Table 2.

Matrix expressions of the irreducible spherical tensors T_l,m in the |i〉 basis. The expressions for T_l,−m are found by $T_{l,-m}={(-1)}^m{T}_{l,m}^{†}$, where † means hermitian conjugation (i.e. simple transposition in the case of real entries of these matrices).

$T_{0,0}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$

$T_{1,0}=\left(\begin{array}{cccc}\frac{3}{2}& 0& 0& 0\\ 0& \frac{1}{2}& 0& 0\\ 0& 0& -\frac{1}{2}& 0\\ 0& 0& 0& -\frac{3}{2}\end{array}\right)T_{1,1}=\left(\begin{array}{cccc}0& -\sqrt{\frac{3}{2}}& 0& 0\\ 0& 0& -\sqrt{2}& 0\\ 0& 0& 0& -\sqrt{\frac{3}{2}}\\ 0& 0& 0& 0\end{array}\right)$

$T_{2,0}=\left(\begin{array}{cccc}\sqrt{\frac{3}{2}}& 0& 0& 0\\ 0& -\sqrt{\frac{3}{2}}& 0& 0\\ 0& 0& -\sqrt{\frac{3}{2}}& 0\\ 0& 0& 0& \sqrt{\frac{3}{2}}\end{array}\right)T_{2,1}=\left(\begin{array}{cccc}0& -\sqrt{3}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \sqrt{3}\\ 0& 0& 0& 0\end{array}\right)T_{2,2}=\left(\begin{array}{cccc}0& 0& \sqrt{3}& 0\\ 0& 0& 0& \sqrt{3}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

$T_{3,0}=\left(\begin{array}{cccc}\frac{3}{2\sqrt{10}}& 0& 0& 0\\ 0& -\frac{9}{2\sqrt{10}}& 0& 0\\ 0& 0& \frac{9}{2\sqrt{10}}& 0\\ 0& 0& 0& -\frac{3}{2\sqrt{10}}\end{array}\right)T_{3,1}=\left(\begin{array}{cccc}0& -\frac{3}{\sqrt{10}}& 0& 0\\ 0& 0& 3\sqrt{\frac{3}{10}}& 0\\ 0& 0& 0& -\frac{3}{\sqrt{10}}\\ 0& 0& 0& 0\end{array}\right)T_{3,2}=\left(\begin{array}{cccc}0& 0& \frac{3}{2}& 0\\ 0& 0& 0& -\frac{3}{2}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)T_{3,3}=\left(\begin{array}{cccc}0& 0& 0& -\frac{3}{\sqrt{2}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$