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Rovibrational computations for He$_2^+$ X~$Σ_\mathrm{u}^+$ including non-adiabatic, relativistic and QED corrections

Edit Mátyus, Ádám Margócsy

TL;DR

This work delivers a comprehensive, high-precision rovibrational analysis of the He2+ X^2Sigma_u+ state by combining a refined non-relativistic BO energy with post-BO, relativistic, and QED corrections. The authors employ a variational fECG basis to generate a broad potential energy curve over $\rho$ and compute coordinate-dependent masses, delivering rovibrational energies with an estimated accuracy of $0.005\,\mathrm{cm^{-1}}$. Regularization techniques (IT and Drachmanization) and a Bethe-logarithm approximation underpin the accurate evaluation of singular operators and QED terms, while DVR-based solutions yield the bound-state spectrum that agrees well with experiment and improves upon prior theory. The results enable more reliable tests of subtle physical effects in small, calculable molecules and guide future work on non-adiabatic relativistic coupling and magnetic spin-rotation interactions.

Abstract

We report the potential energy curve, the diagonal Born-Oppenheimer, non-adiabatic mass, relativistic, and leading-order QED corrections for the ground electronic state of the helium dimer cation; the higher-order QED and finite-nuclear size effects are also estimated. The computations are carried out with an improved error control and over a much broader configuration range compared to earlier work [D. Ferenc, V. I. Korobov, and E. Mátyus, Phys. Rev. Lett. 125, 213001 (2020)]. As a result, all rovibrational bound states are reported with an estimated accuracy of 0.005 cm$^{-1}$.

Rovibrational computations for He$_2^+$ X~$Σ_\mathrm{u}^+$ including non-adiabatic, relativistic and QED corrections

TL;DR

This work delivers a comprehensive, high-precision rovibrational analysis of the He2+ X^2Sigma_u+ state by combining a refined non-relativistic BO energy with post-BO, relativistic, and QED corrections. The authors employ a variational fECG basis to generate a broad potential energy curve over and compute coordinate-dependent masses, delivering rovibrational energies with an estimated accuracy of . Regularization techniques (IT and Drachmanization) and a Bethe-logarithm approximation underpin the accurate evaluation of singular operators and QED terms, while DVR-based solutions yield the bound-state spectrum that agrees well with experiment and improves upon prior theory. The results enable more reliable tests of subtle physical effects in small, calculable molecules and guide future work on non-adiabatic relativistic coupling and magnetic spin-rotation interactions.

Abstract

We report the potential energy curve, the diagonal Born-Oppenheimer, non-adiabatic mass, relativistic, and leading-order QED corrections for the ground electronic state of the helium dimer cation; the higher-order QED and finite-nuclear size effects are also estimated. The computations are carried out with an improved error control and over a much broader configuration range compared to earlier work [D. Ferenc, V. I. Korobov, and E. Mátyus, Phys. Rev. Lett. 125, 213001 (2020)]. As a result, all rovibrational bound states are reported with an estimated accuracy of 0.005 cm.

Paper Structure

This paper contains 14 sections, 53 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. The non-relativistic electronic energy and wave function
  4. Post-BO corrections
  5. Relativistic & QED corrections
  6. Computational details
  7. Converging the electronic energy; PEC generation
  8. Post-BO corrections
  9. Relativistic and QED corrections
  10. Rovibrational energies and comparison with experiment
  11. Summary, conclusion, outlook
  12. Acknowledgement
  13. On the operator relation between the two-electron Fermi contact and delta function terms
  14. Numerical Drachmanization of the Araki-Sucher term

Figures (2)

  • Figure 1: He$_2^+$$\text{X}\ ^2\Sigma_\text{u}^+$: BO potential energy curve, $U$; diagonal BO correction, $U_\text{DBOC}$; rotational and vibrational mass corrections, $\delta m^\text{rot}$ and $\delta m^\text{vib}$; relativistic correction, $U_\text{rel}$; leading-order QED correction, $U_\text{lQED}$; and estimate for higher-order QED corrections, $U_\text{hQED}$, and the finite nuclear-size effect, $U_\text{f.nuc.}$. The PEC points (and the corrections) have been computed at several points in the $\rho\in[0.3,100]$ bohr interval; the computed dataset is available in the Supplementary Material.
  • Figure 2: All bound states of He$_2^+$$\text{X}\ ^2\Sigma_\text{u}^+$ computed in this work improve upon earlier results reported in Ref. Ma18he2p (Comp.2018) referenced to the zero-point vibrational energy (ZPVE) in both computations, $\Delta \tilde{\nu}(v,N)=\tilde{\nu}(v,N)-\tilde{\nu}(0,0)$. The ZPVE has improved by $\tilde{\nu}(0,0)_\text{This Work}-\tilde{\nu}(0,0)_\text{Comp.2018}=0.799\ 8~\text{cm}^{-1}$.