We report a detailed theoretical investigation on electrochemical capacitan
ce of a nanoscale capacitor where there is a de coupling between the two co
nductors. For this "leaky" quantum capacitor, we have derived general analy
tic expressions of the linear and second-order nonlinear electrochemical ca
pacitance within a first-principles quantum theory in the discrete potentia
l approximation. Linear and nonlinear capacitance coefficients are also der
ived in a self-consistent manner without the latter approximation and the s
elf-consistent analysis is suitable for numerical calculations. At linear o
rder, the full quantum formula improves the semiclassical analysis in the t
unneling regime. At nonlinear order that has not been studied before for le
aky capacitors, the nonlinear capacitance and nonlinear nonequilibrium char
ge show interesting behavior. Our theory allows the investigation of crosso
ver of capacitance from a full quantum to classical regimes as the distance
between the two conductors is changed. [S0163-1829(99)03548-1].