The orbital a ligand cannot reach

An octahedron with a strong π channel — a π channel turned up to twice what a carbonyl supplies — shows the eighteen-electron count of an octahedron unmoved: the counted orbital’s metal share approaches a half from above and never crosses, and only the level ordering could flip it. The case that ought to behave differently is the square plane:

A square-planar complex counts sixteen because one of the metal’s nine orbitals — the d(x²−y²) partner — is pushed too high to fill, which is a statement about an energy rather than about symmetry matching. So the sixteen-electron rule is the one that should be sensitive to a π channel, in the way the eighteen-electron rule turns out not to be.

It is not sensitive either, and the reason is better than the one it was supposed to fail for.

The gap that does not move #

A square-planar ML₄ has four σ-bonding combinations at the bottom, four d orbitals in the middle and d(x²−y²) at the top, pushed up by 3eσ. Filling from the bottom closes at eight electrons and again at sixteen, and the gap above sixteen is what makes sixteen special.

That gap is 3eσ less the energy of the highest of the four filled d orbitals. With no π interaction the four sit at 0, 0, 0 and eσ — the last being d(z²) — so the gap is 2eσ.

Turn the π channel up or down and three of the four move. d(xy) is in the plane and shifts by 4eπ; d(xz) and d(yz) shift by 2eπ each. d(z²) does not move at all, at any strength, in either direction.

Not approximately two. Two exactly, to the last bit of a double, over a range of π strengths wider than any ligand offers — and the exactness is the finding. A gap that varied by a per cent would be a gap with a weak dependence on eπ, which is what insensitive usually means; a gap that is bit-for-bit identical at eπ = −0.4 and at eπ = +0.24 is a gap that does not contain eπ at all.

Why one orbital cannot be reached #

The ligands of a square-planar complex all lie in one plane. The π functions they offer are perpendicular to their own bonds, which gives two kinds: those in the plane, and those perpendicular to it.

Reduce those in the plane and they contain b₂g, which is d(xy)'s species. Reduce those out of the plane and they contain eg, which is d(xz) and d(yz). Neither set contains a₁g, which is d(z²)'s — and it cannot, because a₁g is the totally symmetric species and a set of functions perpendicular to their own bonds has no totally symmetric combination.

This is the same kind of statement the count of eighteen rests on and it is being made about a different object. There, the argument is that nine metal orbitals and six ligand σ combinations match in a particular way; here, that four ligand π combinations and one metal orbital fail to match, which is the same arithmetic read as an absence.

So the gap above sixteen electrons has no eπ in it, as a matter of symmetry. That is a much stronger statement than the sensitivity is small: it is a zero, and it holds however hard the ligands push.

The expectation was reasonable and rested on the wrong feature. It is true that sixteen is set by an energy rather than by a symmetry match — d(x²−y²) is pushed too high to fill, and nothing forbids filling it. What it does not check is which orbital sits below the gap, and that one is protected by symmetry even though the count is not.

Where the protection ends #

It ends at a stated place. d(xy) rises by 4eπ and d(z²) sits at eσ, so the two cross when

$$4e_\pi = e_\sigma, \qquad \text{that is} \qquad e_\pi = e_\sigma/4.$$

Beyond that d(xy) is the highest filled orbital, the gap is 3eσ − 4eπ, and it falls at four times the π strength — steeper than anything in the octahedron.

And the expression beyond the threshold is the octahedron’s, exactly: 3eσ − 4eπ in both. That is not a coincidence — past the threshold both counts are capped by an orbital that takes 4eπ of π stabilisation from the ligand set, and the σ antibonding partner is 3eσ above in both geometries.

How close a real ligand gets #

It is worth saying what those numbers are and are not. They are angular overlap parameters extracted from spectra of octahedral complexes, quoted as ratios to eσ, and they carry the usual caution attached to a fitted eπ: the fit that produces them has a sign and a size and the size is the less reliable half. What matters here is the ordering and the rough magnitude, both of which are solid — the halides are π donors at a sixth to a fifth of their σ strength, and nothing in the series is at a third.

None crosses, and the margin is a fifth rather than a factor. So the honest statement is that the sixteen-electron count is protected for every ligand parameterised here, by a margin small enough that a stronger π donor — or a different metal, since eσ varies as much as eπ does — would end it.

That is a boundary rather than a failure, and it is the kind of boundary worth having: it says what to look for, and it says it in a currency an experiment could reach. A square-planar complex of a very strong π donor is the case in which the sixteen-electron count should first start to move, and the model says which orbital’s rise does it.

The count and the gap are different claims #

It is worth separating two things this essay has been treating together, because the separation is where the expectation went wrong.

The count is sixteen, and it is a statement about how many orbitals lie below the highest one. Nothing here moves it: d(x²−y²) is the top orbital at every π strength tried, in both directions, so the level scheme always has eight orbitals below it and always counts sixteen. Sixteen is also a count established that from the symmetry alone.

The gap is how strongly that count is preferred — the energy between the highest filled orbital and the first empty one, which is what makes a closure a closure rather than an arbitrary place to stop filling. That is the quantity a π channel could move, and the original question was about it.

The two come apart at the threshold in an instructive way. Past eπ = eσ/4 the gap has shrunk and the count is unchanged, because d(xy) has risen past d(z²) within the filled set. A reordering inside the filled orbitals changes the gap and cannot change the count, which is why a count is robust and a closure’s depth is not.

What the two protections have in common, and what they do not #

The octahedral finding and this one are both the count is insensitive to a π channel, and they are different results.

The octahedron’s protection is quantitative. The counted orbital’s metal share falls towards a half and never reaches it, so the count is safe by an inequality — and at a π coupling of 16,000 cm⁻¹ nearly three tenths of what the rule attributes to the metal is on the ligands. The rule survives while the thing it counts moves a great deal.

The square plane’s protection is exact. The orbital setting the gap has no π partner at all, so the gap is independent of eπ rather than weakly dependent on it — and then it stops being protected altogether at a threshold.

Those are opposite shapes: one degrades smoothly and never fails, the other does not degrade at all and then fails at a point. A rule can be robust in two ways and the difference matters for what would break it: the octahedron’s needs an enormous coupling, the square plane’s needs a modest one applied to the right orbital.

What a symmetry protection is worth #

There are now three quantities protected by symmetry rather than by size, and they are worth naming together because the protection has the same shape and different strength in each.

An overlap that symmetry forbids comes out at 10⁻¹⁷ — not small, zero — and no separation, charge or basis can make it otherwise. That is the strongest kind: the quantity is zero as an identity.

A depolarisation ratio of three quarters holds for every mode that is not totally symmetric, whatever the bond polarisabilities are, and degrades only when the molecule is distorted away from the symmetry that gives it.

And this gap is independent of eπ because the orbital setting it has no partner of its symmetry — until a different orbital becomes the one setting it, at which point the protection is not violated but bypassed.

The third is the weakest and the most interesting, because the failure mode is one the other two do not have. The symmetry statement stays true; what changes is which quantity the statement is about. A protection that names an orbital rather than a quantity ends when the orbital changes place, and nothing in the symmetry argument warns of it.

Whether a real ligand reaches a quarter #

A threshold is only interesting if something can cross it, so it is worth asking where real ligands sit relative to a quarter — and the answer is that they sit close enough for the boundary to be a chemical statement rather than a formal one.

The threshold is a ratio, $e_\pi/e_\sigma$, and that is the useful form. Both parameters scale with the metal: a second-row metal has larger values of each than a first-row one, a higher oxidation state raises both, and a shorter bond raises both. What the parameterisation assumes, and what makes the tabulated values transferable at all, is that their ratio is much more nearly a property of the ligand than either is alone.

So the question is which ligands have a π channel worth more than a quarter of their σ one, and the halides are the family to ask. Fitted values for chloride put $e_\sigma$ in the region of five to six thousand wavenumbers and $e_\pi$ between one and one and a half thousand — a ratio somewhere between about 0.17 and 0.27, with the heavier halides higher because their donor orbitals are more diffuse and less committed to the σ direction.

The threshold sits inside that range. Chloride is at or just below it, bromide and iodide above. Whatever the exact values, this is not a boundary at some unreachable coupling: it is a boundary running through the middle of the commonest π-donor ligands in inorganic chemistry.

Which turns the finding into a prediction with somewhere to look. Square-planar complexes of the light halides and of the σ-only ligands — fluoride, ammonia, the amines — should have a sixteen-electron gap that is exactly $2e_\sigma$ and completely indifferent to their π properties. Square-planar complexes of the heavy halides should not: their gap should fall below $2e_\sigma$, by an amount that grows with the ratio, and the sixteen-electron count should become correspondingly less robust as the halide gets heavier.

Two cautions keep that honest. The ratio is fitted, and different compilations disagree about it by enough to move chloride from one side of the threshold to the other — which is the same difficulty every fitted parameter in this collection carries. And the tabulated ratios are mostly from octahedral complexes, where the fitting is best conditioned, so applying them to a square plane assumes a transferability that has not been checked here.

What survives both is the shape of the claim: the protection measured here is not a symmetry protection, it has a boundary, and the boundary is at a coupling real ligands supply rather than beyond one.

What is quoted, and what is computed #

Two things are quoted. The angular overlap parameters of the ligands — eπ as a fraction of eσ, from a standard table — and nothing else. There is no complex, no metal and no measured splitting here.

Everything else is computed. The d levels come from diagonalising the angular overlap matrix for each geometry at each π strength, not from the textbook expressions; the expressions are then checked against the diagonalisation, so the model does the work and the formula is the check. The threshold is found by bisection on which orbital is fourth from the bottom.

The σ-bonding set is placed at −3eσ, which is the antibonding partner’s mirror image and is the same convention used for the octahedron. Nothing in the sixteen-electron gap depends on it.

What this cannot say #

The angular overlap model has no electron repulsion in it. A square-planar d⁸ complex is a closed shell in this model at sixteen electrons and the real question of why it is square planar rather than tetrahedral is a competition between a splitting and a pairing energy, which is a separate question about pairing.

eπ and eσ are not independent in a real ligand. They are quoted here as a ratio because the threshold is a ratio, but a ligand that is a strong π donor is usually a weak σ donor, and the correlation between them is exactly what would decide whether any real complex reaches the threshold. The ratio is treated as free here.

The four filled orbitals are not the same four throughout. At every π strength the count is sixteen and the set of filled orbitals is the same five-minus-one, but their order changes, and an order change is a real physical difference — it changes which orbital a d–d transition comes from and which one a distortion couples to. Nothing here follows that up, and reading a transition off these levels is where it would matter.

And the count is of a model’s orbitals. Sixteen electrons in eight orbitals is a statement about a level scheme with four σ combinations and five d orbitals in it; a real square-planar complex has ligand π electrons, metal p orbitals and a solvent, none of which appear. What is established is that a stated model’s count is protected by a stated symmetry, which is the kind of statement that survives being wrong about everything else.

What the square plane requires #

The gap is the same number at every π strength that leaves d(z²) on top — checked as an equality across forty-one values rather than as a small variation, because the claim is a zero.

And that number is 2eσ exactly.

The threshold is a quarter, to eight decimal places, found by bisection rather than derived and then compared against the derivation.

Beyond it the gap falls at every step, which is the check that the protection really has ended rather than merely changed form.

And the octahedron’s gap is 3eσ − 4eπ at every π strength, which is the octahedral expression re-derived here from the diagonalisation. It had to hold: the comparison between the two counts is worthless if one side of it is not the octahedral quantity.

Still open: the metal, and the fifth d orbital #

The obvious open question is the metal, which is held fixed here. The threshold is a ratio of two ligand-field parameters, and both of them scale with the metal — a second-row metal has larger eσ and eπ than a first-row one, and whether the ratio moves is a question the spectrochemical parameters could answer for a second metal and cannot for one. Finding a ligand and a metal whose ratio exceeds a quarter would turn this boundary into a prediction about a real complex.

The nearer question is the fifth d orbital. Everything here counts sixteen because d(x²−y²) is not filled, and the model puts it at 3eσ above the σ set with no π contribution of its own — which is right for a square-planar σ framework and stops being right the moment the ligands are allowed to bend out of the plane. Computing the same levels along a path from square planar to tetrahedral, which a repulsion minimisation already traverses, would say at what distortion the sixteen-electron gap closes — and that is the geometry question the count is usually invoked to answer.