Electron transfer reactions in condensed phase: effect of reversibility

We propose a generalized one-dimensional kinetic equation for multidimensional reversible electron transfer (ET) reaction with a nonequilibrium situation as the initial condition. The rate constant for the forward reversible ET reaction obtained here consists of the rate for the corresponding irreversible ET reaction, and an extra term due to reversibility of the ET process which includes the rates of diffusion dynamics in the reactant and product wells. In order to understand the effect of reversibility, we consider back ET reaction in a system consisting of an electron donor-acceptor pair in a solvent modeled through low frequency solvent collective coordinates (multidimensional) characterized by the orientational polarization and slowly relaxing one-dimensional vibrational mode. We propose here a new generalized polarization energy functional corresponding to the extension of the continuum version for the same, which has opened up the possibility of inclusion of molecular nature of the solvent into the solvent reorganization energy. We then derive an exact expression for the ET rate for this model system. The numerical results calculated by using the proposed one-dimensional approach are shown to be in good agreement with the available experimental results. Non-Marcus free energy gap dependence of the rates observed here for the reversible and irreversible ET reactions are very close to each other in the barrierless region, while for other situations, the rate for the former process is found to be less than the latter. The extra term, which makes the difference between the rate constants for irreversible and reversible ET reactions, is found to be contributed by the diffusion dynamics from both reactant and product wells but the dominating contribution is provided mainly by the product well.