The fluorines' range decides the ring

The ring held flat by its long bonds found iodine heptafluoride’s equatorial ring flat in the repulsion model, and flat by very little. The ring’s pucker is the arrangement’s softest motion by a factor of a hundred with equal bonds. IF₇’s ring bonds are four per cent longer than its axial ones, which makes the pucker three times stiffer, and that holds the ring flat under every power law softer than about 1/r^4.4.

Every point in that calculation was a bond domain. The five fluorines round the equator were not in it, and they are the obvious omission: at IF₇’s bond lengths two neighbouring ring fluorines are 2.184 Å apart, a ring fluorine and an axial one 2.577 Å, and two closed shells of fluorine repel each other hard at that distance. The last essay named the question: put the end atoms in as a second set of repelling points, and see whether they push the pucker over the edge the domains alone stop short of.

The first thing the question needs is a statement of what a second set of points can change. That turns out to be less than it sounds.

A second set of points on the same rays is the first set again #

A fluorine sits on its bond’s ray at the end of the bond. The model’s bond domain sits on the same ray, at a distance proportional to the bond’s length. Every calculation since the sites were given lengths has put the domain at the full bond length, but nothing in the model says where along the bond it belongs — a domain is a region of electron density, and its centre is somewhere between the two nuclei.

Suppose the domains sit at some fraction d of their bonds and the end atoms at the ends. If both repel by the same power law, $1/r^p$, then every distance between two domains is exactly d times the distance between the corresponding end atoms, whatever the arrangement. So the domains’ energy is the end atoms’ energy multiplied by $d^{-p}$, for every arrangement at once. The two sets of points carry the same function of shape, and adding one to the other multiplies that function by a constant.

Under 1/r and under 1/r¹² the rescaled curvature is 1.000 at every one of nine fractions from 0.6 to 1. Where a point sits along its bond is invisible to a power law, and so is the difference between a domain and the atom at the end of its bond. More of the same repulsion changes no angle, no curvature and no verdict — a result the law sweep already implied without saying, since it found the arrangements for four to six domains unmoved by a factor of twelve in the exponent.

So the fluorines can only matter if they repel by a different law from the domains. And that is exactly what the physics of closed-shell contact says they do. The repulsion between two filled shells comes from the exclusion principle forcing their overlapping electrons into higher levels — the antibonding level rises by more than the bonding one falls — and the contact calculation for the noble gases built it as the square of an overlap that decays exponentially. It has a length scale of its own, which a power law does not, and it is much steeper at contact than the Coulomb repulsion the domains carry.

The exponential curve in the figure shows what a length scale does. Moving the points of exp(−r/0.25 Å) inward along their bonds changes which contacts it can reach, and at 0.723 of the bond its verdict on the pucker changes sign. That number is not independent of the ones below: moving every point to a fraction d of its bond is exactly the same, for an exponential, as leaving them at the ends and stretching the range to ρ/d. And 0.25 ÷ 0.723 is 0.346 Å, which will turn out to be the range that decides everything.

How little contact it takes #

The calculation that follows is therefore the domain model’s Coulomb energy plus a contact term of a chosen law, carried by the end atoms, with a weight. The weight is the part the model cannot supply. The domains’ energy is in arbitrary units, and no measurement fixes how hard two ring fluorines repel in those units. What can be computed is the weight at which the flat ring stops being a minimum, and it is a single division. The curvature along the pucker is a sum of the two terms’ curvatures, linear in the weight, so it crosses zero where the contact term’s negative curvature cancels the Coulomb term’s positive one.

The weight is reported as a share: the contact term’s energy at the flat structure divided by the Coulomb term’s. That is the one normalisation that does not depend on the units of either.

A steep contact puckers IF₇’s ring with a few thousandths of the energy. A 1/r⁶ contact needs 2.8 per cent, 1/r⁸ needs 0.68 and 1/r¹² needs 0.16. An exponential contact with a range of 0.2 Å needs 0.24 per cent. The end atoms do not have to repel strongly compared with the domains; they have to repel at all, with a steep enough law.

The long ring bonds still do their work, and the open circles measure it. With equal bond lengths every one of these laws needs four to eight times less — 1/r¹² tips an equal-bond ring at 0.038 per cent. IF₇’s four per cent of extra ring length buys a factor of about four in how much fluorine repulsion the ring can absorb before it goes. What it does not buy is a margin that any plausible contact term would respect.

The two laws at the top and bottom of the list are the exceptions, and they are not the same exception. A 1/r⁴ contact holds the ring flatter at any weight, which the domain-only calculation already required: its critical exponent at IF₇’s ratio was 4.38, and a contact law’s own curvature has the sign that calculation gave. The long-range exponentials are the surprise.

Which contacts pull, and which push back #

The reason a contact term can favour the pucker at all is geometric, and it can be read off by splitting the term’s curvature by kind of contact.

In every law the only contribution in the pucker’s favour comes from the five pairs of adjacent ring atoms. A pucker sends alternate ring atoms up and down, so each adjacent pair separates. The price is paid by the ten ring-to-axis pairs: every ring atom that rises moves towards one axial atom, every one that falls moves towards the other. The across-ring pairs, 3.53 Å apart, resist a little under 1/r and hardly at all under a steep law. The axial pair does not see the pucker, since neither axial atom moves.

So the sign of any contact term is a competition between two distances: 2.184 Å, where the pucker relieves, and 2.577 Å, where it costs. The contacts that relieve are 18 per cent closer. A law steep enough to weight them heavily wins; a law that barely distinguishes them loses, because there are twice as many ring-to-axis pairs. Under 1/r the relieving term is −3.31 and the resisting ones sum to +3.38, a difference of two per cent — which is the domain model’s whole margin, seen from inside. Under 1/r¹² the across-ring pairs drop out and the relieving term wins outright, −4.34 against +2.58. Fluorine’s own contact, the third row, is steeper still at these distances.

The range decides the sign #

An exponential contact is the law closed-shell repulsion actually approximates, and its range is the parameter that matters. The range sets how strongly the law separates 2.184 Å from 2.577 Å: a short range makes the nearer contacts dominate, a long one makes the two nearly equal.

The share needed rises steeply with the range — 0.09 per cent at 0.15 Å, 0.68 at 0.25, 2.5 at 0.3, 5.5 at 0.32 — and diverges at 0.346 Å, where the exponential’s own pucker curvature passes through zero. Beyond it the sign is reversed. At 0.35 Å the contact term’s curvature is +1.8 × 10⁻⁴, at 0.4 Å +3.3 × 10⁻³, and a contact of that range makes the ring stiffer the harder it pushes. The range of the fluorines’ repulsion decides whether they pucker the ring or hold it flat; the size of the repulsion decides only how soon.

The crossing falls within a thousandth of an ångström of a famous number. Born and Mayer, fitting the lattice energies of the alkali halides in 1932, took the range of the repulsion between closed-shell ions to be 0.345 Å for all of them. Nothing in this calculation knows about ionic crystals, and the agreement is a coincidence of two unrelated problems. It is still worth noting, because it means the one range a chemist is likely to reach for sits exactly on the fence, and a pucker argued from it would be argued from the one value that gives no answer.

Fluorine’s own orbital does give one. Treat the repulsion between two fluorine atoms as the noble-gas contact calculation did — proportional to the square of the overlap of their filled shells — and take the 2p shell with the exponent Slater’s rules assign it: 2.60 for the neutral atom, 2.425 for the anion, since the fluorines in IF₇ carry a partial negative charge. One σ pair and two π pairs, with Mulliken’s closed forms for their overlaps, checked here against a direct numerical integration to five decimals. The law this gives is not a single exponential, but its local range at the ring-neighbour distance is 0.159 Å with the atom’s exponent and 0.178 with the anion’s. Both are about half the critical range, far on the puckering side, and the shares they need are 0.11 and 0.16 per cent. The green rows of the second figure show them.

So with a contact law built from fluorine’s own orbital, IF₇’s ring puckers in this model unless the fluorines’ repulsion is less than about a thousandth of the domains’. Nothing in the model says whether it is. What it does say is that the flat ring of the last essay was a verdict about bond domains alone, and that the omission it named was not a correction but the deciding term.

What the ring does once it goes #

Past the threshold the flat ring is not a minimum, but it is not far from one either.

At half the threshold weight the energy along the pucker is an ordinary well. At the threshold its bottom is flat, held only by the quartic term, which is the shape the inverse-square law gave the equal-bond ring — a coincidence of two different parameters reaching the same boundary. Above it the flat ring is a small hump between two symmetric minima. At three times the threshold the minima are about six degrees out of the plane and 2.7 × 10⁻⁴ below the flat ring, in a total energy near fourteen.

The depth grows much faster than the weight. Fully relaxed, with every site free to move, the 1/r¹² ring gains 6.2 × 10⁻⁵ at one and a half times the threshold, 7.6 × 10⁻⁴ at three and 3.3 × 10⁻³ at six. The ring’s most displaced atom reaches 5°, 9° and 12° from the plane, and the axis bends by at most six degrees. Fluorine’s own contact law gives nearly the same numbers at the same multiples: 5.0°, 9.1° and 12.5°.

But a pucker in a pentagon has a property that neither the profile nor the gains show. It is a degenerate pair — two phases of one wave running twice round the ring — so a puckered ring has a phase as well as a size. Any ring atom can be the one that rises furthest, and between those five choices lie all the intermediate ones. Two phases are special. The envelope has one atom out of the plane on one side and a mirror through it and the axis. The twist has a twofold axis lying in the ring plane through one atom, which stays in the plane while its neighbours go opposite ways.

Each is relaxed from a seed of its own symmetry, and each keeps it: the envelope comes back with mirror symmetry only, $C_s$, and the twist with a twofold axis only, $C_2$. The twist is lower in every one of fifteen cases. By how much is the result: 2.4 × 10⁻¹⁰ at one and a half times the threshold, against a depth of 6.3 × 10⁻⁵, for fluorine’s own law — a difference of four parts in a million of the pucker’s depth. At six times the threshold, with the ring twelve degrees out of plane, it is still five parts in ten thousand.

Moving the pucker from one phase to the next — the rising atom handing its role to its neighbour — costs almost nothing. The model’s puckered IF₇ does not have a puckered structure. It has a circle of equivalent puckered structures, joined by a path with essentially no barrier, and a molecule on that circle is pseudorotating: its ring is never flat and never puckered in any fixed direction, and averaged over the motion it looks flat.

The reason is symmetry, and the numbers confirm it to the exponent. The pucker is a wave that runs twice round a ring of five, so turning the ring by one site advances the wave’s phase by two fifths of a turn; a term in the energy can depend on the phase only if it is unchanged by that, and the lowest power of the pucker that is unchanged is the fifth. The ring’s mirror plane then forbids the fifth, since reflecting through the plane reverses every elevation and so reverses an odd power. What is left is the tenth. The envelope–twist difference should grow as the tenth power of the pucker’s size, and between one and a half and three times the threshold it grows as the 10.0th power for fluorine’s own law, the 10.1th for 1/r¹² and between the 10.3th and 10.8th for the others. At small amplitude a tenth power is as close to nothing as a term can be.

That is the second place in this collection where a five-membered ring’s floppiness is a pseudorotation rather than a flip between minima. Cyclopentane’s is the famous one, and it arises the same way — a pucker in a ring of five has a phase the energy barely sees. It is a different kind of floppiness from the exchange of sites at five-coordination, which runs between distinct minima through a genuine barrier.

What was computed and how #

Every structure is seven points on a unit sphere, as in the first repulsion calculation, each scaled by its bond’s length relative to the axial bond — 1.858 and 1.786 Å, the measured values — with an energy summing 1/r over all pairs for the domains and a weighted contact law over all pairs for the end atoms. Because every end atom sits on its domain’s ray, both terms are functions of the same seven directions; the same-rays identity above is what makes that legitimate for the Coulomb term, and it is checked to four decimals at nine placements for two power laws.

The pucker’s curvature is a finite difference along the wave that runs twice round the ring, with two step sizes combined to remove the quartic term’s contribution, as in the last essay. It is linear in the contact weight, so the threshold is a single division, and the critical range is found by bisection on the sign of the exponential’s own curvature. The split by kind of contact sums each kind’s pairs separately; the four parts add to the whole to six decimals.

The fluorine contact law is the sum of the squares of one 2pσ and two 2pπ overlaps between Slater functions of one exponent, from Mulliken’s closed forms; those forms are checked against the site’s own three-dimensional quadrature of hydrogenic 2p functions of charge 2ζ at 2 and 4 bohr, and agree to two parts in a hundred thousand. Its local range is −1 / (d ln f / dr) at the contact distance.

The relaxed structures descend the total energy on the sphere with a backtracking step to a gradient below 10⁻¹⁰. Near the pucker’s flat valley the energy changes stop resolving below the rounding of a total near fourteen, so a step whose energy change is inside that rounding is accepted when it lowers the gradient instead. Symmetry labels come from the same finder that labels every molecule here, at a tolerance of 0.02.

The last column answers a question the new term could have reopened. The measured placement of IF₇’s two short bonds, both on the axis, was the domain model’s lowest of four. Under every contact law in the table, taken alone at the ideal geometry, it is still the lowest, so the fluorines move the ring’s shape without disturbing which sites the short bonds take.

The claims are stated where they can fail: that a power law’s curvature scaled by $d^p$ is constant along the rays; that an exponential’s points at a fraction d of the bond give exactly the curvature of the range ρ/d, and that the fraction where exp(−r/0.25 Å) changes sign times the critical range returns 0.25 Å; that 1/r⁴ has no threshold and every power from 1/r⁶ needs under five per cent, less for each steeper law; that the critical range lies between 0.30 and 0.35 Å; that only the ring neighbours favour the pucker; that fluorine’s own law needs under half a per cent; that every case past the threshold relaxes out of the plane into an envelope of $C_s$ and a twist of $C_2$ whose difference is under a hundredth of the depth, and every case at 0.8 of it returns flat; that the twist is the lower phase every time and the difference between the phases grows as a power of the amplitude between 9 and 11.5. The refusal is the exponential fed to the same-rays identity: a law with a length of its own must fail it, and does — at six tenths of the bond its rescaled curvature is −5.9 where a power law’s is 1.

What the second set of points cannot say #

The weight is not computed. Everything here is a threshold on a ratio the model has no way to fix. The domain energy is in arbitrary units and the contact term’s prefactor is unknown in those units, so the essay’s verdict is conditional: IF₇’s ring puckers in this model if the fluorines’ contact repulsion is above about a thousandth of the domains’ repulsion, and it is not known whether it is. The threshold is small enough that a flat ring is the less likely reading, but a small threshold is not a measurement.

The contact law is a one-electron overlap model. The square of an overlap captures why closed shells repel and roughly how steeply; it leaves out exchange and correlation, and it takes one exponent for a fluorine whose charge in IF₇ lies somewhere between the atom’s and the anion’s. The two exponents bracket that, and both are far from the critical range. A real repulsion curve would move the threshold, not the side of 0.346 Å it falls on.

Nothing here is a frequency or an amplitude. A barrier of a millionth of a curvature says the pucker runs round the ring freely; it does not say how fast, because the model’s points have no masses and its energy no units. And a model that puckers the ring at a weight of its own choosing cannot predict the pucker’s size either: the six to twelve degrees above are what particular multiples of the threshold give.

The measured molecule is averaged. Electron diffraction of IF₇ reads it as a pentagonal bipyramid with a large-amplitude ring motion. Adams, Thompson and Bartell’s analysis of 1970 interpreted that motion as a pseudorotation of the puckered ring, and later work by Christe, Curtis and Dixon in 1993 described it as dynamic puckering. A pseudorotating ring and a flat ring with a very soft pucker look much alike in a vibrationally averaged structure. What this calculation adds is that the model can produce either, and that which one it produces depends on the term the previous calculation left out.

Two verdicts that were about the domains #

The domain model gave IF₇ the right shape and gave its flatness a margin: a pucker three times stiffer than with equal bonds, surviving any law softer than 1/r^4.4. That margin was measured against the domains’ own repulsion. Measured against a contact term that the atoms at the ends of the bonds certainly have, it is a margin of about a thousandth of the energy, and it is gone for any contact law steep enough at 2.18 Å, which fluorine’s is.

The generalisation is that a repulsion model’s end atoms enter only through the shape of their law, never through its size. A second set of points on the same rays under the same law is invisible; a second set under a law with a length of its own is a new term, and its range can reverse its sign. That is true for any molecule put through the repulsion model, and it bears on the long bond that goes to the crowded site, where the reading of the crowding count had the end atoms and the domains pulling in opposite directions. The count’s end-atom half is a statement about a contact law, and its conclusion inherits whatever that law’s range does.

And the verdict the model now gives IF₇ is closer to what the molecule is reported to do. The flat ring held by long bonds was a flat minimum with a very soft pucker. The ring with its fluorines in is either that or a pucker with no preferred phase, running round the ring, which averages to flat — two readings that a diffraction experiment finds hard to tell apart, and the lone-pair version of the same question, for xenon hexafluoride, still to be asked.

Still open: the weight, and a seventh domain that is not a bond #

The obvious open question is the weight. Everything here is a threshold on the ratio of two repulsions the model cannot put in the same units. One way to fix it is from a measurement the domain model already reaches: the pucker’s curvature is a force constant once the model’s energy has a scale, and the lowest ring-deformation frequency of IF₇ is measured. Calibrating the domain energy against a stiffer mode that the contact term barely touches — the ring’s in-plane breathing, say — and then reading the contact weight off the pucker’s frequency would turn the threshold into a comparison, and would say whether IF₇ sits on the flat side or the pseudorotating one.

The nearer question is xenon hexafluoride. Its seventh domain is a lone pair, which has no atom at its end, so the contact term reaches only six of the seven directions; the same-rays argument says the lone pair’s domain position is invisible under one law, but the contact term sees its absence. Whether a missing end atom makes XeF₆’s fluxionality the same pseudorotation or something else is the same calculation with one fewer contact, and it is the other famously floppy seven-domain molecule.