A simple variational approach to perturbation theory

A simple variational approach to perturbation theory is presented which contains as special cases the Goldhammer-Feenberg refinement of the Brillouin-Wigner series, the corresponding refinement of the Rayleigh-Schrödinger series, as well as the continued-fraction expansion of the resolvent matrix element. The approach is very effective and works for arbitrary coupling strengths. The method is applied to one-electron atoms, taking the whole Coulomb attraction $-\frac{Z}{r}$ as a perturbation. Already with the first-order wave function, one obtains 99.3% of the exact ground-state energy for all $Z$. With the second-order wave function one obtains 99.96% of the exact ground-state energy and 99.6% of the exact energy of the first excited state, again for all $Z$. In the present example the numerical effort is comparable to the one needed to perform the Rayleigh-Schrödinger expansion to the corresponding order of the wave function.