Theory of Proton Discharge on Metal Electrodes: Electronically Adiabatic Model

Abstract

The first step of the hydrogen evolution reaction, an important reaction for the storage of renewable energy, is the formation of a surface-adsorbed hydrogen atom through proton discharge to the electrode surface, commonly known as the Volmer reaction. Herein a theoretical description of the Volmer reaction is presented. In this formulation, the electronic states are represented in the framework of empirical valence bond theory, and the solvent interactions are treated using a dielectric continuum model in the linear response regime. The nuclear quantum effects of the transferring proton are incorporated by quantization along the proton coordinate. The ground and excited state electron−proton vibronic free energy surfaces are computed as functions of the proton donor−acceptor distance and a collective solvent coordinate. In the fully adiabatic regime, the current densities and Tafel slopes are computed from the ground state vibronic free energy surface. This theory is applied to the proton-coupled electron transfer reaction involving proton discharge from H 3 O + in aqueous solution to a gold electrode. This theoretical model opens the door for future studies, including examination of the effects of vibronic nonadiabaticity, electronic friction, and solvent dynamics.

Advances and challenges for experiment and theory for multi-electron multi-proton transfer at electrified solid–liquid interfaces

“…This reaction can be described in terms of a diabatic electronic state for the reactant, corresponding to the AH + covalent bond, and a set of diabatic electronic states for the product, corresponding to the MH covalent bond associated with the continuum of energy levels in the metal electrode. 255 The free energies of these diabatic electronic states depend on the proton coordinate r, a collective solvent coordinate X, and the distance of the proton-donating acid from the electrode R (i.e., the proton donor-acceptor distance). 255 Quantization of the proton produces a set of electron-proton vibronic states for the reactant and the product with free energies that depend on X and R. For a fixed distance R, reorganization of the solvent leads to an intersection between a pair of reactant and product vibronic free energy surfaces.…”

“…255 The free energies of these diabatic electronic states depend on the proton coordinate r, a collective solvent coordinate X, and the distance of the proton-donating acid from the electrode R (i.e., the proton donor-acceptor distance). 255 Quantization of the proton produces a set of electron-proton vibronic states for the reactant and the product with free energies that depend on X and R. For a fixed distance R, reorganization of the solvent leads to an intersection between a pair of reactant and product vibronic free energy surfaces. In the vibronically nonadiabatic regime, a nonadiabatic transition can occur at the intersection point with a probability depending on the square of the vibronic coupling between the reactant and product vibronic states.…”

“…242,243,251 In the vibronically adiabatic regime, the reaction occurs on the vibronic ground state free energy surface, which smoothly transitions from the reactant to the product. 255 In both regimes, the resulting rate constant k(R,E) depends on the distance R from the electrode and the applied electrode potential E. The current density is obtained by averaging this rate constant over the distance R, weighting by the concentration c HA +(R) of the acid:…”

“…16). 255 Quantization of the transferring proton coordinate results in a vibronic free energy surface that depends on only X and R, as well as the applied electrode potential E. For each value of E, the free energy barrier corresponding to the saddle point relative to the reactant minimum on this two-dimensional vibronic free energy surface can be computed. 255 Within a transition state theory framework, the rate constant depends exponentially on this free energy barrier, and the transfer coefficients can be computed.…”

“…This vibronically adiabatic approach was applied to the Volmer reaction in acidic aqueous solution. 255 In this application, an empirical valence bond model was developed to generate the ground electronic state free energy surfaces at each applied potential E, followed by quantization of the proton and calculation of the free energy barrier on the vibronic ground state free energy surface. Most of the parameters in this model can be computed using first-principles simulation methods, although in some cases experimental data may be used as a guide.…”