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 $$\begin{equation*} \begin{array}{ll} &\\ \langle(\tilde{\alpha}_p\tilde{\alpha}_q )LSJ'T\vert\hat{v}_{k}\vert (\tilde{\alpha}_r\tilde{\alpha}_s )L'S'J'T\rangle&= {\displaystyle\frac{1}{\sqrt{(1+\delta_{\tilde{\alpha}_p\tilde{\alpha}_q})(1+\delta_{\tilde{\alpha}_r\tilde{\alpha}_s})}}} (-1)^{J'}\hat{k}\left\{\begin{array}{ccc}L&S&J'\\S'&L'&k \end{array}\right\} \\&\\ &\times {\displaystyle\sum_{J}}(-1)^{J}\hat{J}\left\{\begin{array}{ccc}L&S&J\\S'&L'&k \end{array}\right\} {\displaystyle \sum_{\alpha_p \alpha_q \alpha_r \alpha_s}} \langle \alpha_p\alpha_q|LSJ\rangle \langle \alpha_r\alpha_s|L'S'J\rangle \\&\\ &\times\sqrt{(1+\delta_{\alpha_p\alpha_q})(1+\delta_{\alpha_r\alpha_s})}\langle(\alpha_p\alpha_q)JT\vert\hat{v}\vert (\alpha_r\alpha_s)JT\rangle .\end{array} \end{equation*}$$