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

Electrochemical systems play a decisive role in, e.g. clean energy conversion but understanding their complex chemistry remains an outstanding challenge. Constant potential and grand canonical ensemble (GCE) simulations are indispensable for unraveling the properties of electrochemical processes as a function of the electrode potential. Currently, GCE calculations performed at the density functional theory (DFT) level require fixing the Fermi level within the simulation cell. Here, we illustrate that this method is inadequate when modeling outer sphere reactions and a biased two-electrode cell. For these systems, the Fermi level obtained from DFT calculations does not accurately present the experimentally controlled electrode potential or describe the thermodynamic independent variable in GCE-DFT. To address this limitation, we developed and implemented a constant inner potential (CIP) method offering a more robust and general approach to conducting GCE-DFT simulations of electrochemical systems under constant potential or bias conditions. The primary advantage of CIP is that it uses the local electrode inner potential as the thermodynamic parameter for the electrode potential, as opposed to the global Fermi level. Through numerical and analytical studies, we demonstrate that the CIP and Fermi level GCE-DFT approaches are equivalent for metallic electrodes and inner-sphere reactions. However, CIP proves to be more versatile, as it can be applied to outer-sphere and two-electrode systems, addressing the limitations of the constant Fermi-level approach in these scenarios. Altogether, the CIP approach stands out as a general and efficient GCE-DFT method simulating electrochemical interfaces from first principles.