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Inverse Problems of Vibrational Spectroscopy
 9783110943269, 9783110363968

Table of contents :
Introduction
Chapter 1. Physical model of molecular vibrations
1.1. Classical theory of small vibrations. Preliminary considerations
1.2. Simplified quantum mechanical statement of the problem
Chapter 2. Full statement of the vibrational problem
2.1. Vibrational-rotational interaction. Classical consideration
2.2. Centrifugal distortion. Classical consideration
2.2.1. Linear molecules
2.3. Quantum mechanical model
2.4. Adiabatic approximation
2.5. Vibrational-rotational interaction. Quantum mechanical consideration
2.6. Centrifugal distortion. Quantum mechanical consideration
2.7. The adiabatic theory of perturbations
Chapter 3. Consideration of the mathematical model for molecular vibration analysis. Direct and inverse problems
3.1. Introduction. Parameters of the model
3.2. Ab initio methods
3.3. Results and difficulties of using ab initio methods
3.4. Semiempirical methods
3.5. Empirical methods
Chapter 4. Vibrational problem in internal coordinates. Use of the redundant coordinate system
4.1. Models of a molecular force field
4.2. Choice of generalized coordinates
4.3. Construction of the auxiliary matrices
4.3.1. The matrix of kinematic coefficients
4.3.2. Bond stretching
4.3.3. Valence bond angle bending
4.3.4. Angle between a bond and the plane of an atomic triple
4.3.5. Angle between two planes (torsion coordinate)
4.3.6. An example of constructing the matrix B for the torsional coordinate
4.3.7. An example of constructing matrix G
4.3.8. Construction of the matrix H
4.3.9. The other matrices
4.4. Use of redundant coordinates
Chapter 5. Vibrational problem in symmetry coordinates
5.1. Use of molecular symmetry
5.2. Use of symmetry in the calculation of molecular vibrations
5.3. Calculation of the molecular constants in symmetry coordinates
Chapter 6. Ill-posed problems and the regularization method. Regularizing algorithms for constructing force fields of polyatomic molecules on the base of experimental data
6.1. Well-posed and ill-posed problems
6.2. Ill-posedness of the problem of constructing force field on the base of experimental data (inverse vibrational problem)
6.3. Mathematical formulation of the inverse vibrational problem
6.4. Ill-posedness of the problem of searching for a normal pseudosolution of the linear algebraic equation system (LAES)
6.5. Regularizing algorithms for constructing a normal pseudosolution of LAES
6.6. Nonlinear ill-posed problems
6.7. Inverse vibrational problem for a single molecule
6.8. Joint calculation of force field in a series of related molecules
Chapter 7. Numerical methods
7.1. Searching for eigenvalues and eigenvectors
7.2. Minimization of functional. Simple constraints
7.3. The linearization method
7.4. Calculation of the functional gradients
7.5. Estimate of operator error
7.6. Estimate of the measure of incompatibility
7.7. Choice of the regularization parameter
7.8. Projection of gradients in symmetry coordinates
Chapter 8. Analysis of band intensities in vibrational spectra of polyatomic molecules
8.1. Classical and quantum mechanical consideration of intensities in molecular vibrational spectra
8.2. The mathematical model
8.3. Statements of the inverse electrooptical problem
8.4. Computational aspects
Chapter 9. Numerical implementation of algorithms for solving problems of vibrational spectroscopy
9.1. Principles of construction of the software package
9.2. Structure of the input file
9.3. Detailed structure and description of the software package of programs
9.4. Symmetry analysis package
9.5. Some other useful options
Chapter 10. Examples of molecular force field calculations on the basis of experimental data
10.1. Some preliminary remarks
10.2. Force field of the water molecule
10.3. Force fields of the transition metal oxotetrafluorides
10.4. Force field of fiuoroform
10.5. Example of joint treatment of force fields of two molecules
Chapter 11. Joint treatment of ab initio and experimental data in molecular force field calculations with Tikhonov's method of regularization
11.1. Ab initio force fields in regularizing procedures
11.2. Computational details and practical aspects of calculations
11.3. Examples of using ab initio data in force field calculations
11.4. Regularization procedure for empirical scaling of quantum mechanical force constants
Appendix A. Systems of units used in vibrational spectroscopy
A.l. International System (SI)
A.2. Measurement of frequencies in spectroscopy
A.3. CGS-based system
A.4. Measuring force field in frequency units
A.5. Measuring force field in atomic units
A.6. Conversion between different systems
A.7. Electrooptical parameters
Bibliography

Citation preview

INVERSE A N D ILL-POSED PROBLEMS SERIES

Inverse Problems of Vibrational Spectroscopy

Also available in the Inverse and Ill-Posed Problems Series: Volterra Equations and Inverse Problems A.L Bughgeim Elements of the Theory of Inverse Problems AM. Denisov Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularizaron, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P. Tanana Inverse and Ill-Posed Sources Problems Yu.E Anikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P. Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A. Asanov and ER. Atamanov Formulas in Inverse and Ill-Posed Problems Yu.E. Anikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.E Anikonov Ill-Posed Problems with A Priori Information V.V. Vasin and A.L Ageev Integral Geometry of Tensor Fields V.A. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

Related Journal: Journal of Inverse and Ill-Posed Problems Editor-in-Chief: M M Lavrent'ev

INVERSE AND ILL-POSED PROBLEMS SERIES

Inverse Problems of Vibrational Spectroscopy AG.Yagola, I.V. Kochikov, GM. Kuramshino and Yu.A. Pentin

IUM SP II I

Utrecht, The Netherlands, 1999

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax: +31 30 693 2081 E-mail: [email protected] Home Page: http://www.vsppub.com

© V S P BV 1999 First published in 1999 ISBN 90-6764-304-1

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands by Ridderprint bv,

Ridderkerk.

Contents

Introduction

1

Chapter 1. Physical m o d e l of molecular vibrations

9

1.1. Classical theory of small vibrations. Preliminary considerations

.

10

1.2. Simplified quantum mechanical statement of the problem

18

Chapter 2. Full s t a t e m e n t of the vibrational problem

27

2.1. Vibrational-rotational interaction. Classical consideration

27

2.2. Centrifugal distortion. Classical consideration

32

2.2.1. Linear molecules

33

2.3. Quantum mechanical model

34

2.4. Adiabatic approximation

40

2.5. Vibrational-rotational interaction. Quantum mechanical consideration 2.6. Centrifugal distortion. Quantum mechanical consideration 2.7. The adiabatic theory of perturbations

43 ....

45 46

Chapter 3. Consideration of t h e mathematical m o d e l for molecular vibration analysis. Direct and inverse problems 53 3.1. Introduction. Parameters of the model

53

3.2. Ab initio methods

55

3.3. Results and difficulties of using ab initio methods

58

3.4. Semiempirical methods

62

3.5. Empirical methods

64

vi

A. G. Yagola. Inverse problems of vibrational spectroscopy

Chapter 4. Vibrational problem in internal coordinates. Use of the redundant coordinate system 4.1. Models of a molecular force field 4.2. Choice of generalized coordinates 4.3. Construction of the auxiliary matrices 4.3.1. The matrix of kinematic coefficients 4.3.2. Bond stretching 4.3.3. Valence bond angle bending 4.3.4. Angle between a bond and the plane of an atomic triple . 4.3.5. Angle between two planes (torsion coordinate) 4.3.6. An example of constructing the matrix Β for the torsional coordinate 4.3.7. An example of constructing matrix G 4.3.8. Construction of the matrix H 4.3.9. The other matrices 4.4. Use of redundant coordinates Chapter 5. Vibrational problem in symmetry coordinates 5.1. Use of molecular symmetry 5.2. Use of symmetry in the calculation of molecular vibrations . . . . 5.3. Calculation of the molecular constants in symmetry coordinates .

71 71 74 77 77 79 80 81 82 83 84 85 86 87

101 101 Ill 119

Chapter 6. Ill-posed problems and the regularization method. Regularizing algorithms for constructing force fields of polyatomic molecules on the base of experimental data 125 6.1. Well-posed and ill-posed problems 125 6.2. Ill-posedness of the problem of constructing force field on the base of experimental data (inverse vibrational problem) 130 < 6.3. Mathematical formulation of the inverse vibrational problem . . . 133 6.4. Ill-posedness of the problem of searching for a normal pseudosolution of the linear algebraic equation system (LAES) . . . . 137 6.5. Regularizing algorithms for constructing a normal pseudosolution of LAES 142 6.6. Nonlinear ill-posed problems 146 6.7. Inverse vibrational problem for a single molecule 150 6.8. Joint calculation of force field in a series of related molecules . . . 155

Contents

vii

Chapter 7. Numerical methods

161

7.1. Searching for eigenvalues and eigenvectors

162

7.2. Minimization of functionals. Simple constraints

167

7.3. The linearization method

171

7.4. Calculation of the functional gradients

172

7.5. Estimate of operator error

174

7.6. Estimate of the measure of incompatibility

179

7.7. Choice of the regularization parameter

180

7.8. Projection of gradients in symmetry coordinates

181

Chapter 8. Analysis of band intensities in vibrational spectra of polyatomic molecules 187 8.1. Classical and quantum mechanical consideration of intensities in molecular vibrational spectra

188

8.2. The mathematical model

192

8.3. Statements of the inverse electrooptical problem

198

8.4. Computational aspects

200

Chapter 9. Numerical implementation of algorithms for solving problems of vibrational spectroscopy 205 9.1. Principles of construction of the software package

205

9.2. Structure of the input

207

file

9.3. Detailed structure and description of the software package of programs

214

9.4. Symmetry analysis package

221

9.5. Some other useful options

223

Chapter 10. Examples of molecular force field calculations on the basis of experimental data 227 10.1. Some preliminary remarks

227

10.2. Force field of the water molecule

231

10.3. Force fields of the transition metal oxotetrafluorides

242

10.4. Force field of

244

fluoroform

10.5. Example of joint treatment of force fields of two molecules . . . . 250

viii

A. G. Yagola. Inverse problems of vibrational spectroscopy

Chapter 11. Joint treatment of ab initio and experimental data in molecular force field calculations with Tikhonov's method of regularization 259 11.1. Ab initio force fields in regularizing procedures 259 11.2. Computational details and practical aspects of calculations . . . 264 11.3. Examples of using ab initio data in force field calculations . . . . 267 11.4. Regularization procedure for empirical scaling of quantum mechanical force constants 271 Appendix A. Systems of units used in vibrational spectroscopy A.l. International System (SI) A.2. Measurement of frequencies in spectroscopy A.3. CGS-based system A.4. Measuring force field in frequency units A.5. Measuring force field in atomic units A.6. Conversion between different systems A.7. Electrooptical parameters

281 282 282 283 284 285 285 286

Bibliography

289

Preface We present to the reader the expanded and revised edition of our book which was published in Russian by Moscow University Publishers in 1993. The main goal of the Russian edition was to show new stable numerical methods for determining molecular force fields in the framework of purely empirical approach. After the publication of this book, new opportunities arose for applying modern quantum chemical methods for calculations of harmonic molecular force fields. Rapid progress in quantum mechanical calculations of harmonic force fields for moderate size molecules with inclusion of electron correlation at MP2 and DFT levels, provides fundamental new possibilities for more accurate interpretation of experimental data as well as improved methods for empirical force field calculations. This has led the authors to introduce the concept of regularized quantum mechanical force fields (RQM FF) and new formulations of inverse problems. In the last chapter (Chapter 11), written especially for this English edition, this new approach toward solving inverse vibrational problems is illustrated with numerical applications. Quantum mechanical calculations were performed with the program GAUSSIAN 92/DFT on computers of the University of Wisconsin (Madison, Wisconsin, USA). We express our appreciation to Professor Frank Weinhold (Department of Chemistry, University of Wisconsin) for his helpful cooperation. Financial support of the Fulbright Program, that allowed two of the authors to spend the spring semester of 1994 at the University of Wisconsin, is gratefully acknowledged. The authors are also thankful to the Russian Foundation of Basic Research (grants No 95-01-00486, 95-0308268, and 98-03-36166) for partial financial support.

Introduction Many natural science problems relate to inverse problems, in which through known experimental data of the object we need to determine some of its properties, based on a certain model that connects these properties with measured characteristics. The mathematical relation between the object's properties and its experimental display can be defined in the form of an operator equation Az = u, where z, u are elements of metrical spaces Ζ and U, respectively. A is operator acting from Ζ into U. Here z, stands for all properties of the object sought for, and u for all experimental data taken into consideration. The form of the operator A is defined by the choice of the mathematical model for the object. Usually, most of the inverse problems are ill-posed, and classical methods cannot be used to solve them, since the inevitable errors in experimental data can lead to large (sometimes overwhelming) variations of the solution. Nevertheless, there is a practical need for solving such kinds of problems with given approximate data (note that an operator A can be approximately defined too) in a manner that stimulates the development of the theory and provides suitable numerical methods for solving the ill-posed problems. The works of Prof. A. N. Tikhonov are fundamental in this field, subsequently developed further by M. M. Lavrent'ev and V. K. Ivanov and their schools. Many authors have mentioned that ill-posedness of the problem can be connected with its incomplete physical definition, so the idea of constructing stable approximations to the exact solution for such problem is based on the idea of using additional a priori information about the sought-for solution. In this respect there are two possible approaches, depending on the amount and kind of a priori information available. If the information makes

2

A. G. Yagola. Inverse problems of vibrational spectroscopy

it possible to pick up a subsequently narrow (compact) set of possible solutions M Ç Z, then we can choose as an approximate solution of the ill-posed problem any element of this set which fits the experimental data within the limits of precision requested. However, the α priori known information is often not sufficient to enable us pick out the compact in Z. Then it occurs that one can use less restrictive assumptions concerning the kind of solution (belonging to some functional space or proximity to some element defined beforehand, etc.) This approach is based on the ideas of a regularizing algorithm or regularizing operator. Using this notion, A. N. Tikhonov and his school have constructed the branching theory and created numerical algorithms for solving ill-posed problems. At present there are a great number of monographs and original articles devoted to the general theory of regularization and the applications of stable numerical methods to concrete classes of problems, and in a number of cases the numerical algorithms have been developed to the level of standard programs for computers. At present, both the theoretical and practical basis exist for constructing regularizing algorithms in the majority of practically important problems. Since 1980, research on the correctness of some classes of inverse problems in vibrational spectroscopy has been carried out in Moscow State University. These problems include finding (from experimental data) the molecular force fields and electrooptical parameters, characterized frequency and intensity distributions in vibrational (IR and Raman) spectra, and creating stable numerical methods and software package for molecular vibrational spectra analysis. The notion of a molecular force field arises when a molecule is considered as a quantum mechanical system, consisting of two kinds of charged particles: electrons and nuclei, differing greatly in their masses (10 _3 -10~ 5 times). Because of this difference we can treat electron and nuclear motions separately, and that is a basic premise of the adiabatic approximation. In terms of the Born - Oppenheimer approximation for the perturbation theory for a nuclear subsystem of a molecule, there is some effective force field (called the adiabatic potential) created by the electron subsystem. The force field of a molecule determines many of its important physical properties. For example, the force field minimum (with respect to nucleus coordinates) assigns the equilibrium geometry of the molecule, and second derivatives of the potential with respect to nuclear coordinates in the point of equilibrium (a force constant matrix) determine vibrational

Introduction

3

properties (including infrared absorption and Raman spectra, and, also, vibration-rotational characteristics, thermodynamical functions, etc.). For all that, accurate quantum mechanical calculation of an adiabatic potential V(qi,..., qn), where qi,..., qn are the relative nuclear coordinates, is possible in the more or less simple cases. For large molecules, main source for determining the force field parameters still lies in the processing of experimental data from vibrational spectra (infrared and Raman). In spite of their quite different mechanism of formation, both infrared and Raman spectra are connected with the transitions between vibrational states, which we can observe in a fairly narrow range of wavelengths (wave numbers): in between 1 and 1000 μπι (10000 to 10 cm - 1 ). For complete analysis of the molecular vibrational spectrum one has to investigate both infrared and Raman spectra, since these complement each other due to the differences in selection rules. Molecular vibrational frequencies depend on atomic masses, geometrical parameters and forces, binding atoms together in a molecule. If the atomic masses and the geometric parameters are known we can estimate the forces acting in a molecule from the vibrational frequencies, measured in infrared and Raman spectra. Conversely, if we know, besides atomic masses and geometrical parameters of a molecule, the characteristics of intramolecular forces, then a theoretical vibrational spectrum for the adopted molecular model can be estimated. In order to solve both these problems using the information extracted from experimental vibrational spectra, we apply the theoretical apparatus of classical and quantum mechanics and established techniques of calculation. Besides providing information about the forces acting in a molecule, vibrational spectra can give information on intermolecular forces in a substance, since vibrational frequencies are very sensitive to the molecular surroundings and depend on the aggregate state of the substance. In this case of strong intermolecular interactions (for example, in the formation of hydrogen bonds) frequency shifts in comparison with the spectrum of isolated molecule can reach several hundred cm - 1 . For complete solution of the vibrational problems, which are concerned with the charge distribution in a molecule and its changes due to nuclear vibrations, it is necessary to use the information on band intensities in IR and Raman spectra. The infrared intensities depend on changes of the molecular dipole moment under vibrations; Raman intensities are determined by changes of the molecular polarizability ellipsoid. In this book we consider the problems of constructing molecular force

4

A. G. Yagola. Inverse problems of vibrational

spectroscopy

fields using infrared and Raman experimental data, gas electron diffraction data etc., and vibrational spectra intensity analysis problems as well. In constructing new regularizing algorithms for solving inverse problems of vibrational spectroscopy, we have tried to formalize evident and not so evident model assumptions concerning the solving of these problems, introduced earlier, in terms of the general regularization theory of ill-posed problems. In this manner, α priori assumptions are incorporated into the regularizing algorithms created for the solution of inverse problems of vibrational spectroscopy. Now, let us glance at the book content. The first chapter is devoted to the theory of molecular vibrations in the simplest approximation, when translational and rotational movements are considered separately from vibrations themselves. This approximation is considered both from classical and quantum mechanical points of view and is sufficient for the majority of applications. In the second chapter a more rigorous approach is given, where vibrations of a molecule are separated on the basis of an adiabatic approximation. We deduce here the formulae that connect experimentally measured values (vibrational frequencies, Coriolis constants of vibration-rotational interactions, centrifugal distortion constants, mean square amplitudes) with the parameters of a molecular force field. The third chapter consists of formulating the mathematical model, which is necessary for the posing of the inverse problem. Several (empirical, semiempirical and ab initio) methods of force fields calculations are considered and discussed. The fourth chapter is devoted to the analysis of various physical models of a molecular force field, used in calculations of molecular vibrations. Special attention is given to the introduction of the redundant system of generalized (internal or symmetry) coordinates. The application of such systems is justified by their convenience, in particular, when we take into account the symmetry of a molecule, as well as by their more transparent representation of the force field model. As an important practical example we consider the case of using Cartesian coordinates of molecular nuclei as generalized coordinates. In the fifth chapter the main elements of the symmetry point group theory are described, as it is applied to the problem of molecular vibrations. We give here some examples of constructing the symmetry coordinate system for several molecules. In the sixth chapter the foundation of the ill-posed problem regular-

Introduction

5

ization theory is presented, including problems associated with an approximately given operator. It is shown that the problem of calculating molecular force field on the basis of experimental data is an example of an ill-posed problem and the three conditions of correctness are not satisfied. The difficulties related to the nonuniqueness of the solutions and instability of the solution with respect to the perturbations of experimental data had been discussed by previous workers, and various model assumptions had been proposed based on researcher's intuition. The instability and nonuniqueness of the solution often leads to significant differences in the force field parameters obtained by various investigators for the same molecule. For this reason the possibility of comparing or transferring force fields in the series of related compounds is doubtful and the physical interpretation of the experimental data is rather obscure. It is very important to diminish the arbitrariness in the calculated force constants to search the stable solutions of the inverse vibrational problem which have some given properties. What do we mean by requiring some given properties of force constants, or in other words, what additional limitations for the values of the force constants can be added while solving the inverse problem? In practical terms, molecular spectroscopists make of use model assumptions arising from the classic theory of chemical structure involving concepts of bonded and nonbonded interactions, bond orders, monotone changing of the physico-chemical properties in series of related molecules, and the preservation of the properties of separate molecular fragments in various compounds, taking into account the first environment of a fragment. It is the the model of so-called valency-force field and transferability of the force constants that are related with these properties. We propose to formalize these model notions and use them in the force field calculation. All the necessary model assumptions may be taken into account by the choice of a priori given matrix of force constants, F°. Then we can formulate the problem of constructing the matrix F of the force constants which would be nearest in some given metric to matrix F°. For solving the inverse vibrational problem within this statement, stable numerical methods have been created which are based on the Tikhonov's regularization theory. In the seventh chapter we describe some numerical methods required for the numerical realization of regularizing algorithms: the solution of the eigenvalue algebraic problem, minimization of non-quadratic functional with simple restrictions, the minimization of quadratic functionals on the set of non-convex constraints, the conjugate gradient method, the Pshenichny

6

A. G. Yagola. Inverse problems of vibrational spectroscopy

linearization method, and some auxiliary algorithms having significance for solving a variety of vibrational spectroscopy problems. We consider some questions of the projecting functional gradients for use in conjunction with symmetry coordinates. The estimations of error of an approximate operator and measure of incompatibility of exact problem, both in the obvious form, have been made. In the eighth chapter we give a preliminary discussion of the theory of vibrational intensities and the formulation of the mathematical problem, which connects experimentally measured intensities with electrooptical parameters. Various reformulations and statements of inverse problems are considered and the regularizing algorithms for their solution and discussion of computational aspects are given. In the ninth chapter we discuss the principles of organization and describe the structure of the software package of programs adapted to IBM PC/AT for theoretical analysis of vibrational spectra of polyatomic molecules This includes programs for solving the inverse problems of molecular force field calculation and for the analysis of intensities in vibrational spectra, created on basis of the regularizing algorithms. We consider the general requirements of the software package. Some examples are given of the formation of a standard input file (which can be prepared by means of the special utility program), the analysis of the symmetry point group and construction of symmetry coordinates, and the final output file containing the results of force field calculation for the W O F 4 molecule. In chapter 10, examples are given of concrete force field calculations of several molecules by using the regularizing algorithms. As an example we present results for the water molecule, a number of oxotetrafluorides of transition metals of Group V I ( W O F 4 , M0OF4, CrOF4), and fiuoroform (with the expanded set of experimental data, including the frequencies of three isotopomeres, Coriolis constants and centrifugal distortion constants). The example of joint calculation of the force fields for two conformers of CF2CICF2CI molecule is given as well. In chapter 11, the stable numerical algorithms based on Tikhonov's regularizaron method are applied to joint treatment of ab initio and experimental data in molecular force field calculations. In this approach, the ab initio quantities serve to "regularize" the initially ill-posed problem, leading to variationally stable and unique force field parameters that optimally mimic overall patterns of the (approximate) ab initio quantities, but exactly reproduce the available experimental data within specified experimental precision. In this manner, ab initio and experimental data can be jointly combined

Introduction

7

to produce more stable and reliable force fields (improvable to any degree through higher level ab initio treatment, additional experimental data, etc.) that could be attained by theoretical or experimental methods alone. The proposed procedure allows use of any system of generalized coordinates, including redundant systems of internal or symmetry coordinates, simplifying the transfer and comparison of force constants in series of related molecules. Two transformation procedures are employed into software package for conversion of the ab initio force constant matrix from Cartesian to redundant internal coordinates. The procedure is illustrated with numerical application to hexa-fluorosubstituted ethane, C 2 F 6 at HF/6-31G* and MP2/6-31G* levels of theory, demonstrating the stability and consistency of force fields obtained from different levels of theoretical input. In the appendix we represent the unit systems, which are commonly used in vibrational spectroscopy. Relations between various unit systems are also described.

Chapter 1. Physical model of molecular vibrations

The vibrational spectrum of a substance appears as a result of interaction of infrared electromagnetic radiation with the substance. This interaction changes the vibrational component of the total molecular energy (Vilkov and Pentin, 1987). The vibrational spectrum of a molecule is a characteristic and complicated function of its geometric structure, nuclei masses and electron density distribution, i.e. of intramolecular forces. The physical consideration of vibrational spectra and their models may use two approaches - classical and quantum mechanical. In other words, molecules belong to the quantum systems whose behavior in certain cases may be described by the classical theory. To solve vibrational problems, one can often use simple approximations of classical mechanics. The classical theory considers the vibrational spectrum of a molecule as small vibrations of a system of linked material points. At the same time, some problems of the molecular spectroscopy, for example, the intensity of vibrational bands, the appearance of combined frequencies and overtones cannot be properly formulated and solved by means of classical physics. In these cases, it is necessary to apply the approximations of quantum mechanics (Herzberg, 1945; Volkenshtein et ai, 1949, 1972; Mayants, 1960; Wilson, et ai, 1955; Landau and Lifshits, 1965, 1977; Califano, 1976).

10

A. G. Yagola. Inverse problems of vibrational spectroscopy

1.1.

CLASSICAL THEORY OF SMALL VIBRATIONS. PRELIMINARY CONSIDERATIONS

It is well known that classical mechanics, generally speaking, is inapplicable to the treatment of atoms and molecules because this theory cannot explain the fact of their stability. Indeed, a resting or uniformly and linearly moving system of charged particles (nuclei and electrons) cannot have a stable equilibrium. On the other hand, a dynamic system of charged particles (of the solar system type) also cannot exist because, according to classical electrodynamics, the accelerating charges have to radiate energy, which immediately leads to falling of the electrons into the nuclei. Thus, we may consider atoms and molecules as systems of charged material points only in quantum mechanics. Nevertheless, some essential peculiarities of vibrational motion of a molecule may be derived from the simplified classical analysis. In many cases this analysis yields results which coincide with those obtained in quantum mechanics, although are more descriptive. In addition, the classical analysis allows us to simplify essentially the quantum mechanical considerations. Hence, let us consider a molecule as a classical system of Ν material points ("atoms" or "nuclei") with the masses M\, M2, · · ·, Μ χ placed in a certain force field such that the potential energy of the system is a function U(R[,..., R;v)· Here R· (i — 1,2,..., N) are the radius vectors of material points in a certain fixed inertial frame of reference (laboratory system). The force field includes electrostatic interaction between nuclei and a certain effective field created by all the electrons of the molecule. In this statement of the problem we do not consider individual electrons. The sense of the field U will be completely clarified in the quantum mechanical statement. Here we shall only note the following.

1. The potential energy U(R^,... ,R'^) has to be independent of rotations and motion of the coordinate system (it is the corollary of the fact that the field is created by the particles of the system themselves). 2. The equilibrium configuration {R°,..., R^} must exist where the potential energy U is minimal (this reflects the fact of existence of the stable molecule). Further, let us agree to denote the vector by a column of three numbers

11

Chapter 1. Physical model of molecular vibrations (its Cartesian components), for example

We introduce the scalar product of two vectors 3

(x,y) = x*y = ^ x a y a ,

Q=1 where the sign * denotes transposition (i.e. x* is a row vector), and, also, the vector product

r = [x,y], defined by relations η = X2V3 - X3V2,

r2 = X3V1 - xm,

r3 = Xiy2 - X2V1-

We shall also use vectors of arbitrary dimensions, treating them as columns. For consideration of motion of the nuclei we should compose the Lagrange function L

1 N = ö¿ Σ M Ä i=l

«

V

II

El v"=2 Ev„

v"=l

=

_

v"=0

Ely

E" Fig. 2: Energy levels of a molecule Besides these three types of spectra, there are also spectra of electronspin resonance and nuclear-magnetic, nuclear quadruple and gamma-resonance, which are not considered here (Vilkov and Pentin, 1987, 1989). In an exact treatment, besides electron, vibrational and rotational components of the total molecule energy there arise terms connected with electronic-vibrational, electronic-rotational and vibrational-rotational interactions. The complete quantum mechanical statement of the problem will be given in Sec. 2.3. In Sec. 2.4, we present methods for approximate separation of molecular interactions. Here we shall consider the simplified quantum mechanical statement of the problem of molecular vibrations. We shall assume that the rotational motions and translations of the molecule's nuclei

Chapter

1. Physical model of molecular

23

vibrations

have no influence on vibrations, and electronic motion is taken into account as creating a certain equivalent force field. Thus, we are to solve the same problem as in Sec. 1.1 but for the quantum mechanical model. Considering this simplified statement is useful since the results obtained remain valid in more general cases. For stationary vibrational states, the Schrödinger equation is written as follows: HVV

EvVv,

=

where the Hamilton operator is obtained from (1.10) by substituting the momenta pk by corresponding operators pk = ( h / i ) d / d q k ; Φ = Φ(?ι, • • ·, qn) are the wave functions, and Ev are the vibrational energy values. The substitution leads to the equation 1

n

2 Σ GkiPkPlV + V(qi, • • •, 9η)Φ = .ΕΨ. k,l=l

To solve this equation, it is useful to transfer to the normal coordinates. As a result we obtain =

¿

(124)

k=1

Here, we have used the representation of V(q) in the form (1.11), as was done in Sec. 1.1. Thus, we have to solve η independent problems ¿Pj!*k

+ \ 3, the vibrations are accompanied by the appearance of vibrational angular momentum. Further, we shall see that the vibrational moment always exists for the multiple frequencies (and only in this case). 2.3.

QUANTUM MECHANICAL MODEL

In consideration of a molecule as a quantum system, we restrict ourselves to the nonrelativistic Schrödinger equation, and we shall not consider spins of particles (except for certain conditions imposed by spins on the parity of wave functions). Such a treatment holds for molecules consisting of not too heavy atoms. So, let a molecule consist of Ne electrons and Ν nuclei. We assume them to be point particles with masses m and M¿ (i = 1,2,..., Ν) respectively and charges —e and +Z¡e (Z¡ is the atomic number of the i-th nucleus). Choose an inertial coordinate system and denote the radius vectors of electrons and nuclei by r'k (k = 1,..., Ne) and R¿ (i — 1,..., Ν). The Hamiltonian of such particle system is as follows A

=èië(Pk)2+Σ k=l

t=l

¿ (1 p ' ) 2 + ·

··,,rì,...RU

where U is the electrostatic interaction energy: U = Uee +

+ Unei

(2.11)

35

Chapter 2. Full statement of the vibrational problem

v3= 2349 cm A Fig. 3: Normal vibrations of CO2 molecule where Π -

1

V

β2

TT -

1

ZjZke2

V

π

y

^e2

were determined in Sec. 1.2; the operators p^ and P^ are represented as p'k = (h/i)d/dr'k, P'fc = (h/i)d/dR'fc; h = h/2n is the Planck constant. For the considered stationary state of the molecule it is necessary to solve the eigenvalue equation =

(2.12)

It means that Φ = Φ(γ'1, . . . , r'^ , R' x ,... R'^) are the eigenfunctions of operator (2.11).

36

A. G. Yagola.

Inverse

problems

of vibrational

spectroscopy

Before discussing the problems arising when solving this equation, we shall separate, as in the classical case, the different types of particle motion. For this purpose, we fix an arbitrary configuration of nuclei {R?, · · ·, Rjv) (assuming it is nonlinear) and attach a new coordinate system to it (as we have done in Sec. 2.1). The coordinates of nuclei and electrons in this new system, R¿ (i = 1 , . . . , Ν) and (k = 1 , . . . , Ne) respectively, are related to the coordinates in the initial system by the formulas η Κ; = R +

C(0)

R = R + C(0)(R? +

r'k = R + C(9)rk

(k = 1 , . . . ,

Aikqk) fc=l

(i =

l,...,N),

(2.13)

Ne).

Here, the rotation matrix C is determined by the Eulerian angles θ = {θι,θ2,θ3}, Caß — (e Q ,i^); βι,62,63 are the unit vectors of the laboratory coordinate system, and 11,12,13 are the unit vectors of the molecular coordinate system. Note that R and θ are determined only by the positions of the nuclei. To obtain the Hamiltonian for the new variables {R, Θ, q}, we need to substitute formulas (2.13) into Hamiltonian (2.11). In the expression for the potential energy we may simply change r'k to r*., R¿ to Ru since the potential energy depends only on the following arguments: lrj

— r

fcl — \ r j

—r

fc|i

IrJb-Ril = h f c - R i l

and

|Rj -

= |R¿ - R ;

The kinetic energy is represented as the Laplace operator h23(N+Ne)

if we introduce χ = (χχ, X2,..., X3(n+nc)) χ = (\/mr'{,...,.1*

d2

as

, v ^ R ' i * , · · ·, V ^ R ' / v ) ·

New coordinates

allow us to obtain (Braun and Kiselyev, 1983) ^ ( ^ e )

_

d

(2.14) j,k=l

Chapter 2. Full statement of the vibrational problem where, as in Sec. 2.1, s — g

,

3 {N+Ne) Σ k=1

37

= det