Accurate optimized-potential-model solutions for spherical spin-polarized atoms: Evidence for limitations of the exchange-only local spin-density and generalized-gradient approximations

Abstract

We present accurate optimized-potential-model (OPM) results for spherical spin-polarized atoms emphasizing the precise construction of the OPM exchange potential from the numerical solution of the OPM integral equation, especially for large r. The results are used to discuss the quality of the local spin-density approximation (LSDA) and a generalized-gradient expansion (GGA) [A. D. Becke, Phys. Rev. A 38, 3098 (1988)] for describing these atoms. It is shown that the LSDA can produce substantial errors (beyond what is known from unpolarized atoms) for quantities which are directly related to the spin polarization of these systems. In particular, the LSDA overestimates the magnetization density in the interior of Cu by a factor of 2. While the GGA improves integral quantities like total ground-state and exchange energies, remarkably it is less successful for energy differences like ${\mathit{E}}_{\mathit{x}\mathrm{\uparrow }}$-${\mathit{E}}_{\mathit{x}\mathrm{\downarrow }}$. Most important, however, it is not able to reduce the LSDA’s errors for local quantities like the difference between spin-up and spin-down exchange potentials and magnetization densities significantly nor does it reverse the LSDA’s incorrect ordering of the two highest occupied majority-spin eigenvalues of Cr and Cu.