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

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.