Spin below zero

A three-orbital radical keeps its unpaired electron’s level fixed when one end’s overlap with the middle changes, and puts the spin in the squared ratio of the two overlaps. Coupling the two ends as well dissolves that separation: the level feels the asymmetry, the spin feels the coupling, and the middle orbital acquires spin, growing as the square of the coupling. Both essays read the spin from one orbital — the singly occupied one — and both said what that leaves out.

The allyl radical, the molecule the trio caricatures, has a proton on its central carbon whose hyperfine coupling is positive: +4.06 gauss, where the terminal protons’ are about −14. A proton’s coupling has the opposite sign to the spin on its carbon, so the central carbon’s spin is negative. One determinant cannot produce that. Every doubly occupied orbital carries as much spin up as down, and the singly occupied orbital’s density is positive or zero everywhere.

The missing piece is a second determinant — a paired electron promoted into the singly occupied level — and to mix it in, electrons have to repel one another. With that added and solved exactly, the middle orbital goes negative at once, and the coupling between the ends acts on it at first order instead of second.

Keeping everything else the same #

The one-electron part is exactly the trio of the earlier essays: three orbitals at −13.6 eV, overlaps of 0.25 between each end and the middle, the Wolfsberg–Helmholz rule with K = 1.75, and optionally an overlap raised on one end and an overlap between the two ends. It is carried into the Löwdin-orthogonalised basis, where each of the three sites is a function of its own and overlaps no other, and an on-site repulsion U is added to each: a pair of electrons on one site costs U.

Three electrons with two spins up and one down have nine configurations, so the problem is solved exactly — every configuration kept, the doublet ground state found by diagonalisation. The spin on each site is the difference between its up and down occupations.

The first check is that nothing else changed. At U = 0 the exact spins equal the earlier essays’ one-determinant Löwdin spins to the ninth decimal, at every pair of overlaps tried — 0.236 and 0.764 on the ends for the unequal radical, 0.0116 on the middle when the ends are coupled as well. What follows is the second determinant’s doing and nothing else’s.

The middle orbital goes negative #

In the symmetric radical, with no coupling between the ends, the singly occupied orbital has a node on the middle and one determinant puts exactly zero spin there. Switch on the repulsion and the middle goes negative immediately: −0.060 at 2 eV, −0.140 at 5, −0.200 at 8, −0.251 at 12.

The mechanism has a standard description and the calculation shows it working. The unpaired electron sits on the two ends. A paired electron in the bonding orbital, which has weight on the middle, avoids it better if its spin is parallel — it cannot occupy the same site as a parallel electron at all — so the bonding pair rearranges: more up-spin density goes to the ends where the unpaired up-spin electron is, and the down-spin density left behind is on the middle. The total stays one. The ends gain what the middle goes below zero: 0.570 each at 5 eV against a half.

Against allyl #

The measured couplings fix the sign; a number needs McConnell’s relation, $a = Q\rho$, with $Q$ for a carbon’s own proton between about −27 and −22.5 gauss depending on who calibrated it. The central coupling of +4.06 gauss then gives a spin of −0.150 to −0.180 on the central carbon, and the terminal couplings, 13.93 and 14.83 gauss, give 0.53 to 0.64 on each end — consistent with the three summing to one.

The trio reaches that band at a repulsion between 5.5 and 6.9 eV. That is about six tenths of a free carbon atom’s difference between ionisation energy and electron affinity, 10.0 eV, which is what an on-site repulsion would be with nothing screening it. A model with repulsion only on the site, and three electrons only, is expected to want less than the bare value: the σ electrons and the repulsion between neighbouring sites, both absent here, soften it. So the number is plausible without being a measurement of U — the trio has one parameter left and allyl fixes it.

That one comparison is what this model can offer as evidence, and it is worth being exact about what it tests. It tests that the sign and size of the negative spin come out right for a repulsion of reasonable size. It does not test the overlaps, which are the earlier essays’ round numbers rather than allyl’s, and it cannot test anything about the coupling between the ends, because allyl’s end-to-end overlap is small and its effect on the central spin, computed below, is a few thousandths.

The coupling acts at first order #

The question the earlier essay left was whether the negative spin grows with the coupling between the ends, as the one-determinant leak does, or ignores it. It grows, and at a lower order than the leak.

In the symmetric radical the one-determinant spin on the middle is exactly zero at every coupling, because the singly occupied orbital’s node there is protected by the symmetry between the ends, and coupling them preserves that symmetry. The polarisation is not protected. At 5 eV the middle orbital’s spin falls from −0.140 with no coupling to −0.151 at an end–end overlap of 0.1, and it falls in proportion: the rate at zero coupling is −0.134 per unit overlap, and the curvature is small beside it.

The likely reason is the one the earlier essays already measured from the other side. The coupling between the ends moves the singly occupied level at first order and leaves its shape alone, and the weight of the second determinant depends on where that level sits relative to the paired one. That link is not separated out here — no calculation holds the level fixed while the coupling changes — so it is offered as the reading of the numbers rather than as their proof.

The leak and the polarisation, in one bent radical #

The earlier essay’s leak needs both changes at once: an unequal pair of end overlaps, and a coupling between the ends. So the two effects meet in the case a real bent radical is — one end closer to the middle than the other, both ends near enough each other to interact.

They have opposite signs and start at different orders. The leak is positive and starts as the square of the coupling, with a curvature of 2.1 per unit overlap squared; the polarisation is negative and starts in proportion. So as the coupling grows the middle orbital’s spin first falls — the polarisation’s first-order term wins while the coupling is small — reaches its lowest point, and rises as the leak’s square overtakes it.

At 5 eV the lowest point is at an end–end overlap of 0.043, and the spin returns to zero at 0.198. At 2 eV the turn comes earlier, at 0.026, and the leak wins outright at 0.148. At 8 eV the spin is still negative at 0.2. So the two effects cancel rather than reinforce, and where the cancellation happens is set by the repulsion — which makes the middle orbital’s spin in a bent unequal radical a quantity whose sign depends on a parameter no one-determinant calculation contains.

The unequal ends drain the polarisation #

There is a second interaction between the two effects, and it runs the other way. The asymmetry that makes the leak possible also reduces what there is to cancel it. With one end’s overlap raised from 0.25 to 0.45 and no coupling between the ends, the middle orbital keeps between a third and two fifths of the symmetric radical’s negative spin — −0.050 against −0.140 at 5 eV, −0.070 against −0.200 at 8.

The first-order effect of the coupling is far less affected. At 5 eV its rate is −0.134 per unit overlap with equal ends and −0.116 with unequal ones — an eighth less, where the spin it acts on is two thirds less. The coupling’s effect on the polarisation is nearly a property of the coupling; the polarisation’s size is a property of the asymmetry. Both rates are largest between 7 and 10 eV and fall at higher repulsion, where the electrons are frozen onto their sites and have less freedom to rearrange at all.

The level the count held, no longer held #

This argument began from a level that could not move. With one end’s overlap raised, the trio’s middle level stays exactly at the free-atom energy, because the rows belonging to the two ends stay proportional and a count of orbitals forces one level to sit there. At three electrons that level is the radical’s ionisation energy and its attachment energy at once, and both were constant across the whole sweep while the spin moved from a half to under a quarter.

The exact solution can ask the same question of the charged states, since the two- and four-electron ground states are as easy to solve as the three-electron one. With no repulsion the answer is the old one to the ninth decimal: both energies are 13.6 eV at every asymmetry.

With repulsion it moves. At 5 eV the ionisation energy rises by 0.25 eV as the overlap is raised from 0.25 to 0.45, and the attachment energy falls by about as much; at 2 eV the rise is 0.13; at 8 eV the ionisation energy first dips by two hundredths and then rises by 0.21. The count still holds a one-electron level at 13.6 eV — the Wolfsberg–Helmholz problem underneath has not changed — but no measurable energy is that level any more. Removing an electron now also changes how the remaining two avoid each other, and how well they can avoid each other depends on how the orbitals are shared between the sites, which is exactly what the asymmetry changes.

So the separation the earlier essays drew — an energy held by a count, a spin set by a ratio — was a separation of one-electron quantities, and repulsion couples them from both sides. The spin acquires a negative part the ratio knows nothing about, and the energy acquires a dependence on the ratio that the count forbade. Neither effect is small at repulsions that describe carbon: a quarter of an electronvolt in an ionisation energy is larger than many substituent effects chemists measure, and a fifth of an electron’s spin on the wrong side of zero is what a hyperfine spectrum reports first.

How the numbers were made #

The Wolfsberg–Helmholz Hamiltonian and overlap matrix of the earlier essays are transformed by the inverse square root of the overlap matrix into an orthogonal basis; an on-site repulsion is added in that basis; the nine configurations of three electrons with a total spin projection of a half are built with fermion signs from the order of occupied sites; and the lowest state is found by Jacobi diagonalisation. The rates are central differences in the end–end overlap at zero, the turning points a golden-section search, and the allyl repulsion a bisection on the measured band.

The checks, made wherever these figures are drawn: at U = 0 every site spin equals the one-determinant Löwdin spin to 10⁻⁹; the three spins sum to one at every setting; the middle orbital is negative at every repulsion above zero and every coupling in the symmetric radical; its rate in the coupling is negative at every repulsion; the leak alone starts with no linear term and a curvature above 1.5; the unequal radical turns at an interior coupling between 0.02 and 0.08 for repulsions from 2 to 8 eV; the asymmetry leaves under two fifths of the polarisation; and allyl’s band is reached between 5 and 7 eV. The two- and four-electron energies equal the free-atom level at every asymmetry with no repulsion, and move by more than a fifth of an electronvolt at 5 eV. The refusal is the symmetric radical without repulsion, whose middle orbital must carry no spin at any coupling — one determinant cannot put negative spin anywhere, and its node there is exact.

What the three sites leave out #

Repulsion on the site only. A pair of electrons on neighbouring sites repels too, and in a π radical that repulsion is not small. It would compete with the on-site term — a neighbour repulsion favours charge on alternate sites — and the band of U that matches allyl would move. The model’s one parameter absorbs everything the model leaves out, which is why the match is a consistency check rather than a determination.

Three electrons in three orbitals. The σ framework is frozen, and it is polarised by the π spin too; the central proton’s coupling is transmitted through a σ bond, which is what McConnell’s Q summarises. A different Q moves the band by a fifth.

Löwdin sites. The spin is read on orthogonalised functions, and the earlier essay showed that Mulliken and Löwdin spins part as soon as the middle orbital’s energy differs from the ends’. Here the three are equal, so the two partitions agree at U = 0. With repulsion the state is built on the Löwdin sites and the numbers are Löwdin’s by construction; a Mulliken reading would need the density carried back onto the original, overlapping functions, which is not done here.

And no geometry. An end–end overlap of 0.043 and an asymmetry of 0.2 are positions in the trio’s parameter plane, not angles. A real bent radical moves along one line in that plane as it bends, and where that line meets the turning point is a question about a molecule.

The zero was a property of one determinant #

The earlier essays found the middle orbital’s spin exactly zero in the symmetric radical and traced it to a node the symmetry protects. The node is real. The zero was the node’s only while the spin was the node’s orbital’s density: one determinant, and the spin is whatever that orbital’s shape says. With a second determinant the spin is a property of the whole state, the node no longer decides it, and the middle orbital’s spin becomes the most sensitive number in the radical — it changes sign between models, it responds to the coupling at a lower order than anything else, and it is the one a hyperfine spectrum measures with its sign attached.

That is the same shape as the hole two electrons dig around each other without any repulsion, seen from the opposite side: there, a correlation that looks like repulsion turned out to be the Pauli principle alone; here, a spin that looks like an orbital’s turns out to need the repulsion to exist.

Still open: the line a real radical follows, and the neighbour repulsion #

The obvious open question is still geometry. A bent allyl-like radical ties the asymmetry and the end–end coupling to its angles and bond lengths, and with overlaps computed from those the trio would trace one curve through the plane drawn here. Whether that curve passes the turning point — whether a real distortion ever sees the middle orbital’s spin fall and then rise — is a calculation with overlaps from a stated geometry, each of which is one integral between two orbitals at a stated distance.

The nearer question is the neighbour repulsion. Adding a repulsion V between adjacent sites is one more diagonal term in the same nine configurations, and it would say whether the band of U that matches allyl survives a second parameter or moves to meet the free-atom value — which decides whether the six electronvolts found here describe carbon or describe the model’s omission.