Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Addition theorems can be constructed by doing three‐dimensional Taylor expansions according to f ( r + r ′)=exp ( r ′⋅∇) 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 ′∂/∂ x )exp ( y ′∂/∂ y )exp ( z ′∂/∂ z ) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp ( r ′⋅∇) involving powers of the Laplacian ∇ 2 and spherical tensor gradient operators {article}{empty}{document}Yᵐ_(∇){document} , 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.