The Exact Muffin-Tin Orbital Method

Since the language of the Exact Muffin-Tin Orbital (EMTO) method is phrased in terms of wave functions and the solution to the Schrodinger equation, and not in terms of Green s functions, I will also start with this formulation, and then later make the transition into Green s functions when appropriate. Some papers with a good account of the methodology are Refs.[58, 64, 65] 

Our main goal is, of course, to solve the one-electron Kohn-Sham equations, as discussed in Chapter 2  

We will also use the so called muffin-tin approximation for the potential  

To solve Eq.(4.1) one expands the wave function in a complete basis set  

Here L stands for the (l,m) set of quantum numbers, and will do so through out this thesis. The aRL are the EMTOs, which will be defined below. 

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Before going into the theory for the Exact Muffin-Tin Orbital method, I want to give a short historical overview of muffin-tin based methods. This is done in order to put the method, and the development of the quantities used in the method, in a historical context. 

A short introduction to electronic structure theory is included as well as a chapter on Density Functional Theory, which is the underlying method behind all calculations presented in the accompanying papers. Multiple Scattering Theory is also discussed, both in more general terms as well as how it is used in the methods employed to solve the electronic structure problem. One of the methods, the Exact Muffin-Tin Orbital method, is described extensively, with special emphasis on the slope matrix, which energy dependence is investigated together with possible ways to parameterize this dependence. 

Furthermore, a chapter which discusses different ways to perform calculations for disordered systems is presented, including a description of the Coherent Potential Approximation and the Screened Generalized Perturbation Method. A comparison between the Exact Muffin-Tin Orbital method and the Projector Augmented-Wave method in the case of systems exhibiting both compositional and magnetic disordered is included as well as a case study of the MoRu alloy, where the theoretical and experimental discrepancies are discussed. 

Not until the so called third generation of LMTOs [58], was there a way to properly include the interstitial part of the muffin-tin potential and perform calculations without the ASA, in fact it was possible to perform calculations for exact muffin-tins using the Exact Muffin-Tin Orbitals (EMTO) method. Since the structural dependent part is called the slope matrix in the EMTO method, this is the name I will use for the rest of this thesis when discussing the EMTO method. 

Recently, we have shown that the accuracy of the CPA is greatly improved via an implementation within the basis set of the so-called exact muffin-tin orbitals (EMTO). The EMTO theory has been developed by Andersen . We have shown that the EMTO-CPA method, combined with the full charge density (FCD) formalism , allows one to calculate the energy of a random alloy with the same accuracy, as that of modem full-potential methods in the case of pure elements or ordered compounds. In this paper we present a detailed description of the FCD-EMTO-CPA method. 

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

If we approximate the crystal potential by an array of non-overlapping muffin-tin wells as in (5.2), the energy-dependent muffin-tin orbitals (5.13) may be used in conjunction with the tail-cancellation theorem to obtain the so-called KKR equations. These have the form (1.21) and provide exact solutions for muffin-tin geometry. Computationally, however, they are rather inefficient and it is therefore desirable to develop a method based upon the variational principle and a fixed basis set, which leads to the computatinal-ly efficient eigenvalue problem (1.19). 

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