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

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 $Z \leq 20$.