Let us compare a system of \( N \)-independent electrons with pseudo-independent electrons described by the mean-field Hartree-Fock model. Fill the table below. Use Koopmans theorem when appropriate. Define any notations you use, i.e.:
\[
\begin{array}{r}
h_{i} \equiv h\left(x_{i}\right)=-\frac{1}{2} \nabla_{i}^{2}-\sum_{A} \frac{Z_{A}}{\left|R_{a}-r_{i}\right|} \\
v_{i j} \equiv v\left(x_{i}, x_{j}\right)=\frac{1}{\left|r_{i}-r_{j}\right|} \\
\Phi_{k}=\mid \phi_{1} \phi_{2} \ldots \not \chi_{K} \ldots \phi_{N}> \\
\Phi_{i}^{a}=\mid \phi_{1} \phi_{2} \ldots\left\{\not \phi_{a}\right\} \ldots \phi_{N}>
\end{array}
\]
\begin{tabular}{||l|l|l||}
\hline & Independent electrons & Pseudo-independent electrons \\
\hline Hamiltonian & \( H=\sum_{i=1}^{N} \hat{h}_{i} \) & \( H=\sum_{i=1}^{N} \hat{h}_{i}+\frac{1}{2} \sum_{i j=1}^{N} v_{i j} \) \\
\hline Wfn \( \left|\Phi_{0}\right\rangle= \) & \( \left|\phi_{1} \phi_{2} \ldots \phi_{N}\right\rangle \) & \( \left|\phi_{1} \phi_{2} \ldots \phi_{N}\right\rangle \) \\
\hline\( \phi_{i} \) are solutions of: & & \\
\hline Total energy is: & & \\
\hline \begin{tabular}{l}
Energy required to remove \\
electron from \( \phi_{k} \)
\end{tabular} & & \\
\hline \begin{tabular}{l}
Energy for attaching \\
electron to \( \phi_{a} \notin \Phi_{0} \)
\end{tabular} & & \\
\hline \begin{tabular}{l}
Energy difference between \\
\( \Phi_{k} \) and \( \Phi_{j} \)
\end{tabular} & & \\
\hline \begin{tabular}{l}
Energy difference between \\
\( \Phi_{0} \) and \( \Phi_{i}^{a} \)
\end{tabular} & & \\
\hline
\end{tabular}