Six electronvolts was a difference

The three-orbital radical, solved exactly with a repulsion between two electrons on the same site, puts negative spin on its middle orbital — which one determinant can never do — and reaches the allyl radical’s measured central spin at an on-site repulsion U between 5.5 and 6.9 eV. That essay compared the number with a free carbon atom’s ionisation energy minus its electron affinity, 10.0 eV, and read the difference as screening: the σ electrons and the repulsion between neighbouring sites, both absent from the model, soften the on-site value. It named the second of those as the thing to add next.

It is one more diagonal term in the same nine configurations. Added, it does not soften anything. The repulsion allyl needs rises almost one for one with the neighbours’ repulsion, and with the neighbour values π-electron theory has always used, the on-site repulsion lands on carbon’s own. The six electronvolts was a difference between two repulsions, not a screened value of one.

The band, closed without McConnell’s constant #

The measurement first, because the band that the model was fitted to is wider than it needs to be. The allyl radical’s central proton has a hyperfine coupling of +4.06 gauss and its terminal protons −13.93 and −14.83. McConnell’s relation, $a = Q\rho$, turns a coupling into a π spin density on the carbon beside it, with $Q$ somewhere between −27 and −22.5 gauss depending on the calibration. Read that way the central spin is −0.150 to −0.180, a spread of a fifth.

But the three couplings share one $Q$, whatever it is, so their ratio is the ratio of the spins. The central coupling is −0.282 of the mean terminal one. In a π model the three spins sum to one, two ends and a middle, so the middle spin is that ratio divided by two plus it: −0.164, with no $Q$ in it at all. It sits inside McConnell’s band, and it says what $Q$ must have been, −24.7 gauss, which is near the middle of the calibrations.

That turns a band into a line. The previous comparison found U between 5.5 and 6.9 eV; against the fixed value the same trio needs 6.1. Everything below uses both — the band, because it is the published statement, and the line, because it is the sharper one.

A neighbour repulsion makes no negative spin of its own #

The term being added charges the product of two sites’ occupations: $V_1$ between each end and the middle, $V_2$ between the two ends. It treats two electrons with parallel spins exactly as it treats two with opposite spins. What it does to the middle orbital’s spin is therefore not obvious in advance, and alone it does nothing useful.

With no on-site repulsion, a repulsion between each end and the middle leaves the middle with no spin at all, to the last digit, at every value up to five and a half electronvolts. One between the two ends puts positive spin on the middle — 0.11 at three electronvolts, 0.20 at five and a half — the opposite sign to allyl’s. Pushing the ends’ charge apart pushes the unpaired electron towards the middle, and a one-determinant spin is where the unpaired electron is. Above about six electronvolts, with nothing to stop two electrons sharing an end, an end–middle term large enough drives the charge off the middle altogether and the ground state changes character, which is a different question and is not followed here.

With an on-site repulsion present the two act together, and the neighbour term works against the on-site one. At U = 6 eV the middle orbital’s spin is −0.162; switching on both neighbour terms, in the proportion the allyl geometry gives them, shrinks it to −0.102 at three electronvolts and −0.050 at five and a half. A negative spin on the middle is made by electrons avoiding each other on one site. A neighbour repulsion reduces how much there is to gain by avoiding, because an electron that leaves a site to avoid its partner arrives next to someone else.

The U allyl needs follows the V #

So the U that reproduces allyl has to rise as V does, and it rises steadily. With the end–end repulsion held at the proportion the Mataga–Nishimoto form gives it at allyl’s distances — 0.72 of the end–middle value — the Q-free U goes from 6.11 eV with no neighbour term to 7.86 at $V_1 = 2$, 9.67 at 4 and 11.54 at 6. It climbs about nine tenths of an electronvolt for each one of $V_1$. The band moves with it: 5.5–6.9 eV with no neighbours, 9.9–11.5 at $V_1 = 5$.

Drawn as a map, every line of constant middle spin runs through the plane at nearly the same slope. To this quantity the two repulsions are very nearly one number, their difference: the on-site repulsion minus most of the neighbour’s. That has a reading. What polarises the bonding pair is the cost of two electrons sharing a site compared with the cost of their sitting on neighbouring ones, since those are the two things an electron can choose between. The earlier fit found that difference, and called it U because the model had nothing else to call it.

The slope is not exactly one, and the lines are not exactly straight. The difference $U - V_1$ that matches allyl without $Q$ runs from 6.1 eV with no neighbours to 5.6 at $V_1 = 5$ and 5.5 at 7; the ends’ own repulsion, which is in the calculation too, and the three-site problem’s higher orders take up the rest. It is a near-invariance, not an identity, and the map draws what it is.

Why a difference, and why nearly one for one #

The near-invariance has a simple origin, and it is worth having because it says when to expect it. The negative spin on the middle comes from mixing configurations in which an electron has hopped from one site to its neighbour. What such a hop costs is not U alone. Moving an electron from an end onto the already occupied middle puts two electrons on one site, which costs U; but the electron was already a neighbour of the middle’s electron before it moved, paying $V_1$ for the privilege, and after the move that interaction is replaced by the on-site one. The energy of the ionic configuration a hop creates, measured from the covalent one, is $U - V_1$ to first order, with smaller corrections from the charge left behind on the end and from the repulsion between the ends.

Every quantity the polarisation depends on at leading order is a hopping squared over that energy, so a model that sees only the on-site term reads $U - V_1$ and reports it as U. The same combination is the effective repulsion of the extended Hubbard model that polymer and solid-state physics uses, and the correlation essays’ ring of six found a neighbour term reversing a quantity rather than adjusting it — a reminder that the combination is leading-order only, and that what the neighbour term does beyond leading order depends on what is being asked. A root that looked like one crossing turned out to be two in that ring for the same reason: a second repulsion does not merely rescale the first.

Here the corrections are visible and small. The slope of U against $V_1$ is 0.9 rather than 1, because the end–end repulsion — pinned at 0.72 of the bond value — also enters the ionic configurations’ energies, with the opposite sense, and because at the size of these repulsions the hops are not small. None of that changes the reading: the parameter fitted to allyl was a difference of two repulsions, and it should never have been compared with an atom’s I − A, which is an on-site quantity.

Carbon’s own repulsion, with carbon’s own neighbours #

π-electron theory has a standard answer for both repulsions. Pariser’s valence-state value for a trigonal carbon’s π electron — its ionisation energy minus its electron affinity in the valence state a conjugated molecule uses — is 11.13 eV. The repulsion between electrons on sites a distance R apart is interpolated between that at R = 0 and the bare Coulomb $e^2/R$ at large distance, and there are two interpolations in common use: Mataga and Nishimoto’s $e^2/(R + e^2/U)$ and Ohno’s $e^2/\sqrt{R^2 + (e^2/U)^2}$. The allyl radical’s C–C bond is about 1.39 Å and its C–C–C angle about 124°, so the ends are 2.45 Å apart.

At U = 11.13 eV with the on-site term alone, the middle orbital’s spin is −0.242 — half as large again as allyl’s, which is the fit’s six electronvolts seen from the other side. With the Mataga–Nishimoto neighbour terms, 5.37 eV between bonded sites and 3.84 between the ends, it is −0.168, two per cent from the value the couplings fix without Q. With Ohno’s larger terms, 7.58 and 5.19 eV, it is −0.126, a quarter too small.

Turned round, the question is what U reproduces allyl exactly. With the Mataga–Nishimoto neighbours fixed at their 11.13 eV values the answer is 10.94 eV; with them recomputed from each trial U, so that the model has one parameter as Pariser–Parr–Pople theory does, it is 10.89 eV. Both are within a quarter of an electronvolt of Pariser’s 11.13. With Ohno’s form recomputed the same way the answer is 13.8 eV, well above it.

The ends agree too, which is not automatic: the three spins sum to one, but the fit matched only the middle. With the neighbour terms each end carries 0.584 against the 0.582 the couplings imply; with the on-site term alone at the same U, 0.621.

What this does and does not establish #

The agreement at 11.13 eV is close, and it would be easy to read it as a determination of carbon’s on-site repulsion from one radical’s hyperfine spectrum. It is not that, and the reasons are specific.

The neighbour form is chosen, and it decides the answer. The two interpolations differ by two electronvolts in the neighbour repulsion at a bond length, and they move the U that matches allyl by nearly three. Mataga–Nishimoto was introduced because it reproduced the spectra of aromatic molecules better than the alternatives, so choosing it here is choosing the form with the better record, not a free choice made to fit. But what the calculation shows is that with that form Pariser’s U is consistent with allyl’s spin — not that allyl measures U.

The one-electron part is the earlier essays’, not allyl’s. The three orbitals, their overlap of 0.25 and the Wolfsberg–Helmholz rule give a hopping of 2.91 eV between each end and the middle after orthogonalisation. Pariser–Parr–Pople theory usually takes 2.4 eV for a C–C bond. Since the spin depends on the repulsions measured against the hopping, a smaller hopping brings the U down: scaled to 2.4 eV and solved again with the Mataga–Nishimoto neighbours recomputed from U, the trio matches allyl at 9.6 eV. The agreement is with this trio’s hopping, which was never adjusted to allyl.

Three sites are not a molecule’s π system alone, and the σ electrons are still frozen. The two kinds of correlation a bond has are both present in allyl, and the trio carries only the kind three orbitals can hold. McConnell’s Q summarises how the π spin reaches the proton through a σ bond. The Q-free central spin removes the uncertainty in Q’s value but not the assumption that one Q serves both kinds of carbon, and a calibration that distinguished a central from a terminal CH would move the line.

So the claim is narrower and firmer than a determination: the shortfall the earlier fit found was exactly the size of the neighbour repulsion it left out, and the standard π-electron parameters, with nothing adjusted, give the allyl radical’s central spin to two per cent.

How the numbers were made #

The one-electron trio is the one whose level the overlap count held fixed: three orbitals at −13.6 eV, overlaps of 0.25 between each end and the middle, the Wolfsberg–Helmholz rule with K = 1.75, orthogonalised by the inverse square root of the overlap matrix. On those sites — a calculation of the kind a single on-site term turns a Hückel matrix into, with nine configurations rather than thirty-six — the on-site repulsion U and the neighbour repulsions $V_1$ and $V_2$ are added as diagonal terms, and the nine configurations of three electrons with two spins up are diagonalised exactly. The U that matches a target spin is bisected between 0 and 40 eV; the one-parameter version recomputes both V from each trial U before solving.

The checks, made wherever these figures are drawn: with no neighbour repulsion the spins are the on-site trio’s to 10⁻¹²; the three spins sum to one at every setting; an end–middle repulsion alone leaves no spin on the middle and an end–end repulsion alone leaves positive spin; at fixed U the neighbour terms shrink the negative spin monotonically; the U matching allyl rises with $V_1$ with $U - V_1$ between 4.7 and 7 eV up to $V_1 = 5$; the ratio of couplings fixes −0.164 inside McConnell’s band with Q = −24.7 G; at 11.13 eV the Mataga–Nishimoto middle spin is within five per cent of allyl’s and the U that meets it exactly within a quarter of an electronvolt of 11.13, and the Ohno U is above 12.5. The refusal is the trio with no neighbour term, whose band must be the earlier essay’s 5.5 to 6.9 eV.

Who found what #

McConnell related proton couplings to π spin densities in 1956, and the positive central coupling of allyl was one of the first measurements to require negative spin density — the observation that one-determinant pictures could not reproduce. Pariser and Parr, and Pople, set up the π-electron model with on-site and neighbour repulsions in 1953; Pariser’s valence-state value of 11.13 eV for carbon’s on-site repulsion is from the same year. Mataga and Nishimoto’s interpolation is from 1957, Ohno’s from 1964.

What is computed here is the same trio with neighbour repulsions, the band of U against them, the map of equal middle spin, the three standard parameter sets, the Q-free central spin and the one-parameter solution.

A spin measured with its sign, and what it weighs #

The pattern is one this argument has met in another form. Two changes to the trio that each touched one quantity stopped separating as soon as both were on, and the same is true of two repulsions. Two electrons in a bond dig a hole around each other with no repulsion at all, because the Pauli principle does it; a spin density that looks like an orbital’s turned out to need repulsion to exist. Here the repulsion it needs turns out to be a comparison between two repulsions, and the quantity that reads that comparison most directly is the one number in a hyperfine spectrum that carries its sign.

It is also a warning about fitted parameters in a model with an omission. A one-parameter model fitted to one observable absorbs whatever it leaves out into that parameter, and the fitted value then looks like a physically reasonable screening of the true one. Six tenths of carbon’s I − A was plausible, and every textbook reason for expecting a screened value supported it. The value was not screened. It was a different quantity, and it became the right one as soon as the omitted term was put back.

Still open: a geometry, and a hopping from allyl itself #

The obvious open question is still the geometry — now with the neighbour repulsions in place, which change where a bent, unequal radical’s middle spin turns over. A real distortion ties the overlap asymmetry, the end–end coupling and the end–end repulsion to its angles; with all three computed from a stated geometry, the path a bending radical follows through the model would be one curve, and whether it passes the turning point is a single calculation.

The nearer question is the hopping. The trio’s 2.91 eV comes from round overlaps, not from allyl, and the one thing the calculation cannot rule out is that the agreement at 11.13 eV rests on it. Computing the hopping from carbon 2p functions at 1.39 Å — one overlap integral and the same Wolfsberg–Helmholz rule — and repeating the one-parameter solution would say whether the agreement survives the one number in the model that was never allyl’s.