Eighteen electrons, from a reduction

One of the figures on a d shell in a field: What a set of ligands does to five degenerate orbitals — computed twice, from an integrated point-charge potential and from an angular overlap matrix, which agree on every ratio.

14 essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

Eighteen is a count

The count as a computation. The six ligand σ orbitals of an octahedral complex are reduced in its own point group, matched against the nine metal valence orbitals by symmetry species, and what lies below the antibonding set is counted: six bonding orbitals and three non-bonding, which hold eighteen electrons.

The same count taken to systems with more than one metal. Six ligand σ orbitals in Oh reduce to a₁g ⊕ eg ⊕ t₁u — three species for six combinations — and a cluster is the same bookkeeping repeated with metals in place of some of the ligands. The count survives; what stops surviving is the assignment of any particular pair to any particular bond.

The square-planar count. Four ligand σ orbitals span a₁g ⊕ b₁g ⊕ eu; four metal orbitals find partners and five do not; and of those five, four lie low enough to hold electrons while one does not. Eight orbitals, sixteen electrons.

Sixteen is also a count

The nine metal orbitals of a square-planar complex sorted by what the four sigma donors do to them: four are used in bonding, four are left low enough to fill, and one is left far too high. Eight filled orbitals is sixteen electrons.

The same geometry with the whole molecular-orbital diagram rather than the d shell alone. Four ligand combinations match four of the metal’s nine orbitals, one metal orbital is left far too high to use, and eight orbitals filled is sixteen electrons — the count read off a reduction rather than off a rule.

However hard the π channel is driven, the counted orbital stays the metal’s. Nothing lies above or below the plane, so nothing there interacts with an orbital pointing that way — and the count is therefore stable under a change that moves a great deal of charge, which is the property a rule needs.

The bonds are what is left over

Thirteen carbonyls, the total valence electron count, the bonds that leaves over from eighteen per metal, and the bonds the crystal structure has. They agree for every cluster up to five metals.

The eighteen-electron count for a single octahedral metal, from a reduction of the ligand set rather than from a mnemonic. Nine metal orbitals, six of them matched by ligand combinations and three left non-bonding, and the count follows. This is the argument the cluster rule extends, and it is an argument about one centre.

Where the share of a shared pair crosses a half, and where it does not. The counting rules take a carbonyl as a two-electron donor and the actual share is a function of the ligand’s levels — the two coincide only where the metal and ligand levels cross, and everywhere else the count is a convention rather than a measurement of anything.

The count that is not always eighteen

The octahedral level diagram as the counting argument builds it: six ligand combinations reduced in the complex’s own group and matched against the metal’s nine orbitals by symmetry. The three t2g orbitals are what is left over, and everything in this essay is about whether they are worth filling.

An integer nobody measured

Where the eighteen come from, by symmetry: six ligand combinations match six of the nine metal orbitals, leaving three metal orbitals with nothing to pair with. Nine orbitals filled is eighteen electrons, and that arithmetic is what the rule is.

The count that cannot be broken by strength

The metal’s share of the orbital the rule counts, against the π coupling. It falls and it never crosses a half.

The same share against where the ligand level sits. The crossing is at the coincidence of the two energies and is independent of how strongly they are coupled.

The count itself, from the reduction: six ligand σ combinations matched against the metal’s nine orbitals by species, and the nine filled ones that follow. Everything in this essay is what happens when the π set is added to that picture.

The square that wastes an orbital

Every arrangement of every case, with the group recovered from the arrangement’s own coordinates. The three warned rows are the disagreements, and none of them is a shape the molecule adopts.

The three disagreements set out. Same shortfall, same size, same reason.

Six ligands laid flat. The perpendicular orbital changes sign across the plane and no σ combination does, so it cannot be used and one more ligand pair is left over.

Expensive is not the same as unadopted

Every arrangement of the census, by how many ligand combinations are left without a partner and by how far its repulsion sits above the best arrangement of that many points.

Every distinct arrangement ranked by repulsion, with whether a molecule adopts it and whether the formula holds.

A square plane of four ligands on methane and on xenon tetrafluoride: identical geometry, identical repulsion, opposite verdicts.

A count that changes at one point

The repulsion energy and the orphan count along the path from a tetrahedron to a square plane.

Every geometry on the path, with the group found, the orphan count in it, the repulsion energy, and how far one ligand has moved from its tetrahedral position.

The last few geometries before the plane, by how far a ligand still sits from where the plane would put it.

The group nobody wrote a table for

The repulsion along the twist and, underneath it, which geometries have an orphan count at all. Four of the twenty-one do.

Every geometry on the twist: the group the finder identifies, the count reduced in it, the repulsion, and how far one ligand still sits from its octahedral position.

The operations actually assembled at three points. The halfway geometry has six, and they hold to four parts in 10¹⁶.

The count the table was hiding

The whole of what was missing: three classes, three irreducible representations, and two sums that have to come out at six.

The count along the twist, with the same sweep before the table underneath it. Four answered geometries became fifteen.

The six σ combinations reduced in each group the twist passes through. The species differ in name, number and dimension; the count does not.

The leftover changes sides

The spare metal orbitals along the Bailar twist. Three at every geometry the finder can name, in three point groups, with the species changing completely.

Both counts along the same path. Two orphaned ligand combinations at a four-orbital centre, three spare metal orbitals at a nine-orbital one, and neither moves.

Every arrangement in the census at a nine-orbital centre. Fourteen match every ligand combination; one does not.

The gap found on purpose

Every path, with each geometry coloured by what its group turned out to be. Two cells in two hundred and seventy are a finite group with no character table.

The two hundred and seventy geometries sorted into what “no answer” can mean. The three are unrelated failures.

The Bailar twist’s five bands, after the missing table was written. Three answer and two never will, and this survey asks the same question of every path at once.

Folding the ring does not give the orbital back

For rings of four to eight ligands, the orphan count’s excess over n + L − 4 at every polar angle from 60° to 120°, bare and with a ligand on the axis.

Six ligands round a centre, side on: flat, folded to 110°, folded and capped, and folded with one lone pair.

The character tables of $C_{5v}$ and $C_{6v}$ as generated from the rotation angle, with the sums that fix their size.