Recursion relations on irregular electric multipoles (excited atom spectroscopy)

Recursion relations are derived for the non-relativistic Coulomb matrix elements of negative integer powers of r. Explicit relations are given for the bound-free elements but generalization to the bound-bound case and to the free-free case is straightforward. Using the analytically known values for two monopole matrix elements and a directly computed value for one quadrupolar matrix element, one is then able to get for each negative integer power of r the Coulomb matrix elements with all allowed values of angular momentum. The numerical efficiency of the method is checked and turns out to be excellent.