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'.del), which generates f(r + r') by doin g a three-dimensional Taylor expansion of f around r. In atomic and molecul ar quantum mechanics, one is usually interested in irreducible spherical te nsors. In such a case, the application of the translation operator in its C artesian form e(x'partial derivative)/(partial derivative x)e(y'partial der ivative)/(partial derivative y)e(z'partial derivative)/(partial derivative z) leads to serious technical problems. A much more promising approach cons ists of the use of an operator expansion for e(r'.del) which contains exclu sively irreducible spherical tensors. This expansion contains as differenti al operators the Laplacian del(2) and the spherical tensor gradient operato r y(l)(m)(del), which is an irreducible spherical tensor of rank l. Thus, i f y(l)(m)(del) is applied to another spherical tensor, the angular part of the resulting expression is completely determined by angular momentum coupl ing, whereas the radial part is obtained by differentiating the radial part of the spherical tensor with respect to r. Consequently, the systematic us e of the spherical tensor gradient operator leads to a considerable technic al simplification. In this way, the originally three-dimensional expansion problem in x, y, and z is reduced to an essentially one-dimensional expansi on problem in y. The practical usefulness of this approach is demonstrated by constructing the Laplace expansion of the Coulomb potential as well as t he addition theorems of the regular and irregular solid harmonics and of th e Yukawa potential. (C) 2000 John Wiley & Sons, Inc.