The reach is the molecule's

The floor was in the bookkeeping found that the reach of a heteroatom’s relaxation had been reported with a floor in it — a floor that belonged to the bookkeeping rather than to the molecule — and re-measured it against a shared reference. It closed by naming what its own design could not separate.

Every measurement so far puts the heteroatom on the end ring, and the reason is practical: an end gives the longest unbroken run of rings to measure a decay over. But the response of a molecule to a perturbation at its end is not the response to one in its middle, and ring currents turn on exactly that distinction. The reach measured so far might be the end’s.

Moving the heteroatom inward costs one argument. The calculation takes a ring index, and it has only ever been given the end one.

Two directions, one decay #

From an interior position the profile falls away on both sides, and the two sides fall at the same rate: 0.687 rings on the short side and 0.707 on the long one, a difference of under three per cent.

And both agree with the end’s. Measured from ring one, the decay is 0.719 — within two per cent of the long side and four of the short.

Step by step the three sequences track each other. The short side gives ratios of 4.00, 4.86, 3.78; the long side 3.96, 4.80, 4.08, 3.47; the end 4.11, 4.72, 3.48. They wander together rather than drifting apart, and the wander is the same on all three — a ratio near four, alternating a little, which is the fused system’s own two-ring period showing through.

So the answer to the question is that the reach is the molecule’s. The reaches reported so far were not artefacts of measuring from an end, and they stand.

The one place the three sequences part company is the last step of the short side, where the ratio drops to 3.78 against the long side’s 4.08 at the same distance. That is the molecule running out: three rings from the peak, the short side has reached the end of the chain and the profile has nowhere left to decay into. It is a boundary effect and it appears exactly where a boundary is.

There is a reason to have expected the two directions to differ, and it is worth setting out because its failure is the result rather than a footnote.

A perturbation on the fifth ring of twelve has four rings on one side and seven on the other. If the response were a standing pattern — something that fills the molecule and takes its shape from the whole of it — the two sides would carry different amounts of it, and the side with more molecule would decay more slowly because it has more to decay into. That is how a particle in a box behaves, and a fused chain’s frontier orbitals are not far from one.

They do not. The decay is local: each ring’s response is set by its neighbour’s and by the gap, and the molecule’s total length does not enter. That is what a decay length means when it is a real length rather than a fitted one, and it is the property the magnetic reach could not establish, where the fitted number moved with the window it was fitted over.

So the finding has a second half worth keeping. The reach is the molecule’s, and it is also local — which are different claims, and the two-directional measurement establishes both at once because a non-local response could not give the same rate on two sides of different length.

The amplitude is not the molecule’s #

The peak response is 3.763 × 10⁻² with the heteroatom on the end ring and 1.942 × 10⁻² with it on any interior ring — a factor of 1.94, and then flat. Positions three, four, five and six give the same amplitude to three figures.

So there are two quantities here and only one of them is a property of the molecule. The decay is; the amplitude is a property of where the heteroatom was put, and every earlier measurement put it in the one place that doubles it. A perturbation on an end ring has only that ring to sit on, while one placed inside shares itself between two, and that difference is the whole of the factor.

The cause is geometric and it is worth stating carefully because it is not a physical effect at all. The perturbed carbon is chosen as the one furthest from the molecule’s centre in its own ring. In the end ring that carbon belongs to that ring and to nothing else. In an interior ring, the carbon furthest from the centre lies on a bond that ring shares with its neighbour — so the perturbation sits on two rings at once, and its response is split between them.

That is why the peak is shared. Two adjacent rings carry the response to within a per cent of each other, and each gets about half of what a single end ring gets. The total is conserved; only its distribution over the ring index changes.

Seen together the three profiles are one shape translated, with a single peak at the end and a shared pair inside. Nothing about the slope of any of them differs.

What this means for the numbers already reported #

The table is the audit. Every earlier measurement quoted a decay length and an amplitude from the first row, and the audit says the first of those transfers and the second does not.

That is a mild correction rather than a retraction, and it matters most for one specific use. A decay length can be quoted for the molecule; an amplitude has to be quoted with the position, and comparing amplitudes between molecules whose heteroatoms sit differently is comparing a factor of two of geometry. The floor was in the bookkeeping removed one such artefact from these numbers; this is the second, and it is in the other quantity. Nothing in the earlier measurements did that — they all measured from the end — but nothing in them said it could not be done either.

What a chemist would do with a local reach #

The locality is the part worth carrying out of the model, because it is what licenses the way these numbers are usually used.

A substituent effect quoted as “falls off by a factor of four per ring” is a statement that only makes sense if the factor does not depend on how much molecule is left. On a naphthalene, an anthracene and a pentacene the same substituent at the same kind of position should then perturb its neighbour by the same fraction — and the two-directional measurement is the evidence for that, because it is the same molecule tested against two different amounts of remaining chain.

What the measurement does not license is transferring the size. Two checks have now found an amplitude that belongs to the setup rather than to the chemistry: first a floor put there by an unshared reference, now a factor of two put there by which carbon carries the perturbation. Both were invisible in the reported number and both were found by changing something the earlier measurement had held fixed without saying so.

The practical form is a rule about reporting. A decay length can be quoted alone. An amplitude has to be quoted with the position of the perturbation, the choice of reference, and the number of rings the perturbed carbon belongs to — and if that reads as a lot of conditions for one number, that is the finding rather than a complaint about it.

What was computed, and how #

The system is a chain of fused six-membered rings with a staggered site energy, relaxed to a fixed point in which each bond’s length responds to its own bond order and each bond order responds to the lengths. Two systems are relaxed — the plain molecule and the same molecule with one carbon’s site energy shifted by one unit of β — and every bond’s order is differenced.

Both relaxations use the plain molecule’s reference bond order, which is the bookkeeping repair and the reason these numbers differ from the unrepaired ones. Without it the reported response flattens at a floor belonging to the bookkeeping.

The response of a ring is the mean of the absolute difference over its six bonds. The peak is the largest of those, and the “shoulder” is every ring above two fifths of it — which is one ring at the end and two inside, and is measured rather than assumed.

The decay length is a least-squares slope on logarithms against distance, taken over the rings outside the shoulder so that both directions start from the same place in the profile. That exclusion matters: including a shared peak in one direction and a single one in the other would compare two different things and would produce exactly the asymmetry being tested for.

Four things are checked. The interior case must have two rings in its shoulder and the end case one. The two interior decays must agree within twelve per cent. The end’s amplitude must exceed the interior’s by more than 1.6. And the end’s decay must agree with the interior’s within fifteen per cent — which is the finding, stated so that it fails if the reach is the end’s.

Why the profile alternates #

The step ratios are not constant, and the pattern in them is the same on all three sequences: 4.00, 4.86, 3.78 on one side and 3.96, 4.80, 4.08 on the other. High, low, high — an alternation of about a fifth around a mean near four.

That is the fused chain’s own two-ring period. A linear acene’s bonds come in two kinds — the ones shared between neighbouring rings and the ones on the outside — and a ring’s mean bond order response mixes them in a proportion that alternates as the ring index does. So the alternation is a property of the geometry rather than noise in the measurement, and it appears identically in every direction because every direction walks the same alternating structure.

It matters for one practical reason. A decay length fitted over an even number of steps averages the alternation away and one fitted over an odd number does not, so two fits over windows differing by a single ring can disagree by a few per cent for no reason at all. The three lengths above are all fitted over the rings outside the shoulder, which happens to be three or more in every case, and the residual few per cent between them is largely this rather than any difference between the directions.

That is worth knowing before any of these numbers is quoted to better than about five per cent, and it is why the agreement above is reported as these are the same number rather than as a measurement.

Where the model stops #

This is Hückel theory with a relaxed geometry: one π electron per carbon, one hopping integral that depends on a bond length, and a length that depends on a bond order. It has no repulsion and no σ framework, and the “heteroatom” is a shifted site energy rather than an atom.

The molecules are linear chains of twelve rings. An angular chain has a different geometry and the calculation accepts one, but nothing here is measured on one — and a bend is known to change a fused system’s response by a great deal, so the transfer is not safe. What an angular chain does to a delocalisation rather than to a current is untested, and the ring with a twist in it is the only non-planar case treated at all.

The decay lengths are under one ring. That is a fast decay measured over three or four points, and the three per cent agreement between directions should be read as these are the same number rather than as a measurement good to three per cent. A finer statement would need longer chains, where the response falls below what the relaxation converges to.

And the amplitude’s factor of two is a consequence of how the perturbed carbon is chosen. A different rule — the carbon nearest the ring’s own centre, say — would put the heteroatom on an unshared carbon in every ring and would remove the effect. That would be a different measurement rather than a better one, and which is wanted depends on what a real heteroatom is like; the point here is that the choice has a factor of two in it and was never stated.

The generalisation #

A measurement made from a boundary contains the boundary until something is measured away from it. The end-ring measurements had no way to know which of their two numbers was a property of the molecule, because both were measured in the one configuration available, and the way to find out was to move the perturbation rather than to improve the fit. The same lesson appears in an end effect with two signs, where a scatter attributed to a boundary turned out not to be one, and it is the same discipline in reverse: there a boundary explanation failed, here it survives for one quantity and not the other. One integer, and everything it changes is the version where moving one feature along a molecule was the whole measurement.

And a shared vertex is not a small detail. The factor of two above is not a physical effect; it is the difference between a perturbation carried by one ring and by two, and it appears in the amplitude of every number reported from an end. Anything defined per-ring on a fused system inherits it, and the tell is easy to check: an interior peak shared between two rings and an end peak that is not.

Who found it, and when #

The response of a conjugated system to a site perturbation is elementary Hückel theory, and that it decays over a length set by the gap is the standard reading. Nothing in the mechanism is new here.

What was asked for, and what this supplies, is an audit: a length and an amplitude had been reported from one configuration and had no evidence about which of them transferred. Both were plausible either way, the test cost one argument, and the answer is one of each — which is the sort of result that only exists because the question was asked rather than assumed away. Delocalisation is stabilising, and other things that are false in general is where distrust of such summaries starts.

Still open: the closed form, and the angular chain #

The obvious open question is the closed form, left untouched here. Two responses agreeing on an exponent of −0.68 while an argument says −1 is either a fact about the model or a fact about the fits, and a one-dimensional chain with a staggered site energy has a band structure that can be written down. Its complex band at the gap gives a decay to compare against a measurement made exactly the way these were, and with the position confound now removed the comparison is between two numbers rather than three.

The nearer question is the angular chain. Everything here is a straight chain of twelve, and a single bend is known to change a fused system’s response by a factor of four and that the effect depends on where the bend is. The same measurement on a chain with one bend, with the heteroatom on each side of it in turn, would say whether a bend is a boundary of the same kind as an end — and if it is, the reach measured here is a property of a straight chain rather than of a fused system.