The angular factor belongs to the chain

A magnetic flux through one ring of a fused chain of benzene rings changes the current in every other — and not evenly — and the size of the response between two rings falls with the number of fusions between them — unevenly. Pairs the same number of fusions apart can differ by a factor of six, and neither the fusion count nor the distance between centres accounts for it. The scatter counts angular rings found what most of that unevenness is made of, on twenty-seven chains of nine rings: one constant for each separation, then a factor for every angular ring that lies between the pair’s two rings, 0.602, and one for every angular ring the pair itself sits on, 0.655. Together they explained 93.5 per cent of the variation in the logarithm of every pair’s response, against 78 per cent for the separation alone.

That essay ended on two questions about what kind of numbers those factors were. Its nearer one was the ring count. Every chain had nine rings, and a quantity quoted without the molecule it was measured on is how a factor becomes a constant it never was — the straight chain’s reach had already turned out to grow with every ring added. Its other question was position: a single angular ring scattered a fifth more near an end than at the centre, which a single factor per angular ring cannot express.

Both have answers. The factors change with the chain’s length, by a fifth between seven rings and eleven, while the share of the variation they explain does not change at all. And a factor that depends on position explains almost nothing more.

Every chain of three lengths #

The first fit used a chosen set, built after a family of turned fusions turned out to vary two things at once: every single angular ring and every adjacent pair, eight spaced pairs, three runs of three and one of four. A refit at other lengths needs a rule rather than a choice, or the lengths will differ in which chains were picked as well as in how many rings they have. The rule here is every chain of the length with at most two angular rings, wherever they are, every run of three adjacent angular rings, the straight chain and the zigzag. At seven rings that is 20 chains and 300 pairs; at nine, 35 and 910; at eleven, 54 and 2,160.

The nine-ring set is then a complete enumeration of the kinds the first fit sampled, which is a check on that fit before any other length is looked at. The factors come out at 0.593 and 0.630, against the chosen set’s 0.602 and 0.655. The factor between a pair moves by a hundredth with the choice of chains, the factor under it by a fortieth. That is not nothing, and it is the first thing the original numbers lose: a factor fitted on a sample of chains carries a few hundredths that belong to the sample.

Each chain is built as before. The fusions are given directions from a list of angular rings, the angular rings are read back from the built geometry 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. Pairs more than half the chain apart are left out, as in every earlier measurement, so the separation classes run to three fusions at seven rings, four at nine and five at eleven.

The form transfers #

The first result is the one that makes the second interpretable. Fitted separately at each length, every model explains the same share of the variation at all three. One constant per separation class gives $R^2$ of 0.790, 0.791 and 0.790. Adding whether the pair touches an end gives 0.800, 0.795 and 0.792 — the end term, which was supposed to be the leading correction, is worth less on longer chains. Adding the two angular counts gives 0.938, 0.939 and 0.939.

That is a strong statement about the counting. Three lengths, chain sets of three different sizes, separation classes of three different extents, and the same two terms remove the same share of what the separation leaves. Whatever angular rings do to a pair’s response, they do it in the same form on seven rings as on eleven: a multiplication per ring passed, and another per ring stood on. If the scatter were a property of nine-ring chains — some resonance of the length with the positions sampled — the share explained would have moved with the length. It does not, to a thousandth.

The numbers do not #

The factors themselves move, steadily and in one direction. Each angular ring between a pair multiplies the response by 0.656 on seven rings, 0.593 on nine and 0.542 on eleven. Each angular ring the pair sits on multiplies it by 0.681, 0.630 and 0.587. Both fall by about 0.06 for every two rings added, and neither shows any sign of settling over the range measured.

How much that matters depends on how many angular rings a pair spans, because the factor enters as a power. Using the nine-ring factor between a pair on an eleven-ring chain overstates a pair spanning one angular ring by 9 per cent, three by 31 per cent, and five by 57 per cent — and an eleven-ring chain has pairs spanning five. Pooling all three lengths with one pair of factors, 0.561 and 0.609, and a separate set of separation constants for each length, gives $R^2$ of 0.9397, against 0.9411 with a pair of factors for each length. The difference is small in the share of variation because most pairs span zero or one angular ring; it is large for exactly the pairs where the counting was supposed to do its work.

So the answer to the lead is that a factor per angular ring is a property of the form and not of the number. The angular ring always multiplies; what it multiplies by depends on the chain it is in.

Two decays, one of which does not move #

A factor that drifts with length means something else in the fit drifts with length and the factor is measured against it. The fit has two kinds of parameter: the factors, and the separation constants they are measured against, which are fitted mostly by pairs with no angular ring between them — by linear fusions. The cleanest place to see what each is doing is the two chains where every fusion is of one kind.

In the zigzag, where every interior ring is angular, the response two fusions apart is 0.3166, 0.3153 and 0.3148 of the response one fusion apart, at seven, nine and eleven rings. That is the same to half a per cent: the zigzag’s decay per fusion is a property of the zigzag, and adding rings to it does not change how fast its response falls. Its scatter does not change either — the zigzag’s worst spread within a separation class is 1.373, 1.367 and 1.368.

In the straight chain the same ratio at two fusions is also stable, 0.531 to 0.534, but further out it is not. Three fusions against two gives 0.590, 0.606 and 0.612; at eleven rings, five against four is 0.685. The straight chain’s response falls more slowly the further it reaches, and more slowly still the longer the chain is. Its scatter grows with it, 1.332, 1.520 and 1.652.

The reason is the one the reach that has no length found for a heteroatom’s reach along an acene. A response that falls with distance falls at a rate set by how far the occupied and empty levels are apart, and the two chains differ completely in that. The straight chain’s gap — the acenes’ — falls from 0.828 β at three rings to 0.268 at seven, 0.180 at nine, 0.128 at eleven and 0.096 at thirteen, closing towards zero. The zigzag’s — the phenacenes’ — falls from 1.210 to 0.916, 0.870, 0.843 and 0.824, each step smaller than the last, settling near 0.8.

Put the two together and the drift has a reading. The separation constants are set by linear passages, whose decay lengthens as the chain’s gap closes. An angular ring’s passage decays at a rate that does not depend on the chain, because it belongs to a local arrangement whose gap stays open. A factor per angular ring is the ratio of the second to the first — and when the denominator grows with length and the numerator does not, the ratio falls. That is a reading rather than a derivation: the factor is a fitted number in a model with a separation constant per class, and the zigzag’s step ratio of 0.315 is not the fitted factor of 0.54 to 0.66. But the direction and the cause agree, and they explain the most surprising feature of the result, which is that a chain’s angular rings appear to get more expensive as the chain gets longer when nothing about them has changed. What changed is the price of a linear passage.

An end or the middle #

The second lead was position. A single angular ring scattered a fifth more near an end than at the centre on nine rings, and a factor that depends on whether the angular ring is next to an end might absorb that. The test is direct: split each count in two, angular rings on ring one or on the ring before the last, and angular rings further in, and fit four factors instead of two.

The angular rings between a pair barely care. Next to an end and further in, the factor between is 0.655 and 0.656 on seven rings, 0.615 and 0.589 on nine, 0.587 and 0.537 on eleven. The angular rings under a pair care more: next to an end they cost 0.710, 0.696 and 0.689, further in 0.669, 0.616 and 0.571. An angular ring at the edge of the molecule, which has only an end ring beyond it, takes less of a response that passes through it than one with more molecule on both sides — and that difference grows with length, which is the same direction as everything else here, since the end ring’s neighbour has less linear chain beyond it to cheapen.

But almost no pairs sit on such a ring, so almost nothing changes. $R^2$ rises by 0.0006, 0.0010 and 0.0014. Adding the end term on top of the split adds another 0.0000 to 0.0002. The answer to whether the end and the angular ring are two causes or one is that they are two, and that the second is small: the end-adjacent angular ring is genuinely different, and it is not what the residual is made of.

What is left over #

What is left after the counting grows with length too. The worst residual spread over every chain is 1.84 on seven rings, 2.18 on nine and 2.65 on eleven, against measured spreads up to 3.75, 5.95 and 8.69. The straight chain, which has nothing for the counts to act on, keeps its own spread exactly — 1.332, 1.520 and 1.652 — and that is the refusal the fit is held to at every length. The single angular rings sit above it by a factor between about 1.2 and 1.6, peaking a third of the way along the chain on every length, exactly as on nine rings.

So the residual is not a floor that belongs to the angular rings, and not one that a position term reaches. It grows with length alongside the straight chain’s own end effect, which is the scatter of a chain with no angular ring at all — and that scatter is what the end effect with two signs first measured and what a closing gap makes worse. The count removes what the angular rings do; what remains is the linear chain’s own unevenness, which this model was never asked to explain and which lengthens with the gap that drives everything else.

What was fitted, and how it was checked #

Each fit is ordinary least squares on the natural logarithm of every pair’s response, with one constant per separation class and the model’s terms; a pooled fit has one constant per class and length. The residual spread for a chain is the largest ratio between two of its fitted residuals in one separation class. The angular counts are read from the built geometry, never from the list that built it.

The checks, run wherever these figures are drawn. The refusal is the straight chain at every length: with no angular ring, its residual spread must equal its measured spread to five per cent, or the fit is removing scatter by another route. Every chain must have exactly the angular rings it was built with. The two-term fit must reach $R^2$ above 0.93 at every length and agree across lengths to 0.005. Both factors must fall monotonically with length, the factor between by more than 0.08 from seven rings to eleven. The position split must add under 0.003 to $R^2$ at every length. The zigzag’s two-fusion ratio must agree across lengths to one per cent while the straight chain’s three-fusion ratio rises; the straight chain’s gap must close towards zero and the zigzag’s settle above 0.8; and the complete nine-ring enumeration’s factors must lie within 0.03 of the chosen set’s.

There is a detail of construction worth stating. The per-molecule response is memoised under the molecule’s label, and the first survey of angular rings found every chain returning one chain’s answer because none had a label. Here the label names the length as well as the chain, since a seven-ring and an eleven-ring chain with angular rings at positions one and three would otherwise share a key and a wrong answer.

Three lengths of one model #

Three lengths. The factors fall by 0.06 for each two rings and show no sign of settling between seven and eleven. Whether they reach a limit, as the zigzag’s gap does, or keep falling with the acenes’ closing gap, needs longer chains, and the cost grows quickly: an eleven-ring chain took about four seconds, and a thirteen-ring enumeration would hold about eighty chains.

Hückel currents. The response is a second derivative of a one-electron energy with respect to two fluxes, not a computed magnetic shielding, and the acenes’ closing gap is itself a Hückel feature that a correlated or bond-alternating calculation softens. Aromaticity as a shell closure is the reminder of how far an electron-count picture is from a measurement; what transfers is the structure — a multiplication per angular ring in a form that holds at every length — rather than any value.

A reading, not a derivation. That the drift comes from the linear passages is supported by two measurements — the zigzag’s fixed decay and the straight chain’s lengthening one, with the gaps behind them — and not by a fit that separates the two. A model with an explicit decay per linear fusion and one per angular fusion, with the per-length drift put in the first, would test it.

Chains only. Every ring graph here is a path. A ring fused on three sides has more than one route between two rings, and the count of angular rings between them is no longer a single number.

The same lesson from the other side #

Angular rings rescale what lies beyond them for a heteroatom’s bond-order response in the same chains, whose single bend was the same two-ring step, and which of an angular ring’s carbons carries the heteroatom turned out to matter more than the ring. The ring currents have now given the matching result for the quantity that carries the scatter. In both, the angular ring is the unit and the multiplication is its signature; in both, the number it multiplies by is not a constant of the ring.

The general point is about what a fitted factor is. The share of variation a model explains says whether its form is right; the factor says how big an effect is in the systems it was fitted on. Those can come apart, and here they come apart cleanly: the form is the same to a thousandth at three lengths, and the factor is not the same to a tenth. A factor quoted without its chain length is a factor for a chain of nine.

Still open: whether the factor settles, and a linear passage with its own price #

The obvious open question is the limit. The zigzag’s gap settles near 0.8 and the acenes’ closes; if the factor is the ratio of an angular passage to a linear one, it should keep falling as long as the acenes’ gap keeps closing, and on chains long enough for the gap to be negligible the factor would be set by how a linear passage’s decay behaves when the gap is nearly zero — which is a reach with no length, and might mean no finite limit at all. Chains of thirteen and fifteen rings would say whether the 0.06 per two rings slows.

The nearer question is the reparametrisation. The fit measures an angular ring against a separation constant per class; giving the model one decay per linear fusion and one per angular fusion instead, and asking which of the two carries the length dependence, would turn the reading above into a test. If the angular decay is the same at every length and the linear one is not, the drift is the linear chain’s, and the angular ring’s own price is a constant after all — just not the one the fit reports.