tightbinding

Tight binding is a quantum mechanical method used to calculate the electronic band structure of solids, particularly in condensed matter physics. The approach is based on the assumption that electrons are tightly bound to their respective atoms and that the wave functions of neighboring atoms overlap only slightly.

The method starts with atomic orbitals as the basis set and constructs molecular orbitals through linear combinations of these atomic states. When atoms come together to form a solid, their discrete energy levels broaden into bands due to interactions between neighboring atoms. The tight binding approximation considers only the overlap between nearest neighbors and uses parameters that can be determined either empirically or from more fundamental calculations.

In mathematical terms, the Hamiltonian matrix elements are expressed in terms of atomic orbital overlaps and interaction energies. The diagonal elements represent the energy of an electron on a particular atom, while the off-diagonal elements describe the transfer or hopping of electrons between adjacent sites. These hopping parameters determine the width and shape of the resulting energy bands.

The tight binding method is particularly useful for understanding the electronic properties of materials where localized atomic states play a dominant role, such as transition metal oxides and organic crystals. It provides intuitive insight into how atomic orbitals hybridize to form bonding and antibonding states that determine the material's conductivity properties.

Compared to more sophisticated methods like density functional theory, tight binding is computationally efficient and physically transparent. However, it typically requires empirical parameters and may not capture all electronic correlation effects. Extensions to the basic tight binding model include including multiple orbitals per atom, spin-orbit coupling, and long-range interactions to improve accuracy for specific applications.