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 a is used. Previous calculations of orders m alpha(6) and m alpha(7) are improved by including second-order contributions due to the vibrational motion of nuclei. Furthermore, we evaluate the largest corrections at the order m alpha(8). That allows us to reduce the fractional uncertainty to the level of 7.6 x 10(-12) for fundamental transitions and to 4.5 x 10(-12) for the ionization energies.