The count that cannot be broken by strength

An integer nobody measured found that a metal’s charge runs over 2.06 electrons across the partitions that compute it, that its oxidation state lies outside that range, and that the electron count is eighteen at every point. It ended with the objection that the count’s own derivation invites:

Back-donation puts electrons into orbitals that are not the metal’s. The eighteen-electron rule counts the metal’s nine valence orbitals, and a π channel mixes them with ligand π* combinations — so a strong enough channel ought to be able to make one of those combinations dip below a metal orbital, at which point the rule would be counting the wrong nine.

The model can do that with one parameter turned. The answer is that it cannot happen, and there are two independent reasons.

The count as blocks #

An octahedron’s six ligand σ functions span A₁g + Eg + T₁u; its twelve π functions span T₁g + T₂g + T₁u + T₂u. Both spans come from the reduction formula applied to characters computed from the coordinates rather than from a table. The metal supplies A₁g from its s, Eg and T₂g from its d, and T₁u from its p — nine orbitals in four species.

So the whole complex is four small matrices and two leftovers:

The last two rows are the first reason. T₁g and T₂u have no metal orbital of their own species, so no coupling of any strength touches them: they sit where the free ligand π functions sit, at every value of every parameter.

A ligand combination can only fall below a filled orbital if something pushes it down. Nothing can push these, and they are the only combinations whose falling would add to the count. The intrusion worth worrying about is forbidden by symmetry rather than made unlikely by arithmetic. That is the strongest shape of answer available and it is the same one a selection rule gives: a species without a partner is not a small interaction, it is no interaction.

Turning it up, and what falls #

The T₂g block is the one back-donation acts in, and it does what a two-level problem does: the lower orbital falls, the upper rises, and the lower acquires ligand character.

The last row is a coupling an order of magnitude beyond anything a real ligand supplies, and the orbital is still more the metal’s than the ligands’.

It never crosses, and the reason is a theorem rather than a number. For a two-level problem with the metal below the ligand, the lower eigenvector always carries more of the lower basis function; as the coupling grows the mixing tends to fifty-fifty and reaches it only in the limit. There is no coupling at which the counted orbital becomes majority-ligand, and there could not be.

What does flip it #

The share does cross a half. It just does not cross it by getting stronger.

Sweeping where the ligand π level sits relative to the metal d — from well below it to well above — at a fixed coupling:

It crosses at exactly zero, where the two diagonal energies coincide, and it does so at every coupling, because a two-level problem with equal diagonals mixes exactly evenly whatever the off-diagonal is.

That is a chemically meaningful boundary and it has a name. A ligand whose π level is above the metal d is a π acceptor, and its counted orbital is the metal’s. A ligand whose π level is below — a halide, an oxide, anything with filled π lone pairs — is a π donor, and its counted orbital is the ligands’.

So a hexafluoridocobaltate’s filled T₂g orbital is a fluorine π orbital with a little cobalt in it, and its eighteen-electron count is a count of orbitals that are mostly not the metal’s — and the count is still eighteen.

The two answers are independent, and both are needed #

It is worth separating the two halves, because either alone would be a weaker result.

The symmetry half says no ligand combination can intrude. That would still leave the possibility that the counted orbitals become mostly the ligands’ — the rule would then be counting nine orbitals, correctly, while every one of them was a ligand orbital wearing a metal’s species label. A reader told only this half could reasonably conclude that the count is a bookkeeping trick.

The arithmetic half says the counted orbitals stay majority-metal at any coupling. That would still leave the possibility that some other combination drops in — the counted nine would be the metal’s, and there would be a tenth filled orbital nobody counted. A reader told only this half could reasonably conclude that the count is right for the wrong reason.

Together they say something stronger than either: the nine orbitals the rule counts are nine, and they are the metal’s, at every strength of the interaction the rule was supposed to be vulnerable to. And the one thing that does move the second — the ordering of the levels — leaves the first untouched, so a π donor’s count is nine ligand-dominated orbitals and is still nine.

That the two halves fail in opposite ways is what makes the pair worth stating. A model in which one held and the other did not would produce a rule that worked and was mis-explained, which is the commonest condition of a rule of thumb and is exactly what is worth testing.

Why the count survives both #

The rule counts filled orbitals matched by symmetry species, and a species is not a mixing coefficient. Whether the T₂g bonding orbital is 95 per cent metal or 6 per cent, it is one orbital of T₂g symmetry, it is filled, and it is one of the nine.

That is the whole answer, and it is worth stating as a correction to how the rule is usually taught. The eighteen-electron rule is not a count of electrons on the metal. It is a count of the orbitals a metal’s nine valence functions can be paired with, and the pairing is by symmetry.

The same point holds for the metal’s charge — the count is eighteen while the charge runs over two electrons — and here it is made about the orbitals rather than about the electrons in them. The integral that cannot count electrons is the third member of the family: a quantity that does the work of a count and is not one. Both are the same fact seen twice: the count is invariant to everything that varies, which is why it works and why it says nothing about the quantities people want it to say something about.

The one thing that would break it #

There is a way to break the count and this model can say what it is.

The count is nine matched orbitals, and it fails if a ligand combination has no metal partner and sits low enough to be filled. In an octahedron those are T₁g and T₂u, and they sit at the free ligand π energy — which is high for an acceptor and low for a donor.

For a π donor they are low, and they are filled: a halide’s twelve π electrons occupy T₂g (bonding, six electrons), T₁g and T₂u (six more). Those six are ligand lone pairs, they are not counted by the rule, and nobody counts them — because the rule’s convention treats a halide as a two-electron donor contributing its σ pair only.

So the count already excludes them by fiat, and the reason it can is exactly that they have no metal partner and therefore no claim to be metal orbitals. Sixteen is also a count records the other convention of the same family, where a metal orbital rather than a ligand one is left out. The rule’s convention and the symmetry blocking agree, which is not obvious and is worth having checked.

What is quoted, and what is computed #

Nothing is quoted. The energies are model parameters in wavenumbers, chosen to be of the ordinary size for a first-row complex, and the argument does not depend on their values — it depends on the ordering of two of them and on which species have partners.

The species come from reducing the ligand functions in the molecule’s own point group, which this collection generates from coordinates rather than looking up. The blocks are two-by-two matrices diagonalised by the solver used everywhere here, and the angular-overlap factors — √3 for Eg, 2 for T₂g — are the standard angular sums.

The count is recomputed at every point of every sweep rather than checked once. That matters: the claim is that a number does not move, and a number that is computed once and reported many times has not been shown not to move.

What this cannot say #

Ligand–ligand interaction is neglected. T₁g and T₂u are degenerate with the other π combinations only because the twelve π functions are treated as non-interacting; a real set has direct overlaps between neighbouring ligands, which would split them. Whether that splitting could push one of them low enough to be filled where it is not already is a question this model cannot answer, and it is the only remaining route to breaking the count.

Every block here is two by two. The T₁u block is really three by three — metal p, ligand σ and ligand π all have T₁u components — and treating it as two levels is a simplification that does not affect the T₂g argument this essay is about.

Nothing here is a measurement. The metal’s charge is not observable and the oxidation state is a convention; what is established here is that the count is insensitive to both, which is a fact about the count and not a defence of it. A quantity that cannot be moved by anything is a quantity that reports nothing about what moved.

And a count is not an energy. Nothing here says an eighteen-electron complex is stable or that a sixteen-electron one is not; what the count is a count of is orbitals, and which of them are low enough to fill is a separate question the rule does not ask.

Why the rule’s success is not evidence for the story attached to it #

The sweep establishes that the count survives a quantity that is not small, and that has a consequence for what the count can be used to argue, which is worth separating from what it can be used to predict.

The rule’s original justification was that a metal reaching eighteen has attained the electron configuration of the next noble gas. That story requires the eighteen electrons to be, in some sense, the metal’s. The sweep says they are not: at a coupling twice what a carbonyl supplies, nearly three tenths of the counted density sits on the ligands, and the counted orbital’s metal share approaches a half without ever reaching it.

So the rule works and the story does not. That combination is only possible because the count is insensitive to the very quantity the story is about — and the insensitivity cuts both ways.

It is why the rule is useful. A prediction that survives a threefold change in a parameter nobody has measured is a prediction that can be made from a formula, and that is exactly what a counting rule is for.

And it is why the rule’s success is not evidence. A quantity that comes out eighteen whether the bonding is ionic or strongly covalent cannot distinguish the two, so observing eighteen tells nothing about which. The rule’s long record of working is not support for the noble-gas account, for the ligand-field account, or for any account of what the electrons are doing — it is compatible with all of them, which is the definition of evidence that does not discriminate.

That is an unusual position for a rule to be in and it is worth stating plainly rather than leaving implicit. Most successful rules in chemistry are successful because they are approximately tracking something. This one is successful because it is tracking something that does not move, and what does move — where the electrons actually are — is a quantity the rule was never sensitive to and was historically believed to be a report on.

Which is the sharpest form of the electron-counting argument. Eighteen is a count of orbitals below a gap. It is not a count of the metal’s electrons, it never was, and the reason the mistaken reading survived a century is that no observation of an eighteen-electron complex could ever have refuted it.

What was checked #

The count is eighteen at every π coupling, checked as a set of distinct values with one member.

No ligand combination without a metal partner falls below a filled orbital, at any point of any sweep — the symmetry half of the answer.

The metal character of the filled T₂g orbital falls monotonically with the coupling, so the effect is real and in the expected direction.

And it never crosses a half, approaching it from above — the arithmetic half of the answer, checked as a bound rather than as a value, because the claim is about a limit rather than about the number at any particular coupling.

With no coupling the orbital is entirely the metal’s, to the last bit, which is the check that the mixing is the only thing moving it.

The share does cross a half when the ligand level crosses the metal’s, and it crosses within a thousandth of the coincidence — so the boundary is the ordering and not some coupling-dependent place.

And the count is eighteen on both sides of that crossing too, which is the finding stated in the form that matters: the thing that changes what the counted orbitals are made of does not change how many there are.

Still open: ligand–ligand splitting, sixteen electrons, and a metal population #

There is one more number worth recording first. At a π coupling of 16,000 cm⁻¹ — about twice what a carbonyl supplies and the strongest anything reasonable does — the counted orbital is 71 per cent the metal’s, so nearly three tenths of what the rule attributes to the metal is sitting on the ligands. That is a large fraction, it is real, and it is exactly the population that moves between partitions. The count is insensitive to a quantity that is not small. Both halves of that sentence matter, and the second is why the insensitivity is worth demonstrating rather than assuming.

The obvious open question is the ligand–ligand splitting the last caution names. Twelve π functions on six ligands are not twelve degenerate levels: the neighbouring ones overlap, and that overlap splits T₁g, T₂g, T₁u and T₂u by an amount computable from the same geometry the angular-overlap factors come from. If it splits them far enough to put one below a filled orbital, the count changes and the rule acquires a boundary in a parameter nobody has measured — and if it does not, the count is safe against everything this model contains.

The nearer question is the sixteen-electron case. A square-planar complex counts sixteen because one of the metal’s nine orbitals — the $d_{x^2-y^2}$ partner, in D₄ₕ terms — 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, and the same calculation reaches it by changing which geometry the ligand functions are reduced in.

And there is a third thing worth doing with the same calculation, which is to point it at the count’s cousin. The count is the population turns a count of orphan combinations into a charge; the same step applied here would turn eighteen filled orbitals into a metal population, and would say in one number what the sweep above says in a curve — that the count is stable and what it counts is not. The obstacle is the same one a label that prices nothing meets: a symmetry species is a complete answer only where it appears once.