On 1/Z expansion, the critical charge for a two-electron system, and the Kato theorem

The 1/Z expansion for the ground state energy of the Coulomb system of an infinitely massive center of charge Z and two electrons (two-electron ionic sequence) is studied. A critical analysis of the 1/Z coefficients presented in Baker et al. (Phys. Rev. A, 41, 1247 (1990)) is performed and its numerical deficiency is indicated, leading, in particular, to unreliable decimal digits beyond digits 11–12 of the first coefficients. We made a consistency check of the 1/Z-expansion with accurate energies for Z = 1–10: the weighted partial sums of the 1/Z expansion with Baker et al. coefficients reproduce systematically the ground state energies of two-electron ions with Z ≥ 2 up to 12 decimal digits and for Z = 1 up to 10 decimal digits calculated by Nakashima and Nakatsuji (J. Chem. Phys. 127, 224104 (2007)) with unprecedented accuracy. This rules out the presence of non-analytic terms at Z = ∞ contributing to the first 10–12 decimal digits in the ground state energy; it agrees with the Kato theorem about convergence of the 1/Z expansion within that accuracy. The ground state energy of two-electron ions Z = 11 (Na9+) and Z = 12 (Mg10+) is calculated with 12 decimal digits. This study can be considered as the independent confirmation of the correctness of 10 decimal digits in all 401 coefficients of 1/Z-expansion printed in Baker et al. (Phys. Rev. A, 41, 1247 (1990)).