The blindness needed a single top level

A square-planar complex with sixteen valence electrons keeps its highest filled d level and its lowest empty one a fixed distance apart, and the sixteen-electron count leans on that gap. Fold the four ligands out of the plane by a common angle and the gap shrinks. Lift each ligand by its own angle instead and the gap’s gradient has four equal parts, so every distortion whose four lifts add to zero leaves the gap unchanged to first order: the antisymmetric pair fold, both tilts of a trans pair, and any combination of them, a three-dimensional space of directions the gap cannot see.

That argument was made at folds of 5°, 10°, 15° and 17°, and its methods section said why it stopped there. At 17.90° the highest filled level stops being dz² and becomes the degenerate pair dxz, dyz; near the crossing the tilts mix the two through a small denominator and their curvatures grow, and the essay recorded that as “a statement about the crossing rather than about the blindness; symmetry keeps the first order at zero on both sides of it”. Its last section pointed past the crossing and guessed what would happen there: that a tilt might close the gap at first order through the degenerate pair.

Neither statement survives being computed, and the reason is a premise the symmetry argument carried without saying so.

What the argument assumed #

The argument that made the zero-sum space blind is short. At any point where all four ligands are lifted alike the complex has a fourfold axis and four mirror planes, the group $C_{4v}$. The gap is unchanged by every operation of the group, so its gradient — a vector in the space of the four lifts — must be unchanged too, and the only direction in that space every operation leaves alone is (1, 1, 1, 1), all four lifting together. A gradient along that line is perpendicular to every direction whose lifts add to zero.

The step that needs examining is the first: that the gap has a gradient. A gap is a difference of two levels, the fifth d level less the fourth, and a level is a smooth function of the geometry only while it is single. If the fourth level is one member of a degenerate pair, the gap is the empty level less whichever member of the pair is higher — a maximum of two smooth functions, which has a corner wherever they cross. At a corner there is no gradient for symmetry to constrain, and the argument says nothing.

So the blindness was never a property of the four lifts alone. It was a property of the four lifts and a single highest filled level, and past 17.90° the second half is gone.

Past the crossing the pair fold sees #

The computation is direct: from a common fold, move a small amount along a direction, once each way, and take the change in the gap over the distance moved. A smooth function gives two one-sided slopes that are equal and opposite and vanish at a stationary point; a corner gives two that need not be either. With steps of $10^{-4}$ of a degree the difference between the two is unmistakable.

Below the crossing every zero-sum direction behaves as the argument said: both one-sided slopes are smaller than a millionth and shrink in proportion to the step, the signature of a flat, smooth top. Past it the picture splits. Along the pair fold — one trans pair up, the other down — the gap falls by 0.0258 eσ a degree at a fold of 20°, whichever way the pairs move. Along either tilt of a pair the one-sided slopes are still below a millionth. The pair fold has become visible and the tilts have not.

The shape of the change says what is happening. At 15° the gap along the pair fold rises gently either way — a smooth curve with a minimum of the gap at the symmetric point, the opening found in the plane of two folds. At and past the crossing it is a tent: a sharp peak at the symmetric point and straight lines falling away on both sides. Nothing in between is gradual. The corner appears at the crossing, and from there to 25° its slope barely changes, 0.0249 to 0.0262 eσ a degree.

The fold itself already had a corner #

The crossing leaves a mark on the symmetric fold too, and it is the same kind of mark for a different reason. Below 17.90° the highest filled level is dz², which rises slowly as the ligands fold, and the gap closes by about 0.011 eσ for each degree of common fold. Past it the highest filled level is the pair, which rises steeply, and the gap closes by 0.105 eσ a degree — nine times faster. At the crossing the two one-sided rates are those two numbers: the gap as a function of the fold has a corner of its own, because which level it is measured from changes there.

That corner involves no degeneracy split by any distortion. It is simply what a gap defined by ordering does where two levels exchange order, and the distortions that close and open the gap were all measured below it. It sets the scale for the new one. Past the crossing a unit step along the pair fold closes the gap at 0.46 of the rate a unit step along the symmetric fold does — a distortion that was invisible a degree earlier is suddenly half as effective at closing the gap as the distortion that defines the family.

Which directions a degenerate level lets in #

The tent is a split level. Past the crossing the highest filled orbitals are dxz and dyz, a degenerate pair carrying the representation E of $C_{4v}$. A distortion splits such a pair at first order when the distortion’s own representation appears in the pair’s symmetric square, and for E in this group that is

$$E \otimes E = A_1 \oplus A_2 \oplus B_1 \oplus B_2 .$$

The four lifts carry $A_1$ — all four together — plus one B, the pair fold, and an E, the two tilts. B is in the square of E and E is not. So the pair fold splits the filled pair, and the tilts cannot.

Drawn as levels, that is all there is to it. At a fold of 20° the pair fold moves dxz up and dyz down — or the other way, depending on its sign — by equal amounts, linearly. dz², further down, barely moves; the empty level, at 2.34 eσ, moves by less than a thousandth. The gap is the empty level less the upper member of the pair, and whichever member that is, it has risen: the gap closes on both sides at the rate the pair splits. A level degenerate by symmetry, split by a distortion of the right symmetry, always produces a gap with a corner, and the only question is which distortions have the right symmetry.

That also replaces the arithmetic rule the earlier essay left. Below the crossing a distortion’s first-order effect on the gap is the sum of its lifts times one number; it is blind exactly when the sum is zero. Past the crossing the first-order change is its $A_1$ component times the symmetric fold’s slope, less the size of its B component times the pair fold’s rate. Three ligands up and the fourth down by three times as much adds to zero and was blind; it has a B component of 0.577 of its length, and at 20° it closes the gap at 0.0149 eσ a degree, which is 0.577 of 0.0258 to a per cent. The rule is still arithmetic, but it is no longer one sum: it is a projection, and the absolute value in it is the corner.

At the crossing itself the tilts see too #

The guess the earlier essay made about the crossing was not wrong, only misplaced in its mechanism and its scope.

Exactly at 17.903°, where dz² and the pair have the same energy, the top of the filled levels is three-fold: $A_1 \oplus E$. Now a distortion can also act by mixing dz² with a member of the pair, and that needs its representation to appear in $A_1 \otimes E = E$ — which is exactly the tilts. A tilt at the crossing closes the gap at first order, at 0.00145 eσ a degree, a corner like the pair fold’s but seventeen times shallower, and the slope is the same at steps of $10^{-3}$ and $10^{-4}$ of a degree. A twentieth of a degree either side of the crossing it is gone: dz² and the pair have separated, the mixing is second order again, and the tilts’ slopes fall back below $10^{-6}$.

So the tilt does close the gap at first order, but only on the crossing — a set of folds of measure zero — and not through the degenerate pair, which it cannot split, but by mixing the pair with the level crossing it. The direction that closes the gap at first order everywhere past the crossing is the pair fold, at seventeen times the rate.

The energy does not see the corner #

A gap is a proxy. What a complex distorts along, if it distorts, is set by its energy, and the energy of a closed-shell $d^8$ complex in this model is the sum of its four filled levels, each holding two electrons. Along the pair fold past the crossing that sum does not have a corner: one member of the filled pair rises as the other falls by the same amount, and both are full. The filled levels’ energy changes along the pair fold only at second order, at every fold.

What the energy does at second order is a separate story with its own turning point, and it is not at the crossing. At small folds the filled levels’ energy rises along the pair fold — the symmetric arrangement is a minimum. Its curvature falls as the fold grows and changes sign at 13.83°, four degrees before the crossing, and from there the filled levels gain energy as the pairs move apart. The ligand repulsion, the second model these essays have carried, falls along the pair fold at every fold; its curvature there is negative from 5° to 30°. So past 13.8° both quantities fall along the pair fold, and whatever weight they are given relative to each other, a symmetrically folded $d^8$ complex in this model is no longer at a minimum along it. Along the tilts both curvatures stay positive.

The gap marks none of this. At 13.8° its pair-fold behaviour is still the smooth, gently opening curve of the plane of two folds; at the crossing, where its corner appears, the energy’s curvature passes through without a kink. The gap had been the stand-in for the energy throughout this line of essays because, near the square plane, a large gap and a stable singlet go together. Past the first few degrees of fold they part: the gap’s most dramatic event is invisible to the energy, and the energy’s turning point is invisible to the gap.

How the slopes were taken #

The levels are those of the angular overlap model with σ interactions only, four point ligands at unit bond length, each lifted out of the square plane by its own angle — the model every essay in this sequence has used, in which the square-planar gap is exactly 2eσ. The crossing is located by bisection on the energy difference between dz² and dxz along the common fold, to well under $10^{-12}$ of a degree. One-sided slopes are taken with a step of $10^{-4}$ of a degree along unit directions in the space of the four lifts, at sixty-one folds from 10° to 25° and at the crossing itself. Second derivatives of the filled levels’ energy and the repulsion are central differences with a step of 0.05°.

The checks, run wherever these figures are drawn. From 10° to 17.75° every zero-sum direction’s one-sided slopes are below $10^{-6}$. From 18° to 25° the pair fold’s two one-sided slopes are both below −0.024 and equal to $10^{-4}$, while a tilt’s stay below $10^{-6}$, and three up with one down matches its B projection to a per cent at every fold. At the crossing a tilt’s slope is below $-10^{-3}$, the same at two steps to a per cent, and under a tenth of the pair fold’s. Along the pair fold at 20° the filled levels’ energy has no first-order change, and its finite-difference slope grows tenfold with a tenfold step, as a second-order change must. Its curvature turns negative between 10° and the crossing; the repulsion’s is negative throughout and both are positive along a tilt. The refusal is a smooth extremum: at 15° the pair fold’s one-sided slopes must be equal and vanish with the step, or the corner test would report a corner at every flat point.

Where the model stops #

σ only, and point ligands. π interactions split dxz and dyz from dz² differently and move the crossing; they cannot remove it, since the crossing is between levels of different symmetry, and they cannot change which distortions split the E pair, which is group theory. The rates — 0.0258 and 0.00145 eσ a degree — and the turning point at 13.8° are this model’s.

A one-electron energy. The filled levels’ sum is the model’s whole electronic energy; it has no electron repulsion, no pairing energy and no change in bond length as the ligands move. “Both quantities fall along the pair fold” is a statement about two model energies, each with its own units, not about a total energy.

Symmetric folds only. Every working point has all four ligands lifted alike. A complex already distorted along the pair fold has lost the fourfold axis and its E pair is split; the corner belongs to the symmetric line, and the slopes measured are the slopes leaving it.

Who found the rule #

That a degenerate level is split at first order exactly by distortions in its symmetric square is the group theory behind the Jahn–Teller theorem of 1937, which asks the question for a partly filled degenerate level and finds that some such distortion always lowers the energy. Here the degenerate level is completely filled, so the energy feels nothing at first order — the Jahn–Teller argument’s conclusion does not apply — while the gap, which watches only the upper member, feels everything. The angular overlap model is Schäffer and Jørgensen’s. The corner along the pair fold, its rate, the tilts’ first-order response at the crossing and nowhere else, the projection rule, and the energy’s turn at 13.8° are computed here.

A zero proved by symmetry is a zero of a smooth function #

The general point is about what a symmetry argument for a vanishing derivative needs. “The quantity is invariant, so its gradient lies in the invariant subspace” is a statement about gradients, and it applies only where there is one. Quantities defined by ordering — the highest filled level, the gap, the strongest line in a spectrum, the lowest root — are smooth only between the places their order changes, and at and past those places a symmetry that forbade a slope can coexist with a corner. The right question at a degenerate level is not “what is invariant” but “what does the symmetric square of this level contain”, and the answer can be read off a character table before anything is computed.

The other half is the habit the earlier essay named: say what symmetry a zero belongs to before saying which coordinate shows it. Here the symmetry was the same on both sides of 17.90°. What changed was the level the symmetry was acting on, and that was enough to give one of three blind directions its sight back.

Still open: the triplet, and where the pair fold leads #

The obvious open question is the spin state, which the gap has stood in for since the count that is not always eighteen. Past the crossing the gap closes linearly along the pair fold while the filled levels’ energy changes only quadratically, which is the configuration in which a triplet — one electron promoted from the upper pair member to the empty level — gains on the singlet fastest. With a stated pairing energy the model can say where along the fold, and how far along the pair fold, the triplet first falls below the singlet; whether that happens before or after the energy’s own instability at 13.8° decides which of the two a real $d^8$ complex near this geometry would show first.

The nearer question is where the pair fold goes. Past 13.8° both of this model’s energies fall as the pairs move apart, so the symmetric fold is not where a complex folded that far would rest. Following the pair fold from 20° until the filled levels’ energy and the repulsion stop falling would say what geometry the model prefers instead — and whether it is a recognisable one, such as the butterfly or the sawhorse shapes four-coordinate complexes are found in.