Leading order relativistic corrections to the ro-vibrational spectrum of $\mbox{H}^+_2$ and $\mbox{HD}^+$ molecular ions

TL;DR

This paper presents high-precision calculations of relativistic corrections to the ro-vibrational spectra of H2+ and HD+ ions, achieving about ten significant digits to support ultra-precise transition frequency predictions.

Contribution

The study provides the first high-precision variational calculations of relativistic corrections for these molecular ions at the leading order, enabling ultra-precise spectral predictions.

Findings

Achieved about ten significant digits in relativistic correction calculations.

Provided theoretical transition frequency predictions with $10^{-12}$ relative uncertainty.

Extended calculations to states with L=0-4 and v=0-10.

Abstract

High-precision variational calculations of the operators for the relativistic corrections in the leading $m\alpha^4$ order are presented. The ro-vibrational states in the range of the total orbital angular momentum $L\!=\!0\!-\!4$ and vibrational quantum number $v\!=\!0\!-\!10$ for the $\mbox{H}_2^+$ and $\mbox{HD}^+$ molecular ions are considered. We estimate that about ten significant digits are obtained. This high precision is required for making theoretical predictions for transition frequencies at the level of $10^{-12}$ relative uncertainty.