High-Order Perturbation Theory and Padé Approximants for a One-Electron Ion in a Generalized Central-Field Potential

The solution to the nonrelativistic Schrödinger equation for a one-electron ion in a generalized central-field potential is investigated using high-order perturbation theory. It is shown that by utilizing a finite expansion of the perturbation-theory wave function in terms of associated Laguerre polynomials, perturbation-theory results can be obtained for any $n$, $l$ state to arbitrarily high order. Results for the wave function and energy are explicitly given to third and fourth order, respectively. It is also shown that by reexpressing the high-order perturbation-theory energy expansion as a series of rational fractions (Padé approximants), accurate eigenvalues are obtained, even for large values of the expansion parameter.