Analytic methods for the calculation of the electronic structure of solids

Andersen’s atomic-sphere approximation has been utilized with approximations based upon linear-combination of atomic orbitals (LCAO) theory to obtain approximate energy-band parameters for solids. Simple analytic expressions for the bandwidth and position of the band center have been derived that require only free-atom wave functions evaluated at the Wigner-Seitz atomic-sphere radius. For convenience, the method has been named the atomic surface method (ASM). The following simple analytic expressions for the band parameters have been derived from the ASM: (i) The bandwidth is equal to the product of ${ħ}^2$/m, the gradient of the electron density at the atomic-sphere radius, and the surface area of the sphere; (ii) the average band energy is shifted from the atomic-term-value energy by an amount given by the product of the bandwidth, electron density at the atomic-sphere radius, and atomic-sphere volume. The theory has been applied without adjustable parameters to the transition metals and f-shell metals with use of tabulated Hartree-Fock wave functions and is in reasonable agreement with full band-structure calculations. The same analysis is applied to atomic core states under compression and is also in reasonable agreement with complete band-structure calculations. The 2s and 2p states of Na and Al have been calculated to the point where they merge with the conduction band as free-electron states. These bandwidths and shifts are also written in terms of the atomic term values by using the asymptotic form of the radial wave function. Finally, the LCAO energy bands of Ni are calculated with use of the ASM parameters. The average d-band energy relative to the conduction-band minimum and the hybridization of the conduction band with the d band are calculated with use of the asymptotic form of the radial wave function, giving approximate energy bands entirely in terms of free-atom parameters.