Tight Binding - Mathematical Formulation

Mathematical Formulation

We introduce the atomic orbitals, which are eigenfunctions of the Hamiltonian of a single isolated atom. When the atom is placed in a crystal, this atomic wave function overlaps adjacent atomic sites, and so are not true eigenfunctions of the crystal Hamiltonian. The overlap is less when electrons are tightly bound, which is the source of the descriptor "tight-binding". Any corrections to the atomic potential required to obtain the true Hamiltonian of the system, are assumed small:

A solution to the time-independent single electron Schrödinger equation is then approximated as a linear combination of atomic orbitals :

,

where refers to the m-th atomic energy level and locates an atomic site in the crystal lattice.