Generalized gradient approximation to the angle- and system-averaged exchange hole

A simple analytic model is proposed for the angle- and system-averaged exchange hole of a many-electron system. The model hole depends on the local density and density gradient. It recovers a nonoscillatory local-spin density (LSD) approximation to the exchange hole for a vanishing density gradient. The model hole reproduces the exchange energy density of the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) for exchange, and facilitates a detailed understanding of the PBE GGA. The hole model is applied to atoms and molecules, and a comparison is made to exact and LSD angle- and system-averaged exchange holes. We find that the GGA hole model significantly improves upon the LSD model. Furthermore, the GGA hole model accurately describes the change in the exchange hole upon the formation of single bonds, but is less accurate for the formation of multiple bonds, where it misses the appearance of a long-range tail.