What a molecule gives up to sit in a crystal

A benzene ring has a six-fold axis. A water molecule has a two-fold axis and two mirrors. Ferrocene, in its eclipsed conformation, has a five-fold axis — and no crystal has a five-fold axis.

Something has to give, and what gives is not the molecule.

The compatibility, and which way it runs #

Put a molecule in a crystal and it occupies a site: a point, with a symmetry of its own. The site’s symmetry is the set of operations of the space group that leave that point exactly where it is — its stabiliser, in the language this collection uses for orbits — and it is a finite group — necessarily so, since an infinite group of operations fixing a point would violate the discreteness a crystal is defined by.

The site’s symmetry must be a subgroup of the molecule’s own point group. An operation fixing the site carries the molecule onto itself, since the molecule is there; and an operation carrying the molecule onto itself is a symmetry of the molecule. So anything the site has, the molecule must have.

The reverse does not hold. A molecule may have symmetry the site does not use. Benzene sits at a site of symmetry 1̄ in many crystals, keeping only an inversion of its six-fold, and nothing is wrong: the extra symmetry is simply not exploited by the packing.

The count for a molecule with an axis #

Take a molecule whose only symmetry is a single n-fold axis. Which sites can it occupy?

The site’s group must be a subgroup of the molecule’s, which is cyclic of order n. So the site’s group must be cyclic, of order dividing n, and generated by a rotation about the molecular axis.

Both halves of that condition are needed and the second is easy to drop. A site of symmetry 222 has three operations, each of order two, so every one of them divides two — and it is not a subgroup of a cyclic group of order two, because it has order four. A test applied to operations one at a time accepts it, and the answer for a two-fold molecule comes out at four instead of two. The test has to be applied to the group.

Where the restriction bites #

The five-fold row is the one this essay exists for.

No site symmetry of any space group contains an operation of order five. That is not a fact about the forty-five groups computed here; it is a consequence of the crystallographic restriction, since every operation of a space group has an integer matrix in the lattice basis and an integer matrix of finite order has order 1, 2, 3, 4 or 6.

Here it is checked rather than quoted: every operation of every site of every group is typed by its order, and the orders available are exactly 1, 2, 3, 4 and 6.

So a molecule with a five-fold axis and nothing else must sit at a general position. Its axis is a symmetry of the molecule and of nothing larger. The crystal has no use for it, does not exploit it, and does not notice it.

What that costs, and what it does not #

The consequences are worth separating, because the loose statement a five-fold molecule cannot crystallise is wrong.

It costs the multiplicity. A molecule at a site of symmetry n contributes one n-th as many independent atoms to the asymmetric unit as one at a general position. A benzene at a site of symmetry 1̄ needs three carbons refined rather than six; at 6/mmm it would need one. A five-fold molecule at a general position needs all of them, so its structure has more parameters and needs more data.

It does not cost the crystal. Ferrocene crystallises perfectly well. So does C₆₀, whose icosahedral symmetry contains six five-fold axes and which sits in a face-centred cubic lattice with its molecules rotating at room temperature, and orientationally ordered at low temperature into a structure whose site symmetry is a crystallographic subgroup of the icosahedral group.

And the icosahedral case is the interesting one. The largest crystallographic subgroup of the icosahedral rotation group is 23 — the tetrahedral rotations, of order twelve, out of the icosahedral group’s sixty. So a C₆₀ molecule in a cubic crystal can have twelve of its sixty rotations be symmetries of the crystal, and the other forty-eight are the molecule’s alone. That ratio, five to one, is the most of an icosahedron a crystal can keep.

The Wyckoff positions, which are the same list under another name #

Everything above is a description of a group’s Wyckoff positions, and it is worth saying so plainly since that is the word a structure report uses.

A Wyckoff position is a class of points with the same site symmetry, up to the group’s own operations. Each has a letter, a multiplicity — how many points of the orbit are in one cell — and a site symmetry. The multiplicity times the order of the site symmetry equals the order of the point group, always, because the orbit and the stabiliser divide the group between them.

So the list computed here is the Wyckoff list, arrived at by a search over grid points rather than from the Tables. The two agree, which is the point of doing it this way: a table can be mistyped and a search cannot, and a search that reproduces a table is a check on both.

What the search adds is the count of grid points at each symmetry, which the Tables do not print and which makes something visible: the general position dominates overwhelmingly. In a group of order forty-eight, almost every point of the cell is fixed by nothing at all, and the special positions are a set of measure zero. A molecule placed at random is at a general position with probability one, and every special position is a coincidence the crystal has arranged.

Which molecular groups are crystallographic at all #

The molecule’s own point group need not be one of the thirty-two, and the ones that are not are worth listing since they are the interesting cases.

The infinite families. A linear molecule has ∞/mm or ∞m, which no crystal has. Carbon dioxide and acetylene are linear and crystallise; their sites have finite symmetry, and the molecular axis is simply not fully exploited.

The five-fold groups. 5, 5m, 5̄m and their relatives — ferrocene eclipsed, cyclopentadienyl rings. None is crystallographic.

The icosahedral groups. 235 and m3̄5̄, of orders sixty and one hundred and twenty. C₆₀, the boron icosahedra in boron carbide, many virus capsids.

Everything else. A molecule whose point group is one of the thirty-two may sit at a site of that symmetry if a crystal offers one, and often does.

The pattern is the crystallographic restriction and nothing else. A molecular point group is crystallographic exactly when its rotation orders are 1, 2, 3, 4 and 6 — which is the same list that every argument in this collection converges on, arriving here as a constraint on chemistry.

How the sites are found #

The computation deserves a paragraph because it is deliberately naive.

Take every point of a grid of twelfths of the cell. Ask, for each operation of the group, whether it moves the point. The ones that do not are the site symmetry. Collect the distinct answers.

The grid’s fineness is the one parameter. Twelfths carry halves, thirds, quarters and sixths, which are the coordinates special positions occupy in almost every group. They do not carry eighths, which is where Fddd’s two origins are, and a grid of twelfths would miss a site at (⅛, ⅛, ⅛) — so the fineness is stated rather than assumed, and a caller who needs eighths asks for twenty-fourths.

No table is consulted anywhere. The site symmetries the International Tables print for each group are the same ones this search finds, and finding them by asking every operation about every point is slower than looking them up and immune to the failure a lookup has.

The local and the global, in one object #

Here the local and the global sit closer together than anywhere else in the subject.

A site symmetry is a local object. It is a statement about one point and its immediate neighbourhood: which operations fix it, what a molecule sitting there must have. Nothing about it refers to the rest of the crystal.

It is also a piece of the global group. The site symmetry is a subgroup of the space group, its order divides the point group’s order, and the multiplicity of the position is the index. The two are not merely related; the local object is a quotient of the global one, and every fact about one is a fact about the other.

Which is why the restriction reaches into a molecule at all. A five-fold rotation is forbidden globally, by an argument about integer matrices acting on a lattice. That argument is about the whole crystal. It reaches down to a single point because the operations fixing that point are operations of the whole group, and they inherit the restriction.

The plane case is the one where all of this can be seen at once. A p3 pattern has three-fold centres at three distinct points of its cell; each of those points is a site, its symmetry is the cyclic group of order three, and that group is a subgroup of the whole plane group. A motif placed at one of them must have a three-fold axis or the pattern is not p3; a motif placed anywhere else need have no symmetry at all, and the group is unchanged. The same three-fold that constrains a motif locally is the three-fold that appears in the group globally, and there is only one of it.

What the count is a count of #

One clarification, since the figure reports a single number per molecular symmetry.

The count is over orders, not over orientations. A three-fold molecule may sit at a site of symmetry 3 only if the site’s axis is the molecule’s axis, and whether a given crystal offers a site with the axis in the right direction is a further question this computation does not ask. What is computed is which orders are available, which is the part the restriction decides.

And the molecules considered are the simplest possible ones: a single n-fold axis and nothing else. A molecule with mirrors as well has more subgroups to offer and can sit at more kinds of site, and the full analysis for a given molecule is a comparison of subgroup lattices rather than of orders.

The refinement the count deliberately does not make is worth naming, because it is where a real structure determination starts rather than stops. A four-fold molecule can sit at a site of symmetry 4, and also at 4/m, 422 or 4/mmm — but only if it has the mirrors or the two-folds those groups require, and a molecule with a four-fold axis and nothing else has none of them. So the honest comparison is between the molecule’s subgroup lattice and the site’s group, not between two integers, and the answer for a particular molecule is a list of which of its own subgroups a crystal offers. The count above is the coarse form of that question: which orders the crystal has at all. It is the part the crystallographic restriction settles, and it is the part that is the same for every molecule of a given order.

What is owned, and what is not #

Owned: the site symmetries of every space group this collection builds, found by search rather than by table; the typing of each by the orders of its operations; the orders available across all of them; and the compatibility count for cyclic molecular groups.

Not owned: any real crystal structure, any claim about which site a particular molecule chooses, and any energetics. Which of the available sites a molecule occupies is decided by packing, and packing is a different question with different machinery — one where the shape of the molecule matters and its symmetry hardly does.

The same argument in reverse #

The compatibility runs one way, and it is worth asking what the other direction would mean.

A site cannot demand symmetry the molecule lacks, because a molecule placed at such a site would be carried onto something it is not. What a crystal does instead, when the packing wants a symmetric site and the molecule is not symmetric enough, is disorder: the molecule occupies several orientations, each with fractional occupancy, and the average over the crystal has the site’s symmetry even though no individual molecule does.

That is a real and common structure rather than a failure of the model. A tetrahedral anion on a site with a three-fold axis but no mirror, a nitrate ion on a six-fold — both resolve the contradiction by averaging, and a diffraction experiment sees the average and reports fractional atoms.

So the rule is not that the two symmetries must match; it is that the site’s symmetry must be a subgroup of the symmetry of what is at the site, which for a disordered structure is the average rather than the molecule. Reading a suspiciously high site symmetry in a structure report as a statement about a molecule is the standard way of getting this backwards.

The number a structure report prints, and what it is saying #

There is a single quantity in every structure report that carries all of this, and reading it is the practical form of the argument above.

Z is how many formula units are in the cell. Z′ is how many are in the asymmetric unit — the fraction of the cell the group’s operations replicate to fill it. The relation between them is the multiplicity arithmetic of this page: a molecule at a general position has Z′ = 1, and a molecule sitting on a symmetry element has Z′ less than one by exactly the order of its site symmetry.

So Z′ = ½ says the molecule sits on an element of order two — an inversion centre, a two-fold axis or a mirror — and the crystallographer has determined half a molecule and generated the rest. Z′ = ⅓ says a three-fold axis. Z′ = ⅙ says the molecule found a site of order six, which is what benzene’s ring is looking for and rarely gets.

And Z′ greater than one says the opposite: several molecules in the asymmetric unit, related by no symmetry at all, each independently determined. That is common, it is not an error, and it usually means the molecules pack better in an arrangement whose symmetry is lower than the packing suggests.

The number is therefore a compact statement about which of this page’s cases a structure is in, printed in every report, and readable without looking at a single coordinate.

What the site does to the molecule’s shape #

There is a consequence running the other way, and it is the one that catches people out, because it constrains chemistry rather than symmetry.

A molecule at a site with an inversion centre must be centrosymmetric as it sits in the crystal. That is not a statement about what the molecule prefers; it is a statement about what the site requires, and a molecule whose lowest-energy conformation is not centrosymmetric has three options. It can adopt a centrosymmetric conformation it would not otherwise choose. It can move to a general position, giving up the multiplicity. Or it can be disordered — two orientations related by the site’s operation, each half-occupied, so that the average has the symmetry and no individual molecule does.

The third is extremely common and it is why so many structure reports contain half-occupied atoms. It is also where a symmetry argument stops describing a molecule and starts describing an average, which is the boundary this collection draws everywhere.

Buckminsterfullerene is the case worth remembering, because it shows both halves. Above about 260 K it crystallises in a face-centred cubic arrangement with the molecules rotating almost freely — the site symmetry is satisfied by an average over orientations rather than by any molecule. Cooled below that, the rotation freezes and the structure orders, with each molecule fixed with one of its three-fold axes along a body diagonal. Sixty rotations offered by the molecule, three used by the site, and the difference disposed of first by rotation and then by a choice.

Where the ladder goes next #

Towards the average. A molecule at a general position has its full symmetry and the crystal has none of it, so the symmetry of the average over many cells is the crystal’s rather than the molecule’s — which is why a diffraction experiment on a ferrocene crystal shows no trace of a five-fold axis anywhere.

And towards disorder, which is what happens when a molecule’s symmetry and its site’s are incompatible in the other direction: a molecule placed at a site with more symmetry than it has, which it cannot satisfy, so the structure resolves the contradiction by occupying several orientations at once. That is common, it is the standard reading of a suspiciously high site symmetry in a structure report, and it is a place where the arithmetic here says what must be true and the crystal finds a way round it.