Second-order corrections of orders $mα^6$ and $mα^6(m/M)$ 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.