1\section{Silicon --- valence and low lying conduction states}
2\label{sec11:silicon}
3
4\begin{figure}[h!]
5\centering
6\includegraphics[width=0.25\columnwidth,trim={45pt 45pt 55pt 55pt},clip]{figure/example11/silicon.png}
7\caption{Unit cell of Silicon crystal plotted with the \xcrysden{} program.}
8\label{fig11.0}
9\end{figure}
10
11\subsection*{Valence States}
12\begin{itemize}
13\item Outline: {\it Obtain MLWFs for the valence bands of silicon.}
14\end{itemize}
15
16
17\begin{itemize}
18	\item[1-5] {\it Inspect the output file {\tt silicon.wout}. The total spread converges to its minimum value after just a
19few iterations. Note that the geometric centre of each MLWF lies at the centre of the Si--Si bond. Note
20also that the memory requirement for the minimisation of the spread is very low as the MLWFs are
21defined by just the $4\times4$ unitary matrices $U(\mathbf{k})$.}
22
23Below a snippet from the {\tt silicon.wout} output file
24  \begin{tcolorbox}[sharp corners,boxrule=0.5pt]
25  {\small
26\begin{verbatim}
27 Final State
28  WF centre and spread    1  ( -0.674701,  0.674701, -0.674701 )     1.59185520
29  WF centre and spread    2  ( -0.674701, -0.674701,  0.674701 )     1.59185520
30  WF centre and spread    3  (  0.674701,  0.674701,  0.674701 )     1.59185520
31  WF centre and spread    4  (  0.674701, -0.674701, -0.674701 )     1.59185520
32  Sum of centres and spreads ( -0.000000,  0.000000,  0.000000 )     6.36742081
33
34         Spreads (Ang^2)       Omega I      =     5.801375426
35        ================       Omega D      =     0.000000000
36                               Omega OD     =     0.566045385
37    Final Spread (Ang^2)       Omega Total  =     6.367420811
38 ------------------------------------------------------------------------------
39\end{verbatim}
40}
41\end{tcolorbox}
42Memory estimates may be found in the {\tt MEMORY ESTIMATE} section of the {\tt silicon.wout} file.
43  \begin{tcolorbox}[sharp corners,boxrule=0.5pt]
44  {\small
45\begin{verbatim}
46 *============================================================================*
47 |                              MEMORY ESTIMATE                               |
48 |         Maximum RAM allocated during each phase of the calculation         |
49 *============================================================================*
50 |                        Disentanglement            1.57 Mb                  |
51 |                            Wannierise:            0.47 Mb                  |
52
53\end{verbatim}
54}
55\end{tcolorbox}
56Converged values for the total spread functional and its components are shown in Tab.~\ref{tab11.1}.
57
58\begin{table}[t!]
59\centering
60\caption{Converged values of the components of spread functional and their sum, given in \angsqd{}.}
61\begin{tabular}{@{} lllll @{}}\toprule[1.5pt]
62MP mesh & $\Omega$ & $\Omega\tinysub{I}$ & $\Omega\tinysub{OD}$ & $\Omega\tinysub{D}$ \\\midrule
63$4\times4\times4$ & 6.3674  &  5.8014 & 0.5660 & 0.0000 \\\bottomrule
64\end{tabular}\label{tab11.1}
65\end{table}
66
67\item {\it Plot the MLWFs}
68The four MLWFs with $\sigma$ character describing the valence manifold of Si are shown in \Fig{fig11.1}(a),(b) and (c) respectively.
69	\begin{figure}[h!]
70	\centering
71	\subfloat[1]{\includegraphics[width=0.25\columnwidth,trim={500pt 0pt 500pt 0pt},clip]{figure/example11/silicon_valence_1.png}}
72	\centering
73	\subfloat[2]{\includegraphics[width=0.25\columnwidth,trim={500pt 0pt 500pt 0pt},clip]{figure/example11/silicon_valence_2.png}}
74	\centering
75	\subfloat[3]{\includegraphics[width=0.25\columnwidth,trim={500pt 0pt 500pt 0pt},clip]{figure/example11/silicon_valence_3.png}}
76	\centering
77	\subfloat[4]{\includegraphics[width=0.25\columnwidth,trim={500pt 0pt 500pt 0pt},clip]{figure/example11/silicon_valence_4.png}}
78	\caption{Four MLWFs for the valence manifold of Si.}\label{fig11.1}
79	\end{figure}
80
81\end{itemize}
82\newpage
83\subsection*{Valence + Conduction States}
84\begin{itemize}
85\item Outline: {\it Obtain MLWFs for the valence and low--lying conduction-band states of Si. Plot the
86interpolated bandstructure. Apply a scissors correction to the conduction bands.}
87\end{itemize}
88
89\begin{itemize}
90	\item {\it Inspect the output file {\tt silicon.wout}. The minimisation of the spread occurs in a two-step procedure. First, we minimise $\Omega\tinysub{I}$ -- this is the extraction of the optimal subspace in the disentanglement procedure. Then, we minimise $\Omega\tinysub{D} + \Omega\tinysub{OD}$.}
91
92	Converged values for the total spread functional and its components are shown in Tab.~\ref{tab11.2}.
93	The two groups of four MLWFs with $sp3$ character are shown in \Fig{fig11.2}
94
95	\begin{tcolorbox}[sharp corners,boxrule=0.5pt]
96	{\small
97	\begin{verbatim}
98                   Extraction of optimally-connected subspace
99                   ------------------------------------------
100 +---------------------------------------------------------------------+<-- DIS
101 |  Iter     Omega_I(i-1)      Omega_I(i)      Delta (frac.)    Time   |<-- DIS
102 +---------------------------------------------------------------------+<-- DIS
103       1      12.97640155      12.44630235       4.259E-02      0.00    <-- DIS
104       .		.					.				.			 .
105
106      79      12.33580893      12.33580893      -6.531E-11      0.23    <-- DIS
107      80      12.33580893      12.33580893      -5.241E-11      0.23    <-- DIS
108
109             <<<      Delta < 1.000E-10  over  3 iterations     >>>
110             <<< Disentanglement convergence criteria satisfied >>>
111
112        Final Omega_I    12.33580893 (Ang^2)
113
114 +----------------------------------------------------------------------------+
115	\end{verbatim}
116	}
117	\end{tcolorbox}
118
119	\begin{tcolorbox}[sharp corners,boxrule=0.5pt]
120	{\small
121	\begin{verbatim}
122	 Final State
123  WF centre and spread    1  (  1.807167,  1.807167,  1.807167 )     2.01695824
124  WF centre and spread    2  (  1.807167,  0.891636,  0.891636 )     2.01695823
125  WF centre and spread    3  (  0.891636,  1.807167,  0.891636 )     2.01695823
126  WF centre and spread    4  (  0.891636,  0.891636,  1.807167 )     2.01695824
127  WF centre and spread    5  (  0.226733,  0.226733,  0.226733 )     2.37014516
128  WF centre and spread    6  (  0.226733, -0.226733, -0.226733 )     2.37014508
129  WF centre and spread    7  ( -0.226733,  0.226733, -0.226733 )     2.37014515
130  WF centre and spread    8  ( -0.226733, -0.226733,  0.226733 )     2.37014514
131  Sum of centres and spreads (  5.397608,  5.397608,  5.397608 )    17.54841346
132
133         Spreads (Ang^2)       Omega I      =    12.335808933
134        ================       Omega D      =     0.177593840
135                               Omega OD     =     5.035010692
136    Final Spread (Ang^2)       Omega Total  =    17.548413465
137 ------------------------------------------------------------------------------
138 	\end{verbatim}
139 	}
140
141	\end{tcolorbox}
142
143
144	\begin{table}[t!]
145	\centering
146	\caption{Converged values of the components of spread functional and their sum, given in \angsqd{}.}
147	\begin{tabular}{@{} lllll @{}}\toprule[1.5pt]
148	MP mesh & $\Omega$ & $\Omega\tinysub{I}$ & $\Omega\tinysub{OD}$ & $\Omega\tinysub{D}$ \\\midrule
149	$4\times4\times4$ & 17.54841 &  12.3358 & 5.03501 & 0.17759 \\\bottomrule
150	\end{tabular}\label{tab11.2}
151	\end{table}
152
153	\begin{figure}[h!]
154	\centering
155	\subfloat[1]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_1.png}}
156	\centering
157	\subfloat[2]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_2.png}}
158	\centering
159	\subfloat[3]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_3.png}}
160	\centering
161	\subfloat[4]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_4.png}}\\
162	\centering
163	\subfloat[5]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_5.png}}
164	\centering
165	\subfloat[6]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_6.png}}
166	\centering
167	\subfloat[7]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_7.png}}
168	\centering
169	\subfloat[8]{\includegraphics[width=0.25\columnwidth,trim={300pt 0pt 500pt 0pt},clip]{figure/example11/silicon_v+c_8.png}}
170	\caption{Eight MLWFs with $sp3$ character, four on each Si atom in the unit cell.}\label{fig11.2}
171	\end{figure}
172\item {\it Plot the bandstructure.}
173
174The interpolated bandstructure is given in \Fig{fig11.3}.
175\begin{figure}[h!]
176\centering
177\includegraphics[width=0.7\columnwidth]{figure/example11/silicon_bandstructure.pdf}
178\caption{Bandstructure of silicon from DFT calculation (solid black) and from Wannier interpolation (solid red).}\label{fig11.3}
179\end{figure}
180\end{itemize}
181
182\subsection*{Further ideas}
183\begin{itemize}
184	\item {\it Compare the Wannier-interpolated bandstructure with the full pwscf bandstructure with a finer $k$-point grid.}
185
186    Result for a $8\times8\times8$ mesh is shown in \Fig{fig11.4}.
187
188	\begin{figure}[h!]
189	\centering
190	\includegraphics[width=0.7\columnwidth]{figure/example11/silicon_bs_DFT_vs_W90_finer_grid.pdf}
191	\caption{Bandstructure of silicon from DFT calculation (solid black) and from Wannier interpolation with a $4\times4\times4$ mesh (solid red) and $8\times8\times8$ mesh (solid blue).}\label{fig11.4}
192	\end{figure}
193
194	\item {\it Compute four MLWFs spanning the low-lying conduction states.}
195
196	The \MLWFs{} spanning the 4 low-lying conduction states are shown in \Fig{fig11.5}. The initial projections were 4 $sp3$ on the Si atom at (0,0,0).
197	\begin{figure}[h!]
198	\centering
199	\subfloat[1]{\includegraphics[width=0.25\columnwidth,trim={0pt 0pt 0pt 0pt},clip]{figure/example11/silicon_conduction_1.png}}
200	\centering
201	\subfloat[2]{\includegraphics[width=0.25\columnwidth,trim={0pt 0pt 0pt 0pt},clip]{figure/example11/silicon_conduction_2.png}}
202	\centering
203	\subfloat[3]{\includegraphics[width=0.25\columnwidth,trim={0pt 0pt 0pt 0pt},clip]{figure/example11/silicon_conduction_3.png}}
204	\centering
205	\subfloat[4]{\includegraphics[width=0.25\columnwidth,trim={0pt 0pt 0pt 0pt},clip]{figure/example11/silicon_conduction_4.png}}
206	\caption{}\label{fig11.5}
207	\end{figure}
208\end{itemize}
209