  We report a detailed theoretical investigation on electrochemical capacitance
of a nanoscale capacitor where there is a DC coupling between the two
conductors. For this ``leaky'' quantum capacitor, we have derived general
analytic expressions of the linear and second order nonlinear electrochemical
capacitance within a first principles quantum theory in the discrete potential
approximation. Linear and nonlinear capacitance coefficients are also derived
in a self-consistent manner without the latter approximation and the
self-consistent analysis is suitable for numerical calculations. At linear
order, the full quantum formula improves the semiclassical analysis in the
tunneling regime. At nonlinear order which has not been studied before for
leaky capacitors, the nonlinear capacitance and nonlinear nonequilibrium charge
show interesting behavior. Our theory allows the investigation of crossover of
capacitance from a full quantum to classical regimes as the distance between
the two conductors is changed.
