Density-functional band theory

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen 

In the pseudopotential method, core states are omitted from explicit consideration, a plane-wave basis is used, and no shape approximations are made to the potentials. This method works well for complex solids of arbitrary structure (i.e., not necessarily close-packed) so long as an adequate division exists between localized core states and delocalized valence states and the properties to be studied do not depend upon the details of the core electron densities. For materials such as ZnO, and presumably other transition-metal oxides, the 3d orbitals are difficult to accommodate since they are neither completely localized nor delocalized. For example, Chelikowsky (1977) obtained accurate results for the O 2s and O 2p part of the ZnO band structure but treated the Zn 3d orbitals as a core, thus ignoring the Zn 3d participation at the top of the valence region found in MS-SCF-Aa cluster calculations (Tossell, 1977) and, subsequently, in energy-dependent photoemission experiments (Disziulis et al., 1988). 

The LCAO methods can treat all electrons and need not make shape approximations to the potential. However, as for Hartree-Fock band calculations, there is a very large number of electron-electron repulsion integrals, and care must be taken in truncating their sums. A number of 

Note that, in the above discussion, we have neglected methods that generate band structures from empirical data. Most band calculations before the 1970s were of this type. The considerable contributions to knowledge made through use of the empirical pseudopotential approach, for example, have been discussed by Cohen (1979). Such approaches have 

Sonne l- rom Hathaway et a , 1985 see text and Appendix C for information on methods. 

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The focus then shifts to the delocalized side of Fig. 1.1, first discussing Hartree-Fock band-structure studies, that is, calculations in which the full translational symmetry of a solid is exploited rather than the point-group symmetry of a molecule. A good general reference for such studies is Ashcroft and Mermin (1976). Density-functional theory is then discussed, based on a review by von Barth (1986), and including both the multiple-scattering self-consistent-field method (MS-SCF-ATa) and more accurate basis-function-density-functional approaches. We then describe the success of these methods in calculations on molecules and molecular clusters. Advances in density-functional band theory are then considered, with a presentation based on Srivastava and Weaire (1987). A discussion of the purely theoretical modified electron-gas ionic models is... 

Table 3.9. Static structural properties for C, Si, and Ge obtained from the ab initio pseudopotential calculations of density-functional band theory compared with experiment...

Table 3.10. Comparison with experimental values of calculated properties for Fe obtained by different density-functional band-theory methods...

However, metals have a fundamentally different electronic structure than do atoms and molecules the electrons are described more realistically by bands than by localized orbitals. In the early 1950s, using a density functional based theory, Lindhard [23-26] developed a successful method for treating solids directly that is still in use today. 

While the assignment in [11] is rather tentative, the fundamentals in [10] were Identified by using the fundamentals calculated by local density functional (LDF) theory as guides. However, the assignment of Vg and Vg may have to be changed. Some combination bands and overtones were also identified force constants calculated by the LDF method were also given [10]. 

The optimised interlayer distance of a concentric bilayered CNT by density-functional theory treatment was calculated to be 3.39 A [23] compared with the experimental value of 3.4 A [24]. Modification of the electronic structure (especially metallic state) due to the inner tube has been examined for two kinds of models of concentric bilayered CNT, (5, 5)-(10, 10) and (9, 0)-(18, 0), in the framework of the Huckel-type treatment [25]. The stacked layer patterns considered are illustrated in Fig. 8. It has been predicted that metallic property would not change within this stacking mode due to symmetry reason, which is almost similar to the case in the interlayer interaction of two graphene sheets [26]. Moreover, in the three-dimensional graphite, the interlayer distance of which is 3.35 A [27], there is only a slight overlapping (0.03-0.04 eV) of the HO and the LU bands at the Fermi level of a sheet of graphite plane [28,29],... 

The experimental evidence, first based on spectroscopic studies of coadsorption and later by STM, indicated that there was a good case to be made for transient oxygen states being able to open up a non-activated route for the oxidation of ammonia at Cu(110) and Cu(lll) surfaces. The theory group at the Technische Universiteit Eindhoven considered5 the energies associated with various elementary steps in ammonia oxidation using density functional calculations with a Cu(8,3) cluster as a computational model of the Cu(lll) surface. At a Cu(lll) surface, the barrier for activation is + 344 k.I mol 1, which is insurmountable copper has a nearly full d-band, which makes it difficult for it to accept electrons or to carry out N-H activation. Four steps were considered as possible pathways for the initial activation (dissociation) of ammonia (Table 5.1). 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

Heaton, R.A., Harrison, J.G. and Lin, C.C. (1983) Self-interaction correction for density-functional theory of electronic energy bands of solids, Phys. Rev., B28, 5992-6007. 

Although dimethylberyllium is a coordination polymer in the solid state,27 it has long been known to be monomeric in the gas phase.28 It has also been found to be monomeric when synthesized from the co-condensation of laser-ablated beryllium atoms and a methane/argon mixture at 10 K.11 Formed in conjunction with several other species, including hydrides (see Section 2.02.2.4), (CH3)2Be was identified from its infrared absorption bands, which were compared to DFT-calculated frequencies (DFT = density functional theory). 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

Recent calculations of hyperfine parameters using pseudopotential-density-functional theory, when combined with the ability to generate accurate total-energy surfaces, establish this technique as a powerful tool for the study of defects in semiconductors. One area in which theory is not yet able to make accurate predictions is for positions of defect levels in the band structure. Methods that go beyond the one-particle description are available but presently too computationally demanding. Increasing computer power and/or the development of simplified schemes will hopefully... 

Using perturbation theory. Hammer and Nprskov developed a model for predicting molecular adsorption trends on the surfaces of transition metals (HN model). They used density functional theory (DFT) to show that molecular chemisorption energies could be predicted solely by considering interactions of a molecule s HOMO and LUMO with the center of the total d-band density of states (DOS) of the metal.In particular. 

Abstract In the light of quantum chemistry, sulphide minerals and the interaction of them with reagents are investigated in this chapter. With the density functional theory pyrite is first researched including its bulk properties about energy band and frontier orbitals and the property of FeS2 (100) surface. 

Keywords molecular orbital energy band density functional theory(DFT)... 

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