Density functional theory and the band gap problem

How can the fundamental band gap of an insulator be predicted? As a difference of ground-state energies, the fundamental gap seems to fall within the reach of density functional theory, yet the predicted gaps from band structure calculations within the local density approximation (LDA) are about 40% too small. It is argued here that even the exact Kohn-Sham potential veff(r), which generates the exact density in a self-consistent-field calculation, generates a band structure which underestimates the gap. Within the context of the band gap problem, several recent developments in the density-functional theory of many-electron systems are reviewed: (1) The Langreth-Mehl approximation to the Kohn-Sham exchange-correlation energy and potential, based upon the Langreth-Perdew wavevector analysis of the density gradient expansion. This functional leads to more accurate ground-state energies and densities than those of the LDA with little change in the calculated band structures of solids. (2) The derivative discontinuity of the exchange-correlation energy, which is responsible for substantial underestimation of the fundamental gap by even the exact Kohn-Sham potential. (3) The self-interaction correction, which yields accurate gaps in insulators only by virtue of its orbital-dependent potential. (4) The density response function of the uniform electron gas, which suggests that the LDA gives a good estimate of the exact Kohn-Sham potential for a semiconductor with a weak periodic potential. In short, several very different (but admittedly approximate) numerical calculations suggest that most of the error in the LDA fundamental gap would persist in the gap of the exact Kohn-Sham band structure. This error would persist in any attempt to calculate the gap from LDA total energy differences for clusters of increasing size.