Neither of the two separations

A reach that has no length measured how far a ring current’s response reaches between the rings of an acene, discovered that the bare molecule has no length at all — the fitted value grows with every ring because the gap that would set it is closing — and recovered one by holding a gap open with a staggered site energy. With the gap fixed the reach settles at 0.636 rings by the third molecule and stops moving.

That number is a length in a unit that had to be chosen. Rings, not ångströms. In a linear acene the two are the same statement: rings $i$ and $j$ are $|i-j|$ fusions apart and their centres are $\sqrt{3}\,|i-j|$ units apart, so every fit against one is the same fit against the other, with the same residuals and the same coefficient of determination to the last bit. No acene could ever say which of the two the response decays in.

Bending the molecule separates them. Phenanthrene has anthracene’s formula and its rings are fused at an angle, so two rings two fusions apart have centres 3.000 units apart instead of 3.464 — and in a chain whose fusions are mixed, two pairs can be the same number of steps apart and different distances apart, or the same distance apart and reached by different routes.

The answer is that the response decays in neither, and the way it fails is more useful than either answer would have been.

A gauge that does not know what shape the molecule is #

The acene calculation put a phase factor on each of the transverse bonds and read each ring’s flux off the difference between consecutive ones. That construction is exact and it is specific: it needs a molecule whose rings are strung along a line and share a set of parallel bonds. A phenacene has no such set.

The general construction is a thin solenoid at each ring’s centre. Its vector potential is the gradient of the polar angle about that centre, so what a bond picks up is the change in that angle along it, wrapped into half a turn either side. Summed around any closed loop, that change is exactly one turn if the loop encloses the centre and zero if it does not — so ring $k$ carries the flux it was given and every other cycle carries none, whatever the shape.

The wrap is safe here, and it is worth saying why rather than hoping. A bond of a ring subtends sixty degrees at that ring’s own centre; every other bond is further away and subtends less. Nothing in a fused benzenoid comes near the half turn at which the wrap would take the wrong branch.

The two agree to $6 \times 10^{-8}$ on entries running up to $10^{-2}$, which is the noise of a second derivative taken at a step of a thousandth. A gauge written for generality is worth nothing unless it is the same gauge, and this is the only way to know.

The molecule that cannot be asked #

Running both fits on the linear molecule gives the same number twice.

The coefficients of determination are identical to twelve figures, because the horizontal axis of the second panel is $\sqrt{3}$ times the first. What changes between them is only the number the length is quoted in: 0.6430 rings, or 1.1137 units.

The angular molecule separates them, and it separates them weakly — a zigzag chain’s centre distances are 1.732, 3.000 and 4.583 at one, two and three steps, which is not proportional to the step count but is still a function of it. So the two fits differ, and they differ only in curvature.

Fitted against steps the angular molecule gives $r^2 = 0.98105$; against distance, 0.97514. That is a preference and it is a small one.

What a mixed chain does to both #

A chain whose fusions are neither all straight nor all alternating breaks the remaining tie. In a chain of six hexagons fused straight, straight, angled, angled, straight, the four pairs of rings two steps apart are at two different distances, and pairs at the same distance sit in different parts of the molecule.

Rings 1–3 and 3–5 are both 3.4641 units apart and respond at $2.3312 \times 10^{-3}$ and $1.5247 \times 10^{-3}$. Rings 2–4 and 4–6 are both 3.0000 units apart and respond at $1.9744 \times 10^{-3}$ and $2.6580 \times 10^{-3}$.

The largest of the four is at the shorter distance and the smallest is at the longer one, with one of the shorter-distance pairs between them. A quantity that a variable does not even order cannot be said to decay in it.

The same molecule refutes the other candidate as well, less dramatically. All four pairs are two steps apart and they span a factor of 1.74, where one step of decay is a factor of about five — so the scatter inside a single separation class is worth about a third of a step.

What the scatter is #

It sorts almost perfectly on one thing: whether the pair includes an end ring. Of the four pairs at two steps, the two largest are 1–3 and 4–6, and each of them reaches an end.

That is the acene fit’s own exclusion arriving from a different direction. It restricted the fit to separations of at most half the molecule, because at the largest separation there is exactly one pair — the two ends — and its entry is an end effect. The restriction is right and it is not enough: excluding the longest separation leaves every other end-involving pair in the fit, and those are systematically high.

The spread within a separation class is 1.194 on the four-ring linear molecule and 1.596 on the eight-ring one, so it grows with the molecule rather than washing out. That is what an end effect does when the fit is over the pairs a small molecule has: a larger molecule has more interior pairs, but it also has more pairs at each separation, and the end ones stay as far out as they were.

That reading has a test the previous paragraph does not: if the excess is about the end, it should be a property of the outer ring rather than of the pair, and the same excess should appear at every separation. It does. On the linear molecule the largest entry at one step is between rings 1 and 2 and the smallest is between 3 and 4; at two steps the largest are 1–3 and 4–6 and the smallest are 2–4 and 3–5; at three steps the largest are 1–4 and 3–6. Every one of those is the pair that reaches an end.

Why an end ring should respond more is what the current does not divide already implies: a ring’s response is not its own property, because the current in it depends on the flux through its neighbours, and a ring at the end has one neighbour instead of two. Less of the molecule is available to take its share.

Which of the two is the better description anyway #

Taking the two families’ fits at face value: the per-step length is 0.6347 linear and 0.5976 angular at eight rings — six per cent apart. The per-unit length is 1.0994 and 0.8584 — twenty-two per cent apart.

So if a single number has to be quoted, it should be quoted per fusion. The bond count is roughly four times better than the distance at being invariant between the two shapes, which is the answer the question was asked for, and it comes with the caution that neither is invariant and that the residual six per cent is the same size as the scatter the fit averages over.

That is not a surprising conclusion for a Hückel model, and it is worth saying that it is not surprising. Nothing in this model knows about space at all: the Hamiltonian is a graph, the flux enters through which cycle it threads, and the centre coordinates exist only because they were used to place the solenoids. The finding is not that the model prefers topology — it must — but that the magnitude of the preference is small enough to be swamped by an end effect in every molecule of the size anybody computes.

One more comparison is worth making because it is free. The angular family’s decay per step is faster than the linear family’s — 0.5976 against 0.6347 at eight rings — while its rings are closer together. A picture in which the response leaks from ring to ring through the shared bond would have predicted the opposite: the same bond, the same leak, the same length in steps. What differs between the two families is which bonds the middle ring shares, and a fusion that turns puts the two shared bonds one edge apart rather than opposite, which changes what a cycle through both of them looks like. That the change is only six per cent is what makes the topological description nearly right rather than exactly right.

What was computed, and how #

Each response matrix entry is a mixed second derivative of the π energy with respect to two rings’ fluxes, taken as a four-point stencil at a step of one thousandth of a flux quantum. The energy at each flux is the sum of the occupied levels of a complex Hermitian matrix, diagonalised through the standard real embedding of a complex Hermitian matrix.

Every molecule carries a staggered site energy of $\pm 0.6$, which opens a gap that stays open as the molecule lengthens. Without it there is no length to measure, which is the acene finding and is why the control is inherited rather than re-derived. The stagger is checked to be a stagger: the graph is required to be bipartite before it is applied, so what it does is open a gap rather than shift a band.

The chain builder places hexagon centres $\sqrt{3}$ apart in one of six directions and generates each ring’s six vertices, deduplicating on coordinates rounded to six places. Two fused hexagons therefore share exactly two carbons, and the carbon and bond counts are checked against $4m + 2$ and $5m + 1$ rather than trusted.

The refusal is benzene. One ring has no pairs, so there is no decay to fit, and the fit must be refused rather than returned from a line through one point.

Where the model stops #

And the two separations are the only two the graph offers. A molecule has a third notion of distance — through-space, in ångströms, along a line that need not lie in the molecule at all — and nothing here computes it, because the model has no geometry beyond the positions the builder assigns. On a linear acene the three orderings coincide; on a molecule that folds back on itself they do not, and the scatter measured here would be a different scatter.

There is no geometry in the Hamiltonian. Two rings whose centres are three units apart and two whose centres are three and a half apart are, to this model, distinguished only by which carbons they share with what — so the finding that distance does not order the response is a statement about a graph model and cannot be carried to a calculation that has a real potential in it. What can be carried is the shape of the failure: a response tabulated against separation, in a molecule where two definitions of separation exist, will disagree with itself.

Whether a ring current is the right thing to be measuring at all is a separate question with its own answer: two rules for aromaticity share no arithmetic and agree anyway, and a magnetic criterion is one of them. Nothing here bears on that agreement; what is measured is how one of the two criteria falls off inside a molecule, which is a question the criterion itself does not settle.

The stagger is an artificial gap. It is the right control for the question — the same molecule, the same graph, one number changed — and it is not a chemical situation. A real molecule with a gap that stays open as it lengthens is a different molecule.

And a mixed chain of six hexagons fused straight, straight, angled, angled, straight is not a compound anybody has made. It is an instrument built to separate two variables that no real acene separates, which is what an instrument is for, and no claim here is a claim about a substance.

The generalisation #

The useful statement is about tabulation rather than about rings. A quantity that has only ever been tabulated against one variable, in systems where two variables coincide, has not been measured against either.

The acene fit is the clean example: it fitted a decay against ring separation on molecules where ring separation and centre distance are proportional, and no amount of care inside that fit could have revealed which it was measuring. A structural length measured the same way has the same exposure, and on a chain there is only one separation to measure it against. The repair is not a better fit. It is a system where the two come apart — and the systems that do that are usually not the ones the quantity was defined on.

The same shape turns up twice more. A constant that cancelled a coupling was a constant of one lattice, because the argument that produced it did not mention the property the lattice happened to have. A local search calibrated on sixteen sites reported a strength that belonged to sixteen sites. In all three, the demonstration is sound and the range it was demonstrated over is silent about a variable.

Who found it, and when #

Ring-current models of fused benzenoids go back to Pauling in 1936 and to London in 1937, and the observation that the current in a fused system does not divide between its rings the way it would between resistors is old enough to have been argued about at length. The specific question here — whether an inter-ring response is a function of topological or of geometric separation — is the same question the nuclear-magnetic-resonance literature asks about ring-current shielding, where it is usually settled by fitting a geometric expression because a geometric expression is what a chemist needs at a proton some distance from the molecule.

That is a different question with a different answer, and the difference is worth keeping: a proton outside a molecule sees a field, which is geometric by construction. A ring inside it sees a graph.

Still open: rings with cycles, and the end effect #

The obvious open question is a molecule that is not a chain. Every system here is catacondensed — each ring shares one bond with the next and the ring adjacency is a path — so the topological separation is unambiguous. Pyrene and coronene are not: their ring adjacency has cycles in it, and two rings can be two steps apart by one route and three by another. Whether the response follows the shorter route, the number of routes, or neither is a question a chain cannot pose, and the same construction builds those molecules unchanged.

The nearer question is the end effect, which has been identified here and not measured. If the scatter within a separation class is entirely a matter of how much molecule sits on either side of the pair, then it should collapse against that quantity — the number of rings outboard of the pair, or the pair’s distance from the nearer end — and the collapse would turn the scatter into a second term rather than a nuisance. Every number needed for it is already in the matrices computed here, and the same variable is what a reversal in the acene series turns on, which would put two responses on one axis at last.