Mutual exclusion across every group here

One of the figures on representations: Character tables generated from a molecule's own operations, bases reduced in them, and what a descent in symmetry does to a level.

Four 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.

Mutual exclusion does not prove a centre

Fifteen point groups: which representations carry a coordinate, which carry a product of coordinates, and whether any carries both. The eight with a centre all exclude, which is the theorem. Six of the seven without one show coincidences. The seventh is the interesting row.

Mutual exclusion across four molecules and three groups, two of which have a centre of inversion and one of which does not. The two conformers of ferrocene are the pair the essay turns on: one shows the exclusion and one does not, and only the second has a centre — so the rule’s converse is refuted by a molecule that is not exotic and is not a counterexample constructed for the purpose.

The two conformers side by side. Both have 57 vibrations; the eclipsed form’s split across the D5h representations and the staggered form’s across D5d. One of the two has a centre of inversion and the other has not, and neither has a single representation carrying both a coordinate and a product of coordinates.

One table, three groups

The vibrational species of the two ferrocene conformers, sorted by infrared and Raman activity. Fifty-seven modes each and the same character matrix; the eclipsed conformer has sixteen infrared-active and twenty-six Raman-active modes with none in both, and the staggered one has coincidences. Identical tables, opposite selection rules.

A dipole is not what an infrared spectrum sees

The other half of the same rule, from the symmetry field: in a centrosymmetric molecule no mode can be both infrared and Raman active, because no species carries both a linear and a quadratic function. Carbon dioxide’s silent stretch is not silent in a Raman spectrum, which is how it is measured.

The one intensity symmetry does fix

The mutual-exclusion census: every tabulated group, with whether it excludes and where its coincidence lies. The trace is a totally symmetric quadratic in every one of them without exception, which is why the three quarters has no counterexample of the kind that mutual exclusion turned out to have.