The half of the square a ring of four cannot show

The warning a cheap calculation gives costs nothing. A composite scheme computes an expensive correction on a cheap reference system and adds it to a cheap calculation on a different one, and whether that is allowed had previously been knowable only afterwards. The mean field’s own spin polarisation says beforehand: it collapses at a definite site-energy modulation, and the transfer fails where it goes.

It also said what it had not done. Everything was measured across a change of site energy at a fixed repulsion, and the other half of the same square — a change of repulsion at a fixed site energy — was named and not run. A diagnostic that works on one axis and not the other is a coincidence; one that works on both is a rule.

It works on both. The rule is not one rule, and the system the warning was first measured on is the one that could not have shown either fact.

The other axis has a threshold, on the other side #

A broken-symmetry solution exists below a critical site-energy modulation: turn the modulation up and it collapses onto the symmetric one. On the repulsion axis the inequality runs the other way — it exists above a critical repulsion, because the repulsion is what makes the symmetric solution unstable in the first place.

So the two halves of the square are mirror images and not the same statement twice. The site-energy axis fails by turning something up; the repulsion axis fails by turning something down.

Both thresholds are located by stepping over the bifurcation rather than bisecting onto it, and the reason is the same on both axes: the self-consistency is a fixed-point iteration, and exactly at the bifurcation the two solutions meet and there is no answer to converge to. A bisection walks straight into it.

A chain of four breaks above a repulsion of 1.6725 and a ring of six above 2.3550. A ring of four has no threshold at all: its half-filled shell is degenerate to $3 \times 10^{-17}$, so the symmetric solution is unstable at any repulsion however small, and the broken one survives down to 0.25 with a polarisation of 0.516.

That is the system the warning was first measured on. Its site-energy axis has a collapse and its repulsion axis has none, so half the square is invisible from it and nothing measured on it could have said so.

The diagnostic does predict the other failure #

Run the same transfer on a system that does have a threshold, and the diagnostic behaves exactly as it did on the first axis.

Below the threshold the composite removes −1,538 per cent, −237 per cent and +4 per cent of the low level’s error at repulsions of a half, one and one and a half. A negative share is a composite that is worse than the cheap calculation it was meant to improve — the failure the practice’s own assumption forbids — and at a repulsion of a half it is sixteen times worse.

Above the threshold it removes 81, 95, 97, 89, 87 and 92 per cent on the way up to the reference, and 80, 57 and 33 per cent on the way past it.

The ring of six gives −2,581, −493, −106, −24 and +29 per cent below its threshold and 90, 90, 84, 90, 77, 53 and 28 per cent above it. Two systems, two thresholds, the same behaviour on each side of each.

So the answer to the question is yes. The diagnostic is not a coincidence of the site-energy axis; the mechanism — the transfer holds while both systems are broken and fails once the target is not — is the same on both.

And it is not one rule #

Being right on both axes is not the same as being one number that means one thing. The way to ask that is to put both axes’ targets on one pair of coordinates.

The two marked points are the worst case. A target reached along the repulsion axis sits at a polarisation difference of 0.12546 and removes 96.5 per cent of the low level’s error. A target reached along the site-energy axis sits at 0.13945 — the same diagnostic to within a ninth — and removes 39.2 per cent. Their errors differ by a factor of 46.

That is a diagnostic that ranks and does not calibrate. Within one axis it is a scale: the further the polarisation has moved, the worse the transfer, monotonically. Across the two it is at best a switch — broken or collapsed — and reading its numerical value as a measure of risk would give a practitioner ninety-six per cent confidence in a case that is about to lose sixty.

What the two failures are made of #

The two axes fail differently and the difference is legible in the shapes above.

On the repulsion axis, the failure on the weak-coupling side goes negative — the composite is worse than the low level — while the failure on the strong-coupling side merely degrades, from 100 per cent to 33 as the repulsion is tripled. The reason is that the correlation energy the reference carries is a large number and the target’s is a small one: at a repulsion of a half the exact and mean-field energies differ by 0.026, and a correction of the reference’s size dropped onto it is a much larger error than the one it replaced.

On the site-energy axis the failure is the reverse in an important way: the low level’s error stays large while the composite’s grows, so the share removed falls without going negative until very late. The site-energy measurement found exactly that and warned about the denominator — the share removed improves at one point while the absolute error falls, because the low level has stopped being wrong.

So the honest summary of a composite scheme’s risk needs two numbers and not one: how far the mean field’s symmetry breaking has moved, and which way the correlation energy itself is going. The first is the diagnostic; the second is what turns it into a size.

Why a degenerate shell has no threshold #

The ring of four’s exemption is worth deriving rather than reporting, because it is what decides which systems this whole diagnostic can be measured on.

A mean field breaks spin symmetry when the energy gained by putting the two spins on different sites beats the energy lost by no longer occupying the best symmetric orbitals. The loss is set by the gap between the highest occupied and lowest unoccupied one-electron levels; the gain is set by the repulsion. So the condition is a competition, and there is a critical repulsion where the two balance.

Unless the gap is zero. A ring of four at half filling has two levels at exactly the same energy — its one-electron spectrum is $2, 0, 0, -2$, and the two zeros are the shell the last two electrons go into — so there is no loss to beat and any repulsion at all breaks the symmetry. The same degeneracy is what makes a ring of four distort and what makes its Hückel treatment predict a triplet ground state; it is the most-used fact about that system, and its consequence for a symmetry-breaking threshold had not been noticed.

A chain of four has levels at $\pm 1.618$ and $\pm 0.618$, so its gap is 1.2361 and its threshold is 1.6725. A ring of six has levels at $2, 1, 1, -1, -1, -2$, a gap of 2.0000, and a threshold of 2.3550. The thresholds are not proportional to the gaps — 1.35 and 1.18 times them — so the competition is not a one-line inequality in this model; but the ordering is right, the larger gap needs the larger repulsion, and the zero gap needs none.

That also settles which systems the warning could have been measured on. Of the three here, only the ring of four is small enough for the exact solver and degenerate at half filling, and it was chosen for the first reason. A degeneracy at the Fermi level is the condition for an instability in every such model, and it is the condition for this one too — which means the system chosen for being the simplest is the system in which the instability is unconditional.

That last point is worth sitting with, because it inverts the reason the system was picked. A ring of four is the standard demonstration system for correlation — it is four sites, its exact ground state fits on a page, and its mean field breaks. Every one of those properties is why it was chosen, and none of them was chosen with a threshold in mind. But the property that makes it tractable, its size, is the same property that puts two levels at exactly zero, and a threshold is a competition against a gap that this system does not have. The demonstration system is the one system on which half the diagnostic is invisible.

There is no way to fix that by measuring the ring of four more carefully. The exemption is exact and it is structural: the threshold is not small, not hard to resolve, and not hidden by the scan’s resolution — it is zero, and a scan of four hundred repulsions between zero and three finds the symmetry already broken at the first of them. The only remedy is a second system, which is why the chain of four and the ring of six are here at all, and why the two of them rather than one: two points do not establish the relation between a gap and a threshold, but they do establish that there is one and that the ring of four sits at its degenerate end.

What was computed, and how #

Everything is at half filling on four or six sites, with the whole configuration space written down and diagonalised: nothing truncated, nothing selected, fermion signs kept and checked by requiring the matrix to be symmetric. The low level is the unrestricted mean field with the symmetric solution allowed to break, and the high level is that exact diagonalisation, so the correction transferred is the difference between two calculations of the same system by the two methods.

Each threshold is a scan of four hundred repulsions rather than a bisection, and the repulsions at which the self-consistency refuses to converge are counted and skipped. A refusal there is not a failure of the solver: it is the bifurcation, and stepping over it is the only way to bracket it.

The composites on six sites are cached between builds, and the cache verifies what it restores by recomputing one exact ground state from its own arguments — which is the half of the calculation a stale entry could be wrong about without any sign.

The refusal is the reference itself. At the reference repulsion the transferred correction is the target’s own, so the composite must be exact and the polarisation difference must be exactly zero. A scheme that gave anything else there would be assembled the wrong way round, and every measurement downstream would be of the assembly.

Where the model stops #

The correction being transferred is also a single number rather than a functional form. Nothing here interpolates, extrapolates or reweights it: the difference between the two methods is computed once at the reference and added to the low level everywhere else, which is the crudest composite anybody would build and is exactly the one the warning priced. A scheme that scaled the correction by a ratio of gaps would behave differently on both axes, and the failure measured here is not evidence about it.

Four and six sites are four and six sites. Whether a ring of four’s degenerate shell is a peculiarity of that size or the ordinary state of a system with an open shell is not answered here — what is answered is that it is the wrong system to have measured a threshold on, which is a statement about the original measurement rather than about chemistry.

The mean field here is the unrestricted one, and an unrestricted solution that breaks spin symmetry is not an eigenfunction of the total spin. Everything a broken-symmetry solution buys is bought with that, and the polarisation used as a diagnostic is a property of a wavefunction that is, strictly, of the wrong symmetry.

And a Hubbard repulsion is one number. A real composite scheme moves between systems in many ways at once — a different basis, a different geometry, a different element — and the square measured here has two axes because the model has two parameters. A rule that fails to be one rule across two axes has no reason to become one across ten.

The generalisation #

Two things are worth carrying, and they are about method rather than about correlation.

A diagnostic must be tested on the axis it was not built on. The warning was built on a site-energy axis and it holds on a repulsion axis, which is real evidence that it is about the physics — and the same test showed that the calibration does not transfer, which no amount of further work on the first axis could have revealed.

And a system chosen because it is small enough may be small enough to hide the phenomenon. A ring of four is the smallest system that can carry a composite scheme, which is why it was chosen, and its degenerate half-filled shell removes one of the two thresholds the question is about. That is the same shape as a constant measured on the one lattice whose spectrum is symmetric and a search calibrated on the one size where the landscape has two basins: the convenient case is convenient because something has degenerated, and what degenerated is usually the thing being asked about.

Who found it, and when #

The instability of a symmetric mean field above a critical interaction is Stoner’s, from 1938, and its molecular form — the point at which an unrestricted Hartree–Fock solution departs from the restricted one — is the Coulson–Fischer point of 1949. That the threshold is at zero interaction when the shell is degenerate is the standard statement of why an open-shell system has no restricted solution worth having.

Composite methods in the modern sense date from the Gaussian-n schemes of the early nineteen-nineties, and their transferability has been argued about ever since, mostly by benchmarking. The observation that the mean field’s own symmetry breaking is a cheap diagnostic for it comes from the site-energy measurement, and what is added here is the boundary: it is a diagnostic on both axes and a calibration on neither.

Still open: both axes at once, and a second number #

The obvious open question is the corner. Everything here moves along one axis at a time, and a real change of system moves along both — a different repulsion and a different site energy. If the diagnostic is a switch rather than a scale, then the two-dimensional version should be a region rather than a curve, and its boundary is where the broken solution disappears in the plane. That boundary is a line the same scan can trace, and the interesting question is whether a target inside it transfers well however far inside it sits.

The nearer question is the second number the shapes above ask for. If the size of a transfer failure needs the change in the correlation energy as well as the change in the polarisation, then the product or ratio of the two ought to collapse the scatter that the polarisation alone leaves. Both numbers are already computed for every point here, so the test is arithmetic rather than a calculation — and a diagnostic that collapses both axes onto one curve would be the rule the warning was looking for.