On the connection of semiclassical instanton theory with Marcus theory for electron transfer in solution

We present a derivation of Marcus theory of electron transfer in solution starting from semiclassical instanton theory. The conventional semiclassical instanton theory provides an inadequate description of the electron transfer process in the inverted Marcus regime. This has been attributed to the lack of backscattering in the product region, which is represented as a semi-infinite continuum of states. For electron transfer processes in condensed phase, the electronic states in the acceptor well are bound, which violates the continuum assumption. We show by detailed analysis of the minimum action path of a model system for electron transfer that the proper tunneling coordinate is a delocalized, "bead-count" mode. The tunneling mode is analytically continued in the complex plane as in the traditional derivation. Unlike the traditional analysis where the method of steepest descent is used, the tunneling coordinate is treated as a quasi-zero mode. This feature allows including the influence of backscattering in the acceptor well and leads to the recovery of the Marcus formula for the rate of electron transfer. The results have implications on the performance of ring polymer molecular dynamics for the study of electron transfer dynamics.