The ring held flat by its long bonds

The crowded site takes the long bond. At five-coordination the crowded site is axial, so a long bond goes axial; at seven, in the pentagonal bipyramid, it is equatorial, because an equatorial site has two neighbours at only 72°. Phosphorus pentafluoride’s longer bonds are axial and iodine heptafluoride’s are equatorial, and one reading of crowding predicts both signs.

That argument placed one odd bond at a time. IF₇ does not have one odd bond. It has five long ones round its equator — 1.858 Å — and two short ones on its axis at 1.786, a ratio of 1.040. And it has a property the single-bond calculation could not touch: its equatorial ring is reported to pucker dynamically, the five fluorines rising and falling out of the plane in a motion fast enough that the molecule is described as fluxional rather than as having a puckered structure.

So there are two questions a whole composition raises and one odd bond does not. Where does the model put two short bonds among seven sites — and once it has put them somewhere, is the ring flat?

Where the model puts two short bonds #

A pentagonal bipyramid has two kinds of site, so two short bonds can be placed four ways: both on the axis, one on the axis and one in the ring, or both in the ring — next to each other or across it.

Both on the axis is the lowest. That is the measured structure, and it is what the single-bond result predicts: a short bond prefers the uncrowded axial site, and there are exactly two of them. The placement is a genuine minimum, and relaxes to $D_{5h}$ with nothing asked of it.

The unit the others are priced in is the cost of one short bond placed in the ring rather than on the axis, computed in a composition of six long bonds and one short: 2.25 × 10⁻³, in a total energy near fourteen. If two short bonds simply added, a placement with one of them in the ring would cost one unit and a placement with both in the ring would cost two.

Across the ring they add. Two short bonds 144° apart cost 2.017 units, within one per cent of two.

Close together they do not. One on the axis and one in the ring, 91° apart, cost 1.147 units rather than one. Two in the ring side by side, 73° apart, cost 2.335 rather than two — and that placement is not even a minimum: a kick of three hundredths lowers its energy and it slides into a structure of mirror symmetry only. So the penalty for a short bond in the ring depends on what else is nearby, and it is largest when the other short bond is one of its 72° neighbours, which is exactly the contact the single-bond argument identified as the source of the ring’s crowding.

That is worth having because it is the first place in this argument where the crowding count has to be applied to a pair rather than to a site. The count says a short bond is worse off at a crowded site; the pair result says two short bonds are worse off still when each is part of the other’s crowding.

The ring is flat, and barely #

The lowest placement has a flat ring. Relaxed from the ideal pentagonal bipyramid and from three random kicks, it returns to $D_{5h}$ every time. By the test the single-bond calculation used — does a small disturbance lead anywhere lower — IF₇’s ring is a minimum, and the model does not predict a puckered molecule.

That test answers yes or no. It does not say how firmly the answer is held, and for a molecule described as fluxional the firmness is the whole question. The way to ask it is to compute every curvature of the energy at the flat structure: the second derivative in each of the independent ways the seven sites can move on their sphere.

Seven points on a sphere have fourteen coordinates, three of which are rigid rotations of the whole arrangement and must cost nothing; they come out at zero to the precision of the arithmetic, which is the check that the curvatures are being measured rather than manufactured. The other eleven are the arrangement’s internal motions.

With equal bonds the lowest pair is 0.0092. The next is 0.963. The pucker is a hundred times softer than anything else the arrangement can do. It is a degenerate pair because a pentagon has two independent ways to pucker — two phases of the same wave running twice round the ring — and those are the motions the molecule is described as making.

IF₇’s longer ring bonds raise the pair to 0.0272, three times stiffer, while every other motion softens a little. The pucker is still thirty-four times softer than the next motion. The molecule’s ring is flat in the model, and it is flat in the way a coin balanced on a slightly concave table is flat.

What holds it flat #

The single-bond argument explains why the pucker is soft. An equatorial site has two neighbours at 72°, closer than anything an axial site has, and a pucker moves every ring site away from its two nearest neighbours: half of them up and half down, so that each pair of adjacent sites separates. The price is that each ring site moves towards an axial site. At equal bonds the two effects almost cancel, and the small remainder is the 0.0092.

It is the pucker specifically, not any out-of-plane motion. The ring has another way to leave its plane: tilt as a whole, one side rising and the other falling, the pattern that runs once round the ring rather than twice. Its curvature at equal bonds is 2.65, more than a hundred times the pucker’s 0.023 measured the same way. A tilt moves the ring sites on one side towards an axial site without separating any adjacent pair, so it pays the full price and gains nothing. The pucker alternates round the ring, which is what separates neighbours, and that alternation is only possible in a ring with an odd number of sites by leaving one pair on the same side — the envelope shape the unstable cases below settle into.

And because the pucker is a degenerate pair, it has no preferred phase at this order: a wave running twice round five sites can start at any site, and every starting point costs the same. That is the harmonic content of the word fluxional for this molecule. At five-coordination the exchange of axial and equatorial sites runs between two distinct minima through a barrier, and the group that describes a molecule doing it is larger than its structure’s. A soft degenerate pair is a different kind of floppiness — no second minimum, and a continuous family of equivalent distortions round the ring — and it is the kind the model gives IF₇.

Lengthening the ring bonds moves the ring sites outward and apart, which relieves exactly the crowding the pucker would relieve, so there is less for a pucker to gain. Shortening them does the opposite.

Under the Coulomb law every shape in this collection is computed with, the flat ring stays a minimum down to a ratio of 0.983. A ring whose bonds were two per cent shorter than its axial bonds would pucker in this model, and one whose bonds are four per cent longer, as IF₇’s are, sits comfortably on the flat side — comfortably by the standard of a motion that is still thirty-four times softer than the next.

The curve crosses zero at a ratio that depends on the law. Under 1/r³ the crossing moves to 1.017. Under 1/r² it is at exactly one.

The law at which the ring changes character #

That last number deserves a figure of its own, because it is not approximately one.

Under 1/r the energy rises as the square of the pucker: slope two on logarithmic axes, as for any ordinary minimum. Under 1/r² the slope is four. The quadratic term is missing entirely — the coefficient that should multiply the square of the pucker comes out below 10⁻⁸ — and the ring is held flat only by the fourth power. At the smallest pucker computed, the energy divided by the pucker’s fourth power is 5.937 and not moving; divided by its square it is 0.00015 and falling towards zero.

This connects to something already on record. Swept across repulsion laws, the seven-point minimum is a pentagonal bipyramid under 1/r and 1/r², a low-symmetry arrangement under 1/r³ and 1/r⁴, and a threefold arrangement under 1/r⁶ and steeper. That was a statement about which structure a descent lands on. The curvature says where the change happens and why: the equal-bond pentagonal bipyramid stops being a minimum exactly at the inverse-square law, and the soft mode that goes first is the ring’s pucker. The inverse-square law is not one of the laws in the first regime; it is its edge.

A marginal mode at a round exponent is the kind of result that suggests an identity rather than a coincidence, and nothing here proves one. It holds to the precision the arithmetic carries, at a ratio of exactly one and at no other.

How steep a law IF₇’s bonds can survive #

IF₇’s ring bonds stiffen the pucker under 1/r, and they do so under steeper laws as well, but a steeper law weighs the ring’s 72° contacts more heavily, so the bond-length ratio needed to hold the ring flat rises with the exponent.

The critical ratio climbs from 0.983 at 1/r through exactly one at 1/r² and 1.017 at 1/r³ to 1.066 at 1/r⁶ and 1.148 at 1/r¹². IF₇’s ratio of 1.040 crosses the curve at an exponent of 4.38. Under any repulsion softer than about the 4.4th power, IF₇’s own bond lengths hold its ring flat in this model; under any steeper one they do not.

That turns the measured fluxionality into a statement about the model rather than a vague agreement with it. A molecule whose ring is flat on average and puckers dynamically is a molecule whose flat structure is a minimum with a very soft mode — which is what the model gives under the softer laws. A molecule whose ring were statically puckered would be one the model could only reproduce under a steep law. IF₇ is the first kind, so if bond-pair domains repel by any power law at all, the molecule’s own structure says the power is not steep.

What a ring does when it cannot stay flat #

The cases on the wrong side of the curve are worth relaxing, to see what the model offers instead of a flat ring.

A ring two per cent too short under 1/r — a ratio of 0.95 — settles into an envelope: one site rises 8.3°, its two neighbours drop 6.6°, and the far pair rise 2.5°. The arrangement keeps one mirror plane and nothing else. The energy gained is 2.6 × 10⁻⁴, a hundredth of the placement penalties above. Equal bonds under 1/r³, and IF₇’s own ratio under 1/r⁶, give the same envelope at a similar size.

Two cases go further. At a ratio of 0.9 under 1/r the envelope deepens to 33°, and the axis itself bends to 140° as the two axial sites lean away from the rising site. Equal bonds under 1/r⁶ do not stop at an envelope at all: the axis bends to 155° and the arrangement becomes a capped octahedron with a threefold axis, which is the structure a single very long bond was found to produce and the structure the law sweep finds at that exponent. A pucker is where the instability starts, and at the steep end it is a way out of the pentagonal bipyramid rather than a variant of it.

What was computed and how #

Every structure is seven points on a unit sphere, each scaled by its bond’s relative length, with an energy that sums $1/r^p$ over all pairs; $p = 1$ unless a law is named, as throughout these essays. The placements relax by 40 000 steps of descent from the ideal pentagonal bipyramid, then by 80 000 more, and are kicked three times by up to 0.015 in each coordinate and relaxed again, exactly as the single odd bond was.

The curvatures are finite differences, and the one along the pucker takes two step sizes and removes the error that grows with the square of the step — which matters, because at 1/r² that error is the quartic term and would otherwise be mistaken for a curvature of 1.2 × 10⁻⁵. The full set of curvatures is the eigenvalues of the fourteen-by-fourteen matrix of second derivatives in two tangent coordinates per site. Critical ratios and the critical exponent are found by bisection on the sign of the pucker’s curvature. Symmetry labels come from the same finder that labels every molecule here.

The table carries the two halves of the result side by side. Reading down the first column, the equal-bond ring goes from barely stable to exactly marginal to unstable as the law steepens; reading down the second, IF₇’s own ring survives three laws that the equal-bond ring does not survive two of. The tilt column never approaches zero, which is the check that the instability is a property of the pucker and not of the ring’s whole out-of-plane freedom.

The claims are stated where they can fail: that the axial placement is the lowest of the four and a minimum; that two short bonds across the ring cost two single penalties within four hundredths while the two close placements exceed their count by more than a tenth; that the pucker is more than fifty times softer than the next motion at equal bonds under 1/r; that its curvature at equal bonds vanishes under 1/r² while the energy’s ratio to the fourth power settles; that every steeper law gives a negative curvature at equal bonds; and that IF₇’s ratio has a critical exponent between four and five. The refusal is the three rigid rotations, whose curvatures must be zero.

Where the model stops #

A curvature is not a frequency. The model’s energy has no units and its sites have no masses, so the pucker being a hundred times softer than the next motion says nothing directly about wavenumbers. It says the pucker is the motion to expect first and largest, which is what the molecule’s large-amplitude ring motion is; it does not predict how large.

The domains are points. An iodine–fluorine bond is represented by a point on a sphere at a distance proportional to its length, repelling every other point by one law. Fluorine atoms at the ends of the bonds, which crowd each other in the ring as much as the bond domains do, are not in the model; the single-bond argument found that end atoms read the same crowding the other way round, so a model that included them would have a second term pulling the ring towards the pucker.

The measured puckering is dynamic and its amplitude is not quoted here. Electron diffraction and vibrational analyses of IF₇ agree that it is a pentagonal bipyramid with a large-amplitude ring motion — the reading given by Christe, Curtis and Dixon in 1993 — and a vibrationally averaged structure is not a static one. What the model can be held to is the sign: a flat minimum with a very soft pucker, not a puckered minimum.

The generalisation #

The single-bond calculation asked which site a long bond suits and answered with a count of neighbours. This asks how firmly the answer is held, and answers with a curvature — and the two readings disagree in a way that matters. The count says the ring is crowded, so a pucker should help. The curvature says the pucker helps almost exactly as much as it costs, so that whether it helps at all is decided by a four per cent difference in bond length and by the exponent of a law nobody states.

A structure that the model predicts can be right for a reason that holds by very little. IF₇ has the geometry the repulsion model gives it, and its flatness is held by one coordinate in eleven, a hundred times softer than the rest, which the molecule’s own bond lengths make three times firmer. It is also a different kind of agreement from the one the sites are not the same size began with, where the question was which site a bond suits and the answer was a sign. Here the sign is not in doubt — the ring is flat — and what the molecule tests is a magnitude: the curvature of one mode relative to ten others. Above six coordination the arrangements were already known to sit close together in energy; this puts a number on how close the pentagonal bipyramid sits to its own neighbours, and finds the number set by the bond lengths the model is given rather than by the repulsion it computes.

That is a stronger agreement between model and molecule than a correct symmetry label, because it predicts the fluxionality as well as the shape, and a more fragile one, because a steeper repulsion would take both away.

Still open: the end atoms, and a ring that is shorter than its axis #

The obvious open question is the fluorines. The end atoms of the five ring bonds sit 72° apart on a larger sphere, and at IF₇’s bond lengths they are closer together than the bond domains are. Adding a second set of repelling points at the bond ends — the calculation that would test the bulky-atom half of the crowding argument at seven — would say whether the end atoms push the pucker over the edge that the domains alone stop short of, and whether IF₇’s softness is a property of its bonds or of its fluorines.

The obvious comparison is xenon hexafluoride, the other famously fluxional seven-domain molecule, whose seventh domain is a lone pair rather than a bond. The same curvature analysis, with the lone pair’s weight in place of a bond length, would say whether its softness is the same pucker or a different mode entirely.

The nearer question is the other side of the curve. A pentagonal bipyramid whose ring bonds are shorter than its axial bonds is on the puckered side of the model under every law, and the model says it should be puckered statically, not dynamically, with an envelope of mirror symmetry. Seven-coordinate species with a short equatorial ring would be the sharpest test of that — the model’s first prediction of a distortion that is not driven by a lone pair.