The long bond goes to the crowded site

The repulsion model of molecular shape places bond domains on a sphere and lets them push each other apart. Given one bond of a different length, at five-coordination, it puts a longer bond axial — the opposite of the familiar rule that a bulkier group goes equatorial, “where there is more room”. That result came with a reconciliation. The atoms at the ends of the bonds see the same geometry differently: an axial substituent has three neighbours at 90° against the equatorial one’s two, so a bulky atom goes equatorial, while a long bond goes axial. Two levels of the same picture answer two different questions.

Stated that way, the result looks like a fact about five-coordination. It is a fact about which site is crowded, and five-coordination is only the case where that site is axial. The distinction matters because above six the arrangements change character: seven sites settle into a pentagonal bipyramid, whose axial and equatorial sites are crowded in the opposite way. If the reconciliation is right, every sign should reverse at seven.

One reading of crowding, two ways to use it #

A longer bond carries its domain further from the centre, which moves it away from all its neighbours at once. How much that relieves depends on how close those neighbours were: a domain surrounded by near neighbours gains more from moving out than one whose neighbours were already distant. So a long bond is worth most at the most crowded site. A short bond does the opposite and wants the least crowded one.

An atom at the end of a bond is different. It does not move its neighbours away by being bulky; it adds to the crowding it finds. So a bulky substituent costs most at the most crowded site and goes to the least crowded one. The two rules use the same count and give opposite answers, which is why the long bond and the bulky group went to opposite sites at five.

Neither rule is about axial or equatorial as such. Each needs only the neighbour count of each kind of site, and the neighbour count is geometry.

At five, an axial site has three neighbours at 90° and one at 180°, and an equatorial site two at 90° and two at 120°. The axial site is the crowded one under every distance law: its sum of 1/r is 2.621 against 2.569, and of 1/r¹² 0.0471 against 0.0340.

At seven the picture turns over. An axial site of the pentagonal bipyramid has five neighbours at 90° and one at 180°. An equatorial site has two neighbours at only 72° — its neighbours around the pentagon — plus the two axial sites at 90° and two more at 144°. Those two at 72° are closer than anything the axial site has. The equatorial site is the crowded one: by 3.3 per cent at 1/r, 9 per cent at 1/r², and a factor of 4.07 at 1/r¹², against the 39 per cent by which the axial site is crowded at five.

The 72° neighbours are not an accident of this arrangement that some other seven-site shape avoids. Five points in a ring are 72° apart however the ring is drawn, and any arrangement that puts five sites around an axis pays that price in its equator. The axial sites of the same arrangement are the only ones in it with no neighbour nearer than 90°. At five the trade runs the other way: three sites around an axis are 120° apart, which is roomy, and it is the axis that is pressed by three neighbours at 90°. The ring size decides which site is crowded, and a ring of five is past the point where the equator stops being the comfortable place.

So the prediction is sharp. At seven a long bond should go equatorial, a short one axial, and a bulky end atom axial.

At seven every sign reverses #

At five, the minimisations reproduce what was found before. A bond 1.1 times the others is lower by 2.8 × 10⁻³ axial; at 1.2 by 5.8 × 10⁻³; at 1.5 by 1.4 × 10⁻². A bond 0.9 times the others is lower equatorial, by 2.3 × 10⁻³, and at 0.8 by 3.7 × 10⁻³. The curve crosses zero at equal lengths and runs downward.

At seven it runs the other way. A bond 1.02 times the others is lower equatorial by 1.4 × 10⁻³; at 1.1 by 7.7 × 10⁻³; at 1.3 by 2.8 × 10⁻². A bond 0.9 times the others is lower axial, by 5.1 × 10⁻³, and at 0.8 by 6.5 × 10⁻³. The seven-site curve is the five-site curve reflected, and steeper — the pentagonal bipyramid’s two kinds of site differ in crowding by more, so a given difference in length is worth more.

Both curves bend back at the short end. At five, a very short bond returns to the axial site once it has been pulled so far in that it stops being the crowded one; at seven, at a ratio of 0.6, the short bond is lower in the equator again. The reversal holds over the range a real bond-length ratio occupies, from about 0.7 to 1.3, and both coordinations have a window beyond which the simple reading gives out.

What molecules do with their long bonds #

A repulsion model that places a single odd bond is making a statement about which kind of site suits a longer bond. The cleanest test is a molecule whose bonds are all to the same ligand but whose two kinds of site have different lengths anyway.

Phosphorus pentafluoride is a trigonal bipyramid with axial bonds of 1.577 Å and equatorial bonds of 1.534 Å: the longer bonds are axial, which is the crowded site at five. Iodine heptafluoride is a pentagonal bipyramid with axial bonds of 1.786 Å and equatorial bonds of 1.858 Å: the longer bonds are equatorial, which is the crowded site at seven. The two molecules differ in the sign of the difference, and the model predicts both signs from one reading of crowding.

The sizes are worth a sentence, because they are small. The energy differences between placements are a few thousandths to a few hundredths of a total near 6.5 at five sites and 14.5 at seven — a fraction of a per cent. The model is not claiming that a molecule strains hard to put its long bonds in the right place. It is claiming that, of two ways to arrange unequal bonds, one is slightly lower, and that the lower one changes with the ring size. A small preference is still a definite sign, and the sign is what the two measured structures record: nothing forced PF₅’s axial bonds to be longer rather than shorter, and nothing forced IF₇’s to be shorter rather than longer. Both went the way the crowding count says.

Short, multiply bonded ligands point the same way. Thionyl tetrafluoride, a trigonal bipyramid, carries its short sulfur–oxygen bond in the equator — the uncrowded site at five. Rhenium oxide hexafluoride, a pentagonal bipyramid, is reported with its short rhenium–oxygen bond on the axis — the uncrowded site at seven. A multiple bond is also a heavier domain, which the model treats with a separate knob, so these two are consistent with the length reading rather than a test of it; the two fluorides, with one ligand throughout, are the test.

What neither case shows is the bulky-atom half at seven. It predicts that a large substituent on a pentagonal bipyramid sits axial, where its neighbours are all at 90° or more — the opposite of the five-coordinate rule of thumb, again for the same reason.

At seven the avoided site is not a minimum #

At five, both placements of an odd bond are genuine structures. Relaxed from a bond placed axially, the minimisation stays axial; nudged and relaxed again, it returns to the same energy. That is what makes “the higher-energy isomer” a meaningful phrase, and it is consistent with five-coordinate molecules exchanging their axial and equatorial substituents by a definite rearrangement between two real structures.

At seven that stops being true.

Every placement was relaxed, relaxed again without disturbance to confirm it had converged, and then relaxed from three small random nudges; a structure whose nudged energy falls below its converged one is not a minimum. At five, no placement at any length falls. At seven, a long bond set on the axis falls at every length from 1.05 upward — by 1.4 × 10⁻⁵ at 1.05, 2.7 × 10⁻⁴ at 1.1, and at 1.2 by 1.7 × 10⁻², which is the whole difference between the two placements: the axial structure slides into the equatorial one. A short bond set in the equator falls too, by a few millionths at 0.7, 0.8 and 0.9. Only at 1.02, a two per cent difference, are both placements minima.

So at seven there is no disfavoured isomer to speak of. A bond put on the site it does not suit is not in a higher well; it is on a slope. That is the same flatness that makes seven-coordinate arrangements depend on the repulsion law when five and six do not, and it has a direct consequence for how seven-coordinate molecules should be described: a structure with a ligand in the “wrong” site is not something to be looked for.

The two coordinations differ here in kind and not just in degree. At five, exchanging an axial and an equatorial substituent is a rearrangement between two minima through a barrier, and a molecule doing it quickly enough is described by a group larger than its structure’s. At seven the model has no second minimum to exchange with: a disturbance that moves a long bond towards the axis is simply undone, and a disturbance that moves a short bond into the equator is undone the same way. What a seven-coordinate molecule with unequal bonds is doing when its ring puckers and unpuckers is therefore not isomerisation in the five-coordinate sense. It is motion about a single structure on a surface flat enough that the two kinds of site are one valley rather than two.

A long enough bond reshapes the arrangement #

At a ratio of 1.5, neither placement survives. Nudged, both fall — the equatorial one by 8.5 × 10⁻⁴ — into a structure that is no longer a pentagonal bipyramid.

The new structure is a capped octahedron, of symmetry $C_{3v}$, with the long bond as the cap: its three nearest neighbours sit at 69° and the other three at 128°. That arrangement is one of the three seven-site shapes a minimisation can find, close in energy to the pentagonal bipyramid, and a single long bond is enough to choose it. At five, the trigonal bipyramid carried a bond of 1.5 in its axial site without changing shape at all.

The ordering of the four panels is the argument in pictures. A short bond on the axis is at home. A long bond in the equator is at home. The same long bond set on the axis slides into the equator. And a long enough bond remakes the arrangement around itself, taking the one site of the new arrangement that has three close neighbours.

How the placements were computed #

Every structure is a set of points on a unit sphere, one per bond, with one point’s distance from the centre multiplied by the stated ratio; the energy is the sum of 1/r over all pairs, the law every shape here has been computed with from the start. Placements start from the ideal trigonal bipyramid or pentagonal bipyramid with the odd bond at the stated site, relax by 40 000 steps of descent, and then by 80 000 more from the result; the energy compared is the second. The nudge test adds a random displacement of up to 0.015 in each coordinate to every point, renormalises, relaxes for 80 000 steps, and keeps the lowest of three independent nudges. A drop below 10⁻⁹ counts as none; the smallest drop reported as a fall is 2.5 × 10⁻⁶, more than three orders of magnitude above that threshold.

The neighbour sums are computed on the ideal arrangements with equal radii. Point groups of the relaxed structures come from the same symmetry finder that assigns the point group of every molecule in these essays.

The checks, run wherever these figures are drawn. The axial site is the more crowded at five and the equatorial site at seven, under all four distance laws. At five a longer bond is lower axial and a shorter one equatorial at every length, and every placement is a minimum. At seven a bond of 0.7, 0.8 or 0.9 is lower axial and is not a minimum in the equator; a bond of 1.02 to 1.2 is lower equatorial; on the axis, a bond of 1.05 to 1.3 is not a minimum. At 1.5 the equatorial placement relaxes to $C_{3v}$ with the long bond’s three nearest neighbours under 75°. The longer measured bonds of PF₅ and IF₇ are at the site the model sends a long bond to. The refusal is equal lengths: at a ratio of one the two placements must be the same structure, to 10⁻⁹, or the comparison would carry a bias of its own.

Where the reading stops #

One law. Everything here uses 1/r between domains. At seven sites the arrangement itself depends on the law, so a steeper repulsion changes the reference structure before it changes the placement; the neighbour sums keep their sign at every law tried, but the minimisations were run only at the one.

One odd bond. IF₇ has five longer bonds and two shorter ones, not one odd bond. The model’s statement is about which kind of site suits a longer bond, and the molecule agrees with it; a minimisation with five long and two short would be a second calculation, and the measured structure of IF₇ is known to be fluxional, with its equatorial ring puckering.

Lengths are input. As at five, the model produces no bond lengths. It says where a longer bond should sit, and the measured lengths say where they do; the agreement is between a sign and a sign.

And nine sites were not settled. The tricapped trigonal prism, the nine-site minimum, has two kinds of site as well, and the neighbour sums call its caps the crowded ones. But its placements migrate between kinds during relaxation often enough that a clean table could not be made from the same procedure, and nothing about nine is claimed.

A rule stated in positions is a rule about one arrangement #

“Bulky groups equatorial” and “long bonds axial” are both stated in the names of positions, and both are true of one arrangement only. The quantity underneath them is crowding, and crowding is a property of the arrangement rather than of the words axial and equatorial: the trigonal bipyramid is crowded on its axis and the pentagonal bipyramid in its equator. A rule carried to a new arrangement has to be carried in the variable it depends on, not in the label it was first written with.

The second lesson is about isomers. The idea of a less stable placement assumes the placement is a structure, and at five it is. At seven the landscape is flat enough that a bond in the wrong place has nowhere to rest, and “the isomer with the long bond axial” does not exist in the model at all.

Still open: the bulky atom at seven, and more than one odd bond #

The obvious open question is the other half of the reversal. The atoms at the ends of the bonds should now prefer the axis, because the equatorial site has two neighbours at 72°. A seven-coordinate complex with one large ligand among small ones — an iodine or rhenium centre with a single heavier substituent — would say whether real structures follow the atom count or the bond length when the two disagree, which at seven they do in the opposite sense from five.

The nearer question is IF₇ itself. Five long bonds and two short ones is a composition the minimiser can take directly, and whether it holds a planar pentagon or puckers it — the measured molecule is reported to pucker dynamically — is one more minimisation of the same kind, with a measured structure to hold it against.