On the equivalence of grand-canonical and capacitor-based models in first-principle electrochemistry

First principles-based computational and theoretical methods are constantly evolving trying to overcome the many obstacles towards a comprehensive understanding of electrochemical processes on an atomistic level. One of the major challenges has been the determination of reaction energetics under a constant applied potential. Here, a theoretical framework was proposed applying standard electronic structure methods and extrapolating to the infinite-cell size limit where reactions do not alter the potential. More recently, grand-canonical modifications to electronic structure methods which hold the potential constant by varying the number of electrons in a finite simulation cell have gotten increasingly popular. In this perspective, we show that these two schemes are thermodynamically equivalent. Further, we link these methods to ones based on capacitive models of the interface, in the limit that the capacitance of the charging components (whether continuum or atomistic) are equal and invariant along the reaction pathway. We further benchmark the three approaches with an example of alkali cation adsorption on Pt(111) proving that all three approaches converge in the cases of Li, Na and K. For Cs, however, strong deviation from the ideal conditions leads to a spread in the respective results. We discuss the latter by highlighting the cases of broken equivalence and assumptions among the approaches.