The cage that averages like a sphere

A localisation is free to mix the occupied orbitals among themselves, because that mixing changes nothing physical. It is not free to mix an occupied orbital with an empty one, and where the occupied set takes only some members of a degenerate shell, the members it takes are an arbitrary choice made by the eigenvalue routine. Six of the cage family’s thirty-three inputs are cut that way. Re-orienting their cut shells five ways moved the best Pipek–Mezey functional on five of them by up to twenty-one per cent, and flipped the classification three times.

The sixth did not move at all. The icosahedron at four electrons takes two of the three members of its first excited shell, and all five orientations gave three orbitals of 5.347 centres, a best Pipek–Mezey functional of 0.561111 and a best Boys functional of 0.888889, identical to the last digit. The octahedron at four electrons has exactly the same shape of cut — two of a three-fold shell, three occupied orbitals — and moved by fifteen per cent.

The natural explanation is symmetry, and it was offered and doubted in the same paragraph: the three-fold shell transforms like the three directions of space, choosing two of its members chooses a plane, and if the cage’s symmetry could carry any plane onto any other the choice would not matter. The icosahedral group has 120 operations. A plane through the centre has two degrees of freedom. A finite group cannot carry every plane onto every other, so the explanation was recorded as incomplete and the measurement as measured.

A sum that does not care how the cage is turned #

The explanation is not a symmetry of the cage. It is a property of how the cage’s vertices average.

The members of the icosahedron’s first excited shell are the vertex coordinates. The Hückel matrix of a cage with this much symmetry has, as eigenvectors of that three-fold level, the functions x, y and z evaluated at the twelve vertices. That is measured here rather than assumed: every member of the shell lies, to one part in a hundred million, in the span of the degree-one polynomials sampled at the vertices, and in no smaller span.

So choosing two members of the shell is choosing two directions $\mathbf n_1$ and $\mathbf n_2$, and the occupied orbitals are the constant function and the two functions $\mathbf n_1\cdot\mathbf r$ and $\mathbf n_2\cdot\mathbf r$ on the vertices. Re-orienting the shell is turning $\mathbf n_1$ and $\mathbf n_2$. And since every orthogonal re-orientation of three coordinate functions is a rotation or reflection of the coordinates, re-orienting the shell is the same as leaving the orbitals alone and turning the cage.

Now look at what the Pipek–Mezey criterion computes. For each localised orbital it squares the orbital’s population on each vertex and adds over vertices. An orbital is a combination of the three occupied functions, so on each vertex it is a polynomial of degree one in that vertex’s coordinates; its population is the square, degree two; and the criterion adds the squares of populations, degree four. The whole functional, for any mixing of the occupied set, is a sum over the vertices of a polynomial of degree four in the vertex coordinates.

A set of points on a sphere over which every polynomial up to degree t sums to exactly what the sphere’s own average predicts is called a spherical t-design. On a t-design, the vertex sum of a polynomial of degree t or less does not change when the points are rotated, because the sphere’s average does not. So if the cage is at least a 4-design, turning the cage leaves the Pipek–Mezey functional of every mixing unchanged — the whole landscape the basin search climbs is the same landscape — and so is its maximum, and so is every participation number, which is also a degree-four sum.

Which cages average like a sphere #

The design strength of a set of points can be read off without writing a single harmonic. By the addition theorem, the degree-l harmonics all average to zero over the points exactly when the average over every pair of points of the Legendre polynomial of degree l, in the angle between them, is zero.

For the octahedron the sums vanish at degrees one, two and three, and at degree four the average is 0.58. For the icosahedron they vanish at one through five, and at degree six the average is 0.44. So the octahedron is a 3-design and the icosahedron a 5-design, which are the classical values.

The same computation on every cage in reach: the tetrahedron is a 2-design, the octahedron and cube 3-designs, the icosahedron and dodecahedron 5-designs. The family’s three irregular cages, of nine, ten and eleven vertices, are 1-, 1- and 0-designs — the eleven-vertex cage’s centre of mass is not even exactly at its centre. The icosahedron clears the four that Pipek–Mezey needs. The octahedron falls one short.

That is the fifteen per cent, explained.

The prediction nobody had tested #

The argument makes a second prediction immediately, and it was not made before the calculation.

The Boys criterion is built differently. It computes, for each localised orbital, the centroid of its population — the population on each vertex times that vertex’s position, added over vertices — and maximises the sum of the squared centroid lengths. On a degree-one shell the population is degree two and the position adds one: each centroid is a vertex sum of a polynomial of degree three. On a 3-design those sums turn with the cage rather than changing, so every centroid’s length is unchanged and so is the functional. Boys needs a 3-design where Pipek–Mezey needs a 4-design. The octahedron is a 3-design. It should not move under Boys at all.

It does not. The survey that found the fifteen per cent had computed both criteria on all six inputs and had reported the Pipek–Mezey movement, which was the one that mattered to the classification it was checking. Read under Boys, the octahedron’s five best functionals are 0.888889, 0.888889, 0.888889, 0.888889 and 0.888889 — the same to every digit carried. The same cut, the same cage, the same five orientations: one criterion moves by fifteen per cent and the other does not move.

So the question “is this input determined?” does not have a yes-or-no answer that belongs to the input. It has one answer per criterion, and the answers differ by exactly one degree of polynomial.

What it does to the labels the survey read #

The inputs the survey grouped into thirty-three calculations were classified by one number each — the relative spread between the best and worst descriptions a basin search finds — and three of the six cut inputs had that label flip under re-orientation. The rule says which flips were possible before any were seen. The octahedron at four electrons flipped under Pipek–Mezey and could not have flipped under Boys; the ten-vertex cage at sixteen flipped under Boys, and nothing about its averaging protects it under either; the icosahedron at twenty flipped under Pipek–Mezey, on a shell of degree three that no design can protect.

The description counts are covered by the same argument, and that matters because the count was read as a property of a cage in the essay that found three shapes from one search and chased to four thousand starts. Descriptions are grouped by their participation numbers — how many vertices each localised orbital spreads over — and a participation number is the reciprocal of a sum of fourth powers of an orbital’s coefficients: a vertex sum of degree four again. On a 4-design with a degree-one cut, the populations of descriptions cannot depend on the orientation, which is why the icosahedron at four electrons reported the same count every time. On the octahedron they can, and did.

So the rule does not only say whether the best functional moves. It says whether anything a localisation reports can move, for any quantity that is a vertex sum of low enough degree — and every quantity these essays have read off a localisation is.

Twenty predictions, and the ones that did not have to come true #

The rule has two conditions, and both are needed. The cage must be a design of the criterion’s degree — four for Pipek–Mezey, three for Boys. And the cut shell must be of degree one, because only then is every re-orientation of it a rotation of space.

On the family’s six inputs, twelve predictions: the icosahedron at four electrons safe under both, the octahedron at four safe under Boys only, and the other four moving under both — the ten-vertex cage because it averages almost nothing, and the icosahedron at ten, twelve and twenty electrons because their cut shells are of degree two and three. Every one is right. The icosahedron at twelve and twenty electrons move under Boys by only 1.0017 and 1.0012, which is small, and it is not one.

The second condition is the refusal, and the icosahedron at ten electrons is its test. It is the same 5-design that protects the degree-one cut. It cuts its five-fold shell, whose members are the degree-two polynomials sampled at the vertices, and re-orienting a five-dimensional shell is a five-by-five orthogonal matrix — ten parameters, of which the rotations of space supply three. Almost every re-orientation is not a rotation of anything, the argument does not apply, and the functional moves by 1.207. A rule that looked only at the cage would have called it safe.

Four cages the family never contained #

Twelve correct predictions on the cases that suggested the rule are a description of those cases. The test is cages that played no part in it, chosen before any was run: the tetrahedron at four electrons, the cube at four and at ten, and the dodecahedron at four. Each has a cut shell; the cube at ten cuts its second shell, of degree two.

The dodecahedron — twenty vertices, a 5-design, a degree-one shell cut — does not move under either criterion. The cube at four electrons, a 3-design, moves by 1.051 under Pipek–Mezey and not at all under Boys, exactly as the octahedron does. The tetrahedron, a 2-design, moves under both: 1.311 and 1.510. The cube at ten, with a degree-two cut, moves under both. Eight predictions, eight right, and together with the family’s twelve, twenty of twenty.

The connection worth carrying is where the rule comes from. Spherical designs were defined in the 1970s by Delsarte, Goethals and Seidel as the finite point sets that integrate polynomials on the sphere exactly, and they are the reason quadrature grids for angular integrals are built on the icosahedron and its relatives. A localisation criterion evaluated on a cage’s vertices is a quadrature of a polynomial over those vertices. The property that makes the icosahedron a good integration grid is the property that makes its localisation indifferent to an arbitrary choice.

What was computed and how #

The cages are the family’s own, built by minimising repulsion on a sphere and taking the 3n − 6 shortest distances as edges, and the Platonic solids from their standard coordinates with edges at the shortest distance. Each is diagonalised as a Hückel graph with no repulsion. A shell is a run of eigenvalues equal to a part in a billion; it is cut where the orbitals with any occupation take some of its members and not others, the same test the family survey used.

The design strength is the largest t for which the average of $P_l(\cos\gamma)$ over every pair of vertices vanishes, to 10⁻⁹, for every degree from one to t. The degree of a shell is the smallest l such that every eigenvector in it lies within 10⁻⁸ of the span of the monomials of degree up to l sampled at the vertices. The re-orientations are the family survey’s own: an independent random rotation of every pair of members inside each shell, four seeds, plus the orientation as returned. For the family’s six inputs the best functionals are the survey’s, from sixty starts per orientation; for the four new cages the same basin search is run with forty starts.

The claims are stated where they can fail: that the five Platonic solids have design strengths two, three, three, five and five; that for every input and each criterion the functional is unmoved exactly when the shell has degree one and the design strength reaches the criterion’s degree; that the octahedron at four electrons moves under Pipek–Mezey and not under Boys; that the dodecahedron is unmoved under both. The refusal is the icosahedron at ten electrons, whose degree-two cut on a 5-design must move.

Where the rule stops #

It is proved in one direction and measured in the other. A degree-one shell on a design of the criterion’s degree cannot move, and that is an argument about polynomial sums, not a measurement. That every other case does move is what twenty computations show; it is what one expects of a sum that has no reason to be invariant, and it is not a theorem. A special cut of a higher shell could in principle happen to be invariant for a reason this does not see.

It is about Hückel orbitals on vertices. The argument uses the orbitals’ being exact polynomials in the vertex coordinates, which holds for a degree-one shell on a cage whose three-fold level is its coordinate representation. A basis of atomic functions with overlap, or orbitals from a calculation with repulsion, would add terms that are not vertex sums of low-degree polynomials, and the design strength would stop being the whole story.

And an invariant answer is not a determined state. The same cages sit on both sides of other boundaries, and the lesson there applies here: a localisation is a description of an occupied space, and a description that cannot tell two spaces apart has not shown that they are the same. The icosahedron at four electrons gives the same localisation from every orientation because its localisation criterion cannot tell the orientations apart. The orientations are still different wavefunctions, with different densities; a self-consistent calculation would still break the degeneracy and pick one. What the rule says is that for this cage and these two criteria, the choice has no consequence for the description.

An invariance is sometimes an average #

The habit this suggests is to ask, when a result is unexpectedly independent of an arbitrary choice, what the result is computed from, before asking what symmetry the object has. Here the result was a vertex sum of a quartic, and the question “which point sets make a quartic’s sum rotation-free?” has a classical answer that has nothing to do with the cage’s point group. The icosahedral group was a red herring with the right answer attached: the icosahedron is a 5-design because it is so symmetric, but the octahedron is extremely symmetric too, and its symmetry buys it three degrees of averaging and not four.

It also turns the family survey’s twenty-seven of thirty-three unaffected into a rule. The inputs whose shells are taken whole are unaffected by construction. Of the six that cut a shell, one is unaffected under both criteria and one under Boys alone, and which ones could have been said before any localisation was run, from two numbers: the degree of the shell and the design strength of the cage. Whether the number of descriptions a cage has is a property of the cage is, on the cut inputs, a question about those two numbers.

Still open: choosing the orientation, and the cages between designs #

The obvious open question is the one this argument set aside: for the inputs that do move, the orientation of the cut shell can be chosen rather than inherited, by maximising the localisation functional over it — one angle per pair of shell members, a nested maximisation on four inputs. The design rule says which inputs need it and which do not, which halves the work; what it cannot say is whether the maximum over orientations sits at the top of the ranges measured or above them.

The nearer question is what happens between designs. A cage that is nearly a 4-design — an octahedron distorted slightly toward a shape with more averaging, or a deltahedron between two that are designs — should move under Pipek–Mezey by an amount that shrinks as the degree-four Legendre sum does. If the movement is proportional to that sum, the sum is a quantitative predictor of how undetermined an input is, not just a yes or no, and it can be computed for any cage in a line.