The density-gradient expansion of the fermion exchange hole is analyzed in real space. Unlike the local-density approximation, the second-order gradient-expansion approximation is found to violate two important properties of the exact hole: The exact hole is negative everywhere, and represents a deficit of one electron. Imposition of these exact constraints leads to an accurate new density functional for the exchange energy. Residual errors in the exchange energy for atoms are about 1% of this quantity. The new functional approximation may be generalized to include correlation.

• Received 11 July 1985

DOI:https://doi.org/10.1103/PhysRevLett.55.1665