Category:Wannier functions

Wannier functions [math]\displaystyle{ |w_{m\mathbf{R}}\rangle }[/math] are constructed by a linear combination of Bloch states [math]\displaystyle{ |\psi_{n\mathbf{k}}\rangle }[/math], i.e., the computed Kohn-Sham (KS) orbitals, as follows:

[math]\displaystyle{ |w_{m\mathbf{R}}\rangle = \sum_{n\mathbf{k}} e^{-i\mathbf{k}\cdot\mathbf{R}} U_{mn\mathbf{k}} |\psi_{n\mathbf{k}}\rangle. }[/math]

Here, [math]\displaystyle{ U_{mn\mathbf{k}} }[/math] is a unitary matrix which can be generated using different approaches discussed below, [math]\displaystyle{ m }[/math] is an index enumerating Wannier functions with position [math]\displaystyle{ \mathbf{R} }[/math], [math]\displaystyle{ n }[/math] is the band index, and [math]\displaystyle{ \mathbf{k} }[/math] is the Bloch vector. Generally, one starts with an initial guess for [math]\displaystyle{ U_{mn\mathbf{k}} }[/math] that is built from [math]\displaystyle{ A_{mn\mathbf{k}} }[/math]. The latter can be built from projections onto some localized-orbital basis.

• Comprehensive instructions on how to construct Wannier orbitals.

One-shot singular-value decomposition (SVD)

In one-shot SVD, [math]\displaystyle{ A_{mn\mathbf{k}} }[/math] is computed by projecting the KS orbitals onto localized orbitals basis [math]\displaystyle{ \phi_{m\mathbf{k}} }[/math] that is specified by the LOCPROJ tag:

[math]\displaystyle{ A_{mn\mathbf{k}} = \langle \psi_{n\mathbf{k}} | S |\phi_{m\mathbf{k}}\rangle, }[/math]

where

[math]\displaystyle{ \phi_{i\mathbf{k}}(\mathbf{r}) = e^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}} Y_{lm}(\hat{r})R_n(r). }[/math]

Note that [math]\displaystyle{ i }[/math] encodes the quantum numbers [math]\displaystyle{ n }[/math], [math]\displaystyle{ l }[/math], and [math]\displaystyle{ m }[/math]. Thus, in [math]\displaystyle{ A_{mn\mathbf{k}} }[/math], [math]\displaystyle{ m }[/math] is not the magnetic quantum number.

Then, VASP performs one-shot SVD for each k point

[math]\displaystyle{ A_{mn\mathbf{k}} = [D \Sigma V^*]_{mn\mathbf{k}} }[/math]

to obtain the unitary matrix

[math]\displaystyle{ U_{mn\mathbf{k}} = [DV^*]_{mn\mathbf{k}}. }[/math]

Selected columns of the density matrix (SCDM)

The SCDM method ^[1] is switched on using LSCDM. It has the advantage that the specification of a local basis in terms of atomic quantum numbers is omitted.

Maximally localized Wannier functions using Wannier90

The interface of VASP with the Wannier90 code^[2][3] is mainly controlled by LWANNIER90 and LWANNIER90_RUN. First, the initial guess for [math]\displaystyle{ A_{mn\mathbf{k}} }[/math] can be created by providing the projections block in the wannier90.win file (also see WANNIER90_WIN) and setting LWANNIER90=True.

In order to obtain maximally localized Wannier functions, [math]\displaystyle{ U_{mn\mathbf{k}} }[/math] is constructed in a second step. For this, [math]\displaystyle{ A_{mn\mathbf{k}} }[/math] could be created using any projection method in the first step, i.e., single-shot SVD method (LOCPROJ), SCDM method (LSCDM), or Wannier90 (LWANNIER90). Then, Wannier90 can be executed directly or through VASP with the LWANNIER90_RUN tag.

References