The vector spherical harmonics $\vec{X}_{\ell,m}(\theta,\phi)$ are generalizations of spherical harmonics for vectors fields. \[\vec{X}_{\ell,m}(\theta,\phi) = \frac{-i(\vec{r}\times\nabla)Y_{\ell,m(\theta,\phi)}}{\sqrt{\ell(\ell + 1)}}\]