The Linear Method

Abstract

In this chapter we shall use the Rayleigh-Ritz variational principle in conjunction with energy-independent muffin-tin orbitals (MTO’s) to derive the LMTO equations, which will have the form (1.19) of a generalised eigenvalue problem and hence provide solutions to the energy-band problem in a computationally efficient manner. First we define energy-dependent muffin-tin orbitals similar to those used in Sect.2.1. Then we present that choice of augmented spherical waves which will make the muffin-tin orbitals simultaneously orthogonal to the core states and energy independent. The augmentation is based upon the ϕ, ϕ • formalism developed in Chap.3. Both kinds of muffin-tin orbitals obey a convenient expansion theorem, whose expansion coefficients are the structure constants which contain all the necessary information about the crystal symmetry. The final derivation of the LMTO Hamiltonian and overlap matrices is based upon muffin-tin orbitals with tails which, before augmentation, are solutions of the Helmholtz wave equation rather than the Laplace equation, as in the atomic-sphere approximation (Chap.6). Thereby the results of the present chapter may be applied not only to the case of energy-band calculations, where the atomic-sphere approximation has the necessary accuracy, but also to cases of molecules and clusters where one may need basis functions more versatile than those given within the atomic-sphere approximation.