The lone pair has no bond to stretch

The model lengthens the wrong three gave xenon hexafluoride’s six bonds two lengths by hand — three long, three short — and asked where the seven-domain repulsion model would put them. Wherever the capped octahedron was its lowest structure, it put the long three opposite the face the lone pair opens, which is the reverse of what electron diffraction and correlated calculations report. Only above a lone-pair weight of about 0.725 did the lone pair’s outward push on its neighbours win, and by then the face had given way to an edge.

That argument chose the lengths. Its closing question was the obvious repair: give every bond a natural length and a stiffness, let the repulsion stretch them, and minimise over directions and lengths together. Then the lengths are outputs, and the model has to say for itself which three are long and by how much.

Doing it turns up a question the fixed-length model never had to answer, and the answer to it decides everything else.

Seven domains, six springs #

The model is the one the lone pair goes through a face and its successors used: seven points around the xenon, one a lone pair, repelling each other as one over their separation, with every pair involving the lone pair weighted by q. Below a weight of 0.657 the lowest structure is the capped octahedron, the lone pair through one face of an octahedron of fluorines — the $C_{3v}$ structure the molecule is observed in.

Each of the six fluorines now sits at the end of a harmonic bond: its distance from the xenon may change, at a cost of half k times the square of the change. The repulsion pushes every domain outward, so every bond stretches until its spring balances the push. Lengths are in units of the unstretched bond.

The stiffness has no value in the model. The repulsion carries no energy scale — it is a rule about directions, and one over a distance in units of a bond is not a number of kilojoules — so a spring constant measured in the laboratory cannot be converted into one here. The honest way to enter it is as a free parameter and to report it through the one thing it controls: how far the six bonds stretch. At k = 1000 they stretch by a fifth to a third of a per cent; at k = 100 by two or three per cent; at k = 20 by seven to twelve; at k = 5 by a quarter to a third.

The question the fixed lengths never asked #

The fluorines are held by springs. The lone pair is not held by anything, because it is not at the end of a bond. When the six fluorines move outward, something has to be said about where the seventh domain goes, and the model contains no physics that says it.

Three answers are defensible.

The lone pair can keep its place among the bonds: its distance from the xenon is always the mean of the six bond lengths, so that the seven domains expand together. This is the reading closest to the fixed-length model, where every domain sat at one distance.

It can be left where it was, at the unstretched bond length, while the fluorines retreat.

Or it can sit on a bond’s spring of its own, pushed outward by the other six and held by the same stiffness as a bond — as though it were a ligand with no atom.

None of the three is the model’s. VSEPR describes a lone pair as a region of charge near the central atom, larger in angle than a bonding pair and closer in; it says nothing about how that region’s distance responds when the bonds lengthen.

At a weight of 0.5 and a stiffness of 20, with the bonds stretched by 8.7%, the three readings put the lone pair at 1.000, 0.925 and 0.970 of the mean bond. Left where it was, it ends up relatively nearer the xenon than the fluorines are; on its own spring, it moves out less than they do, because every repulsion it takes part in is weighted by q and at a weight below one it is pushed outward less than a fluorine is.

Three verdicts at one weight #

The split is the near three bonds’ mean length over the far three’s, minus one: positive when the long three are round the face the lone pair opens, which is what the molecule does.

Kept among the bonds, the lone pair reproduces the fixed-length verdict: negative below a weight of 0.725, positive above. Left where it was, the split is positive at every weight on the grid, from 0.05 to 2 — the long three are round the face throughout. On a bond’s spring it is positive at low weight and turns negative above about 1.3: the orientation the fixed-length model found, upside down.

At a weight of 0.5 the three readings give −0.010%, +0.084% and +0.027%. Two say the model gets the molecule’s orientation right and one says it gets it wrong, with the same seven domains, the same repulsion, the same springs and the same weight. The only difference is a statement about a domain with no atom at its end.

The mechanism is not obscure. A lone pair nearer the xenon than the fluorines makes a smaller angle with the near three at the same separation and so pushes them outward harder, relative to the far three, than a lone pair at the bond distance would. The long bond goes to the crowded site found the same rule in five-coordination: a bond that is pushed harder lengthens. The readings that pull the lone pair in relative to the bonds make the near three the most crowded, and the near three then take the long bonds.

The one reading the stiffness cannot move #

The lone pair kept among the bonds has a property the other two lack, and it can be stated before any number is computed. A repulsion that goes as one over the distance has no length scale of its own. Expanding all seven domains together by any factor scales every force by the same amount and leaves the directions where they were. So whatever the springs do to the overall size of the molecule, the balance between the near three and the far three is the balance at fixed lengths, scaled.

The computation agrees exactly. At six stiffnesses from 5 to 1000 — bonds stretched from a fifth of a per cent to a quarter — the split crosses zero at 0.7253 every time, and the scaled curves lie on top of one another. That is the fixed-length model’s crossing to four figures, and the stiffness does not move it.

So this reading adds nothing to the fixed-length answer. It is the fixed-length answer, recovered from a calculation in which the lengths were free.

The other two do not settle as the springs stiffen #

It would be reasonable to expect every reading to converge on the fixed-length verdict as the bonds become stiff, since in that limit nothing moves and the lone pair’s distance hardly matters. They do converge, but slowly, and the crossing they produce in between is anywhere.

Left where it was, the lone pair produces no crossing at all for any stiffness up to 50: the near three are long at every weight. At k = 100, with the bonds stretched by under two per cent, the crossing appears at 0.295. At k = 1000, stretched by under a fifth of a per cent, it is at 0.673 — still 0.05 from 0.725, with every bond within two parts in a thousand of its natural length.

On a bond’s spring, the crossing is at 1.06, 1.12 and 1.33 for the three softest stiffnesses, in the reversed orientation; vanishes at 50; reappears at 0.556 at 100 with the ordinary orientation; and reaches 0.714 at 1000.

The sensitivity is the point. At k = 1000 the lone pair left where it was sits at 0.998 of the mean bond — closer to the bonds than the precision of any measured Xe–F length — and it still moves the crossing by 0.05. A verdict that depends this much on the position of a domain nobody can locate does not belong to the model; it belongs to the convention.

A lone pair weaker than a bond put the capped octahedron in a window of weights below 0.657 and said the window was the model’s. The fixed-length analysis put the long three opposite throughout that window and said the orientation was the model’s too. With lengths free, the second statement does not survive. Whether the first does was not asked here: every structure was relaxed from the capped octahedron and kept its face, and the competing structures were not re-relaxed with springs.

How large the split is #

The orientation question has been argued for two essays as though the model’s split were the right size and only its sign were in doubt. It is not the right size.

The survey above six-coordination recorded the molecule’s distortion; electron diffraction puts xenon hexafluoride’s Xe–F lengths over a spread of about 0.08 Å around 1.89 Å, a split of 4.3%. Correlated calculations of the capped octahedron give 1.856 and 1.972 Å, a split of 6.25%.

The model’s splits are measured in hundredths of a per cent. At a stiffness of 100, where every bond stretches by between 1.6 and 2.9 per cent — the kind of stretch a modest ligand push would produce — no reading at any weight splits the two sets by more than 0.03%. That is a hundred and fifty times smaller than the diffraction spread.

Softening the springs makes the split grow, and it grows with the stretch rather than faster. The best ratio anywhere on the grid — split over mean stretch — is 0.046, from the lone pair left where it was at a weight of 2 and a stiffness of 5. At that ratio, reproducing the diffraction spread would need every bond stretched by about 94%. At the softest stiffness tried the bonds have already stretched by 31% and the split is 1.4%.

So the repulsion model does not get xenon hexafluoride’s bond lengths backwards or forwards. It does not produce them. The orientation the fixed-length analysis argued about is the sign of an effect two orders of magnitude smaller than the one in the molecule, and the sign of that small effect is set by a convention.

Why imposing the lengths hid this #

The fixed-length analysis imposed the measured ratio — three bonds at 1.972 and three at 1.856 Å in proportion — and asked which arrangement of them the repulsion preferred. That is a legitimate question and it has an answer. But it asks the repulsion to rank two structures that differ by a 6% split, when left to itself the repulsion produces a split of 0.01%. The energy differences being compared were differences between structures the model would never reach.

That is the general hazard of imposing an output and asking a model to adjudicate it. The adjudication can be perfectly well defined and still be about a regime the model does not inhabit, and the only way to see that is to free the output and look at where the model puts it. What a lone pair is worth measured the lone pair’s weight against angles the model does produce; lengths are a different kind of output, and the one this model produces is too small to carry the argument made about it.

What the springs leave out #

The springs are harmonic and identical: one stiffness for all six bonds, no anharmonicity, no coupling between stretches. A real Xe–F bond softens as it lengthens, and the fluorines round the face could have a different stiffness from those opposite. A softer near three would lengthen more, which would favour the molecule’s orientation without changing the size argument, since nothing within reason makes a 0.03% effect a 4% one.

There is no contact term. The atoms at the ends of the bonds were shown elsewhere to matter for seven-coordination, and a fluorine–fluorine contact would push the near three, which are crowded by one another as well as by the lone pair. The fixed-length analysis found the contact made the orientation worse; whether a contact with free lengths does the same is open, and it would bring its own range parameter.

And the lone pair is a point. A real lone pair is a distributed region whose centroid, size and shape all respond to the bonds; the three readings here are three ways of collapsing that response into one number, and the truth need not be any of them.

A domain without an end #

The seven-domain model treats a lone pair as a bond with no atom, and for directions that works: a point on a sphere has the same meaning whatever is or is not at the end of the ray. For lengths it does not work, because length is precisely the thing a bond has and a lone pair lacks. The model can say how hard the lone pair pushes; it cannot say where the lone pair is once the pushing has moved everything else, and that turns out to be the quantity the bond-length verdict depends on.

The same gap runs through the VSEPR rules as computed more generally. The rules are about angles and say lone pairs are bigger; bigger is an angular statement, and the radial statement that would go with it — closer to the nucleus, by how much, responding how — was never part of the model. In the fixed-length world nothing needed it. With lengths free, it is the first thing needed.

Who measured what #

The valence-shell electron-pair repulsion model is Gillespie and Nyholm’s, from 1957, and its treatment of xenon hexafluoride as a seven-domain case is Gillespie’s. Bartell and Gavin’s electron-diffraction study of 1968 established the molecule’s distortion from octahedral and the spread of its bond lengths; the correlated $C_{3v}$ lengths are MP2 values from studies of the xenon fluorides’ ionisation. The free-length minimisation and the three readings of the lone pair’s distance are new here.

How the lengths were found #

Seven unit directions and six bond lengths were relaxed together from the capped-octahedron seed by steepest descent with backtracking. The radial steps were scaled by one over the stiffness so that stiff springs did not throttle the angular relaxation. Each structure was taken as converged when the torque on the directions fell below 10⁻⁶ and the residual radial force over k — the length still unrelaxed — fell below 10⁻⁸ bond lengths.

Under the first reading the lone pair’s distance is the mean of the six and its radial force is shared among them; under the second it is fixed at one; under the third it is a seventh spring.

The grid was ten weights from 0.05 to 2 and six stiffnesses from 5 to 1000, under each of three readings. Crossings were found by bisection on the weight.

Several things had to hold. Every structure kept three equal near bonds and three equal far ones. The first reading’s crossing had to sit within 0.01 of the fixed-length value at every stiffness. The other two had to be closer to it at k = 1000 than at k = 100. The three readings had to disagree in sign at a weight of 0.5 and stiffness 20. At k = 100 every split had to be under 0.04%, and nowhere on the grid could any split reach the diffraction spread.

Still open: a lone pair with a radius of its own, and a contact #

The lone pair’s distance is the parameter this calculation found and could not fix. A model that gave the lone pair a radius determined by something physical — the centroid of a lone-pair hybrid on the xenon, the quantity the carbonyl’s lone pairs had computed in closed form — would replace three conventions with one value. Whether that value favours the molecule’s orientation is the obvious question, and the answer would need the hybrid’s centroid to respond to the bond lengths as well.

The other is the contact between fluorines, with lengths free. The fixed-length analysis found a contact made the orientation worse at every share it tried. Free lengths might change that, because a contact pushes the crowded near three outward directly, and it is the one term in the model that acts on fluorines as atoms rather than as the ends of rays. Nothing here suggests it could close a factor of a hundred and fifty in size.