On the comparison between crude and adiabatic Born-Oppenheimer coupling elements

A comparison is made between the coupling elements arising from the crude and the adiabatic Born-Oppenheimer approximation. It is shown, that if one includes the differences between the vibrational states in these approximation, it is possible, within the harmonic approximation, that the crude coupling elements are equal to, or even less than, the adiabatic ones. Further, it is shown, which quantities should be known to decide for the better basis set.