Point groups from coordinates

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.

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 $\mathbf{r}$ goes to $-\mathbf{r}$.

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 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 can be carried out mechanically rather than matched against a table — so a molecule nobody has classified before still gets an answer.

The round trip #

Every molecule figure here is checked against its group by a round trip: build the structure, forget the label, recover the group from the positions, and require the two to agree.

Twelve molecules are checked this way before any of these pages exists, from water at C₂ᵥ to benzene at D₆ₕ and sulfur hexafluoride at Oₕ. A molecule assigned the wrong group throws.

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.

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.

From the symbol to the group #

Everything above stops at the symbol, and the symbol is a label. The uses in the previous section need the group itself: not “this molecule is Td” but the twenty-four matrices, sorted into classes.

Getting there takes one further step and it is the step that makes the symbol worth having. The operations the search found are multiplied together, and the products are kept, and their products are kept, until the set stops growing. That is closure, and it matters because a group is not a list of the elements somebody thought to look for. A closed set of matrices that all map the molecule onto itself is the group, whether or not the enumeration was complete.

The classes come next, and they are computed rather than read off. Two operations belong to the same class when some third operation carries one to the other, and testing that is a search over the group. So the columns of a character table exist before any table is opened.

The checking is worth describing because it is what keeps the table from being an act of faith. A character table has to satisfy four relations that involve nothing outside itself: the class sizes sum to the group order, the squared dimensions sum to the group order, there are as many representations as classes, and the rows are orthonormal while the columns are orthogonal under the class-weighted inner product. All four are checked before the table is used. They catch real errors: a mistyped character in a table as large as D₆ₕ’s breaks at least one of them, and would otherwise be invisible in any figure that used it.

Water’s is small enough to read entirely: four operations, four classes of one operation each, four one-dimensional representations — and that is the whole of what symmetry can say about a C₂ᵥ molecule, before any bonding argument is made.

There is one honest wrinkle, and it is a naming convention rather than a physical fact. Which of two indistinguishable classes is called C₂′ and which C₂″, and which mirror planes are σᵥ rather than σ_d, is a choice. This site takes C₂′ to be the class whose axes pass through atoms and σᵥ to be the planes containing those axes, and decides both by measuring the molecule rather than by assuming an orientation. The multiset of representations a basis spans never depends on the choice. The labels attached to them do.

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.

What it costs, and the bug the tolerance hid #

The search is cheap. Candidate axes come from the structure — through atoms, through the midpoints of pairs, perpendicular to pairs — each is tried at every rotation order from two to eight, as a mirror and as an improper rotation, and what survives is kept. For the molecules here that is a few thousand coordinate transformations and takes milliseconds.

Closure is where it went wrong, and the failure is worth recording because it was invisible until the very last step.

The tolerance in the section above is right for deciding whether an operation exists and wrong for using one. Ammonia’s threefold axis is not exactly the z axis in the search’s answer; it came out tilted by about $5\times10^{-5}$ radians, because the candidate direction that was tried first was a combination of atom positions rather than the axis itself, and a rotation about that tilted line still maps the molecule onto itself to well within six hundredths of an ångström. Perfectly good as a verdict. Disastrous as a matrix. The products of two such operations drift, no two of them match, and the set grows without ever closing — the run stopped at a two-hundred-element ceiling on a group of order six.

The repair is to stop treating the approximate matrix as an answer and treat it as a question. It still identifies unambiguously which atom goes where, and that permutation is exact: it is a fact about labels, not about numbers. The operation is then recomputed as the orthogonal matrix that best realises that permutation, by a few steps of Newton’s iteration on the polar factor, and what comes back is orthogonal to about $10^{-15}$. Products of such matrices stay that accurate, and — the property closure actually needs — the refined matrix is a function of the permutation alone, so two long chains of products that move the same atoms to the same places come out bit-identical.

What that does not do is make the input exact. The coordinates here are quoted to four decimal places, so ammonia’s hydrogens sit about $5\times10^{-5}$ ångströms off a perfect threefold arrangement and no orthogonal matrix reproduces them exactly. That residual is reported rather than hidden, and demanding exactness of a rounded structure is the tempting fix — it rejects every rotation ammonia has.

A second version of the same problem was better hidden. A planar molecule’s own plane is a mirror that moves no atom at all, so its permutation is the identity’s, indistinguishable from doing nothing. Keying operations on the permutation alone collapses benzene’s twenty-four operations to twelve — silently, with every one of the twelve correct, and with a symbol of D₆ₕ still reported because the symbol comes from the axis search rather than from the count. Only checking the order against the character table exposes it.

Where the model stops #

Four 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.

The linear groups are refused rather than approximated. Carbon dioxide and hydrogen cyanide get their symbols here — D∞h and C∞v — and stop there. Their groups have infinitely many operations, and every use in the section above divides by the group order. A calculation that quietly truncated the rotation axis at some large finite order would produce plausible answers for the wrong group, so the search declines instead and says why.

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.

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.

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.

Why the search terminates #

Calling the assignment decidable is a strong word, and it is earned by a fact about where symmetry operations can be rather than by any cleverness in the search.

Every operation of a point group leaves the centre of mass where it is, and every one permutes atoms of the same element among positions the same distance from that centre. So a candidate operation is not an arbitrary rotation of space: its axis has to pass through the centre of mass, and it has to map some atom onto some other atom of the same kind.

That makes the candidates countable and few. A rotation axis must pass through the centre of mass and either through an atom, or through the midpoint of a pair, or through the centroid of a set of equivalent atoms; a mirror plane must contain the centre of mass and either contain atoms or bisect pairs of them. For a molecule of a few dozen atoms that is a list of tens or hundreds of candidates, not a continuum.

So the search enumerates a finite list, tests each candidate by applying it and checking whether the atom positions are reproduced, and stops. It cannot miss an operation, because every operation is on the list; and it cannot run for ever, because the list is finite.

That is what separates this from looking a group up. A lookup requires recognising the molecule; the search requires only the coordinates, and its answer is complete rather than as complete as the person doing the recognising.

Where to read on #

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.