Linking the monopole part with Hartree-Fock theory, angular momentum
We can rewrite this equation in an angular coupled basis (\( jj \)-coupled basis) as
$$
\begin{equation}
\epsilon_{\alpha_p}= \langle \alpha_p|\hat{h}_0|\alpha_p\rangle+\frac{1}{2j_p+1}\sum_{\alpha_i\le \alpha_F}\sum_{J} (2J+1)\langle (\alpha_p\alpha_i)J | \hat{v} | (\alpha_p\alpha_i)J \rangle,
\tag{23}
\end{equation}
$$
or
$$
\begin{equation}
\epsilon_{\alpha_p}= \langle \alpha_p|\hat{h}_0|\alpha_p\rangle+\frac{1}{2j_p+1}\sum_{\alpha_i\le \alpha_F}\sum_{J} (2J+1)\langle (\alpha_p\alpha_i)J | \tilde{v} | (\alpha_p\alpha_i)J \rangle,
\tag{24}
\end{equation}
$$
where the first equation contains a two-body force only while
Eq.
(24) includes the medium-modified contribution
from the three-body interaction as well.