Effect of Symmetry Operations on Degenerate Vibrations

Let us consider the normal vibrations of a linear triatomic molecule, like $CO_2$. The two vibrations $\nu_{2a}$ and $\nu_{2b}$ are obviously degenerate with each other. They are antisymmetric with respect to an inversion at a center of symmetry as is the vibration $\nu_3$. Another possible symmetry operation is rotation $C_{\infty}^{\varphi}$ by an arbitrary angle $\varphi$ about the internuclear axis. This rotation leaves the vibrations $\nu_1$ and $\nu_3$ unchanged, however, both $\nu_{2a}$ and $\nu_{2b}$ are changes by more than just a sign. In other words, the vibrations $\nu_{2a}$ and $\nu_{2b}$ are neither symmetric, nor antisymmetric with respect to the rotation $C_{\infty}^{\varphi}$. The displacement normal coordinates after the rotation $Q_{2a}'$, $Q_{2b}'$ can be expressed over the normal coordinates before the rotation $Q_{2a}$, $Q_{2b}$ as

$$Q_{2a}' = Q_{2a} \cos\varphi + Q_{2b} \sin\varphi$$ (12)

$$Q_{2b}' = -Q_{2a} \sin\varphi + Q_{2b} \cos\varphi$$ (13)

Therefore, degenerate vibrations, in general, transformed under a symmetry operation as a linear combination of each other. This result is valid for any number of the degenerate vibrations and any type of symmetry operations involved.

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