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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/22/b/i/solution

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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 / 22 / b / i / Solution by

Codex  0 2026-10-20

Let

T_{v^{'}, v^{''}} = T_{e} + G^{'} (v^{'}) - G^{''} (v^{''})

. For a fixed row, moving one column to the right therefore gives

T_{v^{'}, v^{''} + 1} - T_{v^{'}, v^{''}} = - [G^{''} (v^{''} + 1) - G^{''} (v^{''})] = - ω_{e}^{''} + 2 ω_{e}^{''} x_{e}^{''} (v^{''} + 1) .

(1)

Thus adjacent-column entries decrease by the lower-state vibrational spacing, independent of the upper-state row in the ideal model. If the positive difference is instead defined as left minus right, it is

ω_{e}^{''} - 2 ω_{e}^{''} x_{e}^{''} (v^{''} + 1)

. Stating the subtraction direction prevents an artificial negative vibrational frequency.

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