Theory for the Rydberg states of helium: Results for $2 \le n \le 35$ and comparison with experiment for the singlet and triplet $P$-states

TL;DR

This paper presents highly accurate variational calculations for helium Rydberg states up to n=35, including relativistic and QED effects, and compares results with experimental data to test quantum defect extrapolation methods.

Contribution

The study provides the most precise theoretical energies for helium Rydberg states up to n=35, incorporating relativistic and QED corrections, and compares these with recent experimental measurements.

Findings

The calculated ionization energy agrees with experiment within 17 kHz.

A significant 9σ discrepancy is confirmed with previous theoretical results.

The results serve as a benchmark for quantum defect extrapolation methods.

Abstract

High precision variational calculations in Hylleraas coordinates are presented for all singlet and triplet $P$-states of helium up to principal quantum number $n = 35$ with a uniform accuracy of 1 part in $10^{22}$ for the nonrelativistic energy. Mass polarization, relativistic and quantum electrodynamic effects are included to achieve a final accuracy of $\pm$1 kHz or better for the ionization energy of the Rydberg states of $^4$He in the range $24\le n \le 35$. The results are combined with 11 transition frequency measurements of Clausen et al. Phys. Rev. A 111, 012817 (2025) to obtain complementary measurements of the ionization energy of the $1s2s\;^3S_1$ state that do not depend on quantum defect extrapolations to the series limit. The result from the triplet spectrum yields an ionization energy of 1152 842 742.728(6) MHz, which agrees with but is larger than the experimental value by 14 $\pm$17 kHz. However, it confirms a much larger 9$\sigma$ discrepancy of $0.468\pm0.055$ MHz with the theoretical ionization energy of Patk\'o\v{s} et al. Phys. Rev. A 103, 042809 (2021). The results provide a test of the quantum defect extrapolation method at the level of $\pm$17 kHz. This revised version contains an additional table of spin-dependent matrix elements of the Breit interaction in the appendix for $24\le n\le 35$. (12 pages, 1 figure).