General band-structure methods

Since the early days of solid state theory, a number of approaches have been introduced to solve the single-particle hamiltonian and obtain the eigenvalues (band structure) and eigenfunctions. These methods were the foundation on which modern approaches for electronic structure calculations have been developed. We review the basic ideas of these methods next. 

Cellular or Linearized Muffin-Tin Orbital (LMTO) method This approach, originally developed by Wigner and Seitz [44], considers the solid as made up of cells (the Wigner-Seitz or WS cells), which are the analog of the Brillouin Zones in real space. In each cell, the potential felt by the electrons is the atomic potential, which is spherically symmetric around the atomic nucleus, but its boundaries are those of the WS cell, whose shape is dictated by the crystal. Due to the Bloch character of wavefimctions, the following boundary conditions must be obeyed at the boundary of the WS cell, denoted by r,  

Taking matrix elements of the hamiltonian between such states creates a secular equation which can be solved to produce the desired eigenvalues. Since the potential cannot be truly spherical throughout the WS cell, it is reasonable to consider it to be spherical within a sphere which lies entirely within the WS, and to be zero outside that sphere. This gives rise to a potential that looks like a muffin-tin, hence the name of the method Linearized Muffin-Tin Orbitals (LMTO). This method is in use for calculations of the band structure of complex solids. The basic assumption of the method is that a spherical potential around the nuclei is a reasonable approximation to the true potential experienced by the electrons in the solid. 

Augmented Plane Waves (APW) This method, introduced by Slater [45], consists of expanding the wavefunctions in plane waves in the regions between the atomic spheres, and in functions with spherical symmetry within the spheres. Then the two expressions must be matched at the sphere boundary so that the wave-functions and their first and second derivatives are continuous. For core states, the wavefunctions are essentially unchanged within the spheres. It is only valence states that have significant weight in the regions outside the atomic spheres. 

Orthogonalized Plane Waves (OPW) This method, due to Herring [47], is an elaboration on the APW approach. The trial valence wavefunctions are written at the outset as a combination of plane waves and core-derived states  

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General band-structure methods Appendix B) which obey the following equation ... 

General band-structure methods which, with the definition... 

We remark that memory function methods have been extensively applied to perfect crystals, where they are alternative tools to the traditional band structure methods. In the presence of translational symmetry and long-range order, however, the use of the memory function techniques is not essential but rather a matter of convenience (or taste) in a number of situations. We thus focus on the study of the electronic structure in systems that are aperiodic because of impurities our unorthodox way of looking at defects is to consider them as a source of a given frozen-in disorder in an otherwise perfect crystal lattice. The more general case of the presence of stochastic disorder will be discussed in Section VI. 

Another standing topic during the last two decades has been to evaluate the electronic structure of solids, surfaces and adsorbates on surfaces. This can be done using standard band structure methods [107] or in more recent years slab codes for studies of surfaces. An alternative and very popular approach has been to model the infinite solid or surface with a finite cluster, where the choice of the form and size of the cluster has been determined by the local geometry. These clusters have in more advanced calculations been embedded in some type of external potential as discussed above. It should be noted that these types of cluster have in general quite different geometries compared with... 

A general discussion of band structure methods is given by N. W. 

Any of the existing band structure methods can be adapted for use as a semiempirical scheme, or an interpolative scheme to facilitate the calculation of quantities which depend on interband integrals and the like. Tight binding theory, reduced to its bare essentials, with the overlap parameters used to fit experimental data or as an interpolation scheme in band structure calculation is generally referred to as Slater-Koster theory. Pseudopotential theory used in this way has been dubbed the empirical pseudopotential method (EPM) and has been the subject of a recent comprehensive review. Some comparisons of parameters t>(g), which have been fitted to experiment, with theoretical calculations have already been shown in Figure 12. 

The primary reason for interest in extended Huckel today is because the method is general enough to use for all the elements in the periodic table. This is not an extremely accurate or sophisticated method however, it is still used for inorganic modeling due to the scarcity of full periodic table methods with reasonable CPU time requirements. Another current use is for computing band structures, which are extremely computation-intensive calculations. Because of this, extended Huckel is often the method of choice for band structure calculations. It is also a very convenient way to view orbital symmetry. It is known to be fairly poor at predicting molecular geometries. 

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

A very simple and effective method of increasing the intensity of Ti So transitions by several orders of magnitude was introduced by Evans 53-s5) Either the pure liquid or concentrated solutions of an aromatic hydrocarbon in chloroform were saturated with oxygen or nitric oxid at high pressures. The newly appearing absorption bands are proportional to the applied gas pressures from 0—100 atm. The So absorptions are in general well structured and the posi-... 

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. 

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... 

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