Inverse Problems of Vibrational Spectroscopy

by Yagola, A. G.; Kochikov, I. V.; Kuramshina, G. M.; Pentin, Yu. A.

• ISBN13: 9789067643047
• ISBN10: 9067643041
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: De Gruyter

Table of Contents

Introduction	1	(8)
Physical model of molecular vibrations	9	(18)
Classical theory of small vibrations. Preliminary considerations	10	(8)
Simplified quantum mechanical statement of the problem	18	(9)
Full statement of the vibrational problem	27	(26)
Vibrational-rotational interaction. Classical consideration	27	(5)
Centrifugal distortion. Classical consideration	32	(2)
Linear molecules	33	(1)
Quantum mechanical model	34	(6)
Adiabatic approximation	40	(3)
Vibrational-rotational interaction. Quantum mechanical consideration	43	(2)
Centrifugal distortion. Quantum mechanical consideration	45	(1)
The adiabatic theory of perturbations	46	(7)
Consideration of the mathematical model for molecular vibration analysis. Direct and inverse problems	53	(18)
Introduction. Parameters of the model	53	(2)
Ab initio methods	55	(3)
Results and difficulties of using ab initio methods	58	(4)
Semiempirical methods	62	(2)
Empirical methods	64	(7)
Vibrational problem in internal coordinates. Use of the redundant coordinate system	71	(30)
Models of a molecular force field	71	(3)
Choice of generalized coordinates	74	(3)
Construction of the auxiliary matrices	77	(10)
The matrix of kinematic coefficients	77	(2)
Bond stretching	79	(1)
Valence bond angle bending	80	(1)
Angle between a bond and the plane of an atomic triple	81	(1)
Angle between two planes (torsion coordinate)	82	(1)
An example of constructing the matrix B for the torsional coordinate	83	(1)
An example of constructing matrix G	84	(1)
Construction of the matrix H	85	(1)
The other matrices	86	(1)
Use of redundant coordinates	87	(14)
Vibrational problem in symmetry coordinates	101	(24)
Use of molecular symmetry	101	(10)
Use of symmetry in the calculation of molecular vibrations	111	(8)
Calculation of the molecular constants in symmetry coordinates	119	(6)
Ill-posed problems and the regularization method. Regularizing algorithms for constructing force fields of polyatomic molecules on the base of experimental data	125	(36)
Well-posed and ill-posed problems	125	(5)
Ill-posedness of the problem of constructing force field on the base of experimental data (inverse vibrational problem)	130	(3)
Mathematical formulation of the inverse vibrational problem	133	(4)
Ill-posedness of the problem of searching for a normal pseudosolution of the linear algebraic equation system (LAES)	137	(5)
Regularizing algorithms for constructing a normal pseudosolution of LAES	142	(4)
Nonlinear ill-posed problems	146	(4)
Inverse vibrational problem for a single molecule	150	(5)
Joint calculation of force field in a series of related molecules	155	(6)
Numerical methods	161	(26)
Searching for eigenvalues and eigenvectors	162	(5)
Minimization of functionals. Simple constraints	167	(4)
The linearization method	171	(1)
Calculation of the functional gradients	172	(2)
Estimate of operator error	174	(5)
Estimate of the measure of incompatibility	179	(1)
Choice of the regularization parameter	180	(1)
Projection of gradients in symmetry coordinates	181	(6)
Analysis of band intensities in vibrational spectra of polyatomic molecules	187	(18)
Classical and quantum mechanical consideration of intensities in molecular vibrational spectra	188	(4)
The mathematical model	192	(6)
Statements of the inverse electrooptical problem	198	(2)
Computational aspects	200	(5)
Numerical implementation of algorithms for solving problems of vibrational spectroscopy	205	(22)
Principles of construction of the software package	205	(2)
Structure of the input file	207	(7)
Detailed structure and description of the software package of programs	214	(7)
Symmetry analysis package	221	(2)
Some other useful options	223	(4)
Examples of molecular force field calculations on the basis of experimental data	227	(32)
Some preliminary remarks	227	(4)
Force field of the water molecule	231	(11)
Force fields of the transition metal oxotetrafluorides	242	(2)
Force field of fluoroform	244	(6)
Example of joint treatment of force fields of two molecules	250	(9)
Joint treatment of ab initio and experimental data in molecular force field calculations with Tikhonov's method of regularization	259	(22)
Ab initio force fields in regularizing procedures	259	(5)
Computational details and practical aspects of calculations	264	(3)
Examples of using ab initio data in force field calculations	267	(4)
Regularization procedure for empirical scaling of quantum mechanical force constants	271	(10)
Appendix A. Systems of units used in vibrational spectroscopy	281	(8)
A.1. International System (SI)	282	(1)
A.2. Measurement of frequencies in spectroscopy	282	(1)
A.3. CGS-based system	283	(1)
A.4. Measuring force field in frequency units	284	(1)
A.5. Measuring force field in atomic units	285	(1)
A.6. Conversion between different systems	285	(1)
A.7. Electrooptical parameters	286	(3)
Bibliography	289