 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2b/section-b/34/b/solution

Write

s = v + 1/2

E (s) = h c ω (s - x_{e} s^{2}) = \frac{h c ω}{4 x _{e}} - h c ω x_{e} (s - \frac{1}{2 x _{e}})^{2} .

The finite bound-level sequence approaches the dissociation limit at the vertex of this extrapolated parabola. Thus the bond potential-well depth is

D_{e} = \frac{h c ω}{4 x _{e}} .

The continuous vertex is a convenient way to identify the limit; it is not an additional normalizable bound level at the threshold.

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