Exchange energy density in density-functional theory via the Dirac density matrix for a nonrelativistic 10-electron atomic ion compared with Becke’s proposal for a gradient-corrected local-density-approximation result

Abstract

Especially in atomic systems, it is now well established that exchange energy generally dominates correlation effects. Therefore we focus here on the exchange energy density ${\ensuremath{\epsilon}}_{x}(r)$ as given in terms of the idempotent Dirac density matrix. This is then brought into contact with the one-parameter form of Becke's functional, which corrects the local-density-approximation form $\ensuremath{-}{c}_{x}{[n(r)]}^{4/3}$ with $n(r)$ as the ground-state electron density, ${c}_{x}=(3/4){e}^{2}{(3/\ensuremath{\pi})}^{1/3}$, by terms involving the dimensionless gradient ratio $|\ensuremath{\nabla}n(r)|/{n}^{4/3}(r)$. A particular nonrelativistic model of the 10-electron Ne-like atomic ions, with large atomic number $Z$, is then compared to Becke's approximation to ${\ensuremath{\epsilon}}_{x}(r)$.