Dynamics of reversible electron transfer reactions

The dynamics of reversible electron transfer reactions in Debye solvents are studied by employing two coupled diffusion-reaction equations with the rate constants depending on the reaction coordinate. The equations are solved analytically in four limiting cases: fast and slow reactions as well as wide and narrow reaction windows. A general solution for the survival probabilities is obtained by employing a decoupling approximation similar to the one used by Sumi and Marcus [J. Chem. Phys. 84,4896 ( 1986) ] for nonreversible reactions; our solution verifies the existence of four limiting cases and also predicts the behavior between these limits. Interpolation between long and short time approximations to the general solution, leads to survival probabilities with a single exponential time dependence and rate constants ki satisfying the relation k/k,! = exp ( PAGO), where AGO is the standard free energy change for the reaction. Multiexponential behavior of the survival probabilities is exhibited when higher order terms are included in the evaluation of the general solution, but this deteriorates to a single exponential, governed by a first order rate constant, at long times. In the narrow reaction window limit the multiexponential solution is exact when both the forward and reverse reactions are barrierless, and the behavior at long times is determined by a rate constant k = 0.83 r~ ’ where rL is the longitudinal relaxation time. Similar behavior is found when the forward reaction alone is barrierless and the barrier for the reverse reaction is large (BAG T = 0, BAG : $1)) except that the forward rate constant k, z 7; ’ [ 0.6 + (r//3AG : ) “2] ’ depends on the barrier height for the reverse reaction which has a small rate constant. Our solutions reduce to those of Sumi and Marcus when the reverse reaction is ignored. They are also compared with numerical solutions to the diffusion reaction equations. The extension to non-Debye solvents is briefly discussed.