Second-order corrections of orders $m\alpha^6$ and $m\alpha^6(m/M)$ to the spin-averaged energy in the HD$^+$ and H$_2^+$ ions

TL;DR

This paper calculates high-order relativistic corrections to the energy levels of hydrogen molecular ions, achieving high numerical accuracy to improve theoretical predictions of transition frequencies beyond 1 part-per-trillion.

Contribution

It provides the first detailed calculation of second-order relativistic corrections at orders $meta^6$ and $meta^6(m/M)$ for HD$^+$ and H$_2^+$ ions, with an estimated uncertainty below 10^{-5}.

Findings

Numerical convergence of the corrections was thoroughly studied.

The results enable future theoretical predictions with sub-ppb accuracy.

Uncertainty in the calculations is estimated to be less than 10^{-5}.

Abstract

The relativistic second-order corrections at orders $m\alpha^6$ and $m\alpha^6(m/M)$ in hydrogen molecular ions are calculated. Convergence of numerical results is studied, which allows estimating the relative numerical uncertainty to be less then $10^{-5}$. This accuracy is sufficient to enable future improvement of theoretical predictions beyond the 1 ppt (part-per-trillion) precision level for the ro-vibrational transition frequencies.