Molecular Geometry & Bonding
what the shape decides
Concept

Coordination complex — where it appears

A metal ion surrounded by bound molecules or anions. Their arrangement fixes the energies of the metal's d orbitals, and the pattern of that splitting follows from the point group before any strength is quoted.

Named by 16 essays across 2 fields — 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 One metal orbital, two ligands competing for it. Metal–ligand bond orders in a three-orbital model as the left-hand ligand's interaction is turned up. Its own bond order rises and the bond order to the ligand opposite falls, from 0.62 at equal strengths to 0.42 at the strongest. Nothing else in the model can carry the effect: switch the second bond off and it vanishes exactly.

The trans influence is an overlap argument

Two ligands on opposite sides of a metal both bond through the same metal orbital, and there is only one of it. Strengthen one and the bond order to the other falls — computed exactly on three levels, and measured as a bond length that grows by a tenth of an ångström.

applied · Overlap A coupling that is second order in the hopping. The singlet–triplet splitting of a two-site Hubbard model, and the same quantity multiplied by U. The product settles on −4t² — -4 at U = 64 — which is what makes the coupling a second-order effect rather than a term somebody put in.

What couples two spins

Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.

applied · Magnetism 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 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 Why d⁸ is square planar and nothing else is. The ligand field energy a square plane buys over a tetrahedron, for every d^n, each geometry in its own ground state. The maximum is at d⁸ (1.73 in units of eσ), it is exactly zero for an empty shell and a full one, and d⁸ is one of only two fillings where the plane is diamagnetic and the tetrahedron is not.

VSEPR does not reach a transition metal

Four ligands minimising their repulsion give a tetrahedron, whatever the metal. Half the four-coordinate complexes of the platinum group are square planar, which has larger repulsion, and the term that overrules it is largest at d⁸ and exactly zero at d⁰ and d¹⁰ — which is where the repulsion rule works again.

wrong · VSEPR 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 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.

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 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 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 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.

Electron countd orbitalsLigand fieldModel limitAngular overlapEighteen-electron ruleDegeneracyIrreducible representationsSplittingClosed-shell configurationsLigand repulsionPi-acceptor

All concepts