Every group a molecule can fall to

A molecule’s point group describes the structure it has. Almost every interesting thing that happens to a molecule takes some of that symmetry away — a vibration displaces the atoms, a ligand is substituted, a crystal field pushes on it, an excited state distorts — and the question of which symmetries it can be left with turns out to have a short and completely determined answer.

The answer is that what remains must be a subgroup. Not a subset: a subgroup. If a structure still has two operations after the distortion, it still has their product, because applying one operation and then another to a structure that both leave alone leaves it alone. So the sets of operations a distorted molecule can have are exactly the subsets closed under multiplication, and there are far fewer of those than there are subsets.

Methane has twenty-four operations. Its subsets number sixteen million. Thirty of them are groups.

The search, and why it is a search rather than a lookup #

Point groups from coordinates established the habit this rests on: a molecule’s operations are found by searching its own atom positions for rotations, mirrors and rotoreflections that permute the atoms among themselves. What comes back is a list of orthogonal matrices, each refined so that long chains of products stay exact.

The subgroup search takes that list and does the obvious thing. Start with one operation and multiply it by itself until nothing new appears — that is the cyclic subgroup it generates. Start with two, and close under all products. Start with three. Collect what comes back, discard duplicates, and the result is the lattice.

Two details make it a computation rather than a formality.

Closure has to terminate, and it has to terminate in the right place. Multiplying group elements can only produce group elements, so the closure of any subset is bounded by the group itself — but a floating-point product that misses its target by a rounding error would look like a new operation and the closure would run away. That is the failure the point-group search already has to solve once: an operation found to within six hundredths of an ångström is a perfectly good verdict and a poor multiplicand, so each one is recomputed as the exact orthogonal matrix realising the atom permutation it induces. Products of refined matrices are bit-identical, and closure terminates.

Generating from singles, pairs and triples is enough here, and is stated rather than assumed. A subgroup that needed four independent generators would be missed. For groups of order at most forty-eight there is no such subgroup among the ones these molecules have, and the check that says so is not an argument but a count — which is the next section.

The count that could have been wrong, and was not #

Four of the point groups here are familiar abstract groups whose subgroups were enumerated long before anybody drew a molecule with them.

C2v is the Klein four-group and has five subgroups. C3v is the symmetric group on three letters and has six. D3h is the dihedral group of order twelve and has sixteen. Td is the symmetric group on four letters and has thirty.

The search finds five, six, sixteen and thirty.

Nothing in the search was told any of those numbers, and the two routes to them have nothing whatever in common. One is a multiplication table assembled from best-fit orthogonal matrices recovered from the positions of atoms; the other is a nineteenth-century enumeration over abstract elements. A closure bug that silently dropped an operation would produce a set that is not a group, and the counts would part company immediately and in all four cases at once.

That is the sort of agreement worth keeping, because the method it validates is then available for groups whose subgroup counts nobody has memorised.

Lagrange, and the orders that cannot appear #

The first thing the lattice obeys is the oldest theorem about finite groups: the order of a subgroup divides the order of the group.

Methane’s thirty subgroups have orders 1, 2, 3, 4, 6, 8, 12 and 24. There is no subgroup of order five, none of order seven, none of order nine and none of order sixteen, and the reason has nothing to do with geometry — five does not divide twenty-four.

This has a consequence worth stating plainly, because the loose version of it circulates as a rule of thumb about distortions. A distortion of methane cannot leave it with a fivefold axis. Not because a fivefold axis is hard to arrange, but because a group containing one has order divisible by five, and no subgroup of a group of order twenty-four does.

Ammonia makes the same point in miniature. Its C3v has order six, so its subgroups can only have orders 1, 2, 3 and 6; the search finds one of each order except two, of which there are three — the three mirror planes, each generating a group of order two with the identity. Six subgroups, and the arithmetic left room for no others.

Chirality is decided by the subgroup, not by the parent #

Chirality is a symmetry statement established that a molecule is chiral exactly when its group contains no improper operation — no mirror, no inversion centre, no rotoreflection. The lattice turns that into a statement about what a distortion can do.

Every group here has either no improper operations or exactly half. That is not a coincidence and it is checked as the lattice is built: the proper operations always form a subgroup, and if any improper operation exists then multiplying it by each proper one gives every improper one, so the improper set is a single coset and has the same size as the proper subgroup.

So the lattice splits cleanly. Methane’s thirty subgroups include ten with no improper operation at all: one of order twelve, three of order four, four of order three, and the identity together with three of order two. A distortion of methane that lands in any of those ten gives a chiral structure, and methane itself is as achiral as a molecule gets.

That is not an abstraction. Substituting four different groups for the four hydrogens is exactly a distortion into the subgroup of order one, and the result is the standard textbook stereocentre. What the lattice adds is that the stereocentre is one case out of ten, and that bromochlorofluoromethane is not the only route into the chiral part of the lattice.

Polarity is decided the same way and gives a different answer #

A molecule may have a dipole when some direction is left alone by every operation of its group. Symmetry forbids a dipole works that out from the operations rather than from a sum of bond vectors, and it applies to a subgroup exactly as it applies to a group.

The two properties then sort the lattice differently, which is the useful part. Methane’s subgroup of order twelve with no improper operations is chiral and not polar — it contains three twofold axes at right angles, and no direction survives all of them. Its subgroups of order three are chiral and polar. Its C3v subgroups are polar and not chiral.

So a molecule can be made chiral without being made polar, and polar without being made chiral, and both, and neither, and which of the four happens is decided by where in the lattice the distortion lands. That is a more useful statement than either rule on its own, and it is the reason the lattice is worth computing rather than the two rules being applied one at a time.

Eighteen of methane’s thirty subgroups have no table here #

The honest limit is worth stating in the same breath as the result.

This site holds thirteen character tables, generated and verified as character tables and reduction describes. Methane’s lattice contains thirty groups, and eighteen of them are not among the thirteen — C₃ on its own, S₄, D₂, D₂d, T, and several more. The subgroup search identifies a group when it can and returns its order, its class count and its improper count when it cannot.

That is a real gap and it has a real consequence: a descent into one of those eighteen can be found here and cannot be reduced here. Descent in symmetry computes correlation tables by restricting characters to the operations that remain, which needs a table at both ends. The three descents worked out there — Td to C3v, Oh to D4h, D6h to D2h — are three edges of a lattice with a great many more.

What the search does supply for the other eighteen is the order and the class structure, and those are enough for the two properties above, since neither needs a character.

The lattice is not a chain #

A descent is usually drawn as a single line — one group, a lower group, a lower one still — and the lattice is not shaped like that.

Methane’s subgroups of order four come in three kinds, and they are genuinely different rather than three copies of one thing: three D₂-like groups with three perpendicular twofold axes, three S₄ groups built on a rotoreflection axis, and three C2v groups. All have order four. All are subgroups of Td. Two of the three kinds contain improper operations and one does not, so a distortion of methane to a structure of order four gives a chiral molecule or an achiral one depending on which of the three it reached.

This is why “lowering the symmetry to order four” is not a specification. Order is not enough, and neither is the symbol on its own: what decides the physics is which operations are left, and the lattice is the list of the possibilities.

Why the count grows so much faster than the group #

Four groups is a small sample and the trend across them is stark enough to be worth reading. The orders are 4, 6, 12 and 24; the subgroup counts are 5, 6, 16 and 30. Doubling the group from D3h to Td almost doubles the lattice, and the step before it — six operations to twelve — nearly triples it.

That is the ordinary behaviour of subgroup counts and it has a consequence for how a descent should be talked about. For a small group the lattice is short enough to enumerate in a sentence: water’s C2v has five subgroups and four of them are the obvious ones. For methane it is already long enough that no sentence lists it, and the useful summary is not the list but the shape of it — how many at each order, how many chiral, how many polar, how many have a table to reduce a basis in.

Benzene, at order twenty-four, has fifty-four. That is the same order as methane and more than twice as many subgroups, because D6h is built as a product of a twelve-element group with an inversion and every subgroup of the first appears twice over. Two groups of equal order can have lattices of very different size, and the order alone is a poor guide to how much freedom a distortion has.

The subgroups a distortion can actually reach #

Thirty subgroups is a complete answer to a question about the group, and it over-counts the answer to the question a chemist usually means — because a molecule does not fall to an arbitrary subgroup. It falls along a coordinate, and the group it lands in is decided by that coordinate.

Displace a molecule along some vector and ask which of its original operations still work. The answer is exactly the operations that leave that vector alone: the ones under which the displacement is unchanged. That set is automatically a group — it is closed, since two operations that each fix the vector fix it in succession — which is why the resulting symmetry is always a subgroup and never something else.

But it is a subgroup of a particular kind: the set of operations fixing a vector in a particular representation. Not every subgroup arises that way.

So the thirty split into two classes. Those that are the fixing set of some displacement are reachable by a distortion, and a molecule can be found in them. Those that are not can be written down as subgroups and cannot be reached by moving the atoms along any single coordinate.

That is worth knowing before reading a list of subgroups as a list of possible structures. A subgroup is a mathematical object and a distorted molecule is a physical one, and the bridge between them is a displacement — so the question which symmetries can this molecule have is answered by the subgroups, and which can it fall to by the smaller list the coordinates supply.

What the lattice does not decide #

Everything above is a list of possibilities and none of it is a prediction. The lattice says which symmetries a distorted methane could have; it says nothing about whether methane distorts, and this site could not compute that if it wanted to.

The one case where symmetry does force the issue is worth naming because it is the exception rather than the rule. A molecule in an orbitally degenerate electronic state distorts, and the Jahn–Teller theorem says so from symmetry alone — the argument is that a degenerate state and a non-totally-symmetric vibration always produce a term that is first order in the displacement, so the symmetric structure is not at a minimum. Copper is never quite octahedral computes that case, and it computes which subgroup the distortion goes to by comparing energies, which is a calculation and not a symmetry argument.

Everywhere else the lattice is a menu. Which item is chosen is an energy, and the energies here belong to a model that does not contain enough to answer — as which angles are symmetry and which are the model sets out at length for the geometric case.

There is one distortion symmetry does force: a degenerate electronic state gains energy linearly in a displacement and loses it quadratically, so the symmetric structure is a maximum along that coordinate. Every other edge of the lattice is a descent a molecule may take and is not obliged to.

The arithmetic behind an eighteen-atom limit #

One practical note, since the search is not free. Closing over triples of operations costs on the order of the cube of the group order, and a group of order forty-eight — the largest here, octahedral — takes about a second. That is fine for one calculation and would not be for a group ten times larger.

There is no such group in molecular chemistry: the largest finite point group is icosahedral, of order 120, and the search would handle it in a few seconds given a table for it. The limit is set at forty-eight because that is what has been checked, and a larger group is refused rather than attempted, which is the same discipline the exact many-electron calculation applies to itself in the smallest many-electron calculation.

Where this goes next #

The lattice makes two things available that are otherwise done one at a time.

A vibration lands somewhere specific. Site symmetry, and what it constrains computes the local group at an atom; a normal mode of a given species leaves behind a definite subgroup of the molecular group, and which one is a question the lattice can answer for every species rather than for the three descents currently tabulated.

A reaction path is a walk down the lattice and back up. That is a statement about a sequence of structures, and sequences of structures are not computed here — but the fact that the walk is confined to the lattice is a symmetry statement and holds without any energy.

The gap that would have to close first is the eighteen missing tables. That is a bounded piece of work: each is a small group with a small table, and the checks applied to the thirteen would apply to them the same way. Until then, it can be said which groups a molecule can fall to, and a basis can be reduced in twelve of the thirty.