Semiempirical Band-Structure Calculations

Table 2.5 presents the positions and widths of the valence and conduction bands of poly(TNCQ) and poly(TTF) obtained with the help of the above-described procedure. The most striking result in the table, as in the case of the previous semiempirical band-structure calculations,< > is that the valence band of poly(TTF) [from which the charge transfer (CT) occurs] is comparatively broad ( 0.3 eV) and the conduction band of poly(TCNQ) (to which the charge is transferred) is broad ( 1.2 eV), while both the valence band of poly(TCNQ) and the conduction band of poly(TTF) (which do not take part in the Cl process) have widths less than 0.1 eV. One should also note that the positions of the conduction band of poly(TCNQ) and the valence band of poly(TTF), respectively. 

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. 

The COOP idea has found a plethora of applications in the framework of a popular semiempirical (see Section 2.11.1) method, and COOP is probably the decisive quantum-chemical tool that has convinced many solid-state (materials) chemists to eventually strive for band-structure calculations. Before the introduction of COOP, chemical bonding discussions were too often restricted to the simplistic "ionic model", no matter whether or not the chemical bonding was, in fact, ionic in nature. 

After a number of semiempirical crystal-orbital (CO) calculations (extended Hvickel and CNDO/2 CO calculations for further references see Table III of Andre ) Otto et performed the first ab initio band-structure calculations on different fluorinated polyethylenes. The calculations were conducted for all six different polyfluoroethylenes, namely (CFH-CHj), (CF2-CH2), (CFH-CFH),, (CFH-CFH), (CF2-CFH)jt, and (CF2), which can be obtained from polyethylene through fluorine substitution. 

It can be seen from Table 2.6 that the physically most important valence and conduction bands in the later ab initio calculations are much broader (0.435-0.789 and 0.245-0.820 eV, respectively) than those obtained by application of different semiempirical crystal-orbital methods. With the simple PPP-CO approximation, the corresponding values for the highest filled bands are 0.218-0.299 eV and those for the conduction band are about 0.109 eV. Energy-band-structure calculations for the base stacks taking into account the effect of the other valence electrons with the aid of the CNDO/2 CO method give again broader bands (valence bandwidths of 0.136-0.490 eV and conduction band-widths of 0.109-0.245 eV), while the MINDO/2 CO results indicate somewhat less-broad bands (valence bandwidths of 0.027-0.299 eV and conduction bandwidths of 0.027-0.163 eV). For futher details on the semiempirical crystal-orbital calculations see also Chapter 3. 

Most of the semiempirical tight-binding methods for nanostructures are based on the parametrization of bulk systems. It consists of an iterative fitting procedure, performed on the tight-binding parameters, to match the bulk silicon band structure calculated using the most advanced techniques [21]. The as-calculated parameters are then applied to the study of the electronic properties of silicon nanostructures. When the nanostructures are well passivated, the surface is expected to play a minor role, and the main electronic and optical properties are determined by the nanocrystal core. 

Ht should be stressed that HF-SCF method with semiempirical INDO Hamiltonian used at band structure calculation [55] overestimates bonding character and consequently band-width, which means that Mgh-enetgy effects can hardly be studied, but for low-energy physics (like gap opening, kink formation,...) the method is reliable enough at least in a qualitative way. 

Due to the central role of DNA and proteins in biochemistry and biophysics the computation of the electronic structure of periodic polymers built from nucleotide bases, base pairs, nucleotides and amino acids, respectively, had been of high interest since about twenty years. Early calculations of the band structure of DNA related periodic polymers have been performed with the crystal orbital (CO) method on the basis of different semiempirical levels (1). Recently the results of ab initio Hartree-Fock CO (2, 3) band structure calculations for the four nucleotide base stacks (4-6), the two Watson-Crick base pair stacks (6), the sugar-phosphate chain (4,5) and the three nucleotides cytidine (4,5), adenylic acid and th3nnidine (6) have been reported. These computations represent a significant progress but the following improvements are required for a more accurate description of the electronic structure of real DNA and its transport properties ... 

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. 

With the help of these experimental techniques the study of semiconductors in recent years has been a highly successful marriage of theory and experiment. The trends of the important parameters of band structure from one element or compound to the next were found to follow rules which are remarkably simple when one considers the potpourri of effects and approximations involved in band structure calculations. This led Phillips to construct a much simpler semiempirical picture based on bonds rather than bands—but that is another storytold elsewhere in this volume (Chapter 1). 

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... 

Local density functional (LDF) quantum mechanical calculations for materials science. deMon for density functional calculations. Turbomole for Hartree-Fock and MP2 ab initio calculations. ZINDO for extended Fliickel, PPP, CNDO, and INDO semiempirical molecular orbital calculations and prediction of electronic spectra. Plane Wave for band structures of semiconductors. ESOCS for electronic structure of solids. Silicon Graphics and IBM workstation versions. 

We then proceed with a semiempirical i.s,px,py,pz) tight-binding (ETB) approach, the parameters of which were fitted to a large LDA data base of carbon molecules and solid structures. Details of this Hamiltonian are published elsewhere [9]. For Ceo its predictions agree well with LDA results (e.g., the overall bandwidth, the density of states, and the symmetry of states near the gap). With this Hamiltonian we not only calculate band structures, but we also obtain the approximate deformation potentials for electronic states. The additional ingredient here... 

If we know the translational symmetry of an extended solid (Bloch s theorem) and also have a trustworthy strategy on how to deal with the nuclear potential and the electron-electron interactions (by, say, a semiempirical method or by DFT), we are ready to explicitly calculate the band structure of any real material. Although we have done this before for idealized systems (see sketches in Section 2.6), let us now attack the problem once again, but in more general terms. For real materials, one needs to solve SchrBdinger s equation using the true potential v r), namely. 

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