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\begin_layout Title
Band weights --- coherent mixtures
\end_layout

\begin_layout Author
Klaus Koepernik
\end_layout

\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents

\end_inset


\end_layout

\begin_layout Standard
We will show how to create band weights for coherently mixed orbitals.
 This is different from adding band weights, since the band weights are
 already 
\begin_inset Quotes eld
\end_inset

magnitude-squared
\begin_inset Quotes erd
\end_inset

.
 The example is chosen such that we can have a little tour through the GUI.
\end_layout

\begin_layout Standard
The solution will be found in the 
\family typewriter
solution
\family default
 directory.
 You will work in the 
\family typewriter
task
\family default
 directory.
 
\end_layout

\begin_layout Standard
The basic converged calculation is alredy setup in 
\family typewriter
task
\family default
.
 It is MgB
\begin_inset Formula $_{2}$
\end_inset

 Fig(
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-structure-of"

\end_inset

).
 In the following we will refer to boron atoms 
\begin_inset Formula $\mathrm{B}_{0\ldots3}$
\end_inset

, which denote the atoms in the right panel.
 Keep this in mind.
\end_layout

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\begin_inset Float figure
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\begin_layout Plain Layout
\noindent
\align center
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	filename pics/mgb2strtop.jpg
	lyxscale 10
	width 50col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:The-structure-of"

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The structure of MgB
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.
 Left: unit cell.
 Right: usefull view.
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\end_inset


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\begin_layout Plain Layout

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\begin_layout Section
The structure view
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\begin_layout Standard
We need a particular structure representation to continue.
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\begin_layout Standard
So, we created the right panel of Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-structure-of"

\end_inset


\end_layout

\begin_layout Itemize
First open the structure
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\begin_deeper
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xfplo =.in
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Open the boundary dialog 
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In the 
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 selectbox choose 
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.
 
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Click on the arrow of the 
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MI{load}
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 button and choose 
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\backslash
MI{Hexagonal Faces}
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.
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\begin_layout Itemize
At the bottom enter the origin 
\begin_inset Formula $\left(\frac{2}{3},\frac{1}{3},0\right)$
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.
 Use fractionals as input: 
\family typewriter
2/3
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.
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\begin_layout Itemize
Hit 
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\backslash
MI{enter}
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 (
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\backslash
MI{Accept}
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 button).
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\begin_layout Itemize
Close the dialog.
\end_layout

\begin_layout Itemize
Click on the structure view to make it the current window.
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\begin_layout Itemize
Switch on the cartesian frame 
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	scale 50

\end_inset

 (hotkey: 
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status collapsed

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\backslash
MI{a a}
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) for better orientation.
 
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\begin_layout Itemize
Switch to top view 
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	filename pics/view.png
	scale 70

\end_inset

 (hotkey: 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{v t}
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)
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Hit the 
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MI{reset}
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 button or 
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MI{Ctrl-R R}
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 in the structure view (SV) .
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Now, open 
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.
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Check the 
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MI{visible}
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 checkbox.
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\begin_layout Itemize
Check out the content of the 
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\backslash
MI{Type}
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\end_inset

 combobox.
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\begin_layout Itemize
Hit the 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{Load default}
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\end_inset

 button, followed by 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
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\end_inset

.
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\begin_layout Itemize
Hit the 
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\backslash
MI{Set}
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 button and select 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{Unchecked}
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 from the menu and hit 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
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\end_inset

.
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\begin_layout Itemize
Now, click on the central atom (
\begin_inset Formula $\mathrm{B_{0}}$
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) in the SV to select the corresponding label in the 
\begin_inset ERT
status collapsed

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\backslash
MI{User defined}
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 table.
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Check the checkbox in the 
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MI{first column}
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 and edit the 
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MI{Label}
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 to read 
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\backslash
MI{B
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$
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_0
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$.}
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 (use 
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MI{double click}
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 or 
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 on the label).
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\end_deeper
\begin_layout Itemize
Repeat the last point for the other three atoms according to Fig.
 
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LatexCommand ref
reference "fig:The-structure-of"

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.
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Edit the 
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 line-edit and enter 
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 to 
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.
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Hit 
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 and close the dialog.
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Save the structure setup in 
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: 
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 or 
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MI{File 
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RA{} Save}
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\end_layout

\begin_layout Itemize
Close the SV (
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\backslash
MI{Ctrl-W}
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).
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\begin_layout Section
The default band weights
\end_layout

\begin_layout Standard
First we calculate the default fatbands, since they tell us the first fact
 about this system.
\end_layout

\begin_layout Itemize
Open 
\family typewriter
fedit
\family default
 and switch on 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{bandplot 
\backslash
RA{} bandstructure plot}
\end_layout

\end_inset

 and 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{bandplot 
\backslash
RA{} weights}
\end_layout

\end_inset

.
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\begin_layout Itemize
run 
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\begin_deeper
\begin_layout LyX-Code
fplo
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status collapsed

\begin_layout Plain Layout


\backslash
version{}
\end_layout

\end_inset

-x86_64 | tee out
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\end_deeper
\begin_layout Itemize
Search 
\begin_inset Quotes eld
\end_inset

Atom sites
\begin_inset Quotes erd
\end_inset

 in the output file 
\family typewriter
out
\family default
 to establish that the boron sites are 2 and 3.
\end_layout

\begin_layout Itemize
run
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\begin_deeper
\begin_layout LyX-Code
faddwei
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\backslash
version{}
\end_layout

\end_inset

-x86_64
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\end_deeper
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and answer yes to create =.addwei
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edit 
\family typewriter
=.addwei 
\family default
to look like this
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position "t"
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height_special "totalheight"
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LatexCommand verbatiminput
filename "../solutions/=.addwei"

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Now run
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faddwei
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\backslash
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-x86_64
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Run
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xfbp +bwsum
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go to the 
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 dialog and it's 
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MI{Weights}
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 tab.
 In the 
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MI{Weights}
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 table select all three with the mouse.
 Then click on the arrow of the buttom above the 
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\begin_layout Plain Layout


\backslash
MI{Weights}
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 table and select 
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.
 Hit 
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MI{enter}
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\end_inset

 or click the 
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MI{Apply}
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 button.
 In the main window hit the 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{autoscale all}
\end_layout

\end_inset

 button 
\begin_inset Graphics
	filename pics/worldbutton.png
	scale 70

\end_inset

 or press 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-A}
\end_layout

\end_inset

.
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\end_deeper
\begin_layout Itemize
Print the picture as 
\begin_inset Quotes eld
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bwei.png
\begin_inset Quotes erd
\end_inset

 (
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status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-P}
\end_layout

\end_inset

 or 
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status collapsed

\begin_layout Plain Layout


\backslash
MI{File 
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RA{} Print}
\end_layout

\end_inset

)
\end_layout

\begin_layout Itemize
Close xfbp (
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status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-Q}
\end_layout

\end_inset

 or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{File 
\backslash
RA{} Quit}
\end_layout

\end_inset

).
\end_layout

\begin_layout Standard
Your must see Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-default-B"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/bwei.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:The-default-B"

\end_inset

The default B 2s, 2
\begin_inset Formula $\mathrm{p}\sigma$
\end_inset

 and 2
\begin_inset Formula $\mathrm{p}\pi$
\end_inset

 fatbands.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
We have added all in-plane p orbitals (
\begin_inset Formula $\mathrm{p}_{x}$
\end_inset

 and 
\begin_inset Formula $\mathrm{p}_{y}$
\end_inset

) and called them 
\begin_inset Formula $\sigma$
\end_inset

 (which will become clearer later).
 The remaining 
\begin_inset Formula $\mathrm{p}_{z}$
\end_inset

-orbitals form 
\begin_inset Formula $\pi$
\end_inset

-bonds along the hexagonal boron ring.
 Last, the boron 
\begin_inset Formula $s$
\end_inset

-orbitals are also printed.
 We see that the s and 
\begin_inset Formula $\mathrm{p}\sigma$
\end_inset

 orbitals hybridize while the 
\begin_inset Formula $\mathrm{p}\pi$
\end_inset

 bands are mostly criss-crossing the s, 
\begin_inset Formula $\mathrm{p}\sigma$
\end_inset

-bands without much hybridization.
 This is typical for hexagonally coordinated carbon-ring like structures
 and indicative of the usefulness of the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids.
 Note further, that the 
\begin_inset Formula $\mathrm{p}_{x}$
\end_inset

 and 
\begin_inset Formula $\mathrm{p}_{y}$
\end_inset

 orbitals are transforming into each other under a fourfold rotation.
 The threefold rotation present at the boron sites does not have the same
 symmetry as the 
\begin_inset Formula $\mathrm{p}\sigma$
\end_inset

-orbitals.
 This is not really an issue for the calculations but it bugs a chemists's
 mind.
 The set of three 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids on the other hand will have manifest 
\begin_inset Formula $C_{3}$
\end_inset

 rotational symmetry.
\end_layout

\begin_layout Section
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids on all B sites
\end_layout

\begin_layout Standard
We will construct the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids at the central 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

-atom.
 At the other inequivalent boron site in the unit cell a similar set of
 3 
\begin_inset Formula $180^{\circ}$
\end_inset

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

-hybrids exists.
\end_layout

\begin_layout Standard
Now, lets count a bit.
 We have roughly 3 occupied s-
\begin_inset Formula $\mathrm{p}\sigma$
\end_inset

 bands (red and green in Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-default-B"

\end_inset

).
 In total we have 2 boron, which makes 6 such orbitals.
 Obviously, there must be 3 unoccupied bands of the same orbital character.
 If we find 3 equivalent orbitals per site, we span the same Hilbert space.
 The well known answer is the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrid
\begin_inset Formula 
\begin{eqnarray*}
\phi_{1} & = & \alpha s+\beta p_{x}\\
\phi_{2} & = & \alpha s+\beta C_{3}p_{x}\\
\phi_{3} & = & \alpha s+\beta C_{3}^{2}p_{x}
\end{eqnarray*}

\end_inset

where we need to choose the orientation of the 
\begin_inset Formula $\mathrm{p}_{x}$
\end_inset

 lobe to point along the connection of two closest boron atoms.
 Obviously
\begin_inset Formula 
\[
C_{3}\phi_{i}=\phi_{i+1\mod3}
\]

\end_inset


\end_layout

\begin_layout Standard
Compare Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Left:-a-sketch"

\end_inset

 to Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-structure-of"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/sp2-orbitals.eps
	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:Left:-a-sketch"

\end_inset

Left: a sketch of the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 hybrids on the 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

 and 
\begin_inset Formula $\mathrm{B}_{3}$
\end_inset

 sites.
 The open symbols denote the other two orbitals of site 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

.
 Right: two 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 hybrids of two closest boron sites can form a bond centered bonding orbital.
 The remaining two bond centered orbitals are depctied as open symbols.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
We still need to chose proper coefficients.
 There is the matter of sign and magnitude.
 First we discuss the magnitude.
 Assuming, that we have orthogonal orbitals (which is true at least for
 orbitals sitting on the same site.) we can determine the overlap of two
 orbitals at the same site
\begin_inset Formula 
\begin{eqnarray*}
\left\langle \phi_{1}\mid\phi_{2}\right\rangle  & = & \alpha^{2}\left|s\right|^{2}+\alpha\beta\left\langle s\mid C_{3}p_{x}\right\rangle +\alpha\beta\left\langle p_{x}\mid s\right\rangle +\beta^{2}\left\langle p_{x}\mid C_{3}p_{x}\right\rangle \\
 & = & \alpha^{2}+\beta^{2}\left\langle p_{x}\mid C_{3}p_{x}\right\rangle \\
 & = & \alpha^{2}+\beta^{2}\left\langle p_{x}\mid\cos\left(\gamma\right)p_{x}+\sin\left(\gamma\right)p_{y}\right\rangle ,\quad\gamma=120^{\circ}\\
 & = & \alpha^{2}+\beta^{2}\left\langle p_{x}\mid-\frac{1}{2}p_{x}+\frac{\sqrt{3}}{2}p_{y}\right\rangle \\
 & = & \alpha^{2}-\frac{1}{2}\beta^{2}
\end{eqnarray*}

\end_inset

This must be zero, so 
\begin_inset Formula $\alpha^{2}=\frac{1}{2}\beta^{2}$
\end_inset

.
 The norm of an orbital should be one 
\begin_inset Formula 
\begin{eqnarray*}
\left|\phi_{1}\right|^{2} & = & \alpha^{2}\left|s\right|^{2}+\alpha\beta\left\langle s\mid p_{x}\right\rangle +\alpha\beta\left\langle p_{x}\mid s\right\rangle +\beta^{2}\left|p_{x}\right|^{2}\\
 & = & \alpha^{2}+\beta^{2}
\end{eqnarray*}

\end_inset

Hence, 
\begin_inset Formula $1=\alpha^{2}+\beta^{2}=\frac{3}{2}\beta^{2}$
\end_inset

, and
\begin_inset Formula 
\[
\alpha=\sqrt{\frac{1}{3}}\approx0.577
\]

\end_inset


\begin_inset Formula 
\[
\beta=\sqrt{\frac{2}{3}}\approx0.816
\]

\end_inset


\end_layout

\begin_layout Standard
Another argument would be to request the following identity 
\begin_inset Formula $\sum_{i}\left|\phi_{i}\right|^{2}=3\alpha^{2}\left|s\right|^{2}+3\beta^{2}\left|p\right|^{2}\equiv\left|s\right|^{2}+2\left|p\right|^{2}$
\end_inset

 in order to obtain a proper density with one s and two p-electrons occupied
 (according to the band structure).
 Hence 
\begin_inset Formula $2\left|\alpha\right|^{2}=\left|\beta\right|^{2}$
\end_inset

.
 Normalization then gives the known result.
 
\end_layout

\begin_layout Standard
However, we are lazy and use 
\begin_inset Formula $\left|\alpha\right|=\left|\beta\right|=\frac{1}{\sqrt{2}}$
\end_inset

.
 In general, one has to make guesses for these pre-factors, since the optimal
 factors with respect to the final projection outcome usually depend on
 the hybridizations (on the Hamitonian).
 We should, however, at least choose coefficients in such a way that the
 resulting projected DOS is normalized to approximately two/one electrons
 for non-spinpolarized/spinpolarized calculations, respectively.
\end_layout

\begin_layout Standard
Now, if we pick 
\begin_inset Formula $\phi_{1}=\frac{1}{\sqrt{2}}\left(s+p_{x}\right)$
\end_inset

 for the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 -hybrid pointing from 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

 to 
\begin_inset Formula $\mathrm{B}_{2}$
\end_inset

 in Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-structure-of"

\end_inset

 we would expect the outcome to be the blue orbital at 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

 in the left panel Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Left:-a-sketch"

\end_inset

 (assuming blue is positive and red negative), right? Well not so fast.
 
\end_layout

\begin_layout Standard
You need to remember this: the orbital is (real) spherical harmonics times
 radial function.
 In FPLO the sign of the radial function is determined by the starting condition
 at the nucleus (all radial functions start positive at the radius 
\begin_inset Formula $0^{+}$
\end_inset

).
 Then the number of nodes determines the sign of the big bulge of the radial
 functions in the valence region of the atom.
 This bulge in turn determines the sign of matrix elements and it affects
 the sign choice for our hybrid orbitals, since we want the blue blob of
 Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Left:-a-sketch"

\end_inset

.
 
\end_layout

\begin_layout Itemize
Open the 
\family typewriter
fedit
\family default
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{options}
\end_layout

\end_inset

 menu and check 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{PLOT BASIS}
\end_layout

\end_inset

 (hotkey 7).
 Save and quit 
\family typewriter
fedit
\family default
.
\end_layout

\begin_layout Itemize
run
\end_layout

\begin_deeper
\begin_layout LyX-Code
fplo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
version{}
\end_layout

\end_inset

-x86_64 | tee out
\end_layout

\begin_layout Standard
(We could interrupt the run after the initial segment, but then it runs
 fast enough, so let it run.)
\end_layout

\end_deeper
\begin_layout Itemize
Have a look at the first component (the usual large component) of the radial
 valence functions for Wyckoff position 2 (boron)
\end_layout

\begin_deeper
\begin_layout LyX-Code
xfbp +fval.002.1
\end_layout

\begin_layout Standard
You will see (Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-large-component"

\end_inset

) that the 2s radial function is negative around 2 
\begin_inset Formula $a_{B}$
\end_inset

, while the 2p function is positive.
 Consequently we will need to use 
\begin_inset Formula 
\begin{equation}
\phi_{1}=\frac{1}{\sqrt{2}}\left(-s+p_{x}\right)\label{eq:sp2hybrid_b0_b2}
\end{equation}

\end_inset

 to obtain the desired 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-orbital.
 For clarity see Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Red:-negative.-Blue:"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/orbarithmetic.eps
	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:Red:-negative.-Blue:"

\end_inset

Red: negative.
 Blue: positive.
 The orbitals sit at the same site!
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/fval.png
	lyxscale 30
	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:The-large-component"

\end_inset

The large component of the radial semi-core and valence functions for boron.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Ideally, you switch the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{PLOT BASIS}
\end_layout

\end_inset

 option off now.
\end_layout

\begin_layout Standard
We are now prepared to create 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrid orbitals on both boron sites.
 We will create all three 
\begin_inset Formula $\phi_{1\ldots3}$
\end_inset

 at 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

 and then use a trick to get the orbitals at the second site.
 There are two boron sites per unit cell one of them being 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

.
 By looking at the structure you should understand that B
\begin_inset Formula $_{1\ldots3}$
\end_inset

 are all the second site.
 By placing one 
\begin_inset Formula $\phi$
\end_inset

 on each of those atoms we get our three second site orbitals.
 This example also emphasizes that it does not matter in which unit cell
 the projector is defined.
\end_layout

\begin_layout Standard
We need one more input, which is the connection vectors between the central
 boron atom and the other three.
 It is easiest to measure these and use them as local axes of a 
\begin_inset Formula $\mathrm{p}_{x}$
\end_inset

-orbital, thus defining the p-orbital contribution to the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

.
\end_layout

\begin_layout Itemize
Execute
\end_layout

\begin_deeper
\begin_layout LyX-Code
xfplo =.xstr
\end_layout

\end_deeper
\begin_layout Itemize
Press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{m}
\end_layout

\end_inset

 or use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Tools 
\backslash
RA{} Measure}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
In the SV first click on the center atom 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

 and then on 
\begin_inset Formula $\mathrm{B}_{1}$
\end_inset

 to get Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-measurement-of"

\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/measure.png
	lyxscale 30
	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:The-measurement-of"

\end_inset

The measurement of the vector 
\begin_inset Formula $\mathrm{B}_{1}-\mathrm{B}_{0}$
\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
You will convince yourself by looking at the table in the measurement tool
 that 
\family typewriter
dif
\family default
 is the vector pointing from the first atom in the table to the second.
 Write down the difference vector for 
\begin_inset Formula $\mathrm{B}_{1}-\mathrm{B}_{0}$
\end_inset

 and continue with the other two vectors.
 
\end_layout

\begin_deeper
\begin_layout Itemize
Either click on the second atom again to remove it from measurement 
\end_layout

\begin_layout Itemize
Or use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{escape}
\end_layout

\end_inset

 to delete the whole list
\end_layout

\begin_layout Itemize
You can click on three atoms to measure angles.
\end_layout

\begin_layout Itemize
Press the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{m}
\end_layout

\end_inset

 key to close this tool.
\end_layout

\begin_layout Itemize
Also take notice of the box on the bottom of the tool.
 This is plain text for easy copying.
\end_layout

\begin_layout Standard
Well, here is the table.
 Note, that we are not very precise.
 For the fatbands that is sufficient.
 For Wannier functions similar difference vectors must be often much more
 precise.
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Float table
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="4">
<features rotate="0" tabularvalignment="middle">
<column alignment="right" valignment="top">
<column alignment="right" valignment="top">
<column alignment="right" valignment="top">
<column alignment="right" valignment="top">
<row>
<cell alignment="right" valignment="top" bottomline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $i$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $y$
\end_inset


\end_layout

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

\begin_layout Plain Layout
\begin_inset Formula $x$
\end_inset


\end_layout

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

\begin_layout Plain Layout
1
\end_layout

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

\begin_layout Plain Layout
-1.68
\end_layout

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

\begin_layout Plain Layout
-2.91
\end_layout

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

\begin_layout Plain Layout
0
\end_layout

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

\begin_layout Plain Layout
2
\end_layout

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

\begin_layout Plain Layout
3.36
\end_layout

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

\begin_layout Plain Layout
0
\end_layout

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

\begin_layout Plain Layout
0
\end_layout

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

\begin_layout Plain Layout
3
\end_layout

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

\begin_layout Plain Layout
-1.68
\end_layout

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

\begin_layout Plain Layout
2.91
\end_layout

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

\begin_layout Plain Layout
0
\end_layout

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

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:The-difference-vectors"

\end_inset

The difference vectors 
\begin_inset Formula $\mathrm{B}_{i}-\mathrm{B}_{0}$
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
We are finally prepared.
 
\end_layout

\begin_layout Itemize
Close the measurement tool (press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{m}
\end_layout

\end_inset

)
\end_layout

\begin_layout Itemize
Choose menu entry 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Activities 
\backslash
RA{} FatBands}
\end_layout

\end_inset

 to open the band weight editor (BWE), Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-empty-band"

\end_inset

.
 Place the BWE and the SV side by side.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/xfplobw.png
	lyxscale 30
	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:The-empty-band"

\end_inset

The empty band weight editor
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
In the BWE click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{new molecular}
\end_layout

\end_inset

 button and type 
\family typewriter
sp2(3,1)
\family default
.
 The first number is the site number which you get by hovering with the
 mouse pointer over the central atom.
 In the status bar it will read: 
\family typewriter
B(S3,W2,T5)
\family default
, 
\family typewriter
S
\family default
=site, 
\family typewriter
W
\family default
=Wyckoff position, 
\family typewriter
T
\family default
=type (by default the nuclear charge, but it can be changed in the symmetry
 dialog).
 The second number in the name is an index for the orbital.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{New Contrib}
\end_layout

\end_inset

 button to open the molecular orbital contribution dialog (MOCD, Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Molecular-orbital-contribution"

\end_inset

) 
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/molcontribdlg1.png
	lyxscale 30
	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:Molecular-orbital-contribution"

\end_inset

Molecular orbital contribution dialog
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
In the MOCD click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on the central atom 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

.
\end_layout

\begin_layout Itemize
Set 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{orbital}
\end_layout

\end_inset

 to 
\family typewriter
2s+0
\family default
 and the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 to 
\family typewriter
-0.707
\family default
 (
\begin_inset Formula $\frac{1}{\sqrt{2}}$
\end_inset

).
 We do not need to set 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{axes}
\end_layout

\end_inset

, since an s orbital is spherical.
\end_layout

\begin_layout Itemize
Hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{OK}
\end_layout

\end_inset

 button).
\end_layout

\begin_deeper
\begin_layout Standard
\noindent
\align center
\begin_inset Box Boxed
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "50col%"
special "none"
height "1in"
height_special "totalheight"
status collapsed

\begin_layout Plain Layout
You can always use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

 or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{double click}
\end_layout

\end_inset

 on the items in the BWE to re-open the corresponding dialog.
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
Now, you have one contrib.
\end_layout

\begin_layout Itemize
Copy it by hitting the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Copy}
\end_layout

\end_inset

 button (or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Alt-y}
\end_layout

\end_inset

).
\end_layout

\begin_layout Itemize
The copied contrib has the focus, so hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

 and change the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{orbital}
\end_layout

\end_inset

 into 2p+1 (
\begin_inset Formula $\mathrm{p}_{x}$
\end_inset

) and the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 into 
\family typewriter
0.707
\family default
.
 In the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{x-axis}
\end_layout

\end_inset

 line-edits enter the first difference vector (the one to 
\begin_inset Formula $\mathrm{B}_{1}$
\end_inset

).
 See Tab.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:The-difference-vectors"

\end_inset

.
\end_layout

\begin_layout Itemize
Close the MOCD and compare with Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-first-sp2-projector"

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/firstbwesp2.png
	lyxscale 30
	width 90col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:The-first-sp2-projector"

\end_inset

The first 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 orbital projector
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
Are you convinced that we have defined a projector, which looks like the
 the blue blob of Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Left:-a-sketch"

\end_inset

 (just rotated)? Well, then let's add the rest.
\end_layout

\begin_layout Itemize
In the BWE single click on the first line for focus and click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Copy}
\end_layout

\end_inset

 button.
\end_layout

\begin_layout Itemize
Press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

 and change the last number to 2.
 Then press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Go to the second contrib (last line), hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

 and change the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{x-axis}
\end_layout

\end_inset

 line-edits to contain the second difference vector of Tab.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:The-difference-vectors"

\end_inset

.
\end_layout

\begin_layout Itemize
Close the MOCD.
\end_layout

\begin_layout Itemize
Click on the second molecular orbital (second green line) and hit the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{copy}
\end_layout

\end_inset

 button.
\end_layout

\begin_layout Itemize
Change the name (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

).
\end_layout

\begin_layout Itemize
Go to the last line, press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

 and enter the last difference vector into the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{x-axis}
\end_layout

\end_inset

 line-edits.
\end_layout

\begin_layout Itemize
Close the MOCD and in the BWE 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{File 
\backslash
RA{} Save}
\end_layout

\end_inset

 the current table.
\end_layout

\begin_layout Standard
We have defined all 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 orbitals of the first boron site, Fig 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-three-sp"

\end_inset

.
\end_layout

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

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/first3bwesp3.png
	lyxscale 30
	width 90col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:The-three-sp"

\end_inset

The three 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 orbital projector of the first boron site.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Let's define the three MOs of the second boron.
 All we need to do is to copy each MO, select the proper atom for all contributi
ons and to invert the sign of the coefficient of the 
\begin_inset Formula $\mathrm{p}_{x}$
\end_inset

-orbital.
 This is so, since the orbitals at the second site are 
\begin_inset Formula $180^{\circ}$
\end_inset

 rotate, which gives a factor of 
\begin_inset Formula $-1$
\end_inset

 for 
\begin_inset Formula $\mathrm{p}$
\end_inset

 but not for 
\begin_inset Formula $\mathrm{s}$
\end_inset

.
\end_layout

\begin_layout Itemize
Copy the first molecular orbital and type 
\begin_inset Formula $2\times$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-down}
\end_layout

\end_inset

 or click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{down}
\end_layout

\end_inset

 button 
\begin_inset Graphics
	filename pics/downbutton.png
	scale 75

\end_inset

 2 times.
\end_layout

\begin_layout Itemize
Rename this orbital into sp2(2,1) [site number 2 in the structure].
\end_layout

\begin_layout Itemize
Select it's first contribution, hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on 
\begin_inset Formula $\mathrm{B}_{1}$
\end_inset

 in the structure view.
 Then in the MOCD press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Select it's second contribution, hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on 
\begin_inset Formula $\mathrm{B}_{1}$
\end_inset

 in the structure view.
 In the MOCD, change the sign of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 and press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
Now, we copy-modify the second MO.
\end_layout

\begin_layout Itemize
Copy the second molecular orbital and type 
\begin_inset Formula $2\times$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-down}
\end_layout

\end_inset

 or click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{down}
\end_layout

\end_inset

 button 
\begin_inset Graphics
	filename pics/downbutton.png
	scale 75

\end_inset

 2 times.
\end_layout

\begin_layout Itemize
Rename this orbital into sp2(2,2) [site number 2 in the structure].
\end_layout

\begin_layout Itemize
Select the it's first contribution, hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on 
\begin_inset Formula $\mathrm{B}_{2}$
\end_inset

 in the structure view.
 The in the MOCD press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Select the it's second contribution, hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on 
\begin_inset Formula $\mathrm{B}_{2}$
\end_inset

 in the structure view.
 In the MOCD, change the sign of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 and press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
And, the last copy-modify action to get the the third MO of the second boron
 site.
\end_layout

\begin_layout Itemize
Copy the third molecular orbital and type 
\begin_inset Formula $2\times$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-down}
\end_layout

\end_inset

 or click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{down}
\end_layout

\end_inset

 button 
\begin_inset Graphics
	filename pics/downbutton.png
	scale 75

\end_inset

 2 times.
\end_layout

\begin_layout Itemize
Rename this orbital into sp2(2,3) [site number 2 in the structure].
\end_layout

\begin_layout Itemize
Select the it's first contribution, hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on 
\begin_inset Formula $\mathrm{B}_{3}$
\end_inset

 in the structure view.
 The in the MOCD press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Select the it's second contribution, hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Click the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Get Atom}
\end_layout

\end_inset

 button and click on 
\begin_inset Formula $\mathrm{B}_{3}$
\end_inset

 in the structure view.
 In the MOCD, change the sign of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 and press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:All-sp2"

\end_inset

 shows the table of all six 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-MO projectors.
\end_layout

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

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/allsp2.png
	lyxscale 30
	width 90col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

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

\end_inset

All six 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 molecular projectors.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Standard
Finally,
\end_layout

\begin_layout Itemize
in the BWE 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{File 
\backslash
RA{} Save}
\end_layout

\end_inset

 the current table and close everything: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-Q}
\end_layout

\end_inset

/
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Activities 
\backslash
RA{} Quit all}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
We are in the position to use these projectors now
\end_layout

\begin_layout Itemize
Open fedit and switch on 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{bandplot 
\backslash
RA{} Bandstructure plot}
\end_layout

\end_inset

, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{bandplot 
\backslash
RA{} Weights}
\end_layout

\end_inset

 and set 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Weights def File}
\end_layout

\end_inset

 to 
\family typewriter
=.bwdef
\family default
.
\end_layout

\begin_layout Itemize
Then run
\end_layout

\begin_deeper
\begin_layout LyX-Code
fplo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
version{}
\end_layout

\end_inset

-x86_64 | tee out
\end_layout

\end_deeper
\begin_layout Itemize
We got 
\family typewriter
+bdos/+bdos.b001
\family default
\SpecialChar \ldots{}
 
\family typewriter
+bdos/+bdos.b006
\family default
 and 
\family typewriter
+bweights_bwdef
\family default
.
 
\end_layout

\begin_layout Itemize
Execute
\end_layout

\begin_deeper
\begin_layout LyX-Code
xfbp +bweights_bwdef
\end_layout

\end_deeper
\begin_layout Itemize
Go to the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Edit 
\backslash
RA{} Sets}
\end_layout

\end_inset

 dialog, select the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Weights}
\end_layout

\end_inset

 tab.
\end_layout

\begin_layout Itemize
Select all items in the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Weights}
\end_layout

\end_inset

 table with the mouse.
 Click on the arrow of the button above the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Weights}
\end_layout

\end_inset

 table and select 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Checked}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
In the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{properties of all}
\end_layout

\end_inset

 box increased 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 to 8 and set the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{style}
\end_layout

\end_inset

 to individual.
\end_layout

\begin_layout Itemize
The weights should still be all selected.
 If not select all: click into the weights table and type 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-A}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
In the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{individual properties}
\end_layout

\end_inset

 box uncheck the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{fill}
\end_layout

\end_inset

 checkbox and set 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{skip}
\end_layout

\end_inset

 to 5.
\end_layout

\begin_layout Itemize
Hit 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{enter}
\end_layout

\end_inset

, close the dialog and autoscale all (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-A}
\end_layout

\end_inset

) to obtain Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:All-sp2-weights."

\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/bweisp2.png
	lyxscale 10
	width 50col%

\end_inset


\begin_inset Graphics
	filename pics/bdossp2.png
	lyxscale 10
	width 50col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:All-sp2-weights."

\end_inset

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

 weights.
 Left: fatbands.
 Right: DOS.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
You might notice that only two colors are visible.
 This is due to the high symmetry between the two boron sites.
 However, by switching on one pair of weights at a time you can convince
 yourself that the first three and the last three weights are different
 among each other (the last three seem to be identical to the first three,
 but they differ from the first three by their site number anyhow), which
 means that we at least are sure that we do not have some identical projectors.
\end_layout

\end_deeper
\begin_layout Itemize
Let's proceed to show the MO projected dos.
 Execute
\end_layout

\begin_deeper
\begin_layout LyX-Code
xfbp +bdos/*
\end_layout

\begin_layout Standard
You will realize that all six bdos look the same, which reflects the symmetry
 of our projector choice.
\end_layout

\begin_layout Standard
\begin_inset Box Boxed
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "80col%"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
For some taste
\end_layout

\begin_layout Itemize
Open 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Edit 
\backslash
RA{} Script/Transformations}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Select 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Mode 
\backslash
RA{} Python}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Write
\end_layout

\begin_deeper
\begin_layout LyX-Code
G[1].Gr[1].S[1].integrate()
\end_layout

\end_deeper
\begin_layout Itemize
Hit the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Apply}
\end_layout

\end_inset

 button.
 ONLY ONCE! Otherwise you integrate the integral.
\end_layout

\begin_layout Itemize
Close the dialog 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-W}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-A}
\end_layout

\end_inset

.
 
\end_layout

\begin_deeper
\begin_layout Plain Layout
You will see that the integrate DOS reaches around 0.949 electrons at the
 Fermi level (
\begin_inset Formula $E=0$
\end_inset

).
 This makes 5.694 electrons for all 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 orbitals.
 If we add the two occupied 1s electrons per boron (makes 4) and the 0.7
 occupied 
\begin_inset Formula $\mathrm{p}_{z}$
\end_inset

 electrons per boron (makes 1.4) we get a total of 11.094 electrons or 5.547
 per boron, which compares well with the population analysis.
 Hence, we chose the correct normalization for our projectors.
\begin_inset Newline newline
\end_inset

(The 0.7 occupied 
\begin_inset Formula $\mathrm{p_{z}}$
\end_inset

 electrons stem from an LDOS calculation, which we did not perform here.)
\end_layout

\end_deeper
\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
You can close the program: press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-W}
\end_layout

\end_inset

.
\end_layout

\begin_layout Section
Bonding and anti-bonding 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 hybrids
\end_layout

\begin_layout Standard
Finally, we get to the bonding and anti-bonding combinations of 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids.
 Since, we already setup all six hybrids, the final step is a lesson in
 copy-and-paste.
 One of the bonding (anti-bonding) combinations are obtained from 
\begin_inset Formula 
\[
\phi_{B}=\frac{1}{\sqrt{2}}\left(\phi_{1}\left(\mathrm{B}_{0}\right)\pm\phi_{1}\left(\mathrm{B}_{1}\right)\right)
\]

\end_inset

where 
\begin_inset Formula $\phi_{1}\left(\mathrm{B}_{0}\right)$
\end_inset

 it the 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrid at 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

, which points towards 
\begin_inset Formula $\mathrm{B}_{1}$
\end_inset

 and 
\begin_inset Formula $\phi_{1}\left(\mathrm{B}_{1}\right)$
\end_inset

 sits at 
\begin_inset Formula $\mathrm{B}_{1}$
\end_inset

 and points towards 
\begin_inset Formula $\mathrm{B}_{0}$
\end_inset

.
\end_layout

\begin_layout Standard
Please, tightly follow the steps.
\end_layout

\begin_layout Itemize
Execute
\end_layout

\begin_deeper
\begin_layout LyX-Code
xfplo =.bwdef&
\end_layout

\end_deeper
\begin_layout Itemize
Go to the first MO.
\end_layout

\begin_layout Itemize
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Copy}
\end_layout

\end_inset

 it followed by 
\begin_inset Formula $5\times$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-down}
\end_layout

\end_inset

 to make it the last MO in the table.
\end_layout

\begin_layout Itemize
Press 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{space}
\end_layout

\end_inset

 to rename it into 
\family typewriter
sp2 B1
\family default
.
\end_layout

\begin_layout Itemize
Select the first contribution of the 4-th MO (mouse click or cursor keys)
 and 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{copy}
\end_layout

\end_inset

 it.
 Press 
\begin_inset Formula $10\times$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-down}
\end_layout

\end_inset

 to make it the last line.
\end_layout

\begin_layout Itemize
In the same manner copy the second contribution of the 4-th MO and use 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-down}
\end_layout

\end_inset

 to make it the last line.
\end_layout

\begin_layout Itemize
Now, for each of the four contributions set the magnitude of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 to 0.5, but 
\series bold
keep the sign
\series default
.
\end_layout

\begin_layout Standard
We could now apply similar copy past procedure to duplicate MO 2(3) and
 add the contributions of MO 5(6) to create the other three bonding 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids as shown in the right panel of Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Left:-a-sketch"

\end_inset

.
 But we skip this part.
 Instead we construct the anti-bonding combination.
\end_layout

\begin_layout Itemize
Go to MO number 7.
\end_layout

\begin_layout Itemize
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Copy}
\end_layout

\end_inset

 it.
\end_layout

\begin_layout Itemize
Rename to 
\family typewriter
sp2 AB1
\family default
.
\end_layout

\begin_layout Itemize
Invert the sign of the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{factor}
\end_layout

\end_inset

 of the last two contributions of MO 8 to get Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:The-bonding-and"

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/BABsp2.png
	lyxscale 30
	width 90col%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:The-bonding-and"

\end_inset

The bonding and anti-bonding MOs.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
in the BWE 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{File 
\backslash
RA{} Save}
\end_layout

\end_inset

 (or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-S}
\end_layout

\end_inset

) the current table and close everything: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Ctrl-Q}
\end_layout

\end_inset

/
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
MI{Activities 
\backslash
RA{} Quit all}
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Now, run
\end_layout

\begin_deeper
\begin_layout LyX-Code
fplo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
version{}
\end_layout

\end_inset

-x86_64 | tee out
\end_layout

\end_deeper
\begin_layout Itemize
We got 
\family typewriter
+bdos/+bdos.b001
\family default
\SpecialChar \ldots{}
 
\family typewriter
+bdos/+bdos.b008
\family default
 and 
\family typewriter
+bweights_bwdef
\family default
.
 
\end_layout

\begin_layout Itemize
Execute
\end_layout

\begin_deeper
\begin_layout LyX-Code
xfbp +bdos/*
\end_layout

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

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename pics/bdosBAB.png
	lyxscale 10
	width 50col%

\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-and-DOS"

\end_inset

Bonding, anti-bonding and 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

 DOS.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
You will notice that in Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Bonding,-anti-bonding-and-DOS"

\end_inset

 the bonding (gray) and anti-bonding (orange) DOS are twice as tall as the
 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-DOS.
 This is due to the fact that only 3 bonding MOs are fully occupied and
 6 
\begin_inset Formula $\mathrm{sp}^{2}$
\end_inset

-hybrids are half-occupied.
 At the end it is the same number of electrons.
\end_layout

\end_deeper
\end_body
\end_document
