Molecular Geometry & Bonding
what the shape decides
Where the atoms go

Five sites are not alike

Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.
10 min read 6 figures Counted, not quotedSymmetry settles it

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 angles each arrangement givesEvery distinct angle subtended at the centre, for each number of sites, measured off the minimised arrangement. Five is the case with three distinct angles, which is the arithmetic signature of sites that are not all equivalent.sitesshapeangles measured2linear180.0°×13trigonal planar120.0°×34tetrahedral109.5°×65trigonal bipyramidal90.0°×6 120.0°×3 180.0°×16octahedral90.0°×12 180.0°×3each row is a separate minimisation
Fig. 1 The distinct angles each arrangement produces. Every row has one or two — except five, which has three. That is not a quirk of the drawing; it is the signature of positions that are not all the same.

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.

Five sites are not five of a kindThe minimised arrangement of five points, with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, so the two kinds of position are genuinely different places — which the shape's name does not convey.axial — 2 sitesneighbours at 90°, 90°, 90°, 180°equatorial — 3 sitesneighbours at 90°, 90°, 120°, 120°the two are not equivalentneighbour angles measured, not assumed5 sites
Fig. 2 The five-point minimum with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, which is what makes them different places rather than five of a kind.

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.

phosphorus pentafluoride — D3hThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.FFFPFFD3hprincipal axis C34 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates6 atoms
Fig. 3 Phosphorus pentafluoride at its equilibrium geometry: two axial fluorines and three equatorial ones, at measurably different bond lengths. The structure is not in doubt; what a room-temperature spectrum shows is a single environment.

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.

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.

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 asserts it — a version of this site 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 an assertion is the point.

What the other counts look like

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

4 sites, minimisedThe arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.tetrahedral109.47° × 6repulsion minimised, angles measured off the result4 sites
Fig. 4 Four sites: one angle, 109.47 degrees, every site equivalent to every other. Any one can be carried onto any other by a rotation of the arrangement.
6 sites, minimisedThe arrangement of 6 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.octahedral90.00° × 12180.00° × 3repulsion minimised, angles measured off the result6 sites
Fig. 5 Six sites: two angles, 90 and 180, and every site still equivalent. The octahedron has enough symmetry that all six positions are alike, which the trigonal bipyramid does not.

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.

Where the model stops

Two 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.

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 eight, nor most numbers above six — 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 angles each arrangement givesEvery distinct angle subtended at the centre, for each number of sites, measured off the minimised arrangement. Five is the case with three distinct angles, which is the arithmetic signature of sites that are not all equivalent.sitesshapeangles measured2linear180.0°×13trigonal planar120.0°×34tetrahedral109.5°×65trigonal bipyramidal90.0°×6 120.0°×3 180.0°×16octahedral90.0°×12 180.0°×3each row is a separate minimisation
Fig. 6 The pattern in one place. Reading down, the arrangements are regular until five and regular again at six, and the row with three angles is the one where geometry runs out of symmetric answers.

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.

Where the ladder goes next

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.

Where this goes next

tetrahedral109.47° × 6repulsion minimised, angles measured off the result4 sites VSEPR, computed The tetrahedral angle is not 109.5 degrees because a textbook says so. It is arccos(−1/3), and it falls out of minimising the repulsion of four points on a sphere without ever being written down. HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms Point groups from coordinates A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not. HHOC2vprincipal axis C22 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms Why water is bent The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.