## Analyzing our results, decomposing the Hamiltonian

Similar to the
decomposition in the \( jj \)-scheme, the \( LS \)-coupled matrix element of a
given component \( k \)
are related to the corresponding matrix elements of the total
interaction in the \( jj \)-scheme by
$$
\begin{equation*}
\begin{array}{ll}
&\\
\langle(\tilde{\alpha}_p\tilde{\alpha}_q )LSJ'T\vert\hat{v}_{k}\vert (\tilde{\alpha}_r\tilde{\alpha}_s )L'S'J'T\rangle&=
{\displaystyle\frac{1}{\sqrt{(1+\delta_{\tilde{\alpha}_p\tilde{\alpha}_q})(1+\delta_{\tilde{\alpha}_r\tilde{\alpha}_s})}}}
(-1)^{J'}\hat{k}\left\{\begin{array}{ccc}L&S&J'\\S'&L'&k
\end{array}\right\}
\\&\\
&\times {\displaystyle\sum_{J}}(-1)^{J}\hat{J}\left\{\begin{array}{ccc}L&S&J\\S'&L'&k
\end{array}\right\}
{\displaystyle \sum_{\alpha_p \alpha_q \alpha_r \alpha_s}}
\langle \alpha_p\alpha_q|LSJ\rangle
\langle \alpha_r\alpha_s|L'S'J\rangle
\\&\\
&\times\sqrt{(1+\delta_{\alpha_p\alpha_q})(1+\delta_{\alpha_r\alpha_s})}\langle(\alpha_p\alpha_q)JT\vert\hat{v}\vert (\alpha_r\alpha_s)JT\rangle
.\end{array}
\end{equation*}
$$