Expensive is not the same as unadopted

Ten molecules put into every arrangement their ligand counts allow give a census: recover the point group from each arrangement’s own coordinates, reduce the ligand σ set in it, and count how many combinations find no central orbital of their species. Twenty-six arrangements; the formula $n + L - 4$ right for twenty-three and wrong for three.

The census invites a reading, and it is fair to call it only that. Every failure is a planar arrangement of four or more ligands, and every failure is a molecule that does not adopt that arrangement — so perhaps the formula is reliable because chemistry stays away from where it breaks. The test is plain: every arrangement has a repulsion energy that can be computed, so put the two on one axis. If the failures are also the expensive arrangements, the two accounts agree.

Energy does not sort them #

The three failures sit 4.20, 6.29 and 9.80 per cent above their own repulsion minimum. The arrangements the formula gets right run from exactly zero to 26.47.

So the most expensive arrangement in the whole census is one the formula survives, and it is not close: a bent arrangement of two ligands, at 26.47 per cent, against a worst failure at 9.80.

The ranking makes the point harder. The second most expensive arrangement here is a T-shaped three, at 10.52 per cent — more expensive than every failure — and it is what chlorine trifluoride actually is. So “expensive” does not even mean “unadopted”, let alone “where the formula breaks”. Two of the eight arrangements molecules actually adopt are among the four most expensive in the census, and both of them are cases where lone pairs make the ligand-only energy the wrong quantity to be ranking by in the first place.

The reading is refused, and it is refused in both of its halves. The failures are not the expensive arrangements, and expense is not what keeps a molecule away from an arrangement.

There is a reason the correlation looked convincing and it is worth naming, because it is the trap rather than the answer. Among the arrangements a molecule actually adopts, the formula never fails — that much is true. But the arrangements a molecule adopts are eight of twenty-six, and the other eighteen are hypothetical geometries put there to test the count. Restricting attention to the real ones leaves too few cases to distinguish “the failures are expensive” from “the failures are planar”, because on the real molecules those two happen to agree.

The census exists precisely to escape that restriction, and the energy is the second axis it needed. With both, the two candidate explanations separate immediately, and they separate in favour of the one that was harder to see.

The same arrangement, two verdicts #

The sharpest single test is not a correlation across twenty-six rows. It is one arrangement asked twice.

A square plane of four ligands is 4.197 per cent above a tetrahedron. That number is a property of four points on a sphere and is the same for both molecules, to nine figures.

The formula fails on methane in that arrangement and holds on xenon tetrafluoride. Same arrangement, same energy, different answer — so whatever decides is not in the arrangement, and cannot be in the energy. It is not a near-identity either: the two excesses agree to nine figures, because they are the same eight numbers summed in the same order.

What differs is the lone pairs. Xenon tetrafluoride has two and methane has none, and the two lone pairs occupy the central orbital that a square plane cannot reach: the p perpendicular to the plane. With that orbital occupied there is no spare left, and the count comes out where the formula says.

What does sort them #

It is tempting to say that a centre cannot have both — either the four valence orbitals run out, leaving combinations unmatched, or the combinations run out, leaving orbitals spare. That is true of twenty-three arrangements and false of three, and the three are exactly the failures.

The mechanism is a symmetry, not an energy. A planar arrangement of four or more ligands has all its σ combinations even under the plane’s own mirror, and the central p orbital perpendicular to the plane is odd under it. So that orbital cannot match anything the ligands offer: it is spare by symmetry, at any energy, and the combination it might have matched goes unpartnered.

That is why the arrangement’s expense is irrelevant. A T-shaped three is expensive and has only three combinations to place among four orbitals, so nothing is orphaned however awkward the geometry. A square plane is cheaper and has four combinations, one of which the perpendicular p can never reach.

The formula $n + L - 4$ counts orbitals against combinations and assumes every match that the numbers allow is a match the symmetry allows. Where a plane forbids one, the count is out by exactly the number of orbitals the plane cannot reach — which is one, in all three cases.

What the energy is good for #

The energy is not useless; it is answering a different question, and it answers that one cleanly.

Every molecule with no lone pairs sits at the repulsion minimum exactly: boron trifluoride trigonal planar, methane tetrahedral, phosphorus pentafluoride trigonal bipyramidal, sulfur hexafluoride octahedral, all at zero to six figures. That is a strong check on the arithmetic — four independent point sets, four independent minimisations, four exact agreements.

And every molecule that departs from the minimum has lone pairs: water bent at 26.47 per cent, ammonia pyramidal at 5.15, chlorine trifluoride T-shaped at 10.52, xenon tetrafluoride square planar at 4.20. No exception either way.

So the bare-ligand repulsion predicts a geometry exactly when there are no lone pairs and fails exactly when there are — which is not a defect of the calculation but a statement of what it counts. It is the same content as the lone-pair clause of the shape rules, arrived at from the other side: the departures from the ligand-only minimum are the lone pairs’ doing, and they are large.

Water’s 26.47 per cent is the largest number in the whole census, and it is the price two lone pairs charge for bending a molecule that two ligands alone would keep linear. That the two are not equivalent to each other — one in the plane, one perpendicular — is a separate finding about shape, and it is invisible to a count that treats them as a number.

What the formula is worth now #

Three calculations have now taken the same counting formula apart, and it is worth saying what is left of it, because the answer is more than nothing.

$n + L - 4$ counts a centre’s four s and p orbitals against the ligand combinations and the lone pairs competing for them, and it is right whenever every match the numbers permit is a match the symmetry permits. That is twenty-three of twenty-six arrangements here, and — the part that matters for anyone using it — every arrangement any of the ten molecules actually adopts. As a rule for real molecules it has not failed once.

What it cannot do is survive being applied to a geometry rather than to a molecule. Asked about methane in a square plane it gives the wrong count, and a reader who does not know why would have no way of telling that case from the twenty-three. The condition it needs is not “the molecule is real” but “no central orbital is forbidden by the arrangement’s own symmetry from matching anything” — which is checkable in advance, takes one reduction, and is what the census does.

So the formula’s status is that of a shortcut with a stated precondition rather than a rule with mysterious exceptions. That is a modest upgrade and it is the honest one: the exceptions were never mysterious, they were unexamined, and examining them costs a character table.

What was computed, and how #

The energy is unit charges on a unit sphere at the arrangement’s own directions, summed over pairs as the reciprocal separation, against the minimum for that many points — which this collection’s repulsion minimiser already has and which is checked there against published Thomson minima. Nothing is fitted and there is no distance scale, because every arrangement of a given count is compared with the others at the same radius.

The orphan count is the census’s, unchanged: the point group recovered from the coordinates, the ligand σ set reduced in it, the lone pairs assigned to central orbitals in order, and the remaining combinations counted. The one part that is a Lewis structure rather than a computation is the lone-pair assignment, and that is stated openly.

Five things are checked. Three arrangements must break the formula, which is the census’s own count re-derived. The most expensive arrangement must be one the formula gets right — which is the refusal of the reading, stated so that it fails if the reading is true. Every failure must have both a spare orbital and an orphan. The square plane must give different verdicts on the two molecules at identical energies, which is checked as an equality of the two excesses to nine figures. And every real arrangement with no lone pairs must sit at the minimum exactly, which is the arithmetic’s own control.

The last of those is what would catch a broken energy. A sign error or a missed pair would move all four of the lone-pair-free molecules off their minima together, and they would fail before anything else did.

Why the two accounts had to be separate #

It is worth being precise about why an energy could never have explained the count, because the answer is structural rather than empirical and would have saved the measurement.

The orphan count is an integer. It comes from a reduction — how many times each irreducible representation appears in the ligand σ set, matched against how many times it appears among the central s and p orbitals — and a reduction returns whole numbers or the character table is wrong. Nothing about it is continuous, and it changes only when the point group changes.

The repulsion is a real number that varies smoothly with every coordinate. Two arrangements a degree apart have almost the same energy and can have different point groups and therefore different counts.

So the two quantities cannot be functions of one another. An integer that steps at symmetry boundaries and a smooth function of position agree only by coincidence over a finite sample, and twenty-six arrangements is a finite sample. The reading was a coincidence found in one, and the way to see it was to notice that the two quantities are of different kinds — which is cheaper than computing either.

That does not make the measurement wasted. What it produced is the right sorting variable, and the right one is also an integer: whether the arrangement leaves a central orbital that no ligand combination can reach. Integer against integer, and it sorts all twenty-six.

Where the model stops #

The repulsion is between ligands only. It has no lone pairs in it, no bond lengths, no charges other than one apiece and no distinction between a fluorine and a hydrogen. So it is a comparison of shapes rather than a calculation of any molecule’s energy, and the per cents above are not energies anybody could measure.

That restriction is why the last figure works and also why it is limited. Saying “water is 26 per cent above the two-ligand minimum” is saying that a bare pair of ligands would be linear, which is true and is the whole content of it. It does not say how much of water’s bend the lone pairs are worth in kilojoules, and this model cannot.

The census is ten molecules and twenty-six arrangements, and the arrangements are a fixed list of named geometries rather than a continuum. A distortion halfway between a square plane and a tetrahedron is not in it, and the orphan count along such a path is not a count — it changes discontinuously where the symmetry does.

And the counting model has s and p only, which is the whole premise of the counting argument. A centre with accessible d orbitals has more to match with and a different count, and none of the three failures would be failures if the perpendicular d were included — which is exactly the argument being refused here, and the refusal rests on those orbitals being too high rather than on their symmetry — which is hypervalency without d orbitals in one sentence.

The generalisation #

A rule’s reliability and a rule’s correctness are different questions, and answering the first does not settle the second. The reading was a good one: a formula that only fails where nothing goes is a formula that can be used safely. What makes it wrong here is that the reason a molecule avoids an arrangement is not the reason the formula fails on it — the first is energetic and the second is a symmetry, and they happen to coincide on three cases out of twenty-six. A coincidence over three cases is not a mechanism, and the way to find out was to compute the other quantity.

And a control that shares everything but one variable beats a correlation over the whole set. Twenty-six rows with energy on one axis is suggestive at best; one arrangement on two molecules at identical energy, giving opposite answers, settles it in a line. The same shape appears in two nets sharing a coordination and a kurtosis and in a square plane and a chain reaching to its second neighbours: when a census suggests a cause, look for the pair that holds it fixed.

Who found it, and when #

That a planar molecule’s perpendicular p orbital is odd under its own mirror plane and cannot mix with an even ligand set is elementary group theory, and it is the reason a square-planar complex has a non-bonding orbital at all — which is what counting electrons in a complex has been about from the start, and what four is all s and p can match established here. Nothing in the mechanism is new.

What is new is the test and the answer it gives. The reading being refuted was offered honestly and with its status marked, and it is the kind of reading that would have survived indefinitely if the second quantity had not been computed — because the three failures really are unadopted arrangements, and the correlation really is perfect on the cases that exist. It takes the arrangements that do not exist to break it.

Still open: six coordination #

The obvious open question is six coordination, which nothing here touches. An octahedron and a trigonal prism both leave two combinations over, so the orphan count cannot tell them apart — and the repulsion can: 0.00 against 2.43 per cent. So the two accounts are silent and vocal on exactly opposite comparisons, and putting them together is a way of asking which of a molecule’s choices each one is entitled to speak about. The arrangement a count cannot pick is another version of the same silence.

The nearer question is the path between arrangements. The orphan count jumps where the symmetry does, and the repulsion is smooth, so along a distortion from a square plane to a tetrahedron one quantity steps and the other slides. Where the step falls relative to the energy’s minimum decides whether a molecule sitting slightly off a plane has the plane’s count or the tetrahedron’s — and a rule that depends on an exact symmetry is a rule with no answer for a real molecule, which is never exactly anything.