Useful alternative to the multipole expansion of 1/r potentials - art. no.052509

Few-body problems involving Coulomb or gravitational interactions between p airs of particles, whether in classical or quantum physics, are generally h andled through a standard multipole expansion of the two-body potentials. W e develop an alternative based on an old, but hitherto forgotten, expressio n for the inverse distance between two points that builds on azimuthal symm etry. This alternative should have wide applicability throughout physics an d astronomy, both for computation and for the insights it provides through its emphasis on different symmetries and structures than are familiar from the standard treatment. We compare and contrast the two methods, develop ad dition theorems for Legendre functions of the second kind and a number of u seful analytical expressions for these functions. Two-electron "direct" and "exchange" integrals in many-electron quantum systems are evaluated to ill ustrate the procedure, which is more compact than the standard one using Wi gner coefficients and Slater integrals.