Application of the Rayleigh-Schrödinger Perturbation Theory to the Hydrogen Atom

The motion of a single electron in the electrostatic field of a nucleus is treated by the Rayleigh-Schrödinger perturbation method, the whole electrostatic potential being considered as the "perturbation." The contribution of the first approximation to the energy vanishes. The second approximation gives a finite ionization energy which is, however, incorrect numerically. The first approximation also vanishes for potentials $\sim r^{-n}$ with $0<n<\mathrm{}3\mathrm{}$ but the second approximation is finite only for $n=1\mathrm{}$: it vanishes for $n>\mathrm{}1\mathrm{}$, and is infinite for $n<\mathrm{}1\mathrm{}$.