Construction and application of an accurate local spin-polarized Kohn-Sham potential with integer discontinuity: Exchange-only theory

An accurate spin-polarized exchange-only Kohn-Sham (KS) potential is constructed from a consideration of the optimized-effective-potential (OEP) method. A detailed analysis of the OEP integral equation for the exchange-only case results in a set of conditions which are manifestly satisfied by the exact OEP; these conditions are employed to construct an approximate OEP, ${\mathit{V}}_{\mathit{x}\mathrm{\sigma }}$, and therefore an approximate KS exchange-only potential as a functional of KS orbitals. Further, it is shown that this ${\mathit{V}}_{\mathit{x}\mathrm{\sigma }}$ can be derived analytically based on a simple approximation of the Green’s functions in the OEP integral equation. The constructed potential, although approximate, contains many of the key analytic features of the exact KS potential: it reduces to the exact KS result in the homogeneous-electron-gas limit, approaches -1/r as r→∞, yields highest occupied-orbital energy eigenvalues ${\mathrm{\varepsilon }}_{\mathit{m}\mathrm{\sigma }}$ that satisfy Koopmans’s theorem, and exhibits an integer discontinuity when considered as a function of fractional occupancy of the highest-energy occupied single-particle state of a given spin projection σ. In addition ${\mathrm{\varepsilon }}_{\mathit{m}\mathrm{\sigma }}$ nearly exactly satisfies Janak’s theorem. The approximate OEP is a simple but remarkably accurate representation of the exact, numerically derived exchange-only OEP.