On 1/Z expansion for two-electron systems

TL;DR

This paper revisits the 1/Z expansion for two-electron Coulomb systems, correcting previous numerical issues, and achieves highly precise calculations of ground state energies for ions with Z=1 to 12.

Contribution

It introduces a minor modification to the 1/Z expansion coefficients, enabling highly accurate energy calculations for two-electron ions across a range of nuclear charges.

Findings

Corrected higher order coefficients in 1/Z expansion.

Achieved 15th decimal digit accuracy for Z=1 to 10.

Calculated energies for Z=11 and 12 with 14 decimal digits.

Abstract

The $1/Z$-expansion for the Coulomb system of infinitely massive center of charge Z and two electrons is discussed. Numerical deficiency in Baker et al, {\em Phys. Rev. \bf A41}, 1247 (1990) is indicated which continue to raise doubts in correctness of their calculations of the higher order coefficients in $1/Z$-expansion expressed in Refs.[4-5]. It is shown that a minor modification of a few first coefficients found in Ref.[3] allows to calculate the ground state energies at $Z\ =\ 1, 2,\ldots \ 10$ (as well as at $Z > 10$) with a portion of 15th decimal digit in comparison with highly accurate calculations by C. Schwartz and by Nakashima-Nakatsuji. Ground state energies of two-electron ions $Z=11\ (Na^{9+})$ and $Z=12\ (Mg^{10+})$ are found with 14 decimal digits.