VSEPR does not reach a transition metal

This site’s shape field is built on a repulsion model, and the model works. VSEPR, computed minimises the mutual repulsion of points on a sphere and recovers the standard arrangements without consulting a table; what a lone pair is worth extends it to unequal domains and gets the bond angles of water and ammonia; five sites are not alike finds the trigonal bipyramid and the two inequivalent positions in it.

It works for the main group. Applied to a transition metal it produces a definite, confident and often wrong answer, and this essay is about where the wrongness comes from.

The failure is specific rather than general, which is what makes it worth computing. The repulsion argument does not become vague for transition metals; it stays exact and gives an answer that is contradicted by half the four-coordinate compounds of the platinum group.

What the minimiser does with four points #

The repulsion model takes the ligands as points on a sphere and minimises the sum of their mutual repulsions. For four points the answer is a tetrahedron, and a numerical minimisation finds it from random starts without being told.

That last sentence is the whole problem. The minimiser has no input describing the central atom. It knows how many domains there are and how much each repels, and a metal with a partly filled d shell is a different sort of centre in a way the model has no representation of.

For the main group that omission is harmless, because a filled or empty inner shell is spherical and contributes nothing to the angular preference. For a transition metal it is not, because a partly filled d shell is not spherical, and its asymmetry has a preferred orientation relative to the ligands.

The term VSEPR does not have #

The ligand field stabilisation is that missing term, and it is computable for every filling.

Take four ligands, put them in a square plane and in a tetrahedron, compute the d levels in each, fill them in whichever spin state is lower, and take the difference. The result is the bar chart above, and three features of it are worth naming.

It is exactly zero at d⁰ and at d¹⁰. With no d electrons or a full shell there is nothing to stabilise: the levels sum to the barycentre and a full shell occupies all of them equally. The computed values are zero to arithmetic precision, not merely small.

It is largest at d⁸, at 1.733 in units of $e_\sigma$. The next largest are d⁴ and d⁹ at 96 per cent of that, and those two are the Jahn–Teller configurations that get most of the same stabilisation from a small distortion instead, as copper is never quite octahedral computes.

It is a competition, not a veto. The square plane has to overcome the repulsion penalty, so a small ligand field advantage is not enough. What decides is whether the electronic gain exceeds the repulsion cost, and the repulsion cost is roughly the same for every metal.

So the prediction is not that d⁸ complexes are square planar. It is that d⁸ complexes are square planar when the ligand field is strong enough, and that d⁰ and d¹⁰ complexes are tetrahedral always — because for those two the term that could overrule repulsion is exactly nothing.

Why a square plane costs more repulsion #

The claim that the square plane is the worse arrangement for repulsion deserves a number rather than a bare claim, because a reader might reasonably suspect that four ligands in a plane are somehow more spread out.

A tetrahedron has six ligand–ligand angles, all 109.47°. A square plane has four at 90° and two at 180°. Repulsion falls with distance, so what matters is the close pairs: the tetrahedron’s closest approach is 109.47° and the square plane’s is 90°, which puts four pairs of ligands substantially nearer each other than any pair in the tetrahedron.

Summing an inverse-distance repulsion over the pairs makes the tetrahedron the lower-energy arrangement by a clear margin, and a minimiser confirms it operationally: started from random points, it never finds the square plane, because the square plane is not a minimum at all. It is a saddle — flattening a tetrahedron is downhill in one direction and uphill in another, and the minimiser rolls off it.

That is the term the repulsion argument does not have, drawn as a coordinate rather than as two end points. A minimiser on ligand–ligand repulsion alone rolls monotonically down this path from left to right; the electronic energy does not, and where the sum has its minimum depends on a d count that the repulsion model has no place to put.

So a square-planar complex is a structure sitting at a point that pure repulsion actively rejects. That is a strong statement about how large the electronic term must be, and it is why the effect appears in a minority of four-coordinate complexes rather than in most of them.

The two nickel compounds #

The cleanest test uses one metal in one oxidation state with one coordination number and two different ligands.

The tetrachloridonickelate ion is tetrahedral. It is paramagnetic, with two unpaired electrons and a magnetic moment near 3.2 Bohr magnetons, and it is blue.

The tetracyanidonickelate ion is square planar. It is diamagnetic — no unpaired electrons at all — and it is yellow.

Both are nickel(II), both are d⁸, both have four ligands. What differs is the ligand: chloride is a weak-field π donor near the bottom of the spectrochemical series, and cyanide is a strong-field π acceptor near the top. The stronger field makes the square plane’s gap large enough to pay both the repulsion penalty and the cost of pairing the two electrons that were unpaired in the tetrahedron.

Both structures are in this collection with their groups found from their coordinates rather than assumed: the tetrachloridonickelate ion is Td, four chlorides in the arrangement repulsion would have chosen anyway, and its platinum analogue is D4h, with two of the six ligand–metal–ligand angles at 180° and four at 90° — a worse arrangement by every repulsion measure, and the observed one.

There is a version of the level pattern that has no orbitals in it at all, and it is worth drawing because it is the model VSEPR’s repulsion argument is a cousin of.

The magnetic measurement is what makes this decisive rather than merely descriptive. A shape argument can be argued about; a magnetic moment is a number, and the two compounds differ by the whole of it. The geometry and the spin state are the same fact, because a square plane’s fourth level is far enough above the other three to force pairing and a tetrahedron’s is not.

It is worth noticing that a two-set field cannot produce that fact at all. In an octahedral-style pattern of three below two, d⁸ has no spin choice: both fillings give the same two unpaired electrons, and the pairing energy decides the moment shows the four fillings that do have a choice, of which eight is not one. The square plane changes that by producing four levels rather than two, with one of them far above the rest — a shape no two-set model can represent, which is why the spin state and the geometry are one fact here and not two.

Six sites minimised give an octahedron, and that is the case where the repulsion model and the ligand field agree — six ligands want an octahedron on almost any account. So the disagreement this essay is about is confined to four coordination, which is where two candidate arrangements are close enough on repulsion for a d-electron term to decide between them.

Where the repulsion argument keeps working #

It would be wrong to leave the impression that VSEPR simply fails for transition metals, because the computation says something more useful than that.

Six-coordinate complexes are octahedral, and the repulsion model gets them right for the same reason it gets them right for main-group SF₆: six points on a sphere want an octahedron on almost any account, and the ligand field for the common fillings does not prefer anything else strongly enough to overrule it. The exception is the Jahn–Teller distortion, which is a small deviation from the octahedron rather than a different arrangement.

d⁰ and d¹⁰ metals behave exactly as the main group does. Titanium(IV) chloride is tetrahedral; permanganate and chromate are tetrahedral; zinc(II) and copper(I) complexes are tetrahedral. All are d⁰ or d¹⁰, all have a spherical d shell, and all have a ligand field stabilisation of exactly zero — so the model with nothing in it about the centre is the right model for them.

That is a satisfying kind of failure. The rule works precisely where its missing term vanishes, and the missing term is computable, so the rule’s domain of validity is an output rather than a caveat.

Five coordination is the case that shows the same competition without the same clean answer. Repulsion on five points produces an arrangement with two kinds of position — five sites are not alike minimises it and finds the trigonal bipyramid — and five-coordinate transition-metal complexes run the same argument between that and the square pyramid, with the ligand field again supplying the term repulsion has not got and the two arrangements often within a few kilojoules of each other.

The angles, and which of them are symmetry #

There is a second reason to be careful with the repulsion argument here, and it is one that already applies to the main group.

Which angles are symmetry and which are the model separates two kinds of number in a VSEPR prediction. The 109.47° of a tetrahedron is fixed by symmetry: given four equivalent ligands and a tetrahedral arrangement, no model has any freedom about the angle. The 104.5° of water is a model output, sensitive to how the repulsions are weighted.

A square-planar complex’s 90° is a symmetry number in the same sense. What is not fixed by symmetry is which arrangement occurs, and that is what the ligand field decides. So the disagreement between VSEPR and observation here is never about an angle — it is about which of two symmetric arrangements is chosen, which is a question about energies.

Above six, where both arguments get harder #

The repulsion model’s own limits are worth recalling here, because they are not the same limits.

Above six coordination found that the minimising arrangements stop being the ones a reader would guess: eight points on a sphere give a square antiprism rather than a cube, and nine give a tricapped trigonal prism with three distinct angles. Those are results about repulsion alone and they are correct as far as they go.

A high-coordinate transition-metal complex has both problems at once. The repulsion answer is unfamiliar, and the ligand field term is added on top of it — and for eight coordination the two candidate geometries, the antiprism and the dodecahedron, differ by very little on either count. The observed structures accordingly depend on the ligand, on the counter-ion and on the crystal packing, which is a way of saying that nothing dominates.

That is the honest end of this line of argument. Where one term is much larger than the others a rule works; where two are comparable, a rule of thumb is replaced by a calculation; and where three are comparable, the answer stops being a property of the molecule at all.

Why the failure is worse for the heavy metals #

The bar chart is drawn against $d^n$ and says nothing about which row of the periodic table the metal is in. The observation it is being compared with does: nickel gives four-coordinate complexes of both shapes, and palladium and platinum give square-planar ones almost without exception. All three are d⁸.

The two competing terms respond very differently to going down a group, and that is the whole of it.

The repulsion term barely moves. It is a statement about four points around a centre, and four points do not know which element they are around. Changing the metal changes the bond length, which scales the whole comparison, and changes nothing about which arrangement wins.

The ligand field term grows steeply. For the same ligands and the same charge, the splitting rises about half again from the first transition series to the second, and about a fifth again from the second to the third — the hexaammines of cobalt, rhodium and iridium are the standard series, at roughly 22,900, 34,100 and 41,000 cm⁻¹. That is a factor of 1.79 from top to bottom, from larger and more diffuse d orbitals overlapping the ligands better.

So the two terms are not merely of comparable size — their ratio changes by nearly a factor of two down one column, and it changes in the direction that makes the repulsion argument worse.

That converts the essay’s finding into a prediction with a boundary in it. At the first row the competition is real, which is why nickel supplies one compound of each shape and why a magnetic measurement is needed to say which is which. At the second and third rows it is not a competition, because the term the repulsion model lacks has grown by three quarters while the term it has has stayed still. A tetrahedral d⁸ complex of platinum is not merely unusual; on this arithmetic it should not exist at all with ordinary ligands, and it does not.

It also says where the repulsion rule recovers, and it is not only at d⁰ and d¹⁰. The rule is at its worst exactly where the ligand field is strongest, and the strength runs with the row as well as with the filling — so a rule of thumb that is unreliable for nickel is useless for platinum, and the two failures have the same cause measured on two different axes.

What this says about rules of thumb #

Three observations, all of which apply beyond this case.

A rule with no input for a variable cannot depend on it. VSEPR takes a count of domains and their relative sizes, and produces an arrangement. Nothing in its inputs distinguishes nickel from carbon, so nothing in its outputs can. That is a structural fact about the model rather than a defect to be patched, and it says exactly where to look for failures: at centres whose inner shells are not spherical.

The failure is largest where the missing term is largest. The bar chart is not a list of exceptions; it is a computed measure of how wrong the repulsion answer can be for each filling, and it is zero exactly where the rule is safe. That is the same service Bent’s rule, computed performs for hybridisation and above six coordination for the arrangements themselves. A rule whose failures can be predicted is more useful than one whose failures have to be memorised.

A wrong prediction from a correct model is a scope error, not an arithmetic one. Nothing in the repulsion minimisation is miscomputed for a nickel complex: four points still settle into a tetrahedron, the angle is still 109.47°, and the energy reported is still the energy of that arrangement under the stated repulsion law. What is wrong is the assumption that the quantity being minimised is the whole of what the molecule minimises. That distinction matters because the two failures need different repairs — an arithmetic error is fixed inside the model, and a scope error is fixed by adding a term and saying when it matters.

The right answer needs both terms. Neither model alone gets the four-coordinate complexes right. The repulsion term is real — square-planar complexes do pay a penalty for it, which is why weak-field d⁸ complexes are tetrahedral — and the ligand field term is real, which is why strong-field ones are not. The shape is decided by their sum, and the interesting compounds are the ones where the two are comparable.

What a shape rule is for #

It is worth asking what the repulsion model was ever supposed to do, because the answer changes what counts as a failure.

VSEPR is a prediction rule: given a formula and a count of lone pairs, it names an arrangement, and it does so in seconds and without a computer. That is an enormous amount of value from a very small model, and it is why the rule is taught first and remembered longest. A numerical minimisation is the better version of it precisely because the rule’s content is an energy comparison that can be performed rather than recited — the position VSEPR, computed takes.

Ligand field theory is not a prediction rule of the same kind. It needs a geometry before it can compute anything: the levels are computed for an arrangement, so the comparison between two arrangements requires both to be proposed first. What it supplies is the term that decides between candidates, not the candidates.

So the two are complementary in a way that is easy to miss when one is presented as a correction to the other. The repulsion model generates the shortlist and the ligand field chooses from it, and a four-coordinate complex is exactly the case where the shortlist has two entries and the choice matters. The failure of VSEPR here is not that it computed something wrongly; it is that it was asked to do the second job with the tool for the first.

Who found it, and when #

Nevil Sidgwick and Herbert Powell proposed the repulsion argument in 1940 and Ronald Gillespie and Ronald Nyholm turned it into the modern rule in 1957 — and the paper that did so was explicit that it applied to non-transition elements, with the d-shell case set aside as unsolved. The restriction was in the rule from the start and did not survive the journey into teaching.

The ligand field account of square-planar d⁸ came from the same decade, out of the crystal field arithmetic being applied to spectra and magnetism. So the two halves of the story are contemporaries, and the reason they are not usually taught together is that they belong to different courses: shape belongs to the introductory one and ligand fields to the inorganic one, and the compound that needs both is met in the first.

That is a recurring shape in this site’s wrong field. The mistaken version is rarely a mistake anybody made; it is a correct rule with its scope removed, taught to people who will not meet the scope for another two years.