Appendix.tex

\begin{appendix}
\section{3D Plots}\label{sec:appendix}
We will now show the energy's dependencies on the variational parameters $\alpha$ and $\beta$ first for the case of two interacting electrons and then for the case of six interacting electrons in a harmonic oscillator potential.
\subsection{Two electron perturbed case}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.36]{ptwo1}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for two interacting electrons at the harmonic oscillator frequency $\omega=1$}
    \label{fig:ptwo1}
\end{SCfigure}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.36]{ptwo07}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for two interacting electrons at the harmonic oscillator frequency $\omega=0.7$}
    \label{fig:ptwo07}
\end{SCfigure}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.36]{ptwo05}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for two interacting electrons at the harmonic oscillator frequency $\omega=0.5$}
    \label{fig:ptwo05}
\end{SCfigure}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.36]{ptwo028}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for two interacting electrons at the harmonic oscillator frequency $\omega=0.28$}
    \label{fig:ptwo028}
\end{SCfigure}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.36]{ptwo001}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for two interacting electrons at the harmonic oscillator frequency $\omega=0.01$}
    \label{fig:ptwo001}
\end{SCfigure}
\FloatBarrier
\subsection{Six electron perturbed case}\label{sec:app_six_electron}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.48]{psix1}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for six interacting electrons at the harmonic oscillator frequency $\omega=1$}
    \label{fig:psix1}
\end{SCfigure}
\begin{SCfigure}[25][h]
    \centering
    \includegraphics[scale=0.53]{psix05}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for six interacting electrons at the harmonic oscillator frequency $\omega=0.5$}
    \label{fig:psix05}
\end{SCfigure}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.35]{psix028}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for six interacting electrons at the harmonic oscillator frequency $\omega=0.28$}
    \label{fig:psix028}
\end{SCfigure}
\begin{SCfigure}[30][h]
    \centering
    \includegraphics[scale=0.5]{psix001}
    \caption{Plot of $\alpha$- and $\beta$-dependencies of the ground state energy for six interacting electrons at the harmonic oscillator frequency $\omega=0.01$}
    \label{fig:psix001}
\end{SCfigure}
\end{appendix}