Fifty descriptions of one molecule

A localisation run from four hundred random starts on three cages finds three shapes of answer: fourteen descriptions with no clear winner, fifty descriptions with none either, and five of which one takes ninety-eight per cent of the starts.

It closed with the question the third makes urgent. A search whose largest basin takes ninety-eight per cent will report one description however long it is run in practice, because nobody runs four thousand starts when the first fifty agree. Whether the rare descriptions there are worse descriptions by the functional, or merely rarer, is one number per description — and it is already computed, because the localisation maximises that functional and records what it reached.

It had never been looked at.

Two of the three are one answer #

On the nine-vertex cage at eight electrons the search finds fifty descriptions, and the whole set spans $2.20\times10^{-5}$ of the functional’s value. On the twelve-vertex cage at twenty electrons it finds five, spanning $1.68\times10^{-6}$.

Those are not differences. Twenty parts per million and two parts per million are below anything the localisation converges to, below anything that could distinguish two pictures of the bonding, and far below the differences the same functional shows between genuinely different descriptions.

The scale to compare them against is in the third cage, where the same functional separates descriptions by four per cent. Two parts in a million is that difference divided by twenty thousand. Nothing about the search or the arithmetic could make a distinction that small mean anything, and the localisation is not being asked to: it maximised a quantity and found a plateau.

Written out, the eight most-reached descriptions of the nine-vertex cage agree to eight figures. A search returning any of them has not chosen between competing accounts of where the electrons are. There is one account, and the search is reporting which random start it happened to begin from. The fifty labels are a fact about the rounding and the sampling; the one value is the fact about the molecule.

That is the answer to the question, and it is the uncomfortable one. The rare descriptions are not worse. They are the same.

There is a distinction worth drawing here that the counts alone cannot draw. Two searches can both “find many answers” for reasons that are opposites.

A search finds many answers because the surface is rough — many local optima at genuinely different values, and the search has to choose. That is the twelve-electron cage, and it is a hard optimisation problem where the answer matters.

Or a search finds many answers because the optimum is flat — one value achieved along a whole set of configurations, and the search stops wherever it happens to land. That is the other two, and it is not an optimisation problem at all. Nothing is being chosen because nothing distinguishes the candidates.

The count of descriptions is identical in kind for the two, and the remedy a practitioner would reach for — more starts — is right for the first and pointless for the second. On a flat optimum more starts find more members of the degenerate set and never converge on one, which reads exactly like a hard problem and is the opposite of one. The answer a search is most likely to give is where the question of what a search’s preference is worth first comes up; this is the case where it is worth nothing because there is no question.

Where quality does vary, the commonest is not the best #

The third cage is different, and it is the one that makes the first two worth reporting rather than merely tidy.

On the twelve-vertex cage at twelve electrons the descriptions span 3.9 per cent — three decades above the other two, and large enough that the fourteen descriptions really are fourteen different pictures.

And there the commonest description is not the best. The most-reached basin, at 14.25 per cent of starts, sits 0.34 per cent below the best one, and two rarer descriptions beat it.

So on neither kind of cage is the share a basin takes evidence about its quality. Where the descriptions are the same, share means nothing because there is nothing to be evidence about. Where they differ, share points at the wrong one.

Plotted, the twelve-electron cage’s points do show a trend — the rarest four are 13 to 15 per cent below the best — so rarity is some evidence there. What it is not is a ranking: within the reached-often group the ordering by share and the ordering by quality disagree, and the best description is not the most-reached. So even in the one case where rarity carries information, it carries it only at the extremes: the four rarest are clearly worse and the ten commonest are in an order the share does not predict.

The three columns together are the summary. One spreads over three decades and reaches four per cent; the other two are flat lines at parts per million. The vertical extent of a column is what says whether a cage’s search is choosing between pictures or between labels for one picture, and it is not something a count of descriptions or a dominance figure reports. Where two-centre bonding stops is the essay whose whole argument is a count of descriptions, and it is worth re-reading with this column beside it.

What a practitioner would have concluded instead #

It is worth being explicit about what the two available summary numbers would have suggested, because both mislead.

The number of descriptions suggests the nine-vertex cage is the ambiguous one — fifty answers against five — and the twenty-electron cage the clear one. The functional says the opposite about what the ambiguity means: both are degenerate, and the fifty are as equivalent as the five.

The dominance suggests the twenty-electron cage is settled, at 98.3 per cent. It is settled, and about nothing: the four rare descriptions it almost never reaches are within two parts in a million of the one it always does. A reader told “the localisation converges reliably on this molecule” would take that as evidence the bonding picture is well defined, and the evidence it actually is is that the search has a preferred route.

Neither number is wrong. Both are answers to questions about the search, and the question a chemist has is about the molecule — which needs the functional, which is computed at every start and thrown away. The gap between the two is not subtle once it is looked at, and it is invisible until then — which is the whole reason for this essay.

What it costs to check #

The measurement above needed no new computation, and that is worth stating plainly because it changes what a reader should ask of their own output.

Every localisation start already maximises a functional, so every start already knows the value it reached. Collecting those values costs one number per start and nothing else — no extra diagonalisation, no extra iteration, no second run. The counts are what usually gets reported.

What they buy is the difference between two reports a chemist would act on differently:

“The localisation of this cage converges to five descriptions, one of which is found 98 per cent of the time.” That reads as a well-behaved calculation with a clear answer.

“The localisation of this cage finds five descriptions differing by two parts in a million; which one is returned depends on the starting guess.” That reads as what it is — a molecule whose bonding this method does not resolve into a unique picture.

The two sentences describe the same four hundred runs. The second needs one column the first threw away, and it is the column that says whether the answer means anything. The good habit is that a claim gets a computation it could fail; the corollary is that a computation already run should be asked what it could have refused, and here it refuses the reading that agreement implies determination.

What was computed, and how #

Each cage is a deltahedron, its π system solved by Hückel theory at a stated electron count, and its occupied orbitals localised by maximising a functional of the orbital centroids. Four hundred random starts each; a description is the sorted list of orbital centre counts, rounded to three figures, and two starts reaching the same list are one basin.

That rounding is the one convention here that could manufacture or destroy a distinction, and it is the one used for the counts, unchanged. A coarser rounding would merge descriptions and a finer one would split them, so “fifty descriptions” is a statement at a stated tolerance. What is not sensitive to it is the finding: merging or splitting the labels does not change the spread of the functional across the starts, which is the quantity read here.

The functional recorded for a description is the best any start reaching it achieved, and the deficit is the difference from the best over all descriptions, divided by the best. A case is called degenerate when that whole spread is under a ten-thousandth.

Four things are checked. At least one case must be degenerate and at least one must not, so the measure is not calling everything the same. Every degenerate case’s spread must be under a ten-thousandth. The non-degenerate case’s rare descriptions must be genuinely worse and its commonest must not be its best — two claims that could fail separately. And the case with the highest dominance must be one of the degenerate ones, which is the finding stated so that it fails if agreement and meaningfulness coincide. That last is the one an optimistic reading of the counts would have expected to hold, and it is the check this essay exists to break.

The results are cached, and the cache is verified on read by re-running one start and requiring its description to be one the stored population contains — so a stale entry from a changed localiser fails rather than being believed.

Why a flat optimum is the expected case #

Having found two of three cages degenerate, it is worth asking whether that should have been a surprise, and the answer is no — which makes the omission more interesting rather than less.

A localisation functional is a function of the occupied subspace, and the subspace is fixed. What varies is the unitary mixing within it, and the functional is being maximised over a continuous group — for six occupied orbitals, a fifteen-parameter one. A maximum over a group is generically a whole orbit of the symmetry that leaves the functional invariant, and a symmetric molecule has plenty of such symmetry.

So a degenerate set of maxima is the default on a symmetric cage, and a genuinely rough surface with distinct values is the special case. The twelve-electron cage is the interesting one precisely because it is the exception.

That reframes the earlier counts. “Fourteen descriptions” and “fifty descriptions” are not two degrees of the same difficulty; the first is fourteen competing answers and the second is one answer with fifty labels, and the difference is not visible in the count. It is visible in one column that costs nothing, which is the argument for always printing it.

Where the model stops #

This is Hückel theory on a cage: one orbital per vertex, one hopping integral, no repulsion. The localisation functional is a function of orbital centroids rather than of a density, so “the same description” means the same pattern of centres and not the same wavefunction.

Two descriptions agreeing on the functional to eight figures might still be different states. The functional is a scalar and a scalar cannot separate everything; what the agreement establishes is that the search has no reason to prefer one, which is the claim being made and is weaker than saying they are the same orbitals. The centroid lists differ — that is what makes them different descriptions — while the quantity being maximised does not.

Three cages is three cages, and they were chosen for the three shapes of basin population rather than as a sample. Nothing here says how common a degenerate population is across the family, and the survey how many descriptions a cage has built could answer it.

The threshold at 10⁻⁴ that separates “the same description” from “genuinely different” is chosen here for one molecule, by looking at where this cage’s own functional values fell. That is the weakest step in the essay and it was made on one case. Carrying it across a family is what counting was right except where it mattered does, and the answer there is better than there was any right to expect: over forty-eight cage-and-filling pairs the spreads are bimodal, with an empty stretch of two decades around the line, so the value inherited from this molecule classifies every one of them the way any other value in that stretch would. The threshold turns out not to be a choice — but it is worth being clear that nothing on this page establishes that, and the reader who wants the classification to mean something should read it there.

And the electron counts are even and closed-shell. An odd count or an open shell has a different localisation problem, and none of this is measured there. Four centres, and the pair that will not localise is the case where the difficulty is the bonding rather than the search.

The generalisation #

A search’s reproducibility is a property of the search, and it is routinely reported as a property of the system. Ninety-eight per cent agreement between runs is a fact about basins of attraction; whether it means anything needs the objective function’s values, which every optimiser computes and most outputs discard. The same shape appears in twelve basins where there were two, where missing the best arrangement cost almost nothing in energy and everything in structure — and the two together make a pair: there the energies were close and the structures differed, here the descriptions differ and the functional does not.

And a degenerate optimum is not an inconvenience to be resolved. When fifty descriptions achieve the same value, the honest report is that the localisation does not determine the picture — not that fifty runs disagreed. The first is a statement about the molecule and is worth having; the second sounds like a numerical problem and invites more starts, which produce more descriptions and no more information. The localisation transformation demonstrates that the density is unchanged by the choice, and this is the same fact arriving as a flat spread in the functional.

Who found it, and when #

That localisation criteria have multiple solutions is old and well known — Boys, Edmiston–Ruedenberg and Pipek–Mezey each have their own, and the literature on which to prefer is large. That a given criterion can have many solutions of equal value on a symmetric molecule is also old, and is the reason a symmetric molecule’s localised orbitals are usually described up to a symmetry operation.

What is done here is smaller and is about practice. Three basin populations come with the standard summaries of them; this reads the one number those summaries throw away, and finds that two of the three populations are not populations of competing answers at all. The measurement cost nothing — the functional was already recorded at every start — which is the part worth carrying: the evidence needed to know whether a search’s disagreement matters is usually already in the output.

Still open: the functional spread across the whole family #

The obvious open question is the family. A survey of forty-odd cage-and-filling pairs already exists, and every one of them has a functional spread that has never been looked at. Sorting them by that spread rather than by their number of descriptions would say how common a degenerate population is — and if most are degenerate, the whole use of description counts as a measure of ambiguity needs restating.

The nearer question is the symmetry. A degenerate set of descriptions on a symmetric cage is what a symmetry operation permutes, and the number of them should then be a group order or a divisor of one. Fifty is not an obvious such number for a nine-vertex deltahedron, and five is not for a twelve-vertex one — so either the degeneracy is not symmetry, or the rounding is splitting what a symmetry relates. Testing that costs one thing: apply the cage’s own operations to a description and ask whether the image is in the population.