A strictly diabatic electronic basis is defined as one for which all components of the nuclear momentum coupling vanish. We examine the possibility that such a basis may exist, and we find that, in general, it does not. The only important exception is for diatomic states of the same symmetry. We also consider some conditions for the definition of an approximately diabatic electronic basis. For molecular systems with three or more nuclei, one can obtain useful approximate diabatic basis sets if the transverse (solenoidal) part of the coupling is negligible; this may occur, for example, if the part of the coupling due to the internuclear-distance dependence of the configurational wave functions is negligible as compared to that due to the internuclear- distance dependence of the configurational coefficients. We derive a criterion showing that such approximations may be useful and accurate if the role of the coupling is important over regions of sufficiently small linear dimensions.