The frame that was allowed to relax

Which numbers carry a frame built a table. Nine quantities, four changes of frame — shifting the energy zero, scaling the resonance integral, changing the reference against which a stabilisation is measured, changing the electron count — and a column for each saying which quantities survive.

Three quantities came out immune to everything. The dimensionless ratio was immune to both exact symmetries and infinitely sensitive to the reference. And a bond order sat quietly in the immune column, unmoved by anything.

That table ends by naming why. Every quantity here was computed at one geometry with every resonance integral equal, and a real conjugated system alternates. Letting the geometry relax adds a fifth column to the table, and it is the one that would move a bond order — since a bond order is a property of the eigenvectors and the eigenvectors depend on the matrix, which depends on the geometry.

That is the column, and it does what the sentence said.

Why the first four could not touch it #

The reason the earlier changes left bond orders alone is worth stating precisely, because it is what makes the fifth different in kind.

Shifting the energy zero adds a multiple of the identity to the matrix. Scaling β multiplies it by a constant. Neither operation changes an eigenvector — the first shifts every eigenvalue by the same amount and the second scales them all — and a bond order is built entirely out of eigenvectors, which is how it was defined here in the first place:

$$p_{ij} = \sum_{m} n_m\, c_{mi}\, c_{mj}.$$

Changing the reference does not touch the molecule at all; it changes what is subtracted afterwards. Changing the electron count changes which eigenvectors are summed over, which does move a bond order — and the frame table records that it does.

So the four changes tested were three that act on the scale of the matrix and one that acts on the filling. None of them changes the matrix’s structure, and the structure is what an eigenvector is a property of.

A matrix that depends on its own eigenvectors #

The fifth change does. A bond with more π bonding in it is shorter, and a shorter bond has a larger resonance integral, so the matrix depends on the answer it produces.

The relation used is the standard linear one,

$$k_{ij} = 1 + \lambda\,(p_{ij} - \bar{p}),$$

with $\bar{p}$ the value the uniform calculation gave, so that λ = 0 is exactly the ordinary fixed-β calculation and nothing moves for a system whose bond orders are all equal. It is iterated to a fixed point, and the residual is reported rather than assumed: every system here converges to better than 10⁻¹².

λ is a parameter and nothing here pretends otherwise. What is being asked is not how large the effect is but which quantities feel it at all, and that question has an answer that does not depend on λ.

Benzene does not move, and that is the check #

The first thing to test is a system that must not move, because a scheme that broke a symmetry would be reporting its own arithmetic.

Benzene’s six bond orders are equal by symmetry: the molecule’s own point group relates every bond to every other, so the eigenvectors give six identical values and the feedback gives six identical resonance integrals. There is nothing for the iteration to do. It converges in one step, the spread stays below 10⁻⁹, and the π energy does not move by 10⁻⁹.

That is not a small check. A self-consistency loop with a slightly asymmetric starting point, or with an eigensolver that ordered degenerate eigenvectors inconsistently, would drift — and drifting would look exactly like a physical result, since bond alternation in benzene is a claim the literature has argued about for decades. The hexagon is the frame’s doing is where that argument is settled, and this calculation has to agree with it.

The chains alternate more #

Butadiene and hexatriene do the expected thing and it is worth having the numbers.

Butadiene’s spread runs from 0.447214 to 0.562408 — the strong bonds get stronger and the weak one weaker, which is the feedback amplifying what was already there. Hexatriene runs from 0.387685 to 0.509002, with its three distinct bonds going 0.8711, 0.4834, 0.7849 to 0.9127, 0.4037, 0.8469.

The gaps open with them: butadiene’s from 1.23607 to 1.4605, hexatriene’s from 0.89008 to 1.11651. That is the same arithmetic a chain cannot stay even rests on — alternation opens a gap at the Fermi level and the gap is what pays for the alternation — arriving in a molecule rather than in a solid.

The π binding rises in every case — 4.472136 to 4.618772 for butadiene, 6.987918 to 7.181048 for hexatriene — which it must, because the iteration is finding a lower energy in a larger space of matrices. A relaxation that raised the energy would mean the feedback had been wired backwards.

Naphthalene refuses #

The fourth system does something the first three do not, and it is the result.

Naphthalene’s bond-order spread narrows: 0.206330 to 0.205622. Every bond moves — the largest move is 0.024676, which is far above any convergence tolerance — and the spread comes down.

The reason is which bond is the weakest. In the uniform calculation naphthalene’s eleven bonds run 0.5547, 0.7246, 0.6032, 0.7246, 0.5547 around each ring with 0.5182 for the bond the two rings share. That shared bond is the weakest, and relaxation raises it: 0.5182 to 0.5386.

A shared bond is fed by two rings and is not an isolated bond with a feedback loop on it. Weakening it would cost both rings, and the arithmetic finds that the two rings between them push it up rather than down — while the outer bonds alternate more, which is why the maximum rises from 0.7246 to 0.7442 at the same time as the minimum rises from 0.5182 to 0.5386.

That is worth putting in the same sentence as the chains. The feedback amplifies existing differences in a chain and does not in a fused system, and the difference is a matter of connectivity rather than of the parameter. A rule inferred from polyenes — that relaxation makes bond alternation worse — is wrong for the first non-polyene it meets.

The fifth column, filled in #

Set against the frame table, the new column says this.

The three quantities immune to everything — a bond order among them — are no longer three. A bond order moves under relaxation and moves in a direction that depends on the connectivity, so its immunity was an immunity to four particular changes rather than a robustness.

The gap, which survived the shift and not the scaling, moves under relaxation and opens in every case. The π energy, which was immune to the shift and sensitive to the scaling, rises. The delocalisation energy against a fixed reference inherits both.

The one thing that does not move is the thing symmetry fixes. Benzene’s bond orders are equal in every column of the table, because they are equal by a theorem rather than by a calculation — which is a distinction worth keeping: symmetry conclusions are exact and everything else is approximate and says so.

What a self-consistent bond order is a property of #

There is a general shape here worth separating from the four molecules.

An ordinary Hückel bond order is a property of a graph: the matrix has ones on the edges and zeros elsewhere, so two molecules with the same connectivity have the same bond orders whatever their atoms are. That is why conjugation and its limits could be argued about with adjacency matrices and nothing else.

A relaxed bond order is a property of a fixed point, and a fixed point is a property of the graph and of the feedback strength. Two molecules with the same connectivity still get the same answer, so the graph is still doing most of the work — but the answer is no longer a linear function of anything, and the superposition arguments that make Hückel theory pleasant stop applying.

The practical consequence is small and worth knowing. A relaxed bond order cannot be predicted by inspection, and the alternation pattern of a long polyene under relaxation is not the uniform pattern with a constant added: the end bonds move most and the middle bonds least, so the pattern has a shape as well as an amplitude.

There is another way to make a Hückel matrix unequal, and it is worth distinguishing: substituting an atom changes the diagonal while relaxation changes the off-diagonal. The first introduces a fitted parameter for the substituted atom; the second introduces one for how strongly a bond’s integral responds to its own order, and neither is determined by the graph.

What relaxation does to every quoted delocalisation energy #

The fifth column moves bond orders, which is what it was introduced to do. It also moves something the fixed-β calculations treat as settled, and the direction is the same for every molecule — which makes it a systematic correction rather than a complication.

A delocalisation energy is the molecule’s π energy less a reference built from isolated double bonds. In a calculation with every resonance integral fixed at $\beta$, both sides are measured in the same $\beta$ and the answer is a pure number. Once the integrals depend on the bond orders they produce, the two sides stop sharing a $\beta$.

An isolated double bond has a bond order of one — the largest any bond can have. Benzene’s have two thirds. So in a self-consistent scheme the reference’s bonds are the stronger ones: each of them adjusts to a shorter length and a larger resonance integral than any bond in the ring it is being compared against.

The reference therefore gains from relaxation and the delocalised molecule gains less, so every delocalisation energy computed this way is smaller than its fixed-$\beta$ counterpart, at every ring size, for aromatic and antiaromatic systems alike. Benzene’s 2β is an upper bound rather than a value.

That is not a defect in the arithmetic; it is a real physical term with a name in the older literature. A delocalised ring pays for its uniformity: its double bonds are longer than they would like to be and its single bonds shorter, and the σ frame charges for both. The measured hydrogenation cycle contains that cost already, which is part of why the calorimetric numbers sit below the naive computed ones.

It is worth separating this from the other correction of the same family. The vertical resonance energy holds a molecule at its delocalised geometry and constrains its wavefunction, which is a question about electronic structure and needs a many-electron calculation. The correction here holds the wavefunction alone and lets the geometry answer back on both sides of the subtraction. They are different quantities, they point the same way, and neither is the other.

Two consequences follow for how fixed-β results should be read.

The ordering probably survives. Every ring’s reference is built the same way, and the reference’s advantage is roughly the same in each case, so a comparison between two rings of the same size is protected by the same cancellation every such comparison relies on.

The absolute numbers do not. Benzene’s 2β, cyclobutadiene’s exact zero, cyclooctatetraene’s 1.657β — all three are computed with the reference held at a strength no isolated double bond would choose, and all three shrink when it is released. The exact zero is the interesting one: it is exact only in the fixed-$\beta$ frame, and in the relaxed frame cyclobutadiene’s ring comes out worse than two ethenes rather than equal to them.

Which is the fifth column doing to a stabilisation what the frame table did to the energy scale: not overturning a result, but naming the frame it was computed in.

What this cannot say #

λ is not measured. A realistic value is fitted to bond lengths in polyenes and 0.4 is used here because it is of that order. Every magnitude above scales with it; every sign does not, and the signs are the argument.

There is no elastic term. A real relaxation pays for shortening a bond, and the balance between the π gain and the σ cost is what decides how far a real molecule alternates. The critical force constant is where that balance is computed, and putting it into this loop would turn a fixed point into a minimisation.

One geometry per molecule, still. The relaxation changes the resonance integrals and never the coordinates, so no length is ever computed. Turning a bond order into a length needs a relation between them, and that relation is a second parameter not introduced here.

No repulsion, still. A Hückel matrix has no electron–electron term, and the received view is that the alternation of long polyenes is set by correlation as much as by the σ framework.

And a fixed point is not a minimum. The iteration converges to a matrix consistent with its own bond orders, which is a stationary condition rather than a variational one. A different starting point could in principle find a different fixed point, and for the systems here it does not.

What was checked #

The iteration reaches a fixed point with a residual below 10⁻¹², for every system — because a relaxation that had not converged would be a number that depends on how long it ran.

λ = 0 changes nothing at all, exactly, bond by bond, for two systems — the statement that this is the ordinary calculation with one term switched off.

Benzene’s bonds stay equal to better than 10⁻⁹ and its π energy does not move, which is the symmetry check.

Every other system moves a bond order by more than a thousandth, opens its gap and raises its π binding.

Butadiene and hexatriene spread further apart, which is the feedback amplifying what is there.

And the refusal: naphthalene’s spread narrows and its weakest bond gets stronger, so the rule the chains suggest fails on the first fused system it is asked about.

Which quantity is worth quoting #

Four arguments have now asked what survives a change of frame, and it is worth saying what the answer amounts to for somebody who has to quote a number.

Quote an energy difference between two states of one molecule. A gap, an excitation, an ionisation: those survive the shift, survive the reference entirely because there is none, and move under scaling and relaxation by amounts that are the model’s honest uncertainty rather than an artefact.

Do not quote a stabilisation without its reference. The reference is half the number and it is a construction rather than a molecule.

Quote a bond order with the caveat relaxation supplies. It is a good quantity — dimensionless, computed from eigenvectors, immune to two exact symmetries — and it depends on a geometry that the calculation quoting it usually holds fixed.

And quote a symmetry-fixed quantity without hedging. Benzene’s six equal bond orders survive every column of the table because a theorem puts them there, and such conclusions may be stated plainly.

Still open: a shared reference, and naphthalene along the acenes #

Three arguments now turn on one question — what a computed number is a property of — and the fifth column is the last of the frame changes that is cheap. The next one is not: it is the reference, and a stabilisation is measured from somewhere ended by asking how to construct a reference that is the same reference for two different molecules. Relaxation makes that harder rather than easier, because a relaxed molecule ought to be compared against a relaxed reference and an isolated double bond has nothing to relax.

The nearer question is naphthalene’s. If the shared bond behaves differently because two rings feed it, then the size of the effect should scale with how many rings share a bond — and the acene series is straightforward to build. Running the relaxation up anthracene and tetracene would say whether the central bonds go on strengthening, and whether the spread goes on narrowing, which would turn one anomaly into a trend or expose it as an accident of naphthalene’s particular topology. Six electrons in a ring that is not all carbon is the other direction the same question runs in: what a change to one site does to the bonds everywhere else.