1%!TEX root=./user_guide.tex 2\chapter{Methodology}\label{sec:method} 3\wannier\ computes maximally-localised Wannier functions (MLWF) 4following the method of Marzari and Vanderbilt 5(MV)~\cite{marzari-prb97}. For entangled energy bands, the method of 6Souza, Marzari and Vanderbilt (SMV)~\cite{souza-prb01} is used. We 7introduce briefly the methods and key definitions here, but full 8details can be found in the original papers and in 9Ref.~\cite{mostofi-cpc08}. 10 11First-principles codes typically solve the electronic structure of 12periodic materials in terms of Bloch states, $\psi_{n{\bf k}}$. 13These extended states are characterised by a band index $n$ and crystal 14momentum ${\bf k}$. An alternative representation can be given in terms 15of spatially localised functions known as Wannier functions (WF). The WF 16centred on a lattice site ${\bf R}$, $w_{n{\bf R}}({\bf r})$, 17is written in terms of the set of Bloch states as 18\begin{equation} 19w_{n{\bf R}}({\bf r})=\frac{V}{(2\pi)^3}\int_{\mathrm{BZ}} 20\left[\sum_{m} U^{({\bf k})}_{mn} \psi_{m{\bf k}}({\bf 21 r})\right]e^{-\mathrm{i}{\bf k}.{\bf R}} \:\mathrm{d}{\bf k} \ , 22\end{equation} 23where $V$ is the unit cell volume, the integral is over the Brillouin 24zone (BZ), and $\Uk$ is a unitary matrix that mixes the Bloch 25states at each ${\bf k}$. $\Uk$ is not uniquely defined and different 26choices will lead to WF with varying spatial localisations. We define 27the spread $\Omega$ of the WF as 28\begin{equation} 29\Omega=\sum_n \left[\langle w_{n{\bf 0}}({\bf r})| r^2 | w_{n{\bf 30 0}}({\bf r}) \rangle - | \langle w_{n{\bf 0}}({\bf r})| {\bf r} 31 | w_{n{\bf 0}}({\bf r}) \rangle |^2 \right]. 32\end{equation} 33The total spread can be decomposed into a gauge invariant term 34$\Omega_{\rm I}$ plus a term ${\tilde \Omega}$ that is dependant on the gauge 35choice $\Uk$. ${\tilde \Omega}$ can 36be further divided into terms diagonal and off-diagonal in the WF basis, 37$\Omega_{\rm D}$ and $\Omega_{\rm OD}$, 38\begin{equation} 39\Omega=\Omega_{\rm I}+{\tilde \Omega}=\Omega_{\rm I}+\Omega_{\rm 40 D}+\Omega_{\rm OD} 41\end{equation} 42where 43\begin{equation} 44\Omega_{{\rm I}}=\sum_n \left[\langle w_{n{\bf 0}}({\bf r})| r^2 | w_{n{\bf 45 0}}({\bf r}) \rangle - \sum_{{\bf R}m} \left| \langle w_{n{\bf 46 R}}({\bf r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle \right| ^2 47 \right] 48\end{equation} 49\begin{equation} 50\Omega_{\rm D}=\sum_n \sum_{{\bf R}\neq{\bf 0}} |\langle w_{n{\bf 51 R}}({\bf r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle|^2 52\end{equation} 53\begin{equation} 54\Omega_{\rm OD}=\sum_{m\neq n} \sum_{{\bf R}} |\langle w_{m{\bf R}}({\bf 55 r})| {\bf r} | w_{n{\bf 0}}({\bf r}) \rangle |^2 56\end{equation} 57The MV method minimises the gauge dependent spread $\tilde{\Omega}$ 58with respect the set of $\Uk$ to obtain MLWF. 59 60\wannier\ requires two ingredients from an initial electronic 61structure calculation. 62\begin{enumerate} 63\item The overlaps between the cell periodic part of the Bloch states 64 $|u_{n{\bf k}}\rangle$ 65\begin{equation} 66M_{mn}^{(\bf{k,b})}=\langle u_{m{\bf k}}|u_{n{\bf k}+{\bf b}}\rangle, 67\end{equation} 68where the vectors ${\bf b}$, which connect a given k-point with its 69neighbours, are determined by \wannier\ according to the prescription 70outlined in Ref.~\cite{marzari-prb97}. 71\item As a starting guess the projection of the Bloch states 72 $|\psi_{n\bf{k}}\rangle$ onto trial localised orbitals $|g_{n}\rangle$ 73\begin{equation} 74A_{mn}^{(\bf{k})}=\langle \psi_{m{\bf k}}|g_{n}\rangle, 75\end{equation} 76\end{enumerate} 77Note that $\Mkb$, $\Ak$ and $\Uk$ are all small, $N \times N$ 78matrices\footnote{Technically, this is true for the case of an 79 isolated group of $N$ bands from which we obtain $N$ MLWF. When 80 using the disentanglement procedure of Ref.~\cite{souza-prb01}, 81 $\Ak$, for example, is a rectangular matrix. See 82 Section~\ref{sec:disentangle}.} that are independent of the basis 83set used to obtain the original Bloch states. 84 85To date, \wannier\ has been used in combination with electronic codes 86based on plane-waves and pseudopotentials (norm-conserving and 87ultrasoft~\cite{vanderbilt-prb90}) as well as mixed basis set techniques such as 88FLAPW~\cite{posternak-prb02}. 89 90\section{Entangled Energy Bands}\label{sec:disentangle} 91The above description is sufficient to obtain MLWF for an isolated set 92of bands, such as the valence states in an insulator. In order to 93obtain MLWF for entangled energy bands we use the ``disentanglement'' 94procedure introduced in Ref.~\cite{souza-prb01}. 95 96We define an energy window (the ``outer window''). At a given 97k-point $\bf{k}$, $N^{({\bf k})}_{{\rm win}}$ states lie within this 98energy window. We obtain a set of $N$ Bloch states by 99performing a unitary transformation amongst the Bloch states which 100fall within the energy window at each k-point: 101 \begin{equation} 102| u_{n{\bf k}}^{{\rm opt}}\rangle = \sum_{m\in N^{({\bf k})}_{{\rm win}}} 103U^{{\rm dis}({\bf k})}_{mn} | u_{m{\bf k}}\rangle 104\end{equation} 105where $\bf{U}^{{\rm dis}({\bf k})}$ is a rectangular $N^{({\bf k})}_{{\rm win}} \times N$ 106 matrix\footnote{As ${\bf U}^{{\rm dis}({\bf k})}$ is a rectangular 107 matrix this is a unitary operation in the sense that $({\bf U}^{{\rm 108 dis}({\bf k})})^{\dagger}{\bf U}^{{\rm dis}({\bf k})}={\bf 1}_N$.}. The 109 set of $\bf{U}^{{\rm dis}({\bf k})}$ are obtained by minimising 110 the gauge invariant spread $\Omega_{{\rm I}}$ within the outer energy 111 window. The MV procedure can then be used to minimise $\tilde{\Omega}$ 112 and hence obtain MLWF for this optimal subspace. 113 114It should be noted that the energy bands of this optimal subspace may 115not correspond to any of the original energy bands (due to mixing 116between states). In order to preserve exactly the properties of a 117system in a given energy range (e.g., around the Fermi level) we 118introduce a second energy window. States lying within this inner, or 119``frozen'', energy window are included unchanged in the optimal 120subspace. 121