## Analyzing our results, decomposing the Hamiltonian

The transformation from an $$LS$$ basis to a $$jj$$-coupled scheme is then given by the relation $$\begin{equation*} |(\alpha_p\alpha_q)J\rangle = \sum_{LS}\langle \alpha_p\alpha_q|LSJ \rangle |(\tilde{\alpha}_p\tilde{\alpha}_q)LSJ\rangle, \end{equation*}$$ where the symbol like $$\tilde{\alpha}_{p}$$ refers to the quantum numbers in an $$LS$$ basis, that is $$\tilde{\alpha}_{p}=(n_{p},l_{p},s_{p},t_{z_{p}})$$.

To derive Eq. (28), we have used the fact that the two-body matrix elements of $$\hat{v}_k$$ can also be interpreted in the representation of the $$LS$$-coupling scheme.