VSEPR, computed

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.

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 $n$ points on a sphere so as to minimise the sum of $1/r_{ij}$ 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 every time it is run.

The answers #

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

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: $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$. The angle between two of those vectors has cosine equal to their dot product over the product of their lengths — that is $(1 - 1 - 1)/3 = -1/3$.

So the angle is $\arccos(-1/3) = 109.4712...$ 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.

What the reader can turn, and what does not turn with it #

The arrangement figures on this page carry a slider, and it is worth saying what it does — because until it was corrected it did the wrong thing, and the wrong thing is the more obvious one.

A rotation figure normally turns the camera. That makes a pleasant picture and demonstrates nothing, because no quantity is recomputed as it moves: the number in the caption was measured once, before the turning began, and would read the same whatever the slider did. On a site whose premise is that a figure states what it measures, a slider that moves the view and reports nothing is exactly backwards.

So these figures turn the arrangement instead, about a tilted axis, with the camera fixed. Every frame’s angle spectrum is measured again from the turned coordinates, and the readout is that measurement rather than a caption.

An angle between two directions is invariant under rotation, which everybody knows, and which is worth watching hold while the picture moves. What makes it a demonstration rather than a decoration is that it could come out otherwise — the number of distinct angles and each of their values are checked at every step of the turn, and a different spectrum would mean an error.

There is a second reason for turning the object rather than the camera, and it is about the drawing rather than the argument. A camera fitted to the projected outline rescales the arrangement as it turns, growing when the shape lies across the frame and shrinking when it points at the viewer — which is a change in the thing being measured rather than in the view of it. Fitting instead to the widest extent over every orientation gives one scale that holds throughout, so what moves on the screen is the arrangement and nothing else.

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.

The sphere is the strong assumption, not the potential #

The previous section blamed the potential and let something larger pass. Almost any repulsive potential gives these arrangements, so the potential is not where the model’s content is. The content is in the sphere.

Putting every region at one radius is an assumption, it is stated nowhere in the usual presentation, and it is false in three separate ways that matter.

Lone pairs are not at the same radius as bonding pairs. The standard extension to the rule says as much — a lone pair is held closer to the nucleus — and then goes on using an arrangement computed with everything at one radius. The two halves are not consistent, and the inconsistency is where the “a little under 109.5” comes from.

Different ligands are at different distances. Phosphorus pentafluoride’s axial bonds are longer than its equatorial ones by about a tenth of an ångström, which is a five per cent difference in radius between two sets of positions the model treats as identical. In a mixed-substituent phosphorane the spread is larger still.

Multiple bonds are not one region. Counting a double bond as a single region is a rule bolted on afterwards, and it works because the extra density lies roughly along the same direction — which is an argument about geometry that the sphere model cannot itself make.

None of these is fatal, and all three are the same defect: a minimisation over directions has been asked to stand in for a minimisation over positions. Allowing the radii to vary would let the model reproduce the axial–equatorial length difference and would also destroy its main virtue, which is that it can be done in the head.

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.

What it costs #

The minimisation is a few hundred steps of gradient descent on a sum of inverse distances, started from several random configurations to avoid settling into a local minimum, and it converges in milliseconds. That is the entire computational cost of every arrangement on this page.

What it costs to do honestly is a little more, and the additions are the interesting part.

Several starts, not one. A repulsion minimisation on a sphere has local minima, and the five-point case is the one that has them: a square pyramid is a genuine stationary point that a single unlucky start can find and stay in. Running from several seeds and keeping the lowest is what makes the trigonal bipyramid an answer rather than a coincidence, and it is why five is the case worth staring at.

The angles are measured off the result, not read from a table. The tetrahedral angle is never put in. It emerges as $\arccos(-1/3)$ from four minimised points, and the figure prints what it measured — which means a minimisation that had quietly converged somewhere else would print a different number rather than the expected one.

The count of distinct angles is checked. Every arrangement here must produce the number of distinct angles its shape implies, and five must produce three where four and six produce fewer. That check carries the essay’s sharpest claim and is capable of failing: a minimiser that returned a square pyramid for five points would produce a different count.

And the energy is checked against somebody else’s answer. This is the strongest of the three, because convergence is not correctness. The Thomson problem — point charges on a sphere, minimising Coulomb repulsion — has been solved to many decimal places, so every arrangement here has an independent value to be compared with.

That is the check worth having, and it is the one an energy tolerance would have missed. A descent can converge perfectly well into a flat valley and stop short of the floor, reporting an energy nobody would question and an arrangement that is subtly the wrong shape. Only a comparison with an answer obtained by other means catches it, and the repair — finishing the descent with an adaptive step rather than a fixed schedule — is what the shapes above six then depend on.

The honest bill is what all this does not buy. The potential is a choice. Using $1/r$ rather than $1/r^{12}$ or anything else changes the answer for five points by a fraction of a degree and changes it for none of the others, which is exactly why the agreement with real molecules is weak evidence for the mechanism. A model whose predictions barely depend on its central assumption is a model whose success does not confirm that assumption.

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.

Between three and six the sequence passes through the two cases already drawn above, and comes out the other side with the symmetry restored. Four gives the tetrahedron and its single angle; five gives the trigonal bipyramid and its three; and six gives back a single pair of angles on a figure where every position is equivalent again, which is the last count at which that happens.

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.

The same angle without the cube, and where the argument stops #

The cube derivation needs the cube, which has to be produced from somewhere. There is a shorter route that needs nothing but the symmetry, and its value is that it says exactly which coordination numbers it works for.

Suppose $n$ unit vectors are all equivalent to one another — every pair separated by the same angle $\theta$. Then their sum has no direction it could point in, so it is zero. Square it:

$$0 = \left|\sum_i \mathbf{v}_i\right|^2 = n + n(n-1)\cos\theta \qquad\Longrightarrow\qquad \cos\theta = -\frac{1}{n-1}.$$

Two sites give $\cos\theta = -1$ and 180°. Three give $-1/2$ and 120°. Four give $-1/3$ and 109.47°. The tetrahedral angle arrives in one line, with no coordinates, no potential and no minimisation — and the reason it is not a round number is now visible as the reason it is not a round fraction: it is a reciprocal of an integer, one place along from the two angles that are.

Then it stops. Five sites would need $\cos\theta = -1/4$, which is 104.5°, and no five directions in three dimensions are mutually separated by 104.5°. Six would need 101.5°, and an octahedron’s angles are 90° and 180°.

So the formula holds for two, three and four and for nothing above. What fails is the premise rather than the algebra: from five upwards the sites cannot all be equivalent, so there is no single $\theta$ for the argument to solve for. That failure is not a technicality of this derivation — it is the whole subject of the five-point problem, arriving here as an equation with no solution rather than as an observation about a molecule.

Five sites, water, and the point group #

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.