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

Let $u=r-r_e$ measure displacement from the equilibrium bond length and put the minimum of potential energy at zero. The harmonic model and the Morse potential areThe harmonic approximation is a symmetric parabola and has no dissociation limit. The Morse potential has a steep compressed side and a shallower extended side, approaching $D_e$ as $r\to \infty$. Matching the curvatures at the minimum gives $k=2D_e{\alpha }^2$. With $\tilde{\omega }$ denoting a spectroscopic wavenumber rather than an angular frequency, the energy levels areHere $x_e$ is the dimensionless spectroscopic anharmonicity constant. The harmonic spacing is constant, while the Morse oscillator spacing is $E_{v+1}-E_v=hc\tilde{\omega }[1-2x_e(v+1)]$ and decreases as the levels approach dissociation. Only finitely many Morse oscillator levels are bound. The exactly solvable potential was introduced to model this anharmonic behaviour.^[ref]

For the harmonic model put $q=\sqrt{\mu \omega /\hslash }u$, with $\omega =2\pi c\tilde{\omega }$, and $N={\pi }^{-1/4}$. The first three normalized reduced-coordinate wavefunctions areTheir numbers of nodes are zero, one and two, and their parities are even, odd and even. The $v=2$ nodes are at $q=\pm 1/\sqrt{2}$. The diagrams below use analytic curves and arbitrary illustrative scales, with both potentials given the same local curvature.

Write $s=v+1/2$ and complete the square in the Morse oscillator energy:The finite bound-level sequence approaches the dissociation limit at the vertex of this extrapolated parabola. Thus the bond potential-well depth isThe continuous vertex is a convenient way to identify the limit; it is not an additional normalizable bound level at the threshold.

The ground-state dissociation energy starts at the lowest vibrational level, rather than the minimum of the potential. The zero-point energy of the Morse oscillator isSubtracting it from the bond potential-well depth givesThis distinction matters whenever a dissociation energy is inferred from a vibrational spectrum.

The strong band is the $v=0\to 1$ vibrational fundamental, and the weaker band near twice its wavenumber is the $v=0\to 2$ first vibrational overtone. At room temperature these high-frequency oscillators have overwhelmingly ground-state vibrational populations. In the harmonic-potential, linear-dipole limit the infrared selection rule is $\Delta v=\pm 1$; anharmonicity and nonlinear dipole dependence make the overtone weakly allowed. A $1\to 2$ hot band would lie near, and slightly below, the fundamental wavenumber, rather than near twice it.

Let $f$ and $o$ denote the measured fundamental and first-overtone wavenumbers, and put $y=\tilde{\omega }{x}_e$. Differences of the Morse oscillator levels giveConsequently $x_e=y/\tilde{\omega }$, $D_e/(hc)={\tilde{\omega }}^2/(4y)$ and $D_0/(hc)=D_e/(hc)-(\tilde{\omega }/2-y/4)$. Substituting the two pairs of measured bands gives the following derived quantities:The conversion is $1\mathrm{c}{\mathrm{m}}^{-1}=0.0119626566\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, after multiplication by $N_{A}hc$ and consistent length units. Therefore the requested ground-state dissociation energies are $D_0(\mathrm{A})=448.4\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and $D_0(\mathrm{B})=388.5\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The corresponding potential-well depths are $D_e(\mathrm{A})=464.1$ and $D_e(\mathrm{B})=402.2\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. These are extrapolations under the stipulated Morse oscillator model; two low-lying bands do not establish the accuracy of the real potential near dissociation.

The molecule with the larger inferred dissociation energy is assigned the stronger hydrogen–halogen bond: $\mathrm{A}=\mathrm{HBr}$ and $\mathrm{B}=\mathrm{HI}$. Moving from bromine to the larger iodine atom generally lengthens the bond and reduces effective overlap with the compact hydrogen $1s$ orbital, so hydrogen iodide is more weakly bound. Independently, the tabulated ground-state harmonic constants near $2649\mathrm{c}{\mathrm{m}}^{-1}$ for hydrogen bromide and $2309\mathrm{c}{\mathrm{m}}^{-1}$ for hydrogen iodide agree with the inferred assignment.^[ref][ref] The ordering is the identification criterion; the low-level Morse oscillator extrapolations above should not be presented as measured dissociation energies.