Editing Wannier function (section)

===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>