Tight-binding method basis functions

At the lowest level of sophistication of quantum treatments, the tight-binding method and the semi-empirical HF method reduce the complexity of the interacting electron system to the diagonalization of an effective one-electron Hamiltonian matrix, whose elements contain empirical parameters. The electronic wave functions are expanded on a minimal basis set of atomic or Slater orbitals centered on the atoms and usually restricted to valence orbitals. The matrix elements are self-consistently determined or not, depending upon the method. 

Within density-functional theory, a linear combination of overlapping non-orthogonal orbitals from first principles may be utilized to arrive at at full-potential local orbital (FPLO) method [213], and this k-dependent LCAO approach comes close to full-potential APW-based methods (see Sections 2.15.3 and 2.15.4) in terms of numerical accuracy, although FPLO is much faster simply because of the locality of the basis set. Even faster, due to a strongly simplified potential, is a parameter-free (density-functional) tight-binding method called TB-LMTO-ASA, derived through localization of a delocalized basis set (see Section 2.15.4). 

The Harris functional provides not only relatively accurate non-self-consistent estimates of energies and structures, it also provides a basis from which the success of other non-self-consistent approximate methods can be understood. One of the more widely used of these, the tight binding method, is discussed in the next section. 

J. C. Slater together with G. F. Koster developed a method called the Slater-Koster tight-binding method, which builds directly on Figure 16.1. This tight-binding model is a simplified MO model where the crystal orbitals are expressed in terms of atomic functions, as in the Hiickel model. Later, basis sets of the same type as in quantum chemical calculations on finite systems have also come to be used in infinite systems. 

The quantitative determination of the magnitude of the tensor component is based on the explicit knowledge of the spatial dependence of the electronic wavefunctions. As pointed out in Section 2 the Z , w, II) functions transform as the basis functions of the irreps of the symmetry group G and can be obtained with the modified projector technique method [23] in tight binding approximation. The so obtained symmetrized wavefunctions preserve the transformation properties as dictated by the irreps of the symmetry group G. 

In the "tight-binding" or LCAO method [1.12] one uses as basis functions the eigenfunctions xn m of The bound states of the free atom, and the wave function for an electron in the solid is then expressed in terms of the Bloch sums... 

Three different approaches have been followed to solve this clue, and they form the backbones of all existing band-structure methods in terms of their nuclear potentials. Somewhat simplified, one may either ignore the core functions (empirical tight-binding approaches), one may modify the potential, thereby also ignoring the core functions (pseudopotential approach), or one may modify the basis sets and split the functions into core and beyond-core functions (cellular approaches and successors) [210]. 

---