Editing Wannier function (section)

==Modern theory of polarization==
Wannier functions have recently found application in describing the [[Polarization density|polarization]] in crystals, for example, [[Ferroelectricity|ferroelectrics]]. The modern theory of polarization is pioneered by Raffaele Resta and David Vanderbilt. See for example, Berghold,<ref name=Berghold>{{cite journal | last1=Berghold | first1=Gerd | last2=Mundy | first2=Christopher J. | last3=Romero | first3=Aldo H. | last4=Hutter | first4=Jürg | last5=Parrinello | first5=Michele | title=General and efficient algorithms for obtaining maximally localized Wannier functions | journal=Physical Review B | publisher=American Physical Society (APS) | volume=61 | issue=15 | date=15 April 2000 | issn=0163-1829 | doi=10.1103/physrevb.61.10040 | pages=10040–10048| bibcode=2000PhRvB..6110040B }}</ref>  and Nakhmanson,<ref name=Nakhmanson>{{cite journal | last1=Nakhmanson | first1=S. M. | last2=Calzolari | first2=A. | last3=Meunier | first3=V. | last4=Bernholc | first4=J. | last5=Buongiorno Nardelli | first5=M. | title=Spontaneous polarization and piezoelectricity in boron nitride nanotubes | journal=Physical Review B | volume=67 | issue=23 | date=10 June 2003 | issn=0163-1829 | doi=10.1103/physrevb.67.235406 | page=235406|arxiv=cond-mat/0305329v1| bibcode=2003PhRvB..67w5406N | s2cid=119345964 }}</ref> and a power-point introduction by Vanderbilt.<ref name=Vanderbilt>[http://www.physics.rutgers.edu/~dhv/talks/rahman.pdf  D Vanderbilt] ''Berry phases and Curvatures in Electronic Structure Theory''.</ref> The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density:
:<math>\mathbf{p_c} = -e \sum_n \int\ d^3 r \,\, \mathbf{r} |W_n(\mathbf{r})|^2 \ , </math>
where the summation is over the occupied bands, and ''W<sub>n</sub>'' is the Wannier function localized in the cell for band ''n''. The ''change'' in polarization during a continuous physical process is the time derivative of the polarization and also can be formulated in terms of the [[Berry phase]] of the occupied Bloch states.<ref name=Bohm/><ref name=Resta>{{cite book |author=C. Pisani |title=Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials |isbn=978-3-540-61645-0 |year=1994 |publisher=Springer |edition=Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society |page=282 |url=https://books.google.com/books?id=5ak5TwSLreAC&dq=%22Berry+connection%22&pg=PA282}}</ref>