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

Electrochemical interfaces and reactions play a decisive role in 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, constant electrode potential calculations at the density functional theory (DFT) level are carried out by fixing the Fermi level of the simulation cell. However, the Fermi level from DFT calculations does does not always reflect the experimentally controlled electrode potential or describe the thermodynamic independent variable in GCE-DFT i.e the electrochemical potential of an electron reservoir. Here we develop and implement the constant inner potential (CIP) method as a more robust and general approach to GCE-DFT simulations of electrochemical systems under constant potential or bias conditions. The CIP is shown to directly control the reservoir electron electrochemical potential making 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. The CIP method is shown to be applicable in simulating also semiconductor electrodes, outer sphere reactions, and a biased two-electrode cell for which the Fermi level approach does not reflect the experimental electrode potential. Unlike the Fermi level method, CIP does not require any electronic structure information as only the inner potential is needed, which makes the approach more compatible with classical force field or machine learning potentials. The CIP approach emerges as a general GCE DFT method to simulate (photo)electrochemical interfaces from first principles.