Definition

Wannier functions (WFs) are a dual representation of Bloch wavefunctions from the periodic potential problem. They are defined as Fourier transform’s of Bloch wavefunctions (i.e. a linear combination with plane-wave weights)

\[ {w_{n\mathbf{R}_j}}=\frac{1}{N}\sum_{\mathbf{k}}e^{-i\mathbf{k}\cdot\mathbf{R}_j}{\psi_{n\mathbf{k}}}, \]

where \(\mathbf{R}_j\) is the lattice vector, \(\mathbf{k}\) the crystal momentum, and \(n\) the band index; we assume a discrete \(\mathbf{k}\)-point mesh, and thus \(N\) represents the number of mesh points in the Brillouin Zone.

\[ {w_{n\mathbf{R}_j}}=\frac{1}{N}\sum_{\mathbf{k}} \]