Simplified LCAO Method for the Periodic Potential Problem

TLDR

The LCAO (tight‑binding) approximation for solids is presented as an interpolation method that complements more accurate cellular or orthogonalized plane‑wave calculations. The authors propose using pre‑computed integrals as constants so that the tight‑binding method matches accurate calculations at Brillouin‑zone symmetry points and can then be applied across the entire zone. They provide a general discussion and tables of energy matrix components for simple cubic, face‑centered cubic, body‑centered cubic, and diamond structures. Applications to Ni, Cu, and fcc structures illustrate the method; for bcc chromium the band splitting due to an antiferromagnetic alternating potential is derived, and energy‑state distributions for non‑magnetic cases are shown; diamond results agree with orthogonalized plane‑wave calculations, and InSb properties fit the band‑picture framework.

Abstract

The LCAO, or Bloch, or tight binding, approximation for solids is discussed as an interpolation method, to be used in connection with more accurate calculations made by the cellular or orthogonalized plane-wave methods. It is proposed that the various integrals be obtained as disposable constants, so that the tight binding method will agree with accurate calculations at symmetry points in the Brillouin zone for which these calculations have been made, and that the LCAO method then be used for making calculations throughout the Brillouin zone. A general discussion of the method is given, including tables of matrix components of energy for simple cubic, face-centered and body-centered cubic, and diamond structures. Applications are given to the results of Fletcher and Wohlfarth on Ni, and Howarth on Cu, as illustrations of the fcc case. In discussing the bcc case, the splitting of the energy bands in chromium by an antiferromagnetic alternating potential is worked out, as well as a distribution of energy states for the case of no antiferromagnetism. For diamond, comparisons are made with the calculations of Herman, using the orthogonalized plane-wave method. The case of such crystals as InSb is discussed, and it is shown that their properties fit in with the energy band picture.