The rotational spectra of diatomic molecules are often described with the rigid-rotor model. In this
approximation, the two atoms are treated as point masses connected by a bond of fixed length. The molecule can rotate,
but the bond length does not stretch. This simple model explains why microwave rotational spectra show regularly spaced
lines and why the key molecular parameter is the rotational constant \(B\).
Moment of inertia
The first quantity needed is the moment of inertia:
Rigid-rotor inertia.
\[
\begin{aligned}
I &= \mu r^2.
\end{aligned}
\]
Here \(\mu\) is the reduced mass of the two-atom system and \(r\) is the bond length. A larger reduced mass or a
longer bond gives a larger moment of inertia. Physically, that means the molecule is harder to rotate, so the
rotational level spacing becomes smaller.
Rotational constant
The rotational constant can be written in several equivalent ways depending on the units you want:
Rotational constant in different units.
\[
\begin{aligned}
B_E &= \frac{\hbar^2}{2I}, \\
B_\nu &= \frac{B_E}{h} = \frac{h}{8\pi^2 I}, \\
\tilde{B} &= \frac{B_\nu}{c} = \frac{h}{8\pi^2 c I}.
\end{aligned}
\]
In spectroscopy, \(\tilde{B}\) is often reported in cm\(^{-1}\). In microwave work, the frequency version \(B_\nu\) is
also convenient, often expressed in GHz. These forms describe the same constant; they are just different unit systems.
Rotational energy levels
The rigid-rotor rotational levels are
Rotational levels.
\[
\begin{aligned}
E_J &= B_E\,J(J+1),
\qquad
J = 0,1,2,\dots
\end{aligned}
\]
or, in spectroscopic units,
\[
\begin{aligned}
\frac{E_J}{hc} &= \tilde{B}\,J(J+1).
\end{aligned}
\]
The factor \(J(J+1)\) means the levels are not equally spaced in energy. Instead, the gap between successive levels
increases with \(J\). That is why the transition frequencies also increase linearly with the lower rotational quantum
number.
Allowed transitions and line spacing
For a simple rotational spectrum, the main selection rule is
\[
\begin{aligned}
\Delta J &= \pm 1.
\end{aligned}
\]
If we consider the absorption transition \(J \to J+1\), then the line position becomes
Rotational transition positions.
\[
\begin{aligned}
\tilde{\nu}_{J\to J+1}
&= \frac{E_{J+1}-E_J}{hc} \\
&= \tilde{B}\left[(J+1)(J+2)-J(J+1)\right] \\
&= 2\tilde{B}(J+1).
\end{aligned}
\]
This is one of the most important rigid-rotor results. It shows that the rotational lines appear at
\(2\tilde{B}, 4\tilde{B}, 6\tilde{B}, \dots\). The spacing between neighboring lines is therefore constant:
\[
\begin{aligned}
\Delta \tilde{\nu}_{\text{adjacent}} &= 2\tilde{B}.
\end{aligned}
\]
Sample calculation: CO
For carbon monoxide, a common approximation is \(\mu \approx 6.857\ \mathrm{amu}\) and
\(r \approx 1.13\ \mathrm{\AA}\).
Step 1. Convert and compute the 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
\[
\begin{aligned}
\tilde{B} &\approx 1.93\ \mathrm{cm^{-1}}.
\end{aligned}
\]
Therefore the adjacent rotational spacing is
\[
\begin{aligned}
2\tilde{B} &\approx 3.86\ \mathrm{cm^{-1}}.
\end{aligned}
\]
In frequency units, this corresponds to about
\[
\begin{aligned}
2B_\nu &\approx 115\ \mathrm{GHz}.
\end{aligned}
\]
That is firmly in the microwave part of the spectrum, which is why rigid-rotor spectra are so important in microwave
spectroscopy.
Physical meaning
The rotational constant tells you how strongly the molecule resists rotation. Small, light molecules with short bonds
usually have larger \(B\) values and more widely spaced rotational lines. Heavier molecules or molecules with longer
bonds have smaller \(B\) values and more closely packed lines. That is why measuring a rotational spectrum is a direct
way to learn about molecular structure.
Advanced note
The rigid-rotor model is an excellent starting point, but real molecules are not perfectly rigid. At higher rotational
speeds the bond stretches slightly, which increases the moment of inertia and shifts the lines. This effect is called
centrifugal distortion. In an advanced treatment, the exact line positions are no longer perfectly
equally spaced. Even so, the rigid-rotor rotational constant remains the central first parameter in diatomic
rotational spectroscopy.
| Concept |
Main relation |
Meaning |
| Moment of inertia |
\(I = \mu r^2\) |
Rotational inertia of the diatomic molecule |
| Rotational constant |
\(\tilde{B} = h/(8\pi^2 c I)\) |
Spectroscopic rotational constant in cm\(^{-1}\) |
| Frequency form |
\(B_\nu = h/(8\pi^2 I)\) |
Rotational constant in frequency units |
| Rotational levels |
\(E_J/hc = \tilde{B}J(J+1)\) |
Rigid-rotor energy ladder |
| Transition positions |
\(\tilde{\nu}_{J\to J+1} = 2\tilde{B}(J+1)\) |
Rotational absorption or emission line positions |
| Adjacent spacing |
\(\Delta \tilde{\nu} = 2\tilde{B}\) |
Equal spacing of neighboring rigid-rotor lines |