 Solution

ID: 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/ii/solution

Substituting

y_{i} = q_{i} / m_{i}

gives

E = \frac{1}{2} \overset{q}{˙}_{a}^{2} + \frac{1}{2} \overset{q}{˙}_{b}^{2} + \frac{k _{1}}{2} (\frac{q _{b}}{m _{b}} - \frac{q _{a}}{m _{a}} - \overset{r}{ˉ})^{2} .

The shifted equilibrium coordinates satisfy

\overset{q}{ˉ}_{b} / m_{b} - \overset{q}{ˉ}_{a} / m_{a} = \overset{r}{ˉ}

. Expanding about them replaces q by displacements without changing the kinetic metric or the Hessian.

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