Fourth-order series expansion of the exchange hole

Abstract

Approximate functionals for the exchange-correlation energy of electrons often draw on explicit or implicit models for the exchange-correlation hole. Here we focus on the spherically averaged exchange hole ${\ensuremath{\rho}}_{\mathrm{X}}(\mathbf{r},u)$, which depends on the reference point $\mathbf{r}$ and on the electron-electron distance $u$. We extend the well-known [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989)] second-order Taylor-series expansion in $u$ to fourth order and we show that the fourth-order term can add important additional information that is particularly relevant for molecules compared to atoms. Drawing on these findings, we explore exchange functionals that depend on the fourth-order term of the expansion of ${\ensuremath{\rho}}_{\mathrm{X}}(\mathbf{r},u)$. We also find that Gaussian basis set expansions, frequently used in electronic structure codes, result in unsatisfactory representations of the fourth-order term.