The characters of C6, and the levels they are

One of the figures on hückel systems: Adjacency matrices diagonalised: levels, coefficients, bond orders, and the shell closures that decide which rings are stable.

Five 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 ring's levels are its group's characters

The six representations of a six-ring’s rotation group, drawn as points on the unit circle, with the levels read off as twice the horizontal coordinate. A representation and its conjugate have the same real part, so the levels pair up automatically. Nothing here has been diagonalised.

Every level of every ring from three to twelve, from the characters and from the eigensolver, drawn together. The dots are the group-theoretical values and the bars are the eigenvalues; they agree to nine figures at every size. Beneath each ring is the electron count of its first closed shell.

One level per representation, with the representation printed beside it. The degenerate pairs are the conjugate pairs of characters, and which real combination of such a pair gets drawn as an orbital is a convention. The integer is not a convention: the k-th representation carries k nodes, so the orbital pictures and the energies are ordered by the same number.

One spectrum, a line of models

And the reason: a ring’s levels are its group’s characters, so their ordering and degeneracies are fixed by the symmetry before any parameter is chosen. What a parameter sets is the scale of the level pattern, not the shape of it.

Two rules that share no arithmetic

The six characters of the rotation group of a six-ring, as points on the unit circle. Each one’s real part, doubled, is a level. Threading a flux through the ring rotates every point by the same angle — the flux, in quanta, times a full turn divided by the ring size — which is the whole of what a magnetic field does to this calculation.

A parameter that never finds a value

Why one of the two ties is exact rather than close: a ring’s Hückel levels are the characters of its rotation group, so benzene’s highest occupied level sits at 1 for a reason with no parameter anywhere in it. Ethene’s sits at 1 because a two-site system has levels at ±1. Two unrelated arguments, one number, and no fitting in either.

The current does not divide

Why a single ring’s answer is exact: its levels are its rotation group’s characters, so a flux is a shift of the character index and the response follows in closed form. A fused system has no such group, which is why the matrix above had to be computed.