The warning a cheap calculation gives

A composite scheme — a correlation correction computed on a cheap reference system and added to a cheap calculation on a different one — can be measured for what it is worth. Near the reference it removes 99.74 per cent of the low level’s error; far from it, it is 15.5 times as wrong as the low level on its own.

That is a useful measurement and a useless rule, because it says what happened after the expensive calculation was run. What a practitioner wants is a warning, available beforehand, that the reference is about to stop being a reference.

There is an obvious candidate for a warning. A broken-symmetry mean-field solution is a signal that a single determinant is struggling, and it costs nothing to look at. This essay asks whether it works.

It does, and it works for a sharper reason than expected: the broken solution does not weaken as the two systems diverge. It disappears, at a definite value, and the transfer fails where it goes.

The setup: one correction, carried across #

Four sites in a ring, half filled, with an on-site repulsion $U$ and a site-energy modulation $\varepsilon$ that raises every other site and lowers the rest. The reference system is $\varepsilon = 0$; the target is the same ring at some non-zero $\varepsilon$.

The composite answer is the target’s mean-field energy plus the reference’s correlation energy — where the correlation energy is the difference between an exact diagonalisation and a mean field, both computed here rather than quoted. Its error is measured against the target’s own exact answer, and the share of the low level’s error it removes is the quantity reported.

The diagnostic is one number from the mean field: the spin polarisation of the broken-symmetry solution, which is how far the up and down densities differ from site to site. It requires no diagonalisation of a many-electron Hamiltonian, and on any system where a composite scheme would be used it is already available as a by-product of the low-level calculation.

That asymmetry is the whole reason the question is worth asking. The exact calculation on four sites is a small matrix and on anything real it is the expensive step — the one a composite scheme exists to avoid. A diagnostic is only a diagnostic if it can be read before paying for what it predicts, and a great many proposed ones cannot: a correlation energy is defined by a subtraction, so measuring it means doing both calculations, at which point there is nothing left to predict.

The collapse #

The first finding is about the diagnostic itself rather than about what it predicts.

Turn up the modulation at fixed repulsion and the polarisation falls, slowly at first, then faster, and then reaches exactly zero and stays there. It is a bifurcation and not a decay: below the critical modulation the self-consistency has a spin-polarised solution and above it the only solution is the symmetric one.

The critical field rises with the repulsion, which is the check that the mechanism is what it looks like — the modulation has to overwhelm the repulsion before the broken solution goes:

It is located here by a scan and reported as a bracket of a tenth rather than by bisection, and that is not fussiness. The self-consistency stops converging exactly at the bifurcation, where the two solutions meet, so a bisection walks into the one field at which there is no answer and stops. A scan steps over it, records how many probes were refused, and returns the interval.

What it predicts #

Sorting the thirty-two systems by how far the target’s polarisation has moved from the reference’s:

Where the two are alike — a difference below 0.01 — the transfer removes between 93.2 and 99.9 per cent of the low level’s error. Twelve systems, all of them good.

Where they are unlike and the solution survives — a difference above 0.05, but the target still broken — the transfer removes between 39.2 and 76.4 per cent. Two systems, both of them poor.

Where the target has collapsed the transfer is erratic, from removing 81 per cent to being five hundred and seventy times as wrong as the low level.

There is a gap of nearly seventeen percentage points between the worst of the first group and the best of the second, so the diagnostic is a rule rather than a tendency: it says which side of a line a case falls on. And across the whole grid the two quantities rank together at 0.9025.

Within each repulsion, before the collapse, the relation is monotone. Every step of the field makes the transfer worse, without exception, so the diagnostic is a scale as well as a switch — up to the point where the thing being measured ceases to exist.

The number that improves for the wrong reason #

One row in the grid is worth separating out, because it is a trap the share-removed statistic walks into.

At a repulsion of 8, the transfer removes 39.2 per cent of the low level’s error at a modulation of 6 and 81.4 per cent at a modulation of 8. On that statistic the transfer got better. Its absolute error over the same step fell from 0.545 to 0.081.

Both numbers are correct and both are misleading, because the denominator moved. The low level’s own error at the two points is 0.896 and 0.432: the mean field stopped being wrong, so the correction had less to do and did a smaller share of a smaller job.

A “percentage of the error removed” is a ratio of two things that both move, and it is worth watching for exactly the reason two wrong numbers can make a right difference: a quantity defined as a difference is a quantity whose behaviour is the behaviour of two other quantities. The absolute error is the safer figure and it is the one the correlation above is computed on.

The underlying quantity — how far a single determinant is from the exact answer — grows with the repulsion, and it grows at a rate that can be written down before anything is diagonalised.

Why the diagnostic works #

The mean field breaks symmetry when a single determinant cannot describe the state — when the exact ground state has weight on more than one configuration and the best single determinant compromises by localising the spins. That is the same condition under which the correlation energy is large and system-specific, because it is made of a near-degeneracy rather than of the smooth short-range correlation that transfers well.

The reason a single determinant compromises rather than failing outright is worth stating, because it is what makes the signal appear at all. A determinant built from spatial orbitals that the two spins share cannot describe two electrons sitting on opposite sides of a ring; a determinant that lets the two spins use different orbitals can, at the price of no longer being an eigenfunction of the total spin. The mean field takes that price whenever it buys enough energy, and a mean field cannot get out of the way by any other route.

That is worth being precise about, because the diagnostic’s whole value is that it costs nothing and a reader is entitled to ask what it costs instead. What it costs is the spin symmetry of the reference, and the essay does not spend the broken solution’s energy on anything: the energy is used nowhere, the polarisation is used everywhere, and the two are not the same claim.

So the polarisation is not a proxy for the correlation energy’s size. It is a proxy for its kind. The two kinds can be separated: the smooth part is nearly the same in two similar systems and the near-degenerate part is not, and only the second is what a transfer gets wrong.

Two systems with the same polarisation have the same amount of the awkward kind, whatever their total correlation energies are, and the correction moves between them. Two systems on opposite sides of the collapse do not have the same kind of correlation at all, and no amount of it transfers.

What a practitioner should take from it #

Look at the low-level solution before trusting a composite. A broken-symmetry solution on the reference and a symmetric one on the target — or the reverse — means the correction is being carried across a bifurcation, and there is no reason it should survive.

A small change in polarisation is a licence and a large one is a refusal. The line here falls between 0.01 and 0.05 on a four-site ring; the numbers are the model’s, the shape of the rule is not.

And the collapse is where a calculation should be checked rather than corrected. Near it the low level is at its worst, the transfer is at its worst, and — the awkward part — the share of error removed can be rising. It is the one region where every summary statistic points the wrong way at once.

None of the three costs anything. The polarisation of a mean field is printed by every program that computes one, and the comparison between two of them is a subtraction. What makes it useful is not that it is clever but that it is free and prior, which is a rare combination: the alternative diagnostics for a composite scheme are all versions of running the expensive calculation and looking. Two corrections computed separately do not sum to the correction computed together, and knowing that in advance would have saved the same kind of trouble.

One more thing happens as the repulsion grows and it is worth naming because it is what the broken solution is reaching for: a charge gap opens that no one-electron model has a term for, computed in the insulator band theory cannot see. The broken-symmetry solution is a mean field’s attempt at that gap, which is why its disappearance is a statement about what the electrons are doing rather than about the solver.

From a rank correlation to a decision, and what the decision costs #

A rank correlation of 0.9025 is a statement about ordering, and nobody has to make a decision about an ordering. What has to be decided is whether to trust one number, so the diagnostic is worth restating as a rule with a threshold on it and then charged for its mistakes.

The rule the grid supports is a conjunction rather than a cut-off. Transfer the correction when the reference and the target are both on the broken side of the collapse and their polarisations differ by less than 0.05; otherwise run the expensive calculation. Both clauses are needed: a target that has collapsed has a polarisation of exactly zero, so a rule reading only the magnitude of the difference would treat a strongly asymmetric collapsed system and a mildly asymmetric broken one as the same case.

Applied to the thirty-two systems the rule admits twelve. Every one of those twelve removes between 93.2 and 99.9 per cent of the low level’s error, so among the cases it licences there is no failure at all — not a small failure rate, none. It refuses the two unlike-but-broken cases, which remove 39.2 and 76.4 per cent and deserve refusal, and it refuses all eighteen collapsed ones, which is where the cost sits.

That cost is real and worth naming rather than hiding, because a diagnostic that only ever refuses is free to be perfect. Some of the refused eighteen would have been fine — one of them removes 81 per cent — so the rule pays for its clean record in wasted calculations, and on a grid this small it wastes them on more than half the systems.

The trade is nonetheless the right way round, and the reason is the shape of the two errors rather than their frequency. A false refusal costs an expensive calculation whose result is correct. A false licence costs a wrong answer with nothing to indicate it is wrong — the failure documented above has no internal symptom, and a number 15.5 times worse than the uncorrected one looks exactly like a number that is right. Wasted work announces itself and a silent error does not, so an asymmetric rule is the correct response to an asymmetric penalty.

Which also settles what the 0.9025 is for. It is evidence that the diagnostic tracks the thing it is supposed to track across the whole range, and it is not the rule. The rule is a threshold and a conjunction, and its performance is not a correlation coefficient but two counts: twelve licences, none of them wrong.

What is quoted, and what is computed #

Nothing is quoted. There is no molecule in this essay and no measurement. The ring, the filling, the repulsions and the modulations are the model’s parameters; every mean field, every exact ground state, every correlation energy, every transfer error, the collapse fields and the rank correlation are computed.

The exact answers come from a full diagonalisation of the many-electron Hamiltonian on four sites, which is small enough to do without approximation and is the only reason the errors here can be quoted at all. That is the same smallest honest calculation that every exact correlation energy here rests on.

What this cannot say #

Four sites. A bifurcation on four sites is a sharp event and on a large system it is a crossover with a width. The diagnostic would then have a grey band rather than a line, and how wide it is is not something this can answer.

And the reference is one system rather than a set. A real composite scheme has a family of reference systems and picks the nearest, which is a much weaker demand on any one of them than transferring from a single fixed reference across a whole range. Which reference is chosen decides what the correlation energy even is, so a scheme with a choice of them has a diagnostic problem of a different shape: not will this transfer but which of these will transfer best.

One kind of perturbation. The target differs from the reference by a site-energy modulation and by nothing else. A real composite scheme transfers a correction between molecules that differ in geometry, in basis and in what atoms they contain, and whether the polarisation tracks those differences is a separate question.

No orbital relaxation in the correction. The correlation energy carried across is a number rather than a functional of the density, which is the crudest version of a composite scheme and the one measured in the composite test.

And a broken-symmetry solution is not a wavefunction. It has the wrong spin symmetry, its energy is a variational bound on nothing in particular, and it is used here purely as an indicator. Nothing about the numbers it produces is claimed to mean anything on its own.

What the test requires #

The grid contains cases on both sides of the collapse, or there would be nothing to separate.

Every target whose symmetry breaking is like the reference’s transfers better than every one whose is not, with more than ten percentage points between the two groups. Measured: 93.2 against 76.4.

The transfer error rises at every step of the field while the broken solution survives, at every repulsion — so the diagnostic is monotone before it is a switch.

And over the whole grid the two rank together above 0.6. Measured: 0.9025.

The broken solution collapses at a definite field at every repulsion, and the field rises with the repulsion, which is the check that the collapse is the modulation overwhelming the repulsion rather than a failure of the solver.

The refusal is a repulsion of zero. With nothing to repel there is no symmetry to break, so the polarisation must be zero at every field — and there is nothing to transfer either, because a mean field is exact for electrons that do not interact and the low level’s error must be zero to eight decimal places. Both halves are needed, because the first alone would pass on a calculation that had lost the correlation entirely.

Two things this argument leans on are computed elsewhere and are worth pointing at rather than redrawing. The near-degeneracy the diagnostic detects is two many-electron states approaching each other as the repulsion grows — the smallest many-electron calculation plots both — and that is exactly the situation a single determinant cannot describe. The polarisation is a mean field’s way of noticing it without ever computing the second state, which is the whole of why it is free.

And the modulation has a chemical reading. It is a difference in site energy, so it moves charge between unlike sites, and the repulsion opposes the move — a difference does not make a transfer computes how much moves at four repulsions. The collapse in this essay is the field at which the modulation wins that argument outright, and once it has won there is nothing left for the two spins to disagree about.

Still open: transferring across the repulsion #

The obvious open question is the other variable. Everything here transfers a correction across a change of site energy at fixed repulsion, and the diagnostic was tested against that one axis. Transferring across a change of repulsion at fixed site energy is the other half of the same square — named but not yet run — and the interesting question is whether the polarisation predicts that failure too, or whether it is specific to the modulation it was measured against. A diagnostic that works on one axis and not the other is a coincidence; one that works on both is a rule.

The nearer question is about the grey band a larger system would have. The collapse is sharp here because four sites is small enough for the mean field to have exactly two solutions, and on a longer ring the polarisation could go continuously to zero rather than stopping. Running the same scan at six and eight sites would say which — and if the collapse survives, the field it happens at is a number that can be compared across sizes, which would turn a diagnostic into a line on a phase diagram.