  The spherical tensor gradient operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$,
which is obtained by replacing the Cartesian components of $\bm{r}$ by the
Cartesian components of $\nabla$ in the regular solid harmonic
${\mathcal{Y}}_{\ell}^{m} (\bm{r})$, is an irreducible spherical tensor of rank
$\ell$. Accordingly, its application to a scalar function produces an
irreducible spherical tensor of rank $\ell$. Thus, it is in principle
sufficient to consider only multicenter integrals of scalar functions: Higher
angular momentum states can be generated by differentiation with respect to the
nuclear coordinates. Many of the properties of ${\mathcal{Y}}_{\ell}^{m}
(\nabla)$ can be understood easily with the help of an old theorem on
differentiation by Hobson [Proc. London Math. Soc. {\bf 24}, 54 - 67 (1892)].
It follows from Hobson's theorem that some scalar functions of considerable
relevance as for example the Coulomb potential, Gaussian functions, or reduced
Bessel functions produce particularly compact results if
${\mathcal{Y}}_{\ell}^{m} (\nabla)$ is applied to them. Fourier transformation
is very helpful to understand the properties of ${\mathcal{Y}}_{\ell}^{m}
(\nabla)$ since it produces ${\mathcal{Y}}_{\ell}^{m} (-\mathrm{i} \bm{p})$. It
is also possible to apply ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ to generalized
functions, yielding for instance the spherical delta function
$\delta_{\ell}^{m} (\bm{r})$. The differential operator
${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can also be used for the derivation of
pointwise convergent addition theorems. The feasibility of this approach is
demonstrated by deriving the addition theorem of $r^{\nu}
{\mathcal{Y}_{\ell}^{m}} (\bm{r})$ with $\nu \in \mathbb{R}$.
