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

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.

6 pages