On an angular ring the carbon decides

A heteroatom on a chain of fused benzene rings perturbs the bond orders round it, and the perturbation dies away along the chain. Counting angular rings one at a time found that each angular fusion the perturbation passes through rescales what lies beyond it, and that the kink of two angular rings treated earlier as a single bend did something curious at its own rings: with the heteroatom on the kink’s first angular ring, that ring’s response fell to 0.811 of the straight chain’s; on the second, only to 0.969. The explanation offered was the heteroatom’s carbon. On ring five it sat beside the long run of unshared bonds an angular fusion leaves on one side of a ring; on ring six, the same position faced the short side.

The question that essay left was the obvious one. An angular ring, it said, has four outer carbons in two inequivalent pairs, and putting the heteroatom on each would say whether the fifth belonged to the long run or to a carbon’s distance from the nearest fusion.

Asking it starts with counting the carbons, and the count does not come out as four.

An interior ring has two hydrogens #

A catacondensed chain of m six-membered rings has 4m + 2 carbons, and 2(m − 1) of them are shared between two rings. Twelve rings have fifty carbons, twenty-two of them junctions. The other twenty-eight carry a hydrogen, and they are distributed unevenly: each terminal ring has four, and each of the ten interior rings has exactly two. That is true of a straight chain, of a kinked one and of a fully angular one alike; fusion direction changes where the two sit, not how many there are. On a ring fused para, the two are opposite each other, like anthracene’s 9 and 10 positions. On a ring fused meta they are adjacent and bonded to each other, like phenanthrene’s 9 and 10.

So “four outer carbons in two pairs” describes the terminal ring and nothing else. On an angular ring there is one pair, and the other four positions are junctions in two pairs of their own.

Which raises the question of what the earlier placements were. The heteroatom was placed by a geometric rule: on the chosen ring, the carbon furthest along the chain’s axis from the molecule’s centre. On the terminal ring that rule lands on a CH carbon, because the end of the chain is made of them. On an interior ring, the carbon furthest along the axis is one the ring shares with its neighbour.

On every interior ring of the straight chain and of the kink, the rule picks a junction. The comparison of heteroatoms placed along the chain knew this and said so — it traced a factor of two in the interior amplitude to the perturbed carbon lying on a shared bond. The comparison across a bend and the angular-ring count both described the same placements as “an outer carbon”. The numbers in both are right; the carbon they name is not the carbon they used.

The fifth sat at the bay #

Once the carbon is known to be a junction, the explanation built on it can be checked bond by bond rather than by description. An angular ring has two bonds it shares with neighbours, and four it does not: one short unshared bond between the two fusions, at what in phenanthrene would be the bay, and a run of three round the other side.

On ring five the ringed carbon is the junction that ends the short unshared bond. Its three bonds go to ring four’s outer carbon, along the bond ring five shares with ring four, and across the short side to the junction ring five shares with ring six. Nothing about it is next to the long run. On ring six the ringed carbon ends the long run: one of its bonds is the first of the three unshared bonds round the far side. The published account had them exactly reversed — the carbon that lost a fifth faces the short side, and the one that lost three per cent sits beside the long run.

That reversal is worth stating plainly because it decided what the next question was. A loss blamed on the long run invites a sweep along the long run. A loss at a bay junction invites a different question: what each kind of carbon on an angular ring does, which is the one asked here.

Matching a carbon, not an index #

Comparing a heteroatom on one carbon of a kinked chain with one on “the same carbon” of the straight chain needs a definition of same, and the obvious one fails.

The chain is a bipartite graph, and the model holds its gap open with a staggered site energy: every carbon of one sublattice is shifted down by 0.6 in units of the hopping and every carbon of the other up by 0.6. The stagger is there for a reason — without it the gap of a long acene closes and the decay length has nothing to be measured against — but it means two carbons of the same kind on the same ring are not alike. A heteroatom that adds one unit of site energy to a carbon already at −0.6 is a different perturbation from one added to a carbon at +0.6.

So each carbon is matched with the straight chain’s carbon on the same ring that has the same kind and the same sublattice. Kind is fixed by which rings a carbon belongs to: a CH carbon belongs to one, a junction is shared with the ring before or the ring after. On a straight ring there are exactly six such combinations, one per carbon, so the match is unique, and the bay and long-side junctions of an angular ring are matched to the straight ring’s junction on the same side. The earlier ratios survive this: the bay junction of ring five, at −0.6 and shared with ring four, gives 0.811 against its partner, and the long-side junction of ring six gives 0.969. They were matched correctly; they were simply junctions.

A comparison by position in the ring’s vertex list, which is the other natural choice, would have matched a junction on the kink with a CH carbon on the straight chain whenever the kink moved the fusions round the ring — and on ring six it pairs a carbon at −0.6 with one at +0.6. Neither is a comparison of anything.

The carbon decides more than the ring #

With six carbons on each ring and each one matched, the question the angular-ring count asked can be answered on the carbons rather than on one of them.

The control comes first. Ring three of the kinked chain is fused para on both sides and sits two rings before the kink. All six of its carbons give between 0.993 and 1.045, so a ring that is itself straight barely notices a kink two rings away, whichever carbon carries the heteroatom.

On ring five the six give 0.811, 0.981, 1.005, 1.025, 1.081 and 1.446. On ring six, 0.869, 0.935, 0.969, 1.104, 1.111 and 1.242. On a lone angular ring at five, with every other fusion para, 0.780 to 1.075. The spread on one angular ring is wider than the difference between angular and straight rings that the comparison set out to measure. “An angular ring costs a fifth” is a statement about one junction, the bay junction on the sublattice shifted down; the long-side junction on the same sublattice of the same ring gains 45 per cent.

The two CH carbons are where the picture is simplest. On every angular ring examined, both lie within eleven per cent of the straight ring’s partner: 1.005 and 1.081 on ring five, 0.935 and 1.104 on ring six, 0.950 and 1.075 on the lone angular ring. Measured from the carbons a real aza-arene would substitute, an angular ring’s own response is nearly a straight ring’s.

What a CH carbon sends the rest of the chain #

The ring’s own response is only one number, and the earlier essays measured it because the heteroatom’s ring is where the response is largest. What the rest of the chain receives is a separate quantity, and for the CH carbons of an angular ring it is the one that moves.

The response summed over every other ring of the chain, divided by the same sum for the straight chain’s matched carbon, separates the twenty-four placements into three clusters with nothing between them. The para ring’s six sit at one on both axes, between 0.976 and 1.010. The angular rings’ junctions send more than a straight ring’s do, 1.05 to 1.36. Their CH carbons send between 0.47 and 0.63 — about half — while their own ring stays where it was.

So a heteroatom on a CH carbon of an angular ring is a perturbation the ring keeps. It is not smaller where it starts; it is smaller everywhere else.

The profiles show the half is not a single factor applied at one fusion. From the CH carbon at −0.6 on the kink’s first ring, every ring behind it, from zero to four, receives between 0.48 and 0.50 of the straight chain’s, and every ring beyond, from six to eleven, between 0.43 and 0.64. The lone angular ring is cleaner still: 0.41 to 0.44 behind and 0.40 to 0.59 beyond. The factor applies in both directions at once, from the source rather than at a fusion the response crosses. Ring currents in fused chains show a factor of similar size for a different reason — each angular ring between two rings multiplies their coupling by about 0.6 — but that is a factor per fusion crossed, which is exactly what this is not.

That matters for the directional picture the angular-ring count drew, in which each angular fusion between a ring and the source multiplies that ring’s response and fusions on the far side do almost nothing. The picture was built with the source on the terminal ring, where the placement rule does pick a CH carbon, and it stands there. What it does not cover is a source on an angular ring, where the ring’s own geometry acts on everything that leaves it.

The bond that holds it #

The two CH carbons of an angular ring are bonded to each other, and that bond is special in a way chemistry has known about for a long time.

In the plain chain, before any heteroatom, it is the strongest bond in the molecule: a bond order of 0.660 on the kinked chain, against a mean over all of its bonds of 0.529 and a strongest bond of 0.645 anywhere on the straight chain. In unrelaxed, unstaggered Hückel theory on phenanthrene itself the same bond — the 9,10 bond — has order 0.775, the largest in that molecule too. It is the bond that makes phenanthrene’s middle ring behave less like an aromatic ring than like an alkene held between two benzenes: it adds bromine across itself, and in Clar’s sextet picture of the molecule it is the one fixed double bond, with the two outer rings carrying the sextets.

A heteroatom on one end of that bond spends much of its effect on it. Measured directly, the change in that one bond’s order is between 42 and 47 per cent of the heteroatom’s whole ring’s response and between 28 and 34 per cent of the response of the entire twelve-ring chain. The perturbation is polarising an isolated double bond, and an isolated double bond is, by construction, not well connected to the rest. The same localisation that gives phenanthrene its one reactive alkene-like bond is what halves how far a heteroatom placed on it reaches.

That is a connection between two things usually kept apart: a reactivity rule in the chemistry of polycyclic aromatics, and a decay measurement in a one-electron model. The model knows nothing about bromine. It knows that the bond between an angular ring’s two hydrogens carries more π bonding than any other, and that is enough.

What stands, and what does not #

Stands, as computed. The angular-ring count from the terminal ring — its saturation ring by ring, the decay length’s eleven per cent against the responses’ factors, and the one-ring backward reach — used a CH carbon on ring 0 throughout. The kink’s two ratios, 0.811 and 0.969, are correct ratios on correctly matched carbons. The placement comparison along the chain identified its interior placements as shared carbons and drew its amplitude conclusion from exactly that.

Corrected. That the placements on rings five and six of the kink were outer carbons, and that the first sat beside the long run while the second faced the short side. Both were junctions, and the sides were reversed. That an angular ring has four outer carbons. The angular-ring essay and the comparison across a bend have been amended to say what carbon each interior placement used, and point here.

Moved. The fifth was offered as a property of a heteroatom position relative to an angular fusion. It is a property of one of six positions, on one sublattice; the other five on the same ring range from a two per cent loss to a forty-five per cent gain.

How each placement was computed #

Each chain has twelve fused six-rings, specified by which interior rings are angular and built so that exactly those rings have meta fusions; the angular rings are read back from the geometry. The kink is rings five and six; the lone angular ring is ring five with every other fusion para. The π system is a Hückel model with a heteroatom as a shift of one unit in one carbon’s site energy, with the staggered site energy of ±0.6 that holds the gap open and a relaxation that makes bond orders and bond lengths self-consistent through a linear dependence of the hopping on the bond order, the plain and doped chains sharing the relaxation’s reference so that no floor enters the difference. Bond orders are read from the eigenvectors. A ring’s response is the mean absolute change in its six bond orders; every other ring is that quantity summed over the remaining eleven rings.

A carbon’s kind is read from the geometry: the rings it belongs to, and on an angular ring whether its in-ring unshared neighbour is a junction of the other fusion (bay) or not (long side). Its partner on the straight chain has the same side and the same sublattice; every carbon has exactly one.

The checks, run wherever these figures are drawn: every interior ring of the straight, kinked and fully angular chains has two CH carbons and each terminal ring four; the placement rule picks a junction on every interior ring of the straight chain and the kink; the earlier 0.811 is reproduced at ring five’s bay junction on the ring-four side and 0.969 at ring six’s long-side junction on the ring-seven side; both CH carbons of each angular ring stay within eleven per cent; ring five’s six carbons span more than 0.6; the para control’s six stay within five per cent; and every relaxation converges below 10⁻⁸. The refusal is the straight chain matched against itself, which must give exactly one on every ring for every carbon.

The limits of a site-energy heteroatom #

A junction heteroatom is a real but unusual object. A nitrogen at a ring fusion has three σ bonds and no hydrogen, and as part of a neutral π system of this kind it exists mainly as a cation, as in the quinolizinium ion. The junction placements answer a question about the model’s response to a perturbation at that site; they do not describe the common aza-arenes, which put nitrogen where a CH was. The CH results are the chemically ordinary ones.

A site energy is half of a heteroatom. A pyridine-type nitrogen also changes the hopping to its neighbours, and it brings a lone pair in the plane that this π model does not see. The sign and rough size of the shift is the standard first approximation; the ratios here are ratios of a model response.

The stagger is a control, not a molecule. The ±0.6 site energies exist to give the chain a gap and a decay length, and they are the reason the sublattice match is needed. A real chain’s gap comes from its own bond alternation, which the relaxation supplies only in part. A real phenacene is also non-planar at its bays once it is long.

A junction belongs to two rings. For a heteroatom on a junction, “its own ring” is the ring the placement named; the ring on the other side of the shared bond responds about as strongly, which is why the interior amplitude halved in the first place.

Who found what #

Bond order as a quantity read from Hückel coefficients is Coulson’s, from 1939. The localised 9,10 bond of phenanthrene and its alkene-like chemistry are long known, and Clar’s sextet rule, which assigns phenanthrene two benzenoid sextets and one fixed double bond, is from his monograph of 1972. The same bond is the K-region of the Pullmans’ work on the carcinogenicity of polycyclic hydrocarbons. What is computed here is new arithmetic in a relaxed, staggered model: the count of CH carbons against the placement rule, the response of each of six carbons on an angular ring against a sublattice-matched partner, and the halving of what reaches the rest of the chain from a CH carbon.

A placement is a choice of carbon #

The general lesson is about rules that choose for a measurement. “The carbon furthest along the axis” is a perfectly good rule — deterministic, symmetric, easy to state — and on the one ring it was designed for it picks the obvious carbon. Carried to the interior, it picks a different kind of carbon without saying so, and the words attached to its choice were carried along unchanged. Nothing in the numbers was wrong; a name was, and as before, the name decided which question came next.

The second lesson is about which number to watch. The ring’s own response was the natural measurement and, from a CH carbon, it says an angular ring is almost a straight one. The response summed over the rest of the chain says it is half of one. A local measurement can be robust and still miss where the difference went.

Still open: a branch, and a nitrogen that is more than a site energy #

The question the angular-ring count named second is untouched. A ring fused to three neighbours — the central ring of triphenylene — offers a perturbation two routes, and whether the responses on the far side of the branch add or interfere is a question no unbranched chain can ask. The finding here sharpens it: triphenylene’s central ring has no CH carbon at all, so a heteroatom there would be a junction placement of the kind just shown to behave least like the straight chain, and its outer rings’ CH carbons sit on the same strong localised bonds.

The nearer question is the heteroatom itself. Half of what makes a pyridine-type nitrogen different from a carbon is the change in hopping to its two neighbours, and on an angular ring’s CH carbon one of those two bonds is the strongest bond in the chain. Whether a hopping change on that bond halves what reaches the chain as the site energy does, or undoes it, is the same sweep with one more parameter.