Constant inner potential DFT for modelling electrochemical systems under constant potential and bias

Electrochemical interfaces and reactions play a decisive role in \textit{e.g.} clean energy conversion but understanding their complex chemistry remains an outstanding challenge. Constant potential or grand canonical ensemble (GCE) simulations are indispensable for unraveling the properties of electrochemical processes as a function of the electrode potential. Currently, GCE calculations at the density functional theory (DFT) level are carried out by fixing the Fermi level of the simulation cell. Here we show that this method is inadequate for modelling semiconductor electrodes, outer sphere reactions, and a biased two-electrode cell; for these systems the Fermi level obtained from DFT calculations does not reflect the experimentally controlled electrode potential or describe the thermodynamic independent variable in GCE-DFT. To remedy this deficiency, we developed and implemented a constant inner potential (CIP) method as a more robust and general approach to carry out GCE-DFT simulations of electrochemical systems under constant potential or bias conditions. In CIP-DFT the electrode inner potential and hence the thermodynamically relevant electron bath electrochemical are directly controlled which makes the method widely applicable in simulating electrochemical interfaces. We demonstrate that the CIP and Fermi level GCE-DFT approaches are equivalent for metallic electrodes and inner-sphere reactions but CIP is also applicable to systems for which the constant Fermi level approach fails. A key advantage of CIP is that, unlike the Fermi level method, it does not require any electronic structure information. This is because only the inner potential of the systems is needed, CIP is also more compatible with classical force field or machine learning potentials. Altogether, the CIP approach emerges as a general and efficient GCE-DFT method to simulate (photo)electrochemical interfaces from first principles.