An angular ring rescales what lies beyond it

A heteroatom on the end of a chain of fused benzene rings perturbs the bond orders around it, and the perturbation falls off along the chain with a characteristic length close to 0.7 rings. Where along the chain the heteroatom sits changes the amplitude and not that length, and a bend in the chain was found to be a much weaker boundary than an end: a fifth off the response on one ring, and nothing off the decay length. The question left was what several bends do — whether the effect accumulates, saturates, or does nothing until every fusion in the chain is angular.

Asking it properly needs a count, and the count exposes something about the bend. A chain here is built from one fusion direction per junction, and a straight chain is every direction the same. The bend was a single different direction among the rest, which means the chain turns and then turns back: the ring before the different direction and the ring after it are both fused across meta bonds rather than para ones. What was measured as one bend was a kink of two adjacent angular rings. So the unit counted here is the angular ring — a ring whose two shared bonds sit one position apart — and a chain is described by which of its ten interior rings are angular.

The heteroatom’s own ring does not notice #

Put the heteroatom on the terminal ring and make the interior rings angular one at a time, starting from the ring beside it. Across every count from one angular ring to ten, the heteroatom’s own ring stays within one and a half per cent of the straight chain’s response, and the ring next to it within two and a half. Angular fusion does not reach back to the source.

What it does reach is everything further along. With one angular ring, ring 1, the rings beyond it fall to between 0.68 and 0.80 of the straight chain’s. With two, the third ring is at 0.49. With four, the sixth ring is at 0.31. And with all ten interior rings angular — the zigzag chain of the phenacenes — the rings from the fourth onward sit between an eighth and a third of the straight chain’s response, alternating from ring to ring as the geometry alternates.

The effect saturates, and it saturates ring by ring. The third ring’s response is set once rings one and two are angular: 0.493 with two, 0.482 with three, 0.453 with ten. The sixth is set once the rings before it are: 0.307 with four, 0.313 with ten. Making a ring angular beyond a given ring adds nothing to that ring. So the accumulated effect of many angular fusions is a sequence of factors, each fixed by the fusions between that ring and the heteroatom.

A slope and a level #

The earlier bend changed the fitted decay length by under two per cent, and that was read as the bend changing nothing along the chain. The count says why that reading was available and why it was wrong.

The decay length falls by at most eleven per cent — from 0.719 rings on the straight chain to 0.638 on the fully angular one. The responses fall by factors of two to three. A decay length is fitted to the logarithm of the response against distance, so it measures the slope of that logarithm. Angular fusion changes the level: it multiplies the response on the rings beyond it by a factor and leaves the rate at which the response then falls nearly as it was.

On a logarithmic plot spanning five or six decades — which is how a decay is naturally drawn — a factor of 0.4 is a small vertical shift and two profiles with the same slope look like one. That is exactly how the profile across the kink came to be described as lying on the straight chain’s: ring by ring, the kink put the ring just past it at 0.39 of the straight chain’s response. The description has been corrected there. The general point is that a single fitted length summarises a slope and discards a level, and a structural change can act entirely through the second.

It is the same distinction a floor in the bookkeeping forced in a different form: a decay fitted over a region where the response has stopped falling reports a length that belongs to nothing. Here the response is still falling everywhere; what the length hides is how much of it arrived.

One angular ring, placed along the chain #

With a single angular ring and the heteroatom on the end, the effect can be read as a function of where the angular ring is.

Every ring two or more places nearer the heteroatom than the angular ring stays within three and a half per cent, and every ring three or more places nearer within one. The ring immediately before an angular ring can move by up to a tenth. So an angular ring’s influence reaches one ring back and every ring forward.

Forward, it is not simply a loss. An angular ring close to the heteroatom lowers the rings beyond it: at ring 1 they fall to between 0.68 and 0.80, at ring 3 to between 0.46 and 0.73. An angular ring further out does something less tidy. At ring 5 the rings beyond range from 0.86 to 1.33 of the straight chain’s; at ring 7 every ring beyond is raised, by between 14 and 41 per cent. A single angular ring rescales what lies beyond it, and the rescaling has both signs, depending on how far along the chain it sits.

That both-signs behaviour is why the fully angular chain alternates. Each ring’s response is a product of factors from every angular ring between it and the source, and the factors are not all below one; they alternate with the parity of the chain’s geometry, and the product alternates with them.

And from the far end #

The same count run from the other end tests the reach backward directly.

Rings 0, 2 and 3 stay within two per cent as angular rings are added from ring 10 down to ring 3. At eight angular rings, the third ring is itself angular and so is every ring beyond it, and its response is 1.012 of the straight chain’s. It falls, to 0.453, only at ten, when rings one and two become angular too. A ring’s own angularity does not decide its response; the angular fusion between it and the heteroatom does.

That makes the picture directional in a way that a bond-order response need not be. The perturbation is introduced at one end, and each angular fusion it passes through changes how much of it arrives at what lies beyond. Fusions it has not yet passed through, or never passes through, do nearly nothing to it. A ring current, by contrast, is a global property of the circuit, which is why one bend could change a ring current fourfold while changing this response by a fifth on one ring.

Reading the six placements again #

The directional picture explains the comparison that started this, including the part of it that was right. The kink sat at rings five and six, and the heteroatom was placed on rings 0, 2, 4, 5, 6 and 8. On each placement only the heteroatom’s own ring was compared.

On ring 0 and ring 2, the kink is three or more rings away and on the far side, and nothing reaches back that far: the ratios were 1.0012 and 0.9991. On ring 4, one ring before the kink, the one-ring backward reach applies and the ratio was 0.9948. On ring 8, two rings after, the kink lies between the heteroatom and the chain’s start, but the ring being measured is the heteroatom’s own and the kink is two rings behind it: 0.9859. On ring 6, the second angular ring, 0.9685; on ring 5, the first, 0.8111.

So “it does not matter which side of the bend the heteroatom is on” was true, and for a reason the comparison could not see. Only the heteroatom’s own ring was measured, and angular fusion changes a ring’s response mainly through the fusions between that ring and the source — of which the source’s own ring has none. Every placement read its own ring, and every own ring is on the source side of everything. Had the comparison read the ring four places from the heteroatom, across the kink, the side would have mattered by more than a factor of two: 0.391 with the heteroatom on ring 2, reading ring 6, and 0.926 with it on ring 8, reading ring 4. The kink is not even symmetric under reversal — ring five is the first angular ring from one end and the second from the other — and a measurement at the source hides that too.

That is a small instance of a pattern with consequences. The reach a heteroatom has was found to belong to the molecule rather than to the heteroatom, and it is worth adding that it belongs to the molecule in a direction: the same heteroatom on the same chain perturbs the rings on one side of an angular fusion and not the rings on the other, and a comparison made at the source cannot tell.

The kink, bond by bond #

The kink that started this is two adjacent angular rings, five and six. With the heteroatom on ring five its own ring’s response was 0.811 of the straight chain’s; with the heteroatom on ring six, 0.969. Both rings are fused across meta bonds, so meta fusion alone cannot be why the first loses a fifth and the second three per cent.

The obvious guess was that the loss belongs to the two bonds a ring shares with its neighbours, because those are the bonds angular fusion rearranges. It does not. On ring five the shared bonds’ summed response rises, from 0.0352 on the straight chain to 0.0434 on the kinked one, and the outer bonds’ falls, from 0.0815 to 0.0513. On ring six both change by under a tenth: shared 0.0352 to 0.0372, outer 0.0815 to 0.0758.

The difference between the two rings is where the heteroatom’s carbon sits relative to the meta pair. A heteroatom on an outer carbon of ring five is adjacent to the long run of four unshared bonds that an angular fusion leaves on one side of the ring; on ring six the same position faces the short side. The response is carried by the outer bonds nearest the heteroatom, and which outer bonds are nearest changes with the kink’s orientation. The fifth is a property of a heteroatom position relative to an angular fusion, not of the fusion.

What was computed, and in what model #

Each chain has twelve fused six-rings. A chain is specified by which interior rings are angular, and its fusion directions are built so that exactly those rings have their two shared bonds meta; the construction is checked by reading the angular rings back from the geometry’s shared bonds, and the single flag at five that defined the earlier bend reads back as rings five and six.

The π system is a Hückel model with a heteroatom and bond-length relaxation, its bond orders read from the eigenvectors: bond orders and bond lengths are made self-consistent through a linear dependence of the resonance integral on the bond order, from a staggered start, to a residual below 10⁻⁸. The heteroatom is a site-energy shift on an outer carbon of the terminal ring, and the plain and doped chains share the relaxation’s reference. A ring’s response is the mean absolute change in the bond orders of its six bonds, and a response is compared by dividing by the straight chain’s on the same ring with the heteroatom in the same place. The decay length is fitted over the rings beyond the heteroatom that are still falling, as before.

The checks, run wherever these figures are drawn: the angular rings read back from the geometry; the earlier bend’s flag reads back as two angular rings; every relaxation converges; the heteroatom’s own ring stays within one and a half per cent at every count; the third ring keeps under half its response and the sixth under a third in the fully angular chain; the decay length moves by at most fifteen per cent; the effect reaches one ring back and no further, over every chain built; on the kink’s first ring the shared bonds gain response and the outer bonds lose it; and the kink’s two rings differ, one losing about a fifth and the other a few per cent. The refusal is the straight chain, whose ratios must be exactly one.

The limits of a relaxed Hückel chain #

One-electron and planar. The bond-order response is a Hückel quantity with a single relaxation parameter, and a real phenacene is non-planar at its bay regions once the chain is long, which this construction ignores. What transfers is the directional structure — factors fixed by the fusions between a ring and the source — rather than the size of any factor.

One heteroatom position. The heteroatom sits on one outer carbon of the terminal ring throughout; on ring five and six of the kink it sits on the equivalent carbon of those rings. The bond-by-bond reading shows that the carbon chosen matters on an angular ring, and a sweep over the ring’s four outer carbons is a calculation not made here.

And twelve rings is twelve rings. Beyond the seventh ring the responses are three or four decades below the terminal ring’s, and ratios of such small numbers are well above the relaxation’s residual but are still ratios of small numbers.

What a bend count needs #

The general lesson is about what a structural parameter counts. “A bend” was a single flag, and a single flag in a list of directions is two geometric events. Nothing in the numbers was wrong; the name attached to them was, and the name decided what the next question was. A count should be of the object the argument is about — here, rings whose fusion is angular — and read back from the geometry rather than from the construction’s flags, because the construction’s convention is exactly the thing that can differ from the concept.

The second lesson is one that keeps recurring: a summary statistic that is robust is robust about the quantity it summarises. The decay length is robust to angular fusion. The response is not. Saying angular fusion does not change the reach is true of the length and false of the response, and which of the two a reader hears depends on whether a number or a picture is shown.

Still open: every outer carbon, and a branch #

The obvious open question is the heteroatom’s position on an angular ring. The kink showed a fifth lost on one ring and three per cent on its neighbour, and the bond-by-bond reading tied the difference to which outer bonds the heteroatom’s carbon is next to. An angular ring has four outer carbons in two inequivalent pairs relative to its meta fusion, and sweeping the heteroatom over all four on a single angular ring would say whether the loss is a property of the long unshared run or of the carbon’s distance from the nearer shared bond.

The nearer question is a branch. Every chain here is catacondensed and unbranched, so a perturbation has one route along it. A ring fused to three neighbours — the branch point of a triphenylene — offers two routes, and the directional picture predicts that each route applies its own sequence of factors and the response on the far side of the branch is a sum. Whether it is a sum, or whether the two routes interfere, is the question a single chain cannot ask.