A lone pair weaker than a bond

Xenon hexafluoride’s lone pair goes through a face — in the repulsion model, with seven domains on a sphere and the lone pair’s repulsion weighted by q — only once a contact between the six fluorines is added. With the domains alone the model put the lone pair on the axis of a pentagonal pyramid at every weight it tried, from 1.05 to 3. That essay closed on the case it could not reach: the stereochemically inactive lone pair, the one that leaves a molecule a regular octahedron as if it were not there. The model cannot put a lone pair nowhere, it said, but it can shrink its weight.

It said towards one, and that is the wrong direction. A weight of one makes the lone pair an ordinary domain — seven equal points, the Thomson problem’s pentagonal bipyramid, with nothing inactive about it. The inactive limit is a weight of zero: a seventh point that repels nothing, leaving six fluorines to repel each other alone. Six points on a sphere make a regular octahedron, and a weightless lone pair sits wherever it was put.

So the walk runs downwards, through weights the earlier survey never tried — a lone pair that repels less than a bond. Every weight below 0.657 puts the lone pair in a face with no contact at all, and the face’s distortion vanishes continuously as the weight goes to zero.

A weight below one #

VSEPR’s clause is that a lone pair repels more than a bonding pair, and it enters the model as q > 1. What a lone pair is worth fitted that weight to real bond angles and found 1.24 for water, 1.15 for ammonia, 2.26 for hydrogen sulfide and 2.87 for phosphine — never one number, but always above one. The earlier xenon survey stayed in that range, which is VSEPR’s own.

An inactive lone pair is the opposite claim. Xenon’s lone pair in the hexafluoride sits largely in its 5s orbital, which is compact and spherical, and a spherical pair repels the bonds equally in every direction — which is the same as not preferring any direction at all. In the weighted model the only way to say repels less than a bond, and less directionally is a weight below one. The model does not know about s character; it knows about weight, and q below one is the region where the clause has been turned round.

Everything else is as before. Seven points on a unit sphere repel as one over their separation, with the lone pair’s ten pairs multiplied by q. Three candidates are relaxed from seeds of their own symmetry — the pentagonal pyramid with the lone pair on its axis, the capped octahedron with it through a face, the edge-capped octahedron with it over an edge — and each is tested for being a true minimum by its eleven internal curvatures. Where the answer matters, a search from nudged starts looks for anything lower of any symmetry.

The inactive limit is reached continuously #

The refusal comes first, and it is the limit itself. At a weight of 0.001 the twelve cis F–Xe–F angles lie between 89.98° and 90.04° — a regular octahedron to six hundredths of a degree — and the lone pair sits 54.76° from three fluorines and 125.27° from the other three, the angles from a face centre of a regular octahedron to its vertices, 54.74° and 125.26°. A weightless point that moved the six fluorines would be a model with a force in it that its energy does not contain.

From there the distortion grows, and at first it grows linearly: the spread of the cis angles is 58.5° per unit of weight at q = 0.001 and 56.3° per unit at 0.05, the same to four per cent. At a weight of a tenth the face the lone pair sits over has opened by about four degrees and the angles spread from 88.4° to 93.8°; at a half, from 83.3° to 104.5°. The distortion keeps growing, more slowly, all the way to the weights the earlier essay used — but by then the structure has stopped being the lowest.

So the lead’s question — whether the model’s distorted octahedron connects smoothly to the regular one or whether the pentagonal pyramid intervenes — has a plain answer. It connects smoothly, and the pyramid never intervenes below a weight of one. Along the whole walk from zero to one the capped octahedron is a single continuous family whose distortion is a smooth function of the weight, and it is a true minimum from zero to 0.821.

Three regions, not two #

The earlier essay found two structures competing: the pyramid, which the domains prefer, and the capped octahedron, which the fluorines’ contact produces. Walked below one, the domains alone produce three.

Below a weight of 0.657 the capped octahedron is the lowest structure — the lone pair in a face. From 0.657 to 1 the edge-capped octahedron is lowest — the lone pair over an edge. From 1 upwards the pentagonal pyramid is, which is the earlier survey’s whole range.

The edge region is new, and it is not a numerical accident. At a weight of exactly one the seven points are the Thomson bipyramid, and the lone pair’s position in it is a label: on the axis it is the pentagonal pyramid, on the equatorial ring it is the edge-capped structure, and the two have the same energy to nine decimals, which was the earlier essay’s refusal. Move the weight off one and the label becomes a choice. A lone pair heavier than a bond takes the axis, where its five neighbours are all at 90°; one lighter than a bond takes a ring site, where two neighbours are at 72° and two at 90° — closer contacts matter less to a domain that repels less. Push the weight lower still and the six fluorines’ own repulsion takes over: they settle towards the octahedron they would make alone, and the lone pair goes to the point of the sphere farthest from all of them that an octahedron leaves — a face centre, 54.7° from its three nearest fluorines, which is as far as any point can get from the nearest vertex of a regular octahedron.

So the sequence the model gives, from inactive to strongly active, is face, edge, axis — and xenon hexafluoride, whose lone pair everyone describes as barely active, is observed at the first of them.

The edge structure is not an octahedron #

The edge region would matter less if the edge-capped structure were only a slightly different octahedron. It is not. At a weight of 0.8, where it is the lowest structure with the domains alone, the lone pair sits 69.2° from two fluorines, 88.8° from two and 143.1° from the last two, and the twelve smallest F–Xe–F angles run from 73.7° to 138.3° — a spread of 65°, against 29° for the capped octahedron at the same weight. Its six fluorines are closer to a pentagonal bipyramid with one ring site taken by the lone pair than to any octahedron: the Thomson bipyramid it becomes at a weight of one is already visible in it.

So the three regions are not three variations on one shape. The face region is an octahedron distorted by an amount that goes to zero with the weight. The edge region is a bipyramid with a gap in its ring. The axis region is the pentagonal pyramid, a bipyramid with a gap at its apex. Only the first looks like the molecule that is observed, and which angles a symmetry fixes and which the model does is exactly the distinction at stake: the face region’s $C_{3v}$ is the observed symmetry, and its angles are the model’s.

This is also where the domain-counting picture that computes VSEPR from repulsion alone meets the inactive pair. Seven domains are seven points, and a pentagonal bipyramid is what seven points do; six domains are six points, and an octahedron is what six do. A lone pair of weight q interpolates between the two — at one it is a full seventh domain, at zero it is not a domain at all — and the three regions are the stages of that interpolation. Five sites are not alike because a bipyramid has two kinds of site; the edge region exists because the seven-point bipyramid does too, and a light seventh point takes the crowded kind, leaving the roomier axial sites to the bonds that repel more.

Where each is a minimum #

Lowest is not the same as stable, and the curvatures separate them. The capped octahedron is a true minimum from zero to 0.821; past that two directions fall away from it. The edge-capped structure is a minimum from 0.601 to 1.028. The pentagonal pyramid is a minimum from one upwards.

So between 0.601 and 0.821 both the face and the edge are minima, and the change of lowest structure at 0.657 is a crossing between two basins rather than one structure sliding into the other. A molecule in that range would have two kinds of structure within about a thousandth of each other in the model’s energy units — against total energies near thirteen — and a barrier between them. That is a different picture of xenon hexafluoride’s floppiness from the earlier essay’s, where — with the contact, at a weight of 1.5 — the edge-capped structure was the saddle the lone pair crossed going from face to face. With a weak lone pair and no contact, the saddle is itself a place the molecule can stop.

Around a weight of one, the edge and the axis overlap only in the sliver from 1 to 1.028, where they are the same bipyramid’s two sites a hair’s breadth apart in weight.

What the fluorines’ contact does #

The contact that made the earlier essay’s face acts on this picture in one direction only: it favours the face. With a Born–Mayer contact at 0.345 Å worth half a per cent of the domain energy, the face–edge crossing moves from 0.657 to 0.781; at one per cent, to 0.954; by three per cent the lone pair is in a face at every weight up to 1.6. The face region, which reaches down to the inactive limit, only grows.

The contact also changes what lies beyond the face. With any contact at all the lowest structure between the face and the axis is no longer the edge-capped octahedron but something with only a mirror plane — the same off-axis, mirror-only structures the earlier essay found sliding between face and axis at a weight of 1.5 — and that intermediate region shrinks and moves to higher weight as the contact grows. With no contact it is the edge; with contact it is a mirror; and in both cases it separates a face region that starts at zero from an axis region that starts at or above one.

Nudged searches over 217 cells of the plane find nothing lower than these four kinds of structure, and none with less symmetry than a mirror plane.

Two routes to a face #

The earlier essay’s route to xenon hexafluoride’s observed shape was a lone pair heavier than a bond, held on the axis by the domains and pushed into a face by the fluorines. This essay’s route is a lone pair lighter than a bond, which needs no push. Both end in a capped octahedron, and the model can say how far each opens it.

By the weak-pair route the capped octahedron first becomes the lowest structure at q = 0.657, with its cis angles spread from 81.8° to 107.4° — 25.6° — and every smaller distortion is available below it, down to none. By the contact route it first becomes lowest at a weight of 1.2 with a two per cent contact, spread 31.6°, and at 1.5 with three per cent, 33.5°. The contact does pull the fluorines back towards an octahedron as it grows — at a weight of 1.5 the capped octahedron’s spread is 40.9° with no contact and 33.5° at three per cent — so a much larger contact might reach a mild distortion from above one as well. Within the shares mapped here it does not; the weak-pair route reaches every distortion from zero upwards with no contact at all.

That is the measurable difference between the two stories, and it is the one the earlier essay pointed at: xenon hexafluoride’s distortion is small. The survey of shapes above six recorded it as a slightly distorted octahedron. In this model a slightly distorted capped octahedron comes most directly from a lone pair lighter than a bond; a lone pair heavier than a bond gives, at every contact share mapped here, a strongly distorted one or none.

What the model cannot say is how slight. Its distortion measure is the spread of angles between bonds of one length on a sphere, and a real molecule’s distortion includes bond lengths that differ round the opened face, which the model holds equal. A measured angle spread would pick out a curve in the plane of weight and contact share rather than a point — the two dials trade against each other along it — and that curve is a calculation away once a spread is chosen to fit.

What the picture cannot show #

The weight is a dial, and below one it is a dial in a region VSEPR never licensed. The model says what a lone pair repelling less than a bond would do; it does not say that xenon’s does. That the region reproduces the observed structure’s symmetry, and a small distortion, is a consistency, not a derivation. The weights fitted for main-group hydrides — 1.15 to 2.87 — all sit above one, and nothing here fits a weight for xenon.

All six bonds are one length. The capped octahedron’s three fluorines round the opened face and the three opposite are held at 1.89 Å. The site a crowded domain occupies takes the long bond, and letting the two sets of three take two lengths is the other half of the earlier essay’s lead, untouched here.

The lone pair is a point on the sphere. An inactive pair in a real molecule is closer to the nucleus than the bonds, not on the same sphere with less weight. A weightless point on the sphere and a pair shrunk to the centre give the same regular octahedron in the limit, which is why the limit is safe; how they differ on the way there is not something this model can tell.

A dial VSEPR turns one way #

VSEPR’s lone-pair clause is directional: lone pairs repel more. Written as a weight, it is a dial that the rules only ever turn up, and every molecule it has been fitted to here agrees. Xenon hexafluoride is the famous case where the rules give the wrong shape, and the usual reading is that the clause breaks down for a lone pair with a lot of s character.

In the weighted model the clause does not break down; it turns round. Below one the same arithmetic that put water’s lone pairs apart and bent it puts a light lone pair in the most open place available — a face, not an axis — and the regular octahedron of the inactive pair is simply the end of that road. What the rules call an exception is the model’s own continuation, and the region of weights where xenon hexafluoride’s shape lives begins at zero and reaches two thirds of a bond. The sharpest statement of the finding is the comparison the earlier essays make possible: phosphine’s lone pair is worth 2.87 bonds, and a lone pair worth less than two thirds of one gives xenon hexafluoride its face.

Who found what #

The seven-point Thomson problem and its pentagonal bipyramid are classical. VSEPR’s clause is Gillespie and Nyholm’s, and the description of xenon hexafluoride’s lone pair as stereochemically weak — with the structure’s fluxionality — comes from the electron-diffraction work of Bartell and Gavin in the late 1960s and the spectroscopy that followed. The inert-pair language for heavy-atom s electrons is older still.

What is computed here is the continuation of the weighted model below one: the continuous approach to the regular octahedron, the three regions and their stability ranges, and the map of what the fluorines’ contact does to them.

Still open: the two bond lengths, and a weight for xenon #

The obvious open question is the bond lengths the model still holds equal. In the capped octahedron the three fluorines round the opened face are closer to the lone pair than the three opposite; letting the two sets take two radii, as the heptafluoride essays already do, would say whether a light lone pair’s small distortion gets smaller still when the near bonds lengthen, and whether the face–edge crossing at 0.657 moves.

The nearer question is a weight for xenon at all. The hydride weights were fitted to single bond angles. Xenon hexafluoride’s structure has more to fit — the opened face’s angles and the opposite face’s — and a fit of weight and contact share together to a measured distortion would say whether the molecule sits in the weak-pair region, the contact region, or on the curve where the two trade off. That needs a measured distortion brought to the model’s single-bond-length form, which is a data step before it is a calculation.