Energy-band structure approximation

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

The energy band structures are qualitatively very similar for all four PDA backbones. The four core bands of practically zero width are situated around —299 eV. The other nine doubly occupied bands lie in the region of —4 to — 30 eV. Both the highest filled and lowest unfilled (valence and conduction) bands have n symmetry and both are crossed by the nearest band dispersions are also relatively stable against various approximations and some quantitative differences between them may play an important role in transport calculations on these polymers ... 

Fig. 9.3 Energy band structure of graphene calculated with tight-binding approximation...

Fig. 9.4 Energy band structure of graphene. Solid thick lines density functional theory with local density approximation circles and dashed lines GW approximation [8]...

Calculations of approximate energy band structure are possible on the basis of the assumption that the energy bands are mostly determined by the short-range order. This aspect will be discussed in Chapter 8. The absence of periodicity on the one hand and presence of extensive polymeric structure in most important glasses on the other, are factors which severely restrict the use of quantum mechanical calculations. Most quantum mechanical calculations are restricted to investigations of small clusters because of the prohibitive increase of the computational cost with size. 

Fig.1.4. Self-consistent energy-band structure for bcc tungsten obtained by the LMTO method within the atomic-sphere approximation (ASA) using local-density theory for exchange and correlation. Relativistic effects are included except spin-orbit coupling which is neglected...

In this section, the ground state properties of the lanthanides are studied with a first principles all-electron total energy band structure method. The LMTO method is employed within the local density (LDA) and local spin density (LSD) functional approximations (Hohenberg and Kohn 1964, Kohn and Sham 1965, Gunnarsson and Lundqvist 1976). The von Barth-Hedin (1972) interpolation formula is used for the exchange and correlation potential with the parameters of Hedin and Lundqvist (1971) and RPA scaling (Janak 1978). 

Yamagami and Hasegawa carried out a self-consistent calculation of the energy band structure by solving the Kohn-Sham-Dirac one-electron equation by the density-functional theory in a local-density approximation (LDA). This self-consistent, symmetrized relativistic APW approach was applied to many lanthanide compounds and proved to give quite accurate results for the Fermi surface. 

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. 

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

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