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On the applicability of a nonlinear Schrödinger equation to the determination of rate constants in Kramers' theory of chemical reactions
Author(s) -
Schuch Dieter
Publication year - 1993
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560450302
Subject(s) - smoluchowski coagulation equation , dissipative system , brownian motion , physics , reaction rate constant , nonlinear system , constant (computer programming) , fokker–planck equation , space (punctuation) , classical mechanics , diffusion , chemistry , thermodynamics , quantum mechanics , differential equation , linguistics , philosophy , computer science , kinetics , programming language
In 1940, Kramers derived an expression for the rate constant of chemical reactions in viscous media. In this theory, chemical reactions are modeled as Brownian processes in the presence of potential barriers. The derivation of the rate constant is based on the solution of the Smoluchowski equation, a Fokker–Planck‐type equation in position space. Kramers' theory has been confirmed for reactions in systems ranging from nuclear processes to biochemical problems. The reaction rate can be obtained from the diffusion currents appearing in the Smoluchowski equation. A Smoluchowski‐type equation in position space, similar to the one applied by Kramers, has been used to find a nonlinear Schrödinger equation ( NLSE ) for dissipative, frictionally damped systems, taking into account wave‐particle duality. For several potentials, the NLSE can be solved exactly, and analytic expressions for the currents, appearing in the Smoluchowski equation and necessary to determine the rate constants, can be obtained. These currents are not only stationary, but can also be time‐dependent. It will be shown that, essentially, the determination of the time‐dependence of the currents can be reduced to the solution of Newton‐type equations of motion. © 1993 John Wiley & Sons, Inc.