One was a symmetry and one was not

A square-planar complex’s sixteen-electron gap has a direction it does not move along, and it is the antisymmetric fold: one trans pair of ligands out of the plane one way, the other pair the other way. The ligand–ligand repulsion falls along the same direction, by about a tenth of a per cent at every amplitude tried, so the distortion looked free in both senses at once and there was a phrase for it — the direction a complex is softest along without paying for it.

That essay’s closing paragraph named the check it had not run. Both results were computed with all four ligands at the same bond length, and whether the repulsion’s fall is a general property of an antisymmetric fold or an accident of four bonds at one length is one sweep of the same sum.

The sweep separates the two completely. One of them is a symmetry and the other is a property of the arrangement, and nothing in the essay before it distinguished them because at equal bond lengths they look identical.

Why the blindness was exact #

At equal bond lengths, exchanging the two trans pairs maps the arrangement onto itself. The gap is a function of the arrangement and nothing else, so it is an even function of the antisymmetric coordinate — the coordinate that swaps the two pairs — and an even function has a vanishing first derivative at its centre.

That is a theorem rather than a result. The gap’s slope along the antisymmetric fold comes out at 2 × 10⁻¹⁶ per degree, which is not a small number but the finite difference’s own floor, and no computation could have made it anything else.

It is the same argument the essay that found the direction used to establish it exactly rather than numerically, and it is worth restating in the vocabulary of the rest of this argument: the antisymmetric fold belongs to a symmetry species that the gap, being invariant, cannot contain a first-order term in. The gap has been made to move along other directions — a two-ligand fold opens it by five per cent and a four-ligand bend closes it — and both of those directions are symmetric under the exchange.

The same argument applies to the repulsion, and this is the part that is easy to over-read. The repulsion is also a function of the arrangement, so it inherits the same parity and its first derivative also vanishes identically along the fold. Both slopes are zero, and both are zero for the same reason, which is why the earlier essay found them agreeing to the last bit.

But a vanishing first derivative says nothing about the second, and the two quantities’ second derivatives are not related by anything.

Two per cent of a bond length removes it #

Make the two trans pairs different lengths and the exchange stops being a symmetry. The second pair sits at a ratio r of the first’s length, its position scales with r, and its σ interaction scales as r to some negative power — so both models see the mismatch, the repulsion through the distances and the gap through the strengths.

At a ratio of 1.02 the gap’s slope is 2.15 × 10⁻³ per degree. At 1.10 it is 9.05 × 10⁻³, at 1.20 it is 1.31 × 10⁻², and at 1.50 it is 2.68 × 10⁻².

Two per cent of a bond length is enough. There is nothing gradual about the onset: the slope is zero at exactly one ratio and non-zero at every other, because the thing producing the zero is a symmetry and a symmetry is either present or absent. The growth after that is roughly linear in the mismatch, which is what a first-order term looks like when the symmetry that forbade it has been broken by a small amount.

The size is worth putting beside something. At a ratio of 1.10 the gap moves by 9 × 10⁻³ of a σ unit per degree along the fold, and the gap itself is 1.72 at that ratio — so a four-degree excursion moves it by two per cent. That is the same order as the five per cent a two-ligand fold opens it by, which is the distortion the gap is supposed to be most sensitive to. A ten per cent bond-length mismatch makes the blind direction about as informative as the direction the gap moves along fastest, which is the end of the blindness as a useful idea.

So the blind direction exists for a square-planar complex with four identical ligands, and for nothing else. That is a real restriction: a sixteen-electron complex with two different kinds of ligand in trans positions — which is most of them — has no direction the gap cannot see.

The fall does not care #

The repulsion’s curvature along the fold is −2.87 × 10⁻⁴ at equal bonds, −2.70 × 10⁻⁴ at a ratio of 1.10, −1.88 × 10⁻⁴ at 1.50, −1.03 × 10⁻⁴ at 2.00 and −4.6 × 10⁻⁵ at 2.50.

Negative everywhere, and shrinking smoothly. The arrangement sits at a maximum of the repulsion along the antisymmetric direction at every ratio in the sweep, so moving either way lowers it — and that is not a symmetry statement at all. It is a statement about where four charges are, and it happens to be robust.

The mechanism is geometric and is worth saying because it explains the robustness. Folding one pair up and the other down takes the two pairs away from each other: the four ligands were coplanar and the two pairs now sit in two planes that have separated. Every one of the four cross-pair distances grows and the two within-pair distances do not change, so the sum of inverse distances falls — and that is true whatever the two pairs’ lengths are, because it is about the angle between two lines and not about how long they are. The mismatch changes how much each cross-pair distance grows and not whether it grows.

What the mismatch does instead is make the fall lopsided. At equal bonds the two directions are worth exactly the same −0.0587 per cent, which is what an even function gives. At a ratio of 1.20 they are −0.036 and −0.077; at 1.50, −0.0008 and −0.096; and at 1.75 the first has turned uphill at +0.027.

So the statement this direction is downhill survives to a fifty per cent bond-length mismatch and then becomes a statement about which sign was chosen. A fifty per cent mismatch between two trans bonds is not a square-planar complex, so the fall is available for every real case and the blindness is available for almost none.

Which of the two the sweep could have changed #

There is a test of whether this reading is right, and it is to sweep the model’s least certain ingredient.

How an angular-overlap σ parameter scales with bond length is a fitted matter, usually quoted as an inverse power between about four and six. Taking it as the inverse third, fifth and seventh power changes the slope at a ratio of 1.20 by up to a half between the three.

It does not move the ratio at which the slope vanishes. That is one under all three, and it is one because a symmetry puts it there — the σ strengths of the two pairs are equal when the two bond lengths are, whatever law relates them, so the exchange is a symmetry whatever law is used.

That is the cleanest available statement of the difference between the two findings. A result that depends on a fitted power is a measurement in a model; a result that the fitted power cannot move is a consequence of the arrangement’s symmetry. The gap’s blindness is the second kind of thing and its disappearance is the first kind, and the essay before it had no way to tell which it had because it never varied anything the symmetry depended on.

What a real complex has #

It is worth putting numbers on how unequal two trans bonds actually are, because the whole reading turns on whether two per cent is a lot.

Two trans ligands in a square-planar complex are related by the inversion centre when they are the same ligand, so their bond lengths are equal by symmetry and the ratio is exactly one — the idealisation is exact for a complex like tetrachloroplatinate. The blindness is therefore real for that case, and the earlier essay was computing something rather than nothing.

The moment the two trans ligands differ the ratio departs from one by whatever the two bond lengths differ by, and that is not a small number: a metal–chloride and a metal–carbon bond differ by ten to twenty per cent, a metal–hydride and a metal–iodide by more than a third. So the ratios at which the blindness is gone are the ordinary ones, and the ratio at which it survives is the special one.

That leaves the finding with a clean statement of its scope. The direction the gap cannot see belongs to a homoleptic square plane. Two of the complexes this argument has counted are homoleptic and most of the interesting ones are not, and nothing in the essay before it said which kind its result was about — because at the level of an angular model the four ligands are four σ strengths, and four equal σ strengths is what was entered.

What the two models are #

The gap is angular overlap, σ only. Five d orbitals, one σ interaction per ligand, and the gap is the interval between the fourth and fifth levels — which for a square plane is the sixteen-electron gap, and which a π channel cannot close because one of the five orbitals has no partner there. A π interaction would change the levels and would not change the symmetry argument, because a π parameter of the two pairs is equal when their lengths are.

The repulsion is unit charges at the ligand positions, summed pairwise as 1/d. It has no fitted parameter at all, which is why its statements are about geometry, and it is the same sum the argument used to price a fifth ligand arriving. What it omits is everything about the ligands except where they are — a point charge is not a carbonyl — and the curvature it reports is therefore the geometric part of a real steric cost.

The two models do not share a parameter, which is the reason the comparison is worth making and also its limitation: nothing here says the two quantities are commensurate, only that one is flat where the other falls. The essay below put a cost per unit of gap change on that, and the ratio is infinite at equal bonds by construction — which is a sign that the ratio was the wrong quantity to report, since an infinite value carries no information about either half of it.

And the fold is rigid. Both pairs stay straight, the metal stays where it is, and the only coordinates are the two fold angles and the length ratio. A real distortion would relax the bond lengths as it folded, which is a coupling neither model here has and which would matter most at exactly the ratios where the two bonds are most unequal.

And the working point is a ten-degree fold of both pairs. The antisymmetric direction is defined at a symmetric working point, which is where it is exactly $(+1, -1)/\sqrt{2}$. At a working point that is not symmetric the two quantities’ null directions have to be found rather than written down, which the earlier essay did and found them nearly perpendicular.

What would have caught it earlier #

There is a cheap test that would have separated the two findings at the moment they were made, and it is worth naming because it costs one line.

A quantity that vanishes by symmetry vanishes to the arithmetic’s floor. A quantity that vanishes for a reason of arrangement vanishes to whatever accuracy the arrangement is specified with. So the size of the zero distinguishes them: the gap’s slope came out at 2 × 10⁻¹⁶ per degree, which is machine precision on a quantity of order one, and the repulsion’s first derivative came out at exactly 0 — also machine precision, because it too is protected.

That is why the size did not separate them here: both are symmetry zeros at equal bond lengths, and what differs is the second derivative, which nothing was looking at. The essay below reported a finite excursion’s effect, which is a second-order quantity read as though it were a property of a direction.

The test that does separate them is the one this essay ran, and it is the general one: perturb the symmetry and see which zero survives. It costs one extra parameter and it is the only way to tell a protected quantity from a small one, because both look like zero.

Two zeros of different kinds #

The habit: when two quantities both vanish, ask whether they vanish for the same reason, and then break the reason.

Here they did vanish for the same reason — one parity argument covering both — and that was exactly what made the coincidence look like a finding. The way to tell a shared cause from a coincidence is to remove the cause and see what survives, and the cause was removable at essentially no cost: one number in a geometry, swept.

The corollary is about what a symmetry-protected zero is worth. It is worth a great deal as an exact statement and almost nothing as a robust one, and those are opposite virtues. A quantity that is zero by symmetry is zero only in the configuration the symmetry describes, and configurations in chemistry are idealisations — four identical ligands at four identical distances is a drawing rather than a molecule. A quantity that is merely small, for reasons of arrangement, is often the one that survives.

Who computed what, and when #

The angular overlap model is Schäffer and Jørgensen’s, from the 1960s, and the inverse-power scaling of its parameters with bond length is a standard fitted relation. The unit-charge repulsion sum is the oldest model of ligand crowding there is. The blind direction and the fall along it are this argument’s own, from the two essays before it.

What is computed here is the antisymmetric fold swept across a family of bond-length ratios, with the gap’s first derivative and the repulsion’s second followed separately, and the σ distance law varied to say which of the two findings a parameter could have moved.

The numbers worth carrying are 2 × 10⁻¹⁶ and −2.87 × 10⁻⁴ — a slope that is zero because of a symmetry and a curvature that is negative because of an arrangement, at the one configuration where they were found together.

Still open: the triplet, and a fold of three #

The obvious open question is the one the gap map named two essays ago and nothing has run: the spin state. Along the symmetric fold the gap falls, and somewhere it becomes small enough that the two frontier electrons occupy both orbitals rather than pairing in the lower one, which is a competition between the gap and a pairing energy. Pairing energies are computable in this model, the antisymmetric direction now has a cost attached to it as well as a gap, and the question can finally be asked properly — at what point on the two-dimensional family does a sixteen-electron complex stop being a singlet, and does the repulsion’s fall carry it there or hold it back. The unequal-length family makes that a three-parameter question rather than a two-parameter one, which is a reason to ask it at equal lengths first.

The nearer question is a fold of three. Everything above folds two pairs, which is the arrangement a square plane’s own symmetry suggests; folding three of the four ligands and leaving one breaks the exchange symmetry without touching a bond length, and it would say whether the gap’s blindness needs the pairing of the ligands or only their equality. Those are different conditions and both are satisfied by the same idealised complex, which is why nothing so far has had to choose between them.