The lone pair goes through a face

The shapes above six coordination reach iodine heptafluoride, whose seven bonds sit at the corners of a pentagonal bipyramid with a ring that will not hold still. Its ring is held flat by its long bonds under every repulsion softer than about 1/r^4.4, and when the fluorines at the ends of the bonds were put back they changed nothing under the same law — they sit on the same rays, and a second set of points on the same rays only rescales the energy — and mattered only as a contact repulsion of a different shape. A contact of short enough range puckered the ring, and the pucker pseudorotated almost freely.

That essay’s nearer lead was xenon hexafluoride, the other famously fluxional molecule with seven domains. Its seventh domain is a lone pair, which has no atom at its end, so a contact repulsion between fluorines reaches six of the seven directions and not the seventh. The same-rays argument says the lone pair’s position is invisible to a second law applied to its own ray; the contact term sees its absence. Whether that makes xenon hexafluoride’s floppiness the same pseudorotation or something else is the same calculation with one fewer contact.

It is something else, and the contact term is what makes it so. Without it the model puts the lone pair on an axis. With it the lone pair goes through a face.

Seven domains, one of them heavier #

The model is the one the essays on iodine heptafluoride used, with two changes. Seven points sit on a sphere and repel each other as one over their separation. Six are bond domains. The seventh is the lone pair, and its repulsion with every other domain is multiplied by a weight q — VSEPR’s clause that a lone pair repels more than a bond, which an earlier essay priced and found not to be one number across molecules. Here q is left as a dial from 1.05 to 3, and every verdict is stated as a function of it.

The second change is the contact term. The six fluorines sit at the ends of their bonds, at xenon hexafluoride’s Xe–F length of 1.89 Å, and each pair of fluorines repels by a contact law with a weight. The laws are the ones the IF₇ essay used: a Born–Mayer exponential with ranges of 0.25, 0.345 and 0.5 Å, and fluorine’s own 2p overlap squared, which has no free range at all. The weight is reported as the contact term’s share of the domain energy, the normalisation that does not depend on the arbitrary units of either. The lone pair has no fluorine, so it takes part in the domain term and not in the contact term.

Three candidate structures are relaxed from seeds of their own symmetry, which descent preserves. The pentagonal pyramid puts the lone pair on the axis of a pentagonal bipyramid, where one of IF₇’s axial fluorines would be: five fluorines in a ring and one opposite the lone pair. The capped octahedron puts the lone pair through one face of an octahedron of fluorines, opening that face. The edge-capped octahedron puts it over an edge.

Read from the lone pair, the three are unmistakable. The pyramid has one fluorine at 180° and five at 93.8°. The capped octahedron has three near 79° — the face the lone pair opens — and three near 130°. The edge-capped structure has pairs at 77.5°, 93.1° and 145.7°.

The refusal comes first. With the lone pair made an ordinary domain, q = 1 and no contact term, the seven points are the Thomson problem’s seven: a pentagonal bipyramid with energy 14.452977, and the edge-capped seed must relax into the same bipyramid. It does, to nine decimal places. A model that told the pyramid and the edge-capped structure apart there would be seeing a difference that does not exist.

With the domains alone the lone pair goes on the axis #

Without the contact term, the answer does not depend on the weight.

At every weight from 1.05 to 3 the pentagonal pyramid has the lowest energy of the three, and it is a true minimum: every one of its eleven internal curvatures is positive. The capped octahedron sits above it by 0.013 at q = 1.05, 0.063 at 1.5 and 0.152 at 3, and it is not even a minimum — it has two falling directions, a degenerate pair, along which it rolls back towards the pyramid. The edge-capped structure is a saddle too.

That is the answer VSEPR computed gives for seven domains, and it is the one VSEPR’s own rules give: seven domains make a pentagonal bipyramid, and a heavier domain goes where it has no neighbour closer than 90° — the axis, whose five neighbours are all at 90°, where a ring site has two at 72°. The model is doing what it was built to do. It predicts a pentagonal pyramid of fluorines, and xenon hexafluoride is not one; the essay on the shapes above six recorded that it is “a slightly distorted octahedron” and that this model had no way to represent that.

Six fluorines move it into a face #

Add the contact repulsion between the six fluorines and the answer changes.

The reason is geometric and needs no calculation to see. In the pentagonal pyramid five fluorines sit in a ring 72° apart, and at a 1.89 Å bond those neighbours are close; in a capped octahedron the six fluorines are near 90° from their neighbours, like an octahedron’s. A contact repulsion that only fluorines feel is lowest when the fluorines are furthest apart, and six points are furthest apart as an octahedron. The domain term prefers the lone pair on the axis; the contact term prefers the fluorines in an octahedron; and the lone pair, which the contact term cannot see, goes wherever the octahedron leaves room — through a face.

At a weight of 1.5 and a Born–Mayer range of 0.345 Å, the capped octahedron’s energy falls to the pyramid’s at a contact share of 2.29 per cent of the domain energy. By 3 per cent it is lower by 0.025, and by 5 per cent by 0.105.

That share is of the same size that puckered IF₇’s ring — between 0.16 and 2.8 per cent of the Coulomb energy for steep contact laws — so the contact term that decides xenon hexafluoride’s shape is not a larger one than IF₇ was already argued to have. It is the same size of term acting on a different geometry.

Every law does it, and the range sets only the price #

The IF₇ essay found that a contact law’s range decided the sign of its effect: an exponential longer than 0.346 Å held the ring flatter and a shorter one puckered it. For xenon hexafluoride there is no such sign.

Under every law tried the capped octahedron takes over. Under Born–Mayer at 0.345 Å the switch comes at 1.0, 2.3 and 4.4 per cent for lone-pair weights of 1.2, 1.5 and 2; at 0.25 Å at 0.7, 1.5 and 3.0; at 0.5 Å at 1.7, 3.8 and 7.2. Fluorine’s own 2p overlap law, whose effective range is near 0.17 Å, needs the least: 0.40, 0.90 and 1.80 per cent. The share rises with the lone pair’s weight under every law, because a heavier lone pair holds the axis harder, and it rises with the range, because a longer-ranged contact term is flatter across the difference between the two arrangements of fluorines.

The difference from IF₇ is the missing end atom. In IF₇ all seven domains had fluorines, and the question was whether the contact term’s curvature along one mode — the pucker — was positive or negative; a long-ranged law reached the ring’s across-neighbours and the axial fluorines and could come out either way. In xenon hexafluoride the question is not a curvature but a choice between two arrangements of six fluorines, one with a crowded ring and one without, and any repulsion between fluorines prefers the uncrowded one.

The lone pair slides off the axis #

The crossing of two energies suggests a jump from one structure to the other. At a lone-pair weight of 1.5 it is not a jump, and what happens instead only shows when the symmetry is allowed to break.

Descent from a symmetric seed keeps its symmetry, so each seed is also nudged off it four times from a fixed sequence of small displacements, and the lowest structure reached from all twelve starts is reported. Up to 2 per cent the pyramid is the lowest structure from every start and a true minimum. By 2.25 per cent it has two falling directions — the lone pair’s tilt off the axis, a degenerate pair — and the lowest structure has left the axis. At 2.5 per cent the lowest keeps only a mirror plane, two pairs of fluorines and two singles as seen from the lone pair, and it lies 0.0073 below the pyramid and below the capped octahedron, which is not yet a minimum either. From 2.75 per cent the capped octahedron is a true minimum and the lowest structure from every start.

So at this weight the lone pair does not jump from the axis to a face. It slides: the pyramid softens along the tilt, the lone pair leaves the axis through structures with less and less symmetry, and it settles into a face once the octahedron of fluorines is stiff enough to hold it there.

Whether it slides depends on the weight #

The same test at the other weights gives a different answer at the top of the range.

At weights of 1.2 and 1.5, where the two symmetric structures’ energies cross neither of them is a minimum: the lone pair is sliding through lower-symmetry structures at the crossing. At a weight of 2 both are minima where they cross. There the heavier lone pair holds the axis firmly enough that the pyramid is still stable when the capped octahedron becomes equally low, and the structure would have to jump between two separate minima with a barrier between them.

That is a qualitative difference the model makes, and it is set by one parameter the model does not fix. The lone pair’s weight was never one number across molecules, and here the kind of transition xenon hexafluoride sits near depends on which side of a weight between 1.5 and 2 it is on.

From face to face over an edge #

Past the switch the capped octahedron is the minimum, and there are eight faces for the lone pair to occupy — eight equivalent minima. The route between two neighbouring ones is the edge-capped structure.

The edge-capped structure has exactly one falling direction at every share past the switch, so it is a true transition state between the two faces that share its edge. Its energy above the capped octahedron is 0.0063 at a contact share of 3 per cent, 0.0080 at 3.5, 0.0096 at 4 and 0.0128 at 5, rising with the contact term that created the preference for the octahedron in the first place. Against a total model energy near seventeen, that is a few parts in ten thousand: the lone pair wanders over the octahedron’s faces almost freely.

That is xenon hexafluoride’s fluxionality in this model, and it is a different motion from IF₇’s — and from the exchange of axial and equatorial sites that makes five-coordinate molecules floppy, where every site is an atom. IF₇’s pucker pseudorotates round a ring of five fluorines, the ring’s shape travelling while every fluorine stays in the ring. In xenon hexafluoride the thing that travels is the lone pair, from face to face, and each passage over an edge changes which three fluorines are opened and which three are closed. The missing end atom is what makes the lone pair the moving part: the contact term fixes the fluorines into an octahedron and leaves the one domain it cannot see free to choose among the octahedron’s faces.

How the claims can fail #

Every statement is checked where the figures are drawn. With the domains alone, the pentagonal pyramid must be the lowest of the three at every weight from 1.05 to 3 and have no falling direction, and each structure must have its seed’s signature — one and five, three and three, two, two and two. Under every contact law the capped octahedron must take over at a share that rises with the weight, and a shorter Born–Mayer range must need a smaller share. At twice the switch’s share the capped octahedron must be a minimum and the edge-capped structure a saddle with exactly one falling direction; at the switch the edge-capped structure must lie within 0.02 of both. Along the share at a weight of 1.5, the lowest structure from any start must be the pyramid up to 1.5 per cent, must have left the axis by 2.25, must keep only a mirror at 2.5 and lie below both symmetric candidates there, and must be the capped octahedron from 3 per cent, with the barrier between faces rising. And at the switch, neither symmetric structure may be a minimum at a weight of 1.5, and both must be at 2.

The refusal is the one stated above: seven equal domains must be the Thomson problem’s bipyramid, at its known energy, reached from two different seeds.

Where the model stops #

A lone pair on the sphere. The model puts the lone pair on the same sphere as the bond domains, pointing somewhere. A stereochemically inactive lone pair — one in an s orbital, occupying no direction — is not representable, and xenon hexafluoride has been argued to sit close to that case, a nearly regular octahedron whose distortion is small. The model’s capped octahedron is a distortion of a definite size, set by the weight and the share, and nothing here says how large the real one is.

Two parameters the model does not fix. The lone pair’s weight and the contact share are both dials. Every verdict above is a statement about a region of the plane they span, and the one measurement that could fix a point in it — the frequency of the motion over an edge, if the model’s energy had units — is not attempted.

A fixed bond length. All six Xe–F bonds are held at 1.89 Å and all six domains at one radius. In a capped octahedron the three fluorines round the opened face and the three opposite are not equivalent, and their bonds differ; a long bond goes to the crowded site, and letting them differ would lower the capped octahedron further.

And an energy with no zero-point motion. Barriers of a few parts in ten thousand of the total are comparable to vibrational energies in a real molecule, so whether xenon hexafluoride sits in one face or is spread over all eight is a question for the vibrational problem on this surface, not for its minima.

A domain no contact can see #

The same-rays result said a second set of points on the bond rays changes nothing under the same law and matters only as a law of another shape. For IF₇ that made the contact term a small correction to a shape the domains had already decided — flat or puckered, and the range decided which. For xenon hexafluoride the contact term decides the shape outright, and the reason is the one thing that differs: a lone pair has no atom at its end.

That turns a question the model could not answer into one it can. The shapes above six listed xenon hexafluoride among the cases where a lone pair “sometimes takes up space and sometimes does not”, with no way to represent either. With the fluorines in, the model represents one of them: the lone pair takes up space, but through a face of an octahedron the fluorines hold, not on the axis the domains alone would give it. Which angles belong to symmetry and which to the model was the question for IF₇’s ring; for xenon hexafluoride the symmetry itself is the model’s output, and it is set by a term between atoms that the lone pair does not have.

Still open: the bonds that differ, and the inactive limit #

The obvious open question is the bond lengths. In the capped octahedron the three fluorines round the opened face sit nearer the lone pair than the three opposite, and a crowded site takes the long bond. Letting the six bonds take two lengths — the same minimisation with two radii, which the IF₇ essays already use — would say whether the distortion deepens, whether the switch moves to a smaller share, and whether the edge-capped transition state becomes relatively higher, which is the difference between a molecule that wanders and one that sits in a face.

The nearer question is the inactive lone pair. The model cannot put a lone pair nowhere, but it can shrink its weight towards one and its radius towards the centre, and the regular octahedron with an inactive pair is the limit where the lone pair’s repulsion vanishes. Tracing the capped octahedron’s distortion as the weight falls towards one would say whether the model’s distorted octahedron connects smoothly to the regular one, or whether the pentagonal pyramid intervenes — and that is a question with a measured answer to hold it against, since the distortion of xenon hexafluoride is small.