No length separates them

An earlier essay closed by naming the bond list as the next thing to check. One length had miscounted three molecules, in both directions; a rule built from per-element bonding radii would decide ferrocene’s iron and hydrogen peroxide’s hydrogens without a special case; and which of the coordinate sets would change under such a rule had not been run.

Run, it is worse than three molecules and the reason is not a badly chosen number.

Two populations that overlap #

The rule in use is one cutoff at 1.85 ångström, with a clause saying a hydrogen keeps only its nearest neighbour. The clause exists because hydrogen peroxide’s O–H bonds are 0.95 ångström and its other O–H distances are 1.82 — inside a cutoff chosen so a long bond is not missed — so the molecule came back with five bonds instead of three, and thirteen internal coordinates where it has five.

The clause repaired hydrogen peroxide. The question is what a cutoff can do at all, and that is answerable without choosing one.

Take every molecule drawn here, write down how many bonds it has, sort its pairs by length, and look at the two that straddle the boundary: the longest pair that is a bond and the shortest that is not. A cutoff has to sit above the first and below the second, in every molecule at once.

The longest bond is 2.3200 ångström — the platinum–chlorine bond of the tetrachloridoplatinate ion. The shortest pair that is not a bond is 1.5144 — water’s two hydrogens.

Those are in the wrong order. A cutoff must be above 2.32 and below 1.51, and there is no such number: the two populations overlap by a factor of 1.53. So the rule is not a rule with a badly chosen constant. It is a rule with no constant that works, and the clause about hydrogen is one patch over the only overlap anybody had noticed.

Divided by two radii #

Divide every distance by the sum of its two atoms’ covalent radii and ask the same question.

Every bond drawn here has a ratio at or below 1.11742 — hydrogen peroxide’s oxygen–oxygen bond, which is the shortest bond relative to its own radii. Every pair that is not a bond has a ratio at or above 1.21120 — bromochlorofluoromethane’s chlorine and bromine, at 2.689 ångström, the closest non-bonded contact in the collection relative to its radii.

Those are in the right order, with a window a factor of 1.0839 wide between them. Every tolerance in it classifies all twenty-three molecules correctly.

The window is closed at the bottom and open at the top, and the reason is the comparison rather than the numbers. The rule keeps a pair whose ratio is at or below the tolerance, so a tolerance of exactly 1.11742 still admits peroxide’s oxygen–oxygen bond and a tolerance of exactly 1.21120 also admits the chlorine–bromine contact that is not one. It is a distinction of no practical consequence — nothing would be set at either endpoint — and it is the difference between a window that has been stated and a window that has been checked.

Both edges are where they are for a reason a single molecule supplies, which is what makes them worth walking past. Below 1.11742 the rule starts losing real bonds and peroxide’s oxygen–oxygen is the first to go; above 1.21120 it starts collecting contacts that are not bonds and bromochlorofluoromethane’s halogens are the first to arrive. Between them nothing changes at all, which is the shape of a rule with room rather than a rule that has been tuned.

And it needs no clause about hydrogen. Peroxide’s long O–H contact is 1.82 ångström against a radius sum of 0.97, a ratio of 1.88 — nowhere near the window, and rejected by the same rule that accepts every real bond. The patch is not needed because the case it patched is not a close call on this scale.

What the cutoff has been losing #

Seven of the twenty-three come back wrong, and five of them come back with nothing at all.

Xenon difluoride and xenon tetrafluoride have every bond at 1.98 ångström. Hexafluoridocobaltate’s are 1.93, tetrachloridonickelate’s 2.26 and tetrachloridoplatinate’s 2.32. The cutoff is 1.85, so all five molecules have zero bonds, and everything built from a bond list has nothing to build from.

Ferrocene is the seventh and is the one named earlier. It finds its twenty carbon–carbon bonds and loses its ten iron–carbon ones at 2.06 ångström each — twenty where it has thirty, which is exactly the shape of the defect that essay suspected.

Nothing failed anywhere, and that is the part worth dwelling on. A molecule with no bonds has no internal coordinates, so it simply does not appear in a census over coordinate sets; the census reports fifteen molecules where there are twenty-three, and the eight missing are missing rather than wrong. The point group is recovered from the atom positions rather than from the bonds, so every symmetry figure of a xenon fluoride is correct. The molecules are drawn, they are assigned, they are counted — and the coordinate computation has never seen five of them.

That is the same failure mode described one level down: an omission where every other check reads something that exists and asks whether it is right — the shape that essay’s own torsion census had.

It is worth being precise about which findings this does and does not reach. Everything established here about symmetry is untouched: a point group comes from atom positions, a character table from the group, a species count from the group and the coordinates. What is reached is everything that counts internal coordinates — the coordinate census, the redundancy count, and the orbit rule for totally symmetric vibrations — because all three read a bond list. Those were established over fifteen molecules, and the five with no bonds were never among them, so no recorded number is wrong. What is wrong is the denominator: fifteen of a collection of twenty-three, presented as the molecules the computation covers rather than the molecules it happened to reach.

Which is the more comfortable of the two ways to find this. A defect that produced wrong numbers would have to be chased through every essay that quoted one; a defect that produced missing rows leaves the quoted numbers alone and makes the census incomplete, which is a thing to extend rather than to correct.

Why a ratio can do what a length cannot #

The two rules differ in one thing and it is worth stating plainly, because the improvement is not a matter of degree.

A bond length is the sum of two atomic sizes plus a correction. Among these molecules the atomic sizes span a factor of four and a half — fluorine’s covalent radius is 0.57 ångström and xenon’s is 1.40 — so bond lengths span from 0.95 to 2.32, a factor of 2.4. A non-bonded contact is set by the geometry of a molecule rather than by the sizes of the atoms in it, and geometry does not scale with the atoms: water’s two hydrogens are 1.51 ångström apart because the H–O–H angle is 104 degrees and the bonds are 0.96 long, and neither of those cares what a platinum–chlorine bond does.

So the bonded population scales with the radii and the unbonded one does not. Dividing by the radii removes the scaling from the first and leaves the second alone — which is why the window exists. It would not exist if the non-bonded contacts also scaled, and it nearly fails where they most nearly do: the tightest contact in the collection is two large halogens on one small carbon, which is exactly the case where a contact is set by the atoms’ own sizes rather than by the molecule’s shape.

That also says where the rule would break. A molecule whose non-bonded contacts are set by its atoms’ sizes — a crowded one, or a solid — would push the lower edge down, and a molecule with a genuinely long bond would push the upper edge up. Neither is in this collection.

Why the window is where it is #

The two edges are worth naming because of what they are not.

The upper edge is an O–O single bond in hydrogen peroxide: 1.475 ångström against a radius sum of 1.32. Oxygen’s radius is tabulated from a large number of ordinary oxygen compounds, and peroxide’s O–O is a slightly long single bond, so the ratio sits high. The lower edge is a chlorine and a bromine on the same carbon in bromochlorofluoromethane: 2.689 ångström against a radius sum of 2.22. Two large halogens on one small carbon are pushed closer together than two such atoms would otherwise come.

Neither edge involves a metal, which matters because the metals are the one genuinely ambiguous entry in the radius table — iron and cobalt have low-spin and high-spin values differing by two tenths of an ångström, and a single-valued table has to choose. The choice moves no edge of the window, because every metal bond and every metal non-bond is far from both.

So the window’s width is a fact about a peroxide bond and a halogen contact, and the rule is decided by main-group chemistry rather than by the hardest radii in the table.

The metals are worth one more sentence because they are where the intuition says the trouble should be. A platinum–chlorine bond at 2.32 ångström is the longest bond in the collection and the one that breaks every length cutoff; on the ratio scale it is 0.975 times its radius sum — shorter than the two radii added together, and among the most clearly bonded pairs there is. The bond that no length rule can accommodate is an ordinary bond once its two atoms’ sizes are put in, which is the whole argument in one pair of numbers.

And the iron–carbon bonds of ferrocene are 2.064 ångström against a radius sum of 2.08 — a ratio of 0.992, again tighter than the radii added. The ten bonds a length cutoff loses sit closer to the middle of the bonded population than the carbon–carbon bonds it keeps, which are at 0.941.

What was computed, and how #

The radii are quoted from the standard tabulation and the bond counts are written down. That second half is deliberate: a test that defined the right answer by a rule would be a test of that rule against itself, so the counts here are what anybody would say the molecules have — two O–H bonds in water, thirty in ferrocene counting the ten to the iron, three in hydrogen peroxide counting the O–O.

Both rules are then measured the same way. Sort each molecule’s pairs by the rule’s own quantity, take the first as many as the molecule has bonds, and record the largest value among those and the smallest among the rest. The rule works for the collection exactly when the first is below the second everywhere, which is a question about a window rather than about a cutoff.

Seven things are checked: that no cutoff on a length separates the two populations, with the two pairs named; that they overlap rather than merely touch; that a cutoff on the ratio does separate them, with those two pairs named; that the window is several per cent wide rather than a single value; that the rule in use gets several molecules wrong; that several of those come back with no bonds; and that the radius rule at the middle of its window gets every molecule right.

The refusal is the overlap itself. Showing that the length rule fails is stronger than showing that some other rule succeeds, because it holds whatever the alternative is — and it is the test that would catch a future molecule making the ratio rule fail too.

One detail of the method is worth recording because it is the part that could have gone wrong quietly. The pairs are sorted and the first k taken, with k the written-down count — so the test never asks whether a particular pair is a bond, only whether the rule’s ordering puts the right number of pairs first. That is weaker than checking each pair and it is the right weakness: a rule that got the ordering right and the threshold wrong would pass, which is exactly the rule being looked for, since the threshold is what the window is about. What it cannot catch is a rule that ordered two pairs wrongly and still got the count right. Checked directly, no such case occurs: the radius rule at the middle of its window returns the same set of pairs as the ordering does, on all twenty-three.

Where this stops #

Nothing here changes a figure yet. The rule in use is what every existing bond list was built from, and swapping it would change the internal coordinates of the seven molecules it gets wrong — which changes their coordinate censuses, their redundancy counts and their totally symmetric species counts. Whether any of the collection’s recorded findings moves under the new rule is a census in itself and this is not it: what is established is that the current lists are wrong on seven molecules and that a rule exists which is not.

Twenty-three molecules is the collection and not chemistry. The window is a fact about these structures, and a molecule with a genuinely long bond or a genuinely tight contact would narrow it. Hydrogen bonds would close it outright — an O–H·⋅⋅O contact is about 1.8 ångström against a radius sum of 0.97, a ratio of 1.9, which is the same place peroxide’s long contact sits, so the rule would call a hydrogen bond a non-bond. That is the right answer for this collection’s purposes and would be the wrong one somewhere else.

And a covalent radius is a fit. The tabulation is a least-squares compromise over thousands of measured bond lengths, so the ratios above are lengths divided by averages, and the window’s width is partly a statement about how good that compromise is. What it is not is a free parameter: the radii were fitted for a different purpose by people who had never seen this collection.

The generalisation #

The habit is to ask whether a classification rule can work before asking what its threshold should be.

A threshold rule has a testable precondition and it is cheap: the two populations it is meant to separate must not overlap in the quantity it thresholds. Testing it needs the right answers, which is the part that feels like work and is the part that makes the test possible — and it returns something better than a threshold, namely a window, whose width says how much room the rule has and therefore how much a later case can move things before the rule breaks.

The corollary is about what a patch means. A clause added to a rule to fix one case is evidence that the rule has an overlap, and the natural response is to fix the case; the informative response is to ask how large the overlap is. Here it is a factor of 1.53 and the clause covered the one instance anybody had looked at. A special case is a measurement of a rule’s inadequacy and is usually read as a repair of it.

Who found it, and when #

Covalent radii as a single additive table are Pauling’s, 1939, in the form used here from the modern tabulation. Bonding by a distance criterion is as old as computational chemistry and the ratio form is what crystallographic software has used for decades. What is computed here is the two populations’ overlap for all twenty-three structures, the window the ratio rule leaves, and which molecules the rule in use has been getting wrong.

The number worth carrying is not 1.0839. It is that water’s two hydrogens are half an ångström closer to each other than a platinum–chlorine bond is long, in a collection whose bond lists are decided by comparing a length to a number.

Still open: what the corrected lists change #

The obvious open question is the census the corrected lists would produce. Five molecules gain a coordinate set where they had none and ferrocene gains ten stretches and the bends that go with them, so the counts of internal coordinates, redundancies and totally symmetric vibrations all move — and the identity between orbits and totally symmetric species established over fifteen molecules, has never been tested on the five that had nothing. Whether it holds for a hexafluoride ion is a run of what already exists, on a bond list that now exists too, and it would take the census from fifteen molecules to twenty-three.

The nearer question is the window’s own robustness to the radii. The two edges are set by main-group atoms, so the transition metals’ ambiguity does not reach them — but the radii are one tabulation of several, and a different one would move every ratio. How far the window survives swapping the table is a re-run with a second set of numbers, and what it would say is whether a factor of 1.084 is comfortable or is the width of the disagreement between two tabulations.