Electrochemical proton-coupled electron transfer: beyond the golden rule

Electrochemical proton-coupled electron transfer rate constant expressions that interpolate between the golden rule and solvent-controlled limits are derived. These expressions include the effects of solvent dynamics and thus are applicable for a wide range of vibronic couplings and solvent relaxation times. The golden rule limit is defined in terms of weak vibronic coupling and fast solvent relaxation, and the solvent-controlled limit is defined in terms of strong vibronic coupling and slow solvent relaxation. In the golden rule limit, the rate constant is proportional to the square of the vibronic coupling and is independent of the solvent relaxation time. In the solvent-controlled limit, the rate constant is independent of the vibronic coupling and increases as the solvent relaxation time decreases. The interconversion between the solvent-controlled and golden rule limits can be induced by altering the proton donor-acceptor mode frequency and the overlap between the reactant and product proton vibrational wave functions, as well as the electronic coupling, the solvent relaxation time, and the overpotential. The kinetic isotope effect behaves differently in the solvent-controlled and golden rule limits and thus provides a unique probe for characterizing electrochemical proton-coupled electron transfer processes. The analogous rate constant expressions for electrochemical electron transfer and homogeneous proton-coupled electron transfer are also presented. The impact of electrode overpotential, solvent relaxation time, and proton donor-acceptor mode frequency on the rate constants are analyzed for model systems.