The count that is not always eighteen

The eighteen-electron rule can be taken apart and put back together as a count: reduce the six ligand σ orbitals in the complex’s own point group, match them against the metal’s nine valence orbitals by symmetry, and eighteen is what comes out. It is a shell closure rather than a preference, and the arithmetic that produces it has no chemistry in it beyond the shape.

That argument has a gap in it, and the gap is the word closure. A level diagram closes wherever there is a gap above a filled set, and an octahedral one has two such places.

The two gaps, in closed form #

The angular overlap model gives both gaps outright. In an octahedron the two eg orbitals are raised by 3eσ and the three t2g orbitals by 4eπ, so

$$\text{gap above } 12 = 4e_\pi + 3e_\sigma \qquad \text{gap above } 18 = 3e_\sigma - 4e_\pi$$

and the two are equal when $e_\pi = 0$.

That is the whole result. A ligand that accepts π density has a negative eπ, which pushes the t2g set down, widens the gap above eighteen and narrows the one above twelve. A ligand that donates π density has a positive eπ and does exactly the reverse.

The expressions are not put in. The two energies are read off a diagonalised angular overlap matrix — the same matrix two models, one ratio checks against an integrated point-charge potential — and the closed forms are checked against what comes out, so a wrong matrix would be caught rather than reproduced.

The justification the rule is usually given #

A transition metal has nine valence orbitals — one s, three p and five d — and nine orbitals hold eighteen electrons. That is the sentence every account starts from, and eighteen is a count already objects to it: it is a restatement rather than a reason. Nothing in it says why a complex should fill all nine, and the same argument applied to a main-group atom with four valence orbitals gives eight, which is a rule with famous exceptions of its own.

The symmetry version is better. Six ligand σ orbitals reduce to a1g ⊕ eg ⊕ t1u in an octahedron; the metal has one orbital of a1g, three of t1u and two of eg; so six of the metal’s nine are matched and become bonding-and-antibonding pairs, and the three t2g are left over. Twelve electrons fill the bonding six, six more fill the leftover three, and the eg* pair is left empty. Eighteen.

What that version still does not say is why the leftover three should be filled rather than left empty. It is an assumption, and it is the assumption this essay’s parameter decides. The symmetry argument fixes where the gaps are; only the π interaction fixes which gap is bigger.

Counting some real compounds #

The prediction is testable against compounds anybody can look up, and it is worth doing because the failures of the eighteen-electron rule are usually listed as exceptions rather than sorted.

Tungsten hexacarbonyl, W(CO)₆: tungsten contributes six d electrons, six carbonyls contribute twelve, total eighteen. Carbon monoxide is the strongest π acceptor in the table, and eighteen is the deep closure. It obeys.

Tungsten hexachloride, WCl₆: tungsten(VI) is d⁰, six chlorides contribute twelve, total twelve. Chloride is a π donor and twelve is the deep closure. It obeys — the other rule.

Titanium tetrachloride is four-coordinate and d⁰ with eight electrons, and the same reasoning applies with a different level diagram: a π-donor set makes filling the metal’s d orbitals unfavourable, so an early metal with halides sits at the σ-only count and stops.

Hexaamminechromium(III), [Cr(NH₃)₆]³⁺: chromium(III) is d³, so the count is fifteen. Ammonia is the ligand with no π interaction, and fifteen is neither closure. The compound is perfectly stable and nobody cites it as an exception, because nobody applies the rule to it — which is the behaviour the tie in the table predicts.

Hexacyanoferrate(II), [Fe(CN)₆]⁴⁻: iron(II) is d⁶, cyanide is a π acceptor, and the count is eighteen. It obeys, and it is the classic textbook case of a complex that is kinetically inert.

Five compounds, three counts, and one parameter sorts them. What is being claimed is not that the rule has been repaired — it is that the rule and its exceptions are the same calculation at two signs of eπ.

The ligands that exist #

The crossover would be a curiosity if real ligands sat on one side of it. They do not.

The pattern reproduces the chemistry with nothing fitted to it. Metal carbonyls obey the eighteen-electron rule so reliably that it is used to predict structures; metal halides and oxides do not, and the early transition metals — found almost exclusively with halide, oxide and alkoxide ligands — are where the rule is usually described as failing.

The usual explanation for that failure is about the metal: early metals have high-lying d orbitals, or too few electrons to reach eighteen. The calculation above says the parameter that decides it is the ligand’s. Tungsten hexachloride has twelve valence electrons and is a perfectly ordinary compound; tungsten hexacarbonyl has eighteen and is equally ordinary. The metal is the same.

Ammonia, which is neither #

The tie in the middle of the table is worth its own paragraph because it is exact rather than approximate.

Ammonia and ethylenediamine have no π orbitals available at the energies that matter, so their eπ is zero and the two gaps come out at 3.000 apiece — the same number to the last bit a double holds. Neither count is special, and the chemistry agrees in the least dramatic way possible: ammine complexes are found at almost every electron count from twelve to eighteen, and nobody uses a counting rule on them.

That is the check that the crossover is a point and not a broad region. If the model produced a band of ligands for which neither count was clearly favoured, the claim would be about a fitted boundary. It produces a tie at exactly one value of one parameter.

What the t2g set is doing in each case #

The mechanism is worth stating in words as well as in parameters, because it is the same one the spectrochemical series turned out to be about.

With a π acceptor, the ligand has empty orbitals of π symmetry above the metal’s d level. The t2g set interacts with them, is pushed down, and becomes weakly bonding. Filling it is then favourable, and the complex wants those six electrons: twelve plus six is eighteen.

With a π donor, the ligand has filled orbitals of π symmetry below the metal’s d level. The t2g set interacts with those, is pushed up, and becomes antibonding. Filling it is now unfavourable, and the complex is better off leaving it empty: twelve electrons and no more.

So the eighteen-electron rule is the statement the t2g set is bonding or non-bonding, and it holds exactly where that is true. What makes it look like a rule about metals is that the ligands which make it true — carbonyl, cyanide, phosphines, alkenes — are the ligands late transition metals are found with.

One thing about the survey above is worth saying explicitly, because it is the whole of why this is a statement about ligands. The order the ligands come in there is the spectrochemical one, and that order is set by the π interaction rather than by charge — chloride and cyanide both carry a single negative charge and sit at opposite ends of it. So the parameter that decides which count closes a shell is the same parameter that decides how large the splitting is, and a rule about electron counts and a rule about colour are two readings of one number.

The closure nobody names #

There is a third closed configuration in the diagram and it never appears in any counting rule: twenty-two electrons, with the eg* pair full as well.

Nothing forbids it arithmetically. What rules it out is that the eg* orbitals are raised by 3eσ, which is the largest energy in the whole diagram — they are the metal’s own σ-antibonding orbitals, and filling them costs the σ bonding the complex was built on. A twenty-two-electron octahedral complex would be one whose metal–ligand bonds have been cancelled.

That is the honest reason the count stops at eighteen, and it is a different kind of reason from the one that chooses between twelve and eighteen. The choice between the two lower closures is a competition between two comparable gaps; the exclusion of the third is a statement that one configuration is far above the others. A rule that lists three closures and says nothing about their scale would suggest three equally plausible counts, and the arithmetic says otherwise.

It also explains why the rule is a maximum in practice. Complexes are found at every count up to eighteen and essentially never above it, and the asymmetry between the two ends of the range comes from the eg* orbitals being much further up than the t2g orbitals are down.

What this model leaves out #

The count is not an energy. A larger gap above a configuration makes it more robust, and nothing here says by how much or compares it with anything else — a bond energy, a reorganisation, a solvation. Two closures differing by 1.44 eσ is a statement about a level diagram, and eσ is a parameter with a fitted value.

The ligand’s own electrons are not in the diagram. A π donor brings filled π orbitals, and those electrons are counted as ligand lone pairs in every counting scheme there is. Including them would move every count by twelve and change no comparison, which is why the convention is harmless — but it is a convention, and a reader comparing these numbers with an ionic-formalism count needs to know which one is in use.

One geometry. Everything above is octahedral. A square plane has a different level diagram and closes at sixteen, which is sixteen is also a count and a different argument — there the leftover orbital is too high to use rather than pushed about by π interactions.

The parameters are fitted. eσ and eπ come from the literature’s fits to measured splittings. What is computed here is what follows from them, and the sign of eπ — which is the whole result — is not a fitted quantity so much as a classification: a ligand with filled π orbitals below the d level has one sign and a ligand with empty ones above has the other.

What was checked #

Three things, and the first is what licenses the rest.

The angular overlap matrix reproduces the textbook expressions. t2g comes out at exactly 4eπ and eg* at exactly 3eσ, at three different values of eπ, from a matrix built by rotating a diagonal set of ligand strengths into the metal’s frame and diagonalising. A wrong rotation or a wrong geometry would break this and nothing else here would notice.

With no π interaction the two gaps are equal, to 10⁻⁹. That is the crossover being a point.

Across the series the sign of eπ decides the count for every ligand, and the charge decides it for none — the anionic ligands split three to one between the two answers. Both halves are checked, because a classification that sorts the data is only interesting if a plausible rival does not.

The square-planar case is worth stating because it is the one where the geometry rather than the ligand decides. One metal orbital there has no partner and lies far too high to use, so eight orbitals are available rather than nine and the closed-shell count is sixteen — a number no ligand parameter can move, because it is a property of the arrangement.

The twelve-electron family, which is most of inorganic chemistry #

The closure nobody names has a name in practice, and identifying it is the strongest evidence that the two-gap account is describing real chemistry rather than a curiosity of a diagram.

A twelve-electron octahedral complex has its six σ-bonding combinations filled and the t2g set empty. Empty t2g means no d electrons at all, so the twelve-electron family is the d⁰ family — and that family is enormous.

Tungsten and molybdenum hexafluoride. Chromate, molybdate and tungstate. Permanganate. Titanium dioxide, zirconium dioxide, tungsten trioxide. The hexafluoro anions of titanium, zirconium and hafnium. Between them those compounds account for a large fraction of the transition-metal chemistry anybody meets outside a catalysis laboratory, and not one of them counts eighteen or comes anywhere near it.

The two families sort exactly as the level diagram says they should, and the sorting is on two variables that travel together.

The ligands. Every compound listed above has oxide, fluoride or chloride as its ligand — π donors, the case in which the essay’s arithmetic puts the deep gap above twelve. Every eighteen-electron complex has carbonyls, phosphines, alkenes or cyanides — π acceptors, where the deep gap is above eighteen.

The metals. The d⁰ compounds are early transition metals in high oxidation states; the eighteen-electron ones are middle and late metals in low ones. That is not an independent fact: a metal in a high oxidation state is a poor back-donor and binds π donors well, and one in a low oxidation state has density to give and binds π acceptors. The ligand preference and the position in the row are the same statement.

So the picture electron counting builds is not eighteen with exceptions. It is two families of roughly comparable size, separated by the sign of one parameter, and the eighteen-electron rule is the rule for one of them. That the other is usually taught in a different course — as oxide and halide chemistry rather than as organometallic chemistry — is why the split is rarely put this way.

One measured consequence is worth adding because it is an integer rather than a trend. A d⁰ complex has no d electrons, so it has no d–d transitions at all, and its optical spectrum cannot contain the weak visible bands that give most transition-metal compounds their colour. Titanium dioxide and zirconium dioxide are duly white, and hexafluorotitanate is colourless.

The exceptions to that prove the point rather than spoiling it. Permanganate and chromate are intensely coloured, far more intensely than any d–d transition manages, and their colour is a charge transfer from ligand to metal — an electron moving into the empty t2g set, which exists precisely because the complex is d⁰. A twelve-electron complex is either colourless or violently coloured, and never faintly coloured, which is the spectroscopic signature of an empty set of orbitals sitting above a filled one with nothing in between.

That is a sharper prediction than the count itself, and it comes free with the level diagram: the same gap that decides where the closure falls decides what an electron has to cross to be excited.

It also identifies the one place the two families meet. A complex with a d count between the two closures — d¹ to d⁵, with the t2g set partly filled — is above one closure and below the other, so it has a gap on neither side and no count protects it. Those are exactly the complexes whose properties are least predictable from a rule and most often discussed in terms of a field strength instead: the ones with unpaired electrons, spin-state choices and the whole apparatus of magnetochemistry attached to them. A counting rule works where a closure is available, and between two closures there is no count to be had.

What sixteen is, in this picture #

A square-planar complex closes at sixteen, and it is worth saying how that closure differs from the two here, because three numbers in one field invites the assumption that they are the same kind of statement.

Sixteen comes from a missing partner: the metal p orbital perpendicular to the plane has no ligand combination of its symmetry to pair with and lies far too high to hold electrons, so eight orbitals are available rather than nine. That is a fact about the geometry and it survives any value of eπ.

Twelve and eighteen are not like that. Both configurations use the same set of orbitals, and which one is preferred is decided by an interaction whose sign is a property of the ligand. So a square-planar complex is sixteen-electron whatever its ligands are, while an octahedral one is twelve or eighteen depending on them.

That difference is testable and the chemistry bears it out: square-planar platinum(II) complexes are sixteen-electron with chlorides, with amines, with phosphines and with alkenes alike, while octahedral complexes split into the two families the table above sorts.

Still open: five-coordinate closures, and distorted complexes #

The obvious extension is the count for a geometry that is neither six-coordinate nor four. Five-coordinate complexes are common and their level diagram has three closures rather than two, so the same calculation would produce a magic number that depends on the π parameter and on which of the two five-coordinate shapes the complex adopts — and those two shapes are close in energy and readily interconverted.

The nearer question is whether the argument survives moving off symmetry. Everything above uses an octahedron, where t2g and eg are exact labels. A distorted complex has neither, and the two closures become gaps in a pattern of five distinct levels — at which point the electron count that closes a shell is not twelve or eighteen but whatever the distortion leaves, which is the same territory copper’s never-quite-octahedral geometry is in.