# Advances in chemical physics - Prigogine I.

### Advances in chemical physics

Author: Prigogine I.Other authors: Rice S.A.

Publishers: Advision

Year of publication: 1970

Number of pages: 167

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ï»¿Advances in

CHEMICAL PHYSICS

EDITED BY

I. PRIGOGINE

University of Brussels, Brussels, Belgium

AND

STUART A. RICE

Department of Chemistry and

The James Franck Institute The University of Chicago Chicago, Illinois

VOLUME XVII INTERSCIENCE PUBLISHERS

A DIVISION OF JOHN WILEY AND SONS NEW YORKÂ» LONDON â€¢ SYDNEY â€¢ TORONTO

Copyright Â© 1970, by John Wiley & Sons, Inc.

All rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher.

Library of Congress Catalogue Card Number: 58-9935

SBN 471 69922 5

Printed in the United States of America

10 987654321

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

B. J. BERNE and G. D. HARP

CONTENTS

I. Introduction...................64

II. Linear Response Theory...............69

A. Linear Systems.................69

B. The Statistical Theory of the Susceptibility.........75

C. The Reciprocal Relations.............79

D. The Fluctuation-Dissipation Theorem..........83

E. Doppler Broadened Spectra.............90

F. Relaxation Times................92

III. Time Correlation Functions and Memory Functions.......94

A. Projection Operators and the Memory Functions.......94

B. Memory Function Equation for Multivariate Processes.....100

C. The Modified Langevin Equation...........102

D. Continued Fraction Representation of Time-Correlation Functions . . 106

E. Dispersion Relations and Sum Rules for the Memory Function . . . 108

F. Properties of Time-Correlation Functions and Memory Functions . . 113

IV. Computer Experiments...............120

A. Introductory Remarks...............120

B. Method Employed................122

C. Data Reduction.................126

D. Potentials Used.................127

E. Summary and Discussion of Errors...........130

F. Liquid or Solid.................132

G. Equilibrium Properties...............136

H. The Classical Limit................138

V. Experimental Correlation and Memory Functions........141

A. Approximate Distribution Functions...........155

VI. Approximations to Time-Correlation Functions........163

A. A Simple Model for Linear and Angular Momentum Correlations . . 163

B. Memory Function Theory of Linear and Angular Momentum Correlations..................167

C. The Martin Formalism..............176

D. The Continued Fraction Approximations.........177

E. Approximate Correlation Functions from Memory Functions . 180

F. Elementary Excitations in Liquids...........186

G. Van Hove Self-Correlation Functions from Computer Experiments . .201

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64

Ð’. J. BERNE AND G. D. HARP

VII. Conclusion..............

Appendix A. Numerical Integration of Differential Equations . Appendix B. The Numerical Solution of the Volterra Equation Appendix C. Properties of the Polynomials HeN(x) .... References................

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I. INTRODUCTION

A large number of experimental methods are currently used to probe the dynamics of molecular motions in solids, liquids, and gases. These include lineshape studies of electronic,1 infrared, and Raman spectra2; studies of the shape of the spectral density function obtained from light- and neutron-scattering experiments3 " 6; lineshape studies in dielectric relaxation7; spin relaxation experiments8; acoustic attenuation9; as well as studies of static and frequency-dependent transport coefficients.10

All of these experimental methods share one characteristic in common. They all use as a probe an external field which is weakly coupled to the system and they all study the response of the physical system to the probe. This is to be expected since a more strongly coupled probe would influence the dynamical behavior of the system and would thereby obscure the fundamental molecular processes taking place.

The experiments can be divided into two categories according to whether the probe is mechanical or thermal. For example, light scattering falls into the first category whereas the measurement of the thermal conductivity falls into the second. The reason for making this division is the fact that the response of systems to mechanical probes is much easier to treat than their response to thermal probes. The interaction between a mechanical probe and the physical system can be described by an interaction Hamiltonian, whereas thermal probe system interactions must be handled differently.

The basic theoretical problem is to describe the response of an equilibrium system to a weak force field be it mechanical or thermal in nature. The solution to this problem is by now well known and there exist many excellent reviews on the subject.11-17 A particularly informative account of this work together with historical comments has been given by Zwanzig.12

The major conclusions of this theory, which is known as linear response theory, can be simply stated as follows. Whenever two systems are weakly coupled to one another such as radiation weakly coupled to matter, or molecular vibrations weakly coupled to molecular motion, it is only necessary to know how both systems behave in the absence of the coupling in order to describe the way in which one system responds to the other. Furthermore, the response of one system to the other is completely describ-able in terms of time correlation functions of dynamical properties.

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