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

Let $T_{v^{\prime },v^{\prime \prime }}=T_e+G^{\prime }(v^{\prime })-G^{\prime \prime }(v^{\prime \prime })$. For a fixed row, moving one column to the right therefore givesThus 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 ${\tilde{\omega }}_e^{\prime \prime }-2{\tilde{\omega }}_e^{\prime \prime }{x}_e^{\prime \prime }(v^{\prime \prime }+1)$. Stating the subtraction direction prevents an artificial negative vibrational frequency.

Averaging the left-minus-right differences across the three rows gives lower-state spacings of about $1613.0$ and $1591.7\mathrm{c}{\mathrm{m}}^{-1}$. Their difference is $2{\tilde{\omega }}_e^{\prime \prime }{x}_e^{\prime \prime }$, soSimilarly, averaging successive down-column differences across the three columns gives upper-state spacings of about $1750.3$ and $1710.7\mathrm{c}{\mathrm{m}}^{-1}$. HenceUsing the first row and first column alone would give approximately $(1637,11)$ and $(1790,20.5)\mathrm{c}{\mathrm{m}}^{-1}$, respectively. The small inconsistency between nominally identical spacings is already present in the entries; it should be treated as limited data/model precision rather than fitted by assigning different lower-state constants to different rows. Values near $(1.63\times 10^3,11)$ for the lower state and $(1.79\times 10^3,20)$ for the upper state express the supported precision.