The gap that only a tetrahedron closes

The orbital a ligand cannot reach expected the sixteen-electron rule to be the fragile one and found it is not fragile at all. The gap above eight d electrons in a square-planar complex is 2eσ exactly — bit-identical from a π strength of −0.4 to +0.24 — because the orbital that sets it is d(z²), which points along the axis perpendicular to the plane, and a square-planar ligand π set contains nothing of that symmetry. The protection ends at a π strength of a quarter of the σ one, which is past every common ligand’s parameters.

That is a statement about a plane. The natural extension is the same level diagram along a path from square planar to tetrahedral, which would say at what distortion the sixteen-electron gap closes, and that is the geometry question the count is usually invoked to answer.

The answer is that it closes nowhere along the path and at the far end exactly, and that the interesting quantity is not the gap but its exactness — which is gone at the first degree.

The path, and why it is the only one #

Four ligands can be taken from a square plane to a tetrahedron by folding two of them up and the other two down, in perpendicular directions. That coordinate keeps a twofold axis and two dihedral mirror planes throughout — the D₂d arrangement — and it is the only path between the two that passes through nothing of lower symmetry.

Along it the six pairwise angles are four of one value and two of another, and the two sets close on 109.47° together. That is the geometry checked rather than assumed: the endpoints are recognised by their angle spectra rather than by being named.

The gap survives most of the path #

At a five-degree fold the gap is 99.98 per cent of its square-planar value; at ten degrees 99.73 per cent; at fifteen — which is more than two fifths of the way to a tetrahedron — 98.65 per cent. It falls below three quarters only past twenty-two degrees, and reaches a third at thirty.

The rearrangement is legible. In the plane the order is three low levels, then d(z²), then d(x²−y²) two eσ above it. At the tetrahedron the order is a low pair and a high triple, and the level that has moved from fourth into the lower pair is d(z²) — the same orbital identified in the plane as the protected one, doing the whole of the work at both ends.

The trace is fixed at $4e_\sigma + 8e_\pi$ whatever the geometry, so nothing here is a total changing: it is a redistribution, and what one level gains another loses.

And it closes only at the end, at first order #

The gap is positive at every angle short of the tetrahedron and exactly zero at it, where the upper three levels are the degenerate t₂ set. There is no interior closing to locate, so the question has no answer in the form it was asked.

What the closing does have is an order, and it is first: the slope of the gap against the remaining angle is 1.013 over a factor of eighty. Half a degree of flattening away from a tetrahedron opens a gap proportional to the half degree, where the same half degree away from the plane moves it in the fifth decimal.

That asymmetry is the useful part. Both ends of the path are symmetric points and they behave completely differently: the plane’s gap is stationary in the distortion and the tetrahedron’s is not. A d⁸ complex near a square plane is therefore insensitive to a few degrees of bending, and a d⁸ complex near a tetrahedron is not near anything at all — which is a statement about why the two geometries are not competitors in the way the textbook comparison suggests.

The protection goes at the first degree #

At the plane the gap’s derivative with respect to $e_\pi$ is exactly zero — that is the planar finding stated as a derivative. At a two-degree fold it is −0.0097; at five, −0.0603; at ten, −0.2340; at fifteen, −0.5000.

Doubling the bend from five degrees to ten multiplies the sensitivity by 3.88, so the π sensitivity grows as the square of the bend. It reaches its largest value of −1.135 at a bend of 18.25° and then falls back to zero at the tetrahedron.

That the far end is protected too is a second symmetry statement, and a different one. At the tetrahedron the gap is zero at every π strength because the t₂ set is degenerate by symmetry, so there is nothing for $e_\pi$ to move. The two ends of the path are the only two places on it where the π channel cannot touch the count, and they are protected by unrelated facts.

So the answer to the expectation is a split. The sixteen-electron gap’s size is robust to a bend of ten or fifteen degrees, which is more than any real square-planar complex shows. Its exactness — the thing that made it more robust than the octahedron’s eighteen-electron gap — is a property of the plane alone and does not survive the first degree of distortion.

Two protections, and why only one of them is a rule #

The two ends being protected for unrelated reasons is worth separating, because only one of the two reasons generalises.

At the tetrahedron the gap is zero by degeneracy. The t₂ set is a three-dimensional irreducible representation of the tetrahedral group, so its three levels are equal whatever any parameter does, and no interaction that respects the symmetry can split them. That is the same kind of statement as a degeneracy being a group theorem, and it holds for every parameter in every model with that symmetry — a π strength, a δ strength, a different radial function.

At the plane the gap is 2eσ because of an absence: the four ligands’ π functions span nothing that transforms like d(z²) in D₄ₕ. That is also a symmetry statement, and it is a much weaker one, because it is about which representations the ligand set happens to supply rather than about a degeneracy of the metal’s own levels. Add a fifth ligand along the axis, or let the four bend, and the ligand set supplies something new.

A protection by degeneracy survives every perturbation of that symmetry; a protection by absence survives only the perturbations that keep the absence. The planar result found the second and read it as though it were the first, and the path measured here is what tells them apart: the tetrahedral end’s derivative is zero at the point and stays zero along the whole family of D₂d geometries at that angle, while the planar end’s is zero at exactly one geometry.

What eight electrons pay for the plane #

At $e_\pi = 0$ the eight occupied electrons sum to $2.0000\,e_\sigma$ in the plane and $5.3333\,e_\sigma$ at the tetrahedron — a difference of exactly $10/3$. That is what a d⁸ complex gains by being planar, and it is what the repulsion between the four ligands has to beat.

It does not, for the metals where square-planar d⁸ is found, and repulsion alone makes the tetrahedron the arrangement four points on a sphere choose when nothing else is in the argument. The square plane exists because the ligand field pays more than the repulsion costs — and the number above is the ligand field’s side of that trade, computed rather than assumed.

Turning the π channel up raises both ends: at $e_\pi = 0.25$ the sum is 6.0000 in the plane and 8.8889 at the tetrahedron, and the difference falls from 3.3333 to 2.8889. So a π-donor ligand set makes the plane less preferred, by about thirteen per cent of the preference, which is a small effect in the direction textbooks assign to it for a different reason.

What a chemist would do with the two numbers #

The two rates have a practical form worth writing down, because they point in opposite directions for the two things the sixteen-electron rule is used for.

As a count, the rule is safer than the planar result made it look for one reason and no less safe for any other. A complex bent by ten degrees still has 99.7 per cent of its gap, so the closed shell at sixteen electrons is intact, and it stays intact well past any distortion a real square-planar complex shows — nickel, palladium and platinum complexes are planar to within a degree or two, and even the strained ones are within ten.

As a quantity, it is not. Anybody using the gap as a number — to predict a transition energy, or to compare two complexes with different ligands — is using the thing that stops being 2eσ immediately. At ten degrees with a π donor at eπ = 0.25 the gap is 1.881 rather than 2.000, a six per cent shift, and that shift is entirely a π effect that the planar analysis says cannot exist.

Six per cent is small, and the point is not its size. It is that the analysis which gave the exact answer gives no estimate of it — an exactness that comes from an absence has no leading correction to quote, so there is nothing to compare three per cent against until the derivative is computed — and there is nothing to compare six per cent against either, since a zero of this kind is not a small number with an error bar on it. The same shape appears wherever a symmetry gives an exact zero: the forbidden quantity is exactly zero at the symmetric geometry and has no natural scale away from it.

What was computed, and how #

Every level is an eigenvalue of a five-by-five angular overlap matrix built from the four ligand directions, with each ligand contributing $e_\sigma$ along its own bond and $e_\pi$ in the two directions perpendicular to it, rotated into the metal’s frame by the d-orbital rotation matrices. The eigenvectors are checked to be the d orbitals themselves at every geometry on the path, which is what says the arrangement was entered correctly — a level that came back as a mixture would mean the directions were wrong.

The trace is $4e_\sigma + 8e_\pi$ at every geometry, which is a completeness statement rather than a result and is the sum rule the angular overlap model is built on.

The π sensitivity is a central difference in $e_\pi$ at a step of 0.05, computed at two step sizes a factor of two apart and required to agree — so a small non-zero number at the plane would have been caught as a step-size artefact rather than reported as physics.

The refusal is either endpoint. At both the derivative must be exactly zero, and a version of this that reported a small non-zero number at one of them would be reporting its own arithmetic.

Where the model stops #

The electron count is fixed throughout and the geometry is idealised throughout, and both of those are doing work. A real cluster relaxes, and a relaxation that lowers the symmetry can open a gap that the idealised geometry says is closed — which is the whole content of a Jahn–Teller argument and is exactly the mechanism this comparison holds still. So the finding is about what the electron count and the symmetry allow, not about what a synthesised cluster would show.

That is the right question for the planar claim, which was a counting claim: it said a particular count closes the gap in every geometry the family offers. A relaxation cannot rescue a count, because the count is what decides which levels are occupied before any nucleus moves.

The angular overlap model has no repulsion between electrons in it, so nothing above decides whether a d⁸ complex is high spin or low spin — and a tetrahedral d⁸ complex is high spin with two unpaired electrons, which means the “sixteen-electron gap” at that end of the path is not a gap any real molecule has eight paired electrons below. The path is a path of one-electron level patterns and the occupation is imposed.

Whether an eighteen-electron count is a count or an energy is the distinction all of electron counting turns on, and it applies here unchanged: the sixteen-electron rule is a count, and everything measured above is the energy underneath it.

The bond lengths are held fixed. A real fold from a plane towards a tetrahedron would relax them, and $e_\sigma$ and $e_\pi$ both depend on the length — so the numbers here are the angular part of a change that has a radial part as well.

And $e_\pi$ is one parameter for both perpendicular directions. That is right for a halide and wrong for a ligand whose π system is one-sided, which is the case the gap that would have to be smaller is about.

The generalisation #

A quantity’s size and its protection are different quantities, and they can fail at completely different rates. Here the size survives a fifteen-degree distortion nearly intact and the protection does not survive one degree. Anybody who read the planar result as the sixteen-electron count is robust would be right about the count and wrong about the reason, and would then be surprised by a π-donor effect in a complex bent by five degrees.

The general form is that an exact statement obtained at a symmetric point is exact at that point, and its neighbourhood inherits nothing unless the derivative is checked. Checking the derivative is one central difference, and it is the whole of the difference between this is protected and this is protected here.

Who found it, and when #

The electron-counting rules in play are Wade’s and Mingos’s, and their standing claim is about which counts a closed cage can take rather than about which geometry a count prefers. Turning that round — fixing the count and asking which geometry it closes a gap in — is the move made here, and it is available only because the whole family can be built and diagonalised rather than looked up.

The square-planar preference of d⁸ complexes and its ligand-field explanation are from the nineteen-fifties, and the D₂d flattening coordinate is the standard one for the square-planar–tetrahedral interconversion, which nickel(II) complexes famously undergo. That the interconversion is the mechanism for their spin-state equilibrium is Holm’s work of the early nineteen-sixties.

The comparison with the octahedron’s eighteen-electron gap — which moves at every π strength — is what made the square plane’s exactness worth measuring. The angular overlap model is Schäffer and Jørgensen’s, from 1965, and the fact that a tetrahedron’s t₂ set is degenerate is Bethe’s from 1929. The specific derivative computed here — how fast the sixteen-electron gap loses its independence from the π channel as the plane is bent — is not a standard quantity and is what is added here.

Still open: a two-ligand distortion, and a fifth ligand #

The obvious open question is the other four-coordinate distortion. The D₂d path holds all four ligands equivalent; a real square-planar complex more often distorts by moving two of them, into a see-saw or towards a butterfly, which lowers the symmetry further and can mix d(z²) with d(x²−y²). Whether the gap’s protection fails faster along that path than along this one is a question of one more set of directions, and the answer decides which distortions a sixteen-electron count should be quoted with a caution about.

The nearer question is the fifth ligand. A square-planar complex that adds one axial ligand becomes square pyramidal, and that is the reaction step the sixteen-electron rule is actually invoked to explain — a sixteen-electron complex is reactive because it can add a ligand. The same level diagram along the approach of a fifth σ donor would say how the gap above eight d electrons behaves as the axial position is filled, and d(z²) points straight at it, so that is the one distortion the protected orbital cannot ignore.