Computing Band Structures

Band structure calculations have been done for very complicated systems however, most of software is not yet automated enough or sufficiently fast that anyone performs band structures casually. Setting up the input for a band structure calculation can be more complex than for most molecular programs. The molecular geometry is usually input in fractional coordinates. The unit cell lattice vectors and crystallographic angles must also be provided. It may be nee- 

As mentioned above, the preferred computational methods for modeling crystals have changed over the years. Below is a list of basis function schemes, with the most often used appearing first  

Green s function method of Korringa, Kohn, and Rostoker (sometimes denoted KKR) 

Any orbital-based scheme can be used for crystal-structure calculations. The trend is toward more accurate methods. Some APW and Green s function methods use empirical parameters, thus edging them toward a semiempirical classification. In order of preference, the commonly used methods are  

The Fermi energy is the energy of the highest-energy filled orbital, analogous to a HOMO energy. If the orbital is half-filled, its energy will be found at a collection of points in /c-space, called the Fermi surface. 

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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. 

Figure 3 Computed band structure of an eclipsed PtH 2- stack, spaced at 3 A. The orbital marked xz, yz is doubly degenerate.

The replacement of main-group atoms in clusters by transition-metal atoms generates a richer structural chemistry superimposed on the cluster basics illustrated by the p-block systems. A logical question arises here. What would a one-dimensional material containing a transition metal look like Well, the d AOs will generate bands in a similar manner as the s and p orbitals. The major novelty will be the introduction of orbitals of 8 symmetry. Let s look at a hypothetical chain composed of equidistant Ni atoms (d = 2.5 A). The computed band structure, DOS and COOP are illustrated in Figure 6.14. The COs at k = 0 and ir/d arc drawn below. As a review of the previous section, we will reconstruct it starting from the Bloch functions associated with the nine Ni AOs. 

Fig.9.21. Predicting (after Roald Hoffmann) the band structure of PtH. (a) The predicted band structure (b) the computed band structure (by Roald Hoffmann) for a = 3 A.

We have developed our model for the simplest possible system, one p orbital per unit cell, and then argued qualitatively about the system with two AOs per unit cell. In fact, there is a rigorous, analytical solution for the generalized problem of a unit cell with any number of MOs in an infinite polymer. This produces the Bloch equations that are the direct analogues of HMO theory as applied to infinite systems. To create computed band structures of the sort shown throughout this chapter, we simply solve the Bloch equations at several points throughout the Brillouin zone and then interpolate between these points to produce the smooth curves shown in band structures. 

Due to its metallic and FM property, similar to double perovskites (see later), Cr02 is of great interest to material scientists and physicists. A computed band structure of Cr02 is shown in Figure 5.6. The Fermi level crosses the majority-spin band while it lies in a gap of the minority-spin density of states (DOS). Therefore, Cr02 is half-metallic on the basis of electronic structure calculations. The same band structure has been reproduced by several groups since it was first demonstrated by... 

E. Wimmer, Computational band structure engineering of ni-V semiconductor alloys, Appl. Phys. Lett., 79 (2001) 368-370. 

Ab initio methods alone have been applied in very limited fashion to CPs other than P(Ac). A representative application to P(Ac) is the work of Tanaka and Tanaka rAppendix 7-2) [232], for Cl- and Na-doping. The computed band structure and DOS representation, discussed at some length in Appendix 7-1 below, shows that for both (n-, p-type) extreme doping cases, HOMO and LUMO nearly merge, with quasi-... 

In some cases, researchers only need to know the band gap for a crystal. Once a complete band structure has been computed, it is, of course, simple to find the... 

Some researchers use molecule computations to estimate the band gap from the HOMO-LUMO energy separation. This energy separation becomes smaller as the molecule grows larger. Thus, it is possible to perform quantum mechanical calculations on several molecules of increasing size and then extrapolate the energy gap to predict a band gap for the inhnite system. This can be useful for polymers, which are often not crystalline. One-dimensional band structures are... 

Extended Hiickel gives a qualitative view of the valence orbitals. The formulation of extended Hiickel is such that it is only applicable to the valence orbitals. The method reproduces the correct symmetry properties for the valence orbitals. Energetics, such as band gaps, are sometimes reasonable and other times reproduce trends better than absolute values. Extended Hiickel tends to be more useful for examining orbital symmetry and energy than for predicting molecular geometries. It is the method of choice for many band structure calculations due to the very computation-intensive nature of those calculations. 

Ah initio calculations of polymer properties are either simulations of oligomers or band-structure calculations. Properties often computed with ah initio methods are conformational energies, polarizability, hyperpolarizability, optical properties, dielectric properties, and charge distributions. Ah initio calculations are also used as a spot check to verify the accuracy of molecular mechanics methods for the polymer of interest. Such calculations are used to parameterize molecular mechanics force fields when existing methods are insulficient, which does not happen too often. 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

A semiempirical crystal band structure program, called BZ, is bundled with MOPAC 2000. There is also a utility, referred to as MAKPOL, for generating the input for band structure calculations with BZ. With the use of MAKPOL, the input for band-structure computations is only slightly more complicated than that for molecular calculations. 

APW (augmented plane wave) a band structure computation method atomic mass unit (amu) atomic unit of mass... 

OPW (orthogonalized plane wave) a band-structure computation method P89 (Perdew 1986) a gradient corrected DFT method parallel computer a computer with more than one CPU Pariser-Parr-Pople (PPP) a simple semiempirical method PCM (polarized continuum method) method for including solvation effects in ah initio calculations... 

It should be noted that a comprehensive ELNES study is possible only by comparing experimentally observed structures with those calculated [2.210-2.212]. This is an extra field of investigation and different procedures based on molecular orbital approaches [2.214—2.216], multiple-scattering theory [2.217, 2.218], or band structure calculations [2.219, 2.220] can be used to compute the densities of electronic states in the valence and conduction bands. 

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 band electronic structure of kl-(BEDT-TTF)2Cu(CF3)4(TCE) was calculated through the use of Hiickel tight binding computations [39] and the infrared properties analyzed [40]. These calculations indicate that the electronic band structure [10, 41] and infrared response [42] is similar to that found in the k-(BEDT-TTF)2Cu(dca)X (X = Cl and Br) salts. Specific heat measurements of kl-(BEDT-TTF)2Ag(CF3)4(TCE) indicate a linear coefficient (y = 50 mJ mol 1 K2), which is a factor of nine greater than expected from a free-electron picture [43],... 

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