Exchange-correlation energy of a metallic surface: Wave-vector analysis. II

The wave-vector analysis $\ensuremath{\gamma}(K)$, whose integral gives the nonuniform part of the exchange-correlation energy, is studied for a planar metallic surface. It is shown (1) that for "exchange only" $\ensuremath{\gamma}(K)\ensuremath{\rightarrow}\ensuremath{-}\frac{0.009662}{{{r}_{s}}^{3}}$ a.u. as $K\ensuremath{\rightarrow}0$ for the infinite barrier surface; (2) that this limit is universal---for all planar surfaces in an exchange-only approximation, $\ensuremath{\gamma}$ approaches this same universal number; (3) how the limit above reverts to that previously derived by the authors [$\ensuremath{\gamma}(K)\ensuremath{\propto}|K|$ as $K\ensuremath{\rightarrow}0$] when electron correlation is included. The method of wave-vector interpolation, which relies on the latter limit for the Coulomb-correlated surface, thus withstands recent questioning based on the "exchange-only" approximation. Finally, the probable small-$K$ behavior of $\ensuremath{\gamma}(K)$ for a system of neutral fermions is discussed.