Four lifts and one that matters

A square-planar complex with sixteen valence electrons keeps its highest filled level and its lowest empty one a fixed distance apart, and that distance is what the sixteen-electron count rests on. Two distortions close it and one opens it, so somewhere between a fold that opens and a fold that closes there is a direction the gap cannot see: lift one trans pair of ligands out of the plane and push the other pair down by the same amount, and the gap does not change to first order.

The essay before this one asked what that blindness was made of and found it was a symmetry. Exchanging the two trans pairs leaves a symmetrically folded complex unchanged, so the gap is an even function of the antisymmetric coordinate and its slope there vanishes identically. Two per cent of difference in bond length between the pairs destroys the symmetry and the blindness with it.

That answer came with a choice built into it. Every distortion in the calculation moved a trans pair together: the plane of two folds has one angle per pair, not one per ligand. So the ligands were paired and they were equal, and the exchange symmetry used both. Whether the blindness needs the pairing, or only the equality, is a question the plane of two folds cannot ask.

Four angles instead of two #

The fix is to give each ligand its own angle. The four ligands sit on the x and y axes in the square plane; each is lifted out of it by an angle $\theta_i$ of its own, keeping its bond length. The two-fold plane is the special case where the two ligands on x share one angle and the two on y share another.

At any point where all four are lifted alike, the arrangement has a fourfold axis and four mirrors. The four lifts, as a set of numbers the symmetry operations shuffle, fall into three kinds: all four moving together; the two on x moving against the two on y; and two combinations in which one pair tilts while the other stays. In the language of the group they are $A_1$, B and E. The gap is unchanged by every symmetry operation, and a quantity unchanged by every operation can only change to first order along the $A_1$ direction.

That is the argument. It needs the four ligands to be equivalent — related by the fourfold axis — and it never mentions pairs.

The computation agrees to the last figure it can carry. At a fold of ten degrees each ligand’s lift changes the gap by −5.400 × 10⁻⁴ per degree, and the four numbers differ from one another by less than four parts in a million million. At five degrees the four parts agree to 1.4 × 10⁻¹¹ of their length, at fifteen to 5.5 × 10⁻¹³. The gradient lies on the line where all four lift together.

Three blind directions #

A gradient on that line is perpendicular to every direction whose four lifts add to zero, and that is not a line but a three-dimensional space.

The antisymmetric pair fold is one: two lifts up, two down, sum zero. The tilt of one trans pair is another — lift one ligand of the pair and lower the other by the same angle, leaving the other pair where it is — and the tilt of the other pair is a third. Any combination of the three is blind as well, which includes a distortion nobody would have tried in the two-fold plane: three ligands up by an angle and the fourth down by three times it. Every one of them changes the gap by less than 4 × 10⁻¹⁵ per degree.

So the blindness was never a property of pairs. The plane of two folds could see exactly one of the three blind directions, because it was the only one that moves ligands in pairs. The tilts move the two ligands of a pair oppositely, and the three-and-one distortion moves them independently.

The tilt deserves a sentence of its own, because at the square plane it has been met before. There, tilting a trans pair rigidly changes nothing at all, because it is a rotation of the whole molecule: two perpendicular pairs always span a plane. Once all four ligands are folded, the pair being tilted is no longer lying in a plane with the axis it tilts about, and the tilt is a genuine change of shape. It is still blind to first order, and now for the symmetry reason rather than because nothing has moved.

A fold of three is not partly blind #

The natural guess about a distortion that lifts three ligands and leaves the fourth is that it sits between the symmetric fold, which closes the gap, and the blind antisymmetric fold, and so should be partly blind. The four equal parts say otherwise.

Each lifted ligand contributes one quarter of the symmetric fold’s change, whichever ligand it is. Three contribute three quarters, 0.750000 to six figures; a trans pair contributes one half; one ligand a quarter. Two ligands side by side contribute the same half as a trans pair. There is no distortion that is a little blind: a direction’s first-order change is the sum of its lifts times the same number, so it is blind exactly when that sum is zero and otherwise fully sighted in proportion to it.

That turns a qualitative question — is this distortion one the gap can see? — into arithmetic anyone can do on the back of the distortion: add up the lifts.

Inside the blind space, the second order decides #

Three blind directions are three directions along which the gap changes only at second order, and the second order need not treat them alike. It does not.

At a fold of ten degrees the gap’s curvature along the antisymmetric pair fold is +6.2 × 10⁻⁴ per degree squared: moving that way opens the gap. Along either tilt it is −5.2 × 10⁻⁵: moving that way closes it. Along three up and one down it is +1.7 × 10⁻⁴, opening, like the pair fold it most resembles.

The ligand repulsion — the second model these essays have kept beside the gap, with nothing in common with it but the four directions — has a gradient on the same line for the same reason, so it is blind along the same three-dimensional space. Its curvatures there have the opposite signs to the gap’s: −1.4 × 10⁻⁴ along the pair fold, where the repulsion falls; +3.4 × 10⁻⁶ along a tilt, where it rises; −4.6 × 10⁻⁵ along three up and one down.

Read together, the two models agree about which blind direction a complex would take. A larger gap is what a sixteen-electron complex wants, and a lower repulsion is what its ligands want. Along the pair fold both happen; along a tilt neither does. The one blind direction the plane of two folds could see turns out to be the one of the three that both models reward, and the two it could not see are the two both models penalise. That the earlier calculation found the interesting direction was, in this sense, luck — it found the only blind direction the plane contained, which happened to be the right one.

The tilt’s curvatures grow with the fold, as they must: at the square plane a tilt is a rotation and costs nothing at any order. From −1.4 × 10⁻⁵ at five degrees the gap’s tilt curvature grows to −1.1 × 10⁻⁴ at fifteen.

The gap’s other curvatures grow the same way, and that is a finding rather than a detail. At the square plane itself every curvature of the gap in the space of four lifts is below 2 × 10⁻⁹ — the gap there is flat to second order in every direction, and changes only at fourth — and at half a degree of fold the pair fold’s curvature is 1.8 × 10⁻⁶, at one degree 7.2 × 10⁻⁶, at two degrees 2.9 × 10⁻⁵: four times for each doubling. The sorting inside the blind space is a property of a folded complex, and it grows as the square of the fold. A square-planar complex has no preference among the three blind directions at all; a complex folded by ten degrees prefers the pair fold over a tilt by a factor of twelve in curvature.

The repulsion behaves differently. Its curvatures along the three blind directions barely move with the fold — along the pair fold −1.39 × 10⁻⁴ at the square plane and −1.43 × 10⁻⁴ at ten degrees — because the repulsion does not have the gap’s fourth-order flatness at the plane. At the square plane the antisymmetric pair fold is the start of the twist towards a tetrahedron, and the repulsion falls along it there already, which is the fall the earlier sweep found to survive every bond-length ratio. So the two models agree about the pair fold from different starting points: the repulsion favours it from the square plane on, and the gap joins it as soon as the complex folds.

What the three look like #

Named as molecules, the three blind directions are recognisable. From a complex folded symmetrically into a shallow umbrella, the pair fold raises the two ligands on one axis further and lowers the other two — the butterfly that carries a square plane through a twisted, $D_{2d}$ arrangement towards a tetrahedron. A tilt swings one trans pair about the axis of the other, like a seesaw pivoting, and leaves the other pair’s folds alone. Three up and one down lifts three ligands further while the fourth drops towards and through the plane, which is the first step from an umbrella towards a trigonal pyramid with the fourth ligand trans to its apex. Only the first is a path sixteen-electron chemistry is known to use, and it is the one both models reward.

Unequal pairs keep two of the three #

The essay before this lengthened one trans pair against the other and found that two per cent was enough to give the antisymmetric fold a first-order slope. With four independent lifts that result can be read more finely.

Unequal pairs remove the fourfold axis and leave two mirrors: one through each pair. The four lifts now fall into two totally symmetric combinations — the two on x together, the two on y together — and two others, the two tilts. The gradient can point anywhere in the first two, so the antisymmetric pair fold, which lies in that plane, picks up a slope: 2.9 × 10⁻³ per degree at a ratio of 1.02, two and a half times the symmetric fold’s 1.2 × 10⁻³, and 1.1 × 10⁻² at 1.1.

The tilts stay blind, to the floor of the difference quotient, at every ratio up to 1.3. Swapping the two ligands of one pair is still a symmetry of the arrangement when the pairs differ in length, and a tilt is exactly the combination that swap reverses. So the blindness that survives unequal bond lengths is the blindness that never needed the pairs to be alike, only the two ligands within each pair. What breaks it is making the two ligands of one pair differ from each other.

What was computed, and how #

The gap is the angular overlap model’s, σ only with eσ set to one, for four ligands at azimuths of 0°, 180°, 90° and 270° each lifted out of the xy plane by its own angle, with the five d levels found by diagonalising the ligand-field matrix and the gap taken as the fifth level minus the fourth. Where one pair’s bonds are lengthened, its σ strength falls with the fifth power of the length, as in the unequal-length family before it. The repulsion is the sum of inverse distances between the four ligand positions.

Gradients and second derivatives in the space of four lifts are central differences with a step of 0.05°, and every named direction is a unit vector in that space. The working points are folds of 5°, 10°, 15° and 17°, all below 17.90°, where the highest filled level stops being dz² and becomes the degenerate dxz, dyz pair. Close to that crossing the tilts mix the two at second order through a small denominator and their curvatures grow sharply, which is a statement about the crossing rather than about the blindness; symmetry keeps the first order at zero on both sides of it.

The claims are stated where they can fail: that at every working fold the gap’s and the repulsion’s gradients lie on the line where all four lifts are equal to better than 10⁻⁸ of their length; that the pair fold, both tilts and three up with one down have slopes below 10⁻⁹ per degree; that a fold of three carries three quarters of the symmetric fold’s change per ligand lifted, and a fold of one a quarter, to a part in a million; that the working folds lie below the level crossing, which falls between 17° and 18°; and that with one pair lengthened the pair fold acquires a slope while both tilts keep none. The refusal is the fold of one: it lifts a single ligand and must carry exactly a quarter — neither the full slope nor zero — or the four parts are not what they claim.

Where the model stops #

One σ parameter and point ligands. The angular overlap model with only σ interactions is the model every calculation in this sequence has used, and it is the one in which the sixteen-electron gap is exactly twice eσ at the square plane. π interactions would add parameters without changing the symmetry argument — the four-equal-parts result follows from the fourfold axis whatever the Hamiltonian is — but they would change every second derivative, and the agreement between the two models about which blind direction is rewarded is a statement about these two models.

Curvatures are not energies. A larger gap is favourable for a sixteen-electron complex in the sense that it keeps the singlet well separated, and a lower repulsion is favourable for the ligands; neither is a total energy, and nothing here says how the two trade off when they disagree. Along the three blind directions they happen not to disagree, which is what makes the reading clean.

And only symmetric folds were tested. Every working point has all four ligands lifted alike, which is where the fourfold axis lives. Away from those points the gradient has no reason to lie on any particular line and the blind space shrinks to what symmetry is left.

A blindness belongs to a symmetry, and a symmetry is bigger than the coordinates used to see it #

The habit this points to is to ask what symmetry is responsible for a zero before asking which coordinate displays it. The plane of two folds displayed one blind direction because it had one coordinate orthogonal to the fold that closes the gap. The symmetry responsible had three directions to give, and the coordinates chosen could hold only one of them. The earlier result was right; it was also a third of the answer.

The same habit runs the other way. When one of those directions stopped being blind — with unequal pairs — the cause was not that the pairs had stopped being equal in some vague sense. It was that a specific operation, exchanging the two pairs, had stopped being a symmetry, while another, exchanging the two ligands of one pair, had not. Naming the operation predicted exactly which blindness would go and which would stay, before either was computed.

Still open: the tilt that costs nothing, and the triplet #

The obvious open question is the tilt near the crossing. Below 17.90° the gap closes slowly along a tilt; approaching the fold at which dz² meets the degenerate pair, that curvature grows sharply, because the tilt mixes the two levels through a vanishing denominator. At the crossing itself a tilt could close the gap at first order through the degenerate pair, and a folded sixteen-electron complex near that fold would be the one place in this family where the blind space becomes a soft one. Locating where the tilt curvature and the repulsion’s curvature cross would say whether such a complex should distort along the tilt, and by how much.

The nearer question is the one the gap has been standing in for all along: the spin state. A gap that closes along a tilt is a singlet approaching a triplet, and pairing energies are computable in this model. Asking where on the symmetric fold, and in which blind direction, the lowest triplet first falls below the singlet is the question the gap has been a proxy for since the count that is not always eighteen, and the three-dimensional blind space says where to look.