Orbit-orbit relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of H2

We report the derivation of the orbit-orbit relativistic correction for calculating pure vibrational states of diatomic molecular systems with electrons within the framework that does not assume the Born-Oppenheimer (BO) approximation. The correction is calculated as the expectation value of the orbit-orbit interaction operator with the non-BO wave function expressed in terms of explicitly correlated Gaussian functions multiplied by even powers of the internuclear distance. With that we can now calculate the complete relativistic correction of the order of α2 (where α=1c). The new algorithm is applied to determine the full set of the rotationless vibrational levels and the corresponding transition frequencies of the H2 molecule. The results are compared with the previous calculations, as well as with the frequencies obtained from the experimental spectra. The comparison shows the need to include corrections higher than second order in α to further improve the agreement between the theory and the experiment.