Crystal-orbital, tight binding

We will limit ourselves here to transition metals. It is well known that in these metals, the cohesive properties are largely dominated by the valence d electrons, and consequently, sp electrons can be neglected save for the elements with an almost empty or filled d valence shelP. Since the valence d atomic orbitals are rather localized, the d electronic states in the solid are well described in the tight-binding approximation. In this approximation, the cohesive energy of a bulk crystal is usually written as ... 

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

In this section we show how the general form of Renner-Teller interaction matrices can be obtained at any order in the phonon variables and with electron orbital functions of different symmetry (p-like, < like, /-like, etc.). For this purpose, we use an intuitive approach [18] based on the Slater-Koster [19] technique and its generalization [20] to express crystal field or two-center integrals in terms of independent parameters in the tight-binding band theory [21] then we apply standard series developments in terms of normal coordinates. 

The band structure of a three-dimensional solid, such as a semiconductor crystal, can be obtained in a similar fashion to that of a polyene. Localized molecular orbitals are constructed based on an appropriate set of valence atomic orbitals, and the effects of delocalization are then incorporated into the molecnlar orbital as the number of repeat units in the crystal lattice is increased to infinity. This process is widely known to the chemical conununity as extended Hiickel theory (see Extended Hiickel Molecular Orbital Theory). It is also called tight binding theory by physicists who apply these methods to calcnlate the band structures of semiconducting and metallic solids. 

As an example of the study of vacancies and self-interstitial impurities by the continued fraction expansion of Eq. (S.2S), we mention the work of Kauffer et al. These authors consider impurities in silicon and set up a model tight-binding Hamiltonian with s p hybridization, which satisfactorily describes the valence and conduction bands of the perfect crystal. A cluster of 2545 atoms is generated, and vacancies (or self-interstitial impurities) are introduced at the center of the cluster. One then takes as a seed state an appropriate orbital or symmetrized combination of orbitals, and the recursion method is started. Though self-consistent potential modifications are neglected in this paper, the model leads to qualitatively satisfactory results within a simple physical picture. 

Only fragmentary and semiqualitative calculations have been published using this line of approach.In the work of Bylander and Rehr, the starting model is a crystal with one orbital per site described by a tight-bind-... 

Anderson s simple model to describe the electrons in a random potential shows that localization is a typical phenomenon whose nature can be understood only taking into account the degree of randomness of the system. Using a tight-binding Hamiltonian with constant hopping matrix elements V between adjacent sites and orbital energies uniformly distributed between — W/2 and W/2, Anderson studied the modifications of the electronic diffusion in the random crystal in terms of the stability of localized states with respect to the ratio W/V. 

Crystal orbital methods have also been used and developed in order to obtain the nonlinear optical responses of stereoregular polymers (Table VI). The rationale is related to the large nonlinear susceptibilities that Ji-conjugated polymers can have. The initial study, carried out by Agrawal et al. [173], employed the Genkin and Mednis [120] formalism within the tight-binding (Hiickel) model and... 

Even though the bonding in metals must be purely covalent, we cannot use the simplified bonding model of the earlier section. That model is appropriate for cases where the delocalized crystal orbitals can be replaced by average localized orbitals. This is not possible for metals, or at least not easy. Actually the tight binding theory at the Hiickel level of approximation has been used for metals in several cases. 

The electronic structure of (CH) has been studied using various approaches within the framework of the one-dimensional tight-binding crystal orbital (CO) method, that is, from the Hiickel to the ab initio Hartree-Fock level (see, e.g., Kertesz, 1982). Some of the calculated results of the energetic stability of the (CH), isomers in Fig. 1 are listed in Table I. 

Transition metals are important materials with intriguing properties and they have been studied with ever improved methods. A major difficulty is posed by the standard one-electron models where the tight-binding model seems appropriate for the narrow, so-called d-bands while near-plane-wave crystal orbitals are adequate for the conduction bands. Canonical Hartree-Fock solutions are awkward starting points for the description of magnetic structures and the use of spin-polarized versions destroys basic symmetry properties. 

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