Advances in CHEMICAL PHYSICS Global and Accurate Vibration

Hamiltonians from High-Resolution Molecular Spectroscopy

Edited by

MICHEL HERMAN, JACQUES LIEVIN AND JEAN VANDER AUWERA*

Laboratoire de Chimie Physique Moltculaire CP 160109 Universite Libre de Bruxelles, Belgium

ALAIN CAMPARGUE

Laboratoire de Spectromttrie Physique (UMR5588) t UniversitC Joseph Fourier de Grenoble. BP 87, France

*Research Associate with the FNRS (Belgium) +Associated with the CNRS

Series Editors I. PRIGOGINE STUART A. RICE

Department of Chemistry

The James Franck Institute The University of Chicago

Center for Studies in Statistical

The University of Texas Austin, Texas

International Solvay Institutes Universite Libre de Bruxelles

Brussels, Belgium

Mechanics and Complex Systems and

and Chicago, Illinois

VOLUME 108

AN INTERSCIENCE It PUBLICATION JOHN WILEY & SONS, INC.

NEW YORK - CHICHESTER * WEINHEIM - BRISBANE * SINGAPORE - TORONTO

ADVANCES IN CHEMICAL PHYSICS VOLUME 108

EDITORIAL BOARD

BRUCE J. BERNE, Department of Chemistry, Columbia University, New York, New York, U.S.A.

KURT BINDER, Institut fur Physik, Johannes Gutenberg-Universitat Mainz, Mainz, Germany

A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A.

DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A.

M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K.

WILLIAM T. Corny, Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin. Dublin, Ireland

F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A.

ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A.

GRAHAM R. FLEMING, Department of Chemistry, The University of Chicago, Chicago, Illinois. U.S.A.

KARL F. FREED. The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A.

PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems. Brussels, Belgium

ERIC J. HELLER, Institute for Theoretical Atomic and Molecular Physics, Harvard- Smithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A.

ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A.

R. KOSLOFF, The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel

RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A.

G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, UniversitC Libre de Bruxelles. Brussels. Belgium

THOMAS P. RUSSELL, Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts

DONALD G. TRUHLAR, Department of Chemistry, University of Minnesota, Minneapolis. Minnesota, U.S.A.

JOHN D. WEEKS, Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U S A .

PETER G. WOLYNES, Department of Chemistry, School of Chemical Sciences, University of Illinois, Urbana, Illinois, U.S.A.

Advances in CHEMICAL PHYSICS Global and Accurate Vibration

Hamiltonians from High-Resolution Molecular Spectroscopy

Edited by

MICHEL HERMAN, JACQUES LIEVIN AND JEAN VANDER AUWERA*

Laboratoire de Chimie Physique Moltculaire CP 160109 Universite Libre de Bruxelles, Belgium

ALAIN CAMPARGUE

Laboratoire de Spectromttrie Physique (UMR5588) t UniversitC Joseph Fourier de Grenoble. BP 87, France

*Research Associate with the FNRS (Belgium) +Associated with the CNRS

Series Editors I. PRIGOGINE STUART A. RICE

Department of Chemistry

The James Franck Institute The University of Chicago

Center for Studies in Statistical

The University of Texas Austin, Texas

International Solvay Institutes Universite Libre de Bruxelles

Brussels, Belgium

Mechanics and Complex Systems and

and Chicago, Illinois

VOLUME 108

AN INTERSCIENCE It PUBLICATION JOHN WILEY & SONS, INC.

NEW YORK - CHICHESTER * WEINHEIM - BRISBANE * SINGAPORE - TORONTO

This text is printed on acid-free paper.

An Interscience '' Publication

Copyright 0 1999 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permission Department, John Wiley & Sons, Inc.

Library of Congress Catalog Number: 58-9935 ISBN 0-471-32843-X

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To the memory of Ikhlef HADJ BACHIR

Series Introduction

CONTENTS xv

I.

11.

General Introduction 1

The Forward Trip: From the Hamiltonian to the Vibration-Rotation Spectrum 4 A. Introduction 4 B. The Forward Trip: An Ab Znitio Approach 5

5 a. Full Molecular Hamiltonian 5 b. Born-Oppenheimer Separation 6 c. Translation-Free Hamiltonian 8 d. Vibration-Rotation Separation 9

1. MOLECULAR AXIS SYSTEM (MAS) 9 2. ROTATION COORDINATES 9 3. ECKART FRAME 11 4. VIBRATION COORDINATES 12

e. Exact Vibration-Rotation Hamiltonian 12 1. VIBRATION-ROTATION KINETIC ENERGY OPERATOR 12 2. POTENTIAL-ENERGY SURFACE 12 3. EXACT VIBRATION-ROTATION SCHRODINGER EQUATION 13

f . Variational Resolution of the Vibration-Rotation Schrodinger equation 13 1. VARIATIONAL PRINCIPLE 13 2. LINEAR VARIATIONAL METHOD 14 3. CONFIGURATION INTERACTION METHOD 16

HAMILTONIAN 16

HAMILTONIAN 17 6. VIBRATION-ROTATION TRANSITION ENERGIES 19

1. Exact Quantum Mechanical Formulation

4. DIAGONALISATION OF THE VIBRATION-ROTATION

5. ENERGY SPECTRUM OF THE VIBRATION-ROTATION

g. Vibration-Rotation Lines Intensities 21 1 . OPTICAL VIBRATION-ROTATION TRANSITION PROBABILITIES 2 1 2. ELECTRIC DIPOLE TRANSITION PROBABILITIES 22 3. DIPOLE MOMENT SURFACE 24

25 25 25

vii

h. Full Ab Znirio Forward Trip

a. Setting the Ab Initio Approach 2. Towards a Converged Ab Znitio Approach

... V l l l CONTENTS

1. NEED FOR A PES

2. GENERAL PROBLEMS WITH PES AND DMS

b. Choice of a Coordinate System 1 . CRITERIA

2. RECTILINEAR VERSUS CURVILINEAR COORDINATES

c. Selected Curvilinear Coordinates 1. CURVILINEAR BOND-ANGLE COORDINATES

2. LOCAL-MODE COORDINATES

3. HELIOCENTRIC-TYPE COORDINATES

4. ADAPTED STRETCHING COORDINATES

5. POTENTIAL-ADAPTED COORDINATES

6. ADAPTED BENDING COORDINATES

7. OPTIMISED COORDINATES

8. CURVILINEAR NORMAL COORDINATES

9. TRANSFORMATION OF COORDINATES

d. Approximate PES from Quantum Chemistry 1 . AB INITIO LEVEL OF CALCULATION

2. ANALYTICAL EXPRESSION FOR THE PES

3. ADJUSTMENT OF AN ANALYTICAL FUNCTION TO AB INITIO CALCULATED POINTS

4. ANALYTICAL VERSUS NUMERICAL DERIVATIVES

CALCULATIONS

5. AB INlTIO DVR APPROACH

e. Ah initio Electric Dipole Moment Surfaces f. Variational Methods

1. CONTRACTION OR DIAGONALIZATION-TRUNCATION

2. GENERAL MEAN-FIELD OPTIMIZATIONS

3. THE VIBRATIONAL MULTICONFIGURATIONAL SCF METHOD

4. DISCRETE-VARIABLE REPRESENTATION (DVR)

5. ADIABATIC APPROACHES

6. MORSE OSCILLATOR RIGID BENDER INTERNAL DYNAMICS

(MORBID) APPROACH

g. Perturbation Theory Methods 1. BASIC FORMULAS

2. TREATMENT OF RESONATING STATES

3. HIGH ORDER CANONICAL VAN VLECK PERTURBATION

THEORY (CVPT)

C. Acetylene: A Laboratory for Intramolecular Advances 1. The Acetylene Molecule

a. Both Simple and Complex b. Symmetry Properties c. Topology of the Ground-State PES

25 26 27 27 27 29 29 29 30 30 30 32 32 33 33 34 34 38

39

42 44 44 45 45 47 49 50 52

52 53 53 54

54 56 56 56 57 58

CONTENTS ix

d. Calculated Properties e. Coordinate Systems

1 . 9D SYSTEMS

2. 6D SYSTEMS

3. SYMMETRY-ADAPTED COORDINATES

4. RECTILINEAR NORMAL COORDINATES

5. PLANAR 5D SYSTEMS

J: Vibration-Rotation Hamiltonian for Acetylene 1. CHOICE OF A COORDINATE SYSTEM

2. KINETIC-ENERGY OPERATOR

3. POTENTIAL-ENERGY SURFACE

2. Forward and Backward Trips Applied to Acetylene a. Refined Quartic Force Fields in Valence Internal

Coordinates 1. REFINEMENT FROM SECOND-ORDER PERTURBATION

THEORY THE STREY-MILLS POTENTIAL

VARIATIONAL CALCULATIONS WITH A MODIFED SM

POTENTIAL

3. SIXTH-ORDER CVPT WITH THE SM POTENTIAL

4. REFINEMENT FROM VARIATIONAL CALCULATIONS:

2.

THE BRAMLEY-CARTER-HANDY-MILLS POTENTIAL

5 . VARIATIONAL CALCULATIONS WITH THE BCHM

POTENTIAL

b. Ab initio Quartic Force Fields 1. SCF QUARTIC FORCE FIELD FROM ANALYTICAL

DERIVATIVES

SD(Q)CI QUARTIC FORCE FIELD AND SDCI CUBIC

ELECTRIC FIELD FROM GRID CALCULATIONS

3. TESTING THE QUALITY OF THE SDCI AND SD(Q)CI

FORCE FIELDS

4. OVERTONE SPECTRUM OF STRETCHING MODES

5 . ANALYSIS OF STRETCH-BEND INTERACTIONS FROM THE

VCASSCF APPROACH

6 . CCSD(T) QUARTIC FORCE FIELD FROM NUMERICAL

DIFFERENCES

2.

c. Bending Dynamics 1. DVR VARIATIONAL CALCULATIONS

2. ADIABATIC VARIATIONAL REFINEMENT

d. Classical and Semiclassical Approaches of Intramolecular Dynamics I . CLASSICAL TRAJECTORIES

2. SEMICLASSICAL APPROACH

62 63 63 63 66 66 67 67 67 69 71 73

73

73

74 75

76

78 79

79

80

80 81

87

87 89 89 91

92 92 93

X CONTENTS

3. NORMAL. LOCAL, AND PRECESSIONAL BENDING MODES

e. Algebraic Approaches

111. The Backward Trip: From the Vibration-Rotation Data to the Hamiltonian A. Introduction

1. Strategy a. Aim b. Selection of Coordinates c. Matrix Image of the Molecule d. Matrix Treatment

a. General Form b. Fractionation

2. Watsonian

B. Vibrational Terms 1. Basic Features 2. Diagonal Terms

a. The Harmonic Oscillator (2:;) b. The Anharmonic Oscillator (HYhdlag)

1. ONE DIMENSIONAL DUNHAM EXPANSION

2. MULTIDIMENSIONAL DUNHAM EXPANSION

3. Off-Diagonal Terms ?hoff-dW a. Anhannonic Resonances ( H

1. TERMINOLOGY

2. MATRIX ELEMENTS

3. BRIGHT AND DARK STATES

b. Quartic Resonances 1. DETAILED CONTENT

2. X-K RELATIONS

3. UNUSUAL QUARTIC RESONANCES

4. Linear Tops a. General Features b. Diagonal Terms

1. HARMONIC CONTRIBUTIONS (fiF(lin)) 2. 1 SUBLEVELS

3. DIAGONAL ANHARMONIC TERMS

c. Off-Diagonal Terms (&Thar('ln)

1. 1 RESONANCE

2. ANHARMONIC RESONANCES

3. ANHARMONIC AND !! RESONANCES

C. Level Clustering 1. Vibrational Polyads

93 94

95 95 95 95 95 96 97 98 98

102 102 102 105 105 108 109 113 115 115 115 117 118 122 122 123 126 128 128 129 129 131 133 135 135 136 139 142 142

CONTENTS xi

a. One-Resonance Polyads b. Multiresonance Polyads c. Pseudo-Quantum Numbers

I . CONSTANTS OF THE MOTION

2. GENERAL PROCEDURE

2. Examples a. Introduction b. Nitrous Oxide

I . LEVEL CLUSTERING

2. BAND INTENSITIES

c. Acetylene 1. ANHARMONIC RESONANCES

2. CLUSTER STRLICTURE

3. PARAMETERS

4. I2C2D2 d. Ethylene

3. Giant Clusters a. V f /-Clusters

1. QUANTUM NUMBERS

2. SPECTRA

b. V f 1 I C-Clusters D. Vibration-Rotation Terms

1 . General Picture 2. Rigid Rotor Hamiltonian ( f i r r i g ) a. Symmetric Tops b. Asymmetric Tops c. Linear Tops

I . 1 DOUBLING

2. e f f LEVELS

3. Vibration-Rotation Hamiltonian a. General Picture b. Diatomic Species c. Linear Polyatomic Species d. Asymmetric Tops e . Coriolis Coupling

I . ASYMMETRIC TOPS

2. ACETYLENE

4. Vibration-Rotation Fits in Acetylene a. MIME for Lower Bending Levels in "C2H2 b. Vibration-Rotation Wavefunctions in "CzH2

1. LOWER-ENERGY BENDlNG ENERGY LEVELS

2. FUNDAMENTAL CH STRETCHING VIBRATION, V j

142 142 145 145 145 147 147 148 148 149 15 1 151 152 156 157 157 160 160 160 161 161 165 165 166 166 167 171 171 173 178 178 179 181 183 184 184 187 188 188 193 193 195

xii C O N T E N T S

c. Lower Bending Levels in "C2D2 d. Rotational Constants in 12C2H2 Overtone Levels

195 197

E. Molecular Vibration-Rotation Spectra 198 I . Spectral Intensity 198 2. Transition Probabilities 199

a. Absorption Induced by Electric Dipole Interactions 200 202

c. Absorption Line Shape 203 d. Line Intensity 205 e. Population Factors 207

I . THERMAL DISTRIBUTION OVER THE ENERGY LEVELS 207 2. TOTAL INTERNAL PARTITION FUNCTION 207 3. DEGENERACY O F THE LEVELS 208 4. STATISTICAL WEIGHT O F THE ROVIBRONIC LEVELS 209

f. Integrated Absorption Coefficient and Cross Section 209 4. Emission 21 1 5. Dipole Moment Matrix Elements for Vibration-Rotation

Transitions 212

7. Vibrational Spectra 214 a. Harmonic Selection Rules 214

I . NON DEGENERATE VIBRATIONS 214 2. DEGENERATE VIBRATIONS 216

Approximation 217 219 220

I . INFLUENCE OF MECHANICAL ANHARMONICITY 220 2. RESONANCES AND INTENSITY BORROWING 22 1

222 b. Vibration-Rotation Transitions 223

1. SYMMETRIC TOP 223 2. ASYMMETRIC TOP 229 3 . LINEAR MOLECULE 234

c. Vibration-Rotation Interaction 243 1. THE EFFECTIVE DIPOLE MOMENT OPERATOR 243

INTENSITIES 246 3. HERMAN-WALLIS EFFECTS IN LINEAR MOLECULES 248

3. Light Absorption 200

6. Absorption Coefficient: Beer's Law

6. Molecular Symmetry 213

6. Vibrational Bands in the Harmonic Oscillator

c. Application to Linear Molecules d. Vibrational Bands for the Anharmonic Oscillator

8. Vibration-Rotation Spectra 222 a. Rigid Rotor Selection Rules

2. INFLUENCE O F VIBRATION-ROTATION RESONANCES ON

CONTENTS

IV. Experimental Overtone Spectroscopy A. Introduction B. Fourier Transform Spectroscopy (FTS)

1. Introduction 2. Overtone Spectroscopy Using FTS

1. Frequency Modulation with Diode Lasers (FMDL) C. Laser Overtone Spectroscopy

a. Method b. Application of FMDL to Spectroscopy

a. Pulsed CRDS

c. Capabilities of CRDS

2. Cavity Ringdown Spectroscopy (CRDS)

b. CW-CRDS

I . SPECTROSCOP\

2. DIAGNOSTICS

3. Intracavity Laser Absorption Spectroscopy (ICLAS) a. Principles b. Sensitivity Limitations c. Seeding Noise d. Spectral Resolution and Calibration e. Capabilities of ICLAS

1 . OVERTONE SPECTROSCOPY

2. DIAGNOSTICS

4. Optoacoustic (OA) Spectroscopy a. Basic Principles b. Application to Overtone Spectroscopy

5. Optothermal (OT) Spectroscopy 6. Comparison of FMDL, CRDS, TCLAS, and OA Methods 7. Other Laser Investigations of Vibration-Rotation Levels

a. Laser Labelling b. Photofragment Spectroscopy c. Stimulated Emission Pumping and Dispersed

Fluorescence

Acknowledgments References

Appendix A. Abbreviations and Symbols Appendix B. Bibliography on acetylene in the ground electronic

state Appendix C. Vibrational Energy Levels of 12C2H2(X'C:)

... xi11

253 253 255 255 257 262 263 263 264 265 265 267 270 270 27 0 27 1 27 1 274 27 5 276 277 278 280 28 1 28 1 282 284 285 288 289 290

290

292 293

33 1

340 363

xiv

Author Index

Subject Index

CONTENTS

389

415

SERIES INTRODUCTION

Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.

I. PRIGOGINE STUART A. RICE

xv

I. GENERAL INTRODUCTION

Over the last 50 years (i.e., since the late 1940s or early 1950s) or so, high- resolution molecular spectroscopy has provided more and more precise, accurate, and reliable information about the internal energy-level structure in gas-phase molecules, with unmatched quality. High-resolution molecular spectroscopy contributed as a major driving force to the advances of chemical physics, and significantly promoted and stimulated connected fields such as quantum chemistry, astrophysics, atmospheric sciences, metrology, and intramolecular dynamics.

By the end of this millennium, among the hot research topics in high- resolution molecular spectroscopy are those considering (1) unstable species, including van der Waals complexes, atmospheric radicals, and astrophysical ions; (2) larger species, such as organic species and clusters; and (3) highly excited molecules, in vibrational overtones and in Rydberg states. They all confirm the fascinating complexity and richness of the pattern of electronic vibration-rotation energy levels in polyatomic species. Interestingly, they simultaneously seem to demonstrate the emergence of some global, universal trends. These are of essential importance in various stimulating perspectives, including predicting the intramolecular dynamics and mastering the related chemistry, at the microscopic level.

In the present review, we focus on a very specific aspect in this general context. We deal with molecular vibration as revealed by high-resolution vibration-rotation spectra in the gas phase. We are concerned with energy and intensity features related to molecules in low and (mainly) highly excited vibrational levels, of particular interest to bond selected chemistry. To avoid disappointing some readers, let us clearly emphasize that our strategy is of spectroscopic rather than dynamical nature, that our interest is restricted to ground electronic states, for asymmetric and linear top molecules, in particular, and that electron and nuclear spins, the influence of external fields, and the role of large amplitude motions are almost completely neglected. Keywords of the present review are coordinate systems, overtone experimental and theoretical spectroscopy, and acetylene.

The concept of global fitting of the pattern of vibration-rotation energy levels was more often raised in the recent (at the time of writing) literature. This trend truly supports the present review. Global fits aim at unraveling the complete vibration-rotation energy pattern in a molecule, by bridging the Hamiltonian and the spectrum. The Hamiltonian, on one side, is a compact mathematical object describing precisely all kinetic and potential-energy couplings in the isolated quantum system considered, specifically, a vibrating-rotating molecule containing electrons and nuclei. The resolution of the Hamiltonian, using standard quantum-mechanical techniques,

1

2 GENERAL INTRODUCTION

generates a complete theoretical spectrum. The experimental spectrum, on the other side, is a complicated object built from the superposition of many pieces, specifically, the spectral lines whose position, intensity, and profile carry information on the eigenstates and on the intramolecular couplings. Their analysis ought to reveal the full Hamiltonian. Two opposite procedures, the forward and the backward trips, can therefore be followed to bridge the Hamiltonian and the spectrum. Both still suffer from many drawbacks and bottlenecks.

Extensive work was devoted over the years to push the investigation of both trips as far as possible. Refinement procedures were developed, sequentially acting on the forward and backward trips to make the experimental and theoretical information converge. Nevertheless, many authors agree (see general discussion hold at the XXth Solvay Conference on Chemistry, 1995, in particular in Ref. 1) that deeper investigation is required to provide a description of the full pattern of the vibration-rotation energy levels allowing reliable predictions to be performed, in particular in the field of intramolecular dynamics.

In one procedure, the exact Hamiltonian expressed in the coordinate or momentum space is the starting point. The aim is to provide a theoretical spectrum through the resolution of the corresponding Schrodinger equation and the calculation of the matrix elements of appropriate transition operators defining the theoretical transitions intensities. This is a direct process, which connects the basic quantum description of the system to its properties. It is furthermore an exact process because it strictly follows the prescriptions of quantum mechanics: the determination of observables from appropriate quantum operators. This forward trip, achieved by quantum chemists, faces problems with the numerical applications that, despite the tremendous developments in computational methods, are not trivial to achieve, by far. These limitations are linked to the well-known bottlenecks occurring in the resolution of n-body quantum systems, facing the management of infinite basis set expansions. It is actually still impossible to derive an exact vibration-rotation Hamiltonian from ab initio calculations and to cope with the huge dimension of the Hamiltonian matrices needed for converging toward solutions of the Schrodinger equation for nontrivial polyatomic systems. This trip is detailed in Section 11, with particular emphasis on the possible selection of coordinate systems. Basic theoretical features are introduced in that section. They are supported by a comprehensive list of symbols provided in Appendix A, actually covering the whole review. Appendix A also lists all abbreviations used throughout the text. Units in Section I1 are atomic units.

In the other procedure, the observed spectrum is the starting point. The aim is, by performing the global treatment of all the experimental

GENERAL INTRODUCTION 3

information available, to recompose the exact Hamiltonian. This is an indirect process as it goes from the properties to the mathematical object generating them. This backward trip, attempted by spectroscopists, is very ambiguous in nature. It relies on a set of information that is restricted in number and quality by experimental factors, and furthermore biased by the interaction of the molecule with the radiation used to produce the spectrum. The full treatment of the data also suffers the inherent ambiguity of the least- squares procedure applied to the inverse secular problem. As a consequence, approximate Hamiltonians must be designed, adapted to the limited quantum information available. They are the so-called effective, algebraic, or spectroscopic Hamiltonians. In Section 111, we build a matrix image of the molecule allowing for global treatments, using well-known theoretical developments. The aim is to provide a reliable way to perform the backward trip, ultimately accounting for the full experimental accuracy and precision now available in high-resolution spectroscopy. Conventional spectroscopic units, with energies in reciprocal centimeters (cm-'), are used in that section. Energy and intensity features are concerned.

The present research subject was dramatically promoted by formidable developments of experimental nature, to which state-of-the-art Fourier transform spectroscopy and laser techniques extensively contributed. Measurement precision and accuracy have increased significantly, the detection sensitivity has been magnified, and broader spectral coverage and spectral decongestion means have become available. A spectacular wealth of new vibration-rotation data have appeared in the literature, shedding light on previously inaccessible regions of the potential energy surface, and stimulating theoretical developments, therefore opening new research strategies. We felt that it was most critical to highlight some of these recent instrumental achievements, as achieved in the final section (IV).

Our strategy throughtout this review is as follows. Basic features are extensively developed for didactic purposes, although they have already been presented in various review papers or books in the literature. Published research results are extensively reviewed, sometimes presented differently than in the original papers, to illustrate the actual trends and perspectives in the subject. Still unpublished information is also included in some places. The consistency between all parts in the review is ensured by focusing on one target molecule, acetylene, which is used as a benchmark of today's status of vibration-rotation high-resolution spectroscopy within the global approach. Its vibration-rotation energy levels indeed constitute a laboratory of thus far unmatched quality, because of its combined richness and accessibility, to merge the backward and forward trips discussed above. The huge literature concerning the ground electronic state of acetylene, which is provided in Appendix B, illustrates the hopefully premonitory evolution of

4 THE FORWARD TRIP

the present research topics, namely, the merging of models unraveling the global vibrational energy pattern, at experimental high-resolution precision, allowing the intramolecular dynamics to be mapped out and bond-selected chemistry to be achieved. As an output of the treatment developed around acetylene, Appendix C provides a comprehensive list of calculated energies of the vibration energy levels in '2C2H2, up to 10,000 cm-', as well as additional information on those levels.

In the present review, we thus aim at building on global concepts a general and didactic introduction to vibration and vibration-rotation spectroscopy, addressing in a systematic way the entire pattern of energy levels. We simultaneously focus on actual research trends in overtone spectroscopy, exemplified with published and sometimes unpublished results on acetylene, eventually illustrated by most recent instrumental advances.

11. THE FORWARD TRIP: FROM THE HAMILTONIAN TO THE VIBRATION - ROTATION SPECTRUM

A. Introduction

The aim of this section is to give a general theoretical background to the review by detailing how to bridge the full molecular Hamiltonian and the corresponding vibration-rotation spectrum. We consider the viewpoint of the theoretician and highlight the bottlenecks to be faced and the approximations to be introduced. As pointed out in the general introduction, the Hamiltorzinri is a compact mathematical object describing precisely all the kinetic and potential-energy couplings in the isolated quantum system: a vibrating-rotating molecule containing n , electrons and n nuclei. Our aim is to detail the steps leading to an hypothetical theoretical spectrum, involving the resolution of the corresponding Schrodinger equation and the calculation of the matrix elements of appropriate transition operators defining the theoretical transition intensities.

An overview of the forward, direct trip is presented in Section ILB, starting in Section II.B.l with a general formulation of the exact quantum- mechanical approach. All basic equations leading from an exact Hamiltonian to an exact vibration-rotation spectrum are provided, ignoring at this stage the severe approximations practically required for computational reasons. This exact formulation provides a reference level toward which practical applications have to converge. Departure from the exact treatment in the forward trip is discussed in Section II.B.2. Of main concern in this analysis is the choice of vibration-rotation coordinates, the definition of approximate Hamiltonians, the finite basis sets to be used, and the alternative methods for getting approximate solutions of the Schrodinger equation. All such features

THE FORWARD TRIP: AN AB INIT70 APPROACH 5

are applied in Section 1I.C to acetylene, the target system adopted throughout this work.

B. The Forward Trip: An Ab Initio Approach

1. Exact Quantum Mechanical Formulation

Let us assume that it is possible to know, from a purely ab initio source, the exact vibration-rotation Hamiltonian. Let us dream some more and assume that this Hamiltonian is numerically tractable. It is then possible to solve the corresponding vibration-rotation Schrodinger equation and to obtain the exact calculated energy spectrum. One can further dream of accessing to exact expressions of transition properties and, therefore, to exact transition intensities. It is the story of such an “exact” forward trip that is detailed, step by step, in this section.

The notations adopted below obey well defined rules. Hamiltonian terms, eigenvalues and eigenfunctions have descriptive subscripts such as “e,” “n,”

r,” and ‘‘u” for electron, nuclear, translation, rotation, and vibration, respectively. Superscripts give additional information on quantum numbers or state labeling, nuclear geometry, and components in a given axis system.

a. Full Molecular Hamiltonian. We consider a molecule made of n, electrons and n nuclei and label these particles from 1 to ntot (ntot = n, + n) . Particle i is then defined by its mass m,, its charge 551 and its coordinates ( X , , Y, , Z , ) in a Cartesian system of coordinates rigidly attached to the laboratory [laboratory axis system (LAS)]. Let R , be the corresponding vector position for particle i. As far as the Coulomb interactions between

“ 3 ) “ t,

these particles are dominant, and relativistic contributions one uses the full nonrelativistic Hamiltonian, which can following compact form:

can be neglected, be written in the

We adopt in this section an usual unit system in theory-atomic units-in which the units of action, mass and charge are Tz, the electron mass me, and the electron charge e, respectively. This means that for electron i the values of Z and mi are - 1 and 1, respectively and for nucleusj, Z j and m, are the corresponding atomic number and mass, respectively.

The nuclear and electron spins are not explicitly described in the above- mentioned electrostatic Hamiltonian. They are nevertheless present and determine, among other properties, the Pauli statistics, whose application to the relative vibration-rotation line intensities is detailed in Section III.E.3.(e).

6 THE FORWARD TRIP

b. Born-Oppenheimer Separation. A first simplification of the preceding Hamiltonian is provided by the usual separation of the electronic and nuclear motions, dictated by the so-called Born-Oppenheimer approximation [2]. This approximation seems indeed fully justified in the present review dedicated to the vibration-rotation spectroscopy of small molecules within an isolated, nondegenerate electronic potential-energy surface. We thus exclude here electronic-vibration-rotation interactions, requiring degener- ated potential-energy surfaces (Renner-Teller, Jahn-Teller, or Herzberg- Teller effects) or two crossing electronic surfaces (conical intersections) [3]. Such cases are dealt with in many papers, including Refs. 4-26. They are exemplified in the study of target molecules such as NO?, presenting a conical intersection between its two low-lying electronic states [27-301, and in some neutral and ionic first-row hydrides such as BHz [311, CHZ 132,331, SiHz [34], NH; [35,36], BH, [37] and CH: [38,39] exhibiting a Renner- Teller effect.

We thus investigate the vibration-rotation motion within an isolated adiabatic potential-energy surface arising from the resolution of the electronic Schrodinger equation:

We now use distinct notations for electrons and nuclei by means of subscripts respectively. The compact notation R e and R n refers to the

whole set of electron and nuclear vector positions, respectively. R e j refers to the vector position of the jth electron. The electronic Hamiltonian in (2.2) is a so-called clamped nucleus Hamiltonian which means that all nuclei have fixed coordinates, which is denoted by R t ) , with the superscript k referring to a given geometry of the nuclear skeleton. The operators 2, and ce represent the kinetic-energy operator of the electrons and the Coulomb potential-energy operator, respectively; the latter term accounts for the attraction between nuclei and electrons and the repulsion between couples of electrons. In atomic units; this can be expressed as

& h e * ? and & G n , 3 l

Equation (2.2) has in principle an infinite number of eigensolutions labeled by the superscript (E). All these states are bound states, in the sense defined by quantum mechanics; that is, they all have square integrable wavefunctions

THE FORWARD TRIP: AN AB mmo APPROACH 7

and negative eigenvalues [40]. Note that this condition is compatible with the fact that some of these states may be unbound with respect to a given dissociation channel of energy Edi s s , which means that at all possible nuclear geometries Rik) the inequality E ( R i k ) ) 2 Ediss holds. To avoid confusion we qualify such states as dissociative rather than unbound.

The total energy of a given electronic state (E) at the clamped nucleus geometry Rik) is given by the sum of the eigenvalue of (2.2) and the nuclear repulsion potential energy:

Focusing on eigensolution ( E ) of (2.2), one defines the corresponding potential-energy surface (PES) by the multidimensional energy function E ( E ) ( R n ) , in which R , holds for the collective variable defining all possible nuclear geometries. Mathematically, all possible Rf) geometries form a 3n- dimensional vector space .@3n associated with an Euclidean space E3" equipped with an inner product [4 I]. In this picture, each vector position Rik) , defining a given molecular configuration, is a point of E3". In practical applications, one is usually restricted to a given domain $3 of 93n, matching nuclear arrangements actually sampled in the application. The PES is the graph of the energy functional E@)(R,), that is, it is mathematicaIly defined as { R , , E ( E ) ( R n ) l R n E 9) c 33n x 9.

In the second step of the Born-Oppenheimer scheme, one has to solve the Schrodinger equation for nuclei moving in the potential created by the electrons:

in which the kinetic-energy operator is:

and the potential-energy operator is simply the PES for the electronic state labeled ( E ) :

The eigenfunctions of equation (2.5) are nuclear wavefunctions describing the 3n degrees of freedom of the nuclei: the translation, rotation and

8 THE FORWARD TRIP

vibration motions. The eigenvalues Eifi) are the total molecular energies including the electronic and nuclear contributions. The corresponding total wavefunctions are simple products of the corresponding electronic and nuclear eigenfunctions:

c. Translation- Free Hamiltonian. The nuclear Hamiltonian in (2.5) con- tains, in addition to a discrete spectrum of levels, a continuum spectrum arising from the center-of-mass motion. In order to eliminate these translation nuclear degrees of freedom, we rewrite the total nuclear Hamiltonian of (2.5) in terms of translation-free coordinates. This transformation can be accomplished exactly; the translation is linearly separable from A'33,' [41,42]. One usually identifies the translation coordinates to those of the nuclear center of mass Ro, defined in the LAS, with the associated total nuclear mass M,. The remaining 3n - 3 coordinates {k,; with i = 1 to tz - 1 and S = X , Y , Z > , accounting for pure rotation and vibration motions, are obtained from the y1 - 1 linear equations:

where R a g are the nuclear coordinates in the LAS and the wgi coefficients obey the formula

n x w y i = 0 (2.10) q= I

for ensuring translation invariance.

translation ?., and vibration-rotation p,,, terms [43]: The total nuclear kinetic operator can now be factored out into pure

Tn = Tt + Tr."

(2.11)

T H E FORWARD TRIP: AN A B INITIO APPROACH 9

d. Vibration-Rotution Separution. A further separation of the 3n - 3 translation-free coordinates { t } is still needed to solve the nuclear Schrodinger equation; this is done by separating the rotation motion (uniform rotation of the molecule as a whole) from the vibration or internal motion (relative motions of the nuclei with respect to each other). The following choices have to be made:

0 Definition of an axis system attached and rotating with the molecule [the molecular axis system or (MAS)].

0 Definition of rotation coordinates, describing the rotation of the MAS with respect to the LAS.

0 Definition of vibrational coordinates in the MAS.

In any case, complete uncoupling of rotation and vibration coordinates is impossible. The remaining coupling terms are of curvilinear nature [41]. As shown below, particular choices, however, tend to minimize the importance of such terms.

I . MOLECULAR AXIS SYSTEM (MAS) Let us define a Cartesian (x, y, z ) axis system fixed to the molecule, which we refer to as the body-Bed or rnoleciifur axis system (MAS). The origin of this frame is selected to be at the nuclear center of mass. Figure 1 represents the two LAS ( X . K 2) and MAS (x, y, z ) systems. The vector position of nucleusj is thus referred to as R, and T, in the LAS and MAS, respectively.

2. ROTATION COORDINATES. Although other choices are possible [44], we consider here the usual Euler angles as rotation coordinates. As represented

Figure 1. Embedding of the molecu- lar axis system (MAS) ( x , y . ; ) in the laboratory fixed system (LAS) (X. K Z). R,, and ?, are the vector positions of nucleusj in the MAS and LAS systems. respec- tively, and R o defines the origin of the MAS in the LAS.

10 THE FORWARD TRIP

Y X

Figure 2. Definition of the Euler angles 0,Cp, and x, with 0 5 0 5 x, 0 5 Cp 5 2x, and 0 5 x 5 2x. OA denotes the intersection of the

c

X X Y and xy planes.

in Fig. 2, following the convention used in [45], angles 8, $ and x, define the orientation of the MAS with respect to the LAS [45-471:

in which the rotation matrix X is the so-called direction cosine matrix, given in Table I.

For a linear molecule having all nuclei lying along the z axis, only two of the Euler angles (0, 4) act as rotation coordinates. The rotation about z , defined by angle x, is an operation of the molecular symmetry point group (Dmfl or C,,,). It therefore does not correspond to an uniform rotation of all nuclei. We are thus left with a number of mint internal or vibrational coordinates (41, q 2 , . . . , qm, , , } with mint equal to 3n - 6 and 312 - 5 for nonlinear and linear molecules, respectively.

TABLE 1 Direction Cosine Matrix, Defining the Orientation of the MAS with respect to the LAS

X Y 2

x

y

cos 8 cos Cp cos x - sin Cp sin x - cos 8 cos Cp sin x - sin Cp cos x

cos 0 sin Cp cos x + cos Cp sin x - sin 0 cos x - cos 0 sin Cp sin x + cos Cp cos x sin 8 sin x

z sin 8 cos Cp sin 8 sin Cp cos e

THE FORWARD TRIP: AN AB INITIO APPROACH 1 1

3 . ECKART FRAME. Given the definition of the rotation coordinates, we have to decide how the molecule is attached to the MAS. We use the well known Eckart frame [42,46,48]. It presents the advantage of minimizing the vibration-rotation coupling terms and is a natural choice when using normal coordinates. More suitable choices do exist, such as when dealing with large- amplitude motion [43,49,50]. The choice of the nuclear frame does not, however, introduce approximations in the exact resolution of the vibration- rotation Schrodinger equation: rather, it, influences the convergence of the calculated vibration-rotation properties when a truncated numerical treatment is applied (see Section II.B.2.f).

Referring to Figs. 1 and 2, the position of a nucleusj can be written as

R; = R o + A - ' . ' j (2.13)

with X the direction cosine matrix defined in (2.12) and Ro the origin of the MAS in the LAS. This equation depends on 3n + 6 variables (3n, 3, and 3 for 7.. Ro, and A, respectively). There are 6 of these free for defining unambiguously the position of the tz nuclei in the LAS. One possible set of 3n linearly independent parameters to be used within the MAS is defined using the Eckart conditions (see, e.g., Ref. 46), for a maximal uncoupling between the different nuclear degrees of freedom. Considering a reference nuclear configuration (0) in the MAS (usually selected to be the equilibrium geometry of the molecule), one can define any other configuration ( k ) by means of displacement vectors s j of each nucleus j with respect to the reference geometry:

(2.14)

Taking the origin of the MAS as the center of mass of the molecule in (0):

(2.15) ;= I

leading to the following two Eckart conditions:

The first condition indicates that the MAS follows the translation of the molecule; the second, that the components of the angular momentum cancel at the reference geometry, meaning that the separation of rotation and vibration is achieved for small-amplitude motions around equilibrium.

12 THE FORWARD TRIP

4. VIBRATION COORDINATES. Now that we have defined how the molecule is attached to the rotating frame, we have still to characterize the coordinates to be used for describing the vibrational motion. Two fundamental classes emerge from the literature: the rectilinear and cuwilinear coordinates. The standard normal mode coordinates are falling in the former class, while all kinds of internal coordinates are in the latter class. Depending on the context in which they are used, internal coordinates are referred to in the literature as internal bond-angle [5 1-55], cuwilinear normal [56-581, geometrically defined [49,59], heliocentric-like [60,61], and local nzode [62-641 co- ordinates. The respective advantages of the different coordinate systems are discussed in Sections II.B.2.b and II.B.2.c and illustrated in Section 1I.C.

e. Exact Vibration -Rotation Haniiltonian

I . VIBRATION-ROTATION KINETIC-ENERGY OPERATOR. It is in principle possible at this point to derive an exact form of the kinetic part of the vibration-rotation Hamiltonian for any choice of the vibration-rotation coordinates (see, e.g., Refs. 42 and 65-70 and references cited therein). The step is straightforward in the case of rectilinear normal coordinates embedded within the Eckart frame, as detailed in Section 1II.A. It is more complicated to achieve when dealing with internal coordinates and non-Eckart frames. We refer to an excellent paper of Bramley et al. [43] and to the references therein, for the specific problems of singularities encountered and for the techniques available for building the operator in specific cases. It is, in any case, possible to derive an exact form of the kinetic-energy operator. This task is greatly simplified by using computer algebra programs [67].

Let us collect the 3n - 3 vibration-rotation variables in a compact notation (0, q} , in which 0 holds for the Euler angles {e,$, x} and q for the whole set of mint vibration coordinates { q l , q 2 , . . . , q,n,nr}, independently of their definition. The label q specifically refers to normal modes in Section 111. A general form of the exact vibration-rotation kinetic-energy operator is

f r u = f v ( q ) + f r ( 0 ) + f r o v i b ( 0 , q ) (2.17)

in which fT, and f r refer to the pure vibration and rotation parts of the kinetic-energy operator, respectively, and frovib gathers all vibration- rotation coupling terms.

2 . POTENTIAL ENERGY SURFACE. The potential Vn(Rn) = E@)(R , ) appearing in the nuclear Schrodinger equation (2.5), corresponds to the PES of the considered electronic state (E ) . It can be expressed directly in the selected set of internal coordinates q. It is indeed invariant under translation and