The groups a single hand may sit in

A protein is built from one enantiomer of each of its amino acids — the left-handed ones, by an accident of biology whose origin is still argued about — and a crystal of that protein contains nothing but that hand. No amount of care in growing it will produce a single crystal with both.

That fact, on its own, removes most of the classification.

A space group containing any orientation-reversing operation cannot describe such a crystal. Apply the operation to a left-handed molecule and a right-handed one appears; the group says that copy is present; and the crystal does not contain it. There is no way round this: it is not a matter of energy, of growth conditions or of which face grows fastest. It is what the group means.

The criterion #

Every operation is a matrix and a translation, and the matrix has a determinant of +1 or −1. Orientation is preserved exactly when the determinant is +1.

So the criterion is: every operation of the group has determinant +1. Groups passing it are called Sohncke groups, after Leonhard Sohncke, who classified the sixty-five of them in three dimensions in 1879 — before the full classification of two hundred and thirty existed, because his question was the easier one.

The test needs no measurement, no structure and no experiment. It is arithmetic on the group’s own matrices, and it partitions the classification completely.

Five of the seventeen, and which five #

The survivors are p1, p2, p3, p4 and p6: the groups whose point group is a rotation group, with no mirror and no glide anywhere.

That list is worth reading twice, because it is short. Twelve of the seventeen contain a reflection or a glide, which means twelve are unavailable to a pattern made of a single handed motif — including every group with a mirror in it and, more surprisingly, every group whose only reversing operations are glides. A glide is as fatal as a mirror: it reverses handedness too, and the fact that it also slides changes nothing about that.

The five that remain are the ones a fabric printer working with an asymmetric motif is confined to, and the ones a chiral molecule in a two-dimensional layer can use. They are also exactly the five whose orbifolds have no mirror boundary.

The check that makes it a claim #

The determinant test is a definition, and a definition can be applied to the wrong object. So it is checked against what the group actually does to a handed motif.

The motif is this collection’s own asymmetric unit — three points in no particular arrangement — and its handedness is the sign of the area of the triangle they make. The orbit is generated, the sign is computed for each copy, and the results are compared with the criterion.

Seventeen groups, seventeen agreements. In the five, every copy has the same sign. In the other twelve, both signs occur — and occur in exactly equal numbers, because the reversing operations form a coset of the rotations and a coset has the same size as the subgroup.

That last detail is worth having: a group with any reversing operation at all has half its operations reversing, never a third and never all of them. Handedness is a homomorphism onto a group of order two, and its kernel has index one or two.

What the five have in common, said three ways #

The same subset of the seventeen arrives from three directions, which is usually a sign that the object is natural rather than the definition convenient.

By determinant: the groups all of whose operations have determinant +1.

By point group: the groups whose point group is a cyclic rotation group — C₁, C₂, C₃, C₄, C₆ — with no dihedral case among them, because a dihedral group contains reflections.

By orbifold: the groups whose quotient has no mirror boundary and no corner points, only cone points. The five orbifolds are the sphere with a few cone points and the torus, and the absence of a boundary is the absence of reflections.

The third description is the one that generalises. Reflections in a group become boundary in the quotient, and a quotient with boundary is a surface with an edge; so asking for a group with no reversing operation is asking for a quotient that is a closed surface or an orbifold with cone points alone. Two of the seventeen fold into a surface with no marked points at all — p1 and pg — and pg is not Sohncke, which is the reminder that free actions and orientation-preserving actions are different conditions.

The same question in three dimensions #

The criterion is unchanged and the enumeration is not.

Among the thirty-two crystal classes, eleven contain rotations only: 1, 2, 222, 4, 422, 3, 32, 6, 622, 23 and 432. That list is complete here, because the thirty-two are built and their matrices examined.

Among the two hundred and thirty space groups, the International Tables record sixty-five passing the criterion. That number is quoted rather than enumerated: this collection builds forty-five space groups, not two hundred and thirty, and a criterion run over forty-five of them is a criterion rather than a classification. Nineteen of the forty-five pass.

Two kinds of handedness, which are not the same #

The word chiral is used for two different things in this subject and conflating them causes trouble.

A Sohncke group contains no reversing operation. Sixty-five of the two hundred and thirty.

A chiral group is one whose mirror image is a different space group. Twenty-two of them, in eleven enantiomorphic pairs — P4₁ and P4₃, P3₁ and P3₂, and so on.

Every chiral group is Sohncke and most Sohncke groups are not chiral: P2₁2₁2₁ contains no reflection and is its own mirror image, because reflecting it and choosing new axes gives the same group back. So a crystal in P2₁2₁2₁ may be built entirely of one hand, and the group cannot say which hand it is.

That distinction is why a structure determination reports a space group and then reports the absolute configuration separately, and why the second is a much harder measurement than the first.

Why the diffraction pattern does not tell #

The criterion is group-theoretical and the experiment is not, and the gap between them is Friedel’s law.

Because the electron density is real, the intensity at −h equals the intensity at h whatever the crystal’s symmetry. So the measured intensities are centrosymmetric even when the crystal is not, and a diffraction pattern from a crystal of pure left-handed material looks exactly like one from pure right-handed material.

Choosing the space group and choosing the hand are separate acts, and the second needs an effect Friedel’s law does not cover: anomalous scattering, where an atom’s scattering acquires a phase near an absorption edge and the two members of a Friedel pair stop being equal. That is Bijvoet’s method, and it is how the absolute configuration of a molecule was first determined, in 1951.

What the criterion does not decide #

Three things, and the second surprises people.

It does not say the crystal is chiral. A racemic mixture — both enantiomers present in equal numbers — crystallises very happily in a centrosymmetric group, with the inversion centre relating the two hands. About ninety per cent of racemates do exactly that. The criterion says which groups a single enantiomer may use, not which groups a chiral molecule may use.

It does not say which of the five, or which of the sixty-five. That is a question about packing, and the four plane groups a molecule packs in is the argument that decides it — of the five Sohncke plane groups, only p1, p2 and pg permit an efficient packing of a lumpy shape, and pg is unavailable to a single hand. So the two arguments together leave p1 and p2, which is a much sharper prediction than either makes alone.

And it says nothing about the molecule’s own symmetry. A chiral molecule may have rotational symmetry of its own and often does; what it may not have is a mirror. The group of the crystal and the group of the molecule are different objects, related only by which of the crystal’s site symmetries the molecule can sit on.

The practical shortlist #

For a protein there are sixty-five groups available and about a dozen that occur, which is a distribution worth a paragraph because it is a measurement rather than a theorem.

The most common by a long way is P2₁2₁2₁, followed by P2₁, then C2, P2₁2₁2 and the tetragonal and hexagonal screw groups. The reasons are the packing argument above — screws and two-folds put a bump against a hollow, and a bare translation does not — together with the fact that a group with more operations needs a smaller asymmetric unit and so a smaller crystal for the same amount of material.

None of that is symmetry. It is a statistic over the structures that have been solved, and it changes slowly as the kinds of molecule people crystallise change. What symmetry supplies is the list of sixty-five, and everything narrowing it further is a fact about the world.

The second of them is worth writing out because it is as small as a space group gets while still being useful. P2₁ has one operation besides the identity — a two-fold screw — so its cell holds two positions, related by a half turn and half a cell along the screw’s axis. There is no reflection to check for and no fixed point anywhere; every operation moves every position, which is what makes the group available to a handed molecule and also what makes its plan almost empty. Most days a protein crystallographer needs no more group theory than P2₁ and P2₁2₁2₁ between them.

Every one of the seventeen has a Sohncke core #

The five are usually presented as the survivors of a test, which makes them sound like a remainder. They are more than that: every group among the seventeen contains one of the five, at index one or two, and nothing else.

The reason is the sentence above about halves. The determinant is a homomorphism from the group onto the two-element group, its kernel is the subgroup of orientation-preserving operations, and a kernel of a map onto two elements has index one or two. So each of the twelve non-Sohncke groups has a distinguished Sohncke subgroup — its own rotations — and that subgroup is necessarily one of the five, because the five are all the Sohncke groups there are.

Running it over the seventeen sorts the twelve into five families by which core they have: three sit over p1 — pm, pg and cm, whose only proper operation is the identity — four over p2, being pmm, pmg, pgg and cmm; two over p3, p3m1 and p31m; two over p4, p4m and p4g; and one over p6, which is p6m. Three, four, two, two and one, and the arithmetic checks: each group’s proper operations number exactly half of its total, and the count matches its core’s exactly.

That map is what a resolution does to a crystal structure. A racemate packing in pgg has both hands, related by the glides; separate the enantiomers and the glides are gone, leaving p2 on the same lattice with half as many operations and the same two positions per cell now of one hand. The enantiopure group is the racemic group’s kernel, so the transition from one to the other is not a change of lattice or of packing motif but the loss of a coset — which is the same index-two descent a twin law is read as, arriving from the other side.

One asymmetry in that map is worth noticing, because it is the reason the five are not evenly used. The core does not determine the group above it: p2 has four groups sitting over it and p6 has one, so knowing that a resolved structure is p2 says nothing about which racemic group it would have been. The map runs one way only, and it runs from twelve onto five rather than from five onto twelve — which is the ordinary situation with a kernel, and the ordinary reason a symmetry argument gives a permission and not a prediction.

The layer between two and three dimensions #

Between the plane’s five and space’s sixty-five sits a case a chemist meets constantly: a molecule confined to a surface or a single layer, which is a two-dimensionally periodic object in three-dimensional space.

There the available operations include ones the plane has not — a two-fold axis lying in the layer, which flips it over, and a mirror plane parallel to it. Both reverse handedness in three dimensions, and both are invisible to a drawing of the layer from above. So a handed molecule adsorbed on a surface is confined more tightly than its two-dimensional picture suggests, and the count of layer groups available to it is the relevant one rather than the seventeen.

This is the standing hazard of a two-dimensional argument about a three-dimensional object, and it is worth the sentence because the plane’s five are so often quoted for a situation that is not the plane’s.

Who counted them #

Leonhard Sohncke published his classification in 1879: the sixty-five groups of motions with no reversing operation, arrived at before Fedorov and Schoenflies completed the full two hundred and thirty in 1891. His question was the one a crystallographer of the time could act on, because the motions he allowed are the ones a lattice of identical particles can have, and the reversing operations were regarded as a complication.

The complication turned out to be most of the subject: a hundred and sixty-five of the two hundred and thirty contain one. But Sohncke’s list is the one that matters for protein crystallography, and it is the reason a structural biologist meets sixty-five space groups rather than two hundred and thirty — and, in practice, meets about a dozen of them, because the rest are rare.

One place the criterion is quietly assumed #

Every pattern figure in this collection is generated from a group applied to a motif, and the motif is deliberately handed — three points making a triangle with a definite sign. That choice was made for a different reason: a motif with any symmetry of its own produces a pattern with more symmetry than the group it was generated from, and the round trip catches it.

A handed motif has a second consequence, visible only now. Every non-Sohncke figure in the collection contains both hands, necessarily, and every Sohncke one does not. That is not a stylistic property of the plates; it is the criterion, showing up in the drawing. A reader who has looked at the seventeen has already seen which five are which, without being told what they were looking at.

The colouring in this essay’s figures makes it explicit by computing the sign rather than by assigning it — so a figure whose colours came out wrong would be reporting a defect in the generator, not a decision about the drawing.

How the hand is actually decided #

The criterion narrows a determination to sixty-five groups and says nothing about which of the two mirror images was measured. That second question is answered by a single refined number, and knowing what it is makes the separation between the two acts concrete.

Include the anomalous scattering in the model and the structure factors of h and −h are no longer equal. Then refine, alongside every other parameter, a quantity x representing the fraction of the crystal that is the inverted structure — so the model predicts intensities as a mixture of the structure and its mirror image, weighted 1 − x and x.

The result is read directly. A value near zero says the structure as refined is the right hand. Near one says the inverse should have been refined and the coordinates should be inverted. And near a half says the crystal is an inversion twin: physically a mixture of both hands in one specimen, which is a real and common thing rather than a failure of the refinement.

What makes it a measurement rather than a guess is its uncertainty, which comes out of the same refinement. A value of 0.03 with an uncertainty of 0.02 settles the question; the same value with an uncertainty of 0.4 settles nothing at all, and the second is what a crystal containing no atom heavier than oxygen typically gives with copper radiation. The determination of the hand therefore has a stated precision and the determination of the group does not, which is exactly the asymmetry between an exact criterion and an experimental one.

The case that has it both ways #

There is an arrangement the criterion permits and intuition does not expect, and it is worth naming because meeting it unprepared leads to the wrong conclusion about a compound.

A racemic mixture normally crystallises centrosymmetrically, with the inversion relating the two hands — that is the easy reading of the criterion, and it is right nearly always. But nothing forbids a racemate from crystallising in a Sohncke group instead, with one molecule of each hand sitting at two independent general positions and no symmetry operation relating them.

Such a structure passes the criterion — it has no reversing operation — and it contains both enantiomers. So the space group being Sohncke does not establish that the compound is enantiopure, which is the inference most often drawn from it.

These are rare and they are documented, and what gives them away is the structure rather than the symmetry: two independent molecules whose coordinates are approximately related by an inversion that is not an operation of the group. That near-relation is the tell, and it is the same shape of observation as everywhere else in this collection — an approximate symmetry visible in the numbers and absent from the group, which is a fact about the arrangement rather than about the classification.

Where the ladder goes next #

The immediate question is what happens when a Sohncke group is put on a bigger cell, and the answer is more interesting than it sounds: a bigger cell, the same group — and sometimes its mirror shows that P4₁ contains P4₃ as a subgroup of index three, so a left-handed screw contains a right-handed one with nothing done to the crystal but a change of description.

The other direction runs into what handedness permits physically. Fifteen may rotate light, and eleven are chiral is the same criterion applied to a property rather than to a structure, and the two lists do not coincide — which is the standing reminder that a symmetry argument gives a permission and never a prediction.

Both cases make the same point about the criterion’s reach. It is a statement about the group, it is exact, and it is silent about the contents — so it cannot say which hand was measured, and it cannot say that only one hand is present. Both of those are determined from the intensities and the coordinates, by measurements with their own uncertainties.