More on the monopole term

The spherical single-particle states, provide an important ingredient for the formation of shells and interplay between spherical configurations and deformation in nuclei. Large shell gaps obtained from a monopole Hamiltonian are a prerequisite to obtain certain magic numbers. Equation (20) can also be expressed in terms of the medium-modified two-body interaction defined in Eq. (19), that is we can have $$ \begin{equation} \tilde{V}_{\alpha_p\alpha_q} = \frac{\sum_{J}(2J+1) \langle (\alpha_p\alpha_q)J | \tilde{v} | (\alpha_p\alpha_q)J \rangle }{\sum_{J}(2J+1)}. \tag{22} \end{equation} $$