Wannier function

Although, like localized molecular orbitals, Wannier functions can be chosen in many different ways, the original, • For any lattice vector ''' R' ''', :\phi_{\mathbf{R}}(\mathbf{r}) = \phi_{\mathbf{R}+\mathbf{R}'}(\mathbf{r}+\mathbf{R}') 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 :\phi(\mathbf{r}-\mathbf{R}) := \phi_{\mathbf{R}}(\mathbf{r}) • The Bloch functions can be written in terms of Wannier functions as follows: :\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}} \sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \phi_{\mathbf{R}}(\mathbf{r}), where the sum is over each lattice vector R in the crystal. • The set of wavefunctions \phi_{\mathbf{R}} is an orthonormal basis for the band in question. :\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} Wannier functions have been extended to nearly periodic potentials as well. Localization The Bloch states ψk(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 eiθ(k) to the functions ψk(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 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 that there is always a unique choice that gives these properties (subject to certain symmetries). This consequently applies to any separable potential in higher dimensions; the general conditions are not established, and are the subject of ongoing research. Contrary to the maximally localized Wannier functions (which are an application of the Foster-Boys scheme to crystalline systems), the Pipek-Mezey Wannier functions do not mix σ and π orbitals. Rigorous results The existence of exponentially localized Wannier functions in insulators was proved mathematically in 2006. ==Modern theory of polarization==

Wannier functions have recently found application in describing the polarization in crystals, for example, ferroelectrics. The modern theory of polarization is pioneered by Raffaele Resta and David Vanderbilt. See for example, Berghold, and Nakhmanson, and a power-point introduction by Vanderbilt. The polarization per unit cell in a solid can be defined as the dipole moment of the Wannier charge density: :\mathbf{p_c} = -e \sum_n \int\ d^3 r \,\, \mathbf{r} |W_n(\mathbf{r})|^2 \ , where the summation is over the occupied bands, and Wn 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. ==Wannier interpolation==

Wannier functions are often used to interpolate bandstructures calculated ab initio on a coarse grid of k-points to any arbitrary k-point. This is particularly useful for evaluation of Brillouin-zone integrals on dense grids and searching of Weyl points, and also taking derivatives in the k-space. This approach is similar in spirit to the tight binding approximation, but in contrast allows for an exact description of bands in a certain energy range. Wannier interpolation schemes have been derived for spectral properties, anomalous Hall conductivity, orbital magnetization, thermoelectric and electronic transport properties, gyrotropic effects, shift current, spin Hall conductivity and other effects. ==See also==