A bend is not an end

Every calculation since a heteroatom was first put in a fused chain has put it on the end ring of a straight fused chain, because an end gives the longest run to measure a decay over. Moving it to the fifth ring of twelve makes the point that the response of a molecule to a perturbation at its end is not the response to one in its middle, and gives two profiles from one calculation.

The decay came out the same: 0.687 rings in one direction, 0.707 in the other, against 0.719 measured from the end. The amplitude did not: it halved, and split across two rings.

So an end is a boundary of a particular kind. It concentrates the response — everything that would have gone the other way comes back — and it leaves the range alone. Which raises whether a bend does the same, because a single bend changes a fused system’s ring-current response by a factor of four, and if a bend is that kind of boundary then the reach measured along straight chains belongs to straight chains rather than to fused systems.

A fifth, on one ring #

The comparison is a chain of twelve rings with one kink in the middle — the chain turns by a sixth of a turn at ring five and turns back at ring six, so both of those rings are angularly fused — against the same chain with none, and the heteroatom placed at six positions: on the end, three rings before the bend, one before, on the bent ring, one after, and three after.

Five of the six placements read within four per cent of the straight chain, and three of them within one. On the end the ratio is 1.0012; three rings before the bend, 0.9991; three rings after, 0.9859.

The exception is the heteroatom on the kink’s first angular ring, where the ratio is 0.8111. That is the largest effect on a heteroatom’s own ring. The kink’s second angular ring is the “one ring after” placement, at 0.9685 — so the two angular rings are not alike, and a fifth is not a property of angular fusion as such. A sweep over the number of angular rings takes that apart.

The size is worth putting beside the end’s. The amplitude on an interior ring is 1.94 × 10⁻² against 3.76 × 10⁻² on the end — a factor of 1.94, which is where “an end halves it” comes from. The bend’s factor is 1.23. So on the same measure of how much a boundary concentrates a response, an end is worth four times what a bend is, and the bend’s effect does not survive moving one ring away while the end’s does.

Against an end, which halves it. So a bend is a boundary of the same sign and less than half the size, and where an end’s effect reaches a ring away — the response splits across two rings — a bend’s does not reach even that far.

The range does not move #

The decay lengths agree to within 10.7 per cent at worst, and most pairs agree to two or three. That number needs a scale beside it, and the scale is the straight chain’s own anisotropy: measuring outward from ring four gives 0.707 and inward gives 0.691, a difference of two per cent, and the middle placement already showed the two directions differing by three.

So the bend moves the range of the response by about as much as the choice of direction does — which is to say, by less than the quantity is worth quoting to.

That matters more than the amplitude result, because the range is the number being reported. Three calculations have quoted a decay length near 0.7 rings as the reach of a heteroatom’s influence, and the worry was that the number belonged to straight chains. It does not: put a bend in and the number does not move.

It is also worth recording what the sweep does not show, since a null result invites the suspicion that the effect was looked for in the wrong place. The bend was put at ring five of twelve, near the middle, so that the heteroatom could be placed several rings on either side of it — and the response was read on every ring of both chains rather than only on the heteroatom’s own. If a bend produced a discontinuity anywhere in the profile it would appear in the next figure whatever the amplitude on the heteroatom’s ring did.

Straight through it #

The clearest form of the result is not a ratio but a profile.

With the heteroatom three rings before the bend, the response falls ring by ring across the whole chain — through the bend and out the other side — at the same fitted rate on the bent chain as on the straight one: 0.697 rings against 0.711. On a logarithmic axis spanning five decades the two profiles are hard to tell apart.

They are not the same profile. Ring by ring, the bent chain’s response just past the kink is 0.39 of the straight chain’s, and the rings beyond it stay between 0.61 and 0.91 of it. The kink multiplies the response down and leaves its rate of decay alone — which is not what a boundary does either, since a boundary would change what happens on its far side rather than scale it, but it is also not “nothing happened at ring five”. The first version of this paragraph read the logarithmic picture as agreement, and a factor of 2.5 is a small distance on that axis.

And it does not matter which side #

A boundary is something a response is on one side of. If the bend were one, the effect of adding it would depend on whether the heteroatom was inside or outside — and the placements here straddle it: two before, two after, one at the end.

The ratios are 1.0012, 0.9991, 0.9948, 0.9685 and 0.9859. They do not change sign as the heteroatom crosses, they do not grow towards the bend on one side and shrink on the other, and the largest of them is under 3.2 per cent. There is nothing here that a boundary does.

What an end does that a bend does not #

It is worth saying what the difference is, because the two changes look similar and are not.

An end removes a neighbour. The last ring of a chain has one fused neighbour where an interior ring has two, so a response that would have propagated in two directions propagates in one, and what would have left comes back — which is why the amplitude concentrates and the range does not change. The arithmetic is a reflection.

A bend changes which neighbour, not how many. A bent ring still has two fused neighbours; the fusion bonds are in a different relative position on the ring, and every ring beyond it still has the two neighbours it had. There is nothing to reflect, and nothing that has to come back.

That reading predicts everything measured above: no change of range, because the number of paths is unchanged; a small local change of amplitude, because the bent ring’s own bonds are differently placed; and no dependence on which side, because the change is not a barrier. It is the reading the numbers support and it was not obvious in advance — the fourfold effect on a ring current is evidence that a bend can matter enormously.

Why the bent ring loses a fifth #

The one real effect deserves a reading even though nothing here settles it.

A ring in a linear chain is fused to its two neighbours across two bonds that are para to each other — opposite sides of the hexagon. A ring at an angular fusion is fused across two bonds that are meta — one position apart. The ring has the same six bonds and the same two neighbours either way; what changes is which of its bonds are shared and therefore which of them the relaxation can move freely.

Two shared bonds close together leave a longer run of unshared bonds on one side of the ring and a shorter run on the other, and a heteroatom on that ring perturbs an environment that is no longer symmetric about it. Whether that is enough to account for a fifth is not established here, and one fact argues that it is not the whole story: the kink’s second ring is fused across meta bonds too, and loses three per cent rather than twenty.

The reading that the kink obstructs the response is not supported either, in the sense of a barrier: the decay rate is the same on both sides. What the kink does to the rings beyond it is a scaling, and whether that is the same scaling a single angular ring applies is a question the kink cannot answer, because it contains two.

What was computed, and how #

A chain of m fused six-rings is built from one flag per fusion — zero for a linear fusion, one for an angular one — so a straight chain is all zeros, the fully angular chain used for ring currents is alternating, and one bend is a single one in a run of zeros. The flags are directions, not turns: a single one turns the chain and the zero after it turns it back, so a single one makes two adjacent angular rings rather than one. Everything else is the straight-chain calculation unchanged: a staggered starting geometry, a self-consistent relaxation of bond orders and bond lengths, and the heteroatom introduced as a shift in one outer carbon’s site energy with the relaxation’s reference shared between the plain and the doped chain.

The response on a ring is the mean absolute change in the bond orders of the bonds inside it. The decay length is fitted over the rings that are still falling — a floor is not a decay, and fitting one as though it were is the error found in the bookkeeping.

Five things are checked: that every relaxation converged, on both chains, since a difference of two unconverged answers is not a response; that the heteroatom is placed on both sides of the bend and on it; that the bend changes the amplitude on the heteroatom’s own ring by less than a quarter; that it changes the decay length by less still; and that its effect on the amplitude does not vary much with placement, which is what says it is not a boundary.

Where the model stops #

One bend is one bend. A chain with several, or with them close together, is a different structure and nothing here says anything about it — and the fully angular chain, where every fusion is bent, is the case the ring-current comparison works with and the case where a large effect was found.

The model is Hückel with a self-consistent bond-order-bond-length relaxation, which has no explicit electron repulsion and no geometry beyond the topology of the fusions and one length per bond. A bend in this model is a change of adjacency; in a real molecule it is also a change of shape, of strain and of overlap, and a ring-current response is sensitive to the shape in a way a bond-order response is not.

And the response measured here is a bond-order change, which is a local quantity. Its reach is the reach of that quantity rather than of delocalisation in general, and the caution carries over.

What survives, and what can now be said #

Three calculations have quoted a reach near 0.7 rings. The number does not depend on where along a straight chain the heteroatom sits, and it now turns out not to depend on whether the chain is straight.

So the number is a property of a fused six-ring system in this model, and can be quoted as one. That is a stronger statement than was available before, and it is the one the question about bends threatened.

What should not be said — though the phrasing invites it — is that the amplitude is similarly robust. The amplitude depends on the end (a factor of 1.94), on the bent ring (a factor of 1.23), and on nothing else measured. Range and amplitude are different quantities with different robustness, and both have now been measured.

The generalisation #

The transferable finding is that two structural probes on the same molecules can disagree about what counts as a boundary, by a factor of twenty.

A single bend changes the ring-current response fourfold and the heteroatom reach by a fifth on one ring. Both are responses of the same π system to the same change of adjacency, computed in models of comparable crudeness. So “is this feature structurally important” has no answer that is not relative to a probe, and a result quoted for one probe should not be carried over to another even when both are about the same molecule and the same feature.

The reason is not mysterious once the two are put side by side. A ring current is a global circulation and a bend interrupts a circuit; a bond-order response to a local perturbation is a local relaxation and a bend is a change of neighbour that leaves the number of neighbours alone. What is a barrier to one is a corner to the other.

The practical version is that a structural sensitivity is a property of a pair — a feature and a probe — and neither alone. That is worth saying because the literature of fused systems talks about “the effect of angular fusion” as though it were a property of the fusion. The same kind of correction applies to what a stabilisation is measured from: a quantity that seems to belong to a molecule turns out to belong to a comparison.

One more consequence is worth drawing for anybody using these numbers. The reach is quoted in rings, and a bend does not change the number of rings between two points — but it changes the distance between them in space considerably, because a bent chain curls. So a reach of 0.7 rings is a topological statement and not a geometric one, and on a bent chain the two diverge. Nothing in this model can tell the difference, since its only geometry is one length per bond; a reader who wants a reach in ångström is being offered a reach in fusions.

The factor of twenty deserves one qualification. It compares a fourfold change to a change of a fifth, and those are ratios of different quantities in different units — a ring-current response and a bond-order reach — so the twenty is a ratio of sensitivities rather than of anything measurable. What it means precisely is that the same structural change moves one probe’s reading twenty times further, relative to that probe’s own scale, than it moves the other’s. That is the comparison the finding needs and it is worth spelling out, because a factor quoted between two dimensionless responses invites being read as a physical ratio, which it is not.

Who found it, and when #

Angular and linear fusion have been distinguished in the chemistry of polycyclic aromatics since Clar, and the ring-current difference between anthracene and phenanthrene is textbook. The relaxation, the reach, the bend and every number above are new arithmetic, and the point is a comparison between two responses to one structural change rather than a report of either.

Still open: how the effect grows with more bends #

Two of this collection’s own habits are visible in that comparison, and they point the same way: an anomaly that is not the first of a series was about reading one number as the start of a trend, and this is about reading one probe’s answer as the structure’s.

The obvious open question is more bends. One bend is a corner; a chain of alternating fusions is a different object, and the ring current’s factor of four is measured on that. Sweeping the number of bends from zero to eleven, with the heteroatom on the end, would say whether the effect accumulates linearly, saturates, or does nothing until the chain is fully angular — and each chain is one sequence of fusion flags.

The nearer question is the bent ring’s own fifth. The one real effect here is that a heteroatom on a bent ring gives a fifth less response than one on a straight ring, and nothing here asks why. A bent ring has the same number of bonds and the same number of neighbours; what differs is which of its bonds are shared. Reading the response bond by bond within that ring rather than as a mean would say whether the fifth is spread evenly or belongs to the two shared bonds, and if it belongs to them then the effect is a property of a fusion bond rather than of a ring, which is a smaller and more transferable statement.