Rate Theory for Electrocatalytic Systems: Fixed Potential Formulation for General, Electron Transfer, and Proton-Coupled Electron Transfer Reactions

<div>Atomistic modeling of electrocatalytic reactions is most naturally conducted within the grand canonical ensemble (GCE) which enables fixed chemical potential calculations. While GCE has been widely adopted for modeling electrochemical and electrocatalytic thermodynamics, the electrochemical reaction rate theory within GCE is lacking. Molecular and condensed phase rate theories are formulated within microcanonical and canonical ensembles, respectively, but electrocatalytic systems described within the GCE require extension of the conventionally used rate theories for computation reaction rates at fixed electrode potentials. In this work, rate theories from (micro) canonical ensemble are generalized to the GCE providing the theoretical basis for the computation reaction rates in electrochemical systems. It is shown that all canonical rate theories can be extended to the GCE. From the generalized grand canonical rate theory developed herein, fixed electrode potential rate equations are derived for i) general reactions within the GCE transition state theory (GCE-TST), ii) adiabatic curve-crossing rate theory within the empirical valence bond theory (GCE-EVB), and iii) (non-) adiabatic electron and proton-coupled electron transfer reactions. The rate expressions can be readily combined with ab initio methods to study reaction kinetics reactions at complex electrochemical interfaces as a function of the electrode potential. The theoretical work herein provides the basis for treating electrochemical kinetics and the inclusion of non-adiabatic and tunneling effects in electrochemical environments widening the scope of reactions amenable to computational studies.</div>