Ligand field — the series · Molecular Geometry & Bonding

A d shell in an octahedral field. The five d energies in octahedral coordination, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

The splitting is a symmetry statement

Put six ligands round a metal and the five d orbitals stop being degenerate. Which of them stay together, and how many sets there are, follows from the point group alone — before any account of what the ligands are made of, and before any number is computed.

part 1 · applied

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.

part 2 · applied

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.

part 3 · wrong

Where the d–d band falls, and where the eye is. Nine ligands' measured splittings as wavelengths, on a logarithmic scale, with the visible range shaded. Three fall inside it; the halides sit in the near infrared and carbon monoxide in the ultraviolet. The splitting decides where the band is and does not decide what is seen.

Where a d–d band falls

A splitting is an energy and an energy is a wavelength, so the ligand field fixes where a complex absorbs. Only three of nine common ligands put that band inside the visible range at all — and the most intensely coloured transition-metal compound in the cupboard has no d electrons to excite.

part 4 · applied

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.

part 5 · applied

The electrons repel each other less inside the complex. Seven chromium(III) complexes. Δ is the first band; B is solved from the second in closed form; β is B against the free ion's 918 cm⁻¹, which is measured on the gaseous ion. Every β is below one — the electrons in a complex repel each other less than the same electrons in the free ion, because they have more room. The two orderings are different: fluoride splits least and reduces the repulsion least, cyanide does both most, and the middle of the two series is not the same middle.

The electrons repel less inside the complex

Two measured bands determine two parameters in closed form, and one of them is the repulsion between the d electrons — which comes out below the free ion's value for every complex, by between two and forty-seven per cent. The ligands that split most are not the ligands that reduce the repulsion most, so a complex is characterised by two numbers rather than one.

part 6 · applied

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.

part 7 · applied

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.

part 8 · applied

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.

part 9 · applied

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.

part 10 · applied

The two overlaps, squared, at the measured bond lengths. For each chromium(III) donor: the σ overlap squared, a metal 3d(z²) against the donor's p(z), and the π overlap squared, a 3d(xz) against its p(x). Everything is computed — the radial functions from Slater's rules, the separation from the measured bond length, the integral by quadrature. Chloride's π overlap is 3.5 times fluoride's, which is the opposite of what the overlap argument required of it.

The overlap the model is not proportional to

Without a computed π overlap, the natural argument reasons about one instead: the denominator over-predicts the halide trend, so the overlap must shrink down the group to cancel part of it. Computed, it grows — 3.5 times from fluoride to chloride. And the fitted parameter changes sign across the series, which no ratio of squared overlaps can do.

part 11 · applied

Two charges, and three ligands no charge reaches. The metal effective charge each ligand would need on its own for the model's ratio to equal the fitted one. The band is the range Slater's rules allow chromium. Two ligands have an answer, both far outside it and 2.30 apart from each other. Chloride needs more than the overlap rule can be trusted to compute. Ammonia's fitted parameter is exactly zero and cyanide's is negative, and a quotient of squares is neither.

A contraction that cannot reach three of them

The angular overlap model's own derivation gives a π/σ ratio that disagrees with the fitted parameters by up to sixfold, and the metal's contraction is the obvious candidate to account for it. The whole range Slater's rules allow moves the ratio by a factor of two. Two ligands need charges far outside it, one needs a charge past where the overlap rule can be trusted at all, and two are unreachable at any charge because a quotient of squared overlaps cannot be zero or negative.

part 12 · applied

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.

part 13 · applied

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.

part 14 · applied