Piecewise nonlinearity and capacitance in the joint density functional theory of extended interfaces

The ab initio simulation of charged interfaces in the framework of density functional theory (DFT) is heavily employed for the study of electrochemical energy conversion processes. The capacitance is the primary descriptor for the response of the electrochemical interface. It is essentially equal to the inverse of the energy curvature as a function of electron number, and as such there appears a conflict with the fundamental principle of piecewise linearity in DFT that requires the energy curvature to be zero at fractional electron numbers, i.e., almost everywhere. To resolve this conflict, we derive an exact expression between the energy curvature and the Kohn-Sham density of states, the local density of states, and the Fukui potential. We find that the piecewise linearity requirement does not hold for the volume- or area-specific energy of extended systems and surfaces. Applied to the joint density functional theory of an electrode-electrolyte interface, including the ionic and dielectric response of the electrolyte, the same expression represents a rigorous basis for the partitioning of the total interfacial capacitance into contributions of the quantum capacitance, space-charge capacitance, and electrochemical double-layer capacitance. It provides insight into the influence of the electrode material, thickness, and temperature on the charging characteristics, as demonstrated by results for a bulk gold electrode, a single-layer gold electrode, and a single-layer graphene electrode.