Two models that disagree about the shape

An orphan count becomes a population. A hypervalent molecule has more ligand σ combinations than the central atom has orbitals to match them with, and the leftovers — the orphans — put charge on the ligands: with no electronegativity difference anywhere, the mean ligand charge is exactly the orphan count divided by the ligand count, in all ten cases tried.

The identity says how much. It says nothing about how it is shared, and the split between phosphorus pentafluoride’s axial and equatorial fluorines came out of the coordinates rather than out of the identity.

So the arrangement that shares it most evenly is a computable thing, and it can be put on one axis with the arrangement the repulsion model picks — two accounts of hypervalent geometry that are usually computed separately and never compared.

Three arrangements, two numbers each #

Five directions on a sphere, three ways: the trigonal bipyramid, the square pyramid, and the pentagonal plane. For each, the ligand charges come from the σ-only model built from those directions, and the repulsion is the Coulomb energy of five unit charges in the same places.

The even sharer is the pentagonal plane, and it is the arrangement the repulsion likes least — by 6.3 per cent, which is far outside anything a small perturbation would move.

The bipyramid, which is what phosphorus pentafluoride is, has two kinds of ligand and a spread of 0.114. So if a molecule chose its shape to share charge evenly it would be planar, and none of them is.

That is a clean disagreement, and it is worth being clear about what it is not. It is not that one model is right; both are crude. It is that the two answer the same question with different arrangements, so charge sharing and repulsion are not two descriptions of one preference.

And they disagree about the amount, too #

The third column has something in it that the argument did not need and is worth keeping.

The bipyramid and the square pyramid transfer almost exactly the same total charge — mean −0.3754 against −0.3760, a difference of two parts in a thousand. The pentagonal plane transfers −0.5155, which is thirty-seven per cent more.

So the two structures a molecule actually interconverts between agree about the total to two parts in a thousand, and the structure it does not adopt disagrees by a third. The identity is stable across the arrangements chemistry uses and not across all arrangements, which is a more useful statement than either half.

The reason is the orphan count. The bipyramid and the square pyramid both use all four central orbitals — s, $p_x$, $p_y$, $p_z$ — and both have one orphan; the plane leaves the perpendicular p unmatched and has two. A count that looked like a property of the molecule turns out to be a property of the arrangement.

The interchange, where the disagreement is sharpest #

Phosphorus pentafluoride does not sit still. The Berry interchange takes it from one bipyramid to another through a square pyramid, and it is fast enough that its fluorines are equivalent on the timescale of a spectrum.

Following that path with both models — one equatorial ligand pivots to become the apex, the two axial ligands close from 90° to 105° from it, and the other two equatorials open from 120° to 105°:

• the repulsion moves by 0.150 per cent across the whole path;
• the spread of the ligand charges moves by 53.6 per cent;
• and the total charge transferred moves by 0.16 per cent.

One model sees almost nothing happening and the other sees a large excursion, on the same path between the same two structures. It is the same kind of mismatch a species label has with a distortion: two instruments pointed at one coordinate, disagreeing about whether there is anything there to measure. That is not a disagreement about which arrangement is best — it is a disagreement about whether anything is going on.

The repulsion’s flatness is the well-known reason the molecule is fluxional: the barrier is tiny because the two arrangements repel almost identically. The σ model’s answer is that the electronic structure is doing something substantial while the repulsion is doing nothing, and that the something is a redistribution rather than a transfer, since the total barely moves.

Why the plane shares evenly, and why that is not a virtue #

The plane shares evenly for a reason with no chemistry in it: all five of its ligands are equivalent under the molecule’s own symmetry. A fivefold axis permutes them, so the charge on one is the charge on all, and no calculation was needed to know the spread would be zero.

That is worth saying because it makes the even-sharing criterion much weaker than it sounds. Share the charge evenly is not a physical preference a molecule can act on; it is a consequence of having a symmetry that makes the ligands one orbit. So the criterion is really adopt the arrangement with the highest symmetry, which is a different claim and one nothing supports.

The bipyramid has two orbits and therefore two charges, necessarily. The square pyramid has two orbits and therefore two charges, necessarily. Neither model is choosing to be uneven; the geometry has already decided how many kinds of ligand there are, and the σ model is only saying how far apart the kinds are.

So the sharper form of the comparison is: given the number of kinds a geometry has, how far apart does the σ model put them, and does that track anything the repulsion model cares about? Along the Berry path it does not — the number of kinds stays at two throughout and the gap between them changes by half while the repulsion does not move.

Which of them a measurement would see #

The two models predict different observables, which is what makes the disagreement useful rather than embarrassing.

The repulsion predicts a geometry and a barrier. Both are measured, and the model does well on both: the bipyramid is right and the barrier is small.

The σ model predicts a charge distribution. That is not directly measurable — a metal’s charge runs over two electrons across the partitions that compute it — but its changes along a coordinate are closer to something: a nuclear quadrupole coupling constant or a core-level shift responds to the field at a nucleus, and a 54 per cent change in the charge spread is a large change in those.

So the honest summary is that the two models are not competitors and should not be scored against each other. The repulsion model chooses the shape; the σ model says what the electrons are doing once the shape is chosen. That the second is doing something large where the first is doing nothing is exactly why the second is worth having.

The case the identity cannot answer #

There is a third result here and it is a boundary rather than a comparison.

At the orphan identity’s own condition — no electronegativity difference anywhere — the planar arrangement’s populations are not determined. The central p perpendicular to the plane has nothing to match, so it sits at exactly the energy of the orphan ligand combinations, and asking which of a degenerate set to fill is not a question with an answer.

The arithmetic says so rather than hiding it: the mean ligand charge comes back at $10^{-17}$ instead of the identity’s −0.2. The bipyramid at the same condition still returns −0.2 exactly, so the failure belongs to the planar case and not to the calculation.

That is why every number above is computed at an electronegativity difference of one rather than zero — not as a modelling preference but because the question is ill-posed otherwise, and a figure drawn at the ill-posed point would be a picture of whatever the eigensolver happened to return.

What the comparison was worth doing #

Two models of one thing that agree are reassuring and uninformative. Two that disagree are useful in proportion to how sharply they disagree, and this pair disagrees in three separate ways:

• about which arrangement is preferred — the plane against the bipyramid, by 6.3 per cent of a repulsion;
• about how much charge moves — the plane transfers a third more than either of the others;
• about whether the interchange is an event — 0.15 per cent against 53.6.

Three disagreements from one comparison is more than a compatibility check would have produced, and each of them is a place a better calculation could be pointed. The third is the most useful: a coordinate along which one model is flat and another is not is a coordinate where the second model is carrying all the information, and it is the coordinate a real molecule spends its time on.

The habit worth taking from this is putting two models on one axis with identical input. Both models here read five unit vectors and nothing else. Neither was fitted, neither was tuned to the other, and the same five numbers went into both — which is what makes the disagreements attributable to the models rather than to the setup. It is the same discipline as computing a quantity twice by independent routes, applied to two theories rather than to two algorithms.

What is quoted, and what is computed #

Nothing is quoted. The three arrangements are constructed from angles, the bond length is a stated 1.6 Å and cancels out of everything, and the electronegativity difference is a stated one.

Every ligand charge comes from diagonalising a matrix built from the directions themselves — the σ overlaps are the direction cosines, so the model reads the geometry and nothing else. Every repulsion is a sum of reciprocal distances between the same directions.

The two models are given identical input, which is the whole point of the comparison: the same five unit vectors go into both, and neither is fitted to anything.

What this cannot say #

The repulsion model has no ligands in it. It is five points on a sphere, and a real ligand has a size and a lone-pair count; which angles are symmetry and which are the model measures how much of an arrangement is decided by the exponent rather than by the count.

The σ model has no π and no repulsion between electrons, so it is subject to the caution a partitioned charge always carries. Its charges are one-electron populations in a model with a single parameter, and the numbers should be read as ratios and signs rather than as charges.

And the pentagonal plane is not a candidate. No five-coordinate main-group molecule is planar, so the arrangement that shares evenly is being used as a probe rather than proposed. The finding is about what the two models rank, not about what phosphorus should do.

Nor is 6.3 per cent a barrier. The repulsion difference between the plane and the bipyramid is a difference in a model energy with no units attached to it, and turning it into kilojoules needs a scale this model does not have.

The molecule the argument is about has its point group found from its coordinates as D₃ₕ, and its two kinds of fluorine are what the identity averages over and what the interchange exchanges. Neither model is told which fluorine is which; both recover the partition.

What the comparison requires #

The arrangement that shares the charge most evenly is not the one the repulsion picks — asserted by name in both directions, so a change to either model that made them agree would fail the check rather than pass it quietly.

The planar arrangement’s five ligands carry exactly the same charge, to a part in a billion, while the bipyramid’s do not — the two halves of the comparison.

The even sharer costs several per cent more in repulsion and puts more charge on each ligand, which is the second disagreement and is independent of the first.

The repulsion is nearly flat along the interchange and the charge spread moves by far more — tested as a ratio between the two ranges rather than as two separate bounds, because the finding is the disagreement rather than either number.

And the ill-posed case is refused. With no electronegativity difference the planar arrangement returns a mean charge of nothing where the identity says −0.2, and the bipyramid still returns −0.2 exactly — so the failure is located rather than merely observed.

Where the two models get their authority #

A disagreement is only interesting if both parties have some claim to be listened to, so it is worth setting out what each of these has earned.

The repulsion model predicts geometries and gets them right. Five points on a sphere minimise their mutual repulsion at a trigonal bipyramid, six at an octahedron, seven at a pentagonal bipyramid — and those are the arrangements. It has one parameter, the exponent, and the arrangements it picks do not depend on it at four, five or six.

The σ model predicts a charge distribution and gets a check nothing else offers. Its identity — the mean ligand charge is the orphan count over the ligand count, exactly — is a statement with no fitting in it, and it holds for ten molecules of four different shapes.

So neither is a strawman, and the disagreement is not one model being tested by another. It is two accounts of the same molecules that were built for different questions and have never been asked the same one. Asking them the same question is the whole content of this essay, and the answer — that they disagree about which arrangement is preferred, by how much charge, and about whether the interchange is an event — is more than a compatibility check would have found.

Where the charge criterion is silent, and why that is a scope rather than a gap #

The comparison worked here because the arrangements being compared have inequivalent ligand sites. It is worth stating the condition, because it decides in advance which questions the charge criterion can be asked.

The criterion measures how evenly the required charge is shared. If every ligand in an arrangement is equivalent by symmetry, the sharing is perfectly even by construction, the criterion returns the same answer whatever the arrangement is, and it cannot prefer one to another.

That covers more cases than it sounds. An octahedron and a trigonal prism both have all six ligands equivalent; a square and a tetrahedron both have all four; a linear arrangement and a bent one with equal bonds both have two. In every one of those pairs the charge criterion is silent and the repulsion model is decisive.

So the two models are not rivals across the board. The charge criterion speaks exactly where the symmetry leaves the ligands inequivalent, which is the five- and seven-coordinate cases and the low-symmetry distortions, and it is mute everywhere else. The disagreement measured here is therefore not a general disagreement — it is located, and its location is a property of the point groups rather than of the chemistry.

Still open: six coordination, the orphan count, and a third model #

The obvious open question is six coordination, where the comparison has a different shape. An octahedron and a trigonal prism are the two arrangements of six, the octahedron wins on repulsion comfortably, and both have every ligand equivalent — so the charge-sharing criterion cannot distinguish them at all. A case where one model is silent and the other is decisive is the complement of the case here, and running it would say whether the two models disagree systematically or only where the symmetry is low.

The nearer question is the orphan count the third section turned up. It is usually treated as a property of the molecule — ligands plus lone pairs minus four — and the planar arrangement shows it is a property of the arrangement, because a central orbital pointing nowhere useful is a central orbital that does not count. Recomputing the census with the count taken from each molecule’s own reduction rather than from the formula would say how many of the ten cases the formula gets right, and it is the kind of check the formula has never been given.

A third direction is to ask what a third model says. A d-orbital splitting can be computed from the same coordinates by two independent routes, and neither is a repulsion or a σ count — so a five-coordinate d-block complex would have three accounts of its shape to put on one axis instead of two, and the integral that cannot count electrons is where the third one’s parameters come from.