Since the HCl molecule has a non vanishing dipole moment, it is allowed to have pure rotational spectrum. If I understand this correctly then it's possible to excite the molecule in such a way that its vibrational energy stays the same while its rotational energy increases by some amount. This can basically be found there: http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/rotrig.html#c4. However in http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/vibrot.html#c2, one reads that the quantum number v must change in a transition according to [itex]\Delta v =\pm 1[/itex] (and [itex]\Delta j = \pm 1[/itex]). So that it seems that the molecule cannot (due to the [itex]\Delta v =\pm 1[/itex] rule) have a purely rotational spectrum. To me this contradicts the first assertion but obviously I'm missing something. What is going on?