LMTO calculations

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

Figure 3. Al L23 edges in FeAI and TiAI (EELS experiments dots) and LMTO calculations for FeAI (line).

Figure 4. Co L23 edges in Ni 43 Co o Al 50 EELS (dots) and LMTO calculation of the Co L23 edge in the Ni7CoA18 supercell (full line).

In this paper we will present a way to improve the evaluation of total energies in LMTO calculations. The Kohn Sham energy functional [1] can be written in the form... 

Consider the electrostatic terms. These are hard to evaluate because the output charge density is a complicated non spherical function in space. In traditional LMTO calculations the charge density is first spheridised before Ees is calculated. In this method p(r) is reduced to a sum of spherically symmetric balls of charge inside each ASA sphere [2]. 

Figure 9. The measured momentum density of an aluminium film. In the left panel we show the measured momentum density near the Fermi level (error bars), the result of the LMTO calculations (dashed line) and the result of these calculations in combination with Monte Carlo simulations taking into account the effects of multiple scattering (full line). In the central panel we show in a similar way the energy spectrum near zero momentum. In the right panel we again show the energy spectrum, but now the theory is that of an electron gas, taking approximately into account the effects of electron-electron correlation (dashed) and this electron gas theory plus Monte Carlo simulations (solid line).

In the central panel we show a measured energy spectrum near zero momentum compared with the LMTO calculation and the LMTO calculation plus simulation of multiple scattering events. 

The same normalisation of theory to the experiment is used as in the momentum density plot. Clearly the Monte Carlo simulation compares better with the experiment than the LMTO calculation by itself, but at high binding energies there is still a significant amount of intensity missing in the theory. 

LMTO calculations on a hypothetical ScZn6 1/1 AC [82] revealed that Sc plays the same role here as in Sc-Mg-Cu-Ga 1/1 AC (Fig. 13). The Sc not only provides valence electrons to push into the pseudogap, but its d orbitals also afford mixing with Zn s, p orbitals to enhance the depth of the pseudogap. This may explain why no Mg-Zn binary or Mg-Cu-Ga ternary Tsai-type QCs exist, but the Sc-Cu-Zn i-QC [24,68] forms, although its discovery was not directed by the pseudogap tuning concept. 

Fig. 7. Calculated and experimental Wigner-Seitz radii for the actinide metals. The Wigner-Seitz radii of the actinide metals from RLMTO calculations, compared with the radii from LMTO calculations and experiment (after Skriver et al. ). Also shown are the Wigner-Seitz radii of the actinide metals from RLMTO calculations (full circles)...

Both, EH and LMTO calculations yield band structures with band overlap at the Fermi level meaning that a metallic conduction is expected. This is displayed in Fig. 3e and 3f. Indeed, Ca7Mg7j5Sii4 shows metallic conductivity. Thus, there are no localized spins but the HOMO states form a conduction band which, according to LMTO-band structure calculations, is exclusively due to x-orbital overlapping between adjacent ecliptically arranged Si 2 moieties along the stacking direction. 

Fig. 14 Band structure of a fully oxygen defective (1 x 1) MgO(lOO) surface along the three symmetry lines J-F-M of the 2D Brillouin Zone, as obtained through the FP-LMTO calculation (Full Potential- Linear MufiSn-Tin Orbital method). The dashed horizontal line represents the Fermi level, black dots (st indicate the energy positions of the filled (empty) Bloch states at F calculated in a (2v x 2- /2) supercell. The dashed line in the gap of the projected bulk bandstructure gives the dispersion of the F, centre band. The dashed-dotted line is used for the surface conduction band of lowest energy (from Ref. 69).

Earlier it was mentioned that the relativistic theory of electronic states in solids in many respects is identical to that of atoms. Since this is well described elsewhere, this section will only deal with some features of specific implementations of the theory in actual calculation methods used for solids, and the importance of relativistic effects — apart from those already discussed — will be illustrated by examples. Although Section 3 did refer to results of LMTO calculations, we did not describe how these included relativity. This section will deal with these items in the form of an overview, and the basic band structure calculations described relate to the density-functional theory [62,63]. Since magnetism is one of the most important solid state physics fields we shall discuss the simultaneous inclusion of spin-polarization and relativistic effects, in particular the spin-orbit coupling. In that context it appears that several of the materials where such effects are particularly large and interesting are those where electron... 

Prior to the work of Ref [84], a series of LMTO calculations had studied UO2. A full-potential, non-spin-polarized LMTO treatment with a gradient-corrected XC model gave a lattice constant (fluorite structure) of 9.81 au, well below the experimental 10.34 au (see [84] for references to the other literature). An approximate LMTO (LMTO-ASA) non-spin-polarized treatment with LDA XC gave 9.92 au and a bulk modulus of 5 = 291 GPa versus the experimental value 207 GPa. Spin-polarized LDA calculations did give anti-ferromagnetic ordering (foimd experimentally) at the experimental lattice constant but the moment vanished at the LDA equilibrium value. All the results were metallic. 

Part of my work has been performed at the laboratory of the National Research Council (NRC) of Canada in Ottawa and at the Kamerlingh Onnes Laboratory in Leiden, and I owe my sincere thanks to my colleagues at these institutions as well as to those at my home base Risj6. I am especially grateful to J.-P. Jan at NRC who, from a desire to interpret his pioneering de Haas-van Alphen measurements on intermetal 1ic compounds, helped perform the first LMTO calculations on alloys in 1975, and who continued to be a true collaborator until his untimely death in 1981. 

If, for some particular values of , E is positioned in the range An-1 a Bn a between the toP the (n " 1H band and the bottom of the nil band, an LMTO calculation may yield a steep band which connects parts of the low-lying (n - 1) band with parts of the high-lying band. The reason is that the LMTO, which uses the smallest possible basis set with only one principal quantum number for each value of, can give only one set of % bands. Hence, when E is chosen between two bands of the same, the LMTO... 

The fact that the LMTO method can give only one set of z bands in any one calculation does not exclude the use of the method in cases where more than one principal quantum number gives rise to a band of z character. In such a situation one simply divides the energy range into panels, performs LMTO calculations with potential parameters appropriate to each panel, and pieces the individual bands together to form the complete band structure. That this in reality is the most efficient way of obtaining such bands may be seen... 

With the combined correction term included in an LMTO calculation, one... 

F i q. 9.1. Illustration of the setup for self-consistent LMTO calculations utiliz ing the programmes listed in Table 9.1... 

In this chapter we list the four standard potential parameters (4.1) for 61 metals as obtained in self-consistent LMTO calculations using the exchange-correlation potential given by von Barth and Hedin [10.1]. Table 10.1 containing parameters for d-transition metals was prepared by O.K. Andersen and D. Glotzel, whom we wish to thank for permission to quote these results. Table 10.2 was prepared by the author. 

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