Rotational lines for three molecules

One of the figures on rotation and vibration: Moments of inertia, normal modes, isotope shifts, and the frequencies a force field does and does not fix.

Six essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The rotational spectrum is a moment of inertia

Three linear molecules on one axis. Hydrogen cyanide’s lines are seven times further apart than carbonyl sulfide’s, because its moment is seven times smaller — it is a light molecule and carbonyl sulfide is not. Carbon dioxide’s lines are drawn dashed: the levels are there and no transition between them can be seen, because the molecule has no dipole.

Three molecules whose constants span a factor of seven. Hydrogen cyanide’s comb is wide and sparse; carbonyl sulfide’s is seven times finer; benzene’s is finer still and is drawn dashed, because benzene has no dipole and shows none of it. The relation is simply one over the moment of inertia, and the moment is a sum over the atoms.

A bond length out of a spectrum

What was measured. Hydrogen cyanide’s lines are seven times further apart than carbonyl sulfide’s because its moment is seven times smaller, and carbon dioxide’s lines are drawn dashed because it has no dipole and shows none of it. Every structural conclusion in this essay came from the spacing of a comb like one of these.

The bond length the spectrum gives, against where in the spectrum it is read. A rigid rotor’s lines are evenly spaced and a real molecule’s are not, so a length recovered from the low-J end and one recovered from the high-J end are different numbers — and the difference is centrifugal distortion rather than experimental error.

A microwave constant predicted from an infrared one, which is the check that the distortion above is what it is claimed to be. The centrifugal term is fixed by the vibrational frequency and the rotational constant together, so it can be predicted from a quite different experiment — and the two agree.

The rotor that stretches

Four diatomics: the rotational constant and the stretching frequency, both measured, the distortion constant predicted from them, and the constant a microwave spectroscopist fits to the line positions. The predicted and fitted values agree within a few per cent, and the quantity being predicted spans three orders of magnitude between nitrogen and hydrogen fluoride.

The bond length a rigid rotor reads out of carbon monoxide’s spectrum, against which pair of lines the spacing was taken from. Seventeen parts per thousand between the lowest lines and J = 40, all in one direction. The lowest lines return the true constant almost exactly, which is why the defect can be missed entirely by an experiment that stops at low J.

The rigid picture, for comparison: three linear molecules, their lines evenly spaced, and the constants read straight off the spacings. Everything above is a correction to this, and every rigid-rotor structure argument is made inside it.

The constant a spectrum cannot see

The rigid rotor’s levels against the stretching rotor’s, for three linear molecules. The departure from even spacing is what a centrifugal term does, and it is the same term that, for a symmetric top, brings K back into the line positions.

The bond length that depends on the isotope

Where the third length comes from: a rigid rotor’s levels, spaced by the rotational constant. That constant is ħ²/2μ⟨r²⟩ in the rigid model and involves ⟨1/r²⟩ once the vibration is included — so a rotational spectrum measures an average of an inverse square, which is a different average from the one diffraction sees.

The coordinate an isotope reports