Generalized gradient approximation for the exchange-correlation hole of a many-electron system

We construct a generalized gradient approximation (GGA) for the density n_xc(r,r+u) at position r+u of the exchange-correlation hole surrounding an electron at r, or more precisely for its system and spherical average 〈n_xc(u)〉=(4π)^-1∫dΩ_u N^-1∫d³r n(r)n_xc(r,r+u). Starting from the second-order density gradient expansion, which involves the local spin densities n_↑(r),n_↓(r) and their gradients ∇n_↑(r),∇n_↓(r), we cut off the spurious large-u contributions to restore those exact conditions on the hole that the local spin density (LSD) approximation respects. Our GGA hole recovers the Perdew-Wang 1991 and Perdew-Burke-Ernzerhof GGA’s for the exchange-correlation energy, which therefore respect the same powerful hole constraints as LSD. When applied to real systems, our hole model provides a more detailed test of these energy functionals, and also predicts the observable electron-electron structure factor. © 1996 The American Physical Society.