TalentDFT

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===== Paper and pen exercises (Wednesday 7/16) =====
  * Continue working on the problem set from {{:exercises_secondquantization.pdf|yesterday}}. 

  * For a local, spin-independent two-body interaction, derive the coordinate space representations of i) the HF single particle Schroedinger equation, and ii) the expression for $E_{HF}$ starting from the general expressions in class. Express answers in terms of $\rho({\bf r})$ and $\rho({\bf r}\sigma,{\bf r}'\sigma')$ as in the lecture.

  * Prove Koopman's theorem, which states that if we approximate the $N-1$ particle eigenstates by $a_i|\Phi^N\rangle$, we can interpret the HF eigenvalues $\epsilon_i$ as removal energies. That is show that  

\begin{equation}
\epsilon_i = \langle \Phi^{N}|H|\Phi^{N}\rangle - \langle\Phi^{N}|a^{\dagger}_iHa_i|\Phi^{N}\rangle
\end{equation}

  * Prove Brillouin's theorem, which states that in the HF basis, the many-body Hamiltonian matrix doesn't connect the $0p0h$ and $1p1h$ sectors:

\begin{equation}
\langle \Phi^{a}_{i}|H|\Phi\rangle = 0 
\end{equation}

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