An anomaly that is not the first of a series

Letting the geometry answer back adds a fifth column to a table of computed quantities: what happens to each of them when the bond lengths are allowed to respond. A bond that comes out strong gets a larger resonance integral, which makes it stronger still, and iterating that to a fixed point is the cheapest possible version of letting a molecule choose its own shape. It is the same feedback a distortion runs on, with the difference that here it is a correction rather than an instability.

It found one thing it could not account for. Naphthalene’s shared bond behaves differently from every other bond in the molecule, and the natural reading is in the phrase shared: two rings feed that bond, and no other bond in a benzene or a butadiene has two rings feeding it. If that is the mechanism, the effect should grow when three rings feed a bond, and grow again at four.

The acene series is already built here. Running it up costs nothing.

The measurement #

An acene’s cross bonds are the bonds perpendicular to its long axis: $m + 1$ of them for $m$ rings, the two at the ends belonging to one ring each and every interior one shared between two. The most interior cross bond is the one with the most molecule on either side of it, and it is the bond the reading above is about.

For each acene, the bond orders are computed at a uniform geometry, the relaxation is run to a fixed point, and the difference is taken. Nothing else changes: the same iteration, the same coupling, the same convergence criterion of a part in $10^{12}$.

Three rings do a third of what two do. Four rings do nothing at all. Five rings do the opposite.

The reading is refused. Whatever makes naphthalene’s shared bond anomalous, it is not that two rings feed it — because three rings feed anthracene’s central cross bond harder and get less. A reading refuted by its own next case is the shape two rules that share no arithmetic warns about: an explanation fitted to one molecule is a description of that molecule.

The ends are the control #

A sequence that falls and reverses could be a computation losing its footing on larger molecules, so the terminal cross bond is worth reading beside it.

The end weakens at every length, in the same direction, by an amount that is settling towards a limit — the steps are 0.0120, 0.0056, 0.0025, halving each time. The end of a long acene stops knowing how long the acene is, which is what a converging quantity looks like and what the middle conspicuously does not do.

So the reversal in the middle is not the arithmetic failing. The two halves of the same calculation behave differently, and only one of them behaves the way a size effect should.

Where the sign change sits, and what it belongs to #

A result that changes sign inside a model’s range invites the obvious suspicion, so the coupling was varied.

Every series falls monotonically and every one crosses zero. Where it crosses moves with the coupling; that it crosses does not. Separating those two is the discipline a fitted exponent needs and for the same reason: a number computed from a model at a stated parameter is not the same kind of statement as a sign that survives every parameter.

That is the right way to hold the result. The crossing point is a number this model produces and a stronger model would produce a different one; the shape — largest at two rings, falling, reversing — is what three different couplings agree about, and it is the shape that refuses the reading.

What is actually happening to the middle #

The middle cross bond’s bond order is falling with molecular size on its own account: 0.5182, 0.4849, 0.4582, 0.4506, heading for the value an infinite acene’s central cross bond has. Two things are competing in the relaxation, and they scale differently.

The cross bond is weaker than average, and the relaxation weakens weak bonds — a bond with a small order gets a smaller integral and becomes smaller still. That pushes the move negative and it gets stronger as the cross bond gets weaker, which is to say as the molecule gets longer.

The cross bond’s neighbours are strengthening, and a bond surrounded by strengthening bonds gains from them through the eigenvectors. In naphthalene the neighbours are the two terminal cross bonds and the four ring bonds, all of them close by; in pentacene the central cross bond’s neighbours are themselves interior and are weakening too.

The first effect grows with length and the second dies away. The sum crosses zero somewhere, and that is all the crossing is. Nothing in it is about two rings sharing a bond.

The spread does the same thing, at the same place #

Relaxation also widens the spread of bond orders — a strong bond gets stronger and a weak one weaker, so the molecule’s bonds become less alike. That is what it does from three rings onwards:

Naphthalene’s spread narrows. Every longer acene’s widens, and by an amount that grows.

Two independent quantities behaving anomalously in the same molecule and normally in every larger one is the strongest form the finding takes. It says naphthalene is a special case of this model rather than the smallest member of a family — and the reason it is special is that it is the only acene whose every cross bond is a terminal one or shared by two rings, with none that is interior to an interior. A bond order predicts a bond length only as far as the bond order is a property of something, and here it is partly a property of where in the molecule the bond sits.

What this does to the question of what a bond order belongs to #

One question runs through four earlier calculations: what a computed number is a property of. Four frame changes left the bond orders alone and the fifth — relaxation — moved them, which is why it is the interesting column.

This adds a caution to that. The fifth column’s effect is not monotone in molecular size, so a comparison of two molecules under relaxation is not a comparison of the same thing at two sizes. Two acenes can differ in the direction their shared bonds move, and a reader comparing published relaxed bond orders for naphthalene and tetracene would be comparing numbers with opposite signs of correction.

The standing difficulty remains the one a stabilisation is measured from somewhere named: a relaxed molecule ought to be compared against a relaxed reference, and an isolated double bond has nothing to relax. Nothing here fixes that, and the reversal makes it harder — the correction the reference would need is not even the same sign for every molecule.

What is quoted, and what is computed #

Nothing is quoted. No measurement appears here. The acenes are graphs whose carbon and bond counts are checked, the coupling is a stated parameter varied over a factor of three, and every bond order comes from the eigenvectors of a matrix.

The relaxation is a fixed point, and its residual is checked below $10^{-12}$ for every molecule — a bond order that depended on how long the iteration ran would make every number here a report about the iteration rather than the molecule.

The three couplings are not a fit. λ is the strength of the feedback between a bond’s order and its resonance integral, it has no measured value in this model, and varying it is what turns a single sign change into a statement about which parts of the result are the model’s.

What this cannot say #

There is no electron repulsion, which is the boundary every one-electron argument works inside. Long acenes are the standard case where a one-electron picture fails: their gaps close, they tend towards open-shell ground states, and a relaxation computed without correlation is describing something a real pentacene is not. The reversal is a property of this model at the sizes where the model is used.

There is no σ frame. The relaxation feeds back only through π bond orders, and a real geometry is set by both. A σ frame that resists compression would damp everything here by an amount that is not computed.

And the cross bonds are not the only bonds. The argument follows the cross bond most interior to the molecule because that is the bond the reading was about; the ring bonds along the long edges do their own thing, and nothing here asks what.

What was checked #

One ring cannot relax. Benzene’s bonds are equal by symmetry and its move must be exactly zero — the check that the scheme reports a chemical effect rather than a symmetry violation.

The move on the shared bond falls with every ring added, at every coupling tried, checked as a monotone sequence for each λ separately rather than for an average of them.

And it changes sign inside the range computed, at every coupling.

Where it crosses depends on the coupling, asserted by requiring the three crossings not to be the same — because three identical crossings would mean the crossing point was a property of the acene, which is a claim this model is not entitled to make.

While the terminal bond’s move settles, with its last step less than half its first — the control that makes the middle’s behaviour a finding rather than a numerical drift.

And every relaxation reached a fixed point below a residual of $10^{-12}$.

What the reversal is not #

Three readings are available and two of them are wrong, so it is worth closing them off.

It is not a numerical artefact. Every relaxation reaches a fixed point with a residual below $10^{-12}$, the sequence is monotone at three couplings, and the terminal cross bond — computed in the same run — behaves exactly as a size effect should. A defect in the iteration would not be so selective.

It is not a statement that the relaxation is a bad model. The relaxation is a one-line feedback between a bond order and a resonance integral, it is crude, and it is the crudest version of something real: bonds that are computed strong do shorten, and shortening does strengthen them. What the reversal says is that this feedback’s effect on one bond is not monotone in the molecule’s size, which is a property of the feedback rather than a failure of it.

And it is not an argument against reporting relaxed bond orders. It is an argument against comparing two molecules’ relaxed bond orders as though the correction were a common baseline. The correction has a sign that depends on the molecule, so two numbers computed the same way are not on the same scale — which is the difficulty a stabilisation measured from somewhere has always had, arriving in a quantity that seemed immune to it.

What the reversal is is a sign change in a computed quantity that no argument anticipated, located to within one ring, and stable against a threefold change in the only parameter the model has. That is worth having on its own, and it is worth having whether or not the mechanism proposed above for it turns out to be right.

What the numbers do after they cross #

The sequence is usually read as a decay that happens to pass through zero: $+0.02036$, $+0.00678$, $-0.00029$, $-0.00603$. Read the magnitudes instead and it is not a decay at all.

They run $0.020$, $0.0068$, $0.00029$, $0.0060$ — falling by a factor of three, then by a factor of twenty-three, then growing by a factor of twenty. A quantity that decays does not turn round, and a quantity that crosses zero on the way to a limit does not accelerate away from it afterwards.

So the interior cross bond’s move looks like a difference between two contributions rather than one contribution dying. One of them settles, exactly as the terminal cross bond’s does, and dominates in the short molecules; the other grows with length and takes over. The crossing is where they are equal, and it has no significance of its own — a difference of two things crossing zero says only that they were briefly the same size.

There is an obvious candidate for the growing term and it is available from elsewhere in this collection. An acene’s gap falls as the molecule lengthens, and a response to any perturbation carries that gap in a denominator: the smaller the separation between the highest occupied and lowest empty levels, the more the occupied orbitals rearrange when the matrix is changed. A relaxation is exactly such a rearrangement, so its size should grow as the gap closes, without limit, in a series whose gap goes to zero.

That reading makes a prediction the four points here do not settle: the magnitude should go on growing at six rings, seven and beyond, and it should grow at a rate that tracks the reciprocal of the gap rather than saturating. Two more members of the series would test it, and a purely local mechanism would fail the test outright — a local effect cannot know how long the molecule is.

It also says what the crossing is not. It is not a length at which the mechanism changes, and there is nothing special about four rings. It is the place where a settling term and a growing one happen to cancel, and its position depends on the coupling — which is why the three couplings in the figure cross at three different places.

The quantity that does behave #

It would be easy to leave with the impression that nothing about the relaxation is dependable, and that is not the case: one of the two quantities followed here does exactly what a size effect should.

The terminal cross bond’s move settles — −0.0247, −0.0367, −0.0422, −0.0447 — with steps that halve, which is the signature of a quantity approaching a limit. That limit is what the end of an infinitely long acene does under relaxation, and four molecules are enough to see it being approached.

So the fifth column is not unreliable. It is local: the correction to a bond depends on how much molecule is on either side of it, and a bond near an end has a fixed and converging amount, while a bond in the middle has an amount that grows. A quantity computed at an end transfers between molecules of different lengths and a quantity computed in the middle does not.

That is a more useful statement than either half alone, and it is the one worth carrying into any comparison of relaxed quantities between molecules: ask where in the molecule the bond is, and how much molecule there is on either side of it. For a small molecule the answer is near an end, always, which is why relaxation looks well behaved in small molecules.

Still open: how far a heteroatom reaches along an acene #

The obvious open question is the other direction: what a change at one site does to the bonds everywhere else. A heteroatom in one ring of an acene is a perturbation with a location, and the relaxation carries it along the molecule — so the question is how far, which is a decay length of exactly the kind a magnetic reach turned out not to have in an acene. Whether a structural perturbation reaches further in a molecule whose gap is closing is the same question asked of a different response, and the two answers can be put side by side.

The nearer question is the angular series. Phenanthrene has anthracene’s formula and a bent frame, so its rings share bonds differently and its most interior cross bond is interior in a different sense. If the reversal is about how much molecule sits on either side of a bond, phenanthrene’s should sit between anthracene’s and naphthalene’s; if it is about ring count alone, it should sit with anthracene’s. Two arrangements of the same atoms disagreeing about the sign of a correction is the cleanest test the reading has left.