Hartree-Fock method band structures

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. 

In practical calculations of the band structure of crystals by the Hartree-Fock method, the one-electron density matrix Prr (k) in the reciprocal space can be calculated only for a rather small finite set of special points (see Sect. 4.2). Let us consider such a set of special points kj, j = 1,2,..., Nq. Then, in the calculation of the density matrix of the infinite crystal the integration over the BriUouin-zone volume Vg is changed by the sum over the special points chosen... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

Recent calculations of the band structure of polyethylene have employed variations of the ab initio method incorporating electron correlation. Sun and Bartlett (1996) utilised many-body perturbation theory to encompass electron correlation in the ab initio framework. Siile et al. (2000) and Serra et al. (2000) have employed variants of DFT. These calculations involved the optimisation of local effective potentials and a local-density approximation respectively. Figure 4.13 shows a comparison of the band structure obtained by Siile et al., Fig. 4.13(d), with those obtained by other ab initio DFT calculations using the Hartree-Fock (HF), Fig 4.13(a), and Slater approaches, Figs. 4.13(b) and (c). 

For the calculation of the Hartree-Fock band structures of polymers a method has been developed including non-local exchange and full-self consistency. It is applicable also in the case of a combined symmetry (e.g., helix). 

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... 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Transition metals are important materials with intriguing properties and they have been studied with ever improved methods. A major difficulty is posed by the standard one-electron models where the tight-binding model seems appropriate for the narrow, so-called d-bands while near-plane-wave crystal orbitals are adequate for the conduction bands. Canonical Hartree-Fock solutions are awkward starting points for the description of magnetic structures and the use of spin-polarized versions destroys basic symmetry properties. 

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