Theory of electrochemical proton-coupled electron transfer in diabatic vibronic representation: Application to proton discharge on metal electrodes in alkaline solution

Proton discharge on a metal electrode is an important step in a wide range of electrocatalytic processes. In this proton-coupled electron transfer (PCET) reaction, also denoted the Volmer reaction, an acid in solution donates its proton to form a metal-hydrogen bond. Herein, a theoretical formulation is developed to describe this PCET reaction as a series of nonadiabatic transitions between reactant and product diabatic electron-proton vibronic states. The product states consist of overlapping continua of vibronic states associated with the continuum of delocalized electrode electronic states, leading to small coupling between each pair of vibronic states. The probability of reaching a crossing point between each pair is influenced by the probability of bypassing all other transitions. The resulting rate constant expression smoothly bridges the adiabatic and nonadiabatic limits for the entire range of electronic coupling and can be interpolated with the solvent-controlled regime. This theoretical formulation is applied to the Volmer reaction on a gold electrode in alkaline aqueous solution. The proposed mechanism involves proton discharge from a water dimer to the electrode surface simultaneously with proton transfer between the two water molecules and adsorption of hydroxide on the electrode. The calculated transfer coefficients and kinetic isotope effects are in qualitative agreement with experimental measurements, predicting a potential-dependent kinetic isotope effect. The decreasing kinetic isotope effect with less cathodic potentials is attributed to increasing contributions from excited reactant vibronic states, which are associated with larger and less isotope-dependent vibrational overlaps. This general theoretical formulation will enable the investigation of heterogeneous electrochemical PCET reactions spanning the adiabatic, nonadiabatic, and solvent-controlled regimes.