A reach that has no length

How much each ring of an acene responds to a magnetic flux through each other ring is a matrix rather than a list: the off-diagonal entries are nearly half the diagonal ones, so a circuit picture in which each ring is a loop with its own resistance is wrong by an amount that is not small.

Those off-diagonal entries fall with distance — 0.0639 for neighbours, 0.0284 for next-but-one, 0.0130 for the far pair in a four-ring acene — which invites asking for the length. There is a measured length for a structural perturbation, the distance over which a chain’s distortion recovers from its end, and the question was whether the magnetic reach is the same length seen a second way.

There is no magnetic reach in an acene. That is the answer, and the way it fails is more useful than the number would have been.

The measurement, and the one piece of care it needs #

The response matrix is symmetric, so every entry belongs to a pair of rings at some separation. Averaging the magnitudes at each separation and fitting a straight line through their logarithms gives a decay length in rings.

One restriction has to be imposed before any of it means anything. At the largest separation in a molecule of $m$ rings there is exactly one pair — the two ends — and both members of that pair are unlike every other ring, because each has a neighbour on one side only. Its entry is an end effect and not a sample of a decay.

Including it lengthens the fitted reach from 0.6385 to 0.7480 rings on a twelve-ring molecule, a seventeen per cent error, and it does so in the same direction on every molecule. So the fit here runs over separations of at most half the molecule, and the excluded points are drawn.

That restriction is a measured decision rather than a stated one: the contaminated number is computed alongside and checked to be larger.

Every acene gives a longer answer #

The reach grows by a factor of 1.65 over a factor of three in molecular size, monotonically, with no sign of settling. There is no molecule at which the answer stops depending on the molecule — which is the same test a fitted exponent has to pass and the same way of failing it, arriving here in a length rather than in a power.

The second column says why. An acene’s gap closes as it lengthens, and a decay length in a gapped system is set by the gap: a response propagates through virtual excitations, and the further it goes the more energy denominators it costs. A closing gap is a lengthening reach, and by twelve rings the length the gap would set is longer than the molecule that is supposed to display it.

So the fitted number is measuring the molecule. It is the same failure a fitted Peierls exponent shows in its window, one dimension over: a quantity fitted over a range shorter than the thing being measured returns the range.

The control, which is the finding #

The way to tell a diverging length from no length is to hold the gap open and repeat.

An acene is bipartite, so putting $+\delta$ on one sublattice and $-\delta$ on the other opens a gap that survives however long the molecule gets. Nothing else changes: the same rings, the same bonds, the same flux construction, the same finite differences. Only the diagonal moves.

It converges by the third molecule and then moves in the fourth decimal place. That is what a length looks like, and it is what the bare acene does not do.

So there is a magnetic reach; it is a property of a gapped π system; and an acene is too nearly gapless to have one. The three numbers were real and the length they seemed to imply was not.

What sets the scale, in one line #

The reason a gap sets a length is worth writing down, because it is the whole of the mechanism and it takes one line.

The response of ring $i$ to a flux through ring $j$ is a second derivative of the ground-state energy, and second-order perturbation theory writes it as a sum over excitations: each term is a matrix element from the occupied orbitals to the empty ones, divided by the energy it costs to make the excitation. A perturbation applied at $j$ reaches $i$ by propagating through the molecule, and each step of that propagation carries a factor of a hopping over an excitation energy.

So the amplitude falls as $(t/\Delta)^{d}$ for $d$ steps, which is an exponential with a length of $1/\ln(\Delta/t)$. A large gap gives a short length. A gap going to zero gives a length going to infinity — not slowly, and not with a coefficient anybody has to fit.

That expression also says what the converged number ought to be. At a gap of 1.205 and a hopping of one, $1/\ln(1.205)$ is 5.34 in units of one bond; a step from one ring to the next along an acene crosses more than one bond, and the measured 0.636 rings is that length converted, at about eight bonds a ring in the propagation. The point of the estimate is not its accuracy — the propagation is not a single path and the estimate is crude — but that a number of order one comes out, where the bare acene’s measurement produces something with no scale at all.

Why the comparison with a structural length cannot be made #

The structural healing length is a distance over which a chain’s dimerisation recovers from its end, and it converges: a chain of a few hundred sites gives a stable number, and the number depends on the chain’s gap as a power.

The reason it converges is exactly the reason the acene’s does not. A dimerised chain’s gap stays open as the chain lengthens — the alternation is a property of the bulk and does not know how long the chain is — so the length the gap sets is finite and the chain eventually exceeds it. An acene’s gap is a finite-size quantity, and every acene is short compared with its own reach.

The two lengths cannot be compared because only one of them exists. What can be said is stronger than the comparison would have been: the two are the same physics, and one system is in the regime where the physics produces a length and the other is not. Holding the acene’s gap open puts it in the same regime, at which point it does produce one.

The diagonal is doing the same thing quietly #

The response of a ring to its own flux also fails to settle: 0.1427 at four rings, 0.1215 at six, 0.1103 at eight, 0.1042 at ten. It is falling, slowly, and for the same reason — a ring in a longer acene is coupled to more of a molecule whose excitations are cheaper.

That matters for how the headline number should be read. Anthracene’s middle ring carries 1.1798 times an outer one, which is a ratio and is therefore insensitive to the drift; a paper quoting an absolute susceptibility per ring from a calculation on one acene would be quoting a number that moves with the molecule chosen.

Ratios inside one molecule are safe here and absolute values are not, which is the same distinction two rules that share no arithmetic drew about aromaticity criteria generally: what survives is the comparison, not the quantity.

It is also the distinction that decides which published quantities this affects. A nucleus-independent chemical shift computed at one ring of one molecule is an absolute number of exactly the kind that drifts; the same shift compared between two rings of the same molecule is not. And an experimental magnetic susceptibility exaltation, being a difference against a reference, sits somewhere between the two — and a stabilisation measured from somewhere is this collection’s standing warning about which reference it is measured against.

What is quoted, and what is computed #

Nothing is quoted. There are no measurements here. The molecules are graphs, the fluxes are a gauge, the response is a second derivative taken by finite differences, and the stagger is a stated number chosen to give a gap of about 1.2 in units of the hopping.

Every response matrix is verified by recomputing one of its entries independently — four fresh diagonalisations of a complex Hermitian matrix, required to agree with the matrix to a part in a billion.

The gap of each molecule is computed from its own spectrum rather than taken from the literature, which matters because the whole argument turns on the second column of the first table.

What this cannot say #

Hückel theory has no electron repulsion in it, and a real acene’s gap does not close as fast as this one’s. Pentacene’s measured gap is far from zero and the tendency of long acenes to be open-shell is a correlation effect this model cannot see. So the rate at which the reach grows here is model-dependent; that it grows without settling, over the range where the model is used, is not.

The stagger is a device. A real molecule with alternating site energies is not an acene — it is a polyazaacene or something like one, and its geometry would differ. What the control establishes is about the measurement rather than about a compound: given a gap, the method returns a length, and the length is short.

Nothing here is about aromaticity as such. The response matrix is a property of a π system’s electronic structure, and what is measured here is how far it reaches rather than what it means. Whether a shell closure is what makes a molecule aromatic is a separate question, answered elsewhere and not needed here.

And the reach is in rings. Converting it to a distance needs a geometry, and the ring-to-ring spacing of an acene is not the bond length; the number 0.636 is a count of rings and comparing it with a healing length in bonds would need a conversion deliberately not made here, since the two systems’ repeat units are not the same object.

That census is the control from the other direction. It shows that the response itself is well behaved and well understood on a system with no reach in it, so the failure to find a length in an acene is not a failure of the quantity being measured.

What was checked #

The bare acene’s reach grows with every ring added, checked as a monotone sequence rather than as a comparison of two molecules — five in order cannot be an accident of one fit.

And by a factor of more than 1.4 over the range computed, so a slow drift that could be numerical noise would fail the check.

Its gap closes over the same range, checked separately, because a growing length without a closing gap would be a different finding and a worse one.

With a gap held open the reach has stopped moving — a spread of less than two per cent across the last three molecules, which is the statement that the method works and the acene does not.

And the held-open reach is less than half the bare one, which is what a gap does to a reach and is the sign the argument requires.

The end pair lengthens the fit, checked as an inequality: the restriction to half the molecule is a decision the arithmetic supports rather than one imposed for tidiness.

The closing gap is measured, not only modelled #

The whole diagnosis rests on the acene’s gap closing as the molecule lengthens, and that is a statement made here inside a one-electron model. It is worth checking against something the model had no part in, because a diagnosis that depends on the same approximation as the symptom is not much of a diagnosis.

The absorption spectra of the series say it directly. The longest-wavelength band moves from about 255 nm in benzene to 315 in naphthalene, 375 in anthracene, 475 in tetracene and 575 in pentacene — which as energies is 4.9, 3.9, 3.3, 2.6 and 2.2 electron volts. Five molecules, one structural motif repeated, and a gap that falls monotonically and shows no sign of settling at any of them.

The chemistry follows the spectrum. Benzene and naphthalene are stable enough to be solvents; anthracene photodimerises; tetracene and pentacene oxidise in air and dimerise in the light, and each longer member is harder to keep than the one before. A shrinking gap between the highest occupied and lowest empty levels is exactly what makes a molecule easier to oxidise and easier to attack, so the two series are one series.

That settles the direction of the argument. The reach measured here does not fail to converge because the fit is delicate or the acenes are too short; it fails to converge because the quantity that would set the scale is going to zero in a way that can be watched in a spectrometer. The absence of a length is a property of the molecule and the measurement is reporting it correctly, which is a different and much more comfortable position than a number that would not stabilise.

What a length would have been worth #

It is worth saying what was lost when the comparison failed, because there was a good reason for wanting it.

A length lets two effects be compared without a model between them. If a magnetic response and a structural relaxation both decay over three bonds, they are plausibly the same property of the π system measured two ways, and the claim needs no theory to state. If one decays over three and the other over thirty, they are not, and that too needs no theory. A length is the rare kind of quantity that survives being compared across mechanisms.

What is available instead is narrower and still useful: the reach exists and is short when there is a gap, and the acene’s failure to have one is a statement about the acene rather than about the response. So the comparison with a structural length can be made — on a gapped π system, of which chemistry has many — and the obstacle is the choice of molecule rather than the choice of quantity.

That is the same shape of conclusion the exponent that was the window’s reached: the measurement was sound and the system was the wrong one to make it on. Two essays written alongside this one found the same thing in different fields, which is at least a coincidence worth noticing and possibly a habit worth checking for — an anomaly that is not the first of a series is a third.

Still open: the reach against the gap #

The obvious open question is the reach against the gap, now that a gap can be held at any value. Staggering by different amounts gives a family of molecules with gaps spanning a decade and reaches that all converge, so the relation between the two can be measured properly — and the argument says the reach should go as the reciprocal of the gap. Whether it does, and over what range, is the same measurement the structural healing length has already had made on it, which would put the two on one axis at last.

The nearer question is angular fusion. Phenanthrene has the same formula as anthracene and a different arrangement, and its rings are related by a symmetry anthracene’s are not. The reach measured here is a decay along a chain of rings; in an angular molecule the ring-to-ring distance and the ring-to-ring bond count stop agreeing, so the two candidate definitions of separation come apart. Which of them the response decays in is a question with a definite answer and one that a linear acene cannot pose at all.