High-precision variational calculations for helium in Hylleraas coordinates are used to obtain a combination of quantum defect expansions for the nonrelativistic energy and $1/n$ expansions for the relativistic and quantum electrodynamic corrections. The extrapolations based on direct calculations for the singlet and triplet $P$ states up to principal quantum number $n=35$ provide ionization energies of the $1snp{}^{1}P_1$ and ${}^{3}P_c$ (centroid) states up to $n=102$ with accuracies better than $\pm 1$ kHz. The calculated ionization energies are combined with 28 measured transition frequencies to obtain values for the ionization energy of the $1s2s{}^{3}S_1$ state. The final result of 1 152 842 742.705(16) MHz differs from theory by $0.474\pm 0.052$ MHz, and provides a strong confirmation of the $9\sigma$ disagreement between theory and experiment obtained previously by quantum defect extrapolation of experimental data to the series limit. An analysis of the quantum defect method is presented, and second-order mass polarization (recoil) terms are identified that vary as $1/n^2$ in lowest order. The nonrelativistic part provides a theoretical justification for the effective reduced-mass Rydberg $R_M^{(+)}$ based on the phenomenological model of a Rydberg electron scattering from a ${\mathrm{He}}^{+}$ core. The Ritz expansion for the nonrelativistic energy is verified to 20-figure accuracy.

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• Atoms
• First-principles calculations
• Perturbative methods
• Schroedinger equation