Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities

The local-density approximation for the exchange-correlation potential understimates the fundamental band gaps of semiconductors and insulators by about 40%. It is argued here that underestimation of the gap width is also to be expected from the unknown exact potential of Kohn-Sham density-functional theory, because of derivative discontinuities of the exchange-correlation energy. The need for an energy-dependent potential in band theory is emphasized. The center of the gap, however, is predicted exactly by the Kohn-Sham band structure.