How far benzene is from its own group, against how far it has been pushed

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.

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.

One coordinate, three point groups

Thirteen structures, one per dihedral angle, each with its point group searched for from the coordinates. The group is C2v where the two hydrogens eclipse, C2h where they are anti, and C2 at every angle between. Below the axis are the two properties the group settles: the molecule may be polar everywhere except at 180°, and it is chiral everywhere except at the two ends.

The same sweep at half the sampling. Seven structures instead of thirteen give the identical three groups and the identical rails, which is what a property decided by exact operations rather than by a continuous quantity looks like: nothing is being interpolated.

How much symmetry is left

Each of the four distortions measured against the whole of D6h, beside the tolerance the search needs before it will use that name. Two of them — the alternating and the stretched structures — are within two per cent of each other in the measure and 1.88 times apart in the tolerance they require.

The same statement as a picture. Two ways of pushing benzene away from D6h — one atom moved, and every atom moved — with the measure against the size of the push. Both curves are quadratic in the distortion and the largest displacement is linear in it, so the two criteria differ by a factor that depends only on how the distortion is spread out.

Six molecules, each distorted in both ways and each scaled until the tolerance test sits exactly on its boundary, with the measure read there. If a tolerance meant one amount of asymmetry, every entry in the two middle columns would be the same number. They run from 0.00249 to 0.14034 — a factor of fifty-six.

The coordinate it was already soft along

The same reading on a molecule with a higher symmetry to lose. The shallowest slope and the steepest differ by more here than in ammonia, because Td has more species to distribute a distortion over — and the softest direction is again one the molecule already moves along at its zero point.

For three molecules, the softest and stiffest vibrations, the species each belongs to, and what a unit distortion along each costs. Every ratio is the ratio of the two frequencies to a part in a million — a check that the pricing is the force field’s rather than something invented for the comparison.

Every vibration of ammonia in order of steepness, with the species each belongs to. Ammonia is the clean case: its softest is A₁, its stiffest is E, and no species appears twice — so for this molecule a label does determine a cost, and for methane it does not.

An estimate that can be wrong by two

What a torsional coordinate is, drawn where it can be computed exactly: hydrogen peroxide at four dihedral angles, with its point group searched for from the coordinates at each. A rotor is a coordinate along which a molecule has distinguishable arrangements, and the count that the ceiling is exponential in is a count of exactly this.