The group of a molecule that will not hold still

Every symmetry argument about a molecule usually starts from a structure — including the enumeration of the groups a molecule can fall to, which distorts a structure and re-searches it. A set of coordinates is handed to a search, the operations that leave it unchanged are found, they are closed into a group, and the group is recovered from the coordinates rather than looked up. That is the right procedure and it has one presupposition, which has not been examined: that there is a structure.

For ethane there is not. Its two methyl groups turn about the C–C bond over a barrier of about 12 kJ per mole, which at room temperature is crossed something like ten billion times a second. That is far faster than the vibrational motions a spectrum resolves, and slower than an electronic transition, which is why the answer depends on which experiment is asking. A spectrum taken over any accessible timescale sees an average, and the average is not a molecule with a shape.

Permutations instead of rotations #

The repair is due to Longuet-Higgins and it changes what the group is made of. A point-group operation is a rotation or a reflection of space; a molecular symmetry group operation is a permutation of identical nuclei, optionally followed by the inversion of every coordinate in the laboratory, written E*.

That definition has three virtues. It does not need a structure. It contains the point group as a special case — a rigid rotation permutes the nuclei, so it corresponds to a permutation, and an improper operation corresponds to a permutation together with E*. And it makes the question which operations are available into a question about barriers rather than about geometry.

The group is then built by closure, exactly as the point groups here are: state the generators, compose them until nothing new appears, and count. Nothing is looked up.

The count that is not a group #

The largest of the three columns is the easiest to compute and the least interesting, and it is worth naming before it is discarded. If a molecule has $n_k$ nuclei of each kind, then permuting identical nuclei arbitrarily and optionally inverting gives $2\prod_k n_k!$ operations. For ethane — six hydrogens and two carbons — that is $2 \times 720 \times 2 = 2880$.

Almost none of them is a symmetry operation of anything. Swapping two hydrogens on the same carbon is a rotation and costs nothing; swapping a hydrogen on one carbon with a hydrogen on the other requires breaking two bonds. The word that separates the two is feasible, and it is the whole of the definition: an operation is feasible if the molecule can be carried into the permuted configuration without surmounting an insuperable barrier.

That is not a mathematical criterion. It depends on temperature, on the timescale of the measurement, and on how large a barrier the experiment can see over. A group that changes with the experiment is unfamiliar, and it is the honest description: a point group already depends on a tolerance, and this is the same kind of dependence one level up.

Ethane, twelve and thirty-six #

Ethane’s rigid group is D₃ₕ for the eclipsed conformer and D₃d for the staggered one, and both have twelve operations. Every one of them is a rigid motion: the threefold rotation turns both methyl groups at once, and no operation in either group turns one of them alone.

The internal rotation does exactly that. Adding it as a generator and closing gives thirty-six.

Thirty-six is $3 \times 3 \times 2 \times 2$: each methyl independently in three orientations, the two ends exchangeable, and the whole thing with or without E*. It is a group nobody would write down by looking at a picture of ethane, and it is the group that labels ethane’s torsional levels — labels of the kind a character table supplies, for a group no table of point groups contains.

The consequence is observable. The torsional motion has energy levels, and because the barrier is finite each of them is split into a pattern of sublevels which no twelve-element group has enough representations to label. The size of the splitting is how the barrier is measured — which makes the barrier height, the group’s order and the spectrum three statements about the same thing.

Ammonia, and the molecule that is never planar #

Ammonia’s rigid group is C₃ᵥ, order six. Its inversion — the umbrella turning inside out — permutes nothing at all and inverts everything, so it is E* itself, and adding it doubles the group to twelve.

Twelve is the order of D₃ₕ, and that is not an accident: D₃ₕ is the group of the planar molecule, which ammonia passes through and never sits at. So the feasible group of a pyramidal molecule that inverts is the point group of a planar molecule that does not exist. The group describes the motion, not the geometry.

The barrier is 24.3 kJ per mole, which is high enough that the molecule spends its time in one well or the other and low enough to tunnel through. Every rotational level is therefore a close doublet, split by 0.794 cm⁻¹, and the transition between the two halves of one such doublet — at 23.786 GHz — was the working transition of the first maser. A microwave device was built on the fact that ammonia’s point group is not its symmetry group.

Methane, and the parity rule nobody puts in #

Methane has no feasible internal motion at all: nothing short of breaking a bond permutes its hydrogens differently from a rotation. So its feasible group equals its point group, and the interest is in which operations they are.

Four hydrogens can be permuted in 24 ways, and each permutation can be taken with or without E*, giving 48. The group has 24 of them. Which 24?

The generators used are a four-cycle and a transposition, both odd, both carrying E*. Everything else in the group is a product of them, and the parity rule propagates: the product of two odd permutations is even, and the product of two E*s is the identity, so the pairing is preserved by composition and every one of the twenty-four obeys it.

That is Td, arrived at by counting permutations. A rotation of a tetrahedron induces an even permutation of its vertices; a reflection induces an odd one; and improper operation and odd permutation turn out to be the same statement about a tetrahedron. The rule was not put in — the generators were, and the closure found it.

An easy mistake is to use a three-cycle and a transposition on the same three letters as generators, which returns a group of order six. That is a correct closure of the wrong generators: they generate the symmetric group on three letters, not on four, and six is what they give. A check comparing the closure against the point group’s order catches it, which is the useful shape of a check — a claim about a relation between two independently computed things, rather than about a single number.

What the extra operations do to a spectrum #

It is worth being concrete about why a larger group is not a curiosity, because the usual objection — the molecule still has the shape it has — is right and beside the point.

A symmetry group’s job is to label states, and it does that through its irreducible representations. A group of order twelve with six classes has six representations to label with; a group of order thirty-six has nine. If the levels of a molecule fall into nine distinguishable kinds and only six labels are available, then some pairs of levels that the molecule keeps apart are being given the same label — and a selection rule derived from the smaller group will forbid transitions the molecule allows.

That is exactly what happens to ethane’s torsional spectrum. The torsion is a motion around a threefold-by-threefold potential, its levels split into components with different behaviour under the relative rotation of the two tops, and no rigid group distinguishes them. The splittings are small — a fraction of a wavenumber — and they are resolvable, and they are how the barrier height is obtained. A barrier is not measured by watching a molecule cross it; it is measured from the fine structure the crossing leaves in the levels.

Ammonia is the same argument with one degree of freedom instead of two. The inversion doubling is a splitting that C₃ᵥ cannot label — the two components differ in their behaviour under E*, and E* is not in C₃ᵥ — so the levels of ammonia have a quantum number that the point group has no name for.

Hydrogen peroxide, where the barrier is low #

The fourth case is the one where the answer is least like a point group. Hydrogen peroxide is non-planar, with a dihedral around 111°, and its rigid group is C₂ — two operations, the smallest non-trivial group there is.

The internal rotation has two barriers, about 29 kJ per mole through the trans configuration and about 4 through the cis one. The lower is crossed freely at any accessible temperature, so the torsion is feasible and the group is of order four rather than two.

That figure — one molecule visiting three point groups as one coordinate is turned — is the clearest statement of the difficulty. Asking for the point group of hydrogen peroxide is asking which point on the curve to take, and the molecule is not at a point.

The tolerance, one level up #

There is a family resemblance between feasible and something every point-group assignment has to decide, and it is worth putting the two side by side.

Assigning a point group needs a tolerance: a real molecule is never exactly symmetric, so whether an operation counts depends on how far an atom is allowed to move before the operation is refused. Setting that tolerance is a decision, four distortions of a hundredth of an ångström can give four different groups at one setting, and the answer is not a property of the coordinates alone.

Feasible is the same decision in the time domain. An operation counts if the molecule can reach the permuted configuration within the timescale of the measurement, and how large a barrier that permits is a matter of temperature and of instrument. In both cases the group is a property of the molecule and of the question, and in both cases pretending otherwise produces a confident answer with an unstated parameter in it.

What is gained, and what is given up #

The gain is that the group describes what the molecule does rather than where its nuclei happen to be, and that is what a spectrum needs. The loss is worth stating plainly.

The operations are no longer geometric. A point-group operation can be drawn: here is the axis, here is the plane. A permutation-inversion operation is a relabelling, and it has no picture. What is lost is the whole visual apparatus that makes point groups teachable.

The group depends on the experiment. Ethane at four kelvin has a twelve-element group, and the same molecule in a room-temperature gas cell has thirty-six. Both are correct for their own measurement.

The point group is still the right tool for a structure. Anything answered by a molecule’s shape — whether it can be polar, whether it can be chiral, what its vibrations transform as at a stationary point — is a point-group question and stays one. The larger group answers questions about states, and the two sets of questions do not overlap as much as they appear to.

A group that depends on how long the looking takes #

The thirty-six-element group has a general construction behind it, and the construction has a feature no other symmetry group in this collection has: its membership depends on the timescale of the experiment.

The construction starts from every permutation of identical nuclei, combined with the operation that inverts all the coordinates. For ethane that is a very large set — the two thousand eight hundred and eighty counted here — and most of its members correspond to rearrangements the molecule never performs: swapping a hydrogen on one methyl with one on the other requires breaking bonds.

The set is then cut down to the feasible operations: those corresponding to a rearrangement the molecule actually achieves, on the timescale of the measurement being described. Rotating a methyl group is feasible, because the barrier is low and the rotation happens billions of times a second. Exchanging hydrogens between the two carbons is not.

That leaves thirty-six, which is the group that describes the spectrum.

The awkward and interesting part is the phrase on the timescale of the measurement. Feasibility is not a property of the molecule alone: it is a comparison between a rate and an experiment’s resolution. A rearrangement that is fast compared with a nuclear magnetic resonance measurement may be slow compared with an infrared one, and the group that describes the two spectra is then different.

So a molecule can have two symmetry groups at once, correctly, for two experiments — and the disagreement is not an error in either.

That is a real departure from ordinary point-group reasoning. A point group is a property of a structure and is settled by coordinates; a molecular symmetry group is a property of a structure and a clock, and reporting one without saying which measurement it describes leaves out half of what determines it.

What the arithmetic guarantees #

Two checks make the counts above more than a list, and both are relations rather than values.

The feasible group’s order divides the conceivable group’s, in every case: 12 of 12, 24 of 48, 36 of 2880, 4 of 8. It has to, because the feasible operations are a subgroup of the permutation-inversion group, and Lagrange’s theorem is not negotiable. A calculation returning 35 for ethane would be visibly wrong without any chemistry being consulted.

And the frozen group’s order divides the feasible one’s, for the same reason one level down. Ethane’s 12 divides 36; ammonia’s 6 divides 12; peroxide’s 2 divides 4. The chain of subgroups is what makes the three columns a nested description of one molecule rather than three unrelated counts.

The refusals the closure carries are the small ones that matter in practice. A generator with a repeated label is not a permutation and is rejected rather than closed on, and a closure that outruns its limit is reported rather than allowed to run — both because the failure mode of a closure algorithm is silence, and a group of the wrong order looks exactly like a group.

Still open: tables for the larger group, and ligand exchange #

The natural open question is the spectrum itself. A molecular symmetry group has character tables like any other group, its representations label the torsional and inversion levels, and the selection rules follow by the same vanishing-integral theorem that governs everything else here. Building them means constructing the classes of a permutation group, which is a different computation from the geometric one used for point groups.

The other direction is the one that makes the whole distinction sharper: the molecules where the rearrangement is not a torsion but an exchange of ligands. Five-coordinate structures are the standard case, and their axial and equatorial substituents exchange freely — which is why the preference between the two sites is so hard to observe directly, and why the group that describes the molecule has to contain permutations no rigid motion supplies.

The claim to carry is one sentence long. A point group is the symmetry group of a molecule at a temperature low enough that nothing moves, and every molecule with a low barrier in it has a larger group which is the one its spectrum obeys.