Self-consistent calculations of atomic properties using self-interaction-free exchange-only Kohn-Sham potentials.

Abstract

The spin-unrestricted optimized-effective-potential (OEP) method and an approximation to it due to the authors (KLI) have been applied to atoms from Li (Z=3) to Ba (Z=56) plus Au (Z=79) and Hg (Z=80) within the exchange-only scheme. Calculations were performed for the term with the lowest energy in the ground-state configuration. For a few transition elements, terms in the low-lying excited-energy-state configuration are also self-consistently calculated. We have compared our OEP and KLI results with each other and with those obtained from the spin-unrestricted Hartree-Fock (SUHF) method and the local-spin-density exchange-only (LSDX) approximation. As the spin-restricted OEP, the spin-unrestricted OEP yields nearly identical results as the corresponding SUHF. The relative difference of the total energy from that of the SUHF is only 40 ppm for $^{3}\mathrm{Li}$ and monotonically decreases as Z increases to less than 2 ppm for $^{80}\mathrm{Hg}$. The highest occupied energy eigenvalues of each spin projection, ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\sigma}}}$, and the various expectation values for the operators ${\mathit{r}}^{2}$, ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$, and \ensuremath{\delta}(r) are all very similar to the SUHF theory.For most of the atoms, even the spin density mimics the SUHF value quite well. Differences appear to be larger for some transition- and noble-metal atoms. Nevertheless, even in these cases, the total energy and the total electron density of the OEP calculations are still very similar to the corresponding SUHF values. For atoms with the highest occupied energy eigenvalues having the same spatial quantum numbers for both spin projections, the spin splitting, ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\downarrow}}}$-${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\uparrow}}}$, corresponding to the difference between the highest occupied eigenvalues of the minority (\ensuremath{\downarrow}) and majority (\ensuremath{\uparrow}) spin states, is also studied. It is found that the OEP and SUHF spin splittings are nearly identical for atoms with the highest occupied eigenstate being a p state. Noticeable differences (of the OEP and HF spin splitting) occur for a few transition elements having an empty (n-1)d\ensuremath{\downarrow} subshell. However, such larger differences are steadily diminished as the (n-1)d\ensuremath{\downarrow} subshell is progressively filled. As an approximate Kohn-Sham (KS) potential, the KLI method shows itself to be a much improved one compared to the LSDX. The exchange potential constructed from this method retains many of the essential properties of the exact OEP that the LSDX approximation lacks and thus always yields better densities, lower total energies, and more accurate eigenvalues for the highest occupied energy state of each spin projection than the LSDX approximation. In particular, the total energies overestimate the OEP results by only 9 ppm for Li, with the overestimate decreasing for increasing atomic number Z to less than 1 ppm for Z>50. In addition, for atoms other than transition elements or noble metals, the ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\sigma}}}$ are within 0.4% of the corresponding OEP results with slightly larger differences for the other atoms. In all cases, the KLI results for the expectation values ${\mathit{r}}^{2}$,${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and the density at the origin are very close to the exact KS results and are a significant improvement over those provided by the LSDX approximation.