Appendix.tex

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\appendix
\appendixpage
\addappheadtotoc
\section{The benzene molecule}\label{benzex}
\begin{wrapfigure}[7]{r}{.3\textwidth}
	\vspace{-2.3em}
	\centering
	\begin{tikzpicture}
		\chemfig{1*6(-2-3-4-5-6-)}
	\end{tikzpicture}
	\caption{Indices of a benzene molecule}\label{benz}
\end{wrapfigure}
As an example the Hamiltonian of benzene is considered. In \cref{benz} one can see the indices of a benzene molecule. Remember that \(\bra{\phi_{\pi}(1)}\hat{H}\ket{\phi_{\pi}(1)} = 0\) and \cref{V}, the Hamiltonian reads:
\begin{align}
	\mqty{                            \\ \\ \\ \vb{H} = V_{pp\pi}\\ \\ \\} \ \mqty{						&  \mqty{1 & 2 & 3 & 4 & 5 & 6} \\
		\mqty{1                           \\ 2 \\ 3 \\ 4 \\ 5 \\ 6} &	\mqty*(0 & 1 & 0 & 0 & 0 & 1 \\
	1 & 0 & 1 & 0 & 0 & 0             \\
	0 & 1 & 0 & 1 & 0 & 0             \\
	0 & 0 & 1 & 0 & 1 & 0             \\
	0 & 0 & 0 & 1 & 0 & 1             \\
	1 & 0 & 0 & 0 & 1 & 0)}\label{BH}
\end{align}
As a helping aid, \cref{BH} shows the atomic indices of the atom on the top and to the left of the matrix. This will give an understanding of how to work with such matrices.
The structure of the benzene molecule is rotationally symmetric and rotating the indices one sixth must yield the same Hamiltonian. Consider the energy eigenvector:
\begin{align}
	\phi = \mqty(c_1 & c_2 & c_3 & c_4 & c_5 & c_6)
\end{align}
There must exist an operator that rotates the indices as such:
\begin{align}
	C_6\phi = \mqty(c_2 & c_3 & c_4 & c_5 & c_6 & c_1)
\end{align}
The rotated Hamiltonian is the same, and thus \(C_6\) and \(\vb{H}\) commute. The rotated vector must be an eigenvector with the same energy and it should be possible to find simultaneous eigenvectors to \(C_6\) and \(\vb{H}\).
\begin{align}
	C_6\phi = \mqty(c_2 & c_3 & c_4 & c_5 & c_6 & c_1) = \lambda\mqty(c_1 & c_2 & c_3 & c_4 & c_5 & c_6)
\end{align}
This operator \(C_6\) is represented with the matrix:
\begin{align}
	\vb{C}_6 = \mqty*(0 & 1 & 0 & 0 & 0 & 0  \\
	0                   & 0 & 1 & 0 & 0 & 0  \\
	0                   & 0 & 0 & 1 & 0 & 0  \\
	0                   & 0 & 0 & 0 & 1 & 0  \\
	0                   & 0 & 0 & 0 & 0 & 1  \\
	1                   & 0 & 0 & 0 & 0 & 0)
\end{align}
It can quickly be shown that the normalised eigenvectors to \(C_6\) are
\begin{align}
	\phi_n = \frac{1}{\sqrt{6}}\mqty(\lambda_n^0 & \lambda_n^1 & \lambda_n^2 & \lambda_n^3 & \lambda_n^4 & \lambda_n^5), \quad \lambda_n = \exp{-i2\pi n / 6}, \quad n = 0,1,2,3,4,5
\end{align}
These eigenvectors are also eigenvectors for \(\vb{H}\) with the eigenvalues:
\begin{align}
	\varepsilon_n = \lambda_n + \lambda_{n-1} = 2 \cos{n\pi/3}
\end{align}
Thus thanks to the rotational symmetry it was possible to find the eigenvectors and eigenenergies for the Hamiltonian.
\section{Additional figures}\label{appfigs}
\begin{figure}[h]
	\centering
	\begin{subfigure}[b]{0.45\textwidth}
		\includegraphics[width=\textwidth]{Figures/BetaimrealTE8.eps}
		\caption{Figure showing a plot of the Green's function at the \nth{8} site}
		\label{4th}
	\end{subfigure}
	~ %add desired spacing between images, e. g. ~, \quad, \qquad, \hfill etc.
	%(or a blank line to force the subfigure onto a new line)
	\begin{subfigure}[b]{0.45\textwidth}
		\includegraphics[width=\textwidth]{Figures/BetaimrealTE11.eps}
		\caption{Figure showing a plot of the Green's function at the \nth{11} site}
		\label{7th}
	\end{subfigure}
	\caption{Two plots showing how the Green's function changes as the site is changed. The \nth{8} and \nth{11} sites are corresponding to atoms of those indices (8, 11) in \cref{inlinepointplot}. Note how the LDOS changes (imaginary part) for the different sites.}\label{siteLDOSplot}
\end{figure}
\begin{figure}[h]
	\centering
	\begin{subfigure}[b]{0.3\textwidth}
		\includegraphics[width=\textwidth]{Figures/FabNPGBS.eps}
		\caption{Normal NPG}
		\label{Fabbs}
	\end{subfigure}
	~ %add desired spacing between images, e. g. ~, \quad, \qquad, \hfill etc.
	%(or a blank line to force the subfigure onto a new line)
	\begin{subfigure}[b]{0.3\textwidth}
		\includegraphics[width=\textwidth]{Figures/paraNPGBS.eps}
		\caption{Para NPG}
		\label{parabs}
	\end{subfigure}
	~ %add desired spacing between images, e. g. ~, \quad, \qquad, \hfill etc.
	%(or a blank line to force the subfigure onto a new line)
	\begin{subfigure}[b]{0.3\textwidth}
		\includegraphics[width=\textwidth]{Figures/metaNPGBS.eps}
		\caption{Meta NPG}
		\label{metabs}
	\end{subfigure}
	\caption{Plot showing band structures in the energy range \SI{-1.5}{\electronvolt} to \SI{1.5}{\electronvolt} for normal, para and meta NPG. The are plotted between symmetry points \(X\) and \(Y\) with respect to the origin \(\Gamma\)}\label{allbands}
\end{figure}

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