Theoretical formulation of nonadiabatic electrochemical proton-coupled electron transfer at metal-solution interfaces

Expressions are derived for the nonadiabatic transition probabilities, the heterogeneous rate constants, and the current densities of electrochemical proton-coupled electron transfer (PCET) reactions. In these reactions, an electron is transferred between a solute complex and a metal electrode concurrently with proton transfer within the hydrogen-bonded solute complex. The reaction is described in terms of nonadiabatic transitions between reactant and product electron-proton vibronic states. The current densities are obtained by explicit integration over the distance between the solute complex and the electrode, thereby accounting for the effects of extended electron transfer. These systematic derivations are based on a series of well-defined approximations, leading to a series of analytical expressions for the heterogeneous rate constants and current densities that are valid in specified regimes. The resulting expressions are compared with the analogous expressions for electrochemical electron transfer (ET), and specific characteristics of the current densities that implicate PCET are identified. The strong dependence of the vibronic coupling on the proton donor-acceptor distance for PCET leads to additional terms in the electrochemical PCET rate constant expressions. The total reorganization energy includes two additional contributions: the reorganization energy of the proton donor-acceptor mode and a coupling term associated with the modulation of the vibronic coupling by this mode. The rate constants also include an additional exponential temperature-dependent prefactor that depends on the frequency of this mode and a parameter characterizing the modulation of the vibronic coupling by this mode. This prefactor leads to non-Arrhenius behavior of the rate constants at higher temperatures. Furthermore, the effective activation energies contain temperature-dependent terms arising from the change in the equilibrium proton donor-acceptor distance upon electron transfer. This term has a different sign for the cathodic and anodic processes, leading to asymmetries of the Tafel plots, even for small changes in the equilibrium distance. These characteristics of electrochemical PCET are illustrated by calculations for model systems. This theoretical formulation is applicable to a wide range of experimentally studied electrochemical PCET reactions.