Molecular Geometry & Bonding
what the shape decides
Concept

Angular overlap — where it appears

A ligand field model in which each ligand acts on a metal's d shell through σ, π and δ channels of stated strength. The geometry enters only as a rotation, so the splitting pattern of any arrangement follows from two parameters and a set of directions.

Named by 14 essays across 2 fields — each of them below, with the objects they name alongside it.

The same d shell in four fields. The five d energies in octahedral, tetrahedral, cubic, square planar coordinations, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

Two models, one ratio

A tetrahedron splits a d shell by four ninths of what an octahedron does. Two models that share nothing but the ligand directions — an integral over a point-charge potential and a rotated diagonal matrix — both produce that number to eight decimal places, and neither was told it.

applied · Ligand field What orders the spectrochemical series, and what does not. Nine ligands' measured octahedral splittings, plotted against charge and against the π parameter that says whether the ligand donates or accepts π density. Charge puts 3 of 20 mixed-charge pairs in the order it predicts; the π parameter puts 35 of 35 in order.

The spectrochemical series is not electrostatics

Ligands can be put in order by how hard they split a d shell, and the order is highly reproducible. It is not the order of charge — three of twenty mixed-charge pairs come out the way a point-charge model predicts, which is worse than tossing a coin — and the two ligands at the strong end are electrically neutral.

wrong · Ligand field Two humps and a dip, which is not what a trend looks like. The measured enthalpy of hydration of the first transition series, in kilojoules per mole, against a straight line fitted through it. The measurements do not fall on the line: they rise and dip at manganese, rise and dip again at zinc. Both dips are at configurations with no ligand field stabilisation — d⁵ high spin and d¹⁰ — and the line alone accounts for only 73 per cent of the variation.

The double hump and what removes it

The hydration enthalpies of the first transition series do not lie on a line — they rise, dip at manganese, rise and dip again at zinc. Subtract the ligand field stabilisation computed from the same model that describes their spectra and what is left is a line, with the one fitted parameter landing inside the range a spectrum measures.

applied · Ligand field 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 splitting against the square of one computed overlap. five chromium(III) complexes: the measured ligand-field splitting against the square of the metal–ligand σ overlap, computed from Slater-type functions at the measured bond lengths. The angular overlap model says the splitting is proportional to that square and to no other power, and the line drawn through the origin is that proportionality with nothing fitted but its slope. Across a series in which the splitting doubles, the ratio varies by 30.75 per cent.

The splitting against something structural

The angular overlap model says a ligand field splitting is proportional to the square of one overlap integral and to no other power. Computing that integral from Slater functions at the measured bond lengths, for five chromium complexes whose splittings run from 13,600 to 26,700 wavenumbers, the ratio varies by thirty-one per cent with nothing fitted. And a power law on the donor's effective charge, at an exponent nobody predicted, does slightly better.

applied · Ligand field What the second channel buys, and what it cannot. Five measured splittings against three models. Adding the π overlap takes the error from 1747 to 1402 cm⁻¹, and telling the model which ligand is an acceptor takes it to 961 — so the fact about occupation is worth more than twice the integral. The two halides are the pair that fixes which model is which, and no single one of the three gets both them and cyanide right.

The integral that cannot count electrons

Adding the π channel to a ligand-field splitting means one more overlap integral over the same two orbitals at the same distance. It removes a fifth of the error. Telling the model which ligand is a π acceptor — one word per ligand, quoted rather than computed — removes forty-five per cent, because an overlap cannot know whether the orbital it reaches is full or empty.

applied · Ligand field 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 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 Every window sits above the value it was meant to reach. For each of the five chromium(III) complexes, the whole range of π/σ ratios the model can produce as the ligand's donor atom is taken through every oxidation state it has — from its bare nucleus to its closed-shell anion — drawn as a bar, with the fitted parameter marked beneath it. The three ligands whose fitted parameter is positive have bars that begin above it and never come down. The other two have fitted parameters of zero and of a negative number, which a quotient of squared overlaps cannot be at any charge.

The correction that moves three of them backwards

Every ligand radial function in the angular overlap sweeps was a neutral atom's, while three of the five donors carry a formal charge. Giving each one the charge it actually has moves three of the five computed ratios — and moves all three away from the fitted parameter, none towards it. The whole window each donor's own oxidation states allow sits above the value it was meant to reach.

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 Four arrangements, and only the first has a symmetry to exchange its pairs. A square-planar centre with both trans pairs folded ten degrees out of the plane, drawn at four ratios of the two pairs' bond lengths. At a ratio of one, exchanging the two pairs maps the arrangement onto itself — so the gap and the ligand–ligand repulsion are both even functions of the antisymmetric fold, and both have a vanishing first derivative along it. At every other ratio that exchange is not a symmetry, and nothing requires either derivative to vanish.

One was a symmetry and one was not

A square-planar complex has a direction the sixteen-electron gap does not move along, and the repulsion falls along it — so the distortion looked free in both senses at once. Both results came from four ligands at the same bond length, and the two behave completely differently when that is relaxed. Two per cent of a bond length gives the gap a first-order slope; the repulsion keeps falling out to a fifty per cent mismatch, and its curvature never changes sign. The blindness was a symmetry; the fall is a property of the arrangement.

applied · Electron count The gap's gradient over four separate lifts has four equal parts. At each fold angle where all four ligands are lifted alike, the rate at which the sixteen-electron gap changes as each ligand alone is lifted a little further, drawn as four bars. They are equal to eleven figures or better: the gradient lies on the one direction in which all four move together, so any distortion whose lifts add to zero leaves the gap unchanged to first order.

Four lifts and one that matters

The direction a sixteen-electron gap cannot see was found in a plane of two folds, where it was the antisymmetric fold of the two trans pairs. That plane hid a choice: the ligands were moved in pairs and they were all alike, and nothing separated which of those the blindness needed. Let each of the four ligands lift by its own angle and the answer is plain. The gap's gradient has four equal parts, so every distortion whose lifts add to zero is blind — three directions, not one — and the pairing never mattered. What sorts the three is the second order, and there the two models agree: the pair fold is rewarded by both and the tilt of a pair is penalised by both.

applied · Electron count Past a fold of 17.90° the pair fold closes the gap at first order and the tilts still do not. The sixteen-electron gap's first-order change along three distortions whose four lifts add to zero, against the fold at which all four ligands are lifted alike. Below 17.90°, where the highest filled level is dz², all three are zero: the blindness. Past it, where the highest filled level is the degenerate dxz, dyz pair, the pair fold closes the gap at −0.0258 per degree whichever way the pairs move, three up and one down at −0.0149, and a tilt still at nothing.

The blindness needed a single top level

Every distortion of a folded sixteen-electron complex whose four ligand lifts add to zero leaves its gap unchanged to first order — while the highest filled level is dz². Past a fold of 17.90° it is the degenerate dxz, dyz pair, and the pair fold, which splits that pair, closes the gap at 0.026 eσ a degree whichever way the pairs move; the tilts, which cannot split it, stay blind. Exactly at the crossing the tilts see too, seventeen times more faintly. And the energy of the filled levels sees none of it at first order: its own turn along the pair fold comes at 13.8°, before the crossing, where nothing in the gap changes.

applied · Electron count The pair fold is the road from a square to a tetrahedron. The flattening angle of the lowest-energy geometry of a sixteen-electron complex, lifting one trans pair of ligands and lowering the other, against the weight given to the filled d levels' energy relative to the ligands' repulsion. With no weight it is the regular tetrahedron at 35.26°; it flattens continuously as the weight rises and reaches the square at 0.0762, beyond which the square is the minimum. The sketches above show the geometry at four weights.

The pair fold was the road to a tetrahedron

Past a fold of 13.8° both of the model's energies fall along the pair fold, and the question left was where a folded sixteen-electron complex goes. It goes nowhere from there, because a folded square is not at rest: both energies rise along the common fold. Applied to the flat square, the same pair fold is the road to a tetrahedron, and on it the minimum of any weighting of the two energies is a flattened tetrahedron that flattens continuously, as a square root, into the square at a weight of 0.0762. The d levels cannot tell the two roads apart.

applied · Electron count

Named alongside it

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

Model limitd orbitalsLigand fieldCoordination complexPi-acceptorPi-donorDegeneracyElectron countSpectrochemical seriesSplittingLigand repulsionOverlap integral

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