SYMMETRY AND ANALYTICAL STRUCTURE OF ADDITION THEOREMS OF SPATIAL FUNCTIONS AND OF MULTI-CENTER INTEGRALS OF ARBITRARY ATOMIC FUNCTIONS

Three-dimensional functions f(r) = g(r) · Yml(θ, ø), which transform like an irreducible tensor, are transformed simultaneously under rotations and translations. The relationships governing the transformation reveal some general properties. If the addition theorem of a function f(r) can be represented by a one-center expansion in terms of surface spherical harmonics Yml, each expansion coefficient is given by a Clebsch-Gordan coefficient and a radial function. Because of these properties, addition theorems are especially helpful for the simplification and evaluation of quantum-mechanical matrix elements and multi-center energy integrals in molecular LCAO calculations. The application of addition theorems has two major advantages: First, because addition theorems are equivalent to translation formulas, the number of centers of an integral can be reduced by translation of orbitals and operators. Second, due to the typical analytical structure of the series expansion representing the addition theorem, the dimensionality of a molecular integral can be reduced, because the integration over the angular variables can be executed. Then, a molecular multi-center integral is represented by a series of one-center integrals over functions of the radial variable only. © 1979 Springer-Verlag.