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I'm studying that if r is the quantum number associated with the total angular momentum of a diatomic homonuclear molecule, then only odd or even values are allowed for r.

Now, the selection rules for r requires a change of one unity, but if my molecule has only odd values for r, then every transition will require at least r --> r+2, and this is not allowed.

So, how can I have a rotational spectrum?

Maybe I have to consider transitions of higher order of approximation than the electric dipole ones? Or should I consider Raman scattering? But I'm reading that Raman scattering allows r --> r+2 if and only if there is an intermediate allowed state for which r-->r+1.