The term \( \langle ab|LSJ \rangle \) is a shorthand for the \( LS-jj \) transformation coefficient, $$ \langle ab|\lambda SJ \rangle = \hat{j_{a}}\hat{j_{b}} \hat{\lambda}\hat{S} \left\{ \begin{array}{ccc} l_{a}&s_a&j_{a}\\ l_{b}&s_b&j_{b}\\ \lambda &S &J \end{array} \right\}. $$ Here we use \( \hat{x} = \sqrt{2x +1} \). The factor \( F \) is defined as \( F=\frac{1-(-1)^{l+S+T}}{\sqrt{2}} \) if \( s_a = s_b \) and we .