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\begin_layout Description
Wannier functions with 
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\begin_layout Description
Klaus Koepernik and Marc H
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ppner
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chapter{Wannier functions with 
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hypertarget{WFs}{}
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label{sec:WannierFunctions}
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{ co-author Marc H{
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textsc{Wannier Functions with 
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 {
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\begin_layout Section
Intro
\end_layout

\begin_layout Standard
Wannier functions can be defined in many ways since there is a gauge freedom
 of choosing a phase transformation.
 One way of fixing the gauge is the requirement of maximum localization
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "Mar97"
literal "true"

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 (ignoring the constraints of the point group).
 This is a tedious algorithm and even more complicated, if one adds point
 group symmetry
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citep
key "Sak13,Run14"
literal "true"

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.
 We do not do this in 
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.
 However, it turns out that our definition
\begin_inset space ~
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\begin_inset CommandInset citation
LatexCommand citep
key "Esch09,Wei02"
literal "true"

\end_inset

 leads to highly localized Wannier functions (WFs) which obey the symmetry
 relations of the original crystal by construction.
 If the WFs turn out to be not well-localized, then in most cases the WF
 is badly chosen.
 The main drawback of our approach is that the user has to decide, where
 the WFs shall sit and which symmetry they shall have.
 However, this is at the same time intended, since for modeling it is exactly
 what one wants to do.
\end_layout

\begin_layout Standard
The WF sitting in cell 
\begin_inset Formula $R$
\end_inset

 and being of type 
\begin_inset Formula $\mu$
\end_inset

 (which denotes the WF center and its symmetry) is defined as
\begin_inset Formula 
\begin{equation}
W_{R\mu}=\int\mathrm{e}^{-ikR}\sum_{n}\Psi_{n}^{k}U_{n\mu}^{k}\label{eq:def_WF}
\end{equation}

\end_inset

where 
\begin_inset Formula $\Psi$
\end_inset

 denotes the Kohns-Sham (KS) functions and 
\begin_inset Formula $U$
\end_inset

 is a unitary matrix.
 If we define less WFs than KS functions, 
\begin_inset Formula $U$
\end_inset

 is a column unitary projector (
\begin_inset Formula $U^{+}U=1$
\end_inset

, 
\begin_inset Formula $UU^{+}=P$
\end_inset

).
 It maps all the KS bands on some few WFs.
 The choice of 
\begin_inset Formula $U$
\end_inset

 is the choice of the gauge.
 In 
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 we use a chemically motivated local orbital basis 
\begin_inset Formula $\Phi$
\end_inset

 to construct KS functions 
\begin_inset Formula 
\[
\Psi_{n}^{k}=\frac{1}{\sqrt{N}}\sum_{Rs\nu}\Phi_{Rs\nu}\mathrm{e}^{ik\left(R+s\right)}C_{s\nu,n}^{k}
\]

\end_inset

(
\begin_inset Formula $s$
\end_inset

 is the atom position and 
\begin_inset Formula $\nu$
\end_inset

 some quantum numbers specifying the orbital).
 These orbitals although non-orthogonal are in a way 'optimally' localized
 by their construction.
 Hence, it is clear that a WF centered at an atom and having a certain orbital
 symmetry has the corresponding orbital as its main contribution.
 This allows the following choice of 
\begin_inset Formula $U$
\end_inset

.
 We project the KS functions on a test function 
\begin_inset Formula $\chi$
\end_inset

, which is an 
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 orbital in the simplest case.
 The resulting number is the square root of the orbital character of the
 KS bands, as plotted in the FAT bands.
 If we now select only the KS functions with a large such orbital character
 in Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:def_WF"

\end_inset

), we will end up with a WF resembling 
\begin_inset Formula $\chi$
\end_inset

 the most and this is where the localization of our WFs comes from.
 If we want WFs corresponding to sub bands of a band complex with a character
 
\begin_inset Formula $\chi$
\end_inset

 we have to project onto a particular energy window as well.
 This happens if there are bonding (B) and anti-bonding (AB) bands of character
 
\begin_inset Formula $\chi$
\end_inset

 and if one wants, say, only WFs of the AB bands.
 So, in total we define a test function or WF projector 
\begin_inset Formula $\chi$
\end_inset

 and an energy window for each WF.
 The WF projector 
\begin_inset Formula $\chi$
\end_inset

 can in principle be a linear combination of 
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 orbitals.
\end_layout

\begin_layout Standard

\series bold
Example
\series default
: suppose we have a Cuprate plane.
 The bands are formed by a linear combinations of Cu
\begin_inset space ~
\end_inset


\begin_inset Formula $3d$
\end_inset

 and ligand O
\begin_inset space ~
\end_inset


\begin_inset Formula $2p$
\end_inset

 orbitals.
 Around certain 
\begin_inset Formula $k$
\end_inset

-points, the upper bands are clearly AB and hence correspond to a certain
 molecular orbital with a certain phase relation between the central 
\begin_inset Formula $3d$
\end_inset


\begin_inset space ~
\end_inset

orbital and the O
\begin_inset space ~
\end_inset


\begin_inset Formula $2p$
\end_inset

 orbitals.
 The lower (B) bands are clearly formed of the same orbitals, but with a
 different phase relation.
 If this is the case, then defining WF-projectors using the AB molecular
 orbital as 
\begin_inset Formula $\chi$
\end_inset

 will automatically yield the anti-bonding bands, given that the band topology
 is dominated by the clear character separation around the considered k-point.
 This would make the energy window obsolete.
\end_layout

\begin_layout Standard
The Wannier transformation can be described in two steps.
 In the first step the unitary projector is build from the users definitions
 of 
\begin_inset Formula $\chi$
\end_inset

 and the energy windows.
 
\begin_inset Formula $U$
\end_inset

 is applied to the KS functions yielding the Bloch sums of the WFs
\begin_inset Formula 
\begin{eqnarray}
W_{\mu}^{k}\left(r\right) & = & \sum_{n}\Psi_{n}^{k}\left(r\right)U_{n\mu}^{k}\label{eq:WF-Bloch-sums}\\
 & = & \frac{1}{\sqrt{N}}\sum_{R}\mathrm{e}^{ikR}W_{R\mu}\left(r\right)\nonumber 
\end{eqnarray}

\end_inset


\begin_inset Formula 
\begin{equation}
W_{R\mu}\left(r\right)=\int\mathrm{e}^{-ikR}W_{\mu}^{k}\left(r\right)dk\label{eq:WF-Bloch-sum-Fouriertrans}
\end{equation}

\end_inset

(Note, that the 
\begin_inset Formula $k$
\end_inset

-integration contains appropriate normalization factors, left out in the
 formulas or being hidden in the integral measure.) The orbital Bloch functions
 are
\begin_inset Formula 
\[
\Phi_{s\mu}^{k}=\frac{1}{\sqrt{N}}\sum_{R}\Phi_{Rs\mu}\mathrm{e}^{ik\left(R+s\right)}
\]

\end_inset

where the additional phase factor 
\begin_inset Formula $\mathrm{e}^{iks}$
\end_inset

 makes live easier by removing the dependence on the coordinate origin.
 From the formulas we get the representation of the Hamiltonian in orbital
 Bloch sums
\begin_inset Formula 
\begin{eqnarray*}
\underline{H}_{s^{\prime}s}^{k} & = & \left\langle \Phi_{s^{\prime}}^{k}\hat{H}\Phi_{s}^{k}\right\rangle \\
 & = & \frac{1}{N}\sum_{R^{\prime}R}\left\langle \Phi\hat{H}\Phi\right\rangle _{R^{\prime}s^{\prime}Rs}\mathrm{e}^{ik\left(R+s-R^{\prime}-s^{\prime}\right)}\\
 & = & \sum_{R}\left\langle \Phi\hat{H}\Phi\right\rangle _{0s^{\prime},Rs}\mathrm{e}^{ik\left(R+s-s^{\prime}\right)}
\end{eqnarray*}

\end_inset

from which we get the KS eigenvalues
\begin_inset Formula 
\begin{equation}
\varepsilon_{n}^{k}=\left(C^{k+}\underline{H}^{k}C^{k}\right)_{n}\label{eq:KS-bandstructure}
\end{equation}

\end_inset

Using the result above, the WF Bloch representation is
\begin_inset Formula 
\begin{eqnarray}
\left\langle W_{\mu^{\prime}}^{q}\hat{H}W_{\mu}^{k}\right\rangle  & = & \sum_{n}U_{n\mu^{\prime}}^{q*}\varepsilon_{n}^{k}U_{n\mu}^{k}\delta_{qk}\label{eq:WF-Bloch_Ham}\\
 & = & \frac{1}{N}\sum_{RP}\left\langle W_{P\mu^{\prime}}\hat{H}W_{R\mu}\right\rangle \mathrm{e}^{ik\left(R-P\right)}\delta_{qk}\nonumber \\
 & = & \sum_{R}\left\langle W_{0\mu^{\prime}}\hat{H}W_{R\mu}\right\rangle \mathrm{e}^{ikR}\delta_{qk}\nonumber 
\end{eqnarray}

\end_inset

with 
\begin_inset Formula 
\begin{eqnarray}
\varepsilon_{0\mu^{\prime},R\mu} & = & \left\langle W_{0\mu^{\prime}}\hat{H}W_{R\mu}\right\rangle \label{eq:WF-hoppings}\\
 & = & \int\mathrm{e}^{-ikR}\left\langle W_{\mu^{\prime}}^{k}\hat{H}W_{\mu}^{k}\right\rangle dk\nonumber 
\end{eqnarray}

\end_inset

which is the WF Hamiltonian in real space representation, which usually
 contains the model we are interested in.
 Its 
\begin_inset Formula $k$
\end_inset

-representation (Bloch sums) Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch_Ham"

\end_inset

) can be diagonalized and will give the bandstructure corresponding to the
 model.
 If the WFs represent the whole Hilbert space spanned by all 
\begin_inset Formula $\Psi_{n}^{k}$
\end_inset

 of course the resulting WF-bandstructure coincides with the original bandstruct
ure Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:KS-bandstructure"

\end_inset

).
 There is a modification one can do, which consists of restricting the matrix
 elements Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-hoppings"

\end_inset

) by removing hoppings with distances above a certain cutoff radius or hoppings,
 which are smaller than a certain threshold.
 The resulting restricted hopping matrix can be Bloch summed and diagonalized
 again.
 Note, however, that this modified Hamiltonian does not strictly correspond
 to the WFs calculated above.
\end_layout

\begin_layout Standard
The whole transformation can be summed up, by defining a transformation
 from the 
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\end_inset

 basis into the WF basis
\begin_inset Formula 
\[
W_{R\mu}\left(r\right)=W_{0\mu}\left(r-R\right)
\]

\end_inset


\begin_inset Formula 
\begin{equation}
W_{0\mu}\left(r\right)=\sum_{R^{\prime}s\nu}\Phi_{R^{\prime}s\nu}D_{R^{\prime}s\nu,\mu}\label{eq:WF-from-Phi}
\end{equation}

\end_inset


\end_layout

\begin_layout Section
The 
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 WF module
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\begin_layout Standard
The Wannier function module in 
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 is currently a postprocessing tool.
 First, one needs a converged calculation.
 Then, the relevant information must be written to the hard disk in order
 to access it conveniently later.
 This might take plenty of disk space (see Sec.
 
\begin_inset CommandInset ref
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reference "subsec:Saving-memory"

\end_inset

)! This information is read afterwards and used in a subsequent 
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 run to calculate the desired Wannier functions.
 In order to use the module a particular file (
\family typewriter
=.wandef
\family default
) must be created by the user.
 Since 
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15 a python script can by used to create default Wannier function definitions
 for a large set of situations (
\begin_inset Flex URL
status open

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../pyfplo/pyfplo.pdf
\end_layout

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).
 If 
\family typewriter
=.wandef
\family default
 is found by a running 
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 process the Wannier module is activated, if the keyword 
\series bold
doit
\series default
 is found on a single line in the file.
 To disable the module change the keyword into something like e.g.
 
\series bold
xdoit
\series default
.
 If 
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 finds the file 
\family typewriter
=.wandef
\family default
 with the keyword it will start dumping data after every Kohn-Sham diagonalizati
on process.
 If the 
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 process comes to convergence (best to start with a converged calculation
 to begin with) it stops like in normal mode.
 The file created after the initial data-dumping is 
\family typewriter
+wancoeff
\family default
 (besides all the usual files).
 
\shape italic
Note
\shape default
: the band structure plot in the fedit submenu should be switched on.
\end_layout

\begin_layout Standard
On restart of 
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 (provided that 
\family typewriter
=.wandef
\family default
 exists and the 
\series bold
doit
\series default
-keyword is set therein) all the data are used and the actual WF-module
 is executed.
 It reads the WF definitions from 
\family typewriter
=.wandef
\family default
 and constructs the WFs accordingly.
 The WFs can be produced in real space for visualization and the (interpolated)
 WF Hamiltonian in Bloch representation Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch_Ham"

\end_inset

) can be written to the file 
\family typewriter
+hamongrid
\family default
 on a (much denser) 
\begin_inset Formula $k$
\end_inset

-space grid.
 The WF hopping integrals Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-hoppings"

\end_inset

) are produced in the output and are written in a convenient format into
 the file 
\family typewriter
+hamdata
\family default
, which is used e.g.
 by the 
\family typewriter
pyfplo.slabify
\family default
 module (also see 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

../pyfplo/pyfplo.pdf
\end_layout

\end_inset

).
 Be aware that the WF Hamiltonian on the 
\begin_inset Formula $k$
\end_inset

-space grid is restricted by the cutoff of the real-space Hamiltonian Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-hoppings"

\end_inset

) according to user input.
\end_layout

\begin_layout Standard

\series bold
Example:
\series default
 We will walk through the example of CaCuO
\begin_inset Formula $_{2}$
\end_inset

 step by step.
 The input-files are provided in the example directory.
 The calculation is done in the ferromagnetic phase of CaCuO
\begin_inset Formula $_{2}$
\end_inset

 (just to have a more complex situation).
 The goal is to create a WF for the anti-bonding 
\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 band.
 In Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:FPLO-fat-bands-CaCuO"

\end_inset

 we show the spin-polarized
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\end_inset


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 fat-bands CaCuO
\begin_inset Formula $_{2}$
\end_inset


\end_layout

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fat-bands as given by 
\begin_inset ERT
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.
 The filled symbols denote the 
\begin_inset Formula $3d$
\end_inset


\begin_inset space ~
\end_inset

majority fat-bands and the open symbols the 
\begin_inset Formula $3d$
\end_inset


\begin_inset space ~
\end_inset

minority bands.
 The 
\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 bands are called 
\begin_inset Formula $3d+2$
\end_inset

 according to the standard convention and orientation of the global coordinate
 system and have magenta color.
 Focusing on the majority bands (filled symbols), you see a bonding and
 anti-bonding band (best seen at the M-point).
 We are interested in a description of the anti-bonding band.
 There is only one Cu atom in the unit cell and according to the list of
 sites in the output file it is site number 2.
 We now guess that the Cu, site
\begin_inset space ~
\end_inset

2, 
\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 orbital will have the largest contribution to a WF describing the anti-bonding
 part of the filled magenta band(s).
 Thus, we choose this orbital as a WF-projector 
\begin_inset Formula $\chi$
\end_inset

.
 We assume that the file 
\family typewriter
=.wandef
\family default
 exists and that the keyword 
\series bold
doit
\series default
 is set and that the file 
\family typewriter
+wancoeff
\family default
 got already created.
\end_layout

\begin_layout Standard
We have to define, which linear combinations of local orbitals are used
 as WF-projectors.
 Note that we have two basic choices: the WF-projector 
\shape italic
and
\shape default
 the energy window.
 In our case of CaCuO
\begin_inset Formula $_{2}$
\end_inset

 we already identified the projector.
 The simplest projector consists of one single orbital centered on site
\begin_inset space ~
\end_inset

2.
 A WF-projector is defined using the keyword 
\series bold
wandef
\series default
.
 A 
\series bold
wandef
\series default
 can have many contributions 
\series bold
contrib
\series default
, each denoting an 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 orbital with a certain weight factor.
 Here comes the example
\end_layout

\begin_layout Standard
\begin_inset listings
lstparams "basicstyle={\ttfamily}"
inline false
status open

\begin_layout Plain Layout

...
\end_layout

\begin_layout Plain Layout

-------- Cu --------------
\end_layout

\begin_layout Plain Layout

wandef
\end_layout

\begin_layout Plain Layout

on
\end_layout

\begin_layout Plain Layout

   name Cu x2-y2
\end_layout

\begin_layout Plain Layout

   emin -4 -1
\end_layout

\begin_layout Plain Layout

   emax -1 3
\end_layout

\begin_layout Plain Layout

   de 1 1
\end_layout

\begin_layout Plain Layout

   contrib
\end_layout

\begin_layout Plain Layout

     site 2
\end_layout

\begin_layout Plain Layout

     difvec 0 0 0
\end_layout

\begin_layout Plain Layout

     xaxis 1 0 0
\end_layout

\begin_layout Plain Layout

     zaxis 0 0 1
\end_layout

\begin_layout Plain Layout
\noindent

     orb 3d+2
\end_layout

\begin_layout Plain Layout

     fac 1
\end_layout

\end_inset


\end_layout

\begin_layout Standard
As can be seen there can be additional lines in the file, and they are considere
d comments as long as they do not start with a keyword.
 In our case 
\family typewriter
'--------- Cu ----------------'
\family default
 is such a comment.
 Subsequently, the keyword 
\series bold
wandef
\series default
 starts a WF-projector section.
 The next keyword is 
\series bold
on
\series default
 or 
\series bold
off
\series default
.
 This allows to have several 
\series bold
wandef
\series default
s in one file, where only some are used (playing around).
 The 
\series bold
wandef
\series default
 has a name, which can be anything after the keyword 
\series bold
name
\series default
 (actually it should not be more than 17 characters).
 The energy window is discussed later.
 In our example the 
\series bold
wandef
\series default
 is made of one 
\series bold
contrib
\series default
.
 The 
\series bold
contrib
\series default
 is an orbital sitting at 
\series bold
site
\series default

\begin_inset space ~
\end_inset

2, being 
\series bold
difvec
\series default

\begin_inset space ~
\end_inset

0
\begin_inset space ~
\end_inset

0
\begin_inset space ~
\end_inset

0 away from the (imagined) center point of the WF, which in this case is
 the position of site
\begin_inset space ~
\end_inset

2.
 The orbital is defined with respect to the local coordinate system, whose
 
\series bold
xaxis
\series default
 is 1
\begin_inset space ~
\end_inset

0
\begin_inset space ~
\end_inset

0 and 
\series bold
zaxis
\series default
 is 0
\begin_inset space ~
\end_inset

0
\begin_inset space ~
\end_inset

1.
 In this local system the 
\series bold
orb
\series default
 is 3d+2 and the 
\series bold
contrib
\series default
 enters the whole 
\series bold
wandef
\series default
 with relative weight 
\series bold
fac
\series default

\begin_inset space ~
\end_inset

1.
 If there are more than one 
\series bold
contrib
\series default
s the 
\series bold
fac
\series default
 of each 
\series bold
contrib
\series default
 can be given.
 The whole thing is normalized automatically.
 If the 
\series bold
wandef
\series default
 is made of several 
\series bold
contrib
\series default
s at different sites it is important to correctly define the difference
 vectors 
\series bold
difvec
\series default
.
 Choose a particular point in space as WF center and express the sites of
 the 
\series bold
contrib
\series default
s by vectors pointing from the WF center to the sites \SpecialChar ldots
 these are the 
\series bold
difvec
\series default
s.
\end_layout

\begin_layout Standard
If there is more than one site belonging to a crystallographic orbit (all
 sites generated by the same Wyckoff position), the user has to give every
 WF-projector separately in the current implementation.
 In this case it is important that the user takes care of all the different
 symmetry-related WF-projectors belonging to the same cryst.
 orbit.
 This means setting up proper 
\series bold
difvec
\series default
, 
\series bold
x/zaxis
\series default
, 
\series bold
orb
\series default
, 
\series bold
fac
\series default
 and so on values.
\end_layout

\begin_layout Standard
In our example the orbital used as WF-projector forms a bonding and an anti-bond
ing part.
 There is only one Cu 
\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 orbital in the simple unit cell and hence it can only form one band.
 However, this orbital forms sigma-bonding hybrids with the plaquette oxygen
 
\begin_inset Formula $2p$
\end_inset


\begin_inset space ~
\end_inset

orbitals and this creates 
\begin_inset Quotes eld
\end_inset

two
\begin_inset Quotes erd
\end_inset

 bands having a shared character of Cu
\begin_inset space ~
\end_inset


\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 and O
\begin_inset space ~
\end_inset


\begin_inset Formula $2p$
\end_inset

.
 Moreover, it is clear from Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:FPLO-fat-bands-CaCuO"

\end_inset

 that the anti-bonding (AB) band is not isolated due to other hybridizations.
 Hence, a well defined Wannier function cannot exist.
 In order to have some approximation for modeling we create a single WF
 of AB type.
 According to the rules of how WFs are created it is evident that we need
 to project away the part of the KS spectrum, which contains the bonding
 part of the corresponding band.
 This is done by specifying an energy window.
 All bands outside this window will be ignored.
 For the majority bands we have the AB band between -5 and 0 eV and the
 B band between -8 and -5 eV.
 For the minority band it is shifted upwards accordingly.
 The energy window could be defined as a sharp cutoff.
 This can lead to weird effects, if the band character specified through
 the WF-projector occurs ONLY outside this window for a particular k-point
 (such things can actually happen).
 This would mean that for one k-point the weight of the WF-projector in
 the considered Hilbert-subspace is zero and this leads to an indefinite
 problem, when calculating the WFs.
 So, it is better to make the window smooth in order to have the interesting
 subspace in the main energy window but allowing to sample outside of it,
 in case we have a character-run-away scenario.
 The energy window is defined by a function being 1 between 
\series bold
emin
\series default
 and 
\series bold
emax
\series default
 and falling of as a Gaussian with a width 
\series bold
de
\series default
 outside this window.
 Of course, the energy window applies to all 
\series bold
contrib
\series default
s of one 
\series bold
wandef
\series default
.
 In spin-polarized cases these three keywords take two values, one for the
 majority and one for the minority bands.
 The user should also pay attention that the energy window has to be the
 same for all 
\series bold
wandef
\series default
s belonging to the same cryst.
 orbit! 
\end_layout

\begin_layout Standard
Suppose that we defined the above described WF projector onto one orbital,
 having one WF per unit cell and that we did not specify the energy window.
 This would lead to a WF whose corresponding band will be pretty dispersionless
 situated at an energy between the B and AB bands in the original band structure.
 This happens because the projector 
\begin_inset Formula $\chi$
\end_inset

 makes one single band out of the B and AB part of the KS-Hilbert space,
 which essentially forms two bands.
 In mathematical terms this will lead to an average of the B and AB KS wavefunct
ions, which will have non-bonding character.
 That should also make clear what happens, if we extend the energy window
 more and more towards the B energies.
 This will mix in more and more of the bonding KS functions, which leads
 to a WF band whose energies get pulled down more and more.
 The user is encouraged to play with these values to get a feeling for the
 construction of specific WFs.
\end_layout

\begin_layout Standard
Now, we have defined all the stuff in the file 
\family typewriter
=.wandef
\family default
 and we can run 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

.
\end_layout

\begin_layout Enumerate
The first thing in the WF module will be that the content of 
\family typewriter
=.wandef
\family default
 is copied to the output, followed by a more condensed printout of the Wannier
 parameters.
 The latter output shows all parameters, also the ones not explicitly set
 in 
\family typewriter
=.wandef
\family default
 (which will have their default values).
 Then follows a section, in which the symmetry of the WF-projectors 
\begin_inset Formula $\chi$
\end_inset

 is checked.
 If this check does not run through properly there is a mistake in the symmetry
 relation between 
\series bold
wandef
\series default
s of 
\begin_inset Formula $\chi$
\end_inset

-s belonging to the same cryst.
 orbit.
 Check all keywords.
 In the moment the energy window is not checked for proper symmetry setting.
 So, if the code runs through, but the output seems weird, check the energy
 windows!
\end_layout

\begin_layout Enumerate
Now, if the file 
\family typewriter
+wancoeff
\family default
 is found it will be loaded.
 If not, the normal 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 execution will continue.
 Reading the data is not the fastest that is why there is an internal loop
 (see below).
 See also Sec.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Saving-memory"

\end_inset

.
\end_layout

\begin_layout Enumerate
Now, the Wannier bands get processed.
 As a result the hopping matrix elements between WFs are printed to the
 output.
 This printout is controlled by user defined restrictions to avoid enormous
 amounts of data.
 Example:
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset listings
lstparams "basicstyle={\ttfamily}"
inline false
status open

\begin_layout Plain Layout

spin 1: WF(Cu x2-y2) -> WF(Cu x2-y2) at relative 
\end_layout

\begin_layout Plain Layout

T=  0.00000  0.00000  0.00000 hop= -2.093458893797475
\end_layout

\begin_layout Plain Layout

T=  0.00000  0.00000  6.04712 hop= -0.061341124468540
\end_layout

\begin_layout Plain Layout

T=  0.00000  0.00000 -6.04712 hop= -0.061341124468540
\end_layout

\begin_layout Plain Layout

T= -7.29434  0.00000  0.00000 hop= -0.463606088774015
\end_layout

\begin_layout Plain Layout

T=  0.00000 -7.29434  0.00000 hop= -0.463606088774014
\end_layout

\begin_layout Plain Layout

T=  0.00000  7.29434  0.00000 hop= -0.463606088774014
\end_layout

\begin_layout Plain Layout

T=  7.29434  0.00000  0.00000 hop= -0.463606088774015
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The WFs are given by their name, and the symbol '
\family typewriter
->
\family default
' means that the vectors, which follow ('
\family typewriter
T= \SpecialChar ldots

\family default
') point from the WF left of '
\family typewriter
->
\family default
' to the one right of '
\family typewriter
->
\family default
'.
 After each vector '
\family typewriter
T=
\family default
' the hopping element in eV is printed.
 If 
\family typewriter
T=0
\family default
 the hopping element is the onsite matrix element.
 The information written to the output is restricted such that only hoppings
 with 
\begin_inset Formula $\left|t\right|>$
\end_inset


\series bold
WF_ham_threshold
\series default
 are considered and only for 
\begin_inset Formula $\left|T\right|\leq$
\end_inset


\series bold
ham_cutoff
\series default
.
\end_layout

\end_deeper
\begin_layout Enumerate
After this the output on the reciprocial grid is performed.
 It is important to realize that the Hamiltonian in 
\begin_inset Formula $k$
\end_inset

-space is constructed from the truncated real space Hamiltonian, i.
 e.
 the Bloch sums of the hoppings Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-hoppings"

\end_inset

) 
\series bold
after
\series default
 cutting off of the hoppings for which 
\begin_inset Formula $\left|T\right|>$
\end_inset


\series bold
ham_cutoff
\series default
 and 
\begin_inset Formula $\left|t\right|<$
\end_inset


\series bold
WF_ham_threshold
\series default
.
 The resulting Hamiltonian is written to 
\family typewriter
+hamongrid
\family default
.
 See the 
\family typewriter
wannier.f90
\family default
 soure code for the order of the data.
\end_layout

\begin_layout Enumerate
The very same hopping data are also written to 
\family typewriter
+hamdata
\family default
 for further processing by 
\family typewriter
pyfplo
\family default
.
\end_layout

\begin_layout Enumerate
Now, 
\family typewriter
+WF_coefficients
\family default
 is written, which contains information about the contributions of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 orbitals to the WFs Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-from-Phi"

\end_inset

).
 This file shows only contributions, which are larger than 
\series bold
WF_coeff_threshold
\series default
 in 
\family typewriter
=.wandef
\family default
.
\end_layout

\begin_layout Enumerate
At the end of the module the WF are written to disk on a real space grid
 into files 
\family typewriter
wfdata...
\family default
.
 Those can be used to visualize the WFs using 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 (also see 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 help screens of the WF/Gridplot dialog in 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

../Xfplo/xfplo.pdf
\end_layout

\end_inset

):
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout LyX-Code
xfplo =.in wfdata001 wfdata005
\end_layout

\begin_layout Standard
or if the 
\family typewriter
use-data-directories
\family default
 option (in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

) was used
\end_layout

\begin_layout LyX-Code
xfplo =.in +wfdata/wfdata001 +wfdata/wfdata005
\end_layout

\end_deeper
\begin_layout Enumerate
After this 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 pauses with the message
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout LyX-Code
CTRL_C for abort, enter for next trial 
\end_layout

\begin_layout Standard
Here, one can type CTRL-C to stop, or one modifies the 
\series bold
wandef
\series default
s (in another window/editor) and hits enter to re-run the module, with re-scanni
ng 
\family typewriter
=.wandef
\family default
 but without re-reading 
\family typewriter
+wancoeff
\family default
 (which is slow, since it is a large file).
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/wanbands.png
	lyxscale 20
	width 80col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Wannier-fit-for-cacuo2"

\end_inset

Wannier fit for CaCuO
\begin_inset Formula $_{2}$
\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Wannier-fit-for-cacuo2"

\end_inset

 shows the result of the CaCuO
\begin_inset Formula $_{2}$
\end_inset

 example.
 Dark yellow and magenta are the 
\begin_inset Formula $3d_{x^{2}-y^{2}}$
\end_inset

 fat bands as produced by 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

.
 Red shows the full Wannier function transformation (without cutoffs) from
 the file 
\family typewriter
+wanband
\family default
 and green shows the model WF bands including cutoffs from the file 
\family typewriter
+wanbandtb
\family default
.
 One can nicely observe how the WF bands interpolate the hybridization gap
 between 
\begin_inset Formula $\Gamma$
\end_inset

 and X, and M and 
\begin_inset Formula $\Gamma$
\end_inset

.
\end_layout

\begin_layout Subsection
Files
\end_layout

\begin_layout Standard
There are three band structure files produced.
\end_layout

\begin_layout Description

\family typewriter
+wanband
\family default
 The Bloch-sums of the WF Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch-sums"

\end_inset

) are obtained by doing the unitary transformation in 
\series bold

\begin_inset Formula $k$
\end_inset


\series default
-space
\begin_inset space ~
\end_inset

(
\begin_inset Formula $U$
\end_inset

).
 The Hamiltonian in WF-Bloch basis in 
\begin_inset Formula $k$
\end_inset

-space Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch_Ham"

\end_inset

) is directly related to the WF-Bloch sums and hence is obtained straight
 from the 
\begin_inset Formula $U$
\end_inset

-transformed KS functions and Hamiltonian.
 The resulting 
\begin_inset Formula $k$
\end_inset

-dependent Hamiltonian has the dimension of the WF-basis defined by the
 user.
 It can be diagonalized to get the band structure belonging to the WF model.
 The result of this is written to 
\family typewriter
+wanband
\family default
.
 If the WFs describe an isolated band complex (one needs as many WFs as
 there are KS bands in the band complex) then the 
\family typewriter
+wanband
\family default
 band structure must coincide with the bandstructure of this band complex
 in the full 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 band structure plot.
 Deviations are possible, if there are not enough 
\begin_inset Formula $k$
\end_inset

-points in the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

-SCF calculation.
 This comes about since a discrete approximation to the 
\begin_inset Formula $k$
\end_inset

-integral in Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:def_WF"

\end_inset

) defines WFs, which have periodic images with a period of the Born von
 Karman torus.
\end_layout

\begin_layout Description

\family typewriter
+wanbweights
\family default
 This file contains the fat-bands corresponding to 
\family typewriter
+wanband
\family default
.
 The band weights are determined with respect to the WF character in the
 corresponding bands.
 Let's make that clearer.
 In a local orbital basis like 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 the overlap of the orbitals with the KS functions determines the fat-bands
 or how much of a certain orbital a band is made of.
 This concept can be applied to the WFs as well thinking of the WFs as a
 basis.
 So, how much of a particular WF makes up a band? This is given by the informati
on in this file.
 Note, that the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 fat-bands and WF fat-bands do not coincide.
 That is the main reason why we do a WF analysis, see Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Comparison-of-FPLO-and-WF-fatbands"

\end_inset


\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Box Frameless
position "c"
hor_pos "c"
has_inner_box 1
inner_pos "c"
use_parbox 0
use_makebox 0
width "25col%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename pic/wan_user/strnio2.jpg
	lyxscale 5
	width 120col%

\end_inset


\end_layout

\end_inset


\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "75col%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open

\begin_layout Plain Layout
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "100col%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename pic/wan_user/fplofatbands.png
	lyxscale 10
	width 80col%

\end_inset


\end_layout

\end_inset


\begin_inset Newline newline
\end_inset


\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "100col%"
special "none"
height "1in"
height_special "totalheight"
thickness "0.4pt"
separation "3pt"
shadowsize "4pt"
framecolor "black"
backgroundcolor "none"
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename pic/wan_user/niowanbands.png
	lyxscale 10
	width 80col%

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Comparison-of-FPLO-and-WF-fatbands"

\end_inset

Comparison of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 fat-bands (upper panel) with WF fat-bands (lower panel) for a NiO
\begin_inset Formula $_{2}$
\end_inset

-chain.
 Some WFs are linear combination of several orbitals, such that the WFs
 are centered between two orbitals.
 One can clearly see that in the WF basis each band has a pure character,
 i.e.
 the bands are completely decoupled by chosing a symmetry adapted basis
 (WFs).
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Description

\family typewriter
+wanbandtb
\family default
 Now, we can go a step further and Fourier transform the WF-Bloch sums Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch-sums"

\end_inset

) into real space Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch-sum-Fouriertrans"

\end_inset

).
 This provides the actual WFs.
 In real space these WFs overlap (ensuring orthogonality) and the expectation
 value of the Hamiltonian in this basis forms the hopping integrals Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-hoppings"

\end_inset

).
 These hopping integrals can be cut off at some distance or if they are
 smaller than a threshold.
 This defines a modified Hamiltonian in real space.
 We transform this modified (model) Hamiltonian back into 
\begin_inset Formula $k$
\end_inset

-space, diagonalize it and get the band structure in this file.
 If the cut off and threshold is moderate the result should equal 
\family typewriter
+wanband
\family default
 otherwise it can differ.
 Especially, for non-isolated bands we can get non-analytic behavior in
 
\family typewriter
+wanband
\family default
, which is unavoidable in some cases.
 It turns out that the real-space cutoff reduces these unwanted effects.
 
\end_layout

\begin_layout Standard
The band structure data files can be visualized using the bandplot-tools
 of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

.
 Besides these files there are a few more.
 If the use 
\series bold
data directories flag
\series default
 in 
\family typewriter
fedit
\family default
 is set most of these files are written into subdirectories for sake of
 keeping the directory clean.
 Exceptions from this rule are 
\family typewriter
+hamdata
\family default
 and the old-style opendx interface files.
\end_layout

\begin_layout Description
+hamdata
\family typewriter
\SpecialChar ldots
 
\family default
contains the WF Hamiltonian in real space after the cutoff procedure (see
 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

../pyfplo/pyfplo.pdf
\end_layout

\end_inset

).
\end_layout

\begin_layout Description

\family typewriter
wfdata\SpecialChar ldots

\family default
 contain the real space Wannier function for visualization with the help
 of 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 or 
\family typewriter
opendx
\family default
.
 These files can be directly loaded into 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

.
 The 
\family typewriter
opendx
\family default
 interface is considered to be obsolete and is no longer switched on by
 default.
 Use the 
\series bold
opendx_interface
\series default
 keyword to switch it on.
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout LyX-Code
xfplo =.in wfdata001 wfdata005
\end_layout

\begin_layout Standard
or if the 
\family typewriter
use-data-directories
\family default
 option (in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

) was used
\end_layout

\begin_layout LyX-Code
xfplo =.in +wfdata/wfdata001 +wfdata/wfdata005
\end_layout

\begin_layout Standard
(also see 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 help screens of the WF/Gridplot dialog 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

../Xfplo/xfplo.pdf
\end_layout

\end_inset

)
\end_layout

\end_deeper
\begin_layout Description

\family typewriter
+WFstat\SpecialChar ldots

\family default
 contains the absolute values of the coefficients in Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-from-Phi"

\end_inset

) as a function of the distance of the corresponding orbitals from the Wannier
 function center.
 Plot this file (e.g.
 
\family typewriter
xfbp ./+WFstat.xpy
\family default
) to have an idea of the localization of the WFs.
\end_layout

\begin_layout Description

\family typewriter
+Rstat\SpecialChar ldots

\family default
 contains the statistics of the position operator matrix elements.
 Plot this to estimate the falloff of this operator (e.f.
 
\family typewriter
xfbp ./Rstat.xpy
\family default
).
\end_layout

\begin_layout Description

\family typewriter
+T\SpecialChar ldots
 
\family default
contains the hoppings from a given WF to the neighbouring WFs corresponding
 to Eq.
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-hoppings"

\end_inset

) as a function of the distance between the WF centers.
\end_layout

\begin_layout Description

\family typewriter
WFstat.xpy
\family default
 use this to plot the orbital contribution statistics (
\family typewriter
+WFstat
\family default
\SpecialChar ldots
) 
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout LyX-Code
xfbp WFstat.py
\end_layout

\end_deeper
\begin_layout Description

\family typewriter
Rstat.xpy
\family default
 use this to plot the position operator statistics (
\family typewriter
+Rstat
\family default
\SpecialChar ldots
) 
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout LyX-Code
xfbp Rstat.py
\end_layout

\end_deeper
\begin_layout Description

\family typewriter
opendxWF.dx
\family default
,
\begin_inset space ~
\end_inset


\family typewriter
WF.net
\begin_inset space ~
\end_inset


\family default
and
\begin_inset space ~
\end_inset


\family typewriter
WF.cfg 
\family default
these are the opndx interface files, which are used in the following way
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout LyX-Code
dx -image WF.net
\end_layout

\end_deeper
\begin_layout Subsection
Keywords
\begin_inset CommandInset label
LatexCommand label
name "subsec:WFKeywords"

\end_inset


\end_layout

\begin_layout Standard
The keywords in 
\family typewriter
=.wandef 
\family default
are explained in the following.
\end_layout

\begin_layout Description
doit switch the WF module on.
\end_layout

\begin_layout Description
keeprunning can be 
\series bold
on
\series default
 or 
\series bold
off
\series default
.
 If this keyword is 
\series bold
on
\series default
 or if it is absent the program will not terminate, which is usefull for
 a re-run of the WF-module after changes to 
\family typewriter
=.wandef
\family default
.
 This avoids the loading of 
\family typewriter
+wancoeff
\family default
(
\family typewriter
+wancoeffbin
\family default
).

\family typewriter
 
\family default
By default the program keeps running.
 So what you want to use to overwrite this behaviour is 
\series bold
keeprunning off
\series default
.
\end_layout

\begin_layout Description
save_spin_info save the spin operator in 
\family typewriter
+wancoeff
\family default
 (only in full relativistic mode).
 This option makes the file bigger and must be set before 
\family typewriter
+wancoeff
\family default
 is created!
\end_layout

\begin_layout Description
save_bfield save the exchange field in 
\family typewriter
+wancoeff
\family default
 (only in full relativistic mode).
 This option makes the file bigger and must be set before 
\family typewriter
+wancoeff
\family default
 is created!
\end_layout

\begin_layout Description
save_position save the invariant position operator matrix elements in 
\family typewriter
+wancoeff
\family default
.
 This option makes the file bigger and must be set before 
\family typewriter
+wancoeff
\family default
 is created! (see 
\series bold
gradorder
\series default
, below)
\end_layout

\begin_layout Description
gradorder can be 1, 3, 5 and 7.
 In the calculation of the position operator a numerical gradient formula
 is needed.
 The position operator converges slowly with the density of SCF 
\begin_inset Formula $k$
\end_inset

-mesh (by its very nature).
 The default is 
\series bold
gradorder 1
\series default
.
 In many cases 
\series bold
gradorder 3
\series default
 and a default mesh-density gives a more accurate result.
 At least it speeds up convergence with the mesh density.
 Possible values are 1,3,5,7 but 5 and 7 seem to not improve the result.
 The current implementation for higher orders is probably not good for strongly
 anisotropic 
\begin_inset Formula $k$
\end_inset

-mesh distances.
 
\begin_inset Newline newline
\end_inset

Note, that if the 
\begin_inset Formula $k$
\end_inset

-mesh subdivision is 1 in some direction(s) the derivative formula will
 be dimensionally reduced, assuming that there is no dispersion in this
 direction.
 This yields better results than the full formula, but only if the dispersion
 is really small.
 It will however, not always yield the same as a a full (isotropic) 
\begin_inset Formula $k$
\end_inset

-mesh, due to the finite nature of the formula.
 Both the full mesh and the reduced mesh will converge to the same result
 for a fine enough mesh, though (hopefully).
\begin_inset Newline newline
\end_inset

In general the position operator is very sensitive to the disentangling
 energy window fall off and the number of SCF 
\begin_inset Formula $k$
\end_inset

-points.
 This is so, since it is used in the MAXLOC-WF algorithm to smooth the Hilbert
 space, which for our maximally projected case must mean strong dependence
 on the energy window.
\end_layout

\begin_layout Description
wandef start a definition of a single Wannier function.
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Description
on/off make the corresponding 
\series bold
wandef
\series default
 active/inactive.
\end_layout

\begin_layout Description
emin the lower bound of the energy window (two numbers if spin-polarized).
\end_layout

\begin_layout Description
emax the upper bound of the energy window (two numbers if spin-polarized).
\end_layout

\begin_layout Description
de the width of the Gaussian tail above and below 
\series bold
emax
\series default
/
\series bold
emin 
\series default
defining a smooth energy window (two numbers if spin-polarized).
\end_layout

\begin_layout Description
delower/deupper like 
\series bold
de
\series default
, but for the lower/upper end of the energy interval.
 Note, that 
\series bold
de
\series default
 will overwrite 
\series bold
delower
\series default
/
\series bold
deupper
\series default
 if specified after the latter!
\end_layout

\begin_layout Description
ubands/lbands band indices, which specify the maximum and minimum band to
 be contained in the projector (energy window).
 (two numbers if spin-polarized) 
\end_layout

\begin_layout Description
contrib add an 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 orbital to the 
\series bold
wandef
\series default
.
 There can be several 
\series bold
contrib
\series default
s in one 
\series bold
wandef
\series default
, one after the other.
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Description
site the site number of the orbital according to the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 output.
\end_layout

\begin_layout Description
name an arbitrary name to identify the WF.
\end_layout

\begin_layout Description
difvec the distance of the contributing orbital from the WF center.
 The 
\series bold
difvec
\series default
 of the first contrib to a 
\series bold
wandef
\series default
 together with the 
\series bold
site
\series default
 of the first 
\series bold
wandef
\series default
 implicitly define the WF center.
 The choice of the WF center decides whether the WF is real or not.
 The WF must be defined such that they are real.
 E.g.
 a single orbital/contrib WF should always have 
\series bold
difvec
\series default

\begin_inset space ~
\end_inset

0
\begin_inset space ~
\end_inset

0
\begin_inset space ~
\end_inset

0.
 (In full-relativistic mode WFs are likely to be complex not matter how
 the center is chosen.)
\end_layout

\begin_layout Description
xaxis the local x-axis expressed in global coordinates
\end_layout

\begin_layout Description
zaxis the local z-axis expressed in global coordinates
\end_layout

\begin_layout Description
sxaxis/szaxis the local x/z-axis for the spin wave functions expressed in
 global coordinates for full relativistic mode.
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
This can be a string or a vector.
 If sxaxis is a vector szaxis must also be defined and a vector.
 If it is a string, szaxis need not be and should not be specified.
 
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
long
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
short
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
meaning
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\family typewriter
'global'
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\family typewriter
'glo'
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
use the global cartesian axes
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\family typewriter
'local'
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\family typewriter
'loc'
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
use the local axes as defined by xaxis/zaxis.
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\family typewriter
'quant'
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\family typewriter
'qua'
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
use the global spin quantization axis as defined in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
By default (if not specified) the spin axis points into the global spin
 quantization axis as defined in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

.
 If you want to construct projectors for full relativistic calculations
 using hand made linear combinations of orbitals the spin axes matter for
 the actual orbital shape as well as for transformation properties, especially
 if local coordinate systems are used.
 Example: the Osmate 
\begin_inset Formula $j_{\mathrm{eff}}=\frac{1}{2}$
\end_inset

 and 
\begin_inset Formula $j_{\mathrm{eff}}=\frac{3}{2}$
\end_inset

 representations for the 
\begin_inset Formula $t_{2g}$
\end_inset

 subspace contain functions like 
\begin_inset Formula 
\[
\left|j_{\mathrm{eff}}=\frac{1}{2},m_{\mathrm{eff}}=\frac{1}{2}\right\rangle =\frac{1}{\sqrt{3}}\left(\left|d_{yz}\downarrow\right\rangle +i\left|d_{xz}\downarrow\right\rangle +\left|d_{xy}\uparrow\right\rangle \right)
\]

\end_inset

which contain specific orbital-spin combinations.
 In order to use the proper coordinate system for the spin part we need
 to specify 
\family typewriter
sxaxis='loc'
\family default
.
 Then we only need to specify the xaxis/zaxis pair to rotate the projector
 in the local coordinate system, including the spin-part.
\end_layout

\begin_layout Standard
The first 
\series bold
contrib
\series default
 of each 
\series bold
wandef
\series default
 is written into the 
\family typewriter
wfdata
\family default
 files to be available for visualization in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Description
orb the orbital (e.g.
 2s+0 or 3d-1).
 In full-relativistic mode the orbitals can either be pseudo non-relativistic
 projections denoted by 
\begin_inset Quotes eld
\end_inset

3d-1 up
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

3d-1 dn
\begin_inset Quotes erd
\end_inset

 or spherical spinors denoted by 3d3/2-1/2 or 3d5/2+3/2.
\end_layout

\begin_layout Description
fac a weight factor, determining the relative weight of the orbital/contrib
 in cases of multiple-contrib wandefs.
 The weights need not to be normalized, this is taken care of.
 
\end_layout

\end_deeper
\end_deeper
\begin_layout Description
automode can be 
\family typewriter
valence
\family default
, 
\family typewriter
all
\family default
 or 
\family typewriter
none
\family default
: This option makes 
\series bold
wandef
\series default
 definitions unnecessary.
 When this mode is not none, 
\series bold
wandef
\series default
s for all valence or all semi-core and valence orbitals are created.
 This leads to a larger Wannier basis and slower calculations.
 It helps however for automated tasks.
 Be aware, that a larger 
\series bold
ham_cutoff
\series default
 might be needed.
 In this context also a larger SCF k-mesh might be needed, since a to small
 SCF k-mesh misses Fourier components of the more extended higher lying
 orbitals.
 This is demonstrated in 
\family typewriter
DOC/Tutorials/wan
\family default
.
 If 
\series bold
automode
\series default
 is 
\family typewriter
valence
\family default
 and if some semi-core bands overlap the valence sector, semi-core orbitals
 are added to the Wannier basis until the totality of resulting bands has
 a clear gap below the lowest band.
 For convenience a python script called 
\family typewriter
makewandeffromauto.py
\family default
 is created.
 To modify this file, please copy it.
\end_layout

\begin_layout Description
coefficients_format 
\begin_inset CommandInset label
LatexCommand label
name "coefficientsformat"

\end_inset

can be 
\series bold
bin
\series default
 or something else.
 Since version 14.00, the file 
\family typewriter
+wancoeff
\family default
 can be converted into binary format for faster loading.
 After the first 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 run with a valid 
\family typewriter
=.wandef
\family default
 present, 
\family typewriter
+wancoeff
\family default
 will have been created.
 A rerun will start the WF creation process.
 If this option is set to 
\series bold
bin
\series default
 the data file will be converted into binary format, if not already done.
 On any further run the binary file 
\family typewriter
+wancoeffbin
\family default
 will be read instead of 
\family typewriter
+wancoeff
\family default
.
 This is faster.
 The user has to take care of deleteing 
\family typewriter
+wancoeffbin
\family default
 whenever 
\family typewriter
+wancoeff
\family default
 got changed due to settings changes by the user.
\end_layout

\begin_layout Description
ham_cutoff restricts output of the real-space WF Hamiltonian in standard
 output and 
\family typewriter
+T\SpecialChar ldots
.
 
\family default
Also restricts the matrix elements used in creating 
\family typewriter
+hamongrid
\family default
 and 
\family typewriter
+hamdata
\family default
.
 Since version 19.00 the reach of the matrix elements is restricted by the
 size of the SC k-mesh (dual R-lattice of the same size).
 This avoids replica and allows to give a large cutoff for safer results
 when 
\series bold
automode
\series default
 is used.
\end_layout

\begin_layout Description
WF_coeff_threshold restricts the output of coefficients in 
\family typewriter
+WF_coefficients
\family default
.
\end_layout

\begin_layout Description
WF_ham_threshold restricts the output of real-space Hamiltonian in standard
 output and 
\family typewriter
+T\SpecialChar ldots
 
\family default
and restricts the hoppings used in creating 
\family typewriter
+wanbandtb, +hamongrid
\family default
 and 
\family typewriter
+hamdata
\family default
.
\end_layout

\begin_layout Description
WF_write_coeff_stats can be 
\series bold
on
\series default
/
\series bold
off
\series default
 and triggers the output of the files 
\family typewriter
+WFstat\SpecialChar ldots

\end_layout

\begin_layout Description
ham_write_t_stats can be 
\series bold
on
\series default
/
\series bold
off
\series default
 and triggers the output of the files 
\family typewriter
+T\SpecialChar ldots

\end_layout

\begin_layout Description
print_T can be 
\series bold
on
\series default
/
\series bold
off
\series default
 and triggers the printing of the 
\family typewriter
T=...
 
\family default
lines on standard output 
\family typewriter
\SpecialChar ldots

\end_layout

\begin_layout Description
do_WF_in_real_space can be 
\series bold
on
\series default
/
\series bold
off
\series default
 and triggers the output of +WFstat and open-dx files
\family typewriter
\SpecialChar ldots

\family default
 everything, which is related to the real space representation of the Wannier
 function.
 (If it is 
\series bold
off
\series default
, the speed increases.)
\end_layout

\begin_layout Standard
For the output of the WFs on the real space grid (visualization) we have
 to define a grid
\end_layout

\begin_layout Description
WF_grid_basis can be 
\series bold
conv
\series default
/
\series bold
prim
\series default
.
 This defines the basis 
\begin_inset Formula $\underline{B}=\left(\begin{array}{c}
\vec{b}_{1}\\
\vec{b}_{2}\\
\vec{b}_{3}
\end{array}\right)$
\end_inset

 of the box-like grid.
 We can use the conventional or the primitive basis vectors.
\end_layout

\begin_layout Description
WF_grid_directions in terms of the basis we can define three vectors forming
 the rows of 
\begin_inset Formula $\underline{V}=\underline{D}\underline{B}$
\end_inset

.
 These three directions span the grid-box.
 The input here is the matrix 
\begin_inset Formula $\underline{D}$
\end_inset

.
\end_layout

\begin_layout Description
WF_grid_subdivision subdivide the box along the directions 
\begin_inset Formula $\underline{V}$
\end_inset

 accordingly.
\end_layout

\begin_layout Description
WF_grid_origin put the origin of the box here.
 If this keyword is commented out (e.g.
 
\series bold
xWF_grid_origin
\series default
) the box will be centered around the WF center.
\end_layout

\begin_layout Standard
For the output of the WF Hamiltonian on the 
\begin_inset Formula $k$
\end_inset

-space grid (file 
\family typewriter
+hamongrid
\family default
) we have to define a grid
\end_layout

\begin_layout Description
k_grid_basis the basis 
\begin_inset Formula $\underline{B}$
\end_inset

 in reciproical space.
 (Note the reciprocial relations between bcc and fcc and the like.)
\end_layout

\begin_layout Description
k_grid_directions the directions 
\begin_inset Formula $\underline{V}=\underline{D}\underline{B}$
\end_inset

 defined here by giving 
\begin_inset Formula $\underline{D}$
\end_inset

.
\end_layout

\begin_layout Description
k_grid_subdivision subdivisions, see above.
\end_layout

\begin_layout Description
k_grid_incl_periodic_points can be 
\series bold
on
\series default
/
\series bold
off
\series default
.
 The Hamiltonian in 
\begin_inset Formula $k$
\end_inset

-space is periodic.
 We can include or exclude the periodically equivalent points at the boundary
 of the box.
\end_layout

\begin_layout Description
opendx_interface if this keyword is found the old-style 
\family typewriter
opendx
\family default
 interface files (
\family typewriter
opendxWF.dx
\family default
, 
\family typewriter
WF.net
\family default
 and 
\family typewriter
WF.cfg
\family default
) are produced.
 In the newer version the real space Wannier function data (
\family typewriter
wfdata...
\family default
) can be loaded into 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 directly.
\end_layout

\begin_layout Subsection
Definition of real spherical harmonics
\end_layout

\begin_layout Standard
We define real spherical harmonics 
\begin_inset Formula $Y_{lm}$
\end_inset

 with magnetic qn.
 numbers 
\begin_inset Formula $m=-l,\ldots,l$
\end_inset

 as 
\begin_inset Formula 
\[
Y_{lm}\left(x,y,z\right)\propto P_{l}^{\left|m\right|}\left(\frac{z}{r}\right)\begin{cases}
\sin\left(\left|m\right|\varphi\right) & \;m<0\\
\cos\left(\left|m\right|\varphi\right) & \;m\geq0
\end{cases}
\]

\end_inset

The sin/cos can be expanded according to the addition theorems, e.g.
 
\begin_inset Formula $\sin\left(2\varphi\right)=2\sin\varphi\cos\varphi$
\end_inset

.
 Using 
\begin_inset Formula $\sin\varphi\propto y$
\end_inset

 and 
\begin_inset Formula $\cos\varphi\propto x$
\end_inset

 we get 
\begin_inset Formula $\sin\left(2\varphi\right)\propto xy$
\end_inset

.
 Additionally we need 
\begin_inset Formula $P_{l}^{\left|m\right|}\left(\frac{z}{r}\right)\propto\mbox{polynomial of degree }l-\underline{m}\mbox{ in }z$
\end_inset

.
 Thus,
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
Y_{2,-2}\propto P_{2}^{2}\left(\frac{z}{r}\right)\sin\left(2\varphi\right)\propto xy
\]

\end_inset


\begin_inset Formula 
\[
Y_{2,-1}\propto P_{2}^{1}\left(\frac{z}{r}\right)\sin\left(\varphi\right)\propto zy
\]

\end_inset


\begin_inset Formula 
\[
Y_{2,0}\propto P_{2}^{0}\left(\frac{z}{r}\right)\cos\left(0\varphi\right)\propto z^{2}
\]

\end_inset


\begin_inset Formula 
\[
Y_{2,1}\propto P_{2}^{1}\left(\frac{z}{r}\right)\cos\left(\varphi\right)\propto zx
\]

\end_inset


\begin_inset Formula 
\[
Y_{2,2}\propto P_{2}^{2}\left(\frac{z}{r}\right)\cos\left(2\varphi\right)\propto x^{2}-y^{2}
\]

\end_inset

For more specific information on the polynomial in 
\begin_inset Formula $z$
\end_inset

 one has to look up the associated Legendre polynomials 
\begin_inset Formula $P_{l}^{\left|m\right|}\left(z\right)$
\end_inset

.
\end_layout

\begin_layout Subsection
Full relativistic spin-part
\end_layout

\begin_layout Standard
In full relativistic mode the spin part is important.
 The real space representation on the grid for visualization with 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 is containing the large and small component for spin up and spin down.
 Usually, you would focus on the large component.
 So, effectively we have a spinor 
\begin_inset Formula 
\[
\left(\begin{array}{c}
g_{\uparrow}\\
g_{\downarrow}
\end{array}\right)
\]

\end_inset

which is expressed via eigenfunctions 
\begin_inset Formula $\chi_{\uparrow}=\left(\begin{array}{c}
1\\
0
\end{array}\right)$
\end_inset

 and 
\begin_inset Formula $\chi_{\downarrow}=\left(\begin{array}{c}
0\\
1
\end{array}\right)$
\end_inset

 of the spin in the global cartesian system.
 In the current implementation a global spin quantization z-axis can be
 defined in 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fedit{}
\end_layout

\end_inset

 which results in a colinear approximation for the xc-field pointing in
 this direction.
 The 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
xfplo{}
\end_layout

\end_inset

 interface offers to display the spinor components (or density 
\begin_inset Formula $n$
\end_inset

 or spin-density 
\begin_inset Formula $m$
\end_inset

, colored by the spin polarization 
\begin_inset Formula $\zeta=\frac{m}{n}$
\end_inset

) in a spin basis which corresponds to the xc-field direction.
 To be specific the Wannier function (the data in the files 
\family typewriter
wfdata...
\family default
) is expressed in global coordinates as 
\begin_inset Formula 
\[
W=\underline{\chi}^{T}\underline{g}=\left(\begin{array}{cc}
\left(\begin{array}{c}
1\\
0
\end{array}\right) & \left(\begin{array}{c}
0\\
1
\end{array}\right)\end{array}\right)\left(\begin{array}{c}
g_{\uparrow}\\
g_{\downarrow}
\end{array}\right)
\]

\end_inset

The xc-field induces a transformation of the form 
\begin_inset Formula $\underline{B}^{T}R=\left(00B_{z^{\prime}}\right)$
\end_inset

, where 
\begin_inset Formula $R$
\end_inset

 is a 
\begin_inset Formula $3\times3$
\end_inset

 rotation matrix.
 The matrix 
\begin_inset Formula $R$
\end_inset

 is obatined as 
\begin_inset Formula $R=\left(\boldsymbol{e}_{x}^{\prime}\boldsymbol{e}_{y}^{\prime}\boldsymbol{e}_{z}^{\prime}\right)$
\end_inset

, since then we get for a field in 
\begin_inset Formula $\boldsymbol{e}_{z}^{\prime}$
\end_inset

-direction 
\begin_inset Formula $\underline{B}=B_{z^{\prime}}\boldsymbol{e}_{z}^{\prime}$
\end_inset


\begin_inset Formula 
\[
\underline{B}^{T}R=B_{z}\boldsymbol{e}_{z}^{\prime T}\left(\boldsymbol{e}_{x}^{\prime}\boldsymbol{e}_{y}^{\prime}\boldsymbol{e}_{z}^{\prime}\right)=B_{z^{\prime}}\left(001\right)=\left(00B_{z^{\prime}}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
The corresponding spin eigenfunctions in this frame are expressed by 
\begin_inset Formula $\underline{\chi}^{\prime T}=\underline{\chi}^{T}D$
\end_inset

, where 
\begin_inset Formula $D$
\end_inset

 is a 
\begin_inset Formula $2\times2$
\end_inset

 spin rotation matrix with the property
\begin_inset Formula 
\[
\underline{\sigma}^{T}R=D\underline{\sigma}D^{+}
\]

\end_inset


\begin_inset Formula 
\[
\underline{\sigma}^{T}R^{T}=D^{+}\underline{\sigma}D
\]

\end_inset

where 
\begin_inset Formula $\underline{\sigma}$
\end_inset

 is the vector if Pauli matrices.
 Let's express the Zeeman term in this spin basis
\begin_inset Formula 
\begin{eqnarray*}
E_{\mathrm{Z}} & = & \left\langle \underline{\chi}^{\prime T}\mid\underline{\sigma}^{T}\underline{B}\mid\underline{\chi}^{\prime T}\right\rangle \\
 & = & D^{+}\left\langle \underline{\chi}^{T}\mid\underline{\sigma}^{T}\underline{B}\mid\underline{\chi}^{T}\right\rangle D\\
 & = & \left\langle \underline{\chi}^{T}\mid D^{+}\underline{\sigma}^{T}\underline{B}D\mid\underline{\chi}^{T}\right\rangle 
\end{eqnarray*}

\end_inset

where the last step is possible because 
\begin_inset Formula $\underline{\chi}^{T}$
\end_inset

 is the unit matrix.
\begin_inset Formula 
\begin{eqnarray*}
E_{\mathrm{Z}} & = & \left\langle \underline{\chi}^{T}\mid D^{+}\underline{\sigma}^{T}\underline{B}D\mid\underline{\chi}^{T}\right\rangle \\
 & = & \left\langle \underline{\chi}^{T}\mid D^{+}\underline{\sigma}^{T}D\underline{B}\mid\underline{\chi}^{T}\right\rangle \\
 & = & \left\langle \underline{\chi}^{T}\mid\underline{\sigma}^{T}R^{T}\underline{B}\mid\underline{\chi}^{T}\right\rangle \\
 & = & \left\langle \underline{\chi}^{T}\mid\underline{\sigma}^{T}\left(\underline{B}^{T}R\right)^{T}\mid\underline{\chi}^{T}\right\rangle \\
 & = & \left\langle \underline{\chi}^{T}\mid\underline{\sigma}^{T}\left(00B_{z^{\prime}}\right)^{T}\mid\underline{\chi}^{T}\right\rangle \\
 & = & \left\langle \underline{\chi}^{T}\mid\sigma_{z}B_{z^{\prime}}\mid\underline{\chi}^{T}\right\rangle 
\end{eqnarray*}

\end_inset

which shows that the field in this basis looks like a field in the 
\begin_inset Formula $z^{\prime}$
\end_inset

-direction.
 Now, the Wannier function in the xc-field frame will be 
\begin_inset Formula 
\[
W=\underline{\chi}^{\prime T}D^{+}\underline{g}
\]

\end_inset

which has components
\begin_inset Formula 
\[
\underline{g}^{\prime}=D^{+}\underline{g}
\]

\end_inset

in this spin frame.
 If the spin polarization energy is larger than the spin-orbit coupling
 this spin frame will show the purest spin character for Wannier functions,
 which have a spin axis along the global quantization 
\begin_inset Formula $z$
\end_inset

-axis, as given by the keywords 
\series bold
sxaxis
\series default
/
\series bold
szaxis
\series default
.
\end_layout

\begin_layout Subsection
Problems 
\end_layout

\begin_layout Standard
The number of 
\begin_inset Formula $\mathbf{k}$
\end_inset

-points used in the SCF calculation influences the Wannier function quality.
 If the 
\begin_inset Formula $\mathbf{k}$
\end_inset

-mesh is not fine enough, the band structure determined by diagonalizing
 the WF Bloch Hamiltonian Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:WF-Bloch_Ham"

\end_inset

) (
\family typewriter
+wanband
\family default
) will not coincide with the corresponding bands of the full 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 band structure (
\family typewriter
+band
\family default
).
 (Coincidence can of course only happen anyway, if the WFs describe an isolated
 band complex.) 
\end_layout

\begin_layout Standard
If the real space Hamiltonian cutoff is smaller than the extend of the WFs
 the fitted Wannier bandstructure (
\family typewriter
+wanbandtb
\family default
) will not conincide with the full WF band structure.
 This may also be due to a bad Wannier definition, which is not localizing
 the WFs sufficiently (check energy window, use molecular orbitals – several
 
\series bold
contrib
\series default
s per 
\series bold
wandef
\series default
 – instead of simple orbitals).
\end_layout

\begin_layout Standard
There might be unnatural spikes at certain 
\begin_inset Formula $k$
\end_inset

-points in the WF band structure.
 This usually means that the energy window is to narrow and that the character
 of 
\begin_inset Formula $\chi$
\end_inset

 is only large outside the window at the corresponding k-points.
\end_layout

\begin_layout Standard
If slabs or chains are calculated, there is an artificial periodicity in
 the irrelevant directions.
 If not enough SCF 
\begin_inset Formula $\mathbf{k}$
\end_inset

-points are used in these directions, we get influences of the artificial
 periodic replica (actually replica are avoided now, but there still might
 be artifacts).
 Hence, either the vacuum spacing is very large, which is detrimental for
 other parts of the code, or one has to use sufficiently many points in
 the irrelevant directions.
 However, always first start with subdivision 1 in this direction and only
 if this fails try more poinst.
 (Only some experimenting can tell.) Especially, when the position operator
 is requested additional points in this direction might be needed.
 This operator really is badly converging.
\end_layout

\begin_layout Subsection
Saving memory (and time)
\begin_inset CommandInset label
LatexCommand label
name "subsec:Saving-memory"

\end_inset


\end_layout

\begin_layout Standard
In order to reduce the size of 
\family typewriter
+wancoeff
\family default
 the user can set the FEDIT option 
\begin_inset Quotes eld
\end_inset

restrict bands to window
\begin_inset Quotes erd
\end_inset

 in the bandplot submenu.
 Then lower and upper energy bounds in the same submenu defines the bands,
 which will be written to 
\family typewriter
+wancoeff
\family default
 for all data contained in this file.
\end_layout

\begin_layout Standard
Use the 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
htmlref{
\backslash
textbf{coefficients
\backslash
_format}}{coefficientsformat}
\end_layout

\end_inset

 option in 
\family typewriter
=.wandef
\family default
 in order to convert 
\family typewriter
+wancoeff
\family default
 to binary format.
 
\end_layout

\begin_layout Section
Examples
\end_layout

\begin_layout Standard
All examples can be found in 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

../WannierFunctions/
\end_layout

\end_inset

.
\end_layout

\begin_layout Subsection
Hexagonal, Graphene, MgB
\begin_inset Formula $_{2}$
\end_inset

: 
\begin_inset Formula $sp^{2}$
\end_inset


\end_layout

\begin_layout Standard
Here we analyse the Wannier function choices for a hexagonal lattice with
 essentially two atoms per unit cell.
 The main issue is to understand the symmetry considerations.
\end_layout

\begin_layout Standard
Have a look at the accompanying example directories for MgB
\begin_inset Formula $_{2}$
\end_inset

.
\begin_inset Separator latexpar
\end_inset


\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/wanhex1.pdf
	width 60col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:real-and-reciprocal"

\end_inset

Real and reciprocal cell of graphene, MgB
\begin_inset Formula $_{2}$
\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:real-and-reciprocal"

\end_inset

 shows the basic lattice structure.
 We have a unit cell given by
\begin_inset Formula 
\[
\boldsymbol{A}=\left(\begin{array}{c}
\boldsymbol{a}_{1}^{T}\\
\boldsymbol{a}_{2}^{T}\\
\boldsymbol{a}_{3}^{T}
\end{array}\right)=\underline{A}\boldsymbol{E},\quad\boldsymbol{E}=\left(\begin{array}{c}
\boldsymbol{e}_{x}^{T}\\
\boldsymbol{e}_{y}^{T}\\
\boldsymbol{e}_{z}^{T}
\end{array}\right),\quad\underline{A}=\left(\begin{array}{ccc}
a_{H} & 0 & 0\\
0 & a_{H} & 0\\
0 & 0 & c_{H}
\end{array}\right)\left(\begin{array}{ccc}
\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right)
\]

\end_inset

and two sites 
\begin_inset Formula $\boldsymbol{s}^{T}=\underline{s}^{T}\underline{A}$
\end_inset

, 
\begin_inset Formula $\underline{s}_{1}^{T}=\left(\frac{1}{3}\frac{2}{3}0\right)$
\end_inset

, 
\begin_inset Formula $\underline{s}_{2}^{T}=\left(\frac{2}{3}\frac{1}{3}0\right)$
\end_inset

.
 The reciprocal space is spanned by
\begin_inset Formula 
\[
\boldsymbol{G}=2\pi\boldsymbol{A}^{-T}=\left(\begin{array}{c}
\boldsymbol{g}_{1}^{T}\\
\boldsymbol{g}_{2}^{T}\\
\boldsymbol{g}_{3}^{T}
\end{array}\right)=2\pi\underline{G}\boldsymbol{E},\quad\underline{G}=\underline{A}^{-T}=\left(\begin{array}{ccc}
\frac{1}{a_{H}} & 0 & 0\\
0 & \frac{1}{a_{H}} & 0\\
0 & 0 & \frac{1}{c_{H}}
\end{array}\right)\left(\begin{array}{ccc}
\frac{2}{\sqrt{3}} & 0 & 0\\
\frac{1}{\sqrt{3}} & 1 & 0\\
0 & 0 & 1
\end{array}\right)
\]

\end_inset

(
\begin_inset Formula $\underline{A}^{-T}=\left(\underline{A}^{T}\right)^{-1}=\left(\underline{A}^{-1}\right)^{T}$
\end_inset

) with orthogonality
\begin_inset Formula 
\[
\underline{G}\underline{A}^{T}=\underline{A}\underline{G}^{T}=1
\]

\end_inset

and closure
\begin_inset Formula 
\[
\underline{G}^{T}\underline{A}=\underline{A}^{T}\underline{G}=1
\]

\end_inset

The two high symmetry points in Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:real-and-reciprocal"

\end_inset

 are
\begin_inset Formula 
\[
\boldsymbol{M}=\left(\frac{1}{2}00\right)\boldsymbol{G}=\frac{2\pi}{a_{H}}\left(\frac{1}{\sqrt{3}}00\right)
\]

\end_inset


\begin_inset Formula 
\[
\boldsymbol{K}=\left(\frac{1}{3}\frac{1}{3}0\right)\boldsymbol{G}=\frac{2\pi}{a_{H}}\left(\frac{1}{\sqrt{3}}\frac{1}{3}0\right)
\]

\end_inset


\end_layout

\begin_layout Standard
The symmorphic space group is 
\begin_inset Formula $191$
\end_inset

 with a maximal point group 
\begin_inset Formula $D_{6h}$
\end_inset

 at 
\begin_inset Formula $\left(000\right)$
\end_inset

 with generators 
\begin_inset Formula $C_{6}\left(z\right)$
\end_inset

, 
\begin_inset Formula $C_{2}\left(0\right)$
\end_inset

 and 
\begin_inset Formula $I$
\end_inset

 (indicated in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:real-and-reciprocal"

\end_inset

).
 Clearly 
\begin_inset Formula $\hat{C}_{6}^{-1}\boldsymbol{s}_{1}=\boldsymbol{s}_{2}$
\end_inset

.
 The local pointgroup at the two sites is 
\begin_inset Formula $D_{3h}$
\end_inset

 with generators 
\begin_inset Formula $C_{3}$
\end_inset

, 
\begin_inset Formula $C_{2}\left(0\right)$
\end_inset

 and 
\begin_inset Formula $\overline{6}=IC_{6}$
\end_inset

.
 For the sake of definiteness let's define operations.
 We understand the operation 
\begin_inset Formula $g$
\end_inset

 to actively transform the basis vectors 
\begin_inset Formula $\boldsymbol{e}_{x,y,z}$
\end_inset

 into a new set according to 
\begin_inset Formula 
\[
g\boldsymbol{E}^{T}=\boldsymbol{E}^{T}R\left(g\right)
\]

\end_inset

where 
\begin_inset Formula $R\left(g\right)$
\end_inset

 is the 3D real space representation matrix of operation 
\begin_inset Formula $g$
\end_inset

.
 Multiplication from the left with a symmetry operation as in 
\begin_inset Formula $g\boldsymbol{E}^{T}$
\end_inset

 denotes the actively transformed object 
\begin_inset Formula $\boldsymbol{E}^{\prime T}$
\end_inset

.
 If we emphasize that something is the operation (as opposed to the representati
on matrix) we use an explicit operator symbol as in 
\begin_inset Formula $\hat{C}_{6}$
\end_inset

, while representation matrices are denoted by plain symbols.
\end_layout

\begin_layout Standard
For example 
\begin_inset Formula 
\[
g=\hat{C}_{6},\quad R\left(C_{6}\right)=\left(\begin{array}{ccc}
\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0\\
\frac{\sqrt{3}}{2} & \frac{1}{2} & 0\\
0 & 0 & 1
\end{array}\right)
\]

\end_inset

transformes the basis 
\begin_inset Formula $\boldsymbol{E}^{T}$
\end_inset

 as
\begin_inset Formula 
\[
\hat{C}_{6}\left(\boldsymbol{e}_{x}\boldsymbol{e}_{y}\boldsymbol{e}_{z}\right)=\left(\boldsymbol{e}_{x}\boldsymbol{e}_{y}\boldsymbol{e}_{z}\right)R\left(C_{6}\right)=\left(\frac{\boldsymbol{e}_{x}+\sqrt{3}\boldsymbol{e}_{y}}{2},\frac{-\sqrt{3}\boldsymbol{e}_{x}+\boldsymbol{e}_{y}}{2},\boldsymbol{e}_{z}\right)
\]

\end_inset


\end_layout

\begin_layout Standard
The coordinates of the actively rotated real space vector 
\begin_inset Formula $\boldsymbol{r}=\boldsymbol{E}^{T}\underline{r}$
\end_inset

 can be obtained from the basis transformation according to 
\begin_inset Formula 
\[
\boldsymbol{r}^{\prime}=g\boldsymbol{r}=g\boldsymbol{E}^{T}\underline{r}=\boldsymbol{E}^{T}\left(R\left(g\right)\underline{r}\right)
\]

\end_inset

which can be abbreviated by
\begin_inset Formula 
\[
g\underline{r}=R\left(g\right)\underline{r}
\]

\end_inset

or 
\begin_inset Formula $g\underline{r}^{T}=\underline{r}^{T}R^{T}\left(g\right)$
\end_inset

.
\end_layout

\begin_layout Standard
Applied to our example we get the following relation
\begin_inset Formula 
\begin{eqnarray*}
\hat{C}_{6}^{-1}\boldsymbol{s}_{1} & = & \boldsymbol{E}^{T}C_{6}^{-1}\underline{s}_{1}\\
 & = & \boldsymbol{E}^{T}C_{6}^{-1}\underline{A}^{T}\left(\begin{array}{c}
\frac{1}{3}\\
\frac{2}{3}\\
0
\end{array}\right)\\
 & = & \boldsymbol{E}^{T}\left(\underline{A}^{T}\underline{G}\right)C_{6}^{-1}\underline{A}^{T}\left(\begin{array}{c}
\frac{1}{3}\\
\frac{2}{3}\\
0
\end{array}\right)\\
 & = & \boldsymbol{E}^{T}\underline{A}^{T}\left(\underline{G}C_{6}^{-1}\underline{A}^{T}\right)\left(\begin{array}{c}
\frac{1}{3}\\
\frac{2}{3}\\
0
\end{array}\right)\\
 & = & \boldsymbol{E}^{T}\underline{A}^{T}\left(\begin{array}{ccc}
0 & 1 & 0\\
-1 & 1 & 0\\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{c}
\frac{1}{3}\\
\frac{2}{3}\\
0
\end{array}\right)\\
 & = & \boldsymbol{E}^{T}\underline{A}^{T}\left(\begin{array}{c}
\frac{2}{3}\\
\frac{1}{3}\\
0
\end{array}\right)\\
 & = & \boldsymbol{s}_{2}
\end{eqnarray*}

\end_inset

A set of functions of the coordinates (orbitals) transforms as 
\begin_inset Formula 
\[
gf_{m}\left(\underline{r}\right)\stackrel{{\scriptscriptstyle \mathrm{def}}}{=}f_{m}\left(g^{-1}\underline{r}\right)=f_{m}\left(\underline{r}^{T}R\left(g\right)\right)\stackrel{{\scriptscriptstyle \mathrm{def}}}{=}\sum_{n}f_{n}\left(\underline{r}\right)D_{nm}^{f}\left(g\right)
\]

\end_inset

where 
\begin_inset Formula $D_{nm}^{f}\left(g\right)$
\end_inset

 are the representation matrices of the group in the space spanned by 
\begin_inset Formula $f_{m}$
\end_inset

.
 In other words the orbitals 
\begin_inset Formula $gf_{m}\left(\underline{r}\right)$
\end_inset

 are also actively transformed like the Cartesian basis vectors 
\begin_inset Formula $\boldsymbol{e}_{x,y,z}$
\end_inset

.
\end_layout

\begin_layout Standard
In our examples we have a light 
\begin_inset Formula $p$
\end_inset


\begin_inset space ~
\end_inset

element sitting at the two sites of lower than maximal symmetry.
 The essential basis consists of one
\begin_inset space ~
\end_inset


\begin_inset Formula $s$
\end_inset

 and three
\begin_inset space ~
\end_inset


\begin_inset Formula $p$
\end_inset


\begin_inset space ~
\end_inset

orbitals.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Bandweights-of-boron"

\end_inset

 shows the band structure of MgB
\begin_inset Formula $_{2}$
\end_inset

, where the boron bands are highlighted.
 (The Mg bands do not play a big role in this energy window.)
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/MgB2/bwfplo.png
	lyxscale 10
	width 60col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Bandweights-of-boron"

\end_inset

Bandweights of boron in MgB
\begin_inset Formula $_{2}$
\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
The irreducible representations of the site symmetry 
\begin_inset Formula $D_{3h}$
\end_inset

 make 
\begin_inset Formula $p_{x,y}$
\end_inset

 an 
\begin_inset Formula $E^{\prime\prime}$
\end_inset

 doublet and 
\begin_inset Formula $p_{z}$
\end_inset

 a 
\begin_inset Formula $A_{2}^{\prime\prime}$
\end_inset

 singlet, while the 
\begin_inset Formula $s$
\end_inset

-orbital represents an 
\begin_inset Formula $A_{1}^{\prime}$
\end_inset

 singlet.
 In total we have 
\begin_inset Formula $8$
\end_inset


\begin_inset space ~
\end_inset

orbitals and hence bands.
 Due to the symmetry the 
\begin_inset Formula $p_{z}$
\end_inset

 bands largely decouple from the other orbitals along high symmetry lines
 in the 
\begin_inset Formula $k_{z}=0$
\end_inset

 and 
\begin_inset Formula $k_{z}=\frac{\pi}{c_{H}}$
\end_inset

 planes.
 At general points they of course couple.
 Furthermore, these bands cross the Fermi level, while the 
\begin_inset Formula $s$
\end_inset

 and 
\begin_inset Formula $p_{x,y}$
\end_inset

 bands form bonding and anti-bonding band complexes of three bands each
 separated by a gap.
 The bonding band complex has exactly three bands, while the AB complex
 shows a high degree of band entangling with other bands.
 The two 
\begin_inset Formula $p_{z}$
\end_inset

 bands cannot be separated into two distinct bands because of their Fermi
 surface and symmetry (see Sec.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Hexagonal-pz"

\end_inset

).
 However the planar-orbital bands can be represented by different kinds
 of Wannier functions.
\end_layout

\begin_layout Standard
The simplest way of defining WFs is to use the atomic 
\begin_inset Formula $s$
\end_inset

, 
\begin_inset Formula $p_{x,y}$
\end_inset


\begin_inset space ~
\end_inset

orbitals sitting at the two sites.
 This more or less reproduces the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo{}
\end_layout

\end_inset

 band characters.
 Another alternative is to try to find WFs for the bonding (anti-bonding)
 bands only.
 There are essentially three bands in each band complex and it is immediately
 clear that atom centered functions cannot fulfill the crystal symmetry
 due to the number of sites (two).
 The bands must be combinations of orbitals from both sites and hence an
 odd (3) number of orbitals from an even number of sites (2) cannot be symmetric.
 We discussed the irreducible representations of the orbitals in the site
 symmetry above.
 We have a singlet 
\begin_inset Formula $\left(s\right)$
\end_inset

 and a doublet (
\begin_inset Formula $p_{x,y}$
\end_inset

) (as is reflected in the degeneracies at the 
\begin_inset Formula $\Gamma$
\end_inset

-point) but all three orbitals are mixed in the bonding bands (no further
 decoupling).
 
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/MgB2/sp2-orbitals.pdf
	width 60col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:sp2-hybrid-basis"

\end_inset


\begin_inset Formula $sp^{2}$
\end_inset

-hybrid basis.
 The left panel shows one 
\begin_inset Formula $sp^{2}$
\end_inset

-orbital at both sites (filled) and the remaining two 
\begin_inset Formula $120^{\circ}/240^{\circ}$
\end_inset

-rotated orbitals at 
\begin_inset Formula $s_{1}$
\end_inset

 (open).
 The right panel shows a bonding bond-centered combination (full) and two
 symmetry related combinations (open).
\end_layout

\end_inset


\end_layout

\end_inset

 Unfortunately, there is hardly a simple linear combination of say the 
\begin_inset Formula $p_{x,y}$
\end_inset

 orbitals from both sites, which will fulfill the full crystal symmetry.
 Hence, we have to resort to reducible representations aiming at constructing
 as highly symmetric orbitals as possible.
 The well known answer is of course 
\begin_inset Formula $sp^{2}$
\end_inset

 hybrids.
 We introduce the following orbitals at each site 
\begin_inset Formula 
\[
\left(\begin{array}{c}
\Phi_{1}\\
\Phi_{2}\\
\Phi_{3}
\end{array}\right)=\frac{1}{\sqrt{3}}\left(\begin{array}{c}
s+\sqrt{2}p_{x}\\
s+\sqrt{2}C_{3}p_{x}\\
s+\sqrt{2}C_{3}^{2}p_{x}
\end{array}\right)
\]

\end_inset

which are depicted in the left panel of Figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sp2-hybrid-basis"

\end_inset

.
 Note, that here 
\begin_inset Formula $C_{3}p_{x}$
\end_inset

 is the 
\begin_inset Formula $C_{3}$
\end_inset

 rotation of the 
\begin_inset Formula $p_{x}$
\end_inset


\begin_inset space ~
\end_inset

orbital with the rotation origin at the site not at the global coordinate
 origin.
\end_layout

\begin_layout Standard
Obviously, 
\begin_inset Formula $C_{3}p_{x}$
\end_inset

 is not parallel to 
\begin_inset Formula $p_{x}$
\end_inset

 and hence contains some 
\begin_inset Formula $p_{y}$
\end_inset

.
 On the other hand 
\begin_inset Formula $\left(1+C_{3}+C_{3}^{2}\right)=3P_{\parallel}$
\end_inset

 where 
\begin_inset Formula $P_{\parallel}$
\end_inset

 is a projector onto the rotation axis.
 For in-plane objects 
\begin_inset Formula $P_{\parallel}p_{x,y}=0$
\end_inset

.
 This consideration shows that the 
\begin_inset Formula $p$
\end_inset

-parts of 
\begin_inset Formula $\Phi_{1,2,3}$
\end_inset

 are linearly dependent, however, with the 
\begin_inset Formula $s$
\end_inset

-admixture they span the same space as 
\begin_inset Formula $s$
\end_inset

 and 
\begin_inset Formula $p_{x,y}$
\end_inset

.
 The factors are chosen such that 
\begin_inset Formula $\left|\Phi_{i}\right|^{2}=1$
\end_inset

 and that 
\begin_inset Formula $\sum_{i}\left|\Phi_{i}\right|^{2}$
\end_inset

 contains one 
\begin_inset Formula $\left|s\right|^{2}$
\end_inset

 density and two 
\begin_inset Formula $\left|p\right|^{2}$
\end_inset

 densities.
 
\end_layout

\begin_layout Standard
Now, we can put three 
\begin_inset Formula $sp^{2}$
\end_inset

 hybrids onto each site (rotated by 
\begin_inset Formula $60^{\circ},$
\end_inset

 
\begin_inset Formula $180^{\circ}$
\end_inset

 and 
\begin_inset Formula $300^{\circ}$
\end_inset

 at the second site) and clearly they posses maximum crystal symmetry as
 a reducible 3-dimensional representation.
 (Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sp2-hybrid-basis"

\end_inset

, left panel).
 The symmetry relations are (rotation origin at 
\begin_inset Formula $\left(000\right)$
\end_inset

)
\begin_inset Formula 
\[
C_{6}\Phi_{s_{2},1}=\Phi_{s_{1},3},\ldots
\]

\end_inset


\begin_inset Formula 
\[
C_{3}\Phi_{s,1}=\Phi_{s,2},\ldots
\]

\end_inset


\begin_inset Formula 
\[
C_{2}\left(0\right)\Phi_{s_{2},1}=\Phi_{s_{2},1},\quad C_{2}\left(0\right)\Phi_{s_{2},2}=\Phi_{s_{2},3},\ldots
\]

\end_inset


\begin_inset Formula 
\[
I\Phi_{s_{2},1}=\Phi_{s_{1},2},\ldots
\]

\end_inset

where we have droped the details which show in which unit cell the transformed
 orbitals end up.
 This information is important for a detailed and complete description of
 the space group symmetry but is not necessary here.
 What these relations show however is that the 
\begin_inset Formula $sp^{2}$
\end_inset


\begin_inset space ~
\end_inset

orbitals at both sites span a basis under the full symmetry.
 The resulting Wannier function fat bands are shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:3-orbital--Wannier-function"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/MgB2/wanbweights_6sp2.png
	lyxscale 10
	width 80col%

\end_inset


\begin_inset space \hspace*{}
\length -10mm
\end_inset


\begin_inset Graphics
	filename pic/wan_user/MgB2/sp2.jpg
	lyxscale 10
	width 30col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:3-orbital--Wannier-function"

\end_inset

Wannier function fit with 3 
\begin_inset Formula $sp^{2}$
\end_inset

-orbital per site.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

The advantage of this high symmetry is that we now can create linear combination
s of one 
\begin_inset Formula $sp^{2}$
\end_inset

 from each site at the bond centers and by construction this additionally
 generates two similar symmetry related combinations (Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sp2-hybrid-basis"

\end_inset

, right panel).
 This would not be possible if we used the irreducible 
\begin_inset Formula $s$
\end_inset

 and 
\begin_inset Formula $p_{xy}$
\end_inset


\begin_inset space ~
\end_inset

orbitals.
 In that way one can separate the bonding from the anti-bonding orbitals.
 This is an example for the fact that in many covalent 
\begin_inset Formula $p$
\end_inset

-electron systems the bonding/anti-bonding bands must be described by bond-cente
red Wannier functions (Carbon chain: 
\begin_inset Formula $sp^{1}$
\end_inset

, diamond 
\begin_inset Formula $sp^{3}$
\end_inset

).
 Once we have separated the bonding Wannier functions, we can of course
 produce either a separate fit for the bonding or anti-bonding bands respectivel
y (Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Bonding/anti-bonding-bond-center"

\end_inset

) or a fit for bonding and anti-bonding bands together (however with clear
 energy separation of the orbital).
\begin_inset Float figure
placement H
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/MgB2/wanbweights_sp2_B.png
	lyxscale 10
	width 80col%

\end_inset


\begin_inset space \hspace*{}
\length -10mm
\end_inset


\begin_inset Graphics
	filename pic/wan_user/MgB2/sp2_B.jpg
	lyxscale 10
	width 30col%

\end_inset


\end_layout

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pic/wan_user/MgB2/wanbweights_sp2_AB.png
	lyxscale 10
	width 80col%

\end_inset


\begin_inset space \hspace*{}
\length -10mm
\end_inset


\begin_inset Graphics
	filename pic/wan_user/MgB2/sp2_AB.jpg
	lyxscale 10
	width 30col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Bonding/anti-bonding-bond-center"

\end_inset

Bonding/anti-bonding bond centered Wannier function fits.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
One last consideration concerns the choice of energy windows.
 For the entangled anti-bonding bands to be fitted better we put a narrow
 energy window onto the occupied bonding bands but allow for a large Gaussian
 tail at the upper side of the energy window.
 This pulls the majority weight from the well-separated bonding bands and
 magically adds the missing anti-bonding weights.
 If we extend the main energy window to encompass the anti-bonding bands
 it would pull in weights from higher lying bands, which also show 
\begin_inset Formula $s$
\end_inset

 and 
\begin_inset Formula $p_{x,y}$
\end_inset

 character, which in turn pulls the fitted bands up in energy.
 This trick can be used rather generally to improve the Wannier fit.
\end_layout

\begin_layout Subsection
Hexagonal, Graphene, MgB
\begin_inset Formula $_{2}$
\end_inset

: 
\begin_inset Formula $p_{z}$
\end_inset


\begin_inset space ~
\end_inset

orbitals
\begin_inset CommandInset label
LatexCommand label
name "subsec:Hexagonal-pz"

\end_inset


\end_layout

\begin_layout Standard
To elucidate the situation with the 
\begin_inset Formula $p_{z}$
\end_inset


\begin_inset space ~
\end_inset

subsystem we perform a little excercise.
 As we stated earlier the 
\begin_inset Formula $p_{z}$
\end_inset


\begin_inset space ~
\end_inset

bands largely decouple from the rest, but have a Fermi surface which prohibits
 the separation of bonding and antibonding bands
\end_layout

\begin_layout Standard
The local 
\begin_inset Formula $p_{z}$
\end_inset

 basis consists of two functions 
\begin_inset Formula $\Phi_{Rs}\left(\boldsymbol{r}\right)=\Phi_{s}\left(\boldsymbol{r}-\boldsymbol{R}-\boldsymbol{s}\right)$
\end_inset

 destinguished by the site label 
\begin_inset Formula $s=s_{1},s_{2}$
\end_inset

.
 This basis can be used as Wannier projectors and allows the construction
 of 
\begin_inset Formula $p_{z}$
\end_inset


\begin_inset space ~
\end_inset

WFs for the 
\begin_inset Formula $p_{z}$
\end_inset


\begin_inset space ~
\end_inset

two-band complex as a whole.
 The corresponding Bloch functions, which form a basis for the eigenstates
 of the Hamiltonian are
\begin_inset Formula 
\[
\Phi_{s}^{\boldsymbol{k}}=\frac{1}{\sqrt{N}}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\left(\boldsymbol{R}+\boldsymbol{s}\right)}\Phi_{Rs}
\]

\end_inset

The convenient additional phase factor 
\begin_inset Formula $\mathrm{e}^{i\boldsymbol{k}\boldsymbol{s}}$
\end_inset

 is a uniform 
\begin_inset Formula $U\left(1\right)$
\end_inset

 phase of each function and does only affect the Boundary condition: 
\begin_inset Formula $\Phi_{s}^{\boldsymbol{k}+\boldsymbol{G}}=\Phi_{s}^{\boldsymbol{k}}\mathrm{e}^{i\boldsymbol{G}\boldsymbol{s}}$
\end_inset

.
 It removes the origin dependence in matrix elements.
\end_layout

\begin_layout Standard
For molecules one can form symmetrized molecular orbitals.
 So we will introduce them for the lattice as well.
 The 
\begin_inset Quotes eld
\end_inset

benzene-ring
\begin_inset Quotes erd
\end_inset

 orbitals around the point 
\begin_inset Formula $\left(000\right)$
\end_inset

 (center point in the left panel of Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:real-and-reciprocal"

\end_inset

) are
\begin_inset Formula 
\begin{eqnarray*}
\Phi_{\pm} & = & \frac{1}{\sqrt{6}}\left[\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{1}}\pm\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{2}}\right]
\end{eqnarray*}

\end_inset

Its Bloch functions are given by
\begin_inset Formula 
\begin{eqnarray*}
\Phi_{\pm}^{\boldsymbol{k}} & = & \frac{1}{\sqrt{N}}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\Phi_{\pm}\\
 & = & \frac{1}{\sqrt{N}}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\frac{1}{\sqrt{6}}\left[\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{1}}\pm\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{2}}\right]_{\boldsymbol{R}}
\end{eqnarray*}

\end_inset

Note that the subscript 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

 at the square bracket means that one first constructs the molecular orbital
 by applying all symmetry operators and then shifts the whole object to
 the lattice vector 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

.
\end_layout

\begin_layout Standard
We use the following detailed transformation properties 
\begin_inset Formula 
\[
C_{3}\underline{s}_{1}=-\underline{s}_{2}=\left(-\frac{2}{3}-\frac{1}{3}0\right)=\underline{s}_{1}-\left(110\right)
\]

\end_inset


\begin_inset Formula 
\[
C_{3}^{2}\underline{s}_{1}=\left(\frac{1}{3}-\frac{1}{3}0\right)=\underline{s}_{1}-\left(010\right)
\]

\end_inset


\begin_inset Formula 
\[
C_{3}\underline{s}_{2}=\left(-\frac{1}{3}\frac{1}{3}0\right)=\underline{s}_{2}-\left(100\right)
\]

\end_inset


\begin_inset Formula 
\[
C_{3}^{2}\underline{s}_{2}=\left(-\frac{1}{3}-\frac{2}{3}0\right)=\underline{s}_{2}-\left(110\right)
\]

\end_inset

to re-express the ring Bloch functions via the local basis Bloch functions
 (of which we know that they form a basis):
\begin_inset Formula 
\begin{eqnarray}
\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{1}} & = & \Phi_{s_{1}}\left(\boldsymbol{r}-\boldsymbol{s}_{1}\right)+\Phi_{s_{1}}\left(\boldsymbol{r}-\boldsymbol{s}_{1}+\left(110\right)\boldsymbol{A}\right)+\Phi_{s_{1}}\left(\boldsymbol{r}-\boldsymbol{s}_{1}+\left(010\right)\boldsymbol{A}\right)\nonumber \\
\left[\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{1}}\right]_{\boldsymbol{R}} & = & \Phi_{s_{1}}\left(\boldsymbol{r}-\boldsymbol{R}-\boldsymbol{s}_{1}\right)+\Phi_{s_{1}}\left(\boldsymbol{r}-\left(\boldsymbol{R}-\left(110\right)\boldsymbol{A}\right)-\boldsymbol{s}_{1}\right)\label{eq:sumC3-of-phi1}\\
 &  & \quad+\Phi_{s_{1}}\left(\boldsymbol{r}-\left(\boldsymbol{R}-\left(010\right)\boldsymbol{A}\right)-\boldsymbol{s}_{1}\right)
\end{eqnarray}

\end_inset

and similarily for 
\begin_inset Formula $s_{2}$
\end_inset

.
 Under the Bloch sum we can redefine the 
\begin_inset Formula $\boldsymbol{R}$
\end_inset

 summation variable to remove the additional lattice vector shifts of Eq.
 (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:sumC3-of-phi1"

\end_inset

)
\begin_inset Formula 
\begin{eqnarray*}
\frac{1}{\sqrt{N}}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\left[\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{1}}\right]_{\boldsymbol{R}} & = & \Phi_{s_{1}}^{\boldsymbol{k}}\mathrm{e}^{-i\boldsymbol{k}\boldsymbol{s}_{1}}+\Phi_{s_{1}}^{\boldsymbol{k}}\mathrm{e}^{i\boldsymbol{k}\left(\left(110\right)\boldsymbol{A}-\boldsymbol{s}_{1}\right)}+\Phi_{s_{1}}\mathrm{e}^{i\boldsymbol{k}\left(\left(010\right)\boldsymbol{A}-\boldsymbol{s}_{1}\right)}
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\Phi_{\pm}^{\boldsymbol{k}} & = & \frac{1}{\sqrt{N}}\sum_{R}\mathrm{e}^{i\boldsymbol{k}\boldsymbol{R}}\frac{1}{\sqrt{6}}\left[\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{1}}\pm\left(1+C_{3}+C_{3}^{2}\right)\Phi_{s_{2}}\right]_{\boldsymbol{R}}\\
 & = & \frac{1}{\sqrt{6}}\left(\left(1+\mathrm{e}^{i\boldsymbol{k}\left(110\right)\boldsymbol{A}}+\mathrm{e}^{i\boldsymbol{k}\left(010\right)\boldsymbol{A}}\right)\Phi_{s_{1}}^{\boldsymbol{k}}\pm\left(1+\mathrm{e}^{i\boldsymbol{k}\left(100\right)\boldsymbol{A}}+\mathrm{e}^{i\boldsymbol{k}\left(110\right)\boldsymbol{A}}\right)\Phi_{s_{2}}^{\boldsymbol{k}}\right)\mathrm{e}^{-i\boldsymbol{k}\boldsymbol{s}_{1}}
\end{eqnarray*}

\end_inset

One can see that the additional phase factors vanish for 
\begin_inset Formula $\boldsymbol{k}=\pm\left(\frac{1}{3}\frac{1}{3}k_{z}\right)\boldsymbol{G}$
\end_inset

.
 Hence on the whole line 
\begin_inset Formula $KH$
\end_inset

 
\begin_inset Formula 
\begin{eqnarray*}
\Phi_{\pm}^{\boldsymbol{k}\in KH} & = & 0
\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Standard
Consequently, the ring functions do not span a basis for the whole BZ.
 This leads to singularity warnings in the WF module.
 If, however the input data for the 
\series bold
wandef
\series default
s is not precise enough such that numerically the phase factor calculated
 above is not exactly zero it looks as if the process worked.
 What you will experience is that the exact WF transform in 
\family typewriter
+wanband
\family default
 might fit the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fplo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 bands while the 
\family typewriter
+wanbandtb
\family default
 fit only fits if a very large 
\series bold
ham_cutoff
\series default
 in real space and a very small threshold 
\series bold
WF_ham_threshold
\series default
 are considered.
 This is so because the WFs defined via ring-functions are essentially non-analy
tic.
\end_layout

\begin_layout Standard
In summary, the ring functions are good for molecules but not for extended
 states.
 In moleculs there is no Fermi surface.
 The MO levels are discrete and hence in this case the MO WFs work.
 One last note: the ring functions do not work since the ring combines orbitals
 of several unit cells with fixed phase factors.
 There are less (2) resulting MOs than unit cells (3) involved in defining
 them.
 This should make clear that a linear combination of the MO Bloch sums cannot
 have the same flexibility and hence Hilbert space size as the individual
 orbitals.
 In truth we left out four of the six possible ring linear combinations.
 However our Hilbert space has 
\begin_inset Formula $\dim=2$
\end_inset

 (two bands or 
\begin_inset Formula $p_{z}$
\end_inset


\begin_inset space ~
\end_inset

orbitals per cell).
 This too tells us that there is something wrong.
 A locally complete ring basis has 
\begin_inset Formula $\dim=6$
\end_inset

 but we use a 
\begin_inset Formula $\dim=2$
\end_inset

 basis in the unit cell to construct this.
 The full phase flexibility requires to use all six ring orbitals but the
 resulting six Bloch bands must be overcomplete.
\end_layout

\begin_layout Standard
Of course, in other cases the ring MOs might work, e.g.
 if the ring is completely contained in a single unit cell and if the other
 four MOs have higher energy.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Due to the symmetry the 
\begin_inset Formula $p_{z}$
\end_inset

 bands decouple from the other orbitals in large parts of the Brillouin
 zone.
 This can be seen from the Bloch functions 
\begin_inset Formula 
\[
\Phi_{s\mu}^{k}=\sum_{R}\Phi_{s\mu}\left(\underline{r}-\underline{R}-\underline{s}\right)\mathrm{e}^{i\underline{k}\left(\underline{R}+\underline{s}\right)}
\]

\end_inset

which transform under a spacegroup operation 
\begin_inset Formula $g=\left\{ \alpha\mid\underline{\tau}\right\} $
\end_inset

, 
\begin_inset Formula $g\underline{r}=\alpha\underline{r}+\underline{\tau}$
\end_inset

 like 
\begin_inset Formula 
\[
g\Phi_{s\mu}^{k}=\sum_{\nu}\Phi_{s_{g}\nu}^{\alpha k}D_{\nu\mu}\mathrm{e}^{-i\alpha\underline{k}\underline{\tau}}
\]

\end_inset

where 
\begin_inset Formula $\underline{s}_{g}$
\end_inset

 is the site obtained by folding 
\begin_inset Formula $g\underline{s}$
\end_inset

 back into the unit cell and 
\begin_inset Formula $\underline{\tau}$
\end_inset

 is a translation about a non-lattice vector (essential in non-symmorphic
 space groups).
\end_layout

\begin_layout Plain Layout
None of the point group operations contained in the space group operations
 transforms a 
\begin_inset Formula $p_{z}$
\end_inset

 orbital into an 
\begin_inset Formula $s$
\end_inset

 or 
\begin_inset Formula $p_{x,y}$
\end_inset

 orbital
\end_layout

\begin_layout Plain Layout
For 
\begin_inset Formula $k$
\end_inset

-vectors without a 
\begin_inset Formula $k_{z}$
\end_inset

 component or with 
\begin_inset Formula $k_{z}=\frac{\pi}{c_{H}}$
\end_inset

 on the BZ border there are two classes of operations, which can belong
 to the group of the 
\begin_inset Formula $k$
\end_inset


\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
-vector.
 The first class turns 
\begin_inset Formula $\underline{k}=\left(k_{x}k_{y}0\right)$
\end_inset

 into itself
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ifx
\backslash
mystandalone
\backslash
undefined
\end_layout

\begin_layout Plain Layout


\backslash
else
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "ref"
options "plain"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
fi
\end_layout

\end_inset


\end_layout

\end_body
\end_document
