Variation-Perturbation Expansions and Padé Approximants to the Energy

The variational solutions to the Rayleigh-Schr\"odinger inhomogeneous equations are derived by simple matrix algebra. It is shown how the approximate perturbation energies and wave functions obtained in this way obey the same type of equations as the exact ones. This permits the formulation of a variation-iteration theory convenient for numerical applications. Results for the He-like ions show that the conventional linear series is less suitable than the present method in obtaining accurate energies. The use of inner projection techniques leads to an efficient calculation of Pad\'e approximants to the energy series, which show remarkable convergence properties.