High-order perturbed corrections with scaling transformation

The performance of the so-called superconvergent quantum perturbation theory (Wenhua Hai {\em et al} 2000 \emph{Phys. Rev.} A {\bf 61} 052105) is investigated for the case of the ground-state energy of the helium-like ions. The scaling transformation $r\rightarrow r/Z$ applied to the Hamiltonian of a two-electron atomic ion with a nuclear charge $Z$ (in atomic units). Using the improved Rayleigh--Schr\"{o}dinger perturbation theory based on the integral equation to helium-like ions in the ground states and treating the electron correlations as perturbations, we have performed a third-order perturbation calculation and obtained the second-order corrected wavefunctions consisting of a few terms and third-order energy corrections. We find that third-order and higher-order energy corrections are improved with decreasing nuclear charge. This result means that the former is quadratically integrable and the latter is physically meaningful. The improved quantum perturbation theory fits the higher-order perturbation case. This work shows that it is a development on the quantum perturbation problem of helium-like systems.