After reviewing some of the progress we have made during the past few years in studying the energy band structure of a large number of semi‐conductors, we consider ways of improving the physical realism and significance of first‐principles band structure calculations. We suggest that the one‐electron binding energy spectrum rather than the one‐electron energy eigenvalue spectrum should be compared with the experimental one‐electron excitation spectrum. In order to calculate the binding energies, it is necessary to approximate the so‐called Koopmans corrections in some manner. A particularly simple method for doing so is outlined. In order to illustrate the procedure, detailed results are reported for atomic krypton. The relative merits of various exchange approximations in the light of this work are also considered. Copyright © 1970 John Wiley & Sons, Inc.