From High Resolution Spectroscopy to Detailed

Molecular Quantum Dynamics

David LuckhausLaboratorium fur Physikalische Chemie, ETH Zurich (HCI), CH–8093 Zurich, Switzerland

present address: Institut fur Physikalische Chemie, Universitat Gottingen, Tammannstrasse 6,D–37077 Gottingen, Germany

1 Introduction

Understanding the mechanisms of energy redistribution at high ro–vibrational excitationis essential to all models of chemical reactions on a microscopic level, statistical or other-wise. From the chemist’s point of view the most intriguing aspect lies in the associationof dynamical properties with structural features, e.g. functional groups, in analogy withthe chromophore concept so well established in spectroscopy. It is the possible transfer-ability of dynamical properties which ultimately establishes the link between molecularstructure and reactivity, arguably the corner–stone of modern chemistry. In this fieldmolecular spectroscopy assumes a particularly prominent role – not only providing themeans to prepare molecular systems in well defined states but also serving as a uniqueprobe for the ensuing molecular motion. In combination with detailed modelling, high res-olution overtone spectroscopy offers an elegant – and often complementary – alternativeto the ’direct’ time resolved approach. The spectral analysis on the basis of microscopicmodels for the molecular motion reveals ”hidden” (approximate) symmetries, couplinghierarchies and ”constants” of the motion on different time scales not easily detected intime dependent experiments [1–3]. While numerous aspects can be analysed in terms ofalgebraic models, often referred to as the ”effective Hamiltonian” approach, only the de-tailed quantum mechanical model in a coordinate representation can relate the dynamicalfeatures to the molecular structure.

2 The Z–Matrix Discrete Variable Representation

Within the Born–Oppenheimer approximation the Hamiltonian for the nuclear motionof a molecule in field–free space takes the following simple form in space–fixed Cartesiancenter of mass coordinates X

H =N∑

a=1

−(h2

2ma

∑α

∂2

∂X2α,a

)+ V (X) (1)

h is Planck’s constant h devided by 2π. N is the number of atoms andma are their masses.α labels the Cartesian axes. V is the potential energy. In principle the Schrodinger

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equation can be solved directly with the Hamiltonian given in the form of Eq.(1), butusually one would like to formulate the Hamiltonian in terms of coordinates that areadapted to the molecule’s motion, i.e. coordinates that come as close as possible toa separable picture by minimizing the coupling between different degrees of freedom.Over the years various schemes have been suggested for the derivation of ro–vibrationalkinetic energy operators [4–6]. Starting from Eq.(1) the ro–vibrational Hamiltonian forany coordinate system can be written in terms of the total angular momentum J andgeneralized vibrational coordinates qk and their conjugate momenta pk = −ıh∂/∂qk [7]:

H =1

2

(Jt, pt

)( I CCt g

)−1 (Jp

)+ u+ V (2)

J and p are column vectors with components Jα and pk, respectively. t indicates thetranspose. I is the instantaneous tensor of inertia and g is the covariant vibrationalmetric tensor. C is the Coriolis coupling matrix. u is a mass dependent pseudo potential,which results from the curvature of the rotational and vibrational coordinates but onlydepends on the latter [7].

u(q1, q2, . . .) =h2

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∑j,k

∂ {Gjk · fk}∂qj

+ fj ·Gjk

4· fk (3)

fj =∂

∂qj`n

∣∣∣∣∣ I CCt g

∣∣∣∣∣ (4)

Gij are the vibrational elements of the contravariant metric tensor and | · | indicates thedeterminant. The only condition the {qj} must fulfill is that they define the moleculargeometry unambiguously, i.e. X(q) must be single–valued. Explicit expressions for thecomponents of I, C, and g follow immediately:

Iαβ =∑a

ma (~eα × ~xa) (~eβ × ~xa) (5)

Cαj =∑a

ma (~eα × ~xa)∂~xa

∂qj(6)

gjk =∑a

ma∂~xa

∂qj· ∂~xa

∂qk(7)

~eα (α = x, y, z) are unit vectors along the Cartesian coordinate axes , and ~xa is theCartesian coordinate vector of atom a, all referring to the (rotating) molecule–fixed centre-of-mass coordinate system. Even if other approaches may be more convenient to deriveanalytical expressions for the metric tensor elements, Eqs.(2–7) are exact and completelygeneral. They are particularly suitable in combination with grid based representations

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of the Hamiltonian where the metric tensor elements can be calculated by numericaldifferentiation and pointwise inversion without any further algebra [8, 9]. The Z–matrixDVR combines this approach with a flexible definition of the geometry in terms of localpolar coordinates and a generalized Chebychev discrete variable representation (DVR,see [10] for a recent review of DVR methods): the result is a method for the treatmentof ro–vibrational dynamics with a very wide range of applicability. It includes the ’exact’treatment of all six vibrational degrees of freedom of tetratomic molecules up to highexcitation energies, not only in the semi–rigid limit (CH2O) but also including largeamplitude torsion (H2O2) and inversion (NH3) dynamics [8]. Most importantly, Eqs.(2–7) are not limited to the exact full–dimensional treatment of intramolecular dynamics.Rather the Z-matrix DVR provides a unified approach to vibrational quantum dynamicsincluding the full–dimensional treatment as a special case. Recent applications includemulti–dimensional models of the torsion–inversion tunneling [11,12] and photodissociationdynamics [13] of hydroxylamine (NH2OH) as well as the intermolecular ro–vibrationaldynamics of CH+

3 –Rg dimers (Rg=He,Ne) [14].

3 Normal Coordinate Models for High Excitations

A particularly simple form of the kinetic energy arises for normal coordinate models. Herethe covariant tensor g is diagonal so that the kinetic energy becomes almost separable aslong as Coriolis couplings can be neglected in Eq.(2):

H = hc

(3N−6∑k=1

ωk

2p2

k

)+ V (q1, q2, ..., q3N−6) (8)

qk = Qk

√ωkhc

h2 (9)

Qk are the usual mass–weighted normal coordinates and ωk are the corresponding har-monic wavenumbers. This formulation has been particularly useful for deriving clampedcoordinate models (i.e. setting qk, pk = 0 except for a few dynamically relevant coor-dinates) to describe the approximate separation of light atom CH stretch–bend ”chro-mophore” dynamics from the remaining vibrations of the molecular frame (see refs.[2, 3, 15, 16] and references cited therein). A physically intuitive justification for thisapproximation lies in the small displacements of the molecular frame coordinates eventhough the chromophore vibrates with fairly large amplitudes. The vibrational angu-lar momentum coupling roughly scales with the inverse tensor of inertia, which is smalland approximately constant for a sufficiently large molecular frame as long as its averageexcitation remains low. An analogous reasoning usually justifies the neglect of pseudo–potential and vibrational angular momenta within the chromophore subspace [15, 16]. Incontrast to local internal coordinate descriptions normal coordinate models account forthe small amplitude vibration of the molecular frame up to quadratic contributions. Byno means limited to semirigid CH chromophores (CHXYZ [2, 3, 15–17], CH2O [18]), thisansatz is generally valid for systems with XH vibrations, including molecules with amino

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(NH) and hydroxyl (OH) groups. Its validity extends to the bound dynamics of moleculeswith ”floppy” degrees of freedom, such as internal rotation or inversion [11, 12, 19, 20],as long as the characteristic time scales associated with the large amplitude motion aresufficiently different from those of the (fast) bound dynamics.

The formulation of the model Hamiltonian in reduced normal coordinates qk (eq.8) al-lows for a useful scaling procedure to correct for small deficiencies typical for ab initiocalculated model potential energy surfaces. In general such calculations tend to overes-timate harmonic wavenumbers. In the harmonic limit this error can be removed exactlyby replacing the reference wavenumber ωk in (eq.8) with

ωscaledk =

(νexp

k

νab initiok

)2

ωk (10)

where νab initiok and νexp

k are the calculated and the experimental fundamental wavenum-bers, respectively. Provided anharmonicities are small (a few percent) this scaling proce-dure offers a simple and efficient way for adjusting ab initio calculated potential energyfunctions with only a single parameter per dimension. As this parameter is directly deter-mined from observed fundamental transitions the method allows for a true prediction of theovertone spectrum. Deviations between calculated and observed transition wavenumbersare typically halved compared with pure ab initio results. The scaled normal coordinatemodel was instrumental in analysing the mode specificity of the ro–vibrational dynamicsunderlying the overtone spectrum of hydroxylamine (NH2OH) from the far infrared up tothe visible range of the spectrum [11,12,21]. Combining this approach with reaction pathmodels for the various large amplitude motions yielded an almost complete picture of theintramolecular energy flow in NH2OH up to excitations close to the dissociation limit.This includes the large amplitude limit of the NOH bending motion, the isomerization toammonia oxide (NH3O) via intramolecular proton transfer.

4 Adiabatic Reaction Path Models

With Eqs.(2–7) vibrational coordinates need not be given in closed form in terms ofthe molecular geometry. An important class of coordinates for the description of largeamplitude motions in larger molecules, are the reaction path coordinates, loosely definedas a path on the potential energy hypersurface connecting reactants and products. Theremaining coordinates required for a complete parameterization of the molecular geometrycan be defined as normal coordinates by projecting the displacement along the reactionpath out of the molecular configuration space at each value of the reaction path coordinate[22]. For an N–atomic non–linear molecule Eqs.(2–7) lead to the following ro–vibrationalHamiltonian in terms of a large amplitude (reaction path) coordinate q and a set of 3N−7normal coordinates Q, their conjugate momenta (Jq and P, respectively), and the angular

momentum Jx,y,z:

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H =(J − π

)tµ(J − π

)+∑k

P 2k + u(q, ~Q) + V (q, ~Q) (11)

µ =(I − C†C

)−1(12)

This form closely resembles the Darling–Dennison normal coordinate Hamiltonian withthe usual definitions [23]. The only difference is that the angular momentum operators Jand π have a fourth element referring to the reaction path coordinate, with correspondingelements of I and C :

Iαq = ~eα

N∑a=1

{(~sa × ~s ′

a) +3N−7∑k=1

Qk

[2(~ak × ~s ′

a

)+

3N−7∑l=1

Ql

(~ak × ~ ′

am

)]}(13)

Iqq = |~s ′ +∑k

~′kQk|2 (14)

Ck,q = ~s ′~k +

3N−7∑l=1

Ql

(~′l

)t ~k (15)

The prime denotes the derivative with respect to q. ~k are displacement vectors with

respect to the mass–weighted Cartesian reference geometry ~s(q), which defines the reactionpath. So far the formalism is exact. Within the RPH model we now introduce theadiabatic ansatz for the eigenfunctions Ψ(n)

m (Q, q) of the full Hamiltonian H:

Ψ(n)m (Q, q) = χ(n)

m (q)ϕn(Q; q) (16)

(〈ϕn|H|ϕn〉Q − E(n)

m

)χ(n)

m (q) = 0 (17)

The semicolon indicates a parametric dependence on q and 〈〉Q the integration over Q.

In the ‘exact’ adiabatic picture ϕn( ~Q; q) are the eigenfunctions of the Hamiltonian forJq = 0. Here we approximate them by a product of 1D harmonic oscillator eigenfunctionsψ(k)

nk(Qk; q) so that n becomes a multi–index.

ϕn(Q; q) =3N−7∏k=1

ψ(k)nk

(Qk; q) (18)

Except for the high OH stretching overtones [24] we completely neglect the Q–dependenceof I and C, Eq.(11). Compared with the neglect of the anharmonicities and the directproduct ansatz of eq.(18) this only leads to minor additional errors but substantially

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simplifies the calculation. Thus eq.(17) reduces to the 1D motion in q with effectiveadiabatic channel potentials εn(q) for each level n.

(〈ϕn|JqµqqJq|ϕn〉 + εn(q) − E(n)

m

)χ(n)

m (q) = 0 (19)

εn(q) =3N−7∑k=1

(nk +

1

2

)ωk +

1

2

∑α,β=x,y,z,q

〈ϕn|παµα,βπβ|ϕn〉 + u(q,~0) + V (q,~0) (20)

χ(n)m (q) are the eigenfunctions associated with adiabatic channel (n). This approach has

been validated for the torsional stereomutation tunneling in hydrogen peroxide bothagainst ’exact’ adiabatic channel calculations and against fully coupled results [25] ona global 6D potential energy hypersurface [26]. It was possible to analyse the pronouncedmode selectivity observed experimentally in the ro–vibrational spectra. The phenomenonwas shown to result from the adiabaticity of the torsional quantum dynamics, which wasfound to be preserved up to very high OH stretching excitations, possibly even beyondthe O–O dissociation limit. With the simplifications of Eqs.(16–20) the adiabatic reactionpath model can be extended to much larger polyatomic systems as demonstrated with itsapplication to the 36–dimensional (36D) inversion tunneling dynamics of aniline [27].

5 Conclusion

The continuing progress in computer technology keeps extending the range of systemswhose quantum dynamics can be treated (numerically) ’exactly’ in all degrees of freedom.But it seems doubtful that much insight is to be gained from such a calculation for -say - a DNA molecule even if it were feasible. True chemical–physical understanding willalways come from simplified models to describe regularities and approximate symmetries.There is a large variety of approaches to choose from of which this short survey couldonly sketch a few. In the quantum dynamics of large amplitude motion, however, acommon feature seems to be emerging with the phenomenon of vibrational adiabaticity.Exploring its dynamical implications – for intramolecular energy redistributions as wellas for reactive processes – will be one of the major challenges in the field of molecularquantum dynamics.

References

[1] David Luckhaus and Martin Quack. The role of potential anisotropy in the dynamicsof the CH–chromophore in CHX3 (C3v) symmetric tops. Chem. Phys. Lett., 205:277–284, 1993.

[2] Martin Quack. Molecular quantum dynamics from high resolution spectroscopy andlaser chemistry. J. Mol. Struct., 292:171–196, 1993.

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[3] A. Beil, D. Luckhaus, and M. Quack. Fermi resonance structure and femtoseconddynamics of a chiral molecule from the analysis of vibrational overtone spectra ofCHBrClF. Ber. Bunsenges. Phys. Chem., 100(11):1853–1875, 1996.

[4] Nicholas C. Handy. The derivation of vibration–rotation kinetic energy operators, ininternal coordinates. Mol. Phys., 61(1):207–223, 1987.

[5] Fabien Gatti, Christophe Iung, Michel Menou, Yves Yustum, Andre Nauts, andXavier Chapuisat. Vector parameterization of the N–atom problem in quantummechanics. I. Jacobi vectors . J. Chem. Phys., 108(21):8804–8820, 1998.

[6] Mirjana Mladenovic. Rovibrational Hamiltonians for general polyatomic moleculesin spherical polar parameterization. I. Orthogonal representations. J. Chem. Phys.,112(3):1070–1081, 2000.

[7] R. Meyer and Hs. H. Gunthard. J. Chem. Phys., 50:353–365, 1969.

[8] David Luckhaus. 6D vibrational quantum dynamics: Generalized coordinate discretevariable representation and (a)diabatic contraction. J. Chem. Phys., 113(4):1329–1347, 2000.

[9] David Lauvergnat and Andre Nauts. Exact numerical computation of a kinetic energyoperator in curvilinear coordinates. J. Chem. Phys., 116(19):8560–8570, 2002.

[10] J. C. Light and Tucker Carrington Jr. Discrete–variabale Representations and TheirUtilization. Adv. Chem. Phys., 114:263–310, 2000.

[11] David Luckhaus. The rovibrational spectrum of hydroxylamine: A combined highresolution experimental and theoretical study. J. Chem. Phys., 106(20):8409–8426,1997.

[12] David Luckhaus. The rovibrational dynamics of hydroxylamine. Ber. Bunsenges.Phys. Chem., 101(3):346–355, 1997.

[13] David Luckhaus, Jacqueline L. Scott, and Crim F. Fleming. An experimental andtheoretical study of the vibrationally mediated photodissociation of hydroxylamine.J. Chem. Phys., 110(3):1533–1541, 1999.

[14] Otto Dopfer and David Luckhaus. Rovibrational Calculations for CH+3 –Rg

(Rg=He,Ne): The Prototype Disk-and-Ball Dimer. J. Chem. Phys., 116(3):1012–1021, 2002.

[15] David Luckhaus and Martin Quack. Spectrum and dynamics of the CH chromophorein CD2HF. I. Vibrational Hamiltonian and analysis of rovibrational spectra. Chem.Phys. Lett., 190(6):581–589, 1992.

[16] T.-K. Ha, D. Luckhaus, and M. Quack. Spectrum and dynamics of the CH chro-mophore in CD2HF. II. Ab initio calculations of the potential and dipole momentfunctions . Chem. Phys. Lett., 190:590, 1992.

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[17] A. Beil, D. Luckhaus, M. Quack, and J. Stohner. Intramolecular vibrational redis-tribution and unimolecular reaction: Concepts and new results on the femtoseconddynamics and statistics in CHFClBr. Ber. Bunsenges. Phys. Chem., 101:311–328,1997.

[18] D. Luckhaus, M. J. Coffey, M. D. Fritz, and F. F. Crim. Experimental and theoreticalovertone spectra of vCH=3,4,5, and 6 in Formaldehyde (H2CO). J. Chem. Phys.,104(10):3472–3478, 1996.

[19] Martin Quack and Martin Willeke. Ab initio calculations for the anharmonic vi-brational resonance dynamics in the overtone spectra of the coupled OH and CHchromophores in CD2H–OH . J. Chem. Phys., 110(24):11958–11970, 1999.

[20] David Luckhaus, Martin Willeke, and Martin Quack. Coupling across bonds: abinitio calculations for the anharmonic vibrational resonance dynamics of the coupledOH and CH chromophores in trans formic acid HCOOH. Z. Phys. Chem., 214:1087–1114, 2000.

[21] J. L. Scott, D. Luckhaus, S. S. Brown, and F. F. Crim. Overtone Spectroscopyof the Hydroxyl Stretch Vibration in Hydroxyl Amine (NH2OH). J. Chem. Phys.,102(2):675–679, 1995.

[22] W. H. Miller, N. C. Handy, and J. E. Adams. Reaction path Hamiltonian for poly-atomic molecules. J. Chem. Phys., 72:99–112, 1980.

[23] D. Papousek and M. R. Aliev. Molecular Vibrational–Rotational Spectra. Studies inPhysical and Theoretical Chemistry. Elsevier Scientific Publishing Company, Ams-terdam, 1982.

[24] Benjamin Fehrensen, David Luckhaus, Martin Quack, and Martin Willeke. Modeselective tunneling in 12CH3OH and 13CH3OH . J. Chem. Phys., 119:5534–5544,2003.

[25] Benjamin Fehrensen, David Luckhaus, and Martin Quack. Mode Selective Stereomu-tation Tunnelling in Hydrogen Peroxide Isotopomers. Chem. Phys. Lett., 300:312–320, 1999.

[26] Bernd Kuhn, Thomas R. Rizzo, David Luckhaus, Martin Quack, and Martin A.Suhm. A new six–dimensional analytical potential up to chemically significantenergies for the electronic ground state of hydrogen peroxide. J. Chem. Phys.,111(6):2565–2587, 1999.

[27] Benjamin Fehrensen, Michael Hippler, and Martin Quack. Isotopomer–selective over-tone spectroscopy by ionization detected IR+UV double resonance of jet–cooled ani-line. Chem. Phys. Lett., 298(4–6):320–328, 1998.

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