Site symmetry, and what it constrains

Phosphorus pentafluoride has five fluorines and two kinds of fluorine. Benzene has twelve atoms and two kinds of atom. Methane has five atoms and two kinds of atom, and the two hydrogens of water are one kind.

Counting kinds is not counting atoms, and the thing that does the counting is not the molecule’s point group but a smaller group attached to each position.

Orbits, and why they are exact #

Take a molecule’s group and act with every operation on one atom. The set of positions the atom visits is its orbit.

Two atoms are in the same orbit when some operation of the group carries one to the other, and in different orbits when none does. That is a partition — every atom is in exactly one orbit — and it is exact in the way symmetry statements are exact: no operation of the group crosses between orbits, and the operations were enumerated by closing a set under multiplication rather than by looking for plausible ones.

Phosphorus pentafluoride’s twelve operations carry the three equatorial fluorines among themselves and the two axial fluorines among themselves, and never between. Two orbits, of sizes three and two.

That is a stronger statement than the angle-spectrum argument makes. The angle argument shows that no operation anybody checked relates the two kinds of site. The orbit decomposition shows that no operation the molecule has does.

The site group #

Now fix an atom and ask which operations leave it where it is.

Those operations form a subgroup — the stabiliser of the position, or in the chemists’ vocabulary its site symmetry. And there is an accounting identity that makes the whole idea useful:

$$|\text{orbit}| \times |\text{site group}| = |\text{molecular group}|.$$

Check it on phosphorus pentafluoride. Twelve operations in D₃ₕ. The equatorial orbit has three members, so each equatorial site has a group of order four; the axial orbit has two, so each axial site has a group of order six. Three times four and two times six both give twelve.

The identity is not a coincidence and it is not chemistry — it is the orbit–stabiliser theorem, which holds for any group acting on any set. What makes it chemistry is that the site group is the group whose representations govern anything localised at that position.

What a site group constrains #

Three things, and each is a statement about that atom rather than about the molecule.

Which local functions are equivalent. An orbital on an atom transforms under the site group. If the site group has a threefold axis, the two p orbitals perpendicular to it are degenerate at that site; if it does not, they need not be.

That first one is worth an example, because it is the one that reads as abstract and is not. Take an equatorial fluorine in phosphorus pentafluoride. Its site group has order four: the identity, the horizontal mirror, the twofold axis running through it, and the vertical mirror containing that axis. Nothing in that list relates its p orbital pointing along the bond to either of the other two, and nothing relates its in-plane perpendicular p to its out-of-plane one. So an equatorial fluorine has three inequivalent p orbitals. An axial fluorine, whose site group of order six contains the molecule’s threefold axis, has two — the pair perpendicular to the axis is degenerate at that site, forced to be so by an operation that fixes the atom and rotates its surroundings.

Two fluorines in one molecule with different numbers of distinct p orbitals, and the difference is decided by which operations happen to leave each one alone.

Whether the atom can carry a local property. A dipole contribution, a hyperfine tensor, a nuclear quadrupole coupling — each is a tensor at a position, and the site group forbids components exactly as the molecular group forbids a molecular dipole. An atom at a site with cubic symmetry cannot carry an electric field gradient at all, which is why cubic sites give no nuclear quadrupole splitting.

How many distinct signals a spectrum can show. This is the practical one. A spectroscopy that probes individual nuclei counts orbits, not atoms — so benzene’s six carbons give one signal and naphthalene’s ten give three.

The counting, done properly #

Benzene is the cleanest illustration and it is worth doing carefully, because the naive count and the right count differ.

Twelve atoms: six carbons and six hydrogens. Twenty-four operations. The carbons form one orbit of six, so each carbon’s site group has order four; the hydrogens likewise.

Two orbits, therefore two environments, therefore one carbon signal and one hydrogen signal — which is what benzene’s spectra show and which was, historically, one of the observations demanding an explanation that alternating single and double bonds could not give.

Naphthalene has ten carbons in three orbits and eight hydrogens in two, so five environments from eighteen atoms. Its spectrum shows exactly that, and the three carbon environments are the same three whose bond orders differ — which is not a coincidence but the same partition seen twice.

The orbit arithmetic checks against the group order directly: benzene’s six carbons times a site group of order four is twenty-four, which is the order of D₆ₕ. That identity — orbit size times site-group order equals group order — is the whole of the bookkeeping, and it holds for every orbit of every molecule here.

Why the local group is a subgroup, and what that costs #

A site group is always a subgroup of the molecular group, and that has an immediate consequence that trips people.

A representation of the molecule reduces to representations of the site. A t₂ set in Td, restricted to a site of C₃ᵥ symmetry, splits into a₁ and e — the threefold degeneracy of the molecule is not a threefold degeneracy at any one position. That descent-in-symmetry argument is how a ligand-field splitting is derived, and it is the same reduction formula applied to a smaller group.

Local symmetry can be higher than the molecule’s. A site can be more symmetric than the molecule containing it, in the sense that the local environment looks symmetric out to some radius. That is where the concept starts being a heuristic rather than a theorem: “the carbon is locally tetrahedral” in a substituted methane is a statement about approximate local geometry, and the actual site group is C₁.

The gap between exact site symmetry and approximate local symmetry is where most of the useful chemistry lives and where most of the confusion does too. A chemist saying “this carbon is tetrahedral” means the local geometry is close to tetrahedral; the site group of that carbon in bromochlorofluoromethane is the trivial group, and nothing whatever is forbidden there.

What was computed, and how #

The orbit decomposition is computed by acting: every operation of the generated group is applied to every atom, and the images are collected.

That requires the group to be complete, which is why it is generated by closure rather than by search. A missing operation would merge two orbits that should be separate — or rather, would fail to merge two that should be one, reporting more environments than the molecule has. The failure is one-directional and quiet.

Two checks guard it. The orbit sizes must divide the group order, by the orbit–stabiliser identity, and the sum of the orbit sizes must be the number of atoms. Both would fail on an incomplete group, and the second caught a real error: when a planar molecule’s own mirror plane was being keyed by the permutation it induces — which is the identity permutation, since it moves nothing — benzene’s group collapsed to twelve operations, and the orbit arithmetic no longer balanced.

The site groups themselves are computed the same way, by testing each operation against each position.

Where the orbits show up in the orbitals #

The partition is not only a bookkeeping device. It appears directly in what the ligand orbitals span, and comparing two molecules makes the mechanism visible.

Now contrast phosphorus pentafluoride. Its five fluorines are two orbits, so their orbitals do not enter one reduction at all — the three equatorial functions span one set of species and the two axial functions another, and no combination mixes across. That is why the two kinds of bond can differ in length and why the two kinds of fluorine can differ in reactivity: they are not competing for the same symmetry species.

A single orbit forces equivalence and two orbits permit difference, and which of the two a molecule has is decided by counting operations rather than by looking.

The surprise: a spectrum counts orbits, and only orbits #

The result worth carrying away is a negative one about what a count of signals establishes.

A carbon spectrum showing three environments says the carbons fall into three orbits. It does not say how many carbons there are, and it does not distinguish between structures with the same orbit structure.

That means two things at once. Symmetry makes spectra interpretable — a single benzene signal is strong evidence for a symmetric ring — and it also caps how much a spectrum can say. Two isomers with the same orbit decomposition are indistinguishable by counting alone, and resolving them needs intensities or couplings or another technique.

The cap is worth knowing because the counting argument is often stated as though it identified structures. It eliminates them, which is a different and weaker operation, and it eliminates them exactly.

What it costs #

The whole computation is a few hundred coordinate transformations: apply every operation to every atom, collect images, check the arithmetic. Milliseconds.

What it costs conceptually is the discipline of keeping two groups apart, and the failure mode is specific. The molecular group governs anything that involves the whole molecule — a total dipole, a vibrational mode, an electronic transition. The site group governs anything localised at a position. Applying one where the other belongs gives answers that are plausible and wrong, and the commonest instance is using a molecule’s group to argue about a local tensor.

There is a second cost, which is that the orbit decomposition throws away everything but the partition. It says the three equatorial fluorines are equivalent and says nothing about how they differ from the axial ones — not which bond is longer, not which is more reactive, not by how much. Symmetry supplies the permission to differ and nothing about the direction or the magnitude.

Where the model stops #

Three limits.

Exact symmetry only. A site group is a group of exact operations. Approximate local symmetry — a carbon that is nearly tetrahedral, a metal centre that is nearly octahedral — is a useful idea and is not this one, and the arguments that hold exactly for the first hold approximately and unquantifiably for the second.

Equilibrium geometry. A vibrating molecule has lower site symmetry almost always, and a molecule that interconverts between structures may show fewer environments than any one structure has. Phosphorus pentafluoride is the standard case: its fluorines are two kinds by symmetry and its room-temperature spectrum shows one, because Berry pseudorotation exchanges them faster than the measurement resolves. The symmetry argument is right and the experiment is measuring something else.

Isolated molecules. In a crystal an atom’s site symmetry is set by the space group rather than by the molecule, and it is generally lower. A molecule with a threefold axis sitting at a general position in a crystal has no threefold axis at all as far as the solid is concerned, which is why solid-state spectra show splittings that solution spectra do not.

The vocabulary is the last thing to settle. “Equivalent atoms” is orbit membership, “local symmetry” is a site group when it is exact and an approximation when it is not, and the two senses of the second are used interchangeably in almost every source. Keeping them apart costs a word and saves an argument.

What a site will not accept, and what happens when it is asked to #

The site’s group constrains the atoms already there, and read forwards it constrains what can be put there — which turns a piece of group theory into an account of one of crystallography’s commonest nuisances.

An object occupying a site must possess at least the site’s symmetry. A position with a threefold axis through it can hold an atom, or a group that is itself threefold symmetric — a methyl, a nitrate, a benzene ring lying flat. It cannot hold a group with only a twofold axis, because the crystal’s threefold operation would have to map that group onto itself and there is no way for it to do so.

When a crystal nevertheless has such a molecule at such a site, something has to give, and what gives is the assumption that every unit cell is identical. The molecule adopts one of the three orientations the threefold axis relates, different cells choose differently, and the diffraction experiment — which averages over every cell in the sample — sees the superposition.

That is disorder, and it is not a defect in the sample or in the refinement. It is the arithmetic of site symmetry: a molecule of the wrong symmetry at a site of the right one, and the average of the orientations having the site’s symmetry even though no molecule does.

The consequence for reading a structure is worth stating, because the symptom is a familiar one. A disordered group appears in a structure as partial atoms — occupancies of a third, or a half — arranged with more symmetry than the group can possess. That pattern is a report on the site rather than on the molecule, and the number of parts the group is split into is the ratio of the site’s order to the molecule’s own.

So a site symmetry is a compatibility condition, and where it is not met the structure does not fail: it averages. The averaged picture has the site’s symmetry, no individual molecule does, and the difference between the two is the whole of what a disorder model is trying to describe.

Who found it, and when #

The orbit–stabiliser theorem is elementary group theory and predates any chemical use by decades — it is a counting argument about group actions with no physics in it at all.

Its arrival in chemistry came through crystallography rather than through molecular spectroscopy. The Wyckoff positions of a space group, tabulated from the 1920s onward, are exactly orbits: each position in a crystal has a site symmetry, the multiplicity of a position times its site-group order is the group order, and the tables are the accounting identity above written out for two hundred and thirty groups.

Molecular chemists imported the idea later and less systematically, mostly as the informal notion of “equivalent atoms”, which is orbit membership under another name. The vocabulary of that informal version is still the one most people use, and it works because the underlying object is exact even when the terminology is not.

Bethe’s 1929 crystal-field paper is where descent in symmetry — a representation of a group restricted to a subgroup — became a chemical tool, and it remains the clearest instance of a site group doing work the molecular group cannot.

Where to read on #

The method is character tables and reduction.

The molecule that motivates the whole idea is five sites are not alike.

The search that produces the group in the first place is point groups from coordinates.

And what the same theorem does to a spectrum is selection rules are one theorem.

The single sentence worth carrying is that a molecule has as many kinds of atom as it has orbits, and the orbits are computed by acting with a complete group rather than by looking at a picture. Everything else in this essay is a consequence of that count being exact.

What the pictures here cannot show. An orbit is a partition of a set, and no figure on this page draws one — the molecule figures show positions and the tables show characters, and the partition itself lives between them. Nothing here can show that phosphorus pentafluoride’s room-temperature spectrum has one fluorine signal rather than two, because that is a statement about a rate, and no rate is computed here.