We’ve made many uses of the idea that atoms are held together by spring-like molecular bonds. We’ve always assumed that the bonds could be modeled as classical springs. The classical model is acceptable for some purposes, but it fails to explain some important features of molecular vibrations. Not surprisingly, the quantum harmonic oscillator is a better model of a molecular bond.

FIGURE 40.23 shows the potential energy of two atoms connected by a molecular bond. Nearby atoms attract each other through a polarization force, much as a charged rod picks up small pieces of paper. If the atoms get too close, a repulsive force between the negative electrons pushes them apart. The equilibrium separation at which the attractive and repulsive forces are balanced is $r_0$, and two classical atoms would be at rest at this separation. But quantum particles, even in their lowest energy state, have $E>0$. Consequently, the molecule vibrates as the two atoms oscillate back and forth along the bond.

FIGURE 40.23 The potential energy of a molecular bond and a few of the allowed energies.

$U_{\text{dissoc}}$ is the energy at which the molecule will dissociate and the two atoms will fly apart. Dissociation can occur at very high temperatures or after the molecule has absorbed a high-energy (ultraviolet) photon, but under typical conditions a molecule has energy $E\ll U_{\text{dissoc}}$. In other words, the molecule is in an energy level near the bottom of the potential well.

You can see that the lower portion of the potential well is very nearly a parabola. Consequently, we can model a molecular bond as a quantum harmonic oscillator. The energy associated with the molecular vibration is quantized and can have only the values

where $\omega$ is the angular frequency with which the atoms would vibrate if the bond were a classical spring. The molecular potential-energy curve is not exactly that of a harmonic oscillator, hence the $\approx$ sign, but the model is very good for low values of the quantum number $n$. The energy levels calculated by Equation 40.49 are called the vibrational energy levels of the molecule. The first few vibrational energy levels are shown in Figure 40.23.

At room temperature, most molecules are in the $n=1$ vibrational ground state. Their vibrational motion can be excited by absorbing photons of frequency $f=\text{Δ}E\text{/}h$. This frequency is usually in the infrared region of the spectrum, and these vibrational transitions give each molecule a unique and distinctive infrared absorption spectrum.

As an example, FIGURE 40.24 shows the infrared absorption spectrum of acetone. The vertical axis is the percentage of the light intensity passing all the way through the sample. The sample is essentially transparent at most wavelengths $(\text{transmission }\approx 100\%)$, but there are two prominent absorption features. The transmission drops to $\approx 75\%$ at $\lambda =3.3\mu \text{m}$ and to a mere 7% at $\lambda =5.8\mu \text{m}$. The $3.3\mu \text{m}$ absorption is due to the $n=1$ to $n=2$ transition in the vibration of a $\text{C}-\text{C}{\text{H}}_3$ carbon-methyl bond. The $5.8\mu \text{m}$ absorption is the $1\to 2$ transition of a vibrating $\text{C=O}$ carbon-oxygen double bond.

FIGURE 40.24 The absorption spectrum of acetone.

Absorption spectra are known for thousands of molecules, and chemists routinely use absorption spectroscopy to identify the chemicals in a sample. A specific bond has the same absorption wavelength regardless of the larger molecule in which it is embedded; thus the presence of that absorption wavelength is a “signature” that the bond is present within a molecule.