Editing Electronic band structure (section)

=== Tight binding model ===
{{Main|Tight binding}}
The opposite extreme to the nearly free electron approximation assumes the electrons in the crystal behave much like an assembly of constituent atoms. This [[tight binding model]] assumes the solution to the time-independent single electron [[Schrödinger equation]] <math>\Psi</math> is well approximated by a [[Linear combination of atomic orbitals molecular orbital method|linear combination]] of [[atomic orbitals]] <math>\psi_n(\mathbf{r})</math>.<ref name=Kittel/>{{rp|pp=245-248}}
<math display="block">\Psi(\mathbf{r}) = \sum_{n,\mathbf{R}} b_{n,\mathbf{R}} \psi_n(\mathbf{r}-\mathbf{R}),</math>
where the coefficients <math> b_{n,\mathbf{R}}</math> are selected to give the best approximate solution of this form. Index {{mvar|n}} refers to an atomic energy level and {{math|'''R'''}} refers to an atomic site. A more accurate approach using this idea employs [[Wannier functions]], defined by:<ref name=Kittel/>{{rp|at=Eq. 42 p. 267}}<ref name=Mattis>{{cite book | author=Daniel Charles Mattis | title=The Many-Body Problem: Encyclopaedia of Exactly Solved Models in One Dimension | year = 1994 | publisher=World Scientific | page=340 | isbn=978-981-02-1476-0 | url=https://books.google.com/books?id=BGdHpCAMiLgC&q=wannier+functions&pg=PA332}}</ref>
<math display="block">a_n(\mathbf{r}-\mathbf{R}) = \frac{V_{C}}{(2\pi)^{3}} \int_\text{BZ} d\mathbf{k} e^{-i\mathbf{k}\cdot(\mathbf{R} -\mathbf{r})}u_{n\mathbf{k}};</math>
in which <math>u_{n\mathbf{k}}</math> is the periodic part of the Bloch's theorem and the integral is over the [[Brillouin zone]]. Here index {{math|''n''}} refers to the {{math|''n''}}-th energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites {{math|'''R'''}} are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the {{math|''n''}}-th energy band as:
<math display="block">\Psi_{n,\mathbf{k}} (\mathbf{r}) = \sum_{\mathbf{R}} e^{-i\mathbf{k}\cdot(\mathbf{R}-\mathbf{r})}a_n(\mathbf{r} - \mathbf{R}).</math>

The TB model works well in materials with limited overlap between [[atomic orbital]]s and potentials on neighbouring atoms. Band structures of materials like [[Silicon|Si]], [[GaAs]], SiO<sub>2</sub> and [[diamond]] for instance are well described by TB-Hamiltonians on the basis of atomic sp<sup>3</sup> orbitals. In [[transition metals]] a mixed TB-NFE model is used to describe the broad NFE [[conduction band]] and the narrow embedded TB d-bands. The radial functions of the atomic orbital part of the Wannier functions are most easily calculated by the use of [[pseudopotential]] methods. NFE, TB or combined NFE-TB band structure calculations,<ref name=Harrison>{{cite book
|author=Walter Ashley Harrison
|title=Electronic Structure and the Properties of Solids
|year= 1989
|publisher=Dover Publications
|url=https://books.google.com/books?id=R2VqQgAACAAJ
|isbn=978-0-486-66021-9
}}</ref>
sometimes extended with wave function approximations based on pseudopotential methods, are often used as an economic starting point for further calculations.