Wannier Functions in a Simple Nonperiodic System

Abstract

This paper defines and analyzes in detail the Wannier functions $a_l$ of a one-dimensional periodic lattice with a point defect. It is shown that these functions have exactly the same exponential localization as the Wannier functions of the perfect lattice and that they approach the latter exponentially as the site $l$ recedes from the defect site. Variational methods for the calculation of the $a_l$ are described. Eigenfunctions of the system can be obtained from the $a_l$ by the solution of a one-band Slater-Koster-type equation, which, however, is exact in the present theory. Moments of the density of states can be obtained directly from the $a_l$ without calculation of the eigenfunctions; so can the total electron density, $n\left(\mathrm{r}\right)$, corresponding to a full "band." It is suggested that for a nonperiodic system the Wannier functions may be easier to compute directly than the eigenfunctions.