Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2A / Section B / 24 / a

The missing kinetic contribution is $\frac{1}{2}{m}_b{\dot{y}}_b^2$. Both atoms contribute independent translational kinetic energy before the coordinates are changed.

Substituting $y_i=q_i/\sqrt{m_i}$ givesThe shifted equilibrium coordinates satisfy ${\bar{q}}_b/\sqrt{m_b}-{\bar{q}}_a/\sqrt{m_a}=\bar{r}$. Expanding about them replaces q by displacements without changing the kinetic metric or the Hessian.

Taking the Hessian of the potential with respect to mass-weighted displacements givesThe equations of motion are $\ddot{\mathbf{q}}=-D_0\mathbf{q}$, so eigenvalues are squared angular frequencies. Normalized eigenvectors and frequencies areBoth vectors are in the q basis, and $\mu =m_a{m}_b/(m_a+m_b)$. The translation corresponds to equal physical displacements $\delta y_a=\delta y_b$. The stretch has $m_a\delta y_a+m_b\delta y_b=0$ and opposite physical displacements with magnitude ratio $|\delta y_a/\delta y_b|=m_b/m_a$. Ordinary frequencies are $\nu =\omega /(2\pi )$; spectroscopic wavenumbers are $\omega /(2\pi c)$.