Derivation of rate expressions for nonadiabatic proton-coupled electron transfer reactions in solution

This paper presents a derivation of rate expressions for nonadiabatic proton-coupled electron transfer (PCET) reactions in solution. The derivation is based on a multistate continuum theory in which the solvent is described by a dielectric continuum, the solute is represented by a multistate valence bond model, and the transferring proton(s) are treated quantum mechanically. In this formulation, a PCET reaction is described as a transition between two sets of diabatic free energy surfaces associated with the two electron transfer states. For PCET reactions involving the transfer of one electron and one proton, these mixed electronic/proton vibrational free energy surfaces are functions of two scalar solvent coordinates corresponding to electron and proton transfer. The Golden Rule is applied to these two-dimensional free energy surfaces in conjunction with a series of well-defined approximations. The contributions from intramolecular solute modes are also included. The final rate expression is similar in form to the standard rate expression for nonadiabatic single electron transfer, but the reorganization energies, equilibrium free energy differences, and couplings are defined in terms of the two-dimensional free energy surfaces. The practical implementation of this rate expression and the calculation of the input quantities are also discussed.