Energies and relativistic corrections for the Rydberg states of helium: Variational results and asymptotic analysis

The results of an extended series of high-precision variational calculations for all states of helium up to n=10 and L=7 (excluding S states above n=2) are presented. Convergence of the nonrelativistic eigenvalues ranges from five parts in ${10}^{15}$ for the 2P states to four parts in ${10}^{19}$ for the 10K states. Relativistic and quantum electrodynamic corrections of order ${\mathrm{\alpha }}^2$, ${\mathrm{\alpha }}^3$, ${\mathrm{\alpha }}^2$μ/M, ${\mathrm{\alpha }}^2$(μ/M${)}^2$, and ${\mathrm{\alpha }}^3$μ/M are included and the required matrix elements listed for each state. For the 1s2p ${}_{}{}^3{}_{}{}^{}$ ${\mathit{P}}_{\mathit{J}}$ states, the lowest-order spin-dependent matrix elements of the Breit interaction are determined to an accuracy of three parts in ${10}^9$, which, together with higher-order corrections, would be sufficient to allow an improved measurement of the fine-structure constant. Methods of asymptotic analysis are extended to provide improved precision for the relativistic and relativistic-recoil corrections. A comparison with the variational results for the high-angular-momentum states shows that the ‘‘standard-atomic-theory’’ and ‘‘long-range-interaction’’ pictures discussed by Hessels et al. [Phys. Rev. Lett. 65, 2765 (1990)] come into agreement, thereby resolving what appeared to be a discrepancy. The comparison shows that the asymptotic expansions for the total energies are accurate to better than ±100 Hz for L>7, and results are presented for the 9L, 10L, and 10M states (i.e., angular momentum L=8 and 9). Significant discrepancies with experiment persist for transitions among the n=10 states, which cannot be easily accommodated by supposed higher-order corrections or additional terms. Finally, the asymptotic analysis indicates that a revision to the quantum-defect method is required for the analysis of high-precision data.