  Dirac sea corrections for bulk properties of finite nuclei are computed
within a self-consistent scheme in the $\sigma$-$\omega$ model. The valence
part is treated in the Hartree approximation whereas the sea contribution is
evaluated semiclassically up to fourth order in $\hbar$. Numerically, we find a
quick convergence of the semiclassical expansion; the fourth order contributing
much less than one percent to the binding energy per nucleon.
