The band-gap problem is studied for an exactly solvable semiconductor model. The exact exchange-correlation potential is constructed and its discontinuity is studied. Depending on the parameter range, a large or a small discontinuity is obtained. The "scissor-operator" method is compared with the local-density approximation for an impurity calculation. In general, the local-density approximation is closer to the exact solution. Some general features of the density-functional formalism are illustrated.

• Received 23 September 1985

DOI:https://doi.org/10.1103/PhysRevLett.56.1968