ammonia's rotational levels, sorted by K

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.

Two 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 constant a spectrum cannot see

Benzene’s rotational lines, with every K value falling on top of every other. That coincidence is the whole difficulty: the transition frequencies of a rigid symmetric top depend on one of its two constants and not at all on the other, so a spectrum with every line resolved still reports one number.

A third symmetric top’s levels, sorted by K. The levels themselves depend on both constants — the K-dependence is where the second one lives — and the differences that a spectrum measures do not, because the selection rule leaves K unchanged and the second constant cancels out of every allowed difference.

Ammonia’s rotational levels up to J = 4, grouped by J. Within each group the levels are pushed apart by (A − B)K², so the second constant is plainly visible as the spacing inside a group. Fifteen levels, every one computed.

The top that reports all three

For comparison, a symmetric top’s levels, labelled by J and K, with every K other than zero doubly degenerate. Those degeneracies are what an asymmetric top splits, and the two-subscript labels are a record of which member of a split pair a level came from.