Renormalized Atoms and the Band Theory of Transition Metals

Abstract

The renormalized-atom approach, first used by Chodorow, is shown to yield quantitative estimates of some of the essential potential-dependent parameters characterizing transition-metal band structures on the basis of essentially atomic considerations. These are the position ${\ensuremath{\Gamma}}_{1}$ of the conduction-band minimum, the mean $d$-band energy, the energies associated with $d$-band extrema, and the degree of $s\ensuremath{-}d$ hybridization as defined within the Heine-Hubbard pseudopotential schemes. The estimates of ${\ensuremath{\Gamma}}_{1}$ and the $d$-band extrema utilize "renormalized-atom" band potentials within the Wigner-Seitz cell in which the interelectronic exchange is taken into account without resort to the ${\ensuremath{\rho}}^{\frac{1}{3}}$ approximation and incorporate the appropriate boundary conditions at the Wigner-Seitz radius ${r}_{\mathrm{WS}}$. The results have comparable accuracy with those obtained from augmented-plane-wave calculations employing the same crystal potential within the muffin-tin approximation. The band results are qualitatively similar to those obtained using more conventional ${\ensuremath{\rho}}^{\frac{1}{3}}$ potentials. The Wigner-Seitz viewpoint is thereby seen to be useful in obtaining quantitative results for certain high-symmetry points in $k$ space aside from ${\ensuremath{\Gamma}}_{1}$ with far less computational effort. In addition, the present scheme may provide a better starting point for dealing with $d\ensuremath{-}d$ exchange-correlation effects. Also discussed are a number of features general to the problem of constructing adequate transition-metal crystal potentials, in particular, how to deal with nonintegral $d$- and conduction-electron counts per atom, and configuration and/or multiplet averaging.