Molecular Geometry & Bonding
what the shape decides
What symmetry decides

Site symmetry, and what it constrains

A molecule's group is not the only group in the problem. Each atom sits at a position with a symmetry of its own, and what that local group permits decides how many kinds of environment a structure really has.
16 min read 7 figures Symmetry settles itCounted, not quotedExactly zero

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.

phosphorus pentafluoride — D3hThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.FFFPFFD3hprincipal axis C34 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates6 atoms
Fig. 1 Phosphorus pentafluoride, D₃ₕ, with the group recovered from the coordinates and searched for again at every step of the slider. Five fluorines; two environments. The molecule turns and neither number changes.

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:

∣orbit∣×∣site group∣=∣molecular group∣.|\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.

The D3h character tableThe irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.D3horder 12E1 found2C32 found3C23 foundσh1 found2S32 found3σv3 foundA1′111111A2′11-111-1E′x, y2-102-10A1″111-1-1-1A2″z11-1-1-11E″2-10-2106 representations for 6 classes · dimensions² sum to 12rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated12 operations
Fig. 2 The D₃ₕ table, with each class heading carrying the number of operations found by generating the group from the coordinates. Twelve operations, six classes — and the two orbits above partition the fluorines into sets that no column of this table can mix.

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.

phosphorus pentafluoride: 12 vibrationsThe vibrational representation, obtained by subtracting the translations and rotations from the full set of Cartesian displacements, with the infrared and Raman activity of each species read off the same character table.D3h · 6 atoms · 3N − 6 = 12Γ(3N) = 2a₁′ ⊕ a₂′ ⊕ 4e′ ⊕ 3a₂″ ⊕ 2e″less translations e′ ⊕ a₂″ and rotations a₂′ ⊕ e″speciesmodesinfraredRamancarriesA1′2 × 1-foldforbiddenactivex²+y², z²E′3 × 2-foldactiveactivex, y, x²−y²A2″2 × 1-foldactiveforbiddenzE″1 × 2-foldforbiddenactivexz, yz8 of 12 modes infrared active · 10 Raman activeno centre of inversion, and 1 species active in both3N − 6 counted, not assumedsymmetry only, no energies
Fig. 3 The same molecule’s motions, sorted into symmetry species. Eighteen vibrations for six atoms, and the species divide along lines the orbit structure sets — modes that move the axial fluorines and modes that move the equatorial ones belong to different species and cannot mix.
benzene — D6hThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHCCHCCHCCHHD6hprincipal axis C67 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates12 atoms
Fig. 4 Benzene, D₆ₕ, turned rather than viewed from elsewhere: the group is searched for again at every step and the twenty-four operations do not depend on the pose. Twelve atoms in two orbits — the six carbons and the six hydrogens — and therefore two spectroscopic environments however many atoms are drawn.
The D6h character tableThe irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.D6horder 24E1 found2C62 found2C32 foundC21 found3C2′3 found3C2″3 foundi1 found2S32 found2S62 foundσh1 found3σd3 found3σv3 foundA1g111111111111A2g1111-1-11111-1-1B1g1-11-11-11-11-11-1B2g1-11-1-111-11-1-11E1g21-1-20021-1-200E2g2-1-12002-1-1200A1u111111-1-1-1-1-1-1A2uz1111-1-1-1-1-1-111B1u1-11-11-1-11-11-11B2u1-11-1-11-11-111-1E1ux, y21-1-200-2-11200E2u2-1-1200-211-20012 representations for 12 classes · dimensions² sum to 24rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated24 operations
Fig. 5 The D₆ₕ table, with class sizes counted off the generated group. Twelve classes for twenty-four operations, and the orbit arithmetic checks against it: six carbons times a site group of order four is twenty-four, which is the identity the previous section stated.

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.

F s on sulfur hexafluoride: a₁g ⊕ eg ⊕ t₁uThe character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.E×18C3×86C4×66C2×63C2×3i×16S4×68S6×83σh×36σd×6χ (the basis)60202000421 × A1g11111111111 × Eg2-100220-1201 × T1u301-1-1-3-1011spans a₁g ⊕ eg ⊕ t₁uone s function on each — 6 functions in allevery multiplicity came out whole, and the reduction rebuilds the character abovecounted, then reducedOh · order 48
Fig. 6 Sulfur hexafluoride’s six fluorines are a single orbit — the forty-eight operations of Oh carry any one onto any other — so all six sigma orbitals enter one reduction and span a₁g ⊕ eg ⊕ t₁u. Three species from one orbit.
F s on boron trifluoride: a₁′ ⊕ e′The character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.E×12C3×23C2×3σh×12S3×23σv×3χ (the basis)3013011 × A1′1111111 × E′2-102-10spans a₁′ ⊕ e′one s function on each — 3 functions in allevery multiplicity came out whole, and the reduction rebuilds the character abovecounted, then reducedD3h · order 12
Fig. 7 Boron trifluoride’s three fluorines are also one orbit, and span a₁′ ⊕ e′. Two species from three functions, which is the smallest interesting case: one totally symmetric combination and one degenerate pair.

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.

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 the ladder goes next

The machinery 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 anywhere on this site.

Where this goes next

Tdorder 24E1 found8C38 found3C23 found6S46 found6σd6 foundA111111A2111-1-1E2-1200T130-11-1T2x, y, z30-1-115 representations for 5 classes · dimensions² sum to 24rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated24 operations Character tables and reduction A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set of orbitals into a statement about how many energies there can be. axial — 2 sitesneighbours at 90°, 90°, 90°, 180°equatorial — 3 sitesneighbours at 90°, 90°, 120°, 120°the two are not equivalentneighbour angles measured, not assumed5 sites Five sites are not alike Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it. HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms Point groups from coordinates A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not.