Useful alternative to the multipole expansion of 1/r potentials

Few-body problems involving Coulomb or gravitational interactions between pairs of particles, whether in classical or quantum physics, are generally handled through a standard multipole expansion of the two-body potentials. We develop an alternative based on an old, but hitherto forgotten, expression for the inverse distance between two points that builds on azimuthal symmetry. This alternative should have wide applicability throughout physics and 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 addition theorems for Legendre functions of the second kind and a number of useful analytical expressions for these functions. Two-electron “direct” and “exchange” integrals in many-electron quantum systems are evaluated to illustrate the procedure, which is more compact than the standard one using Wigner coefficients and Slater integrals.