Grand Canonical Rate Theory for Electrochemical and Electrocatalytic Systems I: General Formulation and Proton-coupled Electron Transfer Reactions

Electrochemical interfaces present a serious challenge for atomistic modelling. Electrochemical thermodynamics are naturally addressed within the grand canonical ensemble (GCE) but the lack of a fixed potential rate theory impedes fundamental understanding and computation of electrochemical rate constants. Herein, a generally valid electrochemical rate theory is developed by extending equilibrium canonical rate theory to the GCE. The extension provides a rigorous framework for addressing classical reactions, nuclear tunneling and other quantum effects, non-adiabaticity etc. from a single unified theoretical framework. The rate expressions can be parametrized directly with self-consistent GCE-DFT methods. These features enable a well-defined first principles route to addressing reaction barriers and prefactors (proton-coupled) electron transfer reactions at fixed potentials. Specific rate equations are derived for adiabatic classical transition state theory and adiabatic GCE empirical valence bond (GCE-EVB) theory resulting in a Marcus-like expression within GCE. From GCE-EVB general free energy relations for electrochemical systems are derived. The GCE-EVB theory is demonstrated by predicting the PCET rates and transition state geometries for the adiabatic Au-catalyzed acidic Volmer reaction using (constrained) GCE-DFT. The work herein provides the theoretical basis and practical computational approaches to electrochemical rates with numerous applications in physical and computational electrochemistry.