Addition theorems as three‐dimensional Taylor expansions

A principal tool for the construction of the addition theorem of a function f is the translation operator e r ′⋅ ∇ , which generates f ( r + r ′) by doing a three‐dimensional Taylor expansion of f around r . In atomic and molecular quantum mechanics, one is usually interested in irreducible spherical tensors. In such a case, the application of the translation operator in its Cartesian form e x ′∂/∂ x e y ′∂/∂ y e z ′∂/∂ z leads to serious technical problems. A much more promising approach consists of the use of an operator expansion for e r ′⋅ ∇ which contains exclusively irreducible spherical tensors. This expansion contains as differential operators the Laplacian ∇ 2 and the spherical tensor gradient operator   l m (∇), which is an irreducible spherical tensor of rank l . Thus, if   l m (∇) is applied to another spherical tensor, the angular part of the resulting expression is completely determined by angular momentum coupling, whereas the radial part is obtained by differentiating the radial part of the spherical tensor with respect to r . Consequently, the systematic use of the spherical tensor gradient operator leads to a considerable technical simplification. In this way, the originally three‐dimensional expansion problem in x , y , and z is reduced to an essentially one‐dimensional expansion problem in r . The practical usefulness of this approach is demonstrated by constructing the Laplace expansion of the Coulomb potential as well as the addition theorems of the regular and irregular solid harmonics and of the Yukawa potential. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 280–295, 2000