  The recently developed semiclassical variational Wigner-Kirkwood (VWK)
approach is applied to finite nuclei using external potentials and
self-consistent mean fields derived from Skyrme interactions and from
relativistic mean field theory. VWK consists of the Thomas-Fermi part plus a
pure, perturbative hbar^2 correction. In external potentials, VWK passes
through the average of the quantal values of the accumulated level density and
total energy as a function of the Fermi energy. However, there is a problem of
overbinding when the energy per particle is displayed as a function of the
particle number. The situation is analyzed comparing spherical and deformed
harmonic oscillator potentials. In the self-consistent case, we show for Skyrme
forces that VWK binding energies are very close to those obtained from extended
Thomas-Fermi functionals of hbar^4 order, pointing to the rapid convergence of
the VWK theory. This satisfying result, however, does not cure the overbinding
problem, i.e., the semiclassical energies show more binding than they should.
This feature is more pronounced in the case of Skyrme forces than with the
relativistic mean field approach. However, even in the latter case the shell
correction energy for e.g. 208Pb turns out to be only -6 MeV what is about a
factor two or three off the generally accepted value. As an ad hoc remedy,
increasing the kinetic energy by 2.5%, leads to shell correction energies well
acceptable throughout the periodic table. The general importance of the present
studies for other finite Fermi systems, self-bound or in external potentials,
is pointed out.
