Density Gradient Expansion of the Electronic Exchange-Correlation Energy, and its Generalization

Abstract

Kohn-Sham density functional theory provides a usefully-accurate description for the ground-state electron density \({\text{n}}\left( {\mathop {\text{r}}\limits_ \sim } \right) \) and energy E of a many-electron system. The exchange-correlation energy has a hierarchy of density functional approximations for a system of slowly-varying electron density. The simplest, the local spin density (LSD) approximations, is reasonably good for real systems. However, the second-order gradient expansion (GEA) is typically worse, due to violation of exact properties which LSD respects. Real-space cutoff of the spurious long-range part of the GEA exchange-correlation hole, with cutoff radii chosen to restore those exact properties, leads to a nonempirlpal generalized gradient approximation (GGA) of the form EXC = ∫d3r f (n↑ n↓, ▽n↑, ▽n↓), which is almost as simple as LSD but usually more accurate. A possible rationale is presented for the successes and failures of GGA.