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

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.
9 min read 6 figures Counted, not quotedTaught confidently, and wrong

Where the atoms go is decidable from a minimisation. Four things repelling each other on a sphere arrange themselves at the corners of a tetrahedron, subtending 109.47 degrees at the centre. That number is usually presented as a fact to be learnt. It is a result, and running the minimisation produces it without anybody having to know it in advance.

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. 1 Four points on a sphere, placed by minimising their mutual repulsion. The angle printed was measured off the result, and it agrees with arccos(−1/3) to three decimal places. Nothing in the code that produced this figure contains the number 109.

The method

VSEPR — valence shell electron pair repulsion — says that the regions of electron density round a central atom get as far from one another as they can, and that the resulting arrangement is the molecule’s shape.

Taken literally that is a minimisation: place nn points on a sphere so as to minimise the sum of 1/rij1/r_{ij} over all pairs. That is a well-studied problem in its own right, the Thomson problem, posed in 1904 for a quite different reason — Thomson was modelling the atom as electrons embedded in a positive sphere, which is the plum-pudding model.

The figures here run it: steepest descent from many random starting configurations, points projected back onto the sphere after every step, with the best of the restarts kept. The starts are seeded so the answer is the same on every build.

The answers

For the small numbers chemistry cares about, the minimisation reproduces the VSEPR table exactly.

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. 2 Every distinct angle each arrangement produces, measured off the minimised configuration rather than quoted. The rows are the familiar shapes and none of them was assumed.

Two points go to opposite poles at 180°. Three form an equilateral triangle at 120°. Four form a tetrahedron at 109.47°. Six form an octahedron with angles of 90° and 180°.

Five is the interesting one and it gets its own essay: the answer is a trigonal bipyramid with three distinct angles rather than one or two, which is the arithmetic signature of sites that are not all equivalent.

Where the tetrahedral angle comes from

Since it is the one number everybody memorises, it is worth deriving.

Put four points at alternate corners of a cube: (1,1,1)(1,1,1), (1,−1,−1)(1,-1,-1), (−1,1,−1)(-1,1,-1), (−1,−1,1)(-1,-1,1). The angle between two of those vectors has cosine equal to their dot product over the product of their lengths — that is (1−1−1)/3=−1/3(1 - 1 - 1)/3 = -1/3.

So the angle is arccos⁡(−1/3)=109.4712...\arccos(-1/3) = 109.4712... degrees, and the reason it is not a round number is that nothing about it was designed to be. It is a consequence of three dimensions and four directions.

Why it works better than it should

Here is the uncomfortable part, and it deserves stating plainly rather than being smoothed over.

VSEPR gets shapes right with remarkable reliability, and the mechanism it proposes for why is not one that survives examination. Electron pairs are not localised objects that repel one another electrostatically; they are regions of a delocalised many-electron wavefunction, and the dominant term in the energy is not the classical repulsion between them.

The better modern account is that the arrangement minimises Pauli repulsion — the exclusion of same-spin electrons from the same region — rather than Coulomb repulsion. That gets the same answers for a reason that is genuinely about quantum mechanics rather than about charged points on a sphere.

So VSEPR is a rule that works, with a rationalisation attached that does not. That is a common and slightly awkward situation in chemistry, and it is worth being honest about, because a student who believes the mechanism will make wrong predictions the moment the rule is pushed past its range.

What the minimisation is and is not

Being precise about the status of these figures.

It is geometry. The calculation places points on a sphere to minimise a simple potential. It knows nothing about atoms, orbitals, electrons or energies.

It is not a molecular calculation. No wavefunction is computed, no energy is evaluated, and the agreement with real bond angles is not evidence that the potential used is the right one.

Where it agrees, it agrees for reasons worth being suspicious about. The arrangements that minimise almost any repulsive potential on a sphere are the same ones, because the constraint doing most of the work is the geometry of the sphere rather than the form of the potential. That robustness is why VSEPR works and also why its success says little about the mechanism.

Lone pairs, and where the rule gets vague

The classic extension is that lone pairs take up more room than bonding pairs, so a molecule with lone pairs has its bond angles squeezed below the ideal value.

Water is the standard example: four regions round the oxygen, two bonds and two lone pairs, so a tetrahedral arrangement, so an angle a little under 109.5 — and the measured value is 104.5.

That works. It also has an adjustable parameter in it — “a little under” — which is doing more work than it looks. Water gets its own essay, because the standard account predicts the right direction and cannot predict the magnitude, and because there is a better rule that can.

water — C2vThe 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.HHOC2vprincipal axis C22 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms
Fig. 3 Water, at its measured geometry. The angle is 104.5 degrees, not 109.5, and the difference is where the simple rule stops being quantitative.

What decides a shape, really

If not electrostatic repulsion between pairs, then what?

The honest answer is that the shape minimises the total electronic energy, and that this is a many-electron problem with no simple decomposition. The terms that matter include nuclear-nuclear repulsion, electron-nuclear attraction, electron-electron repulsion and — decisively — the kinetic energy cost of confining electrons, which is where the Pauli principle enters.

Modern computational chemistry gets shapes right by evaluating that total energy, and it does so without any notion of electron pairs repelling. VSEPR survives because it is a very good heuristic for what that calculation will produce, and because it can be done in the head.

What the model does not reach

Three cases where the simple rule fails, all instructive.

Heavy p-block hydrides. H₂S has a bond angle of 92 degrees, not the 104 that water’s pattern would suggest, and H₂Te is close to 90. The trend is real and VSEPR does not predict it; the usual account is that the heavier central atoms use nearly pure p orbitals, which are at 90 degrees to one another.

Transition metal complexes, where the d orbitals genuinely are involved. VSEPR is unreliable here, because d electrons are not equivalent to the s and p regions the rule counts.

Molecules with delocalised bonding. Where the electrons are genuinely spread over several centres, counting regions round one atom is not well defined. Delocalisation takes that up.

Who proposed it

The rule is Nevil Sidgwick and Herbert Powell’s, in 1940, and it was developed into the form now taught by Ronald Gillespie and Ronald Nyholm in 1957. Gillespie spent much of a long career refining and defending it, and also — to his credit — revising the account of why it works as the quantum-mechanical picture improved.

That is the shape of the thing worth taking away. A rule proposed for one reason, surviving because it predicts well, with its justification quietly replaced underneath it. The predictions are the durable part.

The shapes, one by one

Running the minimisation for each count and looking at what comes out is worth doing, because the answers are more interesting than a table of names.

2 sites, minimisedThe arrangement of 2 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.linear180.00° × 1repulsion minimised, angles measured off the result2 sites
Fig. 4 Two sites: opposite poles at 180 degrees. The only arrangement possible, and the one every linear molecule has.
3 sites, minimisedThe arrangement of 3 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.trigonal planar120.00° × 3repulsion minimised, angles measured off the result3 sites
Fig. 5 Three sites: an equilateral triangle in a plane, at 120 degrees. That the arrangement is planar was not imposed — the minimisation found it, because any out-of-plane displacement brings two of the points closer.
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. 6 Six sites: an octahedron, with neighbours at 90 and opposite sites at 180. Every position equivalent, and the highest symmetry available for six points.

The planarity of the three-point case deserves a moment. Nothing in the calculation restricts the points to a plane, and the answer is planar because the sphere’s geometry makes it so. That is the same kind of emergence as the tetrahedral angle: a number and a shape falling out of a constraint rather than being supplied.

And the sequence 180, 120, 109.47, then three angles, then 90 and 180 is the whole of what VSEPR predicts, arrived at without a table.

Where the ladder goes next

The case that breaks the pattern is five sites, where the arrangement has two kinds of position and the molecule does something about it.

The case that tests the rule quantitatively is water.

And the shape’s consequences follow from the point group, which is decidable from the coordinates the minimisation produces.

What the pictures here cannot show. These are points on a sphere, not atoms, and the minimisation is of a classical potential rather than of a molecular energy. Agreement with real geometries is a fact about the robustness of sphere packings, not evidence that the potential is the right one.

Where this goes next

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 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. 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. 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.