Few-electron atomic ions in non-relativistic QED: the Ground state energy

TL;DR

This paper analyzes the ground state energy of few-electron atomic ions within non-relativistic QED, demonstrating high-accuracy interpolation methods and the potential for near-exact wavefunctions for ions with atomic number up to 20.

Contribution

It introduces accurate interpolation formulas for ground state energies and suggests that optimized trial functions may be effectively exact wavefunctions in NRQED for ions with Z ≤ 20.

Findings

Ground state energy can be interpolated with high precision using meromorphic functions.

Majorana and polynomial formulas accurately reproduce energies for Z ≤ 20.

Optimized trial functions closely match variational energies, indicating near-exact wavefunctions.

Abstract

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for $Z \leq 20$ the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with $Z \leq 20$ the finite nuclear mass effects do not change 4-5 significant digits (s.d.), and the leading relativistic and QED effects leave unchanged 3-4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy not less than 13 s.d. for $Z \leq 12$, and not less than 12 s.d. for $Z \leq 50$ for helium-like as well as for $Z \leq 20$ for lithium-like ions by a compact meromorphic function in ${\lambda}=\sqrt{Z-{Z_B}}$ ($Z_B$ is the 2nd critical charge, see {TLO:2016}), $P_9(\lambda)/Q_5(\lambda)$. It is found that the Majorana formula - a second degree polynomial in $Z$ with two free parameters - and a fourth degree polynomial in ${\lambda}$ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for $Z \leq 20$ in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that $\gtrsim 99.9\%$ of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for $Z \leq 20$, respectively. It may imply that these trial functions are, in fact, {\it exact} wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation - 3 s.d. - for both 2 and 3 electron systems is interpolated by 4th degree polynomial in $Z$ for $Z \leq 20$.