The current does not divide

Two rules for aromaticity agree about which rings are aromatic and share no arithmetic: a count of electrons, and a magnetic response computed by threading a flux through a ring and watching the energy. For a fused system the response is usually computed from the areas of the rings, which is the standard construction — and the construction assumes something nobody checks.

“The natural next question is whether the current divides between them the way a current in a circuit divides between two resistors — a question with a definite answer, since the derivative of the energy with respect to each ring’s flux separately is computable and the two need not be equal.”

Computed, the answer is no twice over. The rings are not independent, and even under a uniform field they do not carry equal currents.

Giving each ring its own flux #

Threading a flux through one ring of an isolated ring is easy: every bond picks up a phase factor and the total round the loop is the flux. A fused system needs a gauge — an assignment of phases to bonds such that each ring encloses the flux it was given and no other loop encloses anything unintended.

A linear acene admits the simplest one there is. Its rings are strung along a line and adjacent rings share a vertical bond, so putting a phase factor on those shared bonds alone gives ring $i$ a flux equal to the difference between the phases of the two bonds it sits between. Setting each shared bond’s phase to the partial sum of the wanted fluxes makes every ring carry exactly its own.

The matrix is then complex Hermitian, which is why this is computable at all: such a matrix has a standard real embedding, and it diagonalises without a complex eigensolver.

Two second derivatives of the total energy — with respect to each pair of fluxes — give the response matrix, and everything below is that matrix.

Why a second derivative is the right object is worth one sentence. The current in a loop is minus the first derivative of the energy with respect to the flux through it, and at zero flux that derivative is zero by time-reversal symmetry — so the response to a small flux is the second derivative, and a matrix of them is the response to a flux anywhere. It is the same construction used for a single ring, with one index instead of none.

What it looks like #

For tetracene, the diagonal runs 0.14082, 0.14266, 0.14266, 0.14082 and the largest off-diagonal entry is 0.06390 — a neighbouring ring’s flux drives nearly half the current a ring’s own flux does.

That settles the first half of the question. A fused system is not a set of loops. The current in one ring depends on the flux through the others, and a picture in which each ring is a resistor with its own conductance has no term for that.

It is worth saying why the coupling is there, because the reason is not a leakage of current between rings. The energy is a single function of all the fluxes, and its mixed second derivative is non-zero because the orbitals are delocalised over the whole molecule — a change of flux anywhere alters every level. What the off-diagonal entry measures is not a current flowing from one loop to another but the fact that the two loops are not separate systems.

And the current is not shared out evenly #

A uniform magnetic field gives every ring the same flux, because every ring has the same area. What each ring then carries is the sum of its row of the matrix.

Naphthalene’s two rings are equal, and they have to be: the molecule has a symmetry exchanging them, so any property of one is a property of the other — which is what a point group settles without any calculation, applied to a response rather than to a dipole. That is a check on the calculation rather than a result — the gauge assigns phases by partial sums from one end and is therefore not symmetric, so the symmetry of the answer is evidence that the gauge is a gauge.

Anthracene’s are not equal, and nothing forbids that: the middle ring is related to the outer ones by no symmetry of the molecule. It carries eighteen per cent more, and in tetracene the inner rings carry twenty-two per cent more, so the difference grows with length.

That is a computed version of something well known qualitatively — the inner rings of an acene are the ones a magnetic criterion calls most aromatic — and it is here without any aromaticity criterion in it at all, from two derivatives of an energy.

It also sits awkwardly beside the other rule. A count of electrons has nothing to say about which ring of anthracene is which: all three have six carbons and the molecule has fourteen π electrons that belong to no ring in particular. Aromaticity as a shell closure is a statement about a molecule, and the magnetic criterion is a statement about a ring — so the two rules that agree about single rings are not even about the same objects once the rings are fused.

Fusing adds rather than divides #

The third finding is about the total, and it is the one that most directly refuses the circuit picture.

If a fixed response were being shared out, four fused rings would respond as much as one ring and no more. If the rings were independent loops, four of them would respond four times as much as one.

Neither. Four fused rings respond at 1.05482 against 0.88889 for four separate benzenes: an enhancement of 1.187, and it grows monotonically with the number of rings — 1.093 for two, 1.149 for three, 1.187 for four.

So fusion makes the whole molecule a better conductor of this response than its parts, which is what a delocalised system should do and is not what any decomposition into rings can produce.

What it means for the area rule #

Computing a fused system’s response from the areas of its rings is standard practice, and this essay says what that assumes.

It assumes the response is diagonal, so that each ring can be given its own conductance. The off-diagonal entries say it is not.

And it assumes the rings are equivalent, so that one conductance does for all of them. Anthracene says they are not.

The area rule is not therefore useless — it gets the total roughly right, because the enhancement is under twenty per cent for the acenes here. What it cannot do is say which ring is carrying what, and a ring-by-ring magnetic criterion is exactly the thing a modern aromaticity argument is usually built on.

That is the same shape of caution as a stabilisation measured from somewhere: a quantity computed for a whole molecule and then divided up between its parts is a quantity whose division is a convention, and the division is where the argument usually happens.

What a fused system is, in this model #

Three findings, and they add up to one description.

The rings are not separate. Their responses are coupled by nearly half the strength of their own, which is what a set of orbitals spread over the whole molecule should give.

They are not equivalent. Identical hexagons in identical surroundings, except that the middle one has two neighbours and the outer ones have one, and eighteen per cent of the response follows from that difference alone.

And the whole is more than its parts. Fusion raises the total, monotonically, so nothing is being divided.

Put together: a fused ring system responds as one object with an internal structure, not as a network of loops. That is the same conclusion this collection reaches whenever it looks at a delocalised system closely — a hexagon is the frame’s doing rather than the π system’s, a stabilisation is measured from somewhere rather than belonging to a bond — and the pattern is worth naming: a quantity that belongs to a whole molecule can be divided between its parts in many ways, and every division is a convention until something forces one.

Here something does force one, which is unusual and is why this calculation is worth having. The flux through a ring is a physically meaningful variable — it is what a field times an area gives — so the derivative with respect to it is a physically meaningful quantity, and the division it produces is not a convention. What the division shows is that the parts are unequal.

What is quoted, and what is computed #

Nothing is quoted. There is no measurement in this essay and no material property. The acenes, their connectivity and their filling are the model’s; every level, energy, derivative and ratio is computed.

The single-ring response is computed twice: once by threading a flux round one ring, and once by the matrix construction here at one ring. They must agree — 2/9 exactly — and that agreement is what says the gauge and the finite difference are doing what they claim.

The off-diagonal terms are one geometric sequence #

The response matrix is not diagonal, which is the essay’s finding. What shape it has instead is a further question, and the four-ring acene answers it with three numbers that were already in hand: a neighbour term of 0.0639, a next-but-one of 0.0284, and a far term of 0.0130.

Take the ratios. $0.0284/0.0639 = 0.444$ and $0.0130/0.0284 = 0.458$. Two ratios agreeing to three per cent across a coupling that has fallen by a factor of five is a geometric decay, and it is worth confirming what it is not: a power law in the separation would need exponents of 1.17 and 1.93 to produce the same two ratios, which is not one power law but two.

So the coupling between two rings falls by a constant factor of about $0.45$ for each ring between them, or

$$R_{ij} \approx R_0\,\rho^{\,|i-j|}, \qquad \rho \approx 0.45,$$

and the essay’s own summary of the diagonal-to-neighbour comparison — nearly a half — is that same ratio at $|i-j| = 1$. The whole matrix, diagonal included, is one sequence.

The decay constant is $-\ln 0.45 = 0.80$ per ring, so the reach is $1.25$ rings: about three ångström along the chain of an acene, and less than the width of the ring it starts in. Two rings with one ring between them are coupled at a fifth of the neighbour value, and three rings apart at a tenth.

Two things follow, one practical and one about what the finding means.

The matrix is effectively tridiagonal. Dropping every term beyond the nearest neighbour loses a twentieth of the total coupling, so a fused system’s magnetic response can be assembled from rings and their immediate neighbours and nothing else. That is a much weaker statement than the circuit picture this essay set out to test — the rings are still not independent, and the neighbour term is half the diagonal rather than a correction — but it is a bounded failure rather than an unbounded one, and it says the enhancement of an inner ring is a local effect with a two-neighbour cause rather than a property of the whole chain.

And the reach is short enough to be a length rather than a shape. A coupling that decays geometrically has a characteristic distance in it, which the fused geometry alone could not have supplied; the areas are all equal and the arrangement is a straight line, so nothing in the construction sets a scale. The scale comes from the π system, and a number of about one and a quarter rings is a small enough reach to be compared against the distance over which any other perturbation of the same π system dies away. Whether the two lengths agree is the question worth asking next, and it now has a number on one side of it.

What this cannot say #

One geometry per ring. Every ring is a regular hexagon of the same size, so the differences above are entirely electronic. Real acenes are not quite that, and their bond lengths alternate in a pattern a relaxation would find. A real acene’s rings differ slightly in geometry, which would add an effect this cannot separate from the one it measures.

Linear acenes only. Angular fusion — phenanthrene against anthracene — puts the rings in a different relationship, and the off-diagonal structure would be different. Whether the inner-ring enhancement survives is not something this answers. The gauge used here is also specific to a linear chain of rings: a general fused system needs a spanning tree and one phase per independent cycle, which is more construction for the same idea.

A one-electron π system. No repulsion, no σ frame, and a resonance integral that does not respond to a bond length — so nothing here can see the competition between the π system and the frame that decides a real acene’s geometry.

No σ frame and no local currents. The response is a derivative of a π energy with respect to a flux, which is a property of the whole ring and not a current density anywhere. Where the current actually flows — round the perimeter, across the shared bonds, or both — is a question about a density that this model has no representation of.

And there is no magnetic field in the model. A flux is a phase factor on a bond, and converting it to a field needs an area, which is the assumption made explicit for single rings and is inherited unchanged.

What was checked #

A single benzene ring’s response is 2/9, which the single-ring flux calculation already gives — two constructions, one number, and the only place in this essay where an independent answer exists to check against.

The response matrix is symmetric, entry by entry, because it is a second derivative of one function. A gauge that had gone wrong would break this before anything else.

And it is not diagonal: the largest off-diagonal is more than a tenth of the smallest diagonal, at every acene length. Measured: nearly a half.

The first and last rings respond identically, as the molecule’s own symmetry requires — and the gauge is not symmetric, so this is a check rather than a tautology.

The inner ring of an acene carries more current than an outer one, above five per cent, at every length from three rings up.

And the difference grows with the length of the acene, which a coincidence of one molecule would not do.

A fused acene responds more than the same number of separate rings would, at every length, which is the statement that fusion adds rather than divides.

And the refusal is a flux list of the wrong length — one flux per ring, or the gauge is not the gauge it says it is, and a padded list would silently thread a zero through a ring that was meant to carry something. That is the failure mode a gauge construction has: it produces a well-formed matrix of plausible numbers about a molecule with the wrong fluxes in it, and no downstream check would notice.

Still open: angular fusion, and the reach of a ring current #

The obvious open question is angular fusion. Phenanthrene has the same formula as anthracene and a different arrangement, and its rings are related by a symmetry that anthracene’s are not — so the response matrix has a different structure and the inner-ring enhancement has somewhere else to go. Comparing the two would separate what the enhancement is a property of: the number of neighbours a ring has, or its position along a chain.

The nearer question is about the off-diagonal entries themselves. They fall with distance — in a four-ring acene the neighbour term is 0.0639 and the next-but-one is 0.0284 and the far one 0.0130 — which looks like a decay with a length in it. Fitting that decay across longer acenes would give a number that is the magnetic analogue of the healing length measured for a structural perturbation, and the interesting question is whether the two are the same length. If they are, a ring current’s reach and a distortion’s reach are one property of the π system measured two ways.