, (13)

with the first term representing the exchange repulsion, the second term the dis­persion attraction, and the last term the electrostatic interactions between effective (fractional) atomic charges Qa and Qb. In the atom-atom Lennard-Jones model the exponential function is replaced by R-;;b12 . The parameters aab, bab, Cab and the ef­fective charges qa and qb can be obtained [25-27] by fitting the potential (12) to an ab initio potential VAB calculated for various distances R and molecular orientations, or they can be determined empirically [28]. In the latter case the potential (12) is used to calculate various measured properties, and the parameters in Eq. (13) are optimized to obtain a best fit to these properties.

The reason that atom-atom potentials are so popular, especially in the study of condensed phases [28] and more complex Van der Waals molecules [29], is that they contain few parameters, while they still describe (implicitly) the anisotropy of the intermolecular potential and they even model its dependence on the internal molecular coordinates. Moreover, they are often believed to be transferable, which implies that the same atom-atom interaction parameters in Eq. (13) can be used for the same types of atoms in different molecules. One should realize, however, that the accuracy of atom-atom potentials is limited by their form (12). Additional inaccuracies are introduced when the atom-atom interaction parameters in Eq. (13) are transferred from one molecular environment to another. Further, Eq. (13) does not include a term which represents the induction interactions, and there is the intrinsic problem that these interactions are inherently not pairwise additive (see below). Numerical experimentation on the C2 Hc C2 H4 and N2-N2 potentials, for example, has taught us [25-27] that even when sufficient ab initio data are available, so that the terms in Eq. (13) can be fitted individually to the corresponding ab initio contributions and, moreover, the positions of the force centers for each term can be optimized, the average error in the best fit of each contribution still remains about 10%. Since the different contributions to the potential partly cancel each other, and the errors for specific molecular orientations are considerably larger, it will be clear that even atom­atom potentials with shifted force centers (called site-site potentials) are of limited value for the description of the detailed anisotropy of the intermolecular potential surface. In some cases one has tried to overcome these problems by including several point charges per atom for the electrostatic interactions and by adopting atomic or bond polarizabilities for the description of induction forces [30,31].

2.3. OTHER ANALYTIC POTENTIALS, RELATIONS

Besides the two basic forms just described, which are employed most frequently in the literature, other expressions for intermolecular potentials have been proposed. Some potentials are given, for instance, by the generalized Lennard-Jones type [32]

[( R )-12 (R )-6] VAB(R, fl) = 4E(fl) p(fl) - p(fl) (14)

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or by exponential functions for the short-range repulsion [33]

(15)

in which the well depth c:(!l) and the non-linear 'range pararpeters' p(!l) and 'slope parameters' a( fl) are assumed to be functions of the molecular orientations !l A and !lB. Such functions may be expanded [33] in terms of the angular basis functions of Eq. (2), which then occur non-linearly in the potential. An advantage of this procedure is [33] that these expansions need fewer terms.

Another way of expressing the intermolecular potential has been advocated by Stone et al. [33,34]. This form, which is intermediate between the two basic forms, involves dividing the molecules into segments, which may be individual atoms or groups of atoms; the inaccuracy inherent in the isotropic atom-atom model is avoided by assuming anisotropic potentials, which have the same form as Eq. (1), between the segments. The expansion in Eq. (1) should converge more rapidly for the segment interactions than it does for the interactions between whole molecules. In particular it has been illustrated [34] that the long range multipole interaction series converges for smaller distances and with fewer multipole moments for segments than for molecules. The arbitrariness in dividing the multipole moments and multipole polarizabilities over the segments can be used to improve this convergence. Such ideas will be partic­ularly useful for larger molecules. The transferability of segment-segment interaction potentials and their extension to induction and dispersion forces are still open to investigation.

Atom-atom or segment-segment potentials, and potentials of the form (14) or (15), usually contain fewer terms and fewer parameters than does the spherical ex­pansion of the intermolecular potential given by Eq. (1). This is useful if the param­eters have to be varied, for instance, to improve agreement with experimental data. On the other hand, the spherical expansion is convenient for several applications of the potential, such as scattering or lattice dynamics calculations. It is always pos­sible [IJ to calculate the spherical expansion coefficients, see Eq. (1), of any known intermolecular potential by numerical quadrature methods. Atom-atom potentials of the form of Eq. (13) can also be transformed analytically into a spherical expansion; the formulas required are given in Ref. [11]. Also segment-segment potentials of the type of Eq. (1) can be analytically converted to a molecular expansion of the same type [33,34].

2.4. OPEN-SHELL MOLECULES: SPIN-DEPENDENT POTENTIALS

The exchange interactions between open-shell molecules, which lead to different po­tential energy surfaces for different spin-couplings between the molecules (see the introduction), can be represented in the form of a Heisenberg hamiltonian [20]

vspin-dependent(R, qA, qB, !lA, !lB, SA, SB) = -2J(R, qA, qB, !lA, !lB) SA . SB, (16)

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where SA and S B are the total electron spin operators of the interacting monomers A and B. Higher terms in the scalar product (SA' S B) will be needed in general [17,35] when the overlap between the molecular charge clouds becomes appreciable, i.e. for distances much closer than the Van der Waals minimum. The Heisenberg coupling parameter J is a scalar function, as is the spin-independent potential, and it can be expanded via Eq. (1). In the case which has been investigated in our group, the OT O2 potential, J appears to be extremely dependent on the molecular orientations, so that rather high terms in Eq. (1) are important. Even the sign of J changes, i.e. the exchange coupling between the triplet O2 molecules alternates between ferromagnetic and antiferromagnetic, when the molecules are rotated about certain axes. This is related to the nodal planes of the antibonding 7r g molecular orbitals which form the open shell. Other terms which may contribute to the spin-dependent potentials between open-shell molecules are spin-orbit and spin-spin interactions, intra- as well as inter-molecular [36]. Such terms are anisotropic in two respects: they depend on the orientations of the molecular axes attached to the nuclear framework, and on the orientations of the molecular spin momenta SA and S B. Only when the intramolecular spin-orbit and spin-spin couplings are much stronger than the intermolecular exchange interactions are the monomer spin momenta fixed with respect to the molecular axes; in general, and for O2-02 in particular, they are not. All the spin-dependent terms in the O2-02 potential have been expressed analytically in Refs. [20] and [36].

3. Additivity of intermolecular potentials

So far we have been concerned only with the interaction potential for a pair of molecules A and B. In most applications it is assumed that the total potential energy can be written as a sum of pairwise intermolecular potentials. In the gas phase, where simultaneous collisions of more than two molecules are relatively rare, this approxi­mation is fairly realistic for most properties. In the condensed phases, however, each molecule is surrounded by several others at Van der Waals distances; this makes it especially important to consider the influence of three- and more-body interactions. This question can be addressed most clearly by considering the basic interaction mechanisms. The first-order electrostatic interactions are exactly pairwise additive, whether they are calculated in the multipole expansion or between continuous charge clouds. The first-order exchange effects are not pairwise additive. The dominant three-body exchange contributions are roughly proportional to the overlap product SA B S BC SAC, while the dominant pairwise exchange energy between three molecules A, Band C increases with S~B + S1c + S~c' For Van der Waals distances between the molecules the overlap integrals S AB, S BC and SAC are of the order of a few hun­dredths, and so the first order exchange non-additivity is also of the order of a few percent. At high pressures this non-additivity will be more important, however, since the overlap integrals increase rapidly with decreasing intermolecular distances.

The second order dispersion interaction is exactly pairwise additive when it is calculated (as is usually done) in the multipole approximation, but not when pen­etration and exchange are included. The second-order penetration and exchange

9

effects are of minor importance, however, and so in general they do not playa major role in the non-additivity. An important non-additive interaction between non-polar molecules is the third-order dispersion energy, the dominant term of which is called the Axilrod-Teller or triple-dipole dispersion interaction [37]. This long range term, which is proportional to RA1RB~RA~' and the first-order exchange term, which is the dominant short-range three-body interaction, have been studied in detail in rare gas solids [38-40]. At normal pressures the three-body contributions to the cohesion energy are estimated to be of the order of 10%; at high pressures they may be more important. For specific distances (pressures) the Axilrod-Teller and the first-order exchange three-body interactions may nearly cancel. In that case the non-additivity is determined by the smaller second-order exchange effects and by third-order contri­butions from higher multipole operators [41]. The review by Meath and Koulis [15] devotes an extensive discussion to these points.

The interaction contribution which lacks pairwise additivity completely is the second-order induction energy. This is easily understood, since the energy of po­larization of a molecule A in the field of two other molecules Band C is given by -~aAF2 = -~aA(FBA + FeA ) . (FBA + FeA ), where aA is the polarizability of molecule A and FBA and FeA are the electric field vectors at the center of molecule A, originating from the (permanent and induced) multipole moments of the molecules Band C. The three-body part of this interaction energy, -aAFBA . FeA , is generally of magnitude similar to that of the pairwise contribution, -~aA(F~A + F6A). In systems with polar molecules and, a fortiori, in ionic systems where the induction energy is relatively important, the total interaction energy is therefore highly non­additive. The second-order induction energy is mainly responsible for this , but the higher-order induction contributions are important too. These can be understood as the (non-additive) polarization energies of the molecules in the fields produced by induced moments on other molecules. The induced moments are proportional to the fields at these molecules, which again originate from permanent or induced multipole moments on other molecules, and so on. A fully self-consistent treatment of the local fields and the induced moments on all the molecules in the system, which is related to the dielectric theory of macroscopic bodies [42J, is necessary to include all these effects. Such a theory should also take into account the higher multipole moments and field gradients.

In practice it is often assumed in calculations or simulations, even of ionic sys­tems, that the interaction energy is pairwise additive. Some justification of this assumption may be given by the fact that in bulk systems the molecules are more or less symmetrically surrounded, so that the local fields at the molecular sites partly cancel and the induced (lower) multipole moments are small (considerably smaller than expected by considering the fields from individual neighbours). However, even in the most perfect crystal of high symmetry this holds only when all the molecules occupy their equilibrium positions, but not when they are (dynamically) displaced. Simplified models have been devised to account for (non-additive) polarization effects in dynamical calculations, such as the shell model for ionic crystals [43].

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4. Obtaining intermolecular potentials from ab initio calculations

The same experimental data listed in the introduction can be used, and have been used in practice, to obtain (semi-)empirical intermolecular potentials. Except for atomic systems and very simple atom-diatom complexes, the number of degrees of freedom, and the corresponding number of parameters needed to describe the long- and short­range anisotropic contributions, is too large, however, to obtain a unique inversion or fit of the experimental data. Hence it is very useful to have supplementary information provided by quantum chemical computations. The best potentials available, even for atomic systems, are at least partly determined by information obtained from such ab initio calculations.

Especially the calculation of the coefficients in the asymptotic R-n expansion of the potential is relatively easy, since these coefficients are completely determined by monomer properties. Explicit formulas for the most general case have been given in section 2, Eqs. (4) to (8). The computation of molecular multipole moments is straightforward, both at the Self-Consistent Field (SCF) level and with the inclusion of electron correlation effects. The calculation of accurate frequency-dependent po­larizabilities is still not a routine job, however. One may use the Time-Dependent Coupled Hartree-Fock method [44] and include correlation corrections by Many-Body Perturbation Theory (MBPT) [45]. Large and flexible basis sets are required, espe­cially to obtain the higher multipolc polarizabilities. Wormer and coworkers [44-49] have thus obtained a collection of ab initio data for a series of small molecules (in­cluding open-shell systems), which can be used to generate the anisotropic induction and dispersion coefficients for all combinations of these molecules. Values for the isotropic part of the coefficient C6 in the leading term of the dispersion interactions are available also from empirical dipole oscillator strength distributions [50].

For the ab initio computation of the short-range behaviour of the expansion coefficients Vf:f:BBL that determine the anisotropic potential a numerical integration procedure has been devised in our group [1,25]. This procedure is based on the fact that the expansion coefficients can be written as

where the integration has to be performed over the angles nA = (aA,,BA,I'A), n B = (aB,,BB,I'B) and (8,<I». Actually, the angles 8,<I> and aA or aB can be con­sidered as external angles and hence fixed at some arbitrary value. This integral may be evaluated by Gaussian quadrature, so that it requires values of the poten­tial V(R,qA,qB,flA,flB ) on a grid of quadrature points for flA and fl B . Those values, which are the short range interaction energies contained in Heitler-London or SCF-type formulas, for instance, can be obtained from ab initio calculations. This procedure has been applied to linear molecules (the N2-N2 [25,51]' O2-02 [20] and CO-CO [52] potentials) and to atom-molecule systems such as Ar-NH3 and

11

0 0 c-o 0 0

if> o-c c-o I I I I 0 0 u u

0 0 0 0 c-o c-o c-o c-o I I I I

u 0 u 0

2.0

1.0

............ 0 E "- 0.0 J ~ '--"

> -1.0

6 7 8 9 10 11 12

R (bohr) Figure 1. Ab initio calculated CO-CO interaction potential for different orientations of the monomers [52].

Ar-H20 [53,54). For dimers of non-linear molecules it is still too expensive. The anisotropy of the CO-CO potential is illustrated in Figs. 1 and 2.

The effects of charge penetration and exchange on the first- and second-order interaction energies can be written explicitly by means of symmetry-adapted pertur­bation theory [55,56]. Calculations of the resulting expressions have been performed for very small systems (He, H2 ) [14]. In practice these effects become important near the Van der Waals minimum, and they are mostly calculated in first order only, by means of a Heitler-London type formula, which already gives the dominant exchange repulsion. Even then, the inclusion of the intramolecular electron correlation in such a calculation is not easy [14,57]. The second-order penetration and exchange effects, which are much less important than the first-order exchange repulsion, are taken into account mainly by the use of (semi-empirical) damping functions [58-60].

12

180

160

140

120

,...... (/J

Q) 100 Q) L... (J'> Q)

80 'U '--'

-< c:::> 60

40

20

160 180

Figure 2. Orientational dependence of the Van der Waals well depth (in kJjmol) in the CO-CO potential [52] for planar geometries (ip = 0°). Orientations without a well (cf. Fig. 1) are found in the upper left hand corner.

Most popular in the ab initio calculation of intermolecular potentials is the so­called supermolecule method, because it allows the use of standard computer pro­grams for electronic structure calculations. This method automatically includes all the electrostatic, penetration and exchange effects. If the calculations are performed at the SCF level, induction effects are also included, but the dispersion energy is not. The latter, which is an intermolecular electron correlation effect, can be obtained by configuration interaction (CI), coupled cluster (CC) methods or many-body pertur­bation theory (MBPT). These calculations are all plagued by basis-set superposition errors (BSSE) [2,3,12-14], however, which are mostly of the same magnitude as the intermolecular interaction energy. For small molecules these errors can now be nearly avoided at the SCF level, but not when the electron correlation is included. More­over, the truncated CI methods, which are the most generally applicable, suffer from a lack of size-consistency [13,14]. Further experimentation with these methods will be required in order to ensure that they will produce reliable potentials. In principle

13

Table 1. Second virial coefficients (in cm3 mol- 1)

of co-co from Ref. [52].

T(K) Belas (1)

Bquant B tot Bexptl

77.3 -319.74 11.36 -308.38 -307.0 90.1 -234.53 6.73 -227.80 -230.0

143.0 -93.11 1.79 -91.34 -92.0 173.0 -61.05 1.12 -59.94 -62.0 213.0 -35.35 0.70 -34.65 -35.0 242.0 -23.08 0.53 -22.54 -22.8 263.0 -16.21 0.46 -15.75 -16.0 273.0 -13.38 0.42 -12.96 -13.0 298.1 -7.27 0.36 -6.91 -8.0 323.2 -2.28 0.31 -1.97 -3.7 348.2 1.85 0.27 2.12 1.1 373.1 5.32 0.24 5.55 4.6 398.1 8.28 0.22 8.49 7.7 423.2 10.85 0.19 11.04 9.6 473.2 15.00 0.15 15.15 14.5 513.2 17.63 0.13 17.76 17.3 573.2 20.75 0.11 20.87 20.5

they are very attractive because they yield complete interaction potentials over the full range of distances.

5. Status of intermolecular potentials, illustrations, applications

In the spirit of the Class I to IV distinction of intermolecular potentials made by Maitland et al. [9] we make a similar classification here. Our criteria are slightly dif­ferent, however. In Class I are the potentials which are considered to be accurately known, as verified on many different observable properties. Examples in this class are the pure rare gas (Rg-Rg) and some of the mixed rare gas (Rg-Rg') potentials. Class II contains the potentials of other atomic and the simplest molecular systems which are probably fairly accurate too, but which have been tested only on a limited set of properties. Examples are the Rg-H2 potentials developed by Le Roy and cowork­ers [22,23]' mainly on the basis of dimer infrared spectra and scattering data, the rare gas - hydrogen halide (Rg-HX) potentials of Hutson et al. [3]' which are deduced from the microwave and infrared spectra of Van der Waals dimers in molecular beams, and, possibly, the H2-H2 potential of Schaefer et al. [61,62]' which is primarily obtained from ab initio calculations, but somewhat improved empirically. Practically all the potentials in this category contain ab initio parameters, especially in their long range

14

part. In Class III we list potentials, mostly from ab initio calculations, which are avail­able as a full anisotropic surface (represented by a spherical expansion), but which are certainly still amenable to improvement. Among this class are the potentials of spe­cific diatom-diatom systems, such as N2-N2 [25,51]' HF-HF [63-65]' CO-CO [52] and O2-02 [20], and certain atom-molecule dimers such as Ar-NH3 [53] and Ar-H2 0 [54]. In the case of Ar-NH3 even one of the internal molecular coordinates has been var­ied [53], viz. the NH3 umbrella angle, in order to study the effects of the /.12 umbrella vibration and the inversion-tunneling of the NH3 monomer in the complex. A semi­empirical potential of this type has been obtained for the Ar-H20 dimer by Cohen and Saykally [66]' from a fit to the far-infrared spectrum. Other fits to such spectra for Ar-NH3 [67] and Ar-H20 [68] so far have yielded only effective angular potentials, which do not depend on the coordinate R. In Class IV we put the more general molecule-molecule potentials, which are known only through crude atom-atom mod­els, sometimes empirical [28], sometimes fitted to ab initio results (as in H2 0-H2 0 [69], NH3-NH3 [70] and C2Hc C2H4 [26]) or through fragmentary ab initio calculations for a limited region of the potential surface (in search for the equilibrium structure, for example, as in C2H2-C2H2 , C2Hc C2H4 [71] and C6H6-C6H6 [72]).

The examples presented are not exhaustive, and they serve merely as illustrations. In the sequel to this section we shall demonstrate some applications of ab initio potentials obtained in our group, and discuss their merits as well as shortcomings. We concentrate on Class III examples, because we feel that in this category most profit can be gained from the cooperation between theorists and experimentalists.

5.1. SECOND VIRIAL COEFFICIENTS

In Table 1 and Fig. 3 we show the second virial coefficients of CO-CO, as calculated with an ab initio potential [52]. At the lowest temperatures the quantum corrections are not negligible. The effect of the anisotropy in the potential is rather important. If we include all these effects, the results lie within the experimental error bars over the whole temperature range from 77 to 573 K. This must be somewhat fortuitous, how­ever, since the potential has been calculated with nearly the same approximations as the N2-N2 potential [51], and the latter required some scaling in order to fit the mea­sured virial coefficients. Figure 4 displays the ab initio calculated [54] and measured virial coefficients of Ar-H2 0. Also here the calculations lie within the experimental error bars. These are considerably wider, however, (as is usual for mixed systems) and the temperature range is much smaller. It is noticeable that the anisotropy is substantially smaller in this case, and that the mixed virial coefficients of Ar-H20 are much closer to the pure Ar-Ar coefficients than to the pure H2 0-H2 0 curve (not shown). Thus, the Ar-H20 interaction potential resembles more the pure Van der Waals potential of Ar-Ar than it does the electrostatically dominated H20-H2 0 potential with its strongly anisotropic hydrogen-bonding features.

50 r-----------------------------------~

o

'j -50

(5 E -100

"'E -150 u

'--" -200 ........

I-m -250

-300

..................................... ---_ .. ---_ ... -_ ....

-3505~O--~~15~O~~-2~5~O~~-3~5~O--~-4~5~O--~-5-5LO~

T (K)

15

Figure 3. Second virial coefficients of CO. The experimental data [73,74] are indi­cated with (estimated) error bars, the ab initio calculations [52] are represented by the closed curve. The dashed curve is the second virial coefficient calculated with the isotropic potential.

20

10

I..., 0 0 ~ I '" Ar-Ar ~ -10 () I '-"

----- -20 I h '-" I:Q

-30

-40 250 300 350 400 450

T(K)

Figure 4. Comparison of the ab initio virial coefficients [54] for Ar-H2 0 (total: -----, isotropic: - - - - -) with experimental values (*: Ref. [75], 0: Ref. [76]) including error bars. The experimental virial coefficients for pure Ar are shown also (0: Ref. [77]).

16

Table 2. Phonon frequencies in a-N2 (in cm-1 ) from Ref. [90J.

Experiment Semi-empirical Ab initio harmonic RPA

Lattice constant a(A) 5.644 5.644 5.699

r (0,0,0)

{ Eg 32.3 37.5 31.0 Librations Tg 36.3 47.7 41.0

Tg 59.7 75.2 68.0

Au 46.8 45.9 47.2

T,=l.tio=l { Tu 48.4 47.7 48.8 vibrations Eu 54.0 54.0 55.6

Tu 69.4 69.5 73.1 M (2!: 2!: 0)

a' a'

{ M12 27.8 29.6 27.6 M12 37.9 40.6 39.1

Mixed M12 46.8 51.8 50.2 M12 54.9 59.0 59.1 M12 62.5 66.4 66.5

R(~,~,~) R- 33.9 34.4 34.4 'fr~lation.l { 1

vibrations R 23 34.7 35.7 35.8 R 23 68.6 68.3 72.3

{ R+ 43.6 50.7 47.9 Librations 1

Rt3 47.2 57.8 50.8

rms deviation of librational frequencies 10.6 5.0

rms deviation of translational frequencies 0.6 2.1

rms deviation of all lattice frequencies 6.1 3.4

5.2. SPECTRA OF VAN DER WAALS MOLECULES

After the ab initio computation of the potentials for Ar-NH3 and Ar-H2 0 in the form of Eq. (11), we have applied them to the calculation of the microwave and far­infrared spectra of Ar-NH3 [67,78-82], Ar-H20 [83,84J and Ar-D20 [82,85J. For the latter system we had first to transform the Ar-H20 potential to a different center of mass. We have calculated the rovibrational states of these complexes for J S; 15, and generated the infrared intensities of all the allowed P, Q and R branch transitions from

17

20.0

-.... 15.0

~

! f).0 '-c.t

5.0

0.0 0.0 5.0 f).0 %5.0 20.0

r(K)

Figure 5. Specific heat of a-nitrogen. The dots represent the experimental val­ues [95], the curves have been calculated in Ref. [92].

Table 3. Optical phonon frequencies (in cm- 1) in solid a-CO from Ref. [27].

Experiment Spherical Site-site expansion potential

Raman 38 35.8 (E) 33.0 (E)

Raman (strong) 44 45.5 (T) { 44.1 (Ttrans) 44.2 (7iong)

Raman 58 56.4 (A) 62.1 (A)

infrared, Raman 49, 50.5, 52 62.0 (T) { 57.0 (Ttrans) 57.2 (l1ong)

Raman 64.5 74.4 (E) 76.6 (E)

infrared 85, 86 87.2 (T) { 90.6 (Ttrans) 90.7 (7iong)

Raman (strong) 90.5 98.3 (T) {119.1 (Ttrans) 119.1 (7iong)

18

Table 4. Macroscopic properties of a-N 2 from Refs. [51,92].

Experiment Calculated (ab initio)

Cll 29.0 28.5±0.6

Elastic constant C12 20.0 22.0±0.6

(kbar)

C44 13.5 13.3±0.6

Compressibility "'T 4.6 4.69 (10- lOm 2N- 1 )

a dipole-surface. The inversion-tunneling of NH3 was originally included by a simple two-state model, but presently we have extended our formalism and computer codes with a basis for the NH3 umbrella coordinate, so that we can include explicitly the V2

vibration and the inversion-tunneling of the NH3 monomer in the Ar-NH3 complex. The results are extensively described in recent papers by Van Bladel et al. [86,87] and compared in detail with experimental data [78-85]. Since the topic of Van der Waals molecules will be covered in this workshop by Hutson and by Brechignac, I have limited myself here to a summary of the main conclusions. It has been found that, indeed, the spectra of these Van der Waals complexes are extremely sensitive to the shape of the potential surface in the entire attractive region. The anisotropy in the Ar-NH3 and Ar-H2 0 potentials is of the same order as the rotational constants of NH3 and H2 0, so that the Van der Waals 'bending vibrations' look more like perturbed internal rotations. Several (small) anisotropic terms in the spherical expansion of the ab initio potential, Eq. (11), have a strong effect on the splittings of the internal rotor states. It was not possible to reproduce the experimental transition frequencies without some scaling of these terms.

5.3. MOLECULAR SOLIDS

In theoretical studies of molecular solids it was common to use rather crude atom-atom potentials. The parameters in these potentials were usually optimized to fit the ex­perimental data: the lattice structures, cohesion energies and, sometimes, phonon fre­quencies. For the simple molecular crystals N2 [88-92]' CO [27], H2 [93] and O2 [36,94]' lattice dynamics calculations have now been performed, however, which use ab initio calculated intermolecular potentials. The anisotropy of these potentials is directly represented by the spherical expansion (see section 2), although site-site models have been given too [25,27]. The lattice dynamics method used in most of these studies

19

Figure 6. Phonon dispersion curves for ortho-D2 (normal pressure hcp phase) cal­culated [93) at T = 0 K. The circles, squares, etc., are neutron scattering data [96,97) at T = 4.2 and 5 K.

Table 5. Static lattice energy (in kcaljmol) of solid a-CO from Ref. [27).

Lattice constant a (A) Exchange Dispersion Electrostatic Total

Spherical expansion

5.658 1.945

-3.673 -0.536 -2.265

Site-site Experiment

potential

5.628 1.933

-3.804 -0.557 -2.428

5.646

-2.480

20

Table 6. Optical (q = 0) libron and magnon frequencies in a­and (3-02 from Ref. [36].

Librons Experiment

a-02: Bg 42.6 cm- 1

Ag 74.2 (3-0 2 : Eg 48.0

Calculated (ab initio)

Including Neglecting Heisenberg term Heisenberg term

39.9 cm-1

72.2 53.6

38.9 cm- 1

50.7 42.9

Magnons Experiment Calculated (ab initio)

6.4 cm-1

27.5 6.7 cm- 1

22.2

is not the standard harmonic method. A Time-Dependent Hartree (TDH) method which can handle larger amplitude (anharmonic) motions has been developed [11,90] for the translational vibrations and the librations in molecular crystals. At T = 0 K this method is equivalent to the Random Phase Approximation (RPA). It is applica­ble even to the quantum crystals H2 and D2, with their nearly free rotor states of the molecules.

Each of the solids listed has some special interest. Solid N 2, with its orientation­ally ordered and disordered phases, has received much attention as a prototype for lattice dynamics studies of molecular solids. Solid H2 and D2 seem even simpler, but they exhibit strong quantum character. This is manifested in the large amplitudes and the corresponding anharmonicity of the translational lattice vibrations, and even more strongly in the rotations. The properties of para-H2 and ortho-D2, with j = 0 molecules, and those of ortho-H2 and para-D2, with j = 1 in the ground state, are completely different. Solid CO is isoelectronic with N2 and its orientationally ordered and disordered phases have similar lattice structures, but it retains head-tail disorder even at the lowest temperatures. Solid O2 combines the properties of a molecular solid with those of a magnetic system and it exhibits magnetic phase transitions as well as structural ones.

Table 2 lists the phonon and libron frequencies of a-nitrogen. The results from the ab initio potential agree to within about 10% with the frequencies obtained from infrared and Raman spectroscopy (for q = 0) and from inelastic neutron scattering (for other wave vectors q). Table 3 shows similar data for a-CO with roughly the same agreement with the scarcer experimental data. Table 4 and Fig. 5 demonstrate that some calculated macroscopic properties of solid N2 are realistic too, and Table 5 shows the same for the lattice constant and the cohesion energy of solid CO. Figure 6

21

compares the phonon dispersion curves of ortho-D2 calculated from the ab initio H2-H2 potential of Schaefer et al. [61,62J with the data obtained from inelastic neutron scattering; the calculated frequencies are just slightly too high. Finally, we show in Table 6 that both the measured libron and magnon frequencies in solid O2 are fairly well reproduced by the ab initio spin-dependent O2-02 potential. It appears that the extremely strong anisotropy of the Heisenberg exchange term in Eq. (16) is responsible for the anomalously large libron splitting which occurs at the magneto­elastic (3 - a phase transition. So, in general, we have found that the properties of (simple) molecular crystals are fairly well described by ab initio potentials. One should realize that improvements on this accuracy will require not only a further improvement of these pair potentials, but also the inclusion of many-body interactions (see section 3).

6. Conclusion

From the experience with the systems described above it can be learned that for some time to come the best multi-dimensional intermolecular potentials will likely be obtained from a combination of ab initio calculations with measured data. The ab initio methods have reached the stage that they yield detailed and fairly reliable information on (parts of) the potential surface, which must still be checked against experimental data, however. In purely empirical potentials one can optimize only a limited number of parameters, and there is always the risk of oversimplification by the use of too crude atom-atom models or by too early truncation of the spherical expansion. The ab initio results are useful in avoiding this risk.

Acknowledgement

An always stimulating co-operation with Paul Wormer in the calculation of inter­molecular potentials and other matters is gratefully acknowledged. The results for Van der Waals molecules are taken from the work by John van Bladel, the results for molecular solids from Wim Briels, Tonek Jansen, Tom van den Berg and Wilfred Janssen.

References

[lJ van der Avoird, A., Wormer, P. E. S., Mulder, F. and Berns, R. M. (1980) Ab initio studies of the interactions in Van der Waals molecules, Topics Curro Chem. 93, 1-5l.

[2J Hobza, P. and Zahradnik, R. (1988) Intermolecular complexes, Elsevier, Amster­dam.

[3J Buckingham, A. D., Fowler, P. W. and Hutson, J. M. (1988) Theoretical studies of Van der Waals molecules and intermolecular forces, Chem. Rev. 88, 963-988.

[4J Miller, R. E. (1988) The vibrational spectroscopy and dynamics of weakly bound neutral complexes, Science 240, 447-453.

22

[5] Faubel, M. (1983) Vibrational and rotational excitations in molecular collisions, Advan. At. Mol. Phys. 19, 345-394.

[6] Buck, D., Huisken, F., Schleusener, J. (1980) Differential cross sections for the j = 0 --t 1 rotational excitation in HD-Ne collisions and their relevance to the anisotropic interaction, J. Chern. Phys. 72, 1512-1523.

[7] Bergmann, K, Hefter, D. and Witt, J. (1980) State-to-state differential cross sections for rotationally inelastic scattering of Na2 by He, J. Chern. Phys. 72, 4777-4790.

[8] Hirschfelder, J. 0., Curtiss, C. F. and Bird, R. B. (1964) Molecular theory of gases and liquids, Wiley, New York.

[9] Maitland, G. C., Rigby, M., Smith, E. B. and Wakeham, W. A. (1981) Inter­molecular forces, Clarendon, Oxford.

[10] Gray, C. G. and Gubbins, K E. (1984) Theory of molecular fluids, Clarendon, Oxford.

[11] Briels, W. J., Jansen, A. P. J. and van der Avoird, A. (1986) Dynamics of molec­ular crystals, Advan. Quantum Chern. 18, 131-206.

[12] Kaplan, I. G. (1986) Theory of molecular interactions, North-Holland, Amster­dam.

[13] Van Lenthe, J. H., Van Duijneveldt-van de Rijdt, J. C. G. M. and Van Duijn­eveldt, F. B. (1987) Weakly bonded systems, Advan. Chern. Phys. 69, 521-566.

[14] Chalasinski, G. and Gutowski, M. (1988) Weak interactions between small sys­tems. Models for studying the nature of intermolecular forces and challenging problems for ab initio calculations, Chern. Rev. 88, 943-962.

[15] Meath, W. J. and Koulis, M. (1991) On the construction and use of reliable two- and many-body interatomic and intermolecular potentials, J. Mol. Struct. (Theochem) 226, 1-37.

[16] van der Avoird, A. (1991) Intermolecular forces and the properties of molecular solids, in Z. B. Maksic (ed.), 'Theoretical models of chemical bonding', Springer, Berlin, part 4.

[17] Matsen, F. A., Klein, D. J. and Foyt, D. C. (1971) Spin-free quantum chemistry. X. The effective spin hamiltonian, J. Phys. Chern. 75, 1866-1873.

[18] Western, C. M., Langridge-Smith, P. R R, Howard, B. J. and Novick, S. E. (1981) Molecular beam electric resonance spectroscopy of the nitric oxide dimer, Molec. Phys. 44, 145-160.

[19] Brechignac, P. H., De Benedictis, S., Halberstadt, N. and Whitaker, B. J. (1985) Infrared absorption and predissociation of NO dimer, J. Chern. Phys. 83, 2064-2069.

[20] Wormer, P. E. S. and van der Avoird, A. (1984) (Heisenberg) exchange and electrostatic interactions between O2 molecules: An ab initio study, J. Chern. Phys. 81, 1929-1939.

23

[21] Brink, D. M. and Satchler, G. R. (1975) Angular momentum, Clarendon, Oxford.

[22] Le Roy, R. J. and Hutson, J. M. (1987) Improved potential energy surfaces for the interaction of hydrogen with argon, krypton, and xenon, J. Chem. Phys. 86, 837-853.

[23] Le Roy, R. J. and Carley, J. S. (1980) Spectroscopy and potential energy surfaces of Van der Waals molecules, Advan. Chem. Phys. 42, 353-420.

[24] Wormer, P. E. S., Mulder, F. and van der Avoird, A. (1977) Quantum theoretical calculations of Van der Waals interactions between molecules. Anisotropic long range interactions, Int. J. Quantum Chem. 11, 959-970.

[25] Berns, R. M. and van der Avoird, A. (1980) N2-N2 interaction potential from ab initio calculations with application to the structure of (N2h, J. Chem. Phys. 72, 6107-6116.

[26] Wasiutynski, T., van der Avoird, A. and Berns, R. M. (1978) Lattice dynamics of the ethylene crystal with interaction potentials from ab initio calculations, J. Chem. Phys. 69, 5288-5300.

[27] Janssen, W. B. J. M., Michiels, J. and van der Avoird, A. (1991) Lattice dynamics of a-CO from an ab initio potential, J. Chem. Phys. 94, 8402-8407.

[28] Pertsin, A. J. and Kitaigorodsky, A. I. (1987) The atom-atom potential method for organic molecular solids, Springer, Berlin.

[29] Hair, S. R., Beswick, J. A. and Janda, K. C. (1988) A quantum mechanical treat­ment of vibrational mixing in ethylene dimer and rare gas-ethylene complexes, J. Chem. Phys. 89, 3970-3982.

[30] Claverie, P. (1978) Elaboration of approximate formulas for the interactions be­tween large molecules: Applications in organic chemistry, in B. Pullman (ed.), 'Intermolecular interactions: From diatomics to biopolymers', Wiley, New York, pp.69-305.

[31J Rullman, J. A. C. and Van Duijnen, P. Th. (1988) A polarizable water model for calculation of hydration energies, Molec. Phys. 63, 451-475.

[32] Gay, J. G. and Berne, B. J. (1981) Modification of the overlap potential to mimic a linear site-site potential, J. Chem. Phys. 74,3316-3319.

[33] Stone, A. J. and Price, S. L. (1988) Some new ideas in the theory of intermolecular forces: Anisotropic atom-atom potentials, J. Phys. Chem. 92, 3325-3335.

[34] Stone, A. J. (1991) Classical electrostatics in molecular interactions, in Z. B. Maksic (ed.), 'Theoretical models of chemical bonding', Springer, Berlin, part 4.

[35] Fuchikama, N. and Block, R. (1982) An effective (exchange) hamiltonian for many-electron systems with degenerate and nonorthogonallocalized orbitals, Phys­ica 112B, 369-380.

[36] Jansen, A. P. J. and van der Avoird, A. (1987) Magnetic coupling and dynamics in solid a- and (3--02 • 1. An ab initio theoretical approach, J. Chem. Phys. 86, 3583-3596.

24

[37] Margenau, H. and Kestner, N. R. (1971) Theory of intermolecular forces, Perg­amon, New York, 2nd edition.

[38] Meath, W. J. and Aziz, R. A. (1984) On the importance and problems in the construction of many-body potentials, Molec. Phys. 52, 225-243.

[39] Loubeyre, P. (1987) Three-body exchange interaction in dense helium, Phys. Rev. Lett. 58, 1857-1860.

[40] Bulski, M. (1989) Ab initio pair and three body potentials in Polian et al. (eds.), Simple molecular systems at very high density, Plenum, New York, pp. 353-388.

[41] Bulski, M. and Chalasinski, G. (1987) On the nonadditivity of the second-order exchange-dispersion energy in the interaction of three helium atoms, J. Chern. Phys. 86, 937-946.

[42] Born, M. and Huang, K. (1954) Dynamical theory of crystal lattices, Clarendon, Oxford.

[43] Cochran, W. (1971) Lattice dynamics of ionic and covalent crystals, CRC Critical reviews in solid state science 2, 1-44.

[44] Visser, F. and Wormer, P. E. S. (1985) The non-empirical calculation of second­order molecular properties by mean of effective states. II. Effective TDCHF spec­tra for NO+, CO, CO2 and C2H2, Chern. Phys. 92, 129-140.

[45] Wormer, P. E. S. and Rijks, W. (1986) An analysis of the correlation effects in molecular second-order time-dependent properties. Application to the dynamic polarizabilities of the neon-atom and the dispersion coefficients of the Ne2-dimer, Phys. Rev. A33, 2928-2939.

[46] Rijks, W. and Wormer, P. E. S. (1988) Correlated Van der Waals coefficients for dimers consisting of He, Ne, H2 and N2, J. Chern. Phys. 88, 5704-5714.

[47] Rijks, W. and Wormer, P. E. S. (1989) Correlated Van der Waals coefficients. II. Dimers consisting ofHF, CO, H20 and NH3 , J. Chern. Phys. 90, 6507-6519; (1990) ibid 92, 5754(E).

[48] Rijks, W., Van Heeringen, M. and Wormer, P. E. S. (1989) The frequency depen­dent polarizability of O2 and the dispersion interaction in dimers containing O2 from an SDT-CI perturbation approach, J. Chern. Phys. 90, 6501-6506.

[49] Hettema, H. and Wormer, P. E. S. (1990) Frequency-dependent polarizabilities and Van der Waals coefficients of half-open shell systems in the time-dependent coupled Hartree-Fock approximation, J. Chern. Phys. 93, 3389-3396.

[50] Kumar, A. and Meath, W. J. (1985) Pseudo-spectral dipole oscillator strengths and dipole-dipole and triple-dipole dispersion energy coefficients for HF, HBr, He, Ne, Ar, Kr and Xe, Molec. Phys. 54, 823-833.

[51] van der Avoird, A., Wormer, P. E. S. and Jansen, A. P. J. (1986) An improved intermolecular potential for nitrogen, J. Chern. PhYfl. 84, 1629-1635.

[52] Van der Pol, A., van der Avoird, A. and Wormer, P. E. S. (1990) An ab initio intermolecular potential for the carbon monoxide dimer (COh, J. Chern. Phys. 92, 7498-7504.

25

[53] Bulski, M., Wormer, P. E. S. and van der Avoird, A. (1991) Ab initio potential energy surfaces for Ar-NH3 for different NH3 umbrella angles, J. Chem. Phys. 94,491-500.

[54] Bulski, M., Wormer, P. E. S. and van der Avoird, A. (1991) Ab initio potential energy surfaces of Ar-H20 and Ar-D20, J. Chem. Phys. 94, 8096-8104.

[55] Chipman, D. M., Bowman, J. D. and Hirschfelder, J. 0. (1973) Perturbation the­ories for the calculation of molecular interaction energies. I. General formalism, J. Chem. Phys. 59, 2830-2857.

[56] Jeziorski, B. and Kolos, W. (1977) On symmetry forcing in the perturbation theory of weak intermolecular interactions, Int. J. Quantum Chem. 12, Suppl. 1, 91-117.

[57] Rijks, W., Gerritsen, M. and Wormer P. E. S. (1989) Computation of the short range repulsion energy from correlated monomer wavefunctions in Van der Waals dimers containing He, Ne, and N2 , Molec. Phys. 66, 929-953.

[58] Douketis, C., Scoles, G., Marchetti, S., Zen, M. and Thakkar, A. J. (1982) Inter­molecular forces via hybrid Hartree-Fock-SCF plus damped dispersion (HFD) energy calculations. An improved spherical model, J. Chem. Phys. 76, 3057-3063.

[59] Tang, K. T. and Toennies, J. P. (1984) An improved simple model for the Van der Waals potential based on universal damping functions for the dispersion co­efficients, J. Chem. Phys. 80, 3726-374l.

[60] Knowles, P. J. and Meath, W. J. (1987) A separable method for the calculation of dispersion and induction energy damping functions with applications to the dimers arising from He, Ne and HF, Molec. Phys. 60, 1143-1158.

[61] Schaefer, J. and Meyer, W. (1979) Theoretical studies of H2-H2 collisions. I. Elastic scattering of ground state para- and ortho-H2 in the rigid rotor approxi­mation, J. Chem. Phys. 70,344-360.

[62] Schaefer, J. and Kohler, W. (1989) Low temperature second virial coefficients of para-H2 gas obtained from quantum mechanical pair correlation functions, Z. Phys. 13D, 217-229.

[63] Hancock, G. C., Truhlar, D. G. and Dykstra, C. E. (1988) An analytic represen­tation of the six-dimensional potential energy surface of hydrogen fluoride dimer, J. Chem. Phys. 88, 1786-1796.

[64] Redmon, M. J. and Binkley, J. S. (1987) Global potential energy hypersurface for dynamical studies of energy transfer in HF-HF collisions, J. Chem. Phys. 87, 969-982.

[65] Bunker, P. R., Kofranek, M., Lischka, H. and Karpfen, A. (1988) An analytical six-dimensional potential energy surface for (HFh from ab initio calculations, J. Chem. Phys. 89, 3002-3007.

[66] Cohen, R. C. and Saykally, R. J. (1990) Extending the collocation method to mul­tidimensional molecular dynamics: Direct determination of the intermolecular

26

potential of Ar-H20 from tunable far-infrared laser spectroscopy, J. Phys. Chem. 94, 7991-8000.

[67] Zwart, E., Linnartz, H., Meerts, W. L., Fraser, G. T., Nelson, D. D. and Klem­perer, W. (1991) Microwave and submillimeter spectroscopy of Ar-NH3 states correlating with Ar + NH3 (j = 1, Ikl = 1), J. Chem. Phys. 95, 793-803.

[68] Hutson, J. M. (1990) Atom-asymmetric top Van der Waals complexes: Angular momentum coupling in Ar-H20, J. Chem. Phys. 92, 157-168.

[69] Niesar, U., Corongiu, G., Huang, M. J., Dupuis, M, and Clementi, E. (1989) Preliminary observations on a new water-water potential, Int. J. Quantum Chem. 823, 421-443.

[70] Sagarik, K. P., Ahlrichs, R. and Brode, S. (1986) Intermolecular potentials for ammonia based on the test particle model and the coupled pair functional method, Molec. Phys. 57, 1247-1264.

[71] Alberts, I. L., Rowlands, T. W. and Handy, N. C. (1988) Stationary points on the potential energy surfaces of (C2H2h, (C2H2 )J, and (C2H4 h, J. Chem. Phys. 88,3811-3816.

[72] Hobza, P., Selzle, H. and Schlag, E. W. (1990) Floppy structure of the benzene dimer: Ab initio calculation on the structure and dipole moment, J. Chem. Phys. 93, 5893-5897.

[73] Dymond, J. H. and Smith, E. B. (1980) The vinal coefficients of pure gases and mixtures, Oxford University Press, Oxford.

[74] Elias, E., Hoang, N. and Sommer, J., Schramm B (1986) The second virial co­efficient of helium-gas mixtures in the region below room temperature, Ber. Bun­senges. Phys. Chem. 90, 342-351.

[75] Rigby, M. and Prausnitz, J. M. (1968) Solubility of water in compressed nitrogen, argon and methane, J. Phys. Chem. 72, 330-334.

[76] Richards, P. and Wormald, C. J. (1981) The enthalpy of mixing of (water + argon) vapour, Z. Phys. Chem. 128, 35-42.

[77] Schramm, B., Schmiedel, H., Gehrmann, R. and Bartl, R. (1977) Die Virialkoef­fizienten der schweren Edelgase und ihrer biniiren Mischungen, Ber. Bunsenges. Phys. Chem. 81, 316-318.

[78] Fraser, G. T., Nelson, D. D., Charo, A. and Klemperer, W. (1985) Microwave and infrared characterization of several weakly bound NH3 complexes, J. Chem. Phys. 82, 2535-2546.

[79] Nelson, D. D., Fraser, G. T., Peterson, K. I., Zhao, K., Klemperer, W., Loves, F. J. and Suenram, R. D. (1986) The microwave spectrum of K = 0 states of Ar-NH3, J. Chem. Phys. 85, 5512-5518.

[80] Gwo, D.-H., Havenith, M., Busarow, K. L., Cohen, R. C., Schmuttenmaer, C. A. and Saykally, R. J. (1990) Preliminary communication: Tunable far-infrared laser spectroscopy of Van der Waals bonds: the jkc = 10 f- 00 ~ bending vibration of Ar-14NH3, Molec. Phys. 71, 453-460.

27

[81] Schrnuttenrnaer, C. A., Cohen, R. C., Loeser, J. G. and Saykally, R. J. (1991) Far-infrared vibration-rotation-tunneling spectroscopy of Ar-NH3: intermolecular vibrations and effective angular potential energy surface, J. Chern. Phys. 95, 9-21.

[82] Zwart, E. and Meerts, W. J. (1991) The submillimeter rotation-tunneling spec­trum of Ar-D2 0 and Ar-NH3, Chern. Phys. 151, 407-418.

[83] Cohen, R. C., Busarow, K. L., Laughlin, K. B., Blake, G. A., Havenith, M., Lee, Y. T. and Saykally, R. J. (1988) Tunable far-infrared laser spectroscopy of Van der Waals bonds: Vibration-rotation-tunneling spectra of Ar-H20, J. Chern. Phys. 89, 4494-4504.

[84] Cohen, R. C., Busarow, K. L., Lee, Y. T. and Saykally, R. J. (1990) Tunable far infrared laser spectroscopy of Van der Waals bonds: The intermolecular stretching vibration and effective radial potentials for Ar-H20, J. Chern. Phys. 92, 169-177.

[85] Suzuki, S., Bumgarner, R. E., Stockman, P. A., Green, P. G. and Blake, G. A. (1991) Tunable far-infrared laser spectroscopy of deuterated isotopomers of Ar-H20, J. Chern. Phys. 94, 824-825.

[86] Van Bladel, J. W. I., van der Avoird, A. and Wormer, P. E. S. (1991) The Van der Waals rovibrational states of the Ar-NH3 dimer, J. Chern. Phys. 94,501-510.

[87] Van Bladel, J. W.I., van der Avoird, A. and Wormer, P. E. S. (1991) Theoretically generated vibration-rotation-inversion spectrum of the Ar-NH3 dimer, J. Phys. Chern. 95, 5414-5422.

[88] Jansen, A. P. J., Briels, W. J. and van der Avoird, A. (1984) Ab initio description of large amplitude motions in solid N2 I. Librons in the ordered Q and"y phases, J. Chern. Phys. 81,3648-3657.

[89] van der Avoird, A., Briels, W. J. and Jansen, A. P. J. (1984) Ab initio description of large amplitude motions in solid N2 II. Librons in the {3-phase and the Q - {3 phase transition, J. Chern. Phys. 81, 3658-3665.

[90] Briels, W. J., Jansen, A. P. J. and van der Avoird, A. (1984) Ab initio description of large amplitude motions in solid N2 III. Libron-phonon coupling, J. Chern. Phys. 81, 4118-4126.

[91] Jansen, A. P. J. (1988) New approach to orientationally disordered molecular crystals, J. Chern. Phys. 88, 1914-1924.

[92] Jansen, A. P. J. and Schoorl, R. (1988) Calculation of thermodynamic properties using the random-phase approximation: Q - N2 , Phys. Rev. B38, 11711-11717.

[93] Janssen, W. B. J. M. and van der Avoird, A. (1991) Dynamics and phase tran­sitions in solid ortho/para hydrogen and deuterium from an ab initio potential, Phys. Rev. 42B, 838-848.

[94] Jansen, A. P. J. and van der Avoird, A. (1987) Magnetic coupling and dynamics in solid Q- and {3 - °2 . II. Prediction of magnetic field effects, J. Chern. Phys. 86, 3597-3601.

28

[95] Scott, T. A. (1976) Solid and liquid nitrogen, Phys. Rep. 27,89-157.

[96] Nielsen, M. and M¢ller, H. B. (1971) Lattice dynamics of solid deuterium by inelastic neutron scattering, Phys. Rev. B 3, 4383-4387.

[97] Nielsen, M. (1973) Phonons in solid hydrogen and deuterium studied by inelastic coherent neutron scatterinq, Phys. Rev. B 7, 1626-1635.

TRADITIONAL TRANSPORT PROPERTIES

W.A. WAKEHAM and V. VESOVIC Department of Chemical Engineering and Chemical Technology, Imperial College, Prince Consort Road, London SW7 2BY, UK

ABSTRACT. Macroscopic properties of a dilute gas or gas mixture that govern the process of relaxation to equilibrium from a state perturbed by application of temperature, pressure, velocity or composition gradients have been studied for 120 years. The present paper reviews the current status of our abilities to measure, calculate and interpret these traditional properties and considers what the next steps in the development of the field might be. It is argued that if such transport properties are to form an important element in the elucida­tion of intermolecular forces for polyatomic systems, as they did for monatomic systems, then there is a need for further measurements of familiar properties and for the development of techniques of measuring less familiar, but no less traditional, properties. In addition, it remains necessary to reduce further the computational time required for the evaluation of the properties from assumed intermolecular potentials.

1. Introduction

The entropy production, <;, in a non-equilibrium system can be written [1] as a sum of products of generalized thermodynamic fluxes, Y i, and forces Xi, i.e.,

(1)

where the 8 implies a full tensor contraction. Intuitively, owing to the notion of cause and effect, it is possible to separate the force and the flux in each term of Eq. (1). For example, the explicit expression for the thermal conduction term is

Xl . Y 1 = - [;2 VT] . q, (2)

where it is usual to regard a quantity proportional to q as the flux, Yl, and an expression proportional to VT as the force, Xl. For small deviations from equilibrium it is possible to linearize the relationship between the fluxes and the forces to lead to a set of linear, phenomenological relationships

n

Yi = 2: L ik 8 Xk,

k=1

29

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 29-55. ©1992 Kluwer Academic Publishers.

(3)

30

where the Lik are constant phenomenological coefficients. Generally, if the system is defined by a set of thermodynamic variables ai ... an, and a fundamental relation F = F(ai ... an), then the fluxes Yi are identified as the time derivatives (daddt) and the forces as the variables conjugate to ai; Xi = [aF! aadcxj' When the vari­ables concerned are those of pressure, temperature, composition and flow velocity we shall term the resulting transport coefficients traditional. Hence, the transport phenomena to be considered include thermal conduction, viscous flow, concentration diffusion, thermal diffusion, the diffusion thermo effect and thermal transpiration. All of the phenomenological coefficients associated with these processes have been mea­sured by direct creation of the force Xi and measurement of the concomitant flux, Yi, in either steady state or transient mode. However, for only a subset of these phenomenological coefficients have there been measurements on a large number of systems over a wide range of conditions, and it is upon this subset that attention is concentrated here. This should not be taken to imply that the other coefficients are of lesser value, but merely that insufficient data exist (owing to the difficulty of mea­surement), to enable a judgement to be made. We further restrict the consideration to the transport properties of dilute gases, for which only two particle collisions are significant. This does not restrict the number of transport properties to be consid­ered in any way, but ensures that there is a formally complete kinetic theory relating the transport properties to the microscopic properties of the molecules and the pair potential governing their interaction.

2. The current status of measurements

The last twenty years have seen a spectacular improvement in the accuracy of mea­surement of transport properties in all thermodynamic states. The improvement amounts to at least an order of magnitude.

2.1. VISCOSITY

For the viscosity the improvement in the accuracy of viscosity measurements around 1970, in conjunction with spectroscopic and beam-scattering data, played a sub­stantial part in the elucidation of the forces between monatomic species over the next decade [2]. During that decade measurements were made on a wide range of monatomic and polyatomic gases and their mixtures at low density within the tem­perature range 70 K to 2000 K, and the accuracy achieved was often better than ±0.5% over much of the range, and as good as ±1.5% at the extremes of the range. Table 1 lists, but not exhaustively, the systems that have been studied to date. Where the upper temperature of the range studied is lower than the maximum attainable with the experimental apparatus, it is determined by the chemical stability of one species involved.

Figure 1 shows the time distribution of the measurements performed, illustrating the strong peak in the 1970's and the subsequent decline. To the authors' knowledge only three instruments are still in active use today for measurements of the same

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Figure 1. Distribution of viscosity measurements.

type; specifically they are the oscillating-disk viscometer of Vogel in Germany [3] and the capillary viscometers of Smith [4] and Matthews [5] in the U.K. It was the refine­ment of these two types of instrument that sparked the revolution in the field in the 1970's. The refinement was the result of an improved treatment of the underlying fluid mechanics and great attention to detail. It is noteworthy that the oscillating-disk viscometer of Kestin [6] was constructed with a significant number of metal compo­nents that led to catalytic degradation of some species at quite modest temperatures. The newer instrument of Vogel [3] is manufactured entirely of quartz, thus obviating the problem to a large extent and increasing the temperature range of the device.

2.1.1. Viscosity data

For pure gases it is straightforward to perform an intercomparison of various indepen­dent measurements and Figure 2 shows the results for the viscosity of carbon dioxide, a relatively well-studied fluid, in the form of deviations from an empirical correla­tion of selected data [7]. The agreement between all of the selected data is generally good near room temperature, degrading at the extremes as the experimental accuracy diminishes, but remaining within estimated error.

For binary mixtures an equivalent direct intercomparison is difficult, but the

33

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TEMPERA TURE / (K)

Figure 2. Deviations of the experimental viscosity data for CO2 from their correla­tion (taken from Ref. [7]).

kinetic theory makes it possible to deduce from measurements of the viscosity of binary mixtures and the two pure gases a single quantity characteristic of the unlike interaction in the mixture which can be used as the basis of the comparison. On the few occasions when this has been done, the results are entirely consistent with those for the pure gases with a marginally greater uncertainty [8J.

2.1.2. Future viscosity measurements

The available information on the viscosity of dilute pure gases and binary mixtures is therefore quite extensive and of an adequate quality to provide an acceptable test ground for theoretical calculations. However, there must remain some doubt as to whether the particular systems studied are those for which theoretical calculations will first be performed. Since there are two operational instruments available, it seems likely that even if the required systems have not yet been studied, measurements could still be performed.

34

2.2. THERMAL CONDUCTIVITY

The development of accurate methods for the measurement of the thermal conduc­tivity of dilute gases lagged far behind that of the viscosity because the measurement processs is intrinsically more difficult [9J. Whenever a finite volume of fluid is subject to a temperature gradient in the earth's gravitational field, there is inevitably a nat­ural convective flow of fluid that causes the heat transfer regime to depart from that of pure conduction. The resolution of this difficulty was not found until the latter part of the 1970's when the pioneering work of Haarman [lOJ established the transient hot-wire technique as the method of choice for the measurement of the thermal con­ductivity of fluids. For this technique the transient temperature rise of a thin (7I1m) platinum wire, immersed in the fluid, is followed over a time which is short compared with the characteristic time necessary to induce significant convective heat transfer. Subsequently, the temperature increase is used to determine the thermal conductivity of the fluid [l1J.

2.2.1. Thermal conductivity data

Initially, measurements with this technique were confined to near-ambient tempera­tures as a function of pressure owing to difficulties with the suspension of the wire but later measurements were extended upwards to 430 K [12J and downwards to 120 K [13J with accuracies between ±0.3 and ±1.5%. However, it must be said that there have been no direct measurements in the dilute gas state and very few measurements reported overall. Table 2 gives a list of the gases studied and the range of conditions of the investigation. Since, for polyatomic gases, the thermal conductivity yields in­formation about molecular interactions not contained in the viscosity, the difference in extent of the information available needs explanation.

First, the theory of the experimental method for work in low-density gases is in­complete owing to unaccounted effects negligible at elevated densities [11 J. Thus, most measurements have been performed at supercritical temperatures and moderate pres­sures and the dilute-gas behaviour inferred by extrapolation [14J. Such a procedure is clearly neither appropriate nor possible at significantly sub-critical temperatures. Secondly, there has been little encouragement for dilute gas measurements since the principal funding has arisen from engineering, rather than scientific, needs. For these two reasons, there is only one transient hot-wire system in the world currently devoted to measurements in the dilute gas phase [12J and it is presently in use for the study of environmentally acceptable refrigerants [15J.

There are, of course, other methods for measuring the thermal conductivity of gases [9,l1J. However, as the direct intercomparison in Figure 3 of the available data for carbon dioxide [7J shows, the results often depart significantly from those obtained with a transient hot-wire system, and the latter are to be preferred.

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TEMPERATURE / (K)

35

Figure 3. Deviations of the experimental thermal conductivity data for CO 2 from semi-theoretical calculations: • transient hot-wire measurements; ... other methods.

2.2.2. Future thermal conductivity measurements

High quality data on the thermal conductivity of pure gases are still relatively scarce and cover quite a narrow temperature range. Given the role that such data could play in the investigation of microscopic features of molecular processes, this is unfortunate. The extension of the temperature range upwards is not intrinsically difficult except for the catalytic effect of platinum in causing decomposition of samples. However, the extension of the measurements to lower temperatures at low densities does require some further refinement of the method. Further, the separation of a single quantity characteristic of the unlike interaction from measurements on a binary mixture is generally more difficult than for the viscosity, so that a comparison with theoretical calculations will always be less direct for such systems [16]. Unfortunately, it is just for such unlike interactions that theoretical results are most likely to become available in the first instance.

2.3. CONCENTRATION DIFFUSION

The binary diffusion coefficient for a two-component mixture is undoubtedly the most direct probe of the unlike interaction between two molecules, depending very weakly

36

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TEMPERATURE • T • K

Figure 4. Deviation of experimental diffusion coefficient data from their correlation (taken from Ref. [11]).

on interactions among similar molecules [2]. However, the diffusion coefficient has proved extremely difficult to measure with great accuracy [9,11] and no single tech­nique has emerged as obviously superior over a wide range of conditions. When data from a variety of methods are combined, a wide range of temperature can be covered, although mostly for the monatomic gases, but the discrepancies between various data are often large, as illustrated in Figure 4. The most accurate diffusion measurements are almost always made near ambient temperature, and the work of Dunlop and his collaborators [11,17] indicates what can be achieved by painstaking effort. However, Dunlop's apparatus is the only instrument known to be operational for the measure­ment of diffusion coefficients in the dilute gas phase so that the production of new data is necessarily limited.

2.4. THERMAL DIFFUSION

Thermal diffusion factors have often been noted as the transport property most sen­sitive to the details of intermolecular forces and collision processes [2]. At the same

37

time, they are also very difficult to determine, since the measurement of the com­ponent separation in a gas mixture that accompanies the imposition of a temper­ature gradient is exceedingly small [11]. There is therefore a considerable body of fragmentary experimental information on a wide range of systems with substantial discrepancies (occasionally even of sign) between different authors' results [11]. Thus, the value of the property for microscopic investigations is currently limited by the accuracy of measurements, and the fact that only one instrument is currently opera­tional [18] makes it unlikely that thermal diffusion factors will be an important source of information in the foreseeable future.

2.5. BULK VISCOSITY

The bulk viscosity of a dilute polyatomic gas is directly related to inelastic collision processes that transfer translational energy to internal energy or the reverse. For this reason, the property is potentially of great significance to the study of inelastic cross-sections. However, there have been few direct measurements of this property since the 1960's, when sound absorption was used for the determination in a frequency regime characteristic of rotational energy relaxation [19,20]' and shock-tube studies were employed for vibrational-energy relaxation [19].

The accuracy achievable in studies of vibrational-energy relaxation times was quite high because of the large difference between the timescales for vibrational and translational relaxation. However, for rotational relaxation the timescale is very close to that characteristic of translational motion, so that the experimental accuracy was much more limited. Advances in acoustic technology might be thought to offer some opportunity for a redevelopment of this field, but it seems that the intrinsic limita­tions imposed by the timescales involved may mean that no greater accuracy can be achieved. In any event, for molecules containing many atoms, the number of possible internal modes of motion is sufficiently large that the interpretation of a bulk viscosity in terms of discrete relaxation of individual modes is both difficult and fraught with uncertainty. Thus, whereas the value of measurements of this type to the study of inelastic processes is clear, it is much less obvious just how they can be performed with sufficient accuracy. There are thought to be no measurements of this type in progress.

2.6. THERMAL TRANSPIRATION

Thermal transpiration is a process whereby the application of a temperature gradient in a closed system induces a small, but measurable, pressure difference [11]. The effect has not been studied very frequently. However, the technique has the unique advan­tage that it yields the translational part of the thermal conductivity, independently of the internal part. This permits the evaluation of several effective cross-sections when combined with other information. The disadvantage of the experiment is that a model of the interaction of molecules with the wall of the containing vessel is required. The most recent measurements by Millat [21] have proved of value in the evaluation

38

of effective cross-sections for a number of gases but there are now no operational instruments of this type.

3. Theory and analysis

3.1. THEORY

The formal kinetic theory of the transport coefficients of polyatomic gases, in the absence of external forces, is well developed and is dealt with in other papers in this workshop. The kinetic theory relates the transport properties of pure gases and mixtures to a number of effective cross-sections [22], 6(;' i, :' i, ). If pqst = p' q' s't',

then 6(;' i' :' i,) == 6(pqst). These cross sections incorporate all of the information about the binary molecular interactions, and hence about the intermolecular potential. It is the purpose of this section to demonstrate how a complete set of results for the 'traditional' transport properties of pure fluids can be used to obtain a consistent set of cross-sections for use in examining the microscopic properties of the molecules.

The viscosity of the pure gas, TJ(T), is given by

(T) _ kT TJ - cr 6(2000) ,

whereas the thermal conductivity of the pure gas, )"(T), is

in which the translational part is

and the internal part is

Here,

1

r = [2Cint/5kJ"2 ,

and

(4)

(5)

(6)

(7)

(8)

(9)

(10)

in which m is the mass of a molecule, Cint the molecular isochoric heat capacity and S is a correction factor to correct for angular momentum-polarization effects, which

39

is expressed in terms of the quantity (~AII/A)sat [22,23]. The latter quantity is the saturation value of the relative change in the thermal conductivity of a gas measured parallel to an applied magnetic field.

Further, the self-diffusion coefficient of the gas can be expressed by

kT D= ,

nmcr 6'(1000) (11)

in which 6'(1000) designates the so-called self-part of the cross section, and n is the number density. A collision number for internal energy relaxation, which is related to the bulk viscosity, can also be defined via

4kT (int = 1[1]Cr 6(0001) . (12)

All of the quantities are, as indicated above, directly measurable, and the transla­tional part of the thermal conductivity is itself directly measurable using thermal transpiration. A further quantity, not yet amenable to direct measurement, is the so-called diffusion coefficient for internal energy [24,25]

(13)

This quantity involves just two effective cross-sections, one of which can be deter­mined from the collision number, so that it provides a reasonably direct route for the evaluation of 6( 1001) if it can be measured directly. The ratio of the two diffusion coefficients, Dint! D, can be related to the viscosity through the kinetic theory and is given by

D

5 nmDint

6A* 1] (14)

where A * is a cross-section ratio very weakly dependent on the pair potential used for its evaluation [2]. For rigid rotors at high temperatures it has been shown [22,26] that for a limited class of anisotropic potentials

(15)

Hence, Dint 6'( 1000) --- ~ --~--~--~~~---------

D 6'(1001) + [3~;2 - ~]6(0001) (16)

at high temperatures and approaches unity from below as T ~ 00. This result, together with a number of exact relationships between the effective cross-sections [22], has been used to great advantage in the analysis of experimental transport property data.

40

Table 3. Effective cross-sections for carbon dioxide and methane. All cross-sections are given in units of 10-20 m 2 .

T/K 6(0001) 6(1001) 6(1010) 6e01O ) 1001 6(2000)

Carbon Dioxide

308.15 15.99 50.06 51.71 11.97 50.69 333.15 13.42 46.05 48.64 10.37 48.91 379.15 10.14 41.91 44.21 8.24 46.22 425.65 8.00 37.73 40.86 6.28 44.06

Methane

309.15 5.15 34.04 32.71 3.69 41.15 337.65 4.41 31.67 31.46 3.28 39.85 379.15 3.58 30.09 30.04 2.82 38.32 425.65 2.89 27.68 28.87 2.44 36.99

3.2. ANALYSIS

The foregoing theoretical results can be used in conjunction with experimental infor­mation on the transport coefficients to deduce an entirely consistent set of effective cross-sections for the gas [27]. However, it is usually necessary to use one further ansatz associated with the independence of the modes of internal motion, so that the overall collision number is related to numbers characteristic for rotational and vibrational modes. Specifically, it is assumed that

Cint Crot Cvib -=-+­(int (rot (vib

a result which permits independent measurements of (rot and (vib to be utilized.

(17)

It can be seen from section 2 that only for a small number of gases over a small temperature range are all of the necessary experimental data available to permit an analysis of this kind. The results of the analysis for two gases are given in Table 3 in the form of a listing of the five cross-sections for carbon dioxide [28] and methane [29]. These results could evidently be used in tests of the ability of proposed anisotropic pair potentials to represent them. It is noteworthy that there are many more cross-sections available for such tests for polyatomic gases than for monatomic systems where there are only two. One result of some practical importance that can be obtained from these results is the behaviour of the ratio Dint! D as a function of temperature. Figure 5 shows the behaviour of the ratio for carbon dioxide, which is essentially a rigid-rotor at the temperatures of interest [7]. It can be seen that the selected data employed for the evaluation lead to a result consistent with the theoretical expectation of Eq. (16). It should be noted that if other, older, thermal conductivity data were employed in the analysis, the same consistency would not be observed [7].

0

" .... c: ... 0

41

1.05~-------------r------------~r-------------~

1.00 - - - - - - - - - - -.- - - ... - - -::-:-:::->------1

• 0.95 •

•

0.90

0.85~--------------~--------------~------------~ 285 385 485

TEMPERA TURE / (K)

585

Figure 5. The ratio D int / D as a function oftemperature deduced from experimental data. -- correlation, (taken from Ref. [7]).

A further result of some interest can also be deduced which refers to an alternative formulation of the theory of polyatomic gases due to Thijsse et al. [30]. Using a different set of expansion vectors, these authors showed that the thermal conductivity of a gas could also be written in the form

>. = 5k2T(1 + r2) 1

2mcr 6( WE) FT

6 2(lOE) :F - 1 _ lOD

T - 6(lOE) 6(lOD) .

(18)

(19)

The cross-sections 6(10E), 6(10D) and 6(ig~) are linear combinations of the cross-sections introduced earlier [22]. These linear relationships make it possible to evaluate the FT term as well as the individual cross-sections and Table 4 contains the results for five gases [31]. It can be seen that the coupling cross-section 6(ig~) is small compared with 6(lOE) and 6(10D) so that FT Rj 1. Thus, for many practical purposes the use of FT = 1 is a very good approximation [30,31]. Table 4 also contains the results for the 6 T (lOE) cross-section, obtained from Eq. (18) by

42

Table 4. Calculated effective cross-sections and related quan-tities. All cross-sections are given in units of 1O-2om2, (taken from ref. [31]).

Gas T/K 6 T (10E) 6(lOE) 6(lOD) 6eOE ) 10D FT

N2 300 24.6 24.8 42.2 3.08 0.9910 400 22.6 22.7 35.7 2.31 0.9934 500 21.4 21.5 32.8 1.81 0.9952 600 20.6 20.7 29.5 1.46 0.9965 700 20.0 20.1 27.7 1.22 0.9973 800 19.5 19.6 26.4 1.03 0.9979 900 19.1 19.2 25.4 0.89 0.9984

1000 18.8 18.8 24.5 0.78 0.9987

CO 300 25.6 25.8 47.3 2.69 0.9941 400 23.0 23.1 38.8 2.53 0.9930 500 21.5 21.6 33.9 2.26 0.9930 600 20.7 20.8 31.0 1.86 0.9947 700 20.1 20.2 29.0 1.55 0.9959 800 19.6 19.7 27.5 1.33 0.9968 900 19.3 19.3 26.4 1.16 0.9974

1000 19.0 19.0 25.5 1.03 0.9978

CO2 300 38.9 39.1 33.7 2.9 0.9966 400 33.3 33.4 48.4 1.7 0.9981 500 30.1 30.2 40.8 1.1 0.9990 600 28.2 28.3 36.5 0.64 0.9996 700 26.9 26.9 33.7 0.33 0.9999 800 26.0 26.0 31.6 0.14 1.0000 900 25.2 25.2 30.2 0.03 1.0000

1000 24.6 24.6 29.0 -0.05 1.0000

CH4 300 30.0 30.0 37.9 -0.15 1.0000 400 26.4 26.4 31.6 0.48 0.9997 500 24.7 24.7 28.8 0.36 0.9998 600 23.6 23.6 27.1 0.27 0.9999 700 22.9 22.9 25.9 0.22 0.9999 800 22.3 22.3 25.1 0.21 0.9999 900 21.7 21.7 24.4 0.20 0.9999

1000 21.3 21.3 23.8 0.20 0.9999

CF4 300 47.5 47.5 65.8 1.20 0.9995 400 43.2 43.2 56.7 0.48 0.9999 500 40.4 40.4 51.9 0.31 1.0000

600 38.5 38.5 48.9 0.30 1.0000

700 37.0 37.0 46.9 0.33 0.9999 800 35.9 35.9 45.5 0.37 0.9999 900 35.0 35.0 44.5 0.40 0.9999

1000 34.3 34.3 43.6 0.44 0.9999

43

1.20r-----------------------r-----------------------,

- --- -

.~/.//-.---... -................ -.-... --..

,//~------------------------------) 1 . 05 .. ;.;.;.;.;.;.;:::-::-: - - --

0.... 1. 00 ...... -; .. ; .. ; ... ~

~

1.15

1.10

--0.90 b-

..-

0.B5~--------------------~--------------------~ :300 450 600

TEMPERA TURE / (K)

Figure 6. Comparison of the behaviour of the ratio Dint / D as a function of temper­ature for several gases. - N2 ; - - - CO2 ; - - - - CH4 ; ••.••• C2H6 .

assuming that FT = 1. From a more fundamental point of view, despite the fact that 6( ~g~) is small, its values are almost exclusively positive. The most frequently used dynamical approximation in the theoretical evaluation of the transport properties of gas, the infinite-order sudden approximation [32] or its classical equivalent, the Mason-Monchick approximation [24,25]' lead to the result that [31]

6(lOE) _ 1" (7rmkT)~ [1-~] lOD - 6(1 + 1"2) 7] 4A* . (20)

For any realistic potential A* ~ 5/4 [2], hence the IOS/MM approximation implies that

(21)

which is in complete contrast to the experimental finding [31]. A second result for which a complete explanation is still lacking concerns the

ratio Dint / D for gases in which the rates of rotational and vibrational relaxation are not widely different, such as methane. Figure 6 shows the experimentally determined value of the ratio D int / D as a function of temperature [7,33], illustrating quite clearly an increase above unity with increasing temperature in contrast to the behaviour for

44

rigid rotors. Since there is no result equivalent to that of Eq. (16) for molecules other than rigid rotors, it is not possible to assert that this behaviour is not in accord with theory, but neither is it consistent with intuition. The result therefore remains a matter of some interest.

3.3. MIXTURES

For multicomponent mixtures of gases the kinetic theory formulation of thermal con­ductivity is equally well developed in both the traditional formulation [22,34] and that of Thijsse et al. [30,34]. However, there is much less complete experimental informa­tion available than we have seen above, and the theoretical formulation involves not just the five effective cross-sections for each pure gas, but also a greater number char­acteristic of the unlike interaction. Thus, it has not yet proved possible to isolate from measurements on binary mixtures all of the various effective cross-sections for the unlike interaction. In the context of the description of the theoretical calcula­tion of cross-sections given later, it can be seen that this circumstance is unfortunate because it is for just those cross-sections between an atom and a molecule that the­oretical results are available [35-38]. However, the argument presented above shows that the comparison with experiment is not possible unless all the cross-sections for the like interaction of the polyatomic species are also available.

4. Theoretical calculations

4.1. FUNDAMENTAL EQUATIONS

All of the effective cross-sections introduced in the previous section can be expressed as functionals of the anisotropic intermolecular pair potential describing the interac­tion of the various species in the gas [22]. The functionals may be obtained from the S-matrix describing the dynamics of the collision and the S-matrix itself obtained from the asymptotic solution of N close-coupled differential equations describing the en­counter of the two species. Full details of these relationships and the numerical meth­ods that can be employed for the solution of the equations are given elsewhere [22,32].

The evaluation of effective cross-sections by this fully quantum-mechanical route is an extraordinarily lengthy calculation [38]. Thus, for both traditional transport properties and other phenomena there has been a search for rather more rapid and simpler, approximate methods for performing the calculations. In one way or another all of these approximations, such as the Mason-Monchick approximation [24,25]' the infinite-order sudden approximation, the centrifugal sudden approximation or the energy-sudden approximation [22,32]' make approximations to the collision dynam­ics. One further computational method, that of classical trajectories [22,32]' treats the collision dynamics essentially exactly, but in a classical, rather than a quantum­mechanical framework.

45

6

" 4 "

*' " " g " " " i= 2 " " " " " <{ I " " 5> • " W • ~ 0 t:. • . ~ ~ t:. !!I 0 r.t • • ~ • • • • • •

-2 100 200 300 400 500

TEMPERATLRE. T/K

Figure 7. Deviations of the cross-sections for the He-N2 system obtained by approx­imate methods from close-coupled results. Classical trajectory method: /:; - diffu­sion cross-section; 0 - viscosity cross-section; MM/IOS approximation: • - diffusion cross-section; • - viscosity cross-section.

4.2. RESULTS

For some time the various dynamical approximations were applied to the com­putation of effective cross-sections for the unlike interaction of an atom with a rigid rotor without a knowledge of the inaccuracies introduced in the process. For this reason extensive calculations of the effective cross-sections for model systems have been carried out to test the accuracy of some of these dynamical approximations [22]. In this paper we shall concentrate on two such studies, namely He-N2 [38,39] and Ar-H2 [40] interactions; the first system is expected to be largely classical, whereas the second is highly quantal. Figure 7 shows the deviations between the essentially exact results from the quantum-mechanical calculations and those computed by the classical trajectory method [41J and by the infinite-order sudden approximation for the viscosity and diffusion cross sections. It can be seen that the discrepancy for the classical trajectory calculation of the viscosity is about 1% at 100 K, and it rapidly decreases to only a few parts in one thousand at 300 K for both cross sections. On the other hand the discrepancy of the results computed by the lOS approximation is higher at all temperatures for both cross sections. While the deviation falls rapidly

46

2.50 +

+ + 2.00

+ + • '$. +

+ + + • Z 1.50 + 0 • i= • <{ • :> 1.00 • W • • 0 • •

0.50

0.00 100 200 300 400

TEMPERATURE. T/K

Figure 8. Deviation of the cross-sections for Ar-H2 system obtained by the MM/IOS approximation from close-coupled results. + - diffusion cross-section; • - viscosity cross-section.

to a few parts in one thousand for the viscosity cross section at higher temperatures, it remains nearer 2% for the diffusion cross section.

For the H2-Ar system the results are similar, as shown in Figure 8 [40]. At first sight it may seem surprising that the behaviour of these two contrasting

systems is so nearly the same, given the very much greater rotational energy spacing in the hydrogenic system. However, this result seems to indicate that although the classical trajectory and the lOS may quite wrongly evaluate the effects of inelastic collisions in highly quantal systems, the overall effect of inelastic collisions on viscosity and diffusion cross-sections is small.

On the basis of these results, it can be concluded that, at least for atom-rigid rotor systems, the classical trajectory method is most certainly adequate for calcula­tions of transport properties with an accuracy commensurate with the experimental uncertainty for both viscosity and diffusion cross sections. On the other hand the infinite-order sudden approximation, while adequate at high temperatures for viscos­ity, is not sufficiently accurate for the diffusion cross section.

It is also of some interest to examine cross sections more sensitive to the inelastic collision processes than viscosity. Figure 9 compares the results of 'exact' quantal calculation of the ratio of the internal energy diffusion coefficient for nitrogen in helium

47

5 I

4 • ~ 0

Z 3 0 • • f= « • :> 2 w 0 • • •

1

0 100 200 300 400 500

TEMPERATURE. T IK

Figure 9. Deviation of the ratio D1int ,2/ D for the N2-He system, obtained by using MM/IOS approximation, from the close-coupled results.

to the mutual diffusion coefficient, D1int,d D12 with the lOS results [40]. Here, the deviations are of the same order as for the diffusion cross section itself.

The evaluation of the thermal conductivity of a system based upon the lOS approximation for the evaluation of the cross-sections would have an error of around 4%, exceeding the experimental uncertainty by almost an order of magnitude.

The routine fully quantal evaluation of the transport properties even of atom rigid-rotor systems is likely to prove prohibitively expensive in computer time for the foreseeable future. However, the results quoted here show two directions for the future. First, that the lOS approximation is sufficiently accurate to be used as a means of coarse screening of proposed anisotropic potential surfaces, and secondly that classical trajectory calculations are adequate for further refinement of the potential surface. The first observation has been used to advantage by McCourt et al. [42] and by Gianturco et al. [43] in a study of a number of anisotropic potentials for He, Ne and Ar with N2 . Figure 10 shows the repulsive and attractive parts of the various potentials employed for the Ne-N2 system together with the deviations of the calculated interaction viscosity [43] from the experimental values [44]. The differences between viscosities obtained using the various potentials are very large despite the fact that each potential has its origin in studies of other properties. The differences

48

.,4

."

102 :> e .... ~ 10' :>

1fIJ

If'

10

C.v D.~"Y

'1" 'f', 2.D

'\ "' \

~ '') \~

C_vt_trr

',,\

~~ .~ , ~

to 2.0 3.Il,/A 0.0 5.0 e.o 10

~-... --... -A-A_.6._AHlSVII

8.0

~

<J 2 ._.~-.-.-.-.-.8TT 500 T/K ._e- --.-_____ .HFDl

~! -8

-,

---.--. -. ----.e-_--__ ef1JSVI

... OJ>

_M3SV' ---M3Syl

····14r03 -'-'~BTT

Figure 10. Ne-N2 potentials used in Ref. [43J and the deviations of the calculated interaction viscosity from the experimental values.

are indeed larger than the expected errors introduced by lOS calculations, justifying the use of this rapid calculation procedure.

At the same time, it should be recognized that once this coarse screening has identified useful potentials, it would be necessary to switch to the computationally more expensive classical trajectory method to obtain further refinement.

All of these conclusions and proposals for the use of calculation methods in the future are, of course, based on the atom-rigid rotor system. It would seem justified

49

to suggest that the same kind of results would pertain to pure polyatomic gases. We thus conclude that we have available computational methods that render the

calculation of transport properties for assumed pair potentials tractable with modern computers. However, there remains the question of the systems to which it would be best to apply them in the first instance.

5. Direct inversion

During the study of the intermolecular forces between monatomic systems, a method was developed by Smith and his co-workers for the direct determination of the inter­molecular potential from measurements of transport properties and, indeed, second virial coefficients [2]. While this method did not, alone, lead to the determination of intermolecular potentials, it most certainly made an important contribution. Since its inception the method has been refined, and an explanation given for the manner in which the method overcomes the lack of uniqueness thought originally to be intrinsic to the problem [2].

The essential features of the inversion method are, that from a set of TJ(T) vs T data, it is possible to deduce the corresponding, spherically-symmetric pair potential by iterative application of the equations

and

7rf~ = 5(mkBT7r)! /16TJ(T)

= ~6(2000) 4

(22)

(23)

in which f'l is a temperature-dependent characteristic distance and G'I(T*) the in­version function for viscosity initially determined from a physically plausible inter­molecular potential function. In the original form of the method the well-depth of the potential, E, had to be prescribed, but later modifications enabled this quantity too, to be defined by the rate of convergence of the process [45].

For systems interacting through a non-spherically symmetric potential, U (r, w), where w represents the angular dependence, the same formal procedure can be ap­plied without change. However, the spherically-symmetric potential obtained as a result, O(f'l) , while capable of reproducing the viscosity data is evidently not the true intermolecular potential as it was in the case of spherically-symmetric systems. The question therefore arises: how is the potential obtained by inversion related to the true potential? Preliminary investigations of the answer to this question have been conducted [46,47] within the IOS/MM approximation, which the argument of the previous section has demonstrated is sufficiently accurate for the purpose.

The investigation has been conducted for a simple purely attractive or repulsive anisotropic interaction for an angular dependent (12-6) potential, as well as for a

50

more complex model potential proposed for the Ar-C02 interaction [48]. For the first two cases the determination of the relationship of the spherically-symmetric potential obtained by inversion to the full anisotropic potential can be performed analytically. It has been shown [47,49] that the potential obtained by inversion (0,1'1)) corresponds exactly to a plot of the root-mean-square of the intermolecular separation along a constant energy contour, GO against 0, in which G is the inversion function.

The application of the same interpretation to a more realistic anisotropic poten­tial shows that the path around the potential energy surface is only approximately one of constant energy. The approximation is extremely good in the purely repul­sive and purely attractive regions of the potential but much less accurate in the well region [47].

It remains an open question as to whether this use of viscosity data for non­spherically-symmetric systems has a greater utility for the investigation of intermolec­ular potentials than the simpler method of direct comparison of experimental viscosity data with values calculated for an assumed potential. That it may have some virtues is illustrated in Figure 11 where the potentials obtained by inversion of viscosity and second virial coefficient data for the Ar-C02 potential model are plotted for the at­tractive region [47,49]. The differences between these two potentials reveal the fact that the two properties lead to different energy contours around the potential. Since the same will be true of other cross-sections, there is some hope that the use of many cross-sections i~ a similar manner may establish some of the features of anisotropic potentials, particularly at distances far removed from the potential well.

6. The Future

At this particular time, the investigation of the traditional transport properties of gases stands at a crossroads. On the one hand, the experimental techniques for the measurement have been refined to a point where measurements of high accuracy can be performed for viscosity, thermal conductivity and diffusion coefficients, but where the active work in the area in the world has declined very greatly. For this reason the extent of the data available with respect to both temperature range and the molec­ular species is more limited than is desirable for future progress in interpretation. In particular, theoretical studies have, for reasons of computational expense, con­centrated upon interactions between atoms and rigid rotors. From the experimental point of view, this is unfortunate because the study of such interactions necessarily imply measurements on mixtures which are both more time consuming and diffi­cult. Furthermore, the isolation of quantities characteristic of the unlike interaction from mixture measurements requires the estimation of some quantities as well as the measurement of a number of pure gas properties. Both of these factors increase the uncertainty in the derived quantities compared with what could be achieved for pure gases.

At the same time, the routine calculation of effective cross sections for anisotropic potentials has now been placed on a secure footing and is within the capabilities of

51

0 x I x "\

I x E\

I /

EI /

/ EI

I /

-100 x d I d

g \ ....... d p x

\

~ \

-200 x

-300~----~------~------~------~

3.00 3.50 4.00 4.50 5.00

~ / (A)

Figure 11. Comparision of the potential obtained in the attractive region -­potential obtained by inverting virial coefficients; - - - potential obtained by inverting viscosity coefficients, (taken from Ref. [49]).

present-day computers. However, to judge from published results this may only be true of the interactions between atoms and rigid rotors.

Finally, and in a more optimistic sense, it may be remarked that sufficient evi­dence has been adduced to show that the traditional transport properties have an im­portant role to play in the study of anisotropic interactions. It may be, however, that it would be more productive to devote attention to those properties more sensitive to the anisotropic part of the potential than the viscosity and diffusion. Specifically, the bulk viscosity and the diffusion coefficient for internal energy Dint are both directly measurable, in principle, but little effort has been devoted to their determination. Ef-

52

forts in this direction, as well as towards an extension of the measurements of thermal conductivity, may be worthwhile, and theoretical calculations might be employed to answer this question and thus provide a guide to experimentalists.

Acknowledgements

The authors are grateful to the U.K. Department of Trade and Industry for financial support of some of the work described in this paper. In addition, we wish to thank our collaborators, A.S. Dickinson, F.R.W. McCourt, G.C.Maitland, J. Millat, M. Mustafa and M. Ross for their invaluable contributions to the work.

References

[1] De Groot, S. R. and Mazur, P. (1962) Non-equilibrium Thermodynamics, North~ Holland, Amsterdam.

[2] Maitland, G. C., Rigby, M., Smith, E. B. and Wakeham, W. A. (1981) Inter­molecular forces: their origin and determination, Clarendon Press, Oxford.

[3] Vogel, E., Bastubbe, E. and Rhode, S. (1984) Wiss. Z., W-Pieck. Univ. Rostock 33, N8 34.

[4] Clarke A. G. and Smith, E. B. (1968) Low temperature viscosities of Ar, Kr, Xe, J. Chern. Phys. 48, 3988~3991.

[4] Hunter, I. Matthews, G. P. and Smith, E. B. (1989) Viscosities of gaseous Ar-H2

mixtures, Mol. Phys. 66 887~895.

[6] Kestin, J. and Leidenfrost, W. (1959), The effect of pressure on the viscosity of five gases in Y. S. Touloukian (ed.), 'Thermodynamics and Transport Properties of Gases, Liquids and Solids', ASME/McGraw Hill, New York, p. 321.

[7] Vesovic, V., Wakeham, W. A., Olchowy, G. A. Sengers, J. V., Watson, J. T. R. and Millat, J. (1990) The transport properties of carbon dioxide, J. Phys. Chern. Ref. Data 19, 763~808.

[8] Kestin, J., Ro, S. T. and Wakeham, W. A. (1972) Viscosity of the binary gaseous mixtures He-Ne and He-N2 in the temperature range 25-700°C, J. Chern. Phys. 56, 5837-5842.

[9] Kestin, J. and Wakeham, W. A. (1988) Transport properties of fluids: thermal conductivity, viscosity and diffusion coefficients, Vol. 1, CINDAS 'Data Series on Materials Properties', ed. C. Y. Ho, Hemisphere Publishing, New York.

[10] Haarman, J. W. (1973) Thermal conductivity measurements of He, Ne, Ar, Kr, N 2 and CO2 with a transient hot-wire method, in J. Kestin, (ed.) 'Transport phenomena - 1973', Am. Inst. Phys. Conf. Proc. 11, 193-198.

[11] Wakeham, W. A., Nagashima, A. and Sengers, J. V. (eds.) (1991) Experimen­tal thermodynamics, Vol. III, Measurement of transport properties, Blackwell Scientific Publications, Oxford.

53

[12] Haran, E. N., Maitland, G. C., Mustafa, M. and Wakeham, W. A. (1983) The Thermal conductivity of Ar, N2 , co in the temperature range 300-430 K at pressures up to 10 MPa, Ber. Bunsenges. Phys. Chern. 87, 657-663.

[13] Mardolcar, U. V., Fareleira, J. M. N. A., de Castro, C. A. N. and Wakeham, W. A. (1985) The measurement of the thermal conductivity of argon, High Temp. High Press. 17, 463-476.

[14] Kestin, J., Paul, R., Clifford, A. A. and Wakeham, W. A. (1980) Absolute deter­mination of the thermal conductivity of the noble gases at room temperature up to 35 MPa, Physica 100A, 349-369.

[15] Ross, M., Trusler, J. P. M., Wakeham, W. A. and Zalaf, M. (1990) Thermal conductivity of R134a over the temperature range 240 to 373 K, in 'Thermophys­ical Properties of Pure Substances and Mixtures for Refrigeration', Inti. Inst. Refrig., Paris, 89-94.

[16] Fleeter, R., Kestin, J., Paul, R. and Wakeham, W. A. (1981) The thermal conduc­tivity of mixtures of nitrogen with four noble gases at room temperature, Physica 108A, 371-401.

[17] Dunlop, P. J., Robjohns, H. L. and Bignell, C. M. (1987) Diffusion and thermal diffusion in binary mixtures of hydrogen with noble gases, J. Chern. Phys. 86, 2923-2936.

[18] Taylor, W. A. (1988) Two-bulb thermal separation factors of helium isotopes as a test of recent interatomic potentials, J. Chern. Phys. 88, 7097-7103.

[19] Lambert, J. D. (1977) Vibrational and rotational relaxation in gases, Clarendon Press, Oxford.

[20] Prangsma, G. J., Borsboom, L. J. M., Knaap, H. F. P., Van den Meijdenberg, C. J. N. and Beenakker, J. J. M. (1972) Rotational relaxation in ortho-hydrogen between 170 and 300 K, Physica 61, 527-538.

[21] Millat, J., Plantikow, A., Mathes, D. and Nimz, H. (1988) Effective collision cross-sections for polyatomic gases from transport properties and thermomolecu­lar pressure differences, Z. Phys. Chemie Leipzig, 269, 865-878.

[22] McCourt, F. R. W., Beenakker, J. J. M., Kohler, W. E. and KU8cer, 1. (1990) Non-equilibrium phenomena in polyatomic gases, Oxford University Press, Ox­ford.

[23) Viehland, L. A., Mason, E. A. and Sandler, S. 1. (1978) Effect of spin polarization on the thermal conductivity of polyatomic gases, J. Chern. Phys. 68, 5277-5282; Coope, J. A. R. and Snider, R. F. (1979) On the contribution of angular momentum polarization to gas phase thermal conductivity, J. Chern. Phys. 70, 1075-1077.

[24] Monchick, L. and Mason, E. A. (1961) The transport properties of polar gases, J. Chern. Phys. 35, 1676-1697.

[25] Mason, E. A. and Monchick, L. (1962) Heat conductivity of polyatomic and polar gases, J. Chern. Phys. 36, 1622-1639.

54

[26] Moraal, H. and Snider, R. F. (1971) Kinetic theory collision integrals for diatomic molecules, Chern. Phys. Lett. 9,401-405.

[27] Millat, J., Vesovic, V. and Wakeham, W. A. (1989) Theoretically based data assessment for the correlation of the thermal conductivity of dilute gases, Int. J. Thermophys. 10, 805-818.

[28] Millat, J., Mustafa, M., Ross, M., Wakeham, W. A. and Zalaf, M. (1987) The thermal conductivity of Ar, CO2 and NO, Physica 145A, 461-497.

[29] Millat, J., Ross, M., Wakeham, W. A. and Zalaf, M. (1988) The thermal conduc­tivity ofNe, CH4 and CF4 , Physica 148A, 124-152.

[30] Thijsse, B. J., 't Hooft, G. W., Coombe, D. A., Knaap, H. F. P. and Beenakker, J. J. M. (1979) Some simplified expressions for the thermal conductivity in an external field, Physica 98A, 307-312.

[31] Millat, J., Vesovic, V. and Wakeham, W. A. (1988) On the validity of the sim­plified expression for the thermal conductivity of Thijsse et al., Physica 148A, 153-164.

[32] Bernstein, R. B. (ed.) Atom-molecule collision theory: a guide for the experi­mentalist, Plenum Press, New York.

[33] Assael, M. J., Millat, J., Vesovic, V. and Wakeham, W. A. (1990) The thermal conductivity of methane and tetrafluoromethane in the limit of zero density, J. Phys. Chern. Ref. Data 19, 1137-1147.

[34] Ross, M., Vesovic, V. and Wakeham, W. A. (1991) Alternative expressions for the thermal conductivity of the dilute gas mixtures, Physica (in preparation).

[35] Heck, E. L. and Dickinson, A. S. (1990) Transport and relaxation cross-sections for He-N2 mixtures, Mol. Phys. 70, 239-252.

[36] Gianturco, F. A., Venanzi, M. and Dickinson, A. S. (1990) Classical trajectory calculations of transport and relaxation properties for Ar-N2 mixtures, J. Chern. Phys. 93, 5552-5562.

[37] Wong, C. C. K., McCourt, F. R. W. and Casavecchia, P. (1990) Classical tra­jectory calculations of transport and relaxation properties for N2-Ne mixtures, J. Chern. Phys. 93,4699-4712.

[38] McCourt, F. R. W., Vesovic, V., Wakeham, W. A., Dickinson, A. S. and Mustafa M. (1991) Quantum mechanical calculations of effective collision cross sections for the He-N2 interaction, Mol. Phys. 72,1347-1364.

[39] Maitland, G. C., Mustafa, M., Wakeham, W. A. and McCourt, F. R. W. (1987) An essentially exact evaluation of transport cross sections for a model of He-N2

interaction, Mol. Phys. 61, 359-387.

[40] Maitland, G. C., Mustafa, M. and Wakeham, W. A. (1987) Close-coupled calcu­lations of transport cross-sections for a highly quantal system, Mol. Phys. 62, 397-409.

[41] Dickinson, A. and Lee, M. S. (1986) Classical trajectory calculations for anisotropy dependent cross-sections for He-N2 mixtures, J. Phys. B 19,3091-3107.

55

[42] McCourt, F. R., Fuchs, R. R. and Thakkar, A. J. (1984) A comparison of the predictions of various model He-N2 potential surfaces with experiment, J. Chern. Phys. 80, 5561-5567.

[43] Gianturco, F. A., Venanzi, M. and Dickinson, A. S. (1988) Computed transport coeffficients from Van der Waals systems via realistic interactions, Mol. Phys. 65, 563-584, 585-598.

[44] Kestin, J., Ro, S. T. and Wakeham, W. A. (1972) Viscosity of the binary gaseous mixturesHe-Ne and He-N2 in the temperature range 25-700°C, J. Chern. Phys. 56, 5837-5842.

[45] Maitland, G. C. and Wakeham, W. A. (1978) Direct determination of inter­molecular potentials for gaseous transport coefficients alone, Mol. Phys. 35, 1429-1442.

[46] Maitland, G. C., Vesovic, V. and Wakeham, W. A. (1985) The inversion of thermophysical properties: non-spherical systems explored, Mol. Phys. 54, 301-309.

[47] Maitland, G. C., Mustafa, M., Vesovic, V. and Wakeham, W. A. (1986) The in­version of thermophysical properties: highly anisotropic interactions, Mol. Phys. 57, 1015-1033.

[48] Parker, G. A., Snow, R. L. and Pack, R. T (1976) Intermolecular potential surface from electron gas method. I. Angle and distance dependence of He-C02 and Ar­CO2 interactions, J. Chern. Phys. 64, 1668-1678.

[49] Vesovic, V. and Wakeham, W. A. (1987) An interpretation of intermolecular pair potentials obtained by inversion for non-spherical systems, Mol. Phys. 62, 1239-1246.

CLASSICAL PATH METHODS FOR LINESHAPE CROSS SECTIONS

JEREMY M. HUTSON l

Joint Institute for Laboratory Astrophysics, Campus Box 440, University of Colorado, Boulder, Colorado 80309, USA

ABSTRACT. A classical path method for calculating lineshape cross sections is tested by comparison with quantal calculations for HCI in Ar, using an accu­rate anisotropic potential energy surface obtained from high-resolution spectra of Van der Waals complexes. The classical path method employed is an M­conserving approximation, using exponential perturbation theory. It is found that the classical path method seriously underestimates contributions from rainbow-like trajectories dominated by the attractive well of the potential. The errors are largest for the lowest rotational line, at collision energies comparable to or a little larger than the well depth. Possible ways of improving the classical path calculation are suggested.

1. Introduction

The study of high-resolution spectra of Van der Waals complexes has provided inter­action potentials for a range of prototype systems such as Rg-H2 [1-3], Rg-HCI [4-6] and Rg-HF [7,8], where Rg indicates a rare gas atom (most commonly Ar). It has been possible to determine up to 20 parameters of interaction potentials by fitting to such data [6], so that the sophistication of the potential forms used is well beyond that usually determined from scattering data. These potentials are generally very well determined in the region of the Van der Waals well, since that is the region sampled by the bound states of the complex. They are usually less reliable in the short-range region, where they rely either on extrapolation from the well region or on constraints based on ab initio calculations.

Properties that depend on inelastic collisions clearly contain information on the short-range potential. These include inelastic molecular beam scattering, relaxation properties and spectroscopic lineshapes. However, quantum scattering calculations are very expensive, and it has not yet been possible to include them inside a least­squares fitting loop for determining many-parameter potentials, though they have been used to make manual adjustments to potential parameters. Quantum scattering

1 Permanent address: Department of Chemistry, University of Durham, South Road, Durham, DH1 3LE, England

57

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 57-72. ©1992 Kluwer Academic Publishers.

58

calculations are particularly expensive for systems containing heavier collision part­ners such as Ar, which are precisely the ones whose potentials are best known from Van der Waals spectroscopy. Various approximate methods, such as classical path approximations, have been used to calculate inelastic scattering processes: they are very much cheaper, but their accuracy has not been properly documented.

Green [9] has recently performed benchmark close-coupling and coupled states calculations of pure-rotationallineshape cross sections for HCI in Ar, using a poten­tial obtained by Hutson [4] from Van der Waals spectroscopy. The agreement with experimental line-broadening cross sections [10-13] was very good for a range of tem­peratures and rotational lines, except for the j = 0 ---> 1 line at the lowest temperature measured (125 K). The purpose of the present paper is to use these benchmark cal­culations to evaluate a simple classical path scattering method for lineshape cross sections.

2. The interaction potential

The Ar-HCl H6(3) potential [4) used in the present work was obtained from molecular beam microwave [14-16] and far-infrared laser [17-22] spectra of the Ar-HCl Van der Waals complex. It is reliable in the well region over the complete range of angle 0, because the far-infrared spectra probe bending and stretching states which sample the whole range. In contrast, earlier potentials for Ar-HCI [23-25] were based principally on data from microwave spectra of the ground Van der Waals vibrational state, which sampled only geometries near equilibrium.

A similar potential, obtained from near-infrared spectra [26,27]' is available for Ne interacting with H CI (v = 1) [5], and potentials based on microwave spectra are avail­able for Kr-HCl and Xe-HCl [25], but far-infrared spectra have not yet been reported for these systems. Near-infrared spectra of Ar-HCl have also been observed [27-30]' and together with the far-infrared spectra have been used to determine the depen­dence of the potential on the HCl stretching motion [6].

The approach taken in the potential determination [4] was to parameterise the intermolecular potential and calculate spectroscopic parameters from a trial poten­tial using (essentially exact) coupled-channel bound-state calculations [31,32). The potential parameters were then optimised, performing a least-squares fit to the ex­perimental parameters. Since the spectroscopic observables are accurate (and could be calculated) to four or more significant figures, very precise potential information was obtained.

The standard coordinate system for atom-diatom systems is used. The vector from the CI nucleus to the H nucleus is denoted r and has length r, while that from the H35 Cl centre of mass to the Ar atom is denoted R, of length R. The angle between Rand r is 0, which is thus zero at the linear Ar-H-Cl geometry. The H6(3) potential is an effective potential V(R,O) for Ar interacting with H35Cl in its v = 0 vibrational state, so that the HCI internuclear distance r does not appear explicitly.

The intermolecular potential V(R, 0) is parameterised in terms of the well depth E and equilibrium distance Rm , both of which are allowed to vary with O. The angle-

dependence is written in terms of Legendre expansions

£(B) = L £>.P>.(cos B), >.

Rm(B) = L Rm>.P>.(cosB). >.

At each angle, the R-dependence of the potential is of the form

8

59

(1)

(2)

V(R,8) = A(B) exp( -jJR) + Vind(R, B) - L Cn(B)Dn(R) R-n. (3) n=6

Induction and dispersion terms up to order R-7 were constrained to theoretical values, and the coefficients C8 (8) and A(B) were chosen (for each value of B) to reproduce the required £(8) and Rm(B) as described below.

For an atom-diatom system such as Ar-Hel, the long-range potential is made up of induction and dispersion contributions. To order R-7 , the induction energy is

and the angle-dependent dispersion coefficients are

where

(5)

(6)

(7)

(8)

(9)

(10)

The notation for molecular properties is that of Buckingham [33]' and the subscripts II and..l apply to properties of Hel. The Hel multipole moments and polarizabilities used in the H6(3) potential are summarized in Table I of Ref. 4. The dispersion expansion was damped at short range, using the damping functions Dn(R) proposed by Tang and Toennies [34]' namely

n (jJR)m Dn(R) = 1 - exp( -jJR) ~ --.

~ m! m=O

(11)

60

The induction energy was not damped, since Knowles and Meath [35] have performed ab initio calculations of induction forces in Ne-HF and found that the induction damping functions are close to unity.

The quantities actually varied in the least-squares fits were the Legendre com­ponents of E( 0) and Rm (0) and the exponent (3. The optimised values are given in Table II of Ref. 4. However, the dynamical calculations require a potential expressed as an expansion in Legendre polynomials at each R, i.e.

V(R,O) = 2: V>. (R)P>. (cos 0). (12) >.

The procedure used in the present work was to evaluate E(O) and Rm(O) at a set of 12 Gauss-Legendre quadrature points; the potential at each angle was then re-expressed in the form of Eq. (3), in terms of A(O) and Cn(O) for n = 6, 7 and 8. C6 (0) and C7 (0) were taken from Eqs. (5)-(10), and C8 (0) and A(O) were chosen to reproduce E( 0) and Rm (0); the equations defining C8 (0) and A( 0) are given in the Appendix of Ref. 4. Finally V(R,O) was evaluated for each R at the twelve quadrature points and the coefficients of the first nine Legendre polynomials were projected out by Gauss-Legendre quadrature [36].

3. Classical path methods

In general terms classical path approximations treat the relative translational motion of the collision partners classically, but handle their internal (vibrational and rota­tional) states quantum-mechanically. The intermolecular potential is decomposed as

V(R, 0) = Veff(R) + V'(R,O), (13)

in which Veff(R) is an effective isotropic potential (usually taken to be the same as Vo(R)). For a given translational energy E and impact parameter b, a classical trajectory is calculated on Veff(R) , providing the intermolecular distance R(t) and deflection angle 8(t) as functions of time t. The part of the potential not included in v"ff(R), V'(t), is then evaluated along the trajectory, and the time-evolution of the internal states under the influence of V'(t) is treated by time-dependent perturbation theory (or some other approximate method). This provides a classical path scattering matrix S(b, E), which can then be processed to provide cross sections of various types.

There is thus a hierarchy of classical path methods similar to the hierarchy of quantum-mechanical scattering methods. Various decoupling approximations (such as the coupled states approximation) can be made, and perturbative treatments anal­ogous to the distorted wave Born approximation and exponential distorted wave ap­proximation are possible.

The most complete form of the classical path approximation was introduced by Neilsen and Gordon [37]. They calculated curved classical trajectories for the isotropic part Vo(R) of the full anisotropic potential, and then solved the first-order matrix

61

differential equations of time-dependent perturbation theory to follow the evolution of the rotational wavefunction along each classical path. The only approximations in this method are those inherent to the classical path approach:

• The (time-independent) quantal translational wavefunctions are replaced by (time-dependent) classical trajectories. Unless different trajectories are allowed to interfere with one another as described by Dickinson [38], this results in the neglect of interference effects between different partial waves, and hence to a loss of rainbow and diffraction oscillations in differential cross sections and a loss of glory oscillations in integral cross sections.

• The classical path is taken to be independent of the rotational state of the molec­ule. This is an approximation, and it might be possible to find an effective po­tential better than Vo(R). It might even be appropriate to make Veff(R) depend on the instantaneous rotational wavefunction, but this would require a separate trajectory calculation for each initial state.

• The classical path approximation allows transfer of energy between translation and rotation, but does not allow such changes to affect the classical path: the translational velocity is assumed to be unaffected. Indeed, it is possible to have collisions that result in more rotational excitation than the available translational energy allows.

It is usually convenient to measure time t with respect to the moment of closest approach, and the deflection angle 8(t) relative to its value at t = 0, so that R(t) and 8(t) are even functions of t. The rotational wavefunctions are conveniently chosen to be quantised along a Z axis defined by the intermolecular vector at t = 0; the X axis is taken to lie in the plane of the collision, and the polar coordinates of r in this axis system are (fJ, a). The intermolecular potential has cylindrical symmetry about the instantaneous intermolecular vector, as in Eq. (12), but its expansion about the quantisation axis involves an additional rotation matrix,

V(,6, a) = LVA(R)CAO(O,O) A

= L VA(R) L CAV(fJ, a)D~o(O, 8(t), 0) A v

= L VA(R)CAV U3, a)CAv (8(t), 0), AV

(14)

where the functions CAV are renormalised spherical harmonics [39]. In a full calcu­lation such as that of Neilsen and Gordon [37], this rotation is properly taken into account. However, an approximation that is commonly made is the "peaking approx­imation" [40]' in which the time-dependence of 8(t) is neglected (and is set equal to zero). This has the effect of making V'(t} (and hence the S matrix) diagonal in the projection quantum number M: it is a good approximation if the collisions are domi­nated by short-range effects that occur near the point of closest approach. This is anal­ogous to the coupled states or centrifugal sudden approximation of quantal scattering

62

theory. However, as discussed by Dickinson and Richards [41], the requirement for M to be conserved is less rigorous: it is only necessary to ignore the terms with v i- 0 in Eq. (14). They thus suggested that, when making M-conserving calculations, it would be advantageous to include an additional factor of C>.o(8(t), 0) = P>.(cos 8(t)) in the matrix elements diagonal in M. This correction will be termed the orbital correction in the following discussion.

Finally, whether or not the peaking approximation is used, the time-evolution of the rotational wavefunction may be approximated in terms of integrals over V'(t) instead of solving the time-dependent perturbation theory equations explicitly. This is analogous to the exponential-distorted-wave method in quantal scattering theory [42J. A phase integral matrix 7](b, E) may be defined in terms of the Fourier transform of V' (t) for each trajectory [40,43]'

1100

UMI7](b,E)Ii'M') = Ii -00 UMIV'(t)Ii'M')exp(iWjjd)dt (15)

where Wjj' is the angular velocity associated with the energy difference between rota­tional states IjM) and Ii'M'). The S matrix may then be approximated as the first term in the Magnus expansion [44J of a time-ordered exponential,

S(b, E) ~ exp(i7](b, E)). (16)

Note that this involves the exponentiation of a matrix, which may be readily achieved by first diagonalising 7](b, E).

Further approximations are possible. Smith et al. [40J have used approximate analytical expressions for the matrix exponential of Eq. (16), and various authors have used straight rather than curved classical paths. This last approximation leads to considerable simplification, but requires the use of an arbitrary cutoff procedure to avoid trajectories in which the colliding particles pass through one another; it is unnecessary with modern computers.

The classical path calculations in the present work used a method similar in spirit to that of Smith et al. [40], in which M was assumed to be conserved, and S matrices were evaluated from the exponential approximation of Eq. (16). In the present cal­culations, however, the phase shift matrix 7] was constructed without approximations in the angular momentum algebra, and the exponentiation was carried out exactly.

4. Lineshape cross sections

S matrices from classical path calculations may be used for a variety of purposes. In the present context we are concerned with lineshape cross sections defined by [37J

K q

j' ) m' j3

(17)

63

where j and l' are the levels connected by the spectroscopic transition and K is the tensor order of the transition, 1 for dipole transitions, and 2 for Raman transitions. Within the M-conserving approximation, ma = m{3 and m~ = m~, so the equation simplifies to

(Jjj,(b,E) =1 _ ~ (j K L -m q

mm'q

l')( j K l') m' -m q m' (18)

x (j'm'IS*(b, E)Il'm'}(jmIS(b, E)ljm}.

The partial cross section (Jjj' (b, E) is a dimensionless quantity. The energy-dependent integral cross section (Jj1' (E) is just the integral of (Jjj' (b, E) over impact parameter,

(19)

and the experimentally observable quantity is the thermal average of this,

(Jjj,(T) = (_1_) 2 r:jj,(E) exp( -E/kBT) EdE. kBT Jo (20)

The real and imaginary parts of (Jjj,(T) are cross sections for pressure broadening and shifting, respectively.

The quantum-mechanical expressions evaluated by Green [9] and in the present work are quite similar, although the S matrices are labelled by the orbital angular momentum I rather than b. In the coupled states approximation, the lineshape cross sections are [45]

7r [ (j K j') (j K j') (Jj1'(E) = k2 L(21 + 1) 1- L -m q m' -m q m' J I mm'q (21)

where the different S matrices are evaluated at the same translational (not total) energy E, and kJ is 2/1,E /1i2. If we associate the orbital angular momentum I with an

impact parameter b through the relationship b2 = 1i2Z(l + 1)/2/1,E, the quantal cross section may be written in the form of Eq. (19), with

(Jjj,(b,E) =1- ~ (j K L -m q

m'Tn'q

l')( j K l') m' -m q m'

(22)

x (j'm'ISI*(E)Il'm'} (jmISI(E)ljm}.

This quantal form of (Jjj' (b, E) is again dimensionless, and allows direct compar­ison between quantal calculations and the results of Eq. (18) at the partial cross

64

section level. In the present work, coupled states partial cross sections were evalu­ated with the MOLSCAT program [46], using the diabatic log-derivative propagator of Manolopoulos [47]. The results obtained were in good agreement with the more extensive calculations of Green [9].

5. Results

It is useful to compare the performance of the classical path method with quantal calculations at a fairly detailed level, even though the ultimate quantities of interest are highly averaged. Classical path partial cross sections (fjjl (b, E) are shown for translational energies of 27 cm -1 , 200 cm -1 and 400 cm -1 in Fig. 1, both with and without the orbital correction, and are compared with coupled states results.

At the lowest energy, the quantal cross sections show very fast oscillations due to resonances as a function of I. This structure is completely absent in the classical path cross section, which nevertheless follows the general outline of the quantal curve. Both quantal and classical curves show a sharp cutoff around b = 7.5 A: this value of b marks the onset of the 3-turning-point region. Classically, collisions with b > 7.5 A bounce off the centrifugal barrier without ever reaching the well region, and so contribute very little to the cross section.

At higher energies, the coupled states partial cross sections for the j = 0 -+ 1 line are quite different from those for higher-j lines. The resonance structure has almost disappeared, and the 0 -+ 1 cross sections are dominated by a large peak around b = 4.5 A, which is slightly larger than Rm for this system. Much smaller peaks at the same value of b occur in the higher-j quantal cross sections at 200cm-1 , but they have almost disappeared by 400 cm -1. Close-coupling calculations also show a large peak in the partial cross section at about b = 4.5 A. The main deficiency of the classical path cross sections is that they underestimate the height of this peak for the o -+ 1 line, and show it persisting to higher energies than in the quantum calculations for higher-j lines. This results in a substantial underestimate of the cross section for the 0 -+ 1 line.

The b = 4.5 A peak is dominated by the attractive part of the potential. In classical terms, it involves trajectories that swing round in the attractive potential, experiencing the attractive anisotropy for quite a long time but without experiencing a hard collision with the repulsive wall; these will be termed "rainbow trajectories" in the following discussion. The corresponding 17 matrix elements are quite large (approaching 7r), so that the exponential approximation of Eq. (16) is suspect. In addition these are the trajectories for which the effect of the anisotropy is not domi­nated by the point of closest approach, and for which 8(t) differs considerably from 0, so that the peaking approximation is also likely to be poor.

It might be expected that the breakdown of the peaking approximation would be alleviated by including the orbital correction. Unfortunately, this appears not to be the case for the rainbow trajectories, as shown in Fig. 1, though there appears to be some improvement in the j = 1 -+ 2 and 2 -+ 3 cross sections at 27 cm -1. For the rainbow trajectories, the effect of the orbital correction is simply to reduce the

65

1.5

1.0

0.5

0.0 L----l_--L._-L._...l.-_.L.:!-._---l

1.5 Coupled stales, E=27 em -1

1.5 CP (Orbital), E=27 em -1

1.0

D.S 0.5

0.0 L----l_--L._-L._-'--_..I..---""'............... 0.0 L----l._--1.._--L_..L-_Ll ___ ---l

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0

1.5 Coupled stales, E=27 em-1 b2/A2

10.0 20.0 30.0 40.0 50.0 60.0 70.0

b2/A2

Figure l(a). Comparison of quantal and classical path line-broadening cross sections for Ar-HCl at E = 27cm- 1 . 0"01: 0,0"12: e, 0"23: D.

66

1.5

&J'" 1.0

~

Ui CP (Peaking), E=200 em-I CP {Peaking}, E=400 em -1

1.5

&J'" 1.0 1.0

~

0.:1

0.0 L_--L __ ...l..-_~I!liiIIII"" __ .......I 0.0 L_-L.._.....JL-":~ __ ..... _-J

1.5 CP (Orbital), E=200 em -1 CP (Orbital). E=400 em-1

1.5

&J'" 1.0 1.0

~

0.5 0.5

L_--L. __ L-~~"'_"'_.J 0.0 .~--L--~!!!.iiilli1l"'_ ... _.J 10.0 20.0 30.0 40.0 50 0.0 10.0 20.0 30.0 40.0 60.0

b2/A2 b'l/A2

Figure 1 (b). Comparison of quantal and classical path line-broadening cross sections for Ar-HCI at E = 200cm- 1 and E = 400cm- 1 . (J01: 0, (J12: e, (J23: D.

67

Table 1. Comparison of energy-dependent integrallineshape cross sections (in A2) for pure rotational lines of HCI in Ar.

Method jmax ReaOl ImaOl Rea12 Im a 12 Rea23 Ima23

E = 27cm-1

Classical path (peaking) 7 160.0 -79.3 131.0 -21.4 116.2 -9.9 Classical path (orbital) 7 142.8 -64.2 121.2 -14.8 110.2 -5.7 Coupled states (this work) 5 199.6 -53.9 138.1 -36.8 115.5 -22.0 Coupled states (Ref. 9) 7 197.5 -50.8 138.3 -34.1 Close-coupling (Ref. 9) 6 159.8 -62.1 129.0 -17.0

E = 200cm-1

Classical path (peaking) 7 71.4 -24.0 65.1 -9.4 49.3 -3.1 Classical path (orbital) 7 55.9 -12.9 54.1 -4.0 44.1 -1.3 Coupled states (this work) 5 96.3 -32.5 58.3 -3.9 50.0 -4.8 Coupled states (Ref. 9) 7 96.2 -31.6 58.0 -4.0 50.2 -4.3 Close-coupling (Ref. 9) 7 99.6 -25.4 52.9 +1.4 46.1 -1.9

E = 400cm-1

Classical path (peaking) 7 53.8 -9.3 50.2 -3.5 39.5 -1.3 Classical path (orbital) 7 48.4 -4.9 44.6 -1.7 37.1 -0.7 Coupled states (this work) 5 79.7 -27.2 46.7 +1.2 40.3 -1.1 Coupled states (Ref. 9) 8 79.6 -27.1 46.5 +1.2 40.5 -0.9 Close-coupling (Ref. 9) 7 78.1 -16.4 46.6 +1.2 38.5 +0.3

low-frequency'T/ matrix elements, and thus to reduce the corresponding cross sections; this is the opposite of the change needed.

In order to distinguish between the breakdown of the M -conserving approxi­mation and breakdown of the exponential approximation, preliminary calculations including the v i= 0 terms in Eq. (14) have been carried out, for a collision energy of 400 cm -1 (see Table 1). The calculations do show a slight enhancement of the peak at b = 45 A but not nearly enough to give agreement with the quantal calculations.

The principal errors thus appear to arise from the exponential approximation. It would, of course, be possible to solve the time-dependent perturbation equations explicitly; this would involve a relatively large amount of computer time, though it would still be cheap compared to the quantal calculations. However, there are also other possible ways of correcting the simple exponential approximation, and these will be investigated in future work.

The integral line-broadening cross sections (which are just 7r times the areas under the curves in Fig. 1) are compared in Table 1. The classical path results are

68

Table 2. Comparison of thermally averaged lineshape cross sections (in A 2) for pure rotational lines of HCI in Ar at 300 K.

Method Re 0"01 1m 0"01 Re 0"12 1m 0"12 Re 0"23 1m 0"23 Re 0"34 1m 0"34

Classical path (peaking) 62.3 Classical path (orbital) 54.8 Quantal (Ref. 9) 86.6

-17.3 -10.0 -21.2

58.4 50.5 52.6

-6.6 -2.9 -0.3

46.3 42.6 44.6

-2.4 -1.0 -1.4

41.0 38.5 39.4

-1.4 -0.6 -2.7

quite poor for the 0 -4 1 line, except at E = 27 cm- 1 , where the rainbow trajectories are relatively unimportant. For the higher- j lines the classical path approximations are much better, especially at higher energies. It appears that there is again some cancellation of errors; the detailed b-dependence of the cross sections is not the same in the quantal and classical path calculations, but the total cross sections are remarkably accurate, as shown in Table 1. This cancellation of errors may not be coincidental, since there may be interference effects in the partial cross sections that are averaged over by the classical path calculations.

The integral line-shift cross sections are also given in Table 1. However, in this case there is little correspondence between the classical path and quantal results, even for the higher-j lines, except perhaps at low energy. This is not surprising, as even the quantal coupled states approximation is known to give poor results for line-shift cross sections.

The thermally averaged line-width and line-shift cross sections at 300 K are com­pared with Green's quantal results in Table 2. The classical path results are quite poor for the j = 0 -4 1 line, but increasingly good for the higher lines.

The method described here can be applied to a range of other properties, in­cluding relaxation cross sections and off-diagonal pressure broadening (line coupling) cross sections. However, care must be exercised in handling the translational part of the phase shift matrix, which is relatively unimportant for line-broadening cross sections, but may be more important for other properties.

6. Conclusions

An M-conserving approximation for calculating line-width and line-shift cross sections has been tested by comparison with accurate quantal calculations for HCI in Ar, using a reliable intermolecular potential derived from high-resolution spectroscopy of Van der Waals complexes. The classical path approximation was found to break down seriously for trajectories that are dominated by the attractive part of the potential. This caused large errors in the calculated cross sections for the j = 0 -4 1 line, and smaller errors for other cross sections. A correction suggested by Dickinson and Richards [41] was found not to result in any improvement in the calculated cross sections.

69

Acknowledgment

The author wishes to thank the Joint Institute for Laboratory Astrophysics for the award of a Visiting Fellowship, which allowed this work to be carried out.

References

[1] Le Roy, R. J. and Van Kranendonk, J. (1974) Anisotropic intermolecular poten­tials from an analysis of spectra of H2 - and D2 -inert gas complexes, J. Chern. Phys. 61, 4750-4769.

[2] Le Roy, R. J. and Carley, J. S. (1980) Spectroscopy and potential energy surfaces of Van der Waals molecules, J. Chern. Phys. 42,353-420.

[3] Le Roy, R. J. and Hutson, J. M. (1987) Improved potential energy surfaces for the interaction of H2 with Ar, Kr, and Xe, J. Chern. Phys. 86, 837-854.

[4] Hutson, J. M. (1989) The intermolecular potential of Ar-HCI: determination from high-resolution spectroscopy, J. Chern. Phys. 89, 4550-4557.

[5] Hutson, J. M. (1989) The intermolecular potential of Ne-HCI: determination from high-resolution spectroscopy, J. Chern. Phys. 91, 4448-4454.

[6] Hutson, J. M. (1991) Vibrational dependence of the anisotropic intermolecular potental of Ar-HF, to be published.

[7] Nesbitt, D. J., Child, M. S. and Clary, D. C. (1989) Rydberg-Klein-Rees inversion of high resolution Van der Waals infrared spectra: an intermolecular potential energy surface for Ar+HF(v = 1), J. Chern. Phys. 90,4855-4864.

[8] Hutson, J. M. (1991) Vibratioonal dependence of the anisotropic intermolecular potential of Ar-HF, J. Chern. Phys., submitted for publication.

[9] Green, S. (1990) Theoretical line shapes for rotational spectra of HCI in Ar, J. Chern. Phys. 92,4679-4685.

[10] Gebbie, H. A. and Stone, N. W. B. (1963) Measurement of widths and shifts of pure rotation lines of hydrogen chloride perturbed by rare gases, Proc. Phys. Soc. (London) 82, 309-314.

[11] Scott, H. E. and Sanderson, R. B. (unpublished): results quoted in Ref. 37.

[12] Van Aalst, R. M., Schuurrnan, J. A. and Van der Elsken, J. (1975) Temperature dependence of the pressure induced width and shift of the rotational lines of HCI, Chern. Phys. Lett. 35, 558-562.

[13] Frenkel, D., Gravesteyn, D. J. and Van der Elsken, J. (1976) Non-linear density dependence of rotational line-broadening of HCI in dense argon, Chern. Phys. Lett. 40, 9-13.

[14] Novick, S. E., Davies, P., Harris, S. J. and Klernperer, W. (1973) Determination of the structure of ArHCI, J. Chern. Phys. 59, 2273-2279.

[15] Novick, S. E., Janda, K. C., Holmgren, S. L., Waldman, M. and Klernperer, W. (1976) Centrifugal distortion of ArHCI, J. Chern. Phys. 65,1114-1116.

70

[16] Hutson, J. M. and Howard, B. J. (1981) High resolution radiofrequency spec­troscopy of Ar ... HCI, J. Chern. Phys. 74, 6520-652l.

[17] Marshall, M. D., Charo, A., Leung, H. o. and Klernperer, W. (1985) Character­ization of the lowest-lying II bending state of Ar-HCI by far infrared laser-Stark spectroscopy and molecular beam electric resonance, J. Chern. Phys. 83, 4924-4933.

[18] Ray, D., Robinson, R. L., Gwo, D.-H. and Saykally, R. J. (1986) Vibrational spectroscopy of Van der Waals bonds: measurement of the perpendicular bend of ArHCI by intra cavity far infrared laser spectroscopy of a supersonic jet, J. Chern. Phys. 84, 1171-1180.

[19] Robinson, R. L., Gwo, D.-H., Ray, D. and Saykally, R. J. (1987) Evidence for a secondary minimum in the ArHCI potential surface from far infrared laser spectroscopy of the lowest ~ bending vibration, J. Chern. Phys. 86, 5211-5212.

[20] Robinson, R. L., Gwo, D.-H. and Saykally, R. J. (1987) An extended study of the lowest II bending vibration-rotation spectrum of Ar-HCI by intra cavity far infrared laser/microwave double resonance spectroscopy, J. Chern. Phys. 87, 5149-5155.

[21] Robinson, R. L., Gwo, D.-H. and Saykally, R. J. (1987) The high-resolution far infrared spectrum of a Van der Waals stretching vibration: the V3 band of Ar-HCI, J. Chern. Phys. 87, 5156-5160.

[22] Busarow, K. L., Blake, G. A., Laughlin, K. B., Cohen, R. C., Lee, Y. T. and Saykally, R. J. (1988) Tunable far infrared laser spectroscopy of Van der Waals bonds: extended measurements on the lowest ~ bend of ArHCI, J. Chern. Phys. 89, 1268-1276.

[23] Holmgren, S. L., Waldman, M. and Klernperer, W. (1978) Internal dynamics of Van der Waals complexes. II. Determination of a potential energy surface for ArHCI, J. Chern. Phys. 69, 1661-1669.

[24] Hutson, J. M. and Howard, B. J. (1981) The intermolecular potential energy surface of Ar-HCI, Mol. Phys. 43, 493-516.

[25] Hutson, J. M. and Howard, B. J. (1982) Anisotropic intermolecular forces I. Rare gas-hydrogen chloride systems, Mol. Phys. 45, 769-790.

[26] Lovejoy, C. M. and Nesbit, D. J. (1988) The near-infrared spectrum of NeHCI, Chern. Phys. Lett. 147,490-496.

[27] Schuder, M. D., Nelson, D. D. and Nesbitt, D. J. (1991) Investigation of internal rotor' dynamics of NeDCI and ArDCI via infrared absor'ption spectroscopy, J. Chern. Phys. 94, 5796-581l.

[28] Howard, B. J. and Pine, A. S. (1985) Rotational pr'edissociation and libration in the infrared spectrum of Ar-HCI, Chern. Phys. Lett. 122, 1-8.

[29] Lovejoy, C. M. and Nesbitt, D. J. (1988) Infrared-active combination bands in ArHCI, Chern. Phys. Lett. 146, 582-588.

71

[30] Lovejoy, C. M. and Nesbitt, D. J. (1988) Sub-Doppler infrared spectroscopy in slit supersonic jets, Faraday Discuss. Chern. Soc. 86, 13-20.

[31] Hutson, J. M. (1991) An introduction to the dynamics of Van der Waals molec­ules, Advances in Molecular Vibrations and Collision Dynamics 1, 1-44.

[32] Hutson, J. M., BOUND computer code, distributed via Collaborative Computa­tional Project No. 6 of the UK Science and Engineering Research Council, on Heavy Particle Dynamics.

[33] Buckingham, A. D. (1967) Permanent and induced molecular moments and long­range intermolecular forces, Adv. Chern. Phys. 12, 107-142.

[34] Tang, K. T. and Toennies, J. P. (1984) An improved simple model for the Van der Waals potential based on universal damping functions for the dispersion co­efficients, J. Chern. Phys. 80,3726-3741.

[35] Knowles, P. J. and Meath, W. (1987) A separable method for the calculation of dispersion and induction energy damping functions with applications to dimers arising from He, Ne and HF, Mol. Phys. 60,1143-1158.

[36] Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions, National Bureau of Standards.

[37] Neilsen, W. B. and Gordon, R. G. (1973) On a semiclassical study of molecular collisions. I. General method, J. Chern. Phys. 58, 4131-4148; Neilsen, W. B. and Gordon, R. G. (1973) On a semiclassical study of molecular collisions. II. Application to HCI-argon, J. Chern. Phys. 58, 4149-4170.

[38] Dickinson, A. S. (1981) Differential cross sections in curved-trajectory impact parameter methods, J. Phys. B 14, 3685-3691.

[39] Brink, D. M. and Satchler, G. R. (1968) Angular Momentum, 2nd edition, Claren­don Press, Oxford.

[40] Smith, E. W., Giraud, M. and Cooper, J. (1976) A semiclassical theory for spec­tralline broadening in molecules, J. Chern. Phys. 65, 1256-1267.

[41] Dickinson, A. S. and Richards, D. (1977) A semiclassical study of the body-fixed approximation for rotational excitation in atom-molecule collisions, J. Phys. B 10, 323-343; Dickinson, A. S. and Richards, D. (1978) On an M-conserving ap­proximation in time-dependent theories of rotational excitation in atom-molecule collisions, J. Phys. B 11, 3513-3528.

[42] Levine, R. D. and Balint-Kurti, G. G. (1970) Opacity analysis of inelastic molec­ular collisions. Exponential approximataions, Chern. Phys. Lett. 6, 101-105; Balint-Kurti, G. G. and Levine, R. D. (1970) Opacity analysis of inelastic molec­ular collisions. Computational studies of the exponential Born approximation for rotational excitation, Chern. Phys. Lett. 7, 107-111.

[43] Cross, R. J. (1968) Semiclassical theory of inelastic scattering: an infinite-order distorted-wave approximation, J. Chern. Phys. 48, 4838-4842.

[44] Magnus, W. (1954) On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649-673.

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[45J Goldflam, R. and Kouri, D. J. (1977) On accurate quantum mechanical approx­imations for molecular relaxation phenomena. Averaged jz-conserving coupled states approximation, J. Chern. Phys. 66, 542-547.

[46J Hutson, J. M. and Green, S. (1986) MOLSCAT computer code, version 9, dis­tributed by Collaborative Computational Project No. 6 of the UK Science and Engineering Research Council, on Heavy Particle Dynamics. Further information can be otbained from the author.

[47] Manolopoulos, D. E. (1986) An improved log derivative method for inelastic scat­tering, J. Chern. Phys. 85,6425-6429.

CROSSED BEAM STUDIES

M. FAUBEL Max-Planck-/nstitut Jur StromungsJorschung Bunsenstrasse 10 D-34 00, Gottingen, Germany

ABSTRACT. Crossed molecular beam experiments yield detailed integral, dif­ferential or internal state-selected cross sections for elementary molecular col­lision processes. Experimental limits are set by finite state and angular reso­lutions. For representative atom-atom, atom-diatom and for linear polyatomic collision partners anisotropic molecular interaction potentials are analyzed by sub-meV resolution, rotationally inelastic scattering cross section measurements.

1. Introduction

Crossed molecular beam experiments explore directly the elementary processes of gas phase phenomena: they allow the measurement of state-selected scattering cross sec­tions with highly resolved scattering angles for well-defined collision energies. General surveys have been given in the monographs [1,2,3] and in several review articles, for example Ref. [4] and Ref. [5].

In an ideally resolved state-to-state cross section for binary collisions,

dai-->f (~E ) ~w, CM, (1)

the initial and final internal molecular states Ii) and If) specify electronic states, vi­brations or a rotational quantum state. They may also represent a change of the molecular species in reactive collisions. In the ordinary range of transport property studies, with translational collision energies ECM from 0.01 to 1 eV, elastic scat­tering and inelastic transitions between rotational states are the dominant collision events. The rotational states for simple diatomic molecules and for linear molecules are specified by the rotational and magnetic quantum numbers j and m

Ii), If) = Ij, m) . (2)

In practice most scattering experiments average over the alignment quantum numbers m, with but a few, noticeable, exceptions [6]. With averaging over m-states, the scattering cross section angular dependence in (1) reduces to a single variable for the centre-of-mass angle w = (8, ll» = 8 CM . The characterization of the rotational states

73

W.A. Wakeham et a1. (eds.), Status and Future Developments in Transport Properties, 73-115. ©1992 Kluwer Academic Publishers.

74

of non-linear, triatomic and larger molecules requires a considerably more complex theoretical effort with a third rotational quantum number, k, for the projection of the rotational angular momentum onto the principal axis of the molecule. The rotational Schr6dinger eigenfunctions are then to be described by linear combinations of the 'rotation matrices' [7]

(xli k m) = D~m(a,{3,'Y), (2a)

rather than the more familiar spherical harmonics

(2b)

which represent the rotational eigenstates of linear molecules. Apart from the experimental problems and challenges of the most complete res­

olution of internal and of translational molecular states, serious limitations for the usefulness and the reliability of experimental conclusions are frequently also set by the availability of appropriate numerical, quantitative theories for cross section cal­culations for the collision system investigated. A survey of the standard numerical theories is given in Ref. [8]. The scattering dynamics of molecular collisions is gov­erned by the molecular interaction potentials. For a given electronic molecular state these depend only upon the interatomic distances between nuclei. For convenience these distances are frequently expressed in Jacobi coordinates as a function of the intermolecular distance R between the centres-of-mass of the colliding molecules and of the intramolecular nuclear coordinates rint

V electronic state (r .) - V(R r· ) nuclei - 'Jnt . (3)

The de facto use of intermolecular forces and potentials is legalized by the Born­Oppenheimer separation. It splits the Schr6dinger equation for the simultaneous motion of the positively charged nuclei and the negative electrons in their mutual electromagnetic fields into two parts. The eigenvalue equation for the relatively fast electronic motion needs then only to be solved for fixed nuclear positions by the meth­ods of quantum chemistry. The eigenvalues are the total binding energy of the collision system at (R, rind. The actual molecular potential is obtained as the difference be­tween two large numbers, one of which is the energy of the 'supermolecule' which consists of all the relevant nuclei and electrons taken together, and the other is the sum of the individual energies of the separated collision partners. Examples of such ab initio potentials will be considered subsequently. But, for the practical requirements of scattering and of transport theory, satisfactory accuracy is, at present, only obtained with phenomenological, fitted potentials, or by empirically corrected potential models with some input of ab initio and of spectroscopic data on the molecules. In the second step of the Born-Oppenheimer approximation, the quantum-mechanical equations of motion for the point-mass nuclei are solved on the Born-Oppenheimer potential sur­face. The cross section is then determined by the asymptotic scattering amplitude of the wave function for the nuclei. The eigenfunctions of the nuclear motion are usually developed into products of translational factors and internal wave functions

75

of the isolated molecules as given, e.g. in Eq.(2). This computational scheme, due to Arthurs and Dalgarno and to Jacob and Wick, leads to an isolated set of coupled differential equations for each total angular momentum, J. The number of coupled equations which have to be solved simultaneously is determined by the number of internal molecular states which may contribute to the scattering under the specific collision conditions. For rotational transitions in linear molecules, j + 1 magnetic m-state alignments have to be carried through in the coupled equations. This leads to a very rapid increase in the size of the computational problem when rotationally inelastic scattering is considered.

At present the numerical effort required restricts the application of the exact theory to the scattering between atoms and molecules, with no more than some few hundred internal states involved. Notable extensions to molecule-molecule collisions exist, e.g. for H2-D2 , and HF-HF scattering [9], in which exceptionally wide spacings of the rotational energy levels relative to room temperature collision energies mean that very few rotational levels are thermally accessible. The practicable cases are thus essentially restricted to the scattering of linear molecules against structureless atoms. In such cases the molecular potential reduces to a function of three relevant variables, namely

V(R,rint) = V(R,T,'Y) (3a)

for molecules which can vibrate in the coordinate T. For rigid molecules, for which 1" = T e , the potential can be represented as

(3b)

in which 'Y describes the angle between the molecular axis and the atom-molecule line. Even this simplest problem still leads to such time-consuming algorithms for calculations from potential to cross section that considerable interest exists in the investigation of more approximate, faster schemes, using classical, semiclassical and quantum approximations of varying - and often unknown - degrees of accuracy [8]. Such approximations, when valid, not only produce quick and cheap cross sections, but also frequently imply compact descriptions of characteristic features of cross sections and simple correlations between particular cross section phenomena and their origin in the potential or in the collision dynamics. Rainbow scattering serves as perhaps the best known such example. We shall emphasize in the following discussion how such characteristic scattering phenomena can serve as a guide for planning crossed beam scattering experiments, and help to provide physical insights.

For an illustration of the present state of crossed-beam experiments we shall first consider the principal techniques and components of molecular beam machines and discuss their specific problems and limitations. Subsequently I will show scattering results, primarily from our own work in Gottingen, on rotationally inelastic collisions of inert gas atoms with linear molecules, and will discuss their theoretical evaluation.

76

v,· al

Figure 1 (a) Idealized crossed beam scattering experiment with uniform molecular beam velocities VI and V2, and with completely defined initial state Ii; Ei) of the target. (b) 'Newton-spheres' velocity diagram of the scattered particle for elastic scattering and for a specific energy loss ~E = Ei - E f in an inelastic collision.

2. Experimental Techniques

2.1 GENERAL CONSIDERATIONS

In the ideal cross section measurement, see Fig. la, two perfectly collimated molecu­lar beams intersect in the scattering centre of a molecular beam vacuum apparatus. The molecules, which are scattered into a freely selectable solid angle (elab, <Plab) , are registered with molecular beam detectors. This measured cross section is called the 'total differential cross section'. When, in addition, both colliding beams are in spec­ified internal molecular states, and the detector is selective to final internal states, we measure the 'state-to-state differential cross section', which has been expressed above in (1). Actual experiments on thermal energy collisions must always compromise on one or another level of experimental averaging introduced by the thermal spreads of beam velocities and by the thermal Boltzmann populations of internal molecular states.

The velocity spread of neutral molecular beams can be narrowed with Fizeau­type rotating slotted-wheel velocity filters at the expense of molecular beam intensity. Magnetic and electric focusing multipole devices are in use for both internal rotational quantum state and individual angular momentum alignment (m-quantum number) selection. Also, for tagging internal states, modulated laser depletion of one molec­ular quantum level of interest has been used. Continuous pumping with a laser for populating electronic molecular states in the scattering centre is also practiced. For enhanced selective population of higher vibrational states dual-laser stimulated Ra­man scattering schemes have been developed. Details and further references can be

77

found in the summaries [1,2,3J. In practice all selection methods lead to losses of total molecular beam flux and

to reduced scattering signals, which then frequently become too low for a meaning­ful differential scattering experiment. It is an unfortunate drawback that molecular beams cannot, at present, be effectively collimated by imaging lens systems, except for electric and magnetic multipole devices which suffer, however, from very long focal lengths, strong chromatic aberrations and small maximum aperture angles. For these reasons, the collimation of molecular beams for higher experimental angular resolu­tion has devastating effects on the scattering intensity of crossed beam experiments: it has to be traded off against the simultaneous reduction of four collimation slits and intensity losses with the fourth power of the angular resolution.

The maximum attainable molecular beam intensity can be estimated from a quite general vacuum argument for free molecular flow conditions. The density in a molecular beam cannot exceed values which could induce beam self-destruction by collisions between nearby beam molecules. A typical value for the mean free path (of air molecules, for example) is 10cm at a pressure of 1O-3mbar, equivalent to a number density n ~ 2.6 x 1013 molecules/cm3 . In a typical laboratory experiment with the beam path of the order of a few centimetres the beam density must necessarily be lower than the equivalent of 1O-3mbar. With a 2° beam collimation, the beam spot 10 cm distant from the (point- ) source has a diameter s ~ 3mm. Thus, the beam presents to the molecules of a second intersecting beam a target density of n s ~ 1013

molecules cm -2. Then, the scattered intensity cannot possibly exceed the value

da Iscatt = 10 n s dw ~w

~ 10-6 10 ,

(4)

(4a)

when we wish to detect a differential cross section of da/dw 1O-16cm2 sterad- 1

with an angular resolution of 2°, or ~w = 10-3 sterad. By the same arguments the primary beam flux for 2° divergence is, at best, after less than 10 cm flight

molecules sec -1 (4b)

for thermal molecules with a typical molecular speed of v ~ lkm· sec-I. One or two orders of magnitude smaller values for the maximum beam density and flux are probably more realistic in order to exclude also the deterioration of beam collimation caused by small-angle scattering in the beam.

2.2 MOLECULE DETECTION TECHNIQUES

The density of scattered molecules at 2° angular resolution and at 10 cm distance from the scattering centre is thus equivalent to a pressure of 10-9 mbar at maximum, and, more likely is equivalent to some 1O-13mbar - or n = 103cm-3 - in a realistic

78

experiment. The primary detector requirement is consequently ultra-high vacuum sensitivity. The principal detection methods are: (i) ionization of the molecule with subsequent ion counting; (ii) laser-induced fluorescence; and (iii) detection of the beam flux energy with cryo-cooled bolometers.

The last technique is surprisingly sensitive, as the molecular flux for the standard example discussed above for 2° divergence corresponds to a scattered power of 10-12

Watt. The sensitivity of bolometers ranges down to the order of 10-14 Watt at liquid He temperature and the liquid helium cryostat is exploited simultaneously as a perfect, and comparably inexpensive, ultra-high vacuum pumping system for the beam detec­tion region. The bolometer method was recently employed for total differential cross section studies on He + C2 H2 [10]' as will be reported below. In combination with a tunable infrared laser for selective pumping of individual vibration-rotation transi­tions, the bolometer also has the potential for perfect state-sensitive detection of e.g. HF, CO and molecules with C-H groups [12,3]. This 'optothermal spectroscopy' is presently used primarily in ultra-high resolution cluster spectroscopy.

Method (ii), laser-induced fluorescence (LIF), has produced impressive, very de­tailed state-to-state differential rotational and vibrational cross section data on the scattering of Na2 molecules in particular, which were studied by Bergmann and by Pritchard [3]. Through the Doppler shift laser fluorescence is also sensitive to the z­velocity component of the scattered molecules. This has led to an interesting variant of measuring, right at the scattering centre, the angular distribution by the Doppler shift (ADDS) [11,1,3]' just by tuning a laser of narrow bandwidth, rv 20MHz, through the Doppler profile of the selected vibration-rotation state. Because of its high sen­sitivity, about 10-2 to 10-1 , the LIF-ADDS method has also been applied in certain reactive scattering cross section studies with, e.g. CsH as the reaction product [13]. But it is not yet readily applied to common gas molecules, which absorb only in the UV or VUV region.

Detection by ionizing methods, (i) achieves fast response and high sensitivity by direct ion counting. By using mass spectrometers they discriminate with high species­selectivity against unintentionally ionized background. According to the ionization mechanism, they can be grouped into:

(a) ionization by lasers, which is species- and highly state-selective. With 2 + 1 resonant multiphoton ionization (MPI), for example, rotationally inelastic scat­tering of NH3 against D2 and against He was studied recently [15]. This system, of particular astrophysical interest, is also now well investigated in conventional time-of-flight studies [14]. The 2 + 1 ionization technique is currently also highly successful in state-analyzed crossed-beam studies of the chemical reaction D + H2 ---> HD + H [16]. It is sensitive to molecular alignment when the angular photoelectron contribution is measured with respect to the laser polarization plane [1]. A drawback with MPI is the low duty-cycle of the pulsed lasers for the generation of frequency-doubled or tripled tunable UV radiation, and the small size of the ionization volume in a :::; O.lmm focal spot. In comparison with the visible LIFand IR absorption detection schemes considered above, the linewidth

79

of the available UV lasers is considerably larger (2:, 0.2 GHz) and restricts at present the use of, e.g. ADDS-type Doppler profile scans for high resolution differential cross section measurements.

(b) In contrast to laser ionization, one of the oldest and most inexpensive detectors is hot-wire surface ionization. It is highly selective and has an efficiency near 1 for the detection of alkali atoms and some of their compounds. It has been used extensively in early crossed molecular beam studies [3,6]. For molecules with a higher electronic work function the efficiency of the method decreases rapidly. Buildup and diffusion of alkali contamination on the hot-wire material cause the noise threshold of the device to deteriorate.

(c) Electron bombardment ionizer mass spectrometers are, at present, still the most universal molecular beam detection devices. Their detection efficiency ranges from 10-4 to 10-6 with an active ionization volume which can be matched to experimental requirements of up to approximately 1 cm3 . Ionizer background partial pressures of 10-15 mbar are routinely achieved for most gases, except for H2 and CO, when turbo-molecular pumps or cryopumping are used for the generation of the detector ultra-high-vacuum, as illustrated in Fig. 2. This is still, even at 1m distance from the scattering centre, of the order of the realistic scattered signal intensity estimated above for 2° angular resolution for cross sections of 1O-16cm2 sr- 1 . The lack of molecular state selectivity, therefore, can be compensated by simple, high-resolution time-of-flight measurements of the velocity distribution of the scattered molecules.

2.3 KINEMATICAL RESOLUTION IN TOF-EXPERIMENTS

In time-of-flight (TOF) studies, and generally in velocity- or energy-changing exper­iments, the sharp collimation and velocity selection of the colliding beams becomes an even more important issue than in the previous considerations for high angular resolution measurements. Conservation of energy and momentum in the molecular collision determines the final velocity V'(~E;e,¢) for an energy change ~E between the final and the initial internal quantum states. The connection is usually repre­sented graphically in a velocity diagram, see Fig. 1 b, which shows the laboratory velocities VI, V2 of the colliding beams, their relative velocity grel = VI - V2 and the construction of the velocity V CM of the common centre of mass. Scattered molecules have velocities in the laboratory system, e.g. v~, ending on a sphere which is centred at the centre-of-mass velocity V CM. The connection illustrated here between CM and laboratory velocities is completed by Jacobian factors for the transformation of the differential cross section between the laboratory and CM systems.

In Fig. Ib, it is obvious that small changes in the speed or in the direction of one of the colliding beams induce changes of varying importance in the position of the centre and of the radii of the Newton velocity-spheres. The consequence is a smearing of the absolute value of the lab-velocities of the scattered molecules, represented by

(Sa)

80

in which OXi is the variation of any of the initial beam coordinates, and Ci is the local Taylor-expansion coefficient of the algebraic expression for the kinematic relations. These initial coordinates OXi can be assumed to be statistically independent, and to be described individually by Gaussian distributions with respective half-widths .:lxi' Then the convolution for (uncorrelated) Gaussian functions results simply in a new Gaussian distribution for the scattered molecules with a half-width

(5b)

as the quantitative, analytical result of the multi-dimensional convolution-integration over the 6 independent velocity variables of the two colliding beams and 2 independent detector aperture angles [17,5]. This half-width quantifies the immanent 'kinematical energy resolution' when it is expressed as an equivalent internal molecular energy level separation. It then provides a handy expression for the systematic optimization of the relevant energy resolution of scattering experiments. Eq.(5b) shows also that energy resolution has to be traded off with dramatic 6th to 8th power intensity losses when the narrowing of the primary distribution is achieved by simple geometrical collimation and by velocity selection of the colliding beams. The numerical evaluation for thermal energy scattering of He on medium-size (m ~ 30) target molecules shows that an energy resolution of ;S 1 meV is attained with beam angle collimations in a range of 0.5° to 2° [5]. This is of a comparable order to that obtained above on the grounds of maximum attainable molecular beam densities.

Finally, Eq.(5b) enters the deconvolution procedures for experimentally obtained time-of-flight spectra, since it predicts the change of the velocity spreads of individual transitions with scattering angle and with inelastic energy loss.

2.4 SUPERSONIC MOLECULAR BEAM SOURCES

Near-ideal molecular beams, close to the maximum flux conditions considered above, are available from supersonic nozzle beam sources. These appeared in the 1950's and 1960's [1,2]. They contrast with the earlier, simple Knudsen, molecular beam sources in that molecular collisions are not avoided at all in the source aperture region. Thus, a 'dense' gas spreads through the nozzle throat in an isentropic hydrodynamic expansion, and experiences an adiabatic cooling of the internal gas temperature T. Simultaneously, the internal gas enthalpy in the nozzle stagnation region,

(6a)

is converted isentropically into translational flow energy of the exhausted gas. The resulting cooled molecular beams have average velocities u, and velocity spreads that are narrower than Maxwellians, and determined by the ratio (T jTN) 1/2 between the

81

internally-cooled temperature T and the original nozzle temperature TN. In the hy­drodynamic flow region the isentropic cooling of the gas proceeds in thermodynamic equilibrium

(6b)

as a function of the expansion ratio V IVN of a streamtube volume element. The compressibility constant is 'Y = 5/3 for a monatomic, and'Y = 715 for a diatomic gas. In a first approximation the streamtube volume element increases with the square of the distance x from the nozzle exit. Thus, replacing VIVN ~ (xl D)2 by the distance expressed in nozzle throat diameters, D, the isentropic cooling of the internal gas temperature yields the approximate relation

for monatomic gases (6c)

and

for diatomic gases. (6d)

With decreasing density, the frequency of gas collisions at larger xl D tends towards zero, and the hydrodynamic flow transforms into free molecular flow, provided that the flow is not destroyed by the onset of shock waves. These can originate from collisions with collimator walls or with residual gas molecules. Experimentally, the nearly unperturbed extraction of a collimated free molecular beam from the core of the jet expansion is best achieved by using truncated cone apertures with sharp-edged 'skimmer' entrance lips.

Qualitative estimates for the location of the continuous flow 'freezing' zone and for the total number of collisions in the nozzle beam expansion, can be obtained from a consideration of the effective number of molecules in the path of a molecule starting, for convenience, at a distance D from the origin of a simplified, radial streamtube cone. Then the density at distance x is

nino ~ (Dlx?, (7a)

relative to a nozzle density no at distance D. The effective number of molecules in the path of one molecule is the integral

LOO ndx ~ no D [molecules cm-2J. (7b)

The path for one molecular collision can be substituted from the Knudsen relation,

(Pref Arer)single ~ 6 . 10-3 [mbar cm], (7c)

82

which holds, e.g. for N2 gas at room temperature. Disregarding for the moment variations of the mean free path Aref with gas temperature, the quotient of Eq.(7b) and Eq.(7c) gives the total number of collisions for a molecule in the expanding jet

Neall ~ (Po D) / (Pref Arer)single

~ 3 X 103 [collisions], (7d)

when representative experimental values of a nozzle diameter D = 0.01 em and a nozzle pressure of Po ~ 2000 mbar are inserted. In this example the total gas flow through the nozzle, proportional to POD2 times the molecule velocity u, is of the order of several mbar L sec-I. This is well handled by standard, 50 em, oil-diffusion pumps.

From Eqs.(7b) and (7c) we can also estimate the distance Xc for a last-but-one molecular collision, as

100 n dx ~ (nref Aref )single Xc

or (7e)

and obtain the approximate result

(7f)

This gives Xc = 2 X 103 D ~ 30 em for Po = 2 bar, D = 0.01 em example. In reality the expansion is already frozen at a distance of Xc ~ 1 to 3cm from the nozzle, as is established by detailed experiments, and by quantitative simulations of the expansion with the appropriate n(2,2) transport integrals [3].

Only the 'parallel' temperature, Til, and its associated velocity distribution in the beam flight direction 'freeze' when this critical freezing zone X f is crossed. The terminal beam velocity resolution is determined by the product Po D. The relations (6b), (6c) yield for the relative velocity spread the estimate: t:::..v /u ~ (x f / D)-2/3 and (Xf/D)-2/5, respectively, for monatomic and diatomic gases. The experimentally attainable values, in addition, are restricted by the onset of condensation in the severely cooled expanding gas. Practical values are 5% to 10% half-width resolution for the rare gases; better than 1 % for the non-condensing helium, and 10% to 20% for most other common gases. For the internal rotational and vibrational degrees of freedom a less complete cooling is accomplished, in accordance with their larger collision numbers, Zeoli for the thermal relaxation of these states (see Lambert [18] and Ref. [3]). The rotation of N2 can be cooled to Trot 2': 10K, CO2 and C2H2 to Trot 2': 20K. The vibrational states of most diatomics have collision numbers of the order ZcolI(vib) 2': 104 and are not cooled at all in a standard molecular beam source nozzle expansion with Neall ~ 3000 according to Eq.(7d).

Of importance for the intensity of molecular beams is, finally, the transverse translational temperature, T.L, of the beam source. It cools beyond the parallel tem­perature freezing radius and, thus is always lower than Til' with experimental values

83

which come close to the isentropic temperature calculated at the position of the beam skimmer entrance. Since a transverse temperature of T.L ~IK still implies an internal beam divergence of the order of 10 , this reduces the transmission through subsequent collimation slits when high angular resolution is required. Eqs.(6b) and (6c) show that for diatomic gases the isentropic expansion proceeds noticeably slower than it does for monatomic gases. This has the experimental implication that diatomic and polyatomic molecular beam intensities are, typically, one order of magnitude smaller than rare gas beam intensities in high resolution experiments.

The angular distribution of the free-jet expansions is approximately a cos2 e dis­tribution, i.e. it has a half-width spread of approximately 60°. The source efficiency for beams collimated to 2° divergence is thus approximately 10-3 . Beam intensi­ties are 10-2 mbar L sec-I, or 1017 molecules per second, for 10 mbar L sec-1 total gas flux. This comes very close to the absolute gas density limit estimated at the beginning of this section.

The overall benefit of nozzle beams is primarily that the hydrodynamic expansion acts as a process which reduces the intrinsic velocity spread and narrows the initial internal state distribution of the molecular beams, at no expense of intensity loss!

2.5. EXAMPLE OF A CROSSED MOLECULAR BEAM APPARATUS

An actual crossed molecular beam scattering apparatus is shown in Fig. 2, with a simplified cross section drawing of the molecular beam machine which we have been using in Gottingen for about a decade for measurements of rotationally inelastic scat­tering cross sections. All principal dimensions are shown here to scale, but major technical parts, such as the main vacuum chamber walls and the gear for the rotation of the detector angles, have been omitted. Separate pumping ports are only indicated by arrows (-+ and @). The two collimated beams propagate along the dashed hori­zontal and vertical lines, and intersect, at right angles, in the scattering centre of the apparatus. The vertical 'target' beam is produced in chamber 'Tl' where an oil dif­fusion pump with 50.000 Lsec- 1 can handle a maximum gas flow of 35 barmLsec- 1 .

The primary beam collimators are skimmer cones with 30° full apex aperture angle. In addition two subsequent collimation stages are needed for reducing, at scattering angles larger than 20°, the primary beam background to less than 10-8 of the direct intensity beam. Either of the beams can be chopped with a single slot, or with a random slot sequence chopper wheel for time-of-flight analysis with resolution down to 5 J.Lsec. The diffuse gas background in the scattering region is kept low by 20,000 L sec- 1 pumping capacity in this chamber. The total residual gas pressure is less than 2 . 10-6 mbar in the scattering region.

Scattered molecules are detected by an electron bombardment 'universal' beam detector with a 10cm radius magnetic mass selector and an open secondary electron multiplier for ion counting. Its overall detection efficiency is 10-5 . The total de­tector pressure in chamber 'D4', typically, is I-2xI0-ll mbar. The partial pressure is ::; 10-15 mbar at most masses of interest. Three additional differential pumping stages with turbo-molecular pumps separate the detector UHV region from the main

84

~Hl

T2

c:=>

Figure 2. Crossed molecular beams apparatus with sub-meV resolution. For time­of-flight analysis the primary beam or the target beam are chopped in chamber P2 or in Tla. The scattered molecules are detected by a mass spectrometer in the UHV­chamber 'D4'. Principal dimensions are approximately to scale, see text.

scattering chamber. An electromagnetic valve with 8mm aperture diameter shuts the detector off from the main apparatus in case of a vacuum system emergency. The length of the time-of-flight path is 140 cm, which allows for a time-of-flight velocity resolution of 1 % to 2%.

3. Representative Experimental Results

3.1 INTEGRAL CROSS SECTIONS

Integral cross section measurements are the simplest type of molecular beam scatter­ing experiment, and also the earliest, starting with M. Born's work with Elisabeth Bormann on the mean free path and on the attenuation of molecular beams by gas collisions [19]. In a first approximation it provides a measurement of the combined diameters of the two molecules. When this simplest, 'hard spheres' (HS) potential (and classical collision dynamics) is a realistic approximation, the total cross section is independent of the collision energy, and given by

(8a)

85

* symmetry to 2 1 Qscillations

c: 0 - 10 u ~ VI orbiting I VI resonances I 0 I L.. I U E-:O.8 --'

1 0 L.. C) aJ -c:

0.1 0.001 0.1 10 100

reduced velocity g*

Figure 3. Integral atom-atom scattering cross sections are closest to the traditional transport coefficients. As a function of the reduced collision velocity g* they show two principal slopes, for scattering in the attractive part of the potential at g* < 1, and for scattering in the r- 12 repulsive potential part at g* > 1 (which is usually at superthermal energies). For further details see text and Van den Biesen in Ref. [3J.

Notably, the differential cross section is then also independent of the eM scattering angle, and given by

(8b)

In contrast to this, all actual integral cross sections, as illustrated in Fig. 3, show a number of pronounced structures and an overall rapid decrease for higher collision energies [3J. The integral cross section is plotted here on a log-log scale as a function of a reduced collision velocity, g*.

For a hypothetical, pure Cs r-s-potential the 'Massey-Mohr' (1933) or 'Landau­Lifschitz' result (d. Ref. [2]) for the velocity dependence of the integral cross section IS

lTtot = const(s) (Cs /g)2/(S-I) . (9a)

This predicts a straight line with slope - 2/ (s - 1) on a log-log cross section plot. In actual integral cross sections, see Fig. 3, it is possible to identify two different slopes. One, for velocities g* > 1, is compatible with a value s ~ 12, and indicates that the dominant interaction is here the repulsive part of the familiar 'Lennard-Jones' C12r- 12 - C6r-6 potential. For low velocities, g* = ng/(Erm) < 1 the average slope

86

gives lsi = 6, as expected, when the importance of the long-range attractive part of the potential increases at low collision energies. The transition near g ~ E . r min is determined by the well depth, E and the location, r m of the potential minimum. For most 'normal' gases this occurs at several thousand m/sec. Thus, the integral, as well as the closely-related transport cross sections, are primarily determined by the attractive long-range part of the potential. Intuitively, this result is somewhat difficult to reconcile with the repulsive barrier dominated concept of a hard sphere model, Eq 8a.

In addition 'glory' undulations are present at these medium collision energies and yield further information on E and rm' At lower velocities 'orbiting resonances' occur when the collision energy coincides with quasi-bound states of the collision partners. At the very lowest energies, in the range of pure s-wave scattering, a 'Ramsauer­Townsend' minimum (RT) is observed. For a full, up-to-date, account we refer to the surveys [1,3].

In the recent literature integral cross section measurements are still a first choice in state-to-state scattering studies, when, e.g. new laser techniques are explored for the selective pumping of highly excited states and either the initial state population or the overall laser-probing sensitivity is not sufficient for fully angle-resolved exper­iments. An example was the REMPI study of NH3-H2 collisions mentioned above. Also, in atom-atom potential studies with difficult to produce species, integral mea­surements were recently employed for, e.g. the interaction of magnetically analyzed fine-structure state selected F atoms and 0 atoms with rare gases and H2 [20].

3.2 ATOM-ATOM SCATTERING

In fully resolved crossed beam differential scattering experiments, both the veloci­ties of the colliding beams and the scattering angle are defined with high precision. Originally, such studies were developed to technical perfection with velocity-selected alkali-atom beams, exploiting the very high efficiency and angular resolution of the Langmuir-Taylor hot-wire detector. The experimental Na-Hg differential cross sec­tions in Fig. 4 illustrate the resolving power achieved [3]. The observed cross section structures are the 'rainbow' scattering, 'supernumerary rainbow' undulations and 'fast diffraction oscillations'. These are just within resolution limits. They are more clearly visible in the theoretical cross section, in the lower part of Fig. 4, which is shown here for Kr-Kr in order to illustrate a different collision system [21].

The Na-Hg rainbow maximum, at 8 R ~ 30° for E = 0.188 eV and at ~ 22° for E = 0.251 eV, occurs at the angle where the classical deflection function folds back from the long-range attraction at larger impact parameters to repulsive wall scattering for small impact parameters. Its location

[rad] (9b)

for a 12-6 potential [3] allows a direct estimate of the well depth E(Na-Hg)~ 45 meV from the experimental rainbow angles in Fig. 4. The additional oscillatory structures

... L.. III

N

-< ..... c .2 -u CD en III III 100 0 L..

0

0 10 -c:: CD L.. CP --0 0.1

a

No - Hg

E = 0.251eV

10 15 20 25 30 35 40 "'cm

20 40

Kr - Kr XC2 I XC3

E = 65.5 meV

60 80 Scattering angle .scm

87

Figure 4. Atom-atom differential scattering cross sections showing the rainbow, supernumerary rainbows and fast diffraction oscillations. They probe both repulsive and attractive parts of the potential well region and/or the repulsive barrier, as is sketched in the schematic trajectory plot for various impact parameters and deflection angles (Na-Hg taken from Buck in Ref. [3], Kr-Kr from Ref. [21]).

at scattering angles smaller than 8R are - in a semiclassical JWKB-type description - quantum interference effects between the three different trajectories which lead here to the same scattering angle. The slowly oscillating supernumerary rainbow structures result from the two solid-line trajectories (1, 2) which run close together in the schematic dynamics illustration in Fig. 4. The fast oscillations originate from the interference of these two with the third solid-line trajectory, which is reflected from the repulsive core of the potential and passes on the opposite side of the molecule with,

88

10'

103

10 2

~

I L-

en 10' 0 5

N e« 103

ft;) --102

o 10 20

10

30

Ar - Ar perp. plane

a)

E:62.1meV

15

40 50

.scm

.......

20

Ar - Ar in plane E: 86.7 meV

60 70

b)

80 90

Figure 5. Ar-Ar experimental scattering cross section with resolved fast diffraction oscillations in the rainbow structure region at 8 ::::: 20°. Symmetry oscillations from interference between forward and backward scattering events, indistinguishable for Bose-statistics spin-O argon particles, dominate the cross section at large scattering angles. [34] Comparison is made with the theoretical cross section predictions of two literature potentials. These are shown as smooth cross section curves. They are (from top to bottom) for the Ar-Ar potentials of Refs. [22,23]. Only the lowest curve has been offset arbitrarily for clarity, and is not averaged over the experimental angular resolution.

correspondingly, much faster changes in phase relations. In principle this second effect can also be seen as a pure wave propagation phenomenon, analogous to the diffraction of light on a small circular obstacle, as will be considered below.

89

Table 1. Predictive abilities of Kr-Kr potentials. t Property Comments' Estimated XC2 XC3 HFD HFD

/range accuracy -B1 -B2

Integral CS Leiden 2% + + + +

Differential CS Chicago /Ber keley + + + +

Second virial a) 273K-573K 1.2mLmol-1 + + + + coefficients b)273K-423K .12mLmol- 1

Viscosity a) 134K-375K 1% + + + + b) 298K -623K 0.1%-0.3% + + + + c) 1l00K-1900K 0.6% +

Thermal a) 328K-468K 0.3% + + + + cond ucti vi ty b) 950K-2000K 1.5% + + + + Diffusion 200K-1050K 9% + + + + Thermal 250K-850K 9% + + + + diffusion

Spectroscopy a) + + + + b) + + +

High energy 3.9ao ~ R ~ 5.5ao 10% + + beam data

t Abridged from Dham et al. [21] * a), b) and c) represent various experimental determinations, as given in [21] rv represents 'almost' agreement.

Direct inversion procedures have been developed for the precision evaluation of spherical interaction potentials from experimental croSs sections. An alternative which is frequently employed is an iterative improvement of test potentials, with forward calculation of cross sections and comparison of theory and experiment. A complete account of atom-atom scattering methods and results is available in reviews by Pauly [8] and by Buck [3].

With improvements in 'universal' molecular beam scattering apparatus, subse­quently rare-gas differential cross sections could also be studied experimentally, in particular by the Lee and coworkers in the early 1970's. The validity of a pot en-

90

tial which is derived from microscopic integral and/or differential cross section mea­surements can be be tested using various macroscopic phenomena in the gas phase, including second virial coefficients, transport constants, relaxation data and spectro­scopic information. For a recent example, the results of such an analysis by Dham et al. [21J are summarized in Table 1 for various realistic potential functions for the Kr-Kr interaction. It was concluded that the potential 'XC 3' proposed in this paper yielded the best fit, giving agreement with almost all available data. It may, perhaps, be noticed that in this comparison all potentials agree with the available scattering data, and only the bulk phase and spectroscopic data seem to be sensitive to finer details. However, this conclusion would be premature because in the available Kr-Kr scattering data the detailed interference structure visible in the theoretical DCS in Fig. 4 has not yet been resolved. This shows the fast diffraction oscillations men­tioned above up to the rainbow, near 8 = 30°. In addition, 'symmetry' oscillations are produced by the superposition of the indistinguishable trajectories for 'forward' scattering, with angle 8, and backward scattering by (1l' - 8) of the identical second collision partner.

We could experimentally observe these oscillations in Ar-Ar scattering (Fig. 5). [34J Argon has only one major isotopic component and is thus better suited for experi­ments. In addition argon has a viz longer de Broglie wavelength. Furthermore, it is probably the most thoroughly investigated bulk gas. We compare theoretical cross sections for two potentials [22,23]' and have averaged the theory over the experimental angular resolution. Fig. 5a shows the fast diffractions superimposed on the rainbow structure. For a wider angular range, with a factor of two lower angular resolution, Fig. 5b shows the symmetry-oscillation dominated region of the differential cross sec­tion. [34bJ Quite significant deviations between experiment and theory appear at the upper end of the rainbow region. In this region from 20° :S 8 CM :S 35°, the diffraction oscillations become comparable in amplitude with the symmetry oscillations. This has the visible result of alternating peak height in adjacent oscillation maxima in this mul­tiple interference region. In this region, with phase information from very different parts of the potential, neither of the two tested potentials fully agrees yet with the experimental results, and maxima appear even with shifted angular positions.

3.3 ROTATIONALLY INELASTIC SCATTERING OF He AGAINST N2

Measurements of rotational state-resolved differential scattering cross sections first became available in the early 1970's. The very first data were time-of-flight spectra for low-energy ion scattering (Li+, H+) on H2 and on D2, which have the widest energy gaps between individual rotational levels, with Ll.EO-2 = 44 meV for H2 and Ll.EO-2 = 22 meV for D2. These were soon followed by neutral, rare-gas atom scattering on H2 and its isotopes, and somewhat later, by laser-induced-fluorescence experiments on rotational and vibrational state resolved scattering of Na2 molecules. We refer here to the discussion above on state-selective detection and to review articles, e.g. [17,5,lJ.

The resolution of all the rotational states of the more common atmospheric gas molecules became possible a decade ago when a resolution of 0.75 meV was attained

91

30800 0) .1.1 4300 c) F.S.T

He-N2 He-OZ 0.39.5° 1-3 0.21.6° E :27.7meV t E=27.1meV

0-2 I

"ii t:: t:: CI .c. u 2'900 .. GI Co

\/I ett - '0300 2700 t:: b) d) ::J He-CO He- CH4 0 0.39.50 0.19.6° u

E = 27.7meV E .31..BmeV

1·2 \·3 0·3 t t t

HOO

1.7 1.8 1.9 2.0 2.1

Flight Time I msec

Figure 6. Time-of-flight spectra with sub-meV (0.75 meV) rotational state reso­lution for the scattering of He atoms against N 2, O2, CO and CH4 molecules at selected laboratory scattering angles. [5] The numbers and arrows assign the locations of individual rotational state-to-state transitions. See the text.

in He-N2 TOF-scattering experiments with the crossed-beam apparatus shown in Fig. 2 and discussed above. Figs. 6a-d show four representative examples of such rotation-resolved time-of-flight spectra at a collision energy near E = 27 meV [5,27]. The target molecules, being prepared in nozzle-beam expansions, are almost com­pletely relaxed (::::: 85%) to the rotational ground states. In He-N2 scattering, Fig. 6a, the 0 -> 2 'ortho' rotational transition with energy spacing fl.EO-2 = 1.5 meV, and the 1 -> 3 'para' nuclear spin modification (33%) of N2 with fl.fl-3 = 2.5 meV, ap­pear as separated peaks, adjacent to the dominant peak for elastic scattering. The deconvolution for the individual transition peak amplitudes is shown (smooth lines) for gaussian distributions, with the peak half-width determined from the kinematic energy resolution formula, Eq. (5b), discussed above. An example of a measurement near the limits of the present resolution is He-CO scattering, Fig. 6b. It shows both fl.j = even and fl.j = odd transitions, which can be separated only partially by de­convolution. The scattering on O2, Fig. 6c, is a particularly well-resolved case since

92

.. 104 L- He- N2 en

N

103 Ecm = 27.7 meV 0«

CC -c 10SA

M3SV 0 - 102 u eu :~::~ 1 experiment en en ++++ en 0

101 L-u

-.I o 0 JDCS c + + -c: 100 eu 1-3 .. L-

eu ~ .. -~~ ......... ".'" - .', .' . 0-4 - . . "C 10-1

I . I

0 10 20 30 40 50 60 scattering angle ~ml deg

Figure 7. Centre of mass total and rotational state-to-state differential cross sections, experiment versus theory, and exact CC-theory versus lOS approximation. Note also the near-proportionality of the 0 -> 2(+) and the 1 -> 3(0) rotationally inelastic cross section and the phase shift of maxima in the diffraction oscillations in the inelastic transitions.

the nuclear spin 0 of the 160 isotope allows for only one modification. The spec­trum shows the elastic, pure 1 -> 1, scattering and the 1 -> 3 rotational transition, only. A new feature, however, is the possibility of fine-structure transitions ('FST') of the 32:; ground state. In the present case they do not significantly broaden the pure rotational peak half-width, and may therefore be neglected. They are observ­able in microwave absorption line broadening, at 60GHz and at 118GHz (0.25 and 0.5 meV) [24]. Finally, Fig. 6d shows the rotational excitation of two of the three modifications of the methane molecule. The gross result of approximately 10% to 30% rotationally inelastic collisions in these spectra is in broad agreement with the macroscopic relaxation collision numbers 3 < Zeoli < 8 for these gas mixtures [18J.

The evaluation of a large number of TOF-measurements at different scattering angles, each 20 to 100 hours accumulation time, then allows the construction of a total

93

differential cross section and state-to-state rotational transition cross sections for the 0-72 and the 1 -73 transitions. The results for He-N2 at E = 27.7 meV collision energy are shown in Fig. 7 for centre-of-mass scattering angles 5 ::; 8 CM ::; 60°, and are compared with theoretical predictions for the currently best 'M3SV' interaction potential [25,26]. For angles 8 > 40° the 0 -7 4 rotational transition cross section could also be determined (~). The prominent cross section features are the diffraction oscillations, which are spaced here by 8°. They correspond, for the collision wave number k >:::i 6.7 A -1, to an approximate diffraction radius of

r = 1f/k8 = 3.4A. (9c)

The rainbow feature is barely present. It appears in the small-angle region of the TDCS at 8 R >:::i 8°. According to Eq.(9b) this implies an effective potential well depth of f >:::i 1.8 meV. As an additional noteworthy feature, the diffraction oscillations in the inelastic cross sectIons for the 0 -7 2 and the 1 -7 3 transitions are both reversed with respect to the total differential cross section, which is dominated by elastic (1 -7 1 and 0 -7 0) scattering [27].

In the detailed comparison of theoretical with experimental cross sections in Fig. 7 we show for one potential surface two results of different cross section algorithms. The continuous lines represent exact, close-coupling, precision cross sections. The broken­line curves are the output of the 'infinite-order-sudden' approximation (IOSA), which is frequently employed in experimental work for convenience, speed and practicality, but is also known for sudden and sometimes hardly foreseeable failures. In the range of the present experimental results the IOSA appears as an acceptable approximation when the deviations of the 0 -7 2 and 1 -7 3 differential cross sections in the rainbow region, 8CM < 10°, are disregarded [26].

The IOSA - when valid - makes two approxim~te statements on atom-linear molecule cross sections which are particularly useful for planning and evaluating the experimental state-resolved cross sections. These are: (i) the 'factorization rule', which states that cross sections for an arbitrary initial rotational state can be con­structed by a weighted superposition of all rotational transitions from an already known initial state, for example ji = 0, namely

l(ji -7 jf; 8) = (2jf + 1) L ( ~ JL

(lOa)

This implies: (ii) the independence (in the CM-system) of the total differential cross section (TDCS) of the initial rotational state (see, e.g. Kouri in Ref. [8]):

TDCS == L l(ji -7lt; 8) for any ji. (lOb) jf

Eq.(lOb) states that all TDCS-measurements are independent of the internal rota­tional state cooling of the target beams in different experiments, while Eq.(10a) ex­plains directly the observation that the 0 -7 2 and the 1 -7 3 rotationally inelastic

94

cross sections, in Fig. 7, are nearly proportional to one another, so long as the 0 -+ 4 cross section is small, and contributes negligibly to the sum for the 1 -+ 3 cross section in Eq.(lOa). The factorization, Eq.(lOa), can be particularly valuable when scrambled TOF spectra of incompletely cooled rotational initial states have to be deconvoluted, as seen in Fig. 6b for He-CO. With a knowledge of the initial rotational populations in the target beam and of the predicted TOF-peak half-width for an isolated transition, the deconvolution can be reduced to the determination of the cross sections for a few transitions from ji = 0 only.

The numerical cross section calculations or the factorization relations, however, give no further physical insights on quantitative connections between the rotational ex­citation cross sections and particular features of the interaction potential. An approx­imate direct analytical cross section relationship can be established for the diffraction on a non-spherical hard-shell intermolecular potential model. This is a generalization of the diffraction on a hard sphere which had given, see Eq.(9c), a reasonable qualita­tive description for the oscillatory part of the observed total differential cross section for He-N2 and similar systems. We can note here only the principal results of this scattering model [28].

The atom-molecule potential is approximated by a deformed hard sphere with the orientation-dependent radius, p developed in a series of spherical harmonics

p((3, I) = L:>5AIL YAIL ((3, I)' (l1a) AIL

In the diffraction region of the He scattering the elastic differential cross section is approximately described by a Bessel function - as is well known from the scattering oflight on a spherical obstacle with average radius ro = boo(47r)1/2. For the 'deformed sphere', Eq.(l1a), a simple analytical scattering amplitude can be derived for 'small' deformations, i.e. bAIL :-:; O.2ro. For 'sudden' inelastic collisions it then yields for the rotationally inelastic cross section for diatomic or linear molecules:

(l1b)

in which x = kroe is the diffraction parameter of Eq.(9c). Its immediate implications are: (i) the inelastic scattering cross section for ilj = even rotational transitions in linear molecules oscillates with opposite phase to the ordinary, clastic Fraunhofer scat­tering cross section - as observed for He-N2 in Fig. 7. It can be shown to change to 'in phase' for large hard-sphere deformations, as seems to be the case for rare gas-D2 scattering, and below for He-C02 at larger scattering angles; (ii) the absolute size of individual rotationally inelastic cross sections is determined here by the square of the expansion coefficient bjo with the index j of the excited final rotational state. Suc­cessive j-states probe successive 'multipole deformation' components; (iii) a corollary in the derivation of Eq. (11 b) implies that Ij, m)-magnetic state-selected experimental cross sections cannot provide additional information on potential parameters.

-CJ.50 ........ > CII-l,.50 g >

-250

-350

2D 40 60 SO R(A) a)

~cw~~~ 'M3SV

-1.00

c) He- N2

E = 27.7 meV

r Bew ::::\exp.

DDD

2.0 I

scattering angle I deg.

4.0 6.0 I I

" .' .:

" e.' ,:

I , , : I

!, M3SV

aD R(A) I

b)

d)

ESMSV

95

Figure 8. Comparison of theory with experiment for two 'almost' fitting, recent He­N2 potentials. Shown are the potential-well region for the 'Y = 0° (a) and'Y = 90° (b) orientations for the ab initio BCW potential and the 'ESMSV' experimental potential, as well as the best-fit M3SV interaction. The comparison of the respective cross sections, in (c) and (d), is analogous to Fig. 7.

The evaluation of the He-N2 total and the 0 -+ 2 rotationally inelastic cross section for Fig. 7 gives, with this model, a quadrupole deformation 820 = 0.65A. With the estimate above for the average radius from the spacing of the diffraction maxima, Eq.(9c), this yields the radii

(llc)

for the two principal alignments of the He-N2 collision system. These hard-shell estimates are compared in Table 2 with the actual potential

parameters for a number of recent potential surfaces for the rigid-rotor He-N2 inter­action [25,29,30]. For the more recent - and most realistic - potentials in the lower

96

1000

c: 10 '"d "-b '"d

1

o 10

E = 31.0 meV em o experiment 1990

- mod. M3SV

20 30 40 50

~LAB/deg.

Figure 9. A new measurement of the He-N2 total differential cross section [33], with a carefully calibrated scattering angle geometry, shows improved agreement with the theoretical prediction using the M3SV-potential. In retrospect, the angular calibra­tion in Fig. 7 appears wrong by an average of 5% of the SCM-value, i.e. wrong by 10

at S = 20°.

half of the table the above hard-shell radii coincide with a contour line of the poten­tial which is close to the zero-crossing, (J contour line of the repulsive barrier, with a slight outward shift toward the potential minimum rm' In particular, the potential 'deformation', tl.a = all - a 1-, in column 7, comes very close to the hard-shell, 'Fraun­hofer' estimate of 0.65A for potentials which yield a realistic rotational excitation cross section, indicated by '+'. The 'rv' - almost - is used in column 10 when the calculated rotational DCS is within a factor two of the experimental value.

The detailed agreement of the, thus far, best-fitting multiproperty potential, the 'mod. M3SV' [25]' with the experimental rotational-state resolved DCS, has already been demonstrated in Fig. 7. In addition this potential, and the slightly more recent BTT potential [2ge], provide a most satisfactory agreement with virial coefficients, with traditional transport properties and with anisotropy-sensitive quantities such as depolarized Rayleigh scattering of light and magnetic-field effects on diffusion, viscosity and thermal conductivity [31,32]. Major effort is currently being invested in the calculation of numerically accurate transport properties which are essentially in full agreement with 'exact' close-coupling quantum calculations.

97

Table 2. Predictive abilities of rigid rotor potential energy surfaces for He-N2 t

Potentials

Keil et al. [29a]

Liu et al. [29b]

Habitz et al. [29c] (HTT)

Fuchs et al. [29d] (HFDl)

Fuchs et al. [29d] (HFD2)

Banks et al. [30a] (BCW) (ab initio

Potential Characteristics* Phenomena used as probes E rm cr b.cr L a/3 b.La/3 B12 T RI

/meV / A / A / A

1.94 3.86 3.39 0.23 2.67 3.60 3.16

1.70 4.24 3.73 0.57 2.80 3.60 3.16

1.05 4.23 3.89 0.74 + 2.44 3.55 3.15

1.24 4.13 3.74 0.87 + 3.64 3.25 2.87

1.03 4.22 3.82 0.83 3.03 3.36 2.98

1.21 4.18 3.76 0.42 1.52 3.65 3.30

DCS DCS

+

+

+

Beneventi et al. [30b] 1.22 4.11 3.65 0.61 + + + (ESMSV)

Gianturco et al. [25] (M3SV)

Gianturco et al. [25] (modified M3SV)

Bowers et al. [2ge] (BTT)

3.08 3.47 3.04

1.55 4.09 3.68 0.60 3.16 3.53 3.08

1.48 4.03 3.62 0.59 + 3.01 3.48 3.03

2.01 3.93 3.56 0.53 + 2.67 3.40 3.03

+ + + +

+ +

t In this table La/3 represents a traditional transport property (see, e.g. Refs. [25, 31]), while b.La/3 = La/3(B) - L<>/3(O) represents an anisotropy sensitive magnetic

field-effect (see, e.g. Ref. [32]), B12 is the interaction second virial coefficient, T DCS is the total differential cross section (see, e.g. Ref. [26, 30b]), and RI DCS is the rotationally inelastic differential cross section (see, e.g. Ref. [26]).

* The upper entry for each characteristic parameter is for the linear geometry and

the lower entry is for the perpendicular geometry.

98

The remaining level of discrepancies in the prediction of inelastic scattering data is illustrated in Fig. 8 for the two nearest competitors to the M3SV potential fit and the previous experimental set of data from Fig. 7. Also shown here are the respective potential wells at "Y = 0° orientation in Fig. 8a, and at "Y = 90° in Fig. 8b. The 'BCW' interaction [30a] is the only fully ab initio-type potential in Table 2. Its well depth is too shallow at 90°. Nevertheless, its anisotropy turns out to be approxi­mately correct, and produces an acceptable 'almost' agreement for elastic and inelas­tic data. The 'exponential-spline-Morse-spline-Van-der-Waals' (ESMSV) potential was derived empirically from precision total-differential cross section measurements at E ~ 70 meV [30b]. In the potential plots it comes very close to the M3SV potential curves, but still produces in an 'almost' agreement unexpectedly large deviations in details of the cross sections.

In view of this advanced level in precision theoretical work on the He-N2 col­lisional interaction we have recently repeated a total differential cross section mea­surement with careful attention to the alignment of beam collimation stages and the calibration of the detector in-plane and perpendicular-plane angles. We used thin copper wires on the respective beam axes in order to exclude misalignment due to diffraction effects in optical calibration methods, and believe that now the definition of the true average scattering angle is more accurate than 1/4 of the overall molecular beam collimation, i.e. < 0.3°. This result, for a collision energy E = 31 meV, shows, in Fig. 9, a near-perfect agreement with the TDCS prediction of the M3SV potential (Schlemmer et al. [33]). Explicitly stated, this new, well-defined, experiment implies that the angular calibration given a decade ago in our first experiment was wrong by about one degree at 8CM ~ 20°. The previous cross section data should, accordingly, be revised with a 5% compression of the 8 c M-scale of Fig. 7 or Fig. 8. The small dis­crepancy of 1 ° in the position of the theoretical TDCS oscillation maxima, visible in Fig. 7, thus has no objective reality, and vanishes. For the predictions of the ESMSV potential in Fig. 8d, as well as for the predictions of the BCW potential in Fig. 8c, however, a small discrepancy still remains with these corrected experimental data.

3.4 ARGON-02 COLLISIONS

Subsequent to a series of He-scattering experiments we could also obtain individual ro­tational state resolution for the scattering of argon atoms on small diatomic molecules. An example for an Ar-02 TOF-spectrum is given in Fig. 10 [34]. It shows distinct TOF-peak maxima for the 1 ---+ 9, the 1 ---+ 11 and the 1 ---+ 13 rotational state-to-state transitions for a laboratory 'perpendicular-plane' scattering angle of 8 LAB = 27.0°. The collision energy, ECM = 97.0 meV, is somewhat higher in these measurements because we had the argon molecular beam source mildly heated in order to find a compromise between higher nozzle beam speed ratio for velocity resolution, and the onset of unwanted cluster formation in the argon beam. Clustering increases also with higher source pressures, but is somewhat delayed at higher source temperatures. The rotational excitation probabilities are much larger here than we had ever noticed before for He-N2, O2 collisions at 27 meV, or even at higher energies of E ~ 75

:::j

.a ... t:J --cu c c t:J

.s::. ~ III -C ::J o U

157 9 11 13 1----------L~7",....I...-...L..---'-----------'. .... -------l ,

\

2.0 2.S

\

• • • . •

3.0

e = 27.00°

Ar-02 . p.p . E=97.0 meV

3.5

flight time Imsec

4.0

99

Figure 10. Ar-02 time-of-flight spectrum showing individual peaks for higher ro­tational transitions up to 1 --; 9, 1 --; 11 and 1 --; 13. The theory prediction of the TOF-envelope (- - -) shows a large, factor of four, discrepancy with predictions from potentials available at the time of the measurement.

meV. The energy gap for the 1 --; 11 rotational transition of O2 is ~El-11 = 23.2 meV. For the 1 --; 13 transition it increases to ~E = 32.1 meV. The large rotational transition probability is quite representative of other partners, like N2-molecules, and also for the TOF-envelope for N2-02 collisions [34b], where individual states cannot be resolved separately. The total differential cross section for Ar-02 is shown in Fig. 11a for the rainbow and diffraction regions. Comparison with the Ar-Ar cross section under identical experimental resolution illustrates the strong quenching of the fast oscillation structures in the cross section for the scattering on the O2 molecule. Fig. 11b covers, with lower angular resolution, the full range of scattering angles. The measurements are in 'perpendicular plane' geometry, for which the maximum Lab-scattering angle is 540 for the Ar-02 kinematics.

The observed strong rotational excitation can be rationalized by a qualitative model which considers the sudden impact of a hard-sphere projectile onto a hard­shell ellipsoid molecule. The recoil, q = n k for a sudden, specular, reflection at a point r on the shell surface is determined by the deflection angle e and the impact momentum p. It is parallel to the surface normal, n, of the shell. This yields for the

100

1 -

Ar-02 • p.p. E=97.0meV

, 10

Lab angle <l>fdeg.

0)

o

15

so 50

Figure 11. Total differential scattering cross section for Ar-02 at E = 97 meV. High angular resolution measurements up to 20°, in the upper part of the figure, show a quenching of Ar-02 experimental cross sections as compared to Ar-Ar. Wide range measurements up to 60° show singularities in the laboratory cross sections at the kinematical cut-off-angles for individual rotationally inelastic transitions. Smooth lines represent lOS cross section calculations, c.f. Fig. 13.

maximum angular momentum the simple 'rotational rainbow' result,

jR = I(r x q)jnl = 2ksin(8j2)I(r x n)1 ~ 2ksin(8j2)(A - B) (12)

for the scattering on a symmetrical ellipsoid molecule with semi-major axes A and B. For heteronuclear molecules it can be shown to produce two separate rotational rain­bow maxima with slightly more complex peak envelopes and a characteristic asym­metry, which is visible in the TOF spectra for the scattering of CO molecules in Fig. 12d. In Fig. 12 we compare unresolved rotationally inelastic TOF spectra in

101

"0 Li+ - Nz Ecm:: 16.8 eV Ar-Oz Ecm::96meV

GI N GI = 600

a E a) ... 0 An c: - Aj -GI c: c:

Li+ - CO a Ecm:: IS.3eV .s= Ar-CO Ecm:: 96meV u ... GI a. e = 60 0

U) b) d) -c: :::J 0 u

8 1S 1.7 1.9 2.1 2.3 energy loss reV] flight time ems)

Figure 12. Illustration of the characteristic 'rotational rainbow' structure in the envelope of unresolved rotational excitation in time-of-flight spectra. The principal difference in the shape of the envelope between heteronuclear and homonuclear molec­ules (see text) is remarkable. The wide range of validity for this phenomenon (and for Eq.12) over a factor of 200 here in collision energy is impressive.

'in-plane' scattering geometry, where the 'rotational rainbow' peaks and the envelope structure are more readily perceived. The hard-ellipsoid model had been originally introduced for an interpretation of K atom scattering results on N2 and on CO at energies near 1 eV. In Fig. 12 we have also taken the opportunity of demonstrating its remarkably wide range of validity with TOF data for the scattering of Li+ ions on N2 and CO, at E = 16 eV, versus Ar-CO and Ar-02 TOF spectra at E = 0.096 eV [35]. The ellipsoid anisotropy can be roughly estimated from the molecular bond length: (A-B) ~ 1/2 re ~ 0.5A for N2 and O2. For the Ar-02 collision wave number k ~ 28.7A-l at E = 0.096 eV, then, the rotational rainbow prediction jR ~ 10 for medium scattering angles agrees well with the position of the excitation maximum visible in Fig. 12c. For Li+ -N2 scattering the collision wave number is approximately 10 times larger and l:!.j ~ 100 rotational transitions are strongly excited. In this case the rotational excitation energies are equivalent to l:!.n = 10 vibrational quantum transitions.

The quantitative theoretical evaluation of the Ar-02 inelastic scattering results is not yet comparable ~ith efforts in the corresponding work on He-oN2. We show in Fig. 11 a comparison of experimental data with lOS-cross section predictions [34]

102

o

cc

Ar-02

E=97.0meV

\ ;\. \\

v\ l~

~ \\.

lOS o 1-3 ,., 1-5 . '-7 . ,-9

101

60 120 'SO scattering angle I deg.

Figure 13. Comparison of exact quantum CC with approximate lOS scattering cross sections for Ar-02 rotationally inelastic scattering at E = 97.0 meV. The elastic cross sections deviate by no more than 15% in the amplitude and in less than 0.3 0

in the position of the diffraction maxima. In the experimentally separable inelastic transitions with large 6.j significant deviations of 200% to 300% are observed.

based upon the multi-property fit potential surface of Candori et al. [36]. The kine­matic singularities in the individual state-to-state laboratory cross sections mark the end of the respective lab-angle range when the edge of the Newton-sphere is ap­proached for this end-state. The theoretical cross sections predict too small excita­tions for the higher rotational state transitions. This leads to notable discrepancies in the total perpendicular-plane laboratory cross section. In a direct comparison of a simulated TOF-spectrum with theoretical lOS transition probabilities versus the measured spectrum in Fig. 10 very large deviations of up to a factor of four appear for individual transitions. The correction for these problems requires an increase of

103

the 'deformation' of the potential by roughly 0.3 A [34]. The prediction of the total cross section in the rainbow region is approximately correct. Nevertheless, slight de­viations can still be noticed in the positions and amplitudes of the strongly quenched diffraction oscillations in the rainbow region in Fig. lla.

Any meaningful improvement of the potential, however, depends crucially on the availability of a test for the reliability of the lOS-scattering approximation employed. As was stated in the introduction, the exact close-coupling treatment is much more difficult to carry out for this case, differing numerically by 3 orders of magnitude from the He-N2 calculations. A single comparison was feasible for us on a CRAY machine, and the results of (136 channels) CC versus the above lOS approximation are shown in Fig. 13 [37]. The (dotted line) 1 ~ 1 elastic cross sections are nearly indistinguishable in both approximations. Also, the average excitation of the f::J.j = 2 to f::J.j = 6 rotational transitions is approximately correct for small angles up to 20°. For the larger energy transfer transitions, which have been prominently excited in our experiments, the lOS gives only qualitatively correct predictions; these values cannot simply be substituted for the exact results. Thus this collision system, much more typical of common gases, is at present awaiting evaluation by faster algorithms and computing machines.

Analogous investigations of exact and approximate calculations for the Ar-N2

differential cross sections (up to E ~ 26 me V only) have recently been reported by Connor et al. [38]. For calculations of the infrared spectrum of the Ar-N2 Van der Waals complex we refer to the work of Tennyson and coworkers [39].

3.5 LINEAR MOLECULE 'ANISOTROPIES': He-C02 , C2H2

The scattering of He on linear polyatomic molecules is the natural extension of the studies of simple diatomic collisions. A quantitative summary of previous work can be found in the review [4]. The scattering algorithms for the cross section evaluation in terms of an interaction potential remain unchanged, and are on a manageable com­putational level for the scattering of He. As new information one may expect to find combination rules for the shape of complex Van der Waals molecules, or corrections for simple additivity of the isolated atom-group potentials. The anisotropy in linear molecules is clearly larger than in diatomic molecules, and promises higher rotational inelasticity on the basis of previous experience with the rotational rainbow model, Eq.(12). The evaluation of very recent, high-resolution, inelastic scattering experi­ments shows that this simple ellipsoid rotational rainbow model has to be significantly revised. These results show that much larger uncertainties exist in all previously de­termined 'experimental' anisotropic potentials of linear molecules than we commonly assume.

The potential for the system He-C02 has been repeatedly investigated. Total differential cross sections are included in a multi-property fit of the potential by Keil and Parker [40]. Revisions, based on average energy loss inelastic scattering data, have been given by Buck et al. [41], and by Beneventi et al. [42]. We have recently repeated He-C02 measurement with sub-meV resolution [33]. We observe

104

1000

,---, r-.

..::: 100 C\I

< '--'

10

1000

o 10

0)

E =28.3 meV em

E = 22.9 meV em c)

b)

. He-C0 2

.[;, \ E,m= 64.8 meV

I ~ I ~ 1

. O' 2 \

\f

~~ : '~o!i;:.~... 1

. .' .. :tr:--r.tfr;l--.-J 20 30

E = 64.0 meV em

20 30 40 50 0 10 20 30

<t> LAS! d e 9 . ~cm!deg,

Figure 14. He-C02 total differential cross sections at three different collision ener­gies. At the medium energy, E = 28.3 meV the potential of Beneventi et al, [42], is clearly superior to that of Keil et al., [40]. The lowest energy shows inconsistencies with the latter potential also. A theoretical cross section plot for E = 64.8 meV shows that the He-C02 total differential cross section for increasing angles is dominated by the inelastic 0 -> 2 and the 0 -> 4 rotational state-to-state transitions.

in total differential cross section measurements made at a collision energy of E = 28.3 meV, as shown in Fig. 14a, significant disagreement with the prediction of the earlier Keil-surface [40], and only modest agreement with the recent Buck/Beneventi potential [42]. The agreement is satisfactory at the higher collision energy, E = 64.8 meV, Fig. 14b, where both beam sources are at room temperature. Considerable deviations occur at the lowest investigated energy, E = 22.9 meV, shown in Fig. 14c. The theoretical cross sections for individual. rotational transitions, shown in Fig. 14d, obtained from the Buck-Beneventi surface, and for E = 64.8 meV, show that most of the oscillatory structure in the total cross section results from the inelastic 0 -> 2 and 0 -> 4 rotational transitions for the third and all higher oscillation maxima. The

OJ c c d .r:: ~ III

c: " o u

1.3

Beneventi '88

1.5 1.7

flight time Imsec

1.9

E= 22.9 rr.eV 0= 0.00

<1>=32.00

105

Kcil '85

1.5 1.7 1.9

flight time Imsec

Figure 15. Sub-meV high resolution TOF spectra for He-C02 at E = 22.9 meV versus the (smooth-line envelopes) inelastic transition predictions of the improved Beneventi et al. [42] and of the earlier Keil et al. [40] interaction potential. Neither is in satisfactory agreement with experiment [33], see text.

first two diffraction structures at small angles are dominated by elastic scattering. In agreement with the prediction of the Fraunhofer inelastic diffraction model for 'small deformation', Eq.(ll), the 0 ---> 2 transition oscillates out of phase with the elastic transition for 8 CM ~ 15°. A change in the phase relation toward the 'in phase' structure for 'large' deformation seems to occur at angles beyond 15°. Thus, in the small-angle region, the shift of the diffraction maxima makes it possible to distinguish the contributions from the two principal final states in the total differential cross section, and has yielded 'the anisotropy of the potential' without explicit inelastic scattering information. We have then studied TOF spectra with a fwhm kinematical resolution ~ 0.6 meV. They are shown in Fig. 15 for three successive 'perpendicular plane' laboratory scattering angles. The first is at 32.0°, in a diffraction minimum of the total cross section in Fig. 14c. The next two TOF spectra are measured at 36.5° and at 39.0°, near the adjacent diffraction maximum. Individual rotation states are

106

1000 He-CZH2

r-;:;' Ecm ;::31.9 meV rJl

......... lOa • E'xppriulI'lLI (II

< -Keil-HFD '--'

c: 1000 '" trunc P4 -HFD "0 ......... b

"0

;

lOa .w· ..•.

- P4-HFD

•• , Keil-HFD

10

1 a 10 20 30 40 50

Lab-scattering-angle /deg.

Figure 16. He-C2H2 total differential experimental cross section versus lOS theory for the literature potential [43] denoted Keil-HFD. We also show the result for a con­siderably varied potential with a large quadrupole deformation term, called P4-HFD (see text). The two are compared in the lower curve and yield quasi-indistinguishable results, demonstrating the serious problems of non-uniqueness of potential inversion from TDCS-data only.

almost resolved separately in this case. The experimental results are compared with theoretical predictions based upon the most recent potential of Beneventi et al., in the left column, and with the earlier Keil and Parker potential in the right column of Fig. 15. The theoretical data have been averaged for an initial CO2 rotational state population for Trot = 20 K. Neither of the two potentials provides a satisfactory fit for all scattering angles, so that the He-C02 potential question is again up for a new theoretical analysis.

The second system to be considered here is He-C2H2 . In the total differential cross section measurement for E = 31.9 meV, shown in Fig. 16, we cannot find noteworthy deviations from the total cross section prediction of the original potential fit of the Keil group, which we shall call 'Keil-HFD' [10,43]. Only near 30° scattering

K-~

• C- '.J!.1

~ 6 - 1411'

C.C C.2 O.~ O.S 0 •• 1.0

u·

£ - l.U.r ffffi cAJ

K-CQ "2

1 £ .. 0.23 .r

~ ' - 1611'

D

O.C 0.2 O.C 0.' 0.'

u·

£-0.23.'1 nTi'i cAl

-3-1-101

X cAl

1.0

107

K-r;H2

., £-1.02.1

'-WI'

0

C.C 0.2 0.4 O.S 0.8 1.0

u·

E -l.CZ ,1 nm cAl

X C~J

Figure 17. Rotationally inelastic scattering of potassium on N2 , CO2 and C2H2 (taken from Velegrakis and Beck [44]). In the first row electron density contour plots of these molecules show the large 'deformation' of the C2H2 molecule. Velocity change spectra in the second row display rainbow structures, analogous to Fig. 12, for N2 and CO2 scattering. These contrast to a strongly anomalous, low inelasticity K-C2H2

velocity change spectrum. The last row shows ellipsoidal hard-shell contours which fit N2 and CO2 and the proposed non-ellipsoidal hexadecapole-anisotropic contour for the correct prediction of the K-C2 H2 spectrum.

angle may a minor deviation of the oscillation maximum be present above the data noise. Inelastic TOF data with sub-meV resolution are shown below, in Fig. 20, and also agree essentially with the Keil potential in the range of scattering angles investigated here. Nevertheless, we did get involved in a theoretical study, showing an unexpectedly large non-uniqueness of this apparently perfect potential fit.

In very recent scattering work with fast potassium atoms, the Beck group ob­served very small rotational inelasticity of the K-C 2H2 system by comparison with K-C02 or with N2 [44]. Fig. 17 from this paper shows in the first column the charge density of the N2-molecule, the (from Fig. 12) familiar K-N2 rotational rainbow struc­ture in the experimental velocity change spectrum for E = 1.19 eV and e = 1400 ,

108

0« 0:::

5

I.

3

2

0

-1

-2

-3

-I.

Keil-HFD -potential P4 - HFD - potential cluster - model

.­/

_..... I / ,/ I 1\ I ·H II·C \ 'xl

--"" \

" "

... " , \/-- \

fi \ C·I) H· )

'j, I

"" I --­/

-5L--L __ J-__ L--L __ ~~L--L __ ~~ __ ~

-5 -4 -3 -2 -1 0 1 2 3 4 5 o

R[A] Figure 18. He-C2H2 contour plots of the O"-contours [V (R, 1') = 0] for the original Keil-HFD potential and for the new, P2/ P4~anisotropic potential, P4-HFD which mimics the Velegrakis and Beck contour for C2H2 in Fig. 17. Note the very large change of the repulsive barrier by 0.3A in the collinear configuration. Also shown is the conventional classical cluster model construction of a He~C2H2 hard-shell contour.

and, at the bottom, the hard-shell ellipsoid contour which provides the best theoret­ical envelope for the measured rotational excitation structure. The second column gives corresponding data on K~C02 collisions. Finally, in the third column, K~C2H2 shows an unexpectedly low inelasticity in view of the quite large deformation of the C2Hz charge-density contour lines. The new explanation approach here is that the standard hard-shell ellipsoid is replaced by a body with additional depressions near l' = 45° alignment, as drawn at the bottom of the C2H2 column in Fig. 17. The authors then find four different rotational rainbow maxima. Three of these are close to the elastic peak. The other is far away to the left, and produces a very small excitation (of several percent max. amplitude, only) over a wide range towards the left of the peak velocity in the measurement.

We have transferred this hard shell K~C2H2 contour to the original Keil-HFD po-

iOO

0.1 o 30

Keil- HFD

60

-iJcm/deg.

109

P4 - HFD

o 30 90

Figure 19. The comparison of individual rotational state-to-state cross sections from the Keil-HFD potential with the results from the P4-HFD potential reveals dramatic changes in the probabilities of rotational excitation with virtually no change in the TDCS.

tential for He-C2H2 by modifying only the shape of the contour line for the minimum of the HFD potential well by the introduction of a new Po and P2 (cos /')-term and an additional P4 (cos/,) term into the equation for rm in Ref. [44]. The absolute potential size, which is not available from the hard-shell fit, was adjusted to fit the position of the previous diffraction maxima. This new hexadecapole-anisotropic 'P4-HFD' po­tential gives a total differential cross section which is barely distinguishable from the original Keil-HFD TDCS, as seen in a comparison in the lower part of Fig. 16. The difference in the potential shapes is, however, quite dramatic as shown in the com­parison in Fig. 18 of the a-contour lines for the zero crossings of the two potentials. The (A - B)-anisotropy in the semi-major axes changes by almost 0.2 A- without a perceptible effect in the total cross section! The conventional ellipsoid contour with these new major axes would produce a visible total cross section shift, shown also in the Fig. 16 comparison as the 'truncated P4-HFD' result. In the contour plot, Fig. 18 we compare, in addition, the standard cluster-model for C2H2 with 7l'-bond C atoms [45], and find significantly better agreement of the new P4-HFD contour with this long-established phenomenological chemical laboratory model.

The individual rotational state-to-state cross sections for the old and new P4-HFD potential are compared in detail in Fig. 19. We note that the disquietingly rapid decrease of the original Keil-HFD elastic cross section disappears for the P4-HFD potential, and the 0-0 cross section is dominant here at large scattering angles.

110

a.> c c Cl

P4 - HFD He-C2H2

E= 32.4 meV 6= 0.00

~=28.2°

Keil-HFD

& ~~~~~~--~~~~~~I:~~~~~~~~~~~~~~~~~ VI -C :l o u

1.2 1.4 1.6

flight time/msec

0= 0.0 0

c[>=32.8°

1.4 1.6

flight time /msec

1.8

Figure 20. He-C2H2 TOF-studies of the rotational inelasticity at scattering angles up to 32.8° show only a very slight improvement of the prediction of the P4- compared with the original Keil-HFD potential. According to Fig. 19, measurements at a few degrees larger scattering angles would result in a much clearer decision for one of the potential surfaces.

The 0-2 excitation shows comparably small changes. The 0-4 excitation is consider­ably increased at smaller scattering angles, and shows typical diffraction-oscillation structure for the P4-HFD potential. This is, of course, expected from the inelastic Fraunhofer theory, Eq.(l1), when a significant 840 deformation term is present in the expansion for the shape of the diffracting obstacle, Eq.(l1a). For larger scattering angles the order of the individual inelastic transitions is also reversed, and the 0-6 excitation probability becomes larger than the 0-4 and the 0-2 transitions near e:=;::j 90°.

Unfortunately, at the time of the He-C2H2 TOF measurements we had not yet anticipated this argument and can only present TOF spectra up to a 'perpendicu­lar plane' angle <PLAB = 32.8° in Fig. 20. This is in a diffraction minimum of the total cross section. A second TOF spectrum is shown for the adjacent diffraction maximum, at <PLAB = 28.2°, and displays comparatively little rotational excitation

111

structure. The comparison with the theoretical spectra for the P4-HFD on the left and the Keil-HFD on the right half of Fig. 20 yields disappointingly small differences. For 32.8° the 30% smaller P4-HFD inelastic structure is, perhaps, marginally more realistic. However measurements for a slightly larger angular range would definitely be much more decisive. The comparison does, however, confirm that the Beck group modification of the C2 H2 interaction potentials is certainly fully compatible with all available differential scattering experiment cross section structures.

4. Conclusions

The evaluation of crossed molecular beam scattering experiments allows the most detailed determinations of anisotropic molecular interaction potentials. Of particu­lar value are sharply defined features in the scattering cross section such as rapid diffraction oscillations, the angular rainbow structure, or rotational rainbows. The uniqueness and precision of the potential studies is basically limited by the extent of experimental averaging over cross section features, and by the statistical noise of the cross section data. We discuss the principal experimental techniques for state­resolved cross section measurements, and obtain a quite general expression for the maximum scattering signal and for its decrease with increasing experimental angu­lar resolution. For practical studies the notorious computational problems with the precise calculation of state-to-state scattering cross sections place additional restric­tions upon the choice of collision systems, essentially limiting serious studies to the scattering of atoms on linear molecules. This makes studies of helium collisions a particularly favourable case for systematic quantitative potential studies.

The He-N2 rotationally inelastic differential scattering and the determination of its anisotropic potential has been reviewed as an illustration of the developments in the past decade. The experimentally fitted potential predicts very satisfactory macroscopic transport constants and rotational excitation scattering cross sections. Semi-ab initio theoretical potential models can yield respectable accuracies, although they seem to need some additional fine tuning for a correct prediction of anisotropy­related properties to within better than 50%.

The heavier collision system Ar-N2 , and similarly Ar-02, shows experimentally a much greater efficiency for the excitation of higher rotational states. This is in good qualitative agreement with the principal expectations of the rotational rainbow model for hard-shell ellipsoids. Exact cross section calculations are now slowly becoming available for developments of accurate potential fits.

The scattering of larger linear molecules is investigated in studies of He-C02

collisions, and for the scattering of He and K atoms against C2H2. They indicate the necessity of substantial modifications of model considerations which had formerly proved successful in atom-diatom scattering. Most importantly, it can be shown in these cases that the determination of potential anisotropies from total differential cross sections is definitely not possible without additional information on the anisotropy. For He-C2H2' and likely for He-C02 as well, it becomes certain that at least two 'anisotropies', consisting of P 2 and P 4 Legendre terms, are to be included in a correct

112

description of the shape of isopotential surfaces. The standard rotational rainbow model, established for one decade, based on a one-parameter ellipsoidal hard shell for diatomic molecules, is found here, for the first time, to be completely inadequate.

As a general result it appears that a considerably higher absolute accuracy in the position of the potential barrier structure is required for the correct description of anisotropy-related phenomena in more complex polyatomic molecules. An accuracy of one percent in the value of Rm represents a very high precision for a spherical atom­atom scattering potential. However, comparable one percent inaccuracies can produce devastating errors in the magnitude of the total nonspherical potential deformation which controls the anisotropy-related collision phenomena in diatomic and polyatomic molecules.

References

[lJ Levine, R. D. and Bernstein R. B. (1987) Molecular Reaction Dynamics and Chemical Reactivity, Oxford University Press, New York.

[2J Fluendy, M. A. D. and Lawley, K. P. (1973) Chemical Applications of Molecular Beam Scattering, Chapman and Hall, London

[3J Scoles, G., Bassi, D., Buck, U. and Laine, D. (1988) Atomic and Molecular Beam Methods, Vol.1, Oxford University Press, New York.

[4J Buckingham, A. D., Fowler, P. W. and Hutson, J. M. (1988) Theoretical Studies of Van der Waals Molecules and Intermolecular Forces, Chern. Rev. 88,963-988.

[5J Faubel, M. (1983) Vibrational and Rotational Excitation in Molecular Collisions, Adv. At. Mol. Phys. 19, 345-394.

[6] Hershbach, D. R. (1986) Molecular Dynamics of elementary chemical reactions, Nobel lecture, December 8, 1986.

[7] Edmonds, A. R. (1958) Angular Momentum in Quantum Mechanics, Princeton.

[8] Bernstein, R. B. (1979) Atom-Molecule Collision Theory, Plenum Press, New York.

[9J Vohralik, P. F., Watts, R. O. and Alexander, M. (1990) HF-HF differential cross sections, J. Chern. Phys. 93, 3983-4002.

[10J Danielson, L. J., McLeod, K. M. and Keil, M. (1987) Damping of total differential cross sections: Observations and empirical anisotropic potentials for HeC2 H2 and HeOCS, J. Chern. Phys. 87, 239-248.

[11J Serri, J. A., Kinsey, J. L. and Pritchard, D. E. (1981) Measuring differential cross sections using the Doppler shift, J. Chern. Phys. 75, 663-668.

[12J Vohralik, P. F. and Miller R. E. (1985) Resonant rotational energy transfer in HF, J. Chern. Phys. 83,1609-1616.

[13] Hermite, J. M., Rahmat, G. and Vetter, R. (1990) The Cs(7P) + H2 ~ CsH reaction: 1 - Angular scattering measurements by Doppler analysis, J. Chern. Phys. 93, 434-444

113

[14J Ebel, G., Krohne, R., Meyer, H., Buck, V., Schinke, R., Seelernan, T., Andresen, P., Schleipen, J., Ter Meulen, J. J. and Diercksen, G. H. F. (1990) Rotationally inelastic scattering of NH3 with H2: Molecular-beam experiments and quantum calculations, J. Chern. Phys. 93, 6419-6432.

[15J Seelernan, Th., Andresen, P., Schleipen, J., Beyer, B. and Ter Meulen J.J. (1988) State-to-state collisional excitation ofNH3 by He and H2 studied in a laser crossed molecular beam experiment, Chern. Phys. 126, 27-45.

[16J Schnieder, L., Seekarnp-Rahn, K, Liedeker, F., Steuwe, H. and Welge, K (1991) The hydrogen exchange reaction H + D2 in crossed beams, Faraday Discuss. Chern. Soc. 91, 14.

[17J Faubel, M. and Toennies, J. P. (1977) Scattering studies of rotational and vibra­tional excitation of molecules, Adv. At. Mol. Phys. 13, 229-314.

[18J Lambert, J. D. (1977) Vibrational and rotational relaxation in gases, Clarendon Press, Oxford.

[19J Born, M. (1920) Eine direkte Messung der freien Wegliinge neutraler Atome, Z. f. Phys. 21,578-582; see also: Przibrarn, K (1929) in W. Wien and F. Harms (eds.) 'Handbuch der Experirnentalphysik', Vol. VIII, 585-591, 612-615, Leipzig.

[20J Aquilanti, V., Candori, R., Cappelletti, D., Luzzatti, E. and Pirani, F. (1990) Scattering of magnetically analyzed Fe P) atoms and their interactions with He, Ne, H2 and CH4 , Chern. Phys. 145, 293-305.

[21J Dharn, A. K, Alnatt, A. R., Meath, W. J. and Aziz, R. A. (1989) The Kr­Kr potential energy curve and related physical properties; the XC and HFD-B potential models, Molec. Phys. 67, 1291-1307.

[22J Parson, J. M., Siska, P. E. and Lee, Y. T. (1972) Intermolecular potentials from crossed-beam differential elastic scattering measurements.IV: Ar-Ar, J. Chern. Phys. 56,1511-1516.

[23J Tang, K T. and Toennies, J. P. (1984) An improved simple model for the van der Waals potential based on universal damping functions for the dispersion co­efficients, J. Chern. Phys. 80, 3726-3741.

[24J (a) Krupenie, P. H. (1972) Spectrum of molecular oxygen, J. Chern. Ref. Data 1, 423; (b) Corey, G. C. (1984) Rotationally inelastic transitions between fine­structure levels of the 3 I:; electronic ground state of O2, J. Chern. Phys. 81, 2678-2683;J (c) Corey, G. C. and McCourt, F. R. (1983) Inelastic differential and integral cross sections for 2S+1 I: linear-molecule 1 S atom scattering: The use of Hund's case (b) representation, J. Phys. Chern. 87, 2723-2730.

[25J Gianturco, F. A., Venanzi, M., Candori, R., Pirani, F., Vecchiocattivi, F., Dick­inson, A. S. and Lee, M. S. (1986) Multiproperty study of the He-N2 potential energy surface, Chern. Phys. 109, 417-429; erratum (1987) ibid 113, 166.

[26J Gianturco, F. A., Venanzi, M. and Faubel, M. (1989) A test of a recently proposed He-N2 potential, J. Chern. Phys. 90, 2639-2650.

114

[27] Faubel, M., Kohl, K. H., Toennies, J. P., Tang, K. T. and Yung,Y. Y. (1982) The He-N2 anisotropic Van der Waals potential, Faraday Discuss. Chern. Soc. 73, 205-220.

[28] Faubel, M. (1984) The Fraunhofer theory of rotational inelastic scattering of He on small molecules, J. Chern. Phys. 81, 5559-5569.

[29] (a) Keil, M., Slankas, J. T., and Kuppermann, A. (1979) Scattering of thermal He beams by crossed atomic and molecular beams. III. Anisotropic intermolecular potentials for He + N2, O2, CO and NO, J. Chern. Phys. 70, 541-551; (b) Liu, W.-K., McCourt, F. R., Fitz, D. E., and Kouri, D. J. (1979) Production and relaxation cross sections for the shear viscosity SEE. II. IOSA results for the N2-He system, J. Chern. Phys. 75, 1496-1508; (c) Habitz, H., Tang, J. T. and Toennies, J. P. (1982) The anisotropic Van der Waals potential for He-N2, Chern. Phys. Letters 85, 461-466; (d) Fuchs, R. R., McCourt, F. R., Thakkar, A. J. and Grein, F. (1984) Two new anisotropic potential energy surfaces for N2-He: the use of Hartree-Fock SCF calculations and a combining rule for anisotropic long-range dispersion coefficients J. Phys. Chern. 88,2036-2045; (e) Bowers, M. S., Tang, K. T. and Toennies, J. P. (1988) The anisotropic potentials ofHe-N2 ,

Ne-N2 and Ar-N2, J. Chern. Phys. 88,5465-5574.

[30] (a) Banks, A. J., Clary D. C. and Werner, H. J. (1986) Vibrational relaxation of N2 by collision with He atoms, J. Chern. Phys. 84,3788-3797; (b) Beneventi, L., Casavecchia, P. and Volpi, G. G. (1986) High-resolution total differential cross sections for the scattering of He by O2, N2 and NO, J. Chern. Phys. 85,7011-7029.

[31] Dickinson, A. S. and Heck, E. 1. (1990) Transport and relaxation cross sections for He-N2 mixtures, Molec. Phys. 70,239-252.

[32] McCourt, F. R. W., Vesovic, V., Wakeham, W. A., Dickinson, A. S. and Mustafa, M. (1991) Quantum mechanical calculations of effective collision cross sections for He-N2 interaction. Part I: Viscomagnetic effect, Molec. Phys. 72, 1347-1364.

[33] (a) Schlemmer, S. (1991) Hochaufgeloste Molekularstmhluntersuchungen der in­elastischen und reaktiven Streuung kleiner Molekiile, Ph.D. Thesis, University of Gottingen; (b) Faubel, M. and Schlemmer, S. (1991) to be published.

[34] (a) Faubel, M. and Kraft, G. R. (1985) Ar-02 rotationally inelastic collisions, J. Chern. Phys. 85, 2671-2683; (b) Kraft, G. R. (1985) Inelastische Streuexper­imente zur Untersuchung der Rotationsanregung von O2-, N2-, CO- und NO­Molekiilen durch Ar und Ne, Ph.D. Thesis, University of Gottingen.

[35] Faubel, M. (1985) Low Energy Atom Collisions in H. Kleinpoppen, H. O. Lutz and J. Briggs, (eds) 'Fundamental Processes in Atomic Collision Physics', 503-520, Plenum Publ. Co., London.

[36] Candori, R., Pirani, F. and Vecchiocattivi, F. (1983) The N2Ar potential energy surface, Chern. Phys. Lett. 102, 412-415

115

[37] Bowers, M. S., Faubel, M. and Tang, K. T. (1987) Close-coupling scattering cross sections for Ar-02 collisions at 97.0 rneV, J. Chern. Phys. 87, 5687-5693.

[38] Connor, J. N. L. Sun, H. and Hutson, J. M. (1990) Exact and approximate cal­culations for the effect of potential anisotropy on integral and differential cross sections: Ar-N2 rotationally inelastic scattering, J. Chern. Soc. Faraday Trans. 86, 1649-1657.

[39] Garcia-Ayllon, A., Santarnaria, J., Miller, S. and Tennyson, J. (1990) Calculated spectra for the N2-Ar Van der Waals complex, Molec. Phys. 71, 1043-1054.

[40] Keil, M. and Parker, G. A. (1987) Empirical potential for the He + CO2 interac­tion: Multiproperty fitting in the infinite-order-sudden approximation, J. Chern. Phys. 822,1947-1966.

[41] Buck, V., Meyer, H., Tolle, M. and Schinke, R. (1986) Rotationally inelastic scattering in CO 2 + He collisions, Chern. Phys. 104, 345-353.

[42] Beneventi, L., Casavecchia, P., Vecchiocattivi, F., Volpi, G. G., Buck, V., Lauen­stein, Ch. and Schinke, R. (1988) Improved potential energy surface for He-C02 ,

J. Chern. Phys. 89, 4671-4679.

[43] Danielson, L. J., Keil, M. and Dunlop, P. J. (1988) Anisotropic intermolecular potentials for HeC2 H2 , HeC2H4 and HeC2H6 and an effective spherical potential for HeCHF3 from multiproperty fits, J. Chern. Phys. 88,4218-4227.

[44] Velegrakis, M. and Beck, D. (1991) Differential translational energy loss distri­butions at 1 eV energies: K-C2H2' J. Chern. Phys. 94,7981-7990.

[45] Stuart, H. A. (1967) Molekiilstruktur, Tab. II4, 3rd edition, Springer Verlag, Berlin.

STATUS OF KINETIC THEORY

F. R. W. McCOURT Chemistry Department University of Waterloo Waterloo, Ontario Canada N2L 3Ci

ABSTRACT. The status of the kinetic theory of polyatomic gases is reviewed. Modern concepts and notation are introduced firstly in the context of the ki­netic theory of monatomic gases, after which the unique features associated with polyatomic gases are considered. In particular, field-effect phenomena are treated in some detail, their role in the analysis of accurate transport property measurements is discussed, and their relationship to relaxation phenomena is indicated. Some of the problems that still remain to be tackled are mentioned, and important recent developments are indicated.

1. Introduction

Transport coefficients measure the response of a physical system to applied thermo­dynamic forces, typically gradients of temperature, flow velocity and concentration. For a gas phase system the response is governed by the Boltzmann equation or one of its generalizations or extensions. If the gaseous system is sufficiently dilute that only binary molecular interactions are of importance, and if it is in a nonequilibrium state that is not far from the equilibrium state, then a linearized version of the Boltzmann or Boltzmann-like equation can be utilized to provide a mathematical description of transport phenomena. Much the same reasoning can be employed for the description of relaxation phenomena associated with the various types of internal states possessed by atoms and molecules. The present discussion will deal exclusively with this dilute gas regime.

The kinetic theory of gases is a venerable field, but at the same time it is one fraught with difficulties. It is relatively easy to formulate expressions for the trans­port coefficients of monatomic gases, and to evaluate these coefficients for simple molecular models, such as that of hard-sphere gases, or of point-centres of repulsion. With modern digital computers it is even relatively simple to evaluate the transport coefficients associated with an arbitrary spherical binary interaction potential with a high degree of accuracy. Indeed, it is safe to claim at the present time that the interaction potentials for the noble gases have been well understood and thoroughly characterized during the past decade [1], and that the temperature dependence of

117

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 117-153. © 1992 Kluwer Academic Publishers.

118

the transport coefficients can now be calculated more accurately than they can be measured, especially at temperatures well above and well below room temperature.

Until very recently it has not been possible to calculate the transport coefficients of polyatomic gases or their mixtures with any significant degree of accuracy. There were three contributing factors that led to the earlier impasse. These were:

• the inclusion of the internal molecular energy in the transport processes required a knowledge of rotational (and sometimes vibrational) relaxation times (or, equiv­alently, cross sections), in order to interpret experimental thermal conductivity data,

• the role played by the rotational angular momentum vector, via so-called po­larization contributions to the transport coefficients complicated the expressions considerably,

• the energy of interaction between two molecules is non-central (or anisotropic), so that scattering calculations have to be made for inelastic as well as elastic collisions.

Each of these factors has had to be overcome before the transport coefficients for poly­atomic gases could be calculated accurately and experimental data could be analyzed unequivocally.

The effects arising from the inclusion of internal state energy in transport phe­nomena were relatively easy to incorporate. Firstly, it led to the introduction of a second viscosity coefficient, now frequently referred to as the volume viscosity 'T)v, which is associated with pure dilatations of the gas. It is normally determined from sound absorption measurements. The volume viscosity can be expressed in terms of the rotational (or rotational and vibrational) relaxation time(s) of the gas. Secondly, the presence of internal states gives rise to an additional major contribution to the thermal conductivity coefficient, associated with the flux of internal energy through the gas. This led to what is referred to as the two-flux approach [2], in which the fluxes of translational and internal energies are treated on the same footing. The collisional coupling between the two corresponding micrscopic fluxes can be expressed in terms of the volume viscosity coefficient, while the bulk of the contribution to the thermal conductivity arising from the transport of internal energy can be interpreted as a diffusion of internal energy through the gas. This is the source of the Hirschfelder­Eucken generalization of the Euken factor relating the shear viscosity and thermal conductivity in a monatomic gas [3]. With the two-flux approach it was not possible to make full use of the thermal conductivity data unless accurate measurements of both shear and volume viscosity coefficients were available. Although accurate shear viscosity values have been available from 1968 onwards, accurate values of the volume viscosity are still difficult to acquire for many gases.

The presence of the molecular angular momentum vector j for polyatomic molec­ules meant that both velocity and angular momentum contributions to the nonequi­librium distribution function for a polyatomic gas had to be taken into consideration. The magnitude of the angular-momentum-dependent contributions were difficult to determine prior to 1962, when field-effect phenomena began to be determined exper­imentally for diamagnetic molecular gases [4]. They were established to be relatively

119

small effects, of the order of the accuracy of the experimental transport coefficient measurements themselves.

The calculation of transport coefficients and their field-effects required extensive development work to be carried out on the molecular collision theory for anisotropic intermolecular interactions. During the past fifteen years considerable progress has been made, both at the quantum mechanical level and at the classical mechanical level. It is now possible, for example to carry out full quantum close-coupled calculations for collisions involving hydrogenic molecules and highly accurate classical trajectory calculations for collisions involving non-hydrogenic molecules [5].

2. Kinetic Theory for Monatomic Gases

The basic equation was formulated in 1872 by Ludwig Boltzmann [6]. For a pure gas it has the well known form [2,7-9]

~~ +c·'\lf= j jde'dc1U'f{-fh)(j(e'--+e,cr)Cr:=CUh) (2.1)

in which f represents the nonequilibrium distribution of molecular velocities c, m is the molecular mass, and C represents the nonlinear collision operator, while primed and unprimed variables denote, respectively, pre- and post-collisional values of dy­namical variables. Further, the subscript 1 refers to the collision partner in a binary collision, (j is the differential scattering cross section for a collision, and the relative velocities before and after the collisional event are designated by

(2.2)

Since the Boltzmann equation is a nonlinear integro-differential equation, :t,s general solution has not been found. The equilibrium solution is defined by the condition CU(O) fiO») = 0, so that the equilibrium solution has the property 1'(0) f{ (0) = f(O) fiO) ,

from which In 1'(0) + In f{ (0) = In f(O) + In fiO), so that In f(O) is a collisional invariant, i.e. a conserved quantity. Even the considerably simplified linearized version,

D Inf(O) := 8Inf(0) + c. '\lIn f(O) = -R¢ Dt 8t '

(2.3)

in which D / Dt is the streaming operator, and the linearized collision operator R is defined by

R¢=- j j de'dcdiO)(¢'+¢~-¢-¢l)(j(e'--+e,Cr)Cr' (2.4)

was not solved until 1916 by Chapman [10]' and then independently in 1919 by En­skog [11]. The linearized version of the Boltzmann equation is based upon the as­sumption that the full nonequilibrium distribution function f = f(O) (1 + ¢), is not far removed from the local equilibrium (Maxwellian) distribution function f(O), given by

(0). _ m m r,t ( )

3/2 { C 2 ( ) } f (c, r, t) - n(r, t) 27rkT(r, t) exp - 2kT(r, t) , (2.5)

120

in which n(r, t), T(r, t) are the local number density, and the temperature of the gas, while C(r, t) = e - v(r, t), with v(r, t) the local stream velocity of the gas, is the peculiar velocity.

It will prove convenient to work with a Hilbert space H generated by the inner product ('!/!ix) = n-1 J de,!/!* j(O)X. Since the collisional invariants for a monatomic gas are five in number (mass, energy, the three Cartesian components of linear mo­mentum), it is also convenient to split the Hilbert space into two parts, one of which is called the hydrodynamic subspace Hh and is spanned by the five eigenfunctions of'R with eigenvalue zero, also known as the summational (or collisional) invariants, while the other subspace is called the nonhydrodynamic subspace Huh and is the complement of Hh.

The Chapman-Enskog solution of the linearized Boltzmann equation is pertur­bative in nature. In the lowest order approximation the equation is solvable only if the pressure, temperature and flow velocity fields p(r, t), T(r, t), and v(r, t) obey the Euler equations of inviscid hydrodynamics. These equations can then be utilized to eliminate the time dependence from the left-hand-side of Eq. (2.3) to obtain

( mC2 5 ) m r=-="1 .-=--0

'Rr/>=- ----kT C·\71nT--CC: \7v. 2kT 2 kT

(2.6)

If we introduce the microscopic heat flux \)!E = C (~mC2 - ~kT) and the microscopic stress \)!7r = m' CC " we can also write Eq. (2.6) in the form

where, in the final step we have introduced also the general symbol X'" for the ther­modynamic forces X E = - \7 In T and X7r = -' \7v " and the symbol 8 to denote a full tensor contraction. A formal solution of Eq. (2.7) is

(2.8)

where the \)!'" are contained in the nonhydrodynamic subspace Huh (spanned formally by the complete set of eigenfunctions of'R which have nonzero eigenvalues).

If we recall that the heat flux vector q and the viscous pressure tensor II are given by the nonequilibrium averages

(2.9a)

(2.9b)

we can also write J" = n(\)!"ir/» (2.10)

121

as a general expression for the flux J<>. By combining Eqs. (2.10) and (2.8), we obtain the expression

J<> = k~ L('w<>IR- l q,i3) 8 Xi3 == LL'>i3 8 X i3 , i3 i3

(2.11)

from which we may extract a formal expression for the linear response coefficients in terms of the inverse of the linearized collision operator, namely

(2.12)

In second order the Chapman-Enskog procedure would involve writing the dis­tribution function as 1 = 1(0)(1 + </P] + q)[2]), with q)[l] the same as q) in the first approximation. By making this approximation in the Boltzmann equation (2.1), and by treating q)[l] as known, we would obtain an inhomogeneous equation for q)[2]

which contains q)[l] in the inhomogeneous term. In this case the Navier-Stokes equa­tions, rather than the Euler equations, serve as the constraints for solvability of the linearized Boltzmann equation. Second Chapman-Enskog approximation solutions typically involve both binary products of gradients, and second gradients, and lead to the so-called Burnett corrections to the hydrodynamic equations.

There are several methods available for solving the linearized Boltzmann equation (2.7), amongst which are the basis set truncation procedures employed in the stan­dard treatises by Chapman and Cowling [6], and by Hirschelder, Curtiss and Bird [7], the moment method of Grad [12] and Waldmann [13], the inverse operator method [2], and the so-called method of kinetic modelling [14]. With the exception of the kinetic modelling method, the essential idea is to approximate the solution q) of Eq. (2.7) by a truncated expansion in terms of a chosen set of basis functions spanning the nonhy­drodynamic subspace of R. The equation is solved traditionally either by optimizing the expansion coefficients according to the Galerkin principle [15], in which the differ­ence between the approximated Rq) and the right-hand-side of Eq. (2.7) is required to be orthogonal to each of the basis functions employed in the expansion, or by ex­panding both q) and the right-hand-side of Eq. (2.7), and then solving the resulting system of equations for the unknown expansion coefficients by matrix inversion [2,7]. The so-called inverse operator method is a modification of this latter procedure, in which maximal use is first made of operator properties, and then the matrix which approximately represents the collision operator is inverted, rather than attempting to find the true inverse of the operator R and then forming its matrix elements. This procedure is akin to procedures often employed in quantum mechanics.

2.1. TRANSPORT COEFFICIENTS FOR PURE GASES

A convenient set of orthonormalized basis functions which span the Hilbert space 'H can be constructed from tensor products of the reduced peculiar velocity W =

122

Table 1. Basis functions <1>PS used for the calculation of .x, T} for pure monatomic gases.

p=1 p=2

<1>10 = V2W <1>20 = V2' WW '

<1>11 = (i)1/2 (W2 - ~)W <1>21 = (~)1/2 (W2 -~) 'ww' <1>12 = (3~)1/2 (W4 _ 7W2 + 345)W <1>21 = (~)1/2 (W4 _ 9W2 + 6;) 'ww'

(m/2kT)1/2C, taken together with associated Laguerre polynomials [16] m namely

<I>PS(W) = (_1)8 8. P .. L~P+2)(W2)'Wp', [ 2P+s '(2 + 1)" ] 1/2 1

p!(2p + 28 + I)!! (2.13)

where (2n+ I)!! = (2n+ 1)!/(2nn!), and 'WP , is the p-fold tensor product of W, made completely traceless. The basis functions are normalized such that

(2.14)

where a (p) is an isotropic tensor of rank 2p, which is separately symmetric and traceless in the first and second p indices. The first few of the basis functions that can be used to calculate 'f/ and .x are listed in Table 1. One good reason for choosing the set {<1>PS} of basis functions is that these functions have been established to be the eigenfunctions of the linearized Boltzmann collision operator for a Maxwell gas [17,18]. It had already been observed by Maxwell in 1867 that the R-4 repulsive interaction, which now bears his name, provided values for the viscosity of air in the vicinity of room temperature that were remarkably close to those observed experimentally by him [19]. We may therefore hope that the basis functions (2.13) will provide a good zeroth-order approximation to the true eigenfunctions of n. The five scalar eigenfunctions represented by <I>0o = 1, the Cartesian components of <1>10 = V2W, and <I>01 = (~) 1/2 (W2 - ~), which are said to span the hydrodynamic subspace of Tt, belong to the fivefold degenerate zero eigenvalue of n associated with the collisional invariants mass, total momentum, energy. The remaining <1>Ps span the so-called nonhydrodynamic subspace (on which the inverse operator n-1 is defined). We also note that the microscopic fluxes \)!E and \)!7r are contained in this latter part of the basis set, and are expressible in terms of the basis functions by

(2.15)

123

so that Eq. (2.12) for the linear response (or transport) coefficients can be written as V,,(3 = La(3 A ((3) ba(3, with the scalar coefficient related to a reduced matrix element of the inverse collision operator by

(2.16)

Note that 0:, f3 represent pairs ps in this notation. We thus see that each transport coefficient is given as a single reduced matrix element of R-1 . This single element is approximated in the present scheme by first forming the matrix representative R of R by constructing the matrix elements

(2.17)

in which cr = (16kT/7rm)1/2 is a mean thermal speed, m is the atomic mass, and 6( ~) is an effective cross section; we then form the inverse of R, and pick out the

appropriate matrix element of R-1 called for in Eq. (2.16). In lowest approximation when a single basis function is chosen, we find that LEE

= )"T and U r7r = 217 are given by

5(kT? )..T= ,

2mcr 6(1l) 2kT

271 = cr 6(20) ' (2.18)

in which 6(ps) == 6(~:) is a short-hand notation for what is referred to as a 'diagonal' effective cross section. For a Maxwell gas q.1l and q.20 are eigenfunctions of R, so that these results would then be exact. For a realistic interaction these functions will no longer be eigenfunctions of R, so that higher approximations may be needed. We shall refer to expressions (2.18) as first Chapman-Cowling approximations to the thermal conductivity coefficient A, and to the shear viscosity coefficient 17 for a pure gas. The second Chapman-Cowling approximations to A and 17 are obtained by adding an additional basis function to each set, namely q.12 for A, and q.21 for 17. The result is that the operator R is approximated in the expression for 'T/ by the effective cross section matrix

(2.19)

so that the element (q.2°IIR-111q.20) needed for example in Eq. (2.16) for 271 is ap­proximated by

q.20 R -1 q.20 '" 1 1 _ 6 (21) { 2 20 }-1

( II II ) - ncr 6(20) 6(20)6(21) (2.20)

A similar argument can be made for A. If we write the first Chapman-Cowling approximation for 17 as [17h = kT/[cr 6(20))' then Eq. (2.20) allows us to write the second Chapman-Cowling approximation to 17 as

(2.21a)

124

where the correction factor 11) is given by

62(;~ ) 11) = 6(20)6(21) (2.21b)

2.2. TRANSPORT COEFFICIENTS FOR BINARY MIXTURES

Thus far we have restricted our discussion to pure gases. By doing so we have reduced considerably the number of transport phenomena to be considered, since the thermo­dynamic forces -V' In T and -' V'v ' are of different irreducible character, and hence do not couple in the first Chapman-Enskog approximation. It is, however, worth mentioning here that coupling of thermodynamic forces can occur in a pure gas in the second Chapman-Enskog approximation, since 'V'V'T' and 'V'v', for example, have the same irreducible character. Gas mixtures possess a richer set of transport phenomena than do pure gases, since in addition to -V' In T and -' V'v " there is an additional set of thermodynamic forces -V' J-li, i = 1,···, N, where J-li is the chemical potential of component i in the general N-component mixture. We shall restrict our discussion to the much simpler two-component mixture, in which the components are labelled by A and B. All essential features of mixtures are covered by this case. Viscous flow is decoupled from the heat and mass fluxes associated with the forces X E = -V' In T, X d = -V' PA, where the partial pressure PA of component A in the binary mixture is employed rather than the chemical potential J-lA [20], and 'd' stands for diffusion. The corresponding diffusive flux is Jd = V A - VB, with v A, VB the average velocities with which atoms A, B diffuse through the binary mixture relative to the barycentric velocity v. The phenomenological equations for a binary mixture are

VA - VB = __ 1_ [DTV'lnT + Dp V'PA] , XAXB

II = -217mix'V'V',

(2.22a)

(2.22b)

(2.22c)

where Amix and 17mix are the thermal conductivity and shear viscosity coefficients of the binary mixture, DT and DT are the Dufour and thermal diffusion coefficients, respectively, while D is !he binary diffusion coefficient. These latter three transport coefficients are all new: DT and DT are equal by Onsager symmetry, and arise because the two vector thermodynamic forces X E and X d have the same irreducible symmetry, and are thus coupled.

Binary mixture expressions for 17 and A can be obtained by replacing \[IE, \[111",

X E, X11", J E, and J11" by two-dimensional arrays (in so-called species space), R by a 2x2 array. The basis functions C{>ps, are also replaced by arrays C{>pslk, k = A, B, with functions C{>pslk = C{>PS(Wk ) in the kth position and 0 in the other position.

125

The effective cross sections have also to be generalized accordingly. In order to avoid notational complexity it is convenient to introduce mole-fraction dependent mixture effective cross sections as

(2.23a)

(2.23b)

in which YA,YB are mass ratios given by Yf = mi/(mA + mB), i = A, B, and the effective cross sections labelled by subscripts AA, BB, AB are those determined by the A-A, B-B, and A-B binary interactions, respectively. The mixture viscosity 1Jmix is, for example now given in the first Chapman-Cowling approximation by [2]

(2.24)

where CAB = (8kT lrrmAB) 1/2 , with mAB the reduced mass of the AB pair. A similar expression is obtained for Amix, while the binary diffusion coefficient is given by

(2.25)

The expression for the thermal diffusion coefficient is considerably more complicated. It is

DT = - XAXBkT {6( 10 I A)AB[6(1lIB) - YAXB 6( 111 A)AB] V2.:l(10E) 11 A YB 11 B

+6( ~~ I ~)AB[YA 6(1lIA) - xA6(g I ~ )AB]} YB

(2.26a)

in which (2.26b)

Note that thermal diffusion involves the collisional coupling of ~101A with ~lllA and ~llIB, and consequently it would be identically zero were ~psIA, and ~pslB eigen­functions of R. Second Chapman-Cowling approximation expressions for the mixture shear viscosity, binary diffusion, thermal conductivity, and thermal diffusion coeffi­cients have also been given in the standard treatises [6,7,21]' and because of their complexity, will not be repeated here. Suffice it to say that in each case the second approximation expressions can be represented by the ratio of two determinants, in which the elements of the determinants are effective cross section ratios or mole frac­tions. These second approximation expressions give results which differ by amounts which vary from as much as 2% to as little as 0.1%, depending upon the system, the composition of the binary mixture, and the temperature. A fairly comprehensive study of these effects for noble gas mixtures has been given by Kestin et ai. [22a], and by Assael et ai. [22b].

126

3. Kinetic Theory for Polyatomic Gases

A number of considerations come into play when we move from simple monatomic gases, such as the noble gases, to the polyatomic gases. One is that polyatomic molec­ules all possess internal states, some of which are electronic in nature, many of which are vibrational in nature, and a very large number of which are rotational in nature. However, unless we are interested in transport processes at high temperatures (> 1000 K), or in the transport of large molecules, excited electronic internal states are of no consequence; even vibrational states do not begin to play any significant role in determining transport properties unless we are considering temperatures signifi­cantly above room temperature. Rotational states do, however, playa rather more significant role in determining transport properties and, indeed, are also associated with new effects that are not found unless rotational states are present. There are two aspects to the role of rotational states. One aspect is simply that these states are readily accessible via molecular collisions, so that inelastic collisional processes become relatively important, and effective cross sections that were elastic are in gen­eral replaced by inelastic ones, although many of them remain predominantly elastic. The second important aspect of rotational states is that they are associated with a rotational angular momentum vector (more correctly a pseudovector), which must be taken into account in the Chapman-Cowling type of expansion of the distribu­tion function, or equivalently, must be included in the construction of a suitable set of basis functions spanning the Hilbert space generated by the inner product (1,blx). Thus, we must consider that the nonequilibrium distribution function f is a function both of the molecular velocity c and of the rotational angular momentum j, so that f = f(c,j; r, t). Considerations of this type actually go back to Pidduck in 1922 [23]' who used the rough sphere dynamical model to study inelastic collisionsal events. But because the behaviour of the transport coefficients could still be explained (within ex­perimental error) by spherical interactions, so that rotational contributions to them could be ignored (except for the thermal conductivity, which was resolved by using a modification of the Eucken factor), and because the model is itself considered to be a pathological one, many of the ideas arising from it were ignored for many years. In 1962 Kagan and Afanas'ev [24] revived the issue of the role played in kinetic theory by the rotational angular momentum, and argued that consideration should be given to this issue. Indeed, Kagan and Maksimov [25] had demonstrated slightly earlier that the angular momentum dependence of f could account in a straightforward manner for the observed effects of magnetic fields on the transport properties of paramagnetic gases, commonly referred to as the SenftIeben effect [26,4].

Of course, the nature of the intermolecular interaction between molecules is dif­ferent from that between atomic e S ground state) species, in that it is no longer central. For the simplest case of noble gas atom- diatomic molecule interactions for example, the potential depends upon both the distance R between the centers-of-mass of the atom and the molecule and the angle "( between the vector R and the molecular figure axis u. Additional angle dependence occurs for the binary interaction between more complicated collision partners. It is the non-central nature of the interaction

127

potential that enables molecular collisions in the presence of a thermodynamic force, such as '\lIn T or 1 '\lv I, to produce nonequilibrium j-dependent polarizations in the gas, and to couple them to the transport phenomena.

Because of the presence of molecular internal states in polyatomic molecules, the Boltzmann equation (2.1) has to be generalized for a polyatomic gas. Its classical mechanical generalization is [2]

~~ +c·'\lj= J J J J J dc~dcldOldO~dOI(j'j~ -jh)c~a(c~,j',j~ -4Cr,j,jl).

(3.1)

The symbols j, t, C, Cl, Cn a and the primes have the same meaning as they did for monatomic gases, while j, j 1 are the rotational angular momenta of the colliding parti­cles in a binary collision, and dO = jdjdl for a linear molecule, where 1 denotes a unit vector directed along j. This classical generalization of the Boltzmann equation can be employed to describe the translational and rotational motion of most molecules. However, it is necessary to have a quantum mechanical generalization of Eq. (2.1) when the molecules are very light, such as the hydrogen isotopes and methane and its deuterated derivatives, or when vibrational internal states are involved. A quan­tum mechanical generalization is also needed in the treatment of nuclear spin relax­ation or spin diffusion of monatomic species. The appropriate generalized equation is known as the Waldmann-Snider equation [27,2]. For the present purposes we shall not require this version of the Boltzmann equation. For small deviations from equi­librium linearization of the polyatomic generalized equation proceeds precisely as for the monatomic one, to arrive at the same expression, Eq. (2.7), except that WlOE

is different for the polyatomic gas, since the internal molecular energy must also be taken into account (see the next section).

3.1. TRANSPORT COEFFICIENTS FOR PURE GASES

In the following we shall focus mainly upon linear polyatomic molecules, so that specific formulae will refer to them. The basis functions (2.13) for the monatomic case must be generalized for the polyatomic case to include the rotational angular momentum vector j and the reduced rotational energy E = B1i2? /kT == E rot (j2)/kT (in the rigid rotor approximation), where B is the rotational constant of the (linear) molecule. The generalization thus requires an additional two indices, one to indicate the tensorial rank inj, the other to denote the scalar dependence of the basis functions on E. These two indices are normally designated by q and t, and the basis functions can be written as

1

~pqst(w ")= [ 2Ps!(2p+ 1)!!(2q+ 1) ]2'WPI"q'L(P+!l(W2)R(q)(E) ,j (q) r=; r=; 1 j S t,

p!(2p+2s+1)!!([Rt J2 jq. jq)2

(3.2a)

128

Table 2. Basis functions c»ps used for the calculation of >., T/, T/v for pure polyatomic gases.

Two-flux Basis Set

c»0010 = (~)1/2 (W2 _ ~)

<}>OOOI = (_k_) 1/2 (£ _ (£)) Crot

c»1010 = (t)1/2 (W2 - ~)W

c»1001 = (~) 1/2 (£ _ (£))W Grot

c»2000 = .j2' WW ' c»0200 - ffi .....,

- [2(j2(j"-4)]l/2.u

.-.;.1200 - y30 W""" '.I" - (j2(j"-4»1/2 .u

Alternate Basis Set

<}>OOE = (W2 _ ~ + £ _ (£))

<}>OOD = (~)1/2 [lW2 -1- _k_(£ _ (£))] 2Cv 3 C wt

c»lOE = (~~ f/2 [W2 - ~ + £ - (£)] W

c»lOD = (~~f/2 [r (W2 -~) - ~ (£ - (£))] W

c»2000 = .j2'ww'

where the R~q)(£) form a set of orthogonal polynomials in £, or equivalently, as

(3.2b)

where the pr(W2 ,j2) are scalar functions which need not be specified at this juncture. Corresponding changes are required in the notation for effective cross sections, with the most general such cross section being written as 6(;:,:.tt') in the case of a pure gas of linear molecules. The simplest of these basis functions are given in Table 2.

3.1.1. The Lowest-Approximation Expressions

Because the linear momentum of a molecule is independent of its internal states, the shear viscosity for a polyatomic gas is described in the first Chapman-Cowling approx­imation in exactly the same way as it is for a monatomic gas. Thus, a single basis func­tion, namely c»2000 = .j2'ww', is chosen, to obtain the result T/ = kT/[cr 6(2000)]' which is formally identical to that for a monatomic gas. Indeed, the only actual dif­ference between this result and that for a monatomic gas is that the effective cross section 6(2000) has small inelastic contributions, whereas 6(20) has only elastic con­tributions.

The total energy of a molecule depends trivially upon the molecular internal states, so that the thermal conductivity for a polyatomic gas differs fundamentally

129

from that for a monatomic gas. The microscopic flux WE is generalized to WE = (~mC2 - ~kT + E rot - (Erot ) )C, and A is given by

(3.3)

There are now two ways in which the expansion can proceed for the first Chapman­Cowling approximation to the thermal conductivity of a polyatomic gas. One is to choose to work with an expansion of the type expressed in Eq. (3.2a), the other is to work with an expansion of the type expressed in Eq. (3.2b). The first choice is the one that has been adopted traditionally [28], and can be referred to as the two-flux approach [2], in which one speaks separately of the translational and rotational heat fluxes, and of the coupling between them. Thus, the two expansion terms traditionally chosen for the first Chapman-Cowling approximation to the thermal conductivity are ~101O = ~ W(W2 _ 2.) and ~1001 = -2-W(E - (E)) where r2 = 2C 15k. The use v'S 2 rv'S ' rot of these two expansion terms give rise to the usual expression for A, namely

A _ 5k2T 6(1001) - 2r6(i~~n + r 26(101O) - 2mcr 6(1010)6(1001) - 62(i~~~) .

(3.4)

Two of the effective cross sections that appear in this expression are predominantly elastic, while the third, 6(~g~~), is completely inelastic. A lowest-order approxi­mation sometimes made to this expression is to ignore the collisional coupling be­tween the translational and rotational energies, and treat the thermal conductivity as the sum of two (largely elastic) contributions. With the exception of the hydro­gen isotopes, this is a very poor approximation: in almost all cases, it it impor­tant to retain the coupling term. In general a complete analysis of thermal con­ductivity data based upon Eq. (3.4) requires additional experimental data or theo­retical input. The coupling cross section 6( ~g~~) is related to the rotational relax­ation cross section 6(0001) which determines the volume viscosity 17v (see below) by [5] 6(~g~O) = -5;6(0001), while 6(1010) is related to 6(2000) and 6(0001) by 6(1010) = i6(2000) + (25r2/18)6(0001), so that if accurate data for 17 and 17v are available, then A data can be used to obtain values of 6(1001). An interesting alterna­tive procedure has been employed by Millat et al. [29], in which thermal transpiration data, determined solely by the translational part of the thermal conductivity, i.e.

5k2T 6(1001) - r6( 10lD) At = __ 1001

2mcr 6(1010)6(1001) - 62(i~~~)' (3.5)

are utilized together with A and 17 to evaluate 6(1010),6(1001) and 6( ~g~~), without requiring values of 17v (which are notoriously difficult to obtain accurately).

A second approach is to consider that it is appropriate in each case to choose the leading expansion term to be proportional to the quantity which is being transported, in this case the total energy E [30]. Hence, the single-term expansion chosen in zeroth order is ~10E, where Cp is the heat capacity of the gas at constant pressure. The

130

zeroth approximation to A is then A = CpkT /[mcr 6(lOE)]. At the next level of approximation in this scheme, a second basis function, orthogonal to ~lOE must be chosen: this function is labelled ~lOD to reflect that it is a difference function. In fact ~lOE and ~lOD can be written in terms of ~lOlO and ~lOOl as

where C1 = (5k/2Cp)1/2 and C2 = (Crot/Cp )1/2. This approximation scheme is much more in the tradition of the usual Chapman-Cowling approximation procedure: thus, the second approximation becomes

A - p 1 _ laD CkT { 62('OE) }-1

- mcr 6(lOE) 6(lOE)6(lOD) , (3.6)

which is completely equivalent to expression (3.3) for A. However, the neglect of the correction term in A which arises from the collisional coupling of the two expansion functions ~lOE and ~lOD has a much less dramatic effect on the value of A than does the neglect of the collisional coupling between the expansion functions ~lOlO and ~lOOl used in the two-flux approach [30,31].

In addition to these two transport coefficients, we now have a third one for pure gases. This is the coefficient of volume viscosity TJv, which arises because the pressure tensor P of the nonequilibrium gas contains a scalar part which is proportional to the thermodynamic force XV = -\7. v, namely

II = -TJv\7· v, (3.7)

wherein TJv is given by

(3.8)

This transport coefficient exists because neither individual translational nor rotational energy is conserved, but only their total is. The volume viscosity does not seem to be measurable directly, and is usually deduced from an analysis of sound absorption data according to the formula [2]

1 [4 b - 1)2 mA ] -3 TJ + 'V -k + TJv ,

2"(Ca d I

(3.9)

where Q is the coefficient of sound absorption, p the gas pressure, "( = Cp/Cv is the usual heat capacity ratio, Cad is the adiabatic speed of sound, and w is the sound frequency. The determination of TJv is difficult because the first two terms, which constitute what is referred to as 'classical' sound absorption (they account entirely for sound absorption in atomic gases), often overwhelm the so-called 'non-classical' contribution to Q that is associated with TJV. This is particularly true for gases like

131

N2 and CO. Accurate values of TJv, and hence of 6(0001), have been obtained from sound absorption studies of the hydrogen isotopes and their binary mixtures with noble gases [33].

At the level of the first Chapman-Cowling approximation the volume viscosity can be expressed in terms of the rotational relaxation cross section 6(0001) as

kGrot kT TJv = G~ cr 6(0001)'

(3.10)

While many of the above expressions have been couched in terms of the rotational en­ergy, rotational heat capacities, and so on, their generalization to include vibrational energy transfer, and vibrational contributions to the heat capacities, is straightfor­ward: the subscript 'rot' should everywhere be replaced by the subscript 'int', to indicate that the total internal energy is involved. Vibrational states are not ex­pected to play very important roles for simple diatomic molecules, such as N2, CO, and so on, until quite high temperatures are reached (say 1500 K or so), but they will start to become important at considerably lower temperatures for heavy diatomic molecules, such as Br2, Iz, and for linear triatomic molecules, such as CO2, HCN and N20, which have low-lying bending modes, as well as for almost all nonlinear polyatomic molecules.

The dependence of the nonequilibrium distribution function f upon internal en­ergy was studied by Van den Oord and Korving [32]. They were able to establish that the relative deviation ¢ of f from the equilibrium distribution f(O) for a heat con­ducting gas depends upon the total internal energy rather than independently upon the rotational and vibrational energies. Further, by making use of the relation

2 k 6(10E) = -6(2000) + --6(0001)

3 2qnt (3.11)

between 6(lOE) and the shear viscosity and volume viscosity cross sections 6(2000) and 6(0001), they obtained the expression

(3.12)

relating ..\ for a pure polyatomic gas to TJ and TJv. Finally, they pointed out that the form of Eq. (3.4) suggests that it may be more natural to use the Prandtl number

TJGp 6(lOE) fp = m..\ = 6(2000)' (3.13)

which is relatively temperature insensitive, rather than the Eucken number fE = m..\/rJCv = Gp/(Gv fp), to relate the shear viscosity and thermal conductivity coef­ficients of polyatomic gases.

132

3.1.2. Second Chapman-Cowling Approximations

The second Chapman-Cowling approximation expressions are more complicated for polyatomic gases than they are for monatomic gases, because there are expansions in two dynamical variables involved, namely W 2 and E (or equivalently, in j2) [34]. For the shear viscosity the second approximation involves the basis functions C[>2000, and C[>201O, C[>200l, while the third approximation will require the addition of C[>2020, C[>2002 and C[>2011. For the thermal conductivity the second approximation involves the basis functions C[>101O, C[>1001 and C[>1020, C[>1002, C[>1011 for the two-flux basis functions, and C[>lOE, C[>lOD and at least one other basis function orthogonal to these two functions for the alternative basis functions. Similarly, the volume viscosity involves <p001O, <pODOl and <p0020 , <p0002, <p00ll . Unlike the second approximation expressions for monatomic gases, those for polyatomic gases are best left in the form of ratios of determinants, with those in the numerators being appropriate signed minors of those in the denom­inators [35].

A classical trajectory calculation of 1']v for pure N2 at room temperature car­ried out by Turfa et al. [36]. This provided the first indication that higher degree polynomials in the internal energy in the expansion of ¢ could playa significant role in transport phenomena which involve the internal energy. In particular for a fairly realistic N2-N2 intermolecular potential [37] they found that the leading nonhydro­dynamic term in the expansion of ¢, namely <pOOD, gave a value of 1']v that differed from the experimental value [38] by about 17%: the inclusion of the second degree expansion polynomials in the translational and rotational energies, and particularly the latter, gave a value of 1']v at 293K that was in substantial agreement with experi­ment. This rather surprising result stood alone: it was supported neither by accurate quantum scattering calculations for the hydrogen isotopes [39] nor by existing rough sphere model calculations [34] (which tend to exaggerate inelastic effects). The re­sults suggested that for phenomena which involve the internal energy the expansion of the nonequilibrium distribution function in terms of polynomials in the reduced internal energy E may be more slowly convergent than the corresponding expansions in polynomials in the reduced translational energy W 2 • This question prompted Van den Oord et al. [40] to probe the role of £ in the expansion of f by carrying out a direct measurement of f in a heat conducting gas using a state-selective laser-induced fluorescence technique. The system chosen for their study was 12 at a temperature of 541K. They found that the contribution of second degree polynomials in £rot and Evib relative to that of the first degree polynomials was 9% for £rot, and 27% for £vib. No contribution from a mixed second degree polynomial in £rotEvib was detected. It is likely that such higher degree contributions to 1']v will have to be taken into account in the analysis of non-classical sound absorption data, so that all existing results will have to be reanalyzed.

3.1.3. Polarization Corrections

The nature of the correction to the transport properties arising from the j-dependence of the nonequilibrium distribution function f can now also readily be demonstrated for

133

the shear viscosity coefficient if we make a second Chapman-Cowling approximation to 11 in which, rather than choosing <t>2000 and <t>20lO to be the two basis functions, we choose <t>2000 and <t>0200. In this case, the linearized collision operator n (whose specific form need not be known) will be approximated by the effective cross section matrix

_ _ (6(2000) ncr6 c:::' nCr 6(0200)

2000

6(~ggg) ) 6(0200) , (3.14)

so that the reduced matrix element (<t>2ooolln-111<t>2000) entering into the expression for 11 will now be approximated by

<t>2000 n -1 <t>2000 ~ 1 _ 0200 1 { 62(2000) }-1 ( II II ) - ncr 6(2000) 6(2000)6(0200) 1

(3.15)

with the consequence that

(3.16)

where 11iso represents the value that 11 would have were the nonequilibrium distribu­tion function independent of the direction of j. A similar calculation made with the total-energy-flux basis consists of employing the two basis functions <t>10E and <t>12E:

the corresponding matrix element (<t>IOElln-111<t>IOE) of n-1 is given approximately by [30,2]

lOE -1 10E ~ 1 _ _ 10E 1 { 5 62('2E) }-1 (<t> lin 11<t> ) - ncr6(10E) 3 6 o(12E)6(lOE) , (3.17)

so that A can be written as

(3.18)

No corresponding polarization correction is required for 11v. Notice that the effective cross section 6( ;~gg) represents the collisional cou­

pling of the microscopic angular momentum polarization tensor 'III ('II : 'II) 1(2 to y'2'ww'; such an effective cross section is referred to as a 'production' cross section, because the linearized collision operator n produces 'ww' from 'II, with a strength which is proportional to 6( ;~gg). More general non-diagonal effective cross sections are referred to simply as 'coupling' cross sections. We shall also refer to diagonal ef­fective cross sections for q =I 0 as 'relaxation' cross sections. No special name is given to effective cross sections of the type 6(pOsO). Use has been made of the equality that is referred to as Onsager-Casimir symmetry [2] of the effective cross sections 6( ~~gg) and 6(~~gg) in obtaining expression (3.15). Moreover, we notice that the effect of this correction is to increase the value of 11 beyond what it would have been had there

134

been no j-dependence of f. Since this contribution depends upon the existence of a collisionally-produced polarization 11, and since the angular momentum j will precess upon application of a strong enough magnetic field, we can also deduce the effect to be had upon "I by the application of a magnetic field across the flowing gas. Precession of j about the field direction will result in the averaging-out of the components of j that are perpendicular to the field direction: this averaging will effectively remove most of the contribution arising from the presence of the 11 microscopic polarization, and result in a decrease in the value of "I. This is what is observed experimentally.

3.2. TRANSPORT COEFFICIENTS FOR BINARY MIXTURES

As for mixtures of monatomic gases, the set of transport properties for binary mixtures containing polyatomic gases is richer than is that for pure gases. Again, we find the additional transport coefficients for binary diffusion, thermal diffusion, and the Dufour effect, associated with the additional thermodynamic force 'iJPA. Here we shall focus solely upon binary mixtures in which component A is a diatomic molecular gas and component B is a noble gas. The expressions for the shear viscosity coefficient T/mix,

the binary diffusion coefficient D, for polyatomic gas mixtures are identical to those for monatomic gas mixtures, the only difference being that the effective cross sections now have inelastic contributions in addition to the elastic contributions that appear in the monatomic case. For the thermal diffusion coefficient DT the structure of the expression for the polyatomic mixture is formally the same as that for the monatomic mixture if the alternate energy flux basis function C)lOE is employed in the expansion of ¢ instead of the two-flux pair of basis functions C)101O, C)lOOl. The actual effective cross sections will, however, be quite different; in particular they will not be obtained simply by adding inelastic contributions to those that appear in the same effective cross section in the monatomic gas mixture expressions.

The binary mixture expression for "I for polyatomic gases is identical to that for monatomic gases, and is hence given by Eq. (2.24), with the two digits '20' for 'ps' replaced by the four digits '2000' for 'pqst'. In addition, the interpretation of the effective cross sections is slightly different than for the monatomic gas mixture, since there will now be small inelastic contributions to them. The expression for the volume viscosity for such binary mixtures is especially simple, namely [2,41]

"Iv = - , C~6(0001IA)

(3.19)

in which C~ = XA(C&)2 +XB(C~?, Mixture shear viscosity results are often presented in the form of the interaction

viscosity, "lAB, which is extracted from measurements of T/mix as a function of mole fraction of one of the two components. This quantity is at the same level as the binary diffusion coefficient for a first Chapman-Cowling approximation, i.e. it is independent of the mixture composition. To obtain "lAB, however, it is necessary to know the ratio of 6(1000IA)AB to 6(2000IA)AB' a value not known a priori:

135

the quantity Ai2' which appears in the mixture expression, depends on this ratio, and thus depends on two predominantly elastic cross sections. It is therefore often approximated by the value obtained from similar noble gas pairs. Such a procedure is not especially satisfactory at the level of accuracy currently available for shear viscosity data (±O.5%), so that comparisons between calculated and experimental shear viscosity data are best made at the level ofo the directly measured Tfrnix rather than at the level of the derived quantity TfAB. Moreover, the interaction viscosity is undefined at the second Chapman-Cowling level of approximation. Finally, it may be useful to make comparisons between measured and calculated values of an 'excess' shear viscosity, D.Tfex, defined via

(3.20a)

which tends to remove the pure gas contributions to Tfrnix. This quantity, however, also causes some problems for analysis at the level of the second Chapman-Cowling approximation.

The expression for the thermal conductivity is simplest in terms of the alternate basis set, for which q>lOEIA, q>lOEIB form the truncated basis. This expression may be written for binary mixtures of a linear molecular gas, A, with a noble gas, B, as [2]

(3.21a)

in which (3.21b)

in which ri = 2C!t/5k. We have seen in §3.1 in dealing with the thermal conductiv­ity of pure gases that the alternate or 'total-energy-fiux' basis provides an excellent single-term approximation for A. There is no reason to expect the situation to be fun­damentally different for binary mixtures containing polyatomic molecules. We may therefore expect Eq. (3.21) to provide a very good first approximation for Amix. Unlike the cases of the thermal conductivity of binary monatomic gaseous mixtures and the shear viscosity of arbitrary binary gaseous mixtures, it has not been found possible to extract an 'interaction' thermal conductivity for mixtures involving polyatomic gases. For this reason thermal conductivity results are often displayed in terms of the excess thermal conductivity AeX, defined, as for the shear viscosity, by

(3.20b)

Meaningful comparisons between calculated and measured results for this quantity are even more difficult to carry out at the level of the second Chapman-Cowling approximation than is the case for D.Tfex.

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Finally, an expression has been developed for DT for binary mixtures of the type being discussed here in the case of the two-flux approach to the thermal conductiv­ity. The expression is quite complicated, and has been written out explicitly when the translational-rotational coupling of the heat fluxes can be neglected [42]. This approximation is, however, not expected to be a very good one for binary mixtures, even when the hydrogen isotopes are involved, since the coupling cross sections have dominant elastic components in mixtures. However, since we may expect the single basis function ~lOE to provide a good single-term representation of the heat flux, we should also employ this same function for the calculation of the thermal diffusion coefficient. The calculation then becomes completely analogous to that for a binary monatomic gas mixture, with ~ll simply replaced everywhere by ~lOE: we thus have the expression

XAxBkT {6(IOdIA)AB[6(lOEIB)- YAXB6(IOEIA) ] V26.(lOE) IDE A YB IDE B AB

+6ug~I~)AB[YA6(lOEIA) - XA6(;g~I~)AB]}} YB

(3.22)

in which 6.(lOE) is given by Eq. (3.21b), and we have also trivially relabelled ~lO = V2W as ~lOd, in which 'd' stands for 'diffusion'.

3.3. EFFECTS OF EXTERNAL FIELDS ON TRANSPORT PHENOMENA

As has been mentioned in §3.1.3, the contributions to the transport coefficients as­sociated with the collisional production of microscopic j-dependent polarizations are small, typically of the order of the ordinary second Chapman-Cowling approximation corrections associated with the scalar dependence of the leading W -dependent terms. Such contributions to 1], A could not be sorted out from the already small higher-order scalar corrections simply by examining the values of the transport coefficients them­selves. The key to their independent determination lies in their dependence upon the direction of j rather than simply upon the scalar j2. Because the application of a sufficiently strong magnetic field B will cause, by precessional motion, an averaging out of those components of j that are perpendicular to the field direction b, it is there­fore possible to measure the polarization contributions differentially by employing a Wheatstone bridge type of arrangement of flow arms in the case of the shear viscosity or of heat conducting cells in the case of the thermal conductivity. The polyatomic gas (or gas mixture), with and without the applied magnetic field, flows through or is placed in one pair of arms, while a monatomic gas, again with and without the applied magnetic field, flows through or is placed in the other pair of arms. This arrangement allows, for example, the difference 6.1] (or 6.A) to be measured directly with the same accuracy with which 1] (or >.) can itself be measured.

It should also be borne in mind that when an external magnetic field is applied, the usual scalar transport coefficients must be replaced by appropriate tensorial trans­port coefficients, e.g. in the case of a pure gas by

II = -2'f1 : 'Vv', (3.23a)

137

q= -,x·VT, (3.23b)

in which TJ is a fourth-rank tensor which is symmetric traceless in both the front and back pairs of Cartesian indices, and ,x is a second-rank tensor. If the direction of the magnetic field is chosen to lie along the z-axis, then ,x, for example can be represented by

(3.24)

in which the coefficients A.l, All are obtained by measuring the heat flux in the direction of VT, with the applied magnetic field perpendicular, parallel, respectively, to VT. The third coefficient, Atr, is obtained by measuring the heat flux in a direction that is mutually perpendicular (transverse) to VT and B. The isotropic thermal conductivity coefficient, A(O) (not to be confused with Aisa), is given by A(O) == :\ = i(2A.l + All). Both longitudinal coefficients, A.l and All' differ from A(O) by small amounts, of the order of the polarization corrections discussed in §3.1.3, while the transverse coefficient Atr starts at zero, and depending upon the nature of the polarization, either falls to a minimum or rises to a maximum which is of the order of 0.5% of A(O), after which it dies off to zero at high field strengths. The shear viscosity tensor has five independent components, denoted by 7Jt, 7Jt, 7Ji, with the superscript '+' denoting longitudinal behaviour, and the superscript '-' denoting transverse behaviour of the components. Various experimental arrangements for measuring these five coefficients have been given by Hulsman and Knaap [43].

The effect of an applied magnetic field on the transport coefficients is the result of competition between binary molecular collisions which, in the presence of an ap­propriate thermodynamic force, 'produce' microscopic j-dependent polarizations from the microscopic fluxes (\]!E, \]!", \]iv and, for mixtures, \]!d), and the precessional av­eraging of these polarizations which results from the action of the applied magnetic field in the time between binary collisions. Since the binary collision frequency is proportional to the gas pressure p, while the precessional frequency is proportional to the magnetic field strength B, the net result is that the field-effects are normally functions of the ratio B / p, which is a measure of the relative effect of these two competing processes. The measurements are naturally represented in the form of the relative changes b..L/ L(O), in which b..L = L(B) - L(O) represents the difference be­tween the transport coefficient as measured directly with the experimental apparatus.

Detailed studies of the characteristic behaviour of the field-effect phenomena (of­ten referred to also as Senftleben-Beenakker effects, or SBE) have been carried out by Beenakker and coworkers [44,2] over a twenty-five year period. They have shown that each transport phenomenon has one (or at most two) polarization(s) that dom­inate the field-effects for that phenomenon. For the shear viscosity field-effect (also referred to as the viscomagnetic effect, or the shear viscosity SBE), the dominant po­larization is the simple 11 polarization, while for the thermal conductivity field-effect, the dominant polarization is the wll polarization. In the shear viscosity case no other polarization contributions can be detected within experimental error, but in the

138

thermal conductivity case there is evidence that the Wj polarization may also play a minor role. This latter statement is cloaked in some ambiguity because the usual expression for the thermal conductivity field-effect is obtained by making an approx­imation, commonly referred to as the 'spherical approximation', which involves the retention of only one of the three irreducible components of the w'If polarization, namely that having rank one, since the rotational invariance of the linearized collision operator R does not allow WE to be collision ally coupled directly to the rank 2, 0 components [2]. A complete analysis for the w'If polarization was first given for the pure gas of diatomic molecules by Tip [45a], and was later extended to other molec­ular gases by Thijsse et al. [45b]. It is possible to fit the data within experimental error either by a single-term expansion using w'If but not making the spherical ap­proximation, or by introducing a second polarization Wj, and making the spherical approximation for both contributions. It has not been found to be possible to dif­ferentiate unequivocally between these two analyses by comparing the nature of the fits to the experimental data. Accurate calculations for realistic interaction potentials will, however, allow this distinction to be made.

3.3.1. Pure Gases

We shall discuss only the expressions for the dominant polarizations present in each case. For the shear viscosity field-effects the dominant polarization for all linear molecular gases is the 'If polarization, for which the governing expressions are

v = 1,2, (3.25)

in which f(x) = x2 /(1 + x2 ), g(x) = x/(l + x2 ), while ~02 and 7./J02 are given by

grotJ-tNkT B ~02 = cr 1i<5(0200) p'

6 2 ( ~~gg) 7./J02 = 6(2000)6(0200)'

(3.26)

with grot the rotational g-factor, and J-tN the nuclear magneton. In addition to the effective cross section 6(2000) which determines TJiso in lowest order, the field-effects depend upon two new effective cross sections, namely 6(0200) which determines the position on the B / p axis at which half-saturation values of the longitudinal effects or extrema of the transverse effects occur, and 6(~~gg) which contributes to the mag­nitudes of the field-effects. Both cross sections can therefore be determined from measurements of the field-effects. Measurement of all five shear viscosity coefficients is required, however, if meaningful conclusions are to be drawn regarding the contribu­tions arising from other polarization terms in the distribution function. In particular 1)t vanishes identically for the 'If polarization, but does not vanish for any of the other polarizations that may contribute to 11, so that deviations of TJt from zero di­rectly indicate the extent to which other polarization terms contribute to 1)(0). The saturation values of f:l.TJt /TJ(O) and f:l.TJi /TJ(O) are the same for the 'If polarization,

139

so that deviations of their difference at saturation from zero also provide indications of the contributions of other polarization terms. Experimental results indicate quite clearly [46] that there is little contribution to !:!."I/rJ(0) coming from other polarization terms.

Although we have introduced Cl>0200 as the dominant polarization for the shear viscosity field-effects without much ado, it is now worth discussing briefly the question of the appropriate scalar factors accompanying the unadorned basis tensors. We have seen in expression (3.2) that the scalar factors accompanying a tensor which has (irreducible) rank p in Wand rank q in j are in general arbitrary. There is no a priori reason why the indices st in the expansion tensor Cl>02st should both be zero. What is clear from the dependence of the field-effects upon the orientation of B relative to the flow direction is that the leading tensorial term is Cl>02st rather than, say Cl>22st. For a number of years there were three suggested zeroth-order expansion terms with 'If providing the second rank tensorial character: the dilemma as to which of the three gave the most appropriate single-term representation of the shear viscosity field-effects was referred to as the 'normalization question', since the precise nature of the scalar factor was unknown. The lack of such knowledge led to the introduction of the labelling 027r, in which the 7r indicated that the shear viscosity was involved. A careful experimental analysis, based upon shear viscosity data, collision-broadening data for the depolarized Rayleigh light scattering spectrum, and flow birefringence data established that it was indeed the unadorned Cl>0200 polarization that should be identified with Cl>021r [47]. Converged quantum close-coupled collisional calculations based upon an accurate ab initio potential energy surface for the HD-He interaction had also indicated earlier that Cl>0200 was the correct basis function for providing the zeroth-order shear viscosity field-effects [48].

The simplest expressions for the thermal conductivity field-effects are obtained when the alternate energy-flux basis function Cl>10E is used for the leading term in the thermal conductivity expansion. The expressions are [2,30]

(3.27a)

Atr 1 EE A(O) = -"2 'l/J12 [g(~d + 2g(~12)], (3.27b)

in which 62 can be obtained from Eq. (3.26) by replacing 6(0200) by 6(12E), while 'l/Jfl is given by

62( lDE) 'l/JEE l2E

12 = 6 o(12E)6(lOE)· (3.28a)

As in the shear viscosity case, this expression contains two effective cross sections, 6 o(12E) and 6( ~~~), in addition to the cross section 6(lOE) that determines Aiso.

Values for both cross sections can be obtained from measurements of anyone of the three components of A. Here too, is it important to determine whether or not there is a single dominant polarization. Such information requires the measurement of both

140

~>'II and ~>'1-. In contrast to the shear viscosity case, in which the polarization 'IJ is readily treated exactly (since it is fully irreducible), the w'IJ polarization is not fully irreducible, so that each of the three irreducible components (of rank 1, 2, 3) can contribute; since the rank 2, 3 components are not directly collisionally coupled to the microscopic heat flux -WE, their contributions are much smaller than that of the rank 1 component. It is not uncommon to use the so-called 'spherical approximation', in which the rank 2, 3 components of w'IJ are ignored, and only the rank 1 component (commonly referred to as the Kagan vector) is retained. In this approximation the ratio ~~ I has the value 3/2. The next most important polarization term, W j, leads

II in the spherical approximation to a value 2 for this ratio, and gives a contribution to ~A and >'(0) which has opposite sign to that from the w'IJ polarization. Significant deviations of ~>'1-/ ~>'II from the value 3/2 are more likely to be due to the presence of the W j polarization than to failure of the spherical approximation.

We have already seen that unambiguous determination of 6(1001) in the two-flux approach requires a knowledge of 17 and 17v in addition to >., or of the translational thermal conductivity>. t, T/ and >.. The strength function 'l/J~E has the structure [30]

EE {6U~1°)[6(1001) - r6U~~n]- 6U~~1 )[6U~~n - r6(1010)]F 'l/J12 = 6 o(12E)[6(1001)6(1010) - 62(;~~~)][6(1001) - 2r6U~~n + r 26(1010)] '

(3.28b)

for the two-flux approach. In this case there are two production cross sections, 6( ~~~1) and 6( ~~}g), whose relative sign is not generally known, in addition to 6 o(12E) and

the cross sections 6(1010),6(1001) and 6(~g~~) which determine >'iso. Again the question of 'normalization' enters. In this case the appropriate scalar factor accompa­nying wlI in (J?12E has been determined from laser fluorescence studies of molecular 12 to be unity, so that it is the unadorned w'IJ polarization that is relevant [49J. This has an important consequence for the analysis of the components of ~>./ >.(0), since it has been shown [50J that 6(~~~g) = - Js6(~~gg), with the consequence that experimental measurements of ~17/T/(0) can be used to eliminate one of the two pro­duction cross sections from Eq. (3.28b), to allow the unambiguous determination of 6(~~~1) from ~>'/>'(O).

We have discussed the field-effects for the shear viscosity and the thermal con­ductivity of pure gases. We should also expect in principle to be able to observe a field-effect for the volume viscosity coefficient. Such a field-effect should show up as a B / p-dependence of the non-classical part of the sound absorption coefficient. No such effect has been found. This may be due in part to the fact that the volume viscosity has no contribution from the simple 11 polarization. It requires a polarization of the type 'WWJ or 'ww''IJ. These polarizations can be expected to contribute only at rather high applied magnetic field strengths. In addition to the effect of magnetic fields on the existing transport coefficients, there is also a field-dependent cross co­efficient between the shear and volume viscosity schemes, again not associated with the 11 polarization [2]. These effects have been considered in detail by McCourt and Snider [51].

141

3.3.2. Binary Mixtures

Except for the case of the volume viscosity 1)v, field-effects have been measured for all the transport phenomena occurring in binary mixtures and discussed in §3.2. The formal structure of the field-effect equations for binary mixtures of a linear molecular gas and a noble gas is the same as that given in Eq. (3.25), in which ';02 is given by Eq. (3.26), but with 6(0200) replaced by 6(0200IA) and 'l/J02 given by [2,52]

[B 20006( 2000 I A) + XB !!!A6( 0200 I A) B]2 ,,/. A 0200 B mB 2000 A A ~=~ ,

6(0200IA)[XABlOOO + xBB§OOO] (3.29)

with BlOOO , B§oOO given by

B 2000 _ 6(2000Ik') - xk,6(~g~gl;, hk'

k - 6(2000Ik)6(2000Ik') - XkXk,62(~ggg I;,)kk' ' k' i= k, k = A, B. (3.30)

The thermal conductivity field-effect in the lowest approximation provided by the total-energy-fiux basis set can be written as

(3.31a)

(3.31b)

.Atr EE 1 EE .A(O) = 'l/Jll g(';l1E) - 2'l/J12 [g(62E) + 2g(262E)], (3.31c)

in which 61E, 62E are given by Eq. (3.26), but this time with 6(0200) replaced by 6(llEIA), 6(12EIA), respectively, and 'l/JftE, 'l/Jrl obtained from Eqs. (3.29) and (30) by replacing '2000' by 'llE', respectively, '12E' in Eq. (3.29). An alternative expres­sion for the components of 6A/.A(0) does exist for the two-flux basis set [53]' but it does not allow an unambiguous analysis to be made of the data, since no relation ex­ists between 6(~~~~1~) and 6(~~;I~) which is equivalent to that between 6(~~~~) and 6n~~~) for pure polyatomic gases. This is essentially because the thermal con­ductivity cross section has an elastic diffusion-like contribution which only exists for the unlike interaction.

Similar field-effect expressions exist for the diffusion and thermal diffusion co­efficients. There is, however, an important difference between the polarization term associated with diffusion and that associated with the thermal conductivity. A care­ful comparison amongst the experimental results for diffusion, thermal diffusion and thermal conductivity field-effects has established that although the tensor nature of the polarizations are identical, the scalar factors for the diffusion and thermal conduc­tivity field-effects cannot be the same [54]. It was not possible to establish the precise

142

nature of the scalar factor for the diffusion field-effects, and it remains unknown at the present time. The field-effect equations for tlD J..I D(O), tlD1l1 D(O) and Dtrl D(O) can be obtained from Eqs. (3.31) by replacing 'lj;fiE, 'lj;f.l, ~llE,62E by 'lj;tt,'Ij;t~, ~lld, 62d, respectively, where 'lj;tt, 'lj;t~ are given by [54,2]

~2( 1000 I A) 'lj;dd = XBC1 pqd A AB pq 6(1000 IA)AB6o(pqdIA) ,

(3.32)

while ~lld, 62d are obtained from Eq. (3.26) in the same way that ~llE, 62E are.

4. Relaxation Phenomena

In previous sections we have discussed transport phenomena and their magnetic field­effects. The latter phenomena are strongly sensitive to the anisotropy of the inter­molecular interaction potential, while the former phenomena are sensitive mainly to the isotropic part of the potential. Another group of phenomena which are sensi­tive to the anisotropic parts of the interaction potential is the class of relaxation phenomena [2]. Included amongst this class are collision broadening and shifting of rotational Raman and Rayleigh lines [55], the collisional broadening and shifting of dipole-allowed spectral lines [56], nuclear spin relaxation in molecules containing nu­clei with spin angular momentum [57], rotational and vibrational relaxation [58], flow birefringence [59], and microwave nonresonant absorption [60]. We have already in­cluded rotational and vibrational relaxation contributions to sound absorption in the previous sections, so that no more will be said here about this topic.

Collision broadening studies of the depolarized Rayleigh line have proven par­ticularly useful in that they complement the results of field-effect studies for the shear viscosity. Data for this phenomenon are available for a number of polyatomic gases [61]. The most readily observed behaviour for the depolarized Rayleigh (DPR) spectrum is a line with Lorentzian shape,

( 4.1)

in which w represents the frequency difference between the scattered and incident light, and tlwl/2 is the half-width at half-height, which in the collision-broadening regime can be expressed as

( 4.2)

with n the number density of the gas, cr the mean relative thermal speed, and 6(0200) an effective cross section for collisional realignment. The 2 appearing in this cross section designates that the second rank tensor polarization involved is 'If 1 (4j2 -3/4) rather than the tensor 'If that appears in the shear viscosity field-effects. This specific form is dictated by the fact that the depolarized component of the scattered

143

Rayleigh light is associated with the diagonal-in-j part of the anisotropic molecular polarizability tensor '(X, namely

= _ 2(all - a-L) ,......, ( 0: )diag - - 4j2 _ 3 JJ , (4.3)

for linear molecules, in which all and a-L represent the molecular polarizability parallel and perpendicular to the figure axis. The DPR collision broadening cross section involves the same scattering S-matrix elements that 6(0200) does, but has a different weighting of the rotational quantum numbers.

The same basic angular momentum polarization occurs in nuclear magnetic relax­ation studies for nuclei which relax either via dipole-dipole or quadrupolar intramolec­ular mechanisms. There are subtle differences between the cross sections, in that the DPR collision-broadening cross section in a pure gas has a contribution from the colli­sion partner, which is absent from the NMR cross section. This difference is indicated by writing 6'(0200) for the NMR cross section, and 6(0200) = 6'(0200) + 6"(0200), in which the primed cross section is called the 'self-part' and the double-primed cross section is called the 'non-self-part', and is associated with the collision partner. For binary mixtures of polyatomic gases with noble gases, the two cross sections (in the infinite-dilution limit) become the same. For molecules with spin-~ nuclei (apart from H2 ) the dominant intramolelcular relaxation mechanism is via the spin-rotation interaction: the relevant relaxation cross section is 6'(0100).

Birefringence is associated with an anisotropic dielectric tensor E'. Normally, an anisotropic dielectric tensor is associated with a crystal in which the crystal structure is non-cubic. However, it can happen that for fluids a sufficiently large flow velocity gradient can give rise to an anisotropic medium through which light can pass. The appropriate constitutive relation is

'E' - -2(3 "ilv , - , ( 4.4)

in which f3 is called the coefficient of flow birefringence. The kinetic theory expression for f3 is [59a]

f3 = - n~a (cJ>0200IiR-lllcJ>2000), 2Eov'15

in which EO is the electric permittivity of free space, ~a == all - a-L, or

~a 6(°200) f3 2000

= - 2Eo\l"3O cr 6(0200)6(2000)'

(4.5)

(4.6)

Notice that flow birefringence measurements allow the sign of 6(~~gg) to be obtained: it has been found to be positive for all polyatomic gases for which flow birefringence has so far been measured.

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5. Discussion, Summary and Conclusions

Ultimately, we are interested in the role that transport phenomena, field-effects on transport phenomena, and relaxation phenomena can play in elucidating the details of the intermolecular potential governing the interaction between pairs of molecules. The field-free transport coefficients, referred to generically as L a{3 (0), are largely in­sensitive to the anisotropy of the intermolecular interaction, but do provide important restrictions on its isotropic component. This is likely the reason why direct inversion procedures, which have been applied successfully to monatomic gas transport property data obtained over a wide range of temperatures [62]' have been much less successful when they have been applied to systems which have anisotropic interactions [63].

The most accurate intermolecular potentials available are those resulting from multiproperty analysis [64] and from the analysis of Van der Waals spectra [65]. Ide­ally, Van der Waals spectral data should be included in the multiproperty analysis. Such data are not always available, especially for molecules that do not contain hydro­gen. The most common gas phase properties employed in such analyses are the second virial coefficient, the shear viscosity and the thermal conductivity for pure gases, and the interaction second virial coefficient, mixture viscosity, mixture thermal conduc­tivity and in addition, diffusion and thermal diffusion for gas mixtures. Very useful, if available, are molecular beam differential and integral scattering cross section data, and ab initio self-consistent-field Hartree-Fock short-range energies. In addition it is important for the determination of anisotropies to have relaxation cross section data, state-to-state inelastic differential scattering cross section data, and field-effect data. In particular, it is now known that the field-effect data provide useful information on the anisotropy high up on the repulsive wall of the potential surface [66].

Shear viscosity, thermal conductivity and diffusion data can be obtained at the present time with an accuracy of ±O.5% (and in some cases even more accurately). This means that quite small errors in the isotropic potential can be detected and corrected in principle. To work at this level of refinement it is necessary to have appropriately accurate expressions for the transport properties: this has been the challenge which modern kinetic theory has had to meet. To be within ±O.5% it is necessary to utilize at least second Chapman-Cowling approximation corrections, and to include the polarization corrections that are probed by the field-effects. It is also necessary to be able to calculate accurately the various effective cross sections that enter into the expressions for the transport coefficients. This we are now able to do, both quantum mechanically for molecules having small moments-of-inertia, and classically for other molecules.

An illustration of the small, yet significant, differences in the potential energy surface to which transport phenomena are sensitive is provided by the NrHe system. There are four accurate potential energy surfaces proposed for the N2-He interaction, all having quite similar isotropic contributions (see Fig. 1 of Ref. [66b]), but different anisotropic contributions (see Fig. 2 of Ref. [66b]). The ordinary transport coef­ficients are quite insensitive to the anisotropies: for example an increase of 17% in the position anisotropy, together with a 31% increase in the depth anisotropy for this

145

Table 3. Classical trajectory results for the N2-He diffusion coefficient* D(O)

T jK Potential energy surface Expt.

ESMSV M3SV BTT HFD1M

77 0.0696 0.0686 0.0696 0.0708

0.0704 0.0693 0.0704 0.0717 0.0725 ± 0.0012a

277 0.6220 0.6177 0.6146 0.6273

0.6351 0.6303 0.6282 0.6407 0.6182 ± 0.0031 b

300 0.7110 0.7174 0.7067 0.7028 0.7261 0.7212 0.7184 0.7327 0.7027 ± 0.0035b

563 2.040 2.030 2.013 2.052

2.084 2.072 2.057 2.094 2.084 ± 0.042c

873 4.254 4.244 4.203 4.256 4.344 4.330 4.291 4.339 4.282 ± 0.086c

* Units of D(O) used in this table are 10-4 m2s-1 . Results from Ref. [66b]. a S. P. Wasik and K. E. McCulloh (1969) J. Res. NBS 73, 207-211. b R. D. Trengove and P. J. Dunlop (1982) Physica U5A, 339-352. c J. Kestin, S. T. Ro and W. A. Wakeham (1972) J. Chern. Phys. 56,5837-4042.

interaction causes a change of 0.6% in the binary diffusion coefficient D(O) at 300 K. The temperature dependence of D(O) for the four potential energy surfaces, designated in Ref. [66b] as the ESMSV, M3SV, BTT and HFD1M surfaces, therefore illustrates the sensitivity of a typical transport coefficient to relatively small differences between the isotropic components of these four potentials. The values of D(O) given in Table 3 were obtained from CT calculations, which provide diffusion cross sections that are within ±0.1% of their exact values.

The upper entry for each potential surface at a given temperature is the result of a first Chapman-Cowling approximation calculation, while the lower entry is the result of a second Chapman-Cowling approximation calculation. These results illustrate both the sensitivity to the potential surface, and the need for accurate calculations in order to make comparisons with precision experimental results.

We have seen that there are three significant differences between the interpreta­tion of transport coefficient data obtained for polyatomic gases and for noble gases. Firstly, the polyatomic gas effective cross sections extracted from the data will have inelastic contributions in addition to the elastic contributions found for monatomic gas cross sections. These inelastic contributions are present even in the case of the shear viscosity and diffusion coefficients, whose structure in terms of effective cross

146

Table 4. Comparison between calculated and measured shear viscosity field-effect cross sections for the N2-He interaction. *

T /K 6/ A2 Potential energy surface Expt.

ESMSV M3SV BTT HFDIM

77 6(0200IA)AB 6.4 7.1 7.3 10.6 8.3 ± 0.9

6( 2000 1 A ) 0200 A AB 0.096 0.110 0.128 0.174 0.14 ± 0.01

293 6(0200IA)AB 2.4 3.0 4.1 5.5 5.6 ± 0.3

6( ~~gg 11 )AB 0.041 0.052 0.073 0.092 0.08 ± 0.01

* Results from Ref. [66b].

sections remains unchanged in passing from the monatomic to the polyatomic gas. The inelastic contributions should not be neglected in the the analysis, since they are always at least of the order of second Chapman-Cowling approximation corrections. Secondly, the thermal conductivity and thermal diffusion coefficients acquire effective cross sections that do not appear in the monatomic expressions. Some of these cross sections are largely elastic, but some of them have substantial inelastic contributions, or are purely inelastic. Thirdly, all these phenomena have polarization contributions that contribute at the level of about 1%. It is now possible to evaluate the field-effects with an accuracy of ±1O%. This is illustrated in Table 4, again for the N2-He system. It is also worth mentioning that there is a new transport coefficient that is associated solely with the existence of internal molecular states, namely the volume viscosity 'f/V.

Finally, it seems worthwhile commenting upon the availability of transport and field-effect data. Measurements of the shear viscosity coefficient exist for a large number of polyatomic gases and binary mixtures containing polyatomic gases: in many cases, the measurements have been carried out over an extensive temperature range, usually from 298-973 K, but in some cases to even higher temperatures [67]. Measurements of the thermal conductivity coefficient are more limited, both in the number of polyatomic species involved, and in the temperature range covered. Indeed, few highly accurate measurements have been carried out at temperatures other than room temperature [68]. There are numerous measurements of binary diffusion coeffi­cients, covering an extensive temperature range [69], but with few exceptions [70]' the accuracy is seldom better than ±1%. Thermal diffusion coefficients have also been measured extensively, but because the experiments are inherently difficult to carry out, their accuracy is not particularly high. Again, there are some exceptions [70].

147

Acknowledgements

I am grateful to my graduate students, Ernie Hanson, Mark Thachuk, and Clement Wong for their unstinting efforts, and their enthusiastic participation in my research program. I am also grateful to Jan Beenakker, Alan Dickinson, Ivan KU8cer, Wing-Ki Lin, Velisa Vesovic, and Bill Wakeham for extensive discussions on various aspects of kinetic theory and molecular collisions. This research has been supported in part by a NSERC of Canada grant in aid of research.

References

[1] Aziz, R. A. (1984) Interatomic potentials for rare-gases: pure and mixed interac­tions, in M. L. Klein (ed) Inert gases. Potentials, dynamics and energy transfer in doped crystals, Springer-Verlag, Berlin, pp.5-86.

[2] McCourt, F. R. W., Beenakker, J. J. M., Kohler, W. E., and KU8cer, I. (1990) Nonequilibrium Phenomena in Polyatomic Gases. Volume 1. Dilute Gases, Ox­ford University Press, Oxford.

[3] (a) Ubbelohde, A. R. (1935) The thermal conductivity of polyatomic gases, J. Chern. Phys. 3, 219-223; (b) Hirschfelder, J. O. (1957) Heat conductivity in polyatomic or electronically excited gases. II, J. Chern. Phys. 26, 282-285.

[4] (a) Beenakker, J. J. M. and McCourt, F. R. (1970) Magnetic and electric effects on transport properties, Annu. Rev. Phys. Chern. 21, 47-72; (b) Beenakker, J. J. M. (1974) Transport properties in gases in the presence of external fields, in G. Kirczenow and J. Marro (eds) Transport Phenomena, Springer-Verlag, Berlin, pp. 414-469; (c) Snider, R. F. (1974) Transport properties of dilute gases with internal structure, op cit, 470-517.

[5] McCourt, F. R. W., Beenakker, J. J. M., Kohler, W. E., and KU8cer, I. (1991) Nonequilibrium Phenomena in Polyatomic Gases. Volume 2. Cross Sections, Rarefied Gases, Oxford University Press, Oxford.

[6] Boltzmann, L. (1872) Weitere Studien iiber das Wiirmegleichgewicht unter Gas­molekiilen, Wien Ber. 66, 275-370. [English translation appears in Brush, S. G. Kinetic Theory. Volume 2. Irreversible Processes, pp. 88-175.]

[7] Chapman, S. and Cowling, T. G. (1970) Mathematical Theory of Nonuniform Gases, Third Edition, Cambridge University Press, Cambridge.

[8] Hirschfelder, J. 0., Curtiss, C. F. and Bird, R. B. (1954) Molecular Theory of Gases and Liquids, Wiley, New York.

[9] Ferziger, J. H. and Kaper, H. G. (1972) Mathematical Theory of Transport Pro­cesses in Gases, North-Holland, Amsterdam.

[10] Chapman, S. (1916) On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction in a non-uniform monatomic gas, Phil. Trans. Roy. Soc. A, 216, 279-348.

148

[11] Enskog, D. (1917) Kinetische Theorie der Vorgiinge in miissig verdiinnten Gases, Almqvist and Wiksell, Uppsala.

[12] Grad, H. (1952) Statistical mechanics, thermodynamics, and fluid dynamics of systems with an arbitrary number of integrals, Commun. Pure and Appl. Math. 5,455-494.

[13] Hess, S. and Waldmann, L. (1966) Kinetic theory for a dilute gas of particles with spin, Z. Naturforsch. 21a, 1529-1546.

[14] Bhatnagar, P. L., Gross, E. P. and Krook, M. (1954) A model for collision processes in gases. I. Small amplitude processes in charged and neutral one­component gases, Phys. Rev. 94,511-525.

[15] Kantorovich, 1. V. and Krylov, V. I. (1958) Approximate Methods of Higher Analysis (transl. C. D. Benster), Interscience, New York, pp. 263-283.

[16] Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions, Dover Publications, New York.

[17] Wang Chang, C. S. and Uhlenbeck, G. E. (1970) The kinetic theory of a gas in alternating outside force fields, eds. J. de Boer and G. E. Uhlenbeck Studies in Statistical Mechanics. V., 76-100, North-Holland, Amsterdam.

[18] Waldmann, L. (1958) Transporterscheinungen in Gasen von mittleren Druck in S. Fliigge (ed) Handbuch der Physik. XII, Springer-Verlag, Berlin, pp. 364-38l.

[19] Maxwell, J. C. (1867) On the dynamical theory of gases, Phil. Trans. Roy. Soc. (London) 157, 49-88; (1868) Phil. Mag. 35, 129-145, 185-217. [Reprinted (1965) in The Scientific Papers of James Clerk Maxwell, Dover Publications, New York.]

[20] De Groot, S. R. and Mazur, P. (1962) Nonequilibrium Thermodynamics, North­Holland, Amsterdam.

[21] Maitland, G. C., Rigby, M., Smith, E. B. and Wakeham, W. A. (1981) Inter­molecular Forces: Their Origin and Determination, Oxford University Press, Oxford.

[22] (a) Kestin, J., Khalifa, H. E. and Wakeham, W. A. (1978) The viscosity and diffusion coefficients of the binary mixtures of xenon with the other noble gases, Physica 90A, 215-228; (b) Assael, M. J., Wakeham, W. A. and Kestin, J. (1980) Higher-order approximation to the thermal conductivity of monatomic gas mix­tures, IntI. J. Thermophys. 1, 7-32.

[23] Pidduck, F. B. (1922) Kinetic theory of a special type of rigid molecule, Proc. Roy. Soc. (London) A101, 101-112.

[24] Kagan, Yu. and Afanas'ev, A. M. (1962) On the kinetic theory of gases with rotational degrees of freedom, SOy. Phys.-JETP 14, 1096-1101 [original Russian Ref. Zh.E.T.P. 41,1536-1545 (1961)].

[25] Kagan, Yu. and Maksimov, L. A. (1962) Transport phenomena in a paramagnetic gas, SOy. Phys.-JETP 14, 604-610 [original Russian Ref. Zh.E.T.P. 41,842-852].

149

[26] Senftleben, H. (1930) Magnetische Beeinflussung des Wiirmeleitvermogens para­magnetische Gase, Physik. Z. 31,901-903.

[27] (a) Waldmann, L. (1957) Die Boltzmann-Gleichung fur Gase mit rotierenden Molekulen, Z. Naturforsch. 12a, 660-662; (b) Snider, R. F. (1960) Quantum­mechanical modified Boltzmann equation for degenerate internal states, J. Chern. Phys. 32,1051-1060.

[28] Mason, E. A. and Monchick, L. (1962) Heat conductivity of polyatomic and polar gases, J. Chern. Phys. 36, 1622-1639.

[29] (a) Millat, J., Plantikow, A. and Nimz, H. (1988) Effective collision cross-sections for polyatomic gases from transport properties and thermomolecular pressure dif­ferences, z. phys. Chemie 269, 865-878; (b) Millat, J., Mustafa, M., Ross, M., Wakeham, W. A., and Zalaf, M. (1987) The thermal conductivity of argon, carbon dioxide and nitrous oxide, Physica 145A, 461-497; Millat, J., Ross, M., Wake­ham, W. A., and Zalaf, M. (1988) The thermal conductivity of neon, methane and tetrafluoromethane, Physica 148A, 124-152.

[30] Thijsse, B. J., 't Hooft, G. W., Coombe, D. A., Knaap, H. F. P. and Beenakker, J. J. M. (1979) Some simplified expressions for the thermal conductivity in an external field, Physica 98A, 307-312.

[31] Millat, J., Vesovic, V. and Wakeham, W. A. (1988) On the validity of the sim­plified expression for the thermal conductivity of Thijsse et al., Physica 148A, 153-164.

[32] Van den Oord, R. J. and Korving, J. (1988) The thermal conductivity of poly­atomic molecules, J. Chern. Phys. 89,4333-4338.

[33] Prangsma, G. J., Borsboom, L. J. M., Knaap, H. F. P., Van den Meijdenberg, C. J. N. and Beenakker, J. J. M. (1972) Rotational relaxation in ortho hydrogen between 170 and 300 K, Physica 61, 527-538.

[34] (a) McCourt, F. R., Knaap, H. F. P. and Moraal, H. (1969) The Senftleben­Beenakker effects for a gas of rough spherical molecules. 1. The thermal conduc­tivity, Physica 43, 485-512; Moraal, H., McCourt, F. R. and Knaap, H. F. P. (1969) The Senftleben-Beenakker effects for a gas of rough spherical molecules. II. The viscosity scheme, Physica 45, 455-468; (b) Condiff, D. W., Lu, W-K. and Dahler, J. S. (1965) Transport properties of polyatomic fluids, a dilute gas of perfectly rough spheres, J. Chern. Phys. 42, 3445-3475.

[35] Maitland, G. C., Mustafa, M. and Wakeham, W. A. (1983) Second-order approx­imations for the transport properties of dilute polyatomic gases, J. Chern. Soc. Faraday Trans. 2, 79, 1425-1441.

[36] Turfa, A., Knaap, H. F. P., Thijsse, B. J. and Beenakker, J. J. M. (1982) A classical dynamics study of rotational relaxation in nitrogen gas, Physica 112A, 18-28.

[37] MacRury, T. B., Steele, W. A. and Berne, B. J. (1976) Intermolecular potential models for anisotropic molecules, with applications to N2 , CO2 , and benzene, J. Chern. Phys. 64, 1288-1299.

150

[38] Prangsma, G. J., Alberga, A. H. and Beenakker, J. J. M. (1973) Ultrasonic determination of the volume viscosity of N2 , CO, CH4 and CD4 between 77 and 300 K, Physica 64, 278-288.

[39] Kohler, W. E. and Schaefer, J. (1983) Theoretical studies of H2-H2 collisions. V. Ab initio calculations of relaxation phenomena in parahydrogen gas, J. Chern. Phys. 78, 6602-6610.

[40] Van den Oord, R. J., de Lignie, M. C., Beenakker, J. J. M. and Korving, J. (1988) The role of the internal energy in the distribution function of a heat conducting gas, Physica 152A, 199-216.

[41) Van Houten, H. and Ten Bosch, B. I. M. (1984) Kinetic theory for the volume viscosity in binary mixtures of polyatomic and noble gases, Physica 128A, 371-376.

[42] Kohler, W. E. and Halbritter, J. (1975) Kinetic theory of thermal diffusion in a magnetic field, Z. Naturforsch. 30a, 1114-112l.

[43] Hulsman, H. and Knaap, H. F. P. (1970) Experimental arrangements for mea­suring the five independent shear-viscosity coefficients in a polyatomic gas in a magnetic field, Physica 50, 565-572.

[44] (a) Hermans, P. W., Hermans, L. J. F., and Beenakker, J. J. M. (1983) A survey of experimental data related to the non-spherical interaction for the hydrogen isotopes and their mixtures with noble gases, Physica 123A, 173-211; (b) Van Houten, H., Hermans, L. J. F. and Beenakker, J. J. M. (1985) A survey of experimental data related to the non-spherical interaction for simple classical linear molecules and their mixtures with noble gases, Physica 131A, 64-103.

[45] (a) Tip, A. (1967) The influence of angular momentum anisotropy on the heat conductivity of diatomic gases, Physica 37, 82-96; (b) Thijsse, B. J., 't Hooft, G. W., Knaap, H. F. P. and Beenakker, J. J. M. (1980) On the Senftleben­Beenakker effect on the thermal conductivity, Physica 102A, 281-297.

[46] (a) Hulsman, H. and Burgmans, A. L. J. (1969) The five shear viscosity coef­ficients of a polyatomic gas in a magnetic field, Phys. Letters 29A, 629-630; (b) Mazur, E., Viswat, E., Hermans, L. J. F. and Beenakker, J. J. M. (1983) Experiments on the viscosity of some symmetric top molecules in the presence of magnetic and electric fields, Physica 121A, 457-478.

[47] Mazur, E., van Houten, H. and Beenakker, J. J. M. (1985) A comparison of data on the viscomagnetic effect, flow birefringence and depolarized Rayleigh line broadening, Physica 130A, 505-522.

[48] (a) Green, S., Liu, W.-K., and McCourt, F. R. (1983) Close-coupled calculations of viscosity transport-relaxation cross sections for HD-He: comparison with ex­periment, Physica 117 A, 616-630; (b) Kohler, W. E. and Schaefer, J. (1983) Ab initio calculation of transport-orientation-phenomena (Senftleben-Beenakker effects) for HD infinitely dilute in He, Physica 120A, 185-212.

151

[49] Van den Oord, R J., De Lignie, M. C., Beenakker, J. J. M. and Korving, J. (1988) Optical determination of the j-dependence of angular momentum alignment in a heat conducting gas, Physica 150A, 77-96.

[50] (a) Moraal, H. and Snider, R F. (1971) Kinetic theory collision integrals for diatomic molecules, Chern. Phys. Letters 9, 401-405; (b) Snider, R F. (1974) On the evaluation of kinetic-theory collision integrals. III. General distorted-wave Born approximation, Physica 78, 387-419;

[51] McCourt, F. R. and Snider, R. F. (1967) Senftleben-Beenakker effect for the viscosity of a dilute gas of diamagnetic diatomic molecules, J. Chern. Phys. 47, 4117-4128.

[52] Burgmans, A. L. J., van Ditzhuyzen, P. G. and Knaap, H. F. P. (1973) The viscomagnetic effect in mixtures, Z. Naturforsch. 28a, 849-861.

[53] (a) Kohler, W. E. and Raum, H. H. (1972) Kinetic theory for mixtures of di­lute gases of linear rotating molecules, Z. Naturforsch. 27a, 1383-1393; (b) Heemskerk, J. P. J., Bulsing, G. F. and Knaap, H. F. P. (1974) The thermal conductivity of gases in a magnetic field: the concentration dependence, Physica 71, 515-528.

[54] Mazur, E., Hijnen, H. J. M., Hermans, L. J. F. and Beenakker, J. J. M. (1984) Experiments on the influence of a magnetic field on diffusion in N2 -Noble gas mixtures, Physica 123A, 412-427.

[55] (a) Knaap, H. F. P. and Lallemand, P. (1975) Light scattering by gases, Annu. Rev. Phys. Chern. 26,59-81; (b) Blackmore, R, Green, S. and Monchick, L. (1988) Polarized D2 Stokes-Raman Q branch broadened by He: a numerical cal­culation, J. Chern. Phys. 88,4113-4119; (1989) Dicke narrowing of the polarized Stokes-Raman Q-branch of the v = 0---'>1 transition ofD2 in He, J. Chern. Phys. 91,3846-3853; Green, S. (1990) Raman Q-branch line shapes as a test of the H2-

Ar intermolecular potential, J. Chern. Phys. 93, 1496-1501; (c) Rosasco, G. J., May, A. D., Hurst, W. S., Petway, L. B. and Smyth, K. C. (1989) Broadening and shifting of the Raman Q-branch of HD, J. Chern. Phys. 90, 2115-2124; Rosasco, G. J., Rahn, L. A., Hurst, W. S., Palmer, R E. and Hahn, J. W. (1989) Measurement and prediction of Raman Q-branch line self-broadening coefficients for co from 400 to 1500 K, J. Chern. Phys. 90,4059-4068.

[56] (a) Rabitz, H. (1974) Rotation and rotation-vibration pressure-broadened spectral lineshapes, Annu. Rev. Phys. Chern. 25, 155-177; (b) Ben Reuven, A. (1975) Spectral lineshapes in gases in the binary-collision approximation, Adv. Chern. Phys. 33, 235-293; (c) Hess, S. (1972) Kinetic theory of spectral line shapes. The transition between Doppler broadening and collisional broadening, Physica 61,80-94.

[57] Armstrong, R. L. (1987) Nuclear magnetic relaxation efects in polyatomic gases, Mag. Res. Rev. 12, 91-135.

[58] (a) Clarke, J. F. and McChesney, M. (1976) Dynamics of Relaxing Gases, Second Edition, Butterworths, London; (b) Ormonde, S. (1975) Vibrational relaxation

152

theories and measurements, Rev. Mod. Phys. 47, 193-258.

[59) (a) Hess, S. (1973) Flow-birefringence in gases. An example of the kinetic the­ory based on the Boltzmann-equation for rotating molecules, Acta Phys. Aus­triaca, Suppl. X, 247-265; (b) Baas, F., Breunese, J. N., Knaap, H. F. P., and Beenakker, J. J. M. (1977) Flow birefringence in gases of linear and symmetric top molecules, Physica 88A, 1-33; Baas, F., Breunese, J. N. and Knaap, H. F. P. (1977) Flow birefringence in gaseous mixtures, Physica 88A, 34-43; Van Houten, H. and Beenakker, J. J. M. (1985) Flow birefringence in gases at room temper­ature; new absolute values, Physica 130A, 465-482; Van Houten, H., Korving, J. and Beenakker, J. J. M. (1985) Flow birefringence in gases: the temperature dependence, Physica 130A, 483-489.

[60) Boudouris, G. (1969) Phenomenes de relaxation dielectrique (absorption non resonnante et dispersion) presentes par Ie gaz dans Ie domaine des microondes, Rev. Nuovo Cim. 1, 1-56.

[61) Keijser, R. A. J., van den Hout, K. D., de Groot, M. and Knaap, H. F. P. (1974) The pressure broadening of the depolarized Rayleigh line in pure gases of linear molelcules, Physica 75, 515-547; Keijser, R. A. J., Jansen, M. Cooper, V. G., and Knaap, H. F. P. (1971) Depolarized Rayleigh scattering in CO2 , OCS and CS2 , Physica 51, 593-600; Van den Hout, K. D., Hermans, P. W. and Knaap, H. F. P. (1980) The broadening of the depolarized Rayleigh line for hydrogen isotopes at low temperatures, Physica 104A, 548-572.

[62) Maitland, G. C. and Wakeham, W. A. (1978) Direct determination of intermolec­ular potentials from gaseous transport coefficients alone. Part I. The method; Part II. Application to unlike monatomic interactions, Molec. Phys. 35, 1429-1442,1443-1469; Maitland, G. C., Vesovic, V. and Wakeham, W. A. (1985) The inversion of thermophysical properties. I. Spherical systems revisited, Molec. Phys. 54, 287-300.

[63) Maitland, G. C., Vesovic, V. and Wakeham, W. A. (1985) The inversion of thermophysical properties. II. Non-spherical systems explored, Molec. Phys. 54, 301-319; Maitland, G. C., Mustafa, M., Vesovic, V. and Wakeham, W. A. (1986) The inversion of thermophysical properties. III. Highly anisotropic interactions, Molec. Phys. 57, 1015-1033.

[64) Pack, R. T, Valentini, J. J. and Cross, J. B. (1982) Multiproperty empirical anisotropic intermolecular potentials for ArSF6 and KrSF6 , J. Chern. Phys. 77, 5486-5499; Pack, R. T, Piper, E., Pfeffer, G. A. and Toennies, J. P. (1984) Mul­tiproperty empirical anisotropic intermolecular potentials. II. HeSF 6 and NeSF 6,

J. Chern. Phys. 80, 4940-4950.

[65) Hutson, J. M. (1990) Intermolecular forces from the spectroscopy of Van der Waals molecules, Annu. Rev. Phys. Chern. 41, 123-154.

[66) (a) Wong, C. C. K., McCourt, F. R. W. and Casavecchia, P. (1990) Classical trajectory calculation of transport and relaxation properties for N2-Ne mixtures, J. Chern. Phys. 93,4699-4712; (b) Beneventi, 1., Casavecchia, P., Volpi, G. G.,

153

Wong, C. C. K., McCourt, F. R. W., Corey, G. C. and Lemoine, D. (1991) On the N2-He potential energy surface, J. Chern. Phys. (in press).

[67] (a) Kestin, J. and Mason, E. A. (1973) Transport properties in gases (comparison between theory and experiment) in J. Kestin (ed.) 'AlP Conference Proceedings No. 11-Transport Phenomena-1973', American Institute of Physics, New York, pp. 137-192; (b) Kestin, J. and Wakeham, W. A. (1988) Transport properties of fluids. Thermal conductivity, viscosity, and diffusion coefficient, Cindas Data Series on Material Properties. Vol. 1-1, C. T. Ho (ed.), Hemisphere Publishing Corp., New York.

[68] Kestin, J., Nagasaka, Y. and Wakeham, W. A. (1982) The thermal conductivity of mixtures of carbon dioxide with three noble gases, Physica 113A, 1-26; Haran, E. N., Maitland, G. C., Mustafa, M. and Wakeham, W. A. (1983) The thermal conductivity of argon, nitrogen, and carbon monoxide in the temperature range 300-430 K at pressures up to 10 MPa, Ber. Bunsenges. Phys. Chern. 87, 657-663; Mustafa, M., Ross, M., Trengove, R. D., Wakeham, W. A. and Zalaf, M. (1987) Absolute measurement of the thermal conductivity of helium and hydrogen, Physica 141A, 233-248.

[69] Marerro, T. R. and Mason, E. A. (1972) Gaseous diffusion coefficients, J. Phys. Chern. Ref. Data 1, 2-118.

[70] Trengove, R. D. and Dunlop, P. J. (1982) Diffusion coefficients and thermal diffusion factors for five binary systems of nitrogen and a noble gas, Physica U5A, 339-352; Trengove, R. D., Robjohns, H. L. and Dunlop, P. J. (1983) Diffusion coefficients and thermal diffusion factors for the systems H 2-N2 , D2 -

N2 , H2-02 and D2-02 , Ber. Bunsenges. Phys. Chern. 87, 1187-1190; Dunlop, P. J. and Bignell, C. M. (1987) Diffusion and thermal diffusion in binary mixtures of methane with noble gases and of argon with krypton, Physica 145A, 584-596; Dunlop, P. J., Robjohns, H. L. and Bignell, C. M. (1987) Diffusion and thermal diffusion in binary mixtures of hydrogen with noble gases, J. Chern. Phys. 86, 2922-2926.

OVERVIEW ON EXPERIMENTAL DATA FROM SENFTLEBEN­BEENAKKER EFFECTS AND DEPOLARIZED RAYLEIGH SCATTERING

L. J. F. HERMANS Huygens Laboratory, Leiden University, P.O. Box 9504, NL 2300 RA Leiden, The Netherlands

ABSTRACT. The effects of external fields on transport properties of polyatomic gases ('Senftleben-Beenakker effects') and Depolarized Rayleigh light scattering ('DPR') result from the non-spherical part of the intermolecular interaction. Their unified description based on the linearized Boltzmann equation gives the results of such experiments in terms of effective cross sections, which are av­erages over velocities and rotational quantum numbers. These can be used for a multi property analysis of potential models. Alternatively, they may be em­ployed for determining various other cross sections directly by making use of theoretical relations between cross sections. This paper serves as an introduc­tion to SBE and DPR and to the vast literature on these phenomena. A brief description is given of the experimental techniques, their possibilities and limi­tations. An overview of the available cross sections for various gases facilitates their retrieval from the literature.

1. Introduction

In the field of intermolecular interactions for rotating molecules, two types of exper­iments in particular have yielded a wealth of information over the last 25 years [1]. One is the influence of external magnetic fields (and, for polar molecules, also electric fields) upon the transport properties, known as Senftleben-Beenakker effects (SBE). The other is depolarized Rayleigh (DPR) light scattering.

These effects have a number of features in common. First, both types of ex­periment are performed on dilute gases at or near equilibrium. The cross sections obtained are therefore essentially velocity averages over the Maxwell distribution, and are consequently often referred to as effective cross sections. In this sense they differ from molecular-beam-type experiments, which can resolve the velocity depen­dence of the cross sections. Second, they depend exclusively on processes connected with the rotational degrees of freedom and therefore on the non-spherical part of the interaction potential. This is in contrast to ordinary transport properties, which are mainly determined by the spherical part of the interaction. Third, the results of SBE and DPR experiments are non-state-specific, i.e. they yield cross sections averaged

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W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 155-174. ©1992Kluwer Academic Publishers.

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over the j and mj quantum numbers. In this sense they differ from optical experi­ments, such as rotational dipolar absorption and rotational Raman scattering, which probe specific rotational levels.

The question of why one would make use of SBE and DPR measurements in the first place may be raised, since molecular-beam experiments, combined with opti­cal detection schemes can provide velocity-selected and state-dependent information. The answer is twofold. First, the techniques of SBE and DPR give macroscopic in­formation, with kinetic theory providing a number of relations between various cross sections [2,3J. This makes it possible to determine certain cross section values directly from other experimentally determined values without having to make the elaborate detour through the intermolecular potential. Obviously, this does not compete di­rectly with the state-selected information obtained by other methods. However, the techniques of SBE and DPR have the additional advantage that they do not depend upon coincidences between molecular transitions and available lasers, nor upon the presence of a sufficiently large transition dipole moment. Instead, they can be ap­plied to any molecule, since each polyatomic molecule has a rotational magnetic dipole (needed to produce precession in SBE) and an anisotropic polarizability (needed for DPR). As such they supply indirect information where state-selected information is not available. Thus, they give a valuable contribution to a multiproperty analysis aimed at obtaining information on the intermolecular interaction. Detailed analyses for the H2-isotopes (e.g. by Kohler et a1., see Ref. [1]) have shown that especially the non-diagonal cross-sections are very sensitive to the details of the interaction.

Other techniques giving information compatible with SBE and DPR data are vol­ume viscosity measurements by ultrasound absorption (rotational relaxation), NMR and experiments on flow birefringence. We will not elaborate on those techniques in this review, but occasionally we shall use some of their results. A key reference to all phenomena mentioned above is the recent monograph by McCourt, Beenakker, Kohler and Kuscer [IJ, in which many details may be found.

2. Notation conventions and cross sections

In this survey, we will limit ourselves to the large class of diamagnetic molecules where the angular momentum fij results from the rotation of the nuclei only (frequently de­noted by fiN). We will therefore ignore molecules like NO (where the electronic ground state is all-state) and O2 (for which the total electronic spin is nonzero), although some experimental data are available also for these molecules [1 J. For the kinetic de­scription of the various phenomena in terms of effective cross sections a basis set of irreducible tensors {~} in the dimensionless peculiar velocity W = C( m/2kT)l/2 (with k the Boltzmann constant, T the temperature, m the particle mass), and the dimensionless angular momentumj is introduced (see Table 1). Various choices have been made for the scalar factors associated with these basis tensors. In the pre-1980 literature the scalar factors were usually expressed in terms of associated Laguerre (Sonine) polynomials in W 2 , and Wang-Chang-Uhlenbeck polynomials in j2. The normalized expansion tensors then are written as ~pqst, where p and q denote the

Table 1. Normalized expansion tensors and their meanings.

Tensor Name 1

cJ>0001 = (~)' (£ - E) Grot

1

cJ>0010 = (~), (W2 _ ~) 1

«>1001 = w (..1B...)' (£ - E) Grot

1

«>1010 = W (~), (W2 -~)

«>20 = y0,'ww'

«>0271" ='If'(!f)~(j2(j2 _ ~))~~P~2 1

«>10E = W ( ~~ )' [(W2 - ~ + £ - E] 1

«>10D = w (~~r [r(W2 -~) - ~(£ - E)]

«>12E = W'If(15)~(j2(j2 _ ~))~~P~2

«>12d = W'If(15)~ (j2(j2 - ~))~~ PJ2 ........ 1 .2 1

«>02 = 'If (25), (j2 - ~)-1(/_4)~' 1

«>01 = v'3j(j2)~'

«>10 = y0,w

a E is the equilibrium average (£)0.

rotational energya

translational energyb

flux of rotational energy E

flux of translational energyb

momentum flux

flow-induced tensor polarizationC

total energy flux

difference energy flux d

heat flow-induced polarization

diffusion-induced polarization

diagonal in j part of 'uu ' e

vector polarization

particle flux

b Note that a definition with opposite sign is used by various authors. C In Ref. [4] the second rank in j tensor is normalized in a different way. d r2 = 2Crot /5R. e u is a unit vector along the direction of the molecular figure axis.

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tensorial rank in Wand j, respectively, and sand t the order of the polynomials in W 2 and j2, respectively. However, around 1980 it was demonstrated [4] that a better description of the SBE was obtained if unspecified scalar factors were introduced in place of the traditional polynomials; in addition, these factors may differ for different transport processes. Thus, the expansion tensors were written in the more recent literature as «>pqs, with the's' referring to a prefactor PJ'q (W2, j2), in which the gen­eral index's' is replaced by 7r in the case of momentum transport, by E (or q) in the case of energy transport, and by d (or j) in the case of mass transport. Fortunately,

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the most recent information obtained from a comparison between flow birefringence and the SBE on viscosity, and from a direct determination of the 'Kagan polariza­tion' 4) 12E, suggests that (at least for heavy linear rotors) the dominant polarizations are 4)027r = 4)0200 and 4)12E = 4)1200; in other words p~2 = 1 = P~2. A detailed discussion of this point can be found in Ref. [1].

Solution of the appropriate linearized Boltzmann equation (viz. the Waldmann­Snider equation) leads to expressions for the macroscopic properties in terms of matrix elements of the linearized collision operator R with respect to the above basis of expansion tensors, 4)pqs or 4)pqst. The effective cross-sections, designated by 6, are then defined by

( 4)pqs I R4)pl qlSI) = nc 6( p q s ) r p'qls' (1)

and similarly for the expansion tensors 4)pqrs. Here cr = (8kT/7rmr)~ is the mean relative velocity, with mr being the reduced mass. Two types of matrix element - or cross section - occur: diagonal ones referring to the decay rates of the basis functions, and non-diagonal ones describing the rate of production of one non-equilibrium func­tion from the other. Short-hand notations are introduced for convenience: equal rows are not repeated in diagonal cross sections, e.g. 6(~!~:) == 6(pqrs), and when scalar factors are absent the last two indices are dropped, e.g. 6(pqOO) == 6(pq). To exem­plify the above connections, the cross section 6(20) refers to the decay of momentum flux, and is thus the ordinary shear viscosity cross section, while 6(027r) refers to the decay (or relaxation) of angular momentum 'IT-type polarization, with the scalar part of the polarization being the type produced in momentum flux; similarly, 6( o;~) describes the production of the latter polarization type from the momentum flux (the symbol ~ denotes a symmetric traceless tensor). In the case of polyatomic gas­noble gas mixtures, further labelling must be used to specify the collision partners. By convention the index A is used for the polyatomic molecule, and B for the noble gas atom. For example the cross section 6(;I:I:,II~)AB refers to the coupling of basis

tensor 4)pqs to 4)p1qI SI, both of the A-species, in A-B collisions. It may be interesting to note that cross sections for collision processes between unlike molecules, e.g. N2 -

Ar collisions, can be extracted not only from experiments on diffusion, but also from other quantities (such as DPR or the shear viscosity SBE), provided that the data are extrapolated to the limit of infinite dilution of the polyatomic molecule in the noble gas. The reason is, as will become clear in the next section, that in this limit the only contribution to the observed effect stems from polyatomic-noble gas collisions.

3. Principle and practice of the Senftleben-Beenakker Effects

3.1 EXPERlMENTAL TECHNIQUES

The basic idea of the field-effects is the following. In any transport process, the anisotropy in the velocity distribution will - through the non-spherical part of the intermolecular interaction - give rise to an anisotropy in angular momentum space,

159

often referred to as a 'polarization'. This, in turn, will influence the velocity distribu­tion. Consequently, the transport coefficients will be slightly different from an imagi­nary case in which this W - j coupling were absent. This difference - which reflects the non-spherical interaction - can be made visible by scrambling the anisotropy in j-space on the time scale of intermolecular collisions. The scrambling is achieved by making use of the precession of the molecular axes around an external field.

For ordinary, diamagnetic, molecules use is made of the (small) magnetic moment which results from molecular rotation, namely

/-Lrot = grot' j /-LN, (2)

where grot is the rotational ('Lande') g-tensor, which reduces to a scalar g-factor for linear molecules, and /-LN is the nuclear magneton. In an external magnetic field, the molecules will precess with a frequency w, which is independent of j, since it is determined by the ratio Jlrot/j. For a detailed discussion the reader is referred to Ref. [1], pp. 118-120.

For molecules having an electric dipole moment along j, the precession can also be achieved by making use of an electric field. The information obtained about the non-spherical potential is essentially the same as is obtained in the case of magnetic fields.

In the presence of an external field symmetry considerations require that the transport coefficients lose their scalar character. In the case of a heat flux q in the presence of a magnetic field B, for instance, one has

(3)

for B = (0, 0, B z ). The physical meaning of the tensor elements is illustrated in Fig. l.

An elegant physical explanation of the occurrence of magnetic-field-induced transverse heat transport in polyatomic gases has been given in Ref. [5].

In practical experiments allowing the determination of a single tensor element, a constant heat flux is generated between parallel plates having a temperature difference 6.T ~ 10K. Gas pressures are of the order of 102Pa (1 Torr) and magnetic field strengths mostly up to 2T (2 x 104 Gauss). Upon switching on the magnetic field, the change in 6.T resulting from the change in A (or, in the case of Atr, the occurrence of a transverse temperature difference 8T across the cell) is measured with a sensitivity of a few times 1O-5 K. This corresponds to a sensitivity in detecting changes in A of a few times 10-6 , if heat losses are small compared with the heat flux through the gas (which is most often the case).

An important aspect of the SBE is that the applied field is so low that, although the gas is no longer isotropic, the collision processes remain unaffected. Therefore, the matrix elements of the collision operator do not lose their isotropic character.

160

Figure 1. Physical meaning of the thermal conductivity components in a magnetic field B [cf. Eq. (3)]. In the field-free case one has A~ = All = A and Atr=O. From Ref. [10].

Consequently, the parity of the j-dependent polarization produced is dictated by that of the driving force, which is a polar vector in the heat-flux case. Therefore a polarization of the simple type j cannot be produced. Polarizations that do occur (e.g. the type C]?12E ex w'I), see Table 1) are found in the kinetic treatment to give their own specific contribution to the observed effects ~AII (== All-A), ~A~(== A~ -A) and Atr. For instance the type C]?12E is found to yield field-induced changes in A of the form

~A~ ( ~;2E (2~12E )2 ) -A- = -¢12E 1 + ~;2E + 21 + (262E F (4)

~AII ~;2E T = -¢12E21+e

12E (5)

Atr _ _ 01. (62E 2 2~12E ) A - 'f'12E 1 + ~;2E + 1 + (262EF '

(6)

where ~12E is an average precession angle W712E, with 712E the decay time of the polarization of type C]? 12E. The behaviour of the three curves (4-6) is displayed in Fig. 2.

Since W is proportional to the field strength Band 7 is inversely proportional to the pressure p, experimental data are usually presented as a function of B/p. The position of the field curves (4-6) along the B/p axis then directly determines the

161

WTI

Figure 2. Behaviour of the three thermal conductivity tensor elements [Eqs. (4)-(6)] as a function of precession angle ~ (or field strength to pressure ratio B / p) for the leading polarization type cp12E. From Ref. [10].

relaxation time 712E, or the cross section 6(12E), using ncr 6(12E) = 7;;1. The magnitude of the effects, symbolized by 1/J12E, is determined by a combination of cross sections and is used to determine the off-diagonal or 'production' cross sections, in this case the production of cp12E from a heat flux (see 3.2).

Field-effects on shear viscosity are measured largely analogously to the thermal conductivity case treated above. However, since the viscosity tensor", couples two second-rank tensors - momentum transport and velocity gradient - it is a fourth-rank tensor in the presence of a magnetic field. All except 5 elements vanish, viz. three even-in-field or longitudinal coefficients, and two odd-in-field or transverse coefficients. The physical meaning and experimental arrangements to measure these effects are described in Ref. [6]. A discussion of the various notations used to identify the different elements of", is given in Ref. [1].

A typical measurement of this 'viscomagnetic effect' employs a flow-Wheatstone bridge arrangement as sketched in Fig. 3.

Similar experiments were performed on polar molecules in electric fields. For a literature survey of both classes of experiments the reader is referred to Ref. [1].

Field-effects on diffusion have also been investigated. The most elegant technique is making use of transverse diffusion, analogous to transverse heat transport illustrated

162

b

Figure 3. Typical set-up for measuring field-induced changes in the viscosity (vis­comagnetic effect). The gas under study flows through a Wheatstone bridge-like arrangement, which is balanced by heating one of the capillaries C3 and C4 . Upon application of the field, the gas flowing through C1 will change its viscosity and the resulting pressure imbalance, Pb - Pa is measured directly by a sensitive differential membrane manometer M.

in Fig. 1. However, since the thermodynamic forces driving heat flux and diffusion flux have the same tensor character - the polar vectors \IT and \lx, with x the mole fraction - cross effects can also occur, viz. thermal diffusion and the Dufour effect. In a magnetic field this gives rise to transverse thermal diffusion and transverse Dufour effect in addition to the 'pure' transverse effects. This is illustrated in Fig. 4.

A special feature of this scheme is that it allows the confirmation of an Onsager reciprocity relation between the phenomena of thermal diffusion and the Dufour effect, viz. DTtr(B) = 'DTtr(B), This was shown to be the case in Ref. [7].

For a more detailed description and references to experiments on diffusion, ther­mal diffusion and the Dufour effect in external fields, see Ref. [1].

3.2 CROSS SECTIONS FROM SBE EXPERIMENTS

The analysis of SBE data in terms of effective cross sections is rather straightforward

163

6x T+AT

'-- t=-nilD~VT ,....-Ox .. r------OT------~

T +AT x+Ax T

OT '-- j tr =_nDtrVx -... ..

x+ Ax T x

'-- qlr = _ ill tlvx -"AKa ~

x

b. Thermal diffusion a. Thermal conductivity and d. Diffusion

c.Diffusion thermo (Dufourleffect

Figure 4. Illustration of the transverse effects which can arise in heat conducting and diffusing gas mixtures under the influence of a magnetic field B. The transverse flow is perpendicular to both the magnetic field and the applied gradient in temperature T or mole fraction x. Left and right panels show 'pure' transverse effects in thermal conductivity and diffusion, middle panels show 'cross' transverse effects.

for the case of the viscosity. Results of such experiments for non-polar diatomic molecules like N2 , CO and HD were found to be almost perfectly described on the basis of a single angular momentum polarization, viz. the type 'If or «)0211" listed in Table 1 (see e.g. Ref. [8]). This was found to be the case even for some non-linear molecules where the anisotropy of the g tensor causes some widening of the curves due to the spread in precession frequencies (see e.g. [9]). This eases the evaluation of these experiments in terms of cross sections. For the leading polarization «)0211",

the relaxation cross section 6(0211") is directly determined from the position of the measured curves along the B/p axis. More precisely, for the case of an isotropic g­factor (e.g. for linear molecules) 6(0211") is related to the B/p value for which the

164

precession angle ~ == WT equals unity by:

(7)

If g is anisotropic (as e.g. for symmetric-top molecules), the spread in precession frequencies gives rise to a slightly more complicated formula, see e.g. Refs. [10] and [11]. Next, the coupling cross section 6(0;~) is determined from the magnitude 1.j;

associated with the observed effects, using the relation

62( 20 ) of. 0211"

0/0211" = 6(20)6(027r)' (8)

where 6(20) is the field-free viscosity cross section found from the viscosity coefficient 1] by

kT 1 6(20) = -=- -.

Cr 1] (9)

Note that the magnitude 1.j; associated with the effects is determined by the square of the coupling cross section 6 (;g1l" ), which leaves the sign of this cross section undeter­mined. However, experiments on flow birefringence [12], which is linear in 6(gg1l")' do yield this sign.

For the thermal conductivity case, agreement between experiment and expres­sions (4-6) with ~ and 1.j; as free parameters is, for most non-polar gases, quite good (see e.g. Refs. [13] and [14]) and almost perfect for CO and HD. Deviations can have two main causes. The first is that polarizations other than the 'Kagan polarization', ~12E may playa role, e.g. the 'Waldmann' type W x j, which has a completely different signature for the set of 3 curves given in Eqs. (4)-(6). The second is that it has been tacitly assumed above that what has been referred to as the 'spherical approximation' is valid, i.e. the decay cross sections for the three irreducible tensor parts of ~12E have been assumed to be equal. This is likely not to be the case for molecules hav­ing a strongly anisotropic interaction (for a detailed discussion of this point see [1]). Either of these two complications separately can account for the observed deviations from Eqs. (4)-(6), but assuming a contribution from the W x j type polarization is usually favoured (see also Ref. [15]). In any case, only for the ~12E type polarizations can cross sections be extracted from the experiments with sufficient accuracy. To this end, the decay cross section 6(12E) is found from the experimental B/p values using Eq. (7) with the indices (027r) replaced by (12E). However, evaluation of the magnitude 1.j;12E in terms of coupling cross sections is more complicated since both the rotational and the translational component of the heat flux can produce the angular momentum polarization. Thus, if the two heat fluxes are treated independently, the magnitude 1.j;12E of the leading contribution to the field-effect on thermal conductivity loses the elegance of its viscosity counterpart expressed in Eq. (8): it contains three coupling cross sections and three decay cross sections (see e.g. Ref. [1], Eqs. (7.2)­(7.4)). Although one can still determine these cross sections separately by invoking

165

exact relations and cross sections from other sources, a simplified approach is usually favoured. This approach, introduced by Thijsse et al. [16], uses as basis functions the total energy flux c})lOE and the difference energy flux c})lOD (see Table 1). It is found that a quite satisfactory approximation is obtained if c})10D is neglected. This yields simple expressions both for the field-free thermal conductivity and the field-effect in terms Of'lj;12E (see e.g. Ref. [1] Eq. (7.2-5)). We will therefore follow this approach and list the cross sections 6(10E) and 6(~~~). An exception may be made for the hydrogen-isotopes. Here, the smallness of the inelastic cross sections relative to the elastic ones allows some simplifications (see Ref. [17] for a discussion). Hence, the separate production cross sections 6( ~~1°) and 6( ~~~l) can be reliably determined. Since the former can be obtained from 6(gg,,) using an exact relation (cf. Eq. (10)), only the latter will be listed in Table 2.

Finally, the magnetic field-effects on diffusion and thermal diffusion have been used to determine cross sections pertaining to polyatomic gas-noble gas collisions. As in the case of thermal conductivity, the dominant contribution is found to arise from the polarization of type w'IJ, i.e. c})12d in diffusion and c})12E in thermal diffusion (cf. Table 1). However, the contribution from the 'Waldmann'-type, W X j is found to be more substantial than in the case of thermal conductivity in pure gases (for a discussion see Ref. [18]); this makes cross section values derived from these experiments somewhat less precise. Even so, the position along the Bjp axis of these data permits determination of the decay cross sections 6(12EIA)AB and 6(12dIA)AB (for notation see section 2). These two cross sections should be equal if the heat flux is collisionally coupled to c})12E only, and a diffusion flux to c})12d only. Within the joint experimental error of about 20% this was found to be the case.

From the magnitude of the field-effect on diffusion the absolute value of the production cross section 6( :gd I ~ )AB can be determined. Data are available for a few systems only, the estimated error being ~ 15% (cf. [15]). From the Bjp position of the field-effect on thermal diffusion, as well as that on thermal conductivity in mixtures in the infinite dilution limit, one can determine the decay cross section 6(12EIA)AB. For the few systems studied, the agreement between the two sources is quite satisfactory (cf. Ref. [15]).

From the discussion above, the main limitations of the experimental data in terms of cross sections now become apparent. First, for molecules having small g-values it is impossible to reach high enough values of ~ with available field strengths without using such low pressures that Knudsen effects become cumbersome. This is the case for molecules like CF4 and SF 6. Second, for molecules having small nonsphericity the value of 'lj; may be too small to perform measurements with high accuracy. Such is the case for H2 and D2. Third, deviations of the experimental data from the single polarization behaviour discussed above create some uncertainty in determining the cross sections from - otherwise accurate - experiments.

For the reliability of the cross sections derived from these experiments, a few rules-of-thumb can now be deduced. First, the decay cross section derived from the position along the Bjp axis (6(12E) in the heat-flux case) has, at best, an uncertainty

166

Table 2. Availability of pure gas effective cross-sections derived from SBE and DPR experiments. *

6(027r) 6( 20 ) 02"

6(12E) 6COE ) 12E 6( 1001)

12E 6DPR

H2 [1], [17] [1],[17] [1],[17] [1],[17] [1], [17]

D2 [1], [17] [1],[17] [1],[17] [1],[17] [1], [17]

HD [1],[17] [1], [17] [1], [17] [1],[17] [1], [17]

N2 [1], [15] [1], [15] [1], [15] [1], [15] [15]

CO [1], [15] [1], [15] [1], [15] [1], [15] [15]

CO2 [1], [15J [1], [15J [1], [15J [1], [15J [15J

OCS [1], [15J [1], [15] [1], [15J [lJ, [15J [15J

CS2 [10J [lOJ [15J

N20 [1], [15J [1], [15]

O2 [lOJ [lOJ [lOJ

HCI [1], [15J [1], [15J

DCI [1], [15J [1], [15J

HCN [1], [15J [1], [15J CH4 [IJ, [10J [1], [lOJ [IJ, [10J [IJ, [lOJ

CD4 [1], [10J [1], [lOJ [IJ, [10J [IJ, [lOJ

CF4 [1], [lOJ [1], [10] [1], [lOJ [1], [lOJ

SF6 [1] [IJ [1] [1]

CH3F [1], [10] [1], [lOJ [1], [10] [1], [lOJ

CHF3 [1], [10] [1], [10] [1], [10] [1], [lOJ

CH3CN [10] [10]

NH3 [10] [10] [10] [10J

NF3 [lOJ [lOJ [lOJ [10]

PF3 [lOJ [lOJ

·Where possible, reference is made to the three most recent review pa-pers [1,17,15], which also give cross-section values derived from field-free transport coefficients, like 6(20) from the viscosity according to Eq. (9). Note that 6DPR is written as 6(02) in Ref.[15].

167

Table 3. Availability of effective cross sections for polyatomic molecule-noble gas

atom collisions. *

6(021rIA)AB 6( g~7r I ~ )AB 6(12EIA)AB 6(12dIA)AB 6U~dl~)AB 6DPR

H2-He [17] [17]

H2-Ne [17] [17]

H2-Ar [17] [17]

D2-He [17]

D2-Ne [17]

D2-Ar [17]

HD-He [17] [17] [17] [17]

HD-Ne [17] [17] [17] [17]

HD-Ar [17] [17] [17] [17]

N2-He [15] [15] [15] [15] [15] [15]

N2-Ne [15] [15] [15] [15] [15] [15]

N2-Ar [15] [15] [15] [15] [15] [15]

* Values are obtained from experiments on SBE for diffusion, as well as from ex­trapolation to infinite dilution in experiments on SBE for viscosity, SBE for thermal conductivity and DPR. As in Table 2, reference is made to review papers.

of 5-10%. The reason is that - even in the absence of the complications discussed above - the shape of the curves (4) stretches over almost three decades of ~-values (in practice logarithmic B/p scales are therefore used). Second, the coupling cross sections derived from the magnitude of the field effects 'lj; suffer not only from the error in 'lj; but also from that in the other cross sections used in the derivation. Therefore, the uncertainty is ~ 10% at best, and larger if the three complications discussed above also enter.

Tables 2 and 3 give a survey of cross sections determined by the various Senftleben-Beenakker effects. In many cases, temperature-dependent cross sections are available in addition to the 300K values. For the convenience of the reader, the tables give references to four surveys, viz. [1,17,15,10]' rather than to the original papers. In using these values, one should bear in mind that various exact and ap­proximate relations may serve to determine values of cross sections not listed here. As an example, the cross section governing the production of Kagan or W1J-type polarization from translational heat flux is related to that determining the production

168

+ o~~~::-~-¥i-c~~<~=~~1 I \.PLOW ANALYZER ~- MONOCHROMATOR PHOlOMULTIPLIER

I, TEMPERATURE FABRY· PEROT II CELL INTERFEROMETER II II II I'

qt> ~-=-U--l I ARGON ION LASER

POLARIZER

Figure 5. Schematic of a DPR experiment. The depolarized Rayleigh component of the light scattered by a polyatomic gas at a pressure of typically 10-100 bar is detected by a photomultiplier. A pressure-tuned, gas-filled Fabry-Perot interferometer provides the desired spectral resolution. From Ref. [20J.

of tensor polarization of type 11 in a viscous flow by

(10)

This makes it possible to find the former value directly from tabulated values of the latter. For a survey of these relations we refer to Ref. [1], section 12.4 and chapter 14.

4. Principle and practice of depolarized Rayleigh light scattering

Rayleigh scattering from a polyatomic gas at intermediate densities has a depolarized component due to orientational fluctuations and the anisotropic polarizability of the individual molecules (see e.g. refs. [19,1]). Intermolecular collisions determine the lifetime of the fluctuations and therefore the line broadening of the DPR line. In a DPR experiment (see Fig. 5) one measures the spectral shape of the scattered light having polarization perpendicular to that of the original laser light [20J.

The Fourier transform of this spectrum yields the auto-correlation function G(t) as a function of time. In the most simple case in which the broadening is purely Lorentzian, the correlation function decays exponentially. Hence, a plot of In G(t) vs. t will give a straight line, the slope of which is found to yield directly 6DPR:

6DPR=-- --1 (dC(t)) nCr dt t=o

(11)

169

In practice, deviations from this idealized single-relaxation-time behaviour are found. Therefore, the value of 6 D P R is derived from the slope of C (t) at t = O. The deviations are attributed to a dependence of the molecular reorientation rate upon the rotational state. For a gas with only a single nonzero rotational quantum number, such as pH2 at temperatures below 300K, no deviations are found within experimental error. For molecules having a few excited rotational levels, such as HD or normal D2 at 300K, the deviations from a single-relaxation-time behaviour are appreciable. However, if many rotational levels are excited, which occurs for molecules having a very small rotational level splitting such as OCS or CO2, th€ deviations become smaller again and an effective relaxation time dominates the behaviour.

In order to have a quantitative measure for the deviations from a single Lorentzian lineshape, the time integral of C(t) is used to yield a second quantity, 6j)PR' Obvi­ously, if C(t) is a pure exponential function of t (i.e. the spectral line shape is purely Lorentzian), one finds 6j)PR = 6 DPR. The relative difference between these two cross sections is found to range from 0% for pH2 to 11% for nD2, with only 6% for OCS and CO2 (for a discussion see [1], chapter 10).

References to literature giving values of 6DPR for the molecules investigated are listed in Table 2 for pure gases and Table 3 for mixtures; in the latter case the cross sections refer to reorientation of the polyatomic molecule in collisions with noble atoms.

5. Discussion

The various SBE-type experiments as well as DPR have yielded a wealth of infor­mation in terms of effective cross sections, an overview of which is seen in Tables 2 and 3. On the one hand, these values can serve as a test ground for potential models in a multiproperty fit. On the other hand, they provide direct insight into the type of collision processes contributing to the various cross sections. In order to illustrate this point, let us consider the hydrogen isotopes. Here the time scales for velocity­changing collisions, reorientation collisions and rotationally inelastic collisions can differ by more than an order of magnitude, due to the small effective nonsphericity combined with the large rotational energy level spacing. This is especially the case for the homonuclear molecules H2 and D2, for which the selection rules t:..j = 0, ± 2 apply.

Easy insight into the role of the various types of collisions is obtained by consid­ering the measured cross sections as a function of temperature. To this end we will discuss the two cross sections obtained from the shear viscosity SBE, viz. 6 (021T) and 6(gg7r)' We will compare their behaviour with that of the purely inelastic cross section 6(0001) for rotational relaxation (Fig. 6), and the essentially elastic cross section 6(20) obtained from the viscosity (dash-dotted curve in Fig. 8).

It is seen from Fig. 6 that the cross section for rotational relaxation is quite small. If, e.g. the value for pH2 at the lower temperatures is compared to the value of 6(20) in Fig. 8, it is found that it takes up to 1000 collisions to achieve rotational energy equilibration. The smallness of 6 (0001) reflects the weak anisotropy in the interaction

170

02r-----.------r-----,------r-----~----~

0.1

O~ ____ ~ ____ ~ ______ ~ ____ _L ____ ~~ ____ ~

o T --- 100 200 K 300

Figure 6. The cross section for rotational relaxation,6(000l) , as a function of temperature. From Ref. [21]. (Note that in Ref.[l]' Fig. 14.1, the vertical scale for HD is in error).

potential, which is in turn caused by the nearly spherical charge distribution. For HD the value of 6(0001) is larger than that for either homonuclear molecule at all temperatures. This reflects the 'loadedness' of the HD molecule, which introduces a considerable contribution from the PI-term in the interaction potential.

The temperature dependence observed in Fig. 6 can be readily explained from the effective rotational energy spacing: as long as it is small compared to kT, the cross section increases with decreasing temperature, in analogy with the behaviour of the viscosity cross section 6(20). At temperatures low enough for the lowest rotational energy jump to become difficult (i.e. j = 0 --+ 1 for HD, 0 --+ 2 for pH2 and oD2 , and 1 --+ 3 for oH2 and pD2 ), the cross section decreases with decreasing temperature.

The behaviour of the production cross section 6( o;~) displayed in Fig. 7 exhibits the same trends: both the temperature dependence and a comparison between the different isotopic species, or between the ortho and para modifications indicate that the dominant contribution to these cross sections stems from inelastic collisions. This

171

0.04 .------.-------,.-----.----,.----.,-----,

0.03

t 0.02

0.01

€) (~~) t 160 G (~5)

'\. /ll-ti -tr... - - ts- __

,Ii HD - - - ---b---_ ----ll---------ir OL-____ L-____ L-I ____ L-____ ~I ____ ~ ____ ~

o 100 200 K 300

Figure 7. The production cross section 6(6g7r) obtained from the SBE on viscosity plotted as a function of temperature. From Ref. [21]. (Note that in Ref. [1], Fig. 14.2, the vertical scale for HD is in error).

is in agreement with the results of a DWBA analysis, which indicates that inelastic collisions contribute to second order in the nonsphericity, while reorientation collisions contribute to at least third order (cf. Ref. [1]).

A completely different picture is seen for the temperature dependence of 6(0211') shown in Fig. 8. Here the temperature dependence follows the pattern of an elastic cross section over the entire temperature range (cf. 6(20) given for comparison). Moreover, the difference in value between HD and the homonuclears is now only a factor 2 to 3, in contrast to the situation in Fig. 7 where it was more than an order of magnitude. In addition the values of pH2 and nH2 (= 75 % oH2, 25 % pH2) are now seen to coincide despite their difference in effective rotational energy spacing. Finally, the values are much larger than those for 6 ( g~7l') shown in Fig. 7. All this indicates that pure reorientation collisions must give the dominant contribution to 6(0211').

Another interesting observation can be made from Fig. 8, viz. the fact that the values for pH2 and nH2 are equal within experimental error over the entire temper-

172

5r--r---.------,------,r------.------.-----~

~

4

3

2

e(02)t

I I I I I \ \ \

~,~ i--_ " HD \ -'F--

1 ~ ":::... ------- ~ 10 (20)/1"· ....... __ _ -- ---------~--_- _____ K

--.-. ~ -----. -'- "-----.

O~ ____ -L ______ ~ ____ ~L-____ -L ______ ~ ____ ~

o 100 200 K 300

Figure 8. The cross section 6(027r) obtained from the SBE viscosity. The field­free viscosity cross section for the hydrogen isotopes is given for comparison. From Ref. [21]. (Note that in Ref. [1] Fig. 14.3, the vertical scale for HD is in error).

ature range between 150 and 300 K. Since the distribution over the j = 0 and j = 2 levels of pH2 changes dramatically over that range whereas in oH2 the distribu­tion remains almost unchanged, the data suggest that j = 0 and j = 2 molecules are approximately equally effective in reorienting j = 2 molecules.

In conclusion it is seen that the various SBE and DPR experiments provide a variety of cross sections related to the non-spherical part of the intermolecular potential. Kinetic theory is found to describe the phenomena quite satisfactorily, and is seen to provide relations between various cross sections. Such relations provide the possibility of cross checks between different experiments and can ease a multiproperty analysis. As illustrated for the case of the H2-isotopes, the data also can provide insight into the type of processes dominating certain cross sections as well as insight into the role of rotational levels in collision processes.

173

Acknowledgements

The experiments described in this paper were performed over a period of some 25 years and involved many people from Leiden University's Molecular Physics group. The author wishes to acknowledge helpful discussions with Jan J. M. Beenakker, whose memory once again proved by far superior to the author's.

This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation of Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Weten­schappelijk Onderzoek (NWO).

References

[1] McCourt, F. R. W., Beenakker, J. J. M., Kohler, W. E. and Kuscer, I. (1990) Nonequilibrium phenomena in polyatomic gases, Clarendon Press, Oxford.

[2] Chen, F. M., Moraal, H. and Snider, R. F. (1972) On the evaluation of kinetic theory collision integrals: diamagnetic diatomic molecules, J. Chern. Phys. 57, 542-56l.

[3] Moraal, H. and Snider, R. F. (1971) Kinetic theory collision integrals for dia­magnetic molecules, Chern. Phys. Lett. 9, 401-405.

[4] Mazur, E., Beenakker, J. J. M. and Kuscer, I. (1983) Kinetic theory of field-effects in nonuniform molecular gases, Physica 121A, 430-456.

[5] Beenakker, J. J. M. (1974) Transport properties in gases in the presence of ex­ternal fields, in J. Ehlers, K Hepp and H. A. Weidenmiiller (eds.), Lecture Notes in Physics, Springer-Verlag, Berlin, pp. 413-468.

[6] Hulsman, H. and Knaap, H. F. P. (1970) Experimental arrangements for mea­suring the five independent shear viscosity coefficients in a poly atomic gas in a magnetic field, Physica 50, 565-572.

[7] Mazur, E., 't Hooft, G. W., Hermans, L. J. F. and Knaap, H. F. P. (1979) The transverse Dufour effect, Physica 98A, 87-96.

[8] Hulsman, H., Van Kuik, F. G., Walstra, K W., Knaap, H. F. P. and Beenakker, J. J. M. (1972) The viscosity of polyatomic gases in a magnetic field, Physica 57, 501-52l.

[9] Mazur, E., Viswat, E., Hermans, L. J. F. and Beenakker, J. J. M. (1983) Ex­periments on the viscosity of some symmetric top molecules in the presence of magnetic and electric fields, Physica 121A, 457-478.

[10] Thijsse, B. J., Denissen, W. A. P., Hermans, L. J. F., Knaap, H. F. P. and Beenakker, J. J. M. (1979) The thermal conductivity of polar gases in a magnetic field, Physica 97 A, 467-514.

[11] Van Ditzhuyzen, P. G., Thijsse, B. J., Van der Meij, L. K, Hermans, L. J. F. and Knaap, H. F. P. (1977) The viscomagnetic effect in polar gases, Physica 88A, 53-87.

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[12] Van Houten, H. and Beenakker, J. J. M. (1985) Flow birefringence in gases at room temperature: new absolute values, Physica 130A, 465~482.

[13] Hermans, 1. J. F., Koks, J. M., Hengeveld, A. F. and Knaap, H. F. P. (1970) The heat conductivity of polyatomic gases in magnetic fields, Physica 50, 410~432.

[14] Heemskerk, J. P. J., Hermans, L. J. F., Bulsing, G. F. and Knaap, H. F. P. (1972) The heat-conductivity tensor for polyatomic gases in a magnetic field at 85 K, Physica 57, 381~389.

[15] Van Houten, H., Hermans, 1. J. F. and Beenakker, J. J. M. (1985) A survey of experimental data related to the non-spherical interaction for simple classical linear molecules and their mixtures with noble gases, Physica 131A, 64~ 103.

[16] Thijsse, B. J., 't Hooft, G. W., Coombe, D. A., Knaap, H. F. P. and Beenakker, J. J. M. (1979) Some simplified expressions for the thermal conductivity in an external field, Physica 98A, 307-3-12.

[17] Hermans, P. W., Hermans, 1. J. F. and Beenakker, J. J. M. (1983) A survey of experimental data related to the non-spherical interaction for the hydrogen isotopes and their mixtures with noble gases, Physica 122A, 173~211.

[18] Mazur, E., Hijnen, H. J. M., Hermans, L. J. F. and Beenakker, J. J. M. (1984) Experiments on the influence of a magnetic field on diffusion in N2~noble gas mixtures, Physica 123A, 412~427.

[19] Keijser, R. A. J., Van den Hout, K. D., De Groot, M. and Knaap, H. F. P. (1974) The pressure broadening of the depolarized Rayleigh line in pure gases of linear molecules, Physica 75, 515~547.

[20] Hermans, P. W., Van Die, A., Knaap, H. F. P., Beenakker, J. J. M., (1985) Measurements on the influence of binary collisions on the depolarized Rayleigh and rotational Raman lines for H2-noble gas systems at low temperatures, Physica 132A, 233~252.

[21] Van Ditzhuyzen, P. G., Hermans, 1. J. F. and Knaap, H. F. P. (1977) The tem­perature dependence ofthe viscomagnetic effect in the hydrogen isotopes, Physica 88, 452~477.

ELASTIC AND INELASTIC CROSS-SECTIONS FROM LASER STUDIES OF SMALL MOLECULES

ANTHONY J. McCAFFERY School of Molecular Sciences University of Sussex Brighton BN19QJ, Sussex, UK.

ABSTRACT. Early laser studies of collisional processes were able to evaluate state to state integral cross sections in small molecules. Refinements in tech­niques have led to spectroscopic methods that yield data of precision approach­ing that attainable in molecular beam experiments. These advances have been achieved through utilising the polarization of laser radiation and the velocity selection that may be achieved via the Doppler effect. These enable the exper­imenter to specify directions and magnitudes of linear and (molecular) angular momentum vectors whilst retaining the relative simplicity of a spectroscopic experiment. This contribution reviews the range of methods currently avail­able using illustrations that emphasise the relation of such studies to transport properties.

1. Introduction

The use of spectroscopic methods to study collisional processes predates the laser era. Atomic lamps were used [1 J to populate a small number of excited quantum states in small molecules. The redistribution of this population resulting from colli­sions is seen in the spectrally resolved emission when collision partners are introduced. Lasers made a significant impact on this field. Using fixed wavelength c.w.ion lasers Bergmann and Demtroder [2J and also Ottinger [3J demonstrated that it was possible to obtain state-to-state cross-sections for vibrationally and rotationally inelastic colli­sions of alkali diatomics with a wide range of collision partners using simple collision cells.

A great advantage of collision cell spectroscopic experiments is the ease with which accurate data sets of rotational and vibrational transfer (RT and VT respec­tively) cross sections may be measured. The introduction of tunable c.w. and pulsed laser sources has led to extensive tabulations of inelastic cross-sections for a wide range of small molecules with many collision partners.

These experiments have the virtue of quantum state selection in the input channel and similarly precise detection in the output channel. This contrasted early molec­ular beam techniques which in other than a small number of noteworthy instances, were characterised by relatively poor internal state resolution. Thus the spectroscopic

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W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 175-188. ©1992Kluwer Academic Publishers.

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results represented a novel view of the collision process and in the first section be­low some of the insights revealed by the determination of the quantities (Jj~j' are reviewed.

As collision experiments however they are relatively unsophisticated since essen­tially they relate to scalar quantities in which, at best, the magnitude of the rotational angular momentum vector is specified with little knowledge of its direction. The use of polarized light in the excitation process and determination of emission polarization improves this experiment since specific m states may be prepared. More commonly, excitation and detection can do no more than specify a range of m states charac­terised by the multipolar moments of the states created or detected. The projection quantum numbers in these optical experiments are referred to lab fixed axes though in conjunction with molecular beam experiments, may be tied to the collision frame. The second section of this contribution describes features revealed through measurements of polarized RT cross-sections, i.e. the (J::~~,.

In scattering experiments a key feature is the determination of angular depen­dence of the cross-section through the change of direction of the relative velocity vector. In recent years it has been demonstrated that information on molecular (and relative) velocity magnitudes and directions can be extracted from the Doppler profile in spectroscopic transitions. Using these methods it is possible to specify the transla­tional order of the collision environment and to investigate collisional behaviour as a function of magnitude and direction of molecular and of relative velocity. With this development spectroscopic measurements have moved close to the ultimate resolution of the collision process formerly achieved only in the most sophisticated of molecular beam experiments. The cross-sections obtained in these velocity-selected experiments,

the (J;:2: --->K' are closely related to the generalised cross-sections of transport theory as demonstrated very recently by Liu and Dickinson [4]. In the third section below, recent experiments yielding cross-sections of this form are described.

It will be apparent that the classification of experiments outlined above follows increasing sophistication through enhanced specification of magnitude and direction of the critical vectors of linear and angular momentum that govern the collision process. This is embodied in the vector correlation concepts introduced to collision dynamics by Herschbach and co-workers [5]. Thus the sequence of experimental methods outlined above and discussed in more detail below follows this pattern of increasing number of vectors correlated in the measurement.

2. Rotationally resolved cross-sections, (Jj--->j'

A wide range of small molecules has been studied at rotational level resolution using both molecular beam and cell spectroscopic methods. The emphasis has largely been on diatomics and by far the greatest number of studies has utilised spectroscopic methods. This has highlighted a discontinuity in theoretical treatment. Rotational transfer is rigorously treated in the Arthurs and Dalgarno [6] total-J scheme through the close-coupled equations. As the number of coupled channels increases, the close

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coupled method becomes computationally intractable and methods have been intro­duced which effect a decoupling through some approximation to the hamiltonian based on the time-scale of the interaction.

The 'sudden' approximations are well known and one aspect of significance to the cell spectroscopic experimental studies was the demonstration by Goldflam et al. [7] that in the sudden regime the scattering amplitude may be factorised to obtain a simple scaling relation for RT rates.

k)2 o (Jj---+o· (1)

This deceptively simple equation embodies several important aspects of the physics of the collisional interaction which will be discussed in appropriate contexts.

A feature of empirical studies has been a search for scaling or fitting relations in the anticipation that, given the quantity and quality of data available, parameters reflecting the intermolecular potential would emerge. The earliest of these,the expo­nential gap law [8] emphasises the statistical nature of the process but for accuracy of fit, was superceded by a power gap law [9] which was able,through suitable choice of parameters to fit many RT data sets. This has yielded, however, little more than the ability to reduce data sets to a functional form and table of parameters. No parame­ters of the potential are derived. The AON model of Brechignac [10] has the virtue of a basis in the dynamics of the collisional interaction and provides a fit to much of the published data from which parameters pertaining to the form of the potential may be derived.

Recently, we [11,12] have found that the transfer of angular momentum provides the dominant mechanism in collisional RT. The distinction between energy and an­gular momentum control is obscured in diatomics but becomes separable in studies on asymmetric rotors. State-to-state RT and VRT studies on NH2 [11] established that RT probabilities fall exponentially with the magnitude of transferred angular momentum. It has been suggested [12J that this reflects the (approximately) expo­nential drop in the anisotropic part of the repulsive portion of the intermolecular potential [12J. If this is verified it could provide the basis of a simple inversion routine to obtain data on the intermolecular potential directly from RT cross-sections.

3. Polarised RT cross-sections, (Jjrn---+j'rn' and (J:!-.~,

Use of polarized light in a laser spectroscopic collisional experiments in theory permits the experimenter to specify the magnetic sub-states of the rotational levels excited. In practice except for the special case of j = 1, the best one can hope to quan­tify are moments of the distribution among degenerate m states expressed as the multipolar moments, K (component Q). It is straightforward to show that polarized (one-photon)) excitation of molecules from an unpolarized ground state leads to an excited array that may be completely prescribed through moments K = 0, 1 and 2 . The physical significance of m-state selection is most readily seen in the case of the

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~ -state diatomic molecule. The projection of j on a laboratory frame axis (e.g. the laser propagation direction) is specified, and thus the molecules are photoselected in the lab frame prior to collision. It is important to distinguish clearly between lab and collision frame projections of j since generally they will be different. Theoretical studies are usually in terms of the latter whilst cell spectroscopy experiments mostly refer to the former. Although the quantum number m will be used for both, the frame of reference will be distinguished as appropriate.

The quantum treatment of polarized RT cross-sections can be usefully cast in the translational-internal (TI) coupling scheme [13] in which initial and final j and I vectors are coupled via the transferred angular momentum k [14]. A further useful development was the expansion of the T-matrix in terms of spherical tensor operators R~(r), S;(R), which operate in the individual positional (R) and orientational (r) spaces respectively [15].

where

y:jm,i'm' = '""'(_I)i'-m-I'-m'+g J ( ., lm,l'm' L-t m'

kg

k q

j ) ( I' -m m'

T1z,jlll = (j] ~ (j' II Rk II j) [I] ~ (l' II Sk Ill).

(2)

(3)

The experimental observables (O"f->j/) are related to the T-matrices via expres­sions of the form [16]

K 7r '""' k{j j' O"j->jl = k2 L..,,( -1) ., J'

J k J ; } [k]-l L I (j'l' II Tk II jl) 12 .

1,1'

(4)

The T-matrices are indexed in k, the transferred angular momentum and it is straight­forward to demonstrate that k is limited to the values

1 '- "I<k<'+ ". J J __ J J

This formulation also indicates how m is changed by the collision and from the 3j symbols above it is apparent that flm = q and that -k :::; q :::; k. This illustrates the value of the TI formulation in discussing polarized cross-sections. Note that the projections referred to in the theoretical description are collision-frame m values. A sudden factorisation of the T-matrix may be employed to obtain an expression for the polarized (or multipolar) cross-sections that is analogous to the form referred to above for scaling population (K = 0) results [17]

O"f->jl = (_I)K(j,j'] L[k]( _1)k 1., { " Kk JJ'.} (Jo' Jo" K J

k)2 o O"Lo· (5)

179

Experimental measurements of the af-+jl have been made both in cell and in beam experiments. This work has recently been reviewed [18). A number of impor­tant features have emerged from these studies and will be discussed briefly. Perhaps the most striking observation has been that cross-sections for m-changing processes are much smaller than those for j-changing. The quantitative formulation of the for­mer process can be expressed in terms of the creation, destruction and transfer of the multipolar moments K = 0, 1 and 2. Cross-sections for these collision-induced events have been measured for elastic, rotationally inelastic and vibrationally inelastic collisions for a wide range of atom-diatom collision partners.

Elastic af-+j cross-sections for the destruction of orientation (K = 1)) are gen­erally in the range 0.1 - 1A2 [18). This contrasts with elastic (population, K = 0) collision cross-sections of 102 - 103 A 2 . Inelastic transfer is also characterised by per­sistence of the original molecular orientation and cross-sections for transfer of the state multipoles af-+j' show that K = 1 and K = 2 are maintained throughout even strongly inelastic processes.

Attempts have been made experimentally and theoretically to quantify this ap­parent propensity to preserve m in collisional processes. Lab frame m conservation can be ruled out [14] (although it appears that ~m = 0 in collision and lab. frames for elastic collisions). Experimentally [19,20] it has been established that 1 ~m 1« ~j in the lab frame and that this applies in the direction of linear momentum trans­fer [20,21] . In the classically impulsive limit it is straightforward to demonstrate that the change in rotational angular momentum must be perpendicular to the change in linear momentum. This implies conservation of the projection of j along the direction of change of linear momentum. This was quantified by Kouri and coworkers [22] in terms of conservation of m along the kinematic apse, ak, defined by ak =Pi-Pj.

That a conservation rule along the kinematic apse may be observed as a propen­sity along some other axis (the lab frame for example) is illustrated in Figure 1. Ex­perimental studies of fluorescence polarization in cells and beam experiments support the interpretation of the RT process as one in which the impulsive limit applies over a wide range of interacting systems and over a range of collision energies [23). Khare et al. [22] noted that the set of differential cross-sections ajm-+j'm' for space-fixed axes could be successfully reproduced by rotating the dominant ajm-+j'm (~m = 0) cross­section from the kinematic apse. A similar approach used by Reid and McCaffery [23), and described in the next section, gives a consistent description of the velocity de­pendence of the polarization ratio in Doppler-resolved experiments. The assumption of m conservation in inelastic processes, rigorous in the classically impulsive limit, may prove to be of value in other contexts. Davis [24) demonstrated application to the case of NH3 - He collisions.

The multipolar cross-sections af-+jl' exhibit an exponential-like falloff with ~j that is similar to the unpolarized cross-sections previously discussed and the scaling relation outlined above may be employed to model them. However, it is found that the parameters needed to fit data sets for K = 0 and K = 1 in the same atom­diatom system are generally different and gives little confidence that the procedure of

180

liP p.

I

--.--~

Figure 1. Demonstration of the conservation of projection of angular momentum vector j along the direction of change of linear momentum (kinematic apse), a con­sequence of an impulsive interaction. Also shown is the projection along an arbitrary axis, z, (e.g. the lab axis) and it is clear that projections along this must change. Conversely a propensity to conserve m along z implies a strong restriction along some axis, e.g. the kinematic apse.

fitting to find the most accurate functional form for RT rates yields insight into the fundamental physics of the process.

4. Velocity and angular momentum polarized RT cross-sections, a-j:2:'K'

In the previous section, spherical collisional symmetry was assumed and in these circumstances it can be shown that the state multipolar moments of the angular mo­mentum evolve independently and do not interconvert. Hence initial and final state K values are unaltered. However, spherical translational symmetry is rare for molec­ules probed either in molecular beams or in cell experiments using laser excitation as Monchick has pointed out [25]. In molecular beam experiments and in cell laser fluorescence measurements there is generally velocity selection and velocity polariza­tion;in the former case it is a feature of experimental design whilst in the latter, its origin lies in the spectral narrowness of the laser and is thus involuntary. As a result

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the strict K conservation assumed in the previous section will not hold when velocity polarized multipole cross-sections ( MCS) are measured.

In this final section, experiments that achieve velocity polarization and thus

measure the O"]~f, K' are discussed. It is these quantities that are of interest III

transport studies since they are related to the generalised cross-sections (GCS) of transport theory through [4]

n o

K') OK'(K')("') _Q O"nK J J . (6)

Liu and Dickinson [4] give the explicit form of the relation of the GCS, O"~~' (K') (j' j), to the T-matrix. The determination of velocity polarized MCS on a systematic basis has barely developed though one hopes that as a result of this NATO meeting there will be more emphasis on such measurements in the future. The main focus in this section will be on the background to velocity selection by spectroscopic methods together with a brief review of experiments that have measured state-to-state effects of velocity polarization due to collisions.

It will be apparent from the foregoing sections that the laser spectroscopic meth­ods described above have great advantage over traditional beam experiments in the relative ease with which good quality collision cross-section data may be acquired and the speed at which the experimenter may switch from system to system. The most significant disadvantage however has been inability to specify molecular or relative velocities in spectroscopic experiments.

Velocity information is of course present in spectroscopic observables, indeed in the form of the Doppler effect it presented a formidable obstacle to high resolution spectroscopy for many years. The functional form of the Doppler profile and its relation to molecular speed distributions is well known and simple calculations show immediately that the range of molecular speeds available through tuning across the Doppler profile (or detuning as it is generally known) can be very wide. A major advance in the use of sub-Doppler techniques for dynamical experiments came with the development by Kinsey [26] of Fourier transform Doppler spectroscopy. This method was applied very effectively to analyse molecular velocity distributions in a molecular beam scattering experiment [27].

In collision cell experiments, a particularly valuable development introduced by Smith, Scott and Pritchard [28] involved selection of a known distribution of molecular velocities from the Doppler profile using a narrow line laser. Collisional transfer data on these velocity and j-selected molecules was transformed to yield the dependence of the state-to-state cross-section on relative velocity using the method of Kinsey. The transformation from molecular to relative coordinates is most precise for high atom:molecule mass ratio though reliable results have been obtained with ratios as low as 1.5:1. This method (known as VSDS) was applied to collisions of Na*, Na~ and Li~ with rare gases. For the favourable Li2 - Xe system velocity resolution was better than 250 ms-1 over a selected velocity range of around 6 : 1 which compares favourably with molecular beam methods.

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In fact, sub Doppler excitation selects molecules by direction as well as by speed and this specification of the molecular velocity vector may be transformed into relative velocity in the manner described by Kinsey [26). The equation that links direction and magnitude of relative velocity (V r ) to laser detuning (via the z-component of molecular velocity (vrnz ) is

(7)

where

and 0: represents the angle between Vr and the laser propagation direction. The VSDS method outlined above may be enhanced by use of polarized light in

selection and in detection. Thus instead of obtaining the cross-sections 0j ..... j' (vr ) of

VSDS we have the potential to evaluate O"J:::' K' also as a function of relative velocity. McCaffery and coworkers [29,30,31) have reported the experimental determination of transfer of state multipoles as a function of laser detuning. Lineshapes for transfer of population and of orientation differ significantly and transform to give differing Vr

dependencies. The effect is most strikingly seen when the polarization ratio is plotted as a function of laser detuning.A marked dip appears in the ratio at line centre. This polarization ratio velocity dependence (PRVD) is shown in Figure 2. The depth of the dip is found to be a function of collision partner, of initial j state and of /lj.

One advantage of optically selected velocity distributions is that it is feasible to identify the moments of the array of velocity vectors and as Monchick has pointed out [25], line centre excitation produces a quadrupolar array (n = 2) of velocities. As a result the collision environment for zero detuning has cylindrical symmetry. As discussed earlier, in velocity symmetries less than spherical the state multipoles of the angular momentum distributions do not evolve independently and they may interconvert. This situation was envisaged by Monchick who wrote explicit expressions for the circular polarization ratio in the cylindrical, line-centre environment in terms

of the O"J:::' K'. In this n takes values 0 and 2 and thus for an experiment in which the state multi poles K = 0, 1 are selected there will be K = 0 ...... K = 2 interconversion induced by the n = 2 velocity polarization.

It would be natural to attribute the PRVD dip of Figure 2 to multipole intercon­version but closer examination suggests that this may not be the case. Close-coupled

calculations of the O"J:::'K' for n = 0 and for n = 2 for the case of N2 - rare gas collisions [32) indicate that the K = 0 ...... K' = 2 terms are around ten percent of the K = K' spherical symmetry cross-sections. However, their energy dependencies ap­pear very similar and the PRVD does not fit this interpretation. Recently [23) we have been able to fit the observed dip using a model that assumes m conservation in the kinematic apse. A multipolar distribution rotated onto the lab frame via an assumed most probable scattering angle and differential scattering cross-section predicts a dip

183

orientation

ratio

2 4 6 8 GHz detuning

Figure 2. Velocity dependence of population (K = 0) and polarization (K = 1) transfer as a function of laser detuning. The system is Li2 - Xe and results are shown for ji = 6, jf = 12. The difference in lineshapes for K = 0 and K = 2 is difficult to see but is displayed dramatically in the polarization ratio curve as a function of detuning where the PRVD dip can readily be seen.

184

at line centre of the kind observed. An important feature in the model is the colli­sion 'stereochemistry' since at line centre the experiment probes a 'head-on' collision whilst in the wings the geometry of the interaction is the T-shaped configuration.

Thus the PRVD dip appears more likely to yield angular variables appropriate to the scattering process than cross-sections related to the GCS. However, polarized

velocity selection experiments of this kind are fully capable of measuring the O";~f,K' for K =f. K' though it is clear that they (the experiments) need to be designed carefully in order that the effect desired may be measured unambiguously.

Finally a number of recent experiments that measure the coupling of order from the translational motion to internal motion are discussed as illustrations of what is currently feasible of relevance to transport in gases. Some of these have been analysed within the framework of the MCS concept as outlined above and others have not. Alexander [33] has developed theoretical expressions appropriate to several experiments in which the velocity-rotational angular momentum coupling may occur.

That transport of non-spherical molecules in gases leads to polarization of the molecular angular momentum has been known for many years through the work of the Leiden group [34]. Molecular alignment was detected in a supersonic beam expan­sion of alkali dimers by Sinha, Caldwell and Zare [35], an effect confirmed by other authors but for some years felt to be anomalous. Very recently Pullman, Friedrich and Herschbach [36] have observed substantial alignment of Iz seeded in expansions of light gases and suggest the phenomenon is more general than previously supposed. Two mechanisms are proposed , the first being bulk alignment due to dependence of cross-section on angle between the relative velocity vector and j. The second is anisotropic rotational cooling which also is a function of the v.j vector relations since this affects the ability of the collision partner to generate torque. The role of velocity slip is important in activating both mechanisms.

Alignment of molecular ions in drift tubes has been probed by polarized laser fluorescence [37]. The Nt - He collision system was investigated and alignment (K = 2) moment determined for the j = 10 rotational state. Results suggested a ratio of 2 : 3 for finding the ions with j vectors parallel and perpendicular to the drift tube field. A theoretical treatment of this phenomenon has been given by Meyer and

Leone [38] in terms of the O")~f, K'. This particular system has a number of interesting but complicating features including the possible effects of the long range potential and the unknown velocity distribution.

The experiments described above have utilised fluorescence intensities (polar­ized and unpolarized) in one-photon experiments to obtain inelastic cross-sections following ficlective excitation. If a second narrow line laser is available the informa­tion content of laser spectroscopic experiments may be greatly enhanced [39,40]. The probe laser,tuned to excite molecules in states populated by collisions, may now de­termine velocity group shift and spread as well as population and polarization. In this way it is possible to relate double resonance lineshapes under certain assumptions, to angular distributions. Recent applications of the double resonance method have been to atomic [41] and to molecular systems [40]. Liu and Dickinson [42] have developed

185

explicit expressions for the lineshape in such double resonance experiments in terms of the GCS. This method is under development and is of considerable promise since rotational and translational relaxation are probed simultaneously. A recent paper [43] describes the use of polarization techniques (selection and detection) in the double resonance configuration which lock in relations between v and j at the outset. The loss of this fixed relationship is probed after the collision thus the experiment represents a four-vector correlation [5] method capable of yielding the collision frame angular dependence of the state-to-state, velocity resolved cross-sections.

5. Summary

In summary we see that laser fluorescence studies of collisional processes yield state to state cross-sections of varying degrees of sophistication. Those most straightforward to measure namely the O"j-->j' are characterised by little vector correlation and prove difficult to relate to parameters of the intermolecular potential. However, there is evidence that relative velocity selected cross-sections, the O"j-->j' (vr ) may be used to obtain a representation of the repulsive anisotropy via an angular momentum model of the RT process [12].

Polarised laser fluorescence becomes a two-vector correlation experiment, namely (jd!), with the projections defined in the laboratory frame. The resultant O"f/-->j' ex­hibit a propensity to conservation that mirrors, in the impulsive limit, a prohibition on m change along the kinematic apse. These studies also provide physical justifica­tion of the basic lOS scaling relation discussed above with its built-in assumption of a planar scattering process.

Velocity selection through molecular beams, gaseous transport or through sub­Doppler excitation gives the opportunity for adding (v.j) correlation to the experi-

ment. The cross-sections evaluated are now the MCS, O"):lf,K' which may be obtained as a function of relative velocity, v r . This becomes a three-vector correlation method and can yield the K +--t K' cross-sections induced by transport processes (those for which n i- 0) which are relateable to the GCS [4]. Indications are that these quanti­ties are readily measured though will require careful experimental design. The PRVD experiment which is in this category appears capable of yielding the differential scat­tering cross-section with certain assumptions.

Double resonance experiments in which a second narrow line laser probes quan­tum state distribution, vector length (and projection) and final velocity distribution represent the latest in the evolutionary chain. The four-vector correlation experiment has been published [43] and one can expect this technique to be widely exploited.

The cross-sections so obtained can be represented by O"]:lf,K' (Vri -+ Vr!) and are as high in specification of the dynamic variables as the best of molecular beam measure­ments. In addition, the high degree of vector correlation implies that angle-dependent information, or dynamical stereochemistry, is a feature of this technique.

186

Acknowledgements

The author wishes to thank the following valued collaborators for their contributions to the ideas and the results described above; M.D. Rowe, S.R. Jeyes, R. Clark, B.J. Whitaker, M.J. Proctor, A.J. Bain, A. Ticktin, C.P. Fell, KL. Reid, Z.T. AlWahabi, M.J. Wynn and T.L.D. Collins. Generous funding from SERC is gratefully acknowl­edged.

References

[1] Steinfeld, J. I. and Klernperer, W. (1965) Energy transfer processes in monochro­matically excited iodine molecules, J. Chern. Phys. 42, 3475-3497.

[2] Bergmann, K and Derntroder, W. (1972) Inelastic collision cross-sections for excited molecules, J. Phys. B 5, 1386-1395.

[3] Ennen, G. and Ottinger, C. (1974) Rotation-vibration-translation energy transfer in laser excited Lb(B'TI2 ), Chern. Phys. 3, 404-430.

[4] Liu, W.-K and Dickinson, A. S. (1990) Multipole cross-sections in polarised flu­orescence for atom-diatom systems, Molec. Phys. 71,1117-1129.

[5] Case, D. A., McClelland, G. M. and Herschbach, D. R. (1978) Angular mo­mentum polarisation in molecular collisions: classical and quantum theory for measurements using resonance fluorescence, Molec. Phys. 35,541-573.

[6] Arthurs, A. M. and Dalgarno, A. (1960) The theory of scattering by a rigid rotator, Proc. Roy. Soc. A 256, 540-551.

[7] Goldflarn, R., Kouri, D. J. and Green, S. (1977) On the factorisation and fitting of molecular scattering information, J. Chern. Phys. 67, 5661-5675.

[8] Polanyi, J. C. and Woodall, K B. (1972) Mechanism of rotational relaxation, J. Chern. Phys. 56, 1563-1572.

[9] Brunner, T. A. and Pritchard, D. E. (1982) Fitting laws for rotationally inelastic collisions, Adv. Chern. Phys. 50, 589-641.

[10] Brechignac, P. and Whitaker, B. J. (1984) Energy dependence of rotationally inelastic cross-sections and fittings laws, J. Chern. Phys. 88, 425-426.

[11] AlWahabi, Z. T., Harkin, C. G., McCaffery, A. J. and Whitaker, B. J. (1989) Stereochemical influences in atom-triatomic collisions, J. Chern. Soc. Far. Trans., 85, 1003-1011.

[12] McCaffery, A. J. and AlWahabi, Z. T. (1991) Mechanism of rotational transfer, Phys. Rev. A 43, 611-614.

[13] Coombe, D. A., Sanctuary, B. C. and Snider, R. F. (1975) Definitions and prop­erties of generalised cross-sections, J. Chern. Phys., 63, 3015-3030.

[14] Alexander, M. H. and Davis, S. L. (1983) M-dependence in rotationally inelastic collisions in cell experiments. Implications of an irreducible tensor expansion, J. Chern. Phys. 78, 6754-6762.

187

[15J Orlikowski, T. and Alexander, M. H. (1984) Quantum studies of inelastic colli­sions of NO with Ar, J. Chern. Phys. 79,6006-6016.

[16J Alexander, M. H. (1979) Tensorial factorisation and rotationally inelastic colli­sions, J. Chern. Phys. 71, 5212-5220.

[17J Proctor, M. J. and McCaffery, A. J. (1984) Scaling of state multipoles in rota­tionally inelastic transfer, J. Chern. Phys. 80, 6038-6046.

[18J For a recent review of cell and beam experiments see McCaffery, A. J., Proctor, M. J. and Whitaker, B. J. (1986) Rotational energy transfer, polarisation and scaling, Ann. Rev. Phys. Chern. 37, 223-244.

[19J Rowe, M. D. and McCaffery, A. J. (1979) Transfer of state multipoles in Li2 following rotationally inelastic with He experiment and theory, Chern. Phys. 43, 35-54.

[20J Mattheus, A., Fischer, A., Ziegler, G., Gottwald, E. and Bergmann, K. (1976) Experimental proof of a t:.m « j propensity rule in rotationally inelastic differ­ential scattering, Phys. Rev. Lett. 56, 712-715.

[21J Treffers, M. A. and Korving, J. (1983) Experimental determination ofmj distri­bution in inelastic scattering of Na2 by He, Chern. Phys. Lett. 97, 342-345.

[22J Khare, V. Kouri, D. J. and Hoffman, D. K. (1981) On Jz preserving propensities in molecular collisions, J. Chern. Phys., 74, 2275-2286, 2656-2657; (1982) 76, 4493-450l.

[23J Reid, K. and McCaffery, A. J. (1991) Spectroscopic determination of the state­to-state differential scattering cross section, J. Chern. Phys., to be published.

[24J Davis, S. L. (1985) M -preserving propensities for rotationally inelastic NH3-He collisions in the kinematic apse frame, Chern. Phys. 95, 411-416.

[25J Monchick, 1. (1981) Generalised reorientation cross-sections for cylindrically symmetric velocity distributions, J. Chern. Phys. 75, 3377-3383.

[26] Kinsey, J. L. (1977) Fourier transform Doppler spectroscopy, a new means of obtaining velocity-angle distributions in scattering experiments, J. Chern. Phys. 66, 2560-2565.

[27J Phillips, W. D., Serri, J. A., Ely, D. J., Pritchard, D. E., Way, K. R. and Kinsey, J. L. (1978) Angular distributions by Doppler spectroscopy, Phys. Rev. Lett. 41, 937-940.

[28J Smith, N., Scott, T. P. and Pritchard, D. E. (1984) Velocity dependence of rota­tionally inelastic collisions, J. Chern. Phys. 81, 1229-1246.

[29J McCaffery, A. J., Proctor, M. J. Seddon, E. A. and Ticktin, A., (1986) Po­larisation ratio velocity dependence, a novel polarisation-sensitive technique for atom-diatom rotational energy transfer cross-sections, Chern. Phys. Lett. 132, 181-184.

[30) Fell, C. P., McCaffery, A. J., Reid, K. L., Ticktin, A. and Whitaker, B. J. (1988) Velocity dependence of rotationally inelastic cross-sections, Laser Chern. 9, 219-240.

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[31] Fell, C. P., McCaffery, A. J., Reid, K. L. and Ticktin, A. (1991) Variation of polarisation ratio for rotationally inelastic collisions with laser-selected velocity, J. Chern. Phys. 95,4948-4957.

[32] McCaffery, A. J., Proctor, M. J., Seddon, E. A. and Ticktin, A. (1986) Velocity dependence of polarised rotational transferi close-coupled calculations on model systems, Chern. Phys. Lett. 132, 185-189.

[33] Alexander, M. H., (1979) Tensorial factorisation and rotationally inelastic cross­sections, J. Chern. Phys. 71, 5212-5220.

[34] Beenakker, J. J. M. (1974) Transport properties in gases in the presence of ex­ternal fields, Lecture Notes in Physics, Springer-Verlag, 413-418.

[35] Sinha, M. P., Caldwell C. D. and Zare, R. N. (1974) Alignment of molecules in gaseous transport.·alkali dimers in supersonic nozzle beams, J. Chern. Phys. 61, 491-503.

[36] Pullman, D. P., Friedrich, B. and Herschbach, D. R. (1990) Facile alignment of molecular rotation in supersonic beams, J. Chern. Phys. 93, 3224-3236.

[37] Dressler, R. A., Meyer, H. and Leone, S. R. (1987) Laser probing of rotational alignment of Nt drifted in helium, J. Chern. Phys. 87, 6029-6039.

[38] Meyer, H. and Leone, S. R. (1988) Steady-state model for the collision induced rotational alignment of molecular ions in electric drift fields, Molec. Phys. 63, 705-717.

[39] Gottscho, R. A., Field, R. W., Bacis, R. and Silvers, S. J. (1980) Simultaneous measurement of rotational and translational relaxation by sub-Doppler optical­optical double resonance spectrospcopy, J. Chern. Phys. 73, 599-611.

[40J McCaffery, A. J., Reid, K. L. and Whitaker, B. J., (1988) Velocity selective double resonancei a novel technique for determining the differential scattering cross-section, Phys. Rev. Lett. 61, 2085-2087.

[41] o 'Callaghan, M. J. and Gallagher, A. (1989) Sodium Doppler-free collisional lineshpaes, Phys. Rev.A 39, 6190-6205.

[42J Liu, W.-K. and Dickinson, A. S. (1991) Theory of the inelastic lineshape for two-step sub-Doppler excitation, J. Phys. B 24, 1259-1271.

[43] Collins, T. D. L., McCaffery A. J. and Wynn, M. J. (1991) Two-colour sub­Doppler circular dichroismi a four vector correlation molecular dynamics exper­iment, Phys. Rev. Lett. 66, 137-140.

ATOMIC ION/MOLECULAR SYSTEMS

LARRY A. VIEHLAND Parks College of Saint Louis University, Cahokia, Illinois 62206, USA

ABSTRACT. The transport coefficients for atomic ions moving in trace amounts through dilute molecular gases under the influence of an electrostatic field can be used to probe sensitively the potential energy surfaces governing ion/molecule collisions. To illustrate this, classical trajectory calculations have been used to compute the transport cross sections for lithium ions colliding with nitrogen molecules in their ground vibrational state, using an ab initio potential energy surface. A recent kinetic theory was then used with these cross sections to compute the gaseous ion mobility and diffusion coefficients. Comparison with experiment indicates some deficiencies in the potential energy surface.

1. Introduction

A thorough understanding of ion/neutral interactions is important in such varied fields as atomic and molecular physics, aeronomy and atmospheric chemistry, gaseous electronics, and laser physics. For this reason the mobility and diffusion coefficients of ions moving in trace amounts through dilute gases under the influence of an electro­static field have been of interest since the turn of the century [1]. The introduction to the field of gaseous ion transport given in this section is an update of the introduction presented in 1989 by Viehland and Kumar [2].

Following the development of drift-tube mass spectrometers about 1960, excellent transport data became available [3-5] for a large number of ion/neutral systems over wide ranges of E / N, the ratio of the electric field strength to the gas number density. Such swarm data are important, because variation of E / N has Toughly the same effect as variation of the gas temperature (both serve to change the mean energy of the ion swarm), and because it is necessary to cover wide ranges in order to probe fully into ion/neutral interactions on the molecular level.

Until the early 1970's it was not possible to exploit fully the available transport data, because theory was capable of dealing only with data at very low E/N [1]. Our understanding advanced considerably with the development of two-temperature and three-temperature methods for solving the Boltzmann kinetic equation for atomic ions in atomic neutral gases [6]. More recently, problems with slow convergence at some values of E / N for some interaction potentials (connected perhaps with partial

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W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 189-204. © 1992 Kluwer Academic Publishers.

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ion runaway [1]) have been overcome [7,8]. It is now possible to compute accurately transport properties for atomic ion/atom systems from an assumed ion/neutral inter­action potential, and to use a comparison of such calculated results with experimental data to assess quantitatively the accuracy of the interaction potential [9-11]. It has even been found possible to invert the data so as to obtain the interaction potential directly, without assuming any specific functional form for it. With mobility data of 1-2% accuracy, potentials with an absolute accuracy of about 10% can be obtained [12] over a wide range of separation, covering the long-range attraction, the well region, and a considerable portion of the short-range repulsive wall. This accuracy equals that of the best ab initio potentials (from CI calculations) and the accuracy of potentials inferred from the best beam-scattering data.

Theoretical understanding of more complicated ion/neutral systems is in a much more primitive state. A theory has been developed and implemented that accounts for the resonant charge transfer that occurs when an atomic ion moves through its parent neutral gas [13], but this situation is not nearly as complex as those that arise when molecular species are involved, since nonspherical interaction potentials and many internal energy states must be considered.

In 1981 a kinetic theory was proposed [14] for polyatomic ion/neutral systems. It was based on the Wang Chang-Uhlenbeck-de Boer kinetic equation, a semi­classical generalization of the Boltzmann equation in which internal states are treated quantum-mechanically but translational states are treated classically. The cross sec­tions that arise in this theory are, unfortunately, so difficult to compute that no actual implementations are possible without introducing approximations of uncertain accuracy [15-18]. Moreover, it is overwhelmingly likely that classical mechanics can provide just as accurate a description of gaseous ion transport properties for all di­atomic systems except those containing diatomic hydrogen or similar species. This is because:

• the electronic and vibrational levels are so widely separated that at the energies accessible in swarm experiments only the lowest levels are populated;

• the rotational levels are so closely spaced that in room-temperature swarm exper­iments, even in the limit oflow E/N, so many rotational levels are populated that the molecules can be closely approximated as classical-mechanical rigid rotors; and

• the transport properties of ions in gases are governed by energy averages of both the elastic and inelastic cross sections, so slight errors due to a classical treatment of the smaller inelastic cross sections are of little importance. To carry the arguments given above to extremes, it might be hoped that gaseous

ion transport coefficients for molecular systems could be calculated with negligible error by treating both the ion and neutral as point particles, by using as the interac­tion potential between them some effective spherical potential, and by neglecting all inelastic collisions. After all, the transport properties of weakly anisotropic, neutral systems can often be treated in this manner, with the effective potential being the spherical average of the true interaction potential [19,20]. However, the spherical com­ponents of ion/neutral interaction potentials fail to reproduce gaseous ion transport

191

data [21]. For neutral, diatomic systems of moderate anisotropy, an effective potential can

be found that reproduces the second virial coefficients and gas-phase transport prop­erties [22], although this effective potential has no obvious connection with the true interaction potential. However, an effective spherical potential is only of limited use for systems with high anisotropy; although the temperature dependence of any partic­ular transport coefficient can be reproduced over small ranges, the effective spherical potential generally has no connection with the true interaction potential and usu­ally cannot be used to predict other transport coefficients [23]. Moreover, there is evidence [24] that this approach would be even less useful for ion/neutral systems in swarm experiments, where the neutral molecules are in equilibrium at a temperature T but where the trace amounts of ions are directly influenced by the electric field and, depending on the value of E/N, can have an effective temperature much greater than T.

As specific support for the use of classical-mechanical techniques, note that Pack has shown [25] that classical mechanics so accurately reproduces the quantum­mechanical, close-coupling cross sections for Ar-N2 that classical results 'are fully accurate enough to be used for relaxation calculations and virtually any other pur­pose' in which energy averaging of the cross sections is performed. Similar results have been obtained by Dickinson and Lee [26,27]' although they could only compare their classical trajectory results with approximate quantal results. Alexander has shown [28] that even in a very bad case, Li+ -H2' the inelastic cross sections con­tribute only about 25% to the momentum-transfer cross section (the energy average that dominates gaseous ion transport) in the energy ranges accessible in swarm ex­periments.

Therefore, if molecular hydrogen and its ionic and isotopic analogues are ex­cluded, it should be possible to calculate by classical-mechanical means (including a classical treatment of inelastic collisions and nonspherical potentials) the gaseous ion transport coefficients with an accuracy of 1 % or better, rivaling the experimental accuracy. To do this it is necessary to combine classical trajectory calculations of the transport cross sections with a recent classical kinetic theory of drift tube experiments involving molecular ion/neutral systems [29]. This theory is based on the kinetic equa­tion of Curtiss [30]' and is thus restricted to atomic ions in non-vibrating (rigid rotor) diatomic gases or to non-vibrating diatomic ions in atomic or non-vibrating diatomic gases.

In Section 2 of this paper classical trajectory calculations of transport cross sec­tions for the Li+ -N2 system are described. Section 3 describes the kinetic theory used to calculate gaseous ion transport coefficients from these cross sections. A comparison of measured and calculated transport coefficients in Section 4 allows conclusions to be drawn about the accuracy of the available potential energy surface for Li+ -N2.

2. Trajectory calculations

The fundamental quantity describing the interaction of atomic ions with diatomic

192

neutrals is the potential energy surface, which varies with the magnitude of the vector connecting the nucleus of the atomic ion to the centre-of-mass of the diatom, and with the angle between this vector and the vector connecting the two nuclei of the diatom. However, accurate information about such surfaces is sparse, except for systems in which the diatom is the hydrogen molecule. This is due, on one hand, to the difficulty of ab initio calculations and, on the other hand, to the difficulty of making accurate measurements of equilibrium, scattering, or transport data and then inferring accurate information about the potential energy surface from them.

Limited information about the Li+-N2 potential energy surface has been ob­tained by Staemmler [31]. His self-consistent field (SCF) calculations show that the potential can be approximated as a combination of chemical interactions (a repulsion at small separations that varies exponentially with separation, combined with a weak 'chemical bond' at intermediate separations) and electrostatic attractions (at large separations). The angle dependence of the interaction potential is due primarily to the angle dependence of the electrostatic terms.

Staemmler carried out limited calculations (with a smaller basis set) that took electron correlation into account. These showed that the SCF points could be inac­curate by 10 to 30%, especially at large separations. Unfortunately, it was beyond the capabilities of computers in 1975 to use a large basis set and to consider a large number of points on the potential energy surface when electron correlation had to be taken into account.

A number of analytical fits to Staemmler's SCF points have appeared. In 1977, Thomas assumed [32] that the rigid-rotor potential could be expanded in Legendre polynomials of the angle, and that the dependence of the expansion coefficients upon the separation could in turn be expanded as a sum of exponeritial and inverse-power terms. The coefficients of the inverse-power terms were fixed to reproduce the ex­perimental values of the dipole and quadrupole moments, in order to maintain the proper long-range behavior of the potential. He then used this analytical potential in a numerical solution of Hamilton's equations of motion to determine differential scattering cross sections from the classical trajectories. Because only a limited num­ber of trajectories were computed, integral cross sections were not determined. The results were in reasonable agreement with the experimental differential cross sections at 4.23 eV (1 eV = 1.6021773 x 10-19 J), so Thomas concluded that the SCF surface and the quasiclassical method provide a reasonably good description of the scattering. It should be noted, however, that this tests only the short-range, repulsive part of the potential. Previous experience with atomic ion/atom systems is that, while the short-range part of SCF potentials generally describe experimental results with fair to good accuracy, the well and long-range parts of SCF potentials are generally poor.

Subsequently, other fits to the SCF potential for Li+ -N2 were used to test the short-range potential, with somewhat contradictory results. Reasonable agreement with experimental cross sections between 1 and 20 e V was obtained by Poppe and Bottner [33] and by Gierz et al. [34] using a larger number of classical trajectories than Thomas, and by Billing [35] using semiclassical techniques. Substantial disagreements with the same data were noted by Pfeffer and Secrest [36] using an infinite-order

193

sudden approximation combined with other techniques. Scattering data in the energy range from 500 to 1250 eV and in the angular domain of 0.3 to 5 mrad were also found to be inconsistent with theoretical results calculated in the sudden approximation [37]. It seems likely that the SCF potential is moderately accurate at separations between about 3 and 4 a.u. (1 a.u. = 5.29177 x 10-11 m), but becomes increasingly inaccurate at shorter and larger separations.

The purpose of the work described here was to test the accuracy of the SCF potential energy surface by comparing experimental values of the gaseous ion trans­port coefficients with values calculated from Billing's fit to the SCF potential. This comparison cannot consider the effect of vibrational excitation, because the kinetic theory used in Section 3 is restricted to non-vibrating diatoms. Consequently the only SCF points of interest are those obtained when the nitrogen atoms were separated by their equilibrium value. This subset of the complete potential energy surface is referred to as the rigid rotor surface, since it only describes the non-vibrating (v = 0) state.

The first step in the present work was to compute from the rigid rotor surface the transport cross sections needed in the kinetic theory program BIMAX [8]:

q(A,V)(E,Eo)=27r dbb- dcosX- d¢>L- d¢jx 1-:'- PA(cos8). 100 1 J1 1 1211" 1111" [( ')V+A/2 ] o 2 -1 27r 0 7r 0 E

(1) Note that the scattering angle 8 and the post-collision kinetic energy E' in Eq. (1) depend upon the pre-collision kinetic energy E, the pre-collision rotational energy EO,

the impact parameter b, the angle X between the rotational angular momentum, j, of the diatom and the angular momentum, L, of the relative motion, the angle ¢L conjugate to L, and the angle ¢j conjugate to j.

The most important transport cross section for determining gaseous ion-transport coefficients is the momentum-transfer cross section, for which>' = 1 and v = 0 in Eq. (1). However, cross sections with values of >. from 1 to 12 and of v from 0 to 6 are necessary to determine the transport coefficients accurately. In this regard note that [38]:

'It is a feature of the classical approach that virtually all of the effort goes into calculation of the trajectories while evaluation of a wide range of cross sections is relatively inexpensive. In contrast, in the quantal case, calculat­ing generalized cross sections involves lengthy multiple sums over S-matrix elements weighted by 6 - j type coefficients; the effort required for this is comparable with that required to generate the S-matrices.'

Unfortunately, the cross sections with high values of >. and v are more sensitive to changes in the trajectory parameters E, EO, b, X, ¢L, and ¢j. Consequently, more trajectories must be sampled in order to determine these cross sections accurately.

A drawback to the use of the cross sections defined by Eq. (1) is that they can become negative at low values of the energy, E, where superelastic collisions (collisions

194

Table 1. Summary of Li+ -N2 trajectories.

Kinetic energy range /10- 3 a.u. <3 3 - 9.5 9.5 - 95 > 95

Number of kinetic energies 2 5 8 8 Number of rotational energies 16 12 10 8 Initial and final separation (a. u.) 150 120 100 80 Number of impact parameters 26 26 25 24 Number of X angles 12 10 8 6 Number of ¢L angles 12 10 8 6 Number of ¢j angles 12 10 8 6 Number of trajectories 1437696 1560000 1024000 331776

involving de-excitation of the neutrals) become common. An alternative set of cross sections which do not suffer from this difficulty are used in the kinetic theory program MOBDIF[2]:

100 1 11 1 1271" 1 171" (7"(>"")(E, EO) =27r db b- dcosX- d¢L- d¢j

o 2 -1 2'if 0 'if 0

(2)

Dickinson and Viehland have modified the Newcastle classical trajectory pro­gram [27] so as to treat atomic ion/diatom systems and determine the cross sections defined by Eqs. (1) and (2). An important difference between such trajectory cal­culations and more conventional calculations with atom/diatom systems is that the long-range potential falls off much more slowly with separation. Consequently, tra­jectories must be started at a much larger initial separation and followed until the particles have separated again by this large distance. Another difference is that the experimental transport data cover a wider range of energy (typically, thermal energy up to one or two electron volts). Consequently, accurate cross sections must be cal­culated at many values of the relative kinetic energy between the colliding particles and of the rotational energy of the diatom. Moreover, the trajectories become suc­cessively longer-lasting as these energies decrease. The trajectories used by Viehland and Maclagan in the present work are summarised in Table l.

Figure 1 shows some of the rotationally-averaged transport cross sections for Li+ ions in N2 gas at T=300 K, as a function of relative kinetic energy. These cross sections are defined by Eqs. (3) and (4), where k is Boltzmann's constant,

Q(>"V)(E) = k~ 100 dEoexp(-:~)q(>"V)(E'EO)' (3)

soo

200

Q (a.u.) 100

50

20

2 10g( ~B T) for T-300 K

3

195

Figure 1. Rotationally-averaged transport cross sections for Li+ ions in N2 gas, as a function of the relative kinetic energy E. The curves are labelled with the values of ..\ and l/ appropriate to the definition in Eq. (3).

and ~(A,v)(E) =...!.- roo dEo exp (_~) a(A,v)(E EO),

kT io kT ' (4)

The behaviour of these cross sections is qualitatively the same as for atomic ion/atom systems, except when superelastic collisions become important.

3. Transport theory

Gaseous ion transport experiments satisfy the following restrictions: (a) a uniform d.c. electric field is present, but there are no magnetic or other electric fields; (b) the velocities of all particles are negligible compared to the speed of light; (c) the dilute, neutral gas is in equilibrium, so its velocity distribution is isotropic and has the well-known Maxwellian form; (d) only trace amounts of ions are present at any time, and no reactions can occur between these ions and the neutral molecules; and (e) the effects of the apparatus walls are negligible, so that a hydrodynamic description (stationary ion velocity distribution and small gradients of ion density) is appropriate.

196

In gaseous ion-transport experiments with atomic ions and neutrals, the ion ve­locity distribution function is governed by the Boltzmann equation of 1872 [1]. For atomic ions in non-vibrating diatomic neutrals, the classical-mechanical generaliza­tion of Boltzmann's equation has been given by Curtiss [30]. The collision term in either equation involves a linear operator, J, which acts on the ion distribution func­tion only through its velocity dependence; that is, J is a local operator in position and time because it is assumed that both the range and duration of collisions are negligible compared to the distance and time between collisions in a dilute gas. The operator J depends functionally on the transport cross sections and upon the neu­tral distribution function, whose isotropic, Maxwellian character guarantees that J is rotationally invariant, i.e. has eigenfunctions proportional to spherical harmonics in ion velocity space. Further general comments about J are given by Kumar et al. [39]

It follows from assumption (e) above that the space-time dependence of the ion distribution function has the form

f(r,v,t) = Lf(j)(v). (-V)jn(r,t), j

(5)

where the functions f(j)(v) are tensors ofrank j, and where the dot indicates a scalar product. The ion number density nCr, t) is described by the equation of continuity which, because of Eq. (5), we can write as a transport equation

(:t -Lw(j) . (-V)j )n(r,t) = O. J

(6)

The quantities w(j) are tensorial transport coefficients of order j. Note in particular that w(O) must be zero in the absence of ion/neutral reactions, and that the drift velocity w(1) must lie along the direction of the electric field axis). The scalar quantity equivalent to the magnitude of w(l) ,

(7)

is customarily referred to as 'the drift velocity' , and the standard (or reduced) mo­bility is defined by the equation

K _ Vd

0- (2.686763x1025m-3)(E/N) ' (8)

Note also that, to second order in gradients of n(v, t), the equation of continuity is the more familiar diffusion equation, with w(2) being the diffusion tensor whose scalar components (the diffusion coefficients parallel and perpendicular to the electric field) are determined by f(1)(v).

If we insert Eq. (5) into the Boltzmann equation, make use of Eq. (6) to remove the time derivative of n( r, t), and then separately equate terms in the same order of

197

gradient of nCr, t), we obtain a hierarchy oflinear equations to solve for the successive components f(j)(v). These components can then be inserted into equations such as Eq. (7) and (8) to determine the transport coefficients that are measured.

Various techniques have been developed for the approximate solution of linear operator equations. Many of them, particularly those that have been used to solve the hierarchy of equations governing the fU)(v), are special cases of the method of weighted residuals [40]. To illustrate the general method of weighted residuals, consider a one-dimensional operator equation of the type

Of (x) = g(x), (9)

where 0 is some known linear operator. Suppose that some basis functions Ti(X) (0 :-:::: i :-:::: N) are chosen and used to make the expansion

N

f(x) = L fm(x). (10) i=O

The choice of some weighting functions, Bj(x) (0:-:::: j :-:::: N), which are then used to multiply Eq. (9) from the left, leads after integration, to the set of linear equations

N

LliOji = gj O:-::::j:-:::: N, (11) i=O

where

(12)

and

(13)

This set of equations can be solved by matrix methods to obtain the expansion coeffi­cients Ii (0 :S i :S N) from which an approximation to the unknown function f(x) can be determined; note that the expansion coefficients are simply related to 'moments' such as that defined by Eq. (7), and hence to the transport coefficients in kinetic theory.

From the equations in the previous paragraph it can be shown that a Galerkin method for solving an operator equation is a special method of weighted residuals in which the basis functions are the same as (or are simply related to) the weighting functions. A moment method is a special method of weighted residuals in which both functions are powers of x. Another special case is the quantum discretization method (QDM) introduced into kinetic theory by Shizgal and coworkers [41,42]. The QDM is an orthogonal collocation method, i.e. a special method of weighted residuals in which the weighting functions are delta functions centered at the roots of an orthogonal polynomial, which roots are used as the quadrature points in evaluating integrals.

198

Numerous other variations have been used in the literature, sometimes imprecisely being called 'moment methods' rather than specialized versions of the method of weighted residuals.

Many studies have shown [40] that using different weighting functions in a method of weighted residuals leads to virtually the same results when a reasonably high order of approximation, N, is used. However, a method of weighted residuals generally works best when the basis functions are a good approximation to the eigenfunctions of the operator. For swarm experiments at very low E/N, the true ion distribution is essentially a Maxwellian characterized by the gas temperature T. Theories used before the early 1970's were based on basis functions that were orthogonal to such a Maxwellian distribution, so in retrospect it is not surprising that they were successful only at very low E/N. The two-temperature approach to gaseous ion transport [43] is successful at higher values of E / N because it uses basis functions that are orthogonal with respect to a Maxwellian distribution characterized by an ion temperature rather than T. At very high E/N, and in order to describe accurately the highly-anisotropic ion diffusion, the three-temperature approach [44,45] is better, because its basis func­tions are orthogonal with respect to a distribution that is displaced along the field direction and is characterized by different ion temperatures parallel and perpendicular to the field. The computer program MOBDIF [2], used to obtain some of the results in Section 4, is based on a three-temperature technique.

Recently [8], a method of weighted residuals has been introduced into the kinetic theory of gaseous ion transport that allows for the possibility of partial ion runaway leading to a true ion distribution function that is a double-humped function of the ion velocity. The true ion velocity distribution function is assumed to be described by the expansion in Eq. (14).

f(O) (v) = 2: f/~s7r-3/2 [9,81 exp( _v2/ ,8d+(I- g),82 exp( _v2/ ,82)] ~t: W/2v). (14) lms

Here 9 is the fraction of the ions characterized by a Maxwellian with a low ion tem­perature T1 , while the remaining ions are characterized by a Maxwellian with a higher temperature, T2 • In addition,

m ,8i = 2kT' ,

(15)

{) = f /,81 + (1 - f) / ,82, (16)

and the ~i~ functions are Burnett functions [1]. This method of weighted residuals reduces to a two-temperature method when 9 = 1, and has the advantage that matrix elements of the collision operator J are not substantially more difficult to calculate than in that method. The computer program BIMAX [8]' used to obtain most of the results in Section 4, is based on this bi-Maxwellian technique.

6

5.5

5

K. 4.5 (cmW.r)

4

3.5

3

• • • • •

........................... .... '" .....

EIN (Td)

199

200 300 SOO

Figure 2. The standard mobility of Li+ ions in N2 gas at 300 K, as a function of the reduced field strength in Td (1 Td = 1 x 10-21 V m 2). The points with error bars are experimental values. The solid curve was calculated from program BIMAX using cross sections at energies above 0.003 a.u. The dotted curve was calculated from program MOBDIF using cross sections at energies above 0.00017 a.u.

4. Comparison with experiment

The mobilities and diffusion coefficients for Li+ ions in N2 gas have been measured by many different groups [46-49]' but because offast clustering reactions at low E IN, only recently have consistent, accurate values been reported [50,51J. These experimental values are compared in Figs. 2 and 3 with values computed from the theory in Section 3 and the cross sections of Section 2. The dimensionless reduced diffusion coefficients in Fig. 3 were defined by Viehland and Mason [9J.

In order to assess the importance of the disagreements shown in Figs. 2 and 3 it is necessary to determine the precision and accuracy of the calculated values. The transport cross sections were carefully monitored at several values of the relative kinetic energy, and the trajectory parameters shown in Table 1 appeared to give cross sections that were precise within at worst a few tenths of a percent. Certainly, a conservative estimate is that these cross sections have an accuracy of ±1 %. A further

200

3

2.S

2

15 1.S

O.S

o

i • . :..,. .... .

'.'~' •••••••••••• ; .0- •

Ein. (Td)

soo

Figure 3. Dimensionless reduced diffusion coefficients of Li+ ions in N2 gas at 300 K, as a function of the reduced field strength. The points with error bars are experimental values [51]. The solid and dotted curves represent the calculated values for the parallel and perpendicular diffusion coefficients, respectively. The curves were not smoothed, and the discontinuities indicate that the precision of the calculated results is approximately the same as the estimated uncertainties in the data.

source of uncertainty, given that low energy trajectory calculations are the most time consuming, is whether the cross sections have been calculated at low enough energies. The two sets of calculated mobilities shown in Fig. 2 agree very well at E / N values above 80 Td, but at lower values (corresponding to lower energies) the difference grows to about 6%. By using the program MOBDIF both with and without cross sections at the lowest energy, we estimate that computing cross sections at even lower energies would change the MOBDIF results by a negligible amount above 80 Td and by no more than 2% at lower E / N. Finally, the kinetic theory calculations using the programs MOBDIF and BIMAX were continued to high enough approximation to ensure at each E / N that the results had the stabilities indicated in Table 2.

201

Table 2. Stabilities of calculated transport coefficients.

E/N Stability Stability of diffusion coefficients range of mobility parallel perpendicular

Td % % %

0- 35 0.10 0.50 0.25 35- 90 0.20 1.00 0.50 90-100 0.30 1.50 0.75

100-120 0.45 2.25 1.13 120-130 0.50 2.50 1.25 139-500 0.55 2.75 1.38

5. Summary

In summary it seems certain that the differences between the calculated and exper­imental values in Figs. 2 and 3 are a reflection of the inaccuracy of the rigid-rotor potential energy surface used in the calculations. The substantial differences between the mobilities at low E / N point to appreciable errors at large separations, and the somewhat smaller differences at high E / N in both Figs. 2 and 3 suggest possible errors at small separations. Given these errors, however, the reasonable agreement at intermediate E / N suggests that the potential is reasonably accurate at intermediate separations, near and just beyond the potential minimum. These results are consis­tent with results obtained from previous tests of the SCF potential, as discussed in Section 1. Work is presently underway by Grice, Maclagan and Viehland to determine a more accurate ab initio potential energy surface and to test it against transport data in the way reported here.

References

[1] Mason, E. A. and McDaniel, E. W. (1988) Transport properties of ions in gases, Wiley, New York.

[2] Viehland, L. A. and Kumar, K. (1989) Transport coefficients for atomic ions in atomic or diatomic neutral gases, Chem. Phys. 131, 295-313.

[3J Ellis, H. W., Pai, R. Y., McDaniel, E. W., Mason, E. A. and Viehland, 1. A., (1976) Transport properties of gaseous ions over a wide energy range, At. Data Nucl. Data Tables 17, 177-210.

[4J Ellis, H. W., McDaniel, E. W., Albritton, D. L., Viehland, L. A., Lin, S. L. and Mason, E. A. (1978) Transport properties of gaseous ions over a wide energy range. Part II, At. Data Nucl. Data Tables 22, 179-217.

202

[5J Ellis, H. W., Thackston, M. G., McDaniel, E. W. and Mason, E. A. (1984) Transport properties of gaseous ions over a wide energy range. Part III, At. Data Nucl. Data Tables 31, 113-15l.

[6J McDaniel, E. W., and Viehland, L. A. (1984) The transport of slow ions in gases: Experiment, theory and applications, Phys. Reports 110, 333-367.

[7J Larsen, P.-H., Skullerud, H. R., Lovaas, T. H. and Stefansson, Th. (1988) Trans­port coefficients and interaction potentials for lithium ions in helium and argon, J. Phys. B 21, 2519-2538.

[8J Ness, K. F. and Viehland, 1. A. (1990) Distribution functions and transport coefficients for atomic ions in dilute gases, Chern. Phys. 148, 255-275.

[9J Viehland, 1. A. and Mason, E. A. (1984) Repulsive interactions of closed-shell ions with He and Ne atoms: Comparison of beam and transport measurements, J. Chern. Phys. 80,416-422.

[IOJ Viehland, L. A. and Mason, E. A. (1984) Repulsive interactions of closed-shell ions with Ar, Kr, and Xe atoms: Comparison of beam and transport measure­ments, J. Chern. Phys. 81,903-908.

[l1J Johnsen, R., Rosh, R. and Viehland, L. A. (1990) Mobility of helium ions in neon: Comparison of theory and experiment, J. Chern. Phys. 92, 7264-7278.

[12J Kirkpatrick, C. C. and Viehland, L. A. (1985) Interaction potentials for the halide ion-rare gas systems, Chern. Phys. 98, 221-23l.

[13J Viehland, L. A. and Hesche, M. (1986) Transport properties for systems with resonant charge transfer, Chern. Phys. 110, 41-54.

[14J Viehland, L. A., Lin, S. 1. and Mason, E. A. (1981) Kinetic theory of drift-tube experiments with polyatomic species, Chern. Phys. 54, 341-364.

[15J Arthurs, A. M. and Dalgarno, A. (1960) The theory of scattering by a rigid rotator, Proc. Roy. Soc. A 256, 540-55l.

[16J Goldflarn, R. and Kouri, D. J. (1977) On accurate quantum mechaical approxima­tions for molecular relaxtion phenomena. Averaged }z-conserving coupled states approximation, J. Chern. Phys. 66, 542-547.

[17J Curtiss, C. F. (1977) Semiclassical theory of rotational transitions in collisions of diatomic molecules, J. Chern. Phys. 67, 5770-5774.

[18J Cross, R. J. (1982) An energy-adjusted sudden approximation for inelastic scat­tering, J. Chern. Phys. 76,931-937.

[19J Monchick, L. and Green, S. (1975) Validity of central field approximations in molecular scattering: Low energy CO-He collisions, J. Chern. Phys. 63, 2000-2009.

[20J Green, S. and Monchick, 1. (1975) Validity of approximate methods in molecular scattering: Thermal HCl-He collisions, J. Chern. Phys. 63, 4198-4205.

[21J Viehland, 1. A. (1988) The spherical compenents of ion-neutral interaction po­tentials fail to reproduce gaseous ion transport data, Chern. Phys. Lett. 144, 552-554.

203

[22] Smith, E. B. and Tindell, A. R. (1982) Gas-phase properties and forces in Van der Waals molecules, Faraday Disc. Chern. Soc. 73,221-223.

[23] McCourt, F. Rand Liu, W.-K. (1982) Anisotropic intermolecular potentials and transport properties in polyatomic gases, Faraday Disc. Chern. Soc. 73,241-256.

[24] Viehland, L. A. and Fahey, D. W. (1983) The mobilities of NO;-, N02, NO+, and Cl- in N2: A measure of inelastic energy loss, J. Chern. Phys. 78, 435-441.

[25] Pack, R. T (1975) Close coupling test of classical and semiclassical cross sections for rotationally inelastic Ar-N2 collisions, J. Chern. Phys. 62, 3143-3148.

[26] Dickinson, A. S. and Lee, M. S. (1985) Classical trajectory calculations of trans­port properties for a model Ar-N2 potential surface, J. Phys. B 18, 3987-4000.

[27] Dickinson, A. S. and Lee, M. S. (1985) Classical trajectory calculations of diffu­sion and viscosity for He-N 2 mixtures, J. Phys. B 18, 4177-4184.

[28] Alexander, M. H. (1976) Inelastic contributions to ion-molecule diffusion cross sections: Li+ -H;, J. Chern. Phys. 64, 4498-4503.

[29] Viehland, L. A. (1986) Classical kinetic theory of drift tube experiments involving molecular ion-neutral systems, Chern. Phys. 101, 1-16.

[30] Curtiss, C. F. (1981) The classical Boltzmann equation of a gas of diatomic molecules, J. Chern. Phys. 75,376-378.

[31] A Staernrnler, V. (1975) Ab initio calculation of the potential energy surface of the system NrLi+, Chern. Phys. 7, 17-25.

[32] Thomas, L. D. (1977) Classical trajectory study of differential cross sections for Li+ -co and N2 inelastic collisions, J. Chern. Phys. 67, 5224-5236.

[33] Poppe, D. and Bottner, R (1978) Inelastic collisions of Li+ with N2 molecules: A comparison of experimental results with trajectory studies, Chern. Phys. 30, 375-386.

[34] Gierz, U., Toennies, J. P. and Wilde, M. (1984) A new look at rotational and vibrational excitation in the scattering ofLi+ from N2 and co at energies between 4 and 17 eV., Chern. Phys. Lett. 110, 115-122.

[35] Billing, G. D. (1979) Semiclassical calculations of differential cross sections for rotational/vibrational transitions in Li + N2, Chern. Phys. 36, 127-134.

[36] Pfeffer, G. A. and Secrest, D. (1983) Rotation-vibration excitation using the infinite-order sudden approximation for rotational transitions: Li+ -N2' J. Chern. Phys. 78, 3052-3070.

[37] Kalinin, A. P., Resandt, R, Wijnaendts, W., Khrornov, V. N., Kleyn, A. W., Los, J. and Leonas, V. B. (1984) Differential cross sections for Li+ scattering by N2 molecules, Chern. Phys. 85,341-347.

[38] Dickinson, A. S. and Lee, M. S. (1986) Classical trajectory calculations for aniso­tropy-depended cross sections for He--N2 mixtures, J. Phys. B 19,3091-3107.

[39] Kumar, K., Skullerud, H. R. and Robson, R E. (1980) Kinetic theory of charged particle swarms in neutral gases, Austral. J. Phys. 33,343-448.

204

[40] Finlayson, B. A. (1972) The method of weighted residuals and variationalu prin­ciples, Academic Press, New York.

[41] Shizgal, B. (1981) A gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems, J. Comput. Phys. 41, 309-328.

[42] Shizgal, B. and Blackmore, R. (1984) A discrete ordinate method of solution of linear boundary value and eigenvalue problems, J. Comput. Phys. 55, 313.

[43] Viehland, L. A. and Mason, E. A. (1975) Gaseous ion mobility in electric fields of arbitrary strength, Ann. Phys. (N.Y.) 91, 499-533.

[44] Lin, S. L., Viehland, L. A. and Mason, E. A. (1979) Three-temperature theory of gaseous ion transport, Chern. Phys. 37,411-424.

[45] Viehland, L. A. and Lin, S. L. (1979) Application of the three-temperature theory of gaseous ion transport, Chern. Phys. 43, 135-144.

[46] Tyndall, A. M. (1938) The mobility of positive ions in gases, Cambridge Univer­sity Press, London.

[47] Takata, N. (1974) Mobilities of Li+ ions in H2, N2, and their mixtures, Phys. Rev. A 10, 2336-2340.

[48] Gatland, I. R., Colonna-Romano, L. M. and Keller, G. E. (1975) Single and double clustering of nitrogen to Li+, Phys. Rev. A 12, 1885-1894.

[49] Koizumi, T., Kobayashi, N. and Kaneko, Y. (1977) Mobilities of Li+, NO+ and ot in N2 and CO 2 gases, J. Phys. Soc. Japan 43, 1465-1466.

[50] Satoh, Y., Takebe, M. and !inuma, K. (1987) Measurements of mobilities and longitudinal diffusion coefficients for Li+ ions in some molecular gases, J. Chern. Phys. 87,6520-6529.

(51] Selnaes, T. D., Lovaas, T. H., and Skullerud, H. R. (1990) Transport coefficients for lithium ions in nitrogen J. Phys. B 23, 2391-2398.

CLASSICAL AND SEMI-CLASSICAL TREATMENT OF ENERGY TRANSFER IN SMALL MOLECULES

GERT DUE BILLING Department of Chemistry, H. C. 0rsted Institute, University of Copenhagen, DK-2100 0, Denmark

ABSTRACT. Some of the problems concerned with a classical mechanical treat­ment of energy transfer processes in molecules are discussed, and possible alter­natives are given.

1. Introduction

Energy transfer in molecules is most conveniently described by using a classical me­chanical treatment of the dynamics. However, it should be remembered that the fundamentally correct description is nevertheless a quantum mechanical one, and that the quantum nature of the problem may manifest itself in various ways, some of which will be described below.

Even today, with easy access to large scale computers and effective numerical algorithms, a full quantum treatment of the collisions of molecular systems is only feasible for the simplest systems. The reason for this is that the number of coupled second-order differential equations which have to be solved increases drastically with energy and with the mass of the system. Thus, for an atom-diatom system the number of open channels increases roughly as N rv I E2 / wh3 , where I is the moment of inertia of the diatomic, E is the energy of the system, and w is the frequency associated with internal motion.

In a classical mechanical description the numerical problem increases only linearly with the number of atoms. In order to obtain observable quantities one has to average over a number of trajectories Nt with randomly selected initial variables - phase angles, initial orientation and impact parameters. The convergence rate to accurate average values is slow, proportional to N;/2. For weak events it is difficult, and for classically forbidden events even impossible, to obtain probabilities or cross sections. In such cases is it advantageous, from both numerical and theoretical points of view, to use a semi-classical description in which the degree of freedom that is only weakly perturbed or excited is treated quantally. For most diatomic and polyatomic molecules these degrees of freedom are normally the vibrational degrees of freedom.

205

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 205-216. © 1992 Kluwer Academic Publishers.

206

2. Classical mechanical treatment

The classical equations of motion are most conveniently integrated using a carte­sian coordinate system, i.e. by introducing, for example, Jacobi coordinates in which the centre-of-mass motion is separated out. However, in order to analyse the trajecto­ries, it is necessary to introduce the appropriate classical action/angle variables. The actions are the classical analogues of the quantum numbers for the internal degrees of freedom, i.e. rotation and vibration. It should be noted that this cannot always be done exactly, since it requires separability of the Hamiltonian or application of procedures such as the adiabatic switching-function method [1].

For an atom-diatom system we introduce coordinates for the relative motion of A with respect to the centre-of-mass of Be, R = (X, Y, Z), and the vector r = (x, y, z) which specifies the orientation of the Be molecule in a coordinate system with origin in the centre-of-mass of Be. The vibration-rotation energy of the diatomic molecule is

P; P EVj = 2mr + 2I' (1)

where I = m rr2 is the moment of inertia and mr = mBme/(mB + me) is the reduced mass. If we now introduce spherical polar coordinates r, 0, rjJ via

x = r sin 0 cos rjJ

y = r sin 0 sin rjJ

z = rcos 0,

(2)

(3)

(4)

random values of x, y, z are obtained by selecting () and ¢ randomly between 0, 7r and 0, 27r, respectively. For the initialization of the momenta we have

Px = Pr sin 0 cos rjJ + t (sin rjJ cos 'T/ + cos 0 cos ¢ sin 'T/ ) r

Py = Pr sin 0 sin ¢ + l.. ( - cos ¢ cos 'T/ + cos 0 sin rjJ sin 'T/ ) r

O j·O· pz = Pr cos - - sm sm'T/, r

(5)

(6)

(7)

where the angle 'T/ is selected randomly between ° and 27r. For the relative motion of the atom we may use the following initialization:

X=o, Y - -JR2 - b2 - 0 , Z=b (8)

Vx = 0, Vy = Vo, Vz = 0, (9)

where Ro is large, and b is the impact parameter.

Table 1. A simple iterative scheme for determination of the rota­tional and vibrational quantum numbers in a diatomic molecule.

1) 2) 3) 4) 5)

6)

7)

2

EVj = 2~r + V(r) V=O Bv = Be - Q e (V + ~) Erot = j(j + l)[Bv - Dej(j + 1)] Evib = EVj - Erot = nwe(v + 1/2) - nweXe(V + 1/2)2

V = ~ [-1 + :. (1 - VI -~ )] if V is changed go to 3

207

The initial value of r may be chosen using the following expression appropriate for a Morse oscillator description of the diatom

1 b 4ac" { [

1 l} r = r e + ,8 In - 2a 1 - (1 -b2) cos 7r~ ,

in which we have used the fact that the vibration-rotation energy is given by

The parameters in Eq.(lO) are given by the formulae

where

and

b = 2De - BL2

C = CL2 - De

A = [1- 3(1- 1/{3re )/{3re]/2mrr;,

B = 2(1 - 3/2{3re )/ {3mrr;

(10)

(11)

(12)

(13)

(14)

(15)

208

Table 2. Energy transfer to the torsional mode of methanol obtained by collision with He atoms. The torsional motion is treated classically or quan­tally [3].

Kinetic energy i::lE (classical) i::lE (quantum) kJ/mol kJ/mol kJ/mol

5 -0.004 0.13 10 -0.008 0.14 20 -0.053 0.24 50 0.035 0.51

The error bars are about 15%

Alternatively, we could start the oscillator at the turning point, r = r ±, with Pr = 0, integrate for a random time, t E [I, T], where T is the vibrational period, and then start the collision at that time.

In the final state analyses the rotational angular momentum is easily obtained as j=1 r x p I. The vibrational quantum number could be found by the iterative scheme given in Table 1 or by using the Bohr-Sommerfeld quantization condition, i.e.

(16)

3. Zero-point energy problems

It is well known that classical mechanics does not preserve the zero-point vibrational energy of a system. Hence, part, or all, of this energy is in principle available for the remaining degrees of freedom. On the other hand it is also known that the inclusion of zero-point energy offers a more realistic description of the dynamics. However, this is not the only problem with the classical description of the vibrational degrees of freedom. The classical distribution function, especially for low vibrational states, is very different from the quantum distribution function.

The problem with conservation of zero-point energy can drastically affect the result obtained for the net energy transfer to a given vibrational mode of a molecule (see for example Table 2). Recently, a simple procedure for avoiding this problem has been suggested [2] according to which a repulsive wall, W( Vk), is introduced which is equal to infinity as soon as the vibrational action Vk becomes negative, i.e. one simply changes the sign of the vibrational phase qk(t) when this happens.

Another approach, which also deals with the problem concerning the initial distri­bution function, would be to introduce a quantum mechanical weighting or sampling

209

function, namely the Wigner function. The Wigner function r is defined by

r( q, p, t) = ~ 100 dy lJ!( q + y, t)*lJ!( q - y, t) exp(2ipy /h). (17)

7fh -00

It obeys the equation

8r(q,p, t) = _ 8H 8r + 8H 8r + O(h2), at 8p 8q 8q 8p

(18)

where H is the classical Hamilton function. Thus we see that by neglecting terms of order h2 and higher we can propagate the Wigner function by using classical trajec­tories. For a harmonic oscillator we can calculate the Wigner function analytically. The result is

2 r v(q,p) = -( -1)v exp( -2E/hw)Lv(4E/hw), hw

(19)

where E = p2/2m + v( q) is the vibrational energy, w the frequency and Lv a Laguerre polynomial. We notice that the Wigner function is normalised, i.e.

(20)

The average energy is equal to the vibrational energy in state v, i.e.

(21)

We notice that the function is defined from zero to infinity, that is also for energies that lie below the zero-point energy. Thus, a collision which brings the classical oscillator from a situation in which the energy is E to one in which it is E' (after the collision) contributes to the state-to-state probability (n to m) with the weight given by the product of the initial and final Wigner functions, i.e.

Pn~m = 4(_I)n+m J dx J dx'N(x,x')exp(-2x - 2x')Ln(4x)Lm(4x') (22)

where N(x, x') is the number of trajectories with initial reduced energy x = E/hw and with final reduced energy x' = E'/hw. In a practical calculation one simply selects the initial energy, either randomly between zero and a large value (less than the dissociation energy), or by using a weighted distribution, records the final vibrational energy and updates the probability for a given transition with the product of the two Wigner functions. Also, the Wigner function for a Morse oscillator can be expressed in terms of known functions [4J. However, for the general case it is most convenient to express the oscillator wavefunction in terms of a sum of distributed Gaussians and then compute the Wigner function for the Gaussian basis set functions [5J. Table

210

Table 3. Comparison of transition probabilities for vibrational ex­citation of a harmonic oscillator by collinear collision with an atom, obtained using classical trajectories and box quantization, Wigner­function weighting and quantum mechanical results (exact).

Transition Classical trajectories Wigner method Exact

0-0 0.000 0.051 0.060 0-1 0.335 0.185 0.218 0-2 0.229 0.310 0.366 0-3 0.244 0.316 0.267 0-4 0.191 0.172 0.089 0-0 0.436 0.390 0.416 0-1 0.262 0.406 0.409 0-2 0.302 0.194 0.150 0-3 0.000 0.037 0.025 0-4 0.000 -0.024

The numbers in the upper part of the table were obtained for a total energy Wnw, with mass and potential parameters msj = 2/3 and a:sj = 0.3 (see ref.[6]). The numbers in the lower part were obtained for a total energy 8.41825nw, with msj = 1/13 and a:sj = 0.1287.

3 shows a comparison between results obtained using classical trajectories, Wigner­function weighted trajectories and exact quantum results.

4. Classically forbidden events

We have seen (Table 3) that events with small probabilities are completely or nearly forbidden in classical mechanics. In such cases many trajectories have to be run in order to obtain accurate numbers.

It is then convenient to introduce an appropriate prior weighting function for the selection of the initial variables. We consider as an example the rate constant k~ (T) for a chemical reaction, with the reactant molecule initially in the vibrational state v. It may be written as

(23)

where x = Ekin/kT, and where the reaction probability PrV(Ekin) « 1 for kinetic energies around the energy threshold.

It has been suggested [7] that a guessed probability

pO = ~Xm-1 r m! m::::: 1,

211

(24)

be used as a weighting function. Thus, we rewrite the above expression for the rate constant as

with

and m 1

U = 1- exp( -x) L 7! . 1=0

We now select the variable u randomly between 0 and 1 and get

(25)

(26)

(27)

(28)

in which Wj = m!/xm - 1(Uj). The value of m should be chosen in some clever way. It has been suggested [7] that the left inflection point of the function Pro (x) be placed at, or close to, the threshold value for the reaction. We notice that m = 1 gives the usual sampling situation. But with a larger value of m it is possible to obtain a factor of three to four reduction in the number of trajectories to be used for weak transitions, i.e. transitions with probabilities less than, say, O.l.

In the Wang Chang-Uhlenbeck theory of transport and relaxation phenomena a similar situation occurs, since the collision integrals for the diatom-diatom case can be written as [8]

({ ... }) = J dxF(U,T)exp(-x),

where x = U / kT and F(U, T) is defined by

(29)

F(U, T) = m~;~~b J 2~~r k~ J djl J d12 J dl jd2l J dj~ J dj~ Pi1h---+j;j~ (U, l){· .. }

(30) in which { ... } denotes a particular quantity determined by the specific transport property which is to be computed. Further, Qa and Qb are rotational partition functions for the two diatomic molecules, mr is the reduced mass for the relative translational motion, and j1, 12, l are rotational and the orbital angular momenta, respectively. The probability for a given rotational transition is calculated classically as

(31)

212

in which Nj,j21 is the number of trajectories with initial values of the angular momenta equal to jl, j2 and l, while the quantity 8i is equal to unity if the final values of j~ and j~ lie within a box of ±1/2 around the value, and zero otherwise. In practical calcu­lations of transport properties we are never interested in state-to-state probabilities and the multidimensional integral in equation (30) is calculated as

F(U T) - 27f1i2 J kT ~ 'max 'maxl ~ "'"' .(i) .(i) lCi) {' "} , - Q Q 2 kTlI 12 max N L..11 12 t, mr a b mr7f t .

t

(32)

where Nt is the number of trajectories and {'i'} denotes the value of {} obtained with the i'-th trajectory. Thus the initial values of the angular momenta are selected randomly, i.e.

where

jl E [0, jf'ax]

h E [0, jrax]

1 E [0, lmax]

_ 1 / 2 lmax - h V 2mrU R o·

The energy U is thus the sum of the rotational energy of the two molecules and the kinetic energy of the relative motion, so that, after the selection of jl and j2, we obtain the kinetic energy as

(33)

Since the initial centre of mass distance Ro is finite, we have to subtract also the orbital energy before we can obtain the momentum for the relative motion as PR = J2mr(Ekin - Eorb). The evaluation of the multidimensional integral by this Monte Carlo procedure gives convergence to within about 5% with 2 to 400 trajectories. It has been observed that the function F(U) is often well approximated by

F(U, T) = A(T)U'/CT) . (34)

Hence one may use this as a prior or sampling distribution so as to obtain faster convergence of the integral (30).

213

5. A semiclassical description

We have mentioned above the problems with a classical mechanical treatment, es­pecially for the vibrational degrees of freedom. In the semiclassical (classical path) approach this has been taken care of by introducing a quantum description of these degrees of freedom.

It can be shown that this separation of the various degrees of freedom into clas­sical and quantum subsets corresponds to making an SCF type of approximation in the quantum wavefunction for the total system. Consider a system with two degrees of freedom denoted by (x) and (y). Hence we assume that the total wavefunction can be written as

(35)

Let y denote the quantum vibrational coordinate and x the classical translational coordinate. For the translational motion we now introduce a wave-packet description, I.e.

<1>1 (x, t) = exp {* [o:(t)[x - x(tW + p(t)[x - x(t)] + ,(t)] } . (36)

Inserting the trial function (35) into the time-dependent Schrodinger equation, ex­panding to second order in [x-x(t)J, and equating similar powers of [x-X(t)Jk<l>l (x, t), for k = 0, 1 and 2, we obtain the following set of equations

.:.() p(t) xt =-, mr

(37)

. aVI p(t) = -(<I>2(y, t) I -a I <l>2(Y, t)), x x(t)

(38)

and

(39) n

We have also used the expansion

(40) n

for the wavefunction in terms of the vibrational wavefunctions ¢n (y) and the ampli­tudes an(t) for specific quantum states. For the width parameter o:(t) and the phase factor ,(t) we obtain

(41)

and

i'(t) = in o:(t) + _1_p(t)2 mr 2mr

(42)

214

where mr is the reduced mass, and the brackets indicate integration over the quantum coordinate y.

The wavepacket (36) represents, in principle, all energies with different weight given by the momentum space distribution

(43)

In order to obtain the contribution from a given initial energy to the S-matrix element for a specific transition I -> n we have to project the wavepacket onto in­coming and outgoing plane waves, and take the ratio between the outgoing flux in channel n and incoming flux in channel I. If we do this, we obtain [9]

outgoing flux ~(t) . . fl = -(-) exp{ -g(t)[Pn - p(tW + g( -t)[PI - p( _t)]2} 1 an(t) 12 (44) mcommg ux g -t

where the amplitude an( -00) = 8nI and PI and Pn are defined as

( 45)

(46)

and the En are vibrational energies. The flux (44) is projected on outgoing waves at time t and incoming waves at time -to The function g(t) is defined by [9]

'Sa(t) g(t) = 2h 1 a(t) 12 (47)

Taking the limits t -> ±oo, this quantity attains constant values g( (0) and g( -(0). In Ref. [9] it is shown that variation of the initial width of the wavepacket leads to state-to-state transition probabilities which are independent of the width parameters for a particular choice of initial momentum, namely p( -00) '" O.5(PI + Pn). This arithmetic mean velocity should be used for the classical trajectory when computing the I -> n transition probability. We notice that in order to obtain this result, it is necessary to make the separability assumption (35). If this breaks down, i.e. when the two sets of variables are strongly coupled, we cannot use (35) as a trial function.

The above approach has been extended to 3D scattering of atom-diatom, diatom­diatom and polyatomic molecules [10). Here the degrees of freedom treated classically are usually the translational and rotational motions. Hence the above symmetrization is introduced such that a total energy, E, is assigned to a given quantum transition I -> n by the formula [10]

(48)

215

in which U is the sum of the translational and rotational energy, i.e. the 'classical energy'. This semiclassical approach has been tested against quantum mechanical calculations - in general successfully (see, for example references [10-12]).

However, it should be remembered that the coupling between the x and y sys­tem is treated approximately, within a simple SCF scheme, leading to the Ehrenfest average potential for the classical degrees of freedom. Thus, if the coupling is strong it is advisable to treat them within the same dynamical description. This may be either a quantum or a classical dynamical one. Thus at low energies where the trans­lational and rotational motions are weakly coupled we can make the above separation between these two degrees of freedom. At higher energies these degrees of freedom are more strongly coupled, and should then both be treated classically, whereas the vibrational motion should be quantized. At still higher energies even the vibrational energy should be treated classically. Thus the classical path approach (in its simple 'SCF version') should be used together with the principle: degrees of freedom which are strongly coupled should be treated within the same dynamical approach (classically or quantally).

Programs for the calculation of energy transfer and state-to-state cross sections for atom-diatom and diatom-diatom systems are available (see Ref. [10]).

Acknowledgement

This research was supported by the Danish Natural Science Research Council.

References

[1] Johnson, B.R. (1985) On the adiabatic invariance method of calculating semiclas­sical eigenvalues, J. Chern. Phys. 83 1204-1217; Billing, G.D. and Jolicard, G. (1989) On the application of the adiabatic invariance method for the identification of 'Quantum Chaos', Chern. Phys. Lett. 155521-526.

[2] Miller, W.H., Hase, W.L. and Darling, C.L. (1989) A simple model for correcting the zero point energy problem in classical trajectory simulations of polyatomic molecules, J. Chern. Phys. 91 2863-2868.

[3] Billing, G.D. (1986) Semiclassical calculation of energy transfer in polyatomic molecules. XII. Organic molecules, Chern. Phys. 104 19-28.

[4] Dahl, J.P. and Springborg, M. (1988) The Morse oscillator in position space, momentum space, and phase space, J. Chern. Phys. 884535-4547.

[5] Hamilton, I.C. and Light, J.C. (1986) On distributed gaussian bases for simple model multidimensional vibration problems, J. Chern. Phys. 84 306-317; Hen­riksen, N.E. and Billing, G.D. On the use of Wigner phase space functions in reactive scattering (to be published).

[6] Secrest, D. and Johnson, B.R. (1966) Exact quantum-mechanical calculation of a collinear collision of a particle with a harmonic oscillator, J. Chern. Phys. 45 4556-4570.

216

[7] Muckerman, J.T. and Faist, M.B. (1979) Rate constants from Monte Carlo quasi­classical trajectory calculations. A procedure for importance sampling, J. Phys. Chem. 83 79-88.

[8] Nyeland, C., Poulsen, L.L. and Billing, G.D. (1984) Rotational relaxation and transport coefficients for diatomic gases: computations on nitrogen, J. Phys. Chem. 88 1216-1221.

[9] Muckerman, J.T., Kanfer, S., Gilbert, R.D. and Billing, G. D. Classical path and quantum trajectory approaches to inelastic scattering, (to be published).

[10] Billing, G.D. (1984) The semiclassical treatment of molecular roto-vibrational energy transfer, Camp. Phys. Rep. 1 237-296.

[11] Billing, G.D. (1984) Rate constants and cross sections for vibrational transitions in atom-diatom and diatom-diatom collisions, Camp. Phys. Comm. 32 45-62; (1987) ibid. 44 121-136.

[12] Billing, G.D. (1986) Comparison of quantum mechanical and semiclassical cross sections and rate constants for vibrational relaxation ofN2 and CO colliding with 4He, Chem. Phys. 10739-46.

GENERALIZED CROSS-SECTIONS FOR SENFTLEBEN­BEENAKKER EFFECTS AND LASER STUDIES OF MOLECULES

WING-KI LIU Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.

ABSTRACT. The Waldmann-Snider kinetic equation is used in the description of both laser studies and external field effects on transport properties of a system of linear molecules diluted in a bath of atoms. By using the irreducible tensor components of the distribution function and the Liouville-space formalism to exploit the rotational invariance of the relaxation mechanism, the collisional processes contributing to both types of experiments can be described in a unified manner in terms of generalized cross sections. Recent scattering calculations of these generalized cross sections for various systems are reviewed, and the need for further calculations of cross sections for laser studies is indicated.

1. Introduction

The determination of interatomic potentials from multi-property fits to experimental measurements of molecular beam scattering, spectroscopic data and bulk properties such as transport and virial coefficients can now produce very accurate results [1]. To carry out the same program for atom-molecule systems is more difficult [2], since the intermolecular potential depends on both the separation and the relative orientation of the atom and the molecule. Because of their weak dependence on the potential anisotropy, ordinary transport coefficients and virial coefficients can be used to deter­mine the isotropic part of the intermolecular potential similarly to the case of atom­atom interactions [3,4]. The anisotropic part of the intermolecular potential gives rise to molecular reorientation and inelastic rotational transitions, which produce colli­sional coupling between the polarizations in angular momentum and velocity spaces created in a nonequilibrium gas. The influence of external fields on the transport properties of molecular gases (the Senftleben-Beenakker effect, SBE), and relaxation phenomena such as nonclassical sound absorption, pressure broadening of microwave, infrared, rotational Raman and depolarized Rayleigh (DPR) spectra, and gas-phase nuclear magnetic resonance, depend on various angular momentum polarization con­tributions [5], and hence in principle, provide direct information on the anisotropic potentials. All these transport-relaxation experiments can be conveniently described in terms of various effective cross sections representing the collisional transfer among

217

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 217-236. ©1992 Kluwer Academic Publishers.

218

the microscopic velocity and angular momentum polarizations carried by the molec­ules.

It has long been recognized that the measurements of the polarization ratios of rotationally resolved collision-induced fluorescence (CIF) [6] can also provide detailed information about rotationally inelastic dynamical processes, which in turn depend on the anisotropy of the intermolecular potential. These experiments are usually described in terms of multipole cross sections [7,8]. The concept of multipole cross sections is also applicable to the analysis of collision-induced rotational alignment of molecular ions drifting in a buffer gas [9,10].

Recently, high-resolution double-resonance lineshapes have been obtained from two-photon sub-Doppler excitation experiments on molecular systems [11,12]. In the limit of low gas density, the inelastic lineshape is directly related to the detailed state­to-state rotational transitions induced by the two-body anisotropic intermolecular potential.

It can be shown that the SBE effective cross sections [13,14]' the multipole cross sections in CIF and drift-tube experiments [15]' and the two-photon inelastic line­shape [16] can be described in a unified manner in terms of generalized cross sections (GCS) [17], suitably averaged over the collision energy. Thus all these experimental properties can be used for systematic testing of anisotropic intermolecular interactions from the computations of these GCSs. The calculation of the GCSs for molecular sys­tems, however, is much more complicated than that of the collision integrals of the atom-atom case, and requires the solutions of the quantal close-coupling equations [18-22].

In this paper, we first discuss the kinetic equation which can be used to describe both the transport-relaxation phenomena and the sub-Doppler spectroscopic experi­ments. The technique of the Liouville-space formalism [23-25] is then applied to the analysis of all these experiments. It is shown that the SBE effective cross sections, the multipole cross sections and the inelastic lineshape can be expressed in terms of matrix elements of a collision super-operator in Liouville space. Hence these phenom­ena can be related in a unified manner to the generalized cross sections. Exact and approximate quantal as well as classical trajectory calculations of the GCS related to the SBE and laser experiments will be reviewed.

2.The Kinetic Equation and Generalized Cross Sections

In all the experiments mentioned in the introduction a non-equilibrium state is es­tablished in the molecular gas system and we use the term polarization to describe the deviation of a physical observable from its equilibrium value. Thus, in transport phenomena, velocity polarizations are set up by the flow due to the application of an external temperature, velocity or density gradient in the gas sample, and angular mo­mentum polarizations are maintained by inelastic and reorientation collisions in the presence of these thermodynamic forces. In sub-Doppler laser excitation experiments, velocity polarizations are created by the selection of a particular velocity component of the molecules using the Doppler effect, and angular momentum polarizations are

219

established because of the selection rule for the rotational states of the molecules in the dipole excitation process. The starting point for the theoretical description of non-equilibrium phenomena in a dilute molecular gas is the Waldmann-Snider equa­tion [26,27]. In this paper, we consider only the simple case (from the point of view of a scattering calculation) of the interaction between a linear molecule and a struc­tureless atom. Experimentally, this corresponds to a gas mixture of linear molecules dilute in a bath of atoms. The Waldmann-Snider equation has also been generalized to the case of a binary mixture [28].

2.1. CROSS SECTIONS IN LASER EXPERIMENTS

We consider first the radiative excitation of a mixture of optically active molec­ules dilute in a bath of inert atoms. Since physical observables are related to the spectroscopic properties of the molecules, the system can be completely described by the Wigner distribution function matrix [29] f(r, Pa , t), which is defined in terms of the single-particle density matrix of the molecules by

f(r, Pa, t) = (21Tn)-3 L eiq-r(ka + ~qlPlka - ~q) q

= (21Tn)-3 J dr'e-iPa-r'/Ii(r + ~r'lplr - ~r'), Pa = nka, (1)

in which r, Pa, ka are the position, momentum, and the translational wave vector of the molecule, respectively. The wave function for the translational states is given by (rlk) = V-l/2eik-r, where V is the volume of the system, so that (klk') = Okk" In the continuum limit, L:ka becomes (21Tn)-3V J dpa for V large; such summa­tion and integration will be used interchangeably. It follows from Eq. (1) that J dpaf(r, Pa, t) = p(r). The equation of motion of the Wigner distribution func­tion for a low-density gas is

af(r,Pa,t) _ ~[H H at - in rot + rad, f(r, Pa' t)] + Pa . aa f(r, Pa , t)

rna r

(2)

where rna, H rot are the mass and the rotational Hamiltonian of an isolated molecule, respectively, and H rad = -d . E with d, E being the dipole moment operator of the molecule and the electic field of the electromagnetic radiation, respectively. Here 'Y is the radiative decay rate and the last term in Eq. (2) represents the collisional rate of change of the distribution function.

Since the spatial variation of physical observables is slow compared to that of the interparticle interaction, the Wigner distribution function possesses only weak spatial inhomogeneities, for which case the collisional rate of change of the Wigner

220

distribution function matrix elements is given explicitly in terms of the transition matrix elements by [27,30)

afF'dr,p~,t)1 -C(f.f) - ,Jb F'['

at coli

(27rli)3 ~ ~ {( , , I ) ( , , == ~ L...J L...J bKK' k F It kF fFI r, Pa , t)!b(r, Pb, t)(kIlk I ) F [ kakbk~

- (k' F'lkF)!FI(r, Pa , t)!b(r, Pb' t)(kIltt Ik' I')

+ 27ri(k' F'ltlkF)!FI(r, Pa , t)fb(r, Pb' t)(kIlttlk' I')b(E - E')} (3)

Here the pre-collision rotational states of the molecule are denoted by I I), I F} and the post-collision states by II'), IF'}, respectively, !b(r, Pb, t) is the di~tribution function of the bath atom at r with linear momentum Pb = likb' and kb, k~ are the pre­and post-collision translational wave vectors of the bath atom. Since the bath atoms do not interact with the electromagnetic radiation the distribution function for the translational states of the bath atoms, !b will be assumed to be Maxwellian:

(4)

where nb, mb are the density and the mass of the bath atoms, (3 == (l/kT), with k the Boltzmann constant and T the temperature. The function !b satisfies the normalization J dpb!b = nb. In Eq. (3) K and k are the centre-of-mass and relative momenta, defined by

(5)

respectively, with mr = mamb/(ma+mb) being the reduced mass of the atom-molecule colliding pair. We make use of the fact that the t-matrix elements depend only on the pre- and post-collision relative momenta and molecular rotational states, and are independent of the centre-of-mass momentum K. The impact approximation [24) is also invoked, where we consider only completed collisions (which give rise to the delta function in Eq. (3), implying the conservation of total energy E' = E in a collision), so that only states satisfying EF - E[ = EF' - El' are included in the sum, and only on­the-energy-shell t-matrix elements are involved. Since the density of the molecules is assumed to be much smaller than that of the bath atoms, molecule-molecule collisions have been neglected in Eq. (3).

We shall consider the case of linear molecules whose quantum states can be rep­resented by lajm} with energy E(a,j), where a is a collective index for the electronic and vibrational quantum numbers, and j, m are the quantum numbers for the rota­tional angular momentum and its projection on a space-fixed z-axis. Both the laser excitation and subsequent fluorescence processes are governed by the dipole-moment

221

operator, d of the molecule. For example, the fluorescence intensity of spontaneous emission from an excited state laU:m~) to the final states lajjfmj) is proportional to

mj

where eF is the polarization unit vector of the fluorescence radiation. If there is a distribution of excited states given by the Wigner distribution function matrix f(r, Pa , t), the fluorescence intensity is then proportional to

IF = J dr J dpa L (ajjfmjleF· dla~j:mD m~,m~'~mJ

(6)

The dipole-moment operator is a tensor operator of rank 1, and hence the matrix element of its spherical components dq (q = 0, ±1) can be written as

(7)

using the Wigner-Eckart theorem [31J, where dji is the reduced dipole-matrix element, which depends only on (aj,jj, ai,ji). This suggests the following definition of the tensor component of the distribution function [25J

/

f CKj)Qj(r P t) = ""' (-I)jj-m f (2K- + 1)1/2 JI,Ji 'a' L..J J

(8)

The dependence of f on a j and ai has not been indicated explicitly. Note that this definition applies to the diagonal matrix elements with (a j, it) = (ai, ji) as well as to the off-diagonal elements. Then the fluorescence intensity of Eq. (6) becomes [15J

(9)

where D~~j) can be written as the product of a geometrical factor and a spectroscopic factor as

0' ~} Idi1j l 12

Ji 1

(10)

in which

i:I>~~j)(eF) = L(-I)Kj +l-q(2Kj + 1?/2 (~ qql

1 K j ) * -q' _Qj eqeql, (11)

222

and eq with q = 0, ±1 are the spherical components of the polarization vector eF.

For the case of resonant excitation of the molecules from the initial levels ladimi) with energy E(ai,ji) to the excited levels lafiJmf) with energy E(af,jf) by a laser with wave vector kL and frequency WL ~ Wfi, where Wfi = (E(af,jf) - E(ai,ji))/fi,

the distribution of the excited states fJ(KJj )Qj can be obtained by considering the tensor f, f

components of the distribution function involving only the states (ai, ji) and (af,jf). Under steady-state conditions and making the rotating-wave approximation [32], the time and spatial dependence of the distribution function matrix elements can be separated out as

(12)

For a weak electric field, the initial state distribution can be represented by its equi­librium value:

(13)

where fao(Pa) is given by an equation similar to Eq. (4) except nb, mb, Pb are replaced by n a, ma, and Pa, respectively, where na (na < < nb) is the density of the molecules. For optical transitions in dilute molecular gas, the collisional rate is much smaller than the radiative decay rate and can be neglected. Under these conditions, Eq. (2) can be solved to give [16]

where E L , eL are the electric field strength and polarization vector of the laser re­spectively, Vaz is the component of the molecule velocity along the z-axis, chosen in the direction of kL, £(kLVaz) is the Lorentzian

(15)

and D~~j)(eL;ji,jf) is defined similarly to Eqs. (10) and (11). Equation (14) shows clearly that excitation by a narrow-linewidth laser produces both angular momentum polarization (non-zero j(Kj)Qj with K j I- 0) and velocity polarization (because of the factor £(kLvaz )) in the excited-state manifold of the molecule.

In CIF experiments, excited-state tensor distribution components f;:j)Qj in the rotational level (a, j) are created by sub-Doppler laser excitation. Collisions with bath

atoms then transfer these to other tensor distribution components f;~;)Qj for differ­ent rotational levels (a, jf) in the same electronic state. Finally, fluorescence to a lower electronic state is detected. Thus the pre-collisional state possesses angular momen­tum and velocity polarizations as given by Eq. (14), while the post-collisional states containing only angular momentum polarization are selected according to Eq. (9).

223

In two-photon excitation experiments, after initial excitation to the level (a, j) and collisional transfer to the level (a, j'), a second laser further excites the molecules res­onantly from (a, j') to a new level (a f' j f) and fluorescence from (a f' it) is detected. In this case, both the pre- and post-collision states possess angular momentum and velocity polarizations created by the two lasers.

It is convenient to introduce the Liouville-space formalism [23-25], in which an operator A in ordinary Hilbert space of the state vectors is considered to be a Liouville vector (L vector) IA)), and the inner product of two L vectors IA)) and IB)) is defined by

((AlB)) = TrAt B.

A complete set of Liouville-basis (L-basis) vectors can be constructed from a Hilbert­space basis set {Ii), U),"'} as {Iij+))}, in which individual basis elements are given by lij+)) == li)(jl. Liouville operators (L operators, or superoperators) can often be expressed as bilinear products of Hilbert-space operators of the form A lSi B*, with matrix elements given by ((ik+IAISiB*ljl+)) = AijBkl . An irreducible L-basis vector for the rotational states can be defined by [23]

(16)

so that the distribution function can be expanded in this basis as

If(r, Pa , t))) = L L L f;~~)Qj (r, Pa , t)lafit(adi)+; KjQj)). (17) KjQj afai ltji

From the above consideration:o and Eq. (3), it can be shown [15,16] that the collisional rates obtained in elF and two-photon experiments can be expressed in terms of the matrix elements of a collision super-operator A (or collision integral) as

((B(k~J; F' I'+IAIA(kaz ); FI+)) =

(21f1i)3 , -~ L L 8KK'fo(Pa)JbO(Pb)B(kaz)A(kaz) kak~ kbk~

x ((k' F', (k' 1')+ I[i lSi t - t* lSi i + 21fit lSi t* 8(E - E')]lkF, (kI)+)),

(18)

where A and B are functions of the post- and pre-collision wave vector components k~z and kaz' such as the Lorentzian of Eq. (15), and i is the identity operator. (The A here corresponds to the iA of Ben-Reuven [23,24].) Now Pa, Pb, kaz' and k~z can be

224

expressed in terms of the relative and centre-of-mass momenta by means of Eq. (5). The plane-wave states Ik} are then expanded in a spherical-wave basis according to

(19)

where l, ml are the quantum numbers for the orbital angular momentum of the rela­tive motion of the atom-molecule pair and its projection along the space-fixed z-axis, respectively, and Ek = fi,2 k2 /2mr is the kinetic energy for relative mot~on. An irre­ducible L-basis vector for the orbital states Ilf(li)+;KI,QI}} can be defined similarly to Eq. (16)

Ilfl;;KIQI}} = L (_1)lrm!f(2KI + 1)~ ( if mlf -ml,

mlJffi l i

Kl ) _QI Ilfmlf} {liml,l;

(20) this basis vector can be coupled with the L-basis vector of Eq. (16) to form the spectroscopically coupled L-basis vector l(lIjt)Kj,(lflt)KI;KQ}} according to the vector coupling scheme

jf - ji = Kj, If -Ii = K" K j + Kl = K. (21)

To take advantage of the rotational invariance of the scattering process, a collisionally coupled L-basis vector l(lIlf)Jf[(jili)Ji]+;KQ)} corresponding to the total angular momentum (J -) coupling scheme

jf+If=Jf, ji+Ii=Ji, Jf-Ji=K (22)

can be constructed from the spectroscopically coupled L-basis vector by employing the Wigner 9-j symbol [16,23]. In this basis, A is diagonal in Jf , h K, Q, and is inde­pendent of Q. Identifying the states IF' 1'+)) and IF J+)) explicitly as 1//+, KjQ)} and IJj+, KjQ}), respectively, the collision integral of Eq. (18) can finally be expressed as [16]

((B(k~z);j'j'+,KjQIAIA(kaz);jj+,KjQ)) = nbcr L J dxxe-xcK;K!(k) K;K!

x ~(_1)K;+K!-K;-Kj(2K + 1) (~ ~l ~) (~ ~; K

(23) where x = j3Ek , cr = (8/7rmr j3)1/2 is the average relative velocity, and the generalized cross section appearing on the RHS of Eq. (23) is defined by [17]

K;Kj(K) ( .,.) _!!... """ .If-l,-lf+i;[l' l'l l·]1/2[J J.] (1K!Kj J J - k2 ~ ~ ~ t f t f t f t

lf l; If I, Jfh

( if' Kl' li') (if Kl li) {!; ~f x 000000 J t

K' K' J I

~~ i } Kl K

(24)

225

in which Sfl',jl is the scattering matrix element [33], related to the on-the-energy­

shell transition matrix element by Sfl',jl = bj'l',jl - 27ritj'I',jl' and [ab· .. ] denotes the product (2a + 1)(2b + 1) .... The coefficient CK;K,(k) in Eq. (23) is the expan­sion coefficient of the double Legendre series of the following integral of the product B( k~z )A( kaz ):

( 2 1/2 00

~~) 100 dKze-/31l2 K;/2M B(k~z)A(kaz)

= L CK;K,(k)PK;(cos(JDPK,(cos(h), (25) K;K,

where M = ma + mb, (Jk, B~ are the angles between the z-axis and the pre- and post-collision relative translational wave vectors k and k', respectively, so that kaz = (mr/mb)Kz + k cos Bk and k~z = (mr/mb)Kz + k'cosB~. The coefficient CK;K,(k) implicitly contains the dependence on the laser frequencies.

In the theory of the inelastic lineshape for two-photon experiments, the functions A and B correspond to the Lorentzian functions of Eq. (15) for the first and second laser, respectively, which give rise to nonequilibrium polarizations in the orbital an­gular momentum space both before and after collision. In CIF experiments, there is no establishment of orbital angular momentum polarization after collision in the fluo-

OK'.(K') rescence process so that B = 1, and hence only the (JK,Kj J cross sections contribute to the depolarization ratios [15].

2.2. SBE CROSS SECTIONS

For transport phenomena in a binary mixture of atoms and molecules, the distribution of the atoms can no longer be assumed to be Maxwellian as in Eq. (4) for the case of laser experiments, and the distribution function of the molecules fa(r, Pa' t) cannot be obtained directly. (In this subsection, we denote the f(r, P a , t) of Section 2.1 by fa(r, Pa' t) for clarity). In this case, we take advantage of the fact that the deviation from the equilibrium state is small, and we use either the Chapman-Enskog or the moment method to find approximate solutions of the Waldmann-Snider equation [5]. Thus, in the moment method, the distribution functions ik, (k = a, b) will be linearized about the corresponding Maxwell-Boltzmann distributions of Eq. (4):

(26)

The perturbation <Pk satisfies the linearized Waldmann-Snider equation

(27)

where iL\<Pk = iWext[jkz, <Pk] describes the precessional motion of the molecules of the kth species in the presence of an external homogeneous field (which gives rise to

226

the SBE), Ck = Pk/mk is the molecular velocity of the kth species, and the linearized collision operator is given by

- iko'Rkk( q,k) = C(iko, ikOq,k) + L C(ikOq,k, ik,o), (28a) k'#k

(28b)

where the collision operator C is defined in Eq. (3). The q,k are then expanded in a set of irreducible tensors, and the coefficients of the expansion series can be shown to be proportional to the nonequilibrium averages of the corresponding tensors (or moments of the distribution). Thus it is only necessary to solve the set of linear equations for the moments (the transport-relaxation equations, TRE) obtained from substituting the expansion of q,k into Eq. (27).

For example, in the description of the viscosity SBE for a binary mixture of atoms and linear molecules, it suffices to use the expansion [34,35]

q,k = '"'(_I)q{a(l)c(l) + a(2)c(2) + b(2)J.(2)8 } L...... k,q k,-q k,q k,-q q -q ak , (29) q

where u~K) is the qth spherical component of the Kth rank tensor operator con­structed from the vector operator U [31,35]' and only the molecules (species a) possess

rotational angular momentum j. The expansion coefficients a~K) are related to the ,q

spherical components of the mean flow velocity Uk, and the viscous pressure tensor P k of species k by

U(I) __ 1_a(l) p(2) _ 2nk a(2) k,q - (3mk k,q' k,q - (32mk k,q' (30)

It is clear that when Eq. (29) is substituted in Eq. (27) to form the TRE, collision

integrals such as J dpa Tra( _1)qj~2) fao'RabCi~~q and J dpa Tra( -1)q j~2) fao'RaajC:~ will appear, where Tra denotes a trace over the rotational states of the molecule. Since the t-matrix elements depend only on the relative velocity of the colliding pair, tensor operators involving molecular velocities, such as Ck(2) , are first written in terms of ,q relative and centre-of-mass velocities, after which the integration over the centre-of-mass momentum in the collision integrals can be evaluated analytically.

In general, SBE collision integrals involving the scattering of a linear molecule and a bath atom can be written as

using the definitions (28) and Eqs. (3) and (18). The irreducible tensor opera­

tors are defined by q, = [g(Kdj(Kj)JCK)g2n(Hrot/B)t, where g = ((3h2/2mr )I/2 k, is the reduced relative velocity, B is the rotational const-ant of the molecule, and [g(Kz)j(Kj)](K) is the irreducible tensor ofrank K formed from the irreducible tensors

227

g(K,) and j(Kj ) corresponding to the vector-coupling scheme Kl + K j = K. The spherical tensor components of W can be expressed in terms of the L-basis vectors corresponding to the total-J coupling scheme of Eq. (22) as

(31)

where

with E j being the rotational energy of level j and the reduced matrix elements are given by [36J

and

The collision integral can then be expressed as a weighted sum over the rotational states of the generalized cross sections, Eq. (24) [36J:

f f {J (')nf+KU2 f f - } ((w'IAlw)) = nbC~d;~jt) dxxde- X : (]"~;~:(K)(jlj) pjd;Kjt) ,

(32) where d = n+n' + ~(Kl +K[) + 1; x, x' are the pre- and post-collision reduced relative kinetic energies given by x = gZ, x' = g'Z, respectively;

C - - Kl·K1· {I'I } l/Z

= cr (2K + 1) [K{KjK1Kj](2Kl _ 1)!!(2K{ _ I)!!

and Pj = Q-1 exp( - f3Ej ), in which Q is the rotational partition function of the molecule.

From Eqs. (23) and (31), we observe that the collision integrals for laser experi­ments and SBE measurements require the computation of different thermal averages of various generalized cross sections, Eq. (24). The SBE collision integrals, however,

228

involve additional sums over the rotational states, and thus provide more averaged in­formation about the collision process. The collision integrals for laser studies, Eq. (23) on the other hand, not only provide state-to-state scattering information, but because of the possibility to select the pre- and post-collision molecular velocity components by tuning the frequencies of the pump and probe lasers across the Doppler width (as reflected in the dependence of the coefficients CK;KI on the laser frequencies), they are expected to yield more details about the intermolecular potential.

3. Scattering Calculations

In order to calculate the generalized cross sections (GCS) of Eq. (24) quantum me­chanically, it is necessary to solve the close-coupling (CC) equations [33] for the atom linear-molecule collision system for the scattering matrix elements. Such CC calcu­lations of the effective cross sections for sound absorption and NMR Tl relaxation processes (which are special cases of the GCS), as well as rotational Raman linewidth cross sections, were performed by Shafer and Gordon [37] almost twenty years ago for the system H2-He using a realistic potential. The first CC calculation of SBE viscosity cross sections was reported in Ref. [18] for the system H2-He employing two ab initio potentials, and it was found that the SBE cross sections are sufficiently sensitive to the intermolecular interaction to distinguish between them.

GCSs with (Kj,K{) i- (Kj,KI ) are called production cross sections, those with

K j = Kj = 0 (and hence K; = Kl = K) are called transport cross sections, and those with (Kj, K{) = (Kj , Kd, K j i- 0 are called relaxation cross sections. When

there is a unique choice of K for given values of (K;, Kj, K l , K j ), the GCS will simply

be denoted by u~!~:. For the shear viscosity SBE, the production cross section

u~6 determines the saturation value, the relaxation cross section ug~ is related to the value of the half-width, and the transport cross sections u}g and u~g determine the isotropic shear viscosity coefficient. The behaviour of these cross sections as functions of collision energy E, and for fixed values of E, the variation with the total angular momentum J of the opacity functions associated with these cross sections have been examined in Ref. [18] and extended to higher energy recently in Ref. [38]. The opacity functions for the relaxation cross section and the transport cross sections are found to be smooth functions of J, and retain the same sign for all values of J, with their magnitude rising to a maximum and then decreasing as J increases. Further, the values for the off-diagonal cross sections (j' i- j) are much smaller than the diagonal ones. The relaxation cross sections, which vanish in the absence of an anisotropic potential, are about two orders of magnitude smaller than the transport cross sections, which have non-zero limits even in the absence of anisotropy. The opacity functions for the off-diagonal production cross sections as a function of J behave in a similar manner, although their magnitudes are much smaller. The opacity functions for the diagonal production cross sections, on the other hand, are oscillatory functions of J, changing sign as J is varied, and hence there are cancellations when summed over J to give the production cross sections at the given energy. Hence, in

229

contrast to the relaxation and transport cross sections, the diagonal (j' = j) and off­diagonal (j' oj:. j) production cross sections are of the same order of magnitude. The temperature dependence of the thermal averages of these cross sections (the effective cross sections) has also been studied. It was found that between 77K and 400K the transport effective cross sections for both para-H2-He and ortho-H2-He are decreasing functions of temperature, the production and relaxation effective cross sections for ortho-H2-He are increasing functions of temperature while those for para-H2-He first increase with temperature up to around room temperature and then decrease as temperature is further raised. Unfortunately, no experimental results are available for comparison.

Similar calculations of the viscosity SBE cross sections have been performed for the system HD-He, using the Shafer-Gordon H2-He potential by the coordinate trans­formation necessitated by the shift of the centre-of-mass of the molecule in isotopic substitution [19]. Excellent agreement with experimental results is obtained for all cross sections at 77.3K and 293K, with the exception of the relaxation cross sec­tion at 293K. Subsequent calculations [39] of the relaxation cross section by including more rotational levels bring the result for the relaxation cross section closer to the experimental value. Furthermore, it was determined in Ref. [19] from a study of the temperature dependence of the relaxation and production cross sections that the un­normalized rotational angular momentum polarization 'If is more appropriate for the description of viscosity SBE than both the pure angular momentum tensor polariza­tion 'If/U2(j2 - 3/4)]1/2 and the DPR tensor polarization 'If/(j2 - 3/4). A compre­hensive CC study of the effective cross sections governing the SBE on viscosity, heat conductivity, diffusion and thermal diffusion, flow birefringence and the depolarized Rayleigh (DPR) linewidths has been carried out for the system HD-He in Ref. [20], in which the HD-He potential is constructed from the ab initio H2-He potential of Meyer, Hariharan and Kutzelnigg [40]. The effective cross sections are formulated in terms of scattering amplitudes describing the j, mj, k ----+ j', mj, k' collisional transi­tions instead of the S-matrix elements in the total-J representation used in Eq. (24), and so comparision of intermediate results between the present formulation and that of Ref. [20J is difficult. Of course, for a given potential, the final results for the effective cross sections using either of these formulations must be identical. The calculations of Ref. [20J are in excellent agreement with experiment, and again it is concluded that the viscosity SBE is governed by the unnormalized angular momentum polarization 'If. The same potential has also been utilized for the CC calculation of cross sections for rotational Raman lineshapes and rotational and NMR relaxation [22]. Again excel­lent agreement with experiment is obtained, and it was concluded that this potential is more accurate than is the semiempirical potential of Shafer and Gordon.

Recent CC calculations of SBE-type cross sections include those of Hutson and McCourt [21] for the system H2-Ar, in which it is found that orbiting resonances contribute significantly to relaxation and linewidth cross sections at low temperature «WOK), the extensive computation by Maitland et al. [41] for the system N2-He, and the study of energy and temperature dependence of cross sections for SBE in viscosity, thermal diffusion and thermal conductivity for the H2-He system by Thachuk and

230

McCourt [38]. In many recent calculations, the MOLSCAT package of Hutson and Green [42] has been used.

Fully converged quantal CC calculations are in principle exact. However, they are extremely time-consuming, and except for hydrogen molecules and their isotopes, cannot be carried out routinely. The computational effort can be reduced consider­ably by introducing decoupling approximations; among these, the centrifugal sudden approximation (CSA) [43], and the infinite-order-sudden approximation (IOSA) [43] have been applied successfully to the calculation of differential and integral cross sec­tions. In the CSA the orbital angular momentum operator is effectively replaced by a constant parameter, and different choices of this parameter lead to the following labelling schemes: the L-initial (LA), the L-final (LF), and the L-average (LA) la­belling. The IOSA involves both the CSA and the energy-sudden approximation. Formal CSA and IOSA expressions for the production, relaxation and transport cross sections for the viscosity SBE have been developed [13]' and in fact the first calcula­tion of SBE viscosity cross sections was carried out using the rOSA for the system N2-He [44]. However, the LA, LF, and 11 schemes produce very different results for the production cross section, indicating that the IOSA is inadequate for this type of cross section. In Ref. [18], comparison between the CC and CSA calculations of vis­cosity SBE cross sections for the H2-He system is made. Only the LA scheme is used in the CSA, since it has been demonstrated to give accurate results for jm -+ j'm' cross sections [45]. The CSA is found to be very accurate for transport cross sections, fairly accurate for the relaxation cross section, and again rather poor for production cross sections.

It appears that the collisional transfer between orbital angular momentum and rotational angular momentum polarizations, which contributes to the production cross section, cannot be handled adequately by the CSA. Recently, an improvement of the CSA was introduced by McLenithan and Secrest [46], which gives much more accurate results than the CSA in the differential cross sections for the HD-Ne system. This new scheme, called the corrected centrifugal sudden approximation (CCSA) in Ref. [38], has been applied to the calculations of cross sections for SBE in viscosity, thermal diffusion and conductivity for the system H2-He [38], and comparison is made with CSA and CC calculations. The CCSA is found to have significant improvement in accuracy over the CSA for transport and relaxation cross sections when the LA scheme is employed. However, even the CCSA(LA) calculations do not yield sufficient accuracy for the production cross sections.

The CSA, CCSA, and IOSA are dynamical approximations. Classical trajectory calculations [47-51] treat the dynamics exactly, and it has been shown recently that the classical kinetic cross sections can be obtained from the quantum mechanical GCSs in the limit n -+ 0 [52]. A stringent test of the classical trajectory calculations of the cross sections for SBE in viscosity [39] and thermal conductivity [36] has been carried out for the very quantal-like HD-He system, and the results are compared with the corresponding CC results. Very good agreement is found for all cross sections at high collision energies, although the agreement deteriorates at lower energies as expected. A hybrid scheme which interpolates between classical results at high energy

231

and quantal results at low energy proves to yield accurate results for temperature dependent cross sections. For heavier molecular systems, the classical trajectory method is expected to be an accurate and practical alternative for the study of the SBE cross sections.

Very few calculations exist for cross sections related to laser experiments. Some CC calculations of the multipole cross sections (MCS) measured in CIF experiments have been performed for the system N2-He [53J. The multipole cross section [6,8J

O"j:2f, --->K' has been shown to be related to the GCS of Eq. (24) by [15J

(n)K--->K' = (_I)n-K+l[KK'J1/2 (K ~~, 0

n

o OK(K)(.,.) K ') " o O"nK J J . (33)

Results of the O"j:2f,--->K' for n = 0,2 have been presented in Ref. [53J over a limited range of energy. It is clear from Eq. (33) that the MCS for n = 0 corresponds to the relaxation cross section. The MCS for n = 2, on the other hand, belongs to the category of production cross sections, and hence it is not surprising to find that it can be positive or negative, similar to the case of SBE production cross sections. However, it is not clear how the results for the N2-He system can be used to infer the behaviour of the Li2(A)-Xe system studied in the experiments. The semiclassical coupled-state sudden approximation has been applied to calculate the MCS for Lb(A)-Ne using an ab initio potential model for the molecule in the excited electronic state [54J. While the CSA is expected to be reasonably accurate for the MCS with n = 0, its applicability to the MCS with n # 0 is doubtful since it fails to provide accurate results for SBE production cross sections. Recently, a detailed CC calculation of the MCS for collision-induced rotational alignment in Nt -He has been reported [IOJ. We are aware of no calculation of the GCSs for two-photon sub-Doppler excitation experiments.

4. Summary and Discussion

In this paper, we have discussed in detail the kinetic equation used in the description of both collision-induced fluorescence and two-photon sub-Doppler double-resonance experiments for a system of dilute linear molecules in a bath of inert atoms. By using the irreducible tensorial components of the distribution function, the m-degeneracy of the rotational levels and their coupling with the electromagnetic radiation can be described in a compact manner. The introduction of the Liouville-space formalism simplifies the description of the collisional contributions by taking advantage of the rotational invariance of the scattering process, leading to a systematic derivation of various collision integrals in terms of the generalized cross sections of Eq. (24).

The description of the Senft leben-Beenakker effects in a binary mixture of atoms and linear molecules employing the linearized Waldmann-Snider equation and its solution using the moment method have been briefly described. Since the collision operator is the same as those used in the theory of laser studies of collision rates, it is not surprising that the SBE collision integrals also involve the GCS of Eq. (24),

232

and thus the relaxation mechanisms of these two different types of experiments can be described in a unified fashion.

Recent exact and approximate quantum mechanical calculations of the cross sec­tions for the SBE and tor collision-induced fluorescence have been reviewed. While there exist a number of studies on SBE cross sections for various atom-linear molecule systems, very few calculations of the CIF multipole cross sections have been reported, and no calculation of the GCS for a double resonance inelastic lineshape has been per­formed. Since the SBE cross sections involve summing over final rotational states and averaging over initial rotational states, while the CIF and double resonance laser exci­tation experiments probe state-to-state collisional transitions with initial and/or final velocity component selection, these latter experiments should provide more informa­tion about the scattering processes, and hence about the intermolecular potentials. Furthermore, while ordinary transport-relaxation experiments measure properties of the ground electronic potential energy surface, CIF and two-photon inelastic lineshape studies probe the electronically excited-state potential energy surface, and hence there is great interest in the calculation of the associated cross sections.

We are thus at a stage where we can test ground-state and excited-state inter­molecular potentials by performing ab initio scattering calculations of the GCS for laser studies, SBE and other transport-relaxation phenomena, and comparison can be made with experiments. For systems where a quantal close-coupling calculation is feasible, the GCSs can be computed by employing the MOLSCAT package [42]. For heavier systems with higher rotational level density, classical trajectory calculations would be a practical alternative.

Acknowledgement

This work is supported by a NSERC of Canada grant in aid of research.

References

[1] Aziz, R. A. (1984) Interatomic potentials for rare gases: pure and mixed inter­actions, in M. Klein (ed.) 'Inert gases: potentials, dynamics and energy transfer in doped crystals', Springer, Berlin, pp. 5-86.

[2] McCourt, F. R., Fuchs, R. R. and Thakkar, A. J. (1984) A comparison of the predictions of various model N2-He potential energy surfaces with experiment, J. Chem. Phys. 80,5561-5567.

[3] Monchick, L. and Mason, E. A. (1961) Heat conductivity of polyatomic and polar gases, J. Chem. Phys. 36, 1622-1639.

[4] Maitland, G. C., Rigby, M., Smith, E. B. and Wakeham, W. A. (1981) Intermolec­ular forces: their origin and determination, Oxford University Press, Oxford.

[5] McCourt, F. R. W., Beenakker, J. J. M., Kohler, W. E. and KU8cer, 1. (1990, 1991) Nonequilibrium phenomena in polyatomic gases, Vols. I, II, Clarendon Press, Oxford.

233

[6] McCaffery, A. J., Proctor, M. J. and Whitaker, B. J. (1986) Rotational energy transfer: polarization and scaling, Annu. Rev. Phys. Chern. 37, 223-244, and references therein.

[7] Fell, C. P., McCaffery, A. J., Reid, K. 1., Ticktin, A. and Whitaker, B. J. (1988) Velocity dependence of rotationally inelastic cross-sections, Laser Chern. 9, 219-240.

[8] Monchick, L. (1981) Generalized reorientation cross section for cylindrically sym­metric velocity distributions, J. Chern. Phys. 75, 3377-3383.

[9] Meyer, H. and Leone, S. R (1988) Steady-state model for the collision induced rotational alignment of molecular ions in electric drift fields, Molec. Phys. 63, 705-717.

[10] Follrneg, B., Rosrnus, P. and Werner, H.-J. (1990) Theoretical investigation of collision-induced rotational alignment in Nt-He, J. Chern. Phys. 93, 4687-4698.

[11] McCaffery, A. J., Reid, K. L. and Whitaker, B.J. (1988) Velocity-selective double resonance: a novel technique for determining differential scattering cross sections, Phys. Rev. Lett. 61, 2085-2087.

[12] Collins, T. L. D., McCaffery, A. J. and Wynn, M. J. (1991) Two-color sub­Doppler circular dichroism: a four-vector correlation molecular dynamics exper­iment, Phys. Rev. Lett. 66, 137-140.

[13] Liu, W.-K., McCourt, F. R, Fitz, D. E. and Kouri, D. J. (1979) Production and relaxation cross sections for the shear viscosity Senftleben-Beenakker effect. 1. Formal expressions and the coupled-states and infinite-order-sudden approxima­tions for atom-diatom systems, J. Chern. Phys. 71, 415-425.

[14] McCourt, F. R W. and Liu, W.-K. (1987) Effective collision cross-sections for the thermal conductivity of a polyatomic-monatomic binary gas mixture, J. Chern. Soc. Faraday Trans. 2 83, 387-401.

[15] Liu, W.-K. and Dickinson, A. S. (1990) Multipole cross-sections in polarized laser fluorescence for atom-diatom systems, Molec. Phys. 71, 1117-1129.

[16] Liu, W.-K. and Dickinson, A. S. (1991) Theory of the inelastic lineshape for two-step sub-Doppler excitation, J. Phys. B 24, 1259-1271.

[17] Fitz, D. E., Kouri, D. J., Evans, D. and Hoffman, D. K. (1981) On CC, CS and 105 generalized phenomenological cross sections for atom-diatom mixtures, J. Chern. Phys. 74, 5022-5030.

[18J Liu, W.-K., McCourt, F. R, Fitz, D. E. and Kouri, D. J. (1982) Close-coupled and coupled-states calculation of shear viscosity SBE cross sections for the HrHe system, J. Chern. Phys. 76, 5112-5127.

[19J Green, S., Liu, W.-K.and McCourt, F. R (1983) Close-coupled calculations of viscosity transport-relaxation cross sections for HD-He: comparison with exper­iment, Physica 117 A, 616-630.

234

[20] Kohler, W. E. and Schaefer, J. (1983) Ab initio calculation of transport-orienta­tion phenomena (Senftleben-Beenakker effects) for HD infinitely dilute in helium, Physica 120A, 185-212.

[21] Hutson, J. M. and McCourt, F. R. (1984) Close-coupling calculations of transport and relaxation cross sections for H2 in Ar, J. Chern. Phys. 80, 1135-1149.

[22] Schaefer, J. and Kohler, W. E. (1985) Quantum calculations of rotational and NMR relaxation, depolarized Rayleigh and rotational Raman line shapes for H2(HD)-He mixtures, Physica 129A, 459-507.

[23] Ben-Reuven, A. (1966) Symmetry considerations in pressure-broadening theory, Phys. Rev. 141, 34-40.

[24] Ben-Reuven, A. (1966) Impact broadening of microwave spectra, Phys. Rev. 145, 7-22.

[25] Omont, A. (1977) Irreducible components of the density matrix. Application to optical pumping, Prog. Quantum Electronics 5, 69-138.

[26] Waldmann, L. (1968) Kinetic theory of dilute gases with internal molecular de­grees of freedom, in E. G. D. Cohen (ed.) 'Fundamental Problems in Statistical Mechanics II', North-Holland, 276-305, and references therein.

[27] Snider, R. F. (1960) Quantum-mechanical modified Boltzmann equation for de­generate internal states, J. Chern. Phys. 32, 1051-1060.

[28] Raum, H. H. and Kohler, W. E. (1970) Kinetic theory for mixtures of dilute gases of linear rotating molecules in an external magnetic field (formal theory), Z. Naturforsch. 25a, 1178-1190.

[29] Imre, K, Ozizmir, E., Rosenbaum, M. and Zweifel, P. F. (1967) Wigner method in quantum statistical mechanics, J. Math. Phys. 8, 1097-1108, and references therein.

[30] Tip, A. (1971) Transport equations for dilute gases with internal degrees of free­dom, Physica 52, 493-522.

[31] Zare, R. N. (1988) Angular Momentum, Wiley, New York.

[32] Sargent, M., Scully, M. O. and Lamb, W. E. (1974) Laser Physics, Addison­Wesley, New York.

[33] Arthurs, A. M. and Dalgarno, A. (1960) The theory of scattering by a rigid rotator, Proc. Roy. Soc. Lond. A 256, 540-551.

[34] Hess, S. and Waldmann, L. (1971) Kinetic theory for a dilute gas of particles with 'spin'III. The influence of collinear static and oscillating magnetic fields on the viscosity, Z. Naturforsch. 26a, 1057-1071.

[35] Liu, W.-K and McCourt, F. R. (1979) A multilevel description of the magne­toviscosity and flow birefringence in polyatomic gases, Chern. Phys. Lett. 62, 489-494.

[36] Liu, W.-K, Dickinson, A. S. and McCourt, F. R. W. (1990) Comparison of quan­tum and classical calculations of thermal-conductivity cross-sections governing

235

the Senftleben-Beenakker effects for HD-He mixtures, Molec. Phys. 71, 1131-114l.

[37] Shafer, R. and Gordon, R. G. (1973) Quantum scattering theory of rotational relaxation and spectral line shapes in H2-He gas mixtures, J. Chern. Phys. 58, 5422-5443.

[38] Thachuk, M. and McCourt, F. R. W. (1990) Use of the corrected centrifugal sudden approximation for the calculation of effective cross sections. I. The H2-

He system, J. Chern. Phys. 93,3931-3949.

[39] Liu, W.-K., McCourt, F. R. W. and Dickinson, A. S. (1989) Quantum and clas­sical mechanical calculations of transport-relaxation cross sections fo the HD-He interaction, Molec. Phys. 66, 565-576.

[40] Meyer, W., Hariharan, P. C. and Kutzelnigg, W. (1980) Refined ab initio calcula­tion of the potential energy surface of the He-H2 interaction with special emphasis to the region of the Van der Waals minimum, J. Chern. Phys. 73, 1880-1897.

[41] Maitland, G. C., Mustafa, M., Wakeham, W. A. and McCourt, F. R. W. (1987) An essentially exact evaluation of transport cross-sections for a model of the helium-nitrogen interaction, Molec. Phys. 61, 359-387.

[42] Hutson, J. M. and Green, S. (1986) MOLSCAT Computer Code distributed by Collaborative Computational Project No.6 of the Science and Engineering Re­search Council, UK, version 8.

[43] Kouri, D. J. (1979) Rotational excitation II: approximation methods in R. B. Bernstein (ed.) 'Atom-molecule collision theory, a guide for the experimentalist' , Plenum, New York, 301-358, and references therein.

[44] Liu, W.-K., McCourt, F. R., Fitz, D. E. and Kouri, D. J. (1981) Production and relaxation cross sections for the shear viscosity Senftleben-Beenakker effect. II. IOSA results for the N2-He system, J. Chern. Phys. 75, 1496-1508.

[45] Fitz, D. E. (1978) On the choice of partial wave parameter for lOS calculations of m-dependent rotationally inelastic cross sections, Chern. Phys. Lett. 55, 202-205.

[46] McLenithan, K. and Secrest, D. (1984) A recoupled states approximation for molecular scattering theory, J. Chern. Phys. 80, 2480-2503.

[47] Dickinson, A. S. and Lee, M. S. (1985) Classical trajectory calculations of trans­port properties for a model Ar-N2 potential surface, J. Phys. B 18,3987-4000.

[48] Dickinson, A. S. and Lee, M. S. (1985) Classical trajectory calculations of diffu­sion and viscosity for He-N2 mixtures, J. Phys. B 18, 4177-4184.

[49] Dickinson, A. S. and Lee, M. S. (1986) Classical trajectory calculations for aniso­tropy-dependent cross sections for He-N2 mixtures, J. Phys. B 19, 3091-3107.

[50] Wong, C. C. K., McCourt, F. R. W. and Dickinson, A. S. (1989) A Comparison between classical trajectory and infinite-order-sudden calculations of transport and r-elaxation cross sections for N2-Ne mixtures, Molec. Phys. 66, 1235-1260.

[51] Dickinson, A. S. and Heck, E. L. (1990) Transport and relaxation cross-sections for He-N2 mixtures, Molec. Phys. 70, 239-252.

236

[52] Liu, W.-K. and Dickinson, A. S. (1990) Classical limits of generalized cross­sections for atom-molecule systems, Molec. Phys. 70, 253-264.

[53] McCaffery, A. J., Proctor, M. J., Seddon, E. A. and Ticktin, A. (1986) Velocity dependence of polarised rotaional transfer rates: close-coupled calculations on model systems, Chern. Phys. Lett. 132, 185-189.

[54] Nyeland, C. and Billing, G. D. (1989) Semiclassical coupled state sudden approx­imation for multipole cross sections in atom-diatom systems, Chern. Phys. 138, 245-252.

FROM LINE-BROADENING TO VAN DER WAALS MOLECULES: COMPLEMENTARY WAYS TO PROBE THE ANISOTROPIC INTER­ACTION

Ph. BRECHIGNAC Laboratoire de Photophysique Moleculaire, Universite de Paris-Sud, Batiment 213, F-91405 ORSA Y CEDEX France

ABSTRACT. The close relationship between ordinary rotationally inelastic scattering, exemplified by state-to-state integral cross sections, and the pres­sure broadening of spectroscopic lines is emphasized, as is the use of pressure broadening data to obtain information about the repulsive part of the interac­tion potential. For example the utility of an analytic fitting law, which provides direct access to the dynamical constraints characteristic of the collision process, as well as to the shape and magnitude of the anisotropic interaction itself, is pointed out. Studies of Van der Waals molecules are normally considered to provide information on the long-range part of the interaction potential, and on the region of the absolute minimum. The use of vibrational predissociation data to access the repulsive wall region of the potential surface is emphasized, particularly for Van der Waals complexes which involve heavy molecules, such as C12 . It is finally shown, taking as an example the complexes of aniline with rare gases, that considerable valuable information on the interaction responis­ble for the formation of such Van der Waals complexes formed under molecular beam conditions can be deduced from their spectra, even for large polyatomic molecules.

1. Introduction

It is widely known that the physical processes governing the lineshapes of molecular spectra are related to perturbation of the transition dipole moment, which occurs during collisional encounters in gas phase media. The leading part in this perturbation is the result of the exchange of internal energy and angular momentum between collision partners, namely the collision-induced rotationally inelastic transitions that are responsible for rotational relaxation.

The outcome of such collisional processes (AB + C) is, of course, governed by the available kinetic energy distribution (temperature) and, more importantly, by the intermolecular interaction. It is essential to note that this interaction has to be anisotropic to allow for angular momentum exchange. Then the kinetic energy and the isotropic part of the potential energy surface (PES) control the way in which the

237

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 237-256. ©1992 Kluwer Academic Publishers.

238

anisotropic interaction is probed. It is obvious that the shape of the same intermolec­ular potential is responsible for the stability, under low-temperature conditions, of the Van der Waals (VdW) molecule AB-C. Any data obtained from such weakly­bound species must contain information on the detailed shape of the relevant PES, particularly for the region of the surface where the bound or quasi-bound states under observation are located. It is the purpose of this article to discuss the specifics and the complementarity of various kinds of observables, i.e. pressure broadening (PB), rotationally inelastic (RI) cross sections, spectroscopy of V dW molecules, vibrational predissociation rates and fragment rotational distributions, in the task of obtaining as accurately as possible an anisotropic PES.

Section 2 illustrates with some selected examples how PB parameters are related to a basis set of RI cross sections, while Section 3 recalls the relationship between these cross sections and the short-range anisotropy of the PES. In Section 4 the high sensitivity of each type of study of Van der Waals molecules to specific features of the PES is stressed. The resulting conclusion is that PB and V dW studies are to be regarded as largely complementary.

2. Pressure-Broadening of Molecular Spectra

The spectral lineshape of molecular absorption lines, in the microwave (MW) or in­frared (IR) range, has long been recognized to be related to rotational relaxation processes. The detailed nature of this interconnection has been made explicit in re­cent years. Within the Liouville formalism PB and RI cross sections may be expressed as different tensorial matrix components of the same super-relaxation operator A [1]. Because these components are not independent, subtle relationships follow.

It is of particular interest to note that, within the range of validity of the fixed­nuclei approximation (FNA), the most general tensorial cross section appears as a weighted sum of a limited number of purely RI cross sections [2]:

x (2n + 1)1/2 (j~ .,

Ja

o ~) (~ .,

lb o

n ., Ja ., Jb

~ } cr(O) (nn; 00)

(1)

This fundamental set of RI cross sections is nothing but the partial efficiencies for the molecule to be scattered out of the rotational ground state, j = 0, which is consistent with the approximation that the molecule has insufficient time to rotate during the collision, i.e. the condition nJ j (j + 1) / I « Cr / R, in which I is the molecular moment-of-inertia, Cr is the relative speed with which the particles are approaching, and Ii is the width of the collision region, applies.

Even more remarkable is the fact that elastic processes like collisional reorien­tation, as special cases of these generalized cross sections, are also described by this simple law. A nice example of this behaviour is presented below.

239

From PB measurements which employ an appropriate combination of tunable IR difference-frequency laser spectroscopy and Stark effects in a symmetric-top molecule, Brechignac [3] has demonstrated that the reorientation cross sections can be expressed simply in terms of inelastic cross sections. Inelastic (in the Stark-split, non-zero field, case), and elastic (in the degenerate, zero-field, case) reorientation processes contribute to the linewidth in different ways. In particular, the inelastic processes give rise to the collisional narrowing associated with line-coupling phenomena: this is referred to as orientational narrowing [4].

Figure 1 displays the results of the linewidth measurements on the R(l,l) transi­tion in CH3F, along with the relevant energy level diagram, showing that the zero-field line is narrower than the Stark-split line. Brechignac [3] gave a theoretical interpre­tion of these results within the tensorial cross section formalism, with the assumption that the inelastic rates are dominated by a single potential term and dipolar selection rules. This approach predicted a rather strong dependence of the PB parameter upon the magnetic quantum number associated with individual Stark components. This dependence was later observed experimentally by Buffa et al. [5].

Similar in essence are the results, also obtained by Buffa et al. [6]' which first demonstrated experimentally the dependence of the PB coefficient upon the nuclear­quadrupole hyperfine-structure (hfs) component of a rotational line. The guidelines of the interpretation are the same: it is the way in which the nuclear angular momen­tum couples with the rotation which governs the relative weightings of the inelastic terms in the various relevant cross sections. The general problem of the effect of the nuclear hyperfine structure on the spectral shape of molecular lines, including both the dependence of PB on hfs components and line-coupling, has also been addressed by Green [7] for the case of HCN.

The major conclusion arising from consideration of these few unusual examples is that PB phenomena are essentially sensitive to the same dynamical interactions that are probed by the RI cross sections.

3. Inelastic rates and anisotropic interaction

The question to be addressed now is double in nature: i) what are the dynamical constraints governing the behaviour of individual RI cross sections?; ii) to which part of the PES are these cross sections sensitive? These two aspects can be examined within the theoretical framework developed by Whitaker and Brechignac [8,9].

In an attempt to explain the physical origin of fitting laws for RI cross sections these authors have derived from a semiclassical collisional approach a very simple expression for the basis set of individual RI cross sections. This expression, called A Over N (AON) because of its analytic form, can effectively be used as a fitting law:

0"1-.0 = C [~ln (~) - ~ + 1] with l = .An (.A = 1 or 2). (2)

It has major advantages over other laws which have been proposed in the same context: it gives direct access to the dynamical constraints which characterize the collisional

240

..

-

1="

,

. AM:O

Alf ,MHZ)

ISO

4

E~O

~

AM: ~ t

11·4 MHz/lOt,

•

II

-I -. o • I

• I

o

o

10

Figure 1 (a) Energy level diagram showing the different Stark components of the R(I,I) transition in II polarizations (j) and .1 polarizations (1). (b) Plot of the measured FWHM iJ.v as a function of CH3F pressure for the M = 1-1 component (0) and for the zero-field (+) R(I,I) line. The full lines correspond to the theoretical result for a Voigt profile with iJ.llDoppler = 190MHz and a pressure-broadening parameter iJ.v/P = 15 and 11.4MHz/Torr, respectively. From Ref. [3].

241

1~10~ ________________ ~

a

1.0 ~

Figure 2. Rotationally inelastic rates for the scattering of 12 (B 3II) molecules out of the state j = 41 by collision with He, as a function of .:l == j f - ji. The bulk gas system is at room temperature. From Ref. [8].

processes, and to the shape and magnitude of the anisotropic interaction responsible for the exchange of internal energy and angular momentum of interest.

Figure 2 illustrates the content of the above formula by displaying on a log-log scale the R1 rates for the scattering of iodine molecules (12 B 3II) out of j = 41 by collisions with He under room temperature conditions. The very good quality of the fit is attested by the small value of the X2 / II coefficient (0.71). Since the cross section goes to zero for n = a, and must be set to zero for n > a, the parameter Aa (A = 1 or A = 2, see below) must be interpreted as the maximum value of the angular

242

momentum which can be transferred during a collision. Two major assumptions were made in the derivation of this law: i) the anisotropic

interaction is dominated by a single term in the Legendre polynomial expansion, VA (R) (>. = 1 for heteronuclear molecules, >. = 2 for homonuclear molecules); ii) this anisotropic interaction is exponentially repulsive with a range parameter TO. Un­der the conditions of validity of this AON law, which appears to hold for a wide variety of atom-diatom systems, knowledge of the two parameters a and c allows the deduction of a value of To = (ac/2)1/2, i.e. the slope of the anisotropic interaction at short range. Simultaneously, the magnitude of this interaction near the classical turning point Rc is given by

(3)

An important feature of the AON approach is that it contains a built-in energy dependence of the cross sections, which can be made explicit in the energy-dependent form of the fitting law [9]' AON(E), namely

2T6 [ ( a) nJEi] O"Z--->O(Et) = - In -- -1 + -- . n nJEi a (4)

It is noteworthy that the kinetic energy EZ to be taken into account (entrance channel for the l-O transition as above) is always in the exoergic direction (i.e. downwards transition). The cross sections for excitation have to be derived from the reverse processes by making use of microscopic reversibility. This is particularly important when considering collisional transitions of large inelasticity, for which the kinetic energy associated with the downwards transitions can become very small. t

An illustration of the adequacy of AON(E) to cope with this particular behaviour is shown in Fig. 3. It reports on a semi-log plot the rate constants for scattering a molecule of esH out of its ground, j = 0 level by collisions with H2 molecules. The solid line represents the best IOS-P fit to the experimental data [10]. It is very clear that the ability of AON(E) at fitting the data is, as expected in such a case, much better than for AON. The appropriate (cell conditions) Maxwell thermally-averaged AON(E) rate constants are in essentially perfect agreement with the experiment [11].

The physical picture behind this description of the RI collisional processes, and its ability to take into account a large number of experimental as well as theoretical data of this kind, gives credit to the main hypothesis underlying the AON approach: that part of the PES to which RI cross sections are sensitive is the short-range anisotropy in the region of the classically allowed repulsive wall.

Are there other observables that would help to yield more complete information on the PES?

t Higher-order interactions are essential in order to allow for large tlj transitions which rely on a single dipolar or quadrupolar anisotropy. One large tlj transition is equivalent to a sum of sequential tlj = 1 or 2 virtual transitions. Then, the longer the interaction time, the larger tlj can be. This is why the slower of the two half-collisions is the more important.

- Q

~l ~­

b

Q

" :3 -I

-2

5 /0

243

o o !

o

o

•

J'f /5

Figure 3. Semi-log plots of the (Jji--->j, cross sections for CsH-H2 collisions as a function of lJ for ji = O. Different values: --- IOS-P; 0 AON; 0 AON(E); 0 (AON(E))Maxwell. From Ref. [l1J.

4. Van der Waals molecules

The increasing amount of data accessible on V dW molecules, formed by the binary association of two stable molecules (or atoms) under low-temperature conditions, such as those achieved in supersonic expansions, also provides information about the intermolecular PES which characterizes the interaction between the two sub-units involved in the V dW association. It is essential in the context of transport properties to have a reasonably good knowledge of which part of the PES to which each set of data is sensitive.

244

zoo

-I J e 35 E

u

ex: JeZ5

> 100

0 ~ 3.0 4.0 5.0

R (A)

Figure 4. Centrifugally modified I-D potentials illustrating the 'rotational RKR procedure' for obtaining the j = 0 potential from j-dependent inner and outer turning points.

4.1 SPECTROSCOPY OF VAN DER WAALS MOLECULES

Extremely good quality spectra of several interesting Van der Waals molecules, par­ticularly HF- or HCI-containing dimers, have been reported during the recent past. Ar-HCI is probably the system which has received the greatest attention [12]. Ne-HF offers an interesting example in which an ab initio generated spectrum was predicted in essentially perfect agreement with experiment [13]. A more complete list can be found in recent reviews [14]. The level of accuracy achieved in this MW or near-IR spectroscopy is so high that when the associated stationary-state quantum mechan­ical calculations are made, the relevant part of the PES is essentially uniquely and accurately determined.

The information extractable from rotational spectra, such as that obtained from MW spectroscopy, can primarily be reduced to a set of moments-of-inertia. This provides the equilibrium geometry of the dimer in its ground state, and establishes the location of the absolute minimum in the PES. If the temperature is such that high- j levels can be observed, centrifugal-distortion constants also become available. It has been shown by Child and Nesbitt [15] that such additional information permits a so-called 'rotational RKR procedure' to be used in order to provide the shape of

245

-39 : ,'Tl- 1 -------'

8®Bl

Figure 5. Three-dimensional plot of the ab initio Ne-HF potential energy surface as a function of the centre-of-mass separation R and the angle f) between the rotor axis and the line joining the HF and Ne centres-of-mass. From Ref. [16].

the PES over an extended region around the absolute minimum. It is remarkable nonetheless that, as is clear from Fig. 4, the locations of the classical inner and outer turning points on the effective potential curves are such that knowledge of the PES is substantially extended only on the long-range side. Thus it is essentially complementary to the RI scattering which, as we have seen, probes the short-range anisotropy.

From the fine structure of these rotational spectra, not only are the energies of the rotational states accessible, but also patterns associated with large-amplitude motions like tunnelling or hindered rotation. These features provide unique information on the anisotropy in the region of the well. The ab initio PES for Ne-HF displayed in Fig. 5 (from Nesbitt et al. [16]) illustrates this point. In this particular case the first bound state lies at an energy which is above the internal rotational barrier. In addition we wish to emphasize the usual superiority of the rotationally-resolved near-IR vibrational spectra (associated with the vibration of an individual monomer

246

v· 1

Figure 6. Sketch of upper and lower potential surfaces to illustrate the vibrational predissociation process.

sub-unit), because, within a limited spectral range, they are able to provide a larger number of transitions, including the Van der Waals low-frequency modes. Again, this results in a better knowledge of the potential shape in the well region. This leads us to address a new question: are VdW molecules insensitive to the repulsive interaction?

4.2 VIBRATIONAL PREDISSOCIATION

Vibrational predissociation of a V dW molecule is the dynamical process through which the dimer excited in a quasi-bound state dissociates into fragment states of the constitutive monomers, while at least one vibrationally-excited molecular (i.e. diatomic) subunit loses one quantum (or possibly several quanta) of vibration. In a classical picture such as that illustrated in Fig. 6, vibrational predissociation can be considered to be a half-collision process in which a point particle evolving in the well of the excited upper surface suddenly switches to the lower surface. It is at that point able to 'feel' the repulsion from the short-range wall. In this manner vibrational predissociation dynamics should therefore be sensitive to the short-range part of the PES. We shall return to this point later.

247

In a perturbative quantum-mechanical description the transition probability is obtained from the Fermi golden rule as the square of a matrix element involving the overlap between the localized quasi-bound state wavefunction and the non-localized continuum wavefunction of the exit channel, namely

(5)

It is remarkable that the shape of this wavefunction depends very much on the kinetic energy release, hence on the particular rotational state in the exit channel (by virtue of energy conservation). There must then be dynamical information contained in the product-state distribution among the various accessible rotational states. This is the reason why effort has been taken to measure such fragment rotational distributions. We shall comment below upon this and its relation to the intermolecular PES using the Rg-Ch system as an example. This same system has been studied in a rather detailed way by Halberstadt, Janda and coworkers [17].

Figure 7 shows the rotational distributions which Cline et af. [17] obtained for the Ch fragment in various vibrational states (B,vf = 11, 10,9) leaving the He atom from the He-Ch dimer initially excited in the region of the (B,v = 12) 8h state. Comparison of the results for these three different vibrational channels ~v = -1, -2, -3 clearly shows that the rotational distribution is nearly independent of the available kinetic energy, which changes drastically from channel to channel. It is very satisfactory that these results are well reproduced by quantum mechanical theoretical calculations (open circles in Fig. 7) [17]. Very similar results have been reported for the case of Ne-CI2 [18].

The drawback of such quantal calculations is that they hide the physical picture of the process. For this reason it is interesting to use a quasi-classical approach, and to compare the results. This approach consists of using an initial distribution of the rare gas atom positions weighted by the wavefunction of the quasi-bound state, I Vfi¢ViI2, and running classical trajectories on the lower (final vf) surface. Figure 8 shows a contour plot of the Ne-Ch potential [19]' together with a 3D representation of the initial ground state wavefunction. It is clear that the relevant classical tra­jectories make a group of half-collision events probing the final-state surface from a limited set of impact parameters. As a consequence of the anisotropic character of this surface, a distribution of final rotational angular momenta is obtained. In agreement with the general findings of both the quantum calculations of RI cross sections [20] and the hard-ellipsoid model [21] a rotational rainbow behaviour, with the characteristic maximum value of the tranferred angular momentum, is observed. It is extremely remarkable that the general trends of this quasi-classical approach are in basic agreement with the experimental and the quantum theoretical results. This gives support to our conclusion that, apart from some specific quantal inter­ference effects, vibrational predissociation can be viewed instructively as a two-step mechanism: vibrational energy transfer occuring for a specified geometry at interme­diate distance and controlled by the vibrational coupling of the PES, followed by a rotationally inelastic half collision.

248

~ ~

.6 ~ (o)6v=-1

J~ B~ ~IO

''It-

o 0 2 4 6 8 () 12 14 1) 18

&;6 (b) 6 v =-2

~N ~ ~~ &~ -~ ~~ cflO ..,

0

0 2 4 6 B 10 12 14 16 18

~ ~

.6 ~ (c) 6 v = -3

.-J~

1~ ~~ ~IO ..

0

0 2 4 6 8 () 12 14 1) 18 Frognent q Rotofund State. j

Figure 7. Rotational state population distributions for dissociation of the j=O level of He35 C12 (B,v' = 12). The ~v = -1 channel (a) accounts for most, while the ~v = -2 (b), ~v = -3 (c) channels account for"'" 5%, < 1%, respectively, of the total fragment population. Experimental data are represented by filled circles and calculated distributions are represented by open circles. From Ref. [17J.

249

Figure 8. Upper Panel. Contour plot of the Ne-CI2 potential surface. The la­belled interaction energy curves are: 1: -60cm- 1 ; 2: -45cm- 1 ; 3: -30cm-1 ; 4: -15 cm-1 . Lower Panel. Three-dimensional representation of the lowest bound state wavefunction of Ne-Ch.

From the above picture it appears that, in addition to spectroscopic information on the well region of the intermolecular potential, Van der Waals molecules can also help, through vibrational predissociation dynamics, to enhance our knowledge of the short-range anisotropy. As initially stated, we see that they provide ways to probe the anisotropic interaction in a manner which is complementary to the use of pressure broadening and rotationally inelastic cross section data.

250

T : O. 5 K 1:DJ

Figure 9. LiF excitation spectrum for the aniline-neon comlex formed in a supersonic free jet. Fitting of the rotational contour leads to a rotational temperature of 0.5K.

4.3 THE CASE OF ANILINE COMPLEXES WITH RARE GASES AND HYDROGEN

The superiority of V dW molecules over the collisional observables is complete in the case of molecular systems larger than the well-studied atom-diatom case. Indeed, for polyatomic molecules the degree of anisotropy of the interaction PES can be very high, and not simple to represent in mathematical forms like Legendre polynomial expansions. Nonetheless spectroscopic and/or dynamical studies of VdW molecules are still capable of providing access to intermolecular forces. In the following para­graphs we shall illustrate this point with the V dW complexes of an aromatic molecule with rare gases and hydrogen.

The experimental source of information here is a set of spectra of these species, obtained either by laser-induced fluorescence (LIF) excitation (Ne, H2 , D2 ) or two­photon resonant ionization (Ar). Figure 9 shows the rotational contour of the An-Ne complex. Although the spectrum is not fully rotationally resolved, it is possible to simulate it by using an asymmetric rotor program. This provides simultaneously the rotational temperature, which is as low as 0.5K in this case, and the moments-of-

251

b

1+1

~ a

1+0 -125 -tIKI -75 -58 -25 9 2S 51 75 1111 or-'

Figure 10. Resonant two-photon two-colour photoionization spectra of An-Ar and An-Ar2 clusters. The Van der Waals vibrational frequencies are b~o = 22 cm- 1 , s!o = 40.7cm- 1 , b;o = 49.0cm-1 , and f3/i = 15.0cm-1 , f31J = 30.0cm-1 , eT01 = 37.5cm-1 .

From Ref. [23].

inertia of the VdW ensemble, i.e. the equilibrium geometry or, put into different words, the location of the absolute minimum of the PES: the Ne atom sits nearly above the centre of the phenyl ring at a distance of 3.5 A [22].

Figure 10 shows the R2PI spectra of the first two aniline/argon clusters [23,24]. The aniline-Ar spectrum is characterized by a relatively intense og band, shifted by 53.2 cm-1 to the red of the og band of pure aniline. A series of smaller peaks extending to higher energy is attributed to Van der Waals modes of the 8 1 state of the complex. This pattern is roughly repeated 54.2 cm-1 further to the red for the An-Ar2 spectrum. The observed red shift additivity law indicates the existence of two equivalent binding sites on either side of the aromatic plane. It should be noted that the red shift observed in aniline is the largest observed for monocyclic substituted aromatics (21cm- 1 for benzene-Ar and 36cm-1 for phenol-Ar). The large red shift in aniline is most certainly due to a relatively large change in the 7r­

electron distribution between So and 8 1 , or, put another way, resonance effects. The dipole moment of aniline changes from 1.54 to 2.38 Debye on going from the ground to the excited state [25].

252

The frequencies of the distinct peaks, interpreted as excitations of symmetric Van der Waals modes in the 8 1 state for both the An-Ar cluster and the An-Ar2 cluster are indicated in Fig. 10. For the An-Ar cluster there are three independent motions of the argon atom: one 'stretching mode', essentially perpendicular to the ring (sz) and two 'bending modes', one within the symmetry plane (bx ), and one perpendicular to the symmetry plane (by). By symmetry the fundamental of the latter mode b~o is forbidden, and only the overtone b;o may be observed. For the An-Ar2 cluster we have only two allowed fundamentals: the symmetric stretching mode (aJ) and the symmetric bend (!3J ) corresponding to the motion of both argons parallel to the ring within the symmetry plane. The assignment of the An-Ar2 spectrum first given by Schmidt et al. [26] is unquestionable. They argued that the Van der Waals frequencies of the symmetric bend and the symmetric stretch of the An-Ar2 complex can, by virtue of the symmetry of the potential, be related to the frequencies of the stretch and the bend in An-Ar by scaling according to the ratios of the reduced masses of the corresponding motions. Inverting this procedure, the predicted value for the An-Ar stretch is 44.5 cm- 1 to the blue of the og band, just midway between the two prominent lines. A second important point is that, on the basis of reduced masses alone, b~o would occur at a frequency rather higher than b~o. Our (81 ) potential surface yields frequencies of 25cm- 1 and 18cm-1 , respectively, for these motions. Perhaps the key argument for assigning the peak at 41 cm- 1 to the stretch is that it is the most intense line in the spectrum, in keeping with the relatively large intensity of the An-Ar2 stretch. From this point we can assign the peak at 49 em -1 to the overtone, b;o. It is hoped that close-coupled quantum mechanical calculations on a good quality An-Ar potential will help to confirm the attribution of this spectrum [26].

A potential model which has been constructed to describe these clusters gives results totally consistent with the observed shifts and frequencies [24]. We have chosen simple potential forms first proposed by Ondrechen et al. [28] for the dispersive and repulsive parts of the aniline-argon interaction, supplemented by a dipole-induced­dipole portion to account partially for electrostatic terms.

It is instructive to look at a cut through the potential hypersurface for the An­Ar molecule. In Fig. 11, we show, superimposed on a stick drawing of the aniline molecule, a contour plot of the interaction potential of Ar with aniline at a constant height (= 3.2 A) above the plane of the ring. It is important to note the presence of a minimum near the centre of the ring. This is in fact the position of the absolute minimum of the potential. Two apparent minima appear in the contour plot centered in the 'arms' formed by the H-C-C-N-H skeleton. These are only apparent minima of the An-Ar hypersurface, since the contour plot has been produced with the constraint of fixed height. It has been shown, however, that these features mark the positions of atoms in the larger clusters [24].

Interesting spectra have also been obtained in the case of aniline-H2 and aniline­D2 complexes [29]. The new feature in this case is that the (nearly free) hindered rotational motion allows recovery of the anisotropy of the surface in terms of the H2 rotation.

163.65

-118.49

-400.67

....... .. '-

253

-loll ~_:':""'_';:::::===--~"':"~~--.,.L=---..L.-l

-l.n x(!) l.ll

2.20

0.73

X V, 1 -0.73

-2.20 3.96

I."

Figure 11. a) Contour diagram of a constant height (3.2 A) cut through the potential surface of An-Ar. Nuclear positions are indicated by open circles of different sizes. The contour energies range from -276cm-1 to -36cm-1 in 40cm- 1 steps. b) An­Ar potential energy in the mirror plane cutting the ring (y=O). The global minimum is indicated by an asterisk. Note the asymmetry of the bending potential in the x-direction due to the presence of the -NH2 group.

254

5. Conclusion

We have presented selected experimental and theoretical results to illustrate the com­plementary character of various kinds of observables towards the acquisition of knowl­edge of the intermolecular PES, especially the anisotropic part of the interaction.

A close relationship has been found between PB and RI cross sections. Since an inelastic transition is essentially a transfer of energy and angular momentum, it is dynamically related to elastic processes such as reorientation and transfer between different nuclear spin (hfs) components. The potential 'window' of observables related to inelasticity is the short-range anisotropy.

Van der Waals molecules have been found to offer very great promise for the study of intermolecular potentials. This is primarily because, thanks to spectroscopy, Van der Waals molecules can provide data of high accuracy. Moreover, they give access to a 'double window' on the PES: the region of the well from bound or quasi­bound state spectroscopy, and the region of the wall from vibrational predissociation dynamics.

References

[1] Pickett, H. M. (1974) General rotational relaxation matrix: its properties, M dependence and relation to experiment, J. Chem. Phys. 61,1923-1933; Liu, W.-K. and Marcus, R. A. (1975) On the theory of the relaxation matrix and its application to microwave transient phenomena, J. Chem. Phys. 63, 272-289.

[2] Launay, J. M. (1980) Sudden approximation relations between tensorial cross sections for the collision of two diatomic molecules, Chem. Phys. Lett. 72, 152-155.

[3] Brechignac, P. (1982) Reorientation and pressure broadening of IR or MW lines: new results in CH3 F, J. Chem. Phys. 76, 3389-3395.

[4] Brechignac, P. (1985) Lineshapes in molecular spectra and rotationally inelastic transitions, in F. Rostas, (ed.) 'Spectral Lineshapes' 3, 699-723, Walter de Gruyter.

[5] Buffa, G., Di Lieto, A., Minguzzi, P., Tarrini, O. and Tonelli, M. (1986) Pressure broadening of molecular lines in a Stark field, Phys. Rev. A, 34 1065-1072.

[6] Buffa, G., Di Lieto, A., Minguzzi, P., Tarrini, O. and Tonelli, M. (1988) Nuclear­quadrupole effects in the pressure broadening of molecular lines, Phys. Rev. A 37, 3790-3794.

[7] Green, S. (1988) Effect of nuclear hyper fine structure on microwave spectral pres­sure broadening, J. Chem. Phys. 88, 7331-7336.

[8] Whitaker, B. J. and Brechignac, P. (1983) A new fitting law for rotational energy transfer, Chem. Phys. Lett. 95,407-412.

[9] Brechignac, P. and Whitaker, B. J. (1984) Energy dependence of rotationally inelastic cross sections and fitting laws, Chem. Phys. 88, 425-436.

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[10] Ferray, M., Visticot, J. P. and Sayer, B. (1984) Radiative and collisional properties of the CsH (A I r;+) state. III. Rotational energy transfer by collisions with H2 , J. Chern. Phys. 81,3009-3013.

[11] Brechignac, P. and Whitaker, B. J. (1986) Fitting form for rotational transitions of large inelasticity, J. Chern. Phys. 84, 2101-2103.

[12] See, for instance: Hutson, J. M. (1988) The intermolecular potential of Ar-HCl: determination from high resolution spectroscopy, J. Chern. Phys. 89,4550-4557; Lovejoy, C. M. and Nesbitt, D. J. (1988) Infrared-active Combination Bands in ArHCl, Chern. Phys. Lett. 146, 582-588.

[13] ONeil, S. V., Nesbitt, D. J., Rosrnus, P., Werner, H. J. and Clary, D. C. (1989) Weakly bound NeHF, J. Chern. Phys. 91,711-721; Clary, D. C., Lovejoy, C. M., O'Neil, S. V. and Nesbitt, D. J. (1988) Infrared spectrum of NeHF, Phys. Rev. Lett. 61, 1576-1579.

[14] Miller, R. E. (1986) Infrared laser photodissociation and spectroscopy of Van der Waals molecules, J. Phys. Chern. 90, 3301-3313; Sub-Doppler resolution infrared spectroscopy of binary molecular complexes, pp. 131-140, in 'Stucture and Dynamics of Weakly Bound Molecular Complexes', A. Weber (ed.), NATO ASI Series, Series C, Vol. 212 (Reidel, 1987).

[15] Child, M. and Nesbitt, D. J. (1988) RKR-based inversion of rotational progres­sions, Chern. Phys. Lett. 149, 404-410.

[16] Nesbitt, D. J., Lovejoy, C. M., Lindeman, T. G., O'Neil, S. V. and Clary, D. C. (1989) Slit jet infrared spectroscopy of NeHF complexes: Internal rotor and j-dependent predissociation dynamics, J. Chern. Phys. 91, 722-73l.

[17] Cline, J. I., Reid, B. P., Evard, D. D., Sivakurnar, N., Halberstadt, N. and Janda, K. C. (1988) State-to-state vibrational predissociation dynamics and spectroscopy of HeCl2 : Experiment and theory, J. Chern. Phys. 89,3535-3552.

[18] Cline, J. I., Sivakurnar, N., Evard, D. D., Bieler, C. R., Reid, B. P., Halber­stadt, N., Hair, S. R. and Janda, K. C. (1989) Product state distributions for the vibrational predissociation ofNeC}z, J. Chern. Phys. 90, 2605-2616.

[19] Reid, B. P., Janda, K. C. and Halberstadt, N. (1988) Vibrational and rota­tional wavefunctions for the triatomic Van der Waals molecules HeCI2 , NeC}z and ArC}z, J. Phys. Chern. 92, 587-593.

[20] Schinke, R., Muller, W., Meyer, W. and McGuire, P. (1981) Theoretical inves­tigation of rotational rainbow structures in X-Na2 collisions using CI Potential Surfaces. I. Rigid-rotor X-He scattering and comparison with state-to-state ex­periments, J. Chern. Phys. 74,3916-3928.

[21] Bosanac, S. (1980) Two-dimensional model of rotationally inelastic collisions, Phys. Rev. A 22, 2617-2622.

[22] Coutant, B. and Brechignac, P., (1989) unpublished.

256

[23] Brechignac, P. and Coutant, B. (1989) Microsolvation of aniline by argon: fluo­rescence excitation and two-photon resonant ionization of An-Arn in a supersonic jet, Z. Phys. D 14, 87-88.

[24] Hermine,P., Parneix, P., Coutant, B., Amar, F. G. and Brechignac, P. Resonant two-photon two-color photoionization (R2P2CI) spectra of Aniline-Arn clusters: isomer structures and solvent shifts, Z. Phys. D, in press.

[25] Lombardi, J. R. (1969) Dipole moments of the lowest singlet II* f- II states in phenol and aniline by the optical Stark effect. J. Chem. Phys. 50, 3780-3783.

[26] Schmidt, M., Mons, M. and Le Calve, J. (1990) Intermolecular vibronic spec­troscopy of small Van der Waals clusters: phenol-and aniline-(Argonh complexes, Z. Phys. D 17, 153-155.

[27] Halberstadt, N., Parneix, P., Amar, F. G. and Brechignac, P., work in progress.

[28] Ondrechen, M. J., Berkovitch-Yellin, Z. and Jortner, J. (1981) Model calcula­tions of potential surfaces of Van der Waals complexes containing large aromatic molecules, J. Am. Chem. Soc. 103,6586-6592.

[29] Coutant, B. and Brechignac, P. (1989) Coupling of hindered internal rotation with Van der Waals vibration in aniline-H2 (D2 ) complexes, J. Chem. Phys. 91, 1978-1986.

CALCULATION OF PRESSURE BROADENED SPECTRAL LINE SHAPES INCLUDING COLLISIONAL TRANSFER OF INTENSITY

SHELDON GREEN NASA Goddard Space Flight Center, Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA.

ABSTRACT. Spectral line shapes, including widths and shifts of isolated lines and collisional transfer of intensity among overlapping lines, can be described within the impact approximation by generalized (kinetic theory) collision cross sections. Theoretical determination of line shapes thus requires accurate molec­ular scattering calculations to obtain collisional S-matrices and this, in turn, requires detailed knowledge of the intermolecular forces. Advances in computa­tional abilities now permit rather accurate calculations, at least for some simple systems. This is illustrated by a review of recent studies of broadening and shifting of isolated lines of CO and D2 in He and of HCI in Ar. Line-coupling cross sections, which describe collisional transfer of intensity, have received much less attention although they are quite important for several practical applica­tions and are no more difficult to compute. Recent work for CO in He demon­strates the ability to obtain an accurate theoretical description for both" micro­windows" in the infrared fundamental band and for the Raman Q-branch.

1. Introduction

It is well known that spectral lines are affected by molecular collisions at all but the lowest pressures. The present discussion considers molecular rotation and vibration­rotation spectra, which occur mainly at microwave and infrared frequencies, respec­tively, in gas phase samples at moderate pressures (fractions of an atmosphere to several atmospheres). In such cases, isolated spectral lines are generally found exper­imentally to be well described by a Lorentzian function,

I(w) = 'Y , [(w - wo - 8)2 + 'Y2]

(1)

where I(w) is proportional to the intensity at frequency w. The half-width at half maximum of this function is 'Y, and the maximum is shifted by an amount 8 from the 'zero pressure' resonance frequency, woo The line width and line shift are both found to vary linearly with the number density of the gas (i.e. linearly with pressure for

257

W.A. Wakeham et al. (eds.), Status and Future Developments in Transport Properties, 257-282. ©1992 Kluwer Academic Publishers.

258

an ideal gas). At pressures where lines are broadened enough to overlap, however, collisions can transfer intensity among them, profoundly altering the line shape.

Knowledge of line broadening parameters is important for a number of practical applications, including the following: modelling the infrared opacities of planetary atmospheres and, hence, their thermal budgets (' greenhouse' effects); retrieving at­mospheric temperature profiles from satellite remote sensing data; and using spectro­scopic probes (e.g. coherent anti-Stokes Raman spectroscopy, (CARS)) to determine pressures and temperatures in hostile environments such as flames and internal com­bustion engines. Line shapes also provide information about fundamental molecular processes, in particular about rotational relaxation, and therefore about molecular collision dynamics and the underlying intermolecular forces.

Theoretical considerations of this phenomenon have a long history. The current framework within which most calculations are done can be traced to the work of An­derson [1], who used a semi-classical approach for molecular scattering (i.e. classical collision trajectories) and of Baranger [2], who developed a fully quantum formalism which also included transfer of intensity among overlapping lines. Spectral line shapes can be described in the weak radiation field limit as a Fourier transform of the corre­lation function of the dipole moment of the system (for usual, electric dipole spectra; for other types of spectra the correlation function of the appropriate multipole op­erator must be used). The difficulty, of course, is that the spectrum is of a bulk sample, and several approximations are required to reduce this many-body problem to the interaction of a single spectroscopically active molecule with a thermal bath of perturbers and, ultimately, to represent the latter in terms of binary collisions with a single molecule of the bath. This theoretical framework is outlined in Section 2.

Applications require calculation of molecular collision dynamics and this, in turn, requires knowledge of the intermolecular forces. It is only the great increases in com­putational capabilities of the last two decades which now make it feasible to do rea­sonably accurate calculations. Intermolecular forces are determined by solving for the electronic energy by standard quantum chemical methods. To some extent infor­mation about the intermolecular forces can also be extracted from experiment. An impediment is that most experiments measure some effect of the potential averaged over molecular collisions; while such experiments provide good tests of postulated potential energy surfaces, it is difficult to extract the potential directly from the data. Recently, however, spectral data for weakly bound Van der Waals complexes have begun to yield precise information about intermolecular forces, at least in the region around the equilibrium structure. Dynamics are treated by solving the quan­tum coupled-channel equations to obtain the collisional S-matrix. Current abilities to predict widths and shifts of isolated lines from theoretical or experimental inter­molecular potentials and quantum scattering calculations are illustrated in Section 3 by reviewing recent work on CO in He [3-6], D2 in He [7,8], and HCI in Ar [9].

Whereas widths and shifts of isolated spectral lines have received much attention, both experimental and theoretical, much less is understood in the case where lines overlap and collisions can transfer intensity among different spectral lines. This is unfortunate, as many of the applications noted above are quite sensitive to this effect.

259

Line coupling makes profound changes (often an order of magnitude from the predic­tions of simple line additivity) for absorption beyond the band head, in the Q-branch gap of parallel transitions, in 'micro-windows' between individual rotational lines of a vibrational band, and in Raman Q-branch spectra.

The formalism outlined in Section 2 is believed to provide a proper description of this phenomenon. Despite this and the importance of such effects, only a few theoretical calculations have been performed to date and, with the exception of re­cent work on CO in He [5,6]' all of these suffered from use of approximate molecular scattering methods and potentials of questionable accuracy. For lack of rigorous theo­retical treatments, most studies of line coupling have adopted semi-empirical models. These generally equate the line coupling relaxation rates to more familiar state-to­state rotational relaxation rates - which is often poorly justified - and use simplistic parameterizations for the latter, often in terms of rotational energy gaps, with pa­rameters which are then obtained by fitting experimental line shapes. The danger in such a procedure, of course, is that the parameters are merely empirical - they no longer correlate with physical quantities - and may give erroneous results when extrapolated to other pressures or temperatures, as is generally necessary for practical applications. Fortunately, the only accurate calculation done to date, that for CO in He, was found to have good predictive abilities when compared with experimental data for micro-windows in the infrared fundamental [5] and also for Q-branch collapse in CARS spectra [6]. These points are discussed in Section 4.

2. Theory

For reasonably weak fields the absorption of radiation is proportional to the square of the electric dipole (or other appropriate multipole) moment matrix element between initial and final states of the system. For isolated molecules this gives sharp reso­nance lines corresponding to allowed transitions between quantum energy levels. For a bulk sample, however, the initial and final states are those of the whole interacting many-body system. Description of the line shape in that case has been approached in a variety of ways including consideration of the (dipole) auto-correlation func­tion [10]' generalization of the quantum kinetic equations [11,12], and rate equations connecting the superposition states which correspond to spectral transitions [13,14]. To make headway in achieving a tractable formalism all of these make essentially the same approximations to reduce the problem to a single spectroscopically active molecule interacting with a thermal bath and to approximate the latter in terms of completed, sequential, binary collisions (the impact approximation). The 'modern' form appropriate for molecular vibration-rotation spectra traces to Ben-Reuven who noted in his 1966 paper [15]: 'No attempt is made at any quantitative evaluation of the relaxation parameters. Even under vastly simplifying assumptions, such eval­uations encounter formidable difficulties, enhanced by our too scarce knowledge of intermolecular forces.' Nonetheless, Shafer and Gordon demonstrated the possibility of accurate calculations in their 1973 study of Hz-He [16]. Only a brief outline will be

260

given here of the theoretical development, as the existing literature is quite extensive. The review by Ben-Reuven [17] is particularly recommended.

The spectral intensity is written as the Fourier transform of an auto-correlation function

lew) = (27r)-1 1: exp (iwt) G(t) dt, (2)

where the auto-correlation function is given by

G(t) = (JL(O) . JL(t)) = Tr{p JL(O) . JL(t)}, (3)

with ( ... ) an average over the ensemble, equivalent to the trace in the second equality, and p the initial (equilibrium) density matrix. The time evolution of JL can be written, as usual, in terms of the Hamiltonian, H, for the entire system:

JL(t) = exp (iHt) JL(O) exp (-iHt). (4)

This can be formally simplified by introducing the Liouville operator, L [10,16]. Then,

JL(t) = JL(O) exp (-iLt) (5)

so that the line shape can be written as

(6)

The total Hamiltonian can be separated according to

(7)

where the first two terms are Hamiltonians for separate spectroscopic and bath molec­ules and V is molecule-molecule interactions, including those between bath molecules. The Liouville operator then separates in the same way,

(8)

As shown by Fano [10] the line shape can then be written in terms of an operator (Mc(w)) which, being averaged over the bath, operates only in the space of the spec­troscopic molecule variables.

(9)

where Trs indicates averaging over only the variables of the spectroscopic molecule. Implicit in Eq. (9) is the assumption that the total density matrix is the (uncorrelated) product of density matrices for the spectroscopic molecule and the bath,

(10)

261

All the difficulties are now placed in evaluating (Mc(w)}. This is done by invoking the impact approximation, that is, the duration of each collision is assumed short compared with the time between collisions so that only completed collisions need be considered. To lowest order in the density, n, one then has [16]

where the average is over bath variables and ta(Ea) is the scattering theory t-matrix matrix operating on spectroscopic level a. Note that there is no longer any dependence on frequency. This comes from considering only completed collisions; i.e. no tran­sient effects, so that only on-the-energy-shell t-matrices (or S-matrices) are needed. However, this also limits validity to frequencies near line centres; the frequency range over which it is valid can be related to the range in energy off-the-energy-shell over which the t-matrix can be considered approximately constant [17,18].

Evaluating Eqs. (9) and (11) over a basis in 'line space', i.e. each element corre­sponds to a spectral transition a ----> b, and written in convenient matrix notation,

I(w) = _71"-1 Imd· (w - wo - in(vu})-l. p. d, (12)

where ( ... ) indicates a thermal average over collision velocity, v, and all the matrices except the cross section matrix, u, are diagonal. Diagonal elements of ware just the frequency, w; of Wo, the line frequencies; of d, the line intensities; and of P, the population in the initial level.

The line broadening cross section matrix is given in terms of binary collision S-matrices (with notation simplified to that for a structureless bath species) [16J

aqh~'"'f~'"'fa'"'fb;E) = (71"/k2) L [JaJ[Jb][j~F/2/[jaF/2 {~a q ~} {~~ Jah>'>" a ..\ b a

X [8ha'"'f~)8hb'"'f~)8(..\..\/) - b~..\/ISJa(Ea)ba..\}b~..\/ISJb(Eb)bb..\}·] .

q ..\'

(13)

Here '"'fa and '"'fb are quantum numbers for the two spectral levels whose total angular momenta are ja and jb; ..\ is the partial-wave collisional angular momentum; the rotor angular momentum j and the orbital angular momentum ..\ are combined to give a total angular momentum J; primes indicate post-collision values; q is the tensor order of the radiative interaction (e.g. q = 1 for dipole spectra); k is the collision wavevector; {:::} is a six-j symbol; and [jJ = 2j + l.

The S-matrices required for the right-hand side of Eq. (13) are obtained by solv­ing a standard quantum scattering problem for two rotating (and vibrating) molec­ules interacting via a potential that depends on the collision distance, the relative orientations and the internal (vibrational) coordinates [19J. The cross sections must be calculated for relevant (centre of mass) thermal collision energies, E = k2 /2m" where mr is the collisional reduced mass, and the S-matrices must be evaluated at total energies equal to the collision energy plus the molecular energy in the initial levels, i.e.

(14)

262

Equations (12) and (13) summarize the basic computational framework used in the present discussion of widths and shifts of isolated lines and transfer of intensity in overlapping lines. Note, however, that neither the Doppler width nor Dicke narrowing are encompassed in this formalism.

3. Results for isolated spectral lines

To indicate current computational capabilities recent studies of three systems are re­viewed. For two of these, CO-He and H2-He, the interaction potentials were obtained from quantum chemical calculations; for the third, HCI-Ar, the potential was obtained from analysis of spectra of the Van der Waals complex. A discussion is then given of future prospects for treating other, more complex, systems.

3.1. CO-He

Because of its importance for interpreting radio-astronomical observations of inter­stellar matter, rotational excitation of CO by collisions with low temperature He and H2 has received considerable attention. The first calculations [20] used relatively crude approximations for the interaction potential, but treated collision dynamics with essentially exact, converged (with respect to CO rotational basis set) close­coupling calculations. The computational expense of solving the coupled-channel equations increases dramatically with jrnax, the highest rotor level included, limiting early calculations to temperatures below about lOOK, where jmax on the order of eight is sufficient; fortunately, this included temperatures relevant to the early astro­physical observations. To provide some check on the accuracy of these calculations, comparisons were made with line-width measurements from liquid nitrogen to room temperatures for the j = 0-1 and 1-2 microwave transitions [21], finding reasonable agreement (within the 10% estimated experimental accuracy). It was possible to obtain converged cross sections at the higher temperatures here even though all en­ergetically accessible rotationallevelp werr. 'lot included in the scattering calculation; for the pressure-broadening cross sel _.Lv ' {c for the state-to-state cross sections, it appeared to be necessary only to induu.e a sufficient number of rotational levels above the spectroscopic levels of interest.

Subsequently an ab initio potential was calculated for this system using self­consistent field and configuration interaction (SCF-CI) methods [22]. Although spe­cific state-to-state cross sections from this potential differed significantly from those from the earlier, less accurate potential, predictions for pressure-broadening cross sec­tions between 77 and 300K from the two potentials [23] were similar enough that the available experimental data could not clearly distinguish between them. It was noted, however, that predicted pressure broadening cross sections from these two potentials did diverge at lower temperatures. DeLucia and coworkers have recently developed a clever method for measuring line widths at cryogenic temperatures, and their results for CO-He clearly demonstrated the (anticipated) superiority of the SCF-CI interac­tion; the agreement between experiment [24] and theory [3], is, in fact, quite good, as

263

60

50 N 0«

- CO in He c:

~ J= 0-1 u Q) 40

(/)

VI VI 0 ~

U 30

20

2 4 6 8 100 200 300 Temperature, K

Figure 1. Comparison of theoretical and experimental values for the line width cross section of the j = 0-1 microwave line of CO in He as a function of temperature. Theoretical values are indicated by the continuous line, experimental values by vertical bars which indicate estimated experimental uncertainties.

shown for the j = 0-1 line in Fig. 1, although some discrepancies are observed below about 2K. These small discrepancies at very low temperatures suggest that the well in the theoretical potential is probably somewhat too deep [25J.

Line widths in infrared vibration-rotation transitions are also of interest. Ro­tational relaxation dominates line widths at thermal energies, even in vibrational spectra. Calculations which included the CO vibrational dependence in the CO-He interaction (although not with the same accuracy as the SCF-CI interaction) sug­gested very little vibrational dependence for this system [26J, an observation which is substantiated experimentally [27J. Although dependence on vibrational coordinate does not therefore pose a problem in extending calculations to the infrared bands, the fact that much higher rotational levels are of interest does make close-coupling calculations prohibitively expensive. Fortunately, the coupled-states approximation has been found to be quite accurate for this system [4,28J. With this approxima­tion calculations can be done to about j = 30 without taxing current computational capabilities.

264

29 Infrared - theory N 0« Z 27 0 ~ u 25 lLl fJ)

fJ)

\. :r expt.

\~ ! I I ~-!-+-i--i---r--r=;-i--i·--I·--Io -N° ° ,. --- .... -\_----~---- -"""-

fJ) 23 0 a: u

21

-- theory Romon Q-bronch • ex pI.

Figure 2. Comparison of theoretical and experimental line width cross sections for infrared and Raman spectra of CO in He at room temperature. Theoretical values for infrared spectra are shown as a continuous line, for the Raman spectra as a dashed line. Infrared experimental data are shown as the range of values reported for P(j+1) and R(j) transitions in the 0-2 and 0-3 vibrational bands; actual experimental errors may be larger. Experimental data for Raman Q(j) lines are shown as circles.

Figure 2 compares theoretical and experimental [27] pressure-broadening cross sections at room temperature as a function of initial spectroscopic level. Theoretical values were obtained from coupled-states calculations using the SCF-CI interaction, and show the small differences predicted between infrared (P- and R-branch) [29] and Raman (Q-branch) [6] transitions. Other experimental data for the infrared funda­mental as well as for overtone bands are in harmony with the data presented in Fig. 2, but show much more scatter owing to experimental uncertainties, and are hence not included. Note that the value for the pure rotational (microwave) j = 0-1 transition, shown in Fig. 1, is also in harmony with the values shown in Fig. 2.

With only four electrons, H2-He is one of the simplest triatomic systems, and its po­tential energy surface has been the subject of several theoretical investigations. Meyer et al. [30] have done a particularly thorough calculation for the three-dimensional (i.e. including stretching dependence) surface. Recent measurements of line shapes in the Q-branch Raman v = 0-1 spectrum ofD2 in He (which, within the Born-Oppenheimer approximation, has the same electronic potential as H2-He) showed interesting fea­tures, including large line shifts and Dicke narrowing at moderate pressures [31].

Raman Q-branch transitions are described by a tensor of order zero and, in the absence of vibrational dependence in the interaction potential, pressure-broadening

265

8

Shift 7 • • • 0+- • 0

0' 6 • 0 D2 in He E

~ T =298K

IE 5 u ,., b 4 -- theory

• • expt .

3

2

J

Figure 3. Comparison of theoretical and experimental values for line widths and shifts in Raman Q(j) spectra of D2 in He at room temperature.

cross sections can be related to state-to-state collision cross sections. As the latter are real quantities, line shifts, which are given by the imaginary part of the pressure­broadening cross sections, should vanish. The observed shifts are larger than the widths, suggesting the importance of vibrational dependence in the potential, and it

266

was of some interest to see how well the ab initio potential, which was expected to be quite accurate, would reproduce these measurements. Because of its large rotational constant, and the homonuclear symmetry which rigorously decouples even and odd rotational levels, accurate close-coupling scattering calculations are quite feasible for Dz-He, and were done by Monchick and coworkers [7J. Resulting widths and shifts are compared with experiment in Fig. 3.

The theoretical widths are in excellent agreement with experimental values; the shifts are predicted to be about 15% too large. Several effects which are usually ignored in line-shape calculations were examined explicitly in Ref. [7J: vibrational in­elasticity and the effect of centrifugal distortion (i.e. rotational stretching) on energy levels and potential matrix elements. None of these caused changes of more than a few percent; these effects will be even smaller for most other systems, and so can generally be safely ignored. The discrepancy between theoretical and experimental shifts presumably indicates small errors in the vibrational dependence of the theoret­ical potential (or the procedures used to fit it and integrate it over vibration-rotation wavefunctions). Quite small errors in the potential could be responsible; the initial calculations for this system suffered from a programming error which caused an ap­parently minor mistake in the potential, but which changed computed shifts by a factor of two.

The impact approximation, which is fundamental to the Fano-Ben-Reuven­Gordon formulation of line shapes, does not account for Dicke narrowing, which is a narrowing of the Doppler width by velocity-changing collisions. Although several authors have developed more general formulations to account for this effect, some of which require the same scattering S-matrices as the impact approximation theories, the first (and at this writing only) test of these theories which used accurate molec­ular scattering S-matrices and compared with experimental data was recent work on Dz-He [8J. Results were generally satisfactory, as can be seen from comparison of the theoretical and experimental pressure dependences of the width of the vibrational Raman Q(2) line, which is shown in Figure 4.

3.3 HCI-Ar

The relationship between the shape of the intermolecular potential and both molecular collision phenomena and the spectrum of bound Van der Waals complexes has been studied more thoroughly for HCI-Ar than perhaps for any other system. A history of this work can be found in Refs. [9J and [32], and references therein. The latter work, which presents the best potential to date, relies on a careful analysis of rather extensive spectral data for the Van der Waals complex; it is a refinement of earlier potentials, which were derived from less complete spectral data, but which also considered other, collisional, data including line shape parameters and second virial coefficients [33J.

Using the potential of Ref. [32J line-shape parameters were calculated for the lowest six far infrared pure rotational transitions using converged close-coupling and approximate coupled-states S-matrices [9J. Besides testing the potential by comparing its predictions for line broadening against experimental data, this study provided an

.020r---~----.-----~--~----_

o Expt. - Collision Kernel --- Generalized Hess

method .015

-IE u

'--"

~ .010 I 3 I

.005

°0~--~----~4----~6----~8~--~IO"

Density (amagats)

267

Figure 4. Experimental widths as a function of pressure for the Raman Q(2) line of D2 in He at room temperature are compared with predictions of different theories for Dicke narrowing. Based upon Fig. 8 of Ref. [8].

important test of the coupled-states molecular scattering approximation. Because this system is heavier than the collision systems involving He for which most of the earlier calculations were done, higher partial waves are required to obtain converged cross sections, and there have been some questions about the validity of the coupled-states method (which approximates centrifugal coupling terms) in such cases. This system also has a deeper well than most of the systems studied previously, making scattering resonances more important and, again, there is some question about the accuracy of the coupled-states method for such cases. In fact, the computationally cheaper approximation was found to be quite reliable for this system (accuracies better than about 5%).

268

130 f 1]0 130

1<0 1<0 1<0

110 11O

100 T'300 K

z 0 ~ t) W (f)

10 (f) (f)

60 0 a: t)

30

40 40

30 30

<0 20

INITIAL ROTATIONAL LEVEL

Figure 5. Comparison of theoretical (continuous lines) and experimental (symbols and error bars) line width cross sections for far infrared j---> j+ 1 rotational spectra of HCI in Ar. From Ref. [9J.

Predicted line widths agreed quite well with experimental values which are avail­able for temperatures from 125 to 300K (see Fig. 5). A possible exception is the fundamental, j = Q-1line at low temperatures; while the discrepancy at 125K is only 15%, this is five times the stated experimental uncertainty. The measured tempera­ture dependence, especially for the higher transitions, is more erratic than predicted by theory, and the latter is undoubtedly more reliable. It is noteworthy that neither the theoretical nor the experimental temperature dependence is well represented by the inverse power functional form which is generally used when extrapolations are needed for practical applications.

3.4 FUTURE DIRECTIONS

With steadily increasing computational power it will be possible to treat more complex systems, that is, ones with more degrees of freedom and more accessible energy levels. Successful calculations require accurate potential surfaces, but these too will become more amenable to quantum chemical calculations. Rapid progress is also being made in elucidating potential surfaces from spectra of Van der Waals complexes, but caution

269

must be exercised as these do not always provide adequate descriptions of all parts of the surface accessible in molecular collisions. For example, an apparently very accurate surface for H2-Ar obtained from extensive spectral data gave poor predictions for Raman line shifts [34].

Work is already being done for more complex systems than the rare gas atom­diatomic molecules discussed above; examples include H2 0 in He [35] and H2 0 in Ar [36]. However, little progress has yet been made for the important case of di­atomic perturbers (note that N2 is the most important perturber for atmospheric applications). For a system like H20-N2 at room temperature, full close coupling is probably not computationally feasible in the foreseeable future, so it is very im­portant to develop and document the accuracy of approximate methods. Some large benchmark close-coupling calculations for typical systems will be invaluable for this, as comparisons with experiment always suffer from possible errors in the potential energy surface or in the experimental data.

The discussion has so far made little mention of the venerable Anderson theory [1], although it still finds widespread application (generally in somewhat improved ver­sions). In fact, for systems dominated by strong long-range interactions (typically dipole-dipole) this approximation is not unreasonable. However, the atmospheric and remote-sensing applications of much current interest generally involve relatively non-polar perturbers (like N2 ), and Anderson-like theories are problematic. On the other hand, it will probably be very useful to consider hybrid methods, for example, Anderson-like theories (or quantum analogues such as the Born or distorted-wave Born approximation) for the high partial waves which do sample only the long-range part of the interaction, and, perhaps, a sudden approximation for low partial waves [37]. The fact that only fully quantum calculations are discussed here is not meant to indi­cate that other kinds of methods, such as various semi-classical methods, cannot be reliable; tests of such methods against some of the accurate quantum results which are now available would be most welcome.

4. Transfer of intensity in overlapping lines

Much of the (extensive) work in this field has been driven by the necessity of model­ing spectra for two problems of some practical importance. The first involves use of coherent Raman spectroscopies, notably CARS, to obtain temperatures and densities in hostile environments such as flames and combustion engines. A number of rele­vant references can be found in Ref. [38]. Because the major species in many of these systems is atmospheric nitrogen, the convenient Raman Q-branch of the fundamental vibrational band of N2 has been thoroughly studied, both self-broadening and broad­ening by combustion species such as CO2 and H20. Rotational lines in Q-branches are closely spaced and line coupling is very important at typical pressures. Carbon dioxide is a second species which has received much attention because of its role in atmospheric infrared opacities and because it is used for remote measurement of at­mospheric temperature profiles by satellite observations; see, e.g. [39] and references

270

therein. For lack of good theoretical support, these applications have come to rely on semi-empirical models which are discussed in Section 4.1.

From the discussion in Section 2 it should be apparent that off-diagonal pressure­broadening cross sections, which describe transfer of intensity among overlapping lines (line coupling), can be calculated with no more effort than is required for the diagonal cross sections, which describe widths and shifts of isolated lines. The main compu­tational effort in both cases involves determination of the intermolecular forces and subsequent calculation of the collisional S-matrices; appropriate combinations of the latter then provide either diagonal or off-diagonal pressure-broadening cross sections. Considering the amount of effort which historically has been expended on calculating broadening parameters for isolated lines and the high accuracy which can now be achieved, at least, as discussed in Section 3, for some simple systems, it is somewhat surprising that very few calculations have been attempted for line-coupling parame­ters. Gordon and McGinnis [40] used a semi-classical (classical trajectory) theory for the pressure-broadening cross sections and a semi-empirical model interaction poten­tial to study collapse of the P- and R-branches in the infrared fundamental of CO in He at pressures to several hundred atmospheres. Oxygen microwave lines, which are important for atmospheric opacity, were considered by Lam [41] and Smith [42]; these studies used rather simplistic models for the interaction and semi-classical scattering methods. Another system of importance for atmospheric opacity, CO2 , was considered by Braun [43], who calculated line coupling for the bending mode Q-branch using an Anderson-like (semi-classical, straight-line trajectories) procedure, retaining only the dominant long-range quadrupole-quadrupole interaction for CO2-C02 and CO2-N2 .

More rigorous molecular scattering calculations were done recently for CO2-He [39]' although using a rather simplistic model for the intermolecular forces. It appears that the only study to date which employed an accurate interaction potential and rigorous scattering methods is recent work on CO in He, which will be discussed in Section 4.2. Preliminary efforts at a proper understanding of line coupling in infrared spectra, such as bending bands of CO2 , are reported in Section 4.3.

4.1. SEMI-EMPIRICAL MODELS

All of the models start with a spectral intensity function in the impact approximation formulation developed in Section 2. In convenient matrix notation [16]

I(w) ~ 1m d· (w - Wo - in W)-l . p. d, (15)

which is just Eq. (12) with the Boltzmann average of the pressure-broadening cross sections replaced by the W matrix.

(16)

where ( ... ) indicates a thermal average over velocity, V; i refers to the spectral line vaja - vbjb, and primes indicate post-collision values. The pressure enters through

271

the number density, n. Recall that possible frequency dependence of the pressure­broadening cross sections (and hence the W matrix) has been ignored, consistent with the impact approximation. This can be valid only for some restricted range of frequencies near resonance. However, the question of modifications which must be made in cases where this is not justified will be deferred. The full line shape can then be obtained efficiently at any frequency from Eq. (15) with the algorithm described by Gordon and McGinnis [44]. When the effects of line coupling are small (i.e. weakly overlapping lines) a perturbation method for evaluating Eq. (15) is also useful [42,45].

In the case that off-diagonal elements of W can be ignored, Eq. (15) reduces to a sum of Lorentzian terms, Eq. (1), with the width and shift of the i th line given by

"Ii = nReW(i,i)

8i = -nImW(i,i). (17)

These diagonal terms can generally be obtained from measurements at low pressures where the lines do not overlap.

The off-diagonal elements are more difficult to determine. For Q-branch Raman spectra, where the tensor order is q = 0, progress can be made by relating these to state-to-state rotationally inelastic collision rates. If vibrational dependence of the interaction can be ignored, then

W(i,i') = - R(ja --; j~), ja #j~, (18)

where R(ja --; j~) is just the rotationally inelastic (but vibrationally elastic) rate. In the case that vibrational dependence cannot be neglected, this can be gener­

alized in a manner analogous to Eqs. (5) and (6) of[7]:

That is, one averages the rotational state-to-state rates in the upper and lower (spec­troscopic) vibrational levels; but there is then also a (complex) dephasing contribution which represents quantum interference between scattering in the two vibrational lev­els. For vibrational bands of typical diatomic molecules (for example, CO as discussed in Section 3.1 and undoubtedly for the similar N2 ) vibrational dephasing can, in fact, be safely ignored. However, caution should always be exercised before making this as­sumption. For D2 , as discussed in Section 3.2, dephasing contributions dominate line widths; this can be attributed to a combination of small rotational inelasticity and relatively large vibrational motion owing to the small mass of the deuterium atoms.

The models also use the fact that for Q-branch Raman spectra in the absence of vibrational dephasing the diagonal elements of W equal the negative sum of the off-diagonal elements, that is

2:W(i,i') = o. (20) i'

272

The detailed-balance relation

P(i,i) W(i,i') = P(i',i') W(i',i) (21)

between forward and reverse rates places further constraints. Even so, it has not been possible to obtain all the off-diagonal values by fitting to spectra without fur­ther assumptions, and the various models differ in how they reduce the full set of line-coupling rates (taken as just the rotationally inelastic rates) to a few parameters which may then be fit to (low pressure) spectral data via Eqs. (18) and (20). The methods in current use rely on a large body of earlier work which attempts to describe state-to-state rotational excitation rates in terms of a few simple physical parameters such as energy gaps and rotational quantum numbers [46J. Some of these should be viewed merely as (arbitrary) empirical fitting laws; others rely on relationships ob­tained from well-defined dynamical approximations, such as the infinite order sudden approximation [47], which relates the entire matrix of rates to "fundamental" rates out of the lowest rotational level. A good discussion of applications of current models to Raman Q-branch spectra, including extensive references and fits for many systems, was given recently by Millot [38J.

Despite attempts to base these models on reasonable approximations to the actual physics involved, it is not clear whether the parameters which have been derived this way (and which, in many cases, provide good descriptions of the spectra) represent physically meaningful quantities or whether they should be viewed more as empirical fitting parameters. Of course, in principle it is possible to measure state-to-state rotational excitation rates independently, from which it should be possible to predict the line shapes. Such experiments are quite difficult, but a particularly interesting study of N2 was recently reported [48]. The measured rates did, in fact, provide good predictions for Q-branch line shapes and also provided a basis for assessing parameters from the several models which have been used to fit these spectra.

For the case of infrared spectra, including Q-branches of perpendicular transitions such as bending modes in CO2, where the tensor order is q = 1, the situation is complicated by the fact that Eqs. (18) and (20) are no longer rigorous. Numerical calculations for O2 [42J and CO2 [39J suggest that the latter may fail by as much as an order of magnitude. Nonetheless, studies of such spectra have generally used these or similar relationships; see, for example Ref. [49J and references therein. To relate line widths and state-to-state rates for infrared spectra, Eqs. (18) and (20) are generally replaced by a similar approximation for the diagonal pressure-broadening cross sections (sometimes called the random-phase approximation [5J):

(22)

that is, the line-width rate is just the average rate of (rotationally inelastic) rates out of the two spectroscopic levels. Equation (22), along with some scaling law for the rotational excitation rates, can then be used to fit rate model parameters to the

273

c.)

FREQUENCY (c ... ~11 FREOUENCY (em-'I

Figure 6. Experimental and theoretical deviations from a sum of Lorentzian lines for micro-windows in the infrared fundamental vibrational band of CO in He. The solid line gives the ratio of experimental intensity to that expected from a sum of Lorentzian lines. Filled triangles and open circles give corresponding theoretical results from coupled-states calculations and the infinite order sudden approximation, respectively. From Ref. [5].

low-pressure spectrum; in these fits, vibrational dependence of the rates is gener­ally ignored. Use of Eq. (18) to get off-diagonal elements of W is still problematic, especially for P- and R-branches, since each line no longer corresponds to a single rotational level, but some identification, typically with the initial spectroscopic level, is made.

From these considerations it is perhaps not surprising that applications have of­ten required some ad hoc adjustments. For example, work on Q-branches in CO2

bending bands [49] raised questions about 'statistical' weighting functions when com­paring rates in the ground state with rates in the excited bending mode (for use in Eq. (22)) and about possible parity-conserving 'propensity' rules in collision rates. More rigorous theoretical calculations [39] suggested post facto justifications for some of the assumptions needed to account for these data.

4.2. ACCURATE CALCULATIONS FOR CO-He

Infrared spectra in 'micro-windows', i.e. between resonance lines, of CO in He at pres­sures where line coupling is important have been analyzed using methods discussed above, despite the fact that such methods are more appropriate for Q-branch Raman than for infrared P- and R-branch spectra [50,51]. The success of accurate theoretical calculations for predicting shapes of isolated lines in this system (cf. Section 3.1) suggested the utility of confronting the theory with data where line coupling is im­portant. The first attempt [52] used the accurate SCF-CI intermolecular potential

274

but a rather simple molecular scattering method, the infinite order sudden approxi­mation, which, nevertheless, gives a reasonable description for collision dynamics in this system. Resulting cross sections provided a fairly good account of the spectrum at both liquid nitrogen and room temperatures, as shown in Fig. 6, which emphasizes the importance of line coupling by showing the ratio of experimental intensity to that predicted by a simple sum of Lorentzian profiles.

More accurate coupled-channel scattering calculations are quite feasible for CO­He and were done to test the simpler approximation [4]. As expected, the more accurate values also provided a good description of the spectrum (Fig. 6), although the better calculations gave somewhat inferior agreement in the far wing at low tem­peratures; this may reflect a breakdown of the impact approximation, which is funda­mental to both calculations. Figure 7 shows that the theory also accurately predicts spectra at very high pressures where all rotational structure is collapsed.

An interesting result of these coupled-channel calculations was the prediction of surprisingly large imaginary parts for the line-coupling cross sections, which did not satisfy the expected detailed-balance relation, Eq. (21), but rather approximately satisfied a similar relationship involving complex conjugation [5]. There appears to be some confusion about whether infrared line-coupling cross sections, like Q-branch Raman cross sections, should satisfy Eq. (21). The Shafer-Gordon [16] equations, which were often considered to be 'exact,' certainly do not satisfy this relationship. A closer analysis by Monchick [18] shows that the more fundamental formulation of Fano [10] does, in fact, require that line-coupling cross sections satisfy detailed balance regardless of tensor order. However, in restricting calculations to completed collisions, which requires only on-the-energy-shell t-matrices (or, equivalently, S-matrices) and which eliminates frequency dependence from the relaxation matrix, as was done by Ben-Reuven [15] and Shafer and Gordon [16], this property is lost. Comparison of the high-energy limits of both the Shafer-Gordon equation [16] and the more exact formulation [10] suggests that the former is likely to predict imaginary parts for line­coupling cross sections which are too large [18]; fortunately, as discussed in Ref. [5], this effect nearly cancels in actually evaluating the line shape.

Very recently, accurate measurements have been done for the Raman Q-branch of CO in He for a range of pressures [6], and theoretical pressure-broadening cross sections (appropriate for the q = 0 tensor order) have been computed to compare with experiment for this case also. Cross sections for the widths of isolated Raman lines measured at low pressure are compared with theoretical predictions in Fig. 2, along with a similar comparison for infrared values. Agreement is seen to be quite good, although theory appears to underestimate systematically (about 3%) values for the higher rotational levels, which might be significant. Higher pressure spectra are shown in Fig. 8; at 2 atm (Fig. 8a) the lower j lines are strongly overlapped and at 6 atm (Fig. 8b) rotational structure is essentially unresolved. Theoretical spectra, including line coupling, give rather good agreement with experiment. The importance of line coupling for these spectra is emphasized in Fig. 8 by also showing predictions of a simple sum of Lorentzians.

275

Relacive intensity

FREQUENCY (em-')

Figure 7. The experimental infrared band of CO in He at 90 amagat and T = 292K is shown by the solid line. Open circles give the spectrum computed as a sum of Lorentzian lines (i.e., no line coupling) and filled triangles show the profile calculated with theoretical line-width and line-coupling cross sections.

4.3. INFRARED BANDS IN C02

As noted in Section 4.1, analyses of infrared spectra have generally been based on models developed for Raman Q-branch spectra even though assumptions in the latter do not necessarily apply in the case of dipole spectra. In an attempt to develop a more rigorous theoretical understanding for the important case of bending bands in CO2 , a recent study [39] applied the infinite order sudden approximation to molecular rotation. This approximation has two advantages: it is expected to be fairly accurate for CO2 [53], and it generally reduces various cross sections to combinations of a small number of 'dynamical quantities', which highlights relationships among them. For

276

CO:He (0.1 :0.9) 6.0 atm 295K

1.00

0.90

0.80

0.70

~ 0.60 Slim of Lorentzions '01 c 0.50 ~ 0040 oS '" 0.30 p:: < 0.20 u

0.10

0.00

-0.10

-0.20 2137 2139 2141 2143 2145 2147

Raman Shift (em-I)

CO:He (0.1:0.9) 2.0 atm 295K

1.10

1.00

0.90

0.80

~ 0.70 Sum of Lorentzians WI c 0.60 .. ... 0.50 .:: til 0040 p:: (j 0.30

0.20

0.10

0.00

-0.10 2137 2139 2141 2143 2145 2147

Raman Shift (em-I)

Figure 8. Comparison of theoretical (solid lines) and experimental (dots) Raman Q-branch spectra for CO in He at (a) 2 atm and (b) 6 atm. Deviations (theory minus experiment) are shown on the shifted lower axis. The theoretical spectra were obtained from line-broadening and line-coupling cross sections calculated for CO-He, corrected for self-broadening using parameters from a fit to data for pure CO. The importance of line coupling can be seen by comparing with the sum of non-interacting Lorentzian lines which is also shown. Note that, to provide perspective, the sum of Lorentzians has been scaled downward so that it matches the full line-coupling theory at high j values (low frequencies).

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example for the fundamental bending band (000 0-0110) it was possible to clarify the applicability of Eq. (18): cross sections coupling Q(j) and Q(j') are (approximately) the same as rotationally inelastic j -+ j' rates within the excited, 0110, mode, but not those within the ground, 000 0, mode.

Besides a useful formal theoretical framework, the infinite order sudden approx­imation provides a computationally tractable molecular scattering method for many systems of interest. Of course, this also requires knowledge of the intermolecular po­tential, and Ref. [39] adopted a simple pairwise additive atom-atom interaction (for CO2-He), which had been proposed earlier [54] in order to obtain some concrete, if preliminary, estimates for relevant cross sections. From this it appeared that the ap­proximate sum rule, Eq. (22), for example, is rather good, but that the alternate sum rule, Eq. (20), is problematic.

These calculations, of course, predict the whole matrix of line-broadening and line-coupling cross sections, and it is of interest to see how well they might predict experimental data available for this system [55]. Despite the simplicity of the in­termolecular potential, predictions for the (diagonal) line-broadening cross sections agreed with low-pressure measurements to a few percent. At higher pressures, where line coupling is important, however, results were only qualitatively correct. It seems clear that calculations with an improved intermolecular potential are warranted; it should be noted that it is also feasible with current computational abilities to perform coupled-states calculations as well as infinite order sudden approximate calculations for this system [53].

5. Conclusions

The line-shape formalism outlined in Section 2, whose development can be traced through the work of Baranger [2], Fano [10], Ben-Reuven [15], and Gordon [16], ac­counts for widths and shifts in isolated lines and collisional transfer of intensity. Be­cause of the impact approximation, which is fundamental to this theory, it is valid only for frequencies near line centres, in particular, for a range of frequencies small compared with the inverse of a collision duration, which is often a few wavenumbers for typical molecular systems. It must fail in the far wings, as discussed below, but for isolated lines this is usually not a limitation, and this formalism is expected to provide a good description for widths and shifts. For these cases, accurate scattering calculations using accurate intermolecular potentials have now been done for several systems (see, e.g. Section 3); results are in good accord with available experimental data, confirming the expectation that the underlying formalism is valid. In some cases, particularly for species which are difficult to handle and for temperature extremes, theory is probably as reliable as experiment (this is especially true of earlier, 'pre­laser' experiments). As noted in Section 3.4, increasing computational capabilities will permit reliable calculations for more complex systems; nonetheless, development of accurate approximate methods remains an important goal.

For cases where lines are broadened enough to overlap, accuracy of the fundamen­tal formalism away from line centres becomes a more pressing issue. Unfortunately,

278

there have been few good comparisons between accurate theoretical values and ex­periment for such cases. The calculations for CO-He discussed in Section 4.2 show promising agreement, but it would be highly desirable to have additional compar­isons. The problem with detailed balance which arose in this study [5,18]' however, is a warning about potential errors. Since it is now quite feasible to obtain accurate solutions within the current formalism, the time seems ripe to consider calculations which relax some of the basic approximations. In particular, frequency dependence of the relaxation operator (see Eqs. (9) and (11)) can be restored by considering off-the-energy-shell collisional t-matrices. Although such quantities have never been calculated for rotational excitation in molecular collisions, techniques developed for other problems might be adapted [18].

The far-wing problem is quite important in its own right, for example for un­derstanding atmospheric opacity. While 'continuum' (i.e far from strong resonance lines) absorption by water vapour in both the microwave and infrared regions is quite important, it is still poorly understood. Some progress has been made by considering the opposite limit to the impact approximation, the static approximation (see [56] for recent application of this approximation and a review of earlier work). Unfortunately, many regions of the spectrum contain absorption from both nearby (weak) lines and far wings of distant (strong) lines, so that a proper theory must include resonance lines and line coupling, and yet be valid over large frequency ranges. Work to incorpo­rate resonance lines within a static approximation, and based on the auto-correlation function in the time domain (see Eq. (3)), has shown some promise, but it currently relies on unrealistic model interaction potentials and simplistic approximations for molecular dynamics [57]. A major achievement of this work was establishment of the validity (and physical origin) of empirical correction factors which had been postu­lated previously to reduce line strengths in far wings to accord with observations. Again, an approach based on off-the-energy-shell t-matrices within the Liouville for­mulation outlined in Section 2 might provide a more rigorous method for studying these effects [18].

Another area where the current fundamental formalism needs to be extended was alluded to in the discussion of Dicke narrowing in Section 3.2. Formulations based on quantum kinetic (generalized Waldmann-Snider) equations [11,12] yield the same result obtained in Section 2 as a special case, yet they allow for proper inclu­sion of the velocity distribution of the spectroscopic molecule and also for collisional and radiative perturbation of the bath molecules. They can therefore account for Doppler broadening and Dicke narrowing. They can probably also account for the 'inhomogeneous' broadening observed in Raman Q-branches of H2 in Ar [58].

In conclusion, the impact-approximation theory for line broadening, considered by Ben-Reuven in 1966 [15] to be impractical for actual numerical calculations, can now be used to calculate broadening parameters of useful accuracy for many molecular systems of practical interest. This theory includes the effect of collisional transfer of intensity, but there have been few calculations to date which critically test its ability to predict experimental spectra in such cases, and further work is most desirable. In this context it is important to consider possible inaccuracies introduced by ignoring

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the frequency dependence of pressure-broadening cross sections. Generalization of the theory to include velocity effects, Doppler widths and Dicke narrowing is another area which needs further study.

References

[1] Anderson, P. W. (1949) Pressure broadening in the microwave and infra-red re­gions, Phys. Rev.76, 647-66l.

[2] Baranger, M. (1958) Simplified quantum-mechanical theory of pressure broad­ening, Phys. Rev. 111, 481-493; Problem of overlapping lines in the theory of pressure broadening, Phys. Rev. 111,494-504; General impact theory of pressure broadening, Phys. Rev. 112, 855-865.

[3] Green, S. (1985) Calculation of pressure broadening parameters for the CO--He system at low temperatures, J. Chern. Phys. 82, 4548-4550.

[4] Green, S., Boissoles J., and Boulet, C. (1988) Accurate collision-induced line­coupling parameters for the fundamental band of co in He: close coupling and coupled states calculations, J. Quant. Spectrosc. Rad. Transf. 39, 33-42.

[5] Boissoles, J., Boulet, C., Robert, D. and Green, S. (1989) State-to-state rotational phase coherent effect on the vibration-rotation band shape: an accurate quantum calculation for CO-He, J. Chern. Phys. 90, 5392-5398.

[6] Hurst, W. S., Rosasco, G. J., and Green, S. to be published.

[7] Blackrnore, R., Green, S. and Monchick, L. (1988) Polarized D2 Stokes-Raman Q-branch broadened by He: a numerical calculation, J. Chern. Phys. 88, 4113-4119; Green, S., Blackrnore, R., and Monchick, L. (1989) Comment on line widths and shifts in the Stokes-Raman Q-branch ofD2 in He, J. Chern. Phys. 91,52-55.

[8] Blackrnore, R., Green, S. and Monchick, L. (1989) Dicke narrowing of the po­larized Stokes-Raman Q-branch of the v = 0-1 transition of D2 in He, J. Chern. Phys. 91, 3846-3853.

[9] Green, S. (1990) Theoretical line shapes for rotational spectra of HCI in Ar, J. Chern. Phys. 92, 4679-4685.

[10] Fano, U. (1963) Pressure broadening as a prototype of relaxation, Phys. Rev. 131, 259-268.

[11] Snider, R. F. (1960) Quantum-mechanical modified Boltzmann equation for de­generate internal states, J. Chern. Phys. 32, 1051-1060.

[12] Monchick, L. and Hunter, L. (1986) Diatomic-diatomic molecule collision inte­grals for pressure broadening and Dicke narrowing: A generalization of Hess's theory, J. Chern. Phys. 85, 713-718; (1987) Erratum, J. Chern. Phys. 86, 7251; Blackrnore, R. (1987) A modified Boltzmann kinetic equation for line shape functions, J. Chern. Phys. 87, 791-800.

[13] Gordon, R. G. (1968) Correlation function for molecular motion, Adv. Mag. Res. 3, 1-42.

280

[14] Albers, J. and Deutch, J. M. (1973) On the rate equation description of spectral lines, Chern. Phys. 1, 89-98.

[15] Ben-Reuven, A. (1966) Impact Broadening of Microwave Spectra, Phys. Rev. 145,7-22.

[16] Shafer, R. and Gordon, R. G. (1973) Quantum scattering theory of rotational relaxation and spectral line shapes in H2-He gas mixtures, J. Chern. Phys. 58, 5422-5443.

[17] Ben-Reuven, A. (1975) Spectral line shapes in gases in the binary-collision ap­proximation, Adv. Chern. Phys. 33, 235-293.

[18] Monchick, L. private communication.

[19] Arthurs, A. M. and Dalgarno, A. (1960) The theory of scattering by a rigid rotor, Proc. Roy. Soc. London A256, 540-551.

[20] Green, S. and Thaddeus, P. (1976) Rotational excitation of carbon monoxide by collisions with He, Hand H2 under conditions in interstellar clouds, Astrophys. J. 205, 766-785.

[21] Nerf, R. B. and Sonnenberg, M. A. (1975) Pressure broadening of the J = 1-0 transition of carbon monoxide, J. Mol. Spectrosc. 58, 474-478.

[22] Thomas, L. D., Kraemer, W. P. and Diercksen, G. H. F. (1980) Rotational exci­tation of co by He impact, Chern. Phys. 51,131-139.

[23] Green, S. and Thomas, L. D. (1980) On the use of pressure broadening data to assess the accuracy of CO-He interaction potentials, J. Chern. Phys. 73, 5391-5393.

[24] Willey, D. R., Crownover, R. 1., Bittner, D. N. and DeLucia, F. C. (1988) Very low temperature spectroscopy: the pressure broadening coefficients for CO-He between 4.3 and 1.7K, J. Chern. Phys. 89,1923-1928; Willey, D. R., Goyette, T. M., Ebenstein, W. L., Bittner, D. N. and DeLucia, F.C. (1989) Collisionally cooled spectroscopy: pressure broadening below 5K, J. Chern. Phys. 91,122-125.

[25] Palma, A. and Green, S. (1986) Effect of the potential well on low temperature pressure broadening in CO-He, J. Chern. Phys. 85, 1333-1335.

[26] Green, S. (1979) Vibrational dependence of pressure induced spectral linewidths and lineshifts: application of the infinite-order sudden approximation, J. Chern. Phys. 70,4686-4693.

[27] BelBruno, J. J., Gelfand, J., Radigan, W. and Verges, K. (1982) Helium and self­broadening in the first and second overtone bands of 12C 160, J. Mol. Spectrosc. 94, 336-342.

[28] Green, S., Monchick, 1., Goldflam, R. and Kouri, D. J. (1977) Computational tests of angular momentum decoupling approximations for pressure broadening cross sections, J. Chern. Phys. 66, 1409-1412.

[29] Green, S. unpublished results; cf. Ref.[4]

281

[30] Meyer, W., Hariharan, P. C. and Kutzelnigg, W. (1980) Refined ab initio calcula­tion of the potential energy surface of the He-H2 interaction with special emphasis to the region of the Van der Waals minimum, J. Chern. Phys. 73, 1880-1897.

[31J Smyth, K. C., Rosasco, G. J. and Hurst, W.S. (1987) Measurement and rate law analysis of D2 Q-branch line broadening coefficients for collisions with D2, He, Ar, H2 and CH4 , J. Chern. Phys. 87, 1001-1011; Rosasco, G. J. and Hurst, W. S. private communication.

[32] Hutson, J. M. (1988) The intermolecular potential of Ar-HCl: determination from high-resolution spectroscopy, J. Chern. Phys. 89, 4550-4557.

[33] Hutson, J. M. and Howard, B.J. (1981) The intermolecular potential energy sur­face for Ar-HCl, Molec. Phys. 43, 493-516; Hutson, J. M. and Howard, B. J. (1982) Anisotropic intermolecular forces 1. Rare gas-hydrogen chloride systems, Molec. Phys. 45, 769-790.

[34J Green, S. (1990) Raman Q-branch line shapes as a test of the Hz-Ar intermolec­ular potential, J. Chern. Phys. 93, 1496-150l.

[35] Green, S., DeFrees, D. J. and McLean, A. D. (1991) Calculations of H20 mi­crowave line broadening in collisions with He atoms: sensitivity to potential en­ergy surfaces, J. Chern. Phys. 94, 1346-1359.

[36] Green, S. (1991) Pressure broadening data as a test of a recently proposed Ar-H20 interaction potential, J. Chern. Phys. 95, 3888-3890.

[37] DeLucia, F. C. and Green, S. (1988) Recent advances in pressure broadening experiment and theory, J. Mol. Struct. 190, 435.

[38J Millot, G. (1990) Rotationally inelastic rates over a wide temperature range based on an energy corrected sudden-exponential-power theoretical analysis of Raman line broadening coefficients and Q branch collapse, J. Chern. Phys. 93, 8001-8010.

[39] Green, S. (1989) Pressure broadening and line coupling in bending bands of CO2, J. Chern. Phys. 90,3603-3614.

[40J Gordon, R. G. and McGinnis, R. P. (1971) Intermolecular potentials and infrared spectra, J. Chern. Phys. 55,4898-4906.

[41J Lam, K. S. (1977) Application of pressure broadening theory to the calculation of atmospheric oxygen and water vapor microwave absorption, J. Quant. Spectrosc. Rad. Transf. 17,351-383.

[42J Smith, E. W. (1981) Absorption and dispersion in the O2 microwave spectrum at atmospheric pressures, J. Chern. Phys. 74,6658-6673.

[43J Braun, C. (1982) Calculation of the absorption coefficient of the 15 micron V2

band of CO2 using the theory of overlapping lines, J. Mol. Spectrosc. 93, 1-15.

[44J Gordon, R. G. and McGinnis, R. P. (1968) Line shapes in molecular spectra, J. Chern. Phys. 49, 2455-2456.

[45J Rosenkranz, P. W. (1975) Shape of the 5mm oxygen band in the atmosphere, IEEE Trans. Antennas Propag. 23, 498-506.

282

[46] Brunner, T. A. and Pritchard, D. (1982) Fitting laws for rotationally inelastic collisions, in 'Dynarnics of the Excited State', K. P. Lawley (ed.), Wiley, New York, 589-641.

[47] Goldfiarn, R, Green, S. and Kouri, D. J. (1977) Infinite-order sudden approxi­mation for rotational energy transfer in gaseous mixtures, J. Chern. Phys. 67, 4149-4161.

[48] Sitz, G. O. and Farrow, R L. (1990) Pump-probe measurements of state-to-state rotational energy transfer rates in N2 (v = 1), J. Chern. Phys. 93, 7883-7893.

[49] Strow, L. L. and Gentry, B. M. (1986) Rotational collisional narrowing in an infrared CO2 Q branch studied with a tunable-diode laser, J. Chern. Phys. 84, 1149-1156; Gentry, B. and Strow, 1. L. (1987) Line mixing in a N2 -broadened CO2 Q branch observed with a tunable diode laser, J. Chern. Phys. 86, 5722-5730.

[50] Bulanin, M. 0., Dokuchaev, A. B., Tonkov, M. V. and Filippov, N. N. (1984) Influence of line interference on the vibration-rotation band shape, J. Quant. Spectrosc. Rad. Transf. 31, 521-543.

[51] Cousin, C., LeDoucen, R, Boulet, C., Henry, A. and Robert, D. (1986) Line coupling in the temperature and frequency dependences of absorption in the mi­crowindows of the 4.3 micron CO2 band, J. Quant. Spectrosc. Rad. Transf. 36, 521-538.

[52] Boissoles, J., Boulet, C., Robert, D. and Green, S. (1987) lOS and ECS line coupling calculation for the CO-He system: influence on the vibration-rotation band shapes, J. Chern. Phys. 87,3436-3446.

[53] Banks, A. J. and Clary, D. C. (1987) Coupled states calculations on vibrational relaxation in He + CO2 (0110) and He + CO, J. Chern. Phys. 86,802-812.

[54] Clary, D.C. (1982) Ab initio computation of vibrational relaxation rate coefficients for the collisions of CO2 with helium and neon atoms, Chern. Phys. 65, 247-257.

[55] Strow, L. L. and Green, S. unpublished results.

[56] Ma, Q. and Tipping, R H. (1990) Water vapor continuum in the millimeter spectral region, J. Chern. Phys. 93,6127-6139; Ma, Q. and Tipping, R H. (1990) The atmospheric water continuum in the infrared: extension of the statistical theory of Rosenkranz, J. Chern. Phys. 93, 7066-7075.

[57] Boulet, C., Boissoles, J. and Robert, D. (1988) Collisionally-induced population transfer effect in infrared absorption spectra. I. A line-by-line coupling theory from resonances to the far wings, J. Chern. Phys. 89, 625-634; Boissoles, J., Menoux, V., LeDoucen, R, Boulet, C. and Robert, D. (1989) Collisionally­induced population transfer effect in infrared absorption spectra. II. The wing of the Ar broadened V3 band of CO2 , J. Chern. Phys. 91, 2163-2171; Bois­soles, J., Boulet, C., Hartrnann, J. M., Perrin, M. Y. and Robert, D. (1990) Collision-induced population transfer in infrared absorption spectra. III. Tem­perature dependence of absorption in the Ar-broadened wing of the CO2 V3 band, J. Chern. Phys. 93,2217-2221.

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[58] Farrow, R. L., Rahn, L. A., Sitz, G. O. and Rosasco, G. J. (1989) Observation of a speed-dependent collisional inhomogeneity in H2 vibrational line profiles, Phys. Rev. Lett. 63, 746-749.

CONCLUDING REMARKS

J.J.M. BEENAKKER Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

ABSTRACT. A short summary of the principal items of discussion, of the great­est points of interest and the clearest directions for the future that emerged from the workshop is presented.

1. Observations

In summarizing my impressions of the 'state-of-the-field' let me begin by mentioning the climate of optimism that I see with respect to several of the main topics addressed by this NATO advanced workshop. Firstly, there is optimism with respect to our ability to handle potential surfaces. Ab initio calculations, especially in combination with their adaptation to experimental data, seem to lead to promising results, at least for simple molecules. Indeed, the participants received, loudly and clearly, the warning not to proceed without ab initio calculations, since they protect us against overlooking essential aspects of the interaction.

The quality of the information on potentials for dimers, obtainable from spec­troscopy, was another source of optimism. Here we saw how Kacs' question, 'can we hear the shape of a drum?', is positively answered in the statement, 'yes, we can see the shape of a well. '

Optimism also characterized the discussions on scattering calculations, where classical trajectories seem to provide the way forward, especially when they are com­bined with full quantum mechanical calculations at the lower energies and for vibra­tional motions. It appears that the lOS and many other dynamical approximations should be applied only when short cuts are needed.

We were reminded how state- and velocity-selected molecular beams can give valuable information on molecular collisions, and it was demonstrated that refined spectroscopic techniques are beginning to compete. At this point the question arises: do (even the non-traditional) transport properties really compete in helping to deter­mine the potential surface?

Before I try to answer this query, let me make another observation. Not only the contributions, but also the lively discussions, pointed to an important development over the past decade. The quantum chemist, who was originally interested only in how

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to calculate a potential surface, the molecular scattering expert, with his emphasis on the pro's and con's of different approximations, and the Boltzmannologist, usually a gourmet or possibly even a gourmand of techniques to solve his equation, have all been shown to be able to leave their ivory towers and cross the boundaries of their fields. They are now all, in one way or another, active on the same grounds; calculating the effects of molecular collision processes on macroscopic phenomena in the gas phase. This development has contributed in a major way to the success of the meeting. But does this unity in activities mean that we all have the same interest? Here I am less optimistic. The discussion showed that the main interest remains how well can a specific surface or a specific scattering approach yield agreement with macroscopic gas kinetic quantities. However, transport properties were, in the majority of the discussions, only a means to another end. This resulted in the demands repeatedly made to the experimentalist; give us data that better fit our interests. Of course this is a valid point of view. But few of our participants showed an interest in transport properties for their own sake. Therefore, in these concluding remarks, in accordance with the title of the meeting, J wish to draw attention to transport phenomena for their own sake.

2. Transport Properties

As we were shown the kinetic theory of polyatomic molecules is currently at a level of perfection that was reached years ago for the noble gases. The same statement holds also for many of the spectroscopically-induced kinetic phenomena. We have seen the wealth of experimental data by now available on most simple gases and often on mixtures with the noble gases. There are survey papers available on transport phenomena, NMR and light scattering, giving the experimental results in terms of the relevant effective cross sections [1,2]. However, inherent in the Boltzmann equa­tion approach and the availability of large computers is the risk of losing track of the physics of the quantities we calculate. We sum over all possible collisions and, in performing this process, we in fact do little more than repeat nature on our computer. At best the results tell us about the quality of our potential surface and of our way to handle the collisions. But do we really understand more about what is behind the transport phenomenon? For the hydrogen isotopes the spin and isotopic modifica­tions allow us to extract the necessary information on the relevant collision processes, at least in a qualitative sense, directly from the experimental data. But in all other cases the physics of what is happening and what processes determine the individual cross sections remain hidden behind complicated averaging procedures. Kinetic the­ory remains at too high a level of abstraction, so that one needs to go at least one step deeper to learn which are the relevant microscopic processes. Even then there remains a still deeper step; what aspects of the molecular interaction determine these processes? Experimental data can at best show trends in different effective cross sec­tions in going from one molecule to the other. But the real task is for the theorists who can investigate the detailed behaviour of the collision integrals. For this task no new experimental data are necessary. What we need is a well-posed computer

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experiment, say for N2-N2 , on a not too unrealistic potential surface. Questions to be answered are then, for example:

• What is the relative importance of the different types of collisions: elastic, reori­entation, inelastic?

• Is there a dominant group of contributing trajectories? • Is there a dominant aspect of the potential surface?

These questions are intimately related to the concepts of 'intelligent importance sampling' and 'sensitivity analysis' that we encountered in our discussions. Such a programme of work would help to decide to what extent transport properties are relevant for co-determining the potential surface. A computer experiment would also show how far one can trust some relations between different effective cross sections that were obtained in a distorted-wave Born approximation, and that seem to hold rather well in practice. This would allow us, for example, to obtain in an indirect way, rotational relaxation data that are virtually impossible to obtain directly with sufficient accuracy. Further, there are correlations found experimentally between, say, the relaxation of the flux of tensor polarization (as produced in a heat flow) and of the tensor polarization itself (as produced in a flow field); or the influence of a dipole moment on the ratio of the decay of tensor polarization as compared to the decay of a viscous deformation; or the correlation between the production and relaxation cross sections for a tensor polarization for simple molecules and the indication for an effect of large level-splitting on this correlation. All of these correlations have been unambiguously observed but, so far, remain without a clear theoretical basis.

Apart from these questions that deal directly with the physics of transport phe­nomena, there is a group of questions related to the quality of our theoretical treat­ment. Again a computer experiment can provide answers to a few questions:

• How important are the higher order Chapman-Cowling corrections? • How important are tensorial contributions to the non-equilibrium distribution

function other than the dominant JT and wJT polarizations? • Why is it that for some molecular gases, odd-in-j polarizations occur, while they

are normally absent, or too small to observe? • Can we obtain information on the scalar factor of the tensorial deformation of

the distribution function?

3. The Future

As should be clear, there is a programme of work to keep quite a few theorists in business for some time. This brings me to Wakeham's observation that 'measuring transport properties is out of fashion.' My impression is that the experimentalists have done their jobs, so that now is the time of the theorists. Unless new, unexpected, developments come out of their work, no new physical insight is to be gained from new experiments. At such a point experimentalists are no longer driven by physics, and a subject may fall out of fashion, if one wishes to use that expression. The experimentalist then looks for new hunting grounds, such as optical manipulation of

288

the distribution function to yield such fascinating new phenomena as light-induced drift [3].

There is, however, a risk involved in such a situation, because know-how rapidly decays. Hence if, in the future, new data are needed, it will be costly and difficult to obtain them. If, indeed, high quality transport data are in demand in the foreseeable future, possibly for the sake of technical applications, only a European-wide solution will solve Wakeham's problem. I see no other possibility for permanent funding. But there is also a lesson to learn from this situation. If you perform experiments in an active field, it is necessary to realize that it is easy for you to extend the range of your data sampling. You never know who will need the additional data in the future. This is in my opinion a moral obligation towards the future.

This brings me to the end of my concluding remarks. There remains only to express our sincere thanks to the organizers who put so much effort into the realization of their ideas and with such success!

References

[1] Hermans, P. W., Hermans, L. J. F. and Beenakker, J. J. M. (1983) A survey of experimental data related to the non-spherical interaction for the hydrogen isotopes and their mixtures with noble gases, Physica 122A, 173-21l.

[2] Van Houten, H., Hermans, L. J. F. and Beenakker, J. J. M. (1985) A survey of experimental data related to the non-spherical interaction for simple classical linear molecules and their mixtures with noble gases, Physica 131A, 64-103.

[3] Werij, H. G. C. and Woerdman, J. P., (1988) Light-induced drift of Na atoms, Phys. Rep. 169, 145-208.

List of Participants

Prof. M.J. Assael Department of Chemical Engineering Aristotle University of Thessaloniki 54006 Thessaloniki Greece

Prof. A. van der A voird Institute of Theoretical Chemistry University of Nijmegen 6525 ED Nijmegen The Netherlands

Prof. J.J.M. Beenakker Huygens Laboratory Leiden University P.O. Box 9504 2300 RA Leiden The Netherlands

Dr. G.D. Billing Department of Chemistry H. C. 0rsted Institute University of Copenhagen DK-2100 0, Denmark

Prof. Ph. Brechignac Laboratoire de Photophysique Moleculaire, Universite de Paris-Sud, Biitiment 213, F-91405 ORSAY CEDEX France

Prof. C.A. Nieto de Castro Departmento de Quimica Faculdade de Ciencias Universidade de Lisboa 1700 Lisboa Portugal

289

Dr. A.S. Dickinson Dept. of Physics University of Newcastle Newcastle upon Tyne NEI 7RU United Kingdom

Miss. I. Dimaki Department of Chemical Engineering Aristotle University of Thessaloniki 54006 Thessaloniki Greece

Dr. M. Faubel Max Planck Institut fur Stromungsforschung Bunsenstrasse 10 D-3400, Gottingen Germany

Dr. S. Green NASA Goddard Space Flight Cente Institute for Space Studies 2880 Broadway New York, NY 10025, USA

Prof. F .A. Gianturco Dipartmento di Chimica Universita Roma I Citta U niversitoria 00185 Roma Italy

Mr. E. Hanson Department of Chemistry University of Waterloo Waterloo, Ontario Canada N2L 3G 1

290

Dr. L.J.F. Hermans Huygens Laboratory Leiden University P.O. Box 9504 NL 2300 RA Leiden The Netherlands

Dr. J.M. Hutson Department of Chernistry University of Durham Durham DH1 3LE United Kingdom

Mr. L. Karagiannidis Department of Chemical Engineering Aristotle University of Thessaloniki 54006 Thessaloniki Greece

Prof. M. Keil Deptartment of Chemistry University of Alberta Edmonton, Alberta Canada T6G 2G2

Prof. L K uscer Odelak za fiziko Univerza v Ljubljani PO Box 64 61111 Ljubljana Yugoslavia

Prof. W-K. Liu Department of Physics University of Waterloo Waterloo, Ontario Canada N2L 3G 1

Dr. M. Laranjeira Centro de Fisica Molecular Universidade de Lisboa Complexo I LS.T. 1000 Lisbo Portugal

Prof. A.J. McCaffery Department of Chemistry University of Sussex Brighton BN1 9QJ Sussex, UK

Prof. F.R.W. McCourt Department of Chemistry University of Waterloo Waterloo, Ontario Canada N2L 3G 1

Dr. U.V. Mardolcar Department de Fisica Complexo Interdisciplinar Instituto Superior Tecnico 1096 Lisboa Portugal

Dr. J. Millat Universitat Rostock Sektion Chemie Hermannstr 14 DO 2500 Rostock 1 Germany

Dr. M Rigby Deptartment of Chemistry Kings College London WC2R 2LS United Kingdom

Mr. N. Sanna Dipartmento di Chimica Universita Roma I Citta Universitoria 00185 Roma Italy

Dr. J. Schleipen Mol. and Laser Fysics University of Nijmegen 6525 ED Nijmegen The Netherlands

Dr. B. Schramm Physikalisch-Chemisches Institut der Univerisitat 1m Neunheimer Feld 253 D-6900 Heidelberg Germany

Mr. S. Serna Dipartmento di Chimica Universita Roma I citta Universitoria 00185 Roma Italy

Dr. E.B. Smith St. Catherines's College University of Oxford Oxford OXI 3Uj United Kingdom

Prof. R.F. Snider Department of Chemistry University of British Columbia Vancouver, British Columbia Canada V6T 1 Y6

Prof. S. Stolte Faculty of Chemistry Vrije Universiteit De Boelelaan 1083 1081 HV Amsterdam The Netherlands

Dr. F.J. Uribe Department of Physics Universidad Autonoma Metropolitana 09340-Iztapalapa Mexico

Dr. V. Vesovic Department of Chemical Engineering Imperial College London SW7 2AZ United Kingdom

Prof. L.A. Viehland Parks College of Saint Louis University Cahokia, Illinois USA 62206

Prof. W.A. Wakeham

291

Department of Chemical Engineering Imperial College London SW7 2AZ United Kingdom

Index

A Over N 239,242 ab initio calculations 189, 190, 192, 201 action-angle variables 206 Anderson theory 269 angular momentum

rotational 126, 127 polarization 38, 287

Ar-HCI 58, 266 atom-atom scattering 86 atom-diatom 59 atomic ions 189 auto-correlation function 168, 259, 260,

278 basis functions 122, 128, 157 BIMAX 193, 198, 200 binding sites 251 Bohr-Sommerfeld quantization 208 Boltzmann equation 119, 121, 127, 189,

190, 196, 286 linearized 158 generalized 127

Born-Oppenheimer separation 4 bound state 57 bulk (volume) viscosity 37, 39, 51 calculations

ab initio 10, 45 CI190 classical trajectory 44, 132, 145 close-coupling (coupled-channel)

44, 58, 139, 262, 266, 274 intramolecular electron correlation

11 supermolecule method 12

capillary viscometer 31 Casimir-Polder formula; generalized 4 CC-theory vs IOSA cross sections

Ar-0 2 102 He--N2 92

centrifugal barrier 64 coupling 267

293

distortion 266 centrifugal sudden approximation

(CSA) 44, 61, 230 Chapman-Cowling

approximations 123, 125, 126, 128, 129, 132, 133, 135, 145, 146

corrections 287 Chapman-Enskog solution 120, 121,

124 charged systems 189 classical

mechanics 189, 190, 191, 192, 193, 194, 196

path 57, 60, 64, 68 trajectories 44, 45, 46, 47, 60, 61,

209, 210, 214, 230, 232, 270, 285

close-coupling (CC) 58, 182, 228 C02 275 CO-He 261, 273 coherent Raman spectroscopies 258,

269 collision-induced

fluorescence (CIF) 218, 231 rotationally inelastic transitions

237 collision

elastic 190 integrals 226, 228 number 40 stereochemistry 184 superelastic 193, 195 operator 119, 193, 198 operator, linearized 119, 122, 158,

226 collisional

reorientation 238 transitions of large inelasticity 242 transfer of intensity 277, 278

collisions 189, 190, 191, 193, 196 collocation 197

294

coupled-states (CS) approximation 58, 60-64, 68, 263, 266

cross section 57, 60, 67, 175 coupling 133 diagonal (see relaxation) 123, 158 differential 61, 119, 144 effective 40, 42, 123, 125, 128, 129,

130, 133, 139, 166, 167 generalized 176, 181, 218, 224, 226 integral 61, 63, 84, 144 line broadening 261, 277 line-coupling 277 non-diagonal (see production) partial 64 pressure broadening 263, 270, 278 production 133, 156, 158, 161,

165, 228 relaxation 68, 163, 228 state-to-state 262, 265

cross sections 38, 39, 40, 41, 45, 50, 189, 190, 191, 192, 193, 194, 195, 196, 199, 200

experimental averaging 73, 76, 79, 83

diffusion 46 ideally resolved 73 lineshape 63 multipole 208, 231 rotational transfer 180 transport 228 viscosity 46

crossed molecular beams 83, 84 cut-off procedure 62 D2-He 264 damping functions 11 decoupling approximation 60 deexcitation 194 deformed hard-shell model 94, 108 density matrix 265 dephasing 271 detailed balance 272, 275, 277 diatom 190, 191, 192, 193, 194, 196 Dicke narrowing 264, 266, 278 diffraction oscillation 86, 93

inelastic phase shift 92 diffusion 40, 43, 47

anisotropic 198 binary 35 coefficient 125, 144, 189, 196, 198,

199, 200 of internal energy 39,40,43,47,

51 parallel 196, 200 perpendicular 196, 200 self 39 transverse 161

dilute gases 189, 195, 196 discretization 197 dispersion 59 distorted-wave 60, 287 distribution function 126, 195, 196, 198

equilibrium 119 initial 208 nonequilibrium 118, 119, 126, 131,

132, 133 product-state 247 rotational 247 sampling 212 velocity 195, 196, 198 Wigner 209, 219, 221

Doppler broadening 278 profile 181

double resonance lineshapes 184 drift tubes 184, 191 Dufour effect 124, 162 dynamical scattering approximations

centrifugal sudden or coupled states (CS) 44, 58, 60-64, 230, 263, 266, 272, 273, 275, 277

corrected CS 230 energy sudden 44 infinite order sudden (lOS) 43-47,

49, 230, 272, 273, 275, 277 sudden 177

eigenfunction 196, 198 electron correlation 192

electrostatic field 189, 195 energy

transfer 215 vibration-rotation 206, 207 zero-point 208

entropy production 29 equilibrium 191, 192, 195 Eucken

factor 118 number 131

excitation 193 experimental techniques 50 exponential

approximation 64, 67 distorted wave 60, 62 perturbation theory 57, 62

factorization rule, use of 93 far-infrared 58 field-effects 118, 136-142, 144, 146,

158-168 fitting laws 239, 242 flow birefringence 143, 158, 164 fluorescence intensity 221 forces

attractive 190, 192 ion-induced dipole 192 repulsive 190, 192 thermodynamic 120, 124, 127, 130,

162 four-vector correlation 185 Fourier transform 62 fragment rotational distributions 238 frequencies

libron 19 magnon 19 phonon 15, 19

functional equation 196 functional form 190 g-tensor 163 Galerkin methods 197 gaseous ion transport 189 Gauss-Legendre quadrature 60 glory oscillation 61 H 2-He 228

half-collision 246, 247 HD-He 229 heat flux vector 120 Heisenberg exchange 21 high temperature 39 Hilbert space 120, 126

295

hydrodynamic subspace 120 non-hydrodynamic subspace 120,

122 hindered rotation 245 hydrodynamic regime 195 hydrogen 190, 191, 192 impact

approximation 259, 261, 266, 270, 274,277,278

parameter 60, 63 inelastic 57

collision 37, 46, 190, 191 Fraunhofer scattering theory 94 lineshape 225 neutron scattering 21 rates 239

infrared spectra 272, 273, 275 interaction potentials 57,117,145,170,

189, 190, 191, 192, 193, 194, 201

anisotropic 118, 126, 145 isotropic 11 7

interactions dispersion 2, 6, 8, 9, 10 electrostatic 4, 6 exchange 2, 6 induc~ion 2, 59 multipole-multipole 2

intermolecular interaction 237 potential 38, 44, 49, 60, 286 potential, anisotropic 50 non-spherically symmetric 49

internal energy relaxation 39 energy states 190 rotational barrier 245 states 118

296

inversion direct method 49, 89 tunneling 18

ions 189 irreducible

L-basis vector 223, 224 tensor operators 226 tensorial components 231

isotropic distributions 195, 196 potentials 190

Kagan polarization 164, 167 kinematic apse 179 kinematical resolution 79 kinetic

equation 189, 190, 191, 231 theory 38, 189, 190, 191, 193, 194,

197, 198, 200 lattice

dynamics 19 vibrations 2

least-squares fitting 57 Legendre

expansion 59 polynomials 60

Li2(A)-Ne 231 Li2(A)-Xe 231 light scattering

line

depolarized Rayleigh (DPR) 142, 155, 168-169, 286

Raman 142, 271-273, 275

broadening 67 coupling 68, 259, 272, 273 shift 68, 257, 277 width 68, 257, 263, 277

linear equations 196, 197 molecule anisotropies 103

linearized collision operator 226 lineshape 57, 237, 259, 260, 271

Fano-Gordon-Ben-Reuven formu-lation 266

cross sections 63

and molecular spectra 237 Liouville

basis (L-basis) 223, 224 formalism 218, 223, 231, 238 operators (L operators) 223, 260

lithium ions 191 M-conserving 62, 63, 67 Magnus expansion 62 Mason-Monchick approximation 43,

44,49 mass spectrometer 189 maximum value of the transferred an­

gular momentum 241,247 Maxwell

gas 122, 123 distribution 155, 195, 196, 198

microwave 58 mixtures 44, 124, 134, 141 MOBDIF 194, 198, 199, 200 mobility 189, 190, 196, 199 molecular

alignment 184 beam(s) 57, 83, 84, 285 beam sources 80 ions 184

molecule(s) 189 detectors 77 nearly rigid 3 Van der Waals 1, 6, 18, 103, 237,

238, 243, 249, 254 MOLSCAT 64, 230, 232 moment

method 225 of distribution function 197, 198

momentum-transfer 191, 193 Monte Carlo procedure 212 Morse oscillator 207, 209 multipole

cross section (MCS) 218, 231 moments 59, 177

multi-property analysis 144, 156 MW or near-infrared spectroscopy 244 N2-He 94-97, 145, 146, 230 Nt -He 231

Newton velocity diagram 76, 79 nitrogen 191 nonspherical potentials 190, 191 nonvibrating molecules 193 nuclear

quadrupole hyperfine-structure 239

magnetic resonance/relaxation (NMR) 143

number density 189, 196 Onsager symmetry 124, 133 operator equations 196, 197, 198 orbital correction 62, 64 orbiting resonances 229 orientation narrowing 239 orthogonal 197, 198 oscillating-disk viscometer 31 peaking approximation 61, 62, 64 phase

factor 213 shift 62 transitions; magnetic 19

phenomenological coefficients 30 polarized

cross section 178 rotational transfer cross section

176,178 polarizabilities 59

anisotropic 156 frequency-dependent 2, 4

polarization ratio velocity dependence (PRVD) 182

polarization(s) 118, 132, 133, 134, 159, 160, 169, 182, 218, 287

Kagan-type 164, 167 Waldmann-type 164, 165

potential Ehrenfest average 215 effective 58 energy 189, 190, 191, 192, 193,

194, 201 energy surface 57 well depth 58

potential tests

Ar-Ar 88 Ar-02 99, 100 He-C2H2 106 He-C02 104, 105 He-N2 93, 97 Kr-Kr 89

potentials anisotropic 47 additivity of 8 atom-atom 3, 18 Axilrod-Teller 9 C2Hc C2H4 6, 14 Lennard-Jones 6 N 2-N2 6 segment-segment 7 site-site 3 spin-dependent 7 ab initio; HF-HF; H20-H20;

NH3-NH3; C2H2-C2H2 14 ab initio; N2-N2;02-02;CO-

297

CO;Ar-NH3;Ar-H20 10, 14 Prandtl number 131 predissociation 238 pressure

broadening 63, 68, 238 tensor 120

product-state distribution 247 production cross section 156, 158, 161,

165, 228 quadrature 197 quadrupole 192 quantum kinetic equations 259 quantum mechanics 190, 191 quasiclassical 192 radiative excitation 219 rainbow 61, 64, 68

scattering 86 Raman spectra 271-273,275 random-phase approximation 19, 272 rare gas 57 rate equations 259 Rayleigh-Schrodinger perturbation

theory 2 reactions 195, 196, 199

298

red shift additivity 251 relaxation

cross section 68, 228 operator 278 phenomena 142-143, 191 rotational 118, 129, 169 times 37 vibrational 118, 131

resonance 64 rigid rotor 39, 40, 44, 47, 190, 191, 192,

193, 201 rotation 190, 193, 194, 195 rotation matrix 61 rotational

cooling 82 dipolar absorption 156 distribution 247 energy gaps 259 excitation 262 inelasticity 238 inelastic rates 272 rainbow tests 99 Raman scattering 142, 156 relaxation 37, 40, 43, 131, 263 RKR procedure 244 state-to-state rates 259, 271, 272 transfer 175

rotationally inelastic scattering Ar-CO 101 Ar-0 2 99 He + N2 ; CO; O2 ; CH4 91 He-C02 ; C2 H2 105, 110 K-N2; C02; C2 H2 107 Li+-N2 ; CO 101

S-matrix 258, 261 scaling relation for rotational transfer

rates 177 scattering 190, 192, 193 scattering matrix 60 SCF calculations 192, 193 selection 181 self-diffusion 39 semiclassical 192

(classical path) approach 212, 214, 215

Senftleben effects 126 Senftleben-Beenakker effects (SBE) 1,

136, 137, 155, 158-168, 169, 217

sensitivity 145 analysis 287

shock-tube 37 sound absorption 37, 130, 132 spectra

Raman 271-273, 275 Van der Waals 1, 18, 58, 144

spectral intensity 260, 270 static approximation 278 statistical weighting functions 273 stretching mode 252 sub-Doppler laser excitation 222 superoperators 223 supersonic beam 184 swarm 189, 190, 191, 198 symmetry oscillations 88, 90 tensorial cross section 238, 239 thermal conductivity 31, 34-35, 38, 41,

44, 128, 129, 133, 137, 139, 144

data 34 excess 135 experimental apparatus 34 field-induced changes 160, 164,

165 interaction 135 internal part 38 low temperature 35 mixture 124, 135 total-energy flux approach 129 two-flux approach 118, 129, 140 translational part 37, 38, 39

thermal diffusion 36 coefficient 125, 136, 144, 146 factor 35 transverse 162

thermal transpiration 37, 39 thermodynamic

fluxes 121, 124, 218 forces 120, 124, 127, 130, 162

three-vector correlation 185 time-dependent perturbation theory 60 time-evolution 62 translational-internal coupling scheme

178 trajectories 191, 192, 193, 194, 199,

200 transfer of angluar momentum 177 transient hot-wire 34 translation 61, 189 translational

energy 61, 64 motion 60

transport coefficients 30, 38, 49, 117, 121,

123, 124, 126, 134, 159, 189 coefficients, traditional 30, 50, 51 longitudinal 161 phenomena 30, 225 tensorial 136, 137 transverse, 161

transport-relaxation equations 226 tunnelling 245 two-photon inelastic lineshape 218 two-vector correlation experiment 185 Van der Waals

complex 57, 68, 103, 237, 250, 258, 262, 266, 268

dimers 13 interaction 2, 14 minimum 8, 11, 57 modes 251, 252 molecule 1, 6, 18, 103, 237, 238,

243, 249, 254 spectra 1, 18, 58, 144

vector correlation 176

velocity distribution 195, 196, 198 polarization 181

299

polarized rotational transfer cross sections 180

velocity-selected experiments 176 vibration 190, 193 vibrational

coupling 247 inelasticity 266 predissociation 246 relaxation 40, 43 transfer 175

virial coefficients 14, 15, 49, 191 viscosity 38,45, 50

data 31 excess 135 experimental apparatus 31 high temperature 30 interaction 134 low temperature 30 mixture 124, 125, 134 shear 30-33, 118, 128, 129, 133,

137, 144, 164 volume 118, 129, 130, 132, 134,

144 VSDS 181 Waldmann-Snider equation 127, 158,

219,225 linearized 225

Wang Chang-Uhlenbeck theory 211 Wang Chang-Uhlenbeck-de Boer

equation 190 weak-radiation-field limit 258 weighted residuals 197, 198 width parameter 213 Wigner distribution function 209, 219,

221