The LMTO method

The LMTO method [58, 79] can be considered to be the linear version of the KKR teclmique. According to official LMTO historians, the method has now reached its third generation [79] the first starting with Andersen in 1975 [58], the second connnonly known as TB-LMTO. In the LMTO approach, the wavefimction is expanded in a basis of so-called muffin-tin orbitals. These orbitals are adapted to the potential by constmcting them from solutions of the radial Scln-ddinger equation so as to fomi a minimal basis set. Interstitial properties are represented by Hankel fiinctions, which means that, in contrast to the LAPW teclmique, the orbitals are localized in real space. The small basis set makes the method fast computationally, yet at the same time it restricts the accuracy. The localization of the basis fiinctions diminishes the quality of the description of the wavefimction in die interstitial region. 

The LMTO method is the fastest among the all-electron methods mentioned here due to the small basis size. The accuracy of the general potential teclmique can be high, but LAPW results remain the gold standard . 

H.L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984). 

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

Self-consistent energy band calculations have now been made through the LMTO method for all of the NaCl-type actinide pnictides and chalcogenides . The equation of state is derived quite naturally from these calculations through the pressure formula extended to the case of compounds . The theoretical lattice parameter is then given by the condition of zero pressure. 

Skriver, H. L. (1984). The LMTO Method Muffin-Tin Orbitals and Electronic Structure. Berlin Springer-Verlag. 

Fig. 9.29. Step energies for 4d transition metals (adapted from Vitos et al. (1999)). The straight lines reflect the outcome of a model near-neighbor calculation for the step energies on these surfaces. The first-principles calculations were done using the LMTO method.

Surface energies calculated for jellium (Perdew et al., 1990) and using the LMTO method (Skriver and Rosen-gaard, 1992), compared with experiment (Skriver and Rosengaard, 1992). The bold letters indicate the stable crystal structure. 

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. 

J. Kollar, L. Vitos, and H. L. Skriver, Electronic Structure and Physical Properties of Solids The Uses of the LMTO Method, edited by H. Dreysse, Lecture Notes in Physics, (Springer-Verlag, Berlin, 2000). 

Values of the coefficient of the spin splitting proportional to of Fi-conduction band for k [110] as obtained with the LMTO method, the p 16 X 16 Hamiltonian, and k p perturbation theory (PT). Experimental data are from Refs. [34,35] Units eVA . For GaAs, GaSb, and InP experiments only give the magnitude. For InSb also the sign was determined. [35]... 

It is obvious from the current literature that the LMTO method is only one of many techniques which may be used to solve the one-electron problem in crystalline solids. However, the method does combine a certain number of very convenient features which, to the author s mind, makes it one of the most desirable techniques currently available. This is so for several reasons. 

There is, however, a price for this versatility. The LMTO method is one of several linear methods, and like all the other linear techniques it is accurate only in a certain energy range. The present technique in particular should not be used for states too far above the Fermi level. If such states are required one may still solve the self-consistency problem by the LMTO technique and then turn to the Linear Augmented Plane Wave (LAPW) method for accurate calculation of the unoccupied high-lying levels. Furthermore, in... 

Fig.1.4. Self-consistent energy-band structure for bcc tungsten obtained by the LMTO method within the atomic-sphere approximation (ASA) using local-density theory for exchange and correlation. Relativistic effects are included except spin-orbit coupling which is neglected...

This monograph is based almost entirely on the work of O.K. Andersen. It is therefore appropriate to reveal the sources of the material presented, and at the same time give a brief history of the development of linear methods. At present several types of such methods are used, e.g. the linear muffin-tin orbitals (LMTO) method [1.19], the linear augmented plane-wave (LAPW) method [1.19], the augmented spherical-wave (ASW) method [1.20], and the linear rigorous cellular (LRC) method [1.15]. Of these the LMTO method, which was the earliest, will be our main concern. 

The theory underlying linear methods was presented rather extensively in the unpublished Mont Tremblant notes [1.27] and subsequently published in [1.19], which also contains the linear augmented plane-wave method. At the end of 1975 the LMTO method had been used in actual calculations by Kasowski [1.28-31], Jepsen [1.32], and Jepsen et al. [1.33]. The results showed that although the method is in principle approximate it has in practice an accuracy comparable to that normally obtained with the conventional APW and KKR methods. Computationally, the method was found to be orders of magnitude faster than the others in use at that time. 

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