Three counts for each molecule, and the barrier between them

One of the figures on point groups from coordinates: A molecule turned in space with its group recovered from the turned coordinates, and what the group settles on its own.

One essay draws 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 group of a molecule that will not hold still

Four molecules and three counts each. The point group is what a rigid rotation or reflection can do; the feasible group adds whatever a crossable barrier makes available; the conceivable group is every rearrangement of identical nuclei, whether the molecule can reach it or not.

Ethane’s three counts, on a logarithmic scale. Twelve operations without crossing a barrier, thirty-six with, and two thousand eight hundred and eighty conceivable — of which barely one in eighty is available.

Water for comparison, where the two groups agree. Its point group has order four, its feasible permutation-inversion group has order four, and nothing it can do at any temperature connects a version of itself to a different one — there is no low barrier and no tunnelling path, so the structural group is the spectroscopic one and the distinction this essay is about does not arise.