Laplace-type, bipolar, and polypolar expansions of r/sup N/Y/sub L//sup M(theta,phi) and Helmholtz harmonics j/sub L/(kr)Y/sub L//sup M/(theta,phi) for molecular calculations

Relationships of the general form: f/sub α/(r1 + r2 + . . . + r/sub n/) = Σ/sub lambda/Σ/sub μ/ . . . Σ/sub eta/ C/sub lambda μ . . . eta//sup α/f/sub lambda/ (r1)f/sub μ/(r2) . . . f/sub eta/(r/sub n/), are given for functions X/sub N,L//sup M/(r) = r/sup N/Y/sub L//sup M/(theta,phi) and Helmholtz solid harmonics b/sub L/(kr) Y/sub L//sup M/ (theta,phi), where Y/sub L//sup M/(theta,phi) is a spherical surface harmonic and b/sub L/(kr) stands for a spherical Bessel, Hankel, or Neumann function. The monopolar or Laplace-type (n = 2), bipolar (n = 3), and polypolar expansions (arbitrary n) of these functions allow many applications in theoretical chemistry and physics. Expansions of functions X/sub N,L//sup M/ (r), which occur in orbitals and operators, are particularly useful for molecular calculations, especially for the evaluation of multicentered integrals. Expansions of Helmholtz solid harmonics are advantageous for instance in scattering theory