Skip to content

Methodology

wannier90 computes maximally-localised Wannier functions (MLWF) following the method of Marzari and Vanderbilt (MV) 1. For entangled energy bands, the method of Souza, Marzari and Vanderbilt (SMV) 2 is used. We introduce briefly the methods and key definitions here, but full details can be found in the original papers and in Ref. 3.

First-principles codes typically solve the electronic structure of periodic materials in terms of Bloch states, \(\psi_{n{\bf k}}\). These extended states are characterised by a band index \(n\) and crystal momentum \({\bf k}\). An alternative representation can be given in terms of spatially localised functions known as Wannier functions (WF). The WF centred on a lattice site \({\bf R}\), \(w_{n{\bf R}}({\bf r})\), is written in terms of the set of Bloch states as

\[ \begin{equation} w_{n{\bf R}}({\bf r})=\frac{V}{(2\pi)^3}\int_{\mathrm{BZ}} \left[\sum_{m} U^{({\bf k})}_{mn} \psi_{m{\bf k}}({\bf r})\right]e^{-\mathrm{i}{\bf k}.{\bf R}} \:\mathrm{d}{\bf k} \ , \end{equation} \]

where \(V\) is the unit cell volume, the integral is over the Brillouin zone (BZ), and \(\mathbf{U}^{(\mathbf{k})}\) is a unitary matrix that mixes the Bloch states at each \({\bf k}\). \(\mathbf{U}^{(\mathbf{k})}\) is not uniquely defined and different choices will lead to WF with varying spatial localisations. We define the spread \(\Omega\) of the WF as

\[ \begin{equation} \Omega=\sum_n \left[\langle w_{n{\bf 0}}({\bf r})| r^2 | w_{n{\bf 0}}({\bf r}) \rangle - | \langle w_{n{\bf 0}}({\bf r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle |^2 \right]. \end{equation} \]

The total spread can be decomposed into a gauge invariant term \(\Omega_{\rm I}\) plus a term \({\tilde \Omega}\) that is dependant on the gauge choice \(\mathbf{U}^{(\mathbf{k})}\). \({\tilde \Omega}\) can be further divided into terms diagonal and off-diagonal in the WF basis, \(\Omega_{\rm D}\) and \(\Omega_{\rm OD}\),

\[ \begin{equation} \Omega=\Omega_{\rm I}+{\tilde \Omega}=\Omega_{\rm I}+\Omega_{\rm D}+\Omega_{\rm OD} \end{equation} \]

where

\[ \begin{equation} \Omega_{{\rm I}}=\sum_n \left[\langle w_{n{\bf 0}}({\bf r})| r^2 | w_{n{\bf 0}}({\bf r}) \rangle - \sum_{{\bf R}m} \left| \langle w_{m{\bf R}}({\bf r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle \right| ^2 \right] \end{equation} \]
\[ \begin{equation} \Omega_{\rm D}=\sum_n \sum_{{\bf R}\neq{\bf 0}} |\langle w_{n{\bf R}}({\bf r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle|^2 \end{equation} \]
\[ \begin{equation} \Omega_{\rm OD}=\sum_{m\neq n} \sum_{{\bf R}} |\langle w_{m{\bf R}}({\bf r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle |^2 \end{equation} \]

The MV method minimises the gauge dependent spread \(\tilde{\Omega}\) with respect the set of \(\mathbf{U}^{(\mathbf{k})}\) to obtain MLWF.

wannier90 requires two ingredients from an initial electronic structure calculation.

  1. The overlaps between the cell periodic part of the Bloch states \(|u_{n{\bf k}}\rangle\)

    \[ \begin{equation} \label{eq:overlap-matrix} M_{mn}^{(\bf{k,b})}=\langle u_{m{\bf k}}|u_{n{\bf k}+{\bf b}}\rangle, \end{equation} \]

    where the vectors \({\bf b}\), which connect a given k-point with its neighbours, are determined by wannier90 according to the prescription outlined in Ref. 1.

  2. As a starting guess the projection of the Bloch states \(|\psi_{n\bf{k}}\rangle\) onto trial localised orbitals \(|g_{n}\rangle\)

    \[ \begin{equation} A_{mn}^{(\bf{k})}=\langle \psi_{m{\bf k}}|g_{n}\rangle, \end{equation} \]

Note that \(\mathbf{M}^{(\mathbf{k},\mathbf{b})}\), \(\mathbf{A}^{(\mathbf{k})}\) and \(\mathbf{U}^{(\mathbf{k})}\) are all small, \(N \times N\) matrices (see the following note) that are independent of the basis set used to obtain the original Bloch states.

Note

Technically, this is true for the case of an isolated group of \(N\) bands from which we obtain \(N\) MLWF. When using the disentanglement procedure of Ref. 2, \(\mathbf{A}^{(\mathbf{k})}\), for example, is a rectangular matrix. See Section Entangled Energy Bands.

To date, wannier90 has been used in combination with electronic codes based on plane-waves and pseudopotentials (norm-conserving and ultrasoft 4) as well as mixed basis set techniques such as FLAPW 5.

Entangled Energy Bands

The above description is sufficient to obtain MLWF for an isolated set of bands, such as the valence states in an insulator. In order to obtain MLWF for entangled energy bands we use the "disentanglement" procedure introduced in Ref. 2.

We define an energy window (the "outer window"). At a given k-point \(\bf{k}\), \(N^{({\bf k})}_{{\rm win}}\) states lie within this energy window. We obtain a set of \(N\) Bloch states by performing a unitary transformation amongst the Bloch states which fall within the energy window at each k-point:

\[ \begin{equation} | u_{n{\bf k}}^{{\rm opt}}\rangle = \sum_{m\in N^{({\bf k})}_{{\rm win}}} U^{{\rm dis}({\bf k})}_{mn} | u_{m{\bf k}}\rangle \end{equation} \]

where \(\bf{U}^{{\rm dis}({\bf k})}\) is a rectangular \(N^{({\bf k})}_{{\rm win}} \times N\) matrix (see the following note). The set of \(\bf{U}^{{\rm dis}({\bf k})}\) are obtained by minimising the gauge invariant spread \(\Omega_{{\rm I}}\) within the outer energy window. The MV procedure can then be used to minimise \(\tilde{\Omega}\) and hence obtain MLWF for this optimal subspace.

Note

As \({\bf U}^{{\rm dis}({\bf k})}\) is a rectangular matrix this is a unitary operation in the sense that \(({\bf U}^{{\rm dis}({\bf k})})^{\dagger}{\bf U}^{{\rm dis}({\bf k})}={\bf 1}_N\).

It should be noted that the energy bands of this optimal subspace may not correspond to any of the original energy bands (due to mixing between states). In order to preserve exactly the properties of a system in a given energy range (e.g., around the Fermi level) we introduce a second energy window. States lying within this inner, or "frozen", energy window are included unchanged in the optimal subspace.

  1. N. Marzari and D. Vanderbilt. Maximally localized generalized wannier functions for composite energy bands. Phys. Rev. B, 56:12847, 1997. ↩↩

  2. I. Souza, N. Marzari, and D. Vanderbilt. Maximally localized wannier functions for entangled energy bands. Phys. Rev. B, 65:035109, 2001. ↩↩↩

  3. A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari. Wannier90: a tool for obtaining maximally-localised wannier functions. Comput. Phys. Commun., 178:685, 2008. ↩

  4. D. Vanderbilt. Phys. Rev. B, 41:7892, 1990. ↩

  5. M. Posternak, A. Baldereschi, S. Massidda, and N. Marzari. Maximally localized wannier functions in antiferromagnetic mno within the flapw formalism. Phys. Rev. B, 65:184422, 2002. ↩