Vibrational spectra: principles and applications with emphasis on optical activity

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studies in physical and theoretical chemistry 85

VIBRATIONAL SPECTRA" PRINCIPLES AND APPLICATIONS WITH EMPHASIS ON OPTICAL ACTIVITY

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s t u d i e s in physical a n d t h e o r e t i c a l c h e m i s t r y 85

VIBRATIONAL SPECTRA: PRINCIPLES AND APPLICATIONS WITH EMPHASIS ON OPTICAL ACTIVITY by RL. POLAVARAPU

Department of Chemistry Vanderbilt University 7332 Stevenson Center, Nashville, TN 37235 U.S.A.

1998 Amsterdam - Lausanne - New York - Oxford

- Shannon

ELSEVIER - Singapore - Tokyo

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

9 1998 Elsevier Science B.V. All rights reserved.

This work and the individual contributions contained in it are protected under copyright by Elsevier Science B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Rights & Permissions Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also contact Rights & Permissions directly through Elsevier's home page 0mp://www.elsevier.nl), selecting first 'Customer Support', then 'General Information', then 'Permissions Query Form'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 171 436 5931; fax: (+44) 171 436 3986. Other countries may have a local reprographic rights agency for payments. Derivative Works Subscribers may reproduce tables of contents for internal circulation within their institutions. Permission of the publisher is required for resale or distribution of such material outside the institution. Permission of the publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Contact the publisher at the address indicated. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the publisher. Address permissions requests to: Elsevier Science Rights & Permissions Department, at the mail, fax and e-mail addresses noted above. Notice 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

First edition 1998 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.

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Q The paper used in this publication meets the requirements of ANSUNISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

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This Page Intentionally Left Blank

vii

Preface This book originated out of a desire to have the vibrational absorption, Raman scattering, vibrational circular dichroism and Raman optical activity combined in one source. The younger generation students practicing vibrational spectroscopy would undoubtedly come across these different areas and would benefit from having these topics covered in a unified manner. Some of the material presented in this book, especially portions of Chapters 2-4, has been taught as a part of Molecular Spectroscopy and Quantum Chemistry courses for graduate students. The topics covered are generally of an advanced level, so this book would be appropriate for graduate students and practicing scientists in vibrational spectroscopy. To keep the book reasonable in length, the contents are limited and several important related topics are not included. For example, rotational structure associated with a vibrational band is not considered. Similarly, in discussing the applications of symmetry, character tables of point groups are not included as they are available in many text books. It is not possible to write a book on vibrational spectroscopy, without being influenced by the pioneering books of G. Herzberg, Infrared and Raman Spectra, E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, and of S. Califano, Vibrational States. I have also depenOecl on the books of P. W. Atkins, Molecular Quantum Mechanics, and of L. D. Barron, Molecular Light Scattering and Optical Activity, for some of the material covered in this book. It is a pleasure to acknowledge Professor L. D. Barron, who patiently read most of the chapters and made valuable comments; Professor L. A. Nafie for very useful general comments, Professor T. A. Keiderling for suggestions on Chapters 11-14, Professor P. R. Griffiths for comments on Chapters 11-13, and Professor J. Tellinghuisen for help in scanning the figures. The influence of former mentors is a key factor in the development of one's career; so my sincere appreciation goes out to Professors Laurence A. Nafie, Duane F. Burow, Surjit Singh and A. S. N. Murthy, with whom I have conducted research during my formative years. I would like to thank Ms. Swan Go of Elsevier for her patience in letting me complete this book even after missing many deadlines; and Jewell Carter for an excellent job in typing most of the manuscript. Finally, copyright permissions from John Wiley & Sons, Wiley Chichester, The Royal Society of Chemistry, Munksgaard International, Cambridge University Press, The Society of Applied Spectroscopy, and The American Chemical Society are gratefully acknowledged.


ix

CONTENTS 1 2 2.1 2.2 2.2.1 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.5 2.6 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4 4.1 4.2 4.3 4.4 4.5 4.6 5 5.1 5.2 5.3 5.4 5.5 6 6.1 6.2 6.2.1 6.2.2

Introduction Quantum theoretical techniques Interaction with electric field Time independent perturbation theory

Static electric dipole polarizability

1 3 4 7

9

Time dependent perturbation theory

10

Electric dipole-electric dipole polarizability Electric dipole-magnetic dipole polarizability Expressions for general molecular properties Interaction with circularly polarized light

11 14 15 16

Transition rates Spectral intensities Transition polarizabilities Diatomic molecules Vibrational equation Energy levels of harmonic oscillator Energy levels of anharmonic oscillator Energy levels of Morse oscillator Electric dipole transition moments of harmonic oscillator Electric dipole transition moments of anharmonic oscillator Electric dipole transition moments of Morse oscillator Magnetic dipole transition moments of harmonic oscillator Magnetic dipole transition moments of anharmonic oscillator Transition polarizabilities Polyatomic molecules Energy levels of harmonic oscillators Energy levels of anharmonic oscillators Electric dipole transition moments of harmonic oscillators Electric dipole transition moments of anharmonic oscillators Magnetic dipole transition moments of harmonic oscillators Magnetic dipole transition moments of anharmonic oscillators Vibrational analysis Cartesian displacement coordinates Internal coordinates Energy distribution Non-equilibrium geometries Redundant internal coordinates Local modes Hamiltonian in local coordinates Approximations

19 21 23 27 27 30 32 41 43 44 51 53 55 58 61 61 64 73 74 78 78 81 81 86 96 97 99 107 107 110

Independent local oscillators Perturbation treatment of inter-oscillator coupling

110 I 11

7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 8 8.1 8.1.1 8.1.2

8.1.3 8.1.4 8.2 8.2.1 8.2.2

8.2.3

8.2.4 9 9.1 9.1.1 9.1.2

Vibrational frequencies and force constants Molecular orbital theory Energy derivatives First derivatives of energy Second derivatives of energy Higher derivatives of energy Experimental determination of force constants Sum rules Vibrational absorption and circular dichroism Vibrational absorption spectra Experimental vibrational absorption intensities Quantum mechanical methods (A) Nuclear atomic displacement gradient method 03) Numerical electric field gradient method (C) Analytic wavefunction derivative method (D) Analytic energy derivative method (E) Analytic dipole moment derivative method (F) Analytic nuclear electric shielding tensor method Classical models (A) Atomic charge concepts (B) Bond charge and bond moment concepts Sum rules Vibrational circular dichroism spectra Experimental VCD intensities Quantum mechanical methods (A) Localized molecular orbital method 03) Analytic magnetic fieM perturbation method (C) Analytic nuclear electric shielding tensor method (D) Vibronic coupling method (E) Localized orbital-local origin method Classical models (A) Atomic charge concepts (B) Bond charge and bond moment concepts (C) Ab initio bond charge concepts Sum rules Vibrational Raman and Raman optical activity Vibrational Raman spectra Vibrational Raman scattering cross sections Quantum mechanical methods (A) Numerical atomic displacement gradient method 03) Numerical electric fieM gradient method (C)Combined atomic displacement-electric field gradient method (D) Analytic energy derivative method (E) Analytic electric dipole moment derivative method

117 118 124 124 126 130 131 137 143 143 143 146 147 148 149 152 152 153 155 155 157 159 161 162 163 165 167 169 171 173 173 173 173 176 177 183 183 183 188 188 189 189 190 190

xi 9.1.3 9.1.4 9.2 9.2.1 9.2.2 9.2.3 9.2.4 10 10.1 10.2 10.3 10.4 10.4.1 10.4.2 10.4.3 11 11.1 11.2 11.2.1 11.2.2 11.3 11.4 11.5 11.5.1 11.5.2 11.5.3 11.5.4 11.5.5 12 12.1 12.2

Classical models (A) Atom-dipole interaction concepts 03) Bond polarizability concepts Sum rules Vibrational Raman optical activity spectra Vibrational Raman optical activities Quantum mechanical methods (A) Nuclear displacement method 03) Orbital polarizability model Classical methods (A) Atom and bond polarizability concepts (B) Ab initio bond polarizability concepts Sum rules Applications of symmetry Introduction Symmetry operations and point groups Representations Vibrational properties Fundamental vibrations and symmetry species Symmetry of vibrational states Selection rules Principles of spectral measurements Introduction Polarization modulation using a photoelastic modulator Circular dichro ism (A) Sample configuration (B) Calibration configuration Linear dichroism (A) Sample configuration (B) Calibration configuration Polarization modulation using a quarter-wave retarder Double polarization modulation Dispersive spectrometers Dichroism measurements Raman optical activity measurements Time resolved infrared measurements (A) Measurements using boxcar integrator (B) Measurements using lock-in amplifier Dynamic infrared linear dichroism Two dimensional spectroscopy Amplitude division interferometry Introduction Principles of measurements

190 191 193 194 196 196 200 200 201 202 202 203 205 211 211 211 213 221 221 227 231 235 235 237 239 239 241 243 243 245 246 248 249 251 253 256 256 257 258 260 263 263 263

~176

Xll

12.2.1

Transmission configuration (A) Background interferogram (B) Transmission interferogram (C) Dichroisrn interferograms

12.2.2 12.2.3 12.2.4 12.2.5

12.2.6 13 13.1 13.2 13.3 13.3.1 13.3.2 13.4 13.4.1

Reflection-absorption measurements Real time sampling Raman scattering (A) Circular Intensity difference (B) Linear intensity difference Time resolved measurements (A) Rapid sweep method (B) Stroboscopic or synchronous perturbation method (C) Asynchronous perturbation method Further developments Polarization division interferometry Introduction PDIs based on Wollaston prism PDIs based on wire-grid beamsplitters

PDIs with roof-top mirrors PDIs with flat mirrors Principles of measurements

Transmission configuration (A) Residual interferogram (B) Background interferogram (C) Sample transmission interferogram

13.4.2 13.4.3 13.4.4 13.5 13.6 13.7 13.7.1

14 14.1 14.2 14.2.1 14.2.2

(D) Dichroism interferograms (E) Calibration interferograms Reflection configuration (A) Differential polarized reflectance interferograms 03) Polarized reflectance interferograms Raman scattering Emission spectra Additional modulations Time resolved measurements Double polarization modulation interferometers

Transmission configuration (A) Residual interferogram (B) Background interferogram (C) Sample transmission interferogram (D) Dichroism interferograms Applications and molecular structure Vibrational frequencies and absorption and Raman spectra Vibrational circular dichroism spectra

Ab initio theoretical predictions Empirical predictions

265 265 265 265 277 278 279 281 282 282 284 284 286 286 289 289 289 291 291 295 295 296 296 296 297 298 302 304 304 306 307 309 309 311 312 313 313 314 315 316 321 321 333 333 336

xiii 14.2.3 14.2.4 14.2.5 14.2.6 14.2.7 14.2.8 14.2.9 14.2.10 14.2.11 14.3 14.3.1 14.3.2 14.3.3 14.3.4 14.3.5 14.3.6 14.3.7

Amino acids, peptides and proteins Nucleic acids Carbohydrates Transition metal complexes Other Molecules Heine proteins Enantiomeric Excess Kinetics Chiral detection Vibrational Raman optical activity spectra Ab initio theoretical predictions Emp irical predictions Amino acids, peptides and proteins Carbohydrates Nucleic acids Other molecules Enantiome ric Excess

340 353 357 357 358 359 359 360 360 360 360 362 363 370 373 373 374

Appendices Summation convention for vector products Harmonic oscillator integrals with dimensionless p and q operators Some relations among dimensionless p and q operators Polarized light Summation convention for tensor products Vibrational Raman optical activities Classical expressions for Aa~7 and G'ec~ Index

383 384 387 388 394 396 404 407


Chapter 1 INTRODUCTION Vibrational spectroscopy has contributed an enormous amount of information on molecular structures and chemical compositions; it also has served as an important analytical tool. In most of the early vibrational spectral studies, the emphasis has been on vibrational absorption and Raman scattering processes. Excellent books 1-3 are available on these topics and they continue to be invaluable sources on the subject. The presence and identification of characteristic vibrational frequencies associated with chemical groups has popularized the role of vibrational spectroscopy as an important laboratory tool. In another direction, vibrational spectral data have been used for deducing the force constants, and those parameters that reflect atomic and bond properties, although this outlook has changed significantly in the last decade owing in part to the development of quantum mechanical methods. Vibrational circular dichroism (VCD) and Raman optical activity (VROA) have emerged in the last two decades as two new areas of vibrational spectroscopy with emphasis on three dimensional structural characterization of chiral molecules. Two excellent books 4,5 are available on the subject providing different emphases and perspectives. VCD measures the difference in absorption of left versus right circularly polarized light by chiral molecules; VROA measures the difference in Raman scattering for right versus left circulary polarized incident light, again for chiral molecules. Since chirality plays an important role in biology, the implications for the utility of VCD and VROA are significant. The instrumental 6 and theoretical aspects of VCD and VROA have now been developed to the extent that the practice of these areas is becoming almost as routine as the conventional vibrational absorption and Raman spectroscopies. The recent developments, mentioned above, provided the primary impetus for this book. The underlining theories of vibrational absorption, Raman scattering, VCD and VROA are treated here on a common basis. Since VCD and VROA phenomena encompass those of vibrational absorption and Raman scattering, special emphasis has not been placed on the latter two, but they come out naturally during the course of the description of the first two. The same approach is maintained for the instrumentation involved. The outline of the book is as follows. Chapters 2-10 deal with theoretical details, with Chapter 2 providing the needed quantum mechanical machinery. Time independent and time dependent perturbation theories presented in this chapter form the basis for much of

2

Chapter 1

the development of optical activity and anharmonic vibrational properties in later chapters. Chapters 3 deals with vibrational equations and transition moments for diatomic molecules at both harmonic and anharmonic levels. Same aspects are considered for polyatomic molecules in Chapter 4, although the mathematical complexity involved here restricted the length of coverage. Vibrational analysis is presented in Chapter 5. Setting up of vibrational secular equation, various coordinates (normal, internal, and cartesian coordinates) involved, the role of nonequilibrium geometries and that of redundant coordinates are considered in detail in this chapter. Vibrational equations cast in terms of normal coordinates are converted to those in terms of local coordinates in Chapter 6, providing a brief introduction to the local modes. Chapter 7 presents the methods for vibrational frequencies and force constants. In particular, the molecular orbital methods for evaluating the energy derivatives and experimental means of deducing the vibrational properties are summarized. The nature of vibrational absorption and circular dichroism spectral features are presented in Chapter 8 while vibrational Raman and Raman optical activity are presented in Chapter 9. The experimental quantities involved and the quantum mechnical methods to determine them are emphasized in these chapters. The use of molecular symmetry, with group theoretical concepts, is presented in Chapter 10. The next three chapters provide signal analysis for dispersive and FTIR spectrometers. The emphasis in these chapters has been placed on using polarization modulation as a means of measuring the optical activity. Fourier transform IR spectrometers based on Michelson interferometer are discussed in Chapter 12 and those based on polarization division interferometer are discussed in Chapter 13. The final chapter includes a survey of applications with a majority of them pertaining to VCD and VROA. References 1 E . B . Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 2 G. Herzberg, Infrared and Raman Spectra, van Nostrand Reinhold Co., New York ( 1971). 3 S. Califano, Vibrational States, John Wiley & Sons, New York (1976). 4 L . D . Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge (1982). 5 M. Diem, Introduction to Modern Vibrational Spectroscopy, John Wiley &Sons, New York (1993). 6 Commercial VCD instruments are available from Bomem-BioTools Inc. Commercial VROA instruments are not yet available.

Chapter 2 QUANTUM T H E O R E T I C A L TECHNIQUES For a particle with mass rn and velocity Vx in the x-direction, the kinetic energy in classical mechanics is given by the relation T = (1/2) mv

2

X 9

In terms of the linear momentum, Px - mvx, this relation is T =

p2x/2m. In quantum mechanics1,2, the state of the particles is described by a function, referred to as the wavefunction ~t, and the energy E is determined by solving the Schrtidinger equation H ~ = E ~ where H is called the hamiltonian, or the energy operator (energy written in the operator form). The kinetic energy operator, -(h2/8rt2m) ~2/~x2, is obtained by replacing Px with (h/2r~i)O/Ox. The potential energy operator is simply the potential energy written as a function of the positions of the particles. The total energy of a system of particles is the sum of kinetic and potential energies. For a one-electron system, such as the hydrogen atom, the Schr/Sdinger equation can be solved. The acceptable solutions result with discrete energy levels and wavefunctions. So the Schr6dinger equation is written more transparently as H~n = EnXl/n, where En and ~n are respectively the eigenvalue (or energy) and eigenfunction (or wavefunction) of the system in state n. For multielectron systems the Schr6dinger equation cannot be solved analytically due to the complications originating from the interelectron interaction terms. Nevertheless, approaches are available to obtain approximate solutions and this will be addressed in Chapter 7. For the purposes of this chapter we assume that the eigenvalues and eigenfunctions of the system are known or can be determined, so that the necessary foundation for quantum mechanical applications to spectroscopy can be developed. As a useful example, to illustrate the eigenvalues and eigenfunctions of the Schrtidinger equation, we will consider the familiar case of a particle moving freely in a one-dimensional box of length L. The kinetic energy operator, as mentioned above is (-h2/8rc2m)O2/Ox2. For a freely moving particle the potential energy is zero within the box, but since the particle is confined to the box, the potential energy is infinitely high in the walls. That is, the potential energy V(x) = 0 for x greater than or equal to zero and for x less than or equal to L; V(x) = oo for x less than zero and greater than L. To satisfy this potential energy form we require, in solving the Schrtidinger equation, that the wavefunction goes to zero within the walls. Between the walls the wavefunction is continuous and single valued. The equation to be solved with these conditions is (-h2/8r~2m)(O2~t/Ox2) = E~.

4

Chapter 2

The functions ~n(X) "- A sin(nrtx~) and energies En = n 2 h 2 / 8 m L 2, with n representing any non-zero integer (1,2,3...etc) and A respresenting a constant, satisfy this equation. Thus we have n equations of the form H~n = En~n or -(h2/8~;2m)~)2~n/~)x2 = En~n, each describing the system in a different quantized state. The constant A can be determined by requiring that the wavefunctions be normalized. That is, by setting ~ ~n*(X) ~n(X) dx -- 1 one obtains A 2 -- 2/L; ~n*(X) represents the complex conjugate of ~n(X). The wavefunctions are orthogonal, that is ~ ~n*(X) ~n'(X) dx - 5nn', where Kronecker delta function 5nn" = 1 for n = n' and zero otherwise; and this property can be easily verified using ~n(X) - (2/L) 1/2 sin(nr~x~). In the bracket notation1, 2 the above mentioned orthonormal conditions are written as ~ ~n*(X)~n'(X) dx = .

(2.2.2)

Now consider the influence of an external perturbation that is weak and time independent. The total hamiltonian can be written as, H=Ho+HI+H2+...

,

(2.2.3)

where H 1 and H2 represent the first and second order perturbations added to the unperturbed hamiltonian H0. The resulting energies and wavefunctions for each state can be written similarly. That is, 0

f

H

E s - E s + E s + E s +... , f

H

9 s - ~ 0 + ~s + ~s+... ,

(2.2.4) (2.2.5)

where the superscripts ' a n d " on E and ~t represent the first and second order corrections. When the above definitions of H, ~s and Es are substituted into the Schr6dinger equation for the perturbed system, H~s = Es~s ,

(2.2.6)

8

Chapter 2

one finds terms of different orders. These include two zero order terms, H0~s, and E~ four first order terms H0~s ' H I ~ s, Es~ s and Es~s; six second order terms (vide infra); and so on. Equating the terms of like order in Eq. (2.2.6), one gets an equation for each order. The zero order equation is same as Eq. (2.2.1) given above; the first order equation is (H0 - E0)Xl/s = (Es - H1)~s .

(2.2.7)

We assume that the first order correction to the wavefunction ~s can be written as a linear combination of unperturbed wavefunctions, ~s = Z c n ~ n '

(2.2.8)

where c'n are the unknown coefficients and the summation is over all states of the system that satisfy Eq. (2.2.1). Upon substituting Eq. (2.2.8) into Eq. (2.2.7), left multiplying with @s (or its complex conjugate in the case of complex wavefunctions) and integrating, one finds that the orthogonality of zero order wavefunctions, namely (2.2.9) leads to the expression for first order correction to the energy of the unperturbed state s as, E'='s

(2.2.10)

Instead of left multiplying with ~s if we had used ~ o with k = n, but k r s, then the above procedure would lead to the result that c n= < ~nlH1 I xl~s> / ( E 0- E0)-

(2.2.11)

Then from Eqs. (2.2.8) and (2.2.5), wavefunction correct to first order is obtained as, VS=~s + E

[< ~(n)nI H11 ~)s > / (g~s - E(n)n)]~n 9

n:g:s

Similarly let us consider the second order equation,

(2.2.12)

Quantum Theoretical Techniques ( n 0 - E 0 ) ~ s ' = ( E s ' - n 2 ) ~ s + (E~-nl)k~f's ,

(2.2.13)

and express the second order correction to the wavefunction as a linear combination of unperturbed wavefunctions,

Igs'= E C"n~n"

(2.2.14)

Here c"n are a different set of coefficients than the c'n used in the expansion for first order correction. Substituting Eqs. (2.2.14) and (2.2.8) into Eq. (2.2.13), left multiplying with ~s (or its complex conjugate in the case of complex wavefunctions) and integrating, one finds that the second order correction to the energy is given as E"=s

+

E

n~:s

[< ~n'Hl 'V0 > < ~ s l H 1 'V0 > / (~ss- ~nn)] "

(2.2.15) pp

To second order, the total energy of the system is Es - E 0s + E's + E s, and Eqs. (2.2.1), (2.2.10) and (2.2.15) permit the evaluation of this energy. The expressions for the first and second order corrections to the energy and for the first order correction to the wavefunction are of vital importance as these quantities help understand how the system behaves under the influence of a weak external perturbation. One can proceed in the same manner to obtain higher order corrections to both energies and wavefunctions. 2.2.1 Static electric dipole polarizability To show the applications of time independent perturbation theory, let us evaluate the electric dipole moment in state s using the wavefunction corrected to first order. Considering only one component of the perturbing electric field, say Fx', and writing H1 as-t-tx' Fx', the x-component of the electric dipole moment of the system can be obtained using Eq. (2.2.12). i.e., < IXx > = < ~gsl ~ ~gs > =

0-~.~\' (la.x,snl-tx',as + ].tx,nsl.tx',sn) F x ' / ( E 0 - ~ n n ) '

(2.2.16)

n~s

where ~x,sn = < ~ s l ~ x [~n > and < ~x >0 = < ~ s I ~x I ~ s >; [note that x',

10

Chapter 2

representing a component of the field perturbing the system, is distinguished from x, the component of the electric dipole moment being evaluated]. The second order terms in Fx' are ignored in arriving at this equation. For real wavefunctions ~tx,ns - t.tx,sn. We can generalize the above equations by considering all components of electric field, so the above equation can be written using the Einstein summation convention (Appendix 1) as < Ixot>- < ~ >0 + o~aa'Fot' ,

(2.2.17)

where CZczcz' - - 2 Z

< ~s I btcz I ~n > < ~n I I.t~' I ~ts0>/(E~s- E~n) ,

(2.2.18)

ng:s

is the general expression for the static electric dipole polarizability of the system. A second application is to calculate the energy to second order in the presence of the electric field perturbation given as H1 = -l.totFcz. From Eqs. (2.2.4), (2.2.10) and (2.2.15), the energy can be written as Es - E ~S - < bt~ >o Fc, - (1/2) ~otocF2

(2.2.19)

where < l.tcx>0 and cxcxcxare defined above.

2.3 Time dependent perturbation theory The time dependent Schr6dinger equation is written in terms of the wavefunction that depends on time. This wavefunction ~ ~ can be written as the product of two terms as, ~n (t) - ~n e-27tiEn0t/h

(2.3.1)

where h is the Planck's constant and t the time; e -2niE0t/h describes the time dependence. The time dependent Schr/Sdinger equation becomes, H0~n(t) = En0~n(t) = i(h/2rt)(~)~n(t)/~)t) ,

(2.3.2)

where H0 is the unperturbed, time independent hamiltonian of the system. The inclusion of time dependent perturbation Hi(t) modifies the hamiltoinan to H(t). That is,

Quantum Theoretical Techniques H(t) - Ho + Hi(t).

11 (2.3.3)

Assuming that the new wavefunction for state k can be written in terms of the original wavefunctions ~n(t) using time dependent coefficients Cn(t), as

yk(t) = ~ Cn(t) ~n(t),

(2.3.4)

the Schr6dinger equation for the perturbed system becomes, [H0 + H I ( t ) ] [ Z

Cn(t)~n(t)] = i ( h / 2 r : ) ~ [ ~ Cn(t)~0(t)]/0t.

(2.3.5)

The summation index in Eqs. (2.3.4) and (2.3.5) runs over all states of the system that satisfy Eq. (2.3.2). Expanding both sides of this equation, making use of the unperturbed equation, Eq. (2.3.2), left multiplying with @k* (* represents complex conjugate), and using the orthonormal property of @n' one obtains the expression for the time dependent coefficients as, ~)Ck(t)/& = (2rffih)Z Cn(t) < V01Hl(t) I ~n > e-2ni(E0- l~)t/h

(2.3.6)

Now we integrate this equation from time zero to a certain time, noting that at time zero the system is in the initial state s, so Cs(0) = 1 and Cn(0) = 0 for n ~ s. Assuming that the coefficient Cs(t) deviates from 1 only slightly at time t, while other coefficients Cn(t) for n ~ s are very small, the expression for the time dependent coefficient Cn(t) associated with a state n is given as, Cn(t) - (2~;/ih) f < ~n I Hl(t)I ~ )s > 0 c~ dt,

(2.3.7)

where COns = 2rl; ( ~ - E0)/h. This is the most important equation for understanding the temporal behaviour of the system under a time dependent perturbation. 2.3.1 Electric dipole-electric dipole polarizability Let us consider the influence 1,2,8 of a dynamic field perturbation on the system, and assume that the perturbation is weak. The time dependent electric dipole moment is given as ~toc(t) = < ~s(t) I ].tc~I ~gs(t) >,

(2.3.8)

Chapter 2

12

with wavefunction ~s(t) given by the Eq. (2.3.4). In the case of electric field perturbation, the time dependent coefficient is evlauted with H 1 (t) = -g~,Fa,(t) [here again o(, representing the component of the field perturbing the system, is distinguished from ~, the component of the electric dipole moment being evaluated] as Cn(t) - -(2rc/ih) I.ta',ns ~ Fo((t) e k~

(2.3.9)

dt ,

where g~',ns = < @n I g~' I @s >" For the real electric field given as Fa'(t) = Fa' [e ic~ + e -i~

(2.3.10)

= 2 Fa, cos cot ,

where co is the frequency of the applied electric field, the time dependent coefficient becomes,

I'_ .-, iO~nst -iO~nst ] c n (t) - [4~ga,,nst~a,e (COnscos C0t - ic0 sin cot - C0nse )

x [1/h(COZns- co2)]

(2.3.11)

These coefficients, their complex conjugates c*(t) and Eq. (2.3.4) can now be used in evaluating ~s(t) and hence ~ta(t). The resulting expression for t.ta(t) is, ~ta(t ) - < Vs01gulv 0 > + ~[4rl;.o~,,nsgo~,snFc~, x (COns cos cot- ico sin cot- COns e-iC~ n~:s

[

+ ~ 4gg~,,snge~,nsF~, x (COns cos cot + im sin cot- runs

e+i~n~t~1

n~:s •

....

In deriving this equation, the hermiticity relation 1,2, < ~~

,

l ~ ~ >* =

was used. The imsinmt term drops out when the product

Quantum Theoretical Techniques

is

real,

so

it

13 is

equal

to

; the cosmt term drops out when the product < V~

0 > < ~g~

0 > is imaginary, so it is equal to

- < V~ 0 > < xg~ 0 >. The (COnse-+-i~ terms represent oscillations at runs frequency when the perturbation has frequency m. These oscillations can be eliminated by modifying 2 the electric field perturbation as,

Fo,~t,_ Fo,[eiOt+ e-iOtl[1_ e-t',]

(2.3.13)

For times t >>> "t, and (COns4-m)>> ll'r,, this modification results in an additional term, 2 - O)2) , 4rq't~x',nsFa'mns / h(mns

(2.3.14)

to be added to the right hand side of Eq. (2.3.11). Then the general expression for time dependent electric dipole moment becomes, l.ta (t) - [.ta (0) + 2otcx{x,Fa, c o s m t - 2a~a,F a, sinmt + . . . .

(2.3.15)

The frequency dependent polarizabilities cxaa, and cx~xcx, are given by the expressions

oo = ReI nsOns' 4 .o .,s ~ r

ocaa, - Im - ~ 4ztl.tcx,snt.tcx,,nsm / h m2s k n~s

(2.3.16)

1

(2.3.17)

where Re[...] indicates the real part of the terms in parenthesis and Im[...] indicates the imaginary part. The sinmt term can be modified to that of time derivative of electric field, so Eq. (2.3.15) can also be written as, p

ga(t) = ga(0) + o~aa,Fa,(t ) + ~aa,Foc,(t) / m + ... ,

(2.3.18)

Chapter 2

14

where dot on Fa,(t) indicates time derivative. For O~ns>> o~, Eq. (2.3.1 6) reduces to that of static polarizability (Eq. 2.2.18). 2.3.2 Electric dipole-magnetic dipole polarizability The expressions for the electric dipole moment in the presence of the dynamic electric field perturbation given above provide the necessary framework for considering other perturbations 8. To evaluate the electric dipole moment in the presence of a dynamic magnetic field Ba,(t), the perturbation would be changed to H1 (t) = - n ~ , B a , ( t ) , where m is the magnetic dipole moment; then in Eqs. (2.3.9)-(2.3.14) the symbols associated with lxa' should be replaced by ma, while the symbols associated with index ~ remain the same (since this index represents the electric dipole moment component being evaluated). Then, the electric dipole moment can be obtained from Eq. (2.3 15), by replacing Fa,, aa and o~a, with Ba,, Gaa' and G~a, respectively as, p

~

laa(t) - ~ta(0) + Gaa,Ba,(t) + Gac~,Ba,(t) / 03 + ... ,

(2.3.19)

where the electric dipole-magnetic dipole polarizabilities Gaa' and G~a' are given as,

ooo "e[s4"osr,monsOns'(Ons0 )1 Iml 4 osnmor,sO'(Ons

(2.3.20)

(2.3.21)

L nc:s

If both electric and magnetic fields are considered, then the sum of Eqs. (2.3.18) and (2.3.19) will be involved (noting that g~(0) will be counted only once). Similarly, to evaluate the magnetic dipole moment in the presence of dynamic electric field F~,(t), the symbols containing g~ should be replaced by those containing m~ (since the index a ' represents the electric field component while a represents the magnetic dipole moment component to be evaluated) in Eqs. (2.3.18). That is, p

~

ma(t) - ma(0) + Ga,aFa,(t) - Ga,aFa,(t) / co + . . . .

(2.3.22)

In this manner, one can obtain the general expressions for the dynamic electric dipole, magnetic dipole and electric quadrupole moments.

15


2.3.3 Expressionsfor general molecularproperties In the previous two sections we have evaluated the time dependent electric and magnetic dipole moments for the system in a given state. Under the influence of time varying electric and magnetic fields, the expressions obtained can be generalized for any molecular property. This is done by replacing ga or m~ with an appropriate symbol, say Pa representing that property. Then, p

~

Pa(t) = P~(0) + Sec~'F~'(t) + See,Fe,(t) / co + Tee,B~,(t) + T~,Bc((t)/co +... , p

~

(2.3.23)

where, f'~ 1 '

(2.3.24)

1 '

(2.3.25)

~ 1 '

(2.3.26)

T&a,-im[_Z4rcpo~,snmet,,nsm/h(o~2 s n,s L -m2) 1 "

(2.3.27)

See'-ReFZ4gPe,snge,,nsf, Ons/Ln~s

h(f'~

S~~

imI_Z4~;pcx,sngc~,,nsO~/h(co2 s n ~ s

Tcx~

Re[Z4~;Pcx,snmcx,,nsO3ns/Ln~:sh(C~

-~

Note that, P~,sn = < ~0 p~ gOn > with P~ in the integral representing the appropriate operator for the molecular property being evaluated. We will use these expressions in Chapter 7 to evaluate the time dependent fields present at the nuclei in a molecule. As an example, to evaluate the electric quadrupole moment 0ap(t) induced by the electric field Fa,(t), Eq. (2.3.23) gives 0ap (t) - 0ap (0) + A ~ , F ~ , ( t ) ,

(2.3.28)

where Aapa, is the electric dipole-electric quadrupole polarizability,

16

Chapter 2

Ac~pcz'- Re[Ln~#s4~0cxp,snl'tcx''nsO~ns / h(CO2s - (02)1 ' and 0cz~,sn = < 11/0 0czl3 11/0 >.

(2.3.29)

The above two equations can also be

written (by noting that the real values of the products 0 czp,snPa ,ns and l-to(,sn0o~13,nsare equal) as, 0cxp(t) = 0ctp(0) + Act'cxpFcx'(t),

(2.3.28a)

Aa,czp - R e [ E 4~lttec,,snOczp,nsOlns / h(012s - 012)1 9

(2.3.29a)

kn#s

2.3.4 Interaction with circularly polarized light Let us now consider the influence of electric and magnetic fields associated with circularly polarized light incident on the system. Taking the z-axis as the direction of propagation of light, the circularly polarized electric vector, F+(t)(where the superscript represents the handedness, + for left circular and - for right circular) can be written as a sum of the phase shifted electric vector components along the x'- and y'-axes. Thus, F+(t) - 2F0[ficosco(t- z / c) + r

co(t- z / c)],

(2.3.30)

where fiand r162represent unit vectors along x'- and y'-axes and c is the velocity of light. The corresponding components of magnetic field can be obtained from the Maxwelrs relation, (2.3.31)

_(VxF +) - B + .

Finding the x and y components from this vector product and using the relation B0 = Fo/c, B +-(t) is written as B+(t) - ~2coB0[ficosco(t - z / c) + r

co(t- z / c)].

From this expression B---(t) is found to be

(2.3.32)

17


(2.3.33)

B + ( t ) - -2Bo [_+fisin co(t - z / c ) - r162 co(t - z / c)].

It is of interest to note that (2.3.34)

B + (t) - ~ ( ~ / c ) F + ( t ) .

Since F+(t) and B+(t), propagating along the z-axis, have only x' and y' components, the perturbing hamiltonian can now be written as +

+

H~ (t) - -J.tx,F+, (t) - lay,Fy, (t) - mx,Bx, (t) - my,By,(t) , where F+,(t) = 2F0 cos m ( t - z/c),

Fy,(t) = + 2F0 sin m ( t -

(2.3.35) z/c), B +'t" xt )

=-T- 2Bo sin m ( t - z/c) and ~,(t) - 2B0 cos m ( t - z/c). Substituting u these x' and y' components into the above equation one obtains, H~(t) - F0{-~x,[e ic~ +e -ic~ _+i~ty,[e ic~ - e-iC~ ]} -B0{__+imx,[e ic~ - e-it~ + my,[e ic~ + e-i~ ,+

9

(2.3.36)

+

The coz/c terms associated with F~a,(t) and B~t,(t) have been omitted here for the sake of economy, and they will not affect the following equations. The time dependent coefficients for the perturbation given by the term [eimt + e -i~t] have been discussed earlier (Eq. (2.3.10)-(2.3.12)). For the case of a perturbation given by the term iFo~'ga'[e imt - e-lint], the time dependent electric dipole moment is given by the relation, l.ta(t) - l.ta(0) + ~ 4r~gtx,sngcz,,nsFcx,[C0ns sin cot + icocoscot]/h(CO2s-co 2) n~:s

+ E 4/I;got,nsgot,,snFtx,[COns sin cot- icocoscot]/h(CO2s- 0)2). n~:s

(2.3.37)

For the perturbation given by the term iBcz'mcx' [e ir176- e -i~ symbols involving g~, in Eq. (2.3.37) are replaced by the corresponding ones with

18

Chapter 2

ma, and Fc~' by Bcz'. Thus all the ingredients to evaluate the electric dipole moment with H l(t) given by Eq. (2.3.36) are now in place. The resulting electric dipole moment is given as, + t "+ + gfi(t) - l.tr (0) + o t ~ , F ~+, (t) + otot~,F~, (t) / 03 + Gcxcx,B~, (t) + p

-q-

(2.3.38)

G~a,B~, (t) / co. The tensor components otaa', ~c~', Gaa', and G~a, are defined by Eqs. 2.3.16, 2.3.17, 2.3.20 and 2.3.21. The ot~a, and Gaa, tensor elements are non-zero only in special situations, and for most common cases only the o~aa' and G~cz' tensors need to be considered. In that event, Eq. (2.3.34) can be used to rewrite the above equation as 9 l.ta (t) - ~a(O) + (~cxa' -Y-(1 / c)Gaa,)F+,(t).

(2.3.39)

This relation makes the nature of interaction of molecular systems, with circularly polarized electric vectors of incident light, quite transparent. Note that the systems that do not support G~cz' cannot discriminate between the left and right circular polarizations. For a general case, the difference in electric dipole moment induced by right and left circularly polarized incident light is proportional to 2 G~a'. As seen from Eq. (2.3.39), the induced electric dipole moment obtained with right circularly polarized light can be different from that with left circularly polarized light in optically active molecules. So the refractive index of the medium can be different for these two polarizations. Since a plane polarized light vector can be written as an appropriate combination of the left and right circular polarizations, and these polarizations experience different refractive indices in a chiral medium, it can be seen that the plane polarized electric wave can be rotated as it passes through the sample. This optical rotation 0 (in radians) is given as 0 - l.too32Ngll(n 2 + 2) / 3,

(2.3.40)

where N is the number density, g is the sample thickness, go is the permittivity, n is the refractive index of the medium and [3 is related to the mean of the G' tensor a s - c o - k 3 ' - 13. Eq. (2.3.40) is applicable when 13 is expressed in SI units. When [3 is expressed in CGS units, the corresponding expression for optical rotation is

19

Quantum Theoretical Techniques 0 - 4rco~2Ng~(n 2 + 2) / 3c 2 .

(2.3.41)

The experimental optical rotations are most commonly reported as specific rotation [ct] in units of deg/[dm (gm/cc)]. The corresponding theoretical quantity is [tx] = 3600t~Vm/2~M where M and Vm are respectively the molar mass and molar volume. Using these definitions, we obtain a convenient expression for the specific rotation as [ct] = 0.1343x 10-3 ~ ~2 (n 2 +2)/3M,

(2.3.42)

with [3 in units of (bohr) 4, M in gm/mol and ~ (wavenumber at which the optical rotation is measured) in cm-1. Sometimes molar rotation, defined as [M] - [a] M/100, is also reported in the literature. Using the quantum mechanical expression for optical rotation [Eqs. (2.3.40)-(2.3.42)], ab initio calculations have been reported 10 only recently. 2.4 Transition rates The evaluation of time dependent coefficients was discussed previously in the context of evaluating the time dependent moments of the system. When the incident light causes the system to transit from the initial low energy (ground) state to a higher energy (excited) state, this process results in transfer of energy from the incident photons to the system and the absorption of the incident light takes place. When the transition is from a higher energy to a lower energy state, the energy transfer is in the opposite direction and stimulated emission results. For these cases the evalution of the time dependent coefficients follows the procedure discussed below. We will only concentrate on the absorption phenomenon in this section. The perturbing hamiltonian (Eq. (2.3.36)) considered earlier from circularly polarized incident light will be used here, as this is needed for determining circular dichroism intensities and the same can be used for absorption intensities and to generalize for a situation when the incident light is linearly polarized. The influence of [e ir176+ e -i~ terms in the hamiltonian (Eq. (2..3.36)) on the time dependent coefficients (Eq. (2.3.7)) is that they will result in the integrals,

f(e ir176+_e-iC~176

_

e i(~ns+co)t - 1 + _ e i(~176

[

i(~ns + CO)

- 1

i(0~ns - co)

1

(2.4.1)

The first term on the right hand side of Eq. (2.4.1) is negligible for absorption process (since COns- co, denominator is large) and the second

Chapter 2

20

term is negligible for stimulated emission (since cons --CO). So, for the absorption process, time dependent coefficients resulting from the circularly polarized perturbation become,

(

c +n (t) - -2rl;V~,ns e i(r176176 - 1) / h(tOns - r

(2.4.2)

where, ns

(2.4.3)

-gx',nsF0 +lgy',nsF0 +imx',nsB0 - my',nsB0 9

The probability P~n+(t) of finding the system in state n (that is transition from ground state s to excited state n) at time t is given by the product +

+

Cn(t)(Cn(t))*, where * represents the complex conjugate. Thus,

P~ (t)

-

16tl;2V~,ns V ~ n s

sin2 (COns - r

] h 2 (cons - 0)) 2

/ 2.

(2.4.4)

The term V +1,ns(VLns) + * can be written as,

+

+

9 [

]

V~n s (V~n s) - -l.tx,,nsF 0 -T-ll.ty,,nsF0 + imx,,nsB0 - my,,nsB 0 • ,

*

*

*

]

-gx,,nsF0 + igy,,nsF 0 ~ imx,,nsB 0 - my,,nsB0 Expansion of this product gives sixteen terms.

9 (2.4.5)

Of these terms,

-T-igx,,nsgy,,n s cancels with +igy,,nsgx,,n s, gx,,nsmy,,ns cancels with

my,,nsgx,,n s, gy,,nsmx,,ns cancels with mx,,nsgy,,n s and T-imx,,nsmy,,ns cancels with +imy,,nsmx,,n s. The surviving terms can be rearranged to the form, V +1,ns(VT, + ns) 9 -

(gx',snl-tx',ns + I-ty',sngy',ns) F2

+ (mx,,snmx,,n s + my,,snmy,,ns)B 2 T- 2i(gx,,snmx,,n s + gy, snmy,,ns)FOB0 .

(2.4.6)


21

For an isotropic sample, one needs to average over all orientations, so this equation can be rewritten as, + + )* V~ns(V~n s - (2 / 3)[}.tcx,sn~tcx,nsF2 + mcx,snm~,ns B2 + 2 Im(g~,snmoc,n s)FOB 0]

.

(2.4.7)

The terms F~, B~ and FoB0 can be converted to the energy contained in the electromagnetic wave (remember to use Eqs. (2.3.30) and (2.3.33)). Now we have to consider the density of states in the energy range of the transition, and integrate over all such transitions. With these two considerations2,11, and noting that the transition rate is given as , W + = dP+/dt, one obtains the transition rate for left and right circularly polarized incident excitation as, W + _ (8rt 3 / 3h2)[~c~,snl.ta,ns + ma,snmcx,ns + 2Im(l.ta,snmo~,ns)]P(v) =(8rt 3 / 3h2)[la, sn.l.tns+msn.mns+2 Im(~n.mns)]p(v) - B+p(v)

(2.4.8)

where p(v) is the energy density (energy per volume per Hz). This is the fundamental equation for deriving expressions for absorption and circular dichroism intensities. For lineary polarized incident excitation, Eq. (2.3.30) will be modified to contain only one component (say fi cos c0(t- Z/c)) and the rest of the procedure outlined above is repeated. Then, one would see 2,11 that the cross term, 2 Im(l.ta, snma,ns), will not appear in the transition rate. 2.5 S p e c t r a l i n t e n s i t i e s

Consider a sample, with concentration CO (moles per liter), absorbing a photon of energy hv from the incident light of intensity I(v). Then the change in intensity of the light when it exits the sample is proportional to the sample's thickness, or the optical path length g. Using the relation, I(v) = cp(v) in the expression for transition rate, and assuming that the energy difference between states s and n is sufficiently large to have all molecules initially in the groud state s, one obtains 2 (dI(v) / I(v)) + - (B + / c)hvCoNdg,

(2.5.1)

where N is Avogadro's number and B + is defined in Eq.(2.4.8). For the case of normal absorption, the superscript + can be left out, and only the

Chapter 2

22

/.tcz,sn/.I.cz,n s

term in Eq. (2.4.8) is needed. Then,

ln[I0(v) / I(v)] - (8~3vCoNg / 3hc)Dns,

(2.5.2)

where I0(v) is the intensity of light before passing through the sample and Dns = /.tot,sn/.tcx,ns =/~sno~tns,

(2.5.3)

is called the electric-dipole transition strength. Beer's law of absorption is given as I(v) = I0(v)e -r

,

(2.5.4)

with r representing the absorption coefficient. From Eqs. (2.5.2) and (2.5.4), the absorption coefficient becomes, r

(2.5.5)

= (8~3vN/3hc)Dns .

Taking the spectral band shapes into account and integrating this quantity over the band, it is necessary to evaluate the integral J(cz(v)/v)dv. However, since v does not vary very much over the width of the band, v can be replaced by v0 (the frequency at the center of the spectral band). Then the integrated absorption coefficient is given as A = fcz(v)dv - (8rc3v0N/3hc)Dns.

(2.5.6)

Sometimes Beer's law is written as I ( v ) - I0(v) 10 -~(v)C01 and r is related to ~ (v) as r = 2.303 ~ (v). To facilitate the conversion between the experimentally measurable quantity A and theoretically predictable parameters Dns, the following relation can be obtained from Eq. (2.5.6). A = (100.38/4) v0 Dns x 1040

,

(2.5.7)

where the integrated absorption coefficient A is in commonly expressed units of cm/mol, the band centerV0 is in units of cm -1 and dipole strength Dns is in units of esu 2 cm 2. To obtain the circular dichroism intensities, Eq. (2.5.1) can be used to obtain r and hence A-+-. Then the difference A A - A+ - A-, is obtained from Eqs. (2.4.8) - (2.5.1) as


-4(8~3voN/3hc)Rns

23

(2.5.8)

where the rotational strength (some times called rotatory strength) Rns is defined as Rns-Im[l-tsnmns] = Im[~sn.mns].

(2.5.9)

With Rns also expressed in units of esu 2 cm 2, and AA in (cm/mol), the simplified equation, analogous to Eq. (2.5.7), is obtained as AA = 100.38 V0 Rns x 1040 .

(2.5.10)

The ratio & ~ A , referred to as the dissymetry factor can be seen from Eqs. (2.5.7) and (2.5.10) to be AM/A = 4Rns/Dns 9

(2.5.11)

2.6 Transition polarizabilities The expression for time dependent electric dipole moment (see Eq. (2.3.18) obtained in the presence of dynamic electric field indicates that the dipole moment oscillates at the frequency of the electric field (associated with incident light on the sample). An oscillating electric dipole moment emits radiation, so the o~aa'Fc~'(t) term serves as the source for scattered light from a molecule. When the frequency of the scattered electromagnetic wave is the same as that of the incident electromagnetic wave, this light scattering phenomenon is referred to as Rayleigh scattering or elastic scattering. To consider the originl2 o f Raman scattering (or inelastic scattering) where the frequencies of the scattered light components are different from the frequency of the incident light, it is necessary to consider the time dependent electric dipole transition moments, < ~gf(t)llx=l~gs(t ) >. The recipe for evaluating this integral proceeds much the same way as that discussed in Section 2.3.1. To keep the identity of time dependent coefficients associated with the expansion of ~gs(t) and ~gf(t) separate, it would be useful to add an additional subscript to Cn(t). For example Cn(t) in Eq. (2.3.4) can be labelled as Cns(t) when k - s and as Cnf(t) when k = f. Then the transition moment integral will have four contributions. Of these, the contribution which has the products Cns(t)Cnf(t) is of higher order and will be ignored. Then,

24

Chapter 2

< vf(t)ll.ttxlVs(t) > - l.ttx,fselmfst + Z 4gl'ttx,fnbtot',nseiC~

[C~ cos cot - it.osin cotl / h(O~2s - co2)

ng:s

+ Z 4rl;btot,nsbtot',fneimfstFoc' [~ n;ef

cosmt + imsin rot] / h(ol2f - ol 2 ). (2.6.1)

This equation can be rewritten in terms of Fct'(t) and Fa,(t), using Eq. (2.3.10), as

(Ve(t)l t=lvs(t))-

+ [~~']fsFod(t) + [0ta~']fsF~'(t) / f'0}eic~ , (2.6.2)

where [otaa, ]fs - Z 2/1;l'tct,fnl'tod,nst'~ / h(~

- 012 )

ng:s

+ Z 2/1;~ot,nsl'tct',fnOlnf / h(t'O2f - t'O2)' n;ef [0~o~' lfs - Z 2rtil'tcx,fnt'ta',nsm / h(~

(2.6.3)

- c~

ng:s

-

2 m2 Z 2gig~,nsg~',fnm ! h(~nf ) 9 n~f

(2.6.4)

The first term btcx,fse imfst is the oscillation of transition moment between unperturbed states. The terms containing the product e~mfstF~'(t) represent oscillations at co + COfs;that is, the transition moment oscillates at the beat frequencies 03 _+ mrs. As mentioned earlier, an oscillating dipole serves as the source of scattered light, so the scattered light can have higher (co + COfs) or lower ( m - mrs) frequencies than the frequency (03) of the incident light. Thus Eqs. (2.6.1) and (2.6.2) expose the quantum t mechanical origin for Raman scattering. The coefficient of e lmf~ F~,(t) represents the transition polarizability, [ot~c(]fs. Note that when the initial


25

and final states are the same (s = f), this equation reduces to the dynamic polarizability [see Eq. (2.3.16)] discussed in Section 2.3. The coefficient of the product e ic~ Fa,(t) " represents the imaginary part of the transition polarizability (similar to ct'aa' in Eq. (2.3.18)). We will use Eq. (2.6.3) to describe vibrational Raman scattering in later Chapters. References 1 H. Eyring, J. Walter and G. E. Kimball, Quantum Chemistry, John Wiley & Sons, New York (1944). 2 P . W . Atkins, Molecular Quantum Mechanics, Oxford University Press, Oxford (1983). 3 L. D. Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, Cambridge (1982). 4 W.S. Struve, Fundamentals of Molecular Spectroscopy, Wiley, New York (1989). 5 H. Sambe, J. Chem. Phys. 58 (1973)4779. 6 P. Lazzeretti and R. Zanasi, Phys. Rev. A24 (1981) 1696. 7 P.W. Fowler and A. D. Buckingham, Chem. Phys. 98 (1985) 167. 8 A.D. Buckingham, Adv. Chem. Phys., 12 (1967) 107. 9 A.D. Buckingham, P. W. Fowler and P. A. Galwas, Chem. Phys. 112 (1987) 1. 10 P. L. Polavarapu, Mol. Phys. 91 (1997) 551. 11 I. N. Levine, Molecular Spectroscopy, John Wiley & Sons, New York (1975). 12 G. Placzek, The Rayleigh and Raman Scattering, UCRL Translation No. 526(L), translated by Ann Werbin (1959).


27

Chapter 3 DIATOMIC MOLECULES 3.1

Vibrational

equation

Consider a diatomic molecule with its constituent atoms designated as A and B. The nuclear kinetic energy operator TN is obtained from the operator equivalent (see Chapter 1) of linear momentum, (h/2rri) (O/OXA + 0/OYA + t)/t)ZA) = (h/2ni)VA, as TN = - ( h 2 / 8 E 2 ) ( V 2 / m A + V~3/mB) 9

(3.1.1)

Here mA and mB represent respectively the masses of A and B. Using the relative coordinates, R = R A - RB, the relation for center of mass coordinates Rcom = (RAmA + RBmB)/M and the relations among partial derivatives, (~)/~)XA)= (~)x]c)XA)(()/t)x) + (~)Xcom/C)XA)(C)/OXcom),the kinetic energy operator can be transformed (remembering that terms such as O2/Ox2 are written as (t)/t)XA)(~)/t)/XA)) to the form, TN = -(h2/8rl;2g)V 2 - (h2/8g2M)V2om 9

(3.1.2)

Note that M = mA + mB, the reduced mass g = mAmB/(mA + mB) ' V 2R -

-

~2/~R2 and V2com - O2/OR2om. The term (-h2/8r~2M)V2om represents the

kinetic energy operator for translational motion of the molecule and can be ignored as it does not couple to, or influence, the vibrational motion that is of interest here. With electronic, vibrational and rotational states designated respectively by subscripts n, a9 and J the Schrtidinger equation is given as H/l/nagj (r,R) = Ena~Jlgn~J (r,R) ,

(3.1.3)

where II/n~ J (r,R) is the total wavefunction that is dependent on electron

coordinates r and relative nuclear coordinates, R. The total hamiltonian is H = TN + VN + He l, with TN defined by Eq. (3.1.2) and Hel and VN defined below" Hel - _(h 2 ] 8~;2m)~ V 2 _ i

i

ZAe2/riA-ZZBe2/riB i

+ZZe2/rij i j,i

28

Chapter 3

(3.1.4) VN = ZAZBe2/RAB 9

(3.1.4a)

Note that ZAe and ZBe respectively represent the charges of nuclei A and B, V 2 is given for electron i as (c)2/~)x~ + ~)2/~)y2 + ~)2/~)z2), riA is the distance from electron i to nucleus A, rii is the distance between electrons i and j, and RAB is the distance between fiuclei A and B. The nuclei and electrons do not move independently, but the relatively large mass difference between them permits certain approximations to be introduced. For any slight perturbation in the nuclear positions, the electrons reorient their positions instantaneously. So it would appear that the total wavefunction can be written as the product ~lnoJ (r,R) = Vn (r,R) ~l/oj (R) . For the sake of simplicity the parentheses showing dependence on r and R will be dropped, so ~l/n (r,R) ~/1JJ (R) - Xi/nXi/~j;

(3.1.5a)

/~/9j describes the motion of slowly moving nuclei in electronic state n, and ~tn describes the motion of fast moving electrons with parametric dependence on nuclear positions. With this assumption, the Schr6dinger equation leads us to the evaluation of a complicated term, TN ~nV~J- This in turn leads us to (-h2/8~2kt) ~l/l)j (02~n//)R 2) and (-2h2/8~2~t)(/)Vn//)R) (t)~j/~)R). Both of these contain (t)~n/t)R) which describes the way the electronic wavefunction changes with nuclear displacement. When these terms are considered small and ignored, the Schr6dinger equation can be rewritten as (-h2/8/1;2~)Xl/n(C)2xl/agj/t)R2) + VnXl/nXl/ajj+ ~ J H e l ~ n - E Xl/n~l/~j . (3.1.6) We further assume that the electronic part of the Schr6dinger equation He 1~n = EnXl/n ,

(3.1.7)

can be solved, where En is the electronic energy of nth state. Then the nuclear part of the Schr6dinger equation is obtained as [(-h2/8~21.t)V2+ V ( R ) ] v v j = EvJXl/vj ,

(3.1.8)

29

Diatomic Molecules

where V(R) = V N + En. In other words, the electronic equation is solved first for a given nuclear geometry and with the effective potential V(R) at that geometry the nuclear equation is to be solved. The above mentioned assumptions leading to the separation of electronic and nuclear motions are referred to as the Born-Oppenheimer approximation. The nuclear Schr6dinger equation describes the rotational and vibrational motions of the molecule as indicated by the subscripts J and respectively. These two motions again can be separated under certain approximations. The nuclear wavefunction 1-3 can be written as ll/agj(R ) II/agYj/R where ~ v describes the vibrational wavefunction; Y j a r e the Legendre polynomials describing the rotational motion. Since our interest here is to consider the vibrational motion only, we can ignore the rotational motion (or assume that rotational quantum number J - 0) and obtain the vibrational equation as

32 ~ ~ +

8n2g{E - V(R)}}Vv - 0 h2

~

(3 1 9) 9

.

.

The expansion of V(R), around the equilibrium geometry Re, in Taylor series gives V(R) = V0 +

(0V) 1 (O2V'}( R _ Re)2 + ... , ~ (R-Re) + ~\~R2 ]

(3.1.10)

where the first term, V0 is a constant representing the effective potential energy at R = Re; in the second term (OV/OR) represents the negative of force at R = Re and is zero for equilibrium geometry. Thus, ignoring the cubic and higher order terms in Eq. (3.1.10), the potential energy is simplified to

(o v)

V(R) = ~ \~)R2/(R - Re) 2 = ~ K(AR) 2 .

(3.1.11)

The potential energy represented by Eq (3.1.11) is called the harmonic oscillator potential. The frequency of oscillation Ve of a particle, with reduced mass g under a harmonic potential, can be shown to be related to K and t-t via Ve = (1/2rg)(K/~) 1/2.

(3.1.12)

Chapter 3

30

The magnitude of K reflects the nature of potential. For large K the potential varies rather steeply around Re and for small K the corresponding variation is slow. 3.2

Energy levels of harmonic oscillator The vibrational equation (Eq. (3.1.9)) can now be written in the mass weighted coordinate Q, which will be referred to later as the normal coordinate whose importance will become obvious when we consider the polyatomic molecules. Introducing the relations, Q - [I, 1/2 ( R - Re) = [tl/2 (AR), and the relations among partial derivates, (~)/OQ) = (OR/~Q) (0/0R) g-1/E(O/t)R), into Eq. (3.1.9), and using Eqs. (3.1.11)-(3.1.12), one obtains f , ~)2 [ 8rt 2E~0 OQ2 + h2

(3.2.1)

where, O~2

4g 2 4g 2 (4rl;2V2)- (_~)2 -h2 (~)-h2 .

(3.2.2)

The superscript on Eu0 and ~ 0 indicates that these quantities are for the harmonic oscillator and represent zero order quantities. Introducing the dimensionless normal coordinate, q = o~l/2Q, and the relation for the partial derivative, (~)/Oq) = (~Q/~)q)(3/bQ) = ot-1/2 3/~Q, the vibrational equation simplifies to 2 0 ~)q2

2E

+

- q

~

= 0

.

(3.2.3)

hv e

If the energy is expressed in cm -1 units, then the above equation becomes ~ ~)q2

+

Ve

- q

gt ~ = 0 ,

(3.2.4)

where, E 0 = EO/hc and Ve = Ve/C. The solution to Eq. (3.2.4) provides the energy,

Diatomic Molecules

--0

E~

31

(3.2.5)

- Ve(l) + I~) ,

and the wavefunction ~0, ~0

=

NvHaj(q) e-q 2/2 9 I

I

(3.2.6)

I

I

t

4000 -

I

I

I

I

S" 3 0 0 0 ~..2.. E -

I

.)

9i

:

O

I

I

l

l

i

m

e

- ~ . . . . . ~t o

m2000-

:

"

..A.-r

,

r

~

;

1000-

....

o . . .0. . .....

,

,'~" @ " 0 - . . . .

These integrals can be evaluated by using the relations in Appendices 2 and 3. The integral < ~0 1H(00) I vO > gives the contribution of Eq. (3.2.5); < V~ I H~~ I V O > - 0; < V~ I H(2~ I vO > gives the contribution of Eq. (3.3.7). In addition,

< v01i iS1, H(00)] I V0> = i< V01S1H(00)I ~ 0 > - i < V0 1H(00)S1 I V 0 > =0 ,

(3.3.26)

which can be seen from the fact that the operator Slnb,, )rn or -_rnHb~)S1 is of fifth order (collectively in p and q) and the odd order operators step-up or stepdown the wavefunction by an odd integer. For example, the effect of qpq3 on V O is to generate wavefunctions vO+5, ~1/o+3 and vO+I which are orthogonal to ~0. The remaining two terms in Eq. (3.3.25) give,

_

~2 (30l)2 + 30~ +11), 8Ve (3.3.27)

which is same as the second order energy correction from ~0) (see Eq. (3.3.8)). The evaluation of integrals in Eq. (3.3.27) are simplified by the relations,

40

Chapter 3

i[S1,H~0)] (3.3.27a) and

44)

i[SI'H~~ - - i / ~ 2 ) ( p3q3 - q P3 3 + qPq - q Pq 2 ~Ve)

.

(3.3.27b)

The integrals involving these quantities are given in Appendices 2 and 3. The energy from Eq. (3.3.25) is

--(1) Ev -- Ve(1)+l)+0+3V4{(1)+ 1)2+ 1 } + 0 - V2 (30~)2+30~)+11) 8~ e

(3.3.28)

which is equivalent to the one obtained from second order perturbation theory using the non-transformed hamiltonian (see Eq.3.3.9). The effect of second contact transformation can be investigated by comparing Eqs. (3.3.15) and Eqs. (3.3.20). The second contact transformation changes ~(1)to w.(2)_ ~(1) + i [S2, ~(0)]. Thus the energy t-12 1-12

~)(2) resulting from the second contact transformed hamiltonian is obtained to second order by evaluating < lit~ I i [$2, H(0~ I/I/0 > and adding it to Eq. (3.3.28). The operator $2 is given as 8

$2

=

( 9~)e3 )2 (1/32){ 15 (p3q + qp3)+ 9 (q3p + pq3)} . _ ( V4 )(1/32){ 6 (p3q + qp3)+ 10 (q3p + pq3)} .

Ve Using this expression the commutator i [$2, ~(0)] is found to be 13

(3.3.29)

Diatomic Molecules

v~

41

{_ 27(p2q2 + q2p2)_ 18 + 60p4_ 36q4 _

64v e m

45qp2q+27pq2p}+_ v4 {1 8(p2q+q 2 2p2)+12+ 24p4 - 4 0 q 4 - 18qp2q + 30pq2p I

"

(3.3.30)

Substituting the appropriate relations listed in Appendices 2 and 3, one finds that < v ~ ~ is:, n~o~ ~ v ~ > - o

.

(3.3.31)

Thus using second contact transformed hamiltonian, the energy of diatomic molecule obtained to second order remains the same as that obtained from the first contact transformation to the same order; i.e., second order, where E~) is given by Eq. (3.3.28) E•) The= tzvthirdto and higher order transformations can be similarly written down, but due to the complexity involved they will not be discussed here. 3.4

Energy levels of Morse oscillator

In the case of harmonic and anharmonic oscillators that we have discussed earlier, the potential energy was expanded in Taylor series for small displacements q. However, no analytic form was associated with the potential. Several different analytic forms have been proposed for the potential of a diatomic molecule. The Morse oscillator 9 potential function has been the most widely used one due to its simplicity. This function is given, in cm -1 units, as Vm =

[)e [1 -exp [- ~t (R- Re)]] 2 ,

(3.4.1)

where ~)e is the dissociation energy and a is constant (in cm-1 units). Using the dimensionless coordinate q, this equation becomes

~rm =

[)e [ 1 - e x p [-aqq]] 2 ,

where dimensionless aq is given as

(3.4.2)

Chapter 3

42

aq =

h )1/2 ~t 4r~Vel.t

.

(3.4.3)

If we use this potential in place of the harmonic oscillator potential, the vibrational equation can be solved 9 to yield the energy, Ea~,m = Ve (~1 + 1/2) - VeXe (a) + 1/2)2 ,

(3.4.4)

where,

VeXe = (h~t2/8r~2cg) .

(3.4.4a)

Substituting Eq. (3.4.3) in Eq. (3.4.4a), one finds

VeXe = (Ve/2) a 2 , and

(3.4.4b)

Xe =

(3.4.4c)

1

2/2 q

.

It might seem disturbing to see drastically different forms for the anharmonicity constant, when compared to that obtained earlier. However, the anharmonicity constants given by Eqs. (3.3.10a) and (3.4.4b) can be shown to be related as described below. At the minimum of the Morse potential curve, the first derivative with respect to q is zero. Using this condition, differentiation of Eq. (3.4.2) gives

~r2,m _ 1/2 ( ~ V m )q=0 - aq2 I~)e'

~r3,m-

3

1/6 (~~33m)q=0 = - a q [ ) e -

(3.4.5)

-aqV2,m ,

= (7/12) a~ E)e = (7/12) a 2q V2,m 9

(3.4.6)

(3.4.7)

In Eqs. (3.4.5)-(3.4.7), subscript m has been placed to remind us that these equations are for the Morse potential. Substituting these derivatives in place

Diatomic Molecules

43

of V3 and V 4 in Eq. (3.3.10a), and remembering that Ve = 2 V2, Eq. (3.3.10a) becomes

2m2 aqV2,m [7~a2~ 2VeXe - ( - ~ ) ~ - t 8 ) qv2,m - a q Ve/2 '

(3.4.8)

which is same as Eq. (3.4.4b). Similarly substitution of Eqs. (3.4.6) and (3.4.7) in Eq. (3.3.10b), for obtaining the constant A of anharmonic oscillator, reveals that A = 0 for the Morse oscillator. The main difference between the perturbation and Morse oscillator approaches is that the higher order potential terms V3, V4, ... etc. are treated as independent quantities in the former, but they are dependent quantities [see Eqs. (3.4.5)-(3.4.7)] in the latter.

3.5

Electric dipole transition moments of harmonic oscillator In the approximation of separating the electronic, vibrational and rotational motions (see Section 3.1) one does not have to be concerned about the electronic and rotational wavefunctions, to the extent of considering the influence of anharmonicity on vibrational transition moments. Furthermore, the vibrational states being considered here belong to the same electronic state (unless specifically mentioned otherwise). Once the hamiltonian has been defined to the order of accuracy desired, the vibrational wavefunction can be determined and transition moments evaluated. In the zero order approximation, the vibrational equation is that of the harmonic oscillator and wavefunctions are defined by the Hermite polynomials (Eq. (3.2.6)). The electric dipole transition moments for transition from vibrational state ~) to ~' can be given as - < v ~ i.ivo

(3.5.1)

>

The electric dipole moment operator ~ can be expanded at the reference geometry in Taylor series using dimensionless normal coordinates as ,

1

=

bt0 +

q + 2 x,~)q2)"~ + 6 \~)q3)'

=

go + g lq + g2q 2 + ~3q 3 + ... ,

+ "'" (3.5.2)

where Ix0 is the dipole moment at the reference geometry. In the electrical harmonic approximation this series is truncated to the first power in q, i.e.

Chapter 3

44

~

-

II,0 + ( ~ q ) q

9

(3.5.3)

The use of Eq. (3.5.3) with harmonic oscillator wavefunctions is usually referred to as the double harmonic approximation. Substitution of Eq. (3.5.3) into Eq. (3.5.1) gives (see Appendices 2 and 3),

- ~l'0~agag" (~q) { (1)§ 1/2 1/2 ~',a~-I }, -I2 ~',~+1 + (2) (3.5.4) where 5 1)l)" = 1 for v = ~)" and zero otherwise.

The first term in Eq.

(3.5.4) survives only for ~ ' = ~; the second and third terms survive only for v ' - v + 1 and t) - 1 respectively. Thus, in the approximation of using harmonic oscillator wavefunctions with the harmonic electric dipole moment operator, the electric dipole transition moments for v --->v + 1 transitions (called fundamentals) can be non-zero, but those for a9 --> v + 2, v + 3, etc transitions (called overtones) are zero. To account for the occurence of overtone transitions, and to accurately describe the transition moments in general, it is necessary to use the anharmonic wavefunctions as well as the anharmonic electric dipole moment operator. There are two approaches one can take to evaluate the transition moments in the double-anharmonic approximation. One approach involves evaluating the anharmonic wavefunctions by using perturbation theory and using the resulting wavefunctions to evaluate the transition moment integral with the expansion of g containing higher order terms (l~2q2, 11,3q3 etc). The second approach involves the contact transformation of the electric dipole moment operator and using the resulting operator with harmonic wavefunctions to evaluate the transition moment integrals. The perturbation method is more tedious than the contact transformation method. Both approaches are described below.

3.6 Electric dipole transition m o m e n t s of a n h a r m o n i c oscillator The wavefunction corrected (see Section 2.2) to first order is

Diatomic Molecules

45

where ~g~ represents the anharmonic wavefunction resulting from the first order correction to zero order wavefunction ~0. It becomes transparent from Eq. (3.6.1) that the perturbation causes the harmonic wavefunction of a given state v to mix with the corresponding harmonic wavefunctions of higher energy states. Thus the anharmonic wavefunction for a given state can be viewed as a mixture of appropriately scaled harmonic wavefunctions of several states, which can be demonstrated nicely for a two level system. If we consider a hypothetical case where only a9 = 0 and 1) = 3 states exist, the transition between these two levels are not allowed in the doubleharmonic approximation. One should consider electrical anharmonicity that includes, at least the ],l,3q3 term, to obtain non-vanishing transition moment. The mechanical anharmonicity resulting from the perturbation H 1 = VC3q3 causes a9 = 0 and a9 = 3 states of this two level system to mix. This can be seen from Eq. (3.6.1), as follows" a 0 ~0 = ~ 0 -

V3 ( 6 ) 1'2 0 3 Ve 11/3

0 0 ~o - b ~3 '

(3.6.2)

m

a 11/3 -

0 1[/3 +

V3

3 Ve

(6"~1'2 0 0 0 ',8J ~g0 - ~3 + b ~g0 9

(3.6.3)

a has a small contribution from lg 03 and ~ga3 has a small Note that ~g0 0 contribution from ~0" The transition moment will have only one

contribution, that from J.L3q3 (ignoring higher order terms). ].t30 =

0 0 b2 0 q3 0 + . . . .

(3.6.7)

Substituting Eqs. (3.6.5) and (3.6.6) into Eq. (3.6.7), the resulting integrals can be evaluated using the relations in Appendix 2. Thus, a

a

< V 1 I I,ta31VO> ---

3 V3

3 V3

Ve a a =l.tl < Xt/1 I la'lq I qt0 > ~

~ 1+

= 0 ,

(3.6.8)

Ve

(3.6.9)

Ve

~ m

a

a

(3.6.10)

< q/1 I 1"1'2q2 I ~ 0 > = -

~ a a < ~1/1 I ~t3q31~0 > - 3p,3 {1

Ve 347

(~__$3)2 } Ve

(3.6.11)

Diatomic Molecules

10~ Ill0 = ~ +

1

3q + 2~r~ Ve

47

{1 03~1, _ 53_~ (~) 2 jgq3

i9q2

12 3q + 3-44873311} dq3

(3.6.12)

Note that this equation is obtained by considering only V3q 3 terms as the perturbation and the resulting first order wavefunction. One can proceed similarly to higher order perturbations for obtaining the anharmonic wavefunctions. Expressions for transition moments using up to third order perturbation have been reportedl~ 11. The perturbation method for evaluating the transition moments described in the previous paragraphs is laborious due to the need for keeping track of the off-diagonal elements in the anharmonic wavefunction. Contact transformation method described earlier, in the context of anharmonic oscillator energy levels, provides a convenient alternative 7,8,12. This method is described below. The vibrational equation involving the anharmonic terms in the hamiltonian can be written as

~(0)11/~ = (H =(0) -a a 0 + n~0) + H(0) + ...)V; = E~V~,

(3.6.13)

Ea3 is the anharmonic energy for state a3. Using the contact where -a transformation ~(n) = Tn~(0)Tn-1 (Eq. (3.3.22)), and noting that ~(n) is diagonal in the basis of harmonic oscillator wavefunctions, i.e.,

~(n)ll/O = E' -aa311t~ 0,

(3.6.14)

one finds that the vibrational equation modifies to H[(O)(Tn-llltO) = "-a(Tn 1 0 E a J /l/aj) ,

(3.6.15)

and,

1 o

/lt~ = Tn-/lq3. Using this relation, the transition moment integral becomes

(3.6.16)

Chapter 3

48

],Laj'v = - < IltvO I Tnll,(0)Tn-1 I lit0 > = ,

(3.6.17)

where, 11,(n)

_

Tn~L(0)Tn- 1

.

(3.6.18)

The operator ~t(n) represents the n times contact transformed electric dipole moment operator. The attractive feature of Eq. (3.6.17) is that the electric dipole transition moment integral is evaluated with the harmonic oscillator wavefunctions and there is no need to worry about the anharmonic wavefunction (Eq. (3.6.1)). The electric dipole moment operator however must be contact transformed, which is relatively easy. m

Following the contact transformation method for H, one can obtain the

operator ~(n) by simply substituting !~ in place of H. The electric dipole moment operator contact transformed twice becomes, ~(2)

=

1,1,(2)+ I.t]2) + 1,1,(2 2) + ... ,

=

~(o~ = ~o + (O0~q)q ,

(3.6.19)

where

r it?) p,~O) I.I,(2)

=

=

=

= .?

(3.6.19b)

+

(/)2~'~ q2

1/2 \~)q2 )'

(3.6.19a)

'

(1) 2 + i[S2, I,t(oO)] ,

(3.6.19c) (3.6.19d) (3.6.19e)

and g?) =

1/6 (~3~'~ q 3 . ~q31

(3.6.19f)

Diatomic Molecules

49

It is to be noted that the sum go + (~q)q is taken as the zero order term for the electric dipole moment operator. Also there are no restrictions on ~t~1) and I.t2-(2), unlike those on ~1) and ri~(2) 2 It is straightforward to evaluate the commutators involved in the above equations using the definitions for S 1 and $2 [Eqs. (3.3.24), (3.3.29)]. These are found to be, i[Sl, l.t(00)] = -

V3 ( ~ q ) ( 2 p 2 + q 2 )

'

(3.6.20)

Ve

i[S1, I1~0)]

= - -_V3 (~2~'~ (p2q + qp2 + q3) Ve ~q21 '

1/2[$1, [$1, It(~

= (~-~3)2 ( ~ q ) (p2q+ qp2_ q 3 ) ,

(3 6.21) " (3.6.22)

Ve

i[S2, g(o~

{A (3 p2q + 3 qp2) + B (2 q3) } ,

= (~q)

(3.6.23)

where, A = ~-~

_6 Ve

9 (~__33)2 B

=

32

Ve

(~___4_4),

(3.6.23a)

Ve 10

(~r4) Ve

(36.23b) 9

Combining Eqs. (3.6.19)-(3.6.23), the twice transformed electric dipole moment operator becomes, g(2) =

(32~'~q 2 (331-t'~q3 gO + ~ q q + 1/2 \Oq2/ + 1/6 \Oq3~' !

Ve

50

Chapter 3

VC3(bztt'~

qp2 +q3) _ (_V---~3) - ( ~ q2)

_-\~qg),(p2q+

Ve

(p2q + qp2 _q3)

Ve

+ A (~q)(3 p2q+3 qp2) +B (~q)(2 q3) ,

(3.6.24)

Using this operator the transition moments for the fundamental and first two overtones are found to be as follows.

X

[{ (~---~3)2+9 (~--33)2-3(~---4-4)} (0~J~q) Ve

- 5

0 ( 2 ) Ivv> 0 = _

(~q) ;l [_~

Xeag-

111/2

1 (~21,t~ [l)i xe~)-2] 1/2 (1-~11) 2

0q2 )'

"

2

(3.7.12)

This is a simple relationship that can be used to predict transition moments of higher transitions from those of lower transitions.

3.8 Magnetic dipole transition moments of harmonic oscillator Unlike the electric dipole transition moments the magnitudes of magnetic dipole transition moments are considerably smaller and are usually ignored. However, the magnetic dipole transition moments are essential to observe and explain the circular dichroism in vibrational transitions of chiral polyatomic molecules. Therefore we will introduce the treatment 16 for the magnetic dipole transition moment of a single oscillator here. The magnetic dipole moment ma, for a collection of point charges is given (see Appendix 1) by the expression, mot=

~1 ZA

e or

e~A RAI3l~A7 ,

(3.8.1)

where RA and R A are respectively the positional and velocity vectors of a particle A with charge e~A (e being the unit of electron charge); ~ a~7 is the alternating tensor with values of 1,-1 or 0. There are some important properties characteristic to magnetic dipole moments. First, the magnetic dipole moment is origin dependent, as can be seen from Eq. (3.8.1). If the chosen molecular origin is displaced by a constant a, then the positional vector changes from RA to R'A = RA -a, and the new magnetic dipole moment becomes

Chapter 3

54

rn~ = rn~ A ~a- Z

~ al3T e~A I~AT = ma - ~a ~ r

(01a~,/bt) .

(3.8.2)

Second, Eq. (3.8.1) suggests that the magnetic dipole moments are generated by the nuclear velocities and hence are functions of nuclear velocities. Since the time variation of the electric dipole moment can be written as

0).t~/~t = Z (0I.t~/ORAS)(3RAS/0t)=Z A

(0ILtT/0RAS)I~AS,

(3.8.3)

A

it becomes apparent 17 that Eqs. (3.8.2) and (3.8.3) lead to a

0rn~la I~A8 = - ~ e r (O~t~'ORAa). Thus the magnetic dipole moment vector can be expanded as

(3.8.4)

m - m0 +

(3.8.5)

51~A~

I~A~ + . . .

,

where m0, the value of magnetic dipole moment at R A - 0, is zero for a free molecule in non-degenerate electronic state. The higher order terms in the expansion of m~ can be written by noting the properties of mo~ with respect to time reversal; when t is replaced by -t, m~ changes sign (see Eq. 3.8.1) [but its velocity derivative (see Eq. 3.8.4) does not change sign]. Making use of this time asymmetry for m~ one finds that the expansion of rn~ will contain 17 only odd derivatives with respect to I~A6. For the sake of convenience we will now use the dimensionless normal coordinate q, and conjugate momentum p, and write the Taylor expansion of m in q and p as m

03m "~ + ~2\OpOq,/(p q + qp) + 1/4( ~)P0q 2/(q2p + pq2) (03m'~ p3 + + 1/6 \ Op3 J' ....

(3.8.6)

This expansion is somewhat different from the analogous expansion for electric dipole moment in Eq. (3.5.2) and ensures that the magnetic dipole moment for molecules in a non-degenerate electronic state is zero. Further details on this expansion can be found in Chapter 8. The derivatives

Diatomic Molecules

55

involved in this equation are implied to be taken at p = 0 and q = 0. This equation can be succintly written as, (3.8.7)

m - m01 p + m l l (pq+qp) + m21 (q2p+pq2) + m03 p3 + ... , where m01 "- ( i ) m / 3 P )

(3.8.7a)

,

m l 1 = (1/2)(~)2m/~)p~)q) = (1/2)(O2m/~)q/)p) ,

(3.8.7b)

m21 - (1/4)(~)3m/OpOq 2) - (1/4)(/)3m/~)q2Op) ,

(3.8.7c)

m03 - (1/6)(~)3m//)p3).

(3.8.7d)

In these equations, the two subscripts included for m (i.e., m01, etc.) represent the order of differentiation of m in q and p respectively. In the harmonic case, all terms except m01P are set to zero. Thus, the magnetic dipole transition moment in the double harmonic approximation is given as mag'aj = = i(/)m//)p)[((~+ 1)/2)1/2 5a9',~+1 -(x)/2)1/2 5~',a9-1].

(3.8.8) Special attention must be given to the electronic contributions to Om/Op (and higher derivatives) as will be discussed in Chapter 8.

3.9 M a g n e t i c oscillator

dipole

transition

moments

of

anharmonic

In considering the influence of anharmonicity on magnetic dipole transition moments, we defer the discussion on electronic wavefunctions to Chapter 8. For the present purpose, it is sufficient to write the magnetic 0

0

dipole transition moment for the ~ to ~' transition as , with m defined by Eq. (3.8.6). This statement will be elaborated further in Chapter 8. To obtain the transition moments of the anharmonic oscillator, we use the contact transformation method as described earlier for the electric dipole moment operator. For this purpose we introduce the definitions, m (0)

(o (o) r m 6) + m 1 + m 2 ) + ... ,

m(00) - mol P ,

(3.9.1) (3.9.1a)

56

Chapter 3

(Pq + qP) '

(3.9.1b)

m(2o) - m21 (pq2 + q2p) + mo3 p3 ,

(3.9.1c)

m] ~

-

ml 1

and obtain the following commutators:

qp) ,

i[Sl,m(o~

= (V3~e) Ino1 (Pq +

i[S 1,m]~

- (W3/Ve) m 11 (pq2 + q2p _ 4 p3) ,

(3.9.2) (3.9.3)

-1/2 [S1 ' [Sl,m(o0)]] = (V3/Ve)2 toOl {(pq2 2+ q2p) - 2p 3} ,

(3.9.4)

i[S2,m(o0)] - _ 2 A toOl p3 _ 3 B nb 1 (pq2 + q2p) .

(3.9.5)

Using these expressions, the twice contact transformed magnetic dipole moment operator is found to be m (2) = m01 p + ml 1 (pq + qp) + m21 (pq2 + q2p) + m03 p3 + (W3]Ve) m01 (pq + qp) + ('~/3]Ve) mll (pq2 + q2p_ 4 p3)

+ (~3],0e)2 m01 { pq2 + q2p - 2 p3}_ 2 Am01 p3 2 - 3 B m01 (pq2 + q2p),

(3.9.6)

where A and B are given by Eqs. (3.6.23a, 3.6.23b). The magnetic dipole transition moments < ~ -oI m (2) I ~o> are obtained by using the harmonic oscillator wavefunctions, ~o. These transition moments for the first three transitions are as follows: -V(3)2_ V3 2 0 IvO>= i(~)+ 1)1,2[c)m,~ ~+ 1)3/2 a c>b /

Similarly H 2 is the second order perturbation term given as H2-X

(4.1.4)

X X XVabcdqaqbqcqd ' a b>a c>b d>c m

where Vabcd is the fourth derivative of potential energy with respect to normal modes and are referred to as quartic force constants. Note that it is common practice to restrict the summation indices in Eqs. (4.1.3) and (4.1.4) so that off-diagonal elements of Vabc and Vabcd are used only once in the expression. This convention makes the definitions of Vab c i

and Vabcd different from the usual expansion terms of a Taylor series. /

Since the zero order hamiltonian H 0 is a sum of the operators of the individual modes, the zero order wavefunction will be a product of the harmonic oscillator functions and the energy a simple sum of harmonic oscillator energies. In other words, each normal mode represents a harmonic oscillator. Therefore, the zero order vibrational energy 1-3 is --0 --0 --0 EaJ - Eaj1 + Eu 2 + . . . - ZVa(Oa + 1~), a

(4.1.5)

where g0~a is the harmonic oscillator energy and ~a the quantum number of ath oscillator. Similarly the zero order wavefunction is V 0 - vO ~ 0 2 . . . - rlVOa, a where ~ 0

(4.1.6)

is the wavefunction of the harmonic oscillator a. When the

quantum numbers ~a are zero for all oscillators, the resulting vibrational energy, ~o _ ~ Va ! 2, represents the zero point energy of the molecule. a

Polyatomic Molecules

63

Let us consider CHFClBr, a simple chiral molecule to illustrate the vibrational states, and transitions between them, in a polyatomic molecule. The experimental vibrational frequencies4 are 3026, 13 11, 1205, 1078, 788, 664, 427, 315 and 226 cm-1. For molecules without symmetry the vibrational modes are numbered in the decreasing order of frequencies. - 1 - ( I 00000000)

7500

- - -_ -

-

f 7000

6

v

)r,

P

6500

a)

c

a)

-m

6000

2-

c

3-

0 .c

d 5500

.p

> 5000

4-

'-1 .-

5- 6-

---I - --I --:

7(000000000)

-- -

--

a-9-

-----

Fig. 4- 1. Vibrational states and energies of bromochlorofluoromethane with harmonic frequencies 3147, 1311, 1205, 1078,788,664,427, 315 and 226 cm-l. States 1 through 9 are respectively 1100000000>, 1010000000>,1001000000>,100010OOOO>,1000010000>, 10oooO1000>, 100OOOOlOO>, 1000000010> and 1000000001>. The states 10 thru 18 are 1200000000>,..., and 1000000002>; 19 thru 27 are 1300000000>,..., and 1000000003>, etc.

The experimentally observed frequencies are not necessarily the harmonic frequencies, due to the presence of anharmonic contributions. After correcting for anharmonicity, the harmonic frequency of the first mode was determined4 to be 3147 cm-1. The remaining observed frequenicies are assumed to be harmonic frequencies. The vibrational state is defined by the quantum numbers of all nine modes as I U ~ W U ~ U ~ U ~ U S U ~ U ~ U ~ > Fundamental transitions are those when only one vibrational mode is excited and its quantum number changes from 0 to 1. Thus transitions from 1000000000> to I1OOOOOOOO>, 1010000000>, 1001000000>, 1000100000>, 1000010000>, 1000001000>, 1000000100>, 1000000010> and 1000000001> states represent the nine fundamentals. These nine states are

Chapter 4

64

numbered 1 through 9 in Fig. 4-1, where the y-axis represents the vibrational energy [Eq. (4.1.5)] and the x-axis gives the fundamental harmonic frequencies. Transitions such as those from 1000000000> to 1000002000> (state no. 15 in Fig. 4-1) and to 1000003000> (state no. 24 in Fig. 4-1) represent respectively the first and second overtone of the sixth vibrational mode. The first overtones of the nine modes are numbered in decreasing energy order as 10 to 18, second overtone states as 19 to 27 etc. in Fig. 4-1. Transitions such as the one from 1000100000> (state no. 4) to 1001000000> (state no. 3) are referred to as difference transitions; here the transition energy is equal to the difference in harmonic frequencies of modes 3 and 4 (which can be read from the x-axis in Fig. 4-1). Similarly transitions such as the one from 1000000000> to 1001010000> are referred to as combination transitions; here the transition energy is equal to the sum of harmonic frequencies of the modes 3 and 5 (which again can be determined from x-axis values in Fig. 4-1).

4.2 Energy levels of anharmonic oscillators As in the case of a single oscillator (see Chapter 3) the correction --0

terms to zero order energy Ev can be evaluated using either perturbation theory 1 or contact transformation 5-8. Both approaches will be described here. Using perturbation theory, the first and second order energy __P

__PP

corrections, E~ and Ea) respectively, are obtained from Eqs (2.2.10) and (2.2.15), with perturbations H 1 and H 2 given respectively by Eqs. (4.1.3) __P

and (4.1.4). It can be easily seen that the first order correction Ev is zero. --

2

This is because H 1 contains terms of the type _q]' qaqb and qaqbqc which are of odd power in at least one normal mode and such operators do not connect the same vibrational states (see Appendices 2 and 3). Therefore it is necessary to use the expression for the second order energy correction --0

for incorporating the correction to E~. In the expression for the second order energy correction, the contribution arising from H 2 is straightforward to evaluate. The only -

-

4

-

-

2

2

terms which contribute here are of the type Vaaaaqa and Vaabbqaqb. From the relation given in Appendices 2 and 3, the contribution from H 2 is found to be

aaaa[/ a

65


+s s a b>a

+ 1~)(~)b + 1//2). (4.2.1)

The evaluation of contributions from H 1 is more involved. The terms which contribute here are the products of integrals of the type, (lifO, q3 igO)

x

a

(tltO q3a lltO,)

aqb

(/1tO' qa lily)

)

x

(/g 0 qaqb

,

'

o o o X (V o Iqbqc qaqb:VaJ) X (qt~ qaqb q/~'), (qtaJ' qaqb qtv) and (~O,}qaqbqc]~O) X {~Olqaqbqc~O,).

)

,

Contributions from these

terms are found to be as follows:

V2aa ,Z ~0_ ~0, 15V2aa (~)a +1~) 2 - 7V2a ~ 4v a m

(4.2.2)

16v a

m

VaaaVa~ _

,,,,,,

~o_~o,

3VaaaVabb (,Oa + l~)(,Ob + 1~),

(4.2.3)

2v a u

x VaabVbb~ > ~o_~o, 3Va --2 [ 8'V'~,-3'~2 1 2 ('Ob + ~) 16(4~ 2 -Va2) [4Va(4V b -V a )

- Vabb -

VaacVbbc (1)a + 1//2)(l)b + 1//2),

(4.2.9)

2Vc --2 Vabc

(~ ~ ]q aq bq c ]~0 )(~ 0 1q aq bq c ]~0' ) -

_

--2 Vabc(T 1 + T 2 + T 3 + T 4) , where,

(4.2.10)


TI=

67

l)aa)ba) c -(a) a + 1)(a)b + 1)(a9c + 1) 8(Va + Vb + Vc)

T2-

1)al)b(D c + 1 ) - (~)a + 1)(l) b + 1)a9c 8(Va + Vb - Vc)

T3=

T4 =

a)a (agb + 1)agc - (~)a + 1)Db (~c + 1) 8(Va - Vb + Vc) 1)a (1) b + 1)(~ c + 1 ) - (1) a + 1)1)b1) c 8(Va - Vb - Vc) ~PP

The total second order energy correction, E~ is obtained by summing Eqs (4.2.2)-(4.2.10). Let us now consider the contact transformation method 5-8 for evaluating the energy corrections. For this purpose the original nontransformed hamiltonian H given by Eq. (4.1.1) will be redesignated as ~(0); that is,

~(o) _ ~ _ ~o o) + ~ o ) + ~ o ) + . . .

_ no + n~ + H2 +...,

(4.2.11)

and use made of Eqs (3.3.12)-(3.3.20) given for the diatomic case to evaluate the total energy to second order. The operator S 1, for first contact transformation, is 5-7

s -E Z ESabc,

(4.2.12)

a b>a c>b

where the expressions for individual terms of the type Saaa, Saab and ~Sabc are given as follows: Saaa - -Vaaa Va ( } Pa3 + qaPaqa ) ,

Saab =

-Vaab Vb(4V 2 - V 2)

{(2V 2 - V~)qa2pb + VaVb(paqaqb

(4.2.13)

Chapter 4

68

-22

}

(4.2.14)

+qaPaPb) + 2VaPaPb ,

Sabc = -Dabc Vabc {~a(~2 _ ~2 _ ~c2)paqbqc -

- Vc )qaPbqc + Vc

-

V~ -2

- Va )qaqbPc (4.2.15)

-2VaVbVcPaPbPc} 9

The first of the above three terms is same as S 1 for a diatomic molecule. This term makes the off-diagonal integrals of q3a vanish in the contact transformed hamiltonian. Similarly the second term Saab makes the off-

2

diagonal integrals involving qaqb vanish. Note that the operator Saab can be obtained by interchanging labels a and b, and this would make the off-

2

diagonal integrals involving qaqb vanish. The third term Sabc serves the

2

same purpose for qabc. In the expression for Sabc, the definition for Dabc is as follows 1.

Dabc - (Va + Vb + Vc )(Va - Vb - Vc )(Va - Vb + Vc )(Va + Vb - Vc) (4.2.16) In the case of a single oscillator we noted that the non-zero contributions

i[

for the energy correction arise from the terms ~(0) and ~- S 1,

of the

contact transformed hamiltonian. The correction terms ( ~ 0 ~(20)~o) are given by Eq (4.2.1).

For the correction terms {q/OI 89

0) q/~

examining the form of S1 [Eq (4.2.12)] indicates that the non-zero contributions come from terms of the type [Saaa, ~(0)~qaaa]' [Saaa, ~(0)~qabb]' --(o)

[Sabb, Haaa], [Saab, nlbbb], [Sbbb, nlaab], [Saab, ~(0)..maab]' [Sabb, ~(0).qabb]' [Saab, -Hbcc], [Sbcc, ~(0) -llaab], [Saac, ~(0) "'bbc]' [Sbbc, ~(0) "laac ], [Sabb Hlacc], --

~(o)

[Sacc, Hlabb] and [Sabc, **labc ]" The individual contributions from these terms are found to be as follows"

(11/0 I~i [Saaa,~(0)nlaaa]lll/O) - - 15V2aa4~ a ( 1)a+ 1~) 2 - 7VLa 16v a

(4.2.17)


m

69

m

(~01.~.i [Saaa , ~(0) labb][~O)__ 3VaaaVbbb ~Va7 (,Oa + 1/2)(,Ob + 1/2)

= {11/0 I~"i[ ab b , laaa

Hlbbb]]~~

- 3VaabVbbb (1,)a + ]~)(1,)b + 1~)

(4.2.18)

(4.2.18a)

-(~,~ i[Sbb b , ~(0) laab]1~,o),

(~o ].~i [Saab,~(O)laab l I~tO) -

4Vb

--2 3VaabVb 16(4V 2 -V 2)

2Va2abVa

'2i[Sab " b ' ~(0) labb]}~0)_ +

4V2 _ Vb2 (~a + 1//2)2

3V2bbVa

(4V~-V~)(~a + ~/(~u + 1,4),

(4.2.19)

Vabb (l)b + ~ ) 2 4Va 4 v ~ - va-2 --2 2VabbVb (,Oa + 1~)(~)b + 1//2)

(4.2.20)

(4v~ -Va) -~

16(4~b~ -Va) -~ i ] ~o)_ (~o I.~[Saab ' ~(o) lbccj

-

VaabVbcc 2Vb

( l)a + 1//2)('0c + 1/2)

i ,cc , laab]1xl/O), - (,,,o i:[s

(4.2.21)

u

(v~

i [Saac ~(0)

' lbbc]lV~

VaacVbbc

2re

('Oa+ 1//2)('Ob+ 1//2)

= (vO I'~i[Sbb " c ' ~(0)]1~0), laac

(4.2.22)

70

Chapter 4 - - - - .

i[sabb"'lacc

(~/012"

-

m

VabbVacc 2V a (~)b + 1/2)(~)c + 1//2)

= (~1/01~i[Sac " c , ~(0) labb]lllt0),

(v~ 189 ~(~

(4.2.23)

ll ~ - - 2-~abc v2bc { Va (~2 _ ~2 _ ~2)(~) b + 1//2)(1)c + 1//2) + ~b(~) _ ~2 _ ~2)(1) a + 1//2)(1)c + 1//2) + Vc(V2 - V2 _ Va2)(1)a + 1//2)(1)b + 1//2) +VaVbVc !2 } ,

(4.2.24)

Comparing Eqs. (4.2.2)-(4.2.10) obtained from perturbation theory with Eqs. (4.2.17)-(4.2.24) from the contact transformation method, a one to one correspondance can be seen between these two sets. The total energy, to second order, is obtained by combining the zero order contribution with the correction terms as mentioned earlier. Noting the restriction on summation indices in Eq. (4.2.12) the total energy, to second order, obtained from the first contact transformation is written as 1

El) - g~)a'Ob...- Z Va(1)a + 1/2) + Z Z X a b ( 1 ) a

+ ,~'~)(1)b + 1/2) + i 2 ,

a b_>a

(4.2.25)

where, Xaa -

3Vaaaa 2

Xab - Vaabb-

15V2aa _ Z V2ab 8v-2a - 3v-2b ) 4~ a br b 4V-----a2-" V---~ ' 3VaaaVabb

3VaabVbbb

Va

V~

Z c~a,b

f

2VaV2ab

+

2VbV2bb

(4Va~-v~) (4v~,-Va~)

VaacVbb~ --Va~c~c(~c-~-~a ~)t -2 -2 Vc

(4.2.25a)

i-62;

'

(4.2.25b)

71

Polyatomic Molecules 7V2aa ) 16Va

+

m2 _ ZZ16(34V~bVb ) b#aa -

_Z Z Z

V2bcVaVbVc .

a b>a c>b

(4.2.25c)

4Dabc

The quantities Xaa and Xab are anharmonicity constants (in the same units as Va, Vb etc). There are two differences in notation for anharmonicity constants of polyatomic molecules compared to the one we used in diatomic molecules. First we designated the anharmonicity constant in diatomic molecules as VeXe (Xe being a number without units). Second, this anharmonicity contribution is added to the harmonic contribution with negative sign, so VeXe is positive (see Eq. 3.3.10). For polyatomic molecules Xaa and Xab are generally negative. Until now we have concentrated only on the first contact transformation to obtain the energy to second order. Since the second contact transformation also gives a second order energy term, i[S 2, H(O~ it is necessary to consider this term for completely determining the energy to second order. The operator for the second contact transformation 5-7 is $2 - Z Z Z Z

1[s~bC(PaPbPcqd + qdPaPbPc) + 1 2 d a b c d LSabc(qaqbqcPcl +pdqaqbqc)

(4.2.26)

with restrictions on the summation given as aaE~b (4-~a2 - V~)

~

+ ~

+

[VaVb(Paqa +qaPa)]

VbPaqb

79


+ a~ b~>ac>~bDVab:{(~~a limb ( ~ - ~2_ ~2)pbqc +Vc(Vc2 -Va2 - v~)qbPc] ~m' (~a2 ~ 2 - 2 + (~-~b)[Va - Vc)paqc +~c (~2 _ ~ 2 - v~)qaPc] (~m ][~a(~2 _ ~ 2 - ~2)paq b + ~Pc - V a - V c qaPb

9

(4.5.3)

The magnetic dipole transition moment m~,v for transition from a0 to ~)' is

).

then obtained a s . .(W~176176, , The second order contributions resulting from the first contact transformation and those from the second contact transformation can be found in Ref. 9. References

1 2 3 4 5 6

8 9

S. Califano, Vibrational States, John Wiley & Sons, New York (1976). E.B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). M. Diem, Introduction to Modern Vibrational Spectroscopy, John Wiley & Sons, New York (1993). M. Diem and D. F. Burow, J. Phys. Chem. 81 (1977) 476. G. Amat, H. H. Nielsen and G. Tarrago, Rotation-Vibration of Polyatomic Molecules, M. Decker, Inc., New York (1971). M . R . Aliev and J. K. G. Watson, Higher order effects in the vibration-rotation spectra of semirigid molecules in" Molecular spectroscopy, ed. K. N. Rao, Vol. III, Academic Press, New York (1985). D. Berckmans, H. P. Figeys, P. Geerlings, J. Mol. Struct. 148 (1986) 81. A. Willetts, N. C. Handy, W. H. Green, Jr. and D. Jayatilaka, J. Phys. Chem., 94 (1990) 5608; O. B ludsky, K. L. B ak, P. Jorgensen and V. Spirko, J. Chem. Phys. 103 (1995) 10110. K.L. Bak, O. Bludsky and P. Jorgensen, J. Chem. Phys. 103 (1996) 10548.

Chapter 4 This Page Intentionally Left Blank

81

Chapter 5 VIBRATIONAL ANALYSIS 5.1 Cartesian displacement coordinates The vibrational, rotational and translational motions of a molecule can be described by the displacements of the constituent atoms in cartesian coordinates. A linear molecule with N atoms has two rotational motions, one each around the two axes perpendicular to the linear molecular axis, and 3N-5 vibrations. A non-linear molecule has three rotations (one each around three mutually perpendicular principal axes of inertia), and 3N-6 vibrations. Unless specifically mentioned otherwise, the discussion here will pertain to non-linear molecules. The 3N cartesian atomic displacements contain the information on all of the motions, namely 3N-6 vibrational motions, 3 rotational motions and 3 translational motions. In setting up the equations for each of these types of motions, it is first useful to note that these three categories can be separated and treated independently under certain conditions, referred to as Eckart-Sayvetz conditions.I, 2 These conditions specify that during vibrational motions, the atomic displacements are such that the center of the mass of the molecule does not move and no net rotational angular momentum is generated. A mathematical description of the separation of these motions can be found in several standard references. 1-3 Let us consider a molecule whose geometry is specified by the atomic cartesian coordinates

x O, yO and z O. These coordinates are collectively designated (a = x, y or z) and their displacements from X ~

as

0 XAc x

as AXAcx = ( X A a -

xOa). The kinetic energy in atomic cartesian displacement coordinates is given as

2T = EmA(A,

, + Ay2A +

A

EmAA: 2A,

,

(5.1.1)

A

with mA representing the mass of atom A. Using the definition of mass weighted cartesian displacement coordinates as, qAcx - m~AXAcx,

(5.1.2)

the kinetic energy is conveniently expressed in the summation or matrix notation as

Chapter 5

82

2T - ~ 1 2 c t

or

2T - ~Cl 9

A

(5.1.3) ,.,.,

Here, /1 is a column vector of 3N quantities ClAct and q (transpose o f / l ) represents the corresponding row vector. The potential energy, whose functional form need not be specified at this point, can be expanded around a reference geometry as,

/ ~~~ ~+-~~ 2 ~x~x~ ~~~+' A B

A

(5.1.4)

with the derivatives taken at the working geometry. The second derivatives (02V/0XAcc0XBI3) are referred to as cartesian force constants. If the working geometry represents the minimum energy equilibrium geometry then the first order derivatives (0V/0XA(x) are zero, since they represent forces (multiplied by -1) on the atoms and these forces are zero for atoms at the minimum energy equilibrium geometry. V0 is the energy at the reference geometry and this is set arbitrarily to zero since it is a constant. For the sake of convenience let us introduce the abbreviations,

i)V

- -OAo~

c)XA{t )

I o~v

,

(5.1.5)

- fAtx,Bi3 ,

~ A(:zOx B13/

v ) = -OAt ~ , ~)2V

/

t)qActc)qBI3

(5.1.6)

(5.1.7)

- tA(x,Bi3 .

(5.1.8)

Then the instantaneous derivatives of the potential energy are given as -

~q

0

+ ~ tA~,B~ qB~" B

(5.1.9)

Vibrational Analysis

83

The equation of motion in mass weighted cartesian coordinates can be derived from the Newton's law, force equals mass times accelaration, as

d-t ~)dlAa +

~qAa

- 0.

(5.1.10)

For a molecule at its equilibrium geometry, the substitution of Eqs. (5.1.3) and (5.1.9) into the above equation gives, ddlAa + Z t A a , B ~ qB9 - 0. dt B

(5.1.11)

A solution to this differential equation is given as (5.1.12)

qAa - CAa cos(0)t + 5),

where co = 2xv, v is the frequency (in cyc/sec), 5 is an arbitrary phase factor and CAcx is the amplitude of atom A. Differentiating the above equation with respect to time, it can be seen that d/tAa - - ~ 2 q A a. dt

(5.1.13)

Then the equation of motion becomes (5.1.14)

,~(tAa,BI3 - C025Aa,Bfl)qBI3 - 0, B

where the Kronecker delta function (as defined in Chapter 2) ~ A a , B [ 3 - 1 for A - B and a = 13 and zero otherwise. Diagonalization of the t matrix, tlx,lx

tlx,ly

tlx,lz

tlx,2x

.........

tlx,Nz

tly,lx

tly,ly

tly,lz

tly,2x

.........

tly,Nz

t

0OQOOOOOO

tNz,lx

tNz,ly

tNz,lz

tNz,2x

......

tNz,Nz

(5.1.14a)

84

Chapter 5

yields 3N eigenvalues of o~ and 3N eigenvectors with each eigenvector containing 3N elements fA~,k. The eigenvector matrix represents the transformations between mass weighted cartesian displacements qAo~and a set of normal coordinates, Qk. These transformations are given in summation or matrix representation as, q A t x - Z gAtx,kQk

or q - g Q ,

(5.1.15)

k

Qk = Zgk,Acx qAtx or Q - gq.

(5.1.16)

A

Here q and Q are column vectors with elements qAa and Qk respectively. The eigenvectors are orthogonal and normalized, so that gk,Ao~ gAoc,k' - ~kk'

or gg- E .

(5.1.17)

A

Additional properties of the t matrix are given by the relations,4 ZgAtx,km~ - 0 ,

(5.1.18)

A 0 Z % q ] T XAI]gAT, k m A - 0 , A

(5.1 19)

and

~8 gAa'kgB~'kA B S a ~ - k

m~m~6a~ , / ~mA/A

(5.1.20)

- ~a~el3rx; m~m/~X~176 where I ~ is the moment of inertia tensor element. Utilizing Eq. (5.1.2), (5.1.15) and (5.1.16) the corresponding transformations between cartesian displacements and normal coordinates become,

Vibrational Analysis AXAa - Z S A a , k Q k

or

Qk -~Sk~AcxZ~Aa A

or Q - S - I x ,

85

X- SQ,

(5.1.21)

(5.1.22)

where S - M-1/2s

(5.1.23)

S -1 - gM 1~,

(5.1.24) 1/

and M1/2 is 3N x 3N diagonal matrix with m ~ as its elements. The column vector X represents the displacements AXAa. From Eqs. (5.1.21) and (5.1.22), the normal vibration can be visualized as a molecular vibration in which all atoms are displaced from their equilibrium positions. The relative magnitudes and signs of the atomic displacements CAa (Eq. 5.1.12) are reflected by the magnitudes and signs of the g matrix elements corresponding to that normal mode. Since all of the elements of gAa,k in a given kth column correspond to the same eigenvalue co2, all atomic displacements in that normal vibration take place at the same frequency. As the working molecular geometry was assumed (in Eq. 5.1.11) to represent the equilibrium atomic positions (i.e. no residual forces on atoms at that geometry), six of the 3N eigenvalues should be zero for a nonlinear molecule. This is because these six zero frequency modes represent the translational and rotational motions (three each) of the molecule and there is no potential energy associated with the translational and rotational motions of a free molecule (under no extemal perturbation). On the other hand, if the working geometry happens to be a non-equilibrium geometry (with residual forces present on atoms even in the absence of external perturbation) then one need not get six zero frequencies. The reason for this can be visualized from the dependency of cartesian force constants tAa, B[~ on the forces as follows. 5 Consider, for example, the term (~)2V/~q2a), which can be written as,

/ [

( O 2 V / ~ q 2 a ) - ~qAa "OqAa - ~qAa ~k ~Qk

~-q-A-Aa

Chapter 5

86

- j~. OQj

9

OQk

OqAa

OqAa

OqAa

0q2A~

(5.1.25) From the second term on the right hand side of Eq. (5.1.25), it is evident that the second order geometric terms (O2Qk/Oq2Aa) and the residual forces,-(3V/OQk), contribute to the cartesian force constants. Such effects can be eliminated by using the projection or correction methods (vide infra), both of which can be better understood with the introduction of internal coordinates, in the next section. It is useful to note that these effects have been addressed more commonly after the development of quantum mechanical methods for determining the cartesian force constants. In the older literature, where vibrational analyses were carried out by assigning force constants in internal coordinate space (vide infra), the issue of residual forces and their consequences did not arise. At this point it is useful to emphasize that if the objective is to determine the vibrational frequencies and normal modes (in terms of cartesian displacements of atoms) at the equilibrium geometry then the equations given above are all what is needed. The needed force constants (~2V/~XAoc~XBI3) in cartesian space are most conveniently obtained from quantum mechanical calculations. When the working geometry is away from equilibrium, or an analysis in terms of more physically transparent coordinates involving chemical bonds is required, further considerations as described in the following sections are needed. 5.2

Internal

coordinates

The use of internal coordinates in vibrational analyses is both important and complicated. It is important because the quadratic potential energy derivatives in internal coordinate space, referred to as the internal coordinate force constants, can be related directly to the properties of chemical structures as viewed in terms of the chemical bonds and angles. The magnitudes of diagonal force constants (that is of terms such as (~2V/~R2) where R:1 represents the jth internal coordinate), and signs and 9 magmtuc~es of off-diagonal force constants (that is of terms such as (~2V/~Rj~Rk) for two different internal coordinates j and k) provide much insight Into the nature of chemical bonding. A significant amount of discussion on this topic can be found in the literature. 6 The consideration of internal coordinates is also important to understand how the energy is

87


distributed among local portions of the molecule during a vibration, and how the charges or polarizabilites respond to the perturbation of localized structures. At the same time, the use of internal coordinates introduces some additional complexities, due to the nature of the coordinate transformations involved and also due to the interdependancy of some internal coordinates. Four different types of internal coordinates are commonly used. 1,2 These include: (a) stretching coordinate which represents change in length between the atoms A and B forming a chemical bond; (b) angle bending coordinate which represents the change in angle A - B - C , between the chemical bonds A-B and B-C; (c) torsional coordinate which represents the change in the dihedral angle formed by the planes defined by atoms ABC and BCD; (d) out-of-plane bending coordinate which represents the out-of-plane bending of the bond A-B from the plane containing all of the atoms A, B, C and D. Hi

I I

"~.L

Arl

~

,

~

O

..n2

Ar 3

--

I'

IA

&'2

'

Ar 1

Aot2

Ar2

Ar3 02 (a)

(b)

Fig. 5-1. Internal coordinate definitions for (a) non-planar H202 and (b) planar H2CO.

With the non-planar H202 molecule (see Fig. 5-1), as an example, one finds a total of six internal coordinates, with two O-H stretching coordinates (Arl, Ar2), one O-O stretching coordinate (kr3), two H-O-O bending coordinates (AOtl, Aot2) and one H-O-O-H torsional coordinates (A'0. The number of internal coordinates in this case happens to be the same as 3N-6, the number of molecular vibrations. However, this coincidence is not a general situation and one often finds that more than 3N-6 internal coordinates can be defined for most molecules. In such cases, some of the internal coordinates will exhibit interdependencies, and such a set of internal coordinates is referred to as a redundant set. For example, for the planar formaldehyde molecule (see Fig. 5-1) one can

88

Chapter 5

define two C-H stretching (Arl, Ar2), one C=O stretching (Ar3), one H-CH angle bending (Act), two H-C=O bending (All, A~2) and one O=C outof-plane bending (not shown) coordinate yielding a total of seven internal coordinates - one more than the number of vibrations. The changes in three in-plane angles within this molecule can be seen to be interdependent because, At~ can also be represented as -(A~I + A~2). So one may specify only the two H-C=O angle bending coordinates as a part of the 3N-6 internal coordinates. That will eliminate the redundancy problem, but the choice to eliminate Atx is not unique. One could have chosen Atx and A~I-A[~2 as the two in-plane angle bending coordinates, with the former representing the symmetric in-plane H-C=O bending and the latter representing the asymmetric in-plane H-C=O bending. Both approaches are equally correct and there are other equally valid choices. Such ambiguities are quite common for most molecules. In such cases, one can select 3N-6 internal coordinates avoiding those that will lead to redundant relations (such as eliminating Ao~ for H2CO). Alternately, one can make use of the molecular symmetry and formulate linear combinations of the internal coordinates to arrive at the 3N-6 independent symmetrized internal coordinates (see Chapter 10). The relations between internal and cartesian displacement coordinates can be derived in a straight forward manner. 1,2 Consider the bond between atoms A and B with bond length rAB. In terms of the cartesian coordinates of these two atoms, in an arbitrary set of cartesian coordinate system, the bond length is given as r2B _ (x O _xO)2 +(yO _yO)2 +(z O _ zO)2.

(5.2.1)

For an infinitesimal change in the bond length, one may write rABArAB - (xOtx - xOtx)(AXAet - AXBct).

(5.2.2)

Since ArAB represents the stretching internal coordinate, say designated as Ri, the gradients of this internal coordinate with respect to the cartesian displacement coordinates become, 0 _ X B~ 0 )/rAB = Bi,A~, (c)Ri/t)XAt~) = (X Ao~

(5.2.3)

0 0 (t)Ri/t)XBa) - - (XAa - XBa)/rAB - Bi,Ba.

(5.2.4)

Such relations can be formulated for all types of internal coordinates described above. Since excellent descriptions 1,2 to obtain these relations,


89

and standard programs incorporating these relations, are available they will not be rederived here. For the present purposes, we will assume that a set of 3N-6 independent internal coordinates have been selected. From the above equations one can see that the transformation from cartesian displacements to internal coordinates can be expressed as, R i - Z Bi,AsAXA5 A

or

R- BX.

(5.2.5)

Here R is a column vector of 3N-6 internal coordinates Ri, X is the column vector of 3N cartesian displacements AXAc~ and B is the transformation matrix with 3N-6 rows and 3N columns. These internal coordinates with linear relationship (Eq. 5.2.5) to the cartesian displacements are referred to as the rectilinear coordinates. The B matrix elements possess some interesting properties. The sum of B matrix elements, corresponding to x, y, or z component, in any given row add up to zero (see Eqs. (5.2.3) and (5.2.4) for the stretching coordinate considered). That is, Z Bi,A~5 - 0, for 5 - x, y or z. A

(5.2.6)

Similarly the sum over cross products of the positional vectors of atoms, 0 X A, with the B matrix elements for the corresponding atoms, in any given row, is zero. That is, 0

Z etxlS), XA~Bi,Av - 0. A

(5.2.7)

These two properties are equivalent 7 to Eckart-Sayvetz conditions since they ensure the absence of kinetic energy coupling between internal and external coordinates (vide infra). The rotational and translational coordinates, p~r and p~t (three each with tx = x, y and z), are referred to as the external coordinates p, and are defined as follows:

A

Chapter 5

90

-

I1 /1 /

mAAXAa.

m A

(5.2.9)

They can be expressed in terms of the cartesian displacement coordinates X, in analogy to Eq. (5.2.5) as pj

-

EI3j,AS~A8

or

p - 13X,

(5.2.10)

A

where p is a column vector of three rotational P~t and three translational ptcoordinates and 13 is the transformation matrix with six rows and 3N columns. It is convenient to include the internal and external coordinates together and write the transformation from cartesian displacement coordinates as X

.

(5.2.11)

A line between R and p or B and [3 indicates the stacking order of their elements. Since cartesian displacements are dependent on both R and p, the inverse transformation is given as, AXAa - E AAS,i Ri + E ~162 Pj i

(5.2.12a)

j

or

x-

/

mb,

To ensure that the coordinates R and p are independent of each other, it must

be

that 8 c)Ri/~)pj - E(~)Ri/~)XAs)(~)XA8/c)pj)- 0, A

c)pj/~)R i - E(~)pj/C)XA~5)(C)XA~5/~)Ri)- 0. That is, A

and

Vibrational Analysis Z Bi,A~St~A~5,j - 0

91

or

B a - 0,

(5.2.13a)

or

[~A- 0,

(5.2.13b)

A Z~j,ASAAs,i - 0 A

where a and A matrices are defined in Eq. (5.2.12). Furthermore, since each of the internal and external coordinate spaces are complete, - ~ii'

or

BA - E 3 N _ 6 ,

(5.2.14a)

Z~j,AS0~A~5,j, - 5jj,

or

~a-

(5.2.14b)

ZBi,ASAAs,i' A

E6,

A

where E is the unit matrix (diagonal elements equal one and off-diagonal elements zero) with its dimension indicated by the subscript. The A matrix is referred to as the inverse of the B matrix, and a matrix as the inverse of the 13 matrix. Note that the products al~ and AB are not unit matrices and obey the relation, AB + a ~ = E 3 N .

(5.2.15)

From Eqs. (5.2.12) - (5.2.15) the A matrices of isotopic molecules can be seen to be related as 8 A2 = A1 + @1~1A2 ,

(5.2.16)

where the subscripts 1 and 2 represent isotopically substituted molecules. These relations play a critical role in transforming the molecular properties such as dipole and polarizability gradients between internal and cartesian displacement coordinates. The a matrix can be constructed from the following relations. (5.2.17) AXAa = [1/~mA]l/2p t = (1/M)I/2p t

(5.2.17a)

A

Using the Eqs. (5.2.12) and (5.1.1) the kinetic energy can be written in matrix form as

Chapter 5

92

-

(/ MA l XMa

(5.2.18)

The matrix, G -1 = 7tMA,

(5.2.19)

is called the kinetic energy matrix in internal coordinates. Similarly the product @Ma represents the kinetic energy in external coordinates. The products 7tMa and @MA, representing the cross terms of kinetic energy between the internal and external coordinates, vanish as shown below. Let us consider the inverse kinetic energy matrix, which contains as the diagonal blocks G = BM -1 B,

(5.2.20)

inverse kinetic energy matrix in internal coordinates, and 13M-11~, the corresponding term in external coordinates; ~M-II3 and BM- 1~ are the off-diagonal blocks. The off-diagonal blocks can be written as, B M "1 ~ - s

(1 / m A)Bi,Aa 13j,Aa

(5.2.21)

A

Substituting the expressions given for the 13 matrix (see Eqs. (5.2.8) to (5.2.10)) into Eq. (5.2.21) one finds that these cross terms are zero when Eqs. (5.2.6) and (5.2.7) are satisfied. These relations were already mentioned above to be the characteristic of the B matrix. Therefore, the inverse kinetic energy matrix does not have cross terms between the internal and external coordinates. Then it must be that the kinetic energy matrix is also in block diagonal form, so AMa = ~MA = 0. Thus the kinetic energy given by Eq. (5.2.18) becomes,

(5.2.22)

Vibrational Analysis 2T - R G ' I R + b&Moq5 .

93

(5.2.23)

Until now there is no restriction placed on the molecular geometry, and the above equations apply whether or not the working geometry is an equilibrium geometry. In considering the potential energy, however, it is necessary to specify whether or not the working geometry is an equilibrium geometry. Let us consider the case of equilibrium geometry, where by definition there are no residual forces on atoms. Then, the potential energy can be written from Eqs. (5.1.4) and (5.2.12) as, 2 V - (Ri 9)

AfxA IXfxA

Offxa

(5.2.24)

Here fx is the cartesian force constant matrix with elements fAa,Bl3 (see Eq. (5.1.6)). The internal coordinate force constant matrix F is given as F = AfxA.

(5.2.25)

The products ~fxa, Afxot and ~fxA should be zero because they represent respectively the force constants associated with external coordinates and interaction force constants between internal and external coordinates. The potential energy of a free molecule does not change with molecular translations and rotations. That is, ~,fxOt- O,

(5.2.26)

A f x a - ~fxA = 0 .

(5.2.27)

One can substitute the analytic formulae (Eqs. 5.2.17) for ot into Eq. (5.2.26) and derive the translational and rotational invariance conditions for fx. There are situations where the quantum mechanical force constants are calculated at non-equilibrium geometries which result in non-zero forces at the atoms. Then the above equation may not hold and one has to give more careful attention to these terms (vide infra). In order to solve the vibrational secular equation in internal coordinates, let us consider Eqs. (5.2.23) and (5.2.24). The external coordinates can be dropped here after, since the kinetic energy matrix was shown to be block diagonal and the potential energy at the equilibrium geometry is independent of external coordinates. The transformation from normal coordinates to internal coordinates can be written using Eqs.

Chapter 5

94 (5.2.5), (5.1.21) and (5.1.23) as R = LQ,

(5.2.28)

where, L - BM -1/2 t .

(5.2.29)

Then the kinetic and potential energies can be cast in normal coordinates using Eqs. (5.2.23), (5.2.24)-(5.2.27) and (5.2.29) as 2T - ( ) L G ' I L Q ,

(5.2.30)

2V - 0 L F L Q .

(5.2.31)

Noting that kinetic and potential energies are diagonal in normal coordinate space, one obtains the relations [,G -1 L - E3N-6 or G = L [ , ,

(5.2.32)

[,FL = A

(5.2.33)

or F = ~-1 A L - 1 ,

where A is the diagonal matrix with eigenvalues representing the vibrational frequencies, Ok 2. From Eqs. (5.2.32) and (5.2.33) one finds that GFL = L A ,

(5.2.34)

which is called the secular equation in internal coordinates. The G matrix required in Eq. (5.2.34) can be obtained from Eq. (5.2.20). The construction of the G matrix is straightforward, once the B matrix has been calculated using the well known relations 1,2. If the F matrix was obtained from quantum mechanical calculations, it is likely that it was transformed from fx. In such cases, one should take note of the corrections needed if the geometry at which the calculations were done contained residual forces on atoms. The product G F is not a symmetric matrix and therefore a double diagonalization method 9 is commonly used. The symmetric G matrix will have 3N-6 non-zero eigenvalues (provided the chosen 3N-6 internal coordinates are independent), so one can find an orthonormal matrix V such that

Vibrational Analysis ~rGV = '1:,

95

(5.2.35)

where x is the diagonal matrix with 3N-6 eigenvalues of G. Then G can be written as G- WW,

(5.2.36)

where W-

VX 1/2 .

(5.2.37)

Then the secular equation becomes, (WWF) L - L A ,

(5.2.38)

or

~fF L = W-1L A .

(5.2.39)

This can be rewritten as

OYCFW)w - 1 L = W- 1 L A ,

(5.2.40)

and using the abbreviations W -1 L = D ,

(5.2.41)

~'FW = H,

(5.2.42)

one has the equation, H D- D A.

(5.2.43)

The diagonalization of the H matrix provides the needed eigenvalues. The required L matrix is obtained as L- WD,

(5.2.44)

where W was obtained from the eigenvalues and eigenvectors of the G matrix. An alternative to the secular equation (Eq. 5.2.34) is to set up the secular equation in terms of the inverse force constant matrix. From Eqs. (5.2.32) and (5.2.33) one can obtain,

Chapter 5

96

F-1G-1L = L A - 1 .

(5.2.45)

The inverse force constants are known as compliance constants. 1~

5.3 E n e r g y distribution The vibrational eigenvalues, intemal coordinate force constants and the L matrix elements can be seen from Eq. (5.2.33) to be normalized as

i

~ LikLjkFij / A k - 1. j

(5.3.1)

In this double summation, the portion defined as Dik, Dik - ~ L i k L j k F i j / A k , J

(5.3.2)

represents the fractional energy contribution from the ith internal coordinate to the kth normal vibration. This distribution of energy among 3N-6 internal coordinates is referred to as the potential energy distribution. From the kinetic energy expression in normal coordinate space (Eq. 5.2.30), one finds that

i

~ L i k L j k G i j 1 - 1. j

(5.3.3)

Since this expression is also normalized, here again, one can define the energy distribution as Dik - ~ LikLjkGij 1 . J

(5.3.4)

However, it is important to note that these two apparently different definitions can be seen to be equivalent to each other. 11 In view of Eq. (5.2.32) and (5.2.33), the energy distribution given by Eqs. (5.3.2) and (5.3.4) can as well be written as -1 Dik - LkiLik ,

(5.3.5)

-1 since L -1 L = E3N-6. Then the product LkiLik is same as the "kinetic energy distribution" of Eq. (5.3.4) or the "potential energy distribution" of


97

Eq. (5.3.2). The energy distribution is most commonly calculated from Eq. (5.3.2) because the need for additional computation of G -1 or L -1 is avoided.

5.4 Non-equilibrium geometries As mentioned earlier, the potential energy has to be handled carefully when the working geometry is not an equilibrium geometry because such geometries will impart residual forces on atoms which in turn contribute to the force constants through higher order terms (see Eq. (5.1.25)). As a result, the fx matrix need not satisfy the invariance to rotational motions which results in non-zero frequencies corresponding to the rotational motions. To see this further, let us assume that we have a set of non-zero cartesian forces on atoms, i.e. (~V/~XAa) are not zero. Transformation of these cartesian forces to translational and rotational coordinates using Eqs. (5.2.17) gives,

(ov,op )-

2(ov oxa /5 ,

(5.4.1)

A and ( c ) V / ~ p ~ ) - 2 %t[~,(c)V/c)XAa)XO~,/I~. A

(5.4.2)

The translational forces are zero as long as the sum of forces on atoms, along each car-tesian axis, add up to zero. For a free molecule this is true so the translational motions will still give zero frequencies. The presence of non-zero rotational forces will mean that rotational harmonic force constants are not zero, since rotational forces compensate these force constants at a higher level (vide infra). Two different approaches have been used in the literature to deal with this situation. One will be referred to as the correction method and the other as the projection method. In the correction method, 12 the expansion of intemal coordinates in terms of atomic cartesian displacement coordinates is considered up to the second order. That is, R i - Z B i , A c ~ ~ A a + 1 Z Z B iA(x,B~ ~ A a A X B A A B where,

(5.4.3)

Chapter 5

98

BiAa,BI3 -(a2Ri/aXAczaXBI3)

9

(5.4.4)

The potential energy expansion to second order in internal coordinates is given as,

V - V 0 + Z(~)V, ~)Ri)R i + (1)ZZ(c)2v/~)Ri~)Rj)RiR i i j

j 9

(5.4.5)

Setting the constant V0 arbitrarily as zero and substitution of Eq. (5.4.3) in the above equation gives, to second order in AXA~, V - Z AXAe Z (OV / 3Ri)Bi,Acz

A

i

+ (89 E aXA aXB E (aV / ORi)BiA,B A B

i

+(1)Z ZAXAaAXB[3Z(t )2V/c)Ric)Rj)Bi,AaBj,B 13 A B i,j

(5.4.6)

Comparison of this expression with the corresponding expression to second order in the cartesian displacement coordinates (Eq. 5.1.4) gives,

( ~)2V/~XAac)XBI] ) - Z(~)v/c)Ri) B iAa,B[$ i + Z(O2V/()Ri~)Rj)Bi,AaBj,B~ i,j

.

(5.4.7)

Note that this equation is similar to Eq. (5.1.25) given earlier in normal coordinates. Now defining the 3Nx3N matrices B i with elements BiAoc,BI3 for each internal coordinate i, the above relation in matrix notation becomes, fx + E ~ R B ' -

BFB,

(5.4.8)

where ~ R - (0V/0Ri). So the corrected internal coordinate force constant matrix is obtained as


( /

V-,g, fx+~r

B1 A ,

99

(5.4.9)

i

where A, the inverse of B, can be obtained from Eq. (5.2.20) as, A - M-1BG -1 .

(5.4.10)

The second order derivatives, (~)2Rj/c)XAa~)XB[~), although not known as commonly as the first order B matrix elements, can be evaluated 13 from the geometrical parameters. To consider the projection method, 14 let us go back to the transformation from internal and external coordinates to the cartesian displacement coordinates (Eq. 5.2.12). From this equation, it can be seen that the product A R gives the projection of the vibrational part of the cartesian diplacements and a 9 gives the projection of the external (rotational and translational) part of the cartesian displacements to the total atomic cartesian displacements X. The product AB, given as AB = E3N - ot~,

(5.4.11)

is called the projection operator (or projector). To project the external contributions out of the fx matrix a projection operator was defined 14 as P = (A'B)(AB)=fi(Bfi) - 1 B .

(5.4.12)

Then the product PfxP = BFB gives internal coordinate force constants matrix F with external contributions projected out. That is, F = ,g,PfxPA.

(5.4.13)

5.5 R e d u n d a n t i n t e r n a l c o o r d i n a t e s

As mentioned earlier, the number of internal coordinates that one can choose in a given molecule usually exceeds the number of vibrational degrees of freedom. Let the total number of internal coordinates chosen for a non-linear molecule be given as 3N-6+m. In such cases two different problems will be encountered. Due to the interdependency of these internal coordinates, one finds m redundant relations among these internal coordinates given as follows.

1O0

Chapter 5

2ajRj - 0 .

(5.5.1)

Substituting Eq. (5.2.5) into (5.5.1), one finds that 2aj2Bj,AaAXAa j A

- 0.

(5.5.2)

Rearranging the summation indices this equation can be written as 2 AXAtx2 ajBj,Aa - 0 . A j

(5.5.3)

This relation indicates that certain combinations of elements in a column of B matrix become zero. Since the inverse kinetic energy matrix is given

as Gij - Z B i , A a B j , A a / m A ,

(5.5.4)

A the sum of Gij weighted by the coefficients ai becomes,

i ~ a i G i j - ~A Bj,Aa( i~aiBi,Aa/ / m A - 0 ,

(5.5.5)

indicating that certain combinations of G matrix elements in m rows (or columns) will also be zero. In other words, the G matrix becomes singular in a redundant set of internal coordinates. As a result when the G matrix is diagonalized one would find 3N-6 non-zero eigenvalues and m zero eigenvalues. So eq. (5.2.35), written equivalently as GV = Vx becomes, G ( N i Z ) - (N!Z)(t~-~O) ,

(5.5.6)

where N is that portion of V representing the 3N-6 eigenvectors of the non-zero eigenvalues; Z represents the m eigenvectors of the zeroeigenvalues; and t is the diagonal matrix of 3N-6 non-zero eigenvalues. The m columns of Z give the coefficients aj defining the redundant relations. The 3N-6 columns of N give the coefficients of transformation between a redundant set of 3N-6+m coordinates Ri and 3N-6 independent


101

internal coordinates Sj as, Sj - ~ N i j R i

0 - Z ZijRi i

j - 1 , 2 . . . 3 N - 6,

J - (3N - 6 + 1)...(3N - 6 + m ) .

(5.5.7)

(5.5.8)

The summation index i runs from 1 to 3N-6 + m. The independent set of internal combination coordinates Si, can also be derived using the molecular symmetry (see Chapter 1(0) in which case the symbol N is usually replaced by U and referred to as the internal symmetry coordinates. The relations derived from diagonalization of the G matrix can be conveniently handled in automated computer calculations. 15 The transformations between internal combination coordinates and cartesian displacements coordinates become, S - NBX - BsX, X - AsS + a p .

(5.5.9) (5.5.10)

The G matrix in a non-redundant set of internal combination coordinates becomes, G s - BsM-l~s

,

(5.5.11)

where Bs = NB as defined in Eq. (5.5.9). The cartesian force constants can be transformed to non-redundant internal coordinates (assuming that we are working with equilibrium molecular geometry) as, F s - / i , sfxA s .

(5.5.12)

So the vibrational analysis procedure becomes identical to the one we discussed earlier except for the replacement of R, B, G and F by S, Bs, G s and Fs. The matrix As is obtained as As - M-l~sGs 1 .

(5.5.13)

The above procedure is adequate if the intention is to solve the secular equation in the presence of redundant internal coordinates. The internal combination coordinates as determined from the diagonalization of the G

Chapter 5

102

matrix (see Eq. (5.5.9)) will however have a complicated nature due to the presence of several internal coordinates in a given combination coordinate. So the physical meaning of force constants in the S coordinate set is not as transparent as that in the R coordinate set, and the issue of transferability of force constants among different molecules in the S coordinate set becomes complicated. The second inconvenience originating from redundant internal coordinates is the presence of non-zero linear potential energy terms 16 even at the equilibrium geometry. Recall that for the equilibrium geometry, the linear terms (aV/aRj) in the potential energy expansion are set to zero and this would not hold in the presence of redundant coordinates. This requires special considerations, especially in the quantum mechanical calculations of optimized geometries and force constants. To gain insight into the issues here, let us consider the case when there is only one redundant internal coordinate, that is the case when rn = 1. Instead of expanding the potential energy in terms of 3N-6 independent intemal coordinates, let us now expand it in terms of 3N-6+m coordinates (with m = 1): ~)V ~)2V RiRj V - Z. ORi Ri + 1Z.. aRiaRj 1

l,J

OV R m + l a2V 2 + aRm 2 aR___~mRm + I s

a2V RmR i+... aRm0Ri

,

(5.5.14)

1

where the indices i and j run over 3N-6 independent coordinates and m is the extra coordinate introduced leading to one redundancy. If we now write Rm as a Taylor series expansion in terms of independent coordinates Ri, then at minimum energy geometry it should follow that, aV aV aR m ) R i _ 0 . Z1. nqR-====4~, aR m ~)Ri

(5.5.15)

So the linear terms aV/aRi or aV/aRm are no longer zero. As a consequence the meaning of force constants also becomes a bit more complex, because the quadratic force constant Fi"J involving coordinates Ri and Rj, appearing as coefficient terms of RiRj now become, Fij _ aV a2Rm a2V t)2V aR m aR m aR m aRiaR j + aRiaRj ~ aR 2 aR i ~)Rj "

(5.5.16)

103


This expression indicates that the linear term OV/ORm also contributes to the internal coordinate force constant, making the visualization of force constants difficult. The choice of 3N-6 independent internal coordinates for a given molecule is not always uniquely defined. It may be convenient to define all possible internal coordinates for a given molecule, which then leads to several redundant relations and the above mentioned uncertainties in interpretation. It is useful to find a way of defining the force constants in terms of the complete internal coordinate basis set, and projecting out the contributions from redundancies. From the eigenvector matrix N associated with 3N-6 eigenvalues of the G (see Eq. (5.5.6)), one finds that NN

= E -

ZZ,

(5.5.17)

where E is the diagonal unit matrix with dimension 3N-6+m. product, P = N~I ,

The

(5.5.18)

is called the projection operator, projecting out the contribution of redundant coordinates. The force constants in 3N-6+m internal coordinates, but with projected out redundant contributions, is given as, 17 F = NFs~I.

(5.5.19)

An alternate approach involves the use of a generalized inverse method for inverting the singular matrix G in 3N-6+m basis. The generalized inverse 18 of G is given as,

G'I-(N'Z)~o

I0

'

(5.5.20)

where t -1 is the inverse of the diagonal matrix t containing the non-zero eigenvalues of G. The conversion of cartesian force constants to internal coordinate force constants in a 3N-6+m size basis can be achieved 19 from F = AfxA where A is now obtained from Eq. (5.4.10) with G -1 given by the Eq. (5.5.20). For interpretational purposes the F matrix is now in the original 3N-6+m basis. Quantum mechanical calculations of geometry optimizations 19 are often performed in the internal coordinates. In such calculations, the 3N cartesian forces are converted to the internal coordinate basis using the relation,

104

CR - CxM-1BG -1 ,

Chapter 5

(5.5.21)

where Ox and OR a r e the vectors of forces respectively in cartesian and intemal coordinate bases. During the geometry optimization cycles, the bond lengths and angles have to be corrected to minimize the forces. These corrections are determined from the inverse of OR - - FR, so one needs to determine the compliance matrix F -1. This is accomplished f r o m l 9 the relation F -1 - P ( P F p ) - I P ,

(5.5.22)

where P was defined above as N~I and the inverse (PFP) -1 is the generalized inverse, similar to the generalized inverse of G given in Eq. (5.5.20). References 1 E . B . Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 2 S. Califano, Vibrational States, John Wiley & Sons, New York (1976). 3 D. Papousek and M. R. Aliev, Molecular Vibrational-Rotational Spectra, Elsevier, New York (1982). 4 J.K.G. Watson, Mol. Phys. 5 (1968) 479. 5 W.T. King and A. J. Zelano, J. Chem. Phys. 47 (1967) 3197. 6 M. Diem, Introduction to Modem Vibrational Spectroscopy, John Wiley & Sons, New York (1993). 7 R. Malhiot and S. Ferigle, J. Chem. Phys. 22 (1954) 717. 8 B. Crawford, J. Chem. Phys. 20 (1952) 977. 9 T. Miyazawa, J. Chem. Phys. 29 (1958) 246. 10 L. H. Jones, C. Kennedy and S. Ekberg, J. Chem. Phys. 69 (1978) 833. 11 J.C. Whitmer, J. Mol. Spectrosc. 68, 326 (1977). 12 P. Pulay, Direct Use of the Gradient for Investigating Molecular Energy Surfaces, in: Modem Theoretical Chemistry, ed. H. F. Schaefer HI, Plenum, New York (1977). 13 I. Suzuki, Appl. Spec. Rev. 9 (1975) 249. 14 I.H. Williams, J. Mol. Struct. 94 (1983) 275. 15 M. Gussoni and G. Zerbi, Atti. Acc. Naz. Lincei. 40 (1966) 1032 (as quoted in ref. 2). 16 B. Crawford and J. Overend, J. Mol. Spectrosc. 12 (1964) 307; M. Gussoni and G. Zerbi, Chem. Phys. Lett. 2 (1968) 145; I. M. Mills, Chem. Phys. Lett. 3 (1969) 267; V. I. Pupyshev, S. V.


105

Krasnoshchiokov and YU. N. Panchenko, J. Mol. Struct. 131 (1985) 347; H. H. Eysel and B. M. Bussian, Appl. Spectrsoc. 35 (1981) 205; N. Neto, Chem. Phys. 87 (1984) 43. 17 K. Kuczera, J. Mol. Struct. 160 (1987) 159. 18 M. Gussoni, G. Dellepiane and S. Abbate, J. Mol. Spectrosc. 57 (1975) 323. 19 P. Pulay and G. Fogarasi, J. Chem. Phys. 96 (1992) 2856.


107

Chapter 6 LOCAL MODES

6.1 Hamiltonian in local coordinates To understand the local modes and to make a connection with the normal modes of polyatomic molecules we will start with the zero order hamiltonian H 0 that was considered for polyatomic molecules. As discussed earlier, with this zero order hamiltonian the vibrational problem can be treated as a sum of 3N-6 independent equations and the vibrational wavefunction becomes the product of 3N-6 harmonic oscillator wavefunctions. In terms of dimensionless normal coordinates qa and conjugate momenta Pa, (6.1.1) 2 where the summation index a runs over 3N-6 harmonic oscillators. Let us consider each of the two terms on the right hand side of Eq. (6.1.1) separately. The relations p = -i~)/~)q and q = (4n 2 v/h)l/2Q (see Chapter 3) can be used to derive the relation

Va 2 _ ,h a ~ 2 "-pa

~2

(6.1.2)

8/i;2c~a OQa2 '

where Qa is the normal coordinate for harmonic oscillator a. Now using the relation between the normal coordinates Qa and internal coordinates Ri, (see Chapter 5) given as Ri - ~LiaQa ,

(6.1.3)

the operator ~9/,gQabecomes

b

b

= ~_~"Lia . ~)Qa i 3Ri Substituting Eq. (6.1.4) into (6.1.2), one obtains

(6.1.4)

Chapter 6

108

~_~p2 _ "~1 ZL2a a 8~2c " a

t)R2

+

ZLiaLja "" a

~)Ri ~)Rj

" (6.1.5)

Using the relation of the L matrix (see Chapter 5) with the inverse kinetic energy matrix G, LL - G ,

(6.1.6)

Eq. (6.1.5) becomes

~._~_p2_ 8/1;2C __ ~i Gii /)R 2 + . . G IJ~)Ri c)Rj

(6.1.7)

9

Our interest here is to describe the vibrational motion in terms of the localized coordinates such as the internal coordinates Ri. To convert the internal coordinates Ri to dimensionless internal coordinates qi and conjugate momenta Pi we use the relations 1 analogous to those for normal coordinates given in Chapter 3, i.e.,

-

-

4/I;2Vi[.I,i

(6.1.8)

qi,

(6.1.9)

Here vi is the frequency ofthe local oscillator i, given as 2-2y/Fii) 1/2 with

\~ti

reduced mass ~i and force constant Fii (vide infra). Then Eq. (6.1.7) becomes,

~.~TPa2 -

8 9i ~

viktiGiip2+j~i(vivj)l~(kti~j)l~GijPiPJ

'

(6.1.10)

a

where Vi

= Vi/c.

Note that the summation index a on the left hand side of

109

Local Modes

this equation runs over 3N-6 normal coordinates while the index i on the right hand side is over 3N-6 independent internal local coordinates. For the second term in Eq. (6.1.1), we have Va 2 471;2 2 2 aVaQa Za ~ - - q ~ c Za 9

(6.1.11)

Using the relation -1 Qa - ~ L a i R i ,

(6.1.12)

where L -1 is the inverse of L in Eq. (6.1.3), Eq. (6.1.11) becomes Z ~_a qa2 - 2tlc . . Z 4~:2VaLai 2 - 1 -Laj 1 ) RiR j 1 1 i ~. I ( ~ a _4g 2VaLai 2._12)R2+j~i( a

a

(6.1.13) This equation can be simplified by noting (see Chapter 5) that LFL - A,

(6.1.14)

where A is a diagonal matrix with elements A a= 4n2v 2 and Fij are the force constant matrix elements. Using the relation with dimensionless coordinates qi (see Eq. (6.1.8)), Eq. (6.1.13) can be written as Va 2

Fij 9

9

9

]1//2 .

(6.1.15)

Thus the zero order hamiltonian of harmonic oscillators can be written in terms of the local coordinates using Eqs. (6.1.1), (6.1.10) and (6.1.15), as,

[ i{ ioiip i

110

Chapter 6

+1 ~~

(~i~j)l~

(~tigj

)1//2 GijPiPj +

Fij (FiiFjj) 1~ qiqj (6.1.16)

6.2 A p p r o x i m a t i o n s

At this point, we have to consider the individual cases in modifying Eq. (6.1.16). If we restrict ourselves to the bond stretching coordinates then it can be shown that giGii = 1 (see the B matrix elements discussed for stretching coordinates in Chapter 5), and the first term on the right hand side of Eq. (6.1.16) is similar to the harmonic oscillator hamiltonian in normal coordinates. The reduced mass gi is not well defined for nonstretch coordinates, where we may define an effective reduced mass as gi = 1/Gii.. With this approximation, Eq. (6.1.16) can be rewritten as follows: HLO - l ~ v i { p 2

+q2}

+ 1 Z X(~i~j)l~[

Gij

PiPj +

L(GiiGjj)~

i jr - H0,LO + H1,LO

Fij

)1/2 qiqj

(6.2.1)

FiiFjj

9 m

The subscript LO on H indicates that the hamiltonian is for the local oscillators; H0,LO is the harmonic part and H1,LO the coupling part. Further developments of the local mode concept can be made at different levels of approximations: (a) consider only the zero order local oscillator part, H0,LO; (b) evaluate the influence of inter-oscillator coupling, H1,LO on the local oscillator energy levels resulting from H0,LO; and (c) introduce anharmonicity using either the higher order terms of potential (involving q3i, qi4 etc.) or the Morse potential. 6.2.1 Independent local oscillators First let us consider H0,LO. This term is similar to the harmonic oscillator hamiltonian in normal coordinates [see Chapters 3, 4]; the only difference being H0,LO is cast in terms of local coordinates qi and

Local Modes

111

conjugate momenta Pi rather than those of normal coordinates. The eigenvalue problem would then be identical to that of the normal coordinate harmonic oscillator (see Eqs. (4.1.2), (4.1.5) and (4.1.6). That is,

--

HO,LO~LO - ELO~LO,

0

--0

0

(6.2.2)

EOo - E vi (~)i + 89

(6.2.3)

0

vOo - .I/~i/i,LO,

(6.2.4)

1

0 where ~/i,LO represents the local coordinate harmonic oscillator wavefunction with frequency vi (which is different from the normal coordinate harmonic oscillator frequency Va)" 6.2.2 Perturbation treatment of inter-oscillator coupling

The second term H1,LO can be treated as a perturbation using 0 perturbation theory with harmonic oscillator basis functions ~l/i,LO. If we consider all (3N-6) independent internal coordinates, each representing a local oscillator, then the number of non-zero terms in H1,LO will be prohibitively large to provide any qualitative picture. Therefore we will consider only the predominant terms as appropriate for individual systems. Let us consider a X2Y2 non-linear molecule, where the two identical X-Y bonds are coupled through the bond between the two Y-atoms (an example being H202). As this type of molecule has six internal coordinates (two X-Y and one Y-Y stretchings, two X-Y-Y bending and one X-Y-Y-X torsion) one has to consider, in principle, the interaction among these six oscillators. For the sake of brevity we will restrict our consideration to the two X-Y stretches interacting through the Y-Y stretching oscillator. Designating the two X-Y and Y-Y oscillators as 1, 2 and 3 respectively, and noting that Vl - v2, Fll - F22, G11 = G22, G13 - G23, F13 = F23 and G12 = 0, the perturbing hamiltonian becomes

H1,LO- Vl [(~121 ]qlq2 ] + (VlV3)~I (G1

G13 1G33

.1/(PlP3 + P2P3)

112

Chapter 6

+

F13 (qlq3 + )1 (F11F33)1//2 q2q3 (6.2.5)

The dominant contribution in Eq. (6.2.5) depends on the nature of atoms in the molecule. Let us explore the situations separately. First, consider the case when v1F12 / F 11 is negligible compared to the second term in Eq. (6.2.5). Then,

H1,LO - (VlV3)I~I(G1

G13

.~ (PlP3 + P2P3)+

1G33 )

F13 )1~ (qlq3 + q2q3) 1" (F11F33 (6.2.6)

This perturbation causes the excited states within a given 'block' to mix. For example, if the total quanta of excitation is 1, then the excited states that are the members of a 'block' are, (Vl - 1, 1)2 - 0, 1)3 - 0), (a91 - 0 , v2 - 1, v3 - 0 ) and (a91 - 0 , v2 - 0 , v3 - 1), and these states are forced to mix by the perturbation in Eq. (6.2.6). To evaluate the energies of these states after perturbation, we have to diagonalize the following interaction energy matrix which can be derived using the integrals in Appendix 2: lagl = 1 , ~ 2 = 0, xr3 = 0 > = Z Cn2En~ EO,

where E0 is the lowest true eigenvalue of the system. Since En are the true energies (with eigenfunctions ~n) and C2n is positive, the weighted average

~

Cn2En can

never

be

lower

than

the

smallest

n

value of En. That means the energy calculated with an approximate wavefunction should be minimized to reach towards the true energy of the system, although that does not guarantee that the true energy will be obtained. Among different approximate wavefunctions, the one which results in lowest energy can be considered to be closest to the true wavefunction. The coefficients coi can be optimized to achieve this energy minimization by setting (t)Eel/C)Cpi) to zero and solving the resulting equations for the coefficients; i.e., the condition O/OCpi[/<Xltappl~app>] = 0 leads to the set of equations,

E (Fpq - EiSpq)Cqi = 0,

(7.1.18)

where, Fpq = Hpq + X PrsIpqrs ,

(7.1.18a)

r,s

and Ei are the orbital energies. These are called Roothan's equations and Fpq are called the Fock matrix elements. The secular equation is of the form FC = S Cs (~ is diagonal matrix with elements ei) which can be rewritten as (S-1/2FS-1/2)(Sl/2C) = (Sl/2C)g. By diagonalizing the (S-1/2FS -1/2) matrix, the energies of the molecular orbitals ei are obtained as the eigenvalues. The eigenvectors are equal to $1/2C from which the matrix of coefficients C can be extracted. But, since the Fock matrix

124

Chapter 7

elements Fpq themselves depend on the coefficients one needs these coefficients even before the diagonalization of the (S-1/2FS -1/2) matrix. Usually one approximates the FI~q as Hpq, diagonalizes the (S-1/2FS -1/2) matrix for coefficients and uses the resultant coefficients to construct the full Fock matrix (see Eq. 7.1.18a). The diagonalization procedure and the construction of Fpq is repeated until the energy and coefficients obtained in the current step are consistent, within a certain tolerance, with those obtained in the previous step. This cyclic procedure to determine the coefficients and energy is referred to as the self-consistent field (SCF) Hartree-Fock (HF) method. The lack of correlation in the motion of electrons with opposite spins is considered to be a serious deficiency in the HF procedure described above. Different methods are available to correct for this deficiency, but they are beyond the scope of this book. Interested readers are encouraged to refer to the original references on this subject. 6

7.2 Energy derivatives The derivatives of energy are important quantities in vibrational spectroscopy, as reflected by their attributes given below. The first derivative of energy with respect to a nuclear displacement gives the negative of the force at that nucleus. These forces are needed for the geometry optimization process. Variation of these forces with respect to the electric field can be used in deriving the nuclear displacement derivatives of the electric dipole moment. The second derivative of energy with respect to electric field is related to the electric dipole polarizability. The variation of forces with respect to the magnetic field can be used to formulate the nuclear displacement derivatives of the magnetic dipole moment. The second derivative of energy with respect to magnetic and electric fields is related to the electric dipole-magnetic dipole polarizability. The second and higher derivatives of energy with respect to nuclear displacements represent the force constants, which are required for vibrational frequencies. The procedures to evaluate the energy derivatives using the molecular orbital theory are outlined below. 7.2.1 First derivatives of energy 7-14 Direct differentiation of the energy expression (see Eq. 7.1.15) for Eel with respect to a parameter b results in two different types of terms. One involves the derivatives of integrals, i.e., (~Hpq/Ob) and (OIpqrs/Ob), and these derivatives can be handled by directly dit~ferentiating the analytic expressions of integrals involving the basis functions. The second type of terms contain the derivatives of coefficients, i.e., (OCpi/Ob). These are not known a priori, but they can be avoided by using the orthonormality condition among the molecular orbitals as follows. From Eq. (7.1.15), the

Vibrational Frequencies and Force Constants

125

differential of Eel with respect to parameter b is, (7.2.1)

(OEel/~)b) = (OEel/Ob)I + (OEel/Ob)II, where, (OEel/ Ob)I - Z 2Ppq (OHpq / Ob) + Z Ppq Prs (~Ipqrs / Ob) , p,q p,q,r,s

(7.2.1a)

(OEel/ Ob)ii - 22(~)Ppq /Ob)Hpq p,q + 2[(clPpq / ~)b)Prs + Ppq (~)Prs / ~b)]Ipqrs. p,q,r,s

(7.2. lb)

Using the expressions (see Eq. 7.1.18) for Fock matrix elements and secular equation (~Eel/3b)II can be simplified l0,13 to (OEel / c)b)II - Z ~i 2 2[(OCpi / c)b)Spqcqi + (OCqi / Ob)Spqcpi]" (7.2.2) i p,q From the orthonormality relation (Eq. 7.1.16) for molecular orbitals, one obtains Z[(C)CP i / c)b)cqiSpq + (C)Cqi / c)Sb)CpiSpq ] - - Z CpiCqi (c)Spq / c)b). P,q p,q (7.2.3) Substituting Eq. (7.2.3) in Eq. (7.2.2) one obtains, (7.2.4)

(OEe' / c)b)II - - 2 [ Z 2CpiCqil3i](~)Spq / c)b). p,q i

From the above equations, it can be seen that the first derivative of energy with respect to a parameter b can be calculated if the derivatives of integrals, i e, (OHoq/~b), (OJpqrs/Ob), (OKprqs/Ob) and (OSp/3b) are known When b represents a nuclear i:lisplaceme-nt~ these derivative integrals can be evaluated using analytic expressions .Iat may be noted that to obtain the total energy derivative, one should a,~,~ (~Enuc/~b) to (3Eel/~b); Enuc for fixed nuclei is given by Eq. (7.1.2), which can be differentiated with respect to nuclear displacements. 9

9

.

.

.

.

fl

9

Chapter 7

126

7.2.2 Second derivatives of energy7-14 Using the expression for the first derivative of energy, differentiating again with respect to a parameter b' gives, (~)2Eel/~)b~)b') =

2[Ppq (~)2Hpq/~)b~)b') + (~)Ppq/~)b')(~)Hpq/~)b)] p,q {Ppq Prs(()2Ipqrs/()b()b') p,q,r,s + (~)Ipqrs/~)b)[Ppq(~)Prs/~)b') + Prs(/)Ppq//)b')] }

-Z P,q

[(~(Z

I~iCpiCqi)/()b')(()Spq/()b) i

EiCpiCqi)(()2Spq/~)b~)b') ].

- (Z

(7.2.5)

i

The second derivatives of the integrals (~)2Hpq/~)b~)b'), (~92Spq/~Ob~Ob') and (()2Ipqrs/()b()b') can be evaluated using the analytic expressions. But the derivatives of coefficients and orbital energies, i.e., (~coi/~gb') and (~)ei/~)b') are not known. A procedure which enables us to calcialate the first order corrections to Cpi and I~i due to a perturbation from b', will allow us to determine the second derivatives of energy. This procedure, called coupled perturbed Hartree-Fock (CPHF) method is summarized here. For this purpose, we follow the work of Pople and coworkers. 13 The secular equation, (Eq. 7.1.18), that we discussed earlier to obtain the orbital energies, has resulted from using a hamiltonian for the free molecule. If the molecule is under a perturbation, represented by parameter b', then the secular equation would have the same form, but each of the quantities, namely Fpq, Spq, Cpi and ~i would have been a function of b'. That is, the secular equation now can be written with functional dependence on b' as F(b')C(b') = S(b')C(b')E(b').

(7.2.6)

Here s(b') is a diagonal matrix with elements ei(b'), C(b') is a matrix of coefficients with elements Cpi(b'), S(b') is an overlap matrix with elements Spq(b') and F(b') is the Fock matrix with elements Fpq(b'). The goal is to determine Cpi(b') without having to solve the secular equation containing the perturbation b' (because, when b' represents a nuclear displacement solving the secular equation for each nuclear displacement in a molecule is


127

computationally prohibitive). Instead, we can write Cpi(b') in terms of Cpi(0), which are obtained in the absence of the perturbation (b' = 0), as Cpi(b') = Z Cpg(O)ugi(b') or C ( b ' ) = C(O)U(b'). g

(7.2.7)

Note that the summation over g contains not only the occupied molecular orbitals, but also the unoccupied molecular orbitals (commonly referred to as virtual orbitals). In order for this equation to be valid at b' - 0, it is necessary that uei(0) - g~i, i.e., the matrix U(0) containing the elements uei(0) is a unit matrix. Then our goal is to find the first order correction to uei(0) in the presence of perturbation b'. Substituting Eq. (7.2.7) into Eq. (7.2.6) gives, (7.2.8)

F(b')C(0)U(b') = S(b')C(0)U(b')g(b') . Multiplying this equation with C(0) and designating Ft(b') = C(0)F(b')C(0),

(7.2.9)

St(b') = C(O)S(b')C(O),

(7.2.10)

one obtains a transformed equation, Ft(b')U (b') = St(b')U (b')t~(b').

(7.2.11)

Note that, (7.2.12)

St(O) = C(O)S(O)C(O) = E,

where E is the unit matrix. This is because St(0) is the overlap matrix in the molecular orbital basis and the molecular orbitals are orthornormal. Considering only the first order changes, one can write, Ft(b') = Ft(0) + Ft 9

P

= E(0) + Ft; U(b') = U ( 0 ) + ~ = E + U " ,

St(b') - St(0) + St =

E + S t and E(b') = ~(0) + E'. Then Eq. (7.2.11) for first order changes becomes, P

Ft + s

I

P

= I~'+ U'g(0) + Stir(0).

(7.2.13)

128

Chapter

7

Diagonal elements of this equation provide the first order corrections to energy as, 9

p

(7.2.14)

e"e - Ft,gg - St,ggEg(O ) 9 p

The off-diagonal elements provide the first order corrections Ugm as, p

p

p

(7.2.15)

[gm (0) - Eg(0)]ug m - Ft,gm - St,gmg m (0). p

In Eqs. (7.2.14) and (7.2.15), St,gm is obtained from the first order contribution of Eq. (7.2.10) as p

I

St,gm = E Cpg(O)Spqcqm (0), (7.2.16) P,q where S' - (OSpq/Ob') can be obtained via analytic differentiation of Spq. pq p

The evaluation of .Ft,em is more involved. transformed Fock matrix is written as,

From Eq. (7.2.9), the

Ft,gm = E Cpg(~ (b')cqm (0) , (7.2.17) p,q where Fpq(b'), in analogy to the Fock matrix at b ' - 0 (see Eq. 7.1.18a), is written as

Fpq(b')- Hpq(b') +EPrs(b')(2Jpqrs-Kprqs)b' 9 r,s

(7.2.18)

The subscript b' in the second term on the right hand side of Eq. (7.2.18) indicates the integral dependence on b'.

Noting that Prs(b')

=

Cri(b')csi(b') and Cri(b') is given by the expansion in terms of Cri(0) (see 1

Eq. (7.2.7)), Ft,trn is obtained from Eqs. (7.2.17) and (7.2.18), as Ft,gm = E Cpg (0) Hpq(b') Cqm(O) p,q


129

+Z Cpg (0)Cqm(0)Z Z Z Uni (b')uoi(b')cm(0)Cso(0) p,q

r,s i n,o • (2Jpqrs- Kprqs)b'.

(7.2.19)

To extract the first order contribution from this equation, we need to remember that Uni(b'), Uoi(b') and the integrals have to be expanded at b' = 0 and that the summations over n and o run over all molecular orbitals (occupied and virtual). Thus, F't,gm) - ZCpg (0)Hpq(b')cqm(0) ' p,q + ZCpg(O)cqm (0)Prs(0)[c)(ZJpqrs - Kprqs ) [ 3b'] p,q,r,s + ~ Uni Z Cpg(0)Cqm(0)Crn (0)Csi (0)(2Jpqrs - Kprq s) i,n p,q,r,s + ~ Uoi Z Cpg(0)Cqm(0)Cri (0)Cso (0)(2Jpqrs - Kprq s)" i,o p,q,r,s

(7.2.20)

Substituting this expression into that for U~m [Eq. (7.2.15)], one finds that the off-diagonal elements of U' are present on both sides of the equation, so one has to use some kind of iteration procedure to determine the offdiagonal elements of U matrix. The readers may consult Ref. 13 for further details. The diagonal elements u~e, however, can be determined from the orthonormality condition associated with the molecular orbitals. This condition can be written for the molecular orbitals obtained in the presence of the perturbation as (7.2.21)

C(b')S(b')C(b') = E . Using the transformation in terms of

Cpi(0),this equation becomes

[I(b') C(O) S(b') C(O) U(b') = [l(b') St(b') U(b) = E .

(7.2.22)

The first order contribution from this equation for diagonal elements (noting that St(0) - U(0) = E), is given as s ,ee = - 2u e. Since S~,tt can be determined as described earlier (see Eq. 7.2.16), the diagonal elements of the U' matrix can be determined. Once the U' matrix elements are determined, the derivatives ~Cpi/3b' can be obtained, and the second

Chapter 7

130

derivative of energy O2Eel/Ob/)b' calculated. The ab initio force constants for CHFC1Br are listed in Table 1; the vibrational frequencies derived therefrom are given in Table 2.

7.2.3 Higher derivatives of energy The third derivatives of energy such as ~)3Eel/~)b~)b'~b" can be obtained by differentiating the expression for O2Eel/~)bOb' with respect to b". The resultant expression contains coefficient derivatives OCpi/Ob" and O2Cpi/Ob'bb". Readers may consult Ref. (14) for further details. TABLE 1 Force constants(mdyn/A) a in symmetry coordinates for (S)-CHFC1Br

S1 $2 $3 $4 $5 $6 $7 $8 $9

S1

$2

$3

5.63

0.15 6.40

0.05 0.60 3.69

$4

$5

$6

$7

$8

0.03 0.54 0.39 2.74

0.02 -0.10 -0.14 -0.09 0.71

0.04 0.42 -0.15 -0.14 0.06 0.68

0.01 0.00 0.29 -0.27 0.01 -0.01 0.58

0.00 0.16 0.15 -0.21 -0.16 0.03 0.06 0.57

$9 -0.00 -0.18 0.16 0.00 -0.03 -0.04 0.02 -0.02 0.32

aObtained using DZP basis set with electron correlation incorporated via MP2 method. The atomic cartesian coordinates (in Angstrom units), representing the optimized geometry, are" C(-0.4320, 0.4612, -0.6383); H(-1.5179, 0.5752, -0.6709); F(0.1574, 1.6482, -0.8538); C1(0.0564, -0.6854, - 1.8782); Br(0.0216, -0.1643, 1.1408). The symmetry coordinate definitons are as follows: S 1 = ArC-H; $2 - ArC-F; $3 = Arc-c1; $4 = Arc_ Br; $5 - (1/~/-6) (A0~HCF + A0~HCC1 + A~HCBr - A 0~FCCl - A (/C1CBr A0~BrCF); $6 - (1A/-6) (2A0~HCF - A (~HCC1 - A ~nCar); $7 - (1/~/-2) (At~HCC1 - At~HCBr); $8 - (1/~/6) (2AtXFCC1 - A (R21Car - A~BrCF); $9 = (1/~-2) (At~lCar- ACtBrCF). In the construction of B and G matrices, the H-C-X angle coordinates were scaled by C-H bond length, while the F-CC1, C1-C-Br, and Br-C-F angle coordinates were scaled by C-F, C-C1 and C-Br bond lengths respectively.


131

7.2.4 Experimental determination of force constants The force constants can be defined in any of the coordinate systems (i.e., cartesian displacements, internal and normal coordinates) discussed in Chapter 5. However, it is convenient, and intuitively transparent, when the force constants are written in internal coordinates. For this reason we restrict the discussion to these internal coordinate force constants. The simplest example to illustrate the application of experimental data in determining the force constants would be a diatomic molecule. The TABLE 2 Predicted vibrational properties for (S)-CHFC1Br

~a 3209 1372 1266 1151 822 678 439 321 233

Ab

Rc

Sd

6.0 0.5 23.4 -8.1 82.6 7.1 189.7 1.0 174.8 15.1 60.6 - 15.5 1.6 0.7 0.1 -0.4 0.1 0.2

80.7 4.9 3.6 1.3 4.0 9.6 4.0 3.1 4.6

pe ( t ~ • 103)f 0.24 0.75 0.65 0.74 0.63 0.16 0.28 0.52 0.52

-0.16 -0.00 -1.8 0.06 0.72 -2.1 -0.64 0.93 -0.57

(72/f.o)f

(52/o~)f

-0.0264 -0.0141 -0.0074 -0.0036 -0.0725 0.0627 0.0077 0.0054 -0.0104

0.0198 0.0006 -0.0231 -0.0048 0.0186 -0.0107 0.0018 0.0025 -0.0017

aFrequency (cm -1). The major contributions to potential energy ditribution (PED) for each of the vibrational modes are as follows: 3209 cm -1" S1(100); 1372 cm -1" $6(100); 1266 cm -1" $7(100); 1151 cm -1" $2(100); 822 cm -1" $3(85), $8(18); 678 cm -1" $5(59); $4(52); 439 cm-l: S5(27), $8(23), $3(18); 321 cm-l" $9(38), $8(30), $4(27); 233 cm-l" $9(54), $8(21). bAbsorption intensity (km/mol); Crotational strength (10 -44 esu2cm2); dRaman activity (A4/amu); edepolarization ratio; fROA o

o

parameters (A5/amu). The frequencies, absorption intensities, Raman activities and depolarization ratios were obtained with DZP basis set and electron correlation using MP2 method. The remaining properties were obtained at the Hartre-Fock level. vibrational frequency of its fundamental transition can be obtained from the vibrational absorption or Raman spectrum. If we assume that the potential energy is given in the form of that for the harmonic oscillator (i.e., third and higher derivatives of energy in the potential energy

Chapter 7

132

expansion are ignored; see Eq. (3.1.11)), the vibrational frequency Ve is related to the quadratic force constant K = O2V/OR2, as Ve -- (| [ 2gc)(K / g)l/2,

(7.2.23)

where tx is the reduced mass (see Eq. 3.1.12). For the harmonic oscillator the vibrational energy levels are given by Eq. (3.2.5). With ~?e, K and expressed respectively in cm -1, mdyn//~ and atomic mass units, Eq. (7.2.23) can be conveniently written as Ve -- 1302.8(K / it) 1/2.

(7.2.24)

Although it may seem straightforward to determine the quadratic force constant from the experimental vibrational frequencies using Eqs. (7.2.23) and (7.2.24), the underlying harmonic approximation is not necessarily valid for most cases. If the harmonic approximation is valid, then the experimental vibrational frequencies of overtone transitions 0 ---) 2, 0 ~ 3 etc. should respectively be at 20e, 3Ve etc. The observed overtone frequencies are usually found at frequencies lower than these values. So one must consider anharmonicity, i.e. higher energy derivatives in the potential energy expansion. When the cubic and quartic terms are included (see Eq. 3.3.3), the vibrational energy levels are given by Eq. (3.3.10). The frequencies of 0 ~ 1, 0 ---) 2, 0 ---) 3 etc. transitions can then be seen to be, respectively, V e - 2VeXe, 2(Ve- 3VeXe)' 3(Ve- 4VeXe)' etc. Note that for diatomic molecules we defined the anharmonicity constant as VeXe (see Eq. (3.3.10a)). A general expression for the vibrational frequency v 0 ~ v of the 0 ---) ~) transition can be found from Eq. (3.3.10) as V0---~ -- Ve l) -- VeXe

(v2 + v).

(7.2.25)

A plot of ( v 0 ~ / ~) vs (~) + 1) should be a straight line with the slope given as -VeXe and intercept as re. The anharmonicity term VeXe is related to the cubic and quartic force constants via Eq. (3.3.10a), and ve is related to the quadratic force constant via Eq. (7.2.23). From the above discussion it is apparent that the determination of force constants requires the availability of experimental data for several overtone transitions, along with that for fundamental transitions. One may include higher order anharmonicity contributions also and in such cases


133

the energy levels are given by Eq. (3.3.11). These higher order contributions are rarely included. For a polyatomic molecule the situation is more complicated. First let us assume that the experimental fundamental frequencies either represent, or can be converted (vide infra) to, the harmonic frequencies. For small symmetric polyatomic molecules one may confidently use the harmonic frequency data to determine the quadratic force constants, but for the rest one may not be able to obtain a unique solution. To explore this further, first let us recall that the vibrational secular equation (see Chapter 5) in the harmonic approximation is solved from G F L = LA, where G is the inverse kinetic energy matrix (determined from atomic masses and geometry), F is the force constant matrix, L is the eigenvector matrix and A is the diagonal matrix of eigenvalues (or vibrational frequencies). It is useful to note that in the older literature the F matrix based on different force fields (namely central force field, Urey-Bradley force field and valence force field) have been employed. These force fileds will not be discussed here as they are not used frequently (as reflected in the recent literature). For a non-linear molecule with 3N-6 independent internal coordinates, the F matrix would have 1/2(3N - 6)(3N - 6 + 1) elements (note that F is a symmetric matrix). For example, for the non-linear H-OF molecule the three internal coordinates (O-H stretch, O-F stretch and HO-F bend) would lead to six independent internal coordinate force constants. For the non-linear and non-planar HOOF molecule, the six internal coordinates (O-H stretch, O-F stretch, O-O stretch, H-O-O bend F-O-O bend and H-O-O-F torsion) lead to 21 independent internal coordinate force constants. Since HOF and HOOF molecules have only three and six vibrations, respectively, one cannot uniquely determine all of the internal coordinate force constants from the experimental vibrational frequencies. For small symmetric polyatomic molecules such as H20 and H202, however the situation is more favorable. For H 2 0 there are four independent internal coordinate force constants, namely

O2V/OR2

~)2V/OR2, ~)2V/~)RI~)R2, ~2V/OR 1~o~ = O2V/OR2~)~ and ~2V/~)~2 (where R1, R2 and a are internal coordinates representing the changes in two O-H bond lengths and angle). The three fundamental vibrational frequencies of H20 are not enough to determine the four force constants, but one can use the experimental data for isotopic molecules such as D20, HOD, H2180. Recall that the F matrix is invariant to isotopic substitution (but G matrix and hence vibrational frequencies change upon isotopic substitution). With the experimental data for the parent molecule and any one of the isotopic molecules, the total number of available vibrational frequencies increases to six (two greater than the number of unknowns). But the vibrational frequencies of isotopic molecules are related through

134

Chapter 7

the product rules (see next section). In the case of H20 and D20 (or H2180) there are two product rules (in the case of H20 and HOD there is one product rule). Thus there are four independent vibrational frequencies for H20 and D20, which are adequate to determine the four independent force constants. In the H202 case there are 12 independent internal coordinate force constants (the reader can easily verify this by using the equivalence of two O-H groups and the internal coordinates mentioned earlier for HOOF), but only six vibrational frequencies. The six vibrational frequencies of D202 will bring the total number of frequencies to 12, but there are two product rules among these frequencies making the number of independent frequencies to be only 10. So one must have the vibrational frequencies of HOOD as well. The eighteen vibrational frequencies of the three molecules H202, D202 and HOOD, must satisfy one sum rule (see next section) and four product rules (two product rules for H202 and D202, one for H202 and HOOD and one for D202 and HOOD). Then one has 13 independent vibrational frequencies from three molecules H202, D202 and HOOD, which can be used to determine the 12 independent internal coordinate force constants. For larger polyatomic molecules (even with some symmetry) the number of independent internal coordinate force constants would, in general, be so high that it would be impossible to obtain enough independent vibrational frequencies from the experimental data. For this reason, one has to make severe approximations (for example, approximating some force constants to be zero and some to be constrained to certain values guided by chemical intuition etc). In these cases there are no unique solutions for determining the force constants from the experimental vibrational frequency data. The coriolis interaction constants and centrifugal distortion constants are also related to the force constants, so one can augment the experimental vibrational frequency data with this additional information in determining the force constants. But the experimental data on coriolis and centrifugal distortion constants are not available for many polyatomic molecules, in general, and therefore their use is limited. The reader interested in these aspects may refer to a detailed discussion in Ref. (15). Coming to the cubic and higher order force constants in polyatomic molecules, one may be able to determine them only for selected small molecules. The reason for this would be clear when we analyze the needed experimental data for determining these higher order force constants. The vibrational energy levels with anharmonic contributions are given by Eq. (4.2.25). For transitions restricted to the ath vibration (for all other vibrations ~b = 0), the transition frequencies can be given by the general expression,


V0"->aga -- Vaaga + Xaa (1)2 + a)a) + E l)aXab / 2 .

135

(7.2.26)

b,a A plot of (v0-->u~ / ~)a) vs (l)a + 1) gives a straight line with slope given as Xaa and intercept as (Va + ~ Xab / 2). Thus the diagonal anharmonicity b,a

constant Xaa can be determined if the fundamental and at least one of the overtone frequencies of the ath vibration can be measured. The offdiagonal anharmonicity constant Xab and the harmonic frequency Va are lumped together in the intercept of the above mentioned plot, so additional experimental information would be needed to separately determine these quantities. In a combination transition two different vibrations a and b are simultaneneously excited by different quanta 9a and agb. Then from Eq. (4.2.25) the combination frequency is given as Vo,o__>aga,~b -- ~al)a + Vba)b + Xaa(a)2 + I)a) + Xbb(1)~ + I)b) + Xab[(1)a / 2)+(I) b / 2)+ 1)al)b]+ ~(1)aXac + ~)bXbc) / 2. c~a,b

(7.2.27)

Subtracting Eq. (7.2.26), and the analogous equation for V0.._>l)b, from Eq. (7.2.27), one obtains (Vo,o....>,Oa,,Ob -- Vo...~,Oa -- Vo...f,Ob ) -- Xab'Oa'O b .

(7.2.28)

Thus to determine Xab one requires the combination frequency V0,0--->~a,~b and the corresponding individual fundamental frequencies for transitions v0-->~a and V0-->~b" Such data are not available for most polyatomic molecules. Once Xab is determined from Eq. (7.2.28), the harmonic frequency Va of the ath vibration can be extracted from the intercept of the plot (V0__>ua / ~ a ) vs (a)a + 1) mentioned above (see Eq. 7.2.26). Remember that Va is related to the quadratic force constant (see Eq. 3.3.3a) and Xaa and Xab are related to the cubic and quartic force constants (see Eq. 4.2.25), all in normal coordinate space (not in internal coordinate space that we discussed earlier). Once the harmonic frequencies Va are determined for all fundamental vibrational transitions, we go back to the discussion given earlier (for H20 and H202) to determine the quadratic force constants in internal coordinate space (where

Chapter 7

136

the use of isotopic information is needed). In the determination of the anharmonicity constants Xaa and Xab discussed above one has to address the possible resonance interactions between different transitions. Looking at the expressions for Xaa and Xab given by Eqs. (4.2.25a) and (4.4.25b), it can be noted that the magnitudes of Xaa and Xab can become very large in two situations. (a) the first overtone transition frequency 2 Va of the ath vibration is approximately equal to the fundamental transition frequency Vb of vibration b; (b) the combination frequency Va+ Vb of vibrations a and b is approximately equal to the fundamental transition frequency Vc of vibration c. The first case was observed in CO2 by Fermi, 16 so this type of resonance is known as Fermi resonance. In such resonance situations, the second order perturbation theory used to obtain the expressions for Xaa and Xab is no longer valid. This is because the perturbation theory is based on the premise that the second order corrections are small, but Xaa and Xab acquire larger magnitudes in resonance situations. So one has to identify the transitions that are in resonance, remove the corresponding contribution from the expressions for Xaa and Xab, and separately correct the observed frequencies for resonance effects. Let us consider the situation where resonance is occuring from 2V a - Vb = 0, i.e., 2 Va " Vb" That means the states laga = 2, 1)b = 0> and I~a = 0, 1)b = 1> are mixing -

2

each other and the perturbation comes from the cubic term Vaabqaqb (see Eq. 4.2.25). From the integrals in Appendix 2, it can be seen that, = 112. I~a=2,~b=0> (Va+3 Vb)/2. be written as To obtain the

The harmonic energy of the

state is (5Va+Vb)/2 and that of I~a-0,a3b=l> state is Since 2 Va-Vb = A " 0, the above mentioned energies can (5~a+ Vb)/2 = (7 Va- A)/2 and (Va+3 Vb)/2 = (7 Va- 3A)/2. corrected energy levels, we diagonalize the 2x2 matrix

(7 Va - A)/2

Vaab/2

Vaab/2

(7V a - 3A)

- 2 1/2]/2 as the new energies of the two yielding, [(7 Va- 2A) + (A2 + Vaab) levels.

Since the zero point energy is (Va+Vb)/2 = (3V a - A)/2, the

transition frequencies become 2 Va- A/2 _+(A2 + V 2aab)1/212.

137


The energy levels before and after perturbation are sketched in Fig. 71, for the case 2 Va=Vb . In this case the difference between the two observed frequencies of the above mentioned transitions gives the magnitude of the cubic force constant Vgaab, and the contribution from this term should be dropped in the expressions for Xaa and Xab. For the second case, where the resonance occurs from Va + V b - V c = 0, the two states with quantum numbers II)a - 1, ~ b - 1, l)c = 0 > and I~a - 0, lkb -

0 , 1)c =

=0)

1> will be mixed by the perturbing term Vabcqaqbqc- The

. . . . .

( ~ a = O, ~b = 1)

,,,

....

I

, 1

abc 9

1)J

4,.

(~a = O , ~ b = 0 )

~J

(ga = 1,X)b = 1,X)c = 0 ) ' ~ (ga = 0,19b = 0 , 9 c =

' ~%

4| 4

-I ( ~ a = 0, ~b = 0, be = 0)

Fig. 7-1. E n e r g y levels depicting the situation for 2 V a = V b , and V a + V b = V c .

procedure to set up and diagonalize the energy matrix is similar to the one discussed above. For the situation where Va+ Vb=V c the two energy levels will be separated by 2Vabc/ff2, as sketched in Fig. 7-1. So the difference in the observed frequencies of the two transitions can be used to deduce the magnitude of V abc.

7.2.5 Sum rules The vibrational secular equation in the cartesian displacement coordinates (see Eq. 5.1.14) reveals a sum rule relating the vibrational frequencies to the diagonal cartesian force constants. The trace of the t matrix (see Eq. 5.1.14) is equal to the sum of the vibrational frequencies (provided the atoms are at equilibrium geometry, i.e. aV/aXAtx = 0), so one can write the sum rule as,

[(a2v / ax2)+ (a2v / ay2a,,)+(a2v / az2)1/mA A

E492c2V2" a

(7.2.29) This relation can also be obtained from Eq. (5.1.25) (note that the normal coordinates are orthonormal and (aV/aQa) = 0 at the equilibrium

Chapter 7

138

geometry). Since c)2V/c)X2Acxare independent of isotopic substitution one can estimate the vibrational frequency sums for isotopically related A

molecules from a knowledge of cartesian force constants ~)2V/~)XAo~. One can define an effective atomic cartesian force constant as, f 2 _ ~)2V / ~9x2 + ~)2V/~9y2 + ~)2V / ~)z2 '

(7.2.30)

and simplify the sum rule given above as, 17 X f 2 / mA _ X 4rl:2c2v2 A

9

(7.2.31)

a

If the effective atomic cartesian force constants fA are transferable for atoms in similar chemical environments, the vibrational frequency sums can be estimated from the above equation for a molecule of interest. Alternately, for molecules where the experimental vibrational frequency data are available, but the atomic compositions are not certain, the above mentioned sum rule provides a criterion to suggest the atomic compositions. The sum rule given by Eq. (7.2.31) leads to additional sum rules 18 for isotopically substituted molecules. As an example, let us consider Eq. (7.2.31) for H202, D202 and HOOD. These are,

= (1/4rC2c2){2fH/m H +

Va

2fo/mo},

(7.2.32)

H202

{Xa} {Xa} Va

= (1/4x2c2){2fH/m D + 2fo/mo},

(7.2.33)

D202

Va

(1,4.~c~)~f. ,m. + f . , mo + ~o 'mo~

HOOD

Substracting the latter two equations from the first, one obtains

(7.2.34)


Va

-

12fH(1 / m H - 1 / m D),

Va

-

139

H202

D202

(7.2.35)

Va

-

=(1/4n2c2)fH(1/m H-1/mD).

Va

H202

HOOD

(7.2.36) From Eqs. (7.2.35) and (7.2.36) it is easy to see that,

Va

-2 H202

{a

Va

+ HOOD

= 0.

Va

(7.2.37)

D20 2

There is no need to restrict ourselves to H/D substitution, and one could have written similar equations by replacing 16 0 with 18 0 . Experimentally, however, deuterium substitution is much easier than 180 substitution. The product of vibrational frequencies also provides a useful relation, which is commonly referred to as the Teller-Redlich product rule. 19 Although this product rule should not be classified as a sum rule, it is included here due to its importance. The product rule can also be obtained from Eq. (5.1.14) as follows. The determinant of the t matrix (see Eq. 5.1.14) is equal to the determinant of the eigenvalue matrix. The eigenvalue matrix is diagonal, so its determinant is equal to the product f.0102...f.03N . 22 2

The t matrix itself can be written as t = M-1/2fM-1/2 [see Eqs.

(5.1.2), (5.1.6) and (5.1.8)], where M -1/2 is the diagonal matrix with its -1/2 and f is the matrix of elements b2V/bXAaOXB~. elements being m A Then the determinant of t is equal to the determinant of f times the product of inverse atomic masses, m -1 A. Since f is independent of isotopic substitution, the ratio of the product of 3N eigenvalues for two isotopic molecules becomes,

nv~/~vf~2 - nmk/rlmA, k

A

A

(7.2.38)

Chapter 7

140

where primes identify the quantities for an isotopically substituted molecule. In Eq. (7.2.38) the index k runs from 1 to 3N, so the products of eigenvalues include not only those of (3N-6) vibrational modes, but also of three rotational and three translational modes. The above equation can be rewritten as, 17v a / H a

-

a

A

A

j

-

,

(7.2.39)

j

where the index a is for 3N-6 vibrational eigenvalues and j for six rotranslational eigenvalues. If we assume that the translational and rotational eigenvalues have very small magnitudes (as in the presence of a weak external force), they can be related for isotopic molecules as 18-20 . . . . vi 2 / v 2 - M / M'for translations and as vi 2 / vj - Ij / Ij for rotations; here M is the molecular mass and Iithe principal moment of inertia along jth principal axis. Then, Eq. (7.2.3q) can be rewritten as

6

V{V2. V3N-6.

mlm2 .

mN .

\~--Tj

T'x~y-z T' T'

/

"

(7.2.40)

This product rule has been widely used 20,21 in the literature. Note that the product rule equation [Eq.(7.2.40)] can be written down separately for vibrations belonging to different symmetry species (Chapter 10). For example, the vibrations of H 2 0 (and also D20) belong to A1 and B1 symmetry species, so a product rule each for vibrations of A1 and B1 symmetry species can be written down for H20 and D20. Only those rotations and translations belonging to the appropriate symmetry species appear in the product rule equations. References

1 2 3 4 5 6

D.R. Hartree, Proc. Cambridge Phil. Soc. 24 (1928) 89. V. Fock, Z. Physik. 61 (1930) 126. G.G. Hall, Proc. Roy. Soc. London A205 (1951) 541. C . C . J . Roothan, Rev. Mod. Phys. 23 (1951) 69. J . A . Pople and D. L. Beveridge, Approximate Molecular Orbital Theory, McGraw Hill, New York (1970). (a). Electron correlation can be treated at different theoretical levels. A review of these methods can be found in: K. Raghavachari, Ann. Rev. Phys. Chem. 42 (1991) 615; (b). Density functional methods are now becoming popular for treating the electron correlation. Reviews on the subject can be found in" B. B. Laird, R. B. Ross, and T.


10 11 12 13 14

15 16 17 18 19 20 21

141

Ziegler, Chemical Applications of Density-Functional Theory, ACS Symposium Series 629 (1996). R. M. Stevens, R. Pitzer and W. N. Lipscomb, J. Chem. Phys. 38 (1963) 550. D. M. Bishop and M. Randic, J. Chem. Phys. 44, 2480 (1966). J. Gerratt and I. M. Mills, J. Chem. 49 (1968) 1719; 49 (1968) 1730. P. Pulay, Mol. Phys. 17 (1969) 197. T. C. Caves and M. Karplus, J. Chem. Phys. 50 (1969) 3649. R. Moccia, Chem. Phys. Lett. 5 (1970) 260. J. A. Pople, R. Krishnan, H. B. Schelgel and J. S. Binkley, Int. J. Quant. Chem. 13 (1979) 225. Y. Yamaguchi, Y. Osamura, J. D. Goddard and H. F. Schaefer III, A New Dimension to Quantum Chemistry: Analytic Derivative Methods in Ab Initio Molecular Electronic Structure Theory, Oxford Univ. Press, New York (1994). S. Califano, Vibrational States, John Wiley & Sons, New York (1976). E. Fermi, Z. Physik. 71 (1931) 251. W. T. King and A. J. Zelano, J. Chem. Phys. 47 (1967) 3197. E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). O. Reidlich, Z. Phys. Chem. B28 (1935) 371. G. Herzberg, Infrared and Raman Spectra, van Nostrand Reinhold Co., New York ( 1971). N. Mohan and A. Muller, J. Mol. Spectrosc. 80 (1980) 455.


143

Chapter 8 VIBRATIONAL ABSORPTION AND CIRCULAR DICHROISM In the previous chapter, we discussed the methods to determine vibrational frequencies, the x-axis of vibrational spectra, and force constants. Coming to the spectral intensities it is important to note that they are not independent of the force constants which determine vibrational frequencies. The spectral intensities are related to the molecular property (dipole moment and polarizability) derivatives in normal coordinate space, and the normal coordinates are composed of individual atomic displacements in a molecule. The composition of these atomic displacements in a given normal mode, which is important for determining or interpreting the vibrational intensities, is also influenced by the force constants. Equally important are the molecular property derivatives with respect to the individual atomic displacements. Without a reliable description of these properties the prediction or interpretation of vibrational intensities would not be successful. Here also the ab initio molecular orbital theories are becoming the first choice.

8.1 Vibrational absorption spectra The vibrational absorption spectral intensities may be used to derive molecular properties, which include electric dipole moment derivatives, and atomic or bond properties, which include effective charges and their changes during nuclear motions. Alternately, absorption spectral pattern may be predicted by developing methods to predict the above mentioned properties. These approaches are considered below.

8.1.1 Experimental vibrational absorption intensities The relation between the integrated absorption coefficient and the electric dipole strength is given in Section 2.5. For the specific case of vibrational transitions two additional aspects have to be taken into account. In the approximation of separating the electronic, vibrational and rotational motions, the total wavefunction is written as a product of the electronic, vibrational and rotational wavefunctions (see Chapter 3). Since the vibrational transitions considered here are in the same electronic state, we need not be concerned about the electronic wavefunction for now (see next Section). Then the states s and n considered in Section 2.5 can be represented by the products ~ v ~ j and ~ , ~ j , , and the vibrational transition moment integral written as < ~ v , ~ j , l ~ l ~ ~ j > . Converting l.tc~ from the space fixed (non-rotating) axes to the molecule fixed (rotating) axes as laa = uo~a'~', with uaa, representing the transformation matrix element, the

144

Chapter 8

transition moment integral becomes ~l,(t',9'9Ut~',J'J, where kta,,a~,v =

and U ~ ' , J ' J -

< ~ j , lucto~, I ~ j >. The rotational integrals can

be replaced 1 by the averages of direction cosines, in which case the dipole strength becomes l.ttx,,~%kt *~, , ~ ) ' " 0 - I.t2, , ~,~ " The energy difference between vibrational levels is such that at room temperature the population in higher vibrational levels is usually non-zero. Then the transition rate per molecule given in Section 2.5 needs to be multiplied 1 by nv- nv, where n~ and nv, are the number of molecules per unit volume in vibrational states ~ and v'. Then Eq. (2.5.1) becomes, (dI(v)/I(v)) + = (B+/c)hv(nv - nv,)dl .

(8.1.1)

Proceeding further as in Section 2.5, the integrated absorption coefficient for a fundamental transition from l)a to ~)a+l (associated with normal mode a and observed band center V0a) becomes Aa - (4~3V0a/3hcCo)(nv a - naJa+1)(~a + 1)(c)kta/~)qa))2 .

(8.1.2)

In the harmonic oscillator approximation the energy difference between successive vibrational levels is the same, so all hot transitions such as 1---)2, 2---)3, etc., appear at the same location as the first fundamental transition 0---~1. Then Eq. (8.1.2) needs to be summed over all vibrational quantum numbers. Using the Boltzman distribution, n~ a can be expressed

as, nva - n e -hc~a (~)a+l/2)/kT / Z

e-hcva (aga+l/2)/kT,

(8.1 .3)

~a where n is the total number of molecules per unit volume and Va is the harmonic vibrational frequency of normal mode qa. Then the sum ~ ( n ~ a - n~a+l)(~a + 1), with the summation going from ~)a - 0 to co, can be shown 1 to be equal to n or NCo (where Nis Avogadro's number and Co is concentration). As a consequence, the absorption intensities of fundamental transitions, in the harmonic approximation, are expected to be independent of temperature (assuming that the temperature changes do not lead to conformational changes). Using the conversion from the dimensionless normal coordinate qa to the normal coordinate Qa, q2 _ (4rt2va/h)Q2 and taking the observed band center V0a as equal to the harmonic frequency Va, Eq. (8.1.2) can be

Vibrational Absorption and Circular Dichroism

145

written as - (~/3c)(~)jacflOQa)2) .

(8.1.4)

This equation gives Aa in units of cm2/mol.sec, with ~ta in units of Debyes (D). Most of the current infrared spectrometers provide the spectra on the wavenumber (cm-1) axis and integrated absorption coefficient A in cm/mol or km/mol. In such cases the right hand side of Eq. (8.1.4) should be multiplied by (l/c). Substituting the standard values for the constants, one obtains the useful relation, N,a = 42.28(~)gJ~)Qa)2) ,

(8.1.4a)

where Aa is in units of km/mol and (~g~x/OQa) in units of D/(A amul/2). For a diatomic molecule A-B, with internuclear axis taken as the z-akis, only (Ogz/OQ)2 component can be non-zero. Furthermore (see Section 3.2), Q = txl/2ARAB, where g is the reduced mass, mAmB/(mA + mB), and RAB is the bond length of A-B; ~)gz/~)RAB is the same as ~)gAB/~RAB where gA-B is the bond moment (for +eq charge on atom A and -eq charge on atom B, the bond moment gA-B = eqRAB). With these considerations, Eq. (8.1.4a) for a diatomic molecule becomes, A - [42.28(mA + mB)/mAmB](~)gA_B/~)RAB)2 .

(8.1.5)

The derivatives ~)ILtA_B/ORAB are to a good approximation independent of isotopic substitution. Nevertheless, it is apparent from Eq. (8.1.5) that the absorption intensities would be influenced by isotopic substitution (for example, absorption intensities of H-F and D-F vibrational bands would be different). The experimental integrated absorption coefficients enable the determination of the magnitude of (~A-B/c)RA-B), which in turn reveal the magnitude of charge movement (or charge flow) during the vibrational motion of the bond A-B. The direction of charge flow however cannot be determined because the sign of (O~tA-B/ORAB) cannot be determined from Eq. (8.1.5). For polyatomic molecules with symmetry, vibrations can be classified according to the irreducible representation they belong to (see Chapter 10). Depending on the point group and the associated irreducible representation, some vibrations can have only one of the three components (~)l.tj~Qa) non-zero. In such favorable cases individual components of (~)l.tot/~)Qa)2 can be determined from Eq. (8.1.4). Unlike in diatomic molecules, Qa is now dependent on several internal coordinates and therefore (O~ta/OQa) does not have as simple a meaning as (O~tA-B/ORAB) of a diatomic molecule. As a result, (3~a/OQa) has to be reduced to

Chapter 8

146

internal coordinates Rj or cartesian displacement coordinates XAor Such a reduction is complicated by the uncertainty in the signs of (~)l.tcfl3Qa), because this process involves the relations,

(o~l.ta/c3Rj)--- ~_.,(o~laoJc3Qa)(0Qa/c3Rj=)

~(~)l.ta/~)Qa)La] ,

(8.1.6)

a

(~)l-ta/~)XBI3) = ~(~)l.tcfl~)Qa)(~)Qa/~)XBI3) = ~(~l.tcx / ~gQa)Sa, ll3 9 (8.1.6a) a

a

aj and S-1 a,B[~ are discussed in The transformation matrix elements L -1 Chapter 5. The nine elements 691Ja/~OXB[~),with o~ = x, y or z and 13= x, y, or z, are referred to as the atomic polar tensor (APT) elements of atom B. Since the sign associated with each of the (~)l.ta/3Qa) is uncertain, Eq. (8.1.6) has to be evaluated for 2 n possible sign combinations of (~)l-tj3Qa); n is the number of normal modes of the molecule that have non-zero 091.tcd~OQa). In some special cases, the relative signs of (c)gcx/c)Qa) can be determined 2 from the analysis of coriolis effects in vibrational-rotational spectra; but for general cases this sign ambiguity associated with 691.ta/~)Qa) presents a serious problem in further analysis. Quantum mechanical calculations of electric dipole moment derivatives (especially of signs) helped overcome this problem. 3 With the sign ambiguity resolved, the derivatives (c)l.ta/~kRj) or (c)goc/c)XA~) can be further reduced to bond moment parameters using the bond moment model or atomic charge parameters using the atomic charge model (see Section 8.1.3).

8.1.2 Quantum mechanical methods Different quantum theoretical approaches can be identified for evaluating the electric dipole moment derivatives ~)l.tjOQa and hence vibrational absorption intensities using Eq. (8.1.4). In all these approaches it is convenient to separate the electric dipole moment into nuclear and electronic contributions. That is, n u c + go~ el = ~ eZAXAo~-e~ Xio~. goc = goc A i

(8.1.7)

Here eZA is the bare nuclear charge, e is the unit of electron charge, XAo~ is the positional coordinate of atom A and Xia is the positional coordinate of electron i. To evaluate the electric dipole moment of the system in state s, the dipole moment integral ' FA8

(8.1.27)

where Hel is the electronic part of the hamiltonian, for a given nuclear configuration. Using the time independent perturbation theory, ~s can be expressed by Eq. (2.2.12) with the perturbing hamiltonian H1 given as -~ta,Fa,. Then, to first order, F~ 8 _ Fel AS,0 + 2 ~ FA8,sn~a,,nsFct, [ (E 0 - E s0 ) ,

(8.1.28)

n4:s

where

FAS, sn = ,

(8.1.29)

Chapter 8

154

and F el AS,0 is the electronic contribution at Ftx' - 0, given as Fel 0 0 AS,0 - ( ~ s I-(0Hel / ~)XAs) / eZAIXl/s ) 9

(8.1.30)

~nuc ~nuc The nuclear contribution X-A8 can be written as "'A8,0 + F~,8~,8 where nuc F A8,0 is the contribution at Ft~' = 0. If the nuclei are at equilibrium trnuc Fel positions at F~, - 0, then the total contribution, FAS,0 - ~AS,0 + AS,0, adds up to zero; for non-equilibrium positions FA8,0 will be non-zero. The total electric field experienced by the nuclei can be written as,

(8.1.31)

F A 8 - FAS,0 + (8o(8- TA,Sct,)Fct', where TA,8a' - - 2 E FAS,snl.toCns/(E 0 - E 0 ) n:g:s = -2(~0l-(~)nel / c)Xgs) [ eZgl(O~ s / 0Fct, )),

(8.1.32)

is called the nuclear electric shielding (also electric dipole shielding) tensor. From Eqs. (8.1.26) and (8.1.31) one obtains, (8.1.33)

(I/eZA)(t~ILtjt~XAS) = 8ct8- TA,S(~ 9

To obtain the analogous expressions in the presence of a time varying electric field Fct'(t), rather than a static electric field Fa,, it is necessary to use the time dependent perturbation theory. Such expressions were given for general molecular properties in Chapter 2 [see Eqs. (2.3.23)-(2.3.27)]. Replacing P~(t) in Eq. (2.3.23) with FA8(t), the expression for TA,&t' is obtained from the coefficient of Fct'(t). Thus,

TA,8oC - - Re

4rCFAS,sn~tog,nsf.0ns/ h COnsLn~:s

.

(8.1.34)

This equation reduces to Eq. (8.1.32) when (Ons>>O). From the details given above it can be seen that the nuclear electric shielding parameters T A,8o~' can be used to obtain the electric dipole moment derivatives (Eq. (8.1.33)), and hence the absorption intensitites, of


155

harmonic fundamental vibrational transitions [see Eqs. (8.1.4) and (8.1.6)]. To obtain "~A,Sa' using Eq. (8.1.32), Random phase approximation method 10 has been used. It is important to note that Eq. (8.1.27) is valid only for exact wavefunctions.

8.1.3 Classical models In these models a molecule is viewed to consist of a set of effective atomic charges e~A, bond charges eqAB or bond moments gA-B. The molecular dipole moment can be written in terms of these parameters and appropriate derivatives formulated. These classical conceptual parameters were widely investigated, before the ab initio quantum mechanical programs became commonly available. At the present time very little activity is apparent in this area, mainly due to the overwhelming influence of quantum mechanical developments. It is worth noting that there are no quantum chemical analogues for the above mentioned classical parameters, although quantities equivalent in spirit can be defined from the wavefunctions. Nevertheless, it is useful to know how these classical parameters are used in vibrational spectroscopy. (A) Atomic charge concepts Suppose that each atom in a molecule has an effective charge e~A (these effective atomic charges are not equivalent to the charges determined from quantum mechanical calculations using Mulliken 12 or other 13 population analyses). For a neutral molecule the sum of effective charges is zero. The molecular dipole moment can be written in terms of these charges as ga = ~ e~AXAa 9

(8.1.35)

A

The required electric dipole moment derivative is obtained 14 by differentiating this equation. For the electric dipole moment derivatives in cartesian space, the nine quantities ~ta/~XA~5 (c~, 5 = x, y or z) pertaining to atom A are referred to as atomic polar tensor (APT) elements; ~Ia/~XA is referred to as APT of atom A. At the simplest level, the effective atomic charges are assumed to be 'fixed', that is they are assumed not to change as the atoms are displaced. This is referred to as the fixed partial charge (FPC) approximation. In this FPC model, the atomic polar tensor elements become ~l-ta/3XB~- ~ e~,~SABSc~5, A

(8.1.36)

Chapter 8

156

and the atomic polar tensor associated with atom A becomes diagonal with the three diagonal elements being equal. In terms of intemal or normal coordinates, the dipole moment derivatives in the FPC approximation become Z eg~,AAa,j , A

(8.1.37)

~)].td()Qa = E e~ASAa,a , A

(8.1.38)

~176

where the A and S matrices are discussed in Chapter 5. Thus the required electric dipole moment derivative can be obtained from Eqs. (8.1.36)(8.1.38) by assigning a set of effective charges for atoms in a molecule. The resulting values would be only approximate however. At the next higher level, the effective atomic charges are not considered to be fixed and to vary differently as different atoms are displaced. Then the above mentioned derivatives become, O).ta/OXa8 = Z [e~AIiABISa~+ e(C);A/c)XBS)XAo~], A

(8.1.39)

})~oJ()Rj - Z [e~AAAoqj + e(})~A/})Rj)XAc~], A

(8.1.40)

c)].toJ()Qa = E [e;ASA(x,a + e(()~A/c)Qa)XAo~]. A

(8.1.41)

The quantities e(~)~A/~IXBIS)indicate how the charge of atom A 'flows' as the atom B is moved; e(~)~A/~)Rj) indicates the same when internal coordinate (bond length, bond angle, etc.)j is displaced; and e(~)~A/c)Qa) has the corresponding meaning for the normal coordinates. Of these three quantities e(O~A/~)Rj) is much easier to connect with chemical intuition. Eqs. (8.1.39)-(8.1.4I) constitute the atomic charge-charge flux model (this terminology was originally used 14b for symmetry coordinates). To obtain the required electric dipole moment derivative, one has to assign not only the effective atomic charges e~A, but also the charge flux parameters e(~)~A/~)XBIS), e(~)~A/~)Rj) or e(~)~A/~)Qa). Looking at these equations from a different viewpoint, if the electric dipole moment derivatives can be determined experimentally, then one has a method to determine the effective atomic charges and charge flux parameters. But the number of charge flux parameters, in general, is


157

larger than the number of independent electric dipole moment derivatives, so one has to resort to approximations in solving Eqs. (8.1.39)-(8.1.41). This is one reason for the lack of wider utility of these models in vibrational spectroscopy. There are however favorable situations, which can be used to illustrate the importance of these models. Consider the planar molecules and let us assume that the molecule is in the x-z plane. The out-of-plane vibrations are not coupled to the in-plane vibrations, so we can discuss the out-of-plane vibrations independent of in-plane vibrations. The absorption intensity associated with the out-of-plane bending vibration will be determined by the derivative Og/OQo , where 9 . Y. Qop is the normal coordinate for out-of-plane bending vibration. ~t is easy to see that there will be no charge flux contribution to /)gy/OQop [differentiation of gy =~e~AYA with respect to Qop will give two A contributions" one from e~AOYA/OQop and another from (Oe~A/OQo)YA p 9 For a planar molecule in the x-z plane, YA = 0 for all atoms and therefore 3l.t/OQo y p does not have a contribution from the charge flow terms]. Thus the out-of-plane vibrational intensity is determined by the atomic charges e~A and the atomic displacements OYA/OQop. The latter can be determined as discussed in Chapter 5, so one can determine the atomic charges from the out-of-plane vibrational intensity. For example, the atomic charge of fluorine in BF3 and of hydrogen in C2H2, C2H4 and H2CO can be determined in this manner. Using the charges determined in this manner, the absorption intensities associated with the in-plane vibrations can be used to determine the charge-flow parameters. Although it is not possible to determine all individual charge flow parameters, certain combinations of these parameters can be found. For example, in BF3 the combinations (~)e~F/~)rB-F - c)e~F/C)rB-F') and (~)e~F/~t~o - ~e~F/~t~a) can be determined; here Oe~F/~)rB-F indicates how the charge on atom F changes as its distance from B is changed; and ~)e~F//)rB-F' indicates the corresponding charge flow when a neighboring bond B-F' is stretched. The parameters Oe~F/OCto and Oe~F/Ot~a are for the corresponding angle changes (t~o is the angle F'BF', opposite to the bond B-F; ffa is the angle FBF' or FBF"). (B) Bond charge and bond moment concepts Since bond charges are defined from atomic charges, the bond charge and atomic charge concepts are related to each other. For example, consider a A-B-C molecule with B as the central atom and A, C as the end atoms. If the effective charges of atoms A and C are e~A and e~c respectively, then the molecular neutrality defines the effective charge on atom B as e~B = - e~A - e~c. The charges e~A on atom A and - e~A on atom B define the magnitude of bond charge eqA-B in bond A-B as equal to e~A. Similarly the atomic charges e~c on atom C and - e~c on atom B

Chapter 8

158

define the magnitude of bond charge eqc-B in bond C-B as equal to e~c. Thus, with a bond charge eqk in bond k, the bond moment is defined as ].tktx = eqkdka = eqk rk U ka ,

(8.1.42)

where dk is the length vector, rk is the bond length and Uk is the unit vector, all for bond k. If bond k is made up of atoms B and C then dk = XC - XB, U k = ( X c - XB)/rk. The molecular electric dipole moment is then written as a sum of bond moments, so ].ta - ~ laka k

9

(8.1.43)

It is useful to note that in the discussion given above we have only defined the magnitude of bond charge. The sign of the bond charge and the direction of the length vector define the direction of the bond moment. One may choose to define the bond moment vector pointing from + charge t o w a r d s - charge, or vice versa. But the same definition should be maintained for all bonds. Thus, one may define all bond charges as positive and the vectors dk and U k directed from the atom with positive charge towards the negative charge counterpart in that bond. The molecular electric dipole moment derivatives can be obtained 15 by differentiating Eq. (8.1.43) with respect to the desired coordinates. The atomic polar tensor elements, for example, are given as

~)l.ta/~)XA5= ~ [eqkAkA~a8 + e(~)qk/~)XAS)dka],

(8.1.44)

k where AKA = +1 when bond k contains atom A and zero when bond k does not contain atom A. The derivatives/)~ct/~)Rj in internal coordinates, and /)l.ta/~)Qa in normal coordinates, can be obtained similarly using the transformation matrices A and S, as discussed before. The first term in Eq. (8.1.44) represents the fixed charge contribution and the second term represents the charge flow contribution to the molecular electric dipole moment derivatives. Thus the FPC approximation requires the omission of charge flow terms in Eq. (8.1.44). Since the number of independent bond charge parameters (eqk and e~qk//)R') again far exceeds the number of ~)l-ta/~)Qa for a given molecule, it ~s not possible to determine these bond parameters from experimental vibrational spectra without making serious approximations. In favorable cases, one can determine the magnitude of bond charge from the out-ofplane vibrational intensity and certain combinations of charge flow parameters from the in-plane vibrational intensities. This procedure is 9

.

J

.


159

identical to that discussed earlier in the context of the atomic charge model, and no new information is obtained. From that viewpoint no advantage is gained in choosing the atomic charge concept over the bond charge concept or vice versa. However, when we consider the molecular magnetic dipole moment derivatives in Section 8.2.3, the bond moment concept turns out to be more useful. 8.1.4 Sum rules The integrated absorption intensity (see Eq. 8.1.4) is related to (~l.ta/~Qa)2. To facilitate the derivation of the sum rule for absorption intensities let us consider the transformation between cartesian displacement and normal coordinates, as discussed in Chapter 5. Using Eq. (5.1.21), one can write the atomic polar tensor element as

(8.1.45)

c~l'ta / aXA5 - E (~)['ta / c)Qk)Sk~Afi 9 k

The summation index k goes over all 3N normal modes, comprising (3N6) vibrational modes Qa and six rotranslational modes pj. Noting that the S matrix is related to the orthonormal matrix g (see Eq. 5.1.24), one can write the relation, Z[(c3l.t~ / C3XA)2 + (bl.ta / ~yA A

)2

)2 + (~91-ta / ~}ZA ] / mA Y--,(~91-ta / ~)Qk)2" k

(8.1.46)

The right hand side of this equation can be written as two separate sums, one for the vibrational normal modes Qa, and another for the six rotranslational modes 9j. That is,

E[(b~t a ] ~)XA)2 + ( ~ a A

] c)yA)2 + (~ltla / ~)ZA)2] ] mA =

)2 9

E (~)ga / ~Qa)2 + E (c}ga / ~)Pj a

(8.1.47)

j

The contribution from rotranslational modes can be easily calculated. In the case of translational modes p~, (O~ta/3pl) = 0 for neutral molecules. *dr

The derivatives (O~t,/3or) for rotational modes can be determined 16 from

Chapter 8

160

the relation (see the notation in Appendix 1),

/11/2

oalLtoJOaP; - ~:o~13~,l-h,,~l 3

(8.1.48)

9

Substitution of Eq. (8.1.48) into Eq. (8.1.47), gives E(ogc~ /OQa) 2 - E[(ogoc /OXA) 2 + (Ogcz / OYA)2 + (O~to~ / OZA)2 / mA] a A T1/2 (8.1.49)

- %~ g~ /-~f~

9

If the molecule under consideration has no symmetry, then for a given normal mode all three components of 3gtct/~gQa can be non-zero. In such cases, the integrated absorption intensity depends on the sum [(agx/OQa) 2 +(~gy/~Qa)2 +(agz/~Qa)2], and the sum rule becomes, 17 A a - K~ a

(Ogtoc / ~gQa)2 a

=K~[(Oga/Ox A

)2 +(Ogtcz/3YA )2 +(Ogcz/ozA )2 / m A ] -

A K{(gz 2 + gy2) / ix x + (gz 2 + gx 2) / Iyy + (gx 2 + gy2) / Iz z },

(8.1.50) where K - ~ / 3 c 2 (see Eq. 8.1.4). For molecules with enough symmetry, a normal mode can lead to a change in the dipole moment along only one of the three axes, so only one of the three components of/ggoJ~gQa will be non-zero. In such cases Eq. (8.1.50) can be written separately for modes of different symmetry. The atomic polar tensor elements ~ga/~gXA8 are invariant to isotopic substitution. So the isotopic dependence on the sum of vibrational intensities can be seen from Eq. (8.1.49) to arise from the atomic masses mA (which also affect the moment of inertia II~l~). It is of interest to note that the atomic polar tensor elements themselves satisfy some sum rules" 18

E c)/s / ~ A5 - 0 , A

(8.1.51 )


Z ~/1$8 (~)~r [ ~)XA~)XA~ - %t[ie ~e 9 A

161

(8.1.52)

Eq. (8.1.51) is called the translational sum rule and can be obtained from the relation (Oga/OP;) - 0 = ~ ( O g a / OXA[~)(OXA[~ / OPt) and Eq. A

(5.2.17); Eq. (8.1.52) is called the rotational sum rule and can be obtained from (Oga/~)P~) - ~ ( 3 g a / 3XAfi)(3XAfl / 3p~) and Eqs. (8.1.48) and A

(5.2.17). A second intensity sum rule 16 can be obtained by transforming the normal coordinates Qa to internal coordinates Ri. Using Eq. (5.2.28) and Eq. (5.2.33), one can write Z(~)gtx / OQa) 2 / 4~2c2v2 - Z(~)ga / ORi)(~)gtx / ~)Rj)Fij 1 . a i,j

(8.1.53)

The inverse force constants F.1j1 are invariant to isotopic substitution. The values of ~ga/~Rj for isotopically substituted molecules are related as

(8.1.54/ where the subscripts 1 and 2 identify the isotopically substituted molecules. The matrices A and ~ were discussed in Chapter 5. Eq. (8.1.54) can be obtained by left-multiplying Eq. (5.2.16) with 311/3X. The sum of frequency weighted integrated absorption intensities can be written [using Eq. (8.1.53) and (8.1.4)] for different isotopically substituted molecules and related to that of the parent molecule using Eq. (8.1.54). 8.2. Vibrational circular dichroism (VCD) spectra

The magnetic dipole transition moments associated with vibrational transitions were discussed in Chapters 3 and 4. These and the electric dipole transition moments discussed earlier determine the spectral pattern in VCD spectra. 19 In fact the only experimental data currently available for deducing the magnetic dipole transition moments are the vibrational circular dichroism (VCD) intensities. VCD intensities may be used to deduce some molecular properties. Alternately theoretical approaches for

Chapter 8

162

predicting VCD intensities may be used to determine the molecular conformations and absolute configurations. These approaches are discussed below.

8.2.1 Experimental VCDintensities The circular dichroism intensities are determined by the rotational (some times called as rotatory) strength (see Eq. 2.5.8). For vibrational transitions, we proceed with the approximations discussed in Section 8.1.1. Then for a fundamental vibrational transition associated with a dimensionless normal coordinate qa, the rotational strength is given in the double harmonic approximation as, Ra

=

(1/2)(3~ttff/)qa)(/)mJ~)pa) (h/4~) (/)~tx/OQa)(~)rn~//)Qa), =

(8.2.1)

where the relations q2 = ctQ2, p _ 0 , and p2 - 4/i;202]h2~, have been used to write the derivatives in terms of Qa and Pa (see Appendices 1, 3); ~ttx and met are in molecule fixed axes. Following Section 8.1.1, the integrated circular difference absorption coefficient (see Eq. 2.5.8)) can be written as, A ~ - (8x2V0aN / 3c) (/)~ta //)Qa)(/)mtx / t)0a) 9

(8.2.2)

Expressing (O~ttx/OQa) in units of D/(A.amu 1/2) and (bma/~0a) in D.sec/(A.amu 1/2), this equation gives ANa in cm2/mol.sec. To use the z ~ in familiar units of kngmol, the above equation is divided by c, i.e., A.~ - (8~2V0aN / 3c 2 ) (t)~ttx / ~)Qa)(Oma / t)0a ) = (8/1;2V0aN / 3c) (t)~ta / ~)Qa)(/)ma / ~)0a ) ,

(8.2.3)

where V0a and V0a represent the peak position of the circular dichroism band a in Hz (or cyc/sec) and cm -1, respectively. Substituting the constants, one obtains the useful relation, (A.~ / V 0 a ) - 1063.2 (~)l.tct//)Qa)(/)ma / / ) 0 a ) ,

(8.2.4)

where the units are" AAa in km/mol, V0a in cm-1, (~)kta//)Qa) in D/(]k.amu 1/2) and (/)ma/O 0 a) in D.sec/(A.amul/2). The product (/)ktJ/)Qa)(/)mtx/~)0a) is more transparently written as

163


(al~x/aQa)(amx/a0a) + (a~ty/aQa)(amy/a0a) + (al~z/aQa)(amz/a0a). The changes in electric dipole and magnetic dipole moments will occur along the same axis (x, y or z) only for chiral molecules (which do not possess plane of symmetry or inversion symmetry; see Chapter 10). For certain chiral molecules, one can find vibrations which have non-zero values for only one of the three components of the derivatives (~tcx/~Qa) and (~rn~/~(~ a). In such cases Eq. (8.2.4) will contain only one product term, say (iglLtz/~gQa)(igmz/ig0a) and the experimental VCD intensity data permits the determination of (Omz/igl~a), provided the information on (Ol.tz/OQa) can be obtained from vibrational absorption data (see Section 8.1.1) or other means. In the vibrational-rotational spectra the intensities of P, Q and R branches are determined by different components of these derivatives, so vibrational-rotational circular dichroism 20 is one unique source to derive the information on individual product components of Eq. (8.2.4). Since all of the three product components (Ol.ta/OQa)(OmoJO(~a) need not have the same sign, the P, Q and R branches of vibrational-rotational circular dichroism need not have same sign and this in fact has been observed 2~ for the enantiomers of methyloxirane. Just like the electric dipole moment derivatives, the magnetic dipole moment derivatives can also be reduced to the internal coordinate or cartesian coordinate space as,

(a

/a %) -

a

(amdaOa)(a Qa/a Rj))

=

-1 , (3moJig(~ a)Laj

(8.2.5)

fi

-1

B

-

a

(a~/aO a)(aQ a/a:~a 19= ~ (amojaOa)S a,Bl3 a

,

(8.2.6) where the L and S matrices are discussed in Chapter 5. The derivatives in cartesian space, (Oma/OX B[~), are referred to as atomic axial tensor (AAT) elements; the nine elements (o~ = x, y and z and [3 = x, y and z) for a given atom being the AAT for that atom. These derivatives in internal and cartesian coordinate space can be further related to the bond or atomic properties (see Section 8.2.3).

8.2.2 Quantum mechanical methods The evaluation of magnetic dipole moment derivatives (and hence VCD intenstivies) faced some problems in the initial stages of development. Different approaches are now available to overcome these

Chapter 8

164

problems, and these approaches are described below. It is convenient to separate the nuclear and electronic parts of magnetic dipole moment as, ma - manuc + mae l _ (1)EeZAE;o~[3~ ' XA[3J~A~' A

2c1 Eeca[~,Xi~Xi ~, i

(8.2.7)

where the summation index A runs over all atoms and i over all electrons. The derivatives of m~uc are straightforward to evaluate, so c)m~uc/~)XB~5 - ( 1 ) ] [ 2 eZAea[~ XA[~SAB~5~,~5 " A

(8.2.8)

The electronic part requires careful consideration. If we write the total wavefunction as a product of the electronic and vibrational wavefunctions (see Chapter 3, and Eq. 8.1.14), the electronic part of magnetic dipole transition moment integral, for vibrational transition from a~ to a~', in electronic state s, becomes 0

The integral

0

el

~s0 mael ~s0) is zero for molecules in non-degenerate

electronic states. A simple way to understand the reason for this is that the hermitian property requires 1(~0 reel/g0)l* = (/lt01mel 11101. Since m el is an imaginary operator, these integrals satisfy the hermiticity by yielding a zero value. As a consequence the numerical method (Section 8.1.2A) used for electric dipole moment derivatives cannot be used for magnetic dipole moment derivatives, (the electronic part of the magnetic dipole moment (~0 mel Xl/s O) calculated at displaced geometries is also identically zero). A second problem associated with the magnetic dipole moment derivatives is the origin dependence of the calculated rotational strength. Since rotational strength is an observable, it should be independent of which molecular origin one uses to calculate the electric and magnetic dipole moment derivatives. The objective then is to find ways that would restore the electronic contribution to the magnetic dipole moment derivatives, and that would yield origin independent rotational

165


strengths. (A) Localized molecular orbital method 21 In the molecular orbital approach, Eq. (8.2.9) can be rewritten, for a closed shell system, as o o (~sWv'

o /Itso Iliad) - -2(e / 2c)(/lt O,

k

(8.2.10)

where q)k represents a doubly occupied molecular orbital. The instantaneous position of electrons in the kth molecular orbital, Xk~ is written as a sum of their average positions (or orbital centroid Xk0,~) and deviations AXk~ from Xk0,13. Furthermore J~ 1 _ =

and this relation holds for higher order operator products such as Paq~ (see Appendix 2). As a result, one can evaluate the anharmonic vibrational magnetic dipole transition moment, for ~a to ~ a transition, as when m is expanded as in Eq. (3.8.7). (C) Analytic nuclear electric shielding tensor method 27 The situation here is somewhat different from the one encountered for electric dipole moment derivatives. In the presence of a magnetic field the energy of interaction, Eint = - ma'Btx'. Expanding the magnetic dipole moment as a function of nuclear velocities, Eint can be written as Ein t - - ~ (~m a, / C}XAs):~AsBa,. A

(8.2.25)

The time dependence of XA~5and Ba' can be written respectively as X0,A5 cos cot and B0,a' cos cot and the sum J~ AsBa' + XASI3~' can be seen to be -mXO,ASBO,a' sin 2rot. In the static limit, m--,0 so it follows that J~ AsBa' - - XASBa'. Then Eq. (8.2.25) can be written as

Chapter 8

170

(8.2.26)

Eint - Z (t)mtx / t))( AS ) X As /3 oc', A

which suggests that potential energy derivatives with respect to nuclear displacements can be written as a function of 13a,. That is, (8.2.27)

(o~V ] O~XA(5) - (c~V ] O~XA(5)0 + (bm a, / c)]~AS)]3oc,+... ,

where (c)V/t)XAS)0 is the value of (c)V/t)XAS) at ]~oc' = 0. The electric field FA8 at nucleus A is equal to the force at that nucleus per unit charge, so Eq. (8.2.27) can be used to write the expansion of FA8 as, 7'10'27 (8.2.27a)

FAIl- FA8,0 + (0FAS/~)13c() ]3ct' + ... , where FA8,0 is the value of FA8 at Btx' = 0 and

(8.2.27b)

~)FAS/~)13ct' - - (1/eZA)~)rn~'/~)X A8.

We can write F A8 as the sum of nuclear and electronic contributions, Fnuc el respectively, as in the case of an electric field perturbation. A8 and FA8 Using the expressions for general molecular properties, obtained using the time dependent perturbation theory [see Eqs. (2.3.23)-(2.3.27)], the coefficient of B~, can be obtained. For the present case, the symbols containing P~ in Eqs. (2.3.23)-(2.3.27) are replaced by l~Ala. Then the .

coefficient of B~, is given as (3F~8/313ct, )- Kk,8o ~, - Im I - ~ 4rCFA8,snmct',ns /h(CO2s-o)2) k n~:s

where FA8,sn is given by Eq. (8.1.29). equation reduces to

1,

(8.2.28)

In the static limit the above

22]

K'A,8a, - Im[-(h / 11;)n#sZ4g2FA8,snma',ns / h tons

.

(8.2.29)

This equation can be written using Eqs. (2.2.12), (8.2.20) and (8.2.21), as


171

Kk,8cx, - (-h / g)(1 / eZA)Im[((OVs / OXAs)I(OVs / 3Ba,)) ] = (h / re)(1 / eZA)Im[((~)Vs / 3Bo~,)l(~)gts / 3XA8)>] = -(1 / eZ A )(0m~' / 0J~A8 )"

(8.2.30)

The last equality follows from Eq. (8.2.24). From Eqs. (8.2.28) and (8.2.30), or from Eq. (8.2.27) it follows that, (~)F~5 / ~)13a,) - -(1/eZA)(~)m ~, / ~)XA~5) 9

(8.2.31)

The total contribution is obtained by adding Eq. (8.2.31) to the nuclear contribution. Thus the electric field at the nucleus, in the presence of the external dynamic magnetic field, is given as 7 FA8 - FAS,0 + [-(1 / 2c)eu138XAi3 + K~,Sa,]B a, .

(8.2.32)

The electronic contribution K~,sa, shields the nuclei from external field and therefore K~,sa, is called the nuclear electric shielding tensor. The relation between K~ and (Om/OXA), given by Eq. (8.2.30), allows the calculation of magnetic dipole moment derivatives, and hence VCD intensities, using nuclear shielding tensors. (D) Vibronic coupling method28,29 In Section 3.1, the vibrational equation was obtained by assuming that the nuclear kinetic energy operator does not influence the electronic wavefunction. This influence can be treated as a perturbation, so the nuclear kinetic energy perturbation is given, to first order, in dimensionless normal coordinates as

a

where ( 0~-%a/ elec operates only on the electronic wavefunction, and (~)

on nuclear wavefunction (see the discussion following Eq. nuc (3.1.5a)). Note that the nuclear wavefunction includes the vibrational

Chapter 8

172

wavefunction (see the discussion preceeding Eq. (3.1.9)). In writing Eq. (8.2.33) the second order term

elec

is ignored. The wavefunction

corrected to first order (see Eq. (8.1.16)) due to the perturbation given by Eq. (8.2.33) becomes,

0~sO\/ 0 t)~Oa\

~-~a/~ ltl/~a t)qa / 0 0 (V0 End--0~_EO~-~-0_ Es -~--OEaga ~/n~/aga'"

0 0 ~l/s~-- ~ts ~/~ + a~ hVan~s ~

(8.2.34) Assuming that (E~ EOa - E 0s - E 0~a)Can be replaced by (E0 _ Es), o the electronic part of the magnetic dipole transition moment can be obtained as follows. Ignoring the term that is second order in energy, and noting that ipa O/Oqa, one obtains the following equation for the ~a to v' transition: --

ma,~a~ e, a

a

_ +ihVa ~ [ (~01m~l~0) (Xl/0 3~t~ ng:s

E0 -E0

(W~

+ E~ ~

a ]Pal~Oa)

c)qa

O) (~0 t)~O\ (I[[/Oa IPa[VOa)]" (8.2.35)

qa/

On the right hand side of Eq. (8.2.35), the first term is zero; the second and third terms are equal (remember hermiticity and the integrals in Appendix 2). Thus, m~,ajaa~a -ihva[(~O~ IPal~Oa)-]

X '~

oel0(

(Vs-~--~ ImalVn) n~:s En-Es

~o

~)V0 (8.2.36)

With the use of Eq. (8.2.21), one can show that Eq. (8.2.36) is equal to Eq. (8.2.22). The vibronic coupling model emphasizes the use of Eq. (8.2.36), with an explicit evaluation of the summation over excited electronic states. The rotational strength calculated in the vibronic coupling method is


173

origin dependent. To overcome this problem, a distributed gauge origin approach has been implemented. 30 (E) Localized orbital-local origin method 31 An elegant approach to the use of localized molecular orbitals in predicting rotational strengths is the localized orbital/local origin (LORG) method 31. The reader should consult the original work for more details on this method.

8.2.3 Classical models Unlike for molecular electric dipole moment derivatives, the consideration of molecular magnetic dipole moment derivatives requires the use of both atomic and bond charge concepts. In formulating these derivatives it is necessary to verify that the resulting molecular magnetic dipole moment derivatives lead to a rotational strength that is independent of molecular as well as bond origins. Some difficulties were encountered in the initial stages, but the formulation is now well understood. (A) Atomic charge concepts With the effective atomic charges the magnetic dipole moment is written as ma - (1/2c)~ ea[~, e~, XAI3X A~. (8.2.37) A In the fixed partial charge approximation, the derivatives of the molecular magnetic dipole moment can be obtained by straightforward differentiation. For example, the atomic axial tensors are given as c)mec/c)]~B8 - (1/2c)~ ec~13~,e~AXAI38AB878. The derivatives in internal A coordinate space ~ma/O l~j and normal coordinate space Ome~/~(~a, can be obtained similarly using the transformation matrices A and S (see Chapter 5). The incorporation of a charge flow contribution into Eq.(8.2.37) is not as straightforward. First we need to switch over to the bond concept to address this issue. (B) Bond charge and bond m o m e n t concepts 32-34 In order to develop an expression for the magnetic dipole moment with these concepts, it is helpful to consider a triatomic molecule A-B-C. Using the discussion presented earlier for bond charges (see Section 8.1.3B) and noting that e~B = -e(~+~C), Eq. (8.2.37) can be written for a molecule A-B-C, as

174

Chapter 8

mix = (e/2c) eal3Y[~AXA~ XAT+ ~BXB~XBy + ~cXcIIXCy) = (e/2c)

et~fiy[qA-B(XAI3XA? - XB~XB?) + qC-B(Xc~Xcy- XB[SXBy)].

(8.2.38)

For a general case, Eq. (8.2.38) becomes, mo~ = (e/2c) Z qk Z AkAetxl3)XAI3]~ Ay, k A

(8.2.39)

where AkA = + 1 when atom A is contained in the bond k and 0 when bond k does not contain atom A. To express Eq. (8.2.39) in terms of bond moments, let us relate the atomic coordinates to bond coordinates. From the definitions in the Fig. 8-1, it can be seen that the bond coordinate X k and the individual atomic coordinates XA (both with respect to a molecular origin) are related as XA = Xk + dA and dA - d B = dk; the vector dA is from a local bond origin (which can be anywhere along the bond) to atom A in that bond. Substituting these relations along with XA = X k + dA into Eq. (8.2.39) will lead us to an expression in terms of bond moments. For the sake of illustration let us consider only the part pertaining to bond A-B in Eq. (8.2.38). Then,

•

local origin for bond A-B

Dc molecular origin Fig. 8-1. Depiction of the molecular and local bond origins. mix= (e/2c)Eoc~iyqA-B[(dAit- dBIt) ]~ky + Xkl3 (dAy - dBy) + (dAI3ClAy- dalgday)] 9

(8.2.40)


175

Note that the product eqA-B(dAI3- dBI]) is equal to the bond moment gA-B,~ and the product eqA-B(dA~- dB~) is equal (in the fixed partial charge approximation) to ~tA_B,I3. Since the same expression can be obtained for each bond in the molecule, the general expression can be written as 9

o

ma=(1/2c)2 Eo~y[(gk~ Xky +Xkl3 ~tk~,)+ eqkZ Z~kAdAI3aA~,]. k A

(8.2.41)

The term (1/2c)easy eqk~AkAdA~dA~, represents the local magnetic A

moment with respect to the bond origin. In the fixed partial charge approximation, gka is given as ka = eqk clka,

(8.2.42)

and in that approximation Eqs. (8.2.39) and (8.2.41) are equal to each other, so the molecular magnetic dipole moment derivatives are conveniently obtained by differentiating Eq. (8.2.39). Now, from Eq. (8.2.42) the charge flow contribution can be expected to have the form e/1 kdka; then the total contribution to g ka is ~t koc = e q k cl k a + e

dlkdko~

(8.2.43)

9

Therefore the magnetic dipole moment can be written, with charge flow contributions, in a computationally convenient form by adding the charge flow contribution to Eq. (8.2.39) as

mc~= (1/2c) Z eolian,[eqk~ AkA XA[3XA~,+ edlk Xkl3 dk~,] 9 k

(8.2.44)

A

The atomic axial tensor elements now can be written as ~)ma/~)X B~5= (1/2c)~ eoq3~,[eqk~ AkA XAI35AB578 k

A

+ e(0qk/0XB~5) Xkl3 dk,r .

(8.2.45)

The other derivatives, Omd~ll~j and ~)mc~/OQa can be obtained from Eq. (8.2.45) using the transformation matrices A and S (see Chapter 5). These expressions lead to a rotational strength that is independent of the choice

176

Chapter 8

for molecular as well as bond origins. One may transform the bond charge flow contribution in Eq. (8.2.43) into an atomic charge flow contribution, but the resulting expression would not be as simple and convenient as the one obtained with bond charges. Another alternative 35 is to write e~A as a sum of bond currents IAB and hence e~)~A/~)Qa as the sum of C)IAB/C)(~ a. (C) Ab initio bond charge concepts The bond charge parameters represent bond charges and their variations for a given change in the bond length or angle. These parameters are involved in the prediction of absorption and circular dichroism intensities using the bond charge (or bond moment) model. Some estimates for some of these parameters are available 36 in the literature, but they were not optimized to represent (or to reproduce) the vibrational intensities in a general class of molecular systems. Thus a set of generally applicable parameters are lacking at the present time. This deficiency can be overcome by combining the classical concepts and ab initio methods as follows. The molecular geometries and electric dipole moments for numerous molecules have been (or can be) obtained at very high levels of ab initio theories. 37 These data can be transformed in to bond charge parameters with very little extra effort, by noting that the changes in electronic structure from molecule to molecule reflect not only as changes in dipole moments, but also as changes in bond lengths and angles. The molecular dipole moment ~ta can be written in terms of bond charges using Eqs. (8.1.42) and (8.1.43). Expanding bond charges around a reference geometry, e q k - eqk,0 + (Oeqk/~)rk) Ark + ~ (~)eqk/30k) A0k + ... , k

(8.2.46)

where eqk,0 is the bond charge at the reference geometry (at bond length rk,0 and bond angle 0k,0); A r k - rk-rk,0 and A6k = 0k-0k,0 are changes from these reference geometries; the summation in the expansion given above is over angles that contain the bond k. In the above expansion one can also include first order derivatives with respect to neighboring bond lengths and angles that do not contain the bond k; also higher order derivatives can be included. However they can be considered after evaluating how well the first order approxiamtion given by the above expansion performs. Since each single bond C - X will have three angles surrounding it, there will be three parameters (~)eqk/~)0k) involving angles and one parameter (~9eqk/~)rk) involving the length, leading to a total of four parameters. These parameters can be easily optimized from the molecular geometries and dipole moments of a set of related molecules.

177


The geometries and dipole moments needed here can be obtained at the converged level, using the ab initio methods, 37 perhaps employing electron correlation. Ideally, one should have as many molecules in the data set as possible, so that the resulting parameters are more generally applicable. With the development of a set of bond charge parameters in this manner, the capability of these parameters to reproduce the vibrational absorption intensities can be evaluated. Here the bond charge parameters in normal coordinate space, aeqk/3Q, can be obtained as .~ (aeqk/aR)(aR'/aQ)j j where Rj is jth internal coordinate. In the first ordeJr approximation, only those internal coordinates that represent bond length change Ark and angle changes A0k (involving the bond k under consideration) can be used in this summation; corresponding derivatives aRj/aQ can be obtained from the normal coordinate analysis (see Chapter 5). The need for including additional bond charge parameters can be identified by comparing the predicted spectra with experimental spectra. The development of such standard bond charge parameters is important for estimating the absorption and VCD in large size biomolecules (where ab initio calculations may not be feasible). These procedures would be similar to those involved in developing standard sets of force fields 38 for bio-molecules. Combining these data sets (force fields and bond charge parameters) it may become possible that the VCD spectra for bio-molecules could be predicted conveniently using a personal computer. 8.2.4 Sum rules As discussed in Section 8.2.1, the VCD intensity is related to the product (a~ta/aQa)(arn~/a(~ a). Following the transformations discussed in the context of absorption intensities [see Eq. (8.1.45)-(8.1.47)], one can verify that, 39

(a/-t(z / aQ a)(amcz / a(~ a) - ~ (a/tot / aXAa)(am(x / aJ~Aa) / m A a

A

- ~ (a/.t(z / apj)(amcz / alSj).

(8.2.47)

J

As before, the summation index "a" runs over (3N-6) vibrational normal modes. The rotranslational contributions to the magnetic dipole moment derivatives amJOlSj can be determined from the sum rules 7,18,39,40 associated with the atomic axial tensors ama/a X Aa:

ama / A

c)]~A~ - - e(x[$~, ~t~, /

2

,

(8.2.48)

Chapter 8

178

~_.,~,5 (~gma / ~RAI3)XA'y =(e/2mp)gasI88 .

(8.2.49)

A

In the above equations mp is the mass of proton, gas is the so called g tensor element. The electronic part of gas is related to the paramagnetic magnetizability Xc~8via the relation, el

g ~ - -(4mpm e /

e2

(8.2.50)

))(;~ / 188 ,

where me is the mass of electron.

In terms of the localized orbital

centroids Xk0, g ~ can be approximately writtenl8a as el

gas

_

-(mp

/

188

)~-, ( X 2 ~ k0~5r 8 - X k 0 , o c X k 0 , 8 ) k

(8.2.51)

,

where the summation index runs over all occupied molecular orbitals of a closed shell system. The nuclear part of gas is given as, (8.2.52)

g~C _ (mp / I~5)~., Z A (X28o~8 - XAaXAs). A

Using the translational sum rule given by Eq. (8.2.48), and the relation given by Eq. (5.2.17b), the translational contribution (3mc~/~15~) is obtained as

~) -

(~m a //)15

~ ( 3 m a //gXAs)(3XA8 /

o~p~)- -

~a~,~t~, / 2M 1/2 .

A

(8.2.53)

This contribution, however, is not necessary for VCD intensities, because Eq. (8.2.47) contains the product (~ta//gpt)(~gma//91St) and/9~oc/~9pt is zero for a neutral molecule. The rotational contribution 3ma/alS~ is obtained |

as,

(~ma / ~0~) - ~ ( ~ m a A

/

.

.112

~)(AS)(~XA8 / ~p~) - ( e / 2mp)ga~f~ff , (8.2.54)


179

where the a matrix elements given by Eq. (5.2.17a), and Eq. (8.2.49) were used. Using Eqs. (8.1.48) and (8.2.54), we can rewrite Eq. (8.2.47) as Z (~)~a / ~)Qa)(bma / ~)(~a) - Z (~)~a / ~XAf)(~ma / ~):KA5) / mA a A -(e / 2mp) ~al3y gal3 g~, 9

(8.2.55)

The sum of rotational strengths (see Eq. (8.2.1)) is obtained by multiplying both sides of Eq. (8.2.55) with h/4~. The sum of integrated VCD intensities is given from Eqs. (8.2.3) and (8.2.55) as, A ~ /V0a - K { ~ (~)~ta / C)XA~5)(c)ma / c)]KA~5)/ mA a A

-(e / 2mp)(gxyl.t z - gxzl.ty + gyzgx - gyx~tz + gzxgy - gzygx)}

,

(8.2.56)

where K = 1063.2 with the units as defined for Eq. (8.2.4). A second sum rule 41 can be obtained by noting that Z (~)~a / ~)Qa)(~ma / ~)(~a) / 4r~2c2v2 - Z (~)~ta / ~)Ri)(~)ma / ~)l~j)Fij 1" a i,j (8.2.57) The magnetic dipole moment derivatives (/gma/Ol~j) for isotopically substituted molecules can be written, in a manner similar to the electric dipole moment derivatives (see Eq. 8.1.54), as (~9ma / c)l~j) 2 = (~9ma / ~)l~j)1

where the subscripts 1 and 2 represent the isotopically substituted molecules, and the summation i runs over six rotranslational coordinates. The 13 and A matrices were discussed in Chapter 5. As in the case of the integrated absorption intensities, the sum of integrated VCD intensities for isotopically substituted molecules can be related to that of the parent

180

Chapter 8

molecules using Eq. (8.2.3), (8.2.57) and (8.2.58). References 1 E.B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw Hill, New York (1955). 2 C. DiLauro and I. M. Mills, J. Mol. Spectrosc. 21 (1966) 386; J. L. Hylden and J. Overend, J. Phys. Chem. 87 (1983) 103. 3 W . B . Person and G. Zerbi, Vibrational Intensities in Infrared and Raman Spectroscopy, Elsevier, New York (1983). G. A. Segal, R. E. Bruns and W. B. Person, J. Chem. Phys. 50 (1969) 3811. 5 P. Pulay, Mol. Phys. 17 (1969) 197. 6 (a) A. Komornicki and J. W. Mclver, Jr., J. Chem. Phys. 70 (1979) 2014; (b) L. J. Schaad, C. S. Ewig, B. A. Hess, Jr. and D. Michalska, J. Chem. Phys. 83 (1985) 5348. 7 A.D. Buckingham, P. W. Fowler and P. A. Galwas, Chem. Phys. 112 (1987) 1. 8 Y. Yamaguchi, M. Frisch, J. Gaw, H. F. Schaefer III and J. S. B inkley, J. Chem. Phys. 84 (1986) 2262. 9 R.D. Amos, Chem. Phys. Lett 108 (1984) 185. 10 P. Lazzeretti and R. Zanasi, Chem. Phys. Lett. 112 (1984) 103; Phys. Rev. A27 (1983) 1301. 11 P . W . Atkins, Molecular Quantum Mechanics, Oxford University Press, Oxford (1983). 12 R.S. Mulliken, J. Chem. Phys. 23 (1955) 2343. 13 P.-O. Lowdin and B. Pullman, Molecular Orbitals in Chemistry, Physics and Biology, Academic Press (1964). 14 (a). S. KH. Samvelyan, V. T. Aleksanyan and B. V. Lokshin, J. Mol. Spectrosc. 48 (1973) 47; (b). J. C. Decius, J. Mol. Spectrosc. 57 (1975) 348. 15 L. A. Gribov, Intensity Theory of Infrared Spectra of Polyatomic Molecules, Consultants Bureau, New York (1964). 16 B. Crawford, Jr., J. Chem. Phys. 20 (1952) 977. 17 W. T. King, G. B. Mast and P. P. Blanchette, J. Chem. Phys. 56 (1972) 4440. 18 P.L. Polavarapu, Chem. Phys. Lett. 171 (1990) 271; P. J. Stephens, K. H. Jalkanen, R. D. Amos, P. Lazzeretti and R. Zanasi, J. Phys. Chem. 94 (1990) 1811. 19 G. Holzwarth, E. C. Hsu, H. S. Mosher, T. R. Faulkner and A. Moscowitz, J. Am. Chem. Soc. 96 (1974) 251. 20 P.L. Polavarapu, Chem. Phys. Lett. 161 (1989) 485. 21 L.A. Nafie and T. H. Walnut, Chem. Phys. Lett. 49, 441 (1977); T. H. Walnut and L. A. Nafie, J. Chem. Phys. 67 (1977) 1501. 22 P.L. Polavarapu and P. K. Bose, J. Chem. Phys. 93 (1990) 7524; J. Phys. Chem. 95 (1991) 1606.


181

23 P. A. Galwas, Ph.D. Thesis, Cambridge University, Cambridge (1983). 24 P. J. Stephens, J. Phys. Chem. 89 (1985) 748. 25 P. J. Stephens, J. Phys. Chem. 91 (1987) 1712. 26 K. L. Bak, P. Jorgensen, T. Helgaker, K. Ruud, H. J. Aa. Hansen, J. Chem. Phys. 98 (1993) 8873. 27 K. L. C. Hunt and R. A. Harris, J. Chem. Phys. 94 (1991) 6995; P. Lazzeretti, M. Malagoli and R. Zanasi, Chem. Phys. Lett. 179 (1991) 297. 28 L. A. Nafie and T. B. Freedman, J. Phys. Chem. 78 (1983) 7108. 29 R. Dutler and A. Rauk, J. Am. Chem. Soc . . . . . . 989) 6957. 30 D. Yang and A. Rauk, J. Chem. Phys. 97 (1992) 6517. 31 A. E. Hansen, P. J. Stephens and T. D. Bouman, J. Phys. Chem. 95 ( 1991) 4255. 32 S. Abbate, L. Laux, J. Overend and A. Moscowitz, J. Chem. Phys. 75 (1981) 3161. 33 J. R. Escribano and L. D. Barron, Mol. Phys. 65 (1988) 327; J. R. Escribano, T. B. Freedman and L. A. Nafie, J. Phys. Chem. 91 (1987) 46. 34 P. L. Polavarapu, Vibrational Optical Activity, in: Vibrational Spectra and Structure, ed. H. D. Bist, J. R. Durig and J. F. Sullivan, Elsevier, Amsterdam (1989). 35 M. Mosokovits and A. Gohin, J. Phys. Chem. 86 (1982) 3947. 36 M. Gussoni, C. Castigloni, M. N. Ramos, R. Rui and G. Zerbi, J. Mol. Struct. 224 (1990) 445. 37 W. Hehre, L. Radom, P. V. R. Schleyer and J. A. Pople, Ab initio Molecular Orbital Thoery, John Wiley & Sons, New York (1985). 38 A. T. Hagler and C. S. Ewig, Comp. Phys. Comm. 84 (1994) 131. 39 P. L. Polavarapu, J. Chem. Phys. 84 (1986) 542. 40 A. Rupprecht, Mol. Phys. 63 (1988) 955. 41 P. L. Polavarapu, J. Chem. Phys. 87 (1987) 6775.


183 Chapter 9 VIBRATIONAL RAMAN AND RAMAN O P T I C A L ACTIVITY

9.1 Vibrational Raman spectra The time dependent transition moments considered in Section 2.6 provide clues for the origin of dipole moment oscillations at frequencies greater or lower than the frequency of incident electromagnetic wave. Since an oscillating dipole can emit radiation at the frequency of its oscillation and serve as the source for scattered light, the wavelength of the scattered light can be larger or smaller than that of the incident light. Such scattering is referred to as Raman scattering. When the energy difference between the incident and scattered light components corresponds to the energy difference between molecular vibrational levels the process refers to vibrational Raman scattering. Vibrational Raman spectral intensities are related to the molecular electric dipole polarizability derivatives. Quantum mechanical methods to predict these derivatives have been developed, which permit the prediction of Raman spectral patterns and hence structural characterization. The molecular polarizability derivatives may in turn be related to the bond or atomic polarizabilities and their gradients, providing a pathway to obtain the information on local properties in a molecule. These different aspects are considered here. 9.1.1 Vibrational Raman scattering cross sections The vibrational Raman spectral intensities are represented as Raman scattering cross sections. To make a connection between these cross sections and molecular parameters, we will start with the transition polarizabilities given by Eq. (2.6.3). In the approximation that we can write the total wavefunction as a product of electronic, vibrational and rotational wavefunctions, (see Section 3.1), the wavefunctions for state s, f and n in (Eq. 2.6.3) can respectively be written as ll/el]/l)ll/J,/I/ell/ag'll/j' and ~e"~v"gtj", where we imposed the restriction that the initial and final vibrational levels v and v' and rotational states J and J' belong to the same electronic state. As long as the difference between the states II/elt/~ll/J and ~ e ~ ' ~ J ' is much smaller than the energy difference between ~e~v~J and ~e"~"~J", the frequencies COnsand COnfin Eq. (2.6.3) can be considered to be approximately equal and colf-o)2 can be replaced by co Zs-C02. Furthermore, if the energy difference between the states ll/e"/l/9"lltJ" and lllel[/~311/J is much larger than the energy of the incident radiation, i.e., 2).. can be replaced by 1/COns. This is referred to Olns>>O~, then COns/(O~2s-CO as the off-resonance situation. For the present purposes, the rotational

184

Chapter 9

wavefunction need not be explicitly considered, as the rotational substructure associated with vibrational bands will not be pursued. Instead, the rotational part of the integrals can be replaced with classical averaging for all molecular orientations (vide infra). With these considerations the transition polarizability given by Eq. (2.6.3) becomes < ~ , l t ~ , l g t ~ > , where o~aa' is the electric dipole polarizability given by Eq. (2.2.18) [in the present notation, s and n of Eq. (2.2.18) are e and e" respectively]. Writing txaa, as a Taylor series (see Eq. 3.10.1) expansion in dimensionless normal coordinates qa and retaining only the linear term in qa, the transition polarizability becomes [(va+l)/2] 1/2(/)ctatx'/~)qa) for the l)a to 1)a+l transition and (~a/2) 1/2 (/)t~txtx'//)qa) for the ~a to a~a-1 transition. The transition dipole moment given by Eq. (2.6.2) can then be seen to depend on ((~a+l)/2) 1/2 (/)txtxtx'/~)qa)et%avat( elt~ + e_io)t), which is 9

,

.

the source for vibrational Raman scattering. 1,2 The oscillating dipole is known to emit radiation at the frequency of oscillation, so the vibrational Raman scattering occurs at frequencies 03 + (OU'aVaand t.o- o>u'aVa; these are respectively called anti-Stokes and Stokes frequencies. The intensity of light emitted by a dipole oscillating at frequency v(= co/2rt) is given 3 as 2r~3cv4kt2 s i n 2 0 / r 2, where ILtzis the magnitude of the oscillating electric dipole moment oriented along the z-axis, ~? = v/c, 0 is the angle between the direction of dipole (z-axis) and the direction of observation (y-axis) and r is the distance from the point of observation to the location of the dipole. Total intensity of light scattered into area dA is obtained by multiplying the above mentioned intensity with dA. Defining dA/r 2 as the solid angle df~, the intensity scattered into unit solid angle then becomes 2rt3c~41a2 sin 2 0. For simplicity, we can take 0 = 90 ~ In the present case, the oscillating dipole moment is the transition dipole moment given by Eq. (2.6.2) so kt2Z can be replaced by ((aga+l)/2) (~)Ctzaq/)Qa)F2 for the ~atO t)a+ 1transition and (Va/2)(/)OtzaV/)Qa)F2 for ~a to aga-1 transition. Now F 2, can be converted into the intensity of incident light as (8r~/c)I0. The ratio of the intensity of scattered light per unit solid angle to the intensity of incident light is called the differential cross section and designated as d~/df~, where tj is the scattering cross section (scattered intensity divided by the incident intensity). The above considerations have to be augmented with spatial averaging of the polarizability components and the Boltzman distribution of population among different vibrational states. For spatial averaging the polarizability component ctaa' in space fixed axes can be written in

Vibrational Raman and Raman Optical Activity

185

terms of those in the molecule fixed axes as ctcttx' = t~BB' utx~ua'[5' where u is the transformation matrix between the two coordfr/ate systems. Then ct

2

, will contain products such as uo~u~fl'u~'~utx'fl' and the averages of

the elements can be obtained 4 through trigonometrical relations. Then the end result is that the spatial averaging leads to replacing (3txtxa'/3Qa) with (45~ 2 +4[32)/45 for ct= og and with ~2/15 for ct:k tx'; t2a and [32 are the mean and anisotropy (vide infra) of the polarizability derivative tensor (3ot/OQa). To consider the Boltzman distribution, the number of molecules in vibrational state a)a is given by Eq. (8.1.3). In the harmonic oscillator approximation, the energy difference between sucessive vibrational levels is the same, so the s u m ~ na ( ~ + 1), over all vibrational states X)a, gives 3 n(1 - e -hc~a/kT) for X)a to X)a+1 transitions and n(e +hc~a/kT 1) for ~3a to X)a1 transition; n is the total number of scattering molecules. Now we are ready to write down the Raman scattering cross sections, for different geometrical arrangements. Consider the incident light to be propagating along z axis (see Fig. 9-1), with y the scattering observation axis (so, 0 = 90 ~ and the sample placed at O. Then the incident electric field vector can be along x - o r y-axes and the induced dipole moment, that emits radiation in the y-direction, can be along z-axis or along x-axis. Let us consider the individual cases. x

Fx

i -z (a)

m ,~ m a~lR~

v

(b)

Fig. 9-1. Relative orientations of the incident and scattered electric vectors, in relation to the direction of propagation of incident light.

(a) Incident electric vector along x-axis (i.e., Fa, = Fx). Here the incident electric vector is perpendicular to the scattering plane. The intensity of scattered light with its electric vector along the x-axis (that is, scattered light with polarization perpendicular to the scattering plane) can only be originating from the induced dipole moment along x-axis, so l.ta = t-tx. Only the o~ 2 term contributes to the scattering in the y-direction. XX

The

Chapter 9

186

differential Raman cross section becomes 3,4 (after taking averaging and the Boltzman distribution into account),

spatial

(dr3a / d~) - 16~4(v -T-Va )4 (h / 8~2CVa) x [(45~ 2 + 4[~2) / 45] / [+1 -T-e-T-hcva/kT].

(9.1.1)

In Eq. (9.1.1), t~a and [32 are the mean and anisotropy of the polarizability derivative tensor/)t~aa'//)Qa;

1 (~axx 3ayy 3azz ]2 Ct~ = ~ ,, t)ea + ~gea + t)ea" '

=

(9.1.1a)

1 { (3axx

i~ayy~ (baxx 3azz~ + (, _ "~)Qa - ~ Q a " + ~ ~ a - ~Qa" ~)Qa ~ a " [(Oaxy]2 (~ax~2 (~axz]21 } . + 6 t,, OQa j + "OQa j + "OQa j j '

(9.1.1b)

h/4~;2Cga comes from converting qa to Qa and 9 is the wavenumber of the incident light. It is convenient to rewrite Eq. (9.1.1) as IX= K[(9 T- 9a)4/[ga(+l-Y e+-'hc~a/kT)llSa , X

(9.1.2)

where K = (16~4/45)(h/8~:2c), Sa = 45 a- 2 a + 4 [32a is called the Raman activity and Ix is the differential Raman cross section (or as commonly called Raman intensities) for the incident electric vector along the xdirection (superscript) and the scattered electric vector in the x-directon (subscript). Since the incident electric vector and scattered electric vector are parallel to each other (both perpendicular to the scattering plane), this scattering is referred to as polarized Raman scattering. The vibrational Raman bands will appear at Stokes and anti-Stokes frequencies x7 - Va and 9 + Va and are shifted from the incident light frequency 9 by the fundamental vibrational frequency. The ratios of vibrational Raman cross sections appearing at the Stokes and anti-Stokes frequencies can be seen from Eq. (9.1.2) to be


I x (Stokes) I x (anti-Stokes) x

= (_~_9a)4 e+hc~?a/kT "

187

(9.1.3)

v + Va

Knowing the wavenumber 9 of the incident light, and Va from the Raman spectral positions, the above equation permits the determination of the sample temperature. (b) Incident electric vector perpendicular to the scattering plane (F~, - Fx) but the scattered electric vector parallel to the scattering plane. To have the scattered electric vector parallel to the scattering plane, the transition moment should be along the z-axis, i.e., g~ = gz. Then the a 2z x term contributes to the scattering, and the Raman scattering cross section becomes I x = K[(9 -T-9a)4/ga(+l-T- e+-'hcva/kT)](3 [32) .

(9.1.4)

The Raman activity Sa in this arrangement then is 3132. Since the scattered electric vector is orthogonal to the incident electric vector, this scattering is referred to as depolarized Raman scattering. (c) One may consider other relative orientations of incident and scattered electric vectors. For example, both Ixy and I yz also represent depolarized Raman scattering (remember that the superscript on I is for the direction of incident electric vector and subscript for scattered electric vector); I ux - I xx + I xZ ~ (u represents unpolarized)"~ ~ = I xX + I yX and so on.

From

the above discussion it can be seen that the ratio IX/I x = 3 z x - 2a + 7 132a). These ratios are called [321(45 ~- a2 + 4 ~a) and ~ /i ux = 6 132/(450~ depolarization ratios and can be used to identify the symmetry nature of a vibrational band. For vibrations that span the totally symmetric irreducible representation (see Chapter 10), d~a will be non-zero, so the depolarization ratio will be less than 3/4 or 6/7, depending on the type of scattered intensities measured. A more convenient way to analyze the polarization properties of Raman scattering is to evaluate the Stokes parameters associated with scattered light. The representation of polarized light and Stokes parameters are described in Appendix 4.

Chapter 9

188

9.1.2 Quantum mechanical methods To predict the vibrational Raman scattering cross sections we need to determine the mean and anisotropy of the polarizability derivative tensors. They can be calculated using different quantum mechanical approaches. (A) Numerical atomic displacement gradient method In the off-resonance situation, the electric dipole polarizability can be approximated by the static polarizability given by Eq. (2.2.18) [with s and n representing the ground and excited electronic states respectively]. In the near resonance situation, one needs to consider the frequency dependent polarizability given by Eq. (2.3.16). In either case, the polarizability expression (being dependent on the summation over electronic states) posed serious problems in determining accurate polarizabilities. Nevertheless, with suitable approximations to this sumover-states expression, polarizabilities have been evaluated. 5 Development of the CPHF method (Section 7.2.2) eliminated the need to explicity calculate the sum over all excited states. Time independent perturbation theory gives an expression for the correction to the wavefunction (resulting from a perturbation) in terms of the sum over excited states. Using Eq. (2.2.12), with H1 = -lal?,FI3,the derivative of the wavefunction with respect to Ffi becomes, o

-

o

Vn/(E s - En0).

(9.1.5) nCs These derivatives can be evaluated using the CPHF method, without the need for knowing the wavefunctions of excited states. Substitution of this equation into Eq. (2.2.18) gives (9.1.6)

otafl = 2 .

Thus static electric dipole polarizabilities can be calculated without worrying about the excited electronic states using the CPHF method. Once the polarizability has been determined, the polarizability gradients can be obtained numerically by evaluating t~tx[~ at displaced geometries. Thus, -

=

+

XB~5) ,

(9.1.7)

where t~a13 + is the value of o~afl when coordinates XB~i of atom B is increased to X~ 5 and o~tfi is the corresponding value when XB6 is


189

decreased to XB~5. Derivatives determined in this manner are not only time consuming but also suffer from numerical inaccuracy. (B) Numerical electric field gradient method In discussing the quantum mechanical methods for electric dipole moment derivatives, we mentioned that since the energy of interaction in the presence of an electric field is -kttxFa, the differential of energy with respect to Ftx gives -kt~. So, O~ct/OXB~5can be written as OtPBS/t)Ft~where t~B is the force on atom B (see Eq. 8.1.12). Since the induced electric dipole moment is given as t~ctl~Ffl, it becomes clear that the polarizability derivatives can be written as 6 (t)O~tx~/t)XB~i) = (~)2q~BS/t)FI3t)Foc) .

(9.1.8)

As the forces (PB8 can be calculated using analytic expressions, the polarizability derivatives can be determined by numerical differentiation of forces with respect to the electric field (which has to be incorporated into the hamiltonian in solving the Hartree-Fock equations). Better numerical accuracy is obtained by using both positive and negative electric field perturbations, so t)ctotl3/OXB8 was written for t~ r 13as7 t)O~tx~/~XB8 = [ q0B~5(o~Ffi) F+ + -

- + ) - q~B~5(otF~) F +q~BS(FotFi3

+ ~BS(F~F/3)]/(F +

(9.1.9)

+

Here Fa and F~ represent positive and negative electric field values and +

+

g~B~i(F~FI3), for example, represents the force on atom B in the presence of perturbations F + and F;. This approach is computationally more efficient than the numerical atomic displacement method discussed above, but still suffers from inaccuracy due to the numerical derivatives involved. (C) Combined atomic displacement-electric field gradient method An alternate approach 8 is to incorporate the electric field perturbation into the hamiltonian and calculate the electric dipole moment. Then the polarizability o~o~ is obtained as the numerical derivative of g~ with respect to FI3, i.e., =

(9.1.10)

190

Chapter 9

Repeating this calculation at displaced nuclear positions allows determination of ~)txtx~)XB~i using Eq. (9.1.7). This approach is computationally inefficient. (D) Analytic energy derivative method The development of the CPHF method for multiple perturbations (see Section 7 .2 2. and . . 7 2 3) permits the evaluation of ~9ota.~/I/)XB~5 using all analytic expressions. This approach, being most efficient, is that currently used in most quantum mechanical calculations. 9,10 The polarizability derivative is the third derivative of energy, i.e., J)tlot~t)XBi5 = (i93E/JgFct~)F~JgXB8) .

(9.1.11)

So taking the energy expression in terms of the molecular orbitals and differentiating it with respect to XBS, Ftx, and FI3 leads to the derivatives of molecular orbital coefficients, which can be determined with the CPHF method. (E) Analytic electric dipole moment derivative method The polarizability derivatives can be shown to be related to electric dipole moment derivatives as ~)t~a~/)XA~5 = (~)21.tJ~)XA~5~)FI3)- The expression for/)l.ta//)XA8 (see Eq. 8.1.25) can be differentiated 10 with respect to FI3 to obtain the analytic expression for 3t~ct~/OXA~5. The resulting expression contains the coefficient derivatives and these are determined using the CPHF method. The two analytic methods discussed above should give the same results. The ab initio predicted Raman activites and depolarization ratios for CHFC1Br are listed in Table 2 of Chapter 7.

9.1.3 Classicalmodels In much the same way as the molecular electric dipole moment is considered to arise from atomic or bond charges, the molecular electricdipole polarizability can be considered to arise from atom or bond polarizabilities. It is natural to view the isolated atom polarizabilities as being spherically symmetric. Then to explain the anisotropy of molecular polarizability one has to introduce interactions among these spherical atomic polarizabilities so that anisotropy, reflecting the local environment in a molecule, is introduced into the atomic polarizabilities. This is not generally required for bond polarizabilities, since anisotropy is naturally apparent for bond polarizabilities (polarizability along the bond axis is different form that along the perpendicular axes). This is a major distinction between the atom and bond polarizability models. The molecular polarizability derivatives derived from the experimental data

191


may be interpreted in terms of these models. Alternately these models may be used to predict the molecular polarizability derivatives, and hence the Raman spectra. (A) Atom-dipole interaction (ADI) concepts In the presence of a uniform electric field Fct associated with the incident light, one may assume that an electric dipole moment is induced at each atom in the molecule. The effective electric field F~c t at an atom A, is then different from the incident field due to the presence of neighboring dipole moments. This effect can be summarized by the equation 11 F~cx = Fa - ~

TABa~ IXBfl .

(9.1.12)

B~A

In this equation, .[-tB~ is the induced electric dipole moment at atom B, TAB is the dipole Interaction tensor, 2 1 - 3 UAB,x

- 3 UAB,x UAB,y

- 3 UAB,x UAB,z

- 3 u AB,y u AB,x

1 - 3 UAB,y2

-- 3 UAB,y UAB,z

- 3 u AB,z u AB,x

- 3 u AB,z uAB,y

2 1 - 3 uAB,z

1

TAB = .

(9.1.13) 9

X B o c ) / R A B and R A B is the distance between atoms A and B. Using the effective fields, the induced moments at the atoms can be written as w h e r e UAB,oc = ( X A t x -

s

~

m

gAa = CXAFAo~ CXA[Fa Z

(9.1.14)

TABcx~l-tBI3] ,

B~:A

where tXA is the isolated atom polarizability. expression as,

One can rewrite this

(9.1.15) B~A

B

-1 5al~. Inversion of this equation gives, where AAAa~ = O~A

Chapter 9

192

I-tAa = ( ~ BABalS)F[5 , B

(9.1.16)

where B is the inverse of A matrix (these matrices are not same as the B and A matrices discussed in Chapter 5). Thus the polarizability of an atom after sampling the influence from the local environment in a molecule is given as, aAal3 - Z BABal3 , B

(9.1.17)

and the molecular polarizability as a~

- Z aAa~ A

(9.1.18)

9

It is important to note that the molecular polarizability obtained in this manner is symmetric, but the individual atom polarizability tensors IXActl3 need not be. The molecular polarizability derivatives ~gaa~gXA8 can. now.be seen to depend on the corresponding derivatives of the B matrix. Since the product BA gives the identity matrix, the derivative of B with respect to a given coordinate (cartesian, internal or normal coordinate) can be written as B' = -BA'B, where prime indicates the derivative. From Eq. (9.1.15), AA~ =-a p

a

(9.1.19)

~,

9

AABal 3 = TABotl3

(for A r B).

(9.1.20)

The derivatives of T can be obtained in a straightforward manner from Eq. (9.1.13). Eqs. (9.1.17)-(9.1.20) represent the atom-dipole interaction (ADI) model for Raman intensities. 11 In the ADI model, the determination of the molecular polarizability derivative ~gCXall/~XAi5requires a knowledge of atomic parameters IXA and bCtA/OXAIS. While etA (being one parameter for each atom) can be estimated, the derivatives OOtA/~XB8 are not easy to predict. If it is assumed that these derivatives O0tA/~gXB5 are zero, then the model represents a fixed atom polarizability hypothesis (similar to fixed partial charge model for absorption intensities).


193

(B) Bond polarizability concepts Here bonds are associated with local polarizabilties. 12 By transforming these local bond polarizabilities into the molecular coordinate system, the molecular polarizability can be written as a sum of the transformed bond polarizabilities. That is, aaf~

=

~., aka~

(9.1.21)

,

k

(9.1.22)

OCkal3 =OCka'a' Ukaa' Ukl3a' ,

where (Zka'a' represents the diagonal element of the polarizability tensor of bond k in the local bond coordinate system; the subscript a' represents the local axes (oriented along and perpendicular to the bond direction); akal3 is the bond polarizability in the molecular coordinate system. For axially symmetric bonds, aka'a" for the two perpendicular directions of the bond are the same. Then aka[~ can be written as 12b, (9.1.23)

aka[3 = OCks5a~ + (OCkp - OCks)UkotUk[3 ,

where aks and CXkpare the local bond polarizability components (of kth bond), perpendicular and parallel to the bond axis respectively; Uk is the unit vector along the bond k (see below). Restricting ourselves to the axially symmetric bonds, the molecular polarizability derivatives with respect to cartesian displacement coordinates can be written as

c)(toc~/c)XA8 = Z ~)l~kcx~/c)XA8 , k

(9.1.24)

0akalff~)XA~5 = (C]aks/C]XAS)Sttl3 + (t)(akp- (~ks)/c]XAS)UkaUkl3 + (0~kp - ~ks)(Ak/rk){ Uk~(~r

Uk~xUkS) +Uka(5138-Uk~UkS) }.

(9.1.25)

As mentioned before, if bond k with length rk is made up of atoms B and C and Uk = (Xc - XB)/rk, then Ak = 1 for A = C,-1 for A = B and 0 for A representing any other atom. If the local bond polarizability derivatives are ignored in Eq. (9.1.25), then the resulting expression represents the fixed bond polarizability model. The molecular polarizability derivatives in internal and normal coordinates can be obtained by transforming O~xafgOX using A and S matrices (see Chapter 5) as for electric dipole moment derivatives.

194

Chapter 9

For molecules where there is only one totally symmetric stretching vibration, the experimental Raman intensity data permits determination of the absolute value of (~5~/0Q). This can be reduced to O6~/OR, where R represents a stretching internal coordinate. In a large number of molecules the magnitude of 0d~/OR is found to reflect the bond order, demonstrating a useful application 13 of Raman spectral data.

9.1.4 Sum rules The electric dipole-electric dipole polarizability oq~13does not depend on the origin of molecular coordinates. So the derivatives of o~al~ with ~)~~Opt are zero. From respect to translational normal coordinates pt,, / the relation between the cartesian displacement coordinates and translational normal coordinates (see Eq. 5.2.17a), it follows 14,15 that

~., o5~(x[3 / DXA7 - 0. A

(9.1.26)

This relation is referred to as the translational sum rule for ( ( ) ~ / 0 X A T) [the derivative tensors 0~/0XA, 0~/OYA, and 0~)ZA are referred 16 to as the atomic Raman tensors]. Similarly the change of a tensor element due to rotational motions is obtained 15 as,

0o~(~13 / 0pYr - (1 / I w )(Ea~zo~z[3 + el3yZotc~Z),

(9.1.27)

which follows from the relation for a general tensor T, AT = p x T - T x p. From the relation between cartesian displacement coordinates and rotational normal coordinates (see Eq. 5.2.17a), one obtains 14,15

~,~ (()~oc13 / 0XA8)E(sTeXA8 - Eo~YX~X~ + EI3TL~~ A

(9.1.28)

This relation is referred to as the rotational sum rule for (()~al3]()XAy). The polarizability derivatives with respect to cartesian displacement coordinates can be written 16 in terms of those with repsect to 3N-6 normal coordinates and six rotranslational coordinates (just as we have done for the dipole moment derivatives; for example see Eq. 8.1.47). Here it is useful to adopt the tensor summation notation (see Appendix 5), so the 1 anisotropy 13a 2 can be written as ~a = 2 [ 3 ( 0 a ~ / 0 e a ) ( 0 a ~ 1 3 / 0 e a ) -

195


(3otcxdaQa)(OotBB/OQa)]. Then, following the procedure for dipole moment derivatives (see Eq. 8.1.47) one obtains, 2132 - Z 1 3 2 / m A - 2132 , a

A

(9.1.29)

j 1

where 132 = 132 Ax +132Ay +132 - Az and 13~x for example, is 132x -[3(O~cx~/OXA)(O~o~/OXA)- (O~cxoJOXA)(O~I3~/OXA)].The rotranslational contributions (summation over j terms in Eq. 9.1.29) depend on the equilibrium electric dipole polarizability and moments of inertia. They can be formulated in the principal coordinates of inertia as follows" Z ~ 2 - 3[(O~yy - ~ z z ) 2 / i x x J + (~XX- ~yy

*(~zz -~xx)2/Iyy

/ IZZ + ~2y(1 / IXX + 1 / I y y + 4 / IZZ)

+ IX2Z(1 / IXX + 4 / Iyy + 1/IZZ ) + CX2yz(4/Ixx + 1 / I y y + 1 / I z z ) l .

(9.1.30)

Similar expressions can be written for isotropic contributions -2 o~a = (~/3Qa) 2 but here the rotranslational contributions are zero, as the translational and rotational motions do not change the mean of c~a~. So o~a = A ~ ~/mA.

The atomic Raman parameters ~

and 13~ may

have interesting properties just like the effective atomic force constants that we discussed in Section 7.2.4. The sum rules expressions can be extended to the Raman activity by noting that the coefficients of -~a 2 and [32 a in the expression for Raman activity depend on the experimental arrangement (see Eq. 9.1.2). It is to be noted that the experimental Raman intensities depend on the exciting radiation frequency and vibrational frequency, so the above mentioned sum rules for -2cx a and 13a 2 do not apply to the Raman intensities (see Eq. 9.1.2) unless some approximations are made. The polarizability tensor derivatives transformed to internal coordinates provide another useful sum rule. The transformation (Eq. 5.2.28) from normal coordinates Qa to internal coordinates Rk, and the relation of inverse force constants (Eq. 5.2.45), provide the following

196

Chapter 9

relations" 17 ~132/4n2c2v2 - 2.

(3Cxcxl3'JCXcxl3'k- ctcxcx'JCX[313'k)Fj-l"

(9.1.31)

a

In these equations, cxcxl~j-," - Octcx~OR'j and similar relations, for other terms, with subscripts j and k representing the internal coordinates. The internal coordinate derivatives will differ for different isotopically substituted molecules and are related 18 through the relations that are analogous to the ones derived for electric dipole moment derivatives (for example, see Eq. (8.1.54)).

9.2 Vibrational Raman optical activity (VROA) spectra The Raman optical activities are determined 19 by the changes in molecular electric dipole - electric dipole polarizability. . cxcx,13 electric dipole - magnetic dipole polarizability G~f~, and electric dipole - electric quadrupole polarizability Acxl3~,,during vibrational motions. The changes in cxcx6during vibrational motions also determine the Raman activities as discus'sed in Section 9.1. The changes in G ~ and Aal3~, during molecular vibrations however are unique to vibrational Raman optical activity. In addition to providing these fundamental quantities, VROA also provides information on the absolute configurations and predominat conformations of chiral molecular systems.

9.2.1 Vibrational Raman optical activities Vibrational Raman optical activity measurements, 20 just as normal Raman measurements, can be carried out in different experimental geometries, namely 90 ~ right angle scattering, 180 ~ back scattering and 0 ~ forward scattering. Some measurements 2~ modulate the incident monochromatic laser light between right and left circular polarization states and monitor the synchronous difference in the Raman scattering from optically active samples of interest. This approach is referred to as the incident circular polarization (ICP) modulation method. In the ICP modulation method with 90 ~ scattering geometry the depolarized and polarized VROA measurements consist of determining the above mentioned synchronous difference for the scattered light with polarization parallel and perpendicular to the scattering plane (the yz plane in Fig. 9-1), respectively. The magic angle VROA measurement is similar to these two except that the scattered light with polarization vector at 35.2 ~ from the xaxis (which is perpendicular to the scattering plane; see Fig. 9-1) is monitored. In the ICP modulation method, the back and forward scattered VROA measurements are performed without an analyzer in the scattered


197

light. When the intensity difference associated with right and left circular polarization states of the scattered Raman light are measured, with a fixed linear polarization state for the incident radiation, this approach is referred to as the scattered circular polarization (SCP) modulation method. 21 When the intensity difference associated with right and left circular polarization states of scattered Raman light, with in- (or out-of-) phase circular polarization states for the incident monochromatic radiation, is monitored this approach is referred to as the dual circular polarization DCP! (or DCPII) modulation method. 22 The Raman scattering cross sections with exciting light circularly polarized, can be formulated just as was done for ordinary Raman scattering (Section 9.1.1). But here it is convenient to use the Stokes parameters of scattered light for deriving the scattering cross sections, so this discussion is deferred to Appendix 6. The Raman optical activities can be designated as Pa (analogous to Raman activities Sa), which depend on the experimental arrangement. These activities are summarized in Table 1. The terms involved in these activities for the ath vibration have the following definitions:

2 1 ~~ ~O~yy..3G'xx ~9G'yy. ~O~xx ~9O~zz](OG'xx_~G'zz] Y a = 2 { ( ~ a -~Qa)("~a - 3Qa )+('3Qa - ~ a J ~ " ~ a 3Qa" -~yy ~O~zz~(~G'yy_ .~G'zz~ + ("~9Qa - ~9Qa"" ~Qa - ~9Qa~ [0~xy ~3'xy 0G'yx~ 3O~xz(3G'xz OG'zx) +3t~Qa (~Qa + ~Qa" +~Qa "~Qa + 0Qa" 0ayz .~3'yz ~G'zy~] }. + ~9Qa (~gQa + 0Qa ~'J '

(9.2.1)

2 1 co{ (OQa .Oayy - Oaxx~ Oazz~ ~)Ayzx 8a=~ OQa" OAzxy OQa + (.CkXxx ~ a -3Qa" OQa ,~9O~zz ~9~yy.~Axyz ~OCxy.~9Ayyz Azyy OAzxx OAxxz~ + ('~Qa- ~Qa ) ~)Qa + ~Qa (" ~Qa " ~Qa + ~9Qa - ff-Q~a) O0Cxz-OAyzz OAzzy OAxxy OAyxx+~gQa (~Qa - 3Qa + oqQa " 3Qa ) O~yz ,OAzzx OAxzz OAxyy OAyyx. + ohQa t o3Qa r + o~Qa o3Qa ) }; -

(9.2.2)

-

-' 1 (~O~xx OOCyy OO~zz) ,3G'xx ~G'yy ~G'zz) 6~aGa=9 "~Qa + ~gQa+ ~9Qa"( ~gQa + ~Qa + ~Qa ""

(9.2.3)

In the above equations, Qa is the normal coordinate for the ath vibration and r is the angular frequency of the exciting radiation. The definitions

Chapter 9

198

for ~2a

and ~2a are given by Eqs. (9.1. la) and (9.1. lb).

The normal

coordinate derivatives involved in these expressions can be transformed from the derivatives (ao~ai]/aXA~,), o]-l(aC~g,/0XA~) and (aAal]~aXas) using the coordinate transformation discussed in Chapter 5. The expressions given above for the tensor invariants can be written compactly in the tensor summation convention (see Appendix 5). For example, TABLE 1 Relevant Expressionsa for Raman and Raman optical activites i

measurement type

scattering geometry

Raman activity, Sa

Raman optical activity, Pa

i

L I~- Iz' IY- IY

90~

2401 ~ c

12132

IR R -I L L

90~

4[~&aGa - ' + ~-Ya 13 2 "~1 ~ ] ~401 c

2(45(X~+13~2)

90~

401 4[~_ - ' 7 2 1 ~ ] ~c &aGa+ ~ Ta +

2 ( 4 5 ~ + 7 ~ 2)

4001 3c

w 3

I~

[~" 31 -~]2

-

L x x I x, IR - IL

R I, -

I,, IR - IL

90 ~

L u u I~-I u, IR-I L

180~

9601 I'1 2 1 !.~ Ya + ~]

4145~ + 7132]

180~

4801 [~ 2 1 ~ ] c Ta +

24~2

0o

-,, 1601 45&aGa+ c ~Ya g

4145~+7~

0o

16Olc 45 &aGa + ~ Ta-

414 + a

iR -

L

*

*

L IL

L u u - Iu' IR- IL iR _ iLL

&aGa+

~ Ya

c

,

i

+2

a

i,

,,,

a. The definitions for the quantities involved are as follows: Superscripts on the symbol I identify the polarization state of the incident radiation; subscripts identify the same for scattered radiation. R: fight CPL; L" left CPL, *" magic angle; u: unpolarized; x,y,z: linear polarization along these axes with direction of incident light propagating along z-axis, and scattered light propagating along y-axis.

2 ~'a=

1/2

[3 ( a ~ c x l ] ) ( ~G~13 falXo~ ( ~ ) ] aea d e a ) - ~" a e a ) aQa

,

(9.2.4)

199


-

~' ~)ea

~" c)Qa

(9.2.5)

9

2 was used in some An alternate definition 8a=3C0/2 eaT8(~O~a )kt'OA78~) OQa 2

literature papers and when that definition is used 8a in Table 1 should be replaced by 82/3. The experimental VROA intensities, just as Raman intensities (see Section 9.1.1), depend on the exciting frequency through the ~4 dependence for scattering, and the incident laser power. In addition, the instrumental response needs to be taken into account. To avoid these difficulties it is customary to express the experimental quantities as a normalized circular intensity differential (C1D) given as A = ( I ~ - ~ ) [ ( 1 7 + I~) ,

(9.2.6)

where the subscripts and superscripts identify the polarization state of scattered and incident radiation, respectively. The denominator (I~' + I~) II

depends on the mean and anisotropy of (Oc~a~/OQ) tensor, while the numerator (I~~' - 18~) depends on the corresponding invariants

of

inter-

ference products of (Oaaf~/OQ) and o~-l(OC~/OQ) and of (Oaa~/OQ) and (OAa.By/OQ) [see Table 1]. The expressions for A corresponding to different experimental arrangements are given in Appendix 6. As an example, for the 180 ~ back scattering arrangement A is given as

A =

I R- IL u u = iR u + i Lu

24 co ( ~ - l ~ a .

~-1~/3)

c(45 ~2a + 7 ~2a )

"

(9.2.7)

It is to be noted, however, that the CIDs are not usually reported, except for those bands that are free from the overlap of neighboring bands. For simulating the theoretical ROA spectra the ROA intensity is calculated with the expression, (I~ - I~) o~ .

(v 9 Va) 4 h p [ hcval8/~2CVa a ' +l-T-exp -T- kT

(9.2.8)

200

Chapter 9

and the corresponding Raman intensity is calculated by replacing Pa with the corresponding Sa. 9.2.2 Quantum mechanical methods From the above discussion, it becomes apparent that the evaluation of VROA intensities requires the knowledge of the derivatives (~o~a~)XAT), c0-1(3G~~XAT) and (~9Aa~/~)XA~5). The methods to determine the former are discussed in Section 9.1.2, and those for the latter two are presented here.

(A) Nuclear displacement method The expression for Ga~, given by Eq. (2.3.21), can be simplified in the non-resonance case (Ons>>O) to p

~-1 Gc~l3 - _ (h/g) Z [1/( E0- E0) 2] Im { Vn01 E 0 E0 ~ n " 0

n

-

(9.2.10)

s

The analogous expression for ~lts/~)Fa (using H1 = - ~ F a ) is,

~ g~ Illts % 0 = '~" - - i ( 2 ) 1/2 --i -- [(~+1)4(~+2)] 1/2=_ = (~+1/2) - -

-

t_[9(09-1)]41/2 -

- 21ol

=- i [(x)+3)(x)82)(~)+l)] 1/2 =- i aJ+l 3/2 _ i - i 3(--~-)

- - i 3(2 ) 312 =- i = i [~(x)-l~(x)-2)] 1/2= i = - i/2 = - =- i r ~(aJ-1)] 1/2 = <x)-21qpl~> L 4 -- i [(~+3)(a~82)(~+ 1)] 1/2

- i [(a~+l)] 3/2 L 2

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