A count that changes at one point

Hypervalency can be priced by counting orphans: ligand σ combinations that find no partner among the central atom’s s and p orbitals, once the lone pairs have taken theirs. The count is a reduction in a point group, and a point group is a property of an exact geometry.

Put beside a repulsion energy, that count and the energy are silent and vocal on opposite comparisons — the count cannot tell an octahedron from a trigonal prism and the repulsion can; the repulsion cannot say which arrangements are adopted and the count can. That leaves what happens between two arrangements rather than at them. The count jumps where the symmetry does and the energy is smooth, so along a distortion one steps and the other slides, and 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.

The step is not along the distortion. It is at the end of it.

The path #

Four ligands, two pairs, each pair in a plane containing z and one of x or y, at ±α from that axis. At α = 54.7356° every dot product is −1/3 and the four are a tetrahedron. At α = 90° all four lie in one plane and are a square. In between the symmetry is D2d, and — this is the whole content — the two extremes are the only points on the path where it is anything else.

The group is found rather than assumed. Each geometry is handed to the same symmetry finder the rest of the collection uses, which looks for operations that map the molecule onto itself and identifies what it finds. It returns Td at one end, D4h at the other, and D2d for everything in between that it will answer at all.

The orphan count is zero at the tetrahedron. It is zero in D2d — at 56.7°, at 62°, at 74°, at 86°, at 89.5°, at 89.9°, at 89.99°. It is one at 90°.

Zero is the expected value for four ligands: a centre has four valence orbitals, an s and three p, and there are four σ combinations, so with no lone pairs there can be exactly one partner for each. Whether there is depends on the species matching, and on this path they match twice and fail once.

In Td the four σ combinations reduce to A₁ ⊕ T₂, and the centre’s p set is T₂ with its s totally symmetric — every species is met. In D2d both the σ set and the p set split, to A₁ ⊕ B₂ ⊕ E and B₂ ⊕ E, and again every species is met. At the square plane the σ set is A₁g ⊕ B₁g ⊕ Eu while the p set is A₂u ⊕ Eu: B₁g finds nothing on the centre and A₂u, the p orbital perpendicular to the plane, is left with nothing to bond to. That is the waste priced earlier, and it is a property of D4h and of no other group on this path.

How close is 89.99 degrees #

The interesting question is not that the count steps but how narrow the step is, and the way to say that is in a length rather than an angle.

At 89.99° a ligand sits 1.7 × 10⁻⁴ of a bond length from the planar position. The symmetry finder’s own tolerance is 0.06 — three hundred times larger. A diffraction experiment on a small molecule resolves atomic positions to a few thousandths of an ångström at best, which on a two-ångström bond is around 10⁻³ of a bond length, six times larger again. Thermal motion at room temperature is larger than either.

So the geometry that has the plane’s count and the geometry that has the tetrahedron’s are, by every instrument anybody has and by the symmetry search used here, the same geometry. The count distinguishes them and nothing else does.

It is worth being precise about what that does and does not undermine. The count at the square plane is correct: a molecule that really is planar really does have a σ combination with no partner, and the b₁g orbital really is left out. The problem is that “really is planar” is a claim no measurement supports, and the count changes its verdict on a difference no measurement resolves. A quantity can be exactly right about a geometry nobody can establish.

That is the worry, arrived at as a number: a rule that changes its answer only at an exact symmetry has no answer for a real molecule. It does not merely have a poorly located answer; it has an answer at one point of a continuum and the same answer everywhere else.

That is the mechanism of the step and it explains why the step is a step. B₁g exists as a species only in D4h; the moment the four ligands leave the plane, the mirror that distinguishes g from u is gone, B₁g and A₂u are no longer different species, and the two orbitals that could not meet each other can. Nothing about the sizes of any overlap changes between 89.99° and 90° — what changes is a selection rule, and a selection rule is exactly true or exactly false.

So the count is doing what it was built to do. It reports a symmetry-forbidden mismatch, and a symmetry-forbidden mismatch is a property of a symmetry.

And the step is at the maximum #

The question was framed around the energy’s minimum, which is where a molecule would sit. The step is at the other end.

The repulsion is 3.674235 at the tetrahedron and 3.828427 at the plane, rising smoothly and monotonically by 4.20 per cent. The plane is the worst geometry on the path by this measure and the tetrahedron the best, which is the ordinary result and is why four ligands around a bare centre are tetrahedral.

So the one geometry whose orphan count differs from every other is the one geometry the energy most disfavours. A square-planar molecule exists — nickel(II) and platinum(II) make them routinely — but not because the repulsion permits it; it exists because a d shell supplies a term this model has no representation of at all, which is the same limitation the two models that disagree about the shape ran into from the other side. On the path as computed, the count’s distinguishing point and the energy’s least favourable point are the same point.

A band with no count in it #

There is a third regime on the path, and it was not designed for.

Between the tetrahedron and about two degrees off it, the symmetry finder refuses. The tetrahedral operations still map the molecule onto itself within its tolerance — a residual of 3 × 10⁻³ against a tolerance of 0.06 — but cannot be refined to the precision it demands of an operation it is going to keep, so it returns no group rather than choosing one.

That is the symmetry search behaving correctly. A near-symmetry is not a symmetry, and an instrument that reported Td for a geometry that is not Td would be worse than one that reports nothing. But the consequence is that the orphan count has three states along this path rather than two: zero, one, and unanswerable — and the unanswerable band is at the energy’s minimum, which is where molecules are.

What the two quantities are each good for #

Putting the two side by side along one coordinate makes the division of labour concrete, and it is a better statement of the finding than the one reached by comparing arrangements.

The repulsion is a smooth function of the geometry. It has a value everywhere on the path, it orders the geometries, and it says the tetrahedron is best by four per cent over the plane. That is the right instrument for asking which of these shapes a molecule adopts and it is useless for asking what is special about one of them, because it singles nothing out.

The orphan count is a step function on the strata of the geometry space. It has one value on the whole D2d interior and a different one at a single D4h point, so it says nothing at all about the ordering of geometries and everything about what is special at one. That is the right instrument for asking what a symmetry forbids and useless for asking which shape is adopted.

Neither is a weaker version of the other and neither interpolates the other. They are silent and vocal on opposite comparisons across arrangements; along a path the same fact is that one is continuous and the other is not, which is a cleaner way to say it and predicts the arrangements result rather than merely agreeing with it.

What was computed, and how #

Twenty geometries on the path, each four unit vectors. The point group comes from a search for symmetry operations at a tolerance of 0.06 ångström; the reduction of the ligand σ set and of the centre’s s and p follows in whatever group is found, and the orphan count is the number of σ combinations left with no partner. The repulsion is the Coulomb sum over unit vectors used for the arrangement census, so the two quantities are the ones compared across arrangements.

Nine things are checked: that the path’s ends are identified as a tetrahedron and a square plane by their own symmetry rather than by the formula that made them; that the repulsion is lowest exactly at the tetrahedron and highest exactly at the plane; that the two ends differ in orphan count, so there is a step to locate; that the count is constant at every angle short of the plane including the nearest one; that the nearest answered angle is inside a hundredth of a degree, so the statement is about real molecules; that the interior is one group throughout; and the two refusals — that a band near the tetrahedron has no symmetry assigned rather than a wrong one, and that the count there is reported as absent rather than guessed.

Where the model stops #

Unit charges on a sphere are not a molecule, and the repulsion here has no bonding in it at all. What it supplies is a smooth function of the geometry with a minimum in the right place, which is what the comparison needs; it is not a claim about which arrangement any particular compound adopts. That distinction has held since the count and the population were separated.

It is also worth saying that the four ligands here carry no lone pairs on the centre, which is the simplest case and the one where the count is zero throughout the interior. A centre with a lone pair has fewer valence orbitals to offer and the interior count would be one rather than zero — but it would still be constant on the interior and still step only at the plane, because the count is a matching of species and the species are what the group decides.

The path is also one path. A tetrahedron can be flattened many ways, and this one keeps a D2d subgroup throughout by construction. A distortion that broke the symmetry further would pass through C2v or C1, where the reduction is different and the orphan count could take other values — so “the count is constant along the path” is a statement about this path and about the group it preserves. What would not change is the shape of the finding: a count computed in a point group is constant on each stratum of geometries with the same group, and the strata that are not generic have measure zero.

And the two-degree refused band is a property of one tolerance. A finder with a tighter tolerance would refuse a narrower band and a looser one a wider band; no finder can refuse none, because deciding whether an approximate operation is an operation is what a tolerance is for. The band’s existence is general and its width is not.

There is one more thing the sweep does not settle. The refusal near the tetrahedron is a property of the finder, but a molecule two degrees off a tetrahedron has a real point group — C1, or D2d if the distortion respects it — and a better finder would say so. What no finder can do is tell that geometry apart from the tetrahedron by measurement, so the honest reading of the band is not “the count is undefined” but “the count is defined and unknowable”, which is the same practical situation and a different logical one.

The generalisation #

A quantity computed in a point group is a step function on the space of geometries, and it takes its non-generic values on sets of measure zero. That is not a defect of the quantity; it is what a point group is. The defect is in using such a quantity to describe a molecule, which is never at a non-generic geometry and is only ever near one.

The practical version has two parts. First, a symmetry-derived count says nothing about the neighbourhood of the geometry it is computed at — it says one thing at a point and a different thing everywhere around it, and there is no interpolation. Second, and less obvious, the count is the generic value almost everywhere, so a count computed for an idealised high-symmetry geometry is the one value that a real molecule near it does not have.

A corollary is worth stating for anybody reading a symmetry-derived number in a paper. If the number is quoted for an idealised geometry, it is the value at a point; if the molecule is described as “approximately” that geometry, the number quoted is the one it does not have. Both sentences are usually in the same paragraph.

Whenever both are available, the smooth quantity is the one to compare across geometries and the stepped one is the one to compare across topologies. That is very close to what the comparison across arrangements found by a different route — the count and the repulsion being silent and vocal on opposite comparisons — and the path supplies the reason: they are different kinds of function of the same argument.

Who found it, and when #

The reduction of a ligand σ set in a point group is standard group theory and older than any of this. The orphan count and its use as a price for hypervalency are original to these essays, as are the path, the sweep and every number above. The finding is about a method rather than about a molecule — and it is not the only count to turn out to distinguish a point rather than a region; another is the tie a rank cut fell inside.

Still open: a path that breaks the symmetry further #

The obvious open question is a path that breaks the symmetry further. Flattening a tetrahedron along a D2d coordinate is the most symmetric way to do it, so the count’s constancy along the interior may be a property of the coordinate rather than of counts generally. A distortion through C2v — moving three ligands and not four — would pass through a group whose reduction differs from both ends, and if the count takes a third value in the interior then the step structure is richer than one point and the picture here is the best case rather than the typical one.

The nearer question is the six-coordinate case. An octahedron and a trigonal prism both leave two combinations over, so the orphan count cannot separate them and the repulsion can, at 0.00 against 2.43 per cent. The path between those two is the Bailar twist, it is one angle, and running this same measurement along it would say whether the count is constant there too — which on the argument above it must be, since the twist keeps D3 throughout, and a prediction made before the calculation is the best test available.