Exchange Approximations Used in the Energy-Band Calculations of Metals

Abstract

The use of different exchange approximations in the self-consistent energy-band calculations for a metal (Cu) is shown to yield different one-electron eigenvalue spectra. Specifically, when an energy-dependent exchange potential proposed by Liberman was used, the resulting bandwidths ($s-p$, $d$) were much wider than those obtained with the ${\rho }^{\frac{1}{3}}$ exchange operator. Also it is shown that when the energy-dependent exchange operator is screened, as in the Bohm-Pines theory, the resulting bandwidths were considerably narrowed. Although the "${\rho }^{\frac{1}{3}}$" operator and the screened-exchange operator yielded similar results, the variation of the screening parameter did not correspond to a variation of the $\alpha$ parameter of the "${\rho }^{\frac{1}{3}}$" method. Use of the screened-exchange operator appears to yield a reasonable band structure.