A blindness that is inherited

Mapping the sixteen-electron gap over a plane of two independent trans-ligand folds finds the direction its gradient vanishes along — the gap being the one the rule of sixteen rests on — and shows that on the symmetric line that direction is the antisymmetric fold — one pair up, the other down — for a reason that is exact rather than numerical, since exchanging the two pairs is a symmetry of the arrangement.

Then it wrote a sentence with two halves and computed only one of them.

a distortion the gap is blind to is the one along which a complex is softest without paying for it.

The first half is what it measured. The second is a claim about an energy, and the angular overlap model the gap comes from has no term for one: it knows how much σ overlap each ligand direction produces and nothing whatever about whether two ligands are close together. Its own limits section said so — “a valley floor in the gap is not a valley floor in the energy” — and left the second model unrun.

Run it, and the answer is not one thing.

A second model on the same directions #

The arrangement energy used for hypervalent molecules is four unit charges on a sphere with the Coulomb sum taken over their six pairs at unit bond length. It is the object the hypervalency census prices arrangements with, and the same sum the repulsion account of shape minimises to place points on a sphere, and it is evaluated here on the directions this family produces rather than on a named arrangement.

The two models share nothing but those four directions. One is a sum of squared overlaps resolved into d-orbital symmetries; the other is a sum of reciprocal distances. Neither knows the other exists, which is what makes putting them on one axis worth doing.

On the symmetric line, the blindness is inherited #

At p = q = 5, 10 and 15 degrees, the gap’s null direction is the antisymmetric fold at exactly −45 degrees. The repulsion’s null direction is the same direction, and the two agree to the last representable bit.

That is not a measurement and it could not have come out otherwise. The parity argument for the gap makes no reference to the gap. It says: raising the first pair by three degrees and lowering the second by three gives the same set of four directions as doing the reverse, with the two pairs swapped — so any quantity that is a function of the arrangement alone is an even function of the antisymmetric coordinate, and its first derivative there vanishes identically.

The repulsion is a function of the arrangement alone. So is the gap. So is any observable computed from four directions and nothing else. An exact blindness is inherited by every such quantity at once, and checking a second one confirms the argument rather than testing the claim.

The arithmetic says so directly: the repulsion at thirteen and seven degrees is 3.884920 and at seven and thirteen it is 3.884920, to every digit double precision carries.

And the distortion is downhill #

Both first derivatives vanish, so both quantities move quadratically, and the second derivatives are not obliged to have the same sign.

They do not. Along the antisymmetric fold the gap rises — 0.21 per cent from five degrees, 0.76 from ten, 1.43 from fifteen — and the repulsion falls, by 0.091, 0.092 and 0.093 per cent respectively.

So the phrase softest without paying for it understates its own case. The complex is not softest along that direction without paying for it; in this model it is paid to go. A square-planar arrangement folded symmetrically towards a tetrahedron sits, in the repulsion model, on a ridge in the antisymmetric coordinate rather than in a valley, and the coordinate the gap cannot see is the coordinate the repulsion is trying to push it along.

The consistency of the repulsion’s number across amplitudes is worth a moment. A tenth of a per cent at five degrees, at ten and at fifteen, changing in the fourth figure — the fall is almost independent of how far up the symmetric fold the molecule already is, while the gap’s rise grows sevenfold across the same range. Two quantities with vanishing first derivatives and quite different second ones is exactly what “the parity fixes the direction and nothing fixes the curvature” looks like.

Off the line, nothing is inherited #

At p = 10, q = 0 the gap’s null direction is at 90 degrees and the repulsion’s is elsewhere, with an alignment of 0.3425. At p = 15, q = 5 the alignment is 0.0242 — the two directions are all but perpendicular. At p = 20, q = 10 it is 0.8026.

There is no geometry off the symmetric line where they agree, and at those geometries the direction the gap cannot see is one the repulsion climbs: by 0.61 per cent at ten and zero, by 1.30 at fifteen and five. Against gap movements of 0.24 and 0.44 per cent, so the ratio of cost to benefit is the wrong way round by a factor of three.

That is where the phrase is actually wrong, and it is the general half. The generalisation that came with it drew the right distinction and stopped one step short of the consequence:

A direction located by searching for where a gradient vanishes is blind to the precision of the search… A direction fixed by a symmetry is blind identically.

True, and there is more. A symmetry-fixed blindness is not merely robust — it is transferable, to every quantity the symmetry acts on, without computing any of them. A search-found blindness is not transferable to anything, including to a second quantity computed on the same objects. So the two kinds of blind direction differ not only in how reliably they are blind but in how far the blindness reaches, and the second difference is the one that decides whether a result about one model says anything about another.

What the two models agree about, and what they do not #

It is worth setting the two side by side across the whole family, because their disagreement is not uniform and the pattern in it is the finding stated another way.

They agree about the square plane. Both are stationary there in both coordinates, both for the same reason — a reflection through the plane reverses each fold angle and leaves the arrangement alone — and neither model can say anything about a direction there.

They agree about the antisymmetric fold on the symmetric line, exactly, for the reason above.

They agree about nothing else. The gap rises along a two-ligand fold and falls along a symmetric one; the repulsion rises along both, since any fold at fixed bond length brings ligands closer together on one side. Their gradients point in different directions at every geometry off the line, and the angle between them varies with no pattern that can be named here — 70.0 degrees at ten and zero, 88.6 at fifteen and five, 36.6 at twenty and ten. On the symmetric line it is 0.0, which is the same statement as the null directions coinciding and is the only value in the set that an argument rather than a computation produces.

So the shared features are exactly the ones a symmetry produces and none of the ones a calculation produces. That is a stronger statement than “the models disagree”, and it is the one that generalises: two models of the same object agree wherever the object’s symmetry decides the answer, and are unrelated wherever it does not. The first half is why symmetry arguments are worth making at all, and the second is why a result that is not one of them has to be recomputed in every model it is quoted in.

The magnitudes are the other way round #

There is one more comparison the two models make possible, and it points the opposite way from that phrase.

The gap’s gradient is small. At five degrees on the symmetric line it is 1.94 × 10⁻⁴ per degree, against a gap of two, so a degree of symmetric fold moves the gap by a hundredth of a per cent. The repulsion’s gradient at the same point is 4.15 × 10⁻³ per degree against a repulsion of 3.83 — a tenth of a per cent per degree, which is ten times larger in relative terms.

The same ordering holds at ten degrees and at fifteen, and it reverses only at twenty and ten, where the gap’s gradient has grown to 7.61 × 10⁻² and the repulsion’s to 1.31 × 10⁻². By then the arrangement is far from square-planar and the gap is falling steeply towards the tetrahedral value.

So over most of the family the quantity that resists a distortion is the repulsion and the quantity that barely notices one is the gap — which is the reverse of the picture the phrase suggests, where the gap is the thing being protected and the direction it is blind to is the free one. The gap is nearly blind to every direction near the square plane; what distinguishes the antisymmetric fold is that both models are exactly blind to it, and that the repulsion is not merely blind but favourable.

That is worth saying in the form a chemist would use. A square-planar sixteen-electron complex is soft against a symmetric fold in its electronic structure and stiff against it in its ligand–ligand contacts, and the one coordinate where the two agree is the one where neither has an opinion to first order and the contacts prefer to move.

What was computed, and how #

The gap, the family and the null directions are those of the two-fold map, unchanged. The repulsion is a six-term Coulomb sum on the same four unit vectors, with no parameters in it at all.

At each of the seven distorted geometries in the family both quantities are followed along the gap’s null direction and along the gradient, at amplitudes up to five degrees, and the relative change in each is reported. The repulsion’s own gradient is a central difference in the same two angles at a quarter of a degree, and its null direction is that gradient’s perpendicular.

Six results are checked numerically. The family contains geometries on and off the symmetric line, since the finding is a difference between them and a family with only one kind would show half of it. On the symmetric line the two null directions agree to within 10⁻⁹ — the exactness, stated as a bound the arithmetic beats rather than as an equality between floating-point numbers. The repulsion falls along that direction, so it is downhill rather than flat. Off the line the null directions do not agree, at any geometry, which is what says the agreement is the symmetry’s. And off the line the repulsion climbs, at two geometries or more.

The sixth is a refusal, the same one the gap map has to make: at the square plane both gradients vanish, every direction is null in both models, and the comparison has no content — so it is declined rather than made. An angle reported there would be the arithmetic noise in a quantity that is identically zero.

Where the model stops #

Two models, neither of which is an energy. The angular overlap model gives an orbital gap and the point-charge sum gives a ligand–ligand repulsion, and a real complex’s energy contains both plus a metal–ligand attraction — and an arrangement’s cost and its count came apart once already on this collection, a σ-bonding stabilisation, and everything the two models leave out. Nothing here says which arrangement a complex adopts; it says what two of the terms do along one coordinate.

Rigid distortions. Bond lengths are fixed and only directions change, as in the gap map. A real antisymmetric fold would move the bond lengths, and the repulsion is far more sensitive to a length than to an angle — which is the knob a repulsion model was found never to have been given.

Unit charges. The repulsion has no screening, no polarisation, no dependence on what the ligand is. The census that prices arrangements with it carries the same caveat and found the model good enough to order arrangements and not to value them.

And the inheritance argument is about functions of the arrangement. A quantity depending on something else — a bond length that relaxes, a solvent, a counter-ion — is not covered by it, and would not inherit the parity. That is not a weakness of the argument; it is its precise scope, and it is what makes it worth stating rather than assuming.

The generalisation #

The transferable point is about what a result in one model licenses about another, and the answer turns on where the result came from.

A quantity computed in one model and found stationary somewhere is a fact about that model. Whether a second model agrees is an empirical question and the answer here is no — three geometries, three different degrees of disagreement, one of them nearly perpendicular.

A quantity found stationary because a symmetry of the object forces it is not a fact about the model at all. It is a fact about the object, and every model that sees only the object inherits it, exactly, without being run. The two situations are described with the same words — the gradient vanishes there — and they differ in whether a second calculation is worth doing.

That gives a cheap test with a definite payoff. On finding a stationary direction, ask whether there is an operation mapping the distortion to its negative while leaving the object unchanged. If there is, the result transfers to every function of the object and no further computation is needed. If there is not, the result is about the model and says nothing about any other quantity computed on the same objects — which is worth knowing before it gets quoted in a sentence about a molecule.

The second half is about the sentence itself. “Softest without paying for it” attaches a cost claim to a gradient result, and the two came from different places: one was computed and one was reached for. The cost claim happened to be true on the symmetric line, in a stronger form than stated, and false everywhere else in the same family — which is the worst arrangement, because the case it is right about is the one anybody would check. The same shape turns up where a formula is right exactly for the molecules it would be tested on.

Who found it, and when #

The angular overlap model is Schäffer and Jørgensen’s and the point-charge arrangement energy is older than either. The two-fold family, the gap’s blind direction and the parity argument come from the gap map; the repulsion on the same family, the inheritance argument and the finding that the direction is downhill are new here.

The sentence that turned out wrong was the useful one: it named its own missing term — the repulsion — in the same breath, which is what made the comparison possible.

Still open: where the sixteen-electron complex stops being a singlet #

The obvious continuation is the curvature, which is now the only part of the picture nothing explains. Both quantities are second order along the antisymmetric fold and their second derivatives have opposite signs, and the parity argument says nothing about either — it fixes the first derivative and leaves the rest free. Whether the repulsion’s fall is a general property of an antisymmetric fold or an accident of four ligands at unit length is one sweep of the same sum over a family of bond-length ratios, and it decides whether “downhill” is a statement about this arrangement or about this model.

The nearer question is the one the gap map named and nobody has yet run: the triplet. 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 the competition that decides a spin state. Pairing energies can be computed in the same model. The antisymmetric direction now has an energy attached to it as well as a gap, so the question can be asked properly — at what point on the two-dimensional family does a sixteen-electron complex stop being a singlet, and is that point anywhere the repulsion would let it reach.