Abstract
Addition theorems can be constructed by doing three-dimensional Taylor expansions according to f(r + r') = exp(r' . del)f (r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp(x'partial derivative / partial derivativex) exp(y'partial derivative / partial derivativey) ...
Abstract
Addition theorems can be constructed by doing three-dimensional Taylor expansions according to f(r + r') = exp(r' . del)f (r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp(x'partial derivative / partial derivativex) exp(y'partial derivative / partial derivativey) exp(z'partial derivative / partial derivativez) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp(r' . del) involving powers of the Laplacian del(2) and spherical tensor gradient operators y(l)(m/) (del), which are irreducible spherical tensors of ranks zero and l, respectively (Santos, F.D. Nucl Phys A 1973, 212, 341). In this way, it is indeed possible to derive addition theorems by doing three-dimensional Taylor expansions (Weniger, E. J. Int J Quantum Chem 2000, 76, 280). The application of the translation operator in its spherical form is particularly simple in the case of B functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater-type functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of B functions, the addition theorems for these functions can be written down immediately. (C) 2002 Wiley Periodicals, Inc. Int.
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