The agreement was hydrogen's energy

Spin below zero solved a three-orbital radical exactly with a repulsion between two electrons on the same site and found the negative spin on the middle orbital that no single determinant can give — the allyl radical’s measured central spin, reached at an on-site repulsion near six electronvolts. Six electronvolts was a difference added the repulsion between neighbouring sites and found the six was a difference between two repulsions: with the Mataga–Nishimoto neighbour terms, computed from the on-site value at allyl’s geometry, the repulsion that reproduces allyl’s central spin of −0.164 is 10.89 eV, a quarter of an electronvolt from Pariser’s 11.13 for a trigonal carbon.

That essay was careful about the one number that was never allyl’s. The trio’s hopping — the coupling between each end and the middle, 2.91 eV — comes from a round overlap of 0.25 and an orbital energy of −13.6 eV through the Wolfsberg–Helmholz rule, and nothing about either is carbon. Its lead was to compute the hopping from carbon 2p functions at 1.39 Å, one overlap integral and the same rule, and to see whether the agreement survives the one number that was never allyl’s.

It does not, and the reason is not the number it suspected. The overlap turns out to be carbon’s already. The energy is hydrogen’s.

Carbon’s overlap was the round one #

Two carbon 2p functions side by side, at Slater’s effective charge of 3.25, overlap by an amount Mulliken tabulated in closed form. At allyl’s C–C bond of 1.39 Å the value is 0.2484.

That is the trio’s round 0.25 to within a per cent. Whoever chose 0.25 for a model π system chose carbon’s number, whether they meant to or not, and replacing it cannot move anything. The closed form is checked against the two-centre quadrature every overlap here is computed with, which was validated against the 1s–1s closed form long ago; they agree to a part in a million.

The overlap that was left out is the one between the two ends. They are 2.45 Å apart at allyl’s angle of 124°, and there carbon’s 2pπ overlap is 0.032 — small, and set to zero in the trio. It turns out to do something the orthogonalisation hides, which a later section takes up.

One matrix, two inputs, varied one at a time #

The trio’s one-electron matrix is built exactly as before: three orbitals on a common energy α, overlaps between them, off-diagonal elements by the Wolfsberg–Helmholz rule — 1.75 times the overlap times the average of the two diagonal elements — and the whole transformed to an orthogonal basis by Löwdin’s symmetric orthogonalisation. The on-site repulsion U and the neighbour repulsions $V_1$ and $V_2$ are then added and the nine configurations of three electrons solved exactly. The one-parameter version computes $V_1$ and $V_2$ from U by the Mataga–Nishimoto formula at allyl’s two distances, so U alone is solved for.

The hopping has two inputs: the overlap and the energy. The trio used 0.25 and −13.6 eV; carbon’s are 0.2484 (with 0.032 between the ends) and −11.4 eV, the valence-state ionisation energy of a carbon 2p electron as the extended Hückel tables give it. Four combinations separate them.

The trio’s own inputs are run through this calculation first, and must return the neighbour essay’s 10.89 eV exactly — they do, to the last digit. Carbon’s overlaps with hydrogen’s energy give 10.68 eV: two tenths of an electronvolt. The round overlap with carbon’s energy gives 9.66: a whole electronvolt and a quarter. Both of carbon’s together give 9.49. The overlap was never the question.

The energy sets the hopping #

The Wolfsberg–Helmholz rule makes the hopping proportional to the orbital energy, and nothing else about the trio’s electrons knows where that energy came from. The trio’s −13.6 eV was a generic level for a model three-orbital system, chosen so that its levels would sit where hydrogen’s do and its questions — which bond orders a third orbital can create, which levels a symmetry protects — would have round answers. Those questions did not care what the energy was. The allyl question does, because the energy sets the hopping’s size and the repulsion is measured against it. Hydrogen’s −13.6 eV is the ionisation energy of a 1s electron; a carbon 2p electron in a trigonal valence state is bound by about 11.4. Replacing one with the other scales the hopping by their ratio.

With carbon’s inputs the hopping between each end and the middle is 2.34 eV. That is not a number the model was fitted to, and it lands within two per cent of Pariser and Parr’s resonance integral of 2.39 eV, the value π-electron theory adopted from benzene’s spectrum. So the hopping the lead asked for, computed from carbon, is the one π-electron theory has always used. The trio’s 2.91 was a fifth too large, and the whole excess was hydrogen’s energy.

And the one-parameter repulsion follows the hopping almost in proportion.

Sweeping the orbital energy from −9 to −16 eV with carbon’s overlaps held, the Mataga–Nishimoto repulsion rises steadily from 8.1 to 12.0 eV. It passes Pariser’s 11.13 at an energy of −14.4 eV — near hydrogen’s, which is where the trio happened to sit, and three electronvolts from carbon’s. The agreement the neighbour essay found was the agreement between a model with hydrogen’s energy scale and a parameter fitted to carbon’s.

The two neighbour forms bracket Pariser’s value #

The neighbour essay compared two standard ways of computing the neighbour repulsion from U and a distance, and found Ohno’s form overshooting badly: 13.8 eV against Mataga–Nishimoto’s 10.89. With carbon’s hopping the picture changes shape.

With carbon’s hopping, Mataga–Nishimoto needs 9.49 eV and Ohno 12.68. Holding the neighbour terms at the values Mataga–Nishimoto gives for U = 11.13 and solving for U alone gives 9.89. Pariser’s 11.13 now sits between the two standard forms, 1.6 eV above one and 1.6 below the other.

That is a weaker agreement than the one it replaces and a more honest one. π-electron theory has used both forms, and the choice between them was always made by fitting spectra rather than derived; which one allyl’s central spin prefers depends on the hopping, and with the hopping the theory itself uses, it prefers neither. The spin can say that Pariser’s U is inside the range the two forms allow. It cannot, on its own, say that it is the right one.

What the spin actually fixes #

Two forms of the neighbour repulsion, four sets of inputs and fifteen orbital energies give nearly forty values of U that reproduce one spin, from 8.1 to 14.9 eV. That is too much variation for U to be the thing the spin is measuring. The neighbour essay had the clue: its U rose almost one for one with the neighbour repulsion, and it read the six electronvolts of the first essay as a difference between two repulsions. Taken seriously, that says the spin fixes a difference, and a difference of energies in a model whose only other energy is the hopping can only be fixed in units of the hopping.

It is. At the U that reproduces allyl, $(U - V_1)/t$ lies between 1.89 and 2.05 for every set of inputs, both neighbour forms and every orbital energy in the sweep — a spread of eight per cent, against a spread in U itself of more than four fifths. With carbon’s inputs, Mataga–Nishimoto needs U = 9.49 with $V_1$ = 4.95, and Ohno needs U = 12.68 with $V_1$ = 8.02: three electronvolts apart in U, and 4.54 against 4.66 in $U - V_1$. The two forms disagree about how large the neighbour repulsion is at 1.39 Å, Ohno’s being much the larger at that distance, and each makes up the difference with U.

So the spin is a measurement of one dimensionless number, the net cost of putting a second electron on a site rather than beside it, in units of the hopping — about 1.94 for allyl — and U enters only through it. Every agreement with a particular U is an agreement about $V_1$ and t as well, and the neighbour essay’s 10.89 eV was a statement about hydrogen’s hopping and Mataga–Nishimoto’s $V_1$ at the same time. The number the radical supplies is the ratio, and it does not care which form supplied $V_1$.

That also explains the bracket of the previous section without further calculation. At carbon’s hopping, the spin requires $U - V_1$ ≈ 4.6 eV. Pariser’s U of 11.13 would need $V_1$ ≈ 6.5 eV at 1.39 Å to satisfy it — between Mataga–Nishimoto’s value at that U, 5.37 eV, and Ohno’s, 7.58. The two forms bracket Pariser’s U because they bracket the $V_1$ that Pariser’s U would need.

Why the forms differ by so much at allyl’s bond is visible in their algebra. Both interpolate between the bare Coulomb repulsion $e^2/R$ at long range and U at zero separation, through a length a = $e^2/U$ that is 1.29 Å at Pariser’s U. Mataga–Nishimoto adds R and a; Ohno adds them in quadrature. At 1.39 Å the two lengths are nearly equal, which is exactly where a sum and a quadrature sum differ most — by a factor approaching $\sqrt{2}$ — and so the two forms give their largest disagreement at precisely the distance a π bond has. At the ends’ separation of 2.45 Å, where R is nearly twice a, Ohno’s is still a third larger. The disagreement is largest for the one repulsion the spin is most sensitive to, the one between a site and its bonded neighbour.

At Pariser’s repulsion the spin overshoots #

The same comparison, read the other way, holds U at Pariser’s 11.13 and asks what central spin the model predicts.

With the trio’s inputs it is −0.168, the essay’s close agreement. With carbon’s overlaps and energy it is −0.192, seventeen per cent too negative. A smaller hopping means a larger ratio of repulsion to hopping at the same U, which pushes the radical further towards the strongly correlated limit, where the middle orbital’s negative spin grows. So carbon’s hopping at carbon’s repulsion over-correlates allyl, and the repulsion that matches it has to be smaller — which is the 9.49 above, seen from the spin rather than from U.

The ends’ own overlap, and what orthogonalisation hides #

The trio set the overlap between the two ends to zero, and the neighbour essay’s lead did not question it. Carbon’s value is 0.032, and including it has an effect larger than its size suggests.

The trio’s one-electron matrix, orthogonalised, has a coupling between the two ends even when their overlap is zero: Löwdin’s orthogonalisation mixes each end with the middle, and two ends that each contain some of the middle couple through it. With the round overlap and carbon’s energy that induced coupling is 0.60 eV. Include the ends’ own overlap of 0.032 and it falls to 0.29 — more than half of it cancelled, because the real overlap between the ends enters the orthogonalisation with the opposite sign to the one it induces.

And yet the repulsion allyl needs moves by only 0.12 eV, from 9.60 to 9.49. The end–end coupling changes the middle orbital’s spin at second order in the correlation, and at these repulsions its effect is a percent. It is a case of an overlap that is not an interaction in the reverse sense: a large change in a coupling that matters little to the quantity measured.

How the claims can fail #

Every statement is checked where its figures are drawn. Carbon’s 2pπ overlap by quadrature must match Mulliken’s closed form to a part in ten thousand at both of allyl’s distances. The trio’s own inputs, through this calculation, must return the neighbour essay’s one-parameter repulsion exactly. Carbon’s overlap at 1.39 Å must be the trio’s 0.25 to within a per cent. Replacing the overlaps alone must move the repulsion by under three tenths of an electronvolt and replacing the energy alone by more than one. With both of carbon’s inputs the hopping must be within five per cent of Pariser and Parr’s β, the repulsion must fall below 10 eV, and the spin at Pariser’s U must overshoot allyl’s by more than a tenth. The repulsion must rise at every step as the orbital energy’s magnitude grows, and reach Pariser’s value only between −13 and −15 eV. The ends’ overlap must more than halve their coupling and move the repulsion by under two per cent.

Where the model stops #

The Wolfsberg–Helmholz rule. The hopping is proportional to the overlap and to an orbital energy, with a constant of 1.75 that was chosen for σ bonds in transition-metal complexes. π-electron theory’s β is fitted to spectra, not computed from an overlap, and that the rule gives 2.34 against 2.39 with carbon’s inputs is a coincidence worth noting rather than a derivation of β.

A valence-state energy. Carbon’s −11.4 eV is a tabulated valence-state ionisation energy for a 2p electron, not a property of allyl. A different tabulation moves the hopping in proportion, and the sweep above says by how much the repulsion follows.

Three orbitals and one geometry. Allyl is held at 1.39 Å and 124°, the σ framework is absent, and the neighbour repulsions are the two standard closed forms rather than integrals. The bracketing of Pariser’s value by the two forms is a statement about those forms.

An invariant that is not exact. $(U - V_1)/t$ drifts across the sweep — from 1.935 to 1.949 with Mataga–Nishimoto neighbours as the energy’s magnitude grows, and from 2.054 down to 1.946 with Ohno’s — because $V_2$ and the end–end coupling also enter the spin and scale differently. The combination is the leading one, fixed to eight per cent across everything computed, not the only one; a model with a second observable would be needed to separate the smaller terms.

And a spin fixed by a ratio. The target of −0.164 comes from the ratio of allyl’s central to terminal hyperfine couplings with the three π spins summing to one, as the neighbour essay established. Everything above is a statement about reproducing that number and nothing else about allyl.

What a second radical would test #

The ratio has a consequence that can be checked on a different molecule without any new calculation beyond the same nine configurations. If allyl’s central spin fixes $(U - V_1)/t$ near 1.94, then a radical with the same three-orbital topology and a different hopping should need a $U - V_1$ that moves in proportion to its hopping — and a radical whose π bonds are longer, so that its hopping is smaller and its $V_1$ smaller too, should show the two neighbour forms disagreeing less about U, since at longer bonds the sum and the quadrature sum of R and $e^2/U$ come closer together. Neither prediction depends on Pariser’s value or on the Wolfsberg–Helmholz constant. Both depend only on the trio’s structure and on the finding that one combination of its three energies is what the spin measures.

An agreement is a statement about every input #

The neighbour essay’s agreement was not wrong; it was a statement about all of its inputs at once, and one of them had a history the agreement did not show. A round overlap and a round energy were chosen for a model trio long before the trio was asked about allyl, and one of them happened to be carbon’s while the other was hydrogen’s. The test the lead proposed — replace the input that was never allyl’s — is the right discipline, and its answer is that the suspect was innocent and its companion was not.

That is the general shape of a model’s agreement with a measurement: the agreement is evidence about the model only after each input has been traced to where it came from. Here tracing them moved the conclusion from “the repulsion is carbon’s own” to “the repulsion is within the range two standard forms allow” — a smaller claim, and one that survives.

Still open: the neighbour form, and the geometry #

The obvious open question is the one the bracket leaves. With allyl’s own hopping the two standard neighbour forms give 9.49 and 12.68 eV, and allyl’s central spin alone cannot choose between them. A second observable of the same radical — its first ionisation energy, or the splitting between its lowest doublet and quartet — computed in the same nine configurations would add an equation, and two equations in U and the form of V would say which form allyl prefers, or whether it prefers neither.

The nearer question is still the geometry. With the overlaps now carbon’s, a bent or unequal radical’s end–end overlap, its two end–middle overlaps and its neighbour distances all follow from one set of angles, and the path a distorting radical takes through the model is a single curve rather than a sweep over independent dials. The ends’ overlap turned out to matter little at allyl’s own angle; whether it matters at the angles where the middle spin turns over is one calculation away.