Quantum mechanical, non-relativistic, non-Born-Oppenheimer (non-BO) calculations are performed for the rovibrational spectrum of H-2 excited to the second rotational level. The non-BO wave functions of the considered states are expanded in terms of all-particle explicitly correlated Gaussian functions. The dissociation energies and rovibrational transition energies are calculated and compared with experimental values and values obtained in calculations performed by others. The average interparticle distances are calculated and compared with the corresponding values for HD. They show that H-2 is a more "diffuse molecule". The nuclear-nuclear correlation functions are calculated and plotted to visualize the "non-BO molecular structure" of H-2. (C) 2016 Elsevier B.V. All rights reserved.