[tulip11]

Hi all,

I'm looking at the rotational constants of C2H2 which is a linear polyatomic. I've calculated the values of B₀ and B₁ for both the IR active asymmetric stretch and bend. However for the bend (perpendicular transition) both rotational constants were higher then for the stretch (parallel transitions). As it is the same molecule I was expecting the rotational constants to be the same and I was wondering if someone could help me work through why they are different (or confirm my thinking that they should be same and I have miscalculated)!

Thank you!

[Corribus]

In the rigid rotor approximation, the molecular structure is fixed and the rotational constant is... constant, because it is assumed in this approximation that the bonds are rigid and the moment of inertia is always the same. In reality this is not the case. Rotational motion is coupled to vibrational motion and as the molecule vibrates, some of the vibrational energy is transferred into rotational motion - and vice-versa. Therefore you actually have different rotational constants depending on what vibrational state the molecule is in. This is why we express the true rotational constant as a function of the rotational constant in the rigid rotor approximation (B_e) modified by several distortion terms such as D_e and α_e.

Maybe a better way to look at it is something like this: if the molecule goes from a low energy vibration state to a high energy vibration state, the average shape of the molecule changes. For a linear molecule undergoing a symmetric stretch along the molecule axis, a higher vibrational state results in a (on average) longer molecule. This translates into a higher moment of inertia, and lower rotational constant. For polyatomic molecules that have multiple vibration modes, you may convince yourself that excitation of these different vibrational modes will result in the molecule obtain different "average" shapes, which means that excitation of the different modes will result in different rotational behavior as well because the moments of inertia change accordingly. You haven't shown any data, so I wouldn't want to venture too much speculation, but undergoing a bend and a stretch result in very different molecular shapes, so it would make sense that the B₁ and B₀ values are different during these different vibrational transitions. Another way to interpret this is that vibration-rotation motion is more strongly coupled during some vibrational transitions than others. You might be able to infer a qualitative degree of change in the average molecular structure by comparing the robivrational coupling constants that you calculate for these two vibrational modes (i.e., which mode results in a larger or smaller average structural change).

So to sum: I would not expect that the B₀ and B₁ values, as well as the rotational-vibrational coupling constants, should be the same for different vibrational modes. You could apply an ad hoc rationalization for why the B₀ and B₁ values are larger for the bend than stretch based on how you would expect the average structure changes during excitation into these vibrational modes. This may be difficult to predict beforehand, though, particularly as the molecular structure gets more complicated.