Electromagnetic multipole moments and transitions

where we have defined $$ \langle l _{a}\vert\vert [Y^{\lambda -1}(\hat{r})\otimes\mathbf{l}]^{\lambda }\vert\vert l _{b}\rangle=(-1)^{\lambda +l _{a}+l _{b}} \sqrt{(2\lambda +1)l _{b}(l _{b}+1)(2l _{b}+1)} $$ $$ \times\left\{\begin{array}{ccc} {\lambda -1} & {1}& {\lambda}\\ {l _{b}}& {l _{a}} & {l _{b}} \end{array}\right\} \langle l _{a}\vert\vert Y^{\lambda -1}(\hat{r})\vert\vert l _{b}\rangle, $$ with $$ \langle l _{a}\vert\vert Y^{\lambda -1}(\hat{r})\vert\vert l _{b}\rangle=(-1)^{l _{a}} \sqrt{\frac{(2l _{a}+1)(2l _{b}+1)(2\lambda -1)}{4\pi }}\left(\begin{array}{ccc} {l _{a}} & {\lambda -1} & {l _{b}}\\ {0} & {0}& {0}\end{array}\right) . $$