A constant ties the polarisation

A composite method runs a mean field on the system it wants and adds a correlation correction computed somewhere else. The transferability square has asked for essay after essay whether anything the mean field produces for nothing can warn when the borrowed correction will be wrong, and its most recent answer undermined the question. Moving the reference moved the polarisation’s worst ratio by a factor of twenty-two, more than any change of method ever had, because those ratios divide by near-zero errors and take an offset set by the reference.

That essay ended by proposing the repair: score each candidate by something absolute, a rank correlation with the size of the error for instance, which has neither a denominator to game nor an offset to take. It had a number ready, and the number was already a warning. The polarisation’s rank correlation with the error ran from 0.47 to 0.94 across those references. A score that was meant to be free of the reference moved by a factor of two with it.

Why the size of the error cannot be the target #

The reason takes one line, and every grid on the square has always obeyed it. Write $c$ for the correlation energy a mean field misses, the exact energy less the mean-field energy, so that it is negative. Borrow a correction from a reference and add it to a system’s mean field. The error is

$$e(\mathrm{sys})=c(\mathrm{ref})-c(\mathrm{sys}).$$

So the signed error, plotted against the correlation energy the system misses, is one straight line, and all a different reference does is slide it up or down.

That holds to a part in a thousand million at every system of every grid tried. It explains the eight references’ disagreement as soon as an absolute value is taken: $|e|$ folds that line at the system whose $c$ equals the reference’s, and the fold moves as the reference moves. Whichever systems sit near the fold have small errors. The candidates themselves are scored as distances from the reference, so their zero moves too.

So a score can be free of the reference in only one way: drop the reference from it. What a diagnostic should predict is $c(\mathrm{sys})$, which is a property of the system and of nothing else. Any composite built on a diagnostic is, after all, a function $g(d)$ that estimates exactly that. And a score of $g$ against $c$ contains no denominators, no fold and no choice of where anything was fitted. The rank correlation between the polarisation and $c$ is 0.49, a single number, and it sits at the very bottom of the range the reference-dependent scores had covered. The eight references’ higher readings were mostly the fold.

Twenty-seven systems at exactly zero #

Asked that directly, the polarisation does something no ratio on the square could reveal.

The picture is not a cloud. Twenty-seven of the eighty-one systems have a polarisation of exactly zero: every one with a site energy at least as large as its repulsion, where the lowest mean field keeps its two spins together. Their correlation energies run from 0.0001, at a repulsion of one and a site energy of ten, to 0.432, at eight and eight. Those are the least correlated system on the whole grid and the most correlated, and the polarisation reads the same for both. It is not a noisy reading, since the two values are the same number. Every other system, fifty-four of them, sits on a separate arc between 0.57 and 0.99 that turns back on itself as well, with a rank correlation of only 0.28 along it.

The square’s earlier refusal of blank scores met the same thing from another side. A rule that compares systems only where a candidate’s values are comparable has nothing to say about a candidate that does not vary, and scoring such a candidate by ratios had turned a failure to distinguish systems into a question of which pairs were admitted. The second number that was the error rearranged was an earlier warning of the same kind, that a score can be a disguise for the thing it scores.

And it makes physical sense of the shape in the grid itself.

The correlation energy is a ridge. Along it the repulsion and the site energy compete and the ground state is heavily mixed, and off it to either side one of the two wins and the state is simple. A diagnostic that rises steadily with either axis is being asked to follow a ridge with a ramp, and every such diagnostic meets the ridge at some value and comes back down past that value on the other side.

The floor, and what reaches it #

That gives a statement stronger than any score, because it does not depend on how a composite is built. Suppose two systems have nearly the same diagnostic, within a per cent of its range, and correlation energies differing by $\Delta$. Then any function of that diagnostic, fitted to any number of references in any form, gives them nearly the same estimate. One of them must be missed by at least $\Delta/2$.

For the polarisation the floor is 0.216, half the range of the whole grid, set by those same two systems. Now compare what can be done with no reading at all. Choose the reference nearest the middle of the range of $c$ and carry its correction to everything: the largest error is 0.2165. A single well-placed constant, which never looks at the system, ties the polarisation. That is the plainest thing this argument has been able to say about it. The double occupancy’s floor is 0.212 and the product of the two 0.216, for the same reason, and neither can do better than a shade under what a constant does.

Two candidates do have room. The highest occupied level has a floor of 0.173 and the mean-field energy 0.136, and those floors are where something could still be gained.

A second reference #

The lead the last essay offered was to carry a correction interpolated between two references, one with weak correlation and one with strong, in the hope of cutting the largest error from the size of one correction to the size of their difference. Every pair on the grid can simply be tried, with the correction interpolated linearly in each candidate in turn.

It does not do what was hoped. With the natural pair, the published reference and the grid’s weakest system, the largest error is 0.432 interpolating in the polarisation, 0.358 in the double occupancy, 0.269 in the mean-field energy and 1.217 in their product, against 0.351 for the published reference alone. Over every pair of references, some three thousand per candidate, the best anywhere is 0.177, with the highest occupied level between two systems at a repulsion of sixteen. That improves on the best single reference by a little under a fifth, after choosing the best among thousands of attempts on the same eighty-one systems the score then reads.

Below that luck, a second reference actively hurts. Every diagnostic’s median pair, between 0.42 and 0.64, is worse than the median single reference at 0.34. An interpolant given a pair close together in the diagnostic extrapolates wildly across the rest of the grid, and the largest errors among pairs run past five on every candidate and past twenty on one.

The choice of pair is itself a fit, and it was tested as one, by selecting the best pair on one colour of a checkerboard over the grid and scoring it on the other. On the highest occupied level that holds, 0.178 and 0.181. On the mean-field energy it holds tolerably, 0.178 and 0.225. On the polarisation and double occupancy one fold holds and the other does not: a pair that scored 0.138 on its half scores 0.386 on the rest, because the half it was chosen on happened to omit the two systems that set the floor.

So the difficulty is not the number of references. It would be possible for a large number of references to beat two, up to the floor, but the floor is already close. A correction cannot be made to depend on a number that does not tell the systems apart, however many places it is borrowed from.

Why a ratio could not have said this #

It is worth asking why several essays of careful scoring did not find twenty-seven identical readings. The warning first proposed for a composite method read a diagnostic as a function of the error it should predict, and on the first family tried, a line at fixed site energy, it looked like one. The half of the square that a ring of four could not show widened the family, and five failures in five different places showed that each diagnostic failed in its own region of the grid. Each of those was a correct reading of how a distance from the reference behaved, and a distance hides a tie. Every zero-polarisation system sits at one single distance from any polarised reference, so all twenty-seven read as a comparable set whose errors differ widely: a large worst ratio, which was reported and blamed on near-zero denominators, and with some justice. It was also the tie. What the near-zero denominators were amplifying was already in the data at a harmless-looking ratio of the size of the grid.

The reference chosen at a repulsion of eight and no site energy made this harder to see still. It sits on the far arc in the figure above, where the polarisation is large, so nothing about that first reference draws the eye to the vertical line at zero. A reference on that line would have had a distance of zero to twenty-six others, and every candidate’s comparison with them would have been excluded by the rule that drops values below a per cent of the scale. The tie was there at every reference, and on none of them was it the number being printed.

What the scores now say #

The verdicts of the square survive in a form that cannot be gamed, and they sharpen. No candidate passes: the best monotonic function of each leaves at least a fifth of the variance of $c$, and the worst leaves nearly two-fifths. The rank correlations run from 0.49 to 0.66. Nothing here resembles a warning a user could trust on an unfamiliar system.

But the ranking does not survive, and that is the part a user should carry. By rank correlation the highest occupied level is near the top at 0.651, beaten only by the product at 0.655, and by the share of variance explained it is the worst, at 0.610 against the mean-field energy’s 0.788. The earlier essays had ranked the highest occupied level the clear worst, since its medians alone stayed high. The two free scores read different things: a rank correlation counts the many weakly correlated systems at equal weight, and a variance share is dominated by the size of the large ones. Even without a reference a best diagnostic is a statement about what is being asked of it. The floor is a third answer, and it puts the mean-field energy first, by a wide margin, at 0.136.

On the conventional start the same scores are much worse, with floors as high as 0.75 and a best constant of 0.90, because a single system’s missed mean field, 1.497 at a repulsion of six and a site energy of four, supplies its own floor. That has nothing to do with correlation, and it is the argument the essay that found the solver’s worst system already made. The numbers above are on the lowest mean field for that reason.

How it was computed #

The grid is the square’s own: a ring of four at half filling, eighty-one systems from nine repulsions and nine staggered site energies, with the exact ground state and the conventional and lowest mean fields computed as before, and the correlation energy read as their difference. Rank correlations handle ties by midranks, which matter here since twenty-seven values are equal. The monotonic fit is by pool-adjacent-violators in whichever direction gives the smaller residual. A tie for the floor is two systems whose diagnostic differs by under a per cent of its range across the grid, and the floor is half the largest gap in $c$ over such pairs. The two-reference estimate is the straight line through the two references’ $(d,c)$ in one diagnostic, skipping pairs within a per cent of each other. Every claim here is checked where it can fail: the identity at three references to 10⁻⁹; each candidate’s monotonic share under four-fifths; no best pair under its own floor. The refusal is the method’s own zero: given $c$ itself as the diagnostic, an interpolant between any two systems must score exactly nothing, and it does, so that the floor belongs to the diagnostics and not to the method.

Where this stops #

It is one ring of four. Eighty-one systems of one small lattice cannot show what diagnostics do on a real molecule, and a mean field’s symmetry breaking in a ring of four is not chemistry’s. What the argument gives is a form of test that can be run anywhere: compute the tie floor of a proposed diagnostic on a benchmark set. For most of chemistry’s diagnostics that has not been the question asked of them.

The floor is a lower bound, and on a grid this size it is a coarse one. It uses only pairs that happen to fall near a tie. Finer sampling can only raise it, by finding pairs closer together with larger gaps. So the two candidates with room may have less than they appear to.

The linear interpolant is not the only composite. Nothing here rules out a smarter function of the highest occupied level or of the mean-field energy reaching close to its floor; only the floor limits it, and linear interpolation reaches within 0.004 and 0.06 of those.

The grid was not chosen to be fair to anything. Its ranges of repulsion and site energy were fixed long before this question, and a user whose systems lie on one side of the ridge has an easier problem in which the same diagnostics may do far better. That is a real use of a composite, and it just requires knowing already which side of the ridge the system is on.

Who said what #

Pople and his colleagues’ composite methods introduced the practice of adding a correlation correction fitted elsewhere, and diagnostics for when single-reference methods fail, of which $T_1$ is the best known, have been reported against known cases since the late 1980s. Comparison of them on benchmark sets is an established practice, and their disagreement has been known. The Hubbard ring is standard. The tie floor is a small observation in the theory of estimators, that a statistic cannot separate what it maps to one value, and it is not new as mathematics. What has not been seen elsewhere in this form is its use as a figure to report for a diagnostic.

Asking what a score is a score of #

The sequence of scores on this square has been a sequence of finding out what each score was measuring. The worst ratio turned out to measure a solver’s miss, and then the smallest error on the grid. The size of the error turned out to measure where a reference folds a line. In each case the number was correctly computed, and it answered a question somewhat to the side of the one being asked. Writing $e=c(\mathrm{ref})-c(\mathrm{sys})$ on one line sooner would have saved several of those steps, and that identity had been true on every grid since the first.

Still open: two readings at once, and a finer grid #

The obvious open question is what happens with two diagnostics read together, since the ties that set a floor are ties on one number. Requiring a near-tie in both collapses the floor: the highest occupied level with the double occupancy gives 0.022, against 0.173 and 0.212 alone, and the polarisation with the mean-field energy gives 0.052. A floor that falls tenfold is an opening, but it is measured on eighty-one points in two dimensions, where near-ties are scarce and a low floor may just be an empty cell. The test is a denser grid, which the systems are cheap enough to afford.

The nearer question is whether the ridge itself can be read. The correlation energy is largest where repulsion and site energy compete, and a mean field knows both of those, at no cost, in the Hamiltonian it was given. A composite allowed to use the system’s definition rather than only its mean-field outputs may do far better than any of the diagnostics here, and it would raise the awkward question of why those outputs were being asked in the first place.