Diatomic Rotational Spectra

The interactive plot below demonstrates the pure rotational (microwave) absorption spectrum of a diatomic molecule, treated as a rigid rotor with a correction for first order centrifugal distortion. A brief overview of the underlying mathematics is below the plot for completeness. The main ideas are:

You can play with molecule parameters $B$ and $D$, temperature $T$, and the resolved linewidth $\gamma$, to see how the ideal rotational spectrum would look. Parameters and spectra are given in wavenumbers. Mousover text on each slider gives more info.
Intensities for the transition $J \rightarrow J+1$ are given by the Boltzmann population in the level $J$ at the given temperature.
A number of preset molecules can be selected to illustrate typical behaviours. The parameters for these molecules are all experimental catalogued in the CCCBDB, with the exception of HeH which uses calculated parameters.

Note that the plot below is a teaching tool, so it allows some unphysical results to occur to illustrate limitations of models etc. In particular, you can push the $D$ constant far beyond what would be reasonable in any real molecule. Have a play and see what you can do.

Rotational Constant ($B$) = 10.592 cm^-1 Centrifugal Distortion ($D$) = 0.00e+0 cm^-1 Temperature ($T$) = 200 K Linewidth ($\gamma$) = 0.10 cm^-1 Highest $J$ included ($J_{\text{max}}$) Molecule preset Axis options

Normalise Autoscale

Briefly, for a diatomic rigid rotor the rotational energies $E_J$ are given by:

$$ E_J = BJ(J+1) - DJ^2(J+1)^2 $$

where $B$ is the rotational constant and $D$ the centrifugal distortion constant. Transitions from $J$ to $J+1$ lead to transition energies (lines) at:

$$ \Delta E_{J \rightarrow J+1} = E_{J+1} - E_J = 2B(J+1) - 4D(J+1)^3 $$

Each line is given on the plot as a Lorentzian with HWHM $\gamma$. The Lorentzian intensity $I_J$ comes from the Boltzmann population of the lower state given below.

$$ I_J \propto (2J+1)\exp\left(-\frac{hcE_J}{k_B T}\right) $$