Ionization energies for Rydberg $^4 \mathrm{He}$ ($1snp\,^{1,3}P$) states using the correlated B-spline basis function method
We extend the correlated B-spline basis function (C-BSBF) method to high-precision calculations of the ionization energies of helium Rydberg $n^{1,3}P$ states ($n=24$--$35$). Using a unified basis set, we evaluate nonrelativistic energies, relativistic corrections of order $m\alpha^4$ (including finite-mass recoil), QED contributions of order $m\alpha^5$, and partial $m\alpha^6$ terms (singlet-triplet mixing, one- and two-loop radiative corrections). The remaining higher-order contributions are estimated via $1/n^3$ scaling. The resulting ionization energies achieve kHz-level accuracy and are in excellent agreement with independent Hylleraas calculations, thereby providing cross-validation between two distinct theoretical approaches. From these data, the quantum-defect parameters are determined and used to extrapolate the ionization energies up to $n=102$. Combining our Rydberg ionization energies with high-precision experimental $2S \rightarrow nP$ transition frequencies yields the ionization energies for the metastable $2^{1}S$ and $2^{3}S$ states as \num{960332040.533(10)}$_\mathrm{stat}(20)_ \mathrm{sys}$ MHz and \num{1152842742.7274(53)}$_\mathrm{stat}(25)_ \mathrm{sys}$ MHz, respectively. The C-BSBF result for the $2 \, ^1 S$ state is consistent with the experimental ionization energy obtained from Rydberg-series extrapolation, while for the $2 \, ^3 S$ state the difference is 0.019(10) MHz.