Eighteen is a count

Counting electrons is how chemists decide whether a compound should exist. The octet rule is a count; the 4n+2 rule is a count; and for transition-metal complexes the count is eighteen.

The justification usually offered is an addition. A transition metal has nine valence orbitals — one s, three p and five d — and nine orbitals hold eighteen electrons. It is arithmetic, it is memorable, and it explains nothing, because the same nine orbitals are present in every complex of every geometry and the counts are not the same.

The count that matters is not of orbitals a metal has but of orbitals lying below a gap, and that is a different question with a different answer in each geometry. Working it out is a reduction of the kind character tables and reduction built, applied to the ligand orbitals instead of to methane’s hydrogens.

Doing the count #

Six ligands, each offering one σ donor orbital, span a six-dimensional space. Reducing it in the octahedral group gives three pieces.

The metal’s nine orbitals sort themselves by the same species: the s is $A_{1g}$, the three p are $T_{1u}$, and the five d split into $E_g$ and $T_{2g}$, which is the basic ligand-field result.

Now match them. A metal orbital whose species appears among the ligand combinations interacts with it, giving one bonding and one antibonding combination; a metal orbital whose species does not appear has nothing to interact with and stays where it was.

• $A_{1g}$: the metal s meets the totally symmetric ligand combination. Bonding.
• $T_{1u}$: the three metal p orbitals meet the $t_{1u}$ ligand set. Bonding, three of them.
• $E_g$: the two d orbitals pointing at the ligands meet the $e_g$ ligand set. Bonding, two of them.
• $T_{2g}$: the three d orbitals pointing between the ligands have no partner. Non-bonding.

Six bonding orbitals, three non-bonding, and everything else — the antibonding partners — above the gap. Nine orbitals, eighteen electrons.

The important line in that list is the last. The three $t_{2g}$ orbitals are non-bonding not because anybody decided they were, but because no combination of six σ donors in an octahedron has $T_{2g}$ symmetry. That is the same fact the ligand field calculation reports as the lower set sitting at exactly zero in a σ-only model — an exact zero produced by cancellation, in the manner of exactly zero, and not by anything being small.

Sixteen, from the same reduction #

Take four ligands in a square plane and run the identical calculation.

The four ligand orbitals reduce to $A_{1g} \oplus B_{1g} \oplus E_u$. The metal’s nine orbitals in this group are $A_{1g}$ (s), $A_{1g}$ (dz²), $B_{1g}$ (dx²−y²), $B_{2g}$ (dxy), $E_g$ (dxz, dyz), $A_{2u}$ (pz) and $E_u$ (px, py).

Matching consumes the metal s, the two in-plane p orbitals and dx²−y². That leaves five orbitals with no ligand partner: dz², dxy, the dxz/dyz pair — and pz.

Four of those five are metal d orbitals, sitting in the middle of the diagram and available. The fifth is the metal p orbital perpendicular to the plane, which is high in energy for the same reason a p orbital is always high relative to d, and which no complex fills.

So the count is four bonding plus four non-bonding: eight orbitals, sixteen electrons. And sixteen-electron square-planar complexes are exactly what the platinum group provides in quantity — the tetrachloridoplatinate ion in the figure below is one.

The reason this is worth doing rather than quoting is that the nine-orbitals-hold-eighteen argument cannot produce sixteen at all. Nine orbitals hold eighteen electrons in a square-planar complex exactly as they do in an octahedral one. What differs is how many of them lie below the gap, and finding that out required the reduction.

Eighteen again, by a different route #

A tetrahedral complex has four ligands, like the square-planar one, and counts to eighteen, like the octahedral one. The reason is a small piece of bookkeeping worth following.

The four ligand orbitals reduce to $A_1 \oplus T_2$: two species rather than three, because a tetrahedron has no centre to distinguish even from odd.

The metal side has a complication the octahedron did not. There is no inversion, so the p orbitals and three of the d orbitals belong to the same species, $T_2$. Both can interact with the one ligand $t_2$ set, so one combination of them is bonding and the other is left non-bonding — and the one left behind is mostly d, because d lies lower.

Count: one $A_1$ bonding, three $T_2$ bonding, then the leftovers — the $E$ pair of d orbitals and the $T_2$ set that did not get used. Four plus five is nine, and eighteen again.

What “matching by symmetry” is doing #

The step in the middle of all three counts deserves scrutiny, because it is where a reader is most likely to feel that something has been assumed.

Two orbitals interact only if they have the same symmetry species. That is not a convention: it is the vanishing-integral theorem again, the same one behind every selection rule. The interaction between two orbitals is an integral of one times the Hamiltonian times the other, and if the product of their representations does not contain the totally symmetric one, the integral is zero — exactly — and the two orbitals are independent whatever their energies or their spatial extent.

So “the metal’s t₂g has no ligand partner” is a computed statement of the same kind as “this transition is forbidden”. Nothing about the geometry of the lobes was consulted, and nothing needed to be: the ligand basis has no $T_{2g}$ component, so no combination of ligand σ orbitals can interact with a $T_{2g}$ metal orbital, and the interaction vanishes by cancellation in pairs rather than by being small.

That is also why the matching can be done mechanically: reduce the ligand basis, sort the metal orbitals by species, pair them off species by species, and report what is left over — checking that every ligand combination found a partner, which confirms that the metal orbitals and the reduction are talking about the same group.

The tetrahedral ligand basis reduces to a₁ ⊕ t₂ — two species for four orbitals. The metal has two sets of T₂ symmetry and the ligands have one, which is the bookkeeping that leaves an extra non-bonding set and is why a tetrahedral complex can be stable at counts other than eighteen.

Where the count comes from, and where it fails #

The pattern in all three is the same and is worth stating plainly. The count is twice the number of orbitals lying below the antibonding set, and every one of those orbitals is either a metal–ligand bonding combination or a metal d orbital that no ligand combination could reach.

That makes the rule fragile in one specific place: the non-bonding orbitals have to stay non-bonding. They do not always.

When a ligand has empty π orbitals the metal’s lower set becomes genuinely bonding rather than merely unreachable, the gap above it widens, and the count is unmoved by all of it — which is the property the rule depends on.

With π acceptors the rule is at its strongest. Carbon monoxide, cyanide, phosphines: their empty π* orbitals give the metal’s $t_{2g}$ set something to interact with, so those three orbitals become bonding and the gap above them grows. Metal carbonyls follow the eighteen-electron rule with remarkable fidelity, and that fidelity is a consequence of the ligand rather than of the metal.

With π donors it is much weaker. A halide or an oxide pushes the $t_{2g}$ set up, narrowing the gap and making the three orbitals less attractive to occupy. Early transition metals with such ligands routinely sit well below eighteen — and permanganate, with a formal d⁰ metal, is the extreme case.

With a weak field the count can be exceeded. If the antibonding $e_g$ pair is not very high, electrons will go into it. Nickelocene has twenty electrons and exists; it is paramagnetic, with two electrons in orbitals the rule says should be empty, and it is less stable than its eighteen-electron neighbour ferrocene rather than impossible.

So the rule is a statement about the size of a gap, and gaps depend on ligands. That is exactly the position the site reached about aromaticity: a count that works because of a shell closure, and a closure that is only as good as the gap above it.

What the count is used for #

A rule that is right about most of a class of compounds and explicable about the rest earns its keep in two ways, and both are worth naming because they are not the same activity.

As a filter. A proposed structure whose count is far from eighteen, with π-acceptor ligands and no reason for the deficit, is probably wrong — or is a fragment rather than a compound. The count is quick, it uses nothing but the formula and the oxidation state, and it rules out more than it rules in.

As a diagnosis. A stable compound whose count is not eighteen is telling something about its ligands, and the deviation says what. Sixteen with a square-planar d⁸ metal is the geometry talking. Fourteen with an early metal and oxide ligands is the π donation talking. Twenty with a weak field is the size of the gap talking.

The second use is the more interesting and it is the one that requires the reduction rather than the arithmetic. “Eighteen because nine orbitals” gives no purchase on a compound with seventeen; “eighteen because nine orbitals lie below a gap in this geometry with these ligands” says exactly which assumption to check.

That distinction is the same one Bent’s rule, computed draws about hybridisation and VSEPR, computed draws about shape: a rule stated as a number is a mnemonic, and a rule stated as the output of a calculation tells its own exceptions apart from its own failures.

The comparison with the familiar counts #

Three counting rules now sit side by side here, and they are the same idea at three sizes.

The 4n+2 rule is the same question asked of a ring: fill it and ask whether the highest occupied shell came out full. The eighteen-electron rule is that question asked of a metal with nine valence orbitals, and the two are one procedure rather than two rules.

The octet is a count of four valence orbitals, all four of which are involved in bonding, which is why it is so nearly inviolable in the second row and fails as soon as a third-row element has orbitals to spare — the subject of hypervalency without d orbitals.

Hückel’s 4n+2 is a count of filled π shells, computed in aromaticity as a shell closure by filling every ring from three to ten and asking whether the highest occupied shell came out complete. It is not a rule imposed on the answer; it is the answer.

Eighteen is the same question asked about a metal’s nine orbitals in a field of ligands, and it differs from the other two in one respect: the number depends on the geometry, because the geometry decides how many of the nine find partners.

That last difference is the argument for computing it rather than remembering it. An octet is eight whatever the molecule; eighteen is eighteen only when the reduction says nine orbitals lie below the gap.

What the diagram does not show #

The figures here are level diagrams and they leave three things out.

They have no energies in them. The vertical positions are schematic: symmetry says which orbitals interact and says nothing about how strongly. The one part with a computed scale is the d splitting, which comes from the ligand-field model, and even that has a fitted parameter in it.

They have no electron repulsion in them. Filling nine orbitals with eighteen electrons treats the electrons as independent, and they are not. What repulsion does to a count of this kind is visible in the smallest case that can be solved exactly, in the smallest many-electron calculation, where a filled shell and an unfilled one differ by more than the sum of their orbital energies.

They assume every ligand brings exactly one σ orbital. Real ligands bring more: a halide has three lone pairs, and the two that are not used for σ are the π donors this essay has already blamed for weakening the rule. The count is a σ-framework count, and the π framework is the subject of back-bonding is two interactions.

The three counts, side by side #

Putting the three geometries in a row makes the shape of the argument visible in a way no one of them does.

The ligand count goes six, four, four and the electron count goes eighteen, sixteen, eighteen, which is enough on its own to show that the answer is not a function of how many ligands there are. What varies is the leftover: three, four and five metal orbitals with no partner, and one of the square plane’s five disqualified by being a p orbital pointing where no ligand is.

Every row of that table is produced by the same twenty lines of code, run on three molecules whose point groups were recovered from their coordinates. Nothing in it was entered by hand except the metal’s own list of nine orbitals, which is a property of an atom rather than of a compound.

The sixteen-electron count is read off the D₄ₕ table and the metal’s nine orbitals, and nothing else enters it. That table is generated from the molecule’s own operations rather than looked up, which is what makes the count a computation on a structure rather than a rule about a formula.

The same procedure gives eight #

The procedure here takes a central atom’s list of valence orbitals, reduces the ligand σ set in the molecule’s own point group, matches by symmetry and counts what lies below the gap. Nothing in that procedure is about transition metals. Run it on a main-group centre and it produces the other counting rule in chemistry.

Methane is tetrahedral, so its four hydrogen 1s orbitals span $a_1 \oplus t_2$ — the same reduction the tetrahedral row of the table above uses. Carbon brings four valence orbitals: one 2s, of $a_1$ symmetry, and three 2p, spanning $t_2$.

Every ligand combination therefore finds a partner, and every central orbital is used up. Four bonding combinations, no leftovers, eight electrons. That is the octet rule, produced by the same twenty lines that produce eighteen, with one line changed — the list of orbitals the central atom brought.

The comparison with the tetrahedral transition-metal row is the useful part, because the two molecules have the same geometry and the same ligand reduction and different answers:

So the octet and the eighteen-electron rule are not two rules. They are one procedure applied to centres with four and nine valence orbitals, and the difference between eight and eighteen is exactly the ten electrons a d shell holds.

Two things follow that the numerical version of either rule cannot supply.

The leftovers are where the chemistry is. A carbon has none, which is why an octet is rigid and a saturated carbon does nothing until a bond is broken. A tetrahedral metal has five, which is why its complexes have colour, magnetism and a d count at all — every one of those properties lives in orbitals the ligands never reached.

And a count is a statement about a gap, in both cases. The octet is eight because the next orbital up is a bonding partner’s antibonding mate, and eighteen is eighteen for the same reason with a different set. Neither number is a target the centre is trying to reach; both are the number of orbitals below the first large separation, which is what the reduction computes.

Who found it, and when #

Irving Langmuir proposed the rule in 1921, four years before quantum mechanics could explain it, on the basis that a metal atom achieving the electron count of the next noble gas would be stable — an argument by analogy with the octet, which he had also generalised. It worked well enough on the compounds then known to become a design principle.

The molecular orbital account came in the 1950s and 1960s with ligand field theory, and it replaced “the metal reaches a noble gas configuration” with “nine orbitals lie below a gap”. The two agree numerically for octahedral carbonyls and disagree everywhere the geometry differs, which is why square-planar sixteen-electron complexes were a puzzle under the first account and are a computation under the second.

The pattern is a familiar one in older chemistry: a rule discovered empirically, justified by a numerical coincidence, and later explained by something that also says when the rule fails. The coincidence in this case is real — nine orbitals really do hold eighteen electrons — and it is not the reason.