## Analyzing our results, decomposing the Hamiltonian

One possible decomposition of the effective interaction is to express the $$k$$-th component of the interaction $$\langle (\alpha_p\alpha_q)J|\hat{v}_{k}|(\alpha_r\alpha_s)J\rangle$$ in a $$jj$$-coupled basis, where $$\hat{v}_k$$ is related to the matrix elements $$\langle (\alpha_p\alpha_q)J|\hat{v}|(\alpha_r\alpha_s)J\rangle$$ (or $$\langle (\alpha_p\alpha_q)J|\tilde{v}|(\alpha_r\alpha_s)J\rangle$$ through the relation \begin{align} \langle (\alpha_p\alpha_q)J|\hat{v}_{k}|(\alpha_r\alpha_s)J\rangle&=(-1)^{J}(2k+1) \sum_{LL'SS'}\langle \alpha_p\alpha_q|LSJ\rangle \langle \alpha_r\alpha_s|L'S'J\rangle \left\{ \begin{array}{ccc} L&S&J\\ S'&L'&k \end{array} \right\} \nonumber \\ &\times \sum_{J'}(-1)^{J'}(2J'+1)\left\{ \begin{array}{ccc} L&S&J'\\ S'&L'&k \end{array} \right\} \sum_{\alpha_{p}'\alpha_{q}'\alpha_{r}'\alpha_{s}'}\langle \alpha_p'\alpha_q'|LSJ'\rangle \nonumber\\ &\times \langle \alpha_r'\alpha_s'|L'S'J'\rangle \langle (\alpha_p'\alpha_q')J'|\hat{v}|(\alpha_r'\alpha_s')J'\rangle . \tag{28} \end{align}