What a lone pair is worth

VSEPR is two claims wearing one name, and only the first of them is a prediction.

The first is that electron domains around an atom arrange themselves as far apart as possible. That is checkable, it needs no parameters, and it is right: VSEPR, computed minimises the repulsion of four points on a sphere and gets arccos(−1/3) without the tetrahedral angle ever being written down.

The second is that lone pairs repel more than bonding pairs. That is not a prediction until somebody says how much more, and nobody does.

Turning the clause into arithmetic #

The repulsion minimisation places $n$ points on a sphere so as to minimise

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

and the arrangement it returns for four points is the tetrahedron, with every angle at 109.4712°. Every point is identical, so every angle must be.

To make a lone pair different from a bonding pair, give it a weight. Let each domain carry a strength $q_i$, with bonding pairs at one and lone pairs at some $q > 1$, and minimise

$$E = \sum_{i<j} \frac{q_i q_j}{r_{ij}}$$

instead. Nothing else changes. That is the whole of VSEPR’s second clause as a computation, and writing it down is what makes it testable, because $q$ now has to come from somewhere.

Fitting it to water #

Water’s H–O–H angle is 104.5°, which is 4.97° below tetrahedral. Bisecting on $q$ until the computed bonding angle matches gives

$$q_{\text{water}} = 1.244,$$

so the model says a lone pair on oxygen repels about a quarter more strongly than a bonding pair. That is a perfectly reasonable-sounding number, it is in the right direction, and it reproduces the measurement exactly — because it was fitted to it.

A single fitted parameter reproducing a single measurement says nothing whatever. The question is what that $q$ predicts elsewhere.

What it predicts elsewhere #

Take $q = 1.244$ to ammonia, which has one lone pair and three bonds. The minimisation gives 106.75°, against a measured 107.8°. That is an error of about a degree, which is respectable for a model this crude, and it is the result usually offered as evidence that the clause works.

Take the same $q$ to hydrogen sulfide, which has the same shape as water and the same electron count. The minimisation gives 104.50°, against a measured 92.1°.

Twelve degrees is not a discrepancy. It is a different molecule. And it is not fixed by adjusting the weight slightly: reproducing 92.1° needs

$$q_{\text{H}_2\text{S}} = 2.259,$$

which is not a small revision of 1.244. Phosphine, at 93.3°, needs 2.872 — where ammonia needed 1.148.

The spread is the finding #

Four molecules, two of each shape, and the weights their measured angles demand:

A factor of 2.50 from end to end. If a lone pair had a repulsion strength, the number fitted to water would be roughly the number fitted to hydrogen sulfide, because both molecules have two lone pairs on a group-16 atom. It is not roughly that number, and the failure is not random: the second-row hydrides need weights near 1.2 and the third-row hydrides need weights near 2.5.

That pattern is the clue. The weight is absorbing something that varies down a group, and the model has no term for it. A parameter that correlates with the row of the periodic table is a parameter standing in for whatever changes down a row, and in this case the thing that changes is not the lone pair at all.

It is worth noticing that the model is not merely imprecise here — it is imprecise in a structured way, which is the more informative kind of failure. Random scatter in the fitted weights would suggest the measurements or the arithmetic were noisy. A clean split into two groups of two suggests a missing term, and a missing term can be identified.

Why the fit looks good when it is only one point #

It is worth being precise about the logic, because the usual presentation is persuasive and the flaw in it is easy to miss.

The clause is stated qualitatively — lone pairs repel more. It is then applied to water, and water’s angle is smaller than tetrahedral, so the clause is confirmed. It is applied to ammonia, whose angle is also smaller than tetrahedral, so the clause is confirmed again. It is applied to sulfur tetrafluoride, whose lone pair sits equatorial rather than axial, and confirmed a third time.

Every one of those confirmations is a test of the sign of the effect and none is a test of its size. A model that says “smaller than tetrahedral” is right about all three because all three are smaller than tetrahedral, and it would be right about them if the true mechanism were something else entirely — which, as the third row shows, it partly is.

Putting a number on the clause is what converts it from an observation with a direction into a claim with a magnitude, and a claim with a magnitude can be wrong. Twelve degrees is how wrong it is.

What is actually varying #

What varies down the group is how much s character the central atom puts into its bonds, and that is what Bent’s rule, against the substituent series is about.

An ideal sp³ hybrid gives 109.47°. Pure p orbitals, with the s left as a lone pair, give exactly 90°. Oxygen mixes s and p substantially and lands near tetrahedral; sulfur mixes them very little and lands near 90°, because the 3s and 3p orbitals differ more in energy and in size than the 2s and 2p do — a difference what an electron actually feels traces to screening.

So the third-row angles are close to 90° for a reason that has nothing to do with lone pairs pushing harder. They are close to 90° because the bonds are made of nearly pure p orbitals, which are at 90° to each other before any repulsion is considered at all.

The weighted-repulsion model has no way to say that. Asked to reproduce 92.1° it does the only thing it can: it makes the lone pairs enormous. The fitted parameter is not measuring lone-pair repulsion; it is soaking up the hybridisation the model does not contain.

Ammonia’s group is recovered as C₃ᵥ from its coordinates, with one lone pair, three bonds and an angle of 107.8°. The weight fitted to that angle is 1.148 — against phosphine’s 2.872 for the same arrangement and the same rule, which is a factor of two and a half in a parameter that is supposed to describe the same kind of object.

The one prediction the clause makes without a number #

There is a case where the lone-pair clause is genuinely predictive, and it deserves stating because the rest of this essay is about where it is not.

In a five-domain arrangement the sites are not equivalent: three sit around an equator at 120° and two sit on an axis at 90° to all of them. An equatorial site has two neighbours at 90°; an axial site has three. So a domain that repels more strongly should go equatorial, where it has fewer close neighbours — and that is an ordering rather than a magnitude, so it needs no weight at all.

Sulfur tetrafluoride’s lone pair is equatorial. Chlorine trifluoride’s two lone pairs are both equatorial. Xenon difluoride’s three are all equatorial, leaving the two fluorines axial and the molecule linear. Three predictions, all correct, and not one of them requires knowing how much more a lone pair repels — only that it repels more at all.

That is the clause doing honest work, and it is a good illustration of the general rule: an ordering can be predictive where a magnitude is a fit. Five sites are not alike computes the inequivalence the argument rests on.

What was computed, and how #

The minimiser is the same one the rest of this site uses, extended in one place: each site carries a weight, the pair energy becomes $q_iq_j/r_{ij}$, and the force follows. With every weight at one it is the Thomson problem and returns exactly what it returned before — the arrangements and angles on every other page of this collection are unchanged by this essay, which was checked by rebuilding and comparing the output byte for byte.

The fit is a bisection on $q$, and it asserts its own preconditions rather than assuming them: that the bond angle falls monotonically as the weight rises, and that the target angle lies inside the range the search covers. A target outside that range is refused rather than returned as an endpoint.

The finding itself is a test. Fit all four molecules and require the spread in $q$ to exceed 1.5. The test is written in the direction that can fail: if the weights had come out within fifty per cent of each other, this essay would be wrong. That is the shape every quantitative claim ought to take, and it is worth noticing how rarely a fitted model is subjected to it — the usual practice is to fit one molecule, report the agreement, and not fit the next. The same discipline is what turns delocalisation is stabilising, and other things that are false in general from an opinion into an arithmetic result.

The measured angles are quoted, from standard structural data: 104.5°, 107.8°, 92.1° and 93.3°. Those are measurements and should be quoted. The weights are computed from them.

Where the model stops #

This is not a calculation of any molecule. It is a minimisation of point charges on a sphere, and there are no atoms in it, no orbitals, and no energy in any physical unit. The weight is dimensionless and its value depends on the exponent in the repulsion law, which is itself unstated in every account of VSEPR — a matter taken up in which angles are symmetry and which are the model.

Nothing here shows lone pairs are not larger. A lone pair genuinely is more diffuse than a bonding pair, and the electron density of water genuinely is fatter on the lone-pair side. The finding is narrower and harder: whatever that difference is worth, it is not a transferable number, so the clause cannot be used as a prediction.

The comparison holds shape constant on purpose. Water and hydrogen sulfide have the same number of domains of each kind, so every difference between the two fitted weights is a difference the model cannot see. Comparing molecules of different shapes would have let the discrepancy hide in the geometry.

The generalisation #

A model with a free parameter fitted to one measurement always reproduces that measurement. The information is in what happens to the parameter when the same model is fitted to the next system: a constant parameter means the model has found something, and a parameter that moves means the parameter is a place where the model’s ignorance is being stored.

That test has now been run three times with the same outcome. Electronegativity is not one quantity, and the four scales disagree on the direction of ordinary bonds. The lone-pair weight is not one quantity, and it varies by two and a half across four hydrides. In both cases the quantity is spoken of as though it were a property of an atom or of a pair, and in both cases it is a property of a fit.

The third case is $\beta$ in Hückel theory, which this site is careful never to give a number: values fitted to different observables differ by a factor of two, and quoting one would turn a clean statement about a graph into a hidden calibration. Hückel theory and what it gets right is built on that refusal, and Hückel with a heteroatom is what happens when two more fitted numbers are let in.

The distinction is not between good models and bad ones. It is between a model whose parameters stay put and a model whose parameters move to wherever the next measurement needs them, and only the first kind has predicted anything.

Who found it, and when #

Sidgwick and Powell’s 1940 paper set out the arrangement rule, and Gillespie and Nyholm added the lone-pair ordering in 1957 — lone pair against lone pair repels more than lone pair against bonding pair, which repels more than bonding pair against bonding pair. It is an ordering rather than a scale, and it was never proposed as a quantitative model.

That reticence was well judged and has not been maintained by the textbooks, which routinely present the clause as though it explained magnitudes. Gillespie himself later moved the justification onto the Pauli principle acting between same-spin electrons rather than onto electrostatics, which is a better argument and does not supply a number either.

The alternative account — that the angles follow s character, and s character follows the electronegativity of the substituents — is Bent’s, from a 1961 review, and it does predict the trend down the group that the repulsion clause cannot.

The second angle, where two accounts can be compared #

The fitted weight is one number taken from one measured angle, so it cannot be tested by that angle. Neither can its rival: Bent’s rule, computed takes the same bond angle and returns the s fractions of the four hybrids, one number from one datum. Two accounts, one measurement each, no test.

There is a second angle, though, and both accounts predict it without any further input. The angle between the two lone pairs is not measurable directly and both models are obliged to say what it is — the minimisation because its lone pairs are points that land somewhere, the hybrid account because the four s fractions must sum to one. So the two can be set against each other even though neither can be set against an experiment.

Running the same minimisation that produced the fitted weights, and reading the lone-pair separation off the result:

Three of the four agree to within about a degree, and one disagrees by twenty-one.

That is a genuinely surprising result for the first three rows. A classical minimisation of weighted point charges and an orbital argument about how one s orbital is shared among four hybrids have nothing in common — different objects, different mathematics, no shared parameter — and they place the lone pairs within a degree of each other in every case where the central atom is a second-row element. Whatever the two accounts are each getting right, they are getting the same thing right.

And they part company at exactly the molecule the fitted weight already failed on. Hydrogen sulfide’s minimisation puts its lone pairs 129° apart and its s fractions put them 150° apart, because the two models absorb a nearly-unhybridised central atom differently: the hybrid account says the bonds are 3.5% s and the lone pairs take almost all of it, while the minimisation has no way to express that and can only push the point charges further apart. Two accounts that agree wherever the underlying assumption holds and diverge where it fails is the most useful thing either of them has done here.

The dimethyl ether row carries a separate refusal. Its fitted weight is 0.936, below one — a lone pair repelling less than a bonding pair. That is not a poor fit to the clause; it is the opposite of the clause, produced by the clause’s own arithmetic from a measured angle. The model has no way to decline: any angle above 109.47° demands a weight under one, and dimethyl ether’s is above it by a degree and a half.

So the parameter is not merely un-measured, as this essay’s title suggests. It is a quantity that goes below the value its own definition forbids as soon as it is asked about a substituent heavier than hydrogen.

Still open: the repulsion law #

One of VSEPR’s two clauses carries a parameter nobody has measured. The sharper question is about the other clause: the arrangement rule assumes a repulsion law, never states which, and it turns out that for four, five and six domains it does not matter in the slightest — and for seven it decides the answer.