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

The effect of solvent nuclear relaxation dynamics on the rate of two-electr on transfer reaction is investigated. We present here a generalized treatme nt 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 followi ng projection operator formalism. This generalization enables us to treat t he 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 ob tained when all three diabatic surfaces are present in the system with nonz ero electronic coupling elements between them. The reaction takes place on a two-dimensional potential energy surface with two coordinates representin g the solvent polarization. The dynamics are governed by overdamped diffusi on along these polarization coordinates with different solvent polarization time scales. The resulting equations, that can interpolate the situation b etween the nonadiabatic and the diffusion limits of electron transfer, are solved numerically for the choice of parameters that validates the criterio n for solvent dynamics-influenced rate limit. The transfer rates, in this l imit, are found to depend strongly on the multiplicity of the solvent polar ization coordinate used. New dynamical solvent effects on the transfer rate s in solvents with one or more characteristic relaxation time scales are id entified because of the effective participation of all three electronic sta tes in the transfer process. The theoretical recipe developed here is not l imited to two-electron transfer problems and can be applied for multiple el ectron transfer events in solvents with multiple relaxation time scales. (C ) 1999 American Institute of Physics. [S0021-9606(99)51619-X].