The transition from nonadiabatic to solvent controlled adiabatic electron transfer: Solvent dynamical effects in the inverted regime

Abstract

We analyze the effect of dynamical solvent effects on the rate of a nonadiabatic electron transfer (ET) reaction. Starting from a Hamiltonian for a reaction coordinate for motion along the potential surfaces of donor and acceptor species, and a bath representing the solvent dynamical effects, we obtain a system of four coupled reduced equations of motion for the elements of the density matrix of the donor/acceptor system. In this derivation the dynamics along the reaction coordinate are reduced to a classical Fokker–Planck operator since we assume the temperature is high compared with bath frequencies. At temperatures where the nuclear motion describing the transition between the surfaces can be treated classically we show that the ET processes may be viewed as a consecutive reaction scheme with rate constant k=kNA kD/(kD+kNA), the steps are diffusion along the reaction coordinate with rate constant kD followed by crossing between the donor and acceptor surfaces at the point of intersection of the surfaces with rate constant kNA. kNA is given by an activated rate expression and is independent of solvent dynamics, e.g., a dielectric relaxation time. When the nuclear motion must be treated quantum mechanically, as is favored in the inverted regime (where the surfaces have slopes of the same sign at their crossing point), the transition between the surfaces is no longer confined to their crossing point. We obtain an approximate expression of the above form where kNA incorporates the width of the transition. Then kNAdepends on solvent dynamics. When the separation into diffusive and crossing motion is no longer appropriate, we use a basis set expansion method to directly solve the four coupled density matrix equations to obtain k. These results are compared with the approximate formula given above.