The bonds are what is left over

Eighteen is a count: a transition metal with nine valence orbitals filled has eighteen electrons, and a mononuclear complex whose ligands supply exactly the shortfall is the stable one. Sixteen is also a count, for a square planar geometry where one orbital is pushed out of use.

Both are statements about one metal. A cluster has several, and the same arithmetic extends to it in a way that is almost too simple to be believed: each metal still wants eighteen, a metal–metal bond is a shared pair which contributes one electron to each partner, and therefore

$$\text{number of M–M bonds} = \frac{18n - \text{total valence electrons}}{2}.$$

The bonds are what is left over. Nothing about the geometry goes in.

Thirteen clusters, counted #

Counting is deliberately literal. Each metal contributes its group number; each carbonyl contributes two, whether it is terminal or bridging, since a bridging CO donates the same pair to the cluster as a whole; and the charge is subtracted.

The mononuclear entries come to eighteen exactly and leave nothing over, which is the case with no bonds — chromium hexacarbonyl, iron pentacarbonyl, nickel tetracarbonyl. That is the eighteen-electron rule, and it appears here as the $n = 1$ member of a family.

The dimers all come to thirty-four, which leaves one bond, and all three have one: dimanganese decacarbonyl with a single unsupported Mn–Mn bond, dicobalt octacarbonyl with a metal–metal bond and two bridging carbonyls, di-iron nonacarbonyl with three bridges. Three quite different-looking structures, one count.

The triangles come to forty-eight and leave three bonds, which is a triangle’s number of edges. The tetrahedra come to sixty and leave six, which is a tetrahedron’s. And Os₅(CO)₁₆ comes to seventy-two and leaves nine, which is the number of edges of a trigonal bipyramid.

That is a lot of structures predicted from a number, with no geometry consulted at any point.

What the rule is really counting #

Before the failure, it is worth being clear about what makes the rule work at all, because the usual statement of it hides the argument.

An eighteen-electron metal is one whose nine valence orbitals are all either filled or used in a bond. Put two metals together and each of them has nine orbitals to satisfy; a shared pair in a metal–metal bond satisfies one orbital on each, so it counts once towards each metal’s eighteen and once towards the total. The total therefore counts every shared pair twice, and the discrepancy between $18n$ and the actual total is $2 \times$ the number of shared pairs.

That is the whole derivation, and it is a bookkeeping identity rather than a physical claim. What is physical is the premise — that the framework is held by shared pairs, one per contact — and it is the premise that fails.

There is a second premise doing quiet work: that every metal in the cluster achieves eighteen. It need not. Square planar complexes stop at sixteen because one orbital is pushed too high to use, and the same thing can happen to a metal in a cluster with an unusual coordination environment. Where it does, the rule under-counts the bonds by the same arithmetic — and this is worth knowing because it is a second, quite different way the count can be wrong, and it looks identical from the outside.

Where it stops #

Both six-metal clusters break it, and they break it by exactly one bond.

Rh₆(CO)₁₆ has eighty-six valence electrons, so $18 \times 6 - 86 = 22$, and eleven bonds. The structure is an octahedron, and an octahedron has twelve edges. [Co₆(CO)₁₄]⁴⁻ has the same eighty-six and the same octahedron and the same discrepancy.

One bond is the smallest amount a counting rule can be wrong by and the hardest to dismiss. It is not a rounding, it is not an ambiguity about a bridging ligand, and it is not confined to one metal.

Why six is the boundary #

The rule assumes that the electrons holding the cluster together are in two-centre bonds — one pair per edge, shared between two metals. For a triangle, a tetrahedron or a trigonal bipyramid that is a workable description. For an octahedron it is not, and the reason is a count of orbitals rather than of electrons.

Six metals at the corners of an octahedron each present one orbital inward, along the axis toward the centre. Those six radial orbitals combine into cluster orbitals, and the number of bonding combinations among them is not six and not twelve — it is one, plus a set of others whose bonding character comes from the tangential orbitals as well. The framework holds itself together with delocalised multicentre orbitals, and the number of them has no reason to equal the number of edges anyone would draw.

This is the same conclusion borane chemistry reaches from the other side. A cage needs one pair more than it has corners: a closed deltahedron of $n$ vertices is held by $n + 1$ skeletal electron pairs, which is Wade’s rule, and it is a statement about a delocalised framework rather than about edges.

The same total, two rules #

The number eighty-six supports both counts, and this is the part worth carrying.

Take twelve electrons per metal — the ones its own carbonyls and non-bonding pairs use — and what is left is the skeletal count: $(86 - 72)/2 = 7$ pairs. Seven is $n + 1$ for six vertices, which is closo: a closed deltahedron, and an octahedron is the six-vertex deltahedron.

So Wade’s rule gets the octahedron right where the leftover rule fails, and the two are not competing — they are the same total electron count read with different assumptions about what the electrons are doing.

The tetrahedra make the point in reverse. Co₄(CO)₁₂ has sixty electrons, which leaves six bonds by the first rule and six skeletal pairs by the second. Six skeletal pairs for four vertices is $n + 2$, which is nido — a five-vertex closed cage with one corner removed, and removing a corner from a trigonal bipyramid does give a tetrahedron. Both rules are right about a tetrahedron, and both are checkable against the same crystal structure.

Bridging carbonyls, and why they do not matter #

A detail that looks as though it should matter and does not, which is a useful thing to know about a counting rule.

Fe₂(CO)₉ has three bridging carbonyls and six terminal ones; Co₂(CO)₈ has two bridges in the solid and none in one of its solution forms; Fe₃(CO)₁₂ has two bridges and Ru₃(CO)₁₂ has none, despite being the same formula with a different metal. All four come out at the counts above regardless.

The reason is that a bridging carbonyl donates the same pair. A terminal CO gives its lone pair to one metal; a symmetric bridging CO gives the same pair to a three-centre orbital spanning two metals and the carbon. Two electrons either way, and the total is blind to which.

That blindness is a feature of the total and a limitation of the picture. Whether a carbonyl bridges is decided by things the count does not contain — the size of the metal, the crowding, the balance of donation against back-donation — and the same compound can be found both ways in solution and in the crystal. A rule that gives the right answer without needing to know is more useful than one that does not, and it is also a rule that cannot be asked the question.

The one that is odd #

Vanadium hexacarbonyl is the entry that refuses to be counted at all. Five valence electrons on the vanadium and twelve from six carbonyls gives seventeen, which is not an even number, so it does not correspond to any whole number of electron pairs.

The right response is not to halve it. Seventeen is a real count for a real compound: V(CO)₆ is a stable, isolable radical, one of very few, and its chemistry is dominated by the fact that it is one electron short. It is also, for that reason, one of the few carbonyls with an unpaired electron and therefore a magnetic moment worth measuring. It reduces readily to [V(CO)₆]⁻, which has eighteen and is entirely ordinary.

An arithmetic that returns half a bond has been asked a question about pairs and handed something that is not pairs, and the check that would catch it is a parity test rather than a comparison against a structure. That distinction — a check that can fail on the input rather than on the answer — is the kind worth writing, because it fails for the right reason without needing a reference.

What the counts assume #

Four assumptions, each of which fails somewhere.

Every metal is a transition metal with nine valence orbitals. The rule is built on $s$, three $p$ and five $d$, and it is why eighteen rather than eight — the same nine orbitals the octahedral splitting is drawn from.

Every ligand’s donation is a fixed integer. A carbonyl gives two, a halide one or three depending on the convention chosen, a bridging hydride one. The conventions are consistent within themselves and the totals do not depend on which is used, which is worth knowing but is a statement about bookkeeping.

The framework is held by localised pairs. This is the one that fails at six, and it fails for a reason with a name.

Every metal is equivalent. The rule distributes the shortfall evenly and real clusters are often not symmetric, so the count is a total rather than a per-metal statement. Mixed-metal clusters and clusters with an interstitial atom stretch it further, and the failure there is not one bond but several.

The three-metal case, worked #

One cluster in full, because the arithmetic is short and the prediction is real.

Triiron dodecacarbonyl: three irons at eight valence electrons each is twenty-four, twelve carbonyls at two each is twenty-four, and the compound is neutral. Total: forty-eight.

Eighteen per metal would be fifty-four. The shortfall is six, so three shared pairs, so three metal–metal bonds. Three bonds among three metals is a triangle, and there is no other arrangement of three bonds among three atoms — a chain would need only two.

The prediction is therefore that the three irons form a closed triangle, and that is what the structure shows. Ruthenium and osmium give the same count with the same result, and the ruthenium compound has no bridging carbonyls at all while the iron one has two, which the count does not notice and does not need to.

Now the skeletal reading of the same forty-eight: twelve per metal is thirty-six, leaving twelve electrons, which is six pairs. Six pairs for three vertices is $n + 3$, which is arachno — a five-vertex closed cage with two corners removed, and a trigonal bipyramid with two corners removed is a triangle. Both rules agree, from the same total, by different routes.

Off by one is the smallest failure available #

Wrong by exactly one bond reads like a near miss, and it is worth noticing that it cannot read as anything else. A counting rule takes integers in and gives an integer out. Its resolution is one bond — two electrons — and there is no such thing as being wrong by a third of a bond.

That has two consequences, and they pull in opposite directions.

One bond is the minimum detectable error, so a discrepancy of one carries almost no information about how badly the assumption has broken. A cluster whose bonding is marginally delocalised and a cluster whose bonding is wholly delocalised would both be reported as off by one, if the totals happened to land there, and the rule has no way to distinguish them.

And one bond is not a small energy. Two electrons in a metal–metal bonding orbital is a substantial fraction of what holds a small cluster together — tens of kilojoules a mole at least — so the failure is not a rounding error that better bookkeeping would absorb. The rule is not nearly right; it is wrong by an amount comparable to a bond, which is the only amount it can be wrong by.

Putting those together says what the tidiness of both six-metal entries fail by one is and is not. It is not evidence that the two failures have the same cause, because one is the only value available and two independent breakdowns would land on it by default. It is evidence that neither cluster is wildly outside the rule’s reach, because a badly delocalised framework could as easily have come out two or three bonds short.

What would separate those readings is a quantity with a continuum in it, and the clusters supply several. Metal–metal distances become uniform when the bonding delocalises and stay unequal when it does not, and diffraction measures them to thousandths of an ångström. Ionisation energies in a series report how far the frontier orbitals have spread. Either would say whether the octahedral clusters are marginal cases sitting just past a boundary or fully delocalised cages that the count happens to miss by the smallest step it has.

That is the general limitation of any rule whose output is an integer, and it is worth carrying beyond cluster counting. An integer prediction is unusually strong when it is right, because it cannot be nearly right by accident. When it is wrong it is unusually uninformative, because the size of the miss is quantised and says nothing about the size of the physics that caused it.

What a counting rule is for #

It is easy to read a rule that fails at six metals as a rule that has been refuted, and that is not the right reading.

The value of a count is that it is checkable against a structure without doing any calculation. Predicting nine metal–metal bonds for Os₅(CO)₁₆ from a number, and finding a trigonal bipyramid, is a real prediction — the number came from group numbers and carbonyl stoichiometry, and the structure came from diffraction, and nothing passed between them.

The value of knowing where it fails is greater still, because the failure is informative. It says that at six metals the framework has stopped being describable in two-centre terms, and it says so from the count alone, before anyone looks at the cluster. That is what a boundary in a counting rule is: not a defect, but the point at which the rule reports that its own assumption has broken.

Every metal in the clusters above is counted as though its d electrons were simply available, and whether a d⁶ ion is high spin or low spin depends on the field it sits in. The count does not notice: it is the same integer either way, which is one more thing it is not sensitive to and one more reason it survives.

Two ligands opposite each other across a metal compete for the same orbital, and the competition is an overlap argument rather than a charge one. The clusters above have metals in place of some of those ligands and the same competition operates — which is one reason a cluster’s bonds are harder to assign than a mononuclear complex’s, and no reason its count is.

Still open: large clusters, the octahedral miss, and the graph #

The obvious open question is the large clusters, where the counts stop working altogether. Beyond about a dozen metals the surface-to-interior ratio changes, the interstitial atoms appear — carbides, nitrides, hydrides buried inside the cage — and the electron counts converge towards what a small piece of metal would have. That is a crossing from molecular chemistry into what a metal actually is, and it happens at a size where neither rule has anything to say.

There is also a question the table above raises and cannot answer. Both six-metal entries are octahedra and both are wrong by one bond, which is suspiciously tidy: a rule that had simply broken down would be expected to fail by varying amounts. One bond is what a single delocalised orbital being counted twice would look like, and that is a testable guess rather than a result — it would need the level pattern of the octahedral framework worked out properly, with tangential orbitals as well as radial ones, and the count of bonding combinations compared against eleven and twelve. The method for that is standard and the calculation has not been done here.

The other open question has already been opened from the borane end. The skeletal count comes out of a level pattern — a strongly bonding radial combination and a set of tangential ones — and that pattern is the same for a borane cage and a metal one, which is why Wade’s rule crosses between them. The unifying statement is about the graph rather than the element, and it is a statement about where two-centre bonding stops that arrives here from the opposite direction.