Electrified interfaces from first-principles: continuum electrochemistry and grand canonical density-functional theory

Simulating electrochemical interfaces within density-functional theory (DFT) requires incorporating the electrochemical potential. In practice, this is commonly achieved by varying the number of electrons in the Kohn–Sham equations, leading to grand-canonical DFT calculations. The grand-canonical formalism is related to the canonical ensemble through a Legendre transformation, which maps the energy onto the grand potential while leaving the corresponding atomic forces unchanged. Building on these established energy and force relations, we investigate how higher-order derivatives, including vibrational response properties, are modified under grand-canonical conditions. We derive an explicit correction term that relates the grand-canonical and canonical force-constant matrices. The correction is validated using finite-difference calculations for CO adsorbed on Pt(111), where it explains the ensemble dependence of the vibrational Stark tuning rate, S ω = d ω / d U . The derived expression also explains why vibrational modes involving motion perpendicular to the surface exhibit larger ensemble-dependent shifts than modes dominated by parallel motion. It also rationalizes the thermodynamic-limit behavior, in which ensemble differences vanish as the adsorbate coverage decreases. Finally, we discuss the limitations of grand-canonical calculations combined with implicit-solvent models, showing that Stark tuning rates depend sensitively on solvent and electrolyte-cavity parameters.