One integer, and everything it changes

An end effect with two signs compared a straight chain of nine fused rings with a zigzag of nine and found the end’s effect on the ring-current response reversed between them — weaker at an end in the first, stronger in the second. It closed by naming the variable that comparison cannot separate: a straight chain and a zigzag differ in every fusion, so nothing in it says whether the sign is set by the presence of a bend, by how many there are, or by where they sit.

Turning one fusion separates them. A chain of nine rings with one fusion turned a sixth of a turn from the rest is eight molecules — one for each fusion — and every one of them has the same rings, the same carbons, and the same number of fusions between every pair. They differ in one integer.

That integer is not quite one bend, and the difference turned out to matter. Each fusion is given a direction, so a single turned fusion turns the chain and the next fusion turns it back. Turned at the first or last fusion, that leaves one angular ring beside an end; turned anywhere between, it leaves two adjacent angular rings whose turns cancel — a step with a straight envelope. The family is one-ring members at its two ends and two-ring members everywhere else, and counting angular rings rather than turned fusions is what the numbers below turn out to be measuring.

One turned fusion is enough #

A straight chain of nine scatters within a separation class by a factor of 1.520 at its worst. Every chain with a turned fusion scatters by more: 3.052, 4.304, 4.600, 4.065, and then the mirror image of those four.

So the scatter is not a property of a molecule being angular all along its length. A single angular ring beside an end doubles it, a two-ring step in the interior triples it, and the eight molecules span a range from three to four and a half — which is wider than the difference between a straight chain and a zigzag.

The shape along the chain is not the obvious one either. The worst position is the third fusion of eight, not the centre, and the two central positions are a little better than it. That rules out the tidiest available explanation — a turn that divides the molecule most evenly would be the one at the centre — but the jump from the first position to the second is mostly the jump from one angular ring to two, and only the rest of the curve is about position.

Compared with the straight chain’s fan, two things have changed. It is three to four times wider, and the ordering inside a separation is no longer monotone in the depth: two pairs at the same depth, on opposite sides of the turn, sit far apart. In the straight chain deeper was always stronger, and that monotonicity is what made an end correction look plausible in the first place.

There is a reading available for the position dependence, and it is worth putting down with its correction. A turn at the third fusion leaves rings two and three angular, with a straight run on each side. What differs along the family is how many pairs straddle the angular rings: a turn at the centre is straddled by the most pairs and a turn at the end by the fewest, so if straddling were the whole mechanism the worst case would be the centre. It is not — and the reason turned out to be that a pair is weakened by an angular ring it sits on as well as by one it spans, so pairs near the step are penalised in two different ways that do not add up to a simple count of straddlers. The asymmetry of the two runs, which the next figure measures, is where that shows.

The two ends are not the same end #

The straight-against-zigzag measurement reported one number per separation for a pair touching an end. That was enough while every molecule was symmetric end for end. These are not.

In a chain turned at its second fusion the left end pair is 0.331 of the deepest pair’s response and the right end pair is 0.855 — a factor of 2.58 between two pairs at the same separation, the same depth, and in the same molecule.

That is the strongest form the end-effect argument takes. A variable computed from a pair’s position — depth, outboard count, distance from the nearer end — takes one value for both of those pairs, and they differ by a factor of two and a half. No such variable can describe them, and no amount of choosing a better one will help, because the two pairs are at the same value of every variable of that kind.

The two curves cross where the turned fusion is at the centre, which is exactly where the molecule regains its reflection symmetry. That crossing is not an artefact of the plot; it is the one position at which the two ends are required to agree.

Taken over all four separations the asymmetry reaches 4.061, with the third and sixth fusions turned. It is symmetric about the middle of the chain, which it must be — turning fusion $k$ and fusion $m-k$ gives one molecule read two ways — and that symmetry is a check rather than a decoration, because the gauge these responses are computed in is assigned by walking from one end and is not itself symmetric.

The size of the asymmetry is worth weighing against what a chemist would expect. Two ends of a molecule differing in their response is not itself surprising — an anthracene’s outer rings differ from its central one, and everybody knows it. What is surprising is that the difference survives being normalised away: each end pair here is already divided by the deepest pair of its own separation class, so the ratio being reported is what is left after the separation, the depth and the overall size of the response have all been removed. A factor of four at that point is a factor of four in a quantity three different corrections were supposed to have accounted for.

And it is not a small-number artefact. The pairs involved are at one and two fusions’ separation, where the responses are the largest in the molecule and the finite difference is at its most reliable — the same places neither of the two separations drew its cleanest points from.

Where in the range the scatter sits #

A scatter that grew with the separation would be the least alarming kind: the responses at large separation are small, a fit has few points there, and any careful reader already distrusts them.

The straight chain does exactly that — 1.073, 1.312, 1.332, 1.520 — rising monotonically to its worst at the largest separation. It is the only one of the five that does.

The chain turned at its third fusion peaks at 4.600 at three fusions, in the middle of the range, and falls back to 4.061 at four. The one turned at its fourth peaks at 4.065 at two. So the worst scatter sits where a fit carries the most points and has the least reason to be suspected, which is the opposite of the comfortable case.

The whole family, in one table #

Four molecules, four integers, and every column moves. The widest spread runs 3.052, 4.304, 4.600, 4.065. The left end pair at two fusions runs 0.430, 0.331, 1.132, 3.625 — a factor of eight across four molecules that differ in which fusion is turned. The ratio of the two ends runs 0.546, 0.387, 0.868, 1.038.

Nothing in that table is a small correction to a decay. The response of a pair of rings at a stated separation in a nine-ring chain is not determined to within a factor of eight by anything the end-effect description contains, and what determines it, it turns out, is how many angular rings the pair spans and sits on.

It is worth being clear about what survives. The earlier finding — that the response decays with the separation, roughly exponentially, at a rate near one to two fusions — is not in question, and the current’s refusal to divide equally between equal rings is the same fact in the diagonal entries. What has gone is the idea that the decay is a function with a small residual. On these molecules the residual is a factor of four and it has structure in it.

What the family cannot say #

Two things are deliberately absent from the argument above and both would be easy to slip in.

The first is a mechanism. Nothing here explains why the third fusion is the worst place for a turn, and the reading offered above — that the asymmetry of the two runs matters more than the count of straddling pairs — is a description of the numbers rather than a derivation from anything. A derivation would need the response written as a sum over paths through the graph, which Hückel theory permits and nobody has built here.

The second is a claim about magnitude. The spreads reported are ratios within a separation class, and a ratio of four between two small numbers is not the same kind of statement as a ratio of four between two large ones. At four fusions’ separation in a nine-ring chain the responses are a few thousandths of a ring’s own, so what the scatter says is that a small quantity is badly described — which matters for the shape of the decay and matters much less for any total. The fitted decay length is the quantity at risk, and it is at risk by a fifth rather than by a factor of four.

What was computed, and how #

Each molecule is a chain of fused six-membered rings built by naming a direction for each fusion — all zero except one, which points a sixth of a turn away from the rest, so the chain turns at that fusion and turns back at the next. The π system is Hückel’s, and the response of ring $i$ to a flux through ring $j$ is the second cross-derivative of the total energy, taken as a symmetric finite difference at a flux of $10^{-3}$, with the gauge assigned by partial sums of the bond angles about each ring centre.

The depth of a pair is the smaller of its two rings’ distances to their nearer ends, and each pair is reported against the mean response of the pairs at the greatest depth in its own separation class.

That mean is a repair rather than a choice, and it is worth recording why. The first version took the first of the deepest pairs, which is what a reduction over a list gives. Two pairs very often tie on depth — at one fusion’s separation in a chain of nine there are two, at depth three — and picking one of a tied pair is a choice that reversing the molecule does not preserve. It showed up as turning fusion one and fusion eight disagreeing by half a per cent about quantities a reflection requires to be identical, on a measurement whose own arithmetic is good to two parts in ten thousand. The mean is invariant because the tied set is.

The reversal check is the one the whole family rests on. Turning fusion $k$ and fusion $m-1-k$ produce one molecule read two ways, so their spreads must agree and their two ends must swap. Both hold to $3\times10^{-4}$ relative, which is what a second cross-difference of a sum of eigenvalues has at this flux step — measured directly by comparing the two response matrices entry by entry, where the worst disagreement is $1.8\times10^{-4}$. The asymmetries being reported are factors of two to four, three orders larger.

And every turned chain is required to scatter more than the straight one, which is the claim rather than an observation, and would fail if the turn were doing nothing.

Where the model stops #

This is Hückel theory: one π electron per carbon, one hopping integral, no repulsion, and no geometry beyond the graph and the ring centres the gauge needs. A response to a flux is not a response to a field, and turning one into the other needs an area the model does not carry — so these numbers compare with each other and not with a measured susceptibility.

One family of one size. Everything above is nine rings with one turned fusion, and neither the number of rings nor the number of angular rings is varied independently — the family itself changes that count between its end members and the rest, which the description above did not at first see. The end ratio was found wandering with the ring count rather than settling, so there is no reason to expect the shape of the curve against bend position to be the same at eleven rings, and it is not tested. A reach that has no length records what happens when a quantity is quoted without the molecule it was measured on.

The finite difference is at one flux step, chosen because the reversal check and the single-ring closed form both hold there. A step ten times smaller makes the reversal check worse rather than better, which is the round-off in a second difference behaving as it should and is the reason the step is not reduced.

And a bend of a sixth of a turn is one bend angle. A fused six-membered ring system has only that angle available, so the restriction is the chemistry rather than a simplification — but it does mean the family varies a position and not a strength, and nothing here says how the effect scales with the size of the geometric change.

The generalisation #

Two objects at the same value of every variable in a description, behaving differently, is a proof that the description is incomplete — and it is a much stronger form of evidence than a poor fit. A scatter can always be blamed on noise, on a missing higher-order term, or on the range being too short. Two pairs at the same separation and the same depth in the same molecule, differing by a factor of two and a half, cannot be: whatever distinguishes them is not in the list. The same argument appears in two structures with the same neighbours, where two nets agreeing on three spectral moments still differ in binding, and it is the cleanest way a variable can be shown to be missing. The half of the square a ring of four cannot show is the version where the missing variable is a symmetry rather than a shape.

And a family that varies one integer is worth more than a family that varies a molecule. The comparison of a straight chain against a zigzag moved every fusion at once, and could establish that something about the shape matters without saying what. Eight molecules differing in one integer separate the presence of a feature from its position, and they do it without needing the feature to be small — provided the integer changes only the feature it is meant to, which here it did not quite do. That is the same design as a controlled pair sharing a coordination and a kurtosis, one level of ambition up: not a pair but a scan, and the scan’s shape is itself the result.

Who found it, and when #

That the rings of a polycyclic aromatic carry unequal currents is standard, and the dependence on topology — linear acenes against angular phenacenes — is one of the oldest observations in the field, going back to the difference between anthracene and phenanthrene. The pairwise response measured here is defined for this argument rather than taken from the literature, so the finding is about the description rather than about the chemistry.

What the family adds is a separation nobody has occasion to make when the molecules are named compounds. Anthracene and phenanthrene differ in shape and in nothing else, which is the two-molecule version of this; the eight chains here are the scan, and the scan says the answer depends on the angular rings by more than on whether there is a turn at all.

Still open: a second bend, and the ring count #

The obvious open question is the second bend. Two bends have a separation between them, and whether the effects add, cancel or interfere is a question with a definite answer and thirty-six molecules in it. The interesting case is two bends in opposite senses, which returns the molecule to a straight envelope while leaving its rings angular — if the scatter comes back to 1.52 there, the deciding quantity is the overall shape rather than the local geometry, and if it does not, it is the reverse. The interior members of this family were already that case, and the scatter did not come back.

The nearer question is the ring count. Everything here is nine rings, and the curve of spread against the turned fusion has a shape — a rise to the third fusion and a fall — that could be a property of nine or a property of a third of the way along. Chains of seven and eleven would say which, and they cost eight and twelve diagonalisation sweeps of a matrix already built here.