It is customary when computing re-vibrational transitions in molecules to invoke the Born-Oppenheimer separation between nuclear and electronic motion. However, it is known from accurate calculations on H-2(+) and H-2 that the first-order (diagonal adiabatic) and second-order (nonadiabatic) corrections are not negligible and are both important. In the present work, we have made an ab initio implementation of the Bunker and Moss formalism for the nonadiabatic correction and applied it to H-2 and H2O. From comparison to accurate calculations for H-2, we find that we can obtain good results for the nonadiabatic correction using Cl singles to treat the electronically excited states if we scale the results, but we must go beyond the SCF approximation to obtain an accurate diagonal adiabatic correction. For H2O, we find that the first-order correction is more important than the second-order correction for bending energy levels, but the second-order correction is more important than the first-order correction for stretching energy levels. The correction to rotational levels is also significant. Thus, first- and second-order corrections are vital for accurate ab initio predictions of transition frequencies.