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 built from ${\displaystyle A_{mn\mathbf{k}}}$. The latter can be built from projections onto some localized-orbital basis.

Comprehensive instructions on how to construct Wannier orbitals in VASP can be found here.

One-shot singular-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 ^[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 ${\displaystyle A_{mn\mathbf{k}}}$ 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