Coupling of translational and reactive dynamics for a Fokker–Planck model

Abstract

The coupling of translational and reactive dynamics is investigated for a Fokker–Planck description of relative particle motion in a solvent. A division into inner and outer relative coordinate spatial regions separated by a boundary layer is made. The inner region is characterized by a deep potential well associated with bound, or reacted, particles and a potential barrier. The outer zone is characterized by less rapidly varying forces which include solvent structural effects. With this division, a Fokker–Planck source equation for unreacted particles is derived. The source in this equation incorporates the complete inner region dynamics involving both reactive and nonreactive trajectories. This equation is then reduced via projection operator techniques to a spatial Smoluchowski source equation for unreacted particles in the outer region for the case of slow reaction. Validity conditions involving velocity relaxation are given. The sources here include both inner region short range repulsive effects and a reversible chemical reaction. The outer Smoluchowski description includes caging effects. The equivalent boundary condition formulation is given; thus, the radiative boundary condition relating diffusive and reactive fluxes is derived. New expressions for the reaction rate constants are found and examined in terms of more fundamental rate kernels and Fokker–Planck barrier crossing dynamics. The rate constants are evaluated for a Kramers‐type treatment of the inner region. These rate constants include effects of velocity nonequilibrium. They reduce to transition state results under intermediate inner region friction conditions.