For nearly all the states dissociating below the ionic limit [i.e., Rb (5s, 5p, 4d, 6s, 6p, 5d, 7s, and 4f )+H (1s)] in 1Σ+ and 3Σ+ symmetries, we present an adiabatic and diabatic study. Adiabatic results are also reported for 1Π, 3Π, 1∆, and 3∆ symmetries. The ab initio calculations rely on pseudopotential, operatorial core-valence correlation, and full valence configuration-interaction approaches, combined to an efficient diabatization procedure. For the low-lying states, our vibrational level spacings and spectroscopic constants are in good agreement with the available experimental data. Diabatic potentials and dipole moments are analyzed, revealing the strong imprint of ionic state in the 1Σ+ adiabatic states while improving the results. As for LiH, vibrational spacing of the A state is bracketed by our results, with and without the improvement, taking into account the diabatic representation. Experimental suggestions are also given.