The spin the count does not hold

Three orbitals in a row — two ends, A and B, that do not touch, and a middle orbital C that overlaps both — have three levels, and one of them sits at the energy of the free ends exactly. That level was long read as a symmetry statement: the two ends are equivalent, so their antisymmetric combination has nothing to mix with. Changing one end’s energy moves it at once, which looked like confirmation. Then changing one end’s overlap with the middle instead of its energy left the level exactly where it was, at every size of change and every energy of the middle orbital, with no symmetry left anywhere. The level was held by a count, not a group: two rows of the secular determinant stay proportional whatever the overlaps to C are.

That settles the energy. It says nothing yet about the orbital. The combination the count holds is $S_{BC}\,A - S_{AC}\,B$ — the one mixture of the two ends that C cannot see — and its composition depends on the overlaps even though its energy does not. With four electrons the level is full and the composition hardly matters. With three it is the radical’s singly occupied orbital, and its composition is where the unpaired electron is. That is a measurable thing: an electron spin resonance spectrum reads the unpaired spin near each nucleus.

So the same change that leaves the energy exactly alone is a change to the radical’s spin distribution — by how much, and whether anything else moves it, is a calculation on numbers the trio already has.

The level stays and the spin goes #

The trio is the one used throughout: A and B at −13.6 electronvolts, both overlapping C by 0.25, the coupling taken from each overlap by the Wolfsberg–Helmholz rule with K = 1.75, and the overlap kept in the secular problem. A’s overlap with C is raised in steps to 0.45 while B’s stays at 0.25, and the trio is filled with three electrons.

The two outer levels do what levels do when a coupling grows. The lowest falls from −16.26 to −17.07 electronvolts and the highest rises from −8.02 to −2.78. The singly occupied level is at −13.600 at every step, to rounding.

The spin is not. At equal overlaps it sits half on A and half on B, as it must. At 0.30 against 0.25 it is 0.410 on A and 0.590 on B; at 0.35, 0.338 and 0.662; at 0.45, 0.236 and 0.764 — nearly a quarter and three quarters. The electron moves towards the end that overlaps C less, which is what the orbital’s formula says: the amplitude on A is proportional to B’s overlap, and on B to A’s. The spin on A is therefore

$$\rho_A = \frac{S_{BC}^2}{S_{AC}^2 + S_{BC}^2}$$

and the open circles on the lower panel are that expression, sitting on the computed curve to every digit. The rate at the symmetric point is −1/(2S), which at S = 0.25 is two units of spin per unit of overlap: a one per cent change in one overlap moves half a per cent of an electron from one end to the other.

None of it reaches C. The singly occupied orbital has no amplitude on the middle atom at any overlap, so the unpaired spin is entirely on the two ends, divided between them by a ratio the energy cannot see.

One change moves the level, one the spin, one both #

The overlap change is one of three ways to make the ends of the trio different, and the level’s response to all three is already known: nothing for the overlap, half the change for a site energy, and first order at (K − 1)·13.6 = 10.2 electronvolts per unit of overlap for a direct coupling between A and B. The spin is the second reading, and the three changes separate on it just as cleanly.

Take each at a size a substituent or a bend might produce. Raising A’s overlap with C by 0.05 moves the level by nothing and the spin on A by −0.090. Raising A’s site energy by 0.2 electronvolts moves the level by +0.102 eV and the spin by only +0.012 — at a rate of 0.067 per electronvolt, so a site change large enough to move the level by a tenth of an electronvolt shifts about one per cent of an electron. Coupling A to B directly, at an overlap of 0.05, moves the level by +0.537 eV, five times the site change, and the spin by nothing at all.

The last is symmetry doing exactly what it does. An A–B coupling touches both ends identically; the two ends stay equivalent, so the spin split must stay even whatever happens to the level. The overlap change is the mirror image: the count holds the energy and nothing holds the composition. The site energy sits between, moving both, and moving the spin far less per unit of level shift than intuition about “inequivalent ends” would suggest.

So the three quantities an experiment might measure on such a radical — its ionisation energy, the energy of its singly occupied level, and the ratio of spin on its ends — are not three readings of one asymmetry. An end-to-end coupling reads on the energy and not on the spin; an overlap asymmetry reads on the spin and not on the energy. A spectrum showing an even spin split says nothing about whether the ends couple, and a level at the free-atom energy says nothing about whether the overlaps are equal.

Nothing about the middle orbital enters #

The split has no energy in it. The formula contains the two overlaps and nothing else — not C’s energy, not the coupling constant, not the energies of the outer levels — and that can be checked rather than read off.

With A’s overlap at 0.45 and B’s at 0.25, the energy of C is moved from −24 electronvolts, far below the ends, to +2, far above them, at Wolfsberg–Helmholz constants of 1.5, 1.75 and 2.0. Across those twenty-one trios the outer levels range over more than thirty electronvolts, from −26.5 to +6.3. The spin on A is 0.2358 in all twenty-one, and the spin on C is zero in all twenty-one. The split does not care whether C lies below the ends, between them or above them; it cares only how much each end overlaps it.

The reason is the same line of algebra that holds the energy. With both ends at one energy and no coupling between them, the rows of H − αS belonging to A and B each have a single non-zero entry, in C’s column, and that entry is the overlap times a factor that is the same for both — a factor containing C’s energy and the coupling constant, which cancels in the ratio. What survives is the ratio $S_{BC} : -S_{AC}$. That also marks where it stops: a coupling rule in which the two ends’ couplings are not the same multiple of their overlaps would put an energy back into the ratio.

Two partitions, one answer #

A spin density “on an atom” is a partition. The unpaired electron occupies one orbital spread over three overlapping functions, and dividing it among atoms needs a rule for the overlap terms. The two common rules can disagree badly, and ionic weights of a bond have been shown differing by a factor of six between conventions on one wavefunction. Here the Mulliken partition, which splits each overlap term evenly between its two atoms, and the Löwdin partition, which first orthogonalises the basis symmetrically, were both computed at every step.

They agree to every digit printed. That is not luck. The singly occupied orbital has no amplitude on C, and A and B do not overlap each other — the arrangement in which a bond order appears between two atoms that do not interact — so every overlap term in the orbital’s population is a product with a zero in it and there is nothing for a Mulliken rule to share. For Löwdin the statement is sharper: the overlap matrix acting on this orbital returns the orbital unchanged, because the only off-diagonal entries connect A and B to C and the orbital’s components along them cancel. It is an eigenvector of the overlap matrix with eigenvalue one, and so of its square root, and every partition built from the overlap matrix gives the same split.

That makes the spin on the ends of this radical one of the rare atomic populations that is not a convention. It is also a warning about how rare: add a small A–B overlap, or give C a share of the singly occupied orbital, and the partitions part company again.

What a hyperfine ratio would measure #

An electron spin resonance spectrum does not report orbital energies. It reports hyperfine couplings, and in the simplest reading a coupling is proportional to the unpaired spin on the atom it belongs to. The ratio of two couplings is then the ratio of two spin densities, with the proportionality constant cancelled.

For this trio that ratio is the squared ratio of the two ends’ overlaps with the middle. Every point of the sweep lies on a line of slope two: at 0.45 against 0.25 the spin ratio is 3.241, which is 1.8 squared. A spin ratio of the two ends measures the ratio of their overlaps with the middle, squared, and measures nothing about energy. Read the other way, a radical whose ends show equal hyperfine couplings has equal overlaps to its middle to within the square root of the measurement’s error — whatever its ionisation energy says, and whether or not its ends couple to each other directly.

The square root is worth noticing, because it works in the measurement’s favour. A spin ratio known to ten per cent pins the overlap ratio to about five, so the spin is the more sensitive instrument of the two quantities it connects. The energy, which is the quantity usually reported first, is the least sensitive of all: for this change it has no sensitivity whatever, and a radical could have its overlaps pulled apart by a factor of two without its singly occupied level moving by a millielectronvolt.

The radical gets more stable and no easier to ionise #

The total energy does change. The lowest level falls as A’s overlap grows, and it holds two electrons, so the radical’s total orbital energy falls by 1.60 electronvolts across the sweep. So do the energies of the two-electron and four-electron trios, and by the same amount.

The three curves are parallel. Removing the unpaired electron costs the singly occupied level, and adding a second electron to it releases the same level, so at every overlap the gap from two electrons to three and from three to four is −13.6 electronvolts exactly. In the orbital picture the radical’s ionisation energy and its electron affinity are both pinned by the count while its stability grows and its spin moves to one end. A filled shell was shown to carry information its energy hides; the half-filled level carries it in its composition.

How the numbers were computed #

Each trio is the three-by-three generalised eigenproblem Hc = ESc, with the overlaps in S, the diagonal of H at the site energies, and each off-diagonal element of H set to K times the overlap times the mean of the two site energies. It is solved by symmetric orthogonalisation. The levels are sorted and filled two to a level; with three electrons the middle level holds one. Its eigenvector is normalised against S, the Mulliken spin on atom i is $c_i(Sc)_i$, and the Löwdin spin is ((S^½c)ᵢ)², with S^½ built from the overlap matrix’s own eigenvectors. Rates are central differences with a step of 10⁻⁵.

The checks, run wherever these figures are drawn. The singly occupied level stays at −13.6 eV to 10⁻⁹ across the overlap sweep and at all twenty-one settings of C’s energy and K. The spin on A equals the squared-overlap ratio to 10⁻⁹ at every one, the spin on C is zero, and the two partitions agree. At 0.45 against 0.25 the spin on A is under a quarter. The A–B coupling moves the level by more than an electronvolt at an overlap of 0.1 and leaves the spin at a half. The level’s rate against the overlap is zero, the spin’s is −1/(2S), the site energy moves the level at a half and the spin at under a tenth per electronvolt, and the A–B coupling moves the spin at zero. The gaps from two electrons to three and three to four both equal the level, while the radical’s total energy falls by more than an electronvolt. The refusal is the symmetric trio: at equal overlaps both partitions must put exactly half the spin on each end, or the partition would be inventing the asymmetry being measured.

Where the trio stops being a radical #

One determinant, no spin polarisation. The spin here is the density of a single orbital. A real radical’s unpaired electron polarises the paired electrons beneath it, and that puts spin — sometimes negative spin — on atoms the singly occupied orbital never reaches. The allyl radical, the molecule this trio caricatures, shows a hyperfine coupling to the hydrogen on its central carbon, which a spin density of exactly zero there could not produce; spin polarisation accounts for it, and puts negative spin on that carbon where this model puts none. The split between the ends is the orbital’s contribution, not the whole spectrum’s.

The coupling rule is proportional to overlap. The ratio loses every energy only because each end’s coupling to C is the same multiple of its overlap. Overlap and interaction are different quantities, and any rule that lets them vary separately — a different site energy on each end, or a coupling that is not Wolfsberg–Helmholz — puts C’s energy back into the split.

An overlap is not a geometry. Raising one overlap from 0.25 to 0.45 stands for bringing one end closer or turning it towards the middle, and no structure is attached to it here. Which real distortions change one overlap by that much without also changing a site energy or introducing an end-to-end coupling is a question about molecules, and it is the question that decides whether the clean separation above is ever seen cleanly.

Energy is counted and composition is weighed #

The count that holds the level is a statement about ranks — two rows of a matrix staying proportional — and ranks do not see magnitudes. The orbital that satisfies it is a vector, and a vector’s components are magnitudes. The count fixes where the level is and says nothing about what it is made of, and on a half-filled level what it is made of is an observable. Any argument from an unchanged energy to an unchanged distribution skips that difference, and the trio shows it skipping a factor of three in the spin on one end.

The second lesson is that asymmetries are not one thing. Three ways of making the ends of a radical different read on the energy and the spin in three different proportions, from all of one and none of the other to the reverse, and a measurement of either alone assigns a cause it cannot distinguish.

Still open: a coupling between the ends and an unequal overlap together #

The obvious next case combines the two clean changes. An A–B coupling moves the level and not the spin; an overlap asymmetry moves the spin and not the level. With both present the rows of the determinant are no longer proportional and the ends are no longer equivalent, so the level and the spin both move — and whether they move as the sum of the two separate responses or interfere is a two-parameter sweep of the trio already built. A bent radical with one end nearer the middle than the other is that case.

The nearer question is the spin the model leaves out. A second determinant — the lowest excitation that promotes a paired electron into the singly occupied level — is the smallest change that can put negative spin on C, and the trio has only three orbitals to excite among. Whether that negative spin grows with the overlap asymmetry or ignores it, the way the level ignores it, is a question the three-orbital problem can answer exactly.