How many descriptions a cage has

One scale from two centres to a cage ran eight systems up a scale from an isolated double bond to a twelve-vertex cage, asking how many atoms one electron pair holds together. Seven of the eight gave an answer. The twelve-vertex borane gave ten answers, differing by six parts in a thousand in the quantity being maximised, and that result was recorded and left there.

It also said what to do about it. Ten maxima found from twenty-four starts is a sample of an unknown population, and the population is finite and countable. And: whether they are related by a symmetry of the cage, or are genuinely inequivalent, is a question the coordinates would settle.

Both are settled here, and the second answer is the one that matters.

A maximisation reports what it reached #

The occupied orbitals of a molecule can be mixed among themselves without changing anything observable — that is the localisation transformation, and the density it leaves unchanged is checked to 10⁻¹⁶ every time the localisation runs. The freedom is usually spent on making the orbitals as local as possible, by maximising a functional that rewards an orbital for sitting on few atoms.

A maximisation is a search. It starts somewhere, climbs, and stops when it can no longer climb — which reports the maximum it reached and not the maximum there is. Running it once and printing the answer is a claim that the functional has one maximum, and that claim is almost never made explicitly and is often false.

Running until it stops finding things #

The rule for deciding when a search has run dry is the usual one: keep going until several batches in a row add nothing new.

Here a batch is twelve starting points, each a random orthogonal mixing of the occupied set, and the search stops when three batches in a row add nothing. On the twelve-vertex deltahedron that happens at sixty starts.

The discovery curve says two useful things. Ten is a count rather than a sample size: the plateau is flat and three batches wide. And twelve starts was not enough: the first batch found seven of the ten, so a study reporting the best of a dozen runs would have reported a maximum that is not the maximum, without anything to indicate it.

The two controls behave as they should. Two isolated double bonds have exactly one localised description, and so does benzene — although benzene’s is not the one a chemist draws, which is a separate finding about benzene rather than this one.

Are the ten one answer seen from ten sides? #

This is the question the eight-system scale deferred, and it has a clean answer that needs no group theory at all.

A symmetry of the structure permutes its sites. A localised set carried into another by such a permutation has the same orbitals sitting on permuted atoms — so every orbital’s participation number, which counts how many atoms it sits on weighted by how much, is unchanged. The multiset of participation numbers is therefore an invariant of any symmetry operation, whatever the symmetry group is and whether or not anybody has identified it.

The ten solutions here are indexed by exactly that multiset, and all ten differ. So no two of them are related by any symmetry of the cage.

The separation is worth dwelling on because it is the other half of the argument. Two runs that reached the same maximum to different precision would differ by something of the order of the convergence tolerance, which is 10⁻¹³. The two closest of these differ by 9.89 × 10⁻⁶, and the whole set spans 0.0236. These are ten places on a landscape, not ten readings of one place.

What it means to have ten of them #

A twelve-vertex borane has ten inequivalent localised descriptions, and choosing between them is a choice.

That is not a defect of the method. It is a statement about the molecule, and it is the one the electron-deficient cluster literature has been making in words since Lipscomb: a closo borane has many styx structures, no one of them satisfactory, and the received way of handling that is to say the true description is a resonance hybrid of them. A cage needs one pair more than it has corners is where the counting rule for those structures was computed here, and the counting rule is exactly what does not fix the description. What the localisation adds is that the hybrid is not what the calculation returns — the calculation returns one of the ten, chosen by where it started.

The canonical orbitals have no such ambiguity. They are eigenvectors of a matrix and are unique up to degeneracies, and the density they give is the density every one of the ten gives. So the object that is well defined is the delocalised one, and the object a chemist finds intelligible is the one that comes in ten versions — which is the same trade the four-centre cluster shows on a much smaller cluster, where the localisation was unique and the answer was four centres rather than two.

The shape of this in the rest of the collection #

A quantity that is a property of a decision rather than of a molecule is a thread worth pulling.

A stabilisation is measured from somewhere: benzene’s delocalisation energy is 2.0000β against isolated double bonds and 1.0121β against the ring cut open, and the number is a property of the reference. Which numbers carry a frame sorted nine quantities by which changes of frame they survive. A weight that depends on how it is weighed found one wavefunction to be 18.73, 34.74 or 5.88 per cent ionic depending on the partition.

This is the same shape with a different mechanism. There is no reference and no partition; there is a maximisation with several maxima, and the answer is a property of the search. What makes it worse than the others is that the others announce themselves — a reference has to be chosen and written down — while a search reports one number and says nothing about the nine it did not find.

What a search that goes dry does and does not prove #

The plateau is evidence and it is not proof. Three dry batches of twelve starts means no eleventh maximum was found in thirty-six attempts; it does not mean there is no eleventh maximum with a very small basin of attraction.

What can be said is the shape of the evidence. The first batch found seven, the second three more, and the next thirty-six found none — a discovery rate falling from 0.58 per start to 0.25 to zero. A population with an eleventh member of comparable basin size would have shown up; one with a basin a hundredth the size of the others would not have. So the count of ten is a count of the maxima a search of this kind finds, and that is the honest form of it.

The other family of electron-deficient systems — odd chains of centres holding one pair more than they have bonds — has unique localisations, because there is nothing to permute. A cage has faces that the group carries onto one another, and that is where the multiplicity comes from.

Ten answers and one density #

The thing that makes this tolerable rather than alarming is worth stating carefully, because it is the reason a chemist can go on drawing bonds.

Every one of the ten gives the same density. Not approximately — exactly, to the last bits a double holds, because they are related by unitary transformations of the occupied space and a density is invariant under those. So the ten disagree about nothing observable. They disagree about a description, and a description is a human artefact placed on top of an answer that does not need one.

That is the same relation the canonical and localised descriptions have always had here: one object, two vocabularies, identical predictions. What the cage adds is that the second vocabulary is not a vocabulary but ten of them, and the choice among them is not made by the physics.

The canonical levels of a delocalised system are eigenvalues of a matrix and are unique. The ten answers above are ten bases for the same occupied space, so nothing about the molecule is ambiguous — what is ambiguous is which basis to call the bonds.

Methane is the other end of the scale: a molecule whose localised description is unique, obvious and the one everybody draws. The interesting question is not why a cage has ten but why methane has one, and the answer is that its four bonds exhaust the occupied space with nothing left over to rotate.

Nothing about a canonical orbital of a delocalised ring looks like a bond, and nothing about it needs to: the bonds appear only after a mixing that changes no observable. On a cage there are several such mixings, and that is the whole of this essay.

What this cannot say #

The functional is one of several. Pipek–Mezey maximises a sum of squared atomic populations; Boys minimises orbital spreads in space; Edmiston–Ruedenberg maximises self-repulsion. They give different answers for the same molecule, and nothing here says whether the other two have ten maxima, three or one.

No σ framework. The pairs being localised are the skeletal ones only, which is where two-centre bonding stops and not the whole of the molecule’s bonding.

The system is a Hückel cage. One orbital per vertex, no hydrogens, no σ framework. A real B₁₂H₁₂²⁻ has both, and its occupied space is much larger; the count of ten belongs to this model’s twelve-orbital space.

The invariant argument is one-directional. Different multisets prove two solutions are unrelated by symmetry; equal multisets would not prove they were related. Here all ten differ, so the argument is used in the direction it works.

And nothing here says which of the ten to use. The highest functional is a defensible choice and is what a program returns when it converges properly; whether the highest is the most chemically useful is a question about usefulness rather than about arithmetic.

What was checked #

A system with one description finds one, however hard it is looked for: two isolated double bonds and benzene each return exactly one answer from a search run to dryness. A search manufacturing extra maxima out of its starting points would fail here first.

The cage’s search goes dry — three batches in a row adding nothing — rather than being stopped at a number chosen in advance, and the discovery curve is flat at the end.

It has more than one answer, which is the observation about the twelve-vertex cage turned into a count.

Every answer has its own multiset of participation numbers, which no symmetry of the structure can change, so none of them is another one seen from a different side.

And the functionals are separated by far more than the sweep converges to — the closest pair by 9.89 × 10⁻⁶ against a tolerance of 10⁻¹³, which is the check that these are different maxima rather than one maximum measured badly.

What a chemist should take from ten #

The practical question is what to do when a program hands back one localised picture of a cluster, and the answer has three parts.

Ask whether the system is one where the question has an answer. A molecule with as many bonds as pairs — methane, ethene, a saturated hydrocarbon — has a localisation that is unique and obvious, and the picture can be trusted. A structure with more atoms than pairs cannot have one pair per bond, so the localisation has to spread pairs over faces and edges, and the number of ways of doing that is what was counted here.

Ask how the answer was found. A single run reports the maximum it reached. Three runs from different starts that agree are evidence; one run is not. That is a two-line change to a script and it is not what most programs do by default.

And do not average them. Ten localised sets are ten bases for one space, related by unitary transformations rather than by addition, so there is no arithmetic that combines them. The object they are all bases for — the canonical set, the density, every observable — is unique and is what to quote.

The last of those is the one worth carrying furthest. Wherever a collection of pictures is described as a resonance hybrid, the hybrid is not a construction: it is a way of talking about a delocalised answer that the pictures were made out of. The hybrid was there first and the pictures are the approximation to it, which is the reverse of how the phrase is usually read.

What the ten descriptions already agree about #

Before asking what an average of ten descriptions would be, it is worth being precise about what they disagree over — because on most quantities they do not disagree at all, and the agreement is exact rather than close.

A localisation is a unitary transformation of the occupied orbitals. Such a transformation leaves the total density unchanged, identically, at every point in space — not to within a tolerance, but as an algebraic identity, because the density is a sum over occupied orbitals of their squares and a unitary mixing preserves that sum.

So all ten descriptions have the same density. They also have the same total energy, the same dipole moment, the same ionisation energies, the same canonical orbitals and the same everything else a measurement returns. There is nothing to average, because there is nothing that differs.

What differs is the partition. Each description divides the same density into twelve or thirteen pieces and attaches each piece to a set of atoms, and the ten divisions are genuinely different: different groupings, different numbers of centres per pair, different multisets of participation numbers. That is what a chemist draws when drawing a cluster’s bonding, and it is the only thing on which the ten disagree.

Which sharpens the finding considerably. Ten inequivalent localisations is not a physical ambiguity. The molecule has one density and the calculation reproduces it ten times over. It is an ambiguity in the drawing convention: the operation of assigning bonds to atoms is underdetermined, by a factor of ten on this cage, and the ten answers are ten pictures of one thing rather than ten candidates for what the thing is.

It also says why the averaging question is the wrong one to ask. Averaging the ten would be an attempt to recover a quantity none of them got wrong. The quantity they disagree about — the partition — is not an observable, so an average of it is not a better estimate of anything; it is a compromise between conventions.

The honest object is therefore the one the argument started from. The canonical description is the one that makes no partition, and the ten localised ones are ten ways of imposing a structure on it that the molecule does not itself possess.

None of that makes the ten worthless, and it says precisely what they are worth. A partition is a device for transferring a description between molecules — a bond drawn here resembles a bond drawn there, which is why chemistry can be taught at all — so a partition is useful in proportion to how well it transfers. Ten partitions of one cage, all reaching within six parts in a thousand of the same functional, is a statement that on this molecule the device transfers badly: there is no preferred set of bonds to carry to the next cluster, and any one of the ten carried there would be a choice made by a search rather than by the chemistry.

Still open: resonance as an average, and the basin sizes #

The obvious open question is the hybrid. If ten descriptions are inequivalent and the molecule is none of them, the natural object is the average — and averaging localised sets is not a well-defined operation, because they are related by unitary transformations rather than by addition. What is well defined is the average of the densities each one contributes to each atom, and that average is the canonical density, which is where the argument started. So the honest statement is that resonance between localised structures is a way of talking about the delocalised answer rather than a construction that produces it, and demonstrating that on this cage is a figure rather than a paragraph.

The nearer question is the basin sizes. Each of the ten was reached from some fraction of the sixty starts, and those fractions are a measurement of the landscape rather than of the molecule — but a very uneven distribution would say that a single run is far more likely to return one particular answer, which would explain why the literature’s localised pictures of a given cluster are more consistent than ten inequivalent solutions would suggest. Counting how often each was reached costs nothing beyond recording it, and it is the sort of thing that turns an awkward result into a usable one. It would also connect to the band limit, where the same localisation is run on structures large enough that the count of descriptions must be enormous and nobody has asked.