Which numbers carry a frame

A delocalisation energy is a subtraction whose second operand is a choice, and the choice changes the answer: benzene’s is 2β against three isolated double bonds and 1.0121β against the same six carbons with one bond deleted, and cyclobutadiene’s is exactly zero against the first and −0.4721β against the second. The obvious question is whether the same sensitivity infects the other quantities Hückel theory quotes, and there is a natural guess: a bond order has no subtraction in it and should be immune; a charge is a difference against a neutral atom and should not be.

This essay answers that properly, by testing nine quantities against four things that can be changed without changing the molecule. The guess turns out to be right and to have the wrong reason attached, which is the better outcome.

Four changes, and two kinds #

A Hückel calculation is written in terms of α, the zero of energy, and β, the unit. Neither is computed and neither is measured. So:

Move the zero. α → α + c. Nothing observable can depend on it.

Change the unit. β → λβ. Nothing observable can depend on it either.

Those two are exact symmetries of the model. A quantity that moves under either is not a quantity at all: it is a coordinate, and reporting it is reporting where somebody put an origin.

Change the reference. Measure a stabilisation from isolated double bonds, or from the same ring cut open. Both are defended in the literature.

Change the divisor. Divide a “per” quantity by the electrons or by the π bonds. Both are used.

Those two are conventions. A quantity that moves under either is a real quantity whose value has to be quoted with its convention attached.

The four molecules are benzene, cyclobutadiene, butadiene and pyridine. Pyridine is in the list for a specific reason: without a heteroatom every charge in an alternant hydrocarbon is exactly one by the pairing theorem, so “the charge does not move” would be a statement about zero rather than about a reference. Pyridine’s nitrogen carries 0.195, and that is what does not move.

The three that survive everything #

The spread of bond orders, the ratio of two bond orders, and the largest charge on any atom are unchanged by all four, to machine precision — zero, exactly, in every cell.

The reason is one line. Shifting or scaling the eigenvalues does not touch the eigenvectors, and all three of those are sums over occupied eigenvectors with no energy in them anywhere. Changing the reference state or the divisor does not touch them either, because neither appears in their definition.

The immunity extends to anything built the same way. A charge distribution, a participation number, a localisation, the composition of a normal mode: all of them are sums over eigenvectors with occupation numbers, and none of them can see where the zero of energy is. That is a large class of the quantities Hückel theory reports, and it is worth knowing that it is closed under the operations here.

So the guess about bond orders was right, and it was right for a stronger reason than the one given: not merely that a bond order has no subtraction in it, but that it has no energy in it. A bond order is read off the eigenvectors and the eigenvectors do not know where the zero of energy is.

The ones that are coordinates #

The highest occupied level moves by 0.37 β when the zero moves by 0.37 and by 0.83 β when the unit changes by 1.83. It is a coordinate twice over.

The gap between two levels survives the shift — it is a difference, so the shift cancels — and does not survive the scaling. So a gap is a quantity in units of β, and quoting one without saying so is quoting a number in an unstated unit.

The total π energy moves under both, by 2.2 and 7.1 β across these molecules, which is why nobody quotes one. Every use of a total π energy in this collection and in the literature is as an operand of a subtraction — including the shell closure that this field’s counting rule is about, which compares one filling against another.

The object those three rows are read off is an ordinary level diagram — Hückel theory draws benzene’s six levels — and the two exact symmetries are what they look like on it. Moving the zero of energy slides the whole diagram up the page and changes nothing; changing the unit stretches it and changes nothing. Both operations move numbers this collection quotes.

The ones that carry a convention #

The delocalisation energy survives the shift exactly — $1.8 \times 10^{-15}$ across four molecules — because its reference state holds the same number of electrons, so the shift adds the same amount to both operands. That is worth having as a fact rather than as an expectation: it is the reason a delocalisation energy is a better thing to quote than a total energy, and it is a property of the construction rather than of the choice of reference.

It does not survive the change of reference, by up to 1.6 β. And a per-electron figure adds a second convention on top: changing the divisor from electrons to π bonds moves it by up to 0.42 β, which for benzene is from 0.333 to 0.667 — a factor of two.

So a delocalisation energy per electron is a number carrying two independent conventions, and it is easy to notice only one of them.

The number that looks safest #

The last row of the table is the one worth the essay.

A ratio between two molecules’ delocalisation energies is the form in which such a comparison is usually quoted — benzene is so many times more stabilised than cyclobutadiene — and it has every superficial mark of safety. It is dimensionless. It is immune to moving the zero, exactly. It is immune to changing the unit, exactly, because both operands scale together.

Against the classical reference it is infinite, because cyclobutadiene’s delocalisation energy is exactly zero there. Against the cut-open reference it is 2.14.

That is the sharpest form of the reference finding, and it inverts the usual advice. The instinct that a dimensionless ratio is safer than an absolute number is correct about the unit and says nothing about the reference — and for a quantity defined by a subtraction the reference is where the difficulty lives.

Why the zero survives a subtraction and the reference does not #

The two conventions behave differently and the difference is structural rather than accidental, which is worth a paragraph because it explains why one of them is fixable and the other is not.

The zero cancels because the two operands have the same number of electrons. Benzene’s π energy and three isolated double bonds’ π energy are both sums of six one-electron energies, so adding $c$ to every level adds $6c$ to both and the difference is untouched. That is guaranteed by construction, for any reference state built from the same electrons.

The reference does not cancel because the two references are different states. Three isolated double bonds and a cut-open hexatriene are not related by any symmetry of the problem; they are two different molecules, chosen by two different arguments about what “no delocalisation” means.

So a delocalisation energy is immune to one convention for a reason and sensitive to another for a reason, and the two reasons are not the same kind of thing. The first is a theorem. The second is a disagreement.

The input to every number in the table is six rings with their levels and their fillings, which aromaticity as a shell closure draws. Everything a Hückel calculation knows is in a picture like that one; what an essay quotes is a function of that picture and of a frame, and this essay is about which of those functions carry the frame through.

What this means for what gets quoted #

The table sorts the nine into three groups and the sorting has a practical form.

Quote freely: bond orders, ratios of bond orders, charges. They depend on nothing but the matrix, and the matrix is the molecule.

Quote with the unit: gaps, and any difference of levels. In units of β, always, because β is not a number.

Quote with the reference and the divisor: delocalisation energies, per-electron figures, and every ratio built from them. Naming the reference is not pedantry; it is the difference between a finite number and an infinite one.

Do not quote at all: orbital energies and total π energies as absolute numbers. There is no frame in which they mean anything on their own.

That last group is the one to be most careful about, because such numbers are printed constantly — every level diagram is a picture of quantities in the first group. What makes those pictures legitimate is that they are read for their pattern, and a pattern is a set of differences. A shell closure is a statement about which gaps are large; a degeneracy is a statement about which are zero; and both survive every change of frame in the table, although the numbers the diagram is drawn from do not.

The same test applied to what a spectrum measures #

There is a check on the whole exercise available, and it is worth running because it says the table is about the model rather than about arithmetic.

Every quantity in the immune group is something a measurement could in principle return: a charge is measurable, a bond order is inferable from a bond length, a ratio of bond orders from two bond lengths. Every quantity in the sensitive group is either not measurable at all — nobody measures a total π energy — or is measurable only as a difference between two real substances, which is what a thermochemical resonance energy is.

That correspondence is not a coincidence and it is the general rule this essay is a case of: a quantity that moves under an exact symmetry of a model cannot be an observable of that model, because the symmetry is a statement that the two frames describe the same physics. So the first two columns of the table are, read the other way, a test of which of the nine could ever have been measured.

The two convention columns are different in kind. A delocalisation energy is not unmeasurable — it is measured, thermochemically, all the time. What is conventional is which subtraction the measurement is compared against, and that is a choice made when the experiment is designed rather than a property of the model.

Why a hydrocarbon’s answer is a pure number and a heterocycle’s is not #

The difficulty named for a heteroatomic ring can be made precise rather than left as a warning, and doing so explains a spread in the literature that looks like disagreement and is arithmetic.

For an alternant hydrocarbon the subtraction is unusually clean. The molecule’s π energy is $n\alpha$ plus a sum of coefficients times $\beta$; the reference — some count of isolated double bonds — has the same number of electrons, so its energy is the same $n\alpha$ plus a different multiple of $\beta$. Subtract, and $\alpha$ vanishes and $\beta$ factors out, leaving a pure number: benzene’s 2, cyclobutadiene’s 0. That is why such quantities can be quoted as multiples of an unfitted parameter and treat them as results.

Put a nitrogen in and neither cancellation survives intact.

The molecule’s energy now carries a term $h\beta$ for every electron sitting on the nitrogen, and the reference carries the same term for however many sit on its nitrogen. Those two occupancies are not equal, because a heteroatom in a ring does not hold the electrons it brought — pyrrole’s nitrogen keeps 1.720 of the two it contributed. So the subtraction leaves a residue

$$h\beta \times \Delta n_{\text{N}},$$

which for pyrrole is $1.5 \times 0.28 = 0.42$ in units of $\beta$ before anything else is counted, and the coefficients multiplying $\beta$ in the first term depend on $h$ and $k$ as well.

The consequence is structural rather than numerical. A hydrocarbon’s delocalisation energy is a pure number times $\beta$; a heterocycle’s is a function of two fitted parameters times $\beta$. The first can be compared between molecules without deciding anything. The second cannot, because two people using different tabulated values of $h$ get different answers, and they are not answers about the same quantity scaled differently — the dependence is not a common factor.

That accounts for the state of the published figures. Resonance energies for pyridine and the five-membered heterocycles vary between sources far more than the hydrocarbons’ do, and the variation tracks which parameter set was used rather than which reference. It is the same difficulty this essay is about, one level down: the reference is a choice, and for a heterocycle the molecule is a choice too, because two of its matrix entries were fitted.

Which sets a limit on what the last column of the table could ever contain. A frame-free comparison between two hydrocarbons is difficult and conceivable. Between a hydrocarbon and a heterocycle it is not conceivable at all, because one of the two numbers is not a property of a graph.

What this cannot say #

Four changes is not all of them. A Hückel calculation has other conventions in it — whether the overlap is kept in the secular equations, what a heteroatom’s parameters are, whether a ring is drawn regular — and each of them moves some quantities and not others. One spectrum and a line of models is the same investigation for the parameters, and what keeping the overlap does to a bond order is a case where a quantity immune to everything here is not immune to that.

The two exact symmetries really are exact in this model and not in a better one. In a calculation with two-electron integrals in it, α and β are not free parameters and the shift and scaling are not symmetries. What survives from this essay into that setting is the reference sensitivity, which gets worse rather than better.

The four molecules are small and one of them is heteroatomic. Adding a second heteroatomic system would test whether the charge’s immunity is general or is pyridine’s; the theorem behind it says it is general, and the table checks one case.

And a per-quantity has more than two conventions available. Per electron, per π bond, per carbon, per ring: the two tested are the two used, and the spread across all four would be wider.

What the frame test requires #

Three quantities immune to all four changes, each to $10^{-9}$ or better, and a charge present to move — pyridine’s 0.195, so that the immunity is not an immunity of zero.

The highest occupied level moving under both exact symmetries, which identifies it as a coordinate.

A gap surviving the shift and not the scaling, which is what makes it a difference rather than a ratio.

A delocalisation energy surviving the shift and not the reference, and a per-electron figure adding the divisor.

And the ratio immune to both exact symmetries and infinitely sensitive to the reference — the two halves checked together, because either alone would be a different and less interesting claim.

The table required to contain both kinds. At least three quantities immune to everything and at least four sensitive to something, so that the result is a distinction rather than a verdict on the model.

Still open: a relaxed geometry, and a frame-free comparison #

The obvious open question is the frame this essay could not vary. 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.

The nearer question is about what a frame-free comparison of two molecules would even look like. Nothing in the last column of the table survives, and the reason is structural: any comparison of two molecules’ stabilisations is a comparison of two subtractions, and the two subtractions are only comparable if the references are constructed the same way for both. For rings of the same size they are. For a ring against a chain, or a hydrocarbon against a heterocycle, they are not — and how to construct a reference that is the same reference for two different molecules is the question the whole delocalisation-energy literature has been arguing about since Dewar, put in a form that could be tested. A heteroatomic ring is where the difficulty is sharpest, since its reference has to price a nitrogen against a carbon before anything is subtracted.