## The monopole term

An important ingredient in studies of effective interactions and their
applications to nuclear structure, is the so-called monopole
interaction, normally defined in terms of a nucleon-nucleon
interaction \( \hat{v} \)
$$
\begin{equation}
\bar{V}_{\alpha_p\alpha_q} = \frac{\sum_{J}(2J+1) \langle (\alpha_p\alpha_q)J | \hat{v} | (\alpha_p\alpha_q)J \rangle }{\sum_{J}(2J+1)},
\tag{20}
\end{equation}
$$
where the total angular momentum of a two-body state \( J \) runs over all
possible values.
In the above equation we have defined a nucleon-nucleon interaction in a so-called angular-momentum
coupled representation with the symbol \( \alpha_{p,q} \) representing all possible quantum numbers except the
magnetic substates \( m_{j_{p,q}} \)
The monopole Hamiltonian can be interpreted as an angle-averaged matrix element.
We have assumed that the single-particle angular momenta \( j_p \) and \( j_q \) couple to a total
two-particle angular momentum \( J \). The summation over \( J \) with the value \( 2J+1 \) can be replaced
by \( \sum_J(2J+1)=(2j_p+1)(2j_q+1) \) if \( \alpha_p\ne \alpha_q \). If \( \alpha_p=\alpha_q \) we can generalize this equation to, assuming that
our states can represent either protons or neutrons,
$$
\begin{equation}
\sum_{J}(2J+1)=(2j_p+1)(2j_q+1-\delta_{\alpha_p\alpha_q}).
\tag{21}
\end{equation}
$$