Two-electron transfer reactions involving three paraboloidal potential surfaces in solvents with multiple solvation time scales

Abstract

The effect of solvent nuclear relaxation dynamics on the rate of two-electron transfer reaction is investigated. We present here a generalized treatment of the Zusman and Beratan model of two-electron transfer reaction using a theoretical scheme that starts from the Liouville equation of motion for the electronic population wave packets to obtain the transfer rates following projection operator formalism. This generalization enables us to treat the three free energy surfaces (three surfaces for D–A, D+–A−, and D+2–A−2 donor–acceptor pairs) involved in such reactions on an equal footing such that the rates for each one- and two-electron transfer step can be obtained when all three diabatic surfaces are present in the system with nonzero electronic coupling elements between them. The reaction takes place on a two-dimensional potential energy surface with two coordinates representing the solvent polarization. The dynamics are governed by overdamped diffusion along these polarization coordinates with different solvent polarization time scales. The resulting equations, that can interpolate the situation between the nonadiabatic and the diffusion limits of electron transfer, are solved numerically for the choice of parameters that validates the criterion for solvent dynamics-influenced rate limit. The transfer rates, in this limit, are found to depend strongly on the multiplicity of the solvent polarization coordinate used. New dynamical solvent effects on the transfer rates in solvents with one or more characteristic relaxation time scales are identified because of the effective participation of all three electronic states in the transfer process. The theoretical recipe developed here is not limited to two-electron transfer problems and can be applied for multiple electron transfer events in solvents with multiple relaxation time scales.