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., ${\lim }_{\lambda \to \infty }{E}_{\mathrm{c}}^{\mathrm{KS}}\left[n_{\lambda }\right]>-\infty$, where $n_{\lambda }\left(\mathbf{r}\right)={\lambda }^{3}n\left(\lambda \mathbf{r}\right)$. 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.

Figure 1

Errors in the total energy $(E-E_{\text{exact}})$ as functions of nuclear charge $Z$ for the 4-electron $\left[\mathrm{Be}\right]$ series computed with the Hartree-Fock (HF) method and with approximate density functionals for exchange and correlation.

Figure 2

Same as in Fig. 1 for the 10-electron $\left[\mathrm{Ne}\right]$ series.

Figure 3

Same as in Fig. 1 for the 18-electron $\left[\mathrm{Ar}\right]$ series.

Figure 4

Errors in the total energy $(E-E_{\text{exact}})$ as functions of nuclear charge $Z$ for the 4-electron $\left[\mathrm{Be}\right]$ series computed with approximate density functionals for correlation combined with Hartree-Fock (HF) exchange.

Figure 5

Same as in Fig. 4 for the 10-electron $\left[\mathrm{Ne}\right]$ series.

Figure 6

Same as in Fig. 4 for the 18-electron $\left[\mathrm{Ar}\right]$ series.