Theory of electron transfer in the presence of dissipation

An analytic study of the density matrix and Wigner representation equations for dissipative electron transfer is presented. An explicit expression is derived for the off-diagonal Green's function, which shows a very fast relaxation in time if the barrier to reaction is greater than the thermal energy. This fast relaxation invalidates previous attempts to derive coupled equations for the density in the large friction limit. The fast off-diagonal relaxation disallows an adiabatic elimination of the momentum even in the large friction limit. We then show, with the aid of the boundary layer method, how one can use the same analysis to derive a set of two coupled equations for the diagonal densities. These equations are a generalization to phase space of the large friction Zusman equations [Chem. Phys. 49, 295 (1980)]. Adiabatic elimination of the momentum from these generalized Zusman equations is correct in the large friction limit and naturally leads back to the Zusman equations. Numerical solution of the generalized Zusman equations is presented for symmetric electron transfer for weak and strong electronic coupling, moderate and high barriers, and a large range of damping. The numerical results provide new insight into the friction dependence of the rate in the weak damping regime and show that previous analytic expressions for the rate are only qualitative in nature. (C) 2003 American Institute of Physics.