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We calculate ionization energies and fundamental vibrational transitions forH$_2^+$, D$_2^+$, and HD$^+$ molecular ions. The NRQED expansion for the energyin terms of the fine structure constant $\alpha$ is used. Previous calculationsof orders $m\alpha^6$ and $m\alpha^7$ are improved by including second-ordercontributions due to the vibrational motion of nuclei. Furthermore, we evaluatethe largest corrections at the order $m\alpha^8$. That allows to reduce thefractional uncertainty to the level of $7\cdot10^{-12}$ for fundamentaltransitions and to $4\cdot10^{-12}$ for the ionization energies.

Fundamental transitions and ionization energies of the hydrogen molecular ions at the few ppt level