In rovibrational spectra, when exactly is the Q branch forbidden?

I'm aware that the Q branch ($\Delta j = 0$) is usually not observed in [rotational-vibrational spectra](https://en.wikipedia.org/wiki/Rotational%E2%80%93vibrational_spectroscopy) of diatomic molecules in the gas state, however, I've heard different things about when exactly this transition is forbidden.


I've sometimes been told that the transition is not allowed in general, for any diatomic molecule, but I've sometimes heard that this is only true if the electronic state has a $\Lambda \neq0$, so which is the correct selection rule here?


I'd love to have this clarified for me. If it is possible I would also like to know when is a Q branch observed in vibronic spectra of heteronuclear and homonuclear diatomics.




To determine the intensity of a rotation-vibration transition, you need to consider the transition momentum integral $\int \psi\_f \hat{\mu} \psi\_i d\tau$, where $\psi\_f$ is the final state, $\psi\_i$ is the initial state, $\hat{\mu}$ is the transition dipole moment operator, and $d\tau$ indicates integration over all space. Assuming separation of vibrational and rotational motion, this factors into the product of two terms: one that specifies the intensity of the vibrational transition and another that specifies the intensity of the rotational transition. The vibrational integral leads to a selection rule $\Delta v = \pm 1$ (in the harmonic approximation) and the rotational integral leads to a selection rule $\Delta J = \pm 1$. So no Q branch occurs. However, *if* there is some additional source of angular momentum (e.g., from vibrations or electronic motion) then it is possible to have a Q branch because now that extra motion can contribute to the total angular momentum in a way that allows the Q branch to occur.


**So here is the rule that you ask for:** a Q branch will not occur in transitions between $\Sigma$ electronic states of a diatomic molecule, but can occur in transitions between $\Sigma$ and $\Pi$ states. Since vibration-rotation transitions by definition occur within a single electronic state, Q branches do not occur in vibration-rotation spectra of diatomic molecules in a $\Sigma$ state. In a polyatomic molecule, however, vibrational modes can carry angular momentum so even within a single electronic state that carries no electronic angular momentum it may be possible to observe Q branches in a vibration-rotation spectrum so long as the vibration being excited is one which carries nuclear orbital angular momentum (e.g., from a vibrationless ground state to a vibrationally excited bending mode).


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