## 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, $$\begin{equation} \sum_{J}(2J+1)=(2j_p+1)(2j_q+1-\delta_{\alpha_p\alpha_q}). \tag{21} \end{equation}$$