Molecular Geometry & Bonding
what the shape decides
Concept

Eighteen-electron rule — where it appears

The observation that many transition metal complexes hold eighteen valence electrons. It is a shell closure rather than a rule, and which closure is the deep one depends on whether the ligands accept or donate π density.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

Eighteen electrons, from a reduction. The ligand σ orbitals of an octahedral complex reduced in Oh (A₁g ⊕ Eg ⊕ T₁u), matched against the metal's nine valence orbitals by species, and counted. 6 bonding and 3 non-bonding orbitals hold 18 electrons.

Eighteen is a count

The eighteen-electron rule is usually justified by adding up an s, three p and five d orbitals. That is a restatement rather than a reason. Reduce the ligand orbitals in the complex's own point group, match them against the metal's by symmetry, and the number that comes out is the count of orbitals lying below a gap — which is eighteen for an octahedron, sixteen for a square plane, and eighteen again for a tetrahedron for a different reason.

applied · Electron count Sixteen electrons, from a reduction. The ligand σ orbitals of a square planar complex reduced in D4h (A₁g ⊕ B₁g ⊕ Eu), matched against the metal's nine valence orbitals by species, and counted. 4 bonding and 4 non-bonding orbitals hold 16 electrons.

Sixteen is also a count

A transition metal brings nine valence orbitals, and nine filled orbitals is eighteen electrons. A square plane leaves more of those nine unmatched than an octahedron does and still holds fewer electrons, because one of the leftovers is out of reach.

applied · Electron count The bonds are what is left over. Thirteen carbonyls and clusters, with the total valence electron count, the number of metal–metal bonds that leaves over from eighteen per metal, and the number the crystal structure has. They agree for every cluster up to five metals. At six they disagree by one, in both entries tested — and the skeletal count in the last column, which is the rule boranes are analysed with, comes out at n + 1 for both, meaning a closed deltahedron, which an octahedron is.

The bonds are what is left over

Every metal wants eighteen electrons and a metal–metal bond gives one to each of its partners, so the number of bonds in a cluster is what is left over after the counting. It works for every carbonyl cluster up to five metals and fails at six by exactly one bond — where the other counting rule, the one boranes are analysed with, is right.

applied · Electron count Which count closes a shell, ligand by ligand. For each ligand the spectrochemical series has parameters for: its π parameter, the two gaps, and which electron count the deeper one sits above. The π donors close at twelve and the π acceptors at eighteen, and the ligand's charge predicts neither.

The count that is not always eighteen

The eighteen-electron rule is a shell closure, and an octahedral level diagram has two of them — one at twelve electrons and one at eighteen. Which is deeper is decided by the sign of one parameter: with a π acceptor the gap above eighteen is 3.720 and above twelve 2.280, and with a π donor the two swap over exactly.

applied · Electron count The total is the same in both columns; nothing else is. Ten complexes counted by both conventions. The neutral method gives the metal its group number and every ligand what it brings as a neutral fragment; the ionic method assigns an oxidation state and gives every anionic ligand a pair. The two totals agree in every row. The oxidation state and the d count do not agree wherever the oxidation state is not zero, and the moments the two d counts predict differ by as much as 2.83 Bohr magnetons.

The same count, two oxidation states

Count a complex by the neutral method and by the ionic one and the total is the same integer every time — eighteen for ferrocene, sixteen for tetrachloroplatinate, twenty for hexaaquanickel. The oxidation state and the d count are not: hexaaquairon is d⁶ on one convention and d⁸ on the other, and the two predict spin-only moments of 4.90 and 2.83 against a measured 5.40.

applied · Electron count The metal's charge is a coordinate, and the count is not. The metal's charge in an octahedral d6 complex, against how much of each shared pair the ligand is given. Half each is Mulliken's rule and the whole to the ligand is the assumption an oxidation state makes; the answer runs over 2.06 electrons between them. The oxidation state itself is 0, which is off the end of the range, and the electron count is the same number at every point on it.

An integer nobody measured

The oxidation state of chromium in the hexacarbonyl is zero. Its charge, computed from the same wavefunction, is anywhere between −3.04 and −0.98 depending on how the shared electrons are divided — and the integer sits outside that whole range. The electron count, meanwhile, is eighteen at every point on it.

applied · Electron count The sixteen-electron gap, from a plane to a tetrahedron. The gap above eight d electrons as four ligands are folded out of a square plane towards a tetrahedron, at three π strengths. It is 2eσ exactly at the plane whatever the π strength is, and exactly zero at the tetrahedron whatever it is — the upper three levels there are the degenerate t₂ set. In between the three curves separate, and the separation is what the π channel is doing.

The gap that only a tetrahedron closes

The sixteen-electron gap is 2eσ exactly, and a square-planar π set cannot touch it because d(z²) has no partner there. Fold the ligands out of the plane and the gap survives almost intact for fifteen degrees, closes to nothing only at the tetrahedron — and loses its exactness at the very first degree.

applied · Electron count

Named alongside it

The objects these essays reach for when they reach for this one.

Electron countCoordination complexLigand fieldd orbitalsModel limitPi acceptorClosed-shell configurationsAngular overlapIrreducible representationsNon-bonding orbitalsPi-donorReduction formula

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