  We calculate exactly, using finite size techniques, the quantum mechanical
and many-body effects to the self-capacitance of a spherical quantum dot in the
regime of extreme confinement, where the radius of the sphere is much smaller
than the effective Bohr radius. We find that the self-capacitance oscillates as
a function of the number of electrons close to its classical value. We also
find that the electrostatic energy extrapolates to zero when $N=1$, suggesting
that the energy scales like $e^{2}N(N-1)$. This establishes, at least for this
configuration, that the semiclassical description of Coulomb charging effects
in terms of capacitances holds to a good approximation even at very small
scales.
