Which angles are symmetry and which are the model

Every account of VSEPR contains the phrase “electron pairs repel” and no account of it says how. The strength of the repulsion is never given, its distance dependence is never stated, and the rule is applied as though those omissions were details.

They are not details. They are the whole model. And it is worth finding out which of VSEPR’s predictions survive changing them.

The experiment #

The minimisation places $n$ points on a sphere and pushes them apart until the total repulsion stops falling. What it minimises is

$$E = \sum_{i<j} \frac{1}{r_{ij}^{\,p}},$$

and $p = 1$ is the Coulomb case, which is the Thomson problem.

Nothing about the method requires $p$ to be one. Setting it to twelve makes the repulsion so short-ranged that only nearest neighbours matter; setting it to two makes it fall off like an inverse-square law that is not electrostatic. Both are perfectly good repulsions and neither is the physical one, because the physical one is not a classical repulsion at all — the actual origin of VSEPR’s effect is the Pauli principle acting between electrons of like spin, which has no $1/r^p$ form whatever.

So the honest question is: how much of what VSEPR predicts depends on the choice?

Four, five and six: nothing moves #

For four domains the answer at every exponent from one to twelve is 109.4712°, to every decimal place the minimiser resolves.

For five it is 90°, 120° and 180°, again at every exponent. For six it is 90° and 180°. Not approximately; identically. Changing the force law by a factor of twelve in the exponent moves nothing.

The reason is not that these minima are insensitive. It is that they are not decided by the energy at all. For four points there is an arrangement — the tetrahedron — in which every point is equivalent to every other and every pair subtends the same angle. Any repulsion that depends only on distance and decreases with it must be stationary there, because there is no direction in which the arrangement can be deformed that does not break the equivalence and raise the sum. The same holds for the trigonal bipyramid and the octahedron.

The angles, in other words, are consequences of the symmetry of the arrangement rather than of the potential. Once the shape is known to be the maximally symmetric one, arccos(−1/3) follows from geometry.

Six domains give the other case where the answer is fixed by symmetry alone: an octahedron, every site equivalent, and two distinct angles for any repulsion whatever.

Seven: everything moves #

Seven points on a sphere have no arrangement in which all seven are equivalent. There is no Platonic solid with seven vertices, and there is no way to place seven points so that each has the same neighbours at the same distances.

Without that equivalence there is nothing to pin the answer, and the answer duly depends on the law:

• at $p = 1$ and $p = 2$ the minimum is a pentagonal bipyramid — five points round an equator and two on an axis, with four distinct angles;
• at $p = 3$ and $p = 4$ it is a low-symmetry arrangement with ten or more distinct angles, which no textbook names;
• at $p = 6$ and above it is a different arrangement again, with six distinct angles and a threefold axis.

Three regimes, three shapes, from the same rule applied with three different unstated force laws. And the shape most chemistry courses attribute to VSEPR for seven coordination — the pentagonal bipyramid, which is what iodine heptafluoride adopts — is the one that comes out for the Coulomb case only.

Seven is the first count at which the arrangement has no symmetry to fall back on, and its angle spectrum has four distinct entries rather than one or two.

And eight is where the mismatch between the expected shape and the computed one is sharpest, which is worth ending on because the expected shape is the one every reader has drawn.

Why symmetry can pin an angle #

The claim that these arrangements are stationary for any decreasing potential deserves an argument rather than a bare claim, because it is doing most of the work.

Consider four points at the vertices of a regular tetrahedron. Every point has the same three neighbours at the same distance, and the arrangement has the full symmetry of the tetrahedral group: twenty-four operations, all of which permute the four points among themselves.

Now try to lower the energy by moving one point. Whatever direction it moves in, the symmetry maps that displacement onto three others, and the energy change to first order is the sum over all of them — which is a totally symmetric quantity summed over a set that averages to zero. The first-order change vanishes identically, for any potential that depends only on the pairwise distances.

That is a symmetry argument in the strict sense: it follows from the group and not from the function. It says nothing about whether the stationary point is a minimum or a maximum, and nothing about how deep the minimum is. It says that the arrangement cannot be improved by a small deformation, whatever the law.

Five points are subtler, because the trigonal bipyramid does not have all five sites equivalent — that is the point of five sites are not alike. What it does have is enough symmetry to pin each site’s position given the others: the three equatorial sites are equivalent to each other, the two axial ones are equivalent to each other, and no continuous deformation preserves both sets. So the angles are again fixed by the group rather than by the potential, and the sweep confirms it.

The one thing the exponent does change below seven #

The angles do not move, and one quantity does: the energy.

At $p = 1$ the Thomson energy for four points is 3.674, for five 6.475, for six 9.985. At $p = 6$ the same arrangements have energies 0.316, 0.877 and 1.547, and at $p = 12$ they are 0.017, 0.098 and 0.188. These are not comparable numbers — different exponents mean different units — but the ratios between counts are, and they change: the six-to-four ratio is 2.72 at $p = 1$ and 11.3 at $p = 12$.

That matters for the one question the arrangement rule cannot answer on its own, which is which arrangement a given number of domains prefers when there is more than one candidate. Comparing two geometries at the same count is comparing two energies, and the comparison depends on the exponent even where the individual arrangements do not.

Below seven that never bites, because the maximally symmetric arrangement wins under every law tried. At seven and above it bites immediately, which is what the sweep shows.

What this does to the rule #

VSEPR’s status is different on the two sides of that line, and the line falls between six and seven.

At or below six, VSEPR is not really a repulsion model at all. It is a statement that the domains adopt the maximally symmetric arrangement, and the repulsion is a story attached to it afterwards. The predictions are exact, parameter-free, and would be unchanged if the mechanism turned out to be something quite different — which it has, since the modern justification is the Pauli principle rather than electrostatics.

Above six, VSEPR is a genuine model with an unstated parameter. Its prediction depends on the force law, the force law is not given, and the value that reproduces the observed geometries is chosen after the fact. That is the same shape of problem as the lone-pair weight in what a lone pair is worth, and it arrives for the same reason: once the symmetry stops deciding, something has to, and what does is a parameter.

The two failures compound rather than cancelling. A seven-coordinate species with a lone pair — xenon hexafluoride is the notorious example — needs both an unstated force law and an unmeasured weight before the model says anything at all, which is a fair description of why its structure was argued about for thirty years.

This is also why seven-coordinate compounds are genuinely awkward in practice. The energy differences between candidate geometries are small, the compounds are fluxional, and reported structures for the same species differ between crystal forms. The shapes above six coordination takes up what the minimisation says about the higher counts, and finds the same thing from the other direction: the arrangements stop being the ones anybody would name.

What was computed, and how #

Every arrangement here is produced by the same minimiser, which is checked against published Thomson minima to about $2\times10^{-10}$ at $p = 1$ for every count from two to twelve.

The exponent sweep needs one change to a standard minimiser, and it is worth recording because the defect is easy to carry without knowing.

The descent takes a step proportional to the force. The force between two points goes as $p/r^{\,p+1}$, so at $p = 12$ it is four orders of magnitude larger near contact than at $p = 1$, and a step size chosen for the Coulomb case throws every point off the sphere on the first iteration. What came back then was not an error: it was points: null and an energy of infinity, silently, because the loop that keeps the best arrangement had never seen a finite one to keep. The caller then failed several frames later with a message about something else.

Two repairs work. The descent uses the force’s direction with a step length from a schedule whenever the exponent is not one, which converges at every exponent tried up to twenty-four; and the minimiser refuses to return a non-arrangement, so a failed search says so where it failed. At an inverse-distance law nothing changes, which is the check that the repair is a repair and not a different minimisation.

The claim the figure makes is checked where it holds. For any count whose number of distinct angles is the same at every exponent, every angle is required to agree with the $p = 1$ value to within a twentieth of a degree. A count that failed would refute the claim. Seven is exempt not by declaration but by measurement: its angle count changes across the sweep, so the check does not apply to it, and the figure says so on its face.

Where the model stops #

None of these exponents is the right one, and the essay is not an argument that some particular $p$ should be used. The physical origin of the effect is exchange repulsion between same-spin electron pairs, which is not a power law and not even a pairwise potential in any exact sense. Choosing $p$ is choosing a caricature.

The arrangement is not the molecule. These are points on a sphere with no bonds, no atoms and no lengths. A real molecule’s ligands differ in size, its bonds differ in length, and both effects are outside the model entirely.

Symmetry pinning is not a proof of stability. That the maximally symmetric arrangement is stationary for any decreasing potential does not by itself prove it is the minimum — for four, five and six the minimiser confirms that it is, from twenty-four independent starts, and that confirmation is numerical rather than a theorem.

The generalisation #

The useful separation this leaves is between two kinds of prediction that a model makes at the same time and in the same voice.

Some conclusions follow from the symmetry of the situation and are indifferent to the mechanism. That a molecule with a centre of inversion has no dipole moment is one, and it is exact — see symmetry forbids a dipole. That an integral over all space vanishes unless its integrand contains the totally symmetric representation is another, and selection rules are one theorem shows how much of spectroscopy is that single sentence. That four equivalent domains subtend arccos(−1/3) is another. Neither can be broken by a better calculation, because neither depends on a calculation.

Other conclusions follow from the details of the model, and those move when the details do. VSEPR’s seven-coordinate geometry, the lone-pair weight, the value of $\beta$ in Hückel theory: all three are numbers the model needs and does not contain.

This site’s invariants make the distinction a rule rather than a preference: symmetry conclusions may be stated without hedging, and everything else says which model produced it. The exponent sweep is a way of finding out, mechanically, which side of that line a given prediction sits on — vary the thing the model does not specify, and see whether the answer moves.

Who found it, and when #

The Thomson problem — minimising Coulomb repulsion of points on a sphere — is J. J. Thomson’s, from 1904, and predates any of the chemistry by half a century. Its solutions for small $n$ were established well before VSEPR existed and have been re-derived many times since; the general problem remains open, and proofs of optimality are known for only a handful of counts.

The observation that the answers for small $n$ do not depend on the exponent is folklore in that literature rather than anybody’s result: it is a corollary of the arrangement being a spherical design, a concept Delsarte, Goethals and Seidel formalised in 1977. Whether the tetrahedron, bipyramid and octahedron are minima for a general decreasing potential is a question in that language, and for these counts the answer is yes.

Gillespie’s own later writing is careful about the mechanism and less careful about the exponent, which is not stated anywhere in the standard presentations. That silence is what this essay measures.

What the exponent does to a fitted parameter #

The sweep above holds all the domains equal. The lone-pair weight does the opposite — it makes two of the four domains heavier and is fitted to a measured angle — and the two questions meet in a way neither asks on its own. If a geometry is exponent-independent, is the parameter fitted to it exponent-independent too?

It is not, and the answer separates two things that are usually run together.

Repeating the lone-pair fit at each exponent in the sweep, with the same four domains and the same target angles:

Water’s weight rises by 41% across the sweep and hydrogen sulfide’s by a factor of 3.5. The number the textbooks would have to quote, if they quoted one, is not a number until the force law is stated.

And every geometry it produces is unchanged. At each exponent the fitted arrangement puts water’s lone pairs exactly 114.60° apart and hydrogen sulfide’s exactly 128.92°, to the precision of the minimisation, at weights differing by a factor of three and a half. So the exponent-independence this essay measures for equal domains survives unequal ones: what moves is the parameter, and nothing the parameter is for.

That is worth stating as a rule, because it is the useful form of both findings at once. A repulsion model of this kind has a geometry that does not know the force law and a parameter that does nothing but. The fitted weight is a coordinate on the family of models rather than a property of anything in the molecule — which is a sharper version of the lone-pair conclusion, reached without needing a second molecule to disagree.

It also strengthens the lone-pair finding rather than excusing it. The spread between water’s weight and hydrogen sulfide’s is a factor of 1.8 at an inverse-square law and 4.5 at $1/r^{12}$; it never approaches one anywhere in the sweep. A reader who suspected that the two-and-a-half-fold spread was an artefact of an arbitrary force law can be told which force law would remove it: none in the range anybody uses, and the harder the repulsion the worse it gets.

What the VSEPR arguments add up to #

Taken together, these arguments turn VSEPR from a table to be memorised into a minimisation with its assumptions exposed: the arrangement rule is exact where symmetry pins it, the lone-pair clause carries a parameter that does not transfer, and the force law is unstated and matters above six. What none of them does is put an electron anywhere, which is where bonding theory starts.