One spectrum, a line of models

A model with parameters in it is fitted to data, and afterwards the parameters are quoted as though they had been measured. The quantity that was actually measured is the data; what the parameters are is a question about how many of them there were.

Hückel theory has three: the energy of a carbon 2p orbital, the interaction between two neighbouring ones, and the overlap between them. The third is almost always set to zero, which is a decision rather than an approximation anybody is forced into — the problem with the overlap kept is no harder to solve.

Benzene’s photoelectron spectrum supplies two numbers for the π system: the ionisation energies of the doubly degenerate level at 9.25 electronvolts and of the lowest one at 12.30. Two equations, three unknowns, and the solutions are a line.

The family #

For each choice of neighbour overlap there is exactly one pair of the other two parameters that reproduces both measured ionisation energies. The line can be written down without any fitting at all, because the levels of a ring have a closed form in the parameters, but the point is what varies along it.

Three things move across that figure and one does not, and separating them is the whole of this essay.

The resonance integral moves by a factor of 2.5, from −3.05 to −7.66 electronvolts. That range is not an artefact of the choices made here: it is roughly the range of values quoted for β in the literature, which has been assembled by fitting different observables under different overlap conventions.

The Coulomb integral moves too, from −6.20 to −4.37 electronvolts, in the opposite direction.

The empty level moves by nearly eight electronvolts. This is the level nothing was fitted to, and it is the one that makes the family testable.

The delocalisation energy does not move at all. It is 6.100 electronvolts in every column, to the last digit.

Why one quantity is fixed #

The invariant is not a coincidence and it is not deep, and being able to say which is the point of computing it.

Benzene’s π energy is two electrons in the level at $x = 2$ and four in the level at $x = 1$. The reference state the delocalisation energy is measured against is three isolated double bonds, which is six electrons in the $x = 1$ level. Subtract:

$$6E(1) - \big[2E(2) + 4E(1)\big] = 2\big[E(1) - E(2)\big]$$

and $E(1) - E(2)$ is the gap between the two levels that were fitted. So the delocalisation energy is exactly twice the measured splitting — 2 × 3.05 = 6.10 electronvolts — and no parameter appears in it anywhere.

Writing it as “2β” makes it look like a consequence of a fitted parameter. It is a consequence of the data, and the two halves of the phrase are both parameter-dependent in ways that cancel: across the family the same energy is 2.000β, 1.598β, 1.330β, 1.140β, 0.997β, 0.886β and 0.797β.

This is the same shape of finding as where a stabilisation is measured from, approached from the other side: there the reference state was the choice, here the units are.

What the orbitals do, and what the numbers computed from them do #

With the overlap kept, the eigenvalue problem is a generalised one and has to be solved rather than assumed. It is solved here by symmetric orthogonalisation — the overlap matrix is diagonalised, its inverse square root formed, and the transformed Hamiltonian diagonalised — so the vectors that come out are computed from matrices rather than inherited from an argument.

They are the same vectors at every overlap. Normalise each to unit length and compare them member by member: the direction agrees to nine decimal places across the whole family. Every statement that depends on the directions alone — which levels are degenerate, which orbital has a node where, what symmetry species each belongs to — is therefore untouched by the parametrisation.

The numbers computed from the vectors are a different matter, and this is where the essay’s expectations were wrong before the arithmetic was done.

A Coulson bond order is a sum of products of coefficients, and with the overlap kept each orbital is normalised against the overlap matrix rather than against the identity — by a factor that depends on that orbital’s own eigenvalue. So the orbitals point the same way and the numbers computed from them do not.

At an overlap of 0.30 butadiene’s terminal bond order is 0.678 and its central one 0.254, and the ratio has gone from exactly 2.000 to 2.671. A thirty-four per cent change in the quantity a chemist reads off, from a parameter the spectrum does not constrain.

The measurement that picks a member #

The family is a line, and a third measurement is a point that either lies on it or does not.

Benzene’s empty π level is measurable. An electron aimed at benzene is not bound — the anion is unstable — but it is temporarily captured, and electron transmission spectroscopy locates the resonance at 1.12 electronvolts above the vacuum level.

Reading that off the family picks the member at a neighbour overlap of 0.189, where the resonance integral is −5.95 electronvolts and the Coulomb integral −5.05.

Two things about that answer are worth saying, and they point in opposite directions.

The overlap it picks, 0.19, is physically ordinary. Two carbon 2p orbitals in a π arrangement at benzene’s bond length overlap by about a quarter, so a fit that came back with 0.02 or 0.4 would have been evidence that the exercise was meaningless.

The other two parameters it picks are not ordinary at all. A Coulomb integral of −5.05 electronvolts is nowhere near a carbon 2p orbital energy, which is about −11.4; a resonance integral of −5.95 is larger in magnitude than any value in ordinary use. The three-parameter model can fit all three measurements, and the parameters it needs to do so are not the ones the parameters are supposed to be.

That is the honest ending, and it is a stronger result than the underdetermination: a model whose parameters are underdetermined by two measurements is merely unconstrained, while one that fits three measurements with unphysical parameters is telling the fitter something.

The same underdetermination in two orbitals #

The arithmetic is easiest to see in the smallest case the theory has, which is two orbitals rather than six.

With the overlap kept, the bonding level falls by less than the antibonding one rises, and the size of that asymmetry is set by the overlap — the one parameter of the three that a spectrum of two lines cannot separate from the other two.

Two levels give one measurable splitting. Three parameters — two orbital energies and an interaction — already outnumber it, and adding the overlap makes four. Every more elaborate argument about orbital interaction is built on that little system, and the underdetermination is inherited by all of them: the asymmetry between bonding and antibonding is a statement about the sign of an effect, which survives, while its size in electronvolts is a statement about parameters, which does not.

This is why the useful results of one-electron theory are so often inequalities and orderings rather than numbers, and why the numbers that do get quoted tend to be ratios in which the parameters cancel.

What the model is leaving out #

What it is leaving out, on the evidence above, is at least the following.

The σ framework. Ionising a π electron relaxes everything else, and the relaxation differs between the two levels; a model with only π in it books that difference into the π parameters. This is Koopmans’ theorem’s difficulty in a place where it cannot be separated from the fit.

The repulsion. The three fitted numbers are ionisation energies and an attachment energy, and the gap between the highest occupied and lowest empty level in a one-electron model is not the difference between them once the electron count changes. Every question about electron correlation is about that gap.

The reference state. The delocalisation energy of 6.10 electronvolts is itself four times the thermochemical resonance energy of benzene, which is about 1.6, and the gap between those two numbers is a separate argument that a stabilisation is a difference from something somebody chose. That discrepancy is not in the fit at all: it is the vertical delocalisation energy of a model with no σ relaxation, compared with a heat of hydrogenation that includes everything.

Whatever the parameters are worth, the number they multiply depends on what benzene is being compared with. A fitted β and a chosen reference state are two independent conventions, and a resonance energy in kilojoules carries both.

Where the same arithmetic goes right #

None of this makes the model useless, and the distinction is worth drawing sharply because it is the useful conclusion.

Every prediction Hückel theory is actually good at is a statement about eigenvectors and eigenvalue orderings rather than about parameter values: which rings close a shell, which levels are degenerate, where the nodes are, which positions of a substituted ring carry charge. All of those survive the whole family untouched — including the ones the theory is most often praised for, like the shell closure that makes six electrons in a ring special and the bond orders’ pattern along a chain, as opposed to their values.

The eigenvalues themselves are checked against the closed form at every ring size from three to ten, along with both trace relations and pairing in both directions. None of that depends on a parameter — which is the division this essay is about: the eigenvalues are the part of the model that is right.

The predictions that do not survive are the ones stated in energy units: a delocalisation energy in kilojoules, a HOMO–LUMO gap in electronvolts, a bond order to three figures. Those are the ones quoted most often.

Why anybody set the overlap to zero #

The convention has a reason and it is not laziness, which is worth saying before the essay leaves it.

Dropping the overlap makes the eigenvalue problem ordinary rather than generalised, and in 1931 that was the difference between a calculation a person could do and one they could not. It also makes the levels symmetric about the Coulomb integral, so a ring’s levels come in pairs at $\alpha \pm x\beta$ and every statement about electrons and holes becomes a statement about a mirror. That symmetry is exact only at zero overlap, and a good deal of the theory’s neatness rests on it — including the pairing theorem for alternant systems, which is what makes an alternant hydrocarbon’s charges all exactly one.

So the convention buys a genuine simplification and costs a parameter’s worth of honesty. What it should not buy is the right to quote the resulting β as a measured quantity, and that is the only thing this essay is asking for.

What is quoted, and what is computed #

The three measurements are quoted: two photoelectron ionisation energies and one electron attachment energy, all standard. The claim that the resonance integral’s literature range is about a factor of two and a half is also quoted, as a description of the literature.

The family, the parameters at each point on it, the levels, the delocalisation energies in both units, the eigenvectors, the bond orders and the member the third measurement picks are all computed, from a generalised eigenvalue problem solved on the matrices themselves.

What was checked #

Every member reproduces both fitted measurements to nine decimal places, or it is not a family.

The parameters differ across it by more than a factor of two, or there is nothing undetermined.

The delocalisation energy in electronvolts is identical across it and equals twice the measured splitting exactly; the same quantity in units of β varies by more than a factor of two.

The orbitals point the same way at both ends of the family, checked as an overlap of normalised vectors; and the ratio of two unlike bond orders does not, changing by more than twenty per cent between the ends.

The unfitted level moves by several electronvolts, so a third measurement can pick a member out at all.

Why the published values of β disagree #

There is a well-known untidiness in the literature that this family explains, and explaining it is worth more than the family itself, because it converts a long-standing embarrassment into an arithmetic consequence.

Values of the resonance integral in print do not agree. Fitted to spectra, β comes out somewhere between about $-2.5$ and $-3.0$ electronvolts. Fitted to resonance energies — benzene’s stabilisation of about 150 kilojoules a mole set equal to $2\beta$ — it comes out near $-0.8$. A factor of three, from careful workers using the same theory on the same molecule.

The usual explanations are that Hückel theory is crude, or that different observables probe different things, and both are true and neither is specific. The arithmetic here gives the specific version.

A model with more parameters than the data constrains does not have a value of β; it has a family of them. Which member a fit lands on is decided by which observable was fitted, because different observables are constant along different directions of the family. A spectrum fixes one combination of the parameters, a resonance energy fixes another, and the two combinations intersect the family at different places.

So the published values are not two estimates of one quantity with a large uncertainty between them. They are two different members of a one-parameter family, each correctly determined by its own data, and expecting them to agree is expecting a curve to be a point.

That reading has a test and the test is the one already run here. The quantity that is constant along the family — the delocalisation energy in electronvolts, which comes out at twice the measured splitting at every member — is a quantity two workers fitting different observables would agree about. The quantity that varies along it is β itself. So the prediction is that the literature should disagree about β and agree about the energies β is used to compute, and to a good approximation it does: benzene’s delocalisation energy is quoted consistently in kilojoules per mole and inconsistently in units of β.

Which gives a rule for reading any such number. A parameter is not a result; an invariant is. A paper reporting β has reported a coordinate on a family whose other members fit the same data equally well, and the way to tell whether a downstream number inherits that freedom is to ask whether it can be written in terms of the measured quantities alone. If it can, it is determined. If it needs β, it is a member of a family too.

The same rule settles a question worth being careful about in any Hückel calculation: why energies are best quoted in units of β and never converted. The convention looks like fastidiousness and is the only honest option available. A delocalisation energy of $2\beta$ is an exact statement about a graph; the same quantity in electronvolts requires a value of β, the value depends on which observable was fitted, and the conversion therefore smuggles a choice into a number that had none in it. Leaving the parameter as a symbol is not a refusal to commit — it is the statement that the result does not depend on the commitment, which is a stronger claim than any converted number could carry.

Still open: which model is doing more work #

The ionic weight of a valence bond wavefunction depends on how it is weighed; this essay finds that the parameters of a molecular orbital model depend on a convention the data does not fix. Both are the same complaint about the same kind of number, from the two sides of an old argument, and the pair of them suggests the next question to ask.

One route is worth naming here as a contrast: a basis set fitted to an energy is tested on a property it was not fitted to, and the size of the failure is informative. That is the same test as the empty level above, applied to a model whose parameters are functions rather than numbers.

The question left is this: given two models with different numbers of parameters, both fitted to the same measurements, is there a computation that says which of them is doing more work? The information-theoretic answer — count the parameters, penalise them, compare the residuals — is available and is a piece of statistics rather than of chemistry. The chemical answer is the one this essay stumbled into: fit the model to more data than it has parameters and look at whether the parameters it needs are the ones the parameters mean. That test has an outcome here, and it is not the one a defender of the model would want.