Fundamental Transitions and Ionization Energies of the Hydrogen Molecular Ions with Few ppt Uncertainty

We calculate ionization energies and fundamental vibrational transitions for H^{₂ +} , D^{₂ +} , and HD⁺ molecular ions. The nonrelativistic quantum electrodynamics expansion for the energy in terms of the fine structure constant α is used. Previous calculations of orders m α⁶ and m α⁷ are improved by including second-order contributions due to the vibrational motion of nuclei. Furthermore, we evaluate the largest corrections at the order m α⁸. That allows us to reduce the fractional uncertainty to the level of 7.6 ×10^-12 for fundamental transitions and to 4.5 ×10^-12 for the ionization energies.