Molecular Geometry & Bonding
what the shape decides
What symmetry decides

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.
10 min read 7 figures Symmetry settles itCounted, not quoted

A molecule’s point group is usually presented as something to be identified by inspection, with a flowchart. It can be computed instead, and computing it turns a classification exercise into a check.

ammonia — C3vThe 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.HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms
Fig. 1 Ammonia, with its point group recovered from its coordinates rather than looked up. Every candidate operation the structure suggests was applied and kept when it permuted the atoms among themselves.

What an operation is

A symmetry operation is a rigid motion that leaves a molecule indistinguishable from itself — that maps every atom onto an atom of the same element.

For a molecule, which is finite, the motions available are those that fix at least one point. There are five kinds.

E, doing nothing, which every molecule has and which is needed for the collection to be a group.

Cₙ, rotation by 360/n degrees about an axis.

σ, reflection in a plane.

i, inversion through a point: every atom at r\mathbf{r} goes to −r-\mathbf{r}.

Sₙ, improper rotation: rotate by 360/n and then reflect in the plane perpendicular to that axis. This one is unfamiliar and it is the one that matters most for chirality, because σ and i are special cases of it.

Unlike a crystal, a molecule has no lattice to be compatible with, so the restriction to two-, three-, four- and six-fold rotation does not apply. A molecule may have any rotation order at all — benzene has six, ferrocene has five, and a linear molecule has infinitely many.

How the search runs

The procedure on this site is exhaustive over a candidate set drawn from the structure itself.

Centre the molecule on the centroid of its atoms, which every symmetry operation must fix.

Collect candidate axes — the directions to each atom, the directions to the midpoint of each pair, and the perpendiculars to each pair. Any symmetry axis of a molecule is one of those for the structures drawn here.

Test each candidate at every rotation order from two to eight, as a mirror normal, and as an improper axis; and test the inversion separately.

Keep what works. An operation is kept when applying it to every atom lands on an atom of the same element within a small tolerance.

The result is a list of the operations the molecule actually has, from which the symbol follows.

Assembling the symbol

The Schoenflies decision tree is standard and worth writing out, because it turns a list of elements into a name mechanically.

Linear? Then D∞h if it has an inversion centre and C∞v if not.

Two or more axes of order greater than two? A cubic group: Td, Oh or Ih, and their rotation-only relatives.

Otherwise, take the highest-order axis as principal, of order n. If there are n twofold axes perpendicular to it, the group is a D group — Dₙₕ with a horizontal mirror, Dₙd with n dihedral mirrors, Dₙ with neither. If there are not, it is a C group — Cₙₕ, Cₙᵥ, S₂ₙ or Cₙ on the same pattern.

No axis at all? Cs with a mirror, Ci with an inversion centre, C₁ with neither.

That is the whole classification, and it is written out in code on this site rather than matched against a table — so a molecule the site has never drawn still gets an answer.

The round trip

Every molecule figure here asserts its group, and the assertion is the same round trip the fleet uses elsewhere: build the structure, forget the label, recover the group from the positions, and require the two to agree.

Twelve molecules are checked this way on every build, from water at C₂ᵥ to benzene at D₆ₕ and sulfur hexafluoride at Oₕ. A molecule assigned the wrong group throws.

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. 2 Benzene at D₆ₕ. The sixfold axis, the six twofold axes in the plane, the horizontal mirror and the inversion centre were all found by search rather than assumed, and the symbol is what the search produced.

What the tolerance is doing

One honest detail. The test asks whether a transformed atom lands within a small distance of an atom of the same element, and that distance is a parameter.

For idealised coordinates it hardly matters — the operations either work exactly or fail badly. For experimental coordinates it matters a great deal, because a real molecule is never exactly symmetric: it vibrates, its environment perturbs it, and a measurement has error bars.

That is a general and slightly uncomfortable feature of molecular symmetry, and it is worth contrasting with the periodic case. In a crystal the symmetry is decidable in integer arithmetic with no tolerance at all, because the lattice supplies exact coordinates. A molecule has no lattice, so a tolerance has to be chosen, and the geometries used here are idealised precisely so that the argument is about the symmetry rather than about the threshold.

Why the group is worth having

Because two physical properties follow from it directly, with no calculation of the bonding whatever.

A molecule can be polar only if its group leaves some direction fixed — which restricts it to C₁, Cs, Cₙ and Cₙᵥ. That gets its own essay.

A molecule can be chiral only if its group contains no improper operation at all. That gets one too.

What the group settlesFor each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.moleculegroupmay be polarmay be chiralwaterC2vyesno2 σcarbon dioxideD∞hnonohas iammoniaC3vyesno3 σmethaneTdnono6 σboron trifluorideD3hnono4 σhydrogen peroxideC2yesyes—bromochlorofluoromethaneC1yesyes—both columns derived from the symbol, not from the bonds
Fig. 3 Both columns derived from the symbol alone. Nothing in either required knowing what the bonds are made of, and the answers are exact rather than approximate.

Beyond those two, the group decides which orbital combinations may mix — the exactly-zero argument — which vibrational modes appear in an infrared or Raman spectrum, and which electronic transitions are allowed. All of them are the same theorem about integrals of non-symmetric functions.

Molecules against crystals

The comparison is instructive, because the two subjects use the same word for different things.

A point group describes a finite object: operations that fix a point. There are infinitely many possible point groups, since a molecule can have any rotation order.

A space group describes an infinite periodic one, and includes translations. There are exactly 230 in three dimensions, and the rotation orders are restricted to one, two, three, four and six.

The bridge is the crystal classes — the 32 point groups compatible with a lattice. A molecule of any symmetry can exist; a molecule sitting on a special position in a crystal must have a symmetry that the crystal’s lattice permits, which is why crystallography quotes site symmetry as a constraint on structure.

Where the model stops

Three limits.

Idealised geometries. The coordinates used here are constructed rather than measured, so the symmetry is exact by construction. A real structure has a symmetry only to within a tolerance.

A single conformation. A molecule that can rotate about a bond has different symmetry in different conformations. Ethane is D₃d staggered and D₃ₕ eclipsed, and at room temperature it is neither for long. Hydrogen peroxide is C₂ in its gauche form and would be C₂ᵥ or C₂ₕ if it were planar.

No vibrations. Every symmetry statement here is about the equilibrium geometry. Instantaneously, a vibrating molecule has lower symmetry almost always, and that is exactly how formally forbidden transitions become weakly allowed.

Where the name came from

Arthur Moritz Schoenflies published his classification in the 1890s, in parallel with Fedorov’s independent derivation of the space groups, and the two men corrected each other’s errors by correspondence.

Chemistry uses Schoenflies notation and crystallography uses Hermann–Mauguin, for the same objects. The reason is historical rather than principled: chemists want the abstract group type and crystallographers want the directions, and each notation is better at one of those. A reader meeting both should know they are two names for one thing.

The groups in practice

A tour of the ones that turn up, since the classification is more useful once the common cases are familiar.

C₁ — no symmetry at all. Most complex molecules, and the reason most of them are chiral.

Cs — one mirror plane and nothing else. Very common in substituted molecules.

C₂ᵥ — a twofold axis and two mirrors containing it. Water, sulfur dioxide, and a large family of bent triatomics and substituted aromatics.

C₃ᵥ — ammonia, chloroform, and every pyramidal AB₃.

D₃ₕ — boron trifluoride, phosphorus pentafluoride. Planar trigonal or trigonal bipyramidal.

D₆ₕ — benzene and its symmetric derivatives.

Td — methane, and every symmetric tetrahedral AB₄.

Oₕ — sulfur hexafluoride, and every symmetric octahedral AB₆.

boron trifluoride — 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.FFBFD3hprincipal axis C34 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates4 atoms
Fig. 4 Boron trifluoride at D₃ₕ: a threefold axis, three perpendicular twofold axes, a horizontal mirror. The horizontal mirror is what makes it D₃ₕ rather than C₃ᵥ, and it is also what forbids a dipole moment.
sulfur hexafluoride — OhThe 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.FFFSFFFOhprincipal axis C49 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates7 atoms
Fig. 5 Sulfur hexafluoride at Oₕ, the highest symmetry in the set: three fourfold axes, four threefold axes, an inversion centre, nine mirror planes. Everything about such a molecule that symmetry can decide is decided.

The pattern worth noticing is that adding a horizontal mirror to a Cₙᵥ group turns it into Dₙₕ and kills the dipole. Symmetry that seems like a small addition to a diagram is often the operation that changes a property from possible to forbidden.

The elements, one molecule at a time

Two more cases, chosen because they show the two extremes of the classification.

hydrogen peroxide — C2The 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.OOHHC2principal axis C20 mirror planesno inversion centremay be polarmay be chiralgroup recovered from the coordinates4 atoms
Fig. 6 Hydrogen peroxide: C₂. A single twofold axis and nothing else — no mirror, no inversion centre, no improper axis. Four atoms, and the lowest non-trivial symmetry a molecule can have while still having any.
bromochlorofluoromethane — C1The 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.BrClCHFC1no rotation axis0 mirror planesno inversion centremay be polarmay be chiralgroup recovered from the coordinates5 atoms
Fig. 7 And the floor: C₁, the group with only the identity. Every operation was tried and none survived, which is what “no symmetry” means operationally — a completed search rather than an impression.

The C₁ case is worth dwelling on because it is the commonest in real chemistry and the least interesting to identify. Most molecules of any complexity have no symmetry at all, and the classification’s value is concentrated in the small, highly symmetric species where it constrains a great deal.

That is also where the properties it settles matter most: a C₁ molecule may be polar and may be chiral, which is to say symmetry has ruled nothing out.

Where the ladder goes next

The two properties the group settles are polarity and chirality.

The same theorem applied to orbitals is exactly zero.

What the pictures here cannot show. A symmetry operation is a motion, and a still drawing shows a structure. The group printed on each figure is the output of a search over operations, and the drawing is what the search was run on rather than evidence for the result.

Where this goes next

moleculegroupmay be polarmay be chiralwaterC2vyesno2 σcarbon dioxideD∞hnonohas iammoniaC3vyesno3 σmethaneTdnono6 σboron trifluorideD3hnono4 σhydrogen peroxideC2yesyes—bromochlorofluoromethaneC1yesyes—both columns derived from the symbol, not from the bonds Symmetry forbids a dipole Whether a molecule can have a dipole moment follows from its point group alone. The usual argument — adding up bond vectors — gets the right answer for easy cases by a route that does not generalise. BrClCHFC1no rotation axis0 mirror planesno inversion centremay be polarmay be chiralgroup recovered from the coordinates5 atoms Chirality is a symmetry statement A molecule is chiral when its group contains no improper operation at all. The four-different-groups rule is a useful special case that misses molecules with no stereocentre and wrongly condemns some that have several. 1s · 2pxS = 0 exactlycomputed -9.6e-17 — arithmetic noiseseparation 2.8 bohrcontours at 50% of each densitythe contributions cancel in pairs, so the integral vanishesone electron Exactly zero Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.