Theory of reversible electron transfer reactions in a condensed phase

We have derived an exact analytical expression for the average forward rate of a reversible electron transfer reaction, modeled through a reaction coordinate undergoing diffusive motion in arbitrary potential wells of the reactant and the product in presence of a localized sink of arbitrary location and strength. The dynamics of diffusive motion is described by employing two coupled generalized diffusion reaction (Smoluchowski) equations with coordinate dependent diffusivity and delta sink. The average forward electron transfer rate constant obtained here for the system, with equilibrium or nonequilibrium distributions as initial condition, is determined by the forward and backward rate constants calculated based on the transition state theory and the weighted average rate for the well dynamics. We also discuss various limiting cases for the rate of electron transfer reactions corresponding to the different experimental situations. As an illustrative example, we have considered back electron transfer (ET) reaction and shown that the present theory can explain the non-Marcus free energy gap dependence of the rate of ET reactions. More importantly, the approach presented here can easily be extended to systems describing the dynamics of diffusive motion in coupled multipotential surfaces associated with electron transfer reactions.