Five sites are not alike

Minimising repulsion on a sphere reproduces the VSEPR shapes. Minimise the repulsion of two points and they go to opposite poles. Three form a triangle, four a tetrahedron, six an octahedron. In every one of those the sites are equivalent: any one can be carried onto any other by a symmetry of the arrangement.

Five is different, and the difference shows up as a number.

The arrangement #

Five points minimising mutual repulsion form a trigonal bipyramid: two on an axis, opposite each other, and three round the equator at 120 degrees.

An axial site has three neighbours at 90 degrees and one at 180. An equatorial site has two neighbours at 90 and two at 120. Those are different environments, and no rotation or reflection of the arrangement carries one kind onto the other.

Why there is no five-point Platonic solid #

The natural question is why five does not give something regular, and the answer is that regular solids with five vertices do not exist.

The Platonic solids have 4, 6, 8, 12 or 20 vertices. There is no arrangement of five points on a sphere in which every point has the same relationship to every other — the combinatorics simply does not permit it, and it is the same kind of finiteness result as the crystallographic restriction forbidding five-fold periodicity, though the mechanism is different.

So the minimum has to be an arrangement with two kinds of site, and the trigonal bipyramid is it. The next-best candidate, a square pyramid, is very slightly higher in energy — close enough that it matters, as it turns out.

What this does to a real molecule #

Phosphorus pentafluoride has five fluorines round a phosphorus, in exactly this arrangement. So there ought to be two kinds of fluorine — two axial and three equatorial — and a spectroscopic method that distinguishes chemical environments ought to see two signals in the ratio 2:3.

It sees one.

The structure is not wrong. The axial bonds really are longer than the equatorial ones — about 1.58 ångström against 1.53 — and diffraction sees the difference plainly. What resolves the discrepancy is that the fluorines exchange places faster than the spectrometer can distinguish them.

The angle spectrum as evidence #

The claim that the sites differ can be made without looking at the picture at all, which is worth doing because a picture of a trigonal bipyramid does not obviously show two kinds of position.

Take every pair of sites and measure the angle they subtend at the centre. Group the values. For four sites there is one group; for six there are two; for five there are three — 90 degrees appearing six times, 120 three times, and 180 once.

Two of those counts are worth naming carefully, because the claim is easy to overstate and often is. Four gives one distinct angle and six gives two; five gives three, which is more than either of its neighbours. It is not true that five gives more than every other count — seven gives four and nine gives six — and a check written the stronger way fails the moment the higher coordination numbers are included. What holds is the comparison against four and six, which is the comparison the argument needs.

Now ask what environment each site has. An axial site’s neighbours are at 90, 90, 90 and 180. An equatorial site’s are at 90, 90, 120 and 120. Those are different lists, and no rotation or reflection of the arrangement can turn one into the other, because a symmetry operation preserves angles.

That is the whole argument, and it is arithmetic rather than visual. The same style of reasoning recovers a molecule’s point group from its coordinates, and here it distinguishes two sets of positions within one.

The group agrees, and says more #

The angle spectrum shows that the two kinds of site cannot be related by any operation. The molecule’s group says the same thing in a form that then goes further.

Phosphorus pentafluoride is D₃ₕ. Its twelve operations, generated from the coordinates and closed under multiplication, fall into six classes — and the useful question is what the five fluorine positions do under them. They do not form one orbit. The three equatorial fluorines are carried among themselves by the threefold rotation and never onto an axial one; the two axial fluorines are exchanged by the horizontal mirror and by the perpendicular twofold axes. Two orbits, of sizes three and two, and no operation crosses between them.

The D₃ₕ table has twelve operations in six classes and six irreducible representations to match them, all generated from the molecule’s own coordinates. What it says about the two kinds of fluorine is that no operation carries one kind onto the other — which is the group-theoretic form of the statement that the sites are not alike.

The distinction between the two site types is therefore not a matter of degree that a better model might wash out. It is a partition of the ligands into orbits, and orbits are exact.

That has a consequence the angle argument cannot reach. Because the sites fall into two orbits, the five fluorine orbitals span a representation that decomposes accordingly, and the axial and equatorial bonding must involve different symmetry species at different energies — which is why the two bond lengths differ in every measurement of the molecule, and why the difference is not a small perturbation on a common value.

The same group applied to the molecule’s motions gives eighteen vibrations for six atoms, sorted into species and marked with what each spectrum can see. Several are infrared-forbidden and Raman-active, which is why a full assignment needs both techniques — and none of that count changes when the two kinds of site exchange.

Berry pseudorotation #

The exchange has a mechanism and a name, proposed by R. Stephen Berry in 1960.

Take the trigonal bipyramid. Bend the two axial bonds toward one of the equatorial ones, and simultaneously open the other two equatorial bonds apart. Pass through a square pyramid — the near-degenerate alternative — and keep going, and the molecule arrives back at a trigonal bipyramid with two of the former equatorial sites now axial and the two former axial sites now equatorial.

Nothing has been broken. No bond was cut, no atom left, and the molecule has permuted its own substituents by a purely vibrational motion. The barrier is a few kilojoules per mole, so at room temperature it happens perhaps a billion times a second.

That is why the spectrum shows one environment: it is an average over an exchange that is fast on the timescale of the measurement. Cool the sample enough and the two environments separate, which is exactly what is observed.

Why the square pyramid matters #

The mechanism only works because the alternative arrangement is nearly the same energy.

The five-point minimisation finds the trigonal bipyramid, and if the square pyramid were far above it there would be no accessible pathway. The two are close because both are reasonable answers to a question with no perfect one — which is the same fact as there being no regular five-vertex solid, seen from a different side.

So the fluxionality of five-coordinate molecules is a direct consequence of the geometry. Four-coordinate and six-coordinate species do not do this, because their minima are well separated from any alternative.

The general pattern #

Once noticed, the inequivalence explains a family of facts.

Substituents choose sites. In a molecule like PF₃Cl₂, the chlorines do not distribute at random: more electronegative substituents prefer axial positions and bulkier ones prefer equatorial, and the preference is strong enough to be predictive. That is only a meaningful statement because the sites differ.

Bond lengths differ. Axial bonds in a trigonal bipyramid are consistently longer, in every molecule of the type.

Reactions go through it. The same inequivalence explains why electronegative substituents prefer the axial sites. Nucleophilic substitution at phosphorus proceeds through a five-coordinate intermediate, and which sites the incoming and leaving groups occupy determines whether the configuration is retained or inverted. Pseudorotation scrambling the sites is therefore a mechanism for scrambling stereochemistry, and it is a real and studied effect.

What the angle count is doing #

It is worth being explicit about the check this site makes, because it is a small one that carries the essay.

The minimisation is run, the angles between every pair of sites are measured and grouped, and the number of distinct values is counted. For every arrangement in ordinary use that count is one or two. For five it is three, and the figure checks it — a minimisation that produced two would be describing something else.

The neighbour angles of an axial and an equatorial site are then compared and required to differ. That is what “inequivalent” means operationally, and stating it as a computation rather than a claim is the point.

What the other counts look like #

Setting the five-site case against its neighbours makes the anomaly plain.

Four gives one distinct angle. Six gives two — 90 for neighbours and 180 for the opposite site — and every site still has the same environment. Five gives three, and two of its sites have a different environment from the other three.

The difference is a symmetry statement. The point group of a trigonal bipyramid is D₃ₕ, and within it the axial and equatorial positions belong to different sets of equivalent sites. In a tetrahedron or an octahedron the corresponding sets are single: everything is one orbit under the group.

So “five sites are not alike” is not a fact about repulsion at all. It is a fact about which finite groups act transitively on how many points, and the minimisation merely finds the arrangement that fact permits.

What it costs #

Distinguishing the two kinds of site costs one sorted list.

Every pair of the five points subtends an angle at the centre; there are ten such pairs; grouping the values to a tolerance gives the spectrum. That is ten arccosines and a sort, and it is the whole apparatus behind this essay’s central claim. It works for any arrangement, needs no picture, and produces a number — three — that four and six do not produce.

The tolerance is the one judgement in it. Two angles are the same angle when they agree to within a small threshold, and a minimisation that has converged to a few parts in $10^{-9}$ makes that threshold uncontroversial. It would not be uncontroversial for measured coordinates, where the axial and equatorial angles in a real substituted phosphorane are perturbed away from 90 and 120 by the substituents, and a spectrum computed too finely would report five distinct angles rather than three.

The group-theoretical route costs more — generating twelve operations, closing them under multiplication, and sorting them into conjugacy classes by conjugating each by every other — and buys a stronger statement. The angle spectrum shows that no operation of the ones anybody thought to check relates the two site types. The orbit decomposition shows that no operation the molecule has does, because the operations were enumerated exhaustively rather than sampled.

That distinction is small on this molecule and is not small in general. A search that tries a list of plausible axes can miss one; a set closed under multiplication cannot, because a group is not a list of the elements somebody thought to look for.

What five costs that four does not #

The minimisation for four points is untroubled: any start converges to the same tetrahedron, and the answer is the unique minimum.

Five is different, and the difference is the reason this essay exists rather than a numerical inconvenience. The square pyramid is a genuine stationary point of the same potential — the arrangement is symmetric enough that the gradient vanishes there — so a single unlucky start finds it, stops, and reports a shape with two distinct angles instead of three. Nothing about the resulting picture looks wrong.

So the minimisation is run from several random starts and the lowest is kept, and the count of distinct angles is then checked rather than reported. Five must give three; every other case here must give at most two. That single check is what turns “the trigonal bipyramid wins” from an observation about one run into a claim that could fail.

The energy gap between the two arrangements is small, which is the same fact that makes pseudorotation easy — and it is why the minimiser can be fooled at all. A potential with a deep, isolated minimum is easy to find and tells nobody anything; a potential with two nearly degenerate minima is hard to search and is describing a molecule that genuinely moves between them.

Where the model stops #

Three limits.

These are points, not atoms, and the minimisation is geometry rather than a molecular calculation. The minimisation places charges on a sphere. Real bond lengths differ between axial and equatorial positions, which the sphere model cannot produce because it fixes every radius at one.

The barrier is not computed here. That pseudorotation is easy is a statement about an energy surface, and nothing on this page evaluates one. The geometric fact — that two arrangements are close — is suggestive and is not a calculation of the barrier.

The two orbits are exact; their consequences are not. Symmetry says the axial and equatorial fluorines cannot be related by any operation, which is a statement admitting no exceptions. It does not say the axial bonds are the longer ones, and could not — that they are longer by about a tenth of an ångström in phosphorus pentafluoride is a measurement, and an argument that appeared to derive it from the group would be smuggling in a model. What symmetry supplies is the permission for them to differ. Everything about which way and by how much comes from somewhere else.

Six is the last easy one #

A closing observation about where the pattern goes.

Two, three, four and six sites all give arrangements in which every position is equivalent. Five does not. Seven does not either, nor most numbers above six — eight is the exception treated below — and the arrangements become progressively less regular, and finding the true minimum becomes a genuine computational problem rather than a matter of recognising a solid.

That is why coordination numbers above six are comparatively rare and comparatively awkward in chemistry. There is no canonical shape to reach for, several arrangements are close in energy, and the resulting species tend to be fluxional for the same reason phosphorus pentafluoride is.

The Thomson problem itself is still open in general. Minimum-energy configurations are proven only for a handful of small n, and for larger numbers the best known arrangements come from computation with no proof that they are optimal. That a chemistry rule taught to first-year students sits on top of an unsolved problem in discrete geometry is worth knowing.

Which counts have equivalent sites, and one correction #

The section above says that two, three, four and six give arrangements with every position equivalent, and that five, seven and eight do not. The first half is right and the second overstates the case by one, in a way worth fixing because the exception shows what “equivalent” is actually asking for.

The question is whether the arrangement’s own symmetry group can carry any site onto any other — whether the sites form a single orbit. For two, three, four and six the group is that of a regular polygon or a Platonic solid and the answer is immediate.

Eight is the interesting one. Its minimum is not a cube; it is a square antiprism, two squares rotated forty-five degrees relative to each other, and that is the arrangement the essay on high coordination numbers finds. Its group contains an eightfold improper rotation — turn by forty-five degrees and reflect through the equator — and that operation carries a top-square vertex onto a bottom-square one. So all eight sites are one orbit and all eight are equivalent, even though no proper rotation relates the two squares and the solid is not Platonic.

Twelve is equivalent too, on the icosahedron. Five, seven, nine, ten and eleven are not.

So the honest sequence of counts with equivalent sites is 2, 3, 4, 6, 8, 12 — and what five lacks is not regularity in some loose sense but membership of that list. The correction sharpens the argument rather than weakening it: eight sites are all alike and still awkward, because being one orbit is not the same as having one angle, and eight has many.

Where to read on #

The minimisation itself is VSEPR computed, where the tetrahedral angle falls out without being written down.

The symmetry consequences follow from the point group — a trigonal bipyramid is D₃ₕ, and the two site types are two different sets of equivalent positions within it.

What the pictures here cannot show. The figures are static and the phenomenon is dynamic. A molecule undergoing pseudorotation has no single structure on the timescale of a slow measurement, and no still drawing conveys that an arrangement is being visited rather than occupied.