Solvent dynamics effect in condensed-phase electron-transfer reactions

Channel-based reaction-diffusion equations are solved analytically for two electron transfer (ET) models, where the fast inner-sphere coordinate leads to an ET reaction treated by Fermi's golden rule, and the slow solvent coordinate moves via diffusion. The analytic solution has let us derive an ET rate constant that modifies the Marcus-Jortner formula by adding a constant alpha which we call a dynamic correction factor. The dynamic correction factor measures the effect of solvent friction. When the relaxation of solvent dynamics is fast, the dynamic correction can be neglected and the ET rate constant reduces to the traditional Marcus-Jortner formula. If the solvent dynamic relaxation is slow, the dynamic correction can be very large and the ET rate can be reduced by orders of magnitude. Using a generalized Zusman-Sumi-Marcus model as a starting point, we introduce two variants, GZSM-A and GZSM-B, where in model A, only one quantum mode is considered for inner-sphere motion and in model B, a classical mode for inner-sphere motion is added. By comparing the two models with experimental data, it is shown that model B is better than model A. For the solvents that have a relaxation time ranging between 0 and 5 ps, our result agrees fairly well with experimental data; for the solvents that have a relaxation time ranging between 5 and 40 ps, our result deviates from the experimental values. After introducing an adjustable scaling index in the effective time correlation function of the reaction coordinate, good agreement is achieved between the experiment and the theory for model B for all of the solvents studied in this paper.