Energies of isoelectronic atomic ions from a successful metageneralized gradient approximation and other density functionals

We show that the poor performance of approximate Kohn-Sham (KS) density functional theory for highly charged atomic ions is improved dramatically by ensuring that (i) the exchange functional recovers the correct leading term in the $Z$ expansion of the exchange energy ($Z$ is the nuclear charge) and (ii) the correlation functional is bounded under uniform scaling of the density to the high-density limit---i.e., ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{E}_{\mathrm{c}}^{\mathrm{KS}}[{n}_{\ensuremath{\lambda}}]>\ensuremath{-}\ensuremath{\infty}$, where ${n}_{\ensuremath{\lambda}}(\mathbf{r})={\ensuremath{\lambda}}^{3}n(\ensuremath{\lambda}\mathbf{r})$. The performance of several density functionals (BLYP, BP86, VS98, HCTH/407, PBE, PKZB, and TPSS) is compared for the 4-, 10-, and 18-electron atomic series spanning values of $Z$ from 4 to 28. Especially accurate results are obtained with the nonempirical metageneralized gradient approximation of Tao, Perdew, Staroverov, and Scuseria (TPSS). High-$Z$ limits of selected exchange and correlation functionals are evaluated and compared with the exact values.