Electromagnetic multipole moments and transitions

The most common types of transitions are \( E1 \), \( E2 \) and \( M1 \). The \( E1 \) transition operator is given by \( \lambda=1 \) $$ O(E1) = rY^{(1)}_{\mu }(\hat{r})e_{q} e= \sqrt{\frac{3}{4\pi }}\mathbf{r}e_{q} e. $$ The \( E2 \) transition operator with \( \lambda=2 \) $$ O(E2) = r^{2}Y^{(2)}_{\mu }(\hat{r})e_{q} e, $$