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

The relativistic second-order corrections at orders $m{\alpha }^6$ and $m{\alpha }^6(m\mathrm{/}M)$ 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.