Potentials in density functional theory and the importance of sum rules

In an ab initio approach to density functional theory one needs to know the electronic pair-density averaged over the coupling strength of the pair-interaction. As this pair-function is not available without having solved the associated N-electron problem, one has to resort to universal properties of the pair-density that are independent of specific features of the ground-state wavefunction. By exploiting these universal properties and so-called sum rules for the pair-correlation factors we derive very simple approximate spin-dependent expressions for the exchange-correlation energy per particle and for the associated potential in the Kohn-Sham equations. There is some similarity of the resulting density functionals with those obtained from the widely applied local spin density approximation (LSDA) based on electron gas theory. As the application of the latter to exceedingly inhomogeneous gases in realistic systems is very debatable, the manifest similarity seems to suggest that LSDA can consistently be justified only via the above pair-density analysis, but the justification of certain electron gas refinements may remain questionable. We shortly review similar attempts made by other authors and particularly focus on the issue of self-interaction and the “overbinding problem”. We demonstrate for the 3d- and 4d-metals that our approximation yields density of states (DOS), magnetic moments and Stoner parameters that are practically identical with respective data obtained from up-to-date LSDA- or gradient corrected (GGA-)potentials. There is also an excellent agreement of the DOS for the insulators (semi-conductors) C, Si, Ge, and GaAs. We show that our approach yields cohesive energies for these materials that are very close to the GGA-values indicating a distinct improvement over the standard LSDA-values. The calculations have been performed with the aid of the WIEN 97 computer code based on the Full Potential Linear Augmented Plane Wave (FLAPW-) method.