The answer a search is most likely to give

A localisation search on a twelve-vertex cage finds ten inequivalent descriptions of one molecule — ten sets of localised orbitals, each a genuine maximum of the same functional, no two related by any symmetry of the cage, and the closest pair separated by 9.89 × 10⁻⁶.

That result is awkward in a specific way. If a molecule has ten equally defensible localised pictures, then every published picture of a cluster is one of ten, and the literature’s habit of agreeing with itself needs explaining.

The explanation one would hope for is simple. Each description is reached from the starting points inside its own basin, and if one basin is very large then a single run almost always returns the same answer — which would make the pictures reproducible without making any of them unique. Counting how often each was reached costs nothing beyond recording it, and it has not been done.

Done here, on four hundred and eighty starting points, the hoped-for answer is refused, and something worse turns up in its place.

What was counted #

The search is the standard one: the occupied orbitals are mixed by a random unitary transformation, the Pipek–Mezey functional is maximised from there, and the answer is identified by the multiset of participation numbers its orbitals have. What is added is the bookkeeping — which answer each start reached, rather than only which answers exist.

That the starting point can be a random mixing at all is the fact the whole subject rests on: a localised set and a canonical set are one wavefunction written two ways, so any unitary mixing of the occupied orbitals is the same molecule, with the same density and the same energy. The search is not looking for a better wavefunction. It is looking for a nicer way of writing down the one it already has, and nicer is a functional somebody chose.

Four hundred and eighty starts. Fourteen distinct descriptions. The reaches, in order: 65, 60, 55, 49, 46, 44, 43, 37, 37, 27, 8, 4, 3, 2.

The first thing that says is about the count itself. The ten was a lower bound. The original search used a hundred and eighty starts and stopped after three batches added nothing, which is a reasonable rule and a rule about the search rather than about the molecule: three of the four extra descriptions found here are reached by fewer than nine starts in four hundred and eighty, so a shorter run misses them with high probability.

That is worth saying plainly because it changes what the earlier result was. Ten inequivalent descriptions is not a property of the cage; it is a property of a hundred and eighty starts.

The basins are uneven and it does not help #

The largest basin takes 13.5 per cent of the starts and the smallest 0.4 per cent, which is a factor of thirty-two. So the search is emphatically not sampling the descriptions uniformly.

The question is whether that unevenness is enough to make a single run reproducible, and it has an exact answer. Two independent searches return the same description with probability $\sum_i p_i^2$, which is 9.86 per cent here — against 7.14 per cent if all fourteen basins were equal.

A factor of 1.38 above uniform. The equivalent number of equally likely answers is 10.93 out of 14.

So two people running the same search on the same molecule agree about one time in ten. That is not a landscape in which one picture dominates; it is a landscape in which fourteen pictures compete on nearly equal terms and the search picks one of them at random.

The hoped-for explanation is closed. Whatever makes the literature’s pictures of a cluster consistent, it is not the shape of this basin structure.

The part that is worse than awkward #

The commonest description is reached 65 times of 480 and its functional is 3.80134. The description with the largest functional is 3.81413, and it is reached 37 times.

So a single run of the search is nearly twice as likely to return a description that is not the best one as it is to return the best. Seven point seven per cent of starts find the optimum; ninety-two per cent find something else.

That is a different complaint from the first one. The first finding was that the molecule has many descriptions and no reason to prefer one — a statement about the object. This one is that the standard procedure, run once as it is always run, usually returns an answer its own criterion says is not the best available. That is a statement about the method, and it is the kind that does not go away by choosing a better criterion: any criterion with this many maxima has the same problem.

Why it does not show #

A procedure that returns the wrong answer nine times in ten would normally be noticed, and this one is not, for two reasons that are both about what a localised picture is used for.

The functionals are nearly equal. The fourteen values run from 3.664 to 3.814, and the ten most-reached lie between 3.790 and 3.814 — a spread of six parts in a thousand. Nobody reports a Pipek–Mezey functional, and if they did, the difference between the best answer and the usual one would look like nothing.

And the pictures are described in words. “Three-centre bonds on alternate faces” is a description that several of these fourteen satisfy. The multiset of participation numbers separates them; a sentence does not. So two papers can report the same qualitative picture from two different maxima and agree, which is the literature’s consistency explained — not by the search, but by the coarseness of what is being compared.

Both reasons point the same way, and the way is uncomfortable. The agreement is real and it is not evidence of anything about the molecule: it is evidence that the reported quantity is insensitive. The count of pairs a cage needs is a genuinely robust statement about the same object, and it is robust because it is a count of orbitals rather than a description of them.

That is the same resolution as the one a valence bond weight has: the quantity everybody quotes is stable because it is coarse, and the underlying object is not stable at all.

What a reader of localised pictures should take from it #

A localised picture is the output of a search, and the search should be reported. How many starts, what stopping rule, which functional. None of that is usually given, and without it the picture is one draw from a distribution nobody has characterised.

Running the search once is not enough, and running it twice is informative. Two runs that agree are weak evidence of a dominant basin; two that disagree are proof that there is not one. On this cage they disagree nine times in ten.

And the robust statements are the counts. How many pairs a cage needs, how many centres one pair can hold together, how many orbitals are filled — none of those depends on which maximum a search found, because none of them is a property of the localised set. What the fourteen descriptions differ about is exactly the part of the picture that a unitary transformation is free to move.

That is the distinction hybrids are a basis draws in the simplest case, and this is the same distinction at a size where it stops being a matter of taste and becomes a matter of arithmetic: with fourteen maxima, choosing a description is choosing among fourteen, and nobody chooses.

What this cannot say #

One functional. Pipek–Mezey is one of several localisation criteria and the others have their own landscapes. Whether Boys or Edmiston–Ruedenberg has fewer maxima on this cage is a question this does not touch, and the count is a property of the criterion as much as of the molecule.

One cage. Twelve vertices with a particular connectivity, at one filling. A different cluster would have a different landscape, and nothing here says how the count grows with size — though the band limit is where the same search is run on structures large enough that the count must be enormous. A four-centre system, where one pair refuses to localise at all, is the small end of the same phenomenon and has one answer rather than fourteen.

The starts are one distribution. Random unitary mixings of the occupied orbitals is a reasonable way to sample and it is not the only one, and a basin’s size is measured with respect to whatever distribution the starts are drawn from. A procedure that started from a chemist’s guess would have entirely different basins, which is not a criticism of the count but a statement of what it is a count of.

And the model is one-electron. These are Hückel orbitals on a graph. A real cluster’s localised orbitals come from a calculation with repulsion in it, and whether the landscape is rougher or smoother there is not something a one-electron model can answer. One scale runs from two centres to a cage in this model, and the scale is what a repulsion would move.

The arithmetic of agreement #

The number that carries the argument is worth writing out, because it is the one a reader can apply to any search they run.

If a search returns description $i$ with probability $p_i$, two independent runs return the same description with probability $\sum_i p_i^2$. That is the whole of it. It is one when a single basin takes everything, and it is $1/N$ when $N$ basins are equal — so it is a direct measure of how much the unevenness is buying.

Here it is 0.0986 against a uniform value of 0.0714. Inverting it gives the effective number of answers: 10.93, where the count is 14. Three of the fourteen have essentially disappeared into the rounding, and the other eleven are as good as equally likely.

The same quantity read the other way says how many runs it takes to be reasonably sure of finding the best description. At 7.7 per cent a start, thirty independent runs give about a ninety per cent chance of hitting it at least once — and nobody runs thirty.

What is quoted, and what is computed #

Nothing is quoted. The cage, its connectivity and its filling are the model’s; every localisation, every functional, every participation number and every count of starts is computed.

The identification of two answers as “the same” is a comparison of participation-number multisets to three decimal places, which is the original rule, inherited unchanged so that the two counts are comparable. It is a conservative rule in one direction — two genuinely different maxima with the same multiset would be merged — and the separate check that the functionals are resolved is what stops that from mattering.

What was checked #

The cage has at least the ten descriptions found before. Fourteen, from four hundred and eighty starts.

Its search does not reach them equally often, by more than a factor of three. Measured: thirty-two.

So the effective number of answers is smaller than the count of them, by more than one — 10.93 against 14, which is the quantity the basin argument is about.

Two independent searches agree more often than equal basins would give, which is the half of the hoped-for explanation that is true.

And nowhere near often enough to explain a literature that agrees with itself — under 20 per cent, and under twice the uniform value. Both halves are checked, because the first alone would read as a confirmation.

The description a single run is most likely to return is not the best one it can find. That is the finding, and it is checked directly rather than inferred from the shares.

The refusal is a system with one description. Two isolated double bonds have exactly one localised solution, every start must reach it, and two searches of them must agree with certainty — a basin count that reported anything else there would be reporting on its own random starts rather than on a landscape.

Why the literature agrees when the search does not #

The question this essay opens with — if a cage has many equally good localised descriptions, why does the literature agree about its picture — has an answer that the basin sizes do not supply, and the answer is that the literature is not doing this calculation.

Published localised pictures of borane cages are not the output of a Boys localisation run from random starts. They come from a different tradition entirely: a notation that counts three-centre bonds, two-centre bonds and terminal hydrogens and enumerates the topologically allowed arrangements, with chemical judgement selecting among them. The agreement between papers is therefore agreement about a convention, arrived at by argument and repetition, and it does not indicate that an optimisation would converge on the same answer.

Where a localisation is run, the starting point is usually not random either. A calculation begun from a chemically motivated guess — the picture the notation suggests — will converge to whatever maximum lies nearest that guess, and the same guess used by different workers lands in the same basin. That is reproducibility of a procedure rather than uniqueness of an answer, and the two are easy to confuse when the procedure is not stated.

So the consistency has two sources and neither is the landscape being simple.

The pictures mostly predate the calculations, and the calculations are read as confirming them rather than as searching independently.

And the searches that are run are seeded, so they sample one basin rather than the four hundred and eighty starts used here.

That is worth stating because it inverts the puzzle. The agreement is not evidence that the multiplicity measured here is somehow unreal; it is evidence that nobody has been sampling the space the multiplicity lives in. A random-start search is a different experiment from a seeded one, and this is a first version of it — which is why its answer is the surprising one and not the literature’s.

It also supplies the check that would settle it, and the check costs one run. Take the picture the notation gives for this cage, build a starting set of orbitals from it, and localise. If the search converges to the description the literature draws, then the literature’s answer is a basin of this landscape reached by a good guess, and the two traditions are describing the same maximum by different routes. If it converges somewhere else, the drawn picture is not a maximum of this functional at all — which would be a more interesting result and would mean the two traditions have never been talking about the same object.

Still open: the average over descriptions, and the stopping rule #

The obvious open question is one this count makes more urgent. If a single run returns the best answer 7.7 per cent of the time, then the honest object is not any one description but the average over them — and averaging localised sets is not defined, because they are related by unitary transformations rather than by addition. What is defined is the average of the densities each contributes to each atom, and that average is the canonical density, which is where the argument started. Demonstrating that on this cage, with the fourteen descriptions weighted by the basins measured here, would close the circle with a figure rather than a paragraph.

The nearer question is about the stopping rule. A search that stops after three batches add nothing finds ten; four hundred and eighty starts find fourteen. The number of descriptions is therefore not a number at all but a curve against effort, and the useful quantity is where it flattens — or whether it does. Running the count out to several thousand starts, and watching whether the smallest basins keep appearing, would say whether a cage of this size has a finite number of localised descriptions or merely a finite number that anybody has ever had the patience to find.