Infrared spectroscopy of diatomic molecules can often be understood with two standard approximations: the
harmonic oscillator for vibration and the rigid rotor for rotation. In the harmonic
approximation, the two atoms oscillate about the equilibrium bond length as if they were joined by an ideal spring.
In the rigid-rotor approximation, the bond length is treated as fixed while the molecule rotates in space.
Although real molecules are not perfectly harmonic or perfectly rigid, these two models give a clear first picture of
molecular vibration-rotation spectra.
Vibrational energy levels
For a quantum harmonic oscillator, the vibrational energy levels are equally spaced:
Harmonic vibrational levels.
\[
\begin{aligned}
E_v &= \hbar \omega \left(v + \frac{1}{2}\right),
\qquad
v = 0,1,2,\dots
\end{aligned}
\]
The lowest vibrational state is not zero; it contains the zero-point energy \(E_{v=0} = \frac{1}{2}\hbar\omega\).
The spacing between adjacent vibrational levels is constant in this model, so the fundamental transition
\(v=0 \to 1\) has the same spacing as \(v=1 \to 2\).
Spectroscopists often describe the vibration in wavenumber units rather than angular-frequency units. If the
fundamental vibrational wavenumber is \(\tilde{\nu}_e\) in cm\(^{-1}\), then the angular frequency is related by
\[
\begin{aligned}
\omega &= 2\pi c \left(100\tilde{\nu}_e\right).
\end{aligned}
\]
Rotational energy levels
For a rigid rotor, the rotational energy depends on the rotational quantum number \(J\):
Rigid-rotor rotational levels.
\[
\begin{aligned}
E_J &= B\,J(J+1),
\qquad
J = 0,1,2,\dots
\end{aligned}
\]
In spectroscopy, it is convenient to express the rotational constant in wavenumber units:
\[
\begin{aligned}
\tilde{B} &= \frac{h}{8\pi^2 c I},
\qquad
I = \mu r^2.
\end{aligned}
\]
Here \(I\) is the moment of inertia, \(\mu\) is the reduced mass, and \(r\) is the bond length. A larger reduced mass
or a longer bond length increases the moment of inertia, which makes \(\tilde{B}\) smaller. That is why heavier or
more extended molecules usually have tighter rotational spacing.
Selection rules
For an infrared-active diatomic molecule in the basic model, the fundamental vibration-rotation band follows the
selection rules
Selection rules.
\[
\begin{aligned}
\Delta v &= \pm 1, \\
\Delta J &= \pm 1.
\end{aligned}
\]
The \(\Delta J = -1\) transitions form the P branch, and the \(\Delta J = +1\) transitions form the
R branch. In this simplified rigid-rotor plus harmonic-oscillator model, there is no Q branch
for a standard diatomic vibration because \(\Delta J = 0\) is not allowed here.
P and R branch line positions
If the band center is approximated by the fundamental wavenumber \(\tilde{\nu}_0 \approx \tilde{\nu}_e\), then the
line positions are
P and R branch formulas.
\[
\begin{aligned}
\tilde{\nu}_P(J'') &= \tilde{\nu}_0 - 2\tilde{B}J'', \\
\tilde{\nu}_R(J'') &= \tilde{\nu}_0 + 2\tilde{B}(J''+1).
\end{aligned}
\]
The characteristic line spacing is therefore about \(2\tilde{B}\). This is one of the most important features of a
simple vibration-rotation spectrum: the vibrational band center tells you about the bond strength and reduced mass,
while the rotational spacing tells you about the molecule’s moment of inertia.
Sample calculation: HCl
Consider HCl with reduced mass \(\mu \approx 0.9801\ \mathrm{amu}\), bond length \(r \approx 1.286\ \mathrm{\AA}\),
and vibrational wavenumber \(\tilde{\nu}_e \approx 2990\ \mathrm{cm^{-1}}\).
Step 1. Moment of inertia.
\[
\begin{aligned}
I &= \mu r^2
\end{aligned}
\]
After converting the reduced mass into kilograms and the bond length into meters, the moment of inertia leads to a
rotational constant of about
\[
\begin{aligned}
\tilde{B} &\approx 10.4\ \mathrm{cm^{-1}}.
\end{aligned}
\]
Therefore the approximate rotational spacing is
\[
\begin{aligned}
2\tilde{B} &\approx 20.8\ \mathrm{cm^{-1}}.
\end{aligned}
\]
The vibrational band center is simply the harmonic fundamental:
\[
\begin{aligned}
\tilde{\nu}_0 &\approx 2990\ \mathrm{cm^{-1}}.
\end{aligned}
\]
That means the spectrum will show one set of lines to the lower-wavenumber side (the P branch) and one set to the
higher-wavenumber side (the R branch), with a gap around the band center in this simple model.
Physical interpretation
The vibrational part of the spectrum mainly reflects the bond stiffness and the reduced mass. A stiffer bond or a
lighter reduced mass gives a higher vibrational frequency. The rotational part mainly reflects the moment of inertia:
molecules with smaller moment of inertia have larger \(\tilde{B}\) and wider spacing between rotational lines.
Together, the vibration-rotation spectrum gives information about both the bond and the geometry of the diatomic
molecule. That is why these models are central in introductory molecular spectroscopy.
University-level refinements
Real molecules are not perfectly harmonic and not perfectly rigid. At a more advanced level, one includes
anharmonicity, centrifugal distortion, and changes in the rotational constant between
vibrational states. These effects slightly shift the band center and make the line positions depart from the simple
equally spaced pattern. Still, the harmonic plus rigid-rotor model remains the standard starting point and is the most
useful first approximation for interpreting diatomic infrared spectra.
| Concept |
Main relation |
Meaning |
| Vibrational levels |
\(E_v = \hbar\omega(v + 1/2)\) |
Harmonic-oscillator vibrational energies |
| Rotational levels |
\(E_J = B J(J+1)\) |
Rigid-rotor rotational energies |
| Moment of inertia |
\(I = \mu r^2\) |
Rotational inertia of the diatomic molecule |
| Rotational constant |
\(\tilde{B} = h / (8\pi^2 c I)\) |
Controls rotational line spacing |
| Selection rules |
\(\Delta v = \pm 1,\ \Delta J = \pm 1\) |
Allowed fundamental IR transitions in the simple model |
| Branch lines |
\(\tilde{\nu}_P = \tilde{\nu}_0 - 2\tilde{B}J'',\ \tilde{\nu}_R = \tilde{\nu}_0 + 2\tilde{B}(J''+1)\) |
P and R branch positions |