Precise ab initio calculations of $^4$He($1snp \, ^3P_J$) fine structure of high Rydberg states

TL;DR

The paper develops and applies an extended correlated B-spline basis-function (C-BSBF) framework to compute the fine-structure splittings of high Rydberg states in $^4$He with ab initio accuracy. It explicitly accounts for $m\alpha^4$ and $m\alpha^5$ relativistic corrections, singlet–triplet mixing via a $2\times2$ effective Hamiltonian, and estimates $m\alpha^6$ using a $1/n^3$ scaling, achieving kilohertz-level precision for $n=24$–$37$ and extrapolating to $n=45$–$51$ through a $1/n$ expansion. The results show strong agreement with Hylleraas-based ab initio data and quantum-defect theory (QDT), while resolving the $n=34$ discrepancy observed in experiments by aligning with theory; the extrapolated predictions to higher $n$ further corroborate QDT with negligible deviations. This work establishes a robust ab initio foundation for precision tests of QED in two-electron systems and paves the way for including complete high-order QED corrections in the future.

Abstract

High-precision measurements of the fine-structure splittings in helium high Rydberg states have been reported, yet corresponding ab initio benchmarks for direct comparison remain unavailable. In this work, we extend the correlated B-spline basis function (C-BSBF) method to calculate the fine-structure splittings of high Rydberg states in $^4$He. The calculations include the $mα^4$- and $mα^5$-order contributions, the singlet-triplet mixing effect, and estimated spin-dependent $mα^6$-order corrections obtained using a $1/n^3$ scaling approximation. High-precision ab initio results are obtained for principal quantum numbers $n=24$-37 with kilohertz-level accuracy and further extended to $n=45$-51 by extrapolation and fitting. The theoretical results show excellent agreement with quantum-defect theory (QDT) predictions and allow direct comparison with experimental measurements. Additionally, the discrepancy observed at $n=34$ is expected to be clarified with improved experimental precision.

Figures (1)

• Figure 1: Fine-structure energy levels (not drawn to scale) for the $n^3P_J\,(J=0,1,2)$ states of $^4\mathrm{He}$, in GHz.