The scatter counts angular rings

In a chain of fused benzene rings, a magnetic flux threading one ring changes the current in every other — the current does not divide equally even between rings that look alike — and the size of that response between two rings falls off with the number of fusions between them. It does not fall off cleanly. Pairs the same number of fusions apart differ by a third in a straight chain, and the obvious explanation — how far a pair sits from an end — took opposite signs on a straight chain and a zigzag of the same length.

The next step was a controlled family: eight chains of nine rings differing in one integer, which fusion is turned. The scatter jumped from 1.52 in the straight chain to between 3.05 and 4.60, was worst with the third fusion turned rather than the middle one, and made the two ends of a single molecule disagree by up to a factor of four. The family was designed to separate whether a chain has a bend from where the bend is.

It did not quite do that, because of how the chains were built. Each fusion is given a direction, a multiple of a sixth of a turn, and the family set one fusion’s direction to one among zeros. That turns the chain at the turned fusion — and turns it back at the next one, which is still zero. So the family was not one bend moved along; it was two families.

Two families under one integer #

A ring is angular when the two bonds it shares with its neighbours are separated by one bond around the ring, rather than sitting opposite each other. Read that way, the chain turned at its first fusion has one angular ring, ring one, next to the end ring. So does the chain turned at its last fusion, at ring seven. Every chain turned at an interior fusion has two adjacent angular rings — a step, whose two turns cancel and leave the chain’s envelope straight. The spreads that were read as a single curve against position — 3.052, 4.304, 4.600, 4.065 and back — are a one-ring value at each end and a two-ring curve between them.

Building the chains from their angular rings, rather than from turned fusions, reproduces every published spread to four figures, which is the check that the reading is right. It also allows the curve the family was meant to draw. A single angular ring moved along the chain gives 3.052 at ring one, 3.660 at ring two, 3.460 at ring three and 3.029 at the centre, then the mirror image. Two adjacent angular rings give 4.304, 4.600 and 4.065. The two curves do not overlap anywhere: the least a two-ring step scatters is more than the most a single angular ring does.

So the jump from the first family member to the second was mostly a jump from one angular ring to two, and only what happened after it was about position. The position effect is real — a single angular ring scatters a fifth more at ring two than at the centre — but it is the smaller of the two things the family had mixed together.

Which way a ring turns does not matter #

With the chains built from angular rings, the question the family’s own closing section asked becomes directly testable. Two angular rings can turn in opposite senses, as a step does, which leaves the chain’s overall envelope straight; or in the same sense, which bends the envelope by 120°. If the scatter depends on the molecule’s overall shape, those two should differ. If it depends on the local geometry at each ring, they should not.

They do not differ. At every one of seventeen arrangements — the six adjacent pairs, eight spaced pairs, and three runs of three angular rings, where same-sense turns bend the envelope by 180° — the worst spread with all turns in one sense is within 0.81 per cent of the worst spread with alternating turns. The points lie on the diagonal. Whatever the ring-current scatter is responding to, it is not the envelope of the molecule, and nothing about a chain beyond which of its rings are angular appears to enter.

That is the direct answer to the question posed when the family was designed: two bends in opposite senses, returning the envelope to straight, do not bring the scatter back to 1.52. They leave it exactly where two bends in the same sense put it.

Nor how many rings are angular #

The next reading to test is the obvious one: the scatter measures how many angular rings a chain has. One gives three or so, two give four or so, and so on.

It does not. Among chains with two angular rings, the worst spread runs from 2.449, with rings two and six angular, to 5.244, with rings one and three — a factor of more than two at the same count, and the far-apart pairs fall below every single angular ring. Three angular rings in a run give 5.35 to 5.95, but four in a run give 4.99. And the chain with every one of its seven interior rings angular — the zigzag — scatters by 1.367, less than the straight chain’s 1.520.

The ends of that range are the telling part. The two chains whose fusions are all of one kind scatter least, and the chains that mix the two kinds scatter most. A count of angular rings cannot say that, because the zigzag has the most of them.

One count per pair #

What the uniform chains share is that every pair at a given separation spans the same kind of rings: in the straight chain none of them angular, in the zigzag all of them. In a mixed chain, two pairs at the same separation can span different numbers of angular rings, or sit on different numbers. That suggests counting per pair rather than per chain.

For every pair of rings in every chain — 702 pairs in 27 chains — the logarithm of the response was fitted against one constant for each separation class, plus a small set of terms, and the fraction of the variation each model explains was compared. The terms are: whether the pair touches an end of the molecule; how many angular rings lie strictly between the pair’s two rings; and how many of the pair’s own two rings are angular.

The separation alone explains 78.0 per cent. Adding whether the pair touches an end takes it to 78.6 — almost nothing, on a family where the end effect was supposed to be the leading correction. Adding the angular rings between the pair takes it to 87.5. Adding also the angular rings under the pair takes it to 93.5, and putting the end term back on top of that changes the fourth decimal place.

The fitted factors are the physical content. Each angular ring between two rings multiplies their response by 0.602, and each of the pair’s own rings that is angular multiplies it by 0.655. Both are large effects: a pair spanning two angular rings and sitting on one responds at a quarter of what the same separation gives in a straight chain. Both are also consistent with what the heteroatom response in fused chains shows, where an angular ring rescales everything beyond it and leaves the rings before it nearly alone.

That parallel is not a coincidence of wording. The heteroatom measurement was built the same way, with one fusion’s direction set among zeros, and its single bend was the same two-ring step. Two different responses of the same π system — the change in bond orders around a substituted carbon, and the coupling of ring currents between two rings — both turn out to be multiplied down by each angular ring they have to pass, and both were first described as responding to a bend. The angular ring is the common unit, and the bend was the encoding’s word for it.

The count also reads the end effect differently. In a zigzag the end rings are the only rings that are not angular, so a pair touching an end sits on one fewer angular ring than an interior pair at the same separation, and responds more strongly — which is the direction of the zigzag’s “positive end effect”. In a straight chain there is no angular ring to count, and the end effect that remains there is a genuine one that the count does not touch.

What is left over #

The test of a model like this is not only its $R^2$ but what it does to the quantity that started the enquiry: the spread within a separation class.

Measured, the worst spreads run from 1.37 to 5.95. Divided by the angular factors, they run from 1.32 to 2.27. The runs of three angular rings, the spaced pairs and the two-ring steps — every chain that scattered by more than four — fall to between 1.69 and 2.19. The straight chain, which has nothing to divide out, keeps its 1.52 exactly, as it must.

The residual is not nothing. The single angular rings at positions two and three keep the most, 2.20 and 2.27, and a factor of two between two pairs at one separation is still a factor of two. A single number per angular ring cannot distinguish an angular ring next to an end from one in the middle, and the one-ring curve shows that position does matter by about a fifth. But what is left is at the size a straight chain’s own end effect produces, rather than three or four times larger.

How the chains were built and measured #

Each chain has nine six-membered rings fused in a line, and each interior ring is linear or angular. The directions of the eight fusions are generated from a list of angular rings and a sense for each turn, and the angular rings are then read back from the geometry — from whether each interior ring’s two shared bonds are opposite or one position apart — and required to match the list. The π system is Hückel’s with one hopping integral, and the response of ring i to a flux through ring j is the second cross-derivative of the total energy with respect to the two fluxes, taken as a symmetric finite difference at 10⁻³, with the gauge assigned by partial sums of bond angles about each ring centre. Pairs more than four fusions apart are excluded, as in every earlier measurement.

The fit is ordinary least squares on the natural logarithm of each response, with a separate constant for each of the four separation classes. Every pair enters once, and the residual spread for a chain is the largest ratio between two of its fitted residuals in one separation class.

The checks, run wherever these figures are drawn. Every published family member has the angular rings stated — one at the two end positions, two adjacent at the six interior ones — and the chain built from them reproduces its spread to four figures. Turning the rings of every two- and three-ring arrangement all one way changes the worst spread by under 1.5 per cent. Two far-apart angular rings scatter less than any single one, and the zigzag less than the straight chain. The fit with both angular terms exceeds $R^2$ = 0.92 where the end term alone stays below 0.8; both factors lie between a half and three quarters; and after dividing them out no chain exceeds 2.3 where the measured spreads exceed 5.5. The refusal is the straight chain: with no angular ring there is nothing for the counting terms to act on, so its residual spread must stay within five per cent of its measured one.

There is a detail of construction worth recording because it produced a false result first. The per-molecule response calculation caches its answer under the molecule’s label, and chains built from direction lists without a label all shared one — so every chain in the first survey returned the spreads of whichever was computed first, and seven different single angular rings reported identical numbers to four figures. Every chain here carries its own label, and the check that the family’s spreads are reproduced is what would catch a recurrence.

Where the counting stops #

Nine rings. Every chain here has nine, and the fitted factors are for nine — and a quantity quoted without the molecule it was measured on is how a factor becomes a constant it never was. Whether an angular ring costs 0.60 of a pair’s response in a chain of seven or eleven, and whether its position matters more in a shorter chain, is not measured.

One number per angular ring. The fit gives every angular ring the same factor wherever it sits, and the single-ring curve says that is an approximation at the level of a fifth. A factor that depends on distance from an end would absorb some of the residual and would reintroduce the end, in a narrower role.

Hückel currents. The response is a second derivative of a one-electron energy with respect to fluxes, not a computed magnetic susceptibility — the same gap aromaticity as a shell closure leaves between an electron count and a measurement; the argument concerns what describes the pairwise responses of this model, and a measured shielding would need the ring areas and a field the model does not carry.

And chains only. Every molecule here has a ring graph that is a path. A pyrene or a coronene has rings fused on more than two sides, where “between two rings” has more than one route and “angular” is not a two-way distinction.

A construction is an instrument too #

The family of turned fusions was a clean design: one integer, eight molecules, everything else held fixed. The integer changed a second thing that nobody intended to vary, because a list of fusion directions describes turns only through differences between neighbours, and setting one entry changes two differences. A controlled variable is only controlled if its encoding changes one thing, and a check that read the angular rings back from the built geometry — a line of code — would have split the family before any spread was computed.

The count that replaces it is also a warning about summaries. “How many bends” and “which way it bends” both describe the whole molecule, and neither enters. What enters is local and per pair: how many angular rings each pair of rings spans and sits on. The end effect, the envelope and the bend count were each a chain-level proxy for that, and each worked on some chains and failed on others for exactly that reason.

Still open: whether the factor has a position, and rings fused on three sides #

The obvious open question is the residual. The single angular ring scatters a fifth more at ring two than at the centre, and a model with one factor for every angular ring cannot express that. Giving the factor a dependence on how far the angular ring sits from the nearer end is one more term, and whether it closes the residual to the straight chain’s 1.52 — or leaves a floor that belongs to the straight chain’s own end effect — would say whether the end and the angular ring are two separate causes or one.

The nearer question is the ring count. The factors 0.60 and 0.65 were fitted on nine rings, and the earlier measurements found the straight chain’s end ratio wandering with length rather than settling. Refitting on chains of seven and eleven built from the same angular-ring lists would say whether a factor per angular ring is a property of the ring or of a chain of nine.