An end effect with two signs

Which of two separations a fused ring system’s response decays against — the number of fusions between two rings, or the distance between their centres — has no answer within a class: neither accounts for the scatter. Pairs the same number of fusions apart differ from one another by a third, and the scatter is not noise: every number here is a second derivative of an energy computed exactly.

It closed with a candidate. If the scatter is a matter of how much molecule sits on either side of a pair, then it should collapse against a single variable — the number of rings outboard, or the pair’s distance from the nearer end — and the collapse would turn the scatter into a second term rather than a nuisance.

It does not collapse, and the reason is sharper than a failure to fit.

The end effect has a sign, and it is not one sign #

Divide each pair’s response by the response of the pair at the same separation sitting deepest in the molecule, and the question becomes one number per separation: is a pair at an end stronger or weaker than an equivalent pair in the middle?

In a straight chain of nine rings it is weaker, at every separation, and increasingly so: 0.9754, 0.7891, 0.6935, 0.6579 as the separation runs from one fusion to four.

In a zigzag chain of nine rings it is stronger, at every separation: 1.2771, 1.2767, 1.2131, 1.1610.

Those are the same measurement on two molecules with the same number of rings, the same number of carbons, and the same graph distances between every pair of rings. The current does not divide equally between equal rings established the inequality; this is the same inequality asked of a pair rather than of a ring. The only difference is the direction each fusion takes, and it reverses the sign of the end’s effect.

A single variable — depth, outboard count, distance from the nearer end, anything of that shape — takes one value for a pair and therefore predicts one sign of correction. Two shapes with opposite signs cannot both be described by it. The hoped-for collapse is not available, and the obstruction is not that the variable is the wrong one; it is that no variable of that kind can work.

Seen as a fan the structure is plain: a decay against the separation alone would be a single line, and instead each separation carries a spread that widens from a factor of 1.073 at one fusion to 1.520 at four. Within one molecule the ordering is clean — deeper is stronger, monotonically — which is exactly what makes the straight chain look like a case for an end correction. It is the second molecule that refuses it.

A third behaviour, not an intermediate one #

The two chains above have every fusion alike, all linear or all angular. A chain with four linear fusions, one angular ring and four more linear fusions — two straight arms meeting at an angle — mixes the two kinds, and it does something neither of them does. It is still symmetric end for end: a mirror through its middle ring exchanges the arms.

Its ratios are 2.644, 2.984, 2.581 and 1.000 at one fusion’s separation; 2.923, 3.029, 1.022, 1.000 at two. Almost every pair is two or three times the deepest pair’s response, which reads at first as an enormous end effect and is not one — because the numbers are ratios to the deepest pair, and what they are saying is that the deepest pairs are the weak ones.

The anomaly is in the middle of the molecule, at the angular ring where the two arms meet. An end-based variable has no way to express that: the pairs it calls most typical are the ones behaving least typically.

And the sign flips inside one molecule. At four fusions’ separation the end ratio is 0.748 — below one, like a straight chain — while at one, two and three fusions it is above two. So even restricting to a single shape does not give a correction with a fixed sign.

There is a reading of the mixed chain that saves something from the wreck, and it is worth stating because it is what any further account would have to build on. An angular ring weakens the response of any pair that spans it: the deepest pairs are exactly the ones that straddle the join, because the join is in the middle, and counted across twenty-seven chains each angular ring between two rings multiplies their response by about 0.60. That explains why the deepest pairs are weak without rescuing the collapse: the variable it needs is which angular rings lie between and under the pair, which is a property of the molecule’s construction and not of the pair’s distance from an end.

Whether dropping the end pairs helps #

The practical form of the hope is simpler than a collapse: if the end pairs are the problem, throw them away and fit the rest.

On the two uniform chains it works. The straight chain’s coefficient of determination rises from 0.95129 over all twenty-six pairs to 0.98986 over the eighteen interior ones; the zigzag’s from 0.98954 to 0.99783. Both are what an end effect looks like when it is removed.

On the mixed chain it fails: 0.84684 over all pairs, 0.84376 over the interior ones. Dropping the end pairs makes the fit very slightly worse, which is the arithmetic saying the scatter is not at the ends.

There is a cost hidden in the successful cases too. The decay length is 1.9005 fusions over all pairs of the straight chain, 2.0690 over the interior ones and 1.718 over the end pairs alone. Those three numbers differ by a fifth, and nothing measured here prefers one of them. One length per molecule is the usual report; which length it is depends on which pairs are in the fit, and that choice is made by the rule about half the molecule rather than by any argument.

That is the same failure as a decay length fitted over a window, arriving from a different direction: there the ambiguity was in how far along a profile to fit, here it is in which pairs of a molecule to include, and in both the quantity is not a property of the system until the choice is stated. A reach that has no length is the third case, and the three together are the standing argument that a fitted length needs its window quoted beside it.

How big the scatter is, molecule by molecule #

The two uniform chains reach a spread of about 1.5 at their widest separation. The mixed chain reaches 3.029, and it reaches it at two fusions rather than at four — so the worst scatter is not at the largest separation, where a fit has fewest pairs and is most fragile, but in the middle of the range where the fit has most weight.

A decay that depended on the separation alone would put a one on every point of that figure. None of the three shapes comes near it, and the shape that mixes linear and angular fusions is twice as bad as the two whose fusions are all alike. That last comparison is the one worth carrying: the scatter tracks where the molecule’s fusions change kind, which is not a property of any pair’s distance from an end. All three chains are symmetric end for end, so this is not the kind of freedom a degeneracy that a group predicts leaves — a reflection relates the arms of the mixed chain as surely as it relates the ends of the other two — and it is a statement about the molecule’s construction rather than about its size.

The correction does not settle #

If the end effect were a boundary term it would approach a limit as the molecule grew, and a long enough chain would let it be quoted once.

It does not. On straight chains of five, seven, nine and eleven rings the ratio runs 0.820, 0.762, 0.789, 0.839 — down, up, up — and on zigzags 1.512, 1.184, 1.277, 1.244. Both wander by five to twenty per cent with no trend, over a range in which the molecule doubles.

What moves them is not the length. It is which pairs exist at that separation in a molecule of that size, and how the deepest pair — the denominator — is placed relative to the centre. A chain of nine has a pair sitting exactly at the centre and a chain of eleven does not, and the ratio notices.

So there is no asymptotic end correction to quote, and an end correction fitted on one molecule is not transferable to the next even at the same shape. That closes the question in the strongest available way: the scatter is real, it is large, it is not noise, and it is not a function of anything a pair carries.

What was computed, and how #

Each molecule is a chain of fused six-membered rings built by naming a direction for each fusion — all zeroes for a straight chain, alternating for a zigzag, and four of one direction then four of the next for the mixed one, which turns the chain once at its middle ring. The π system is a Hückel one, and the ring-current response is the second derivative of the total energy with respect to the magnetic fluxes through two rings, taken as a symmetric finite difference at a flux of 10⁻³ with the gauge assigned by partial sums of the bond angles about each ring centre.

That construction is unchanged, and it carries the same checks: a single ring reproduces the closed-form response of 2/9, and the matrix is symmetric because it is a second derivative of one function.

The one check added here is the one needed. Every chain here is symmetric end for end, so the pair at depth zero on the left and the pair at depth zero on the right must have identical responses. The gauge is not symmetric — the phases are assigned by walking from one end — so this is a real test rather than an identity, and it holds to five figures at a flux step of 10⁻³, which is what a second difference of a sum of eigenvalues has. The effects being reported are tens of per cent, three orders larger.

The depth of a pair is the smaller of the two rings’ distances to their nearer ends, and the deepest pair at a separation is the one with the largest depth. Where two pairs are equally deep the reflection check above says they agree, so the denominator is well defined.

Pairs more than half the molecule apart are excluded, which is the usual rule, kept for its own reason: at the largest separations there is one pair — the two ends — and its entry is an end effect rather than a sample of a decay. That exclusion is what makes the four separations here four separations rather than eight.

Where the model stops #

This is Hückel theory: one π electron per carbon, one hopping integral, no repulsion and no geometry beyond the graph. A ring current computed this way is a response to a flux rather than to a field, and the relation between the two involves an area that the model does not have — the same gap aromaticity as a computed shell closure leaves between a count of electrons and a measured susceptibility. So the numbers are comparable with each other and are not comparable with a measured susceptibility.

Three shapes is three shapes, and all three are symmetric end for end. The ring with a twist in it is the one fused system here whose geometry is not planar at all, and nothing here says whether the sign survives that. What the mixed chain shows is that mixing linear and angular fusions produces a qualitatively different pattern, and one such molecule is not a survey of them. Chains whose angular rings sit in different places would be the next case, and nothing here predicts what they do.

The finite difference is at one step. A response computed as a second difference has an optimal step and this is not chosen by a convergence study; what licenses it is the reflection check, which would fail first if the step were too small, and the single-ring closed form, which would fail if it were too large.

And every molecule here is catacondensed — each ring shares one bond with the next, and the ring adjacency is a path. Pyrene and coronene are the systems whose ring adjacency has cycles in it, and they are not built here. Whether an end effect even has a meaning there is unclear, since a ring in a coronene has no distance to an end.

The generalisation #

A correction with a sign that depends on the system is not a correction. The whole point of describing a scatter by an auxiliary variable is that the variable’s coefficient can be fitted once and reused; a coefficient that changes sign between two molecules of the same size and the same graph distances has to be refitted on every molecule, at which point it is a description of that molecule rather than a term in a law. The same shape appears in a coefficient that a per-axis split makes worse rather than better: more freedom in the fit, more spread in the answer.

And a quantity that has to be fitted over a chosen subset is not a measurement until the subset is stated. Three defensible choices of which pairs to fit give three decay lengths a fifth apart on one molecule, and the literature convention of using pairs up to half the molecule is a convention rather than a result. The honest form is the one forced twice now: report the length with the window, or report the whole profile and let the reader see the fan.

Who found it, and when #

That the rings of a polycyclic aromatic do not carry equal currents is old and is the standard reading of the nucleus-independent chemical shift maps that came in with Schleyer’s work in the nineties: an anthracene’s central ring is more diatropic than its outer ones. That the pairwise response — one ring’s current from another ring’s flux — should be describable by a separation is a natural extension and, as far as can be told, is not something anybody has claimed in print.

What is new here is the comparison rather than any one number. Measuring the same quantity on a straight chain and a zigzag of the same length, and finding the end’s effect reversed, is a comparison a single molecule cannot make and is the whole of the content here. The mixed chain is what says the reversal is not a two-case curiosity but the absence of a variable.

Still open: ring graphs with cycles #

The obvious open question is the molecule whose ring adjacency is not a path. Pyrene and coronene have cycles in their ring graph, so two rings can be two steps apart by one route and three by another, and the question of which separation the response follows becomes a question with more than two candidate answers. There is now a reason to want it: if the end’s effect is really about how much molecule sits beyond a pair, a ring with no end beyond it in any direction is the cleanest possible test, and coronene’s central ring is exactly that.

The nearer question is what the sign is a function of. A straight chain and a zigzag differ in one thing — the fusion direction — and that is a statement about the shape of the boundary rather than about its existence. A chain of nine with a single bend, placed at each of seven positions along it, would give seven molecules differing in one variable, and the end ratio measured on each would say whether the sign is set by the bend’s presence, its position, or its distance from the end being measured. Seven molecules is seven diagonalisations of a matrix already built.