  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.
