High-order perturbed corrections with scaling transformation

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