Editing Wannier function (section)

==Definition==

[[Image:WanF-BaTiO3.png|upright=1.2|thumb|Example of a localized Wannier function of titanium in barium titanate (BaTiO3)]]

Although, like [[localized molecular orbitals]], Wannier functions can be chosen in many different ways,<ref>[https://cfm.ehu.es/ivo/publications/marzari-psik03.pdf Marzari ''et al.'': An Introduction to Maximally-Localized Wannier Functions]</ref> the original,<ref name=Wannier1937/> simplest, and most common definition in solid-state physics is as follows. Choose a single [[Electronic band structure|band]] in a perfect crystal, and denote its [[Bloch state]]s by
:<math>\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_\mathbf{k}(\mathbf{r})</math>
where ''u''<sub>'''k'''</sub>('''r''') has the same periodicity as the crystal. Then the Wannier functions are defined by
:<math>\phi_{\mathbf{R}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r})</math>,
where
* '''R''' is any lattice vector (i.e., there is one Wannier function for each [[Bravais lattice|Bravais lattice vector]]);
* ''N'' is the number of [[primitive cell]]s in the crystal;
* The sum on '''k''' includes all the values of '''k''' in the [[Brillouin zone]] (or any other [[primitive cell]] of the [[reciprocal lattice]]) that are consistent with [[periodic boundary conditions]] on the crystal. This includes ''N'' different values of '''k''', spread out uniformly through the Brillouin zone. Since ''N'' is usually very large, the sum can be written as an integral according to the replacement rule:
:<math>\sum_{\mathbf{k}} \longrightarrow \frac{\sqrt{N}}{\Omega} \int_\text{BZ} d^3\mathbf{k}</math>
where "BZ" denotes the [[Brillouin zone]], which has volume Ω.

=== Properties ===

On the basis of this definition, the following properties can be proven to hold:<ref name=Bohm>{{cite book |title=The Geometric Phase in Quantum Systems |author=A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger |isbn=978-3-540-00031-0 |publisher=Springer |year=2003 |pages=§12.5, p. 292 ff|doi=10.1007/978-3-662-10333-3 |url=https://cds.cern.ch/record/737299 }}</ref>

* For any lattice vector ''' R' ''',
:<math>\phi_{\mathbf{R}}(\mathbf{r}) = \phi_{\mathbf{R}+\mathbf{R}'}(\mathbf{r}+\mathbf{R}')</math>
In other words, a Wannier function only depends on the quantity ('''r''' − '''R'''). As a result, these functions are often written in the alternative notation
:<math>\phi(\mathbf{r}-\mathbf{R}) := \phi_{\mathbf{R}}(\mathbf{r})</math>

* The Bloch functions can be written in terms of Wannier functions as follows:
:<math>\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \phi_{\mathbf{R}}(\mathbf{r})</math>,
where the sum is over each lattice vector '''R''' in the crystal.

* The set of wavefunctions <math>\phi_{\mathbf{R}}</math> is an [[orthonormal basis]] for the band in question.
:<math>\begin{align}
\int_\text{crystal}  \phi_{\mathbf{R}}(\mathbf{r})^* \phi_{\mathbf{R'}}(\mathbf{r}) d^3\mathbf{r} & = \frac{1}{N} \sum_{\mathbf{k,k'}}\int_\text{crystal} e^{i\mathbf{k}\cdot\mathbf{R}} \psi_{\mathbf{k}}(\mathbf{r})^*  e^{-i\mathbf{k'}\cdot\mathbf{R'}} \psi_{\mathbf{k'}}(\mathbf{r}) d^3\mathbf{r} \\
& =  \frac{1}{N} \sum_{\mathbf{k,k'}} e^{i\mathbf{k}\cdot\mathbf{R}} e^{-i\mathbf{k'}\cdot\mathbf{R'}} \delta_{\mathbf{k,k'}} \\
& = \frac{1}{N} \sum_{\mathbf{k}} e^{i\mathbf{k}\cdot\mathbf{(R-R')}} \\
& =\delta_{\mathbf{R,R'}}
\end{align} </math>

Wannier functions have been extended to nearly periodic potentials as well.<ref name=Kohn0>[http://www.physast.uga.edu/~mgeller/4.pdf MP Geller and W Kohn] ''Theory of generalized Wannier functions for nearly periodic potentials'' Physical Review B 48, 1993</ref>

===Localization===

The Bloch states ''ψ''<sub>'''k'''</sub>('''r''') are defined as the eigenfunctions of a particular Hamiltonian, and are therefore defined only up to an overall phase. By applying a phase transformation ''e''<sup>''iθ''('''k''')</sup> to the functions ''ψ''<sub>'''k'''</sub>('''r'''), for any (real) function ''θ''('''k'''), one arrives at an equally valid choice. While the change has no consequences for the properties of the Bloch states, the corresponding Wannier functions are significantly changed by this transformation.

One therefore uses the freedom to choose the phases of the Bloch states in order to give the most convenient set of Wannier functions. In practice, this is usually the maximally-localized set, in which the Wannier function {{math|''&varphi;''<sub>'''R'''</sub>}} is localized around the point '''R''' and rapidly goes to zero away from '''R'''. For the one-dimensional case, it has been proved by Kohn<ref name=Kohn1>{{cite journal|doi=10.1103/PhysRev.115.809 | volume=115 | issue=4 | title=Analytic Properties of Bloch Waves and Wannier Functions | year=1959| journal=Physical Review  | pages=809–821 | author=W. Kohn| bibcode=1959PhRv..115..809K}}</ref> that there is always a unique choice that gives these properties (subject to certain symmetries). This consequently applies to any [[Separable partial differential equation|separable potential]] in higher dimensions; the general conditions are not established, and are the subject of ongoing research.<ref name=Arxiv-Localization/>

A [[Localized molecular orbitals#Pipek-Mezey|Pipek-Mezey]] style localization scheme has also been recently proposed for obtaining Wannier functions.<ref name=Jonsson2016>{{cite journal|doi=10.1021/acs.jctc.6b00809 | pmid=28099002 | volume=13 | issue=2 | title=Theory and Applications of Generalized Pipek–Mezey Wannier Functions | year=2017 | journal=Journal of Chemical Theory and Computation | pages=460–474 | author=Jónsson Elvar Ö., Lehtola Susi, Puska Martti, Jónsson Hannes| arxiv=1608.06396 | s2cid=206612913 }}</ref> Contrary to the maximally localized Wannier functions (which are an application of the [[Localized molecular orbitals#Foster-Boys|Foster-Boys]] scheme to crystalline systems), the Pipek-Mezey Wannier functions do not mix σ and π orbitals.

====Rigorous results====

The existence of [[Exponential function|exponential]]ly localized Wannier functions in insulators was proved mathematically in 2006.<ref name=Arxiv-Localization>{{cite journal | last1=Brouder | first1=Christian | last2=Panati | first2=Gianluca | last3=Calandra | first3=Matteo | last4=Mourougane | first4=Christophe | last5=Marzari | first5=Nicola | title=Exponential Localization of Wannier Functions in Insulators | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=98 | issue=4 | date=25 January 2007 | issn=0031-9007 | doi=10.1103/physrevlett.98.046402 | page=046402| pmid=17358792 |arxiv=cond-mat/0606726| bibcode=2007PhRvL..98d6402B | s2cid=32812449 }}</ref>