We have performed self-consistent (SC) band structure calculations for the A15 compounds V/sub 3/X and Nb/sub 3/X, X = Al, Ga, Si, Ge, and Sn, using the augmented-plane-wave (APW) method. Relativistic effects (except the spin-orbit interaction) have been included in each SC cycle, along with corrections to the usual muffin-tin approximation. The latter apply the APW wave functions outside of the muffin-tin spheres to compute the interstitial charge densities and potentials. The resulting interstitial potential has full cubic symmetry (no spherical averaging). The final SC potentials were used to generate energies and wave functions on a cubic mesh of 35 k points in 1/48th of the Brillouin zone. These results were interpolated onto a finer mesh of 969 k points using a symmetrized Fourier method; the densities of states (DOS), N (E), were determined using tetrahedral integration. These accurate interpolation methods allow us to determine the DOS on a fine energy scale ( +- 3 mRy) around the Fermi level E/sub F/, where we find large variations for the compounds V/sub 3/Ga, V/sub 3/Si, and Nb/sub 3/Sn. This correlates well with the fact that these compounds have high superconducting transition temperatures (T/sub c/) and anomalous electronically derived properties. The energymore » bands of the A15 materials exhibit significant variations even amongst the isoelectronic compounds. All compounds possess very flat bands that evolve from the GAMMA/sub 12/ state near E/sub F/ and give rise to the sharp peaks in N (E). For V/sub 3/Ga, V/sub 3/Si, and Nb/sub 3/Sn, E/sub F/ falls within several mRy of GAMMA/sub 12/ so that these flat bands are responsible for the sharp structure in N (E) at E/sub F/. We find that Nb/sub 3/Ge and Nb/sub 3/Si have relatively low values of N (E/sub F/), which suggests that the high T/sub c/'s observed in films of the former and predicted for the latter are due to unusual mechanisms.« less