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

TL;DR

This paper critically analyzes the $1/Z$-expansion for two-electron systems, identifies numerical deficiencies in previous coefficients, and confirms the expansion's convergence and accuracy up to 12 decimal digits, supporting the Kato theorem.

Contribution

It provides a detailed correction and validation of the $1/Z$-expansion coefficients and confirms the expansion's convergence and precision for two-electron ions, aligning with the Kato theorem.

Findings

Identified numerical deficiencies in previous $1/Z$-expansion coefficients.

Validated the $1/Z$-expansion's accuracy up to 12 decimal digits for $Z=1$ to $Z=12$.

Confirmed the absence of non-analytic terms at $Z= $ in the first 10-12 decimal digits.

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 $Z \geq 2$ 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 $Z=11\ (Na^{9+})$ and $Z=12\ (Mg^{10+})$ is calculated with 12 decimal digits.