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.