Molecular Geometry & Bonding
what the shape decides
Concept

Back-bonding — where it appears

The π component of a metal–ligand bond in which the metal supplies the electrons and the ligand the empty orbital. It is what makes a carbonyl's bond to a metal stronger than donation from its lone pair alone would give, and what puts the carbonyl's own stretching frequency below its free value.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

However hard the π channel is driven, the counted orbital stays the metal's. The metal's share of the filled T₂g orbital against the π coupling, with the eighteen-electron count drawn beside it. The share falls from 1 to 0.5467 across a coupling range of 80,000 cm⁻¹ and approaches a half from above without reaching it: the lower eigenvector of a two-level problem always carries more of the lower basis function, whatever the coupling. The count is eighteen at every point.

The count that cannot be broken by strength

Back-donation puts electrons into orbitals that are not the metal's, and the eighteen-electron rule counts the metal's nine. Turning the π channel up as far as it will go never breaks it: the counted orbital's metal share falls from 100 per cent to 54.67 and approaches a half from above without reaching it. What does flip it is not strength but order.

applied · Electron count The gap that makes sixteen special does not move. The gap above the sixteen-electron closure of a square plane and above the eighteen-electron closure of an octahedron, against the π strength. The octahedron's is 3eσ − 4eπ and moves at every value; the square plane's is exactly 2eσ until the π strength reaches a quarter of the σ one, because the orbital that sets it is d(z²) and a square-planar ligand set has nothing of that symmetry to offer. Past the threshold the two are the same number, which is not a coincidence: beyond it the square plane's gap is set by d(xy) and the expression is the octahedron's.

The orbital a ligand cannot reach

The sixteen-electron count of a square plane is a statement about an energy rather than about symmetry matching, so it was the count that ought to be sensitive to a π channel where the eighteen-electron one is not. It is not sensitive either — and for a sharper reason. The orbital that sets its gap is d(z²), and a square-planar ligand set contains nothing of that symmetry, so the gap is exactly 2eσ until the π strength reaches a quarter of the σ one.

applied · Electron count What the gap alone would predict, and what was fitted. Each halide's π scale as a multiple of fluoride's: the value fitted to the spectrochemical series, against what the energy denominator alone gives with the metal orbital at the vacuum level — which is the weakest the denominator effect can be. It over-predicts at every ligand, and moving the metal level down makes it worse.

The gap that would have to be smaller

An angular overlap parameter is an overlap squared over an energy denominator, and the usual fit folds the denominator away. Put the measured ionisation energies back in and the denominator alone over-predicts the trend down the halide group at every metal level a donor permits — the smallest it can give is 1.67 against a fitted 1.43. So the overlap has to shrink down the group, which is the opposite of the usual expectation.

applied · Ligand field The acceptors' π* levels, as measured. For each π acceptor: the energy at which a slow electron is temporarily captured, which is the π level above the vacuum, and the π scale this collection's series carries for it. Carbon monoxide's and dinitrogen's resonances are measured on the molecule itself; cyanide's cannot be, because an electron cannot be attached to an anion, so hydrogen cyanide's stands in for it — the same π with a proton where the metal would be.

A denominator that fails both ways

The energy gap an e_π folds away over-predicts the halide trend at every metal level a donor allows, and the acceptors look out of reach because a π* is not an atomic level. It is measurable — a slow electron is captured by it — and on that side the same denominator under-predicts. No metal level fixes either, and the two want it moved in opposite directions.

applied · Ligand field Two interactions, and they push the metal in opposite directions. Each ligand's filled π and empty π against a metal d level, on one energy scale with the vacuum at zero. The π lies below the metal and pushes it up, which is the only interaction the model's derivation has; the π lies above and pushes it down, which is the one it lacks. Both level positions are measured — an ionisation energy and an attachment energy — and the metal's is the single quantity nothing here measures, drawn at -8.0 electronvolts and swept elsewhere.

The channel that points at the metal

Two ligands in the spectrochemical series carry a fitted π parameter no quotient of squared overlaps can produce, because it is negative. Giving the derivation the second interaction it lacks makes both of them negative at every metal level — and the reason is not the energy denominators, which favour the donor channel in all three cases. It is where each orbital keeps its amplitude.

applied · Ligand field

Named alongside it

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

Ligand fieldModel limitConventiond orbitalsIonisation energyAngular overlapApproximationDegeneracyElectron countIrreducible representationsOverlap integralPi acceptor

All concepts