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

Abstract

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, {\em Phys. Rev. \bf A41}, 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 $Z\ge 2$ up to 12 decimal digits and for $Z=1$ up to 10 decimal digits. This rules out the presence of non-analytic terms at $Z=\infty$ contributing into 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 $Z=11(N{a}^{9+})$ and $Z=12(M{g}^{10+})$ is calculated with 12 decimal digits.

11 pages, 2 Tables; the title is slightly modified, the text is essentially rewritten, the emphasis to a relation of presented results to the Kato theorem is given, one Table is removed, new references are added, main conclusions remain unchanged, submitted to Can J Physics

References in corpus (1)

• Nuclear critical charge for two-electron ion in Lagrange mesh method

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