Electronic structure atomic-sphere approximation

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

Electronic structure determinations have been performed using the self-consistent LMTO method in the Atomic Sphere Approximation (ASA). 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

Duthie and Pettifor (1977) have treated d bands and, s bands in the rare earths and have considered the structure dependence by calculating the density of states in detail (within the Atomic Sphere Approximation described in Section 20-D) for different structures. They indicate that they have predicted a sequence of four different structures, which occurs both for increasing pressure and for decreasing atomic number across the rare-earth series. This correlation and the ratio of core volume to atomic volume had been related by Johansson and Rosengren (1975), but Duthie and Pettifor argue that the essential feature is the number of electrons in the d bands and that this is only incidcntly rcllected in the core volume. 

Most methods of band-structure calculation are based on the muffin-tin, atomic sphere approximation (ASA) or Wigner-Seitz construction for the electronic potential and... 

In Chap.6 the atomic-sphere approximation is introduced and discussed, canonical structure constants are presented, and it is shown that the LMTO-ASA and KKR-ASA equations are mathematically equivalent in the sense that the KKR-ASA matrix is a factor of the LMTO-ASA secular matrix. In addition, we treat muffin-tin orbitals in the ASA, project out the i character of the eigenvectors, derive expressions for the spherically averaged electron density, and develop a correction to the ASA. 

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. 

The electronic structure is determined using the ab initio all-electron scalar-relativistic tight-binding linecir muffin-tin orbital (TB-LMTO) method in the atomic-sphere approximation (ASA). The nnderlying lattice, zincblende structure, refers to an fee Bravais lattice with a basis which contains a cation site (at a(0,0,0)), an anion site (at o(j,, )), and two interstitial sites occupied by empty spheres (at a(, 5, h) and a(, , )) which in turn are necessary for a correct description of open lattices . ... 

Due to the muffin-tin approximation of the local multi-centre potential, the applicability of the scattered-wave (SW) method is limited to relatively small and high-symmetry clusters and molecules. On the other hand, it is the prerequisite for generalizing the in-out integration scheme of atomic one-electron wavefiinctions to the multi-centre case. This approach furnishes the multi-centre electronic structure in a particular atom sphere in its single-centre representation. Thus atomic transition-matrix elements can be readily rescaled in terms of near-nucleus electron amplitudes. 

Figure 1 In a QM/MM calculation, a small region is treated by a quantum mechanical (QM) electronic structure method, and the surroundings treated by simpler, empirical, molecular mechanics. In treating an enzyme-catalysed reaction, the QM region includes the reactive groups, with the bulk of the protein and solvent environment included by molecular mechanics. Here, the approximate transition state for the Claisen rearrangement of chorismate to prephenate (catalysed by the enzyme chorismate mutase) is shown. This was calculated at the RHF(6-31G(d)-CHARMM QM-MM level. The QM region here (the substrate only) is shown by thick tubes, with some important active site residues (treated by MM) also shown. The whole model was based on a 25 A sphere around the active site, and contained 4211 protein atoms, 24 atoms of the substrate and 947 water molecules (including 144 water molecules observed by X-ray crystallography), a total of 7076 atoms. The results showed specific transition state stabilization by the enzyme. Comparison with the same reaction in solution showed that transition state stabilization is important in catalysis by chorismate mutase78.

The above results have been generalized in three dimensions for describing the electronic structure of atoms. In that case, the most probable distribution of electrons on a sphere is to be examined. For atoms of the second row of the periodic table, the maximum number of electrons of each spin in the valence shell will be four. Assuming that electrons are approximately equidistant from the nucleus, the most probable electronic configuration of the outer shell of any second-row atom is easy to anticipate. The result obtained for neon is shown in Fig. 4 and is compared to the corresponding Lewis model completed by taking account of electron spin. 

Coordination numbers in different crystals depend on the sizes and shapes of the ions or atoms, their electronic structures, and, in some cases, on the temperature and pressure under which they were formed. An oversimplified and approximate approach to predicting coordination numbers uses the radius ratio, r+/r. Simple calculation from tables of ionic radii allows prediction of possible structnres by modeling the ions as hard spheres. For hard spheres, the ideal size for a smaller cation in an octahedral hole of an anion lattice is a radius of 0.414r. Calculations for other geometries result in the radius ratios and coordination number predictions shown in Table 7.1. 

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