Tight-Binding Method in Electronic Structure
In the linear combination of atomic orbitals (LCAO) method, the one-electron wave function is expressed as a linear combination of Bloch sums. When such an expansion is made to the wave function in the Schrödinger equation, one obtains a set of simultaneous linear equations that has a nonzero solution if the determinant of the coefficients vanishes, that is, The matrix elements in this equation have the form: (Formula presented) and (Formula presented) where Ri and Rj denote the positions of atoms located on orbitals ϕn and ϕm, respectively. The integrals in the above equations can, in principle, be calculated directly. However, the most common practice has been to follow Slater and Koster (SK), who suggested replacing these integrals by adjustable parameters that could be estimated from experiment, but which are generally determined by fitting to more elaborate electronic structure calculations. The size of the matrices H̃ and S̃ is determined by the number of atoms in the unit cell and the number of atomic orbitals on each site. So for face-centered cubic (f.c.c.), body-centered cubic (b.c.c.), and simple cubic (s.c.) lattices with one atom per unit cell, H̃ and S̃ are 9 × 9 matrices representing one s-function, three p-functions, and five d-functions. The f-states have been omitted in most works, although there have been papers that provide extensions of the SK scheme that include f-orbitals. For diatomic lattices, such as hexagonal close-packed (h.c.p.) and diamond and in binary compounds the spd model will result in an 18 × 18....
