Computing Band Gaps

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 

Simply doing electronic structure computations at the M, K, X, and T points in the Brillouin zone is not necessarily sufficient to yield a band gap. This is because the minimum and maximum energies reached by any given energy band sometimes fall between these points. Such limited calculations are sometimes done when the computational method is very CPU-intensive. For example, this type of spot check might be done at a high level of theory to determine whether complete calculations are necessary at that level. 

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 

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Larger differences between the supercell and the cluster scheme may appear when other properties related to the infinite nature of the perturbed crystalline system are analyzed. In Figure 57, on the left, the band structure of defective S32 supercell is shown and, on the right, the energy levels of the CSiggHyg cluster are shown (the energy scale is shifted in such a way that the Is level of carbon of the two systems coincide). In the present example, no defect states are present in the gap the computed band gap is 6.38 eV for the defective S32 supercell, to be compared with the 8.8 eV HOMO-LUMO gap of the cluster. 

Computed Band Gaps for Representative Conjugated Polymers... 

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. 

Jason Green has successfully applied the Selenski method to the synthesis of (+ )-bromoheliane (79, Fig. 4.38).34 In this example, two equivalents of the chiral enol ether are added to the benzaldehyde 77 in diethyl ether (0.1 M) and cooled to —78 °C. Methyl Grignard is then added. The cycloaddition occurs while the reaction warms to room temperature. The benzopyran adduct 78 forms in 80% yield with 50 1 diaster-eoselectivity. DFT calculations and experiments suggest that the diastereoselectivity depends on the magnitude of the HOMO-LUMO band gap. In this instance, the LUMO of the supposed o-QM intermediate is computed to be —2.6 eV, whereas the HOMO of the enol ether is —5.9 eV. A 50 1 selectivity is recorded for resulting 3.3 eV gap. For reactions of 2,5-bis-OBoc-4-methyl-benzaldehyde, where the HOMO-LUMO gap is larger (3.6 eV), a 20 1 ratio of diastereomers is observed. 

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. 

Whilst trae 3D photonic band gap materials operating in the microwave and millimetre range have been produced, those operating in the visible region of the spectrum are still awaited. When this eventually happens the optical computer will no longer be a pipe dream. 

The calculations were performed employing either pure ab initio Hartree-Fock (HF) methods, or hybrid HF-DFT functionals, in particular B3LYP [22]. The hybrid functionals have several advantages. One is that they are commonly applied with great success in computational studies of molecules and clusters, thus making it possible to benefit from the gathered experience from molecular studies. Another is their recently noted ability to accurately model band gaps in semiconductor compounds [57]. 

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