*Collect. Czech. Chem. Commun.*
**2005**, *70*, 1225-1271

https://doi.org/10.1135/cccc20051225

### The Spherical Tensor Gradient Operator

**Ernst Joachim Weniger**

*Institut für Physikalische und Theoretische Chemie, Universität Regensburg, D-93040 Regensburg, Germany*

### Abstract

The spherical tensor gradient operator *Y*_{l}^{m}(∇), which is obtained by replacing the Cartesian components of ** r** by the Cartesian components of ∇ in the regular solid harmonic

*Y*

_{l}

^{m}(

**), is an irreducible spherical tensor of rank**

*r**l*. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank

*l*. 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

*Y*

_{l}

^{m}(∇) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. Math. London Soc.

*24*, 54 (

**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

*Y*

_{l}

^{m}(∇) is applied to them. Fourier transformation is very helpful in understanding the properties of

*Y*

_{l}

^{m}(∇) since it produces

*Y*

_{l}

^{m}(-i

**). It is also possible to apply**

*p**Y*

_{l}

^{m}(∇) to generalized functions, yielding for instance the spherical delta function δ

_{l}

^{m}(

**). The differential operator**

*r**Y*

_{l}

^{m}(∇) 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*

^{v}Y_{l}

^{m}(

**) with**

*r**v*∈ πR.

**Keywords:**
Cartesian components;
Spherical tensor;
Hobson's theorem;
Bossel functions;
Spherical delta functions;
Quantum chemistry.

**References:** 140 live references.