  Using the Wentzel-Kramers-Brillouin method, we derive a modified form of the
Thomas-Fermi approximation to electron density. This new result enables us to
calculate the details of the self-consistent ion cores, as well as the
ionization potentials for the first s-orbital bound to the closed-shell ion
core of the Group III, IV and V elements. Next, we demonstrate a method for
separating core electron densities from valence electron densities. When we
calculate the valence kinetic energy density, we show that it separates into
two terms: the first exactly cancels the potential energy of the ion core in
the core region; the second represents the residual kinetic energy density
resulting from the valence density alone. Furthermore, we show that the kinetic
cancellation and the residual kinetic energy can be derived from a slowly
varying envelope approximation for the valence orbitals in the core region.
This kinetic cancellation in the core region and the residual valence kinetic
energy term allow us to write a functional for the total valence energy
dependant only on a low spatial frequency valence density. In the limit, when
we can freeze the potential of the closed-shell ion cores, assuming that they
are not greatly influenced by the readjustment of the valence electrons, we can
minimize the total valence energy with respect to the valence density degrees
of freedom. The variation of the valence total energy equation provides the
starting point for a large number of atomic, molecular and solid-state
electronic structure problems. Here, we use it to calculate the band structures
resulting from the self-consistent valence density and potential on the
zinc-blende and diamond lattices. We give band structure results for most of
the Group III-V, as well as Group IV, materials.
