Linear Muffin-Tin Orbital (LMTO) Method

Abstract

In this chapter we describe especially those quantities and concepts which will be useful for a tight-binding Green function treatment of the one-electron prob-lems to be discussed in detail throughout the whole book. 2.1 SECULAR EQUATION In the case of atoms, molecules, and solids, where effective one-electron prob-lems can be dealt within the local density approximation (see Section 1. 1.4), numerical methods to solve the one-electron Schrodinger equation [ -b.. + V(r) -E] 1j;(r) = 0 (2.1) are of vital importance. In addition, the complexity of the problem as well as a common need for an understanding of properties of many-particle systems by means of transparent physical concepts often require to adopt a simplified yet realistic model for the one-electron potential V(r) in Eq. (2.1). 2.1.1 Atomic Sphere Approximation One of the most successful models for solids is the muffin-tin form of the po-tential [1 , 2] by which the potential V(r) is approximated by (i) a collection of spherically symmetric potentials inside non-overlapping muffin-tin spheres centered at the individual nuclei, and (ii) a constant potential Va in the in-terstitial region outside the muffin-tin spheres. Solutions of the correspond-ing Schrodinger equation (2.1) are then provided by multiple scattering the-37 I. Turek et al., Electronic Structure of Disordered Alloys, Surfaces and Interfaces © Springer Science+Business Media New York 1997 38 CHAPTER 2 ory [2, 3, 4J which separates the problem into (i) an integration of the radial Schrodinger equation inside each muffin-tin sphere, and (ii) an evaluation of structure constants which in turn depend on the kinetic energy E -Vo of the free electrons in the interstitial region and contain information about the posi-tions (origins) of the muffin-tin spheres. During the last two decades, even a more simplified model, namely the atomic sphere approximation (ASA) [5, 6, 7, 8], became very popular in electronic structure calculations. The essence of the ASA consists in (i) the use of spherically symmetric potentials inside slightly overlapping, space-filling atomic (Wigner-Seitz) spheres centered at the individual nuclei, and (ii) a complete neglect of the electronic kinetic energy E -Vo in the (volumeless) interstitial region. From the numerical point of view, the ASA removes the inconvenient interstitial region and replaces integrals over the whole space by a sum over Wigner-Seitz spheres which may be considered as approximations to the true Wigner-Seitz cells.