The census a bond rule was hiding

A molecule’s internal coordinates are built from its bonds: one stretch per bond, one bend per pair of bonds at a shared atom, and a wag where a centre and its three ligands lie in a plane. From those coordinates come the redundancies — combinations of coordinates that move no atom — and from the coordinates and their redundancies under the molecule’s symmetry comes the count of totally symmetric vibrations. That count obeys an identity: it is the number of orbits of coordinates that survive their own stabiliser, less the number of totally symmetric redundancies, and the usual textbook version without those two corrections is wrong for most molecules.

All of it rested on a bond list, and the bond list was shown not to work. One length cutoff, with a clause about hydrogen, gave five of the twenty-three molecules drawn here no bonds at all and gave ferrocene’s iron none. A rule on the length over the sum of two covalent radii separates every bond from every contact with room to spare. It was established and not adopted, because adopting it changes every set of coordinates built on it, and what changes was worth reading on its own before anything else moved.

What the radius rule changes #

Remarkably little and exactly the right things. Of the twenty-three molecules, fifteen had coordinate sets under the cutoff and four are linear, with an infinite point group that a finite census cannot reduce over. Thirteen of the fifteen get identical coordinate sets under the radius rule — the same bonds, and so the same stretches, bends and wags.

The other six are the molecules the cutoff failed. Xenon tetrafluoride, the hexafluoridocobaltate ion, and the tetrachloride ions of nickel and platinum had no bonds, because every one of their bonds is longer than 1.85 ångström; each now has ten or twenty-one coordinates and enters the census for the first time. Xenon difluoride also had none and now has two bonds, but it is linear and stays outside. And both ferrocenes gain their ten iron–carbon bonds, going from sixty coordinates to a hundred and thirty-five.

The census reaches nineteen molecules where it reached fifteen, and the earlier question of whether any complete set would stop being complete under a better rule has a clean answer: none does. Every set that changed became larger, and the two that were incomplete because of the rule became complete.

The identity holds on the newcomers #

The identity was checked on fifteen molecules, all of them molecules the cutoff happened to handle. Four of those it did not handle include the only octahedral ion among the molecules computed, the only tetrahedral ion and both square-planar centres — shapes where a coordinate set is built very differently. If the identity were an artefact of which molecules had bonds, this is where it would show.

It holds on all four. Each newcomer’s totally symmetric vibrations equal its symmetric orbits less its totally symmetric redundancies, and the count agrees with the one computed independently from Cartesian displacements, which shares nothing with the coordinates.

The usual formula — orbits less redundancies, with no correction for orbits a symmetry reverses or redundancies that are not totally symmetric — does exactly what the earlier census would predict from the shapes. The tetrahedral nickel ion has two orbits and one redundancy and one totally symmetric vibration, like methane, and the formula is right. The octahedral cobalt ion has three orbits and six redundancies and one totally symmetric vibration, like sulfur hexafluoride, and the formula predicts minus three for both. The two square-planar molecules have three orbits and three redundancies, and the formula predicts zero where each has one.

So the formula is right for eight of nineteen molecules where it was right for seven of fifteen, and every one of its successes and failures is predicted by the shape rather than by the element. That is the strongest version of the earlier result available: the correction depends on how a group acts on coordinates, and molecules of the same shape built from quite different atoms fail it in exactly the same way.

Why the shape decides and the element does not #

It is worth saying why the newcomers behave exactly like the molecules they resemble, because the reason is the whole content of the identity.

The correction has two parts. An orbit of coordinates contributes a totally symmetric combination only if its coordinate is not reversed by an operation that maps it to itself — the operations that fix a coordinate are its site symmetry, and a reversal there kills the symmetric combination. And a redundancy is subtracted only if it is itself totally symmetric. Both statements are about how a group acts on a set of geometric objects, and neither contains a mass, a force constant or an element.

So two molecules whose atoms sit in the same arrangement, bonded the same way, carry the same coordinate sets up to relabelling and the same group acting on them, and every number the census produces is the same. The hexafluoridocobaltate ion and sulfur hexafluoride share three orbits, six redundancies of which two are totally symmetric, fifteen vibrations and one totally symmetric vibration. The nickel ion and methane share two orbits and one redundancy. A census is a statement about shapes, and it could only have been checked for the new shapes once the new shapes had coordinates — which is precisely what the cutoff withheld.

The one place an element entered was the bond list, and it entered as a length. That is why a length cutoff failed on exactly the heavy-atom molecules: long bonds are a property of large atoms, and a single length treats a platinum–chlorine bond and a hydrogen–hydrogen contact as the same kind of object. The radius rule removes the element from the one place it was doing harm, and a decision about tolerance is left in its place — which is the right kind of thing to have to decide, because it can be tested.

Ferrocene, finally complete #

Ferrocene was the molecule that exposed incompleteness in the first place. Under the cutoff its iron had no bonds, so its sixty coordinates described two cyclopentadienyl rings and nothing holding them together; they spanned forty-four of its fifty-seven vibrations, and the totally symmetric count they gave was three where the molecule has four. The earlier fix recorded the missing bonds and could not add them without changing the rule for every molecule.

With the ten iron–carbon bonds the set has a hundred and thirty-five coordinates — ten more stretches, and bends at the iron between every pair of its ten neighbours as well as at each carbon between the ring and the iron. They span all fifty-seven vibrations, and the totally symmetric count they give is four, the molecule’s own. The orbits go from five to twelve and the redundancies from sixteen to seventy-eight, which is what a ten-coordinate centre does: forty-five bends at one atom that has three degrees of freedom to bend in.

The staggered form gives the same hundred and thirty-five coordinates, twelve orbits and seventy-eight redundancies, as it should: turning one ring by a tenth of a turn changes the point group from D₅h to D₅d and leaves every count of bonds, bends and their orbits untouched.

The usual formula goes from minus eleven to minus sixty-six. That number is worth having in view because it is the clearest illustration yet of what the formula counts. It subtracts every redundancy, and a heavily coordinated atom produces redundancies quadratically in its number of neighbours; the correction subtracts only the totally symmetric ones, of which there are far fewer.

A gap no bond rule closes #

Two of the newcomers are incomplete, and not for the reason ferrocene was. Xenon tetrafluoride and the tetrachloridoplatinate ion have exactly the right bonds — four each, all found — and coordinate sets spanning seven of their nine vibrations.

What they miss is exactly the two out-of-plane species, A₂u and B₂u. A₂u is the centre moving through the plane of its ligands; B₂u is two opposite ligands rising while the other two fall. Every in-plane species is reached, including both degenerate Eu pairs. The missing motions are the ones in which atoms leave the molecular plane, and a coordinate set made of bond lengths and the angles between bonds in that plane has nothing that changes to first order when they do.

The construction provides a wag — an out-of-plane coordinate — for a planar centre with three ligands, which is how boron trifluoride’s out-of-plane vibration is reached. It provides nothing for four. That is a choice in how coordinates are built, not a property of the bonds, and it means a correct bond rule does not by itself give a complete coordinate set: the square-planar gap was invisible under the cutoff only because those molecules had no coordinates at all.

It joins three gaps already known, and it is a different kind. Hydrogen peroxide lacks its torsion, which is a coordinate that needs four atoms in a chain; ethene lacks its twist about the double bond, and benzene two of its out-of-plane species, B₂g and E₂u. The square-planar gap is the first found at a single centre, and it would apply to any four-coordinate planar metal complex that is ever drawn this way.

The window under a second table #

The radius rule’s own robustness was the other question left open. Its window — every bond below a tolerance, every contact above it — was measured with one tabulation of covalent radii, and radii are tabulated several ways.

Under a second, independent tabulation of single-bond radii the window survives, and it is nearly the same width: from 1.171 to 1.262, a factor of 1.078, against 1.117 to 1.211, a factor of 1.084. Both windows are bounded by the same two pairs: hydrogen peroxide’s O–O bond, the longest bond relative to its radii, and bromochlorofluoromethane’s chlorine–bromine contact, the shortest contact relative to its. That is the reassuring part. The edges are set by the chemistry of two molecules — a weak bond between two electronegative atoms, and two large halogens held close by a small carbon — and not by the quirks of one table.

The unreassuring part is that the second window is shifted up by about five per cent, and the factor in use, 1.163, chosen as the middle of the first window, lies below the second window’s lower edge. Under the second table it would drop peroxide’s O–O bond. A factor between 1.171 and 1.211 works under both. The middle of one table’s window was a sensible choice and not a robust one, and the robust choice is the overlap.

How the census was rebuilt #

The bond list for each molecule is every pair of atoms whose distance is at most 1.163 times the sum of their covalent radii, with no clause about any element. The internal coordinates are built from it exactly as before: a stretch per bond, a bend per pair of bonds at a common atom, and a wag at a centre whose three ligands are coplanar with it. The census is the library’s own — the action of every symmetry operation on the coordinates, the orbits under that action, the redundancy space from the Wilson B matrix, and the reduction of both representations over the point group — with the coordinate set passed in rather than built from the cutoff.

The missing species of an incomplete set are found by reducing the molecule’s vibrational representation from Cartesian displacements and subtracting what the coordinates’ own representation contains. The second window uses the same construction as the first: in each molecule the pairs are ordered by length over radius sum, the chemically correct number are called bonds, and the window runs from the largest bonded ratio to the smallest unbonded one across every molecule.

The checks, run wherever these figures are drawn: the census gains exactly the four non-linear molecules that had no bonds; xenon difluoride stays out; the identity holds on every molecule reached; the usual formula is right for exactly one newcomer, the tetrahedral one, and wrong for the octahedral ion by exactly as much as for sulfur hexafluoride; both ferrocenes become complete with the molecule’s own count; both square-planar molecules lack exactly A₂u and B₂u; the second table keeps the window with the same pairs at its edges; and the factor in use falls below its lower edge. The refusal is every molecule whose bond list did not change, whose census must match the old one integer for integer — otherwise the comparison would be measuring a reimplementation of the coordinates rather than the rule.

What the rule still decides by hand #

The correct bond counts are written down. The window is defined by how many bonds each molecule has, as anybody would count them, and a molecule where that count is itself a matter of convention — a bridging hydride, a metal–metal contact — would need the convention stated before the rule could be tested on it.

The coordinates are built one way. Stretches, bends and a planar wag are the conventional valence set, and the square-planar gap is a property of that convention. Other conventions — Cartesian displacements symmetrised directly, or local symmetry coordinates chosen per site — would not have the gap and would have different orbit and redundancy counts; the identity is about whichever set is used, and the gap is about this one.

A tolerance is not a bond order. The rule decides whether two atoms are bonded, not how strongly, so the census counts a ferrocene iron–carbon bond and a sulfur–fluorine bond as one stretch each.

And two tables are two tables. The windows overlap by four per cent, and a third tabulation could move the shared window again. The two edge pairs being the same under both is evidence that the window is chemistry, not proof.

A convention is tested where it fails #

The pattern this completes is a familiar one. A rule was chosen to handle the molecules in hand, a clause was added when one molecule misbehaved, and the census built on it was checked and passed — on every molecule the rule could handle. The molecules it could not handle were not wrong; they were absent. A check that runs only where its preconditions are met is silent about exactly the cases that would have tested it, and the fix was not a better check but a better precondition.

The same absence hid the square-planar gap, and that one is not the bond rule’s to fix. It lives one step further along, in how coordinates are built from bonds, and a coordinate set that counts every bond can still leave motions out.

Still open: an out-of-plane coordinate, and the projection #

The obvious open question is the coordinate that would close the square-planar gap. A four-coordinate planar centre has two out-of-plane motions, and a natural candidate is the pair of pyramidalisation coordinates built from opposite ligand pairs — the distance of the centre from the line through each pair, signed. Adding them would test whether the set becomes complete and whether the identity survives two coordinates that the other nineteen molecules never needed; the census would then say which other molecules, if any, the new construction changes.

The nearer question is the one every coarse species test has deferred: where a species appears more than once, only a projection onto normal coordinates says which vibration a distortion goes into, and a projection needs a force field. With every coordinate set now either complete or incomplete for a known reason, the coarse test and the projection can be compared on the six molecules that have force fields, and nothing in the count underneath either of them is still in doubt.