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The Journal of Chemical Physics

Applications of the Quantum-Mechanical Green's Functions to the Study of Chemical Reactions. II

R. Paul, G.N. Fowler, W. G. Laidlaw

Published Jan 1, 1968• 13 citations
Publisher

Abstract

Starting from a field theoretical representation of chemical reactions, the rate equation is developed in terms of the quantum mechanical Green's functions. The equation is valid for non-Markowian behavior and the approximations that are necessary in order to obtain the conventional (Markowian) rate equation are investigated. The effect of finite collision times is also discussed.

DOI
10.1063/1.1667991

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Starting from a field theoretical representation of chemical reactions, the rate equation is developed in terms of the quantum mechanical Green's functions. The equation is valid for non-Markowian behavior and the approximations that are necessary in order to obtain the conventional (Markowian) rate equation are investigated. The effect of finite collision times is also discussed.
This book presents an up-to-date formalism of non-equilibrium Green's functions covering different applications ranging from solid state physics, plasma physics, cold atoms in optical lattices up to relativistic transport and heavy ion collisions. Within the Green's function formalism, the basic sets of equations for these diverse systems are similar, and approximations developed in one field can be adapted to another field. The central object is the self-energy which includes all non-trivial aspects of the system dynamics. The focus is therefore on microscopic processes starting from elementary principles for classical gases and the complementary picture of a single quantum particle in a random potential. This provides an intuitive picture of the interaction of a particle with the medium formed by other particles, on which the Green's function is built on.
The applicability of the time-dependent Schrödinger equation to molecular reaction dynamics is discussed from several perspectives. First, with reference to Schrödinger's original paper, we point out that he himself stated that what is now called the time-dependent Schrödinger equation is insufficient to solve the problems of non-conservative systems like chemical reactions. Second, the detonation velocity of some explosives has been demonstrated as a new parameter that directly illustrates the timescales of elementary chemical reactions. The calculated values of ~30 fs are in good agreement with the timescale of 10–100 fs for chemical reactions, as experimentally demonstrated by the Nobel laureate Zewail. Third, we consider the state of an electron in a box when the length of the box suddenly increases from L to 2L, which is often found in textbooks as the simplest example of the application of the time-dependent Schrödinger equation to chemical reactions, and highlight the inconsistency between this model and an actual chemical reaction. Finally, we re-iterate that only quantum steady states can be accurately described by the Schrödinger equation and that it is the most natural to connect changes between adjacent quantum steady states by the variational principle.
Abstract It has been shown in a previous paper that classical mechanics are not adequate to treat the transition of a hydrogen atom or proton across an energy barrier of the dimensions commonly met with in chemical reaction. The treatment given was based on an exact solution of the Schrödinger equation and a type of potential curve having no discontinuities in slope, but owing to the laborious nature of the computations involved, no attempt was made to investigate quantitatively the effect of variations in the heat of activation, the width of the barrier, or the mass of the particle. The present paper describes an approximate treatment leading to simple equations which can be applied directly to investigate these points. In a recent paper, Wigner has given a method of treatment applicable to any form of potential curve. His final equations are, however, only valid for the case in which the difference between the quantum mechanical and classical results can be expressed as a small correction term, and cannot be applied when there are large deviations from classical behaviour.
The formal theory of Green functions is reviewed for many particle problems in nuclear physics or in solid state physics. In particular, one- and two-particle Green functions are discussed in detail. The Feynman amplitudes are also explained together with the effective potential and its applications. This article contains the following sections: §1. Preliminary remarks §2. Definition of Green functions §3. Structure of equations satisfied by Green functions 3.1 One-particle Green function 3.2 Two-particle Green function §4. Perturbation theory and its foundation §5. Spectral representation and its application §6. Connection with reaction matrix §7. Amplitudes and their matrix 7.1 Definitions of amplitudes and effective potential 7.2 Nuclear optical model 7.3 One electron in insulators and semi-conductors Appendix I. Functional differentiation Appendix II. Schwinger's dynamical principle Appendix III. Polaron problem
Computational difficulties aside, nonequilibrium Green's functions appear ideally suited for investigating the dynamics of central nuclear reactions. Many particles actively participate in those reactions. At the two energy extremes for the collisions, the limiting cases of the Green's function approach have been successful: the time-dependent Hartree-Fock theory at low energy and Boltzmann equation at high. The strategy for computational adaptation of the Green's function to central reactions is discussed. The strategy involves, in particular, incremental progression from one to three dimensions to develop and assess approximations, discarding of far-away function elements, use of effective interactions and preparation of initial states for the reactions through adiabatic switching. At this stage we concentrate on inclusion of correlations in one dimension, where relatively few approximations are needed, and we carry out reference calculations that can benchmark approximations needed for more dimensions. We switch on short-range interactions generating the correlations adiabatically in the Kadanoff-Baym equations to arrive at correlated ground states for uniform matter. As the energy of the correlated matter does not quite match the expectations for nuclear matter we add mean field to arrive at the match in energy. From there on, we move to finite systems. In switching on the correlations we observe emergence of extended tails in momentum distributions and evolution of single particle occupations away from 1 and 0.