Vibrational intensities

VIBRATIONALSPECTRAAND STRUCTURE Volume 22 VIBRATIONAL INTENSITIES EDITORIAL BOARD Dr. Lester Andrews University of ...

Author: B.S. Galabov | T. Dudev

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VIBRATIONALSPECTRAAND STRUCTURE Volume 22

VIBRATIONAL INTENSITIES

EDITORIAL BOARD

Dr. Lester Andrews University of Vir#nia Charlottesville, Virginia USA

Dr. J. A. Koningstein Carleton University Ottawa, Ontario CANADA

Dr. John E. Bertie University of Alberta Edmonton, Alberta CANADA

Dr. George E. Leroi Michigan State University East Lansing, Michigan USA

Dr. A. R. H. Cole University of Western Australia Nedlands WESTERN AUSTRALIA

Dr. S. S. Mitra University of Rhode Island Kingston, Rhode Island USA

Dr. William G. Fateley Kansas State University Manhattan, Kansas USA

Dr. A. Miiller Universitiit Bielefeld Bielefeld WEST GERMANY

Dr. H. Hs. Giinthard Eidg. Technische Hochschule Zurich SWITZERLAND

Dr. Mitsuo Tasumi University of Tokyo Tokyo JAPAN

Dr. P. J. Hendra University of Southampton Southampton ENGLAND

Dr. Herbert L. Strauss University of California Berkeley, California USA

V)lliJBRAT lUOl NAIL SPIEEC ? III1:~A N lid STIIII~UCTUIIII~IIIIE A SERIES

OF ADVANCES

VOLUME

JAMES R. DURIG (Series Editor) College of Arts and Sciences University of Missouri-Kansas City Kansas City, Missouri

22

VIBRATIONAL INTENSITIES

Boris S. Galabov and Todor Dudev Faculty of Chemistry, Sofia University, Sofia 1126, Bulgaria

1996 ELSEVIER A m s t e r d a m - Lausanne - N e w Y o r k - O x f o r d - S h a n n o n - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam The Netherlands

ISBN 0-444-81497-3 91996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A,. should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

P R E F A C E TO THE SERIES

It appears that one of the greatest needs of science today is for competent people to critically review the recent literature in conveniently small areas and to evaluate the real progress that has been made, as well as to suggest fi'uitf~ avenues for future work. It is even more important that such reviewers clearly indicate the areas where little progress is being made and where the chances of a si~ificant contribution are minuscule either because of faulty theory, inadequate experimentation, or just because the area is steeped in unprovable yet irrefutable hypotheses. Thus, it is hoped that these volumes will contain critical summaries of recent work, as well as review the fields of current interest. Vibrational spectroscopy has been used to make significant contributions in many areas of chemistry and physics as well as in other areas of science. However, the main applications can be characterized as the study of intramolecular forces acting between the atoms of a molecule; the intermolecular forces or degree of association in condensed phases; the determination of molecular symmetries; molecular dynamics; the identification of functional groups, or compound identification; the nature of the chemical bond; and the calculation of thermodynamic properties. Current plans are for the reviews to vary, from the application of vibrational spectroscopy to a specific set of compounds, to more general topics, such as force-constant calculations. It is hoped that many of the articles will be sufficiently general to be of interest to other scientists as well as to the vibrational spectroscopist. As the series has progressed, we have provided more volumes on topical issues and, in some cases, single author(s) volumes. This flexibiLity has made it possible for us to diversify the series. Therefore, the course of the series has been dictated by the workers in the field. The editor not only welcomes suggestions from the readers, but eagerly solicits your advice and contributions.

James R. Durig Kansas City, Missouri

P R E F A C E TO V O L U M E 22

The current volume in the series Vibrational Spectra and Structure is a single topic volume on the vibrational intensities in the inflared and Raman spectra. The current monograph describes the various models for interpreting intensities and it presents, in a systematic way, the theoretical approaches that are used in analyzing and predicting vibrational intensities. The book is divided into 10 chapters, with each chapter covering a specific topic. The first part of the book deals with the absorption of infxared radiation whereas chapter eight, nine and ten deal with Raman intensities. It is hoped that the consistent notation used throughout the book will facilitate the understanding of this rather complicated topic. The Editor would like to thank the editorial board for suggesting the topic for this volume and the two authors for their contributions and patience, which was required when producing the monograph. The Editor would also like to thank his Administrative Associate, Ms. Gail Sullivan, and Editorial Assistant, Mrs. Linda Smitka for providing the articles in camera ready copy form and quietly enduring some of the onerous tasks associated with the completion of the volume. He also thanks his wife, Marlene, for copy-editing and preparing the author index.

James R. Durig Kansas City, Missouri

P R E F A C E BY T H E A U T H O R S

The present book appears more than ten years since the publication of "Vibrational Intensities in Infrared and Raman Spectroscopy," a volume edited by W. B. Person and (3. Zerbi. It contains comprehensive reviews describing major developments in the field made during the seventies. Though not at a very fast pace, advances in the field, especially in theoretical approaches, have been made during the past fifteen years. In 1988 the monograph of L. A. Gribov and W. J. Orville-Thomas "Theory and Methods of Calculation of Molecular Spectra" was published. This volume presents, in detail, the progress achieved within the valence optical theory of infrared and Raman intensifies. Vibrational intensities in infrared and Raman spectra are important physical quantifies that are directly related to the distribution and fluctuations of electric charges in the molecule. These spectral parameters can be experimentally determined with good accuracy for many molecules. Additionally, infrared and Raman intensities are presently estimated theoretically by advanced analytical derivative ab imtio molecular orbital methods. These fundamental molecular quantifies are being used in structural, and other studies, on a limited basis. A monograph describing the currently available methods and models for interpreting intensities is needed to stimulate wider application of these important molecular quantities that can be obtained without much ditticulty from experiment and quantum mechanical calculations. It is a principal aim of the present book to present, in a systematic way, the theoretical approaches that are used in analyzing and predicting vibrational intensities. The formalisms developed are illustrated with detailed numerical examples. By experience we have realized that a theory cannot be fully understood and appreciated unless concrete applications are performed. Thus, most of the theoretical models described in the book were obtained and then applied to chosen molecules. We hope that the approach adopted will facilitate the easier understanding of seemingly complicated formulations. We have also used a consistent notation in presenting the different theoretical approaches, thus eliminating another barrier in uaderstanding some methods, especially those developed by the Russian spectroscopic school. The book does not aim at completeness in covering the various aspects of the field. This is, in principle, difficult to achieve in the present times of scientific information explosion. The authors would like to acknowledge many extremely helpful discussions with researchers from different countries that have contributed so much to the development of the field: Bryce L. Crawford, Jr., John C. Decius, W. J. Orville-Thomas, Willis B. Person, Lev A. Gribov, Ian M. Mills, John Overend, Giuseppe Zerbi, Mariangela Gussoni, Peter Pulay, Henry F. Schaefer HI, Derek Steele, Donald C. McKean, William M. A. Smit, Salvador

vii

Montero, Keneth B. Wiberg, and many others. In fact, a substantial part of the content of this book reflects theoretical and experimental developments achieved by these fine scientists. We are particularly grateful to Professor W. J. Orville-Thomas. One of the present authors (B. G.) was first introduced to the field of vibrational intensities while a postdoctoral fellow in the laboratory of Professor W. J. Orville-Thomas at the University of SalforcL The collaboration established lasts for more than twenty years. We are also very indebted to the series Editor, Professor J. IL Durig, for his encouragement and support.

Boris Galabov and Todor Dudev Sofia, Bulgaria

~176176

viii

C O N T E N T S OF O T H E R V O L U M E S

VOLUME 10 VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDoweH INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Nafie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry VIBRONIC SPECTRA AND STRUCTURE ASSOCIATED WITH JAHN-TELLER INTERACTIONS IN THE SOLID STATE, M.C.M. O'Brien SUM RULES FOR VIBRATION-ROTATION INTERACTION COEFFICIENTS, L. Nemes

VOLUME 11 INELASTIC ELECTRON TUNNELING SPECTROSCOPY OF HOMOGENEOUS CLUSTER COMPOUNDS, W. Henry Weinberg

SUPPORTED

VIBRATIONAL SPECTRA OF GASEOUS HYDROGEN-BONDED COMPOUNDS, J. C. Lassegues and J. Lascombe VIBRATIONAL SPECTRA OF SANDWICH COMPLEXES, V. T. Aleksanyan APPLICATION OF VIBRATIONAL SPECTRA TO ENVIRONMENTAL PROBLEMS, Palricia F. Lynch and Chris W. Brown TIME RESOLVED INFRARED INTERFEROMETRY, Part 1, D. E. Honigs, R. M. Hammaker, W. G. Fateley, and J. L. Koenig VIBRATIONAL SPECTROSCOPY OF MOLECULAR SOLIDS - CURRENT TRENDS AND FIYI'URE DIRECTIONS, Elliot 1L Bemstein

ix

x

CONTENTS OF OTHER VOLUMES VOLUME 12

HIGH RESOLUTION INFRARED STUDIF_~ OF Srl~ STRUCTURE AND DYNAMICS FOR MATRIX ISOLATED MOLECULES, B. I. Swanson and L. H. Jones FORCE FIELDS FOR LARGE MOLECULES, Hiroatsu Matsuura and Mitsuo Tasumi SOME PROBLEMS ONTHE STRUCTURE OF MOLECULES IN THE ELECTRONIC EXCITED STATES AS STUDIED BY RESONANCE RAMAN SPECTROSCOPY, Aldko Y. Hirakawa and Masamichi Tsuboi VIBRATIONAL SPECTRA AND CONFORMATIONAL ANALYSIS OF SUBSTITUTED THREE MEMBERED RING COMPOUNDS, Charles J. Wurrey, Jiu E. DeWitt, and Victor F. Kalasinsky VIBRATIONAL SPECTRA OF SMALL MATRIX ISOLATED MOLECULES, Richard L. Redington RAMAN DIFFERENCE SPEC'I~OSCOPY, J. I.aane

VOLUME 13 VIBRATIONAL SPECTRA OF ELECTRONICALLY EXCITED STATES, Mark B. Mitchell and William A. Guillory OPTICAL CONSTANTS, INTERNAL FIELDS, AND MOLECULAR PARAMETERS IN CRYSTA[~, Roger Frech RF~ENT ADVANCES IN MODEL CALCULATIONS OF VIBRATIONAL OPTICAL ACTIVITY, P. L. Polavarapu VIBRATIONAL EFFECTS IN SPECTROSCOPIC GEOMETRIES, L. Nemes APPLICATIONS OF DAVYDOV SPLITHNG FOR PROPERTIES, G. N. Zhizhin and A. F. Goncharov

STUDIES OF

CRYSTAL

RAMAN SPECTROSCOPY ON MATRIX ISOLATED SPECIES, H. J. Jodl

VOLUME 14 HIGH RESOLUTION LASER SPECTROSCOPY OF SMALL MOLECULES, Eizi Hirota

CONTENTS OF OTHER VOLUMES

xi

ELEL'TRONIC SPECTRA OF POLYATOMIC FREE RADICALS, D. A. Ramsay AB 1NITIO C ~ T I O N

OF FORCE FIELDS AND VIBRATIONAL SPECTRA, Geza

Fogarasi and Peter Pulay FOURIER TRANSFORM INFRARED SPECTROSCOPY, John E. Bertie NEW TRENDS IN THE THEORY OF INTENSITIES IN INFRARED SPECTRA, V. T. Aleksanyan and S. KK Samvelyan VIBRATIONAL SPECTROSCOPY OF LAYERED MATERIALS, S. Nakashima, M. Hangyo, and A. Mitsuishi

VOLUME 15 ELECTRONIC SPECTRA IN A SUPERSONIC JET AS A MEANS OF SOLVING VIBRATIONAL PROBLEMS, Mitsuo Ito BAND SHAPES AND DYNAMICS IN LIQUIDS, Walter G. Rothschild RAMAN SPECTROSCOPY IN ENERGY CHEMISTRY, Ralph P. Cooney DYNAMICS OF LAYER CRYSTALS, Pradip N. Ghosh THIOMETALLATO COMPLEXES: VIBRATIONAL SPECTRA AND STRUCTURAL CHEMISTRY, Achim Miiller ASYMMETRIC TOP INFRARED VAPOR PHASE CONTOURS AND CONFORMATIONAL ANALYSIS, B. J. van der Veken WHAT IS HADAMARD TRANSFORM SPECTROSCOPY?, R. M. Hammaker, J. A. Graham, D. C. Tilotta, and W. G. Fateley

VOLUME 16 SPECTRA AND STRUCTURE OF POLYPEPTIDES, Samuel Krimm STRUCTURES OF ION-PAIR SOLVATES FROM MATRIX-ISOLATION/SOLVA-TION SPECTROSCOPY, J. Paul Devlin LOW FREQUENCY VIBRATIONAL SPECTROSCOPY OF MOLECULAR COMPLEXES, Erich Knozinger and Otto Schrems

xii


TRANSIENT AND TIME-RESOLVED RAMAN SPECTROSCOPY OF SHORT-LIVED INTERMEDIATE SPECIES, Hiro-o Hamaguchi INFRARED SPECTRA OF CYCLIC DIMERS OF CARBOXYLIC ACIDS: THE MECHANICS OF H-BONDS AND RELATED PROBLEMS, Yves Marechal VIBRATIONAL SPECTROSCOPY UNDER HIGH PRESSURE, P. T. T. Wong

VOLUME 17.4, SOLID STATE APPLICATIONS, R. A. Cowley; M. L. Bansal; Y. S. Jain and P. K. Baipai; M. Couzi; A. L. Venna; A. Jayaraman; V. Chandrasekharan; T. S. Misra; H. D. Bist, B. Darshan and P. K. Khulbe; P. V. Huong, P. Bezdicka and J. C. Grenier SEMICONDUCTOR SUPERLATTICES, M. V. Klein; A. Pinczuk and J. P. Valladares; A. P. Roy; K. P. Jain and R. K. Soni; S. C. Abbi, A. Compaan, H. D. Yao and A. Bhat; A. K. Sood TIME-RESOLVED RAMAN STUDIES, A. Deffontaine; S. S. Jim; R. E. Hester RESONANCE RAMAN AND SURFACE ENHANCED RAMAN SCATTERING, B. Hudson and R. J. Sension; H. Yamada; R. J. H. Clark; K. Machida BIOLOGICAL APPLICATIONS, P. Hildebrandt and M. Stockburger; W. L. Peticolas; A. T. Tu and S. Zheng; P. V. Huong and S. R_ Plouvier; B. D. Bhatmchm3,ya; E. TaiUandier, J. Liquier, J.-P. Ridoux and M. Ghomi

VOLUME 1713 STIMULATED AND COHERENT ANTI-STOKES RAMAN SCATTERING, H. W. Schrrtter and J. P. BoquiUon; G. S. Agarwal; L. A. Rahn and tL L. Farrow; D. Robert; K. A. Nelson; C. M. Bowden and J. C. Englund; J. C. Wright, tL J. Carlson, M. T. Riebe, J. K. Steehler, D. C. Nguyen, S. H. Lee, B. B. Price and G. B. Hurst; M. M. Sushchinsky; V. F. Kalasinsky, E. J. Beiting, W. S. Shepard and tLL. Cook RAMAN SOURCES AND RAMAN LASERS, S. Leach; G. C. Baldwin; N. G. Basov, A. Z. Grasiuk and I. G. Zubarev; A. I. Sokolovskaya, G. L. Brekhovskikh and A. D. Kudtyavt$r

OTHER APPLICATIONS, P. L. Polavarapu; L. D. Barron; M. Kobayashi and T. Ishioka; S. 1L Ahmad; S. Singh and M. 1. S. Sastry; K. Kamogawa and T. Kitagawa; V. S. Gorelik; T. Kushida and S. Kinoshita; S. K. Shanna; J. IL Durig, J. F. Sullivan and T. S. Little


xiii

VOLUME 18 ENVIRONMENTAL APPLICATIONS OF GAS CHROMATOGRAPHY/FOURIER TRANSFORM INFRARED SPECTROSCOPY (GC/FT-IR), Charles J. Wurrey and Donald F. Gufl~ DATA TREATMENT IN PHOTOACOUSTIC FT-IR SPECTROSCOPY, K. H. Michaelian RECENT DEVELOPMENTS IN DEPTH PROFILING FROM SURFACES USING FT-IR SPECTROSCOPY, Marek W. Urban and Jack L. Koenig FOURIER TRANSFORM INFRARED SPECTROSCOPY OF MATRIX ISOLATED SPECIF~, Lester Andrews VIBRATION AND ROTATION IN SILANE, GERMANE AND STANNANE AND T t ~ I R MONOHALOGEN DERIVATIVES, Hans Biirger and Annette Ralmer FAR INFRARED SPECTRA OF GASES, T. S. Little and J. R. Durig

VOLUME 19 ADVANCES IN INSTRUMENTATION FOR THE OBSERVATION OF VIBRATIONAL OPTICAL ACTIVITY, M. Diem SURFACE ENHANCED RAMAN SPECTROSCOPY, Ricardo Aroca and Gregory J. Kovacs

DETERMINATION OF METAL IONS AS COMPLEXES I MICELLAR MEDIA BY UVVIS SPECTROPHOTOMETRY AND FLUORIMETRY, F. Fernandez Lucena, M. L. Marina Alegre and A. R. Rodriguez Fernandez AB INITIO CALCULATIONS OF VIBRATIONAL BAND ORIGINS, Debra J. Searles and

EUak I. yon Nagy-Felsobuld APPLICATION OF INFRARED AND RAMAN SPECTROSCOPY TO THE STUDY OF SURFACE CHEMISTRY, Tohru Takenaka and Junzo Umemura INFRARED SPECTROSCOPY OF SOLUTIONS IN LIQUIFIED SIMPLE GASES, Ya. M. Kimel'ferd VIBRATIONAL SPECTRA AND STRUCTURE OF CONJUGATED CONDUCTING POLYMERS, Issei Hamda and Yukio Furukawa

AND

xiv

CONTENTS OF OTHER VOLUMES VOLUME 20

APPLICATIONS OF MATRIX INFRARED SPECTROSCOPY TO MAPPING OF BIMOLECULAR REACTION PATHS, Heinz Frei VIBRATIONAL LINE PROFILE AND FREQUENCY SHIFt STUDIES BY RAMAN SPECTROSCOPY, B. P. Asthana and W. Kiefer MICROWAVE FOURIER TRANSFORM SPECTROSCOPY, Alfred Bander AB/N/T/O QUALITY OF SCMEH-MO CALCULATIONS OF COMPLEX INORGANIC SYSTEMS, Edward A. Boudreaux C~TED AND EXPERIMENTAL VIBRATIONAL SPECTRA AND FORCE FIELDS OF ISOLATED PYRIMIDINE BASES, Willis B. Person and K~styna Sz~ze-

VOLUME 21 OPTICAL SPECTRA AND LATTICE DYNAMICS OF MOLECULAR CRYSTALS, G. N. Zhizhin and E. I. Mukhtarov

TABLE OF CONTENTS

P R E F A C E T O THE SERIES ..........................................................................................

v

P R E F A C E B Y THE E D I T O R .......................................................................................

vi

P R E F A C E B Y THE A U T H O R S ..................................................................................

vii

C O N T E N T S OF O T H E R V O L U M E S ..........................................................................

ix

CHAPTER 1 A B S O R P T I O N O F I N F R A R E D R A D I A T I O N B Y M O L E C U L E S .......................... 1 I.

Theoretical Considerations .....................................................................................

2

II.

Selection Rules For Infrared Absorption ...............................................................

12

A.

Harmonic Oscillator Selection Rules ............................................................

12

B.

Symmetry Selection Rules ............................................................................

14

lII.

Experimental Determination o f Infrared Intensities .............................................. 17

CHAPTER 2 C O O R D I N A T E S IN V I B R A T I O N A L A N A L Y S I S .................................................. 25 CHAPTER 3 S E M I - C L A S S I C A L M O D E L S O F I N F R A R E D I N T E N S I T I E S ............................ 35 I.

Introduction ..........................................................................................................

II.

Rotational Corrections To Dipole M o m e n t Derivatives ........................................ 40

II1.

36

A.

The Compensatory Molecular Rotation ........................................................ 40

B.

The Hypothetical Isotope Method ................................................................

43

The Bond Moment Model ....................................................................................

51

A.

Theoretical Considerations ...........................................................................

51

B.

Applications .................................................................................................

63

C.

Atomic Charge -- Charge Flux Model .......................................................... 68

D.

Group Dipole Derivatives as Infrared Intensity Parameters ........................... 72

CHAPTER 4 M O L E C U I ~ R D I P O L E M O M E N T D E R I V A T I V E S AS I N F R A R E D I N T E N S I T Y P A R A M E T E R S ...................................................................................

77

I.

Atomic Polar Tensors (APT) ................................................................................

79

A.

79

General Formulation .................................................................................... xv

B.

lnvariants of Atomic Polar Tensors Under Coordinate Transformation ............................................................................................. 83

C.

Symmetry Properties of Atomic Polar Tensors ............................................. 88

D.

Atomic Polar Temors - Examples of Application ........................................ 93

E.

Interpretation of Atomic Polar Tensors ......................................................... 98

F.

Predictions of Infrared Intensities by Transferring Atomic Polar Tensors ....................................................................................................... 105

H~

Bond Charge Tensors...

HI.

Bond Polar Parameters ........................................................................................ 111

................................................................................ 106

A.

General Considerations ............................................................................... 111

B.

Formulation ................................................................................................. 116

C.

Examples of Application ............................................................................. 120

D.

Physical Significance of Bond Polar Parameters .......................................... 126

E.

Prediction of Vibrational Absorption Intensities by Transferring Bond Polar Parameters ................................................................................ 130

W. Effective Bond Charges from Rotation-Free Atomic Polar Tensors ..................... 131 A.

Rotation-Free Atomic Polar Tensor ............................................................. 131

B.

Effective Bond Charges ............................................................................... 132

C.

Applications ................................................................................................ 134

CHAPTER 5 RELATIONSHIP BETWEEN INFRARED INTENSITY F O R M U L A T I O N S ..............

141

CHAPTER 6 P A R A M E T R I C F O R M U L A T I O N S OF I N F R A R E D A B S O R P T I O N I N T E N S I T I E S O F O V E R T O N E AND C O M B I N A T I O N BANDS ........................ 149 I.

Introduction ......................................................................................................... 150

II.

AnharmonicVibrational Transition Moment ....................................................... 151 A.

Variation Method Formulation .................................................................... 151

B.

Perturbation Theory Formulation ................................................................ 152

Ill.

The Charge Flow Model ...................................................................................... 158

IV.

The Bond Moment Model ................................................................................... 160

CHAPTER 7

AB INITIO M O C A L C U L A T I O N S OF I N F R A R E D I N T E N S I T I E S .................... 163 I.

Introduction ......................................................................................................... 164 xvi

H.

Computational Methods ...................................................................................... 165 A.

m.

Numerical Differentiation ........................................................................... 165

B.

Dipole Moment Derivative from the Energy Gradient ................................. 166

C.

Analytic Dipole Moment Derivatives .......................................................... 167

Calculated Infrared Intensities ............................................................................. 169 A.

Basis Set Considerations ............................................................................. 169

B.

Influence of Electron Correlation on Calculated Infrared Intensities ............ 176

W~ Conclusions ......................................................................................................... 187 CHAPTER 8 INTENSITIES IN RAMAN SPECTROSCOPY ...................................................... 189 I.

Molecular Polarizability ...................................................................................... 190

II.

Intensity of Raman Line ...................................................................................... 199

111. Raman Intensities and Molecular Symmetry ........................................................ 205 IV.

Resonance Raman Effect ..................................................................................... 207

V.

Experimental Determination of Raman Intensifies ............................................... 211 A.

Absolute Differential Raman Scattering Cross Section of Nitrogen ............. 212

B.

Differential Raman Scattering Cross Sections of Gaseous Samples ............. 213

CHAPTER 9 P A R A M E T R I C MODELS FOR I N T E R P R E T I N G R A M A N I N T E N S I T I E S ........................................................................................................... 215 I.

II.

Rotational Corrections to Polarizability Derivatives ............................................ 216 A.

Zero-Mass Method ...................................................................................... 218

B.

Heavy-Isotope Method ................................................................................ 219

C.

Relative Rotational Corrections ................................................................... 223

Valence-Optical Theory of Raman Intensities ..................................................... 223 A.

Theoretical Considerations .......................................................................... 224

B.

Valence Optical Theory of Raman Intensifies: An Example of Application ................................................................................................. 232

C.

Compact Formulation of VOTR .................................................................. 235

D.

Compact Formulation of VOTR: An example of Application ...................... 239

111. Atom Dipole Interaction Model (ADIM) .............................. ............................... 245 IV.

Atomic Polarizability Tensor Formulation (APZT) .............................................. 249 A.

APZT: An Example of Application ............................................................. 253 xvii

V~

Relationship Between Atomic Polarizability Tensors and Valence Optical Formulations of Raman Intensifies ...................................................................... 258

VI.

Effective Induced Bond Charges From Atomic Polarizability Tensors ................ 261 A.

Theoretical Considerations .......................................................................... 261

B.

Applications ................................................................................................ 263

C.

Discussion of Effective Induced Bond Charges ........................................... 266

C H A P T E R 10

A B INITIO C A L C U l a T I O N S O F R A M A N I N T E N S I T I E S ................................. 273 I.

II.

Computational Methods ...................................................................................... 274 A.

Finite Field Calculations of Raman Intensities ............................................. 274

B.

Polarizability Derivatives from the Energy Gradient ................................... 275

C.

Analytic Gradient Methods ......................................................................... 275

Calculated Raman Intensifies .............................................................................. 276 A.

Basis Set Dependence ofAb Initio gaman Intensifies .................................. 276

B.

Influence of Electron Correlation on Quantum Mechanically Predicted Raman Intensifies ........................................................................ 278

REFERENCES ............................................................................................................ 283

A U T H O R INDEX ....................................................................................................... 303

SUBJECT INDEX ....................................................................................................... 317

.~176 XVlll

CHAPTER 1

ABSORPTION

OF INFRARED

RADIATION

BY MOLECULES

I.

Theoretical Considerations .................................................................................... 2

II.

Selection Rules For Infrared Absorption .............................................................. 12

m~

A.

Harmonic Oscillator Selection Rules .......................................................... 12

B.

Symmetry Selection Rules .......................................................................... 14

Experimental Determination of Infrared Intensities .............................................. 17

2

GALABOV AND DUDEV

I. T H E O R E T I C A L

CONSIDERATIONS

The probability of absorption of a photon with energy hvn,n- by a molecule per unit of time leading to a transition between a lower energy state n" to higher state n' is given by [1-3]

8~;3 Wn': =-~ (n'[ X ej(u~ rj) J

I

n#)2

P(Vn,n,,).

(1.1)

In expression (1.1) ej and rj are the electric charge and position vector of atom j ill a molecule, rj refers to an arbitrary molecule-fixed Cartesian system, u x is the position vector of the photon with respect to a space-fixed Cartesian system. The quantity p called radiation density is equal to the number of photons with energy hvn,n- per unit volume. It is understood that the Bohr condition En, - F_.n,= hvn,n- must be satisfied. The polarization vector of the photons ux does not affect the molecular wave functions. A quantity called electric transition dipole may, therefore, be defined

P c : =(n'[ E ejrj In">. J

(1.2)

Since the electric dipole moment is given by

p

=

Xejrj,

(1.3)

J

Eq. (1.2) becomes

Pn'n" = ( n ' l p In").

(1.4)

Pn'n" has components along the x, y and z axes of the molecule-fixed Cartesian system. The directions of ux and Pn'n" need not coincide since molecules are randomly oriented. There is, therefore, an angle 0 between the vectors ux and Pn'n"- Eq. (1.1) may then be rewritten as

8~;3 n' [ p [ n") 2 COS2 0 P(Vn,n-).

Wn,n- = - - ~ (

(1.5)

cos20 should be taken as an average over all possible orientations of the molecule in space

ABSORPTION OF INFRARED RADIATION

3

2rc COS2 0 = --L1 f cos 2 0 Sin 0 dO dO = _1. 4~ 0 0 3

i

(1.6)

The transition probability associated with absorption of a photon with energy hvn,n,, is then given by 8/I;3 ) = (V' I pg IV")

(g = x , y , z ) .

(1.53)

Since the transition dipole is a physical observable, it is evident that its value should be independent under symmetry operations. In other words, the intramolecular charge distribution and fluctuations are invariant with respect to symmetry operations. The representation of transition dipole moment element is given by the direct product of the representations of the respective vibrational wave functions and dipole moment component rk = rv, • rpg • r w

(g = x, y, z)

(1.54)

where k is an index of the k-th normal mode. Since the dipole moment is a vector with components directed along the axes of the reference Cartesian system, it is clear that under a symmetry operation its components will transform as the respective x, y and z Cartesian axes. The same arguments hold for the igpx/0Qk,/gpy/~ and C3pz/~ dipole derivatives. Therefore, the symmetry representations of the dipole moment components, as present in expression (1.52) coincide with the representations of the respective Cartesian axes. The symmetry properties of vibrational wave functions are treated in detail elsewhere [3] and will not be discussed here. The representation of the ground vibrational state wave function belongs to the point group of the molecule at equilibrium configuration. The direct product F k coincides with one of the irreducible representations of the molecular point group. The component matrix element (V'[pg[V") will be different from zero only if the resulting F k coincides with the totally symmetric representation of the point group of the molecule F 0. In the case of degenerate vibrations F k is a reducible representation. The infrared active (allowed) transitions must have the totally symmetric irreducible representation in the structure of F k associated with the respective degenerate mode. The synunetry selection rule may also be expressed in alternative ways. An inflated transition is not forbidden only in the case where the direct product of the presentations of the two interacting states Fv, XFv,, coincides with the representation of at least one of the dipole moment Cartesian components. For a fundamental transition (v' k = 1, v" k = 0) the above requirement concerns the irreducible representation of the excited level (v'). The selection rule for such transitions is simply r v, • rpg=

to.

(1.55)

16

GALABOV AND DUDEV

Similar symmetry restrictions also apply for overtone and combination bands. As already discussed, these transitions are not aUowed under the harmonic oscillator selection rules. R should be pointed out that even ff a given transition is not forbidden under both symmetry and harmonic oscillator selection rules, it may have a very low intensity. This will be determined by the particular form of the vibration and the electronic structure of the molecule. The assignment of a given band to infrared active or forbidden transition is, therefore, not always a straightforward task. The derivation of expressions relating observed integrated infrared absorption coefficients with dipole moment matrix elements for the respective transitions shows that the experimental quantifies contain important structural information. It is related with the distribution and dynamics of electric charges in molecules. We should bear in mind, however, that there are a number of restrictions associated with the possibility to determine dipole moment derivatives with respect to normal coordinates. In many regions of the observed infrared spectrum for any molecule of a medium size, a strong overlap of closely situated bands is usuaUy present. Thus, individual intensifies for all fundamental transitions may not be accurately determined. Lately, with the development of advanced software for band deconvolution and curve fitting this difficulty has been, to some extent, overcome. As already mentioned, another formidable problem arises from the necessity to know the exact direction of polarization for each vibrational mode, so that individual Cartesian components of the Op/0Qk derivatives are evaluated. So far, no general approach to solve this problem experimentally for molecules in the gas-phase has been developed. Thus, in most cases, an elaborate molecular analysis of observed vibrational absorption intensities is only possible for higher symmetry molecules. Existing perturbations in the spectra associated with Fermi resonances, Coriolis interactions and strong anhannonicity effects may often hamper the interpretation of experimental intensity data. The sign ambiguity problem for dipole moment derivatives, as already discussed, is also present. Finally, it should be emphasized that the quantifies 0p/0Qk contain in a rather obscure form the structural information sought. This is due to the very complex nature of normal coordinates. It is, therefore, essential to further reduce the experimental 0p//~)k derivatives into quantifies characterizing electrical properties of molecular sub-units -atomic groupings, chemical bonds or individual atoms. Various theoretical formulations for analysis of vibrational intensities have been put forward. The approaches developed a r e quite analogous to the analysis of vibrational frequencies in terms of force constants. As known, force constants may be associated with properties of molecular sub-units. If such a rationalization of intensity data is successfully performed, another important aim of spectroscopy studies may become possible: quantitative prediction of vibrational intensities by transferring intensity parameters between molecules containing the same

ABSORPTION OF INFRARED 1LM)IATION

17

structural units in a similar environment. The analogy with transferable force constants should again be underlined.

III.

EXPERIMENTAL DETERMINATION OF INFRARED INTENSITIES

The theoretical models for interpretation of infrared intensities presented in the subsequent chapters have been largely applied in analyzing gas-phase experimental data. Gas-phase intensities provide an unique opportunity to study in a uniform approach the mterrelatiom between molecular structure and intensity parameters. This is due to the fact that, in contrast to vibrational frequencies, the absorption coefficients depend strongly on the phase state and on solvent effects. Intensities of different modes of the same molecule are not influenced in a systematic way by the solvent. The variations of absorption coefficients may reach tens and hundreds percent. Accurately determined gasphase intensities are, therefore, of fundamental importance as a source of experimental information on intramolecularproperties. Past difficulties in experimental measurements of integrated infrared intensities have been associated mostly with the low resolving power of spectrometers, poor accuracy on the ordinate and absence of computer facilities for band integration, deconvolution and curve fitting in overlap parts of the spectra. It is clear that presently we have far better experimental means for accurate determination of the integrated intensities of individual absorption bands. Still, however, careful considerations of a number of possible sources of errors are needed in order to obtain sufficiently accurate intensity data. Some of these problems will be discussed later on. It is interesting that the methods developed for experimental determination of vibrational intensities in the gas-phase were aimed at resolving problems arising mostly from the low resolution power of the available spectrometers at the time. It may appear that nowdays, when the researchers have access to instruments with resolution of the order of a few hundredths or even few thousandths of a wavenumber, these techniques may be of lesser importance. Although this is partly true, the current experimental approaches for experimental determination of vibrational intensities fully rely on the original developments. This is determined by the fact that these methods not only compensate for the effect of low resolution on intensities but also provide criteria for the accuracy of measurements and the influence of such phenomena as adsorption of sample gas or slow diffusion process. Thus, the extrapolation method of Wilson and Wells [ 11], further developed by Penner and Weber [12], is the standard approach for experimental intensity studies.

18

GALABOV AND DUDEV

Early attempts for experimental measurements of infrared intensities [13,14] resulted in greatly divergent values for the same molecule. It was soon realized that most of the difficulties were associated with the low resolving power of the spectrometers used [15]. The problems arise from the fact that the incident infrared beam emerging from the monochromator is not strictly monochromatic, but contains a band of frequencies around the frequency v' determined by the slit function g(v,v'). There is, thus, a perfectly good chance for the intensity of the transmitted radiation I by a cell I = Io exp (- ~ p 1)

(1.56)

to differ from the value corresponding to a monochromatic beam with frequency v'. This is particularly possible if the measurements are carried over rotational f'me structure of an infrared band where sharp variations of the absorption coefficient ct are expected. In expression (1.56) p is the pressure of the sample gas, 1is the optical path length of the cell and I0 is the intensity of the incident radiation. The quantity of principal interest in intensity measurements is the integrated absorption coefficient A as defined by Eqs. (1.43), (1.44) and (1.47). For gas samples the respective expression is (1.57)

Due to the finite width of the slit function g(v,v') the measured intensity at a setting v' will not be equal to the true absorption intensity I. An apparent intensity T(v') will be in fact determined. It is given by the integral oo

r ( v ' ) = I I(v)g(v,v')dv.

(1.58)

g(v,v') is the portion of the light of frequency v that reaches the detector when the spectrometer is set at frequency v'. The integration can be carried out from --oo to +oo since g(v,v') is different from zero only in the immediate vicinity of v'. The width of the frequency band is of the order of the spectrometer resolution. Analogous expression can be written for T o oo

T0(v') = I Io (v)g(v,v')dv.

The apparent absorption coefficient J3(v') can, therefore, be defined as

(1.59)


]3(v') = (1/pl) In [To(v')/T(v')].

19

(1.60)

The integration over the entire interval of an vibration-rotation band will produce the apparent integrated absorption coefficient B

B : Ibaad~ dr'

:lSb=d pl

(1.61)

The Wilson and Wells [ 11] extrapolation theorem proves that Lira B = A. pl-~

(1.62)

Since both p and 1 can be varied in experimental conditions, the theorem (1.62) provides a convenient way of determining the true absorption coefficient A. The entire derivation of Wilson and Wells [ 11] will be presented since important considerations associated with the accuracy of intensity measurements emerge at different stages of proving the theorem. The difference (A-B) is examined

1 Sbandhl(TIO/dv'= 1 ~bandin( f(v') ~ f(v)g(v,v')_:_v/ It T0I ) p-I ~ g(v, v')dv) d r ' .

A - B = p--~

(1.63)

In expression (1.63) f(v) is f(v) = e x p ( - t x p l )

- I/I o

(1.64)

and I0 is assumed to be constant over the frequency range of the slit width. In presence of foreign gases, such as H20 and CO 2, this assumption may not hold. It is of interest to consider here the function f. Let us suppose that f is constant over the frequency range of the slit width. It becomes a constant multiplier of the inner integral and the entire expression (1.63) vanishes. B will then be equal to A. If, however, the width of individual rotational lines is of the order of the slit width, the exponential function f can vary considerably and B may be very different from A. The limit of A-B with p l y 0 needs now be considered. Expression (1.63) is differentiated with respect to pl to yield [ctfgdv l ~ g d__v1j dr'. Lira(A-B)= Lim I {c~-'.~Tg?~ J dv'= I ~/ct - SIgdv

(1.65)

20

GALABOV AND DUDEV

The function f = exp (- a p 1) approaches unity at the limit. Expression (1.65) will vanish and the theorem proven under certain conditions. First, it will vanish if the absorption coefficient {x is constant over the integration range which is the slit width. Note that expression (1.63) vanishes under the condition that f as an exponent of a, is a constant. In Eq. (1.65) the analogous requirement is for c~. Small variations in ~ can result in much larger changes in f. On the second place, expression (1.65) may vanish if the resolving power of the monochromator is constant over the width of the band regardless of {z. In mathematical terms this condition is expressed as g(v,v') = g(v - v')

(1.66)

g ( v - v') = g(v'- v).

(1.67)

and

If conditions (1.66) and (1.67) are satisfied, we have g(v - v') dv = ~ g(v - v') d(v-v') = G.

(1.68)

G is independent of v'. Another simplification follows f~{xgdvdv' = ~c~fgdvdv' : Gfr

(1.69)

Substituting (1.68) and (1.69) into (1.65) we obtain ~ct dv' - ~ctdv = 0 .

(1.70)

Consequently, Lim B = A pl-,0

(1.71)

if a number of conditions are met. The first is that I 0 is a constant in a resolved range. This can be achieved by removing external absorption fxom atmospheric H20 and CO 2. It is also clear that working under conditions of higher resolution betters the constancy of I0 in the narrower resolved range. In addition, the theorem (1.71) will hold if {z does not vary over the range of the resolution of the spectrometer, or if the resolution is constant over the entire vibrational-rotational band. If appropriate care is taken so that the above


21

o:0r V

J

F Bcl

I

50~1

0

I

I

I

1

1

I

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 c/,equivolent poth length in cm at S.T.P.

Fig. 1.1. The dependence Bpl/pl for the infrared active modes of methane (Reproduced from Ref. [ 16] with permission).

mentioned conditions are satisfied, Eq. (1.71) provides a convenient approach of determining the ~ue absorption coefficient A by extrapolating B to zero value of pl. It should be noted that the requirement for constant resolution over the integration range of a band is a specific property of the insmxment. It may not always be fully satisfied. For small molecules in the gas-phase at low pressure with clearly expressed fine rotational structure, the variations of r and f = exp(-~pl) at each rotational line is so extreme that the conditions for accurate extrapolation measurement may not be present. Following an early approach of Bartholome [14], Wilson and Wefts [11] recommended that for such molecules a foreign non-absorbing gas under higher pressure is introduced in the sample cell. It will induce a collision broadening of rotational lines and at sufficiently high pressure will lead to a collapse of the fine structure. Obviously, under such conditions the extrapolation method may be used with confidence. In the original method of Wilson and Wells the true integrated absorption coefficient A is determined as the slope of tangent at the origin of the dependence Bpl versus pl. Typical plots of this dependence are shown in Fig. 1.1. There is, however, certain arbiuminess in choosing the exact direction of the tangent at the origin. The method also requires that the measurements be done at very low partial pressures of the absorbing gas where the accuracy is low. Considerable errors can, therefore, result. The problem is resolved by working at sufficiently high pressures of the inert transparent

22

GALABOV AND DUDEV

160 iO.lcm CH4 ~i 120

,

.

....

U

T uE 8 0

4.95 cm CH4

O I

o ,e-

d~ 4o

0

.

.

.

.

.

2".50 cm

CH 4

100 200 300 400 Pressure of nitrogen inotmospheres

Fig. 1.2. Dependence between the apparent integrated absorption of the v 3 band of CH4 and the pressure of added nitrogen (Reproduced from Ref. [ 16] with permission). 1.2

O.6 ------,0.6

In(1~

---~1,0.4 I

! ---4o.2

~lOAtrn/

i

~OAt~~ 42Atm~

OAtm 2800

L

___---- ~ ~ ' N ~ 3000

1 3200

1~ ~C m -I

Fig. 1.3. v 3 band of methane under different pressures of nitrogen used as an external gas. The partial pressure of methane is unchanged (Reproduced from Ref. [ 16]

with permission).


23

foreign gas (Ar, N2, etc.) so that the Beer's low plot becomes a straight line passing through the origin. This procedure has been suggested by Penner and Weber [ 12]. Depending on the size of the molecules, the type of rotational fme structure or band shape, the adequate pressure may vary significantly. For larger molecules a pressure of one atmosphere may be sufficient. For small molecules a pressure of up to I00 atm may be needed to reach the linear region of the dependence Bpl/pl. An example is shown in Fig. 1.2. The high pressures used may cause, however, some additional complications. For larger molecules under high pressure a certain amount of the gas sample may deposit on the cell walls, though in absence of foreign gas when the partial pressure is measured, the sample is in a gas-phase. This is a particular property of the molecule under study and each case needs careful consideration so that appropriate conditions for measurements are chosen. Overend [16] has pointed out that pressure-induced absorption can affect the apparent absorption coefficient value. The effect is attributed to intermolecular interaction. It is manifested in the slow rise of the apparent absorption coefficient B as the pressure is increased. The phenomenon is clearly shown in Fig. 1.3. If such effects are present, the pressure-induced absorption has to be eliminated. This is achieved by extrapolating the linear part of the curve to zero pressure of the external gas for each B value determined.

This Page Intentionally Left Blank

CHAPTER 2

COORDINATES IN VIBRATIONAL ANALYSIS

25

26

GALABOV AND DUDEV

Accurate potential force field of the molecule is an essential prerequisite for the molecular interpretation of experimental dipole moment derivatives. The transformation of the 0P//~k dipole moment derivatives into quantifies characterizing the electronic structure of the molecule is only possible if the forms of the vibrations are known with satisfactory accuracy. Vibrational forms are determined in the process of normal coordinate analysis on the basis of data for the atomic masses, molecular geometry and potential force field [3-6]. In solving the mathematical problem for the vibrations of a molecule, a special set of 3N-6 or more coordinates describing the variations of molecular configuration with vibrational motion irrespective with the position or orientation of the molecule in space is needed. Most suitable are the internal or, as also called, natural vibrational coordinates [4,6]. These represent changes of bond lengths, interbond angles, out-of-plane angles and torsional angles [3-6]. For small displacement, in the harmonic approximation, the potential energy is represented in the space of internal coordinates Ri by the expression 11

2V = ~ Fij R i Rj, i,j=l

(2.1)

where n is the number of internal coordinates. The coefficients Fij are the harmonic force constants and, as known, represent the second derivatives of V with respect to the vibrational coordinates. The respective expression for the kinetic energy in terms of R i is I1

2 T = ~ G]] 11~i l~j. i,j=l

(2.2)

/tj are time derivatives of the internal coordinates. The coefficients Gij-1 are determined from data for the atomic masses and molecular geometry. The most rational approach in solving the classical vibrational motion problem includes simplification of the expressions for kinetic and potential energy by determining normal vibrational coordinates, Qk. These represent a special type of displacement coordinate in the basis of which the expressions for the kinetic and potential energy acquire diagonal form with coefficients of the diagonal terms equal to unity. Qk are related with the internal coordinates through the expression [4,6] Ri = ~ Lik Qk, k

i = 1, 2,3,..., n

(2.3)

Q = L-1R.

(2.4)

or in a matrix form R = L Q


27

Here we do not aim at presenting the standard methods of solving the Newton vibrational equations. It should be emphasized that essential results of these calculations are the transformation coefficients Lik that define the relative contribution of each internal coordinate to the respective normal vibrations in the molecule. As underlined, the availability of accurate vibrational form coefficients are needed in intensity analysis. This is determined simply by the fact that vibrational intensities in the infrared spectra of molecules in the gas-phase (at low pressure so that no considerable intermolecular interaction is present) are governed by two principal factors: (1) the intramolecular charge rearrangements accompanying vibrational distortions and (2) the form of the normal vibrations as expressed in the coefficients of the normal coordinate transformation matrix L. The elements of L are determined by solving systems of linear equations of the type [4,6] X. {(GF)ij - X fiij} Lj = 0, J

i = L2,3 ..... n.

(2.5)

2. is the frequency parameter (Xk = 47r2 v 2 ) and 80 the Kronecker delta symbol. The L matrix elements are evaluated with accuracy depending on the reliability of both kinematic coefficients and force constants. Molecular geometry is usually derived from alternative experimental sources such as microwave spectra, X-ray and electron diffraction. Accuracy in vibrational analysis is limited to difficulties in evaluating force constants. As is known, in the inverse vibrational frequency problem, there is a multiplicity of solutions that satisfactorily reproduce the observed vibrational wavenumbers. On the other hand, the number of force constants defining the harmonic force field of a molecule is usually much higher than the number of observed vibrational frequencies in the infrared and Raman spectra. Experience has shown that a satisfactory solution of the inverse vibrational problem for the frequencies can be obtained by using an extensive set of experimental data that depend on the potential field [3-6,17-20]: observed wavenumbers in the infrared and Raman spectra, wavenumbers of isotopically substituted molecules, isotopic wavenumber shifts, constants of vibrational-rotational interaction (Coriolis constants), centrifugal distortion constants, mean amplitudes of vibrations and others. It is obvious that such detailed experimental data can only be obtained for relatively small and symmetric molecules. Since the coefficients of the potential field may, however, be transferable between molecules having the same structural elements, real possibilities exist for the analysis of vibrational spectra of complex molecular systems. Theoretical predictions of force constants from ab imtw molecular orbital calculations have proved extremely useful in deriving reliable force fields for many molecules [21,22]. Considerable simplifications in describing molecular vibrations are attained by introducing symmetry coordinates. These are related to the ordinary internal coordinates by the transformation, in matrix notation [4]

28

GALABOV AND DUDEV

S = UR.

(2.6)

U is an orthogonal matrix. The vibrational problem is solved separately for vibrations belonging to different symmetry species of the molecular point group. In this way the number of independent force constants defining the potential field is considerably reduced. The relation between symmetry and normal coordinates is given by the expression S = LS Q.

(2.7)

Aside from purely mathematical simplifications, the solution of the vibrational frequency problem in the space of internal coordinates offers considerable advantages from a physical point of view. As derivatives of the potential energy with respect to changes in bond lengths and valence angles, the evaluated force constants characterize in a clear manner the forces binding the atoms in a molecule. The relationships between force constants and electronic structural parameters of molecular sub-units are well established [23-26]. The transferability properties of potential force constants between structurally related molecules are also best expressed if defined in terms of internal coordinates. There are, however, certain shortcomings in using internal coordinates that need to be mentioned. These are mostly associated with the redundancies that exist between angular internal coordinates for some structural arrangements of atoms in a molecule, e.g. in the case of a common atom of three or more bonds. It is obvious that individual force constant values defined as derivatives of the potential energy with respect to redundant internal coordinates do not carry a definite physical information about molecular structure. This difficulty can be overcome by expressing the force constants in internal symmetry coordinate basis. Particularly advantageous is the use of local group symmetry coordinates describing the vibrations of atomic groupings such as CH 3, CH 2, CO, ring vibrations, etc. The redundancy problem is treated quite efficiently while, at the same time, preserving the direct association of force constants with basic structural units in a molecule. Earlier [27] and more recent studies [21,28-30] have shown the efficacy in applying local group symmetry coordinates in normal coordinate analysis. We discuss this problem in some length since very similar arguments are valid in vibrational intensity analysis as well. The principal aim in intensity analysis is, again, to transform the observed integrated absorption band intensities into parameters that will, hopefully, characterize molecular structure in a clear and physically sound way. As it happened, this goal became particularly difficult to achieve in analyzing vibrational intensities. Outside the treatment in the space of purely vibrational coordinates, such as internal, symmetry and normal coordinates, vibrational motion can be described on the


29

basis of Cartesian coordinates as well [3-6]. Atomic Cartesian displacement coordinates find an extensive application in theoretical analysis of molecular vibrations, especially in theoretical developments associated with intensities in infrared and Raman molecular spectra. This is determined, very possibly, by the fact that, in contrast to the energies of vibrational transitions, the experimental dipole moment derivatives possess spacedirectional properties. It is, therefore, only natural that all theoretical formulations, aimed at reducing experimental integrated intensities into molecular parameters, employ at certain stages atomic Cartesian displacement coordinates in describing vibrational motion. As mentioned, these coordinates are also used in the vibrational frequency calculations. Defining the potential force field in the space of Cartesian coordinates results, however, in the loss of the simple physical sense of force constants as quantifies characterizing molecular structure. The transferability properties of the potential field coefficients are also obscured. Further in the present section, the transformation relations between various types of coordinates used in vibrational analysis are given. The motion of an N-atomic molecule possessing 3N-6 vibrational degrees of freedom (3N-5 in the case of linear molecules) can be defined by 3N atomic Cartesian displacement coordinates Axa, AYa and Az~ where a is an atomic index. In describing vibrational motion it is necessary to express the vibrational kinetic and potential energies into a basis set of coordinates that are independent of the translation and rotation of the molecule in space. This is achieved by imposing on the 3N Cartesian displacement coordinates conditions that eliminate the coupling between the three types of molecular motion. In explicit form these conditions as defined by Eckart [31] and Sayvetz [32] are Z m a Ax a = 0 Ct

~_~ ma Aya =O Ot

maAz a =0 Ot

ma (YOa Aza -zOa Aya)= 0

(2.8)

Ot

ma(Z~ Axa-x~ AZa)=O Ot

o o 2 ma(xaAya-yaAxa) =0 Ct

In expression (2.8)m a are atomic masses, x O, yO and z O are the equilibrium coordinates of the a-th nucleus. As defined, the Eckart-Sayvetz conditions neglect the vibrational-rotational interactions accounting for a part of the kinetic energy of the

30

GALABOV AND DUDEV

molecule, called Coriolis energy. It has been shown that, compared to vibrational kinetic energy, the Coriolis interaction terms are small and may be neglected. The Eckart-Sayvetz conditions imply that, if during the vibration a small translation of the center of masses is invoked, the origin of the Cartesian reference system is displaced so that no linear momentum is produced. The second Sayvetz condition, expressed in the last three equations of (2.8), imposes the constraint that, during vibrational displacements, no angular momentum is produced. Eq. (2.8) implies that the reference Cartesian system translates and rotates with the molecule in such a way that the displacement coordinates Ax~ Aya and Aza reflect pure vibrational distortions. It is evident that through Eq. (2.8) certain mass-dependency is imposed on the atomic Cartesian displacement coordinates. It is sometimes convenient to treat the vibrational motion in terms of massweighted Cartesian coordinates qax= m ~ a Axa qay = m ~ a Aya

(2.9)

qaz= m~~a Aza. The Eckart-Sayvetz conditions can easily be expressed in terms of the coordinates qag (g = x, y, z). Summarizing, the vibrational motion of an N-atomic molecule with 3N-6 vibrational degree of freedom can be described by 3N nuclear Cartesian displacement coordinates forming a column matrix X. Six degrees of freedom are related with translational and rotational motions of the molecule. These motions can be described by the external coordinates p (three translations and three rotations). In a transposed form the different types of vibrational coordinates may be presented as follows = (Q1--- Qk-.. Q3N-6) =

fi = (Arl ...Am ...ArN_I AVl ...ATm-.. )

= (S1 ...Sj ...$3N_6)

x,~

(2.10)


31

Arn and AYm are stretching and angular internal coordinates. X(=) and q(a) are combinations of the three ordinary and mass-weighted Cartesian displacement coordinates of atom Qt. The analytical expressions for the external coordinates are [3,4]

~x = ~

ma Axa

'UY=Z Ot

maAya

Zz = ~

ma Aza

t~

Rx-Z m. (y".

Ay.)

Ry = Z mot (z O Axa - x O Aza) (t

Rz=X ma( x~ AYa-Y ~ Axa).

(2.11)

0~

The relations between the above sets of coordinates are given by the following matrix transformations R

B - (aR/ax)

A - (ax/aR)

s = tmx

tm = (as/ax)

AU = (ax/as)

p = I~x

~ = (ap/ax)

,~ - ( a x / a p )

R

LQ

X

X

Dq

D = (aR/aq)

R

=

=

=

BX

=

AR

=

(2.12)

ALQ

c = (aq/aR)

a = (a~aq).

All transformation matrices between the various coordinates except B and U are mass dependent. The combination between vibrational and rototranslational parts of expressions (2.12) leads to the following relations [33-37]

x

x : (A:o,

In terms of mass-weighted Cartesian coordinates Eq. (2.13) translates into

(2.13)

32

GALABOV AND DUDEV

(2.14)

Additional relations between the matrices defined in Eqs. (2.12) are AB + r 13 = E3N

Bet = 0

13A = 0

BA = E3N.6

13r = E 6

BM -1 B = G

13M-1 ~ = A

~.M cx = N -1

BM -1 ~ = 0

(2.~5) DC = E3N.6

~Sy= E 6

X = M - 89q

q=M 89

DT=0

8C = 0

DM 89 B

8M 8 9 13 .

E3N is a unit 3Nx3N matrix, 0 is a null matrix, G is the matrix of kinematic coefficients, M is a diagonal matrix containing triplets of the atomic masses. A is a diagonal matrix with element triplets of the reciprocal molecular mass and the reciprocal values of the principal moments of inertia. The matrices B, G and U are determined in the process of normal coordinate analysis [3-6]. The matrix A def'med for the ftrst time by Crawford and Fletcher [38] is obtained according to the expression A = M-1 B G -1

(2.16)

A - M-1 BUG~1 U ,

(2.17)

or

where G s is the symmetrized matrix of the kinematic coefficients. The matrix A can also be given by the expression [4] A = M- - ~ C .

(2.~8)


33

The matrix a with dimensions 3Nx6 defines the relations between Cartesian displacement coordinates and the three normal translations and three normal rotations of a molecule. Vectors of the type X ~ ) (= 3X(a)/0xg) and X ~ ) (= 8X(a)/SRg) with g = x, y, z are formed. By arranging the vectors X(xag) and X (a) Rg in rows the matrices 8X//~g and OX/ORg are formed. Their elements are calculated according to the expressions 0X(a)/C~g = M-1 E3

(2.19)

ax(%aRg=

(2.20)

(( r (a))) [ - 8 9

M is the molecular mass matrix and 1 - 8 9= diag ( Ix- 89 Iy- 89 Iz- 89 Ix, Iy and I z are the principal moments of inertia of the molecule, r(a) is the position vector of the a th atom. By the symbol ((n)) an antisymmetric Cartesian vector is designated of the type

0 -n z

nz 0

ny

-n x

-ny / nx .

(2.21)

0

For each atom a 3x6 array is formed with elements of the derivatives 0X(a)/&g and

axo~)/aRg.

The elements of 13 [Eqs. (2.12)] are calculated according to [33,34] pl(Zx):l~l,3a. 2 = trha/M

131,3a.1 = 0

131,3a = 0

p2(xT):132,3a.2 = 0

132,3a.1 = ma/M

132,3a = 0

p3(Zz):~3,3a.2 = 0

133,3a.1 = 0

~3,3a = rna/M (2.22)

P4(Rx):~4,3a.2 = 0

~4,3ct-1 = -rnctz~/Ix 89

~4,3ct = mixY~/Ix 89

P5(ILy):~5,3a.2 = maz ~/Iy 89

135,3a_1 = 0

135,3a = -rnax~ t/Iy 89

P6(Rz)'136,3a.2 = - r n ~ / I z 8 9

136,3a-1 = max ~/Iz 89

136,3a = 0.

In expression (2.21) a is an atomic index (a = 1, 2, 3, ..., N), x~ t, y~ and z~ are the distances of atom a to the center of mass of the molecule and Ix, Iy, and I z are the inertial moments with respect to the equilibrium principal axes. The Eckart-Sayvetz conditions were explicitly presented with regard to the set of Cartesian displacement coordinates [Eqs. (2.8)]. From the relations R = BX and S = UR it is clear that the conditions of zero linear and angular momenta are also imposed on the coordinates Ri and Sj. Thus, certain mass-dependency is implicit in the definition of

34

GALABOV AND DUDEV

internal and synunetry coordinates. This has important consequences for molecular quantifies possessing space-directional properties such as dipole moment or polarizability derivatives when expressed in the space of these coordinates. In order to satisfy the Eckart-Sayvetz conditions in the cases of vibrational modes belonging to synunetry species that contain rotation, a certain compensatory rotation of the molecule is implicit in the respective internal or symmetry coordinates. These compensatory rotations will not be equal for different isotopic species of the molecule. Thus, it becomes necessary to calculate correction terms for the respective dipole moment or polarizability derivatives in order to account for these rotational contributions. Crawford first showed the relations between dipole moment derivatives in different isotopes [35] and later devised a method for calculating the respective correction terms [36].

CHAPTER 3

SEMI-CLASSICAL

MODELS

OF INFRARED

INTENSITIES

I.

Introduction ......................................................................................................... 36

II.

Rotational Corrections To Dipole Moment Derivatives ....................................... 40

gig.

A.

The Compensatory Molecular Rotation ...................................................... 40

B.

The Hypothetical Isotope Method .............................................................. 43

The Bond Moment Model .................................................................................... 51 A.

Theoretical Considerations ......................................................................... 51

B.

Applications ............................................................................................... 63

C.

Atomic Charge -- Charge Flux Model ........................................................ 68

D.

Group Dipole Derivatives as Infi'ared Intensity Parameters ......................... 72

35

36

GALABOV AND DUDEV

I. I N T R O D U C T I O N The theoretical analysis of the observed wavenumbers of vibrational transitions is aimed primarily at determining the potential force field of molecules and the form of vibrations as reflected in the elements of the normal coordinate transformation matrix L [Eq. (2.4)]. Two important goals are simultaneously achieved: (1) The intramolecular binding forces are quantitatively represented through the force constants matrix F. It is now recognized that the physically most plausible way of expressing inu'amolecular forces is by defining the F matrix coefficients as second derivatives of the potential energy with respect to internal vibrational coordinates; and (2) the normal vibrational coordinates are evaluated by solving the vibrational secular equation. As known, in terms of normal coordinates the complex vibrational motion of a molecule with 3N-6 degrees of freedom is reduced to a superposition of 3N-6 simple linear harmonic oscillator motions along the respective coordinates Qk. Thus, an essential and vital bridge between quantum mechanical and classical treatment of vibraffonal motion is established, and the theory of molecular vibrations acquires an integral and complete form. This unique combination of quantum and classical mechanics integrated treatment provides a solid basis in interpreting, calculating and predicting the frequencies of vibrational modes. The theory developed is also a basis for analyzing many other characteristics of the observed spectra, including vibrational intensities. The theory of the interaction of electromagnetic radiation with vibrating molecules defines a simple relation between absorption intensities in the infrared spectra and derivatives of molecular dipole moment with respect to normal coordinates as given by Eqs. (1.47) and (1.48). The success of the theory of vibrational frequencies is determined by the discovery of physically plausible approaches in describing vibrational motion and in devising efficient mathematical methods for evaluation of the respective molecular parameters. Valuable information about molecular structure has been accumulated over the years [36,17-26]. Numerous successful predictions of spectral properties of molecules confirm the validity and solid physical foundation of the models developed. The theory of vibrational transitional intensities is aimed at finding analogous approaches that would enable one to interpret, in a physically sensible and mathematically straightforward way, the observed intensifies of the absorption bands. Though the theoretical developments on vibrational intensities have now more than fifty years of history [39] the progress has been relatively slow. There are a number of factors that have negatively influenced the field. For many years the nature of the intramolecular charge fluctuation effects determining the intensities of vibrational absorption bands have remained more or less unknown. Only after the development during the 1960's and 1970's of semiempirical and, especially, ab mitio quantum theoretical methods combined with the tremendous progress in computational technology, it became possible for

SEMI-CLASSICAL MODELS OF IR INTENSITIES

37

researchers to study in detail the intramolecular charge reorganizations induced by vibrational distortions that determine the intensities of infrared bands [40-51]. As it turned out, these charge fluctuations are extremely difficult to describe in terms of a much desired simple physical model. In the second place, accurate experimental determination of vibrational intensities was, until recently, a formidable task due primarily to instrumental deficiencies and the presence of strong band overlap in many parts of the spectra. Thus, there were very few reliable experimental data for molecules in the gas-phase that could serve as a basis to test the theoretical formulations put forward. The recent progress of insmunentation and specific computer software developed to deal with overlapping bands have provided reliable approaches in the experimental measurements. Unfortunately, even under these much more favorable conditions, the available gas-phase intensity data are still scarce. In this chapter we shall present theoretical approaches for interpretation and calculation of vibrational intensities in the infrared spectra. The formulations developed are aimed at reducing the experimental dipole moment derivatives into molecular parameters characterizing properties of simple structural sub-units. In spite of many difficulties, a steady progress in analyzing vibrational intensities has been achieved, especially during the past two decades. A volume on the subject comprising papers from different laboratories appeared a decade ago [52]. We were tempted, however, to present in a unified treatment the various theoretical models developed, to discuss the relations between these approaches and also to present the progress achieved during the past decade. Examples of applications are also given so that the sometimes seemingly rather obscure formulations become clearer. The theories developed are best applied to experimental data for isolated molecules. Translated into realistic experimental conditions this refers to molecules in the gas-phase at low partial pressure so that no substantial intermolecular interactions are present. As mentioned, the main reason for these restrictions is that in contrast to vibrational transition frequencies, the intensities of absorption bands are extremely sensitive to environmental variations such as change of phase, intermolecular interactions and solvent effects. A good illustration in this respect is offered by the measured experimental intensities of CH2C12 in the gas-phase, in solution and as a pure liquid as summarized in a review by Person and Steele [53] and given in Table 3.1. The experimental data reveal a strong dependence of the observed intensities on phase state and medium. It is clear that fully reliable information on the relationship between intensities and molecular structure can be derived from gas-phase experimental data. As a consequence, the range of molecules that can be investigated is much limited. Theoretical models for quantitative assessment of the influence of solvents on vibrational absorption intensities have been developed [54-58]. Using these approaches, experimental data determined in solution are transformed to expectation values in the gas-phase. These transformations

38

GALABOV AND DUDEV TABLE 3.1. Integrated intensities (in km mo1-1) of infrared absorption bands in CH2CI2 in gas-phase (Av), solution in CCI4 (As) and as a pure liquid (AL)a

Symmetry class

Vibration

A1

Vl v2 v3 v4

B1

v5 v6

Av

AS

AL

3137 1430 714 283

6.90 0.60 8.00 0.60

4.60 2.10 9.00 0.60

1.90 3.80 19.90

3195 896

1.20

2.60 3.00

4.50 3.80

v7 1268 v8 aReprinted from Ref. [53] with permission.

26.60

27.00

28.80

B2

Wavenumber (cm-1)

are, however, always based on approximations and the reliability of estimated intensities is under question. An alternative possibility is to carry out studies in solvents accepted as standard, e.g. chloroform or carbon tetrachloride. The range of molecules that can be investigated is greatly extended and some of the experimental difficulties encountered in gas-phase measurements are avoided. These opportunities have been extensively explored [59-62]. Many problems, however, remain. From the single example provided in Table 3.1 it is seen that the intensities of vibrational modes do not change in any regular manner in the transitions between gas, liquid and solid phases. Thus, the observed quantifies cannot, in our opinion, be considered as reflecting the intramolecular electronic structure and dynamics of electric charges. In this respect, the treatment of solid-state intensity data appears to offer better opportunity for theoretical considerations because of the structural regularities and constancy of interactions within crystals and other solid-state materials. Provided aU problems associated with band overlaps, sign indeterminacy for ~ / / ~ k derivatives or other factors are overcome, the first step in transforming the experimental data is the evaluation of dipole moment derivatives with respect to symmetry vibrational coordinates. Hereafter in our discussion we shall assume that the molecules treated possess some reasonable symmetry. As already mentioned, for such molecules individual Cartesian components of t h e / ~ / ~ derivatives can be determined. The equations presented are valid for an arbitrary molecule. Realistically, however, the reduction of experimental intensity data into molecular parameters, or solution of the socalled inverse intensity problem, can be performed without introducing considerable


39

additional uncertainties for molecules with higher symmetry only. The /gp//gQk derivatives form a rectangular array PQ with dimensions 3x(3N-6) with the following structure

(0px / 0Q1 PQ =/0Py/OQI / 0QI

0px / t)Q2 ~Py/r ~pz / OQ2

..... .--

0px / t)Q3N-6 r J 9 ~pz / OQ3N-6

(3.1)

The transformation to c3p/tgSjdipole derivatives is given by the relation

ap/asj =2 LZJ(ap/aQ )

(3.2)

k

or in a matrix form (3.3)

PS = PQ L s - I

The matrix LS-I is the inverse of L s defined by Eqn (2.7). The PS matrix has the structure

PS =

/~i/igS1 /~)SI / ~)SI

~x/OS2 ~)Py/~)$2 ~Pz / t)S2

...."'"

igPx/~)$3N-6 / 0py/t)S3N. 6 . 0Pz / ~$3N-6

(3.4)

The elements of LS-1 are determined in the process of normal coordinate analysis. As already underlined, due to relation (3.3) the molecular interpretation of vibrational intensities may be carried out with some confidence only if the force field is accurately determined. At least for now, this, again, is only achievable for relatively small and symmetric molecules. By the transformation (3.3) a substantial step is made in the transition from experimental intensities into quantifies characterizing molecular structure. At the first place, a natural separation between dipole derivatives associated with bond stretchings and angle deformations is achieved. In some cases the ~/3Sj derivatives can be associated with vibrations localized within certain atomic groupings. Such distortions may be described by local group symmetry coordinates. Snyder [27] first applied dipole moment derivatives with respect to group symmetry coordinates as basic parameters in infrared intensity analysis on a series of crystalline n-alkanes. The procedure described in his work will be discussed later in this section. The intensities in infrared spectra are determined by the intramolecular charge shifts accompanying the normal vibrations of molecules. Thus, the measured band envelopes contain quite essential information about the distribution and dynamics of

40

GALABOV AND DUDEV

electric charges in molecules. It is the extraction of this information that is a principal goal of the theoretical formulations of infrared intensities. Appropriately defined molecular parameters are sought that would, hopefully, represent in a plausible way the complex picture of charge reorganizations with vibrational distortions. Since experimental data associated with intramolecular electric charges are of considerable importance for both chemistry and physics, the development of theoretical approaches for deriving this information is, no doubt, a task of prime interest. Experience has shown, however, that in spite of early hopeful developments [39,63] this goal became especially difficult to achieve. This is hardly surprising in view of the complexity of intramolecular charge rearrangements determining intensities in the infrared spectra. A second important aim of theoretical formulations is to provide possibilities for quantitative predictions of intensities by transferring intensity parameters associated with polar properties of basic structural sub-units between molecules having the same fragments. In recent years there has been certain competition between purely theoretical predictions provided by ab imtw MO calculations and the semi-classical approaches based on transferable parameters determined from experimental data. The arguments in favor of one approach or the other are hardly of any relevance and have the same validity as the analogous discussions concerning experimental and quantum theoretical force fields. The answer here is that both purely theoretical and experiment based approaches are complementary to each other, thus significantly enhancing our understanding of molecular spectra and sm~cture.

II. R O T A T I O N A L C O R R E C T I O N S T O DIPOLE MOMENT DERIVATIVES

A. The Compensatory Molecular Rotation The Eckart-Sayvetz equations [Eqs. (2.8)] imposed on a vibrating molecule require that the condition of zero linear and angular momenta is fulfilled. The molecular motion is considered as ff it is purely vibrational. Rotations and translations of the molecule as described by the six external coordinates Xx, Zy, Zz, Rx, Ry and Rz, are ignored. 3N-6 vibrational coordinates are defined with respect to a Cartesian system stuck to the molecule and moving with it. The condition for zero linear and angular momenta stems from the more fundamental theorem for momentum conservation. The momentum of a system with respect to a given axis preserves its value and direction ff the momentum of external forces acting upon the system is zero [64]. It is interesting to recall in this respect the


41

well known example with the falling cat which rotates its tail in direction opposite to that of the body, thus compensating for its angular momentum. In this way the cat preserves zero angular momentum and, as is known, it always touches the ground by its paws. Let us turn now to the vibrating molecule. When it undergoes an antisymmetrical distortion with frequency v i the direction of the molecular principal axes changes and, as a result, the molecule acquires some angular momentum. Then, a compensatory rotation with the same frequency v i and opposite direction is invoked in obeying the angular momentum conservation condition. This rotation has no effect on the frequency of the vibrational line but it may contribute to the overall intensity of the i-th band. This is the case with molecules possessing a permanent dipole moment or non-spherical equilibrium polarizability ellipsoid. The contributions could be quite significant in some cases, especially for small-sized highly polar or anisotropic molecules. Evidently, the contributions arising from the compensatory molecular rotation have to be eliminated in the course of vibrational intensity analysis. This procedure is an essential step in the process of decomposing the experimental band intensities into parameters characterizing molecular properties. Intensity parameters free from contributions from compensatory rotation can only be considered as pure intramolecular quantifies. As already discussed, an important step towards reducing the experimental absorption intensities into molecular intensity parameters is the evaluation of the matrix PS [Eq. (3.3)]. Dipole moment derivatives with respect to symmetry coordinates may, however, contain contributions from molecular rotation for certain vibrations. These conlributions can be eliminated by using the following relation p~orr = PS - R S '

(3.5)

where p~orr matrix contains /gp/aSj derivatives corrected for contributions from the compensatory molecular rotation and R s is the 3 x(3N-6) matrix of rotational contribution terms. Its structure is analogous to that of PS and will be illustrated with some examples in the succeeding parts of this section. The elements of R S can be evaluated by the equation [35] Rj = wj • PO,

(3.6)

where wj is the compensatory rotation accompanying the j-th vibration and Po is the equilibrium dipole moment of a molecule, wj is a vector which could be presented in a pseudo-tensor form as 0

w~

- w .y J

0

(3.7) 0

42

GALABOV AND DUDEV

In Eq. (3.7) w jX, w~ and w jZ are the components of wj, where x, y and z denote the respective axes of rotation. Combining Eqs. (3.6) and (3.7) we obtain Rj = - ((wj)). P0 9

(3.8)

The value of the absolute compensatory rotation arising for a particular non-fully symmetric vibration could be determined if a hypothetical non-rotating isotope of the actual molecule is created (hypothetical isotope method). This hypothetical reference species is characterized with negligibly small compensatory rotation. Historically, nonrotating isotopes of molecules were introduced by Crawford, et al. [35,36] who derived the basic mathematical expressions for treating the problem. It was suggested that the masses of some appropriately chosen atoms of the molecule were set equal to zero. Usually these are atoms non-lying along the main symmetry axis of the molecule. Since difficulties were encountered in applying this approach to some classes of molecules [34], another reference isotope was proposed by Van Straten and Smit in the mid-seventies [34]. According to this method, the non-rotating isotope is built up by multiplying the masses of some atoms by a factor of 1000 or more. The choice of atoms to be weighted depends on the symmetry of the molecule and on the form of the particular vibration which is treated. The application of these two approaches in evaluating rotational correction terms to dipole moment derivatives with respect to symmetrical vibrational coordinates will be illustrated with several examples in the following part. The compensatory rotation wj can be calculated with the aid of the following expression [36] w j = ~ p r A ja.

(3.9)

In this equation superscripts r and a stand for the reference and the actual molecule, respectively, and Aj is the j-th column of the Crawford's A matrix [38] [Eqs. (2.16) and (2.17)]. The rotational part of the 13matrix (lip) used in the calculations is in the form 0

-maz~/Ix

~,-may~/Iz

o

may~/Ix / -max~ t / Iy

maxS:/Iz

0

where a = 1,2,3, ..., N is an atom index. Thus, combining Eqs. (3.8) and (3.9) we arrive at

Rj =-(([3prAja)). po .

(3.11)


43

B. The Hypothetical Isotope Method 1.

The Zero-Mass Approach

The formaldehyde molecule will be used as an example to illustrate the zero-mass approach [35,36] for evaluating absolute rotational correction terms to dipole moment derivatives. H2CO has C2v symmetry and infrared active vibrations belong to A 1, B 1 and B 2 symmetry classes. The non-fully symmetric vibrational distortions of B 1 and B 2 species contain contributions from compensatory molecular rotation. Structural parameters for formaldehyde and symmetry coordinates used in the analysis are presented in Table 3.2. Cartesian reference system and internal coordinates are shown in Fig. 3.1. Following the prescriptions given in Ref. [36] the masses of the two hydrogen atoms are set equal to zero in order to create a non-rotating formaldehyde isotope. The application of Eq. (3.9) yields (in units A -1 for stretching and rad -1 for bending coordinates)

$4 w=~;A~=

0.

$5

$6

o

1 -0.147 0

(3.12)

y, z

or in a pseudo-tensor form

S4 ( ( 1= ~3A ) )_~

0 0 -0.051 0 0 0 0.051 0 0

S5 0 0 0.147 0 0 0 -0.147 0 0

S6 0 0 0 0 0 -0.098. 0 0.098 0

J

(3.13)

The equilibrium molecular dipole moment is directed along z-axis and in vector form can be written as

(0)D

Po = -2.33

Hereafter, the following convention for the dipole moment sign will be used: a negative charge shift towards the positive direction of the Cartesian axis results in a negative value of p or of the respective dipole moment derivative.

44

G A L A B O V AND DUDEV

Table 3.2 Structural parameters, symmetry coordinates and rotational correction terms to dipole moment derivatives for H2CO Geometry a,b rCH = 1.09 A ,

Z H C H = 120 ~

r c o = 1.213

po = -2.33 D S.ynunetry coordinates c

Rotational correction terms (D/A or D/rad)

B1

B2

$4 = (At 1 - At2)/x/2

X

-0.119

S 5 = ( A l l - A~2 ) /

X

0.342

$6 = A0d

Y

--0.228

aFrom Res [65]. blA = 10-1~ m; 1D = 3.33564x 10-30 C m. eInternal coordinates are defined in Fig. 3.1. dOut-of-plane mode.

X

FIG. 3.1. Cartesian coordinate reference system, numbering of atoms and defimtion of internal coordinates for H2CO.


45

Eventually, applying Eq. (3.11) separately for each symmetry vibrational coordinate we obtain the rotational correction term matrix R s (in units of D/A or D/rad) 54

$5

$6

0 0

-0 28

X

RS=

y. Z

These quantifies are given in Table 3.2 as well. The same procedure can be followed in the case of other X2CY molecules with X = D, F, C1, Br and Y = O, S. The rotation-free isotope is created by setting the X-masses equal to zero. Application of the zero-mass approach in evaluating rotational correction terms to polarizability derivatives will be illustrated with an example in the second part of the book.

2.

The Heavy Isotope Approach

The method was introduced by Van Straten and Smit [34] in order to overcome problems arising with the zero-mass approach for certain types of molecules. For bent X 2Y and pyramidal X 3Y molecules the creation of an isotope with zero X-masses results 1in indefinite 13p elements, thus hampering the evaluation of rotational correction terms [34]. No such problems are encountered following the procedure of Van Straten and Smit. It will be illustrated with several examples. a.

Ammonia

Ammonia belongs to the group of X3Y pyramidal molecules possessing C3v symmetry. The vibrations belonging to the doubly degenerate E symmetry class contain contributions from molecular rotation. Application of the heavy isotope method requires that the mass of the nitrogen atom be multiplied by a factor of at least 1000 [34]. Molecular geometry data and symmetry coordinates used are given in Table 3.3. The orientation of the molecule in the Cartesian reference frame and definitions of internal coordinates are shown in Fig. 3.2. r sa for NH 3 is as follows (in units A-1 or rad -1) The compensatory rotation matrix 13pA

S3x

S4x

I~PrA a ( 0 s = 0.046 0

0 0.026 0

S3y

S4y

--0.046--0!26) X y. 0 0 Z

(3.16)

46

GALABOV AND DUDEV

TABLE 3.3 Geometry data, symmetry coordinates and rotational correction terms for NH 3 Geometrya rNH = 1.0116 A ,

ZHNH = 106.67 ~

po=-1.47 D Rotational correction terms

Symmetry coordinates b

(D/A or D/tad) E'

Et!

S3x = (2Ar 1 - Ar2 - Ar3)/,~/'6

X

-0.068

S4x = (2Aot1 - Act2 - Atx3)/,~

X

-0.038

S3y ~ (Ar2- - At3)/~f2

Y

-0.068

S3y = (Atx2 - Atx3)/'~

Y

-0.038

aGeometry data are taken from Ref. [66] and dipole moment value from Ref. [67]. bDefinifions of internal coordinates are given in Fig. 3.2.

z

x

FIG. 3.2. Cartesian reference system and definition of internal coordinates for NH 3.

After presenting this matrix in a pseudo-tensor form and multiplication with the equilibrium dipole moment vector we obtain the rotational correction terms matrix R s (in units of D/A or D/tad)


47

TABLE 3.4 Molecular geometry, symmetry coordinates and rotational correction terms for 1,1-dichloroethylene Geometrya rcc = 1.324 A , rcc I = 1.710 A , rCH = 1.070 A ZHCH = 120", ZC1CC1 = 114.5 ~ Po = 1.34 D Syrmnetry coordinates b gl

B2


57 = (AR l - AR2) /

X

0.124

S8 = (A~I - A~2 ) / ~ f 2

X

0.775

S9 = (At 1 - Ar2)/~f2

X

0.014

SI0 = (Abl - Ab2)/~J2

X

--0.019

S 11 = A0CCI

yC

0.546

S12 = AOcH

yc

0.050

astmetural data are taken from Ref. [68] and dipole moment value from Ref. [69]. bIntemal vibrational coordinates are defined in Fig. 3.3(A). cOut-of-plane vibrations.

S3x

RS-

i:oo,

S4x 0 0

S3y

o

-0.068 0

0ix

S4y

-0.038 0

y. z

(3.17)

These quantifies are tabulated in Table 3.3. b.

1,1-Dichloroethylene

The 1,1-dichloroethylene molecule belongs to C2v point group and has vibrations distributed among the following symmetry species F V = 5A l + A 2+4B1 + 2 B 2 .

(3.18)

Vibrations belonging to B 1 and B 2 have to be treated for contributions from compensatory molecular rotation. The geometric parameters and the definition of symmetry coordinates are given in Table 3.4. Cartesian reference system and definition of internal coordinates are shown in Fig. 3.3(A).

48

GALABOV AND DUDEV

A

R1

~1

bI rl~~ )

) x

z

R1 B

~1

r3 r2

~

a3

FIG. 3.3. Cartesian reference system and definition of internal coordinates for (A) 1,1-dichloroethylene and (B) 1,1,1-trifluoroethane.

The most appropriate non-rotating isotope of the molecule is that containing two heavy carbon atoms. Thus, the C-C bond coinciding with the C2 symmetry axis, will maintain fixed direction during vibrational distortions and the rotational correction terms for both B 1 and B 2 vibrations can be evaluated. A question for the weighting factor magnitude, however, arises. A simple survey of values for weighting factors given in Ref. [34] reveals that very significant weighting is needed for molecules containing relatively heavy atoms (CI, Br, I) which do not lye along the main synunetry axis. In order to determine a proper weighting factor in the case of 1,1-dichloroethylene, the compensatory rotations were calculated by employing different isotopes of the molecule.


49

The carbon atoms masses were multiplied by factors of 103, 104, 105 and 106. Compensatory rotation values wj obtained via Eq. (3.9) are presented in Table 3.5.

TABLE 3.5 Compensatory rotations for 1,1-dichloroethylene obtained by using different weighting factors for the two carbon atoms Symmetry coordinate x 103 B1

B2

Compensatory rotation (in A-1 Or tad-l) x 104 • 105

• 106

$7

Y

-0.095

-0.093

-0.093

-0.093

S8

Y

-0.562

-0.577

-0.579

-0.579

$9

Y

-0.010

-0.010

-0.010

-0.010

Sl0

Y

0.014

0.014

0.014

0.014

S 11

X

0.400

0.408

0.409

0.409

S12

X

0.038

0.038

0.038

0.038

The analysis of the results shown implies that factors of 103 and 104 are insufficient to effectively fix the directions of inertial axes of the molecule when it undergoes deformational distortions of the CCI2 group (symmetry coordinates 8 and 11). It is seen that compensatory rotations approach constant values when factors of 105 or higher are used, indicating that proper weighting of the molecule has been reached. Therefore, a multiplication with factor of 105 or more is recommended in this case. The values given in the third column of Table 3.5 (weighting factor of 105) were used to calculate the rotational contribution terms to dipole moment derivatives. These are presented in Table 3.4. c.

1,1,l-Trifluoroethane The 1,1,1-trifluoroethane molecule possesses C3v synunetry with infrared active

vibrations belonging to A 1 and E symmetry classes. Vibrations belonging to E symmetry class are accompanied by a compensatory rotation. Geometric parameters used in evaluating the respective rotational correction terms and the synunetry coordinates are given in Table 3.6. The orientation of the molecule in the Cartesian space is shown in Fig. 3.303). A heavy isotope of the molecule created by multiplying the masses of the two carbon atoms by a factor of 104 was employed in the calculations. The following compensatory rotation matrix was obtained (in units of A -1 and rad-1).

50

GALABOV AND DUDEV

T A B L E 3.6 Structural parameters, definition of symmetry coordinates and rotational correction terms for 1,1,1-trifluoroethane Geometrya r c c = 1.530 A ,

rCF = 1.335 A ,

Z C C F = 111.03",

rCH = 1.085 A ,

Z C C H = 108-32~ Po = 2.32 D

Symmetry coordinates b


E

E

!

M

S7x = ( 2 A R 1 - A R 2 - A R 3 ) / ' f 6

X

0.020

SSx = (2Aa I - Aa 2 - A a 3 ) / i f 6

X

-0.221

S9x = (2A131 - A[32 - A133) /

X

1.092

Sl0 x = (2Ar 1 - Air2 - A t 3 ) / ' f 6

X

-0.044

S l l x = (2Aa I _ Aa 2 _ Aa3)/,r

X

-0.040

S12x = (2Ab I - Ab 2 - Ab3)/.f6"

X

0.069

S7y = (AR 2 - AR 3 ) / ' 4 2

Y

0.020

SSy = (Aa 2 - Aa3) /

Y

-0.221

S9y = (AI32 - A~3 ) / , J 2

Y

1.092

Sl0y = (Ar2 - At3) [,r

Y

-0.044

S 1 ly = (Aa2 - Aa3) /

Y

-0.040

S 12y = (Ab2 - Ab3) /

Y

0.069

aCnmmetric parameters are taken from Ref. [70] and the dipole moment value from Ref. [69]. bIntemal coordinates are defined in Fig. 3.3(B).

I31~A ~ =

S7x

Ssx

0 -0.009 0

0 0.095 0

S9x

S lOx

S 1ix

S 12x

0 0 --0.471 0.019 0 0

0 0.017 0

0 --0.030 0

S7x

Sgx

S9x

Slox

Sllx

S12x

0 0.009 0

0 -0.095 0

0 0.471 0

0 -0.019 0

0 -0.017 0

0 0.030 0

(3.19)

J


51

Values for rotational correction terms to dipole moment derivatives with respect to symmetry coordinates evaluated via Eq. (3.11) are tabulated in Table 3.6. The hypothetical isotope method is aimed at calculating absolute rotational correction terms. An alternative method for evaluating relative rotational corrections in a series of isotopically related molecules was proposed by Escribano, del Rio and Orza [71]. The formalism of this approach will be briefly presented in the second part of the book.

HI. THE BOND M O M E N T M O D E L

A. Theoretical Considerations The first complete formulation of vibrational intensities was put forward some fifty years ago by Volkenshtein et al. [39,63], and timber developed by other authors [72,73] later. The formulation, known as the valence-optical theory (VOT), is based on the assumption that the molecular dipole moment may be represented as a vector-additive sum of bond moments P = E Ok k

(3.20)

where k is a bond index. The changes of molecular dipole moment determining the intensities of infrared bands are attributed to charge fluctuations associated with the dipole moments of individual bonds. At the start, an important feature from physical point of view is introduced: the intensities in the observed absorption spectra are associated with polar properties of valence bonds. Later, the validity of relation (3.20) has been subject to criticism [47,48,53,74]. It does appear, however, that the hypothesis has some physical grounds. Bond moments derived l~om equilibrium molecular dipole moment data have been shown to predict satisfactorily the permanent dipole moments in certain types of molecules [75]. We need also remember that, at the time, very tittle was known about electron density distribution in molecules and its behavior under vibrational distortions. In the original variant of the valence-optical theory a second significant assumption is made. It is assumed that during small vibrations the bond moments remain constant in magnitude and directed along the bonds. In other words, the changes in dipole moment induced by vibrational distortions have purely geometrical origin. The problem is reduced to an appropriate coordinate description of separate bond distortions. A number of approaches have been put forward aimed at representing the forms of

52

GALABOV AND DUDEV

vibrations in terms of bond coordinates. In describing stretching vibrations the ordinary internal coordinates representing changes in bond lengths are used. In describing the changes in spatial orientations of bonds in the case of deformational modes different displacement coordinates have been employed: changes in bond direction cosines, changes in bond direction angles and polar angles [6,63,72,73,76,77]. Considerable progress, in comparison with the earlier formulations of the valence-optical theory, was achieved by introducing in the equations of the zero angular momentum condition [72,77,78]. Thus, the molecular quantifies determined, called electro-optical parameters (eop), are associated with changes in the dipole moment induced by purely vibrational distortions. Gribov has provided an early detailed description of the valence optical theory in a well known book [72]. The derivation is based on expressing the dipole moment as (3.21)

p = ~ ektt k k

ttk are scalar quantifies - the magnitudes of bond moments, and ek the unit vectors directed along the bonds. Differentiating this expression with respect to internal vibrational coordinates leads to equations of the type: ~p/~)Qi = ]~ ]~ ek ( ~ k / t)Rj) Lji + 2 ~ ~tk (t)ek/t)Rj) Lji. k j k j

(3.22)

Qi are normal coordinates, Rj internal coordinates, and Lji elements of the normal coordinate transformation matrix. From expression (3.22) we can see that electro-optical parameters are the following quantifies: ~tk, magnitudes of the bond moments, and 0gk/0Rj, derivatives of bond moments with respect to all internal coordinates. There is a particular advantage of the local intensity parameters defined above. In contrast to other intensity formulations, the eop's do not have space-directional properties. Provided that the approximations introduced are acceptable and the solution of linear equations (3.22) possible, such type of parameters may serve quite well the principal purposes of intensity calculations. Essential elements in solving the problem are the determination L matrix coefficients [Eq. (2.5)] and the derivatives 0ek/0Rj forming a rectangular matrix (0e/0R). In a matrix form equations (3.22) are expressed as PQ = { '~(r / ~R)+ tt(3e / r

L.

(3.23)

As initial quantifies, instead of 0p/0Q i, the dipole moment derivatives with respect to symmetric vibrational coordinates may be used. This offers certain advantages. Quite often important experimental information on the relative signs of dipole moment


53

derivatives expressed as 0p/0Sj may be obtained when a'eating intensity data for different isotopic species. Averaged values over a number of isotopes are expected to provide more reliable initial sets of dipole derivatives. Besides, symmetric properties of molecular vibrations are introduced in a natural way with all consequent simplifications. In particular, reduction in the number of independent intensity parameters is achieved. Equation (3.21) is transformed into expressions of the form 0 P / 0 S j = E E Z e k ( ~ t k / 0 R i ) U i j + Z Z Z ~tk(0ek/0Ri)Uij 9 (3.24) k i j k i j Uij are elements of the orthogonal matrix U [Eq. (2.6)]. The total number of eop's in a completely defined problem is equal to (N-1) • (n+ l ), where N-1 = m is the number of bonds in an N-atomic non-cyclic molecule and n the number of internal coordinates. In absence of redundancies, the number of eop's is given by (N-1)+(N-1)• It is evident that the total number of eop's exceeds by far the number of experimental observables from which these parameters can be determined. These are integrated intensities of the infrared absorption bands and the equilibrium value of molecular dipole moment. The matrix (0e/aR) defining bond reorientations in space with vibrational distortions has the following structure

(a~/aR)=

Oelx / aR 1

Oeix / ~)R2

...

~)elx/ ~)Rn

~)ely / ~)R1

~)ely / OR2

...

~)ely/ ~)Rn

0elz / ~)R1

~elz / ~R 2

9

~elz / ~)Rn (3 9

~emx / t)R 1

Oemx/ OR2

...

~)emx/ t)R n

0emy / OR1

/gemy/ ~)R2

...

~)emy/ ~)Rn

~)emz / t)R 1

~)emz/ ~)R2

9

Oemz/ ORn

(0e/OR) contains, in fact, derivatives of the bond unit vector components ekg (g =x, y, z) with respect to internal coordinates. The elements of (0e/0R) are calculated according to Ref. [72] (ae/aR)

---

r 1 (AA

-

I E:OI)

(3.26)

where r - 1 is (N-1)x(N-1) diagonal matrix with elements the reciprocal values of bond lengths. The kth element of the 3(N-1)• matrix A is equal to: 0 if atom a is not included in bond k; -1 if ct is the initial atom of bond k; + 1 if tz is the end atom of bond k. A is the matrix defining the relation between atomic Cartesian displacement coordinates and internal coordinates [Eqs. (2 916) and (2.17)]. E is a matrix of the bond

54

GALABOV AND DUDEV

unit vector components with different anangement of the elements ekx, eky and ekz as compared to matrix e. 0 is a null matrix with appropriate dimensions. Expressed in terms of derivatives of the dipole moment with respect to symmetry coordinates Eq. (3.23) transforms into PS = { ~(i~t / aR)+ ~t(Be / aR)} U .

(3.27)

In a more detailed form Eq. (3.27) becomes

PS = { e(911 / c3R)+ I1r-I(AA -I E:0 [) } U.

(3.28)

It should be emphasized that the condition of zero angular momentum is implicit in Eqs. (3.23), (3.27) and (3.28). In particular it is contained in the second term of Eq. (3.28). Thus, the usually troublesome problem associated with the compensatory molecular rotations as required by the Eckart-Sayvetz conditions is treated in an elegant way. Solutions of equations of the type (3.27) are not straightforward. First, all equilibrium bond moment values must be known. In the general case, a correct quantum mechanical definition of a "bond moment" is difficult to produce, if possible at all. Attempts to define bond moments quantum-mechanically are always based on severe approximations [79]. Another formidable problem in the above parametric expression of vibrational intensities arises fi'om the very large number of parameters appearing in Eqs. (3.23) and (3.27). These exceed by far the number of experimental observables. To arrive at a defined inverse electro-optic problem, i.e. to evaluate the set of electro-optical parameters from the observed intensities, a considerable number of elements of the matrix (0~aR) have to be necessarily constrained to zero. While this is physically acceptable for some distant interactions, the practice shows that it is necessary to neglect many close interactions as well. The examples presented later in the text illustrate the problem quite clearly. To overcome these difficulties a least squares approach is usually adopted and a set of initial eop's is optimized to fit the experimental intensities of a molecule or, preferably, of a series of structurally related molecules. While experience and intuition can be of considerable help in the choice of structure and values for the elements of the initial eop matrices, the arbitrariness involved is still quite considerable. The valence-optical scheme can be applied, as often done in the past, in a zeroorder approximation. All derivatives of the type ath~/aRj are constxained to zero except 0gk/0rk, where rk is a change of a bond length. The number of eop is reduced to such a degree that the problem becomes completely defined, provided, of course, that static bond moment values are known. The zero-order approximation, however, has long ago been shown to lead to inconsistent results and has been abandoned [72,74].


55

The bond moment model in the variant formulated by Gribov [72] has been applied in interpreting and predicfin~ vibrational intensities in numerous studies from different laboratories [52,53,60-62,73,80,81]. A comprehensive survey of publications on the subject until 1970 is contained in the monograph of Sverdlov, Kovner and Krainov [73]. The contradictory opinions regarding the reliability of the mathematical approach and the physical sense of intensity parameters obtained in view of the numerous approximations implicit in the formulation and, in particular, in its practical applications, require a more detailed discussion on the subject. The basic molecular parameters employed are bond dipole moments. The remaining parameters are derivatives of bond moments with respect to internal coordinates. It is, therefore, of prime importance to define as fully as possible the physical significance of bond dipoles and their contribution to the dipole moment fluctuations determining the intensities of vibrational transitions. An early and quite satisfactory definition of bond dipole moments is given by Coulson [82,83]. The electric moment of a bond is presented as a sum of three terms (3.29) The first term is determined by the non-equal sharing of bonding electrons, and is defined by the so-called net atomic charges. The second term ~ arises from the asymmetric distribution of electron density around the nuclei in molecules. This term is usually referred to as the atomic dipole. It includes the electric moments due to lone pair electrons. The third term is the homopolar or overlap moment. The term ~ is physically associated with the so-called point charge approximation. Viewed as vectors, gq are bond directed and comply, therefore, with the additive approximation for the molecular dipole moment. The second term ttn is formed by dipole contributions that are not, in the general case, bond directed. For example, the electric moment associated with the lone pair of electrons in ammonia is, obviously, not directed along any of the N-H bonds. Therefore, the bond moments, as determined from the permanent dipole moment of the molecule include projections from the lone pair atomic dipole along the bond axes. It is clear, however, that there are atomic dipole components in perpendicular directions to the bond axes as well. The third term, the overlap moment, for the chemical bond in a diatomic molecule is directed along the bond. In polyatomic molecules, however, overlap moments arise from overlap of orbitals centered on non-bonded atoms as well. The above arguments are substantiated by a detailed LCAO MO analysis on dipole moment contributions to infi'ared intensities carried out by Orville-Thomas and coworkers [41-44,48]. In the framework of the LCAO MO approach a Cartesian

56

GALABOV AND DUDEV

component of the dipole moment for closed shell molecules described by a single determinant wave function may be written as

Px = - e Z A

PvvFvv-XAZA -c Z Z Z P~vF~tv-CZ Z PttvFttv A~B ~t v A ~t#v

(3.30)

with (3.31) and

Pttv= 2 ~ C~Cvs, s(occ)

(3.32)

where O~t are atomic orbitals and Pity elements of the electron density matrix expressed in the usual notation. In expression (3.32) the summation is over occupied orbitals. X A and Z A are the Cartesian coordinate and charge of the Ath nucleus. Expression (3.30) provides a convenient interpretation of the molecular dipole moment in terms of contributions associated with the constituent atoms and atomic orbitals. The first term in Eq. (3.30) defines the contribution to the molecular moment from the net charges assigned to nuclei. It can be regarded as a point charge contribution provided that the basis set consists of orbitals symmetrical with respect to the nuclei (pure s, p, d orbitals). This term may be denoted by pq. The second term gives a measure of the contribution of the dipole moment arising from the overlap density of orbitals on different atoms, usually referred to as the homopolar or overlap moment Po- The third term is associated with dipole integrals from orbitals centered on the same nucleus. It is designated as hybridization or atomic dipole term. For the molecular vibration described by a normal coordinate Qk all three terms may change. Thus, ~x

+...--_

(3.33)

"lf~

~

+ ~Qk

{

A~B v

~t~:v

v


57

Contributions from these three terms to the dipole moment changes determining infrared intensifies will be considered in more detail. The first term in Eq. (3.33) may rigorously be related to bond moments. Therefore, the corresponding contributions to the 0p/0Qk derivatives are correctly described by the bond moment model. Early semiempirical [41,42] and later ab jniao MO calculations [41-44,46-49] have shown that a substantial amount of electronic charge may be transferred between the atoms as the molecule vibrates. The contribution to the overall dipole moment variation due to the atomic charge terms may, therefore, be derived into two parts. First, the change in p during vibration due to the motion of the equilibrium charges, Pqe, and, secondly, changes in p due to the flow of electronic charge induced by vibrational motion, p ~ . These two terms may be expressed as follows: Pqe = - e 2 2 Ptttt (Fl~g- Fttg) A ix

(3.34)

PAq

(3.35)

=

- e 2 ]~ (Pl~.t-P~o) F~j.. A l.t

The superscript ( ' ) denotes that the associated term refers to the molecule in a distorted geometry, while the absence of prime indicates that the term refers to the molecule in its equilibrium geometry. The appearance of a charge flux term does not contradict the bond moment description of the molecular dipole moment, but the corresponding bond moment fluctuations have to be taken into account. The second term in Eq. (3.33) arises from the overlap density Of orbitals on different atoms. As in the former case, the corresponding dipole changes induced by vibrational distortions can be expressed as: AB

Poe = - e

Z E 2 P~tv(F~tv- Fttv) A~eB tt v

(3.36)

AB

PAo = - e

~

~ E (Pgv-Pity) F~tv-

A~B ~

(3.37)

v

These dipole terms cannot immediately be associated with bond moments since Pttv and Fttv are taken over all pairs of atoms in a molecule. It may, however, be argued that the dipole integrals between bonded atoms make the greater contribution to the overlap

58

GALABOV AND DUDEV

dipole term. Thus, in a kind of a "tight-binding approximation" the bond moment representation still retains certain merits as far as this dipole term is concerned. The last term in Eq. (3.33) can also be sprit into a part representing the change in p arising from distortions of equilibrium charges and a part determined by charge density fluctuations. A PTIe = -- e ]~ ~ Pttv (F~tv - Fttv) A

pA,i = - e ~

A Y. (P~v-Pttv) F/iv . A ttc-v

(3.38)

(3.39)

Even equilibrium PTI is not formed by bond directed partial moments. Neither can one predict the change of magnitude and direction of these atomic dipoles during vibrations. Therefore, the expression of PTI in terms of bond moments seems little justified. It has been shown that the role of the atomic dipoles in determining infrared intensities, can be quite substantial [44,84]. The above treatment of the various contributions to the total dipole moment change determining infrared intensities is bound to the LCAO representation of molecular wave functions, and, therefore, not to observable physical quantifies. The various terms defined by Eqs. (3.34) through (3.39) reveal, however, how the dipole moment and its derivatives, as calculated by ab initio MO methods, are affected by vibrational distortions. Let us emphasize here that a lot of arguments in recent literature referring to model description of vibrational intensities are based on ab imtio calculated wave functions and charge densities [44-46,81,85,86]. An alternative, and possibly physically more acceptable, interpretation of the intramolecular factors determining the dipole moment changes associated with vibrational motion may be based on consideration involving the electron density function, the spatial part of which is an observable physical quantity. The function p(R) defining the electron density at position R in LCAO expansion reads ~'R) = ~ Pttv Ott(R) Or(R) 9 ~v

(3.40)

Pity are elements of the charge density matrix. Integration of p(R) over the entire molecular space gives the number of electrons in a molecule. It is possible, in principle, to partition the molecular space into sub-spaces related to the constituent atoms and to integrate ~ ) over these sub-spaces. As a result, estimates of the charges associated with individual atomic sites are obtained. If the atomic sub-spaces are considered as points in


59

the molecular space, we arrive at a point charge model of the electric charge distribution in molecules. There is a continuing argument as to the best way of partitioning the electron charge density and the definition of formal atomic charges [47,49-51,87-90]. We shall later discuss in some detail the point-charge approximation as applied to theoretical formulations of infrared intensities. Here we shall mention only results from electron density function analysis referring to bond moment rather than point-charge model. The two representations of molecular charge distribution are, of course, very close. Wiberg and Wendoloski [47,49] using results from large basis set ab mitio MO calculations for a number of hydrocarbons concluded that infrared intensities are determined by: (1) Motion of static bond moments; (2) Creation of a bent bond moment arising l~om incomplete orbital following; and (3) Changes in bond moments due to charge flux effects. In their studies no example of molecules with lone-pairs of electrons is considered. It may be expected that in such cases other factors would also interfere. Generalizing two different approaches - the first based on analysis of contributions of the LCAO composite dipole terms, and the second based on studies of properties of the electron charge-density f u n c t i o n - reveal a complex picture of intramolecular electronic effects determining band intensities in infi'ared spectra. The application of a bond-moment model in describing this particular molecular property is far from straightforward. Carefully considering the existing problems in the physical definitions of the bond dipole moment, Gribov [72] accepts that the electro-optical parameters are effective quantifies characterizing an atomic grouping, such as CH3, CH2, etc., as a whole. The entire combination of eop's associated with a functional grouping is of significance. Individual parameter values, including those denoted by ttk, do not possess a well-defined physical sense. In order to compensate for some of the above difficulties of the bond moment model, Sverdlov et al. [73,91] have proposed a modification of the original formulation. In their treatment certain adjustments of the physical prerequisites are made by allowing the presence of bond moment components perpendicular to bond directions. The additive representation for the molecular dipole moment is retained: 3

P = ~ Pk = ~ ~ ~ki ei. k

k

(3.41)

i=l

In expression (3.41) the bond moments ~ are expressed by three components (gkl' ~k2' gk3) referring to a bond axis Cartesian system, in which the axis 1 coincides with the bond direction. The vectors ei refer to molecular Cartesian frame. Differentiating (3.41) with respect to internal coordinates and using relation (2.4) one arrives at a set of linear equations defining the relation between the experimental dipole moment derivatives 0p/0Qk and the eleca'o-optical parameters associated with

60

GALABOV AND DUDEV

expression (3.41). With the modifications introduced by these authors the physical foundations of the valence-optical theory appear to be improved since it is clear that the charge distribution around the bonded atoms are determined by local dipoles that deviate from the bond directions. On the other hand, however, the number of intensity parameters is tripled. It becomes necessary to go even fmther on the way of approximations, neglect of parameters, and other assumptions in the mathematical sohtiom. These are, very possibly, the reasons for the limited application of this variant of the theory [73]. As early as 1972 it was recognized that an explicit inclusion of terms associated with the charge-flux effects accompanying molecular vibratiom may offer an opportunity for better understanding of the physical significance of parameters in infrared intensity models based on the bond moment concept [42]. The bond charge parameter formulation developed by Van Straten and Smit [92] describes infrared intensities in terms of static bond charges and bond charge fluxes induced by vibrational distortions. In the mathematical procedure many common features with the apparatus of the valence-optical theory are present. The molecular dipole moment is treated in the additive approximation. Each bond moment is presented as lak = qk rk ek"

(3.42)

The molecular dipole moment is, then, defined as P = ~ qk rk ekk

(3.43)

rk is a bond length, qk an effective bond charge, and ek a unit vector directed from the negative towards the positive end of a polar bond. Differentiating Eq. (3.42) with respect to an internal coordinate leads to the expression

aWo

=

+ (akmR

)qk'k + (

daRi)

rk"

(3.44)

If dipole moment derivatives with respect to symmetry coordinates are used as initial parameters, which in most cases is appropriate, the following matrix equation is obtained

Ps = re(~7"~) U+ qe(br/;~R)U+~ (3e/bR)U.

(3.45)

The structure of the PS array was already given [Eq. (3.4)]. (Oq/OR) is a matrix containing derivatives of the effective bond charges with respect to internal vibrational coordinates. These quantifies reflect the charge reorganizations with vibrational motion.


61

r is a diagonal matrix containing the equilibrium bond lengths, while the symbol ( A ) over r denotes a matrix having triplets of the bond lengths on the diagonal. The matrix e is identical with the respective array in the VOT formulation [Eq. (3.23)]. (0r/0R) contains derivatives of bond lengths with respect to internal coordinates, q is a diagonal matrix with elements the effective bond charges while q with A over it is an extended bond charge matrix having on the principal diagonal triplets of the bond charges. The array (0e/0R) is as defined by expression (3.25). A principal part of the calculations is the evaluation of the matrices (0r/0R) and (0e/0R). Though equation (3.45) appears overcrowded with matrices and is too complex, it can in practice be calculated without much difficulty with standard programs for vibrational analysis. The necessity to determine simultaneously for each bond in molecular static and flux bond charges imposes again the sensitive problem associated with the number of parameters involved. Applications have been limited to small symmetric molecules [93]. These studies have underlined once again the significant role of charge-flux effects in determining infrared band intensities, and the necessity of their proper consideration in intensity models. Other approaches aimed at explicit presentation of charge flow effects associated with xa'brational displacements of individual valence bonds have also been put forward [46,94]. These investigations have confirmed the importance of orbital rehybridization effects that accompany vibrational distortions and are a cause for considerable charge rearrangements. It should be emphasized that the charge-flux effects are implicitly included in electro-optical parameters of the type agk/aR i that appear in the first-order bond moment model [72]. Thus, in standard applications to various molecules it does not seem necessary to extend the original formulation since this would result in further increase in intensity parameters. As mentioned, to deal with the considerable gap in number between experimental dipole derivatives and electro-optical parameters, a least squares approach is usually adopted [72]. From the available literature it does not become entirely clear how individual parameter values, especially equilibrium bond moments, are derived from the sets of linear equations as defined by Eqs. (3.23) and (3.24). It can be assumed that extended applications to a large number of molecules that have the same type of structural elements may enable, by multiple testing, to arrive at physically significant elecr parameters. In fact, a library of such intensity parameters has been created and used together with standard transferable force fields to predict the infrared spectra of a number of aliphatic and aromatic hydrocarbons [60-62]. Comparisons between calculated and predicted spectra do not, however, appear to be satisfactory for many molecules. Thus, the accuracy of both force constants and eop's is under question. On the other side, the main purpose of these libraries of empirical vibrational parameters

62

GALABOV AND DUDEV

q~

a.

-~ (e)t

-0

1.0

2.0

I

(=•

I xs

6r

b.

Xcl

,

F

4.0 !

!/

.I

~r

(•

I

c. 1.5

6.0t _

(lOSdyne cfn-1)

=

2.0 t R~

I

cw,,....~~

Cl

F

FIG. 3.4. Plots of the dependencies between experimental atomic equilibrium charges of halogens in CH3a and (a) difference of electronegativity between halogen and carbon, (b) stretching force constant of the bond r and (c) interatomic distance c~C (Reproduced from Ref. [81] with permission. Copyright [1984] American Chemical Society).

+

q:

~ '~c'"~c,,,+ 3H4

0.10

~cc.

C~H,,~ CzHi

1.o

1:2

1:4

1:6 R:o,,,

FIG. 3.5. Plot of dependence between the experimental atomic equilibrium charges of hydrogen in hydrocarbons and the interatomic distance of the adjacent CC bond (Reproduced ~om Ref. [81] with permission. Copyright [1984] American Chemical Society).


63

appears to be their-application in an artificial intelligence software for spectral-structural correlations where only principal features of the calculated spectra need to match the observed spectra. In such a perspective the project seems to fulfill its principal objectives. More detailed work on applications of the valence-optical theory is produced by Gussoni, Zerbi et al. [80,81,95]. These authors have obtained eop values that appear to correlate very well with independent structural parameters and quantum mechanical data. These results appear to support the validity of some of the basic assumptions of the bond moment model. On the other hand, we have the more reserved statement of Gribov about the physical si~ificance of bond moments and derivatives as effective quantifies [72]. With these words of caution we reproduce in Figs. 3.4 and 3.5 some interesting results reported by Gussoni et al. [81] on the dependencies between bond moments as derived from expe"rtmental infrared intensities and other structural parameters. The atomic equilibrium charges are obtained from the respective bond moment values. Some of the problems and ditticulties associated with applying the bond moment model will be seen fxom the examples of application described next.

B. Applications 1.

H20

The analysis of gas-phase infrared band intensities of water using the valenceoptical theory is a case with the inverse intensity problem completely defined in terms of available experimental data. These are the measured integrated intensities of the three fundamental vibrations and the permanent dipole moment value. Experimental intensity data employed are as determined by Clough et al. [96]. The force field of Mills [23] is used to derive the normal coordinate transformation matrix L S. The respective L S is then applied in calculating 0p/0Sj from the 0p/0Q i dipole moment derivatives. In accordance with 1UPAC recommendations [97] the matrix L S is given together with dipole moment derivatives with respect to normal coordinates so that no ambiguity associated with the phase of normal coordinates is present. The signs of 0p/0Q i are as determined by Zilles and Person [98]. The standard sign convention was defined in the beginning of the present chapter. The L S transformation matrices for H20 obtained from the force field of Mills [23] are (in units of ainu- 89or rad ainu- 89A-1 )

A1

S1 S2

Q1 1.0171 0.0111

Q2 -0.0660 1.5284

B2

S3

Q3 1.0345

(3.46)

64

GALABOV AND DUDEV

7

)

x

FIG. 3.6. Definitions of internal coordinates, bond directions, Cartesian reference system and numbering of atoms for H20.

TABLE 3.7 Observed integrated gas-phase infrared intensifies of H20 and calculated dipole moment derivatives with respect to symmetry coordinates Aia

0p/0Sj b

~mol)

(D/A or D/tad)

A 1 = 2.24

Pl = -0.234

A 2 = 53.6

P2 -

0.726

A 3 = 44.6 P3 = -0.992 a From Ref. [96]. b For the sign choice and force field used see text.

The geometric parameters of H20 are: rOH=0.9572 A, LHOH = 104.5 ~ [99]. The reference Cartesian system and internal coordinates are defined in Fig. 3.6. The symmetry coordinates have their usual form. The non-zero elements of PS obtained are given in Table 3.7. From these the application of Eq. (3.24) produces the following equations relating 0p//~Sj derivatives with eop's for the two symmetry classes [ 100] AI:

-0.866 (aia/0r) - 0.866 (ag/&') = - 0.234 -1.224 (ala/a0) + 0.791 ~ -

B2:

0.726

-1.118 (a~Or) + 1.118 (a~/~') - 0.051 ~ = - 0.992.

(3.47)


65

Bond moments are in Debye units, 0p/at are in units D/~-1 and a~O0 are in units D tad -1. Conversion factors to SI units are given in Table 3.2. An additional equation relating the permanent dipole moment value Po =-1.85 D [34] to the OH bond moment has the form: - 1.224 tx = - 1.85.

(3.48)

Eqs. (3.47) and (3.48) contain four observables and four unknowns. Therefore, all eop's can be uniquely determined. The values obtained are laOH = 1.511D c91a/c30 = 0.383 D rad -l (3.49) a~igr = 0.544 D/~-! ait/&' = - 0 . 2 7 4 D A -1 In the third equation of the set (3.47) the term -0.05 lit is the rotational correction term. It should be noted that for H20 the inverse electro-optic problem is entirely defined, simply because of the very small size of the molecule and its symmetry allowing the OH bond moment to be evaluated from the permanent dipole moment. Obviously, in molecules where at least two non-equivalent bonds are present, experimental determination of the bond moments is not possible and certain assumptions have to be made. 2.

CH3Cl

In spite of the high symmetry and small size of methyl chloride the application of VOT in analyzing infrared intensities of this molecule reveals quite clearly the computational difficulties that are usually encountered. Because of the closeness of all bonds no intramolecular interactions can be neglected on acceptable physical grounds. The experimental gas-phase infrared intensities used are those determined by Kondo et al. [ 101]. The L S matrix is obtained with the force field of Duncan et al. [24]. The definitions of internal coordinates and of the Cartesian reference system are given in Fig. 3.7. Geometric parameters and symmetry coordinates are shown in Table 3.8. The L s matrix for CH3CI is as follows: A1

1.0081 --0.0939 -0.0433

0.0011 1.3897 0.0840

-0.0039 0.1668 0.3202

E

1.0510 0.1131 --0.1012

0.0209 1.5134 0.3207

-0.0013 -0.2284 0.9136

(3.50)

66

GALABOV AND DUDEV

X

~Z o2

R

FIG. 3.7. Definitions of internal coordinates, bond directions, Cartesian reference system and numbering of atoms for CH3C1.

TABLE 3.8 Geometry and symmetry coordinates for CH3C1 Geometrya rCH = 1.095 A, rcc I = 1.778 A, Z H C H = 110.83 ~ S3nnmetry coordinates b A1

S 1 = (At I + Ar 2 + A t 3 ) / ~ f 3 S 2 = a (Act 1 + A ~ 2 + Act3) - b (AI31 + A~32 + A133)

a =0.397249,

b=0.418959

S3 = AiR E'

S4a = (2Ar 1 - Ar2 - Ar3)/,~/6

E"

S4b = (Ar 2 - Ar3)/,r

S5a = (2Ac~1 - Act 2 - Act3) /

S5b = (Act2 - A~3) /

S6a = (2AI31 - AI32 - AI33)/~t'6

S6b = ( A ~ 2 - A~3 ) / ' f 2

a From Ref. [24]. b As defined in Ref. [ 102].

The PS matrix obtained is presented in Table 3.9. The signs of dipole moment derivatives are as determined by ab imtio MO calculations [ 103 ]. Application of Eq. (3.24)yields the following relations between 0p/0Sj and electro-optical parameters [100]


67

T A B L E 3.9 Dipole moment derivatives with respect to symmetry coordinates for methyl chloride A1

AI:

Pl = -0.625

E

P4 -- -0.386

P2 = 0.182

P5 -

0.285

P3 = 2.237

P6 - -0-166

-1.732 (cgM/&) + 0.538 (c01a/0r) + 1.075 (egg/or') = -0.625 1.192 (c9M/c9~) + 1.257 (tgM/013) + 0.370 (c91.t/c9o0 + 0.740 (cgl.t/0a') -0.390 (c9~c913) - 0.780 (0~c913') -

1.196 ~t = 0.182 (3.51)

- (0M/0R) + 0.931 (al.t/0R)= 2.237. E:

1.164 (0~0r) - 1.164 (01x/0r') + 0.038 M - 0.035 kt = -0.386 1.164 (Og/o~) - 1.164 (c3~0o~') + 0.037 M + 0.769 ~t = 0.285 1.164 (al.ffOJ3) - 1.164

(a /af3')

(3.52)

- 0.070 M - 0.695 ~t : - 0 . 1 6 6 .

To these an expression relating Po and ~tk can be added 0.931 la - M = 1.870.

(3.53)

In Eqs. (3.51) through (3.53) M is the C-CI bond moment and ~t the C - H bond moment. Bond moments are in Debye units, eop's of the type a~tk/orj are in units D A -1, while a ~ / ~ i with Yi an angular internal coordinate are in units D rad -1. The total number of electro-optical parameters is 13 while the number of observables that can be used in their evaluation is just 7. It is evident that to solve the sets of linear equations some eop's have to be put equal to zero. Another difficult problem is how to decompose Po into particular bond moment values. Even if g and M are assigned particular values [73] a number of different solutions are obtained depending on which eleca'o-optical parameters are ignored. It is, therefore, of particular importance to discuss in detail the approximations adopted in calculations employing the bond moment model.

68

GALABOV AND DUDEV

C. Atomic Charge- Charge Flux Model The bond moment model of infrared intensities is associated with the assumption that chemical bonds are the most basic molecular units of interest. Indeed, the qualitative distinction between isolated atoms and molecules is, perhaps, best reflected in the chemical bond. The characterization of bond polar properties by an appropriate reduction of intensity data is, therefore, quite welcome. On the other hand, the concept of atoms in molecules also has support since theoretical calculations show that some basic atomic properties, including electronegativity, are partially retained in bonded systems. The NMR phenomenon also quite strongly supports, from the experimental side, the concept of atoms in molecules. Though there is little resemblance in the physical factors determining nuclear shielding in NMR and absorption of infrared radiation, it may be argued that both reflect in one way or another the electronic structure of the molecule. Thus, the possibility of describing the electronic structure of a molecule in terms of electric charges assigned to individual atoms is indeed appealing. In 1926 Dennison [104] introduced the concept of fixed atomic charges in describing the dipole moment variations with vibrational motion. The approximation was, of course, too far reaching to be practically applicable in explaining the observed spectral profiles for even very simple molecules. The atomic charge representation of the dipole moment function applied to theoretical formulation of infrared intensities was reviewed some fifty years later by several authors. King et al. [ 105] defined a quantity called effective atomic charge that is derived from the sum of integrated infrared absorption intensities of a molecule in a formulation based on expressing the dipole function in terms of atomic Cartesian displacement coordinates. The exact physical si~ificance of the King's effective atomic charges was not quite clear at the time, though it was shown to be constant for a number of hydrocarbons. The effective atomic charges of King found considerable application later in the atomic polar tensors representation of infrared intensifies [45,106]. In 1973 Samvelyan, Aleksanyan and Lokshin [107] presented in brief an effective atomic charge model of infrared intensities in terms of mathematical approach strongly correlated with the atomic polar tensors formulation [33,108]. It will therefore, be more appropriate to discuss these developments in conjunction with the atomic polar tensors method. In this part we shall concentrate on the effective atomic charge - charge flux model of infrared intensities (ECCF) developed by Decius [109]. In the basic definitions the Dennison idea of assigning point charges to individual atoms in molecules is retained. By introducing the charge-flux terms, however, a si~ificant step is made on the way to describe more correctly the intramolecular charge variations determining band intensities in infrared spectra. The molecular dipole moment is approximately described as


P = E ~t rot.

69

(3.54)

131,

~a is the effective charge associated with atom a, and rot is the position vector of ct. The summation is over all atoms. To account for the charge, reorganization accompanying vibrational distortion terms related with these effects are introduced. A Cartesian component of the molecular dipole is defined, within the harmonic approximation, as px = px (~ + E ( r ~ ( o ) a x ~ + ~ (x

x~(O)) =

(3.55)

= Px0 + AlPx + A2Pxpx(o) is the equilibrium value of the dipole moment component. The term A 1 Px is associated with the equilibrium atomic charges and A2 Px is the charge flux term. The entire formulation is expressed in terms of internal symmetry coordinates. As stressed before, this is natural since great simplifications are achieved. On the other side, individual Cartesian components of the experimental dipole moment derivatives can only be known if the polarization of the vibrational transition moment is fixed by symmetry. This is not the case for molecules with symmetry point groups C 1, Ci, Cs, C2 and C2h. For such systems the inverse intensity problem cannot be treated explicitly in any intensity formulation. For symmetric distortions the terms A 1 Px and A2 Px are represented as functions of the respective symmetry coordinate A1 Px = Z ~-xz(0) E Axa(J) Sj ct j

(3.56)

A2 Px = X X ~x (j) x~ (0) Sj ot j

(3.57)

ax~O) = aax~/asj

(3.58)

~..~x(J) = O~/OSj

(3.59)

where

Ax~ are atomic Cartesian displacements; ~(0) are the equilibrium atomic charges and ~(J) are the atomic charge fluxes for a symmetric distortion defined by Sj. The solution is reduced to determining the quantifies ~x~(0)and ~a(J) for each atom. It is clear that the number of parameters is much higher than the number of experimental quantifies and explicit solutions are only possible in cases where, due to high symmetry, the equilibrium atomic charges can be immediately determined from the equilibrium dipole moment

70

GALABOV AND DUDEV

TABLE 3.10 ECCF intensity parameters for diatomic molecules a AB

PO dp/dr (D) (D/A) -1.82 -1.68 I-IF -1.085 -0.88 HC1 0.112 -3.10 CO aReprinted from Ref. [ 109] with permission.

r(O) (A) 0.917 1.274 1.128

~A(0) (D/A) 1.98 2.13 -0.099

~A(1) (D/A2) -0.33 -1.2 2.84

value. From the condition of conservation of the total charge an additional relation between the parameters C~x(0)follows. The general equation relating the effective atomic charges and charge fluxes to the experimental dipole derivatives is as follows

apx/aSj = E Ix

+ E xjO)(

;josj).

(3.60)

Ix

From the definitions it can be seen that there is a considerable similarity between the VOT formulation and the ECCF model. The evaluation of intensity parameters meets also comparable difficulties. Explicit solutions are possible only for small symmetric molecules such as diatomics, bent AB 2, pyramidal AB3, etc. Least-squares refinement of parameter values can be easily introduced, thus enabling bigger and less symmetric molecules to be treated. The author of the ECCF model did not, however, propose such an approach. Most clearly, the basic ideas of the Decius model are illustrated in the case of a polar diatomic molecule AB. In such simple systems no assumptions need to be made. The equilibrium atomic charges for the diatomics can be unambiguously derived from the permanent dipole moment. There are two intensity parameters to be determined: ~A(0) (~B(0) =--~A(0)) and 0~A/0S 1 where S 1 = ARAB. Let us assume that the atom B is the center of negative charge and is situated along the positive direction of the X axis of a reference Cartesian system. The permanent dipole will then have a negative value. Application of Eq. (3.60) yields 0px/0S 1 = - (~A(0) + CA(1) r(0)) = (Po/r(0)) - ~A(l) r(0).

(3.61)

r(0) is the equilibrium bond length. Decius [ 109] provides very interesting results for a number of diatomic molecules, some of which are shown in Table 3.10. Similar formulation, based on representing a molecular dipole moment in terms of effective atomic charges, has been put forward by Aleksanyan et aL [ 107,110]. Basically the treatment refers to analogous intensity parameters. These are effective atomic charges and atomic charge fluxes expressed in the space of atomic Cartesian displacement


71

coordinates. We shall use here the notation already adopted. Starting from the expression for the dipole moment as given by Eq. (3.54), the change in p induced by a vibrational distortion is given by N N N Ap = Z ~xz(0)Ara+ X ra(0) Z (O~/Orl3)0Arj3. a=l a=l 13=1

(3.62)

~x(0) is the equilibrium (static) effective charge of atom a and ra(o) is the equilibrium radius vector of ct. The charge flux term is expressed in a Cartesian coordinate space. In a matrix form Eq. (3.62) is given by the expression ap =

E'

[C + xo (a~tax)]

x

(3.63)

.

Ap is a column matrix with components APx, Apy and Apz. The matrix E' has the sm~ct~e

E ' = II E3 . . . E3 II

(3.64)

with appropriate dimension where E 3 is a 3x3 unit matrix. ~ is 3x(3N) matrix containing triplets of the static effective charges (~1

0

0

~2

0

0

...

~N

0

0~

=/00

~1 0

0 ~1

0 0

~2 0

0 ~2

... -.-

0 0

~N 0

0 J . ~N

(3.65)

The matrix X 0 is a 3 x(3N) array with elements the equilibrium atomic coordinates in an arbitrmy Cartesian reference system Xl 0) x2 (0)

Xo=

... XN(0) (3.66)

YI(~ y21~l . . yN . (~ . Zl (0) z 2 ... ZN(0)

The matrix (OUOX)has a block structure

(a4sax)

= II aedOXl I aedaX21 . . .

I oxN II.

(3.67)

Each block has the followingform

a;/ax C-CI > C-Br > C-I. We went into some detail in discussing bond polar parameter values in different molecules since only such an analysis can reveal the physical significance of these molecular quantifies. Let us emphasize here, that no approximations in transforming the

MOLECULAR DIPOLE MOMENT DERIVATIVES

S

100

TORR

129

TORR

T

T

o.g 0.5 0.1

'

" "

l"

'

"

I

|

l

TG 0.9 0.5 o.1 I

|

-

9

I

i'

,

i""

9

I

9

GG

Ud (J : . 0.9

H~ +0.099

---}

C--------- C ~ -0.108 -0.123

H +0.133

It is seen that a dynamic dipole is created for the non-polar at equilibrium C~-C bond. These dipole terms will eventually be reflected in final values for 8C_c. The physical significance of effective bond charges will eventually emerge from applications of the formulation outlined above in analyzing infrared intensities of various types of molecules.

C. Applications The effective bond charge (EBC) formulation is aimed uniquely at interpreting vibrational absorption intensities. It is quite tempting to illustrate the theory with ab initio estimated dipole moment derivatives since these data are free of the usual experimental uncertainties and inaccuracies. As will be shown in the subsequent chapter, however, different levels of ab mitio molecular orbital calculations produce very divergent sets of predicted infrared intensities. Though the overall shape of the spectrmn is reproduced qualitatively, the ratios computed and experimental intensities for a given molecule do not follow any definite trend at a given basis set. Thus, there is no certainty that ab initio intensity results can provide a sufficiently reliable source of data for assessing the physical significance of local intensity parameters in different molecules. Thus, in this section we shall present results for gas-phase experimental intensity data. The transformation of the infrared intensities for water and ammonia into effective bond charges will be followed in detail so that the computational procedure becomes clear. Results from the application of EBC formulation in interpreting intensity data for a number of medium size molecules will then be presented. Comparisons of effective bond charges for different bonds in varying molecular environments will provide a basis for assessing the physical significance of these molecular quantifies.


135

TABLE 4.13 Experimental dipole moment derivatives with respect to symmetry coordinates for ammonia (in units D A -I or D rad-l)a,b AI

Pl = 0.290 P2 = 1.620

E

P3x = -0.195 P4x = -0.366

aFrom Ref. [147l. definition of symmetrycoordinates is given in Table 3.3.

Cartesian reference systems, geometric parameters and symmetry coordinates for H20 and NH 3 are given in Chapter 3. Dipole moment derivatives with respect to symmetry coordinates for H20, evaluated in analyzing experimental absolute infrared intensities, are also presented there, igp/aSj dipole moment derivatives for ammonia used in the present calculations were taken from Ref. [147] and are presented in Table 4.13. The signs of these quantifies have been fixed with the aid of ab initio MO calculations [147]. Elements of the respective R s matrices for both molecules were evaluated by employing the heavy isotope method [34], weighting the respective heavy atoms by a factor of 1000. The rotational correction terms for ammonia are tabulated in Table 3.3. The R s matrix for H20 has the following form (in D A -1 or D rad -1)

RS =

Sl

S2

S3

0

0

0

y

0

0

0

z

(4.148)

Rotation-free atomic polar tensors were calculated using Eq. (4.142). Finally, D(v) matrices were obtained with the aid of the respective C -1 matrices [Eq. (4.146)]. As an example, the structures of C and C -1 arrays for the water molecule are shown below.

C

-1 0 0 -1 0 0

0 -1 0 0 -1 0

0 0 -1 0 0 -1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

(4.149)

136

GALABOV AND DUDEV

C-1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1

0

0

0

0

0

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

(4.15o)

PS U B, R s U B, Px(v) and D(v) matrices for H20 and NH 3 are given in Tables 4.14 and 4.15, respectively. Calculated effective bond charges [Eq. (4.147)] for O-H and N-H bonds are represented in Table 4.16 together with the 5k values evaluated liom analysis of experimental gas-phase infrared intensities of three other series of molecules. These include hydrocarbons (methane, ethene, ethyne and propyne), methyl halides (CH3F, CH3C1, CH3Br and CH3I) and X2CY molecules (H2CO, F2CO, F2CS, C12CO and C12CS). More details concerning the computations for these molecules are presented elsewhere [ 148]. By comparing ak values for O-H and N-H bonds, it can be seen that these are quite close in values. This is in accordance with expectations because of the similarity of the two bonds. It is of particular interest to examine the changes of C-H bond charges in the series of hydrocarbons as a function of the varying hybridization of the carbon atom and changes in environment. Analysis of the data collected in Table 4.16 reveals a tendency towards higher values with increased s-character of the carbon valence orbitals, beginning with 0.162 e for methane (Csp3-H), 0.179 e for the Csp2-H ~k value in ethene, and ending with much higher effective charges for the acidic Csp-H bonds in ethyne and propyne. The Csp3-H effective bond charge in various molecules do not appear to be very sensitive to environmental changes in the series of hydrocarbons and methyl halides with variations in the range 0.129 e (CH3Br) to 0.163 e (CH3F). Much more pronounced variations in ~k values are found for the polar carbon-halogen, carbon-oxygen and carbon-sulphur bonds. The C-X effective bond charge in methyl halides is changing from 0.946 e in CH3F, 0.470 e in CH3CI , 0.329 e in CH3Br to 0.135 e in CH3I. This is in full accord with expectations and the values evidently may be related to the electronegativity of the halogen atoms. This dependence is shown in Fig. 4.9. The C=O effective bond charges determined are higher than the respective C=S values in the sulphur analogs. This finding is also in accord with the lower polarity of the C=S bonds due to lower electronegativity of the sulphur atom.

z

TABLE 4.14

9m

2

Experimental PsUB, RsUB, Px(v) and D(v) matrices for H 2 0 (in units of eIectrons)ahc H2

-0.23 1 PsUB =

0 0

-0.292 -0.0 18

(-0.213

0.115

0 -0.069

0 0 0

-0.089

0.146

0.1 15 0 0.069

0 0 0

0.089 0

0.146

0 0 0

0.009

0 0

0 0 0

0.007 0 0

0.009 0 0

0 0

-0.007

0

0

0 0 -0.292

0.106 0 -0.069

0 0 0

0.082

0.107 0 0.069

0 0 0

-0.082

0 0.146

0.106 =

[-0:69

0 0 0

0.146

r?

rl

D(v)

%

H3

0.082 0 0.146

0.107 0 0.069

0

-0.082

0 0

0.146

aReprinted from Ref. [ 1481 with the kind permission from Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK, Copyright (1995) blD A-1 = 0.2082 e. values given correspond to Cartesian reference frame shown in Fig. 3.6.

L

W

4

TABLE 4.15 Experimental PsUB, RsUB, Px(v) and D(v)matrices for NH3 (in units of electrons)a,b N1

0 0

0.044 0 0.182

0.071 0.031 -0.022 0.031 0.036 0.038 0.056 -0.097 0.182

0.071 -0.031 -0.02 -0.031 0.036 -0.038 0.056 0.097 0.182

0

0.008 0 0

0.009 0 0

-0.547

0

0.009

0 0.009

0

0 0

RsUB =

-0.132

H4

0.018 0 -0.112

0 0

H3

H2

0 -0.547

0

0.089 0

0.009 0

0

0.036

0.080

0

0.062 0.031

-0.112

0

0.182

0.056

D(v) =

[&

0

0.080 0

-0.004 0.007

0

0

0.031 -0,018 0.027 0.031 -0.097 0.182 r2

‘1

0.009

0 0.009

0,036 0 0.182

0.062 0.031 0.056

0.031 -0.018 0.027 0.031 4.097 0.182

0 0

0.009

-0.ocn

0

0

0.062 -0.031 0.027 -0.031 0.056 0.097 0.182

-0.031

‘3

0.062 -0.031 -0.031 0.027 -0.031 0.056 0.097 0.182

aReprinted from Ref. [ 1481 with kind permission from Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK,Copyright (1995). bThe values given correspond to Cartesian reference fiame shown in Fig. 3.2


139

TABLE 4.16 Effective bond charges from experimental infrared intensities (in units of electrons)a Bond

Molecule

5k

O-H N-H C-H

H20 0.210 NH 3 0.231 CH 4 0.162 CH2--CH2 0.179 CH-CH 0.352 0.139 CH#-=C-CH3 ~ ~0.349 CH3F 0.163 CH3C1 0.130 CH3Br 0.129 CH3I 0.139 H2CO 0.267 C-F CH3F 0.946 F2CO 1.096 F2CS 1.064 C-C1 CH3C1 0.470 C12CO 0.866 C12CS 0.879 C-Br CH3Br 0.329 C-I CH3I 0.135 C=O H2CO 0.789 F2CO 0.931 C12CO 1.070 C=S F2CS 0.650 C12CS 0.781 aReprinted from Ref. [ 148] with kind permissionfrom ElsevierSci. Ltd, The Boulevard, LangfordLane, Kidlington OX5 1GB, UK. Copyright(1995).

By comparing 5k data for lower polarity bonds (C-H) to the much polar C=O, C=S, C-F, C-C1, C-Br, O-H and N-H bonds it is seen that higher polarity is reflected in higher effective bond charges, with values varying in most cases with expectations. The analysis of data for effective bond charges as obtained from experimental gasphase intensity data for a number of molecules reveals some definite trends of changes. The variations found may, in most cases, be related to polar properties of the bonds considered. As already stressed, the effective bond charges are solely determined by

140

GALABOV AND DUDEV

F r

0.8

cO ILl r re" ,

(nl~lm>

( n I Cry, I m>

.

(8.63)

A vibrational transition is allowed if at least one of these integrals differs from zero. The symmetry selection rule states that the integral ( n I ~k I m ) is not zero if the direct product of representations for the ith vibrational mode F i = F n x F~k • F m

(j, k =x, y, z)

(8.64)

belongs to the fully symmetric species of the molecular point group. Since the ground state wave function is always fully symmetric, the behavior of the product F i will depend on the symmetry properties of the polarizability tensor components and of the vibrational wave function of the first excited state. Hence, if that wave function belongs to the same symmetry species as the respective polarizability component, the direct product F i will be fully symmetric. In this case, the ith transition will be Raman active. An interesting case are molecules possessing a center of symmetry. According to the symmetry selection rule, a given vibrational transition is infrared active ff the direct product between the two vibrational wave function representations and that of the dipole moment vector component is totally syaunetric (Section 1.2.2). The ground state vibrational wave function, as mentioned, is always totally symmetric. It belongs to even or gerade representation since it is not affected by inversion operation with respect to the center of symmetry. The dipole moment vector, however, changes sign when the inversion operation is executed. It, consequently, belongs to odd or ungerade representation. Thus, it is easy to deduce that the representation of the first vibrational excited state wave function must be odd (ungerade)(belonging to A u, B u, etc. synunetry species) if the fundamental transition is to be infrared active. The molecular polarizability tensor ct is defined in relation with two vectors: the induced dipole ~t and the electric field vector f [Eq. (8.1)]. Hence, the representation of ct can be expressed as a direct product of the representations of the respective vectors [265] F a = F~t • I f .

(8.65)

INTENSITIES IN RAMAN SPECTROSCOPY

207

Since the inversion operation alters the sign of the two vectors, according to Eq. (8.65), ct remains unaffected. It, therefore, belongs to an even symmetry species. It is evident that the excited state wave function must possess the same symmetry properties as ct if the transition is to be Raman active. Thus, for molecules with center of synanetry fundamental transitions to excited states belonging to even syrmnetry species (Ag, Bg, etc.) are only active. The above discussion outlines the so-called rule of mutual exclusion in vibrational spectroscopy which reads: for molecules with a center of synanetry vibrations that are Raman active are infrared inactive and v i c e v e r s a .

IV. R E S O N A N C E

RAMAN EFFECT

The relations and conclusions drawn in the previous sections were deduced for the non-resonance Raman experiment performed with frequency of the exciting light laying far from the frequency of any electronic transition in the molecule. In such a case the differential Raman scattering cross section (do/dD)i depends on the incident light frequency Vo through the term (v0 - v i ) 4 [Eqs. (8.45) and (8.58)]. If the Placzek's conditions (section 8.1) are satisfied, the dependence of molecular polarizability on Vo is negligible. In the treatment presented in the preceding sections, vibrational wave functions of the ground electronic state were only considered. If, however, the excitation radiation frequency approaches to Ore-resonance conditions) or coincides with (resonance conditions) the frequency of some electronic transition in the molecule, the application of the ground-state approach is not justifiable any more. The dependence of molecular polarizability on the incident light frequency has to be taken into account. As was already mentioned, under pre-resonance and resonance conditions some Raman lines are enhanced hundreds and thousands of times. In this section a brief outline of the basic theory of resonance Raman effect is presented. Before proceeding further, it is necessary to introduce the following notation: (a) the ground electronic state is denoted by g; vibrational levels belonging to this state are designated as i (initial vibrational state) and f (final vibrational state). The respective vibronic wave functions will be labeled as gi and gf; (b) the higher-energy electronic levels assume symbols r s and t; transitions g--,e and g--,s are considered allowed and g--,t - forbidden; and (c) vibrational levels belonging to e, s and t are denoted by v, I and u, respectively. In the subsequent derivations all electronic and vibrational states are considered as non-degenerate.

208

GALABOV AND DUDEV

The theory of intensities of resonantly-enhanced Raman lines is based on the Kramers-Heisenberg-Dirac dispersion equation [266,267]. The problem is analyzed in terms of vibronic interactions in the molecule. The JKth matrix component of the molecular polarizability for the gi~gf transition in the Raman spectrum can be expressed as follows [256,257]

(0:JK)gi,gf= E / evg:gi

Eev-Egi-E0

(evlMj ] E[g- i)eEgf (gflvM - E0 Klev)

(8.66) "

Mj and M K are the respective dipole moment operators and E 0 is the excitation light energy. In resonance conditions the first term in Eq. (8.66) becomes dominant since the denominator (Eev - Egi - E0) rapidly decreases. In this case, a phenomenological damping constant F is introduced in the denominator expression. Thus, retaining the resonant term in Eq. (8.66) only, we obtain

( .(ev[M-K!gi)(-gflMJ lev)) . )gi,gf = evg:gi E ~, Eev - Egi - E0 + iF

(8.67)

This equation can be further developed if the electronic wave function is expanded in a Taylor series along the normal coordinates Qa of the molecule. This is known as the Herzberg-Teller expansion [268] and its application to Eq. (8.67) yields [255,256] (ctji0gi,gf = A + B + C ,

(8.68)

A = ~ ~ (Mj)ge ( MK)eg (gf[ ev) (ev Igi) e~g v Eev - Egi - E0 + iF

(8.69)

where

INTENSITIES IN RA_MAN SPECTROSCOPY

X X

E

209

(gflQ~lev) (ev Igi)

~., ( (MJ)gshae(MK)eq

e~g v s~e a

E e - Es

Eev - Egi - E0 + iF (8.70)

(Mj)ge heas(MK)sg

E e ~ Es

C

_

Eev - Egi - E0 + iF

(M j )ge (M K )et h~g

(g~Qalev) (ev[gi)

Eg-Et

Eev - Egi - E0 + iF

_

~ ~v ~a e~g tag

(gfl ev)(eqQalgi) /

J

(8.71)

h~ (Mj)te (M K )eg Eg - E t

(gfl ev)(eqQ~lgi) / Eev - Egi - E0 + iF

J

i

In these equations (Mp)s~ = [~0] Mpl e0] (p = J, K; ~, ~ = g, e, s, t) are the electric transition dipole moments at the equilibrium molecular geometry and lk~a= [~0[ (aH/0Qa)ol so]. (aH/SQa)0 is the vibronic coupling operator for the normal mode a. H is the electronic Hamiltonian of the molecule. The contribution of each term A, B and C to the intensity of a Raman line is considered as follows. A term. One excited electronic state e is included in this term. With v 0 approaching v e the Raman line associated with a totally symmetric mode derives its intensity from this term through the Franck-Condon mechanism. This term does not depend on vibronic mixing of state e with other electronic states of the molecule. If the two electronic states g and e lie far below the next electronic levels and, hence, the probability of vibronic coupling between state e and the other excited electronic states is very low, the A term dominates in magnitude over the other two terms in Eq. (8.68). B term. This term comprises allowed transitions between the ground electronic state and two electronic states e and s (E e < Es). It determines to a great extent the intensity of a Raman line associated with a non-fully symmetric transition. In analyzing the properties of the B term the role of vibronic coupling in generating Raman intensifies becomes clear. Analysis shows that those normal vibrations which can couple two excited electronic states (e and s) undergo a striking intensity enhancement if the frequency of incident radiation is tuned on the frequency of the lowest electronic transition g---~. The nearer the state s to state e, the more pronounced is the effect of interaction. C term. It considers transitions to two electronic states e and t. Transition g~t is

not necessarily allowed. Since, in the general case, E e - E s < E g - E t, the magnitude of

210

GALABOV AND DUDEV

TABLE 8.2 Intensities of some Raman lines of pyrazine relative to the intensity of 940 cm-1 Raman line of benzene-d6 obtained at different excitation fight wavelengths (Reprinted from Ref. [274] with permission) Relative intensity a

Excitation wavelength

(in.m)

703cm-1 v 4 (b2g)

754cm- 1 v5 (b2g)

925cm -1 Vl0a (big)

514.5

0.38

0.041

0.046

457.9

0.47

0.082

0.12

363.8

0.57

0.20

1.14

aIn liquid phase.

this term will be small compared with the B term [Eqs. (8.70) and (8.71)]. Therefore, the contribution of C term to the overall intensity of the Raman line is not expected to be significant. This term is important for molecules possessing low-lying forbidden electronic states. Eqs. (8.68) through (8.71) were first derived by Albrecht [255,256]. His predictions have proved correct in many cases [269-277]. An example will demonstrate the role of vibronic coupling in generating resonance Raman intensities as derived in Albrecht's theory 03 term). It has been shown in a series of publications of Ito et al. [270, 271,274] that the Vl0a non-totally symmetric vibration of pyrazine (C-H out-of-plane bending) demonstrates a remarkable enhancement in Raman spectrum when the exciting radiation frequency approaches the frequency of the lowest-lying electronic transition at 323 nm. The Vl0a vibrational transition belongs to blg symmetry species and appears at the 925 cm-1 in the Raman spectnma. Results presented in Table 8.2 illustrate quite clearly the resonance effect on the intensity of this line. Analysis of the data accumulated reveals that the coupling between the lowest-lying allowed electronic state IB3u (n--+n*) and the next excited electronic state IB2u (rc---~*) through non-fully syuunetric big vibrational mode is responsible for the enhancement of 925 cm-1 Raman line. Low energy separation between the two excited electronic states, as seen from Fig. 8.2, favors this process. Vibronic coupling between IB3u and the two 1Blu electronic states will be less effective due to the higher energy separation. Therefore, vibrational modes (v 4 and v5) that are able to couple these electronic states are not expected to be significantly enhanced. The data collected in Table 8.2 illustrate this conclusion. Recently Okamoto [257] evaluated additional second-order terms to Eq. (8.68), thus expanding the applicability of Albrecht's theory. The emphasis is laid on the role of forbidden electronic transitions in generating resonance Raman intensities. It has been shown that: (1) the Raman line can derive intensity from a forbidden electronic transition


I

211

60700

IBlu (n-~r*), IB2u (~-->x*)

50900

1Blu (n--~r*)

37800

1B2u (a:-~*)

30900

1B3u (n-~x*)

z:

0

Ag

Fig. 8.2. Energy diagram of the low-lying electronic states of pyrazine.

even if the excitation radiation is tuned in resonance with an allowed electronic transition, and (2) enhancement may be detected if the excitation is in resonance with a vibronically allowed, though electronically forbidden, transition.

V. E X P E R I M E N T A L D E T E R M I N A T I O N OF RAMAN INTENSITIES Eq. (8.45) shows that for an ordinary Raman experiment the absolute differential Raman scattering cross sections can be expressed in terms of derivatives of the molecular polarizability invariants ~ and V with respect to normal coordinates. These derivatives contain valuable information about the variation of molecular polarizability with vibrational motion. Gas-phase Raman scattering cross sections are most suited for intensity analysis since at low partial pressure of the sampling gas these quantities are not influenced by effects of intermolecular interactions, thus reflecting properties of individual molecules.

212

G.M.,ABOV AND DUDEV

Direct determination of absolute Raman differential cross sections is quite difficult and tedious work often leading to incorrect results. It is easier to measure cross sections relative to some standard. The absolute differential Raman scattering cross sections of the sample can then be straightforwardly obtained.

A. Absolute Differential Raman Scattering Cross Section of Nitrogen The absolute scattering cross section of the Q-branch of the Raman band of nitrogen at 2331 cm-1 has been chosen as a standard in Raman intensity experiments. The special role of this molecule in gas-phase Raman spectroscopy is based on several reasons: (1) nitrogen is a relatively inert gas and does not react with the gas sample; (2) it quickly forms mixtures with the sampling gas; (3) the region of the vibrational specmma where the nitrogen Raman line appears is very low populated and, for many gases, this line does not overlap with sample gas lines; and (4) since the absorption band of nitrogen lies in the far ultraviolet, laser excitation beams with wavelengths corresponding to near ultraviolet or visible light do not cause resonance enhancement of the nitrogen Raman line. Its intensity depends on the excitation frequency through the term (V 0 - v i ) 4 only. Thus, the differential scattering cross section of nitrogen can be used as a standard for a wide range of excitation laser lines. The absolute differential scattering cross section of the standard needs to be determined as precisely as possible. A number of measurements has been performed over the past forty years [260,278-287]. The introduction of lasers in Raman spectroscopy and of computer processing of spectral data has improved highly the accuracy of Raman intensity measurements. As a result, the absolute differential Raman scattering cross section of nitrogen reported from different laboratories deviates within a few percent only [260,284,285,287]. One of the possible approaches in determining (dtr/d.Q)Q,N2 is to use the absolute differential cross section of the strongest purely rotational Raman line J = 1--->3 (J the rotational quantum number) of hydrogen at 587 em-1 as a standard. It is given by [260] __~] rot,H2

= 24 g4 )4 3(J + 1)(J + 2) 7,y20 4---~ (~0 -Vrot 2(2J + 3)(2J + 1) "

(8.72)

70 is the anisotropy at the equilibrium geometry. The rectangular experimental setup, as shown in Fig. 8.1, is considered. Both experiment and theory have provided a reliable value for the hydrogen anisotropy. Moreover, its dependence on the excitation wavelength has been thoroughly established. As a result, the absolute differential cross section of the hydrogen rotational line (do/d~)rot, H2 has been accurately determined. The estimated value has been employed in determining the absolute differential cross


213

section of the Q-branch of the vibrational Raman line of nitrogen. The following value has been obtained [260] (do/d.Q)Q,N2 = (5.05 4- 0.1)x 10-48 (~0 - 233 lcm-l) 4 cm6 sr-1.

(8.73)

It can be applied for a wide range of laser excitation wavelengths coveting the visible as well as the near ultraviolet up to 330 nm [260].

B. Differential Raman Scattering Cross Sections of Gaseous Samples The differential Raman scattering cross section of the ith line of a gas sample relative to that of the 2331 cm -1 line of nitrogen is given by [260] (do

/ d.Q)i

(v0 - vi )4

2331 cm -1

(~0 - 2331 cm-1)4

Vi (1- e-~ihe/kT)

~_

(do / m)Q,N 2

(8.74) gi [45(~)2 + 7gi ('Y[)2 ]

I45 (~)22

+7ZN2 (Ylq2

)2 I

"

In Eq. (8.74) li is the portion of the anisotropic scattering localized in the Q-branch of the respective Raman line. Its value has been determined for a number of molecules. In the case of linear molecules with small rotational constants gi is 0.25 [280,281]. Since the Bolzmann factor for the nitrogen molecule is very small at room temperature, it has been neglected in deriving Eq. (8.74). It is seen fxom the above expression that the relative differential scattering cross section depends on the excitation light wavenumber and on the absolute temperature. To make the measured quantifies comparable, a relative normalized differential Raman scattering cross section Ei has been defmed [260] (d~ / d'Q)i

.

(~0 -Vi) -4

Ei = (do / dn)Q,N2 (% _ 2331 r

(1-e-Vihr (8.75)

2331 cm-1 Vi

gi [45(~) 2 + 7gi (,y~)2] 7%N2('~N2

]

214

GALABOV AND DUDEV

The integrated intensity of the ith Raman line of a gaseous sample for a rectangular experimental setup can be expressed as I i = I0 p (do/d.O)i ,

(8.76)

where p is the partial pressure of the sampling gas. Combining Eq. (8.76) with the respective expression for the integrated intensity of the 2331 cm -1 Raman line of nitrogen

IN2 = IO

(8.77)

(da/d ) z

the following relation is obtained Ii

~2

=

(do/d.O)i

P

(do/d.O)N2

PN2

.

(8.78)

The relative differential Raman scattering cross section of the ith line of the sample can, therefore, be determined experimentally by employing the expression: (do / d.Q)i (do / d.O)N2

Ii

IN2

PN2 P

(8.79)

The absolute differential Raman scattering cross section of the ith line of the sample can be obtained l~om the relative value by using the absolute scattering cross section of nitrogen as given in Eq. (8.73).

CHAPTER 9

PARAMETRIC FOR INTERPRETING

MODELS RAMAN

INTENSITIES

Rotational Corrections to Polarizability Derivatives .......................................... 216

H.

A.

Zero-Mass Method ................................................................................... 218

B.

Heavy-Isotope Method ............................................................................. 219

C.

Relative Rotational Corrections ................................................................ 223

Valence-Optical Theory of Raman Intensities .................................................... 223 A.

Theoretical Considerations ....................................................................... 224

B.

Valence Optical Theory of Raman Intensities: An Example

C.

Compact Formulation of VOTR ............................................................... 235

D.

Compact Formulation of VOTR: An example of Application ................... 239

of Application ..........................................................................................

232

llI.

Atom Dipole Interaction Model (ADIM) ........................................................... 245

W.

Atomic Polarizability Tensor Formulation (APZT) ............................................ 249 A.

V.

APZT: An Example of Application .......................................................... 253

Relationship Between Atomic Polarizability Tensors and Valence Optical Formulations of Raman Intensities ..................................................................... 258

VI.

Effective Induced Bond Charges From Atomic Polarizability Tensors .............. 261 A.

Theoretical Considerations ....................................................................... 261

B.

Applications .............................................................................................

C.

Discussion of Effective Induced Bond Charges ........................................ 266

215

263

216

GALABOV AND DUDEV

As was shown in Chapter 8, the experimental gas-phase differential Raman scattering cross sections are directly related to the molecular polarizability derivatives with respect to normal coordinates forming the supertensor r [Eq. (8.41)]. In intensity analysis the 0oJ0Qi derivatives are usually further transformed into different types of parameters. The eventual goal is to transform the experimental observables into molecular quantifies reflecting electro-optical properties of simple molecular sub-units. Several formulations for parametric interpretation of Raman intensities have been put forward. In this chapter the basic principles and characteristics of the theories developed will be discussed. The mathematical formalism inherent of each theoretical approach will be illusu'ated with examples. It should be emphasized here that the models for analyzing Raman intensities have seen limited appLication, especially if compared with infrared intensity theories. Several factors are responsible for the relatively slow progress in the field of Raman intensity analysis: (1) Polarizability is a second-rank tensor quantity with six independent components in far-from-resonance conditions. Thus, the number of intensity parameters, if compared with those in the infrared, is doubled. This leads to indeterminacy of the inverse electro-optical problem for even very small and symmetric molecules. (2) Relatively few reliable experimental gas-phase Raman intensity data are available to date. (3) Analytical derivative, expressions allowing efficient and reliable evaluation of molecular polarizability parameters, were introduced in ab mitio quantum mechanical calculations few years ago. The impact of ab mitio methods in Raman intensity analysis is still to come.

I. ROTATIONAL CORRECTIONS TO POLARIZABILITY DERIVATIVES The transformation of vibrational intensities in Raman spectra into molecular parameters involves several calculation stages. An essential initial step is the reduction of intensity data to polarizability derivatives with respect to symmetry vibrational coordinates. As pointed out in previous chapters, the inverse electro-optical problem of vibrational intensities can be performed with success only for molecules possessing sufficient symmetry. The transformation between O~OQi and O~/OSj derivatives is carried out with the aid of the normal coordinate transformation matrix L S according to the expression:

PARAMETRIC MODELS OF KA_MAN INTENSITIES

cO(x/cT~Sj = E

217

(&~cOQi)L~l ,

(9.1)

i

or in a matrix form: as

:aQ

(9.2)

L-~~.

a s is a supertensor with dimensions [3x3(3N-6)] and has the following structure: 3ctxx/t)S1 (zS=

t)ctxy/t)Sl Octxz/t)S1 t)ctyy / t)Sl t)ctyz / t)S1

L syaunmeWical

ONtzz / ~$1

.-. ... ...

(9.3) Octxx / t)S 3N-6 ~)ctxy / t)S3N_6 i)ctxz / ~)S3N-6 ~)ctyy / ~)S3N-6 ~)ctyz / ~$3N_6 symmetrical 0ctzz / ~)$3N-6

J

Polarizability derivatives with respect to symmetry coordinates obtained from Eqs. (9.1) and (9.2) are not always purely intramolecular quantifies since contributions from the compensatory molecular rotation accompanying some vibrations may be present. Such contributions arise in the cases of non-totally symmetric modes of molecules having a non-spherical polarizability eUipsoid. Polarizability derivatives corrected for contributions from molecular rotation can be obtained according to the relation ctSc~

aS - PS ,

(9.4)

where PS denotes the tensor comprising the rotational correction terms to the polarizability derivatives. It has the same structure as a S. The rotational correction to a /~/i3Sj derivative is a 3x3 matrix given by the following expression [35] psj : wj • a 0 - a o • w j .

(9.5)

wj is the compensatory rotation arising when a molecule undergoes particular vibrational distortion, and a 0 is the static molecular polarizability tensor. If the vector wj is presented in a pseudo-tensor form [Eq. (3.7)], Eq. (9.5) can be rewritten as psi : ~ ((wj)) - ((wj)) a0.

(9.6)

The absolute compensatory molecular rotation can be evaluated, as already discussed in Section 3.II.A, by employing the hypothetical isotope approach [34-36]. The hypothetical species obtained by setting the masses of some appropriately chosen atoms equal to zero [35,36] or weighted by factors of 1000 or more [34] are incorporated in the

GALABOV

218

A N D DUDEV

calculations. These isotopes have negligibly small compensatory rotation. A formulation adapting the hypothetical isotope method in evaluating rotational corrections to polarizability derivatives was recently put forward by Dudev and Galabov [288]. The elements of wj can be derived according to Eq. (3.9) (Section 3.H.A). Thus, combining Eqs. (3.9) and (9.6) we obtain PSj

=

tx0 ((~ ~ A j

A j )) r 0

(9.7)

.

The superscripts r and a stand for the reference and actual molecule, respectively. More details about the matrices 13 and A are given in Chapters 2 and 3. Some examples of application of the hypothetical isotope approach in deriving rotational corrections to polarizability derivatives are given below.

A.

Zero-Mass Method

Within the zero-mass approach the reference hypothetical isotope contains atoms with zero masses. Typically, the respective atoms do not lie on the main molecular symmetry axis. As an example, the CH3C1 molecule will be considered. In this case, it is appropriate to set all hydrogen-masses equal to zero. Thus, the C-CI bond will maintain fixed direction during vibrational motion and this hypothetical isotope species will have negligibly small compensatory rotations. Methyl chloride has Raman-active vibrations belonging to A1 and the doubly degenerate E, symmetry species. The E-vibrations are non-totally symmetric and contain contributions from compensatory molecular rotation. Geometry parameters, definition of symmetry coordinates and the orientation of the molecule in Cartesian space are given in Table 3.8 and Fig. 3.7. The equilibrium molecular polarizability tensor of CH3C1 employed in the calculations has the following form [289]:

tt 0 =

/4i o Oo/ 4.03 0 558

A3 .

(9.8)

Application of Eq. (3.9) yields (in units of A -1 or rad-1) S4a

S5a

S6a

~P a / 0 0 0 Aj = 0.038 0.037 -0.070 0 0 0

S4b

S5b

S6b

0.038 0 . 0 3 7 - O ! 7 0 / x 0 0 y 0 0 z

(9.9)

PARAMETRIC MODELS OF RAMAN INTENSITIES

219

a

After Iransforming the vector wj = [~ A j into pseudotensor form and applying Eq. (9.7) for each symmetry coordinate, the PS array is obtained (in units of A2 or A3 rad-1)

(

L

~

0.059

0 0 0

S4a 00.059 0 0 0 0

0 0 0.057

S5a 00.057 0 0 0 0

0 0 -0.108

S6a 0-0.108 0 0 0 0 (9.10)

S4b 0 0 0 -0.059 -0.059 0

no

0 0 0

S5b 0 0 0 -0.057 -0.057 0

0 0 0

S6b 0 0 "~ 0 0.108 0.108 0

J

Heavy-Isotope Method

The heavy-isotope approach to evaluate rotational contributions to polarizability derivatives [288] will be illustrated with calculations on a series of molecules consisting of acetonitrile (C3v symmetry), dichloromethane (C2v symmetry) and acetone (C2v symmetry). Structural parameters and polarizability tensors employed in the calculations are summarized in Table 9.1. Since the axes of the Cartesian reference systems (Fig. 9.1) are chosen to coincide with the respective inertial axes, the static polarizability tensors acquire simple diagonal form. The symmetry coordinates corresponding to vibrations which may contain contributions from compensatory molecular rotation for the three molecules are given in Tables 9.2, 9.3 and 9.4, respectively. The following heavy isotopes are employed: 1. Acetonitrile: The masses of the two carbon atoms for acetonitrile are multiplied by a factor of 104. 2. Dichloromethane: The dichloromethane molecule possesses C2v symmetry and has two groups of vibrations belonging to B 1 and B2 symmetry classes that contain contributions from compensatory molecular rotation. Two heavy isotopes are created in this case: C*H2C12" in evaluating rotational corrections to B 1 class (weighting factor of 103); and C*H2*CI2 in the case orB 2 vibrations (weighting factor of 106). The asterisks mark the heavy atoms in the isotopes. 3. Acetone: The masses of oxygen and central carbon atoms for acetone were weighted 104 times. Rotational correction terms to polarizability derivatives for the three molecules obtained by following the procedure described above are given in Tables 9.2, 9.3 and 9.4, respectively.

220

GALABOV AND DUDEV

xI,'

A

R1 a 2 (~C

@

R2

132

I

Z

~

R1

R2

S

131 RI rl

B

C

R2

r4 C

q

Fig. 9.1. Cartesian reference systems and definition of internal coordinates for (A) acetonitrile, 03) dichloromethane and 03) acetone.

PARAMETRIC MODELS OF R A M A N INTENSITIES

221

TABLE 9.1 Structural parameters and gas-phase static polarizabilities for acetonitrile, dichloromethane and acetone Molecule

Geometry

axx

0%

CZzza,b

(A 3) CH3CN

r c c = 1.460 A,

rCN = 1.158 A,

3.98

3.98

5.49

rCH = 1.092 A, .

x

u

4.0--

3.s3.0_

CH~

CH3OH

_

2.5

,l

~ i l l W | W l

1.35

I

'

~

1.40

l

l

l

|1

|

I

I

l

1.45

'

'

|

l

|

l

t

l

I | f l | l l l ' i

1.50

rc_ x (x 10-Io m )

Fig. 9.6. Plot of the dependence between the effective induced bond charges, 6C_X, and the bond lengths, rc_ x (X = C, N, O, F) for ethane, methylamine, methanol and methyl fluoride.

270

GALABOV AND DUDEV

TABLE 9.16 Effective induced charges for C-X bonds (X = C, N, O, F) from HF/6-31 l+G(d,p) ab initio MO calculations Average atomic Bond polarizability Ax length rk (xl0-10m) a (x 10-40 C.m2/V)b

Bond

Molecule

Effective induced bond charge Ok (x 10-30 C.m/V)

C-C

CH3CH 3

4.900

1.5271

1.958

C-N

CH3NH2

3.625

1.4536

1.224

C-O

CH3OH

3.148

1.3998

0.892

1.3651

0.620

2.849 C-F CH3F aFrom I-IF/6-31l+G(d,p) ab mitio results [333]. bFrom Ref. [334].

TABLE 9.17 Effective induced charges for C=X bonds (X = C, N, O, S) from HF/6-31 l+G(d,p) ab initio MO calculations Average atomic Bond polarizability AX length rk (x 10-10 m)a (x 10--40 C.m2/V)b

Bond

Molecule

Effective induced bond charge Ok (x 10-30 C.m/V)

C=C

CH2CH 2

7.375

1.3184

CH2CHCHCH2

12.032

1.3235

C=N

CH2NH

5.897

1.2492

1.001

C=O

CH20

4.250

1.1797

0.946

C=S

CH2S

7.627

1.5959

3.494

aFromHF/6-31l+G(d,p) ab mitio results [333]. bThe values referto atom X participatingin a doublebond [335].

1.491

PARAMETRIC MODELS OF RAMAN INTENSITIES

271

In Table 9.16 the effective induced charges for the single bonds between carbon and some first row elements are tabulated. Again, good correspondence with the respective calculated bond lengths (Fig. 9.6) and atomic polarizabilities is observed. Double bonds, due to the higher n-electrons flexibility, are expected to have higher polarizabilities as compared with the respective single bonds. It is, therefore, of particular interest to examine effective induced bond charges for some double bonds. Calculated values are given in Table 9.17. Juxtaposing aC=X with the respective ac_ X (X = C, N, O) quantifies shows that, indeed, the double bonds have larger ak values. Thus, crC=C = 7.375x 10-30 C.m/V and aC_ C = 4.900• 10-30 C.m/V, aC=N = 5.897x 10-30 C.m/V and aC_N = 3.625x10 -30 C.m/V, aC=O=4.250• C.m/V and 6C_O= 3.148• 10- 30 C.m/V. A definite trend of lowering ac= X values with shorter bond lengths and lower atomic polarizabilities is found in the series CH2CH 2, CH2NH and CH20. Comparing the structural analogs CH20 and CH2S, it can be seen that the aC= S parameter value (7.627>(10-30 C.m/V) is much higher than the effective induced bond charge for the C=O bond (4.250• 10-30 C.ndV). This observation can be easily explained in view of the lower electronegativity of sulfur, its higher atom radius and atomic polarizability, and the longer C=S bond length. In Table 9.17 the C=C effective induced bond charge for s-trans-butadiene is also presented. This is the simplest hydrocarbon containing conjugated double bond system. As known, the conjugation results in a significant electron charge redistribution across the molecule and increase in the n-electrons flexibility. Higher carbon-carbon bond polarizabilities are expected in such a case. As can be seen from Table 9.17, the conjugation leads to a drastic increase in the aC= C value which jumps from 7.375• 10-30 C.m/V in ethylene to 12.032x10 -3~ C.m/V in s-trans-butadiene. Again, 6 k is in good accord with the general expectations. The discussion presented above outlines some definite trends of changes in the effective induced bond charges evaluated in analyzing ab initio calculated atomic polarizability tensors, a k are closely related with the polarizability properties of the respective bonds and depend strongly on bond lengths, atomic polarizabilities, bond multiplicity and conjugation with other bonds.


C H A P T E R 10

AB INITIO CALCULATIONS OF RAMAN INTENSITIES

Computational Methods ..................................................................................... 274

IIo

A.

Finite Field Calculations of Raman Intensifies .......................................... 274

B.

Polarizability Derivatives from the Energy Gradient ................................ 275

C.

Analytic Gradient Methods ...................................................................... 275

Calculated Raman Intensities ............................................................................. 276 A.

Basis Set Dependence ofAb lnitio Raman Intensities ............................... 276

B.

Influence of Electron Correlation on Quantum Mechanically Predicted Raman Intensities ..................................................................... 278

273

274

GALABOV AND DUDEV

Experimental complete polarizabilitytensor of a molecule of even small size is usually very difficultto obtain. This is a major obstacle in any attempt of consistent nonapproximate interpretationof experimental data. That is, very possibly, the main reason for the predominant application of simple zero order additive models for analyzing or predicting Raman intensities. It has been, therefore, of great importance to derive complete polarizability tensors and compute Raman intensities by ab mitw MO calculations. The remarkable efforts of scientists fzom several leading research groups have resulted in successful realization of these goals [166-168,175,177,178,181,336-341]. Current program systems for ab imtio calculations provide these quantifies as a standard output resulting from application of analytic derivative methods [ 153,170-172].

L COMPUTATIONAL

METHODS

A. Finite Field Calculations of Raman Intensities Hush et al. [336,337,342] have employed finite field perturbation treatment to derive molecular polarizabilities and polarizability gradients from SCF and correlated wave functions. In an uniform electric field f the induced dipole moment of a molecule Pind(f) is given by [336,343]

Pind(O = P(f) - Po = ct f + (1/2) [3 f 2 + (1/6) 7 f 3 + . . . .

(10.1)

f is the field strength, ct - the molecular polarizability,and Po - the dipole moment in absence of electricfield. [3 and ~/are the firstand second hyperpolarizabilitytensors. In the frmnework of the finitefield formalism the perturbation to the Hamiltonian by an uniform electricfield is expressed as [175] H(f) : H o + f ~ .

(10.2)

Ho is the Hamiltonian for the unperturbed system and ~t is the dipole moment operator. From the perturbed wave functions evaluated at SCF or CI level the dipole moment value p(f) is obtained. From the Taylor series expansion [Eq. (10.1)] the coefficients ~ [3 and 7 can be derived from calculations at a number of distortions along particular coordinate and field directions. Molecular polarizability is then obtained from the expression


a = [p(f)- Po] / f -

(1/2)]3f2 _ (1/6) 7 f 3

275

.

(10.3)

B. Polarizability Derivatives from the Energy Gradient Komornicki and McIver [178] have provided an efficient method for deriving polarizability derivatives from the energy gradient. Following expressions (7.2)-(7.4) ~[~/O~ are presented as 3ot~,c; aqi

=

a

alan

aqi

afg

=

03 apn ar~

aqi

=

~)

03

~)E

af~ ar n aqi

.

(10.4)

In Eq. (10.4) Pn is a component of the dipole moment, E - the energy, 8E/c3qi- the potential energy gradient, f[ - a component of an external electric field, q i - an appropriate nuclear coordinate, and ~, ~, n = x, y, z in a Cartesian axis system. Again, the same result can be achieved by the procedure prescribed by Schaad et al. [180] by performing standard ab initio calculations with one or more point charges placed at sufficient distance from the molecule so that the electric field is weak. It is seen from Eq. (10.4) that the method of Komornicki and McIver incorporates double numerical differentiation to obtain the respective component of the polarizability derivative tensor.

C. Analytic Gradient Methods Raman intensities are proportional to the square of the derivatives of molecular polarizability with respect to nuclear coordinates. As can be seen from Eq. (10.4), a is a second derivative of the energy with respect to an external electric field. Raman intensities are, therefore, an important third energy derivative property. The development of analytic first [166,168], second [344,345] and third derivative [346,347] methods has been a major breakthrough in ab initio calculations. The creation of efficient ab initio program systems employing analytic derivative techniques has had, as already emphasized, a great effect on vibrational spectroscopy in general. Vibrational assignments and force field calculations have become much more reliable through the combined use of experimental and theoretical data. The solving of the problem with sign indeterminacy for dipole and polarizability derivatives have greatly contributed to all studies associated with the interpretation of vibrational absorption and Raman scattering intensities. These achievements have greatly contributed to our understanding of the intrinsic physical phenomena determining the main parameters of vibrational spectra of isolated or interacting molecules.

276

GALABOV AND DUDEV

The paramount difficulties in the experimental determination of accurate absolute infrared intensities and differential Raman scattering cross sections seem, however, to persist in the theoretical evaluation of these quantifies as well. As we will see, predicted Raman intensities appear to be more consistent with experiment as compared to infrared intensities. Analytic derivative methods for evaluating polarizability derivatives have been developed simultaneously by the theoretical chemistry groups in Cambridge [340] and Berkley [341]. According to Frisch et al. [341] the analytic evaluation of polarizability derivatives is achieved using the expression

aa~ = a3E =(aD ~ ap /

=

~-~

oq/ pl/( 2p oq

~8~ ~q

-~-

+

+

8~

a~

8q

aq

/

(lo.5)

-

In Eq. (1.0.5) Day~ = ( ~tl ~[ v> are dipole moment integrals, P is the electron density matrix, Gq ( ) denotes the contraction of integral derivatives with a matrix, po is a non-perturbed density matrix, S is matrix of overlap integrals, W is energy-weighted density matrix, h is the core Hamiltonian, ( ) denotes a trace of a matrix, and q is a nuclear coordinate (usually in Cartesian space). As can be seen from expression (10.5) no third derivative integrals appear in evaluating polarizability derivatives. No second derivative for the two-electron integrals are also needed. Thus, polarizability derivative calculations do not require much additional time. Second order couple-perturbed Hartree-Fock (CPHF) equations are solved with respect to the six pairs of electric field variables.

II. CALCULATEDRAMAN INTENSITIES A. Basis set Dependence ofAb

Initio

Raman Intensities

Analytic derivative calculations of Raman intensities and Raman depolarization ratios for H20, NH 3, CH 4 and C2H2 have been reported by Amos [ 175,189]. The author

A B INITIO CALCULATIONS OF RAMAN INTENSITIES

277

has used a variety of basis sets including calculafons near the Hartree-Fock limit. Intensities and depolarization ratios for H20 are presented in Tables 10.1 and 10.2 and calculated Raman intensities for CH4 and C2H2 are given in Table 10.3. Overview of the results presented in Tables 10.1-10.3 shows that SCF calculations provide quite satisfactory results for Raman intensi6es. This is in contrast to vibrational absorption intensifies (Chapter 7), where SCF calculations at the highest level could not match in any quantitative terms the experimental values. Acceptable quantitative agreement with experiment for Raman intensities is achieved, however, only with very large basis sets near the Hartree-Fock limit. This is, perhaps, the reason why the results for H20 and CH4 are in much better agreement with measured Raman intensifies than the ab mitio results for C2H2 employing smaller basis sets. Again, as with the case of infrared intensities, there is no general trend in predicted Raman intensities. Some values are overestimated while others are underestimated. The gradual improvement of basis sets shows very slow convergence towards the Hartree-Fock limit values. High level HF/SCF calculations on ethylene using analytic derivative methods have been reported by Frisch et al. [341]. Their results are presented in Table 10.4. Vibrational frequencies and infrared intensities are also given. Interestingly, even at this high level of SCF theory the frequencies for Vl0(Blu) and Vll(B2g) are predicted with reversed order. There are, however, no difficulties in assigning these bands for the correct normal mode on the basis of symmetry properties of the respective normal modes and the fact that B2g mode is Rmnan active while B lu -infrared active. Both infrared and Raman intensities are relatively well predicted at the SCF large basis set calculations. There are no specific theoretical reasons to expect that polarizability derivatives would be less sensitive to correlation effects. Nevertheless, the data of Amos for H20 and CH4 and to lesser extent for C2H2 [175,189] show that near Hartree-Fock SCF calculations may provide quite acceptable values for the intensities in Raman spectra. In a recent study Souter eta/. [352] have obtained experimental Raman spectra of Ga2H6 and Ga2D6 in low-temperature matrices and compared the experimental frequencies and relative intensities with theoretical ab mitio estimates obtained from extended basis set SCF calculations. 14sllpSd contracted to 10s8p2d basis set is used for gallium [353] and two alternative basis sets for hydrogen: (1) DZP contracted Gaussians (4s/2s) augmented with p-functions on the hydrogens; (2) TZP, (Ss/3s) contracted basis set augmented with p-functions on the hydrogens. The calculations have been performed using analytical third derivatives methods using the program PSI developed by Schaefer et a/. [ 171]. A good agreement between calculated and observed Raman intensities is obtained.

278

GALABOV AND DUDEV

TABLE 10.1 Ab mitio HF/SCF Raman scattering coefficients for H2Oa

(in units of A 4 a.m.u.-1) Basis set

vl(B2)

v2(Al)

v3(Al)

4-31G 6-31G DZ 4..31G(2d,2p) 6-31G(2d, Ep) DZP 6-31G extended [5s4p2d]/[3 sEp] [8s6p3d]/[6s2p] [8s6p4d]/[6sap] [Ss61Md lf]/[6s3p 1d] [8s6p4d2f]/[6s3p 1d] Experiment b

39 25 42 18 24 37 27 29 28 24 25 24 19.2

100 71 91 83 87 75 81 71 74 85 85 85 108

10 l0 12 12 4.5 6.8 2.0 3.5 1.3 0.67 0.67 0.68 0.9

aThe theoretical results are taken from Ref. [189] with permission. bFrom Refs. [348] and [349].

B. Influence of Electron Correlation on Quantum Mechanically Predicted Raman Intensities The relative success of SCF ab initio calclflations in predicting Raman intensities seems to be the main reason for the absence of recent detailed studies on the effect of electron correlation on theoretically estimated Kaman intensities. Most correlated ab imtio calculations have been performed sometime ago [337,354-361]. Amos [354] has used pair replacement MCSCF wave function to calculate electric dipole polarizability and polarizability derivatives for I-IF. Double excitations from the single determinant (Hartree-Fock) wave functions from the valence orbital is used. The MCSCF calculations yielded the following values for the mean polarizability and polarizability anisotropy (in units of A 3) = 0.720 tZzz - tZxx = 0.213.

(10.6)


279

TABLE 1 0 . 2 Ab miao HF/SCF Raman depolarization ratios for H20a Basis set

vl(B2)

v2(A1)

v3(Al)

4-31G 6-31G DZ 4-31G(2d, Ep) 6-31G(2d,2p) DZP 6-31G extended [5s4p2d]/[3s2p] [Ss6p3d]/[6s2p] [8561Md]/[653 p] [8s6p4d 1f]/[6s3p 1d] [ 8s6p4d2f]/[ 6s3 p I d] Experiment b

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

0.720 0.074 0.211 0.669 0.667 0.164 0.080 0.119 0.102 0.069 0.069 0.069 0.03

0.403 0.637 0.412 0.291 0.662 0.521 0.558 0.581 0.670 0.750 0.749 0.75 0 0.74

aThe theoretical results are taken from Ref. [ 189] with permission. bFrom Refs. [348] and [349].

TABLE 10.3 Ab mitio HF/SCF Raman scattering coefficients for CH4a and

C2H2b (in units of A 4 a.m.u. -1) 6-31G(2d,2p)

[5s4p2d]/[3s2p]

[8s6p4d2q/[6s3pld]

Exp.

v 1(F2) v2(Al) v3(E ) v4(F2)

186 138 61 7.2

154 160 35 2.2

151 228 6.6 0.06

128 230 7.0 0.25

Vl(Eg)

53

41

75

v2(Eg )

64

98

125

v4(I-Ig)

6.5

15

4.1

CH4

C2H2

aThe theoretical results for CH4 are taken from Ref. [ 189] with permission and experimental data - from gel. [ 3 0 2 ] . bThe theoretical results for C2H2 are taken from Ref. [175] with permission of John Wiley & Sons, Ltd. Copyright [1987] John Wiley & Sons, Ltd. Experimental data are from Ref. [304].

TABLE 10.4 Analytic inflared intensities and Raman scattering coefficients for ethylene molecule

k0i

from RHF/6-3 1l++G(343p) calculationsa Mode

v I(B2u) v2(B I d v3(Ald v4(B3u) V5(Alg) V6(B3u) v7(Alg) vs(B I d v9(Alu) v lo@ lu) vlI(B2g) VlZ(B2u)

Vibrational frequencies (cm-1)

M a r e d intensities (km/mol)

Theory

Experimentb

Theory

ExperimentC

3367 3338 3285 3263 1813 1590 1472 1343 1137 1080 1097 89 1

3 106 3 103 3026 2989 1623 1444 1342 1236 1023 949 943 826

25.6

26.0

19.4

14.3

12.6

10.4

0 118.1

aThe theoretical results are taken from Ref. [338] with permission. bFrom Ref. [350]. CFrom Ref. (3511. dFrom Ref. [303].

0

Raman scattering coefficients (As/a.m.u.) Theory 113.2 183.8

66.8 172.2

55.8

17.5

61.2 0.34

26.0 2.92 0

0

84.4 7.7

0.004

Experimentd

0.03

1.57

&0

1

82


281

The RHF/SCF results are slightly different = 0.715 (10.7) CXzz - axx = 0.180. Experimental value for the anisotropy is a z z - ~ x = 0.22 A 3 [362]. Polarizability derivatives have been also calculated at MCSCF level. Extended basis set SCF and configuration interaction (CISD) calculations for CO have been carried out by Amos [359] with the aim to assess the effect of electron correlation on calculated molecular polarizability and polarizability derivatives. The basis set employed has been of the type (Ss4p2d) contracted Gaussian function on each atom. The configuration interaction function consisted of single and double excitations of the valence shell orbitals. Polarizabilities have been calculated by the finite field method [ 179]. Polarizability derivatives are evaluated by numerical differentiation. The results are given in Table 10.5. In general terms, both SCF and CI results do not differ drastically. It should be remembered that the SCF results are not very close to the Hartree-Fock limit. Thus, the correlated wave functions also contain inaccuracies from the use of limited basis set. Similar calculations have been carried out on the nitrogen molecule [357]. The data are presented in Table 10.6. It can be seen that the CI results are in better agreement with experiment and differ by approximately 20% from the values estimated at SCF level. Marlin et al. [355] have applied multireference CI calculations to evaluate polarizability derivatives for H2S. The basis set for the sulfur atom consisted of 1 ls7p primitive Gaussian orbitals augmented with an additional diffuse functions on the s and p spaces and with three d type polarization functions. The primitive functions have been contracted to (7s/6p). Three s functions and two sets of p polarization functions have formed the contracted basis for the hydrogen. It is important to note that the multireference CI calculations produce values for the molecular polarizability components approximately 25% higher than the CISD estimates. Analytical derivative ab initio calculations on Raman scattering activities provide currently results that are in fairly good accord with the available experimental data. The impact of the theoretical calculations is even greater in view of the difficulties in obtaining full polarizability derivative tensors from experimental measurements.

282

GALABOV AND DUDEV

T A B L E 10.5 Molecular polarizability and polarizability derivatives for CO from SCF and CI calculations a Method

Equilibrium value (A 3)

First derivative (A 2)

at.= %cx

SCF

1.539

0.756

CI

1.637

0.669

all = tgzz

SCF

1.750

2.674

CI

1.872

2.943

SCF

1.609

1.395

1.715 1.93 8b

1.427 1.501 c 1.551 d

CI Experiment

aThr theoretical results arc taken from Rcf. [359] with permission. bFrom gcf. [363]. CFrom R~. [364]. dFrom Rcf. [279].

T A B L E 10.6 Theoretical a and experimental mean polarizability and anisotropy and their derivatives for N 2 (in units of A 3 and A 2) Method

Equilibrium value

SCF

1.681

.... First derivative b 2.184

CI Experiment c

1.720 1.741

1.752 1.752,1.815

SCF

0.779

2.463

CI

0.640

1.932

Experiment c

0.660

2.067

aT~ tla~mtic.al results are taken from Ref. [357] with permission. bThr first derivative is takga with respect to g = (R - Re)/Rr Re is the internuclear distance at equilibrium. CExpcrinamtal data arc taken flom Refs. [365] and [318] for the equilibrium ~ and '5 from Refs. [278] anti [3641 for ~' and from Rcf. [357] for f.

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1LD.Amos, Chem. Phys. Lett., 70, 613 (1980).

360.

J. Kendrick, Jr. Phys. B., 11, L601 (1978).

361.

G.H.F. Diercksen and A.J. Sadlej, Chem. Phys., 46, 43 (1985).

362.

J.S. Muenter, s Chem. Phys., 56, 5409 (1972).

363.

M.P. Bogaard, A.D. Buckingham, R.K. Pierens and A.H. White, Trans. Farad

Soc. I, 74, 3008 (1978). 364.

B. Fontal and T.G. Spiro, Spectrochim. Acta, 33A, 507 (1977).

365.

A.C. Newall and R.C. Baird, J. Appl. Phys., 36, 3751 (1965).

301


AUTHOR INDEX

Numbers in brackets are reference numbers and indicate that an author's work is referred to although his name is not cited in the text. Underlined numbers give the page on which the complete reference is listed.

-----

Amy, J. W., 50 [70], 286

A-----

Andrews, L., 277 [352], 300

Abbate, S., 55 [80], 63 [80], 142, 143, 144 [151], 224 [298], 242 [319], 287, 290, 29.8, 299

Applequist, J., 221 [290], 245 [289, 321], 247, 248 [321], 249, 297, .299

Ablichs, R., 180 [234], 29_..55 Adamowits, L., 177 [222], 294

Art~ken, G., 82 [119], 83 [119], 110 [119], 119 [1191, 289

Akopian, S. H., 37 [58], 286

Asfle, M. J., 47 [69], 50 [69], 286

Albrecht, A. C., 194 [255, 256, 258], 199 [255], 208 [255, 256], 210, 296

Averbukh, B. S., 78, 82 [118], 106, 109 [129, 130, 131], 110, 111 [118], 132 [129], 142 [131], 164 [129], 252 [129], 288, 289

Aldous, J., 66 [102], 288 Aleksanyaa, V. T., 68, 70, 82, [110], 142 [ 107, 110], 288

B

Bacskay, G., 177 [227], 180 [231, 233], 181 [231], 182 [231], 232 [301], 233 [301], 274 [336, 337, 342], 278 [337], 278 [356], .2.94, 29.5., 298, 300

Aliev, M. IL, 183 [245, 246], 295 Alms, G. R., 241,282 [318], 299 Allen, G., 202 [262], 296

Badawi, H. M., 170 [213], 294

Allen, W. D., 164 [171], 274 [171], 277 [171],291

Bader, R. F. W., 37 [50], 59 [50, 90], 112 [133], 285, 289

Amos, R_ D., 142 [1541, 150 [157], 164 [1541, 165 [175], 166 [1751, 168, 169 [175, 184], 170, 173 [189], 176 [175], 177 [184, 225], 183 [184], 274 [175, 340], 276 [175, 189, 340], 277, 278 [189, 354, 357, 359], 279 [175, 189], 281, 282 [357, 359], 290, 291, 292, 294, 300, 301

Bakshiev, M. G., 37 [56, 57, 58], 286

Amot, G., 152 [162], 153 [162], 159 [1621,291

Barbe, A., 150 [158], 152 [158], 155 [ 158], 183 [239], 291,295

Bailey, W. P., 114 [137], 290 Baird, R. C., 282 [365], 301 Baker, J., 142 [153], 274 [153], 290

Bartholome, E., 18 [ 14], 21,283 303

304

AUTHOR INDEX

Bartlett, R. J., 164 [1721, 177 [222, 223, 224, 226], 179 [224, 2301, 274 [ 172], 291, 29._44

Brandmuller, J., 199 [259], 296

Beers, Y., 63 [96], 64 [96], 173 [961, 179 [961, 181 [96], 182 [961, 185 [96], 288

Brewer, W. E., 170 [212], 29..__44

Behfinger, J., 199 [259], 199 Benedict, W. S., 64 [99], 288 Berckmans, D., 183 [244], 295 Bermejo, D., 235 [302, 307], 238 [302], 2791302], 298 Bemsteirg H. J., 194 [253], 202 [253, 262], 212 [279, 281], 213 [281], 282 [279], ~ 29_.27 Beyer, W. n., 47 [69], 50 [69], 286 Biarge, J. F., 68 [108], 72, 78, 83, 87, 99, 142 [108], 164 [108], 288 Binldey, J. S., 142 [1531, 167 [181], 168 [1821, 169 [1821, 170 [1821, 172 [182, 191], 173 [182], 174 [182], 175 [182], 177 [181], 179 [182], 274 [153, 182, 341], 276 [341], 277 [341], 290, 292. 293, 300 Birge, R. IL, 245 [326, 328, 329], 247, 299 Blanchette, P. P., 68 [105], 83 [105], 85 [ 1051, 87 [ 1051, 288 Bociaa, D. F., 245 [328, 329], 299 Bode, J. H. G., 174 [207], .293. Bogaard, M. P., 249, 282 [363], 299, 301 Boggs, J. E., 165 [ 177], 166 [ 177], 274 [ 1771, 292

Breeze, J. C., 174 [203], 293

Bridge, N. J., 218 [289], 245 [289], 246 [289], 297 Brooks, B. R., 164 [171], 176 [219, 220], 274 [171], 277 [171], 29.__!,294 Bruns, R. E., 37 [431, 55 [431, 57 [43], 285 Buckingham, A. D., 218 [289], 245 [2891, 246 [289], 274 [343], 282 [363], 297, 390, 301 Burnham, A. K., 241 [318], 282 [318], 299 Bttrow, D. F., 245 [323], 248 [323], 29__29

C Califano, S., 2 [31, 5 [3], 7 [31, 15 [3], 26 [3], 27 [3], 29 [3], 31 [3], 32 [3], 36 [31, 113 [3], 150 [3], 152 [3], 283 Carl, J. R_, 221 [290], 245, 247 [290], 297 Carsky, P., 165 [ 1741, 169 [174], 170 [ 174], 292 Castiglioni, C., 55 [81], 58 [81, 85], 62 [81], 63 [80], 102 [126, 127], 103 [127], 287, 289 Chang, R. K., 212 [283, 284], 297 Chantry, G. W., 192 [250], 199 [250], 29.._5.5 Cheam, T. C., 75 [115], 142 [115], 288

Botschwina, P., 180, 182 [236], 295

Cherlow, J. M., 212 [287], 29___77

Bourgin, D. G., 18 [131, 28_..~3

Clifford, A. A., 37 [55], 285

AUTHOR INDEX

305

Clough, S., 63, 64 [96], 173 [96], 179 [961, 181 [96], 182, 185 [96], 2.8..8

[41, 193 [41, 194 [4], 197 [4], 223 [4], 283, 288, 289

Cohen, H. D., 166 [179], 281 [179], 292

Defrees, D. J., 142 [153], 274 [153], 290

Cole, S. J., 164 [172], 274 [172], 291

DeLeeuw, B. J., 183 [247, 248], 184 [247, 248], 185 [248], 186 [248], 295

Colvin, M. E., 169 [186], 176 [218], 177 [2181, 178 [2181, 292, 294 Costain, C. C., 221 [293], 298 Coulson, C. A., 55, 287 Crawford, Jr., B. L., 11 [8], 31 [35, 36], 32, 34, 37 [55], 41 [35], 42, 43 [35, 36], 73 [35, 36], 98, 104 [35, 36], 119 [35, 36], 212 [280], 213 [280], 217 [35, 36], 265 [35, 36], 283, 284, 285, 297 Crawford, M. F., 212 [2781, 282 [278], 297 Crawford, T. D., 164 [171], 183 [248], 184 [248], 185 [248], 186 [248], 274 [171], 277 [171], 291, 295 Cremer, D., 112 [ 133], 289 Cross, P. C., 4 [4], 5 [4], 6 [41, 7 [41, 8 [4], 26 [4], 27 [4], 29 [4], 31 [4], 32 [4], 36 [4], 86 [4], 113 [4], 150 [4], 192 [4], 193 [4], 194 [4], 197 [4], 223 [4], 283 Cyvin, S. J., 27 [17], 36 [17], 283

-----D Daeyaert, F. F. D., 170 [211], 29..3 Davidson E. 1L, 278 [355], 281 [355], 30,,0 Decius, J. C., 4 [4], 5 [4], 6 [4], 7 [4], 8 [4], 26 [4], 27 [4], 29 I4], 31 [4], 32 [4], 36 [4], 68, 70 [109], 72, 82 [117], 86 [4], 88, 90, 91 [117], 113 [4], 143 [109, 117], 145 [117], 150 [4], 158, 192

Dellepiani, D., 210 [276], 29__77 del Olmo, A., 235 [303], 280 [303], 298 del Rio, G., 51, 223 [71], 224 [296], 235 [296], 236 [296], 237 [296], 286, 298 Dementiev, V. A., 38 [60, 61, 62], 55 [60, 61, 62], 61 [60, 61, 62], 286 Dennison, D. M., 68, 288 Depuis, M., 164 [ 170], 291 Dickson, A. D., 31 [36], 34 [36], 42 [36], 43 [36], 73 [36], 98 [36], 104 [36], 119 [36], 217 [36], 265 [36], 285 Dierksen, G. H. F., 177, 278 [361], 294, 30! Dinsmore, H. U, 11 [8], 283 Dinur, U., 37 [51], 59 [51], 103 [51], 285 Dirac, P. A. M., 208, 296 Doggett, G., 59 [88], 287 Domingo, C., 235 [303, 304, 3061, 238 [303, 304, 306], 239 [304], 240 [304], 279 [304], 280 [303], 29_~8 Downs, A. J., 277 [352], 300 Dudev, T., 64 [100], 66 [1001, 75 [112, 114], 95 [124], 118 [141], 119 [142], 120 [142, 143], 127 [141], 130 [143, 144, 145], 131 [146], 133 [146], 136 [148], 137 [148], 138 [148], 139 [148],

306

AUTHOR INDEX

140 [148], 142 [100, 124], 143 [100, 124], 175 [114], 218, 219 [2881, 221 [288], 222 [288], 235 [310, 311], 238 [310], 242 [311], 243 [311], 244 [311], 249 [331], 258, 261, 263 [333], 265 [288], 266 [333], 267 [333], 268 [333], 270 [333], 288, 289, 290, 297., 298, 299 Duncan, J. L., 27 [19], 28 [24], 36 [19, 24], 65, 66 [241, 126 [241, 174 [2051, 239, 240 [263], 241 [314, 315], 242, 244 [316], 284, 293, 299 Dunning, T. H., 277 [3531, 300 Dupuis, M., 169 [ 183], 29__.22 Durig, D. T., 170 [213], 294 Durig, J. 1L, 27 [221, 36 [221, 75 [1141, 118 [141], 120 [1431, 127 [1411, 130 [143], 170, 175 [114], 253 [332], 254 [332], 284, .288, 290, 293, 294, 299

Eugster, C. H., 210 [277], 297 Evans, G. T., 245 [327], 247 [327], 299 Ewig, C. S., 167 [ 180], 275 [ 180], 29__22 Exner, O., 51 [75], 287

F Feng, F. S., 170 [209], 293 Fenner, W. R., 212 [286, 287], 297 Ferigle, S. M., 52 [77], 224 [77], 287 Fermann, J. T., 164 [171], 274 [171], 277 [171], 29! Femandez-Sanchez, J. M., 235 [305], 238 [305], 298 Ferrisco, C. C., 174 [203], 293 Figeys, H. P., 183 [244], 295

------

E

-----

Ebenstein, W. L., 172 [ 197], 293 Eckardt, G., 202 [261], 29__fi6 Eckart, C., 29, 40, 54, 72, 261 [31 ], 284 EdgeU, W. F., 50 [70], 28__.fi6 Eggers, D. F., 278 [355], 281 [355], 390 Elkins, J. W., 174 [200], 293 Elyashberg, M. E., 38 [621, 55 [621, 61 [621, 28___fi6 Elyashevich, M. A., 7 [6], 26 [6], 27 [6], 29 [61, 32 [61, 36 [61, 40 [63], 51 [63], 52 [6, 63], 113 [6], 150 [6], 283, 286 Escdbano, R., 51, 223, 235 [306], 238 [306], 286, 298

Fitzgerald, G. B., 164 [171, 172], 177 [223, 226], 274 [ 171, 172], 277 [ 171], 29 !, 294 Fletcher, W. F., 32, 42 [38], 28.5 Flygare, W. H., 241 [318], 282 [318], 299 Fogarasi, G., 27 [21], 28 [21], 36 [21], 165 [21, 177], 166 [177], 170 [21], 274 [177], 284, 292 Fontal, B., 282 [364], 30_.._21 Foresman, J. B., 142 [153], 274 [153], 29.0 Fouche, D. G., 212 [283,284], 297 Fox, D. J., 142 [153], 274 [1531, 2..90 Franzosa, E. S., 210 [272], 297 Fried, A., 174 [200], 293

AUTHOR INDEX

307

Frisch, M. J., 142 [153], 164 [153], 168 [182], 169 [1821, 170 [182], 172 [182], 173 [182], 174 [182], 175 [182], 179 [182], 274 [153, 341], 276, 277, 290, 292 Fung, K. K., 221 [290], 245, 247 [290], 297 Furer, V. L., 38 [59], 286

G

,,,

Gailar, N., 64 [99], 288 Galabov, B., 37 [42, 44, 48], 51 [48], 55 [42, 44, 481, 57 [42, 44, 481, 58 [44], 60 I42], 61 [94], 64 [100], 66 [1001, 72 [111], 75 [112, 113, 114], 95 [124], 115, 118 [141], 120 [1431, 126 [48], 127 [141], 130 [143, 144, 145] 131, 133 [146], 136 [148], 137 [1481, 138 [148], 139 [148], 140 [148], 142 [100, 124], 143, 164 [140], 175 [114], 218, 219 [288], 221 [288], 222 [288], 249 [3311, 253 [332], 254 [332], 258, 261, 263 [333], 265 [288], 266 [333], 267 [333], 268 [333], 270 [333], 285, 287, .2.88, 289, 290, 297, 299 Gans, P., 27 [20], 36 [201, 284 Gatti, C., 37 [50], 59 [50], 285 Gaw, J., 168 [182], 169 [182, 187], 170 [182], 172 [182], 173 [182], 174 [182], 175 [182], 179 [182], 274 [341], 275 [345, 347], 276 [341], 277 [341], 292, 300 Georgieva, G., 75 [ 112], 288 Gerratt, J., 164 [ 166], 167 [ 166], 274 [ 1661, 275 [ 1661, 291 Geerlings, P., 183 [244], 295 Girin, O. P., 37 [56, 57], 286

Goddard, J. D., 164 [169], 165 [169], 176 [220], 291,294 Golden, D. M., 212 [280], 213 [280], 297 ,, Golden, W. G., 183 [243], 295 Goldman, L. M., 212 [285], 297 Goldsmith, M., 152 [162], 153 [162], 15911621,291 Gonzalez, C., 142 [153], 274 [153], 290 Gordy, W., 46 [671, 221 [2911, 286, 297 Gough, K. M., 115, 239 [313], 290, 29.8 Gounev, T., 75 [113], 288 Gray, D. L., 174 [206], 293 Gready, J. E., 274 [336, 337, 342], 278 [337], 300 Green, W. H., 150 [157], 183, 184 [238], 291, 29_..55 Greene, T. M., 277 [352], 300 Gribov, L. A., 38 [60, 61, 62], 51 [72], 52 [72, 78], 53 [72], 54 [72, 79], 55 [60, 61, 62, 72], 59, 61 [60, 61, 62, 72], 63 [72], 150 [155], 151 [155], 160, 161 [155], 162, 164 [72], 192 [155], 224 [155], 225 [155], 226 [155], 231 [155], 243 [155], 286, 287, 291 Guirgis, G. A., 170 [212], 294 Gussoni, M., 55 [80, 81], 58 [81, 85], 62 [81], 63 [80, 81, 95, 97], 102, 103 [127], 142, 143, 144 [151], 224 [298], 230 [300], 242, 287, 288, 289, 290, 298

308

AUTHOR INDEX

----

He=berg, G., 172 [190, 194], 174 [190, 199], 186 [190], 208, 293, 296

H - - - -

Hagler, A. T., 37 [51], 59 [51], 103 [51], 28.5 Haines, 1L, 249, 299

Hester, R. E., 192 [251], 296

Hameka, H. F., 2 [2], 283 Hamilton, T. P., 164 [171], 274 [171], 277 [171], 29__! Hamaguchi, H., 210 [275, 277], 297 Handy, N. C., 150 [157], 169 [184, 185, 187], 176 [216, 219], 177 [184, 225], 183 [184, 238], 184 [238], 274 [339], 291, 292, 294, 295, 300 Hanson, H., 152 [163], 291 Harada, L., 28 [29], 210 [275], 284, 297 Harris, IL R., 59 [89], .287 Harrison, IL J., 164 [172], 177 [223, 226], 274 [172], 291., 294 Havriliak, S., 169 [186], 176 [218], 177 [218], 178 [218], 292, 294 Hay, P. J., 278 [358], 301 Head-Gordon, M., [153], 29_.._0.0

Hess, Jr., B. A., 165 [174], 167 [180], 169 [ 174], 170 [ 174], 275 [ 180], 292

142 [153], 274

Hirschfeld, T., 194 [252], 195 [252], 29__fi6 Holzer, W., 212 [281], 213 [281], 297 Hornig, D. F., 51 [74], 54 [74], 172 [196], 287, 293 Hoy, A. R., 173 [198], 185 [198], 293. Huber, K. P., 174 [ 199], 293 Hurley, A. C., 180 [233], 295 Hush, N. S., 180 [231, 233], 181 [231], 182 [231], 232 [301], 233 [301], 274 [336, 337, 342], 278 [337, 356], 395, 298, 300 Hutley, M. C., 194 [258], 296 Hyatt, H. A., 212 [286, 287], 297 Hyde, G. E., 172 [196], 293 Hylden, J. L., 150 [156, 161], 152 [161], 155 [161], 156, 158, 160 [161, 165], 291

Hehre, W. J., 172 [ 191], 293 Heisenberg, W., 208, 296 Heitler, W., 2 [ 1], 28__.33 Helgaher, T., 176 [217], 294 Henry, B. R., 14 [91, 113 [9], 283 Herratm, J., 68 [108], 72, 78, 83 [108], 87 [1081, 99 [108], 142 [1081, 164 [108], 288

Ilieva, S., 72 [111], 75 [113, 114], 95 [124], 131 [146], 133 [146], 136 [1481, 137 [148], 138 [148], 139 [148], 140 [148], 142 [124], 143 [124], 175 [114], 261, 288, 289, 290 Inagaki, F., 210 [269], 296 Innes, K. K., 210 [272], 297. Ito, M., 210 [270, 271,274], 296, 297

AUTHOR INDEX

_.__

J._~_

309

Kindness, A., 113 [ 136], 114 [ 136], 115 [136], 289

Jalsovsky, G., 37 [41], 55 [411, 57 [41], 285

King, G. W., 206 [265], 296

Janssen, C. L., 164 [171], 274 [171], 2771171],291

King, W. T., 37 [45], 58 [45], 68 [45, 1051, 78, 83, 85, 87, 101, 102, 285, 288

Jaquet, R., 174 [201 ], 293

Klein, G., 63 [96], 64 [96], 173 [96], 179 [96], 288

Jayatilaka, D., 150 [157], 183, 184 [238], 291,295

Klein, M. L., 37 [40], 2.8.5

John, I. G., 278 [356], 300

K16ckner, H.W., 201 [260], 202 [260], 212 [260], 213 [260], 235 [307], 238 [307], 296, 298

Jorgensen, P., 176 [217], 294

Knowles, P. J., 176 [216], 294

Jouve, P., 150 [158], 152 [158], 155 [158], 183 [239], 291,295

Koga, Y., 65 [101], 288

Jensen, H. J., 176 [217], 294

Kahn, L. IL, 142 [1531, 274 [1531, 290 Kalantar, A. H., 210 [272], 297 Kalasinshy, V. F., 239 [317], 241 [3171, 243 [3171, 299 Kaplan, L., 47 [68], 286 Karasev, Y. Z., 38 [62], 55 [621, 61 [621, 286 Kato, S., 176 [214], 294 Kaya, K., 210 [270, 271], 296, 297 Kellam, J. M., 212 [286], 297 Kemble, D. E. C., 18 [ 15], 283 Kendrick, J., 278 [360], 301 Kessler, M., 221 [291], 297 Keyes, T., 245 [327], 247 [327], 299

Komornicki, A., 166, 167, 169, 177 [ 178], 180 [ 178], 274 [ 178], 275, .292 Kondo, S., 65, 93, 177 [228], 178 [228], 280 [351], 288, 289, 294, 300 Koops, Th., 61 [93], 135 [147], 174 [147, 208], 186 [208], 287, 290, 293 Kotov, S. V., 54 [79], 287 Kovner, M. A., 51 [73], 52 [73], 55, 59 [73], 60 [73], 67 [73], 202 [73], 224 [73], 287 Krainov, E. P., 51 [73], 52 [73], 55, 59 [73], 60 [73], 67 [73], 202 [73], 224 [73], 287 Kraka, E., 112 [ 133], 289 Kramers, H. A., 208, 296 Kfimm, S., 75 [115], 142 [115], 288 Krislman, R., 167 [181], 177 [181, 221], 274 [181], 292, 294 Kubulat, K., 58 [86], 68 [106], 103, 116,287

310

AUTHOR INDEX

Kuchitsu, K., 93 [122], 94 [122], 289 Kutzelnigg, W., 174 [201], 293 ---~

L------

"'

M

Ma, Buyong, 277 [352], 300 Magers, D. H., 164 [ 172], 274 [ 172], 2..9.!

LaBoda, M. L, 183 [241], 295

Mallinson, P. D., 241 [314], 299

Laidig, W. D., 164 [172], 176 [219, 220], 177 [222, 223, 226], 274 [172], 291, 29.4

Marcott, C., 183 [243], 295

Laadanyi, B. M., 245 [327], 247 [327], 299 Lapp, M., 212 [285], 297 Larouche, A., 37 [50], 59 [50], 285 Lee, Min Joo, 170 [213], 294

Martin, J., 202 [263], 204, 235 [263, 3091, 238 [263], 239 [2631, 240 [2631, 242, 296, 298 Martin, R. L., 142 [153], 274 [153], 278 [355], 281, 290 Masseti, G., 210 [276], 242 [319], 297, 299

Lengsfield, B. H., 169, 176 [218], 177, 178 [218], 292, 294

Mast, G. B., 37 [45], 58 [45], 68 [45, 105], 78, 82 [ 117], 83 [ 1051, 85 [ 105], 87 [105], 88, 90, 91 [117], 101, 102, 143, 145 [117], 285, 288, 289

Leonard, D. A., 212 [282], 297

Matsmtra, H., 28 [28, 29, 30], 284

Libov, N. G., 37 [56], 286

Mayants, L. S., 78, 106, 109 [129, 130, 131], 132 [129], 142 [129], 164 [129], 252 [129], 288, 289

Lee, Y. S., 164 [172], 274 [172], 291

Lipscomb, W. N., 177 [224], 179 [224], 294 Little, T. S., 170 [212], 239 [317], 241 [317], 243 [317], 294, 29.9 Liu, Jie, 170 [210], 293 Livington, 1L L., 47 [68], 286 Lokshin, B. V., 68, 70 [107], 142 [107], 288 Long D. A., 52 [76], 224 [76], 28.7 Loytsyanski, L. G., 40 [64], 286 Lurie, A. I.,40 [64], 286

Mayers, D. H., 177 [224], 179 [224], 294 McClellan, A. L., 174 [204], 293 McCullosh, R. D., 241 [314], 299 McDougall, P. J., 37 [50], 59 [50], 285 Mclver, J. W., 166, 167, 169, 177, 180 [178], 274 [ 178], 275, 292 McKean, D. C., 28 [24], 36 [24], 51 [74], 54 [74], 65 [24], 66 [24], 113, 114 [136], 115 [136], 126 [24], 239, 241 [314], 242, 244 [316], 284, 287, 289, 299

AUTHOR INDEX

Melius, C. F., 142 [153], 274 [153], 290 Meyer, W., 180 [232, 237], 295 Miehalska, D., 167 [180], 275 [180], 292 Mikami, N., 210 [270, 271], 296, 297 Mildaailov, V. M., 183 [245], 295 Miller, F. A., 50 [70], 286 Mills, I. M., 14 [10], 28 [231, 31 [36], 34 [36], 36 [23], 42 [361, 43 [36], 63, 66 [102], 73 [36], 98 [36], 104 [36], 113 [110], 119 [36], 164 [166], 167 [166], 173 [198], 174 [205], 182, 185 [198], 186 [249], 217 [36], 265 [36], 274 [166], 275 [166], 283, 284, 285, 288, 291, 293, 295

Mink, J., 28 [261, 36 [261, 284 Miyazawa, T., 210 [269], 296 Moeia, 1L, 275 [346], 300 Montero, S., 202 [263], 204, 224 [296, 297], 235 [263, 296, 297, 302, 303, 304, 305, 306, 307, 308, 310], 236 [296, 297], 237 [296, 297], 238 [263, 310, 312], 239 [263, 304, 312], 240 [263, 304], 279 [302, 304], 280 [303], 296, 298 MorciUo, J., 68 [ 108], 72, 78, 83 [108], 87 [108], 99 [108], 142 [108], 164 [ 1081, 288 Morokuma, K., 176 [214], 294 Morrison, M. A., 278 [358], 301 Muenter, J. S., 172 [197], 281 [362], 293, 301 Mttlliken, R. S., 59 [87], 101, 287

311

Murphy, W. F., 115, 203 [2641, 212 [281], 213 [281], 235 [304], 238 [304], 239 [304], 240 [304], 278 [348,349], 279 [304, 348, 349], 290, 296, 297, 298, 300 ------

N

~

Nafie, L. A., 245 [324], 299 Nakagawa, I., 93, 289 Nakanaga, T., 65 [101], 177, 178 [228], 280 [351], 288, 294, .300 Newall, A. C., 282 [365], 301 Newton, J. H., 31 [33], 33 [33], 68 [33], 78, 79, 80 [33], 85, 86 [33], 87 [33], 142 [33], 164 [33], 284 Nielsen, H. H., 152 [162, 163], 153 [162], 159 [162], 29.1 Nikolova, B., 37 [48], 51 [48], 55 [48], 57 [48], 115 [140], 130 [145], 164 [ 140], 285, 290 NiveUini, G. D., 239, 242, 244 [316], 299 Novoselova, O.V., 38 [61], 55 [61], 61 [61], 286 O

Ogawa, Y., 28 [28, 29], 284 Oka, T., 44 [65], 286 Okamoto, H., 194 [257], 208 [257], 210 [257, 277], 296 Orduna, M. F., 235 [303, 304], 238 [303, 304], 239 [304], 240 [304], 279 [304], 280 [303], 298 Orr, B. J., 274 [343], 300

312

AUTHOR INDEX

Orville-Thomas, W. J., 37 [41, 42, 44, 48, 54], 51 [48], 55 [41, 42, 43, 44, 48], 57 [41, 42, 44, 48], 58 [44], 60 [42], 61 [94], 115 [140], 118 [141], 126 [48], 127 [141], 130 [144, 145], 150 [155], 151 [155], 160 [155], 161 [155], 162, 164 [140], 192 [155], 224 [155], 225 [155], 226 [155], 227 [155], 231 [155], 242 [155], 285, 287, 290, 291 Oz-z~ J. M., 51, 223 [71], 235 [306], 238 [3061, 286, 298 Osamma, Y., 164 [ 169], 165 [ 169], 275 [345], 291., .300 Overend, J., 21 [ 16], 22 [ 16], 23, 116, 150 [156, 159, 160, 161], 152 [159, 160, 161], 155 [161, 164], 156, 158, 159, [164], 160 [161, 165], 183 [240, 24 I, 242, 243], 283, 291, 295 p .,..._...._

Pal, S., 164 [ 172], 274 [ 172], 291 Pang, F., 165 [177], 166 [177], 274 [ 177], 292

Phan, H. V., 75 [ 114], 170 [209], 175 [ 114], 288, 293 Pierens, R. K., 282 [363], 30 ! Pine, A. S., 174 [200], 293 Placzek, G., 194 [254], 202 [254], 296 Plyer, E. K., 64 [99], 288 Politzer, P., 59 [89], 287 Pople, J. A., 142 [153], 167, 172 [191], 177 [181, 221], 274 [153, 181], 29__Q0 Porto, S. P. S., 212 [286, 287], 297 Pulay, P., 27 [21], 28 [21], 36 [21], 164 [167, 168], 165 [21, 168, 177], 166 [177], 169 [168], 170 [21, 168], 176 [168], 274 [167, 168, 177, 338], 275 [168], 280 [338], 284, 291,292, 300 Prasad, P. L., 245 [323, 324], 248 [323], 299 Purvis, G. D., 164 [172], 179 [230], 274 [ 172], 291,295

Pao, C., 47 [68], 286 Parisean, M. A., 155 [164], 159 [164], 291

Q

......_._.._

QuicksaU, C. O., 245 [321], 248 [321], 249, 299

Patat, F., 172 [ 194], 29..3 Pauling~ L., 140 [149], 266 [335], 270 [335], 29.0, .299 Penner, S. S., 17, 23, 283 Penney, C. M., 212 [285], 297 Person, W. B., 31 [33], 33 [33], 37 [43, 531, 38 [531, 51 [531, 55 [43, 53], 57 [43], 58 [86], 63, 68 [33, 106], 78, 79, 80 [33], 85, 86 [33], 87 [33], 103, 105, 116, 142 [33], 164 [33], 165 [53, 176], 169 [1761, 170 [176], 183 [240], 284, 285, 287, 288, 292, 295

---~

Raghabachari, [ 153], 290

R----

K.,

142 [153], 274

Ramos, M. N., 58 [85], 102 [126, 127], 103 [1271, 287, 289 Ratajczak, H., 37 [54], 285 Reiehle, Jr. H. G., 172 [ 193 ], 293 Remington, R., 164 [171], 274 [171], 277 [171], 291

AUTHOR INDEX

313

Rendell, A. P. L., 232 [301], 233 [301], 298

Samvelyan, S.Kh., 68, 70 [107, 110], 82 [ 110], 142 [ 107, 110], 288

Rice, J. E., 142 [1541, 164 [154], 177 [2251, 290, 294

Saxe, P., 176 [219], 275 [344, 345], 294, 300

Riley, G., 37 [441, 55 [441, 57 [441, 58 [44], 285

Sayvetz, A., 29, 40, 54, 72, 261 [31], 284

Ring, H., 221 [2911, 297

Schaad, L. J., 165 [174], 167, 169 [ 174], 170 [ 174], 275, 292

Rittby, M., 164 [172], 274 [172], 291 Robb, M., 142 [153], 176 [215], 274 [153], 290, 294 Robiette, A. G., 174 [206], 293 Rocks, L., 47 [68], 286 Rogers, M. T., 55 [83], 287 Roos, B. O., 180 [235], 295 Roothaan, C. C. J., 166 [179], 281 [ 179], 292 Rothman, L., 63 [96], 64 [96], 173 [96], 179 [96], 288

Schaefer III, H. F., 164 [169, 171], 165 [169], 168 [182], 169 [182, 185, 186, 188], 170 [182], 172 [182], 173 [182], 174 [182], 175 [182], 176 [218, 219, 220], 177 [218], 178 [218], 179 [182], 183, 184, 185 [248], 186 [248], 274 [171, 339, 341], 275 [344, 345, 347], 276 [341], 277 [171, 271, 341, 352], 291,292, 294, .295, 3.00 Schick, G.A., 245 [328, 329], 299 Schlegel, H. B., 142[153], 167 [181], 176 [215], 177 [181, 221], 274 [153, 181], 290, 292, 294

Rousseau, D. L., 210 [273], 297

Schr6tter, H.W., 201 [260], 202 [260], 212 [260], 213 [260], 296

Rupprecht, A., 31 [37], 106, 108 [37], 110 [37], 142 [37], 224 [299], 249 [299], 250 [299], 252 [299], 285, 298

Secroun, C., 150 [158], 152 [158], 155 [158], 183 [239], 291,295

------

S----

Sadlej, A. J., 177, 278 [361], .294, 301 Saebo, S., 177 [227], 294 Sa~ki, S., 65 [ l01], 93, 177 [228], 178 [2281, 221 [2921, 280 [351], 288, 289, 294, 298, 300 Saito, S., 2 l0 [277], 297 Salters, E. A., 164 [172], 274 [172], 291 Sambe, H., 99, 2.8.9

Seeger, R., 142 [153], 274 [153], 290 Segal, G. A., 37 [40], 285 Seidl, E. T., 164 [171], 274 [171], 277 [171], 291 Sexton, G. J., 176 [216], 294 Shaffer, W. H., 152 [ 163], 291 Sherrill, C. D., 164 [171], 274 [171], 2771171],291 Shimanouchi, T., 27 [18], 28 [28, 29], 36 [18], 93, 172 [195], 174 [195], 210 [275], 284, 289, 293,297

314 Siesbahnn, P. E. M., 180 [235], 295 Silberstein, L., 245, 299 Simandiras, E. D., 169, 177 [225], 183 [ 184], 292, 294 Slee, T. S., 112 [133], 289 Stair, W. M. A., 31 [34], 33 [34], 42, 45, 48 [341, 58 [84], 60, 61 [931, 65 [34], 73 [34], 96 [34], 97 [34], 104 [34], 119, 121 [34], 135 [34, 147], 142 [152], 173 [34], 174 [147, 207, 2081, 179 [34], 185 [34], 186 [208], 217 [341, 265 [34], 284, 287, 290

AUTHOR INDEX Stewart, J. J. P., 142 [153], 274 [153], 290 Strey, G., 173 [198], 185 [198], 186 [249], 293, _295 Sufxa, S., 210 [276], 29__27 Sullivan, J. F., 170 [213], 294 Sundberg, K. R., 245 [322], 299 Sutton, L. E., 46 [66], 286 Suzuka, I., 210 [270, 271, 274], 296, 297

Smith, W. V., 46 [67], 286

Suzuki, E., 210 [275], 297

Snyder, R. G., 28 [27], 39, 72, 73, 74, 75, 284

Suzuki, I., 155 [164], 159 [164], 29.._.21

Sosa, C., 164 [ 1721, 274 [ 1721, 291 Sourer, P. E., 277, 300 Speirs, G. K., 28 [241, 36 [241, 65 [241, 66 [24], 126 [241, 284 Spiro, T. G., 282 [364], 301

Suzuki, S., 37 [44], 55 [44], 57 [441, 58 [441, 285 Sverdlov, L. M., 51 [73], 52 [73], 55, 59, 60 [73], 67 [73], 202 [73], 224 [73], 287 . . . . .

Swalen, J. D., 221 [293], .298

Staemmler, V., 174 [201], 293

Swanton, D. J., 180 [231], 181 [231], 182 [231], 295

Stansbury, E. J., 212 [278], 282 [278], 297

Szczepaniak, K., 165 [ 176], 169 [ 176], 170 [ 176], 292

Stanton, J. F., 177 [224], 179 [224], 294.

-----

T

------

Takagi, K., 44 [65], 286.

Steele, D., 5 [5], 7 [5], 11, 26 [5], 27 [5], 29 [51, 32 [51, 36 [5], 37, 38 [53], 51 [53], 55 [53], 75 [113], 113 [5], 150 [5], 165 [53, 173], 192 [5], 283, 285, 288, 292

Tanabe, K., 221 [292], 298

Stepanov, B. L., 7 [61, 26 [6], 27 [61, 29 [6], 32 [6], 36 [61, 52 [61, 113 [61, 150 [ 6 ] , 283

Tasumi, M., 28 [301, 142 [1501, 210 [269, 277], 284, 290, 29.._.66

Stewart, D., 113 [136], 114 [136], 115 [1361, 289

Tang, J., 194 [256], 208 [256], 210 [256], 296

Taylor, P. R., 177 [227], 180 [233, 235], 294, .295

AUTHOR INDEX

315

Teller, E., 208, 29....fi6 Thole, B. T., 245 [325], 247, 29__.29 Thomas, J. R., 183 [247, 248], 184 [247, 248], 185 [248], 186 [248], 295

Volkenstein, M. V., 7 [6], 26 [6], 27 [6], 29 [6], 32 [6], 36 [6, 39], 40 [39, 63], 51 [39, 63], 52 [6, 63], 113 [6], 150 [6], 224, 283, 285, 286

- - - -

W

------

Todorovski, A. T., 38 [60], 55 [60], 61 [601, 286

Waggoner, J., 152 [ 163], 29_...!

Topiol, S., 142 [153], 274 [153], 290

Wagner, W. G., 202 [261], 296

Torii, H., 142 [1501, 29___0.0

Waiters, V. A., 114 [ 137], 29_....0

Trambarulo, R. F., 46 [67], 221 [291], 28_..66,297

Wang, A., 27 [22], 36 [22], 170 [209], 284, 293

Trucks, G. W., 142 [153], 164 [172], 274 [ 153, 172], 29..._00,291

Warsop, P. A., 235 [308], 298

Tubbs, L. D., 172 [192], 174 [202], 29__~3

Watson, J. K. G., 183 [246], 295 Weast, R. C., 47 [69], 50 [69], 286

U

,,

Udagawa, Y., 210 [270, 271], 296, 297

Weber, A., 52 [77], 224 [77], 28_.!7 Weber, D., 17, 23,283 Welsh, H. L., 212 [278], 282 [278], 29.7

----

V-----

Vacex, G., 164 [171], 183 [247, 248], 184 [247, 248], 185 [247, 2481, 186 [248], 274 [ 171], 277 [ 171], 291,295

Wendolski, J. J., 37 [46, 47, 49], 51 [47], 57 [46, 47, 49], 58 [46], 59 [47, 49], 61 [46], 126 [49], 134, 169 [183], 285, 2.9.2

van Dam, T., 58 [84], 287

Wells, A. J., 17, 19, 21,283

van der Kveken, B. J., 170 [210], 293

White, A. H., 282 [363], 301

van Straten, A. J., 31 [34], 33 [34], 42, 45, 48 [34], 60, 65 [34], 73 [34], 96 [34], 97 [34], 104 [34], 119, 121 [34], 135 [34], 142 [152], 173 [34], 179 [34], 185 [34], 217 [34], 265 [34], 284, 290

Whiteside, R. A., 142 [153], 274 [153], 290

Verleger, H., 172 [ 194], .293

Wiberg, K. B., 37 [46, 47, 49, 50], 51 [47], 57 [46, 47, 49], 58 [46], 59 [47, 49, 50], 61 [46], 66 [103], 114, 126 [49], 134, 285, 288, 290

Vincent, M. A., 275 [345], 300

WiUets, A., 150 [157], 2.91

Visser, T., 135 [147], 174 [147, 208], 186 [208], 290, 293

Willetts, S., 183, 184 [238], 295 Williams, D., 172 [192], 174 [202], 293

316

AUTHOR INDEX

Williams, P. F., 210 [273], 297 Wilson, Jr., E. B., 4 [4], 5 [4], 6 [4], 7 [4], 8 [4], 17, 19, 21, 26 [4], 27 [4], 29 [4], 31 [4], 32 [4], 36 [4], 86 [4], 113 [4], 150 [4], 192 [4], 193 [4], 194 [4], 197 [4], 223 [4], 283 Wooster, W. A., 110 [132], 199 [132], 289 X

Xie, Y., 164 [1711, 274 [171], 277 [171], 291

____

y___

Yamaguchi, Y., 164 [169, 171], 165 [169], 168, 169 [182, 188], 170 [182], 172 [lg2], 173 [182], 174 [182], 175, 176 [220], 179, 183 [248], 184 [248], 185 [248], 186 [248], 274 [171, 341], 275 [344, 345, 347], 276 [341], 277 [ 171, 341], 291, .292, 294, 295, 3..00 Yao, S., 150 [159], 152 [1591, 155 [159], 183 [2421, 291, 295 Yoshino, T., 212 [279], 282 [279], 297

Young, C., 172 [ 193], 293 - - -

Z----

Zahradnik 1L, 165 [1741, 169 [174], 1701174],292 Zerbi, G., 55 [80, 81], 58 [81, 85], 62 [811, 63 [80, 81, 971, 102 [126, 127], 103 [127], 210 [276], 224 [298], 242 [319], 287, 288, 289, 297, 298, 299 Zilles, B., 63, 288

SUBJECT INDEX

- - -

anisotropy spectra, 203

A - - -

argon, Ar, 23

ab mitio MO calculations, 40, 66, 112,

135, 142, 164, 203, 242

atom dipole interaction model, 245

ab mitio Raman intensifies, 276

atomic Cartesian displacement coordinates, 29, 30, 69, 249, 262

absolute differential Raman scattering cross section, 202, 214

atomic charge--charge flux model, 68

absolute differential Raman scattering cross section of nitrogen, 212

atomic dipoles, 58 atomic effective charge model, 142

absolute trace and anisotropy Raman differential cross sections, 204

atomic polar tensors, 79, 88, 93, 98, 100, 105, 109, 142, 143, 144, 145, 146

absorption coefficient A, 19 absorption probability, 2

atomic polarizabilities, 247, 258

acetimide, CH3CONH2, 247

atomic polarizability tensor, 249

acetone, 219, 220, 221, 222 ----

acetonitrile, 219, 220, 221

B----

basis sets, 171

acetylene, C2H2, 136, 139, 126, 127, 170, 174, 175, 183,238, 244, 277, 279

benzene, C6H6, 126 Boltzmann distribution, 5, 8

Albrecht's theory, 210

bond charge parameter formulation, 60

alkylacetylenes, 130

bond charge tensors, 106, 142

aUene, CH2=C=CH2, 126, 127

bond direction angles, 52

ammonia, NH3, 45, 135, 136, 136, 139, 170, 174, 175, 183, 263, 266, 267

bond direction cosines, 52 bond displacement coordinates, 132, 262

analytic derivative methods, 167, 177 analytic gradient methods, 275

bond displacement vectors, 106, 110

angular momentum, 41

bond moment, 51, 52, 54, 58, 59, 63, 65,

anharmonic vibrational transition moment, 151 317

318

SUBJECT INDEX

bond moment model 51, 55, 59, 158

CEPA2 (CPA'), 179

bond polar parameter, 111, 126, 128, 130, 143, 146

CEPA3 (CPA"), 179

bond polarizability model 235, 236 bond polarizability tensor, 224 Born-Oppenheimer approximation, 196 trans-butadiene, CH2CHCHCH2, 263, 268, 270, 271 2-butyne, CH3-C~-C-CH3, 127

charge densities, 112 charge flow model, 158 charge flux, 57, 72, 143, 158 charge flux parameters, 145 charge-charge flux overlap model, 100 chloroform, CHC13, 38, 114 CI, 281,282

----

C-----

CISD, 183, 281

calculated infrared intensities, 167, 176

combination bands, 14

calculated Raman intensities, 274, 276

compact formulation of VOTR, 235, 238

carbon dioxide, CO2, 19, 20, 170, 172, 183 carbon monoxide, CO, 70, 100, 170, 174, 175, 281, 282

compensatory molecular rotation, 40, 41, 42, 49, 217, 262 configuration interaction, 176

carbon tetrachloride, CC14, 38

coupled-cluster, 177

carbon tetrafluoride, CF4, 105

cyanogen, C2N2, 179

carbonic dichloride, C12C0, 127, 136, 139, 139

cyclohexane, c-C6H12, 114, 242

carbonic difluoride, F2CO, 127, 128, 136, 139 CASSCF, 179 CCSD, 179, 183

cyclopropane, 112 D

,

depolarization ratio, 193, 202

CCSD(T), 183

dich/oro methane thial, C12CS, 127, 128, 136, 139

CCSD+T(CCSD), 179

dichloromethane, 219, 220, 221,222

CEPA- 1 (ED), 181

differential Raman scattering cross sections, 213

CEPA- 1 (EV), 181

SUBJECT INDEX

319

difluoromethane thial, F2CS, 127, 128, 136, 139

electro-optical anharmonic parameters, 162

digallium hexahydride-do, Ga2H6, 277

electro-optical parameters, 52, 54, 59, 61, 142, 144, 158

dimensionless normal coordinate, 204 dimethyl ether, (CH3)20, 114 dipole moment derivatives, 12, 16, 26, 37, 38, 42, 64, 66, 166 DMC 6/12, 177 DMC 6/6, 177 DZ, 170 DZ and TZ basis sets, 170 DZ+P, 183 DZP, 277 DZP (DZ+P), 177 DZP+dSPD, 177 DZP+dSPD/DMC, 177

electron correlation, 176, 278 energy gradient method, 275 ethane, C2H6, 93, 114, 124, 127, 136, 238, 242, 244, 268, 270, ethylbromide, CH3CH2Br, 114 ethylchloride, CH3CH2C1, 114 ethylene oxide, c-C2H40, 247 ethylene, CH2CH2, 126, 127, 139, 238, 263, 268, 270, 271,277 experimental IR band intensities, 37, 39, 41 experimental determination of infrared intensities, 17 external coordinates, 31, 40

1,1-dichloroethylene, 47, 48, 49 ---~ E - - - -

F----

first-order reop, 237

Eckart-Sayvetz conditions, 29, 30, 34, 54, 72

fluorinated methanes, 130

Eckart-Sayvetz equations, 40

fluoroacetylene, FCCH, 183

effective atomic charge, 99, 132, 145

fluoroform, CHF3, 114, 127, 128

effective bond charge, 60, 132

force constants, 16, 28, 29, 61

effective charges, 72, 143, 158

force field, 26, 36, 39, 63, 65,

effective induced bond charge, 261, 263, 266, 271

formaldehyde, H2CO, 43, 44, 136, 139, 177, 178, 183,254, 268, 270, 271,

Einstein coefficient, 3, 6

formamide, HCONH2, 247

electric charge, 16, 40, 84, 107

fundamental transition, 14, 150

320

SUBJECT INDEX

a

....,,..,-..

hypothetical-mass-isotope method, 261, 265

gas-phase Raman scattering cross sections, 211 group dipole derivatives, 72 group moment derivatives, 74, 75 group symmetry coordinates, 39, 73 ,

H----

harmonic approximation, 8, 13, 26, 133

iso-butane, (CH3)3CH, 114 induced bond charge, 263 induced dipole, 190 integrated absorption coefficient, 6, 10, 16, 18

harmonic force constants, 26

integrated intensity of the infrared absorption band, 6, 8

harmonic force field, 239

intensity sum rule, 85

harmonic frequency, 8, 10

internal coordinates, 26, 44, 45, 46, 47, 52, 55, 64, 66, 94

harmonic oscillator selection rules, 12, 13, 16, 205 heavy isotope method, 45, 219 homopolar moment, 56

hot band, 200 hydrocarbons, 136, 242 hydrogen chloride, HC1, 70, 100, 263, 266, 267 hydrogen cyanide, HCN, 170, 172, 179, 181, 182, 183 hydrogen fluoride, HF, 70, 99, 170, 174, 175, 187, 263,266, 267

intramolecular charge distribution, 15 intramolecular charge reorganization, 37 invariants of atomic polar tensors, 83 ----

M----

many body perturbation theories, 177 mass-weighted Cartesian displacement coordinates, 31 MBPT(2), 177, 179, 183 MC SCF calculations, 177, 278, 281

hydrogen isocyanide, HNC, 183

mean polarizability, 191

hydrogen sulfide, H2S, 263,266, 267, 281

methane, CH4, 21, 114, 127, 136, 139, 170, 174, 183, 238, 242, 247, 263, 266, 277, 279

hypothetical isotope method, 42, 43, 51 hypothetical non-rotating isotope, 42

methanethial, CH2S, 263, 268, 270, 271

SUBJECT INDEX

methanimine, CH2NH, 263, 268, 270, 271 methanol, CH3OH, 114, 263,267, 268, 270 methyl amine, CH3NH2, 114, 263, 267, 268, 270

321

n-pentane, 128, 130 natural vibrational coordinates, 26 nitrogen, N2, 22, 23, 282 normal coordinate analysis, 26, 28, 32, 39

methyl amine-d2, CHD2NH2, 115

normal coordinate transformation matrix, 27, 36

methyl bromide, CH3Br, 114, 127, 128, 136, 139

normal coordinates, 7, 13, 14, 16, 36

methyl chloride, CH3C1, 65, 67, 95, 114, 121, 127, 128, 136, 139, 218

numerical differentiation, 165 ------

methyl ethyl ether, CH3OCH2CH3, 176 methyl fluoride, CH3F, 105, 114, 127, 128, 136, 139, 263, 268, 270

O - - -

overlap moment, 55 overtone and combination band intensities, 150

methyl iodide, CH3I, 114, 127, 139

overtone and combination bands, 16

methylene chloride, CH2C12, 37, 38, 114

overtone transitions, 14

methylene fluoride, CH2F2, 105, 114, 127, 128 methylidyne phosphine, HCP, 179, 182 molecular dipole moment, 51 molecular polarization, 190 molecular principle axes, 41 molecular rotation, 41 MP2, 179, 183 multicontiguration SCF, 176

'"

N

'

n-alkanes, 72, 75, 120, 130 n-butane, 130

____

p _ _ _

perdeutero-polyethylene, 242 perturbation theory formulation, 152 phosphine, PH3, 263, 266, 267 Placzek's conditions, 196, 207 point charge, 56 point charge approximation, 55 point charge model, 59 polar angles, 52 polar tensors, 158 polarizability derivatives, 201

322

SUBJECT INDEX

polarizability derivatives with respect to symmetry coordinates, 217 polyethylene, 242 propane, C3H8, 114, 127, 242 propanol, C3H7OH, 247 propyne, HCCCH 3, 126, 127, 136, 239, 240, 241, 244, pyrazine, 211 ----

rotational correction term matrix, 45 rotational correction terms, 41, 42, 40, 46, 50, 217 rotational correction terms to polarizability derivatives, 22, 216 rotational corrections to dipole moment derivatives, 40 rotational polar tensor, 80, 97 rotational polarizability tensor, 251

R-----

roto-translational coordinates, 79

Raman depolarization rations, 279 Raman electro-optical parameters, 230 Raman line intensity, 199 Raman scattering coefficients, 279 Raman scattering cross section, 200 rational correction terms to polarizabilty derivatives, 221 rational quantum numbers, 5

scattering coefficient, 202 SCF, 183, 281 SDMC 6/6, 177 second order perturbation theory, 177 semi-classical approaches, 40 silane, Sill 4, 263,266

Rayleigh scattering, 198

simple hydrides, 269

relative differential Raman scattering cross section, 214

static bond charges, 60 structure of atomic polar tensors, 91

relative rotational corrections, 223

sulfer dioxide, SO2, 230, 232, 259

reop, 230 resonance Raman effect, 207

symmetry and normal coordinates, 28

RHF/SCF calculations, 170

symmetry coordinates, 27, 33, 38, 40, 44, 50, 52, 65, 94

rotation-flee atomic polar tensor, 131

symmetry selection rules, 14

rotation-flee atomic polarizability tensor, 261, 265 rotation-flee bond polarizability tensor, 265

---T theorem for momen~.m conservation, 40

SUBJECT INDEX

trace spectra, 203 transition dipole, 15 transition dipole matrix, 4, 8

323

- - -

W - - - - -

water, H20, 19, 20, 63, 97, 120, 135, 139, 170, 173, 179, 181247, 263, 266, 277, 279

transition dipole moments, 157, 209 transition probability, 3 trifluorobromo methane, CF3Br, 128 trifluorochloro methane, CF3CI, 128 trifluoroiodo methane, CF3I, 128 trimethyl amine, (CH3)3N, 114 TZ basis sets, 170

Z

~

zero angular momentum condition, 52, 118,261 zero-mass approach, 43, 218 zero-order approximation, 54 zero-order approximation of the bond polarizability model, 236 zero-order electro-optical parameters, 238

TZ+2P, 170 TZ+2P, 183

zero-order polarizability parameters, 240

TZP, 277 1,1,1-trifluoroethane, 48, 49, 50 - - -

-----

V------

valence optical theory of Raman intensities, 223, 232, 258 valence-optical theory, 51, 63 variation method formulation, 151 vibrational atomic polar tensors, 80, 103 vibrational atomic polarizability tensor, 25 I, 261 vibrational intensity analysis, 41 vibrational quantum number, 13, 198 vibronic coupling operator, 209


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