Direct calculation of Wannier functions; Si valence bands

Abstract

A method based on Pauling's covalent bonds is developed for the calculation of valence-band Wannier functions directly in $\overrightarrow{\mathrm{r}}$ space. A single-bond function is expressed as a linear combination of Gaussian $s$, $p$, and $d$ functions. The bond energy is minimized in the crystalline potential subject to an orthogonality constraint between nearest-neighbor bonds which is added to the Schrödinger equation via a Lagrange-multiplier method. More distant orthogonality constraints are satisfied by a cluster sum of single bonds and additional variational adjustments are made to zeroth-, first-, and second-neighbor bonds all with orthogonality conditions effectively satisfied. Energy bands calculated from these Wannier functions are accurate to better than 0.1 eV on the average with a maximum error of 0.2 eV. The final Wannier bond energy, equal to the average valence-band energy, is 0.10 eV lower than the initial single-bond energy eigenvalue. Convergence of the band energies is relatively slow in terms of number of bonds included. Best results were obtained after treating interactions of a given bond with 38 inequivalent or 459 total bonds.