The shapes above six coordination

Every first course stops at six. Linear, trigonal planar, tetrahedral, trigonal bipyramidal, octahedral — five shapes with five names, all memorable, all correct.

Seven and above are usually mentioned and not drawn, and the reason becomes obvious the moment the minimisation is run: the arrangements have no memorable names, because they are not shapes anybody would have thought of.

Eight, where the obvious answer loses #

A cube has eight vertices, they are all equivalent, and it is the most symmetric way to put eight points on a sphere. It is not the minimum.

The square antiprism beats it by about one part in three hundred. That is a small margin and it is a real one, and the reason is worth following because it is the same reason five points give a trigonal bipyramid rather than a square pyramid.

In a cube, each point has three neighbours at 70.5 degrees, three at 109.5 and one antipodal. The three close neighbours are the problem — a cube crowds its nearest neighbours to get its far ones spread out, and repulsion falling as $1/r$ weights the close ones heavily. Twisting one face relaxes the crowding at a cost paid by pairs that were already comfortable.

So the cube’s high symmetry works against it. That is a general and slightly counter-intuitive lesson about optimisation on a sphere: the most symmetric arrangement of $n$ points is very often not the least repulsive one, and it is only for the special counts — four, six, twelve — that the Platonic answer wins.

The consequence for chemistry is direct. Eight-coordinate complexes are square antiprismatic or dodecahedral far more often than cubic, and the cubic arrangement is rare enough that finding one calls for an explanation.

What happens to the angle count #

Below six, the arrangements have one or two distinct angles — three for the trigonal bipyramid, which is the whole reason five is interesting.

Above six, the count explodes.

Seven gives four distinct angles. Eight gives four. Nine gives six. Twelve drops back to three, because the icosahedron is exceptionally symmetric.

That column is the quantitative version of “these shapes have no names”. A shape with two distinct angles can be described in a phrase; a shape with six cannot, and the description has to become a list of coordinates or a symmetry label.

It is also the reason the coordination geometries of the lanthanides and actinides are discussed by symbol rather than by name. There is no useful English for a tricapped trigonal prism that a reader could reconstruct the arrangement from.

Seven, and the shape that is a name #

Seven is the one case above six with a familiar answer, and it is worth showing because the familiar answer is right.

Two orbits again, as with five: two axial positions and five equatorial ones, not interchangeable. Iodine heptafluoride is the standard molecular example and its two kinds of fluorine behave exactly as the arrangement implies.

The exactness of that 90 degrees is worth a paragraph, because it nearly was not exact — see below.

What was computed, and how, and what nearly went wrong #

Every arrangement here is a gradient descent on a sum of inverse distances, run from several random starts, with the lowest energy kept.

Several starts rather than one is not optional. These potentials have local minima — the square pyramid is a genuine stationary point for five points, and the cube is very nearly one for eight — and a single unlucky start finds one and stops. Nothing about the resulting picture looks wrong.

That is only half the safeguard, and the other half is what makes these numbers worth printing. The Thomson problem — $n$ point charges on a sphere, minimising Coulomb repulsion — has been solved to many decimal places for small $n$, so there is an independent answer to check against.

Checking against it found a real error, and the error is the kind this site exists to catch.

The descent’s fixed step schedule halves its step ten times and then stops. For two through six that costs nothing: those minima are sharp and the arrangement is pinned by symmetry. Seven has a very flat valley, and the schedule stopped at an energy $2.5\times10^{-5}$ above the published minimum. That is a gap nobody would question. But the arrangement it stopped at had its axial-to-equatorial angles spread from 87.7 to 92.5 degrees rather than all being ninety — a distorted pentagonal bipyramid, reported as a pentagonal bipyramid, with a plausible energy.

The repair is to finish the descent properly: accept a step when it lowers the energy and grow it, reject and halve it when it does not, and stop when the step underflows. No tolerance to choose and no line search. Every arrangement from two to twelve now reaches its published minimum to about $2\times10^{-10}$, and seven’s ten right angles are right angles.

An energy check would have caught a gross failure. What caught this one was checking against somebody else’s answer, and the difference between “converged” and “converged to the right place” is exactly the $2.5\times10^{-5}$ that separated them.

Nine, where naming gives up entirely #

Nine is worth showing because it is where the pattern completes.

The arrangement has two orbits: six vertices forming a trigonal prism and three more capping its rectangular faces. That decomposition is why the angle list is so long — the six-and-three structure gives prism-to-prism, prism-to-cap and cap-to-cap angles, and each of those splits further by whether the pair is on the same triangular face.

Nine-coordinate complexes are common among the lanthanides, and the tricapped trigonal prism is the arrangement most of them adopt. The alternative — a capped square antiprism — is close in energy and is found often enough that both names appear in the literature for the same compound solved by different groups.

The surprise: the sphere is doing the work, not the potential #

Running the same minimisation with a different repulsive potential — $1/r^{6}$, $1/r^{12}$, anything falling off with distance — gives the same arrangements for almost every count.

That robustness is usually cited as a strength of the VSEPR picture and it is better read as a warning. If the answer barely depends on the form of the repulsion, then the agreement between these arrangements and real molecular geometries is not evidence that electron pairs repel in any particular way. The constraint doing the work is the geometry of the sphere: $n$ points confined to a surface and pushed apart arrange themselves the same way whatever is pushing them.

Above six the robustness weakens, which is the more interesting half. The cube-to-antiprism margin is one part in three hundred, and a steeper potential narrows it — with a hard-sphere potential the two are nearly indistinguishable. So the counts where the model is most confident are the ones where it says least, and the counts where it makes a real distinction are the ones where the distinction depends on the assumption.

That inversion is a fair summary of what a points-on-a-sphere model can be trusted for.

What it costs #

Each arrangement costs a few thousand gradient evaluations across several starts, plus the polish. Twelve points is a hundred and thirty-two pairwise terms per step, and the whole table above builds in about a second.

Three things are checked as it does.

The energy must match the published Thomson minimum, to a part in $10^{6}$. That is the check that found the seven-point problem and it is the only one that could have.

The angle spectrum must have the tabulated number of distinct values, so an arrangement that converged to a different shape would be refused rather than drawn.

The minimum must beat the cube, for eight points specifically — computed on both sides rather than assumed, because a margin of one part in three hundred claimed rather than shown would be unconvincing.

The cost that does not appear in any of those is the modelling cost, and it is the large one. These are points, not atoms; the sphere fixes every one at the same radius; and real molecules have bonds of different lengths to different ligands. The radius assumption is the strong one and it is stated nowhere in the usual presentation.

Where the model stops #

Four limits, and the last is specific to high coordination.

Points at one radius. Phosphorus pentafluoride’s axial bonds are longer than its equatorial ones by a tenth of an ångström; an eight-coordinate lanthanide complex has a wider spread still, and a model that fixes every radius cannot produce any of it.

No lone pairs. Above six, lone pairs become stereochemically inactive with startling frequency — the xenon in XeF₆ has one and the molecule is a slightly distorted octahedron rather than the seven-coordinate shape a naive count would give. That the lone pair sometimes takes up space and sometimes does not is a real phenomenon and this model has no way to represent either case.

Ligand size dominates at high coordination. With eight or nine ligands packed round one centre, the ligands touch each other, and what decides the arrangement is steric contact between ligands rather than repulsion between electron pairs. Two entirely different mechanisms produce the same qualitative answer, which is the over-determination problem again.

The energy differences are tiny. The cube and the antiprism differ by a third of a per cent in this model, which is far below any energy scale a real complex is sensitive to. In practice the crystal packing decides, and a compound that is antiprismatic in one salt is dodecahedral in another.

Why the first course stops at six #

Having gone above it, the boundary looks less arbitrary than it did.

Below seven, every arrangement has a name a reader can reconstruct the shape from, at most three distinct angles, and a single unambiguous answer. Those three properties travel together, and all three fail at once.

There is also a chemical reason. Coordination numbers above six are overwhelmingly the province of the larger metals — the lanthanides, the actinides, the heavier transition metals — and those are where a simple electron-pair picture is least appropriate. The d electrons are not equivalent to the s and p regions the rule counts, the ligands touch each other, and the energy differences between candidate arrangements are smaller than the effects the model ignores.

So the syllabus boundary sits roughly where the model’s usefulness does, which is a happier accident than it might have been. What is lost by not going further is the general lesson: that the tidiness of the first six arrangements is a property of small numbers rather than of the method, and a reader who has only seen those six will over-estimate how much the method delivers.

One further consequence deserves recording, because it inverts a habit. Below seven, a reader who remembers the shape can reconstruct every angle. From seven upward the shape has to be computed to be described at all, and a name like “tricapped trigonal prism” is a recipe rather than a description. The transition from a memorised subject to a computed one happens at a specific coordination number, and it is six.

Who found it, and when #

Thomson posed the problem in 1904, and not as a geometry exercise — it was his plum-pudding model of the atom, with electrons distributed through a sphere of positive charge, and the arrangements were meant to explain the periodic table. The model was wrong within a decade of Rutherford, and the mathematical problem outlived it entirely.

It is now one of the standard hard problems of computational geometry. The minima for $n$ up to a hundred or so are known and were established by extensive numerical search rather than proof; only a handful of cases — 2, 3, 4, 6, 12 — have proofs that the known arrangement is optimal.

The chemical application came much later and by a separate route. Gillespie and Nyholm’s VSEPR rules of 1957 are about electron pairs rather than point charges, and the connection to Thomson’s problem was noticed afterwards. That the two lines of work converged on the same arrangements is the historical form of the robustness observed above.

Kepler’s conjecture about sphere packing, settled only in 2005, belongs to the same family of problems and gives some sense of how hard the general question is. The chemistry is easy; the geometry is not.

The quantity that decides which count a molecule gets #

Everything above answers what shape a given number of ligands takes. It says nothing about how many there will be, and it cannot, because a point on a sphere has no size — which is the same missing length scale that keeps coming back.

There is a classical rule that supplies one, and running it against the arrangements computed here shows both what it buys and where it disagrees with them.

Give the ligands a radius $r$ and the central atom a radius $R$, put every ligand in contact with the centre, and ask when the ligands are also in contact with each other. That happens at one value of $R/r$ per arrangement, and below it the ligands would have to overlap. The thresholds are pure geometry:

Each is a lower bound on how large the central atom must be to carry that many neighbours, and the sequence is the whole content of the radius-ratio rule taught in solid-state chemistry.

And the fourth row is the wrong solid. Every statement of the rule quotes 0.732 for eight coordination, and 0.732 belongs to the cube — the arrangement this essay has just shown is not the minimum. Doing the same calculation on the regular square antiprism, whose edges are all equal at a polar angle of 59.26°, gives a threshold of

$$R/r = 0.645,$$

twelve per cent lower. A central atom between 0.645 and 0.732 can carry eight neighbours in the arrangement repulsion actually prefers, and cannot in the one the rule assumes. The rule assigns that whole window to six-coordination.

That is a small correction to a rule whose real failure is much larger — it predicts the observed coordination number for only about half of simple binary compounds, because ionic radii are not transferable and contact is not the criterion that decides anything. But the direction of the correction is worth noticing. The classical rule inherited the cube from the same intuition this essay refuses, and fixing the shape moves its one testable boundary by more than the precision anybody quotes it to.

Where to read on #

The minimisation and its status is VSEPR computed.

The case where two kinds of site first appear is five sites are not alike.

The symmetry treatment of what makes two positions inequivalent is site symmetry.

And the group that any of these arrangements belongs to comes from point groups from coordinates.

There is a last observation the table makes that no single arrangement does. The count of distinct angles is not monotone in the number of points — it rises from one at four to six at nine and falls back to three at twelve — so a reader cannot predict how complicated an arrangement will be from how many things are in it. Complexity here is a property of how well a number of points happens to fit onto a sphere, and twelve fits extraordinarily well while nine does not. That is a fact about the integers rather than about chemistry, and it propagates into chemistry unchanged.

What the pictures here cannot show. Every figure on this page draws points at one radius, and no real high-coordination complex has all its ligands at one distance. The drawings are of the mathematical problem, and the correspondence with molecules — which holds well enough to be useful — is an empirical observation rather than something these figures establish.