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 / a

In the Condon approximation the intensity from $v^{\prime \prime }=0$ to $v^{\prime }$ contains the Franck-Condon factorThe lower ground-state vibrational wavefunction is localized near its equilibrium bond length. If the two equilibrium lengths are similar, it overlaps strongly with the upper $v^{\prime }=0$ wavefunction, but its overlap with highly excited upper vibrational states is usually small or suffers oscillatory cancellation. If the equilibrium lengths differ enough that a vertical excitation reaches the outer portion of the upper well, high-$v^{\prime }$ transitions can be strong while the $0\to 0$ overlap is small. Thus the two ends of a long progression are not normally both intense. This is an overlap argument, not a ban on either set of transitions; anharmonicity, a varying electronic transition moment and unusual wavefunctions can modify the intensity envelope.

Write the upper Morse oscillator term values in wavenumber units asThe spacing is $G(v+1)-G(v)={\tilde{\omega }}_e-2a(v+1)$. A Birge-Sponer plot therefore falls linearly in this model and can be extrapolated to zero spacing. If its horizontal coordinate is $x=v+1/2$, the plotted line is $y=({\tilde{\omega }}_e-a)-2ax$. Its area from $x=0$ to its zero givesEquivalently a plot against $v+1$ has intercept ${\tilde{\omega }}_e$ and triangle area ${\tilde{D}}_e={\tilde{\omega }}_e^2/(4a)$; subtracting $G(0)={\tilde{\omega }}_e/2-a/4$ recovers ${\tilde{D}}_0$. These conventions distinguish dissociation from the potential minimum and from the lowest vibrational level.

Measured consecutive members fix a local slope and intercept, so unobserved members can be supplied by extrapolation under an explicit spacing law. The upper-state limit on the absorption scale is ${\tilde{\nu }}_{\lim }={\tilde{\nu }}_{00}+{\tilde{D}}_0$. A local high-v fit can likewise estimate the remaining energy above an observed member without knowing its absolute v index. Near dissociation a real potential need not remain Morse-like, so an extrapolated triangle has model uncertainty; missing lines must not be silently assumed consecutive.

Successive differences remove the unknown absolute origin and the unknown initial vibrational quantum number. The five spacings are $329.3$, $296.0$, $261.7$, $226.7$ and $190.7\mathrm{c}{\mathrm{m}}^{-1}$. Plot them at relative mid-indices $x=n+1/2$. A straight-line fit givesThe curve reaches zero spacing at $x_{\ast }$. Since a quadratic Morse progression has a spacing equal to its derivative at the interval midpoint, integrating the fitted line from the first observed relative level $n=0$ to $x_{\ast }$ estimates the energy still needed to reach the continuous limit:Thus the appropriate graphical conclusion is ${\tilde{\nu }}_{\lim }\approx 5.80\times 10^4\mathrm{c}{\mathrm{m}}^{-1}$, not a many-digit threshold. Directly fitting a quadratic to the six measured transition positions puts its vertex at about $58042.8\mathrm{c}{\mathrm{m}}^{-1}$, which checks the arithmetic but not the physical extrapolation law. The slight variation of second differences shows that extrapolating a constant anharmonicity beyond the observed range is approximate. No assignment of the arbitrary index n to an absolute upper-state v is needed.