Generalized exchange local-spin-density-functional theory: Introduction and theory

Abstract

A local-spin-density-functional (LSD) scheme is developed using the normalization conditions for an electron gas around a nucleus of charge Z. The correlation factor for electrons of parallel spin, the Fermi hole, is decomposed into single-particle correlation factors. This leads to an orbital-dependent exchange-only density with no adjustable parameters such as the \ensuremath{\alpha} of the X\ensuremath{\alpha} theory; the parameters ${B}_{1}$, ${B}_{2}$, and ${\ensuremath{\alpha}}^{\mathrm{lim}}$ are constant for all atoms once the shape of the Fermi hole is chosen. These parameters are rigorously calculated without assuming an approximate shape for the Fermi-hole correlation factor. The exchange density using this unspecified Fermi-hole correlation factor reduces exactly to the homogeneous free-electron-gas exchange density at the high electron-density limit. The theoretical relationship of this generalized exchange (GX) scheme to the X\ensuremath{\alpha} and free-electron-gas exchange densities is discussed, and it is found that the X\ensuremath{\alpha} scheme needs a classical approximation to justify the use of a variable \ensuremath{\alpha}. These schemes are compared numerically using the total energies and eigenvalues for the atoms helium to krypton, and it is found that the LSD GX scheme gives the most reliable results.