## Linking the monopole part with Hartree-Fock theory

The single-particle energy $$\epsilon_p$$ resulting from for example a self-consistent Hartree-Fock field, or from first order in many-body perturbation theory, is given by (in an uncoupled basis) $$\begin{equation*} \epsilon_p=\langle p| \tilde{f}|p\rangle = \langle p|\hat{h}_0|p\rangle +\sum_{i\le F} \langle pi|\hat{v}|pi\rangle+\frac{1}{2}\sum_{ij\le F} \langle pij|\hat{w}|pij\rangle, \end{equation*}$$ where we have included the three-body interaction as well.