Microscopic formulation of Marcus’ theory of electron transfer

A microscopic theory for the rate of nonadiabatic electron transfer is developed and its relation to classical Marcus theory is analyzed. The focus is on how the nonlinear response of a molecular solvent to a change in the charge distribution of the donor–acceptor pair influences the rate; quantum mechanical and solvent dynamical effects are ignored. Under these restrictions, the rate is determined by the probability density of the energy gap, which is defined as the instantaneous change in solvation energy upon moving an electron from the donor to the acceptor. It is shown how this probability density can be obtained from the free energies of transferring varying amounts of charge between the donor and acceptor (as specified by a charging parameter). A simple algorithm is proposed for calculating these free-energy changes (and hence the energy gap probability density) from computer simulations on just three states: the reactant, the product, and an ‘‘anti’’-product formed by transferring a positive unit charge from the donor to the acceptor. Microscopic generalizations of the Marcus nonequilibrium free-energy surfaces for the reactant and the product, constructed as functions of the charging parameter, are presented. Their relation to surfaces constructed as functions of the energy gap is also established. The Marcus relation (i.e., the activation energy as a parabolic function of the free-energy change of reaction) is derived in a way that clearly shows that it is a good approximation in the normal region even when the solvent response is significantly nonlinear. A simple generalization of this relation, in which the activation energy is given by parabolic functions with different curvatures in the normal and inverted regions, is proposed. These curvatures are inversely proportional to the reorganization energies of the product and the antiproduct, respectively. Computer simulations of a simple model system are performed to illustrate and test these results and procedures.