The basis a diagonaliser happened to return

Counting the family survey over its inputs rather than its rows turned forty-eight pairs into thirty-three questions, and it did so by noticing what an input is: the set of orbitals a localisation is handed, chosen by asking which orbitals have any occupation at all. That rule has a second consequence, and it is worse than the first.

Where the selected set takes only some members of a degenerate shell, the members it takes are not distinguishable. Two eigenvectors sharing an eigenvalue are fixed only up to a rotation between them, and a diagonaliser returns whichever pair its arithmetic arrived at, and the descriptions it then produces inherit the choice. So the occupied space — and therefore the density, and therefore everything a localisation reports about it — is partly a choice nobody made.

The invariance this is not #

A localisation exists because a wavefunction has a freedom in it. Mix the occupied orbitals among themselves by any unitary matrix and the density is unchanged, every observable is unchanged, and the energy is unchanged — so the occupied space is the physical object and any basis for it is a description. Choosing among those descriptions is what a localisation criterion is for, and the basin search in these essays exploits the freedom directly: it starts from a random unitary mixing of the occupied set, maximises, and records where it lands.

That is an occupied-with-occupied mixing and it is exactly invariant. Nothing here disputes it.

The rotation applied here is a different one. It mixes the members of a shell — a set of eigenvectors that happen to share an eigenvalue — and where the occupied boundary falls inside such a shell, some of the members being mixed are occupied and some are not. Mixing an occupied orbital with an empty one is not a change of description. It is a change of state.

So there are two rotations with a superficial resemblance and opposite consequences, and which one a given input is exposed to depends on where its occupied boundary falls relative to its shells. That is arithmetic on the spectrum and it was already done: six of the thirty-three inputs have a boundary inside a shell.

Four of them are on the icosahedron, at four, ten, twelve and twenty electrons, which is more than every other cage put together. The others are the octahedron at four electrons and the ten-vertex cage at sixteen.

Why the icosahedron carries most of it #

The distribution across cages is not chance and it is the same arithmetic that produced the recount.

A boundary can only fall inside a shell if there is a wide shell for it to fall inside, and the icosahedron’s shells are one, three, five and three — the widest in the family by a long way. Its five-fold shell alone offers four places for a boundary to land badly. The eleven-vertex cage, by contrast, has eight shells of which seven are singletons or pairs, and it contributes none of the six at all; the nine-vertex cage has six singletons among its eight shells and also contributes none.

So the two defects in this survey come from the same source and hit different cages. The collapse of rows onto inputs is a tread — several electron counts giving one selected set — and a tread happens where a boundary finishes crossing a shell. A cut is a boundary caught halfway across one. Both are the same geometry read at different moments, which is why the icosahedron loses the most rows to the first and supplies the most inputs to the second.

That has a consequence for any extension of this family. Thirteen and fourteen vertices were proposed as the way to settle whether cage size favours one criterion, and a deltahedron of thirteen or fourteen vertices has lower symmetry than an icosahedron and therefore narrower shells. So the extension would add inputs that are mostly safe, and the cage most affected by both defects would remain the one already in the survey — which makes the extension a poor way to learn anything about either of them, whatever it says about the criteria.

How far the answer moves #

Each of the six is localised from the orientation the diagonaliser returned and from four re-orientations of its shells. Every other input to the search is identical: the same cage, the same filling, the same number of starts, the same functional.

The icosahedron at ten electrons moves most: the best functional runs 2.5171, 2.0852, 2.1453, 2.1977 and 2.4685, a factor of 1.207. Under Boys the same input moves by 1.118. The icosahedron at twelve electrons moves by 1.125 and the octahedron at four by 1.152.

Twenty-one per cent is not a small number in this argument. Every classification in the family is built from a relative spread — the difference between the best and worst functionals divided by the best — and the threshold separating the two modes of that distribution sits at three parts in ten thousand. A best value that moves by a fifth moves the denominator of the classified quantity by three orders of magnitude more than the classification’s own resolution.

And the orientation the routine happened to return is not privileged. It gives the highest functional on four of the six and is beaten on two, so it is neither reliably the best description available nor a typical one. A single run reports it, and a single run is what the literature reports.

The count that was supposed to be a property #

Two earlier essays read the number of distinct descriptions as a fact about a cage — how many equally good pictures its bonding admits.

On the icosahedron at twelve electrons it is 10, 8, 14, 11 and 13. On the octahedron at four electrons it is 1, 2, 2, 1, 1 — so whether that cage has one localised description or two depends on the orientation. On the icosahedron at twenty electrons it is 2, 2, 2, 1, 1.

Three of the six move. Chasing the count to four thousand starts established when it stops rising, which is a real and necessary check and is a check on the search. It cannot see this, because every one of those four thousand starts was a mixing of one occupied set — the set the diagonaliser’s orientation defined.

And the label flips #

The strongest form of the finding is not that a number moves but that a verdict does.

The octahedron at four electrons gives Pipek–Mezey spreads of 0, 5.54 × 10⁻², 4.94 × 10⁻², 0 and 0. The threshold is 3.12 × 10⁻⁴. Two of the five orientations put it well above and three put it at exactly zero, so the cage is degenerate — every description the same answer — under three orientations and clearly not degenerate under two.

The icosahedron at twenty electrons flips the same way, at 1.68 × 10⁻⁶, 3.86 × 10⁻³, 4.99 × 10⁻¹¹, 0 and 0. The ten-vertex cage at sixteen electrons flips under Boys instead: 0, 7.89 × 10⁻³, 0, 0 and 4.49 × 10⁻⁷ against a Boys threshold of 5.78 × 10⁻¹⁰.

Three of the six, and the label is the one everything rests on. The gap in the distribution of spreads is what makes the classification robust to where the line is drawn, and it is robust to that — an empty factor of two hundred under one criterion and seven decades under the other. It is not robust to this, because this moves the points rather than the line.

The one that does not move #

Not all six move, and the one that does not is the more interesting half.

The icosahedron at four electrons takes two of the three members of its first excited shell and its answer is identical under every orientation: three occupied orbitals, each spread over 5.347 centres, to the third decimal, five times over. Both criteria give a span of exactly 1.0000.

The octahedron at four electrons has the same shape of cut — two of a three-fold shell, three occupied orbitals — and moves by fifteen per cent. So the invariance is not a property of the cut. It is a property of the cage, which is the kind of statement these essays have been trying to produce and have mostly failed to.

Why it holds is not established here. The three-fold shell of an icosahedron transforms as the three coordinate directions, and a two-dimensional subspace of that triple is a plane; the description a localisation reports is a sorted list of participation numbers, which is blind to how a plane is oriented if the cage’s own symmetry can carry any plane onto any other. The icosahedral group is finite and cannot do that for every plane, so the argument is at best incomplete — and the measurement is that all five orientations give 5.347 three times over. It is recorded as measured rather than explained.

What was computed, and how #

The shells are runs of eigenvalues equal to a part in a billion, read from the same Hückel calculation the neighbouring essays use. The rotation applies an independent random angle to every pair of members inside each shell, with a fixed seed per orientation, so an orientation is reproducible and five of them are five different bases for the same eigenspaces.

The basin search is then run on the rotated wavefunction exactly as it is run on the original: sixty starts, each a random occupied-with-occupied mixing, grouped by the sorted participation numbers to three digits. Both criteria are run on all five orientations of all six inputs, on the deltahedra built by minimising repulsion on a sphere.

Six things are checked. That several inputs have a boundary inside a shell, with the fraction stated per input. That re-orienting moves the best functional on most of them. That the worst movement exceeds a tenth, which is the comparison against the classification’s resolution. That the description count moves on at least one, under Pipek–Mezey alone — the Boys count is rounded past what its line search reproduces and nothing in this argument reads it. That the degeneracy label flips on at least two. And that the thresholds a flip is measured against are the ones the family survey placed in each criterion’s own empty stretch, read rather than re-derived, so a flip is a flip of the classification the earlier essays quote.

The refusal is the control, and it is what makes the rest mean anything. An input whose degenerate shells are wholly occupied or wholly empty is put through the identical rotation.

It does not move: 0.1666666667 against 0.1666666667, to ten decimal places. There the rotation is a mixing of occupied orbitals with occupied orbitals, the invariance is exact, and the search reproduces it exactly. Had the control moved too, everything above would be a statement about the search’s numerical noise and not about a choice inside a degenerate subspace.

Where this stops #

The two defects found in this pair of essays are independent, and that is worth saying plainly. None of the six inputs here is one of the three the two criteria classify differently. So the recount did not cause this and this does not explain the recount; the survey had two unrelated problems and they were found by asking the same question about its inputs.

Twenty-seven of the thirty-three inputs are unaffected, and every finding resting only on those is untouched. That includes the gap in each criterion’s distribution of spreads, since a gap is a property of the whole set and the six affected inputs’ spreads sit inside the populated regions rather than in the empty stretch. It includes the finding that many descriptions can all be one answer, which was established on cases with closed shells.

And the defect is a defect of a Hückel model as much as of this implementation. A real open-shell calculation does not hand a localisation an arbitrary subspace, because the self-consistency that produced its orbitals is not indifferent to which members of a degenerate set are occupied: it breaks the degeneracy, chooses, and reports the choice as a symmetry-broken solution. A Hückel matrix has no such mechanism. Its degeneracies are exact and it has nothing that could prefer one plane in a triple to another, so the arbitrariness is genuine rather than a bug in a filling routine.

What that means for the repair is unhelpful. The honest options are to restrict the survey to inputs whose shells are taken whole — twenty-seven of thirty-three, which is most of it — or to average over orientations and report a distribution rather than a value. Neither is what was done earlier, and the second changes what is being reported.

The generalisation #

The habit is to find the place where a method’s invariance stops, and then to check that nothing is standing on the other side of it.

The invariance here is famous and correct: a unitary mixing of occupied orbitals changes no observable. Every localisation paper states it and every localisation method is built on it, and the statement has a precondition hiding in the word occupied. It says nothing about a rotation that crosses the occupied boundary, and where does a boundary cross a set of interchangeable orbitals? Exactly where a degenerate shell is partly filled — which is a common situation, not an exotic one, and which no part of the method announces.

The test is the cheap part and is the shape worth copying. Take the transformation the method claims to be invariant to, apply it in a place where the claim’s precondition fails, and run the same calculation. If the answer moves, the precondition was load-bearing. Running the same test where the precondition holds is what turns a movement into evidence, and it is the step that is easiest to skip, because a control that does nothing produces no number worth printing until the moment it is the only thing separating a finding from a bug.

The corollary is about what a single run of a stochastic search reports. Four thousand starts establish that a maximum has been found; they say nothing about whether the question was determined. Those are different kinds of insufficiency and only the first has a standard remedy.

Who found it, and when #

The unitary invariance of a single determinant is as old as the determinant — Fock, 1930. The arbitrariness of an open-shell configuration’s orbitals inside a degenerate shell is standard and is the reason symmetry-broken solutions exist. Neither is new. What is computed here is which of this family’s inputs have a boundary inside a shell, and what re-orienting the shell does to the best functional, the description count and the degeneracy label the earlier essays classify on.

The number worth carrying is not 1.207. It is three: the number of inputs this survey labels degenerate or not according to which basis an eigenvalue routine happened to return, out of the thirty-three it has.

Still open: whether the orientation can be chosen rather than inherited #

The obvious open question is the choice itself. An arbitrary orientation is a free parameter and a free parameter can be optimised: the orientation of the cut shell that maximises the localisation functional is a well-posed problem, one rotation angle per pair of shell members, and solving it would replace whatever the diagonaliser returned with a definite answer. Whether the resulting best values are the tops of the ranges measured above or above all of them is the first thing to find out, and it decides whether the survey’s numbers are a random draw from the range or a systematic underestimate. It costs one more nested maximisation on six inputs.

The nearer question is what the icosahedron at four electrons is doing. It is the one affected input whose answer does not move, its three orbitals come out at 5.347 centres in every orientation, and the octahedron with the same shape of cut moves by fifteen per cent. If that is a symmetry statement it should be provable from the cage’s automorphism group acting on a two-plane inside a three-fold representation — and if it is provable, it would predict which other cage-and-shell combinations are safe, which would turn twenty-seven of thirty-three are unaffected into a rule rather than a census.