Molecular Geometry & Bonding
what the shape decides
Concept

Splitting — where it appears

The energy separation between the sets a field divides a degenerate shell into. Its pattern follows from symmetry and its size from a model, and it is measured from the position of the first absorption band.

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

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.

applied · Ligand field 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 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.

applied · Ligand field d⁹: what a tetragonal distortion is worth. The electronic energy of d⁹, the elastic cost of the distortion, and their sum, against the fractional elongation of the axial bonds. The best distortion is at 0.14 and it is worth 0.54 in units of eσ; d⁶ in the same field gains 0, which is nothing.

Copper is never quite octahedral

A d⁹ ion in an octahedral field has three electrons in a doubly degenerate pair, which cannot be shared evenly. The energy it gains by distorting is linear in the distortion and the elastic cost is quadratic, so no stiffness holds the symmetric structure — and the four configurations with an even occupation gain exactly nothing.

applied · Peierls distortion The spin-only count against nine measured moments. Each ion's magnetic moment computed from the number of unpaired electrons alone, √(n(n+2)) Bohr magnetons, beside the measured value. The two agree to a hundredth for the first five and the measurement exceeds the count by up to 0.93 for Co²⁺ — always in the same direction, which is what an omission looks like rather than noise.

A moment counts electrons, not orbitals

A magnetic moment is one of the few chemical measurements that returns an integer. Feed the count of unpaired electrons into √(n(n+2)) and nine first-row ions come back within a hundredth for five of them — and the five that miss all miss the same way, which is what a missing term looks like.

applied · Magnetism d⁶: two states, and where they cross. The energy of the high-spin and low-spin fillings of d⁶ against the splitting, in units of the pairing energy. They cross at Δ = P exactly, with 4 unpaired electrons below it and 0 above.

The pairing energy decides the moment

Whether the sixth d electron pairs up in the lower set or goes alone into the upper one is a competition between the splitting and the cost of pairing. Run the filling rules over the whole shell and exactly four configurations have a choice — and every one of them changes state at Δ = P exactly.

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 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 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 Which way the line moves counts the electrons. Three ions, their computed g-values and their measured ones. The shift is −2λ times a sum of squared matrix elements over energy denominators; the matrix elements are exactly two for Lz between dx²−y² and dxy and exactly one for Lx between dx²−y² and dyz, which is why the shift along the axis is four times the shift across it. The last column is what is left over — the orbital reduction factor, which is below one when the electron spends part of its time on the ligands and is a covalency measured with a magnet.

The g-value is the orbital coming back

A ligand field quenches the orbital angular momentum of a d electron, and spin-orbit coupling gives some of it back — upward for a shell more than half full and downward for one less than half full. Which way a resonance line moves counts the electrons, and the size of the move comes from two matrix elements and one optical splitting.

applied · Magnetism 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 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

Named alongside it

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

Ligand fieldd orbitalsDegeneracyAngular overlapCoordination complexSpectrochemical seriesElectron correlationMagnetic momentBarycentreCrystal fieldD d transitionModel limit

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