Recent measurements of the ionization energies of the Rydberg ${}^{1}P$ states of helium for principal quantum number $n=24$ and higher present a new challenge to theoretical atomic physics. A long-standing obstacle to high-precision atomic theory for three-body systems is a rapid loss of accuracy for variational calculations with increasing principal quantum number $n$. We show that this problem can be overcome with the use of a “triple” basis set in Hylleraas coordinates. Nonrelativistic energies accurate to 23 significant figures are obtained with basis sets of relatively modest size (6744 terms). Relativistic and quantum electrodynamic effects are calculated, including an estimate of terms of order $m{\alpha }^6$ from a $1/n^3$ extrapolation, resulting in an estimated accuracy of $\pm 1$ kHz. The calculated ionization energy of 5 704 980.348(1) MHz is in excellent agreement with the experimental value 5 704 980.312(95) MHz. These results establish the ionization energy of the $1s24p{}^{1}P_1$ state as an absolute point of reference for transitions to lower-lying states, and they confirm an $11\sigma$ disagreement between theory and experiment in the triplet spectrum of helium. Results are also given for the $1s24p{}^{3}P_J$ states, in agreement with a recent experiment on the triplet Rydberg series, thereby confirming the discrepancy. Results are also given for the $1s24p{}^3{P}_J$ states which are in agreement with a recent experiment on the triplet-$P$ Rydberg series, thereby confirming a discrepancy of $0.468\pm 0.055$ MHz for the ionization energy of the $1s2s{}^3{S}_1$ state.

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• Atomic & molecular structure
• Atomic spectra
• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions
• Atoms
• First-principles calculations
• Schroedinger equation