The count needed no table

The orphan count prices hypervalency. A main-group centre has four valence orbitals, and a molecule with n σ-bonded ligands and L lone pairs has n + L − 4 ligand combinations left with no central orbital to bond to, each one a three-centre four-electron system rather than a two-centre bond. Reduced in each molecule’s own point group, the formula is right for ten real molecules, and run over twenty-seven arrangements it is right for twenty-four and wrong for three — the flat ones — plus every ring of ligands folded off its plane without something on its axis.

Every one of those counts was a reduction. The ligand σ functions and the centre’s s and p orbitals are each written as a sum of the group’s irreducible representations, and a ligand combination counts as matched when the centre has an orbital of the same species. That needs the group’s character table, and the tables ran out three times. A twisting path passed through $D_3$, which had no table until one was written. A survey of every path found a group of order ten with none. And the flat rings of seven and eight, whose groups are $D_{7h}$ and $D_{8h}$, were left as two empty cells in a grid and argued rather than counted.

The count never needed any of them.

One matrix and its rank #

A σ-only model couples each ligand to the centre through overlaps. The ligand’s σ function overlaps the centre’s s orbital by an amount that depends only on the bond, the same for every direction. It overlaps each p orbital by that p orbital’s projection onto the bond — the component of the unit vector to the ligand along x, y or z — times a second radial overlap. So ligand i contributes one row to a coupling matrix:

$$\big(\,S_s,\; S_p x_i,\; S_p y_i,\; S_p z_i\,\big).$$

The number of central orbitals the ligands can reach at all — the number that can form a bonding combination with something — is the rank of that n × 4 matrix. Multiplying a column by a number does not change a rank, so the two radial overlaps drop out entirely, and what is left is the rank of the rows $(1, x_i, y_i, z_i)$.

That rank is a fact about geometry, and a simple one. It is one more than the dimension of the flat space the ligand direction points span. Four points in general position on a sphere span all of space, and the rank is four. Points that lie in one plane give three. Points on one line give two.

A plane meets a sphere in a circle. So the whole condition can be said without algebra: a set of ligands leaves a central orbital unreached exactly when their directions all lie on one circle of the sphere — a great circle if the plane passes through the centre, a small one if it does not.

For three ligands or fewer this is automatic — any three points on a sphere lie on some circle, any two on a line — and the unreached orbital is simply spare, which is what a centre with fewer ligands than orbitals always has. From four ligands up it is a condition. Four points in general position do not lie on one circle, and when they do, the formula n + L − 4 undercounts the orphans by one.

The flat rings are ligands on a great circle, the equator. The folded rings are ligands on a small circle, a line of latitude. Those were two failures with two different explanations, one about an antisymmetric p orbital and one about a single set of ligands meeting two totally symmetric orbitals. They are one failure.

The rank meets the tables #

A claim like that has to agree with every count already made, before it is trusted on anything new.

It was run on every arrangement counted by table in the essays before this one: the census’s own arrangements, every folded ring of three to eight ligands at nine polar angles bare and capped, the flat rings that have tables, and every point on every straight-line path between two arrangements of the same ligand count at which the point group has a table. That is 451 geometries. One reaches two orbitals by both counts, 181 reach three, 269 reach four, and not one is off the diagonal.

The same computation counts 69 geometries no table could. The linear arrangement, whose group is infinite. The pentagonal pyramid, whose group had no table until the folded rings forced one. The flat rings of seven and eight, the two empty cells: both reach three, which is what the argument from the ring’s mirror said they would, and now it is counted. And 65 points on paths between arrangements where the symmetry finder refused or found a group with no table — including all 21 points of the path from a linear to a bent molecule, which reach two.

The tables were never the source of the count. They were a way of computing a rank that the ligand directions already fix.

On one circle, with or without symmetry #

The circle condition makes a prediction the tables cannot: ligands that lie on one circle but are spaced irregularly round it, with no symmetry at all, should still leave an orbital unreached.

They do. Five ligands at polar angle 100° and azimuths 0°, 50°, 130°, 200° and 290° — a folded ring with its ligands bunched and spread at random — reach three orbitals. So does the same set turned through 23° about an axis in the equator, which is still a circle, merely one no longer centred on z; in the figure it is seen edge-on. So do six ligands irregularly placed on a circle at 110°, and five irregularly placed on the equator. Move one of the five off its circle by twenty degrees and the rank is four.

Nothing about this depends on symmetry. The regular ring has the group $C_{5v}$; the irregular sets on a small circle have none beyond the identity; the irregular set on the equator has only the mirror of its own plane. They reach the same three orbitals.

Where the table counts one too many #

The irregular sets on a small circle are the ones the table gets wrong. Their group is $C_1$, the group of the identity alone, which has one irreducible representation. Every ligand function belongs to it, and so does every central orbital. Matching by species then pairs as many as both sides have — four — and the table reports all four orbitals matched and one ligand combination orphaned out of five.

The rank says three are reached and two are orphaned. The unreached orbital is a definite combination: s mixed with the p orbital normal to the circle’s plane, in the proportion that makes its overlap with every ligand on the circle cancel. That is the same object the folded ring was found to waste — a single lobe pointing out of the open side of the umbrella — and it exists whether or not the ligands are arranged symmetrically on their circle. In the regular ring it is totally symmetric and the table sees it. In the irregular one every orbital is totally symmetric and the table cannot tell it from the others.

Matching by species is an upper bound. Two things of the same species can still fail to couple, if the coupling happens to be zero; symmetry forbids a coupling between different species and is silent about the same species. On every symmetric arrangement counted so far, the bound was reached, because symmetry was doing all the forbidding. On an arrangement whose shortage comes from geometry rather than symmetry, the bound is not reached and the table does not know.

None of the census’s arrangements is like that, and the reason is that they were all chosen for their symmetry. Coplanarity without symmetry is not a shape anyone names, so no census of named shapes would contain it.

Where the count changes #

The circle condition also explains a result that looked like a curiosity. On the path from a tetrahedron to a square plane, the count was found to be the same at every angle up to 89.99° and to change only at exactly 90°, a single geometry out of a continuum. A count that is a rank of a matrix built from coordinates has to behave that way, and the matrix says how.

A rank is the number of non-zero eigenvalues of $A^{\mathsf T}A$, a four-by-four matrix built from the rows. Its smallest eigenvalue is a continuous measure of how far the ligand directions are from sharing a plane. At the tetrahedron it is a third per ligand. It falls smoothly as the four ligands flatten, reaches 9.5 × 10⁻⁴ at 88.2°, touches zero at 90° to the precision of the arithmetic, and rises again as the square folds the other way. Near 90° it goes as the square of the angle away from 90°, to within ten per cent at every point computed within three degrees.

So the count steps where an eigenvalue touches zero, and an eigenvalue can touch zero at a point. That is the ordinary behaviour of a rank along a path, and it is why the step came out at exactly one geometry rather than over a range. It also says what the step is worth. Two degrees from the square, the unreached combination is reached, but through an eigenvalue of about a thousandth — so its bonding is correspondingly weak. The count jumps; the physics does not.

An orbital pair needs a line, and a line holds two #

The last open question the folded rings left was whether any arrangement of a single set of ligands could leave px and py short while matching pz — whether a degenerate pair of p orbitals, rather than a single totally symmetric one, could go unreached. A crown of ligands tilted alternately above and below the equator was the candidate.

The rank settles it without trying a single crown. Two orbitals left short means a rank of two. A rank of two means every ligand direction lies on one straight line. A straight line meets the sphere at no more than two points. And the two unreached orbitals form a degenerate pair only when that line runs through the centre, so that the pair is px and py about it.

So only a molecule with two ligands, on one axis through the centre, can leave a degenerate pair of p orbitals unreached, and those are the linear molecules. A bent pair also reaches only two orbitals, but the two it misses are not a pair of anything. Every crown tried — four, six, eight and ten ligands, tilted by 5° to 45° — reaches all four orbitals, and so do all 400 random sets of four to eight directions. Across every geometry computed here, a rank of two appears only with two ligands, and a rank of three only with three ligands or with more on one circle.

A pair can go short only where hypervalency cannot happen. Two ligands cannot outnumber four orbitals, so nothing is orphaned; the unreached pair is simply spare. That is the shape of the linear molecules, and they can now be counted, which the census could not do because their group is infinite. Carbon dioxide’s two oxygens reach s and the p along the axis; the other two p orbitals are spare, and n + L − 4 is zero. Xenon difluoride’s three lone pairs leave one orbital for two ligands, one ligand combination is orphaned, and n + L − 4 is one. Both agree with the formula, counted for the first time.

What the tables still do #

The rank counts; it does not name. It says how many central orbitals are unreached and it gives the unreached combination as a direction in a four-dimensional space, but the direction depends on the ratio of the two radial overlaps that dropped out of the count — a mixture of s and p whose proportions are a property of the bond. A character table does what the rank cannot: it says which species the unreached orbital belongs to, which is what decides whether a lone pair can go there, whether the orbital can mix with a π function, and how it transforms in a spectrum.

So the two are not rivals. The table’s job was always to label, and on symmetric arrangements the labels happen to add up to the count. The rank’s job is to count, and it counts where there are no labels and where the labels miscount.

What was computed and how #

Every geometry is a set of unit vectors from the centre to its ligands, with bond lengths all one, because a σ-only count cannot see a bond length. The rank is the number of eigenvalues of the four-by-four Gram matrix of the rows $(1, x, y, z)$ that exceed 10⁻⁹ of the largest, computed by the same Jacobi diagonaliser the Hückel calculations use. The table counts are the reductions already published: the census’s own tables, the generated $C_{nv}$ tables for the folded rings, and the tabulated groups wherever the finder identified one on a path. Paths are the straight-line interpolations between two arrangements, aligned by the best rotation and ligand permutation, at 21 points each; there are 19 paths.

The claims are stated where they can fail: that on every geometry with a table the rank equals the table’s count; that more than four hundred geometries are compared; that the flat rings of seven and eight, which have no table, reach three; that every rank below four has a smallest eigenvalue of zero; that every random set of four or more directions and every crown reaches four; that ligands on a circle reach three and one moved off it reaches four; that on the irregular sets with group $C_1$ the table counts four where the rank finds three; that carbon dioxide and xenon difluoride agree with the formula; and that on the flattening path the count changes at 90° and nowhere else, with the smallest eigenvalue vanishing as the square of the distance from it. The refusal is the linear arrangement, which must reach exactly two.

Where the count stops #

It is σ only. A ligand with a π function, or a central atom with d orbitals in play, adds columns to the coupling matrix, and the rank argument extends to them only if their overlaps are modelled. The count here is the bookkeeping the three-centre account of hypervalency runs on and nothing wider.

A rank is all or nothing. An eigenvalue of a thousandth is a reached orbital by the count and a very weakly bonding one in any calculation with energies in it. The count is a statement about what symmetry and geometry forbid, not about what is strong, and near a count change the difference is the whole story.

And the irregular circles are not molecules. Five ligands bunched at random on a small circle round a main-group centre is not a structure anyone reports. The case matters because it is the one where the table’s upper bound is not reached, and a counting method that fails on some geometry fails as a method, whether or not chemistry visits that geometry.

Symmetry was standing in for geometry #

The habit this corrects is treating a symmetry argument as the reason for a count, when the symmetry was a convenient way to compute something else. The σ-only count is a rank, and the rank is decided by whether the ligand directions share a plane. Symmetric arrangements share planes in symmetric ways, so a character table reproduces the rank on every arrangement with a name — and on every one the census, the folded rings and nineteen paths could supply, 451 in all. But the rank is the thing, and on the one arrangement built to separate them, the table is wrong.

The corollary is about what the three flat failures were. They were read as a property of planarity, then as a property of a single set of ligands meeting two symmetric orbitals, and each reading was right about the cases in view. The reading that covers every case is shorter than either: a centre loses an orbital when its ligands lie on one circle. A flat ring is one circle. A folded ring is another. A ring of ligands with anything on its axis is not on one circle, and that is why an apex restores the count.

Still open: the eigenvalue as a strength, and what π adds #

The obvious open question is whether the smallest eigenvalue means anything beyond the count. It is a continuous measure of how nearly an orbital is unreached, and along the flattening path it vanishes exactly where the repulsion energy has its maximum. Putting it beside a σ-only bonding energy along the same paths — a Hückel-like energy with the same coupling rows — would say whether a small eigenvalue is a weak bond in the energetic sense, and whether the arrangements chemistry avoids are the ones with a small eigenvalue rather than the ones with a zero.

The nearer question is π. A ligand π function adds two rows per ligand, each along a direction perpendicular to the bond, and the rank then depends on the ligand directions and the orientation of their π functions together. Whether a coplanar set that loses an orbital to σ gets it back through π — which is what happens chemically when a flat ring of π donors surrounds a centre — is a rank of a matrix with twice as many rows, and it can be computed on the same geometries.