The group nobody wrote a table for

A count that changes at one point ended with a prediction, and it was careful to say that making one before the calculation was the best test it had. It had just walked the orphan count along the path from a tetrahedron to a square plane and found it constant everywhere except at the plane itself. It then named the six-coordinate case it had not run — the Bailar twist, from a trigonal prism to an octahedron — and said the count must be constant along that too, since the twist keeps D3 throughout.

The premise is exactly right. The conclusion has no truth value.

What the twist is #

Take an octahedron and think of it as two parallel triangular faces, one rotated sixty degrees against the other. Rotate one face back towards alignment and at zero degrees it is a trigonal prism, with the two triangles eclipsed. Everything between is a one-parameter family, and the parameter is the twist angle. Nothing else moves: all six ligands stay on a sphere at the same radius, so the only thing changing is the arrangement.

Both ends are arrangements that have already been counted. Expensive is not the same as unadopted found that the octahedron and the trigonal prism leave the same number of ligand combinations with no partner on the central atom — two each — so the orphan count cannot tell them apart, while the Coulomb repulsion can, by 2.43 per cent. That was a silence: a place where a counting rule has nothing to say and an energy does.

This essay walks the path between them and asks where along it the count changes. The measurement is the same one run on the four-coordinate path, applied to a different coordinate, and the method is the same: build the geometry from the angle, hand it to the symmetry finder, take whatever group comes back, reduce the ligand σ set in that group, and subtract what the central atom’s valence orbitals can match. What is left over is the orphan count — the same quantity, computed the same way, that four is all that s and p can match introduced and that the population argument tied to something measurable.

The energy behaves as expected. It is smooth, monotone in the twist, lowest at the octahedron at 9.985281 and highest at the trigonal prism at 10.127052 — a span of 1.42 per cent over the whole path. That is the ordinary result, it agrees with the endpoint comparison already made, and it is the thing the count would have been read against had there been a count to read.

The count exists at four angles out of twenty-one #

Twenty-one geometries were sampled, clustered towards both ends because that is where anything interesting was expected. Four of them produced an orphan count. All four read two.

The four are the trigonal prism at 0°, the octahedron at 60°, and the two angles either side of them at 0.01° and 59.99°. Everything else — seventeen of the nineteen geometries strictly between the ends — produced nothing.

That the counted ones all read two is worth stating on its own, because it is stronger than the endpoint comparison that provoked this essay. It is not merely that an octahedron and a prism happen to agree. It is that every geometry on this path whose group can be counted in reads the same number, and no arrangement of six ligands encountered anywhere along the twist gives the count anything to distinguish. Two orphans is what six bonds and four orbitals predicts for a main-group centre with six σ donors, and the twist does not perturb it anywhere it can be asked. The silence is not between two points; it is over everything that can be asked.

But seventeen geometries produced nothing, and nothing turns out to mean two entirely different things that must not be run together.

Two ways to have no answer #

In two bands, the symmetry finder refuses. From about 0.070° to about six degrees, and again from about fifty-four degrees to about 59.930°, it declines to name a group at all. Its own check is the one that fires: 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. Just off the prism that residual is 1.43 × 10⁻³ Å, over the threshold, so the finder refuses.

This is the same refusal found beside the tetrahedron on the four-coordinate path, and it is the right kind of refusal: the finder would rather say nothing than guess a group. It appears twice here rather than once because both ends of this path are special, and the near-symmetry problem sits beside each of them.

Across the whole middle, something else happens. The finder does not refuse. It succeeds.

At thirty degrees — the middle of the twist, as far from either end as a geometry can get — the finder assembles six operations: the identity, two threefold rotations, and three twofold axes. There is no mirror plane among them and no inversion. That is D3, and it is D3 exactly: the worst residual is 4.0 × 10⁻¹⁶ Å, which is machine precision. The same happens at every one of the eleven interior angles sampled, from six degrees to fifty-four. The geometry is not approximately symmetric, not nearly some other group, not resolved with difficulty. It is precisely D3 and the finder says so.

And then it produces no count, because a count is a reduction and there is nothing to reduce in.

Nineteen tables, and the one this needed #

Reducing a representation means decomposing it against the irreducible representations of a group, which means having the group’s character table. The calculation here computes a great deal from first principles — it finds the symmetry operations rather than assuming them, and checks the tables it holds against conjugacy classes it derives — but the tables themselves are written down. It held nineteen of them.

D3h is there, which is why the trigonal prism has a count. Oh is there, which is why the octahedron does. D3 is not there, which is why the ninety-nine per cent of the path between them does not.

The reason this is worth an essay rather than a footnote is what the list is a record of. Nothing about D3 makes it hard. It has six elements and three irreducible representations, and its character table fits on two lines. It is not one of the awkward cases either — not an infinite group needing to be worked in a finite one, not a non-rigid molecule needing a permutation group. C3v is also of order six and was present. D2, D4, D5d and D6h were present, so it is not that rotation-only groups or D-groups are missing as a class. D3 was absent for one reason: no earlier count had needed it.

That makes the list of tables a map of which questions have been asked, and the gap in it a map of which have not. A count that requires a character table is a count that requires somebody to have written that table down — and the geometries where nobody has are not marked as unusual, difficult or borderline. They simply return nothing, in exactly the way a geometry the finder genuinely could not resolve returns nothing.

The tolerance is an angle #

The four geometries that do have counts are worth looking at more closely, because two of them are not the arrangements they are being counted as.

At 0.01° the finder reports D3h, order twelve — the prism’s full group, including the three mirror planes and the horizontal plane that the twist has already destroyed. At 59.99° it reports Oh, order forty-eight. Neither geometry has those operations. The finder reports them because the residual by which they fail is still under its threshold.

That threshold is in angstroms and the path is in degrees, so the natural question is what angle it corresponds to — and along this twist the answer is exact, because the worst residual is precisely linear in the angle.

The slope is 1.4251 × 10⁻² Å per degree, and it holds across five probe angles to within five parts in 10⁸. Dividing the 10⁻³ Å threshold by it gives 0.0702°, and bisecting for the crossing directly gives 0.070173° — the two agree to better than a part in a thousand, which is the point of computing it twice. At 0.9 of that angle the finder answers, with order twelve. At 1.1 of it, the same geometry twisted a tenth further, it refuses.

So the stretch of this path on which an orphan count exists is 0.0702° at each end of sixty degrees: 0.234 per cent. And on all of it, the group being counted in is not the group the molecule has. The counts that exist are counts of the wrong group, granted by a rounding tolerance; the counts that would be of the right group do not exist, because the right group has no table.

What this does to the prediction #

The prediction was that the count must be constant along the interior because the twist keeps D3 throughout. Both halves of that deserve separate verdicts.

The premise is confirmed, and more sharply than it was stated. The twist does keep D3 throughout, exactly, at machine precision, at every interior angle sampled. That was a claim about geometry made without running the calculation, and the calculation upholds it.

The conclusion is not false. It is untestable. To find out whether the count is constant across the interior one would have to evaluate it at more than one interior geometry, and it cannot be evaluated at any of them. A prediction whose subject does not exist is not refuted by the failure to observe it.

This distinction matters because the easy report would be the orphan count is the same at every twist strictly between the two ends. That statement is true of the data — the count is empty at every interior angle, and empty is certainly the same as empty — and it is worthless. Stating it that way would confirm the prediction using the absence of the thing being predicted. The honest statement is that the interior has a group, has an order, and has no count, and it has to be kept apart from the two bands where the finder refuses outright.

What was computed, and how #

Each geometry is six unit vectors: three at the antiprism angle from the pole forming the top triangle, three at its supplement forming the bottom one, with the bottom triangle rotated by the twist angle. At sixty degrees this is the octahedron and at zero it is the trigonal prism, and both of those are checked against the group the finder identifies rather than against the formula that produced them.

The repulsion is the Coulomb sum over the fifteen ligand pairs, in units where the bond length is one. The orphan count is the ligand σ representation reduced in whatever group comes back, minus what the central atom’s valence orbitals can match, summed over multiplicity. The symmetry finder builds its operations by refining candidate matrices against the structure and closing the set under multiplication, then checks two things about the result: that every operation is orthogonal to 10⁻¹⁰, which is about the arithmetic, and that every operation maps the molecule onto itself to better than 10⁻³ Å, which is about the structure. It is the second that fires in the refused bands.

The tolerance reach is measured rather than assumed. Five probe angles inside the answered region give the residual slope and, by the spread of their ratios, a check that the relationship really is linear before anything is divided by it. The bracket for the bisection is tested at both ends before the first midpoint is taken, so a threshold outside it would return a refusal rather than a number pinned to a bracket endpoint.

Where the model stops #

Three limits, and the third is the interesting one.

The ligand field here is six identical σ donors on a sphere with no π interaction and no bond-length change, so the energy is the arrangement’s repulsion and nothing else. A real Bailar twist in a tris-chelate complex is constrained by the chelate bite angle, which is precisely what makes the twist a mechanism rather than a curiosity, and none of that is here.

The angles sampled are twenty-one and not a continuum, so the band edges quoted at “about six degrees” and “about fifty-four” are the sampled ones. The two edges that are quoted precisely — 0.0702° and 59.930° — are the ones that were solved for rather than sampled.

And the third limit is the one the essay is about. The result is a statement about a calculation that counts from a finite list of tables, not about group theory. Someone with a D3 character table gets an answer at every interior angle; the ligand σ representation in D3 reduces perfectly well. What the calculation establishes is that a count computed this way is defined only where a table has been written, that the boundary of that region has nothing to do with the chemistry, and that nothing signals the difference — a geometry with no table and a geometry the finder could not resolve return the same empty answer.

What the two quantities are each good for #

The energy answers everywhere on the path. It is smooth, it is monotone, it identifies the octahedron as the minimum and the prism as the maximum, and it puts a number on the difference. It also cannot see the count’s concerns at all: it is 1.42 per cent from end to end, which is a small barrier, and it says nothing about orbitals.

The count answers on 0.234 per cent of the path, in a group the molecule does not have. Where it answers it says two, both times, which is consistent with every other count and adds no discrimination.

This is not a criticism of counting. It is the shape counting has, and it shows up from several directions: the count changes at a single point on the four-coordinate path, it cannot separate two six-coordinate arrangements, and here it cannot be evaluated across a path joining them. A rule that reads a discrete answer off an exact symmetry is a rule about the symmetric cases, and a continuum is mostly not symmetric — or rather, it is symmetric in groups nobody has tabulated.

Who found it, and when #

The twist is John Bailar’s, proposed in 1958 as the mechanism by which a tris-chelate complex racemises without any metal–ligand bond breaking, and it has been the standard picture of that process since. The trigonal prismatic transition state it passes through was found in real coordination compounds later, which is why the endpoint comparison was worth running at all.

The counting rules involved are older and were never framed as continuous. Sidgwick’s electron count and the σ-matching argument that produces the orphan number are statements about named arrangements, and the whole tradition of applying them works by identifying a molecule with the nearest idealised geometry and counting there. That identification is exactly the tolerance measured above, done by eye instead of by residual — and this essay’s finding is that the interval it covers, when made precise, is a fourteenth of a degree.

Still open: the missing table, and other untabulated paths #

Two open questions, and one of them is cheap. It has since been answered: the count the table was hiding writes the table and reports what the interior says, so the measurement here is a measurement of what a count needs rather than of the twist itself. That is why the figures here withhold the D3 table explicitly instead of relying on its absence — a finding that held only until somebody supplied the missing table would not be a finding.

The cheap one is to write the D3 character table. It is three irreducible representations and three classes, the conjugacy classes are already computed from the operations found, and a new table can be checked against them rather than trusted. Then the interior of this path gets a count and the prediction becomes testable — and the interesting outcome is not whether the count is constant but whether it is two. If it is, the silence extends over the whole path and is a property of six σ donors rather than of two arrangements. If it is not, then the count distinguishes the interior from both ends, and a rule that separates a molecule from both of the idealised structures it sits between is a rule worth examining — including by whatever it would then say about the square that wastes an orbital, which is the other place where an arrangement’s count came apart from its energy.

The other is to ask the question this essay raises about counting rather than about the twist: how many other one-parameter paths between tabulated arrangements pass through untabulated groups, and is a score of tables an unusually thin covering or an unusually thick one. The tetrahedron-to-plane path passes through D2d, which is tabulated, which is why that measurement produced an answer at every angle and this one did not. That looks like luck, and whether it is can be measured.