The LMTO Formalism

Returning now to our main topic of crystalline solids, we consider a primitive cell with h atoms centred at positions q. Some of these atoms may be of the same type, and we denote the number of type t atoms in the cell h. . In the atomic-sphere approximation each atom is surrounded by a sphere of suitably chosen radius S subject to the constraint 

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Our results demonstrate that the augmented space recursion and the orbital peeling method in conjunction with the LMTO formalism, constitute a viable and computationally feasible approach to the calculation of phase stability in binary substitutionally disordered alloys. ... 

The one-centre expansion (6.15) is specialised to the case where R = 0, and is valid inside the atomic sphere centred at the origin. It may be used to derive the LMTO equations and with the normalisation implied by (6.11) it is consistent with the secular matrices (5.46,47) in the ASA. In linear methods in band theory [6.2] Andersen presented the one-centre expansion in the form (6.15) and derived the LMTO formalism from that assumption. His LMTO formalism is equivalent to that presented here apart from the normalising factor [/S/2 (- -1)] 1 appearing in the definition (6.11) of the energy-dependent muffin-tin orbital. 

The LMTO method has the computational speed and flexibility needed to perform calculations of electron states in molecules and compounds. Therefore in the present chapter we shall generalise the LMTO formalism purely within the atomic-sphere approximation to include the case of many inequivalent atoms per cell. The LMTO method is based on the variational principle in conjunction with energy-independent muffin-tin orbitals but, in addition to this approach, we have also considered the tail-cancellation principle which led to the KKR-ASA condition (2.8). Since the latter has conceptual advantages, we apply the tail-cancellation principle to the simplest possible case of more than one atom, namely the diatomic molecule. After that, we turn to crystalline solids and generalise or sometimes rederive the important equations of LMTO formalism. Hence, in addition to giving the LMTO equations for many atoms per cell, the present chapter may also serve as a short and compact presentation of the crystal-structure-dependent part of LMTO formalism. The potential-dependent part is treated in Chap.3. In the final sections are listed the modifications needed to calculate ground-state properties for materials with several atoms per cell. 

For the description of the random Hamiltonian we employ TB-LMTO formalism in the most tight binding representation . The Hamiltonian for the binary random alloy takes the form ... 

The results are conveniently and clearly expressed in a thermodynamic formalism this is why they find their place in this chapter. They depend however on parameters which are drawn from band-theory, especially from the LMTO-ASA (Linear Muffin-Tin Orbitals-Atomic Sphere Approximation) method. 

With the later introduction of the Linear Muffin-Tin Orbital (LMTO) method [46, 47, 48, 49, 50], a formulation of the multiple scattering problem in terms of Hamiltonians was introduced. This provided another way to gain more knowledge about the KKR method, which, although elegant, was not so easily understood. In the LMTO method one had to use energy linearizations of the MTOs to be able to put it into a Hamiltonian formalism. The two methods (KKR and LMTO) were shown [51] to be very closely related within the Atomic Sphere Approximation (ASA) [46, 52], which was used in conjunction with the LMTO method to provide an accurate and computationally efficient technique. 

In Chaps.7 and 8 it is shown how the LMTO method and the physically simple concepts contained in linear theory may be used in self-consistent calculations to estimate ground-state properties of metals and compounds. Here we treat the local-density approximation to the functional formalism of Hohenberg3 Kohn, and Sham, and the force relation derived by Andersen together with an accurate and a first-order pressure relation. In addition, the LMTO-ASA and KKR-ASA methods are generalised to the case of many atoms per cell. 

Expressions (2.32,33) are clearly the matrix generalisation of the unhybridised scaling relations (2.28,29), and they are much simpler than the conventional LMTO equations. They are, however, also slightly less accurate, and therefore we consider here the more versatile, conventional LMTO formalism. 

A typical example where such a procedure is needed is in the application of the LMTO method to molecules. Furthermore, in this situation Bloch s theorem does not apply and k is therefore not a good quantum number. Instead, the k dependence should be substituted by a Q , Q dependence, where Q is a site index. Formally, the LMTO matrix for molecules may be obtained by substituting... 

The LMTO method as defined in this section may be regarded as an LCAO formalism in which the muffin-tin potential, rather than the atomic potential, defines the set of basis functions used to construct the trial functions of the variational procedure. Consequently, all overlap integrals can be expressed in terms of the logarithmic derivative parameters, and the muffin-tin Hamiltonian can be solved to any accuracy. 

The formalism presented so far is based on a muffin-tin geometry. The potential is assumed to be a muffin-tin potential, Fig.5.1, the partial waves (5.9) and the muffin-tin orbitals (5.13,25) are formed by matching wave functions at the muffin-tin sphere, and the LMTO overlap integrals extend only over the muffin-tin sphere. However, we have already substituted the atomic sphere for the atomic polyhedron as the cell used to divide space into suitable units. 

Brooks, Johansson and Skriver (1984) investigated the band structure of UC and ThC by nonrelativistic and relativistic (based on the Dirac formalism) LMTO methods. They analysed the electron density changes in the compounds as compared with free atoms, as well as the influence of pressure on the band structure. Crystal pressures as a function of lattice constants (equations of state) were calculated as well as theoretical values of the lattice constants. The calculated trends in the variations of lattice constants and bulk moduli agree well with the available experimental data. Some of the most important results of these calculations are shown in Figs. 2.20 and 2.21. 

To summarise, we have presented a way to improve an LMTO-ASA calculation of the electrostatic energy in a crystal. The method is stable and general in its formalism so that it should be applicable to a wide range of systems. In this talk we did not mention the exchange correlation energy. It is possible to make an expansion of the (xc(p(r)) in terms of the SSW s. Then the integral... 

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