Category:Wannier functions: Difference between revisions

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

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

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

One-shot single value decomposition (SVD)

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

${\displaystyle A_{mn\mathbf{k}} = \langle \psi_{n\mathbf{k}} | S |\phi_{m\mathbf{k}}\rangle, }$

where

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

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

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

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

to obtain the unitary matrix

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

Selected columns of the density matrix (SCDM)

The SCDM method 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 is mainly controlled by LWANNIER90 and LWANNIER90_RUN. First, the initial guess for ${\displaystyle A_{mn\mathbf{k}}}$ 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, ${\displaystyle U_{mn\mathbf{k}}}$ is constructed in a second step. For this, ${\displaystyle A_{mn\mathbf{k}}}$ 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