9.2 Molecular spectra  (Page 2/6)

The vibrational energy level    , which is the energy level associated with the vibrational energy of a molecule, is more difficult to estimate than the rotational energy level. However, we can estimate these levels by assuming that the two atoms in the diatomic molecule are connected by an ideal spring of spring constant k . The potential energy of this spring system is

Where $\text{Δ}r$ is a change in the “natural length” of the molecule along a line that connects the atoms. Solving Schrödinger’s equation for this potential gives

Where $\omega$ is the natural angular frequency of vibration and n is the vibrational quantum number. The prediction that vibrational energy levels are evenly spaced $\left(\text{Δ}E=\hslash \omega \right)$ turns out to be good at lower energies.

A detailed study of transitions between vibrational energy levels induced by the absorption or emission of radiation (and the specifically so-called electric dipole transition) requires that

[link] represents the selection rule for vibrational energy transitions. As mentioned before, this rule applies only to diatomic molecules that have an electric dipole moment. Symmetric molecules do not experience such transitions.

Due to the selection rules, the absorption or emission of radiation by a diatomic molecule involves a transition in vibrational and rotational states. Specifically, if the vibrational quantum number ( n ) changes by one unit, then the rotational quantum number ( l ) changes by one unit. An energy-level diagram of a possible transition is given in [link] . The absorption spectrum for such transitions in hydrogen chloride (HCl) is shown in [link] . The absorption peaks are due to transitions from the $n=0$ to $n=1$ vibrational states. Energy differences for the band of peaks at the left and right are, respectively, $\text{Δ}{E}_{l\to l+1}=\hslash \omega +2(l+1)E_{0r}=\hslash \omega +2E_{0r},\hslash \omega +4E_{0r},\hslash \omega +6E_{0r},\text{…}\left(\text{right band}\right)$ and $\text{Δ}{E}_{l\to l\mathrm{-1}}=\hslash \omega -2l{E}_{0r}=\hslash \omega -2E_{0r},\hslash \omega -4E_{0r},\hslash \omega -6E_{0r},\text{…}(\text{left band})\text{.}$

The moment of inertia can then be determined from the energy spacing between individual peaks $\left(2E_{0r}\right)$ or from the gap between the left and right bands $(4E_{0r})$ . The frequency at the center of this gap is the frequency of vibration.

Summary

• Molecules possess vibrational and rotational energy.
• Energy differences between adjacent vibrational energy levels are larger than those between rotational energy levels.
• Separation between peaks in an absorption spectrum is inversely related to the moment of inertia.
• Transitions between vibrational and rotational energy levels follow selection rules.

Conceptual questions

Problems