Category:Wannier functions

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

${\displaystyle |w_{m\mathbf{𝐑}}⟩=\sum _{n\mathbf{𝐤}}{e}^{-i\mathbf{𝐤}\cdot \mathbf{𝐑}}{U}_{mn\mathbf{𝐤}}|{\psi }_{n\mathbf{𝐤}}⟩.}$

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.

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 ${\displaystyle {\varphi }_{m\mathbf{𝐤}}}$ that is specified by the LOCPROJ tag:

${\displaystyle A_{mn\mathbf{𝐤}}=⟨{\psi }_{n\mathbf{𝐤}}|S|{\varphi }_{m\mathbf{𝐤}}⟩,}$

where

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

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

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

to obtain the unitary matrix

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

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