Complete α² relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of HeH⁺

We report the implementation of the complete set of the lowest-order relativistic corrections of the order of α² (where α is the fine structure constant) for calculating vibrational states of diatomic molecular systems within the framework that does not assume the Born-Oppenheimer approximation. To test the accuracy of the approach we have performed calculations for all rotationless vibrational states (also called pure vibrational states or S states) of the HeH⁺ ion in the ground electronic state. For the lowest transitions, where very precise experimental results are available, an excellent agreement with the experimental values has been achieved.