The count the table was hiding

The group nobody wrote a table for ended with an unusual sort of finding: a prediction it could neither confirm nor refute, because the quantity the prediction was about did not exist along most of the path it was made for. The Bailar twist from a trigonal prism to an octahedron keeps the point group D3 at every angle strictly between its ends, exactly, and D3 was one of the handful of point groups without a character table to hand. An orphan count is a reduction, a reduction needs a table, and so eleven of the twenty-one geometries sampled returned nothing at all. The quantity itself is the one the ligands’ side of hypervalency is built on, and the arrangement pair it was first measured between is the octahedron and the trigonal prism at the two ends of this very path.

That essay listed writing the table as the cheap continuation. This is it. The table is nine lines.

Writing it, and checking it #

D3 has six operations — the identity, two threefold rotations about the principal axis, and three twofold rotations perpendicular to it — falling into three conjugacy classes. A finite group has as many irreducible representations as it has classes, so there are three, and the sum of the squares of their dimensions must equal the order, so they are one, one and two. There is no freedom left: the table is forced.

The characters follow as quickly. One one-dimensional representation is the totally symmetric one, 1 on every class; the other is 1 on the rotations about the principal axis and −1 on the three perpendicular twofold axes. The two-dimensional one has character 2 at the identity, and orthogonality to the other two rows then leaves it only −1 on the threefold rotations and 0 on the twofold ones. Every row and every column of that can be checked by hand in under a minute.

Nothing about that is difficult, which is exactly the point worth making. What matters is that a table is not taken on trust. A new table is checked against four things: that the class sizes sum to the order, that the squared dimensions sum to the order, that there are as many rows as columns, and that every pair of rows and every pair of columns is orthogonal to within a part in 10⁹. D3 passes all four.

The symmetry finder needed no change at all. It identifies a group by matching the order and the shape of the conjugacy classes it has computed against the tables available, and it requires that a set of operations match at most one of them. D3’s shape — one identity, a class of two threefold rotations, a class of three twofold rotations — does not collide with C3v’s, which is also of order six but whose third class is three mirror planes rather than three axes. So adding the table was enough, and the check guarding against ambiguity is what says so rather than an inspection.

What the interior says #

Fifteen of the twenty-one geometries now produce an orphan count, against four before, and every one of them reads two.

The prediction made for this path was that the count is constant along the interior. It is. That prediction was made before running anything, from the premise that the twist keeps D3 throughout, and it was careful to say that a prediction made in advance was the best test available. It has now been tested and it holds.

But the reason it gave for it does not carry the weight it was asked to. The argument was: the group is the same all along, therefore the count is the same all along. The group is not the same all along — the two ends are D3h and Oh, which are not D3 and are not each other — and the count is two at those as well. So the constancy extends past the region the argument covers, which means the argument was not what was producing it.

Three decompositions, one number #

What produces it is visible as soon as the three reductions are put beside each other.

In D3h the six ligand σ combinations fall into A₁′ ⊕ E′ ⊕ A₂″ ⊕ E″ — four species. In D3 they fall into A₁ ⊕ A₂ ⊕ 2E — three species, one of them appearing twice. In Oh they fall into A₁g ⊕ E𝑔 ⊕ T₁u — three species again, but of dimensions one, two and three rather than one, one and two-twice.

These are genuinely different decompositions. They differ in how many species there are, in what dimensions those species have, and of course in what they are called. If the orphan count were a fact about irreducible representations, three different sets of them would be expected to give three different answers, and the worry that the count is a property of the labelling would be well founded.

They give the same answer, and the reason is that the count is not asking about labels.

There are six ligands, so the σ space is six-dimensional, at every geometry, in every group. The central atom has four valence orbitals — one s and three p — and at every geometry on this path all four of them find a σ combination of matching symmetry to pair with. Six minus four is two. That subtraction is what the orphan count is, and neither number in it depends on what the species are called.

The irreducible representations are how the matching is done: they are what establishes that a particular combination of ligand orbitals and a particular central orbital transform the same way and can therefore interact. But the question the count asks — how many ligand combinations are left with nothing to pair with — has an answer that survives being asked in a different group, provided the pairing still works out the same way. On this path it does. That the representations themselves are what force degeneracy is not in dispute; what is in dispute is whether they force this particular number, and they do not.

Which orbital pairs with what #

The matching is worth writing out, because the three groups do it in visibly different ways and one of them has a trap in it.

At the prism the central atom’s s orbital transforms as A₁′ and pairs with the A₁′ ligand combination; its $p_z$ transforms as A₂″ and pairs with A₂″; its $p_x$ and $p_y$ transform together as E′ and pair with E′. Four orbitals, three species, four dimensions matched. What is left is E″, a two-dimensional species that no valence orbital of an s-and-p centre transforms as — which is the orphan six bonds and four orbitals counts.

At the octahedron the s orbital is A₁g and the three p orbitals go over together as T₁u, so the matching is two species rather than three, and the leftover is Eg. Different names, different shapes, same arithmetic: 1 + 3 matched, 2 left.

The D3 interior is the interesting one. There the s is A₁, $p_z$ is A₂, and $p_x$ with $p_y$ are E — three species again — and the six ligand combinations reduce to A₁ ⊕ A₂ ⊕ 2E. So the leftover is the second copy of E, and E is also one of the matched species.

That means in D3 the orphan cannot be identified by its label. In D3h it can — nothing matched is E″, so anything E″ is left over. In Oh it can — nothing matched is Eg. In D3 the two-dimensional leftover carries exactly the same irreducible representation as a two-dimensional piece that did find a partner, and the only thing distinguishing them is that E appears twice in the ligand set and once in the valence set. Telling them apart is arithmetic on multiplicities, not reading of symbols.

This is a small point and it is the essay’s thesis in miniature. Somebody counting by inspection — looking down the ligand reduction for species that do not appear in the valence reduction — gets the right answer at both ends of the path and the wrong answer, zero, throughout the interior. The label-free version, which subtracts dimensions, is right everywhere.

Where that stops being true #

It is worth being precise about how much this generalises, because the obvious over-reading is that a symmetry-derived count can never depend on the symmetry, which is false.

The tetrahedron-to-plane path is the counterexample already known. There the count is zero along the whole interior and one at the square plane exactly, and the reason is that at the plane one of the central atom’s valence orbitals stops being able to match anything — the matched number falls, and the count rises to take up the difference. The count changed because the matching changed, which is a fact about the geometry expressed through the symmetry rather than a fact about names.

So the rule is not that the labels never matter. It is that the count is six minus the matched dimension, and a change of group moves the count only when it moves how many central orbitals find partners. Along the Bailar twist it never does: all four match at the prism, all four match through the D3 interior, all four match at the octahedron. Along the flattening path it does, once, at a single geometry.

That is a better statement of what the original prediction was reaching for, and it is testable in a way its version was not. Its version — the group is constant so the count is constant — is a sufficient condition that happens not to hold here. The replacement — the matched dimension is constant so the count is constant — holds exactly where the count is constant, on both paths, and says why the one exception is where it is. There is a second case where an arrangement’s count and its energy came apart for reasons the labels did not show, and it is worth reading beside this one: the square that wastes an orbital.

What the table did not buy #

Six geometries still return nothing, and it would be easy to report the table as having closed the gap. It closes part of it, and the part it does not close is a different kind of thing.

Those six sit in two bands, three at each end, from about 0.070° out to about six degrees and symmetrically at the other end. They fail the symmetry finder’s own check: it assembles a set of operations, each of which nearly maps the molecule onto itself, and then requires that the worst of them does so to better than a thousandth of an angstrom. Near an end the geometry is close enough to the end’s high-symmetry arrangement that the finder builds that arrangement’s operations, and they hold to a residual that grows linearly with the twist — 1.4251 × 10⁻² Å per degree — so past a fourteenth of a degree they stop holding well enough.

No character table addresses that. The failure is a statement about how nearly symmetric a geometry is, and the response to it is a tolerance decision or a better refinement, not a reduction. A collection that had every point group tabulated would refuse exactly these six geometries in exactly the same places.

This is the honest shape of the result and it is why the check behind this essay tests it. The obvious claim after writing a missing table is that the hole is filled; the measurement says the hole had two parts, that they were never the same kind, and that only one of them was ever about tables.

What was computed, and how #

The geometry, the repulsion and the reduction are unchanged from the tableless sweep, so that this extends that measurement rather than a different one. Six unit vectors: three at the antiprism angle from the pole, three at its supplement, the lower triangle rotated by the twist angle. The repulsion is the Coulomb sum over the fifteen pairs at unit bond length. The orphan count is the ligand σ representation reduced in whatever group the finder identifies, minus what the central atom’s valence orbitals can match.

The only change is the table. The earlier measurement stays reproducible by running the same sweep with D3’s table withheld, and that is worth being able to do: the finding that the interior had no count is a claim about what a count needs, not a report of which tables happened to be to hand.

That is a small piece of bookkeeping and it is the kind that decides whether a set of measurements stays honest as it grows. A finding written against the incidental state of a calculation is a finding with an expiry date on it, and nothing announces the expiry.

Where the model stops #

The ligand field is six identical σ donors on a sphere: no π interaction, no bond-length change, no chelate constraint. A real Bailar twist happens in a tris-chelate complex where the bite angle of the ligand is what makes the twist a mechanism rather than an abstraction, and none of that is in here.

The constancy is established at fifteen sampled geometries, not proved. The argument in the section above — that the count is six minus a matched dimension, and the matched dimension does not move — is a reason to expect it everywhere on the path, and it is a reason rather than a proof, because the matched dimension is itself something computed at a geometry rather than derived for a family.

And the central atom is a main-group one with four valence orbitals. A transition-metal centre has nine, the σ-matching arithmetic is different, and nothing here says what happens to the count along a twist for those — which is the eighteen-electron count’s territory rather than this one’s.

Who found it, and when #

The Bailar twist is from 1958 and the character table of D3 is very much older than that; neither is new. What is new here is the measurement: the count taken at twenty-one geometries along the path, the observation that three different reductions produce it, and the decomposition of “no answer” into a missing table and a failed tolerance, which are usually reported as the same thing when they are reported at all.

The habit that produced it is worth naming, because it is not the calculation. The tableless sweep could have quoted the two ends, noted that they agree, and stated the interior as constant on the strength of the group being constant — every number in that report would have been correct and the conclusion would have been unsupported. What it did instead was say that the interior had no count, which looks like a weaker result and is a stronger one, because it is what made the table worth writing and what made the replacement statement about matched dimensions findable.

Still open: transition-metal centres, and other untabulated paths #

The obvious open question is the transition-metal case, which the model-limits section names. Nine valence orbitals against six σ combinations inverts the arithmetic — there are more central orbitals than ligand combinations, so the leftover is on the other side, and what the twist does to that number is not obviously constant. The calculation generalises directly: it is the same reduction against a different valence set.

The nearer question is one this has only sharpened. The usual set of tabulated point groups is finite, and at least one path between two tabulated arrangements goes through an untabulated group. Enumerating the one-parameter paths between the arrangements chemists actually use, and asking which of them pass through groups not on the list, would say whether the D3 gap was the last one or the first one found. That is a search over a list of tables and a list of arrangements, both of which exist, and it would turn a gap discovered by accident into a set of them discovered on purpose.