The model lengthens the wrong three

A lone pair weaker than a bond found that the seven-domain repulsion model puts xenon hexafluoride’s lone pair through one face of an octahedron of fluorines — the capped octahedron, $C_{3v}$, the structure the molecule is observed in — for every lone-pair weight below 0.657, with no contact between the fluorines at all. It held all six bonds at one length, and said so, and its closing question was the obvious one. The three fluorines round the opened face sit nearer the lone pair than the three opposite. Let the two sets take two lengths, and see what the model does with them.

The molecule’s own answer is not in doubt. The survey of shapes above six-coordination recorded the molecule as a slightly distorted octahedron, and the lengths are part of that distortion. The electron-diffraction analysis of Bartell and Gavin found the Xe–F lengths spread over about 0.08 Å, with the longer bonds tending to be the ones next to the region of the coordination sphere the bonds avoid. Correlated calculations of the $C_{3v}$ structure say the same more sharply: MP2 lengths of 1.856 Å for three bonds and 1.972 Å for the other three, the long three round the lone pair. And the rule for which site a long bond takes — a long bond goes where its domain is crowded, because moving it outward relieves the most repulsion there — would seem to predict exactly that, since a lone pair next to three bonds crowds them.

It does not, in this model. Given three long bonds and three short, the repulsion model puts the long three opposite the lone pair wherever the face is its lowest structure, with or without a contact between the fluorines, at every ratio of lengths tried.

Which three are crowded is a question about the weight #

The calculation is the previous one with one change. The seven points still sit on rays from the xenon, the lone pair’s ten pairs still carry a weight q, and every pair still repels as one over its separation — but the three fluorines round the opened face now sit at one distance and the three opposite at another. The lengths are given, not found: the long three at $\sqrt{\rho}$ and the short three at $1/\sqrt{\rho}$, with $\rho$ the ratio, and the lone pair at one. Exchanging the two sets is then the same molecule as inverting the ratio, which is a check worth having and one the calculation passes to nine figures. For each weight the two isomers are relaxed from the capped octahedron’s seed — the long three round the face, and the long three opposite — and their energies compared, at the MP2 ratio of 1.972 to 1.856, which is 1.0625.

With the domains alone the difference is positive — the long three go opposite — from the inactive limit up to a weight of 0.727, and negative above it. The face is the lowest structure only below 0.657. So the window in which the model would put the long bonds next to the lone pair, between 0.727 and the weights where the pentagonal pyramid takes over, is a window in which the model’s lowest structure is not a face at all: it is the edge-capped octahedron up to one, and the pyramid after that. Wherever the model draws xenon hexafluoride as a capped octahedron, it lengthens the wrong three.

The crossing hardly cares about the ratio. At ρ = 1.02 it is 0.7255; at the MP2 ratio, 0.7271; at 1.1, 0.7297. A difference that was large enough to matter would move it, and the fact that it does not says what kind of quantity it is: a first-order one, the sign of a derivative, decided before the lengths differ by enough to change the geometry.

The fluorines’ contact makes it worse, not better. The upper curve is the same difference with a Born–Mayer contact — the shape of repulsion that decided whether IF₇’s ring puckers — at a range of 0.345 Å worth one per cent of the domain energy, the range and the normalisation the lone pair’s route through a face used. It never changes sign. At half a per cent the smallest difference along the whole walk from a weight of 0.05 to 2 is still positive, at 5.5 × 10⁻⁴; at six per cent the smallest is 2.6 × 10⁻³. The contact acts between fluorines, and it pushes on the fluorines that are closest to one another — which are not the ones round the opened face.

Two pushes that nearly cancel #

Why the opposite three should be the crowded ones is visible before any two-length structure is relaxed. Take the equal-length capped octahedron at a weight, and ask how hard the energy pushes outward along each bond — minus the derivative of the energy with respect to that bond’s length. A set of bonds pushed outward harder gains more from being the long set, so the difference between the push on the near three and the push on the far three predicts the sign of the comparison above, to first order in the length difference. It is a second route to the same answer with nothing in common with the first: one equal-length relaxation and a derivative, instead of two relaxations at two lengths and a subtraction.

Split by source, the push has two large parts of opposite sign. The lone pair’s repulsion pushes the three bonds round its face outward harder than the three opposite, by an amount that grows with its weight: 0.27 in the model’s units at q = 0.5. That is the crowding the rule expects. But the six fluorines also repel one another, and in the capped octahedron the three opposite the lone pair are the three closest together — the lone pair opens its own face by pushing those three apart, and the far three close up to make room. Their mutual repulsion pushes the far bonds outward harder, by 0.28 at the same weight.

The two parts cancel to within three per cent. What is left is small and of the fluorines’ sign: at q = 0.5 the net difference is −0.0078, against parts of 0.271 and −0.279. The lone pair’s share rises with q faster than the fluorines’ does, and the two balance at q = 0.725 — the crossing the relaxed isomers put at 0.727, found here without relaxing either. That agreement is the check on the whole calculation, and the closeness of the cancellation is the reason the crossing barely moves with the ratio: the decision is made in the third significant figure of two numbers that are each far larger than it.

So the crowded-site rule is not wrong here. It is answering a question with two candidates, and the crowding that counts is not the crowding a chemist looks for. A lone pair weaker than a bond is, by construction, a weak crowder; three fluorines squeezed together by the lone pair’s own opening of a face are a strong one. Five sites are not alike because a bipyramid has two kinds of site; the capped octahedron has two as well, and which of them is crowded is set by the same weight that sets the shape.

The whole plane says the same #

The two curves above are two rows of a larger survey. The contact share and the lone-pair weight are the model’s two dials, and the earlier essays mapped which structure is lowest over both. The same map, with the preference for long bonds round the face marked on it, is the direct test of whether any setting of the dials gives the molecule’s pattern.

Of the 210 cells — thirty weights and seven contact shares — the face is the lowest structure in 94, and the long bonds go round the opened face in 23. No cell is both. The 23 lie in a single row, the one with no contact, at weights from 0.8 to 3; on that row the face stops being lowest after 0.6, before the dots begin. On every other row there are no dots at all, because the contact keeps the preference on the opposite side at every weight.

The map also shows the contact doing what the earlier essays found: it widens the face’s region, from 0.6 with no contact to 2.3 at six per cent. That was the contact route to xenon hexafluoride’s shape — a lone pair heavier than a bond, pushed into a face by the fluorines’ repulsion. The bond lengths turn that route against itself. The same contact that pushes the lone pair into the face pushes the far three’s bonds outward, and does so more strongly the more of it there is. The larger the contact share needed to make the face, the more firmly the model lengthens the wrong three.

The two routes to a face were told apart by how far each opens the octahedron, and the weak-pair route came out ahead because xenon hexafluoride’s distortion is small. The bond lengths tell them apart in a different way, and neither comes out ahead: both get the pattern backwards.

What the long bonds would have done to the angles #

The lead asked a specific question — whether a light lone pair’s small distortion gets smaller still when the near bonds lengthen — and it has a definite answer, even though the model would not lengthen them.

It does. Moving the three bonds round the opened face outward moves their fluorines away from the lone pair, which then pushes them apart less; the face closes a little and the octahedron is more nearly regular. At a weight of 0.3 the spread of the cis angles falls from 14.24° with one length to 13.87° with the long three round the face. The reverse placement does the reverse: the long three opposite move away from each other, the squeeze on them is relieved, and the lone pair can open its face further — 14.80° at the same weight. Across the face region the two isomers differ by about a degree. Even at the inactive limit neither is regular: a weightless lone pair leaves both about 1.2° from an octahedron, because six points at two distances on fixed rays do not repel each other into the arrangement six points at one distance take.

So the answer to the lead is a split one. If the molecule’s pattern of lengths were imposed on the model, the distortion would shrink, and a lone pair at a given weight would look slightly less active than with equal bonds. Left to choose, the model goes the other way, and the face it prefers is more distorted than the equal-length one. Neither shift is large against the distortion itself — a third of a degree on fourteen — and neither changes which weights give a small distortion; the earlier reading that a mild distortion comes most directly from a light lone pair survives either choice of lengths.

Three long bonds need not keep the symmetry #

The comparison so far has been between the two placements that keep $C_{3v}$: all three long bonds round the face, or all three opposite. A composition of three long and three short can be placed on the six sites in twenty ways, and most of them break the symmetry. Relaxing every one of them from the capped octahedron, and keeping those that stay in the face, sorts them into six kinds — by how many of the long three are round the face, and by whether two of them are trans to each other across the xenon.

With the domains alone the lowest is neither $C_{3v}$ isomer. It is a mer placement: one long bond round the face, its trans partner opposite, and one more opposite — a long axis through the xenon plus one more long bond. It is lowest at every weight from 0.05 to 0.6, by 6.3 × 10⁻⁴ at the small end and 1.7 × 10⁻⁴ near the middle. Its preference has nothing to do with the lone pair: three long bonds on an octahedron would rather put two of themselves on one axis, which is the arrangement six points of two radii take when nothing else is pushing. With a one-per-cent contact the picture is tidier. The $C_{3v}$ isomer with the long three opposite is the lowest from a weight of 0.2, by 2.5 × 10⁻⁴ to 1.3 × 10⁻³ over the next mer, and only at the lightest weights does the mer placement keep its lead.

One placement is never the lowest in either panel: the $C_{3v}$ isomer the molecule has, with the long three round the face. It lies between 2.6 × 10⁻⁴ and 6.8 × 10⁻⁴ above the opposite isomer with the domains alone, and between 8 × 10⁻⁴ and 5 × 10⁻³ with the contact. Asked to distribute xenon hexafluoride’s own bond lengths, the model’s least favourite of the symmetric answers is the true one.

The crossing falls #

The earlier essay’s crossing at 0.657 — where the face gives way to the edge-capped structure as the lowest — was a property of six equal bonds. With two lengths each structure can take its best placement of them, and the edge-capped octahedron has more to choose from: its lone pair sits over an edge with two near fluorines, two at the sides and two far, so three long bonds can be split between kinds of site in more ways than the face allows.

The crossing falls steadily with the ratio. At one length it is 0.657; at ρ = 1.02, 0.615; at the diffraction spread, a ratio of 1.043, 0.572; at the MP2 ratio, 0.539; at 1.1, 0.485. The weak-pair region where the face is lowest shrinks by a sixth at the lengths a calculation reports. It does not vanish, and nothing in it changes the earlier reading that the face reaches down continuously to the regular octahedron at zero weight — but the edge region, which was the novelty of that essay, grows into it from above, and a lone pair at two thirds of a bond that put the molecule in a face with equal bonds puts it over an edge with the bonds a calculation gives.

How the comparison was made #

Each structure is seven unit vectors, moved by steepest descent on the sphere of directions with the lengths held fixed, from a seed of the structure’s own symmetry; descent from a symmetric seed keeps its symmetry, so each $C_{3v}$ isomer is relaxed within $C_{3v}$, and the broken placements are found by starting from every one of the twenty assignments of long bonds. A structure is called a face, an edge or an axis from the angles between the lone pair and the six fluorines after relaxation, and a placement counts as staying in the face only if the three fluorines nearest the lone pair are the three the seed put there. The contact is a Born–Mayer exponential in the true F–F distance at a range of 0.345 Å, weighted as a share of the pentagonal pyramid’s domain energy as before, so the map’s contact axis is the one the earlier maps used. The outward push is the derivative of the full energy along each ray, evaluated on the equal-length structure, and split according to whether a pair includes the lone pair.

The checks, run wherever these figures are drawn. The refusal comes first and is the inactive limit: at a weight of 0.001 the six sites of the octahedron differ only by the lone pair’s vanishing weight, so which three bonds are long cannot matter, and the two isomers must agree to a fiftieth of their difference at fifty times the weight. They agree to 6 × 10⁻⁶ against 2.6 × 10⁻⁴. Exchanging the long and short sets must be inverting the ratio, to nine figures. Both isomers must keep their face at every weight walked. The sign must change once, between 0.657 and one, at every ratio tried, and the relaxed and first-order crossings must agree to a hundredth. Every face-lowest cell of the map must prefer the long three opposite; every positive contact share must prefer them opposite at every weight; the lowest face placement must be mer with the domains alone and the opposite $C_{3v}$ isomer from a weight of 0.2 with a one-per-cent contact; and the face–edge crossing must fall with the ratio from 0.657.

What a fixed bond length cannot say #

The lengths are put in, not found. The model has no term that sets a bond’s length, so it cannot predict that any bond is longer than another; it can only say which placement of given lengths it prefers, which is what the outward push measures. A model with a stretching term — a harmonic bond pulling each fluorine towards a natural length, against the repulsion pushing it out — would let the lengths come out, and the first-order push is exactly the force that would lengthen them. That is why the sign here is a statement about the model’s forces rather than about a choice it was given: in this model’s own forces, the far three are the ones pushed out.

The lone pair is one point with one weight, which is all VSEPR computed as a repulsion has ever given it. It is on the same sphere as the bonds and it repels angles and lengths through the same number. A real lone pair’s effect on its neighbours’ lengths and its effect on their angles need not be set by one parameter, and a model whose lone pair could be weak in the one and strong in the other is a different model, not a correction to this one.

One contact law at one range. Every contact share mapped is Born–Mayer at 0.345 Å. A shorter range acts more on the closest fluorines, which are the far three here, so the direction of the result is unlikely to depend on the choice; the size of the preference does.

Symmetric seeds and the twenty placements. The search does not nudge two-length structures off their seeds the way the equal-length maps did, so a lower structure of no symmetry with two lengths is not excluded.

A crowding count the molecule does not keep #

The sites are not the same size gave the repulsion model a bond length for the first time and found the long bond going axial in a trigonal bipyramid, against the rule the model is usually cited for. The reason was a crowding count, and at seven sites the count reversed: IF₇’s long bonds sit in its crowded equatorial ring, as the molecule’s do. Xenon hexafluoride is the case where the crowding count is decided by the lone pair’s own weight — light, and the crowded set is the one the lone pair squeezes together; heavy, and it is the set the lone pair pushes on directly — and the molecule’s lengths say the second while its shape needs the first.

That is the finding worth carrying. The repulsion model’s account of xenon hexafluoride’s shape and its account of the molecule’s bond lengths need the lone pair’s weight on opposite sides of 0.727. Below it the shape comes out right and the lengths wrong; above it, the reverse — and the weights fitted to main-group hydrides, 1.15 to 2.87, all lie above it, where the model says the lone pair is strong enough to lengthen its neighbours and too strong to sit in a face without a contact that then lengthens the others. No single weight reproduces both halves of the observed structure.

Who found what #

The electron-diffraction analysis of xenon hexafluoride and its distribution of distorted $C_{3v}$-like structures, with the longer bonds next to the avoided region, is Bartell and Gavin’s, from the late 1960s. The MP2 bond lengths quoted here are those a study of the xenon fluorides’ ionisation took from earlier correlated calculations of the $C_{3v}$ structure; pseudopotential SCF studies of the same molecule had already described its distortion as a bend and a stretch together with the long bonds next to the lone pair, and explicitly correlated coupled-cluster calculations later confirmed the $C_{3v}$ form as the lowest conformer. The lone-pair clause is Gillespie and Nyholm’s.

What is computed here is the repulsion model’s own preference for where xenon hexafluoride’s long bonds go, its decomposition into the lone pair’s push and the fluorines’, the map of that preference over weight and contact share, and what two lengths do to the distortion, the placements and the face–edge crossing.

Still open: lengths the model sets itself, and a lone pair with two weights #

The obvious open question is the stretching term. Give each bond a natural length and a stiffness and let the repulsion stretch it, and the lengths become outputs: the first-order push says which three would lengthen, but a self-consistent calculation would say by how much, whether the near three can be made the longer set at any weight once the geometry responds, and what the ratio is at the weight where the face is lowest. That is a minimisation over fourteen angles and six lengths instead of fourteen angles, and the only new parameter is a stiffness that a measured stretching frequency would fix.

The nearer question is the lone pair’s single weight. The angles need it below 0.657 and the lengths need it above 0.727, and the obvious repair is to give the lone pair one weight for the angular repulsion and another for the radial push. Whether a pair of weights exists that reproduces both the capped octahedron and its long bonds round the face — and whether those two weights mean anything beyond a fit — would say whether the repulsion model can describe xenon hexafluoride at all, or only one half of it at a time.