The pair fold was the road to a tetrahedron

A square-planar complex with sixteen valence electrons has a gap of exactly two eσ in the angular overlap model, and a long line of essays has asked how that gap and its energy respond when the four ligands are lifted out of the plane. The last of them, past the crossing at 17.90°, measured two energies along the pair fold — one trans pair of ligands lifted further and the other lowered, around a common fold of all four. The filled d levels’ energy, curving upward along it at small folds, turned downward past 13.8°; the ligand repulsion, the second model these essays have carried, curved downward along it at every fold. So past 13.8° a symmetrically folded complex, it concluded, is not at a minimum along the pair fold under any weighting of the two.

The question it left was where the pair fold goes: follow it from a fold of 20° until both energies stop falling, and see whether the geometry is a recognisable one — a butterfly, say, or the sawhorse four-coordinate complexes are sometimes found in.

The answer starts by asking whether the folded complex was ever somewhere to rest.

A folded square is on a slope #

A curvature along the pair fold says whether a point is a minimum along that direction only if the point is stationary along every other one. The folded square has one other symmetric direction, the common fold itself, and along it the two model energies both rise.

Lifting all four ligands together crowds them towards one pole, so the repulsion rises at every fold. And the filled d levels’ energy rises at every fold up to the tetrahedral angle, as the ligands move off the lobes of $d_{x^2-y^2}$ and onto those of the orbitals that hold the electrons. Under any positive weighting, a folded square rolls back towards the plane. Past 35° the d levels’ energy can fall, as levels reorder, but balancing that against the rising repulsion needs a weight of at least 0.91 on the d levels — more than ten times the weight at which anything interesting happens below.

So the folded square at 20° is not a geometry a complex in this model would be found in, and the negative curvature along its pair fold is a statement about a slope. The question “where does the pair fold lead from 20°” has no answer, because nothing is at 20°.

The same distortion, at no fold #

The pair fold has a meaning that does not need a fold. Applied to the flat square, it lifts one trans pair by an angle and lowers the other by the same angle, and at 35.26° — the angle whose cosine squared is two thirds — the four ligands sit at the corners of a regular tetrahedron. In between are the $D_{2d}$ flattened tetrahedra, two pairs of ligands each opened out from 180° towards 109.5°. The pair fold at zero fold is the road from a square to a tetrahedron, the coordinate every account of four-coordinate $d^8$ complexes argues along.

Along the road the two energies do opposite things. The repulsion falls all the way to the tetrahedron, where four charges on a sphere are as far apart as they can be, and rises beyond it. The filled levels’ energy rises all the way, from two eσ in the square to 5.33 at the tetrahedron, and peaks there with a corner: at the regular tetrahedron the sixteen-electron gap is zero, the two highest levels change places, and the energy turns down beyond it. A peak with a corner in one term and a smooth minimum in the other means that any positive weight on the d levels pushes the minimum off the regular tetrahedron, and pushes it to the flattened side, where the d levels’ energy is lower; the elongated branch beyond 35.26° lies higher at every weight.

Two roads with one set of levels #

Here is the fact that makes the earlier picture coherent, and it is exact. At any angle, a square folded by that angle and a tetrahedron flattened to that angle have the same five d levels.

The reason is a sign. The ligands of one trans pair lie in the xz plane and those of the other in the yz plane. A ligand in the yz plane has no amplitude on dxz, and a ligand in the xz plane has none on dyz. Lowering the yz pair instead of lifting it reverses the sign of its z coordinate, which reverses the sign of its dyz amplitude and leaves every other amplitude as it was. Changing the sign of the dyz orbital throughout undoes that, so the two Hamiltonians are the same matrix written in two bases. The filled levels’ energy and the gap agree to twelve figures at every angle checked, and flipping a single ligand’s elevation, which no sign change of one orbital can absorb, changes them by more than a tenth of an eσ.

So the crossing at 17.90° where dz² meets the dxz, dyz pair, the gap that closes past it, and the filled levels’ energy are the same functions of angle on both roads. The d levels cannot tell a folded square from a flattened tetrahedron. Only the repulsion can, and it reads them in opposite directions: along the common fold the four ligands crowd together and the repulsion rises; along the road they spread towards the tetrahedron and it falls. Every difference between the two distortions in this model lives in the second energy.

That also reads the earlier result again. The filled levels’ energy turned downward along the pair fold past 13.8° on the folded square. On the road, where the same pair fold starts from the plane, it curves upward at the square — 0.0073 eσ per square degree — and nothing turns before the tetrahedron.

The direction that looked free #

This is not the first time the road has appeared in these essays; it is the first time it has been followed. At the flat square, the pair fold is the direction the gap cannot see: exchanging the two trans pairs is a symmetry of the square, the gap is even in the distortion, and its first-order change is zero — on the road the gap reads 1.9997 eσ at 5° and 1.9945 at 10°. With each ligand lifted by its own angle that blindness turned out to belong to every distortion whose lifts add to zero. And along the same direction the repulsion falls, so the distortion looked free in both senses at once: the electron count did not object, and the ligands wanted it.

What the gap could not see, the filled levels’ energy can. A gap is the distance between two levels; the energy is the sum of the four filled ones, and on the road the filled dxz, dyz pair rises as the ligands climb onto their lobes while the gap’s two levels move together. The energy’s curvature at the square is what prices the distortion, and it is what the gap was standing in for. Once the two ligand pairs were allowed different lengths the gap’s blindness was already shown to depend on an equality the real complex need not have; here the blindness holds and simply does not matter, because the quantity that decides the geometry was never the gap.

Where the minimum sits #

With the weight w on the d levels’ energy, in eσ against the repulsion’s unit charges at unit distance, the energy to minimise is w·E + R, and the road can be followed weight by weight.

With no weight on the d levels the minimum is the regular tetrahedron, to a hundredth of a degree. With any weight it flattens: 33.57° at a weight of 0.01, 26.34° at 0.04, 11.18° at 0.07. The square plane becomes the minimum at w = 0.0762, where the d levels’ upward curvature along the road, 0.00731 eσ per square degree, times the weight, balances the repulsion’s downward curvature, 0.000557. Above that weight the minimum is the square, exactly.

A square root, not a jump #

Whether the square and the tetrahedron trade places continuously or by a jump is the question that decides whether flattened tetrahedra are a family or a curiosity.

Over the last two hundredths of weight below 0.0762, the square of the flattening angle falls on a straight line to within five per cent, at 20,184 square degrees per unit of weight. So the angle goes as 142° times the square root of the distance below the critical weight — the form of a continuous transition, in which the minimum leaves the square by flattening into a tetrahedron that is flat at first and opens steadily. There is no weight at which a square and a distinct tetrahedron are both minima on the road with equal energies. The family runs unbroken from the square to the regular tetrahedron and its members are the minima at the weights between.

The gap closes on the way #

The sixteen-electron count is a statement about a closed shell, like the eighteen-electron rule it sits beside, and a closed shell is only the lower state while the gap exceeds the cost of pairing two electrons in one orbital.

On the road the gap of the minimum falls from two eσ at the square to zero at the regular tetrahedron. It stays near two until the minimum passes 17.90° at a weight of 0.0603 — the same angle as the crossing on the folded square, because the d levels are the same — and then falls steeply. With a pairing energy of one eσ, the closed shell at the minimum loses to a triplet below a weight of 0.0403, where the minimum is a tetrahedron flattened to 26.22°; with half an eσ, below 0.0246, at 30.55°; with one and a half, below 0.0519, at 21.94°. Whether an electron pairs is a competition between a splitting and a pairing energy, and the road is where that competition is lost: the closed-shell calculation, followed downhill, walks out of its own domain before it reaches the tetrahedron.

That is the triplet the earlier essay pointed to, and it arrives here from the other side. Real four-coordinate $d^8$ complexes divide along the same line: square-planar ones are diamagnetic, and the tetrahedral ones, such as the tetrahalonickelates, are paramagnetic with two unpaired electrons.

Every minimum the elevations allow #

A road is one line through a four-dimensional space of elevations, and its minimum need not be the space’s.

At every weight sampled, the lowest minimum found is the road’s: the flattened tetrahedron at 0.02, 0.05 and 0.07, at exactly the road’s energy, and the square at 0.08 and 0.15. Two other kinds turn up. One is a family of lower-symmetry near-tetrahedra, with angles between 101° and 116° and gaps under 0.16 eσ at the lower weights — closed shells barely held, and between 0.003 and 0.1 above the lowest. The other is the square itself below the critical weight, reached by eight to sixteen starts in forty: these are squares tipped about one axis, whose flattening would need a ligand’s azimuth to change, which four elevations cannot do. At 0.15 one sawhorse-like minimum appears, with a largest angle of 153° and the rest near 100°, a quarter of a unit above the square. No butterfly or sawhorse is ever the lowest.

How it was computed #

The levels are the angular overlap model’s with σ interactions only, four point ligands at unit distance, each at a fixed azimuth — 0°, 180°, 90° and 270° — and its own elevation, as in every essay of this line. The filled levels’ energy E is twice the sum of the four lowest d levels, in eσ, and the repulsion R is the sum of inverse distances between four unit charges. The minimum on the road at each weight is found by golden-section search over the flattening angle, separately on the flattened and elongated sides of the tetrahedron. The critical weight is the ratio of the two energies’ second derivatives at the square, by central differences with a step of 0.01°. The census descends by steepest descent with a numerical gradient and backtracking from forty seeded random sets of four elevations between −85° and 85° per weight, and classifies each end point by its six ligand–ligand angles, which do not depend on the frame.

The checks, run wherever these figures are drawn: the two roads’ filled levels’ energy and gap agree to 10⁻¹² at six angles, and flipping one ligand changes them; both energies rise along the common fold to 35°, the repulsion at every fold, and no folded square is stationary below a weight of four times the largest used; the minimum is flattened below the critical weight and square above it, with the flattened branch below the elongated one at every positive weight; the angle squared is linear in the weight to five per cent near the critical weight; the regular tetrahedron has no gap and the filled levels’ energy peaks there; and every census’s lowest minimum is the road’s. The refusal is the weightless case, which must return the regular tetrahedron to a hundredth of a degree.

Where the model stops #

Two energies with no exchange rate. The weight w is a free parameter because the model has no common unit for an eσ and a repulsion between unit charges at unit distance. Everything here is stated as a function of it; the model does not say which weight a real complex has, and the critical value 0.0762 is the model’s.

Four elevations, fixed azimuths. The ligands move only up and down their own vertical planes. A full four-coordinate geometry has five shape coordinates and this space has three, plus a rotation. The $D_{2d}$ road and the square lie inside it, which is why the lowest minima found are trustworthy for those two shapes; the tipped squares and the lower-symmetry near-tetrahedra are minima of the restricted space and may not be minima of the full one. A butterfly needs azimuths to move, so its absence from the census is a statement about this space.

σ only, a closed shell, a one-electron energy. π interactions, which split dxz and dyz from dxy differently, and electron repulsion beyond a stated pairing energy would both move the critical weight. The closed-shell energy is used even where the gap is small, which is exactly where the gap figure says it stops being the right state.

Who found the road #

That four-coordinate $d^8$ complexes are square planar when the ligand field is strong and tetrahedral and paramagnetic when it is weak is old chemistry, and the angular overlap model is Schäffer and Jørgensen’s. The idea that a geometry is the minimum of a balance between ligand repulsion and the d electrons’ preference goes back to the ligand-field treatments of the 1950s and 1960s. What is computed here is the road in this model: the identity between its levels and the folded square’s, the critical weight, the square-root flattening, the gap along it, and the census of minima over the four elevations.

A curvature needs a stationary point #

The lesson is about which questions a second derivative answers. “Is this geometry a minimum along this direction” asks for a curvature, and it is only well posed at a point where every first derivative is zero. The folded square was never such a point, and its curvature along the pair fold, carefully computed and correctly signed, was answering a question nobody needed. The same pair fold, moved to a point that is stationary — the flat square — answered the one that was being asked.

Still open: azimuths, and a weight from somewhere #

The obvious open question is the azimuths. A butterfly, a sawhorse with its axial pair bent, and the cis-divacant shapes all need ligands to move round the metal as well as up and down, and the census can only find what the coordinates can reach. Releasing the four azimuths adds four coordinates, removes the tipped squares as distinct minima, and asks whether the lower-symmetry near-tetrahedra are real minima or saddles of a larger space.

The nearer question is the weight. The road is a one-parameter family because the model cannot price a repulsion against an eσ, and a real complex sits at one weight. The square-planar and tetrahedral nickel(II) complexes whose spectra give eσ, and whose structures give their angles, could each be placed on the road by their flattening angle alone — and whether that reading puts the diamagnetic ones above 0.0762 and the paramagnetic ones below the weight where the gap meets their pairing energy would test the road against the line chemistry already draws.