![]() The tight-binding model has a long history and has been applied in many ways and with many different purposes and different outcomes. So the tight-binding model can provide nice examples for those who want to learn more about group theory. When simple systems like the lattices of elements or simple compounds are studied it is often not very difficult to calculate eigenstates in high-symmetry points analytically. High-symmetry points in the Brillouin zone belong to different point-group representations. The reciprocal lattice and the Brillouin zone often belong to a different space group than the crystal of the solid. This can lead to complicated band structures because the orbitals belong to different point-group representations. In general there are a number of atomic energy levels and atomic orbitals involved in the model. The most important elements in the model are the interatomic matrix elements, which would simply be called the bond energies by a chemist. Two of those three kinds of elements should be close to zero and can often be neglected. There are only three kinds of matrix elements that play a significant role in the theory. Though the mathematical formulation of the one-particle tight-binding Hamiltonian may look complicated at first glance, the model is not complicated at all and can be understood intuitively quite easily. The energy of the electron will also be rather close to the ionization energy of the electron in the free atom or ion because the interaction with potentials and states on neighboring atoms is limited. As a result, the wave function of the electron will be rather similar to the atomic orbital of the free atom to which it belongs. ![]() The electrons in this model should be tightly bound to the atom to which they belong and they should have limited interaction with states and potentials on surrounding atoms of the solid. The name "tight binding" of this electronic band structure model suggests that this quantum mechanical model describes the properties of tightly bound electrons in solids. Though the tight-binding model is a one-electron model, the model also provides a basis for more advanced calculations like the calculation of surface states and application to various kinds of many-body problem and quasiparticle calculations. The model gives good qualitative results in many cases and can be combined with other models that give better results where the tight-binding model fails. Tight-binding models are applied to a wide variety of solids. The method is closely related to the LCAO method (linear combination of atomic orbitals method) used in chemistry. In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site.
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