Two kinds of answer to a flux, and only one of them is a curve

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

Seven 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.

Two rules that share no arithmetic

Four rings, with their levels computed twice: once as twisted systems and once as flat rings carrying half a flux quantum. The two lists agree to the last digit carried, for every ring. A twist and half a quantum of flux are the same boundary condition arrived at from two directions, and the Möbius calculation and the magnetic one are therefore checks on one another.

The π binding of three rings against the flux through them, each measured from its own value at zero. Benzene and the ten-ring have a smooth maximum there, so their energies have a smooth minimum and a susceptibility. Cyclobutadiene has a corner: its binding rises in both directions with a slope that does not go to zero, and the two one-sided slopes differ by 4π.

Eighty rings, classified twice. A filled mark is a ring whose energy rises when a flux is put through it; an open one is a ring whose energy has a corner instead. The circle marks the cases whose electron count is 4n + 2. The two classifications agree in all eighty, and they share no arithmetic: one is a derivative of an energy and the other is a count of electrons into shells.

The current does not divide

The current each ring of an acene carries under a uniform field, ring by ring. Naphthalene’s two are equal because a symmetry says they must be; anthracene’s three are not, and its middle ring is the busiest.

The response matrix of a four-ring acene. A circuit of independent loops would have nothing off the diagonal at all.

The single-ring picture: the response of a single ring against its electron count, with the diamagnetic and paratropic cases separated. Everything in this essay is that calculation done once per ring instead of once per molecule.

A reach that has no length

The fitted decay length against the number of rings, for the bare acene and for the same molecule with a gap held open. One of them keeps growing.

The mean response against ring separation, for a twelve-ring acene and for the same molecule with a gap. One is a straight line on a logarithmic axis and one is not, and the open marks at the right are what an end pair does to a fit.

The matrix all of this is measured on: a uniform field does not divide its response equally between equal rings. The off-diagonal entries are the ones whose decay is being chased here.

Neither of the two separations

Six hexagons fused two ways. The same formula, the same number of bonds, and not the same graph — the middle rings’ two shared bonds lie opposite each other on the left and one edge apart on the right.

Every entry of a five-ring linear molecule’s response matrix by both constructions, with the size of the entry across and the disagreement up. The largest disagreement is the finite difference’s own noise.

The response of every pair of rings in the linear six-ring molecule, against the number of fusions between them and against the distance between their centres. The two panels are the same picture with the horizontal axis relabelled.

An end effect with two signs

The end pair’s response divided by the deepest pair’s, at each separation, for three chain shapes of nine rings.

Every pair of a straight chain of nine, by separation and response, coloured by how far the pair sits from the nearer end.

Every pair of the mixed chain, divided by the deepest pair’s response at the same separation.

One integer, and everything it changes

Chains of nine rings differing in which fusion is turned: the widest ratio between two pairs at the same separation, against which fusion it is.

Every pair of the worst case — a chain of nine turned at its third fusion — by separation and response, coloured by depth.

At two fusions’ separation: the pair touching the left end and the pair touching the right, each against the deepest pair.

The scatter counts angular rings

The published chains at the angular rings they actually have, over the curves for one angular ring and for two adjacent ones built on purpose.

The worst spread with alternating turns against the worst spread with every turn in one sense, at the same angular rings.

The worst spread against the number of angular rings, for every chain measured.