We have recently developed an accurate and easily implemented approach to many-electron calculations, based on a modified Thomas-Fermi approximation. Specifically, we derived an electron density approximation, the first term of which is the Thomas-Fermi result, while the remaining terms substantially corrected the density near the nucleus. In a first application, we used the new density 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 demonstrated that the new density expression allows us to separate closed-shell core electron densities from valence electron densities. When we calculated the valence kinetic energy density, we showed that it separated into two terms: the first exactly cancelled the potential energy due to the ion core in the core region; the second was the residual kinetic energy density resulting from the envelopes of the valence electron orbitals. These features allowed us to write a functional for the total valence energy dependant only on the valence density. This equation provided the starting point for a large number of electronic structure calculations. Here, we used it to calculate the band structures of several Group IV and Group III-V semiconductors. We emphasize that this report only provides a summary; detailed derivations of all results are in Reference 5.