significance_of_negative_beta.md

Is there a physical reason for why the coupling constant $\beta < 0$? I tried to read the section in your paper but didn't see anything relevant.

Mathematically, it should be clear that as $\beta > 0$, the symmetric stretch becomes higher in energy than the asymmetric stretch. This is a consequence of how the linear combination of eigenvectors works out.

Physically,

The bend in the $\ce{CO2}$ determines the change in the coupling constant between the local modes, $\beta$. The motion of the central carbon atom is the primary motion that couples the two carbonyls. When they are collinear, the motion of one carbonyl directly influences the other. Bending the $\ce{CO2}$ means the carbonyls are no longer collinear, which means that the projection of one local vibration on the other decreases. This decrease, in turn, decreases the effective coupling constant.

The fit I found was

$$ \beta(\theta) = \pu{-515.7 cm^{-1} + (1.12 cm^{-1} deg^{-1}) \theta}, $$

where $\theta = \theta_{\ce{CO2}} - 180$, the deviation of the CO2 bending angle from linearity. As $\theta$ goes from 180 to 0, $\beta$ goes from -314.1 to -515.7. An increase in the magnitude of beta corresponds to an increase in the coupling. Pictorially, as $\ce{CO2}$ becomes bent, the coupling becomes destroyed and you're left with two independent local/isolated oscillators. Now for connection: $\theta = 180$ doesn't make physical sense, as it corresponds to $\theta_{\ce{OCO}} = 0$, or the two carbonyls lying directly on top of each other. Yet, according to the fit (which is from the small-$\theta$ limit, think perturbation theory here), the coupling constant is still negative.

I sense that if I try and explain this further, I might introduce some circular logic (beta is negative because the antisymmetric combination always has a larger eigenvalue). It might be similar to how the QM postulates result in eliminating the cosine term from the particle in a box eigenfunctions, even though they're mathematically admissible as part of a wave equation. If there's deeper meaning, I don't know it.