Derivation of a Generalized Gradient Approximation: The PW91 Density Functional

Abstract

Real-space analysis decomposes the exchange-correlation energy of a many-electron system into contributions from all possible interelectronic separations u. The density-gradient expansion of the exchange-correlation hole surrounding an electron has a characteristic structure. Its zeroth-order term, the local spin density (LSD) approximation, is a good approximation to both the hole and its cusp at u = 0. When the electron density varies slowly over space, addition of each successive term of higher order in ∇ improves the description of the hole at small u, but worsens it at large u. Starting with the second-order gradient expansion, we cut off the spurious large-u contributions in a way that restores the negativity and normalization constraints on the exchange hole, and the normalization constraint on the correlation hole. This procedure defines numerical generalized gradient approximations (GGA’s) for the exchange and correlation energies, using no empirical input. We report the results of this construction in detail. This numerical GGA satisfies the most important exact conditions respected by LSD, plus several more (but not all) exact conditions currently known. The PW91 functional is an analytic fit to this functional, designed to respect several further exact conditions including the Lieb-Oxford bound.