  In order to obtain a reasonably accurate and easily implemented approach to
many-electron calculations, we will develop a new Density Functional Theory
(DFT). Specifically, we derive an approximation to electron density, the first
term of which is the Thomas-Fermi density, while the remaining terms
substantially correct the density near the nucleus. As a first application,
this new result allows us to accurately calculate the details of the
self-consistent ion cores, as well as the ionization potentials for the outer
s-orbital bound to the closed-shell ion core of the Group III, IV and V
elements. Next, we demonstrate that the new DFT allows us to separate
closed-shell 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 due to the ion core
in the core region; the second represents the residual kinetic energy density
resulting from the envelopes of the valence electron orbitals. This kinetic
energy 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 the valence density. This equation provides the starting point for a large
number of electronic structure calculations. Here, we use it to calculate the
band structures of several Group IV and Group III-V semiconductors.
