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-Schr\"odinger 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 ~r^-n with $0 1$, and is infinite for $n<1$.