Outer-sphere electron transfer in polar solvents: Quantum scaling of strongly interacting systems

Abstract

The spin–boson Hamiltonian model is used to study electron transfer (ET) reactions of strongly interacting systems in polar solvents in the limit of fast dielectric relaxation of the solvent. The spectrum of polarization modes consists of low frequency modes which are treated classically, and high frequency modes which are treated quantum mechanically. A general explicit formula for the rate valid in all orders of perturbation theory in electronic coupling is derived. The rate formula is applicable in a wide range of parameters, including the inverted region of the reaction where the quantum tunneling corrections give the main contribution to the rate. It is found that the quantum degrees of freedom can be effectively eliminated from the model by renormalizing the electronic coupling matrix element. This renormalization results in the following scaling property of the electron transfer systems: a system containing both classical and quantum degrees of freedom is equivalent to a system of lower dimensionality, containing only classical degrees of freedom, with renormalized electronic coupling matrix element. An explicit formula for the renormalization is obtained.