Few-electron atomic ions in non-relativistic QED: The ground state

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for $Z\le 20$ 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 $Z\le 20$ 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 $Z\le 12$, and not less than 12 s.d. for $Z\le 50$ for helium-like as well as for $Z\le 20$ for lithium-like ions by a compact meromorphic function in variable $\lambda =\sqrt{Z-Z_B}$ (here $Z_B$ is the 2nd critical charge (Turbiner et al., 2016)), $P_9\left(\lambda \right)∕Q_5\left(\lambda \right)$. It is found that both 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 $Z\le 20$ in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that $\gtrsim 99.9\text{\%}$ 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 $Z\le 20$, respectively. It may imply that these trial functions are, in fact, 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 $Z\le 20$.