A formula that predicts minus eleven vibrations

Counting how often a symmetry label fails to name a distortion uniquely finds that every molecule in the census has a repeated species, and the pattern comes with a one-sentence explanation:

What decides it is how many orbits of coordinates a molecule has: an orbit of equivalent bonds contributes one totally symmetric combination, so a molecule with two kinds of internal coordinate has A₁ twice and one with one kind has it once.

Then it applied the sentence to methane — four bonds and six angles, two orbits, one redundancy, one A₁ — and moved on.

That is an explanation with three integers in it and it was never checked against anything. Checking it costs one function: sort each molecule’s internal coordinates into orbits under its own operations, count the redundancies, and compare the difference against the multiplicity of the totally symmetric species in the vibrational representation, which can be computed by a route that shares nothing with either.

For seven of fifteen molecules the difference is right. For eight it is not, and four of the eight are not merely wrong: ferrocene has five orbits and sixteen redundancies, so the formula predicts minus eleven totally symmetric vibrations.

What an orbit of coordinates is #

The internal coordinates of a structure are generated rather than named here: every bond is a stretch, every pair of bonds sharing an atom is a bend, and a centre with three ligands lying in their plane gets one out-of-plane coordinate as well.

An operation of the point group permutes the atoms, and that permutation carries each coordinate to another coordinate of the same kind. The orbits are the sets that are closed under doing that with every operation. Boron trifluoride’s seven coordinates fall into three: three B–F stretches, three F–B–F bends, and the out-of-plane displacement of the boron.

An orbit is not a count of coordinates and not a count of atoms. Benzene’s thirty-six coordinates fall into five orbits; ferrocene’s sixty into five as well, which is why the two molecules have similar counts of totally symmetric vibrations despite one being nearly twice the size.

The orbit is also the object site symmetry is about, seen from the other end: the stabiliser of a coordinate is the subgroup that fixes it, the orbit is the group divided by that subgroup, and the two multiply to the order of the group. Boron trifluoride’s three stretches have a stabiliser of order four in a group of order twelve; its one out-of-plane coordinate has a stabiliser of order twelve, which is the whole group, and that is why it is the coordinate with something to go wrong.

The first correction: an orbit can contribute nothing #

The reason an orbit ought to give one totally symmetric combination is that the sum of everything in it is unchanged by every operation — an operation shuffles the terms of the sum and a sum does not care about order.

That argument assumes the coordinates are carried onto each other and not onto minus each other, which is true of a stretch and of a bend and is not true in general. A bond length does not know which of its two atoms was which. An out-of-plane coordinate does: it is a signed perpendicular distance from a plane whose orientation is fixed by naming its three ligands in an order, so an operation that reflects the plane sends it to minus itself.

Boron trifluoride’s horizontal mirror does exactly that. The coordinate is fixed by the operation and reversed by it, so its orbit’s sum is not invariant and contributes nothing to the totally symmetric count. Two of its three orbits are symmetric, not three.

The correct statement is about the stabiliser, not about signs in general, and the difference matters. The tempting weaker rule — no operation anywhere changes a sign — is wrong. Ferrocene’s coordinates suffer a hundred sign changes and all five of its orbits are symmetric, because every one of those changes is an improper operation exchanging the two rings rather than fixing a coordinate. A sign attached to a coordinate that has moved somewhere else can be absorbed into how the other one is written; a sign attached to a coordinate that has stayed put cannot.

The second correction: a redundancy is not necessarily symmetric #

The other half of the formula subtracts one totally symmetric vibration per redundancy, and it has the same shape of flaw. A redundancy is a null vector of BBᵀ — a combination of coordinates describing no displacement of any atom — and it is under no obligation to be totally symmetric.

Methane’s one redundancy is: the sum of its six angles cannot change while the bonds stay put, and that sum is A₁. So methane’s arithmetic works, and methane is the case usually checked.

Sulfur hexafluoride has six redundancies, and they span 2A₁g ⊕ 2Eg — of which two are totally symmetric and four are not. Subtracting six removes four things that were never in the count.

The pattern holds wherever a molecule has many angles at one centre: benzene’s nine redundancies span 2A₁g ⊕ E₂g ⊕ B₁u ⊕ 2E₁u, and each ferrocene’s sixteen contain exactly two totally symmetric ones. The count of redundancies grows with the coordinate set and the count of symmetric redundancies does not, which is why the formula’s error grows with the molecule and eventually takes it below zero.

The identity #

With both corrections the sentence becomes exact:

$$n(A_1 \text{ in } \Gamma_{\text{vib}}) \;=\; (\text{orbits surviving their stabiliser}) \;-\; n(A_1 \text{ in } \Gamma_{\text{red}})$$

and it holds for every coordinate set with no exceptions and no tolerance, because every quantity in it is an integer. What it counts is the totally symmetric vibrations the coordinates describe, and that is the molecule’s count only when the coordinates span every vibration. For three of the fifteen they do not: hydrogen peroxide’s five coordinates describe five of its six vibrations, both ferrocenes’ sixty describe forty-four of fifty-seven, and each comes out one totally symmetric vibration short — which is where five coordinates for six vibrations picks the question up.

That is not a coincidence to be checked one molecule at a time. Γ_vib is Γ_int minus Γ_red by construction whenever the coordinates span every displacement, the multiplicity of A₁ is additive under that subtraction, and the multiplicity of A₁ in Γ_int is the number of orbits whose coordinate survives its own stabiliser — which is Frobenius reciprocity, since each orbit’s contribution is a representation induced from the character its stabiliser acts by. The identity is a theorem, and what the computation adds is that the two corrections are not negligible.

Which seven the old formula gets right, and why nobody noticed #

The seven it gets right are water, ammonia, sulfur dioxide, methane, phosphorus pentafluoride, ethene and bromochlorofluoromethane.

They have something in common. Six of the seven have no redundancy at all or a redundancy that is entirely totally symmetric, and no orbit reversed by its own stabiliser, so both corrections are zero for them. The seventh, ethene, needs both — two of its orbits are reversed and only one of its two redundancies is totally symmetric — and the two happen to cancel. The formula is right where the corrections vanish or offset, and they vanish for small molecules with one central atom and few angles.

Which is to say: the formula is right for the molecules anybody would check it on. Methane, the usual worked example, is in the list. So is water. A rule tested on the two most-drawn molecules in chemistry would pass, and the first molecule with two atoms carrying angles — formaldehyde, at seven coordinates — already breaks it.

This is a shape of failure that keeps turning up from different directions, and it is worth naming rather than merely noting. A rank correlation over eight cases failed because the set that was easiest to assemble was also the set that could not separate two candidates. A fitted exponent failed because the window that was easiest to fit was also the window furthest from the limit. Here a formula fails because the molecules that are easiest to check it on are the molecules whose corrections vanish. In all three the sample and the error are chosen by the same convenience, so the check and the flaw are correlated — and no amount of care applied within the check can see it.

The same run on phosphorus pentafluoride is the case that shows the two corrections are independent. It has five orbits and none of them is reversed, so the first correction does nothing; it has three redundancies and all three are totally symmetric, so the second does nothing either; and the old formula gets it right for that reason rather than by luck. A molecule can need neither correction, either one, or both, and which it needs is decided by two unrelated facts about its coordinate set — whether any coordinate carries a sign, and how the angles at its centres are arranged.

What the census found that was not about symmetry at all #

Hydrogen peroxide arrived in the census with thirteen internal coordinates, which is impossible for a molecule with four atoms and six degrees of freedom unless something has been counted that is not a bond.

Something had. Internal coordinates are generated from a bond list, the bond list is everything closer than 1.85 ångström, and hydrogen peroxide’s two O–H bonds are 0.95 while its two other O···H distances are 1.82. The molecule was carrying five bonds instead of three — in the coordinate set, and in every figure that has ever drawn it.

Nothing failed. The point group is recovered from the coordinates rather than from the bonds, so the group came out C₂ correctly; the redundancy count adjusted itself to the larger set; and the identity above holds for a wrong coordinate set exactly as it does for a right one, because it is a statement about whatever coordinates it is given.

It was found by looking at the census the coordinates produce, which is the only place the number thirteen was ever written down. A hydrogen cannot have two neighbours, so the bond list now gives each hydrogen its nearest one and no other, and that rule changes exactly one of the twenty-three molecules here — checked before it was applied rather than after.

What is quoted, and what is computed #

Nothing is quoted. No frequency, no force constant and no measurement appears here: the input is a set of nuclear positions and the output is a set of integers.

The operations come from closing the molecule’s candidate symmetries under multiplication and keeping those that permute its atoms among themselves. The action on coordinates is worked out combinatorially — which coordinate each one is carried to, and with what sign — rather than numerically, because a numerical projection onto a non-orthogonal coordinate set is not a matrix element and gave characters like 1.87 and 3.30 when it was tried that way.

The redundancies are the null vectors of BBᵀ at a threshold relative to its largest eigenvalue, and their species come from the trace of the coordinate action restricted to that null space. The count of totally symmetric vibrations is also computed independently, from the Cartesian displacements the group acts on, and the two routes are required to agree for every coordinate set that spans the vibrations. Made, that comparison finds three molecules whose coordinate sets span too few.

A refutation that needed no measurement #

Returning minus eleven is worth pausing on as a kind of result, because it is the strongest form of refutation available and it costs nothing to obtain.

The quantity being predicted is a count of vibrations. A count is a non-negative integer, and that is not an empirical fact about molecules — it is what counting means. So a formula returning a negative number has been refuted without any comparison against a spectrum, a calculation or a measurement of any kind.

That is a much better position than being wrong by a large amount would have been. A formula predicting three where the answer is eleven is a formula that might be repaired by adjusting something; a formula predicting minus eleven has a structural fault, because no adjustment of a term’s size turns a negative count positive without changing what the terms are.

It also locates the fault. The formula is a difference — one per orbit, less one per redundancy — and only the second term can drive it negative, so the redundancies are being over-counted. That much follows from the sign alone, before either correction is found.

The general practice is worth stating, because it is cheap and rarely applied. Check a formula against the range its output is allowed to occupy, before checking it against data. A count must be non-negative and integral; a fraction must lie between zero and one; a probability must sum to one across its cases. Each of those is a test that requires no measurement, catches a class of error that a comparison against data may not, and can be run on a formula the moment it is written down.

Eight of fifteen wrong is a result that needed the fifteen molecules. Four of them negative is a result that needed only arithmetic.

What this cannot say #

Fifteen molecules is not a survey. They are the molecules with structures and bond lists in these essays, and they are small, mostly symmetric and mostly built around one central atom. The identity is a theorem so it does not need a survey; the sizes of the two corrections would look different on a set of larger molecules, and would probably look worse.

The coordinate set is a choice. Every count here is about the particular internal coordinates this collection generates — all the bends at every centre, which is the redundant convention. A non-redundant set chosen by hand would have no redundancies to subtract and a different orbit structure, and would give the same answer, because the answer is a property of the molecule — provided the set spans every vibration, which three of the sets here do not. That is worth stating plainly: the identity relates two quantities that both depend on the coordinate set to one that does not.

The redundancy species depend on a numerical threshold. A null vector of BBᵀ is one whose eigenvalue is below a stated fraction of the largest, and at 10⁻⁸ the counts here are stable — ferrocene’s sixteen is sixteen at 10⁻⁶ and at 10⁻¹⁰. A molecule with a nearly redundant coordinate set would not be, and the census would then be reporting the threshold. Nothing here is near the boundary, which is a fact about these molecules rather than a property of the method.

And the out-of-plane coordinate is the only signed one here. The first correction is demonstrated on the one kind of coordinate here that carries a sign. A torsion is another, and a molecule with torsions and a symmetry that reverses them would be the sharper test — none of the fifteen has one.

What was checked #

The identity, for every molecule separately. Not for a mean and not for a majority: fifteen checks, each of an equality between integers. And the count each coordinate set gives, against the count from the Cartesian displacements: equal wherever the set spans the vibrations, and one short for hydrogen peroxide and both ferrocenes, where it does not.

And the equivalence in both directions: an orbit gives a totally symmetric combination exactly when no operation fixing its coordinate reverses it. The reverse half is what fails if the rule is stated as no sign changes anywhere, and it fails on ferrocene rather than on anything contrived.

Water is the refusal. Three coordinates, two orbits, no redundancy, two totally symmetric vibrations — so a formula that counted coordinates rather than orbits would report three, and the check names all four numbers rather than the last one.

Still open: projecting onto normal coordinates #

The obvious open question is the repair that needs a force field. Where a species appears more than once, the projection onto normal coordinates is what settles which mode a distortion goes into, and that needs a force field — which exists here for six molecules. Running the coarse test and the projection side by side on those six would say how far apart their answers are. The orbit arithmetic above says something useful about that in advance: the ambiguity in labels is largest exactly where the orbit count is largest, so the six molecules with force fields are not a random sample of the difficulty.

The nearer question is the torsion. The first correction rests on one kind of signed coordinate, and a molecule whose symmetry reverses a torsion would test it against a coordinate of a different shape — hydrogen peroxide has one, and its C₂ axis does not reverse it, which is why it does not serve. A ring with a twofold axis through two bonds would, and generating torsional coordinates is the one addition the coordinate generator needs to reach it.