Evolution of stochastic chemical rate theory from the dynamics of molecular reactant and product nuclear modes coupled to a continuous manifold of heat bath oscillators

Abstract

The Fokker-Planck equation for the system population in potential wells which represent reactants and products of chemical processes in solution, is most frequently taken as the starting point for stochastic approaches to chemical rate theory. In this work we have derived the Fokker-Planck equation on the basis of simple one- and two-dimensional systems spanned by “reactive” modes the damping of which is aroused by weak linear coupling to a manifold of bath oscillators. In the limit of continuous bath oscillator distributions (the Weisskopf-Wigner approximation) the system time evolution is handled by the formalism of coherent states and leads to precise correspondence between the damping coefficients and the diffusion coefficients in the potential wells. In this way dynamic and stochastic features can be viewed in a unified fashion. Several important limiting rate equations are obtained, depending on the time range and coupling constants. Initially equilibrated single-mode systems only contain the damping features implicitly, by additional activation free energy terms which account for the coupling in the transition and initial equilibrium states. On the other hand, single-mode systems initially prepared in non-equilibrium configurations explicitly reveal the damping features by the appearance of the damping coefficient in the pre-exponential factor of the rate constant. The behaviour of systems with two or more reactive modes is more complex and gives both exponential and non-exponential time evolution, and either diffusive or dynamic rate constants in different time and damping coefficient ranges.