Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 23 / c / iii / Solution

$k\simeq 5.10\times 10^2\mathrm{N}{\mathrm{m}}^{-1}$ and $\gamma \simeq 0.638\mathrm{N}{\mathrm{m}}^{-1}$. With $m=m_{\mathrm{u}}$ and ${\omega }_0=2\pi c{\tilde{\nu }}_0$, the dilute-gas stretch givesUse $c$ in $\mathrm{cm}{\mathrm{s}}^{-1}$ here. The shifted line is the higher totally symmetric mode; the other molecular stretch remains at ${\omega }_0$. To first order,The ratio $\gamma /k=1.25\times 10^{-3}$ validates the weak-coupling expansion. Inverting the exact eigenvalue instead, with $L=m{\omega }_{+}^2$, givesa negligible change at the precision of the model.