Semiempirical tight-binding

The sum over i in eqns (4.57) and (4.58) runs over the number of atoms N in the solid, while the sum on a is over the number of orbitals per site. The atomic-like orbital of type a centered on site i is written as i, a) = (/>Q.(r — R,). Indeed, the nature of these orbitals and the size of the parameter n strikes right to the heart of the minimalist character of the semiempirical tight-binding method. 

Mestres and Scuseria have combined semiempirical tight-binding potentials with a GA to find the global minima of small molecular clusters. Their chromosome is an adjacency matrix. The adjacency matrix element is equal to 1 if atoms i and / are neighbors, and 0 otherwise. A method is described to translate from the adjacency matrix representation to internal coordinates. Obviously, it is also important to restrict the number of Is in the chromosome to avoid making impossible structures. They successfully locate global minima for small carbon clusters. 

Fig. 1 Size dependence of optical gaps of silicon nanocrystals calculated using quantum Monte Carlo (QMC), time-dependent local-density approximation (TD-LDA), Hartree-Fock configuration interactions (HF-CI), and semiempirical tight binding (TB). The inset shows schematically the bandgap enlargement due to reducing die nanocrystal size...

The band structures of ZrS2, ZrSe2, HfS2 and HfSe2, based on a semiempirical tight-binding method, have been presented by Murray et al [609]. 

The semiempirical tight-binding approach is a computationally light, well-established tool to study semiconductor nanocrystals [19]. The starting point of the method is the expansion of the nanocrystal wavefunctions into a localized basis set of atomic orbitals. 

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. 

This article is devoted to the description of how this linearization can be accomplished for some of the most popular electronic structure methods, ranging from semiempirical tight-binding and quantum chemical methods, to the first-principle Hartree-Fock method (HF) and density functional... 

As a consequence of the size limitations of the ab initio schemes, a large number of more-approximate methods can be found in the literature. Here, we mention only the density functional-based tight binding (DFTB) method, which is a two-center approach to DFT. The method has been successfully applied to the study of proton transport in perov-skites and imidazole (see Section 3.1.1.3). The fundamental constraints of DFT are (i) treatment of excited states and (ii) the ambiguous choice of the exchange correlation function. In many cases, the latter contains several parameters fitted to observable properties, which makes such calculations, in fact, semiempirical. 

Under some simplifications associated with the symmetry of fullerenes, it has been possible to perform calculations of type Hartree-Fock in which the interelec-tronic correlation has been included up to second order Mpller-Plesset (Moller et al. 1934 Purcell 1979 Cioslowski 1995), and calculations based on the density functional (Pople et al. 1976). However, given the difficulties faced by ab initio computations when all the electrons of these large molecules are taken into account, other semiempirical methods of the Hiickel type or tight-binding (Haddon 1992) models have been developed to determine the electronic structure of C60 (Cioslowski 1995 Lin and Nori 1996) and associated properties like polarizabilities (Bonin and Kresin 1997 Rubio et al. 1993) hyperpolarizabilities (Fanti et al. 1995) plasmon excitations (Bertsch et al. 1991) etc. These semiempirical models reproduce the order of monoelectronic levels close to the Fermi level. Other more sophisticated semiempirical models, like the PPP (Pariser-Parr-Pople) (Pariser and Parr 1953 Pople 1953) obtain better quantitative results when compared with photoemission experiments (Savage 1975). 

AMI Semiempirical Austin method 1 CSA Conformational space annealing DFT Density functional theory DFTB Density-functional-based tight binding EA Evolutionary algorithm... 

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... 

Because they are so computationally intensive, ab initio and semiempirical studies are limited to models that are about 10 rings or less. In order to study more reahstic carbon structures, approximations in the form of the Hamiltonian (i.e., Schrodinger equation) are necessary. The tight-binding method, in which the many-body wave function is expressed as a product of individual atomic orbitals, localized on the atomic centers, is one such approximation that has been successfully applied to amorphous and porous carbon systems [47]. 

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