The tolerance is a decision

Point groups from coordinates opens this field with a claim that has held for every essay since: a molecule’s symmetry is not a label to be looked up but something decidable from the atom positions. Search for the operations that permute the atoms among themselves, close the set under multiplication, and read off the symbol.

Every word of that is true and one word of it is doing work that has never been examined. Permute: an operation is accepted when it sends every atom onto an atom. Onto, exactly? No structure that anybody has ever measured is exactly symmetric. A diffraction experiment returns coordinates with uncertainties in the third decimal place; a calculation returns coordinates converged to whatever gradient threshold it was stopped at; a molecule in a crystal is squeezed by its neighbours into a shape its own symmetry would not choose.

So the test is within some tolerance, and a symmetry finder always has one whether or not anybody chose it. The one used for every structure here is six hundredths of an ångström, and like most defaults it was set once and never examined.

That number is the whole of the answer to a question that is seldom asked. This essay asks it.

Four distortions, chosen rather than jiggled #

The distortions are named and deterministic, and that is deliberate. A random jiggle of a stated size would make the results look like noise, and the finding is that the direction of the error matters as much as its size.

Alternate moves the ring atoms in and out alternately: the Kekulé distortion, which takes a regular hexagon to a bond-alternating one. Pyramidal lifts every hydrogen out of the plane. Stretch pulls the whole molecule along one in-plane axis. One atom moves a single carbon outwards.

At a tolerance of two hundredths of an ångström — smaller than the distortion in the first three cases — the four are assigned D3h, C6v, D2h and C2v. Four different groups, from four errors of the same size in a molecule that is D6h when it is left alone.

Each of those is right. A ring with alternating bonds really is D3h: the sixfold axis is gone and a threefold one remains, and the same reduction is what a chain cannot stay even computes the energetics of. A ring with its hydrogens bent up really is C6v: the horizontal mirror has gone and the vertical ones have not. The point is not that any verdict is wrong. It is that there are four of them, and which one is returned depends on a number that appears in no published structure.

What grows and what does not #

There are two quantities here and it is worth separating them sharply, because one behaves and the other does not.

What the search finds can only grow. Each candidate operation is accepted when it moves no atom by more than the tolerance, so a larger tolerance accepts everything a smaller one accepted and possibly more. Counted before any merging or classifying, the number of accepted operations rises monotonically across every sweep — 7 to 15 to 19 for the stretched molecule, 9 to 31 to 55 for the alternating one — and it could not do anything else.

What it is called does not. The classification asks whether there are $n$ twofold axes perpendicular to the principal one, and whether there are $n$ vertical planes, and a structure part way between two symmetries has some of each. So the symbol can move anywhere.

The sharpest case is the stretched molecule. It is D2h up to a tolerance of four hundredths, C6h at eight hundredths, and D6h at fifteen. The middle verdict is the interesting one: at that tolerance the sixfold axis has been accepted and the horizontal mirror with it, but only two of the six twofold axes have — so the decision tree, asked for $n = 6$ perpendicular twofolds and finding two, falls through to $C_{6h}$.

D2h has order 8. C6h has order 12. Eight does not divide twelve, so by Lagrange’s theorem the group named at the tighter tolerance is not a subgroup of the one named at the looser. The sequence of verdicts is not a chain, and nothing about the molecule changed between them.

That is not a defect in the tree so much as a statement about what a symbol is. A Schoenflies symbol names a group; a set of accepted operations is not a group until it is closed; and the map from one to the other is not order-preserving.

Three traps that only an imperfect molecule springs #

A straightforward group finder, swept across tolerances, fails in three ways, and all three stay hidden for as long as every molecule it is given is exact. They are worth setting out because they share a shape: each is a piece of arithmetic that is exactly right when the structure is exact and quietly wrong when it is not.

Benzene comes back icosahedral. Suppose candidate axes are treated as distinct whenever their directions differ by more than $10^{-6}$, which is far finer than any tolerance an operation is tested at. For an exact molecule the several candidate directions that are really the same axis coincide to $10^{-15}$ and are merged. For a molecule bent by a hundredth of an ångström they do not: the sixfold axis arrives seven times over, as seven directions half a degree apart, every one of which genuinely is a sixfold axis at that tolerance. The decision tree then sees several axes of order greater than two, concludes a cubic group, and — if its test for the icosahedral case asks only whether any axis has order five or more, and six satisfies that — reports Ih.

The repair is not a finer merge. Half a degree at the radius of a hydrogen is exactly the tolerance, so any distance-based test sits on its own boundary. What settles it is not how close two axes are but whether they do the same thing: two rotations of the same order that send every atom to the same atom are one operation found twice. The permutation identifies the operation, provided an axis and its opposite — which induce inverse permutations — are counted once.

That identification fails for mirrors, and the failure is instructive: identifying mirrors by their permutation collapses benzene’s twenty-four operations to twelve, because a planar molecule’s own plane moves no atom and so shares its permutation with the identity. Mirrors therefore have to be merged geometrically, and they do not need a fine version of it.

The cubic branch is decided by presence rather than count. The three cubic families are separated by counts — six fivefold axes, three fourfold, four threefold — and a test that asks instead whether two high axes exist and whether any has order five or more is the one that turned benzene icosahedral. Counting cannot make that mistake, and it changes no verdict for a molecule that really is cubic: methane has four threefold axes, sulfur hexafluoride three fourfold ones. A structure matching none of the three counts should fall through to the axial tree rather than be guessed at, which is the other half of the repair — a presence test returns something cubic for anything that enters it.

Perpendicular tested at a hard thousandth. An operation accepted because it moves no atom by more than $\tau$ may itself sit an angle $\tau/R$ off true, and asking afterwards whether it is exactly perpendicular to the principal axis throws it away again. Benzene bent by a hundredth and searched at three tenths finds all six of its twofold axes, calls four of them perpendicular and two of them not, and reports C6h for a molecule already found to have a centre of inversion and seven mirror planes. The perpendicularity test belongs at the tolerance the search used, with a floor that leaves every exact structure — where the dot products are $10^{-16}$ — behaving exactly as before.

Why exact structures hide all three #

Idealised structures are written down exactly. Water’s coordinates are three numbers chosen to make a $104.5°$ angle; methane’s are $(\pm t, \pm t, \pm t)$; ferrocene’s ring radius is computed from its bond length. Their symmetry operations are exact to fifteen decimal places, so every trap above evaluates to the right answer by a comfortable margin.

This is a recurring shape and worth naming. Two structures, two spectra works from exact geometries, degeneracy is a group theorem counts on exact characters, and an infinite group, worked in a finite one needed a whole apparatus precisely because one case was not exact. A method tested only on exact input is tested on the case where the hard part does not arise.

The general lesson is that a procedure’s hard cases have to be manufactured, because nature supplies them only in forms nobody writes down. A check made only of ideal inputs certifies a method on the one case where it cannot fail.

What a crystallographer does instead #

The literature does not pretend this problem away; it has a vocabulary for it. A structure is described as having approximate or pseudo symmetry, deviations are quoted as root-mean-square displacements from the idealised positions, and software that assigns point groups takes a tolerance as a required argument rather than a default.

What is unusual is how rarely the number travels with the conclusion. A paper states that a complex is “essentially octahedral” and gives the cis angles as $88.4°$ to $91.7°$; both statements are there, and the first is a verdict at a tolerance the second is enough to compute. The two are not usually connected.

There is a reasonable defence, and it is the one this whole field runs on: the consequences of a group are what matter, and they are often robust to the assignment. If a complex is octahedral to within a degree and a half, its d orbitals split into a set of three and a set of two to within a corresponding small mixing, and the splitting is a symmetry statement is not overturned. A near-symmetry gives near-consequences.

But not always, and the exceptions are exactly the interesting cases. A degeneracy is not robust: copper is never quite octahedral is a molecule whose whole behaviour is that a distortion of a few hundredths of an ångström removes a degeneracy that the idealised group requires. A selection rule is not robust either: an integral that vanishes exactly in D6h is small rather than zero in D3h, and what an absence proves turns on the difference between the two.

So the answer is not that the tolerance does not matter. It is that whether it matters depends on what is being asked, and the honest form of a symmetry claim carries the number it was decided at.

What a tolerance ought to be, if it were chosen #

The sweep says the number matters and leaves open what it should be. Three considerations bound it, and none of them gives a single answer.

Below the measurement’s precision is pointless. A tolerance of a thousandth of an ångström on a structure whose coordinates are known to five thousandths is asking a question the data cannot answer, and the C1 it returns is a statement about the noise.

Above the distortion of interest is worse. A tolerance of a tenth forgives a Jahn–Teller distortion of a few hundredths, which is exactly the distortion copper is never quite octahedral is about — so a search at that tolerance would report Oh for a molecule whose entire chemistry is that it is not.

And the useful value depends on the consequence being drawn. A tolerance appropriate for deciding whether a spectrum will show two bands or three is not the one appropriate for deciding whether a dipole moment vanishes, because the two consequences are sensitive to different operations.

So there is no defensible default, and six hundredths is a convention rather than a choice — a number that works for exact structures, which is to say a number that has never been tested. The remedy is not a better default. It is to make the default visible, to sweep beside it, and to state a claim about a group with the tolerance attached.

Where the model stops #

A tolerance is not an uncertainty. The sweep above forgives a fixed displacement for every atom; a real measurement has different uncertainties for a heavy atom and a hydrogen, and a hydrogen’s position from an X-ray structure is barely determined at all. A tolerance that respected that would be a weighted one, and none is used here.

The distortions here are single modes. A real structure is off its idealised symmetry in every direction at once, and the staircases would be correspondingly messier. What the single modes buy is the finding that direction matters — three distortions of the same size giving three groups at one tolerance — which a mixture would have hidden.

Nothing here says which group is right. The verdict at a large tolerance is not more correct than the verdict at a small one; it is an answer to a different question. Whether the useful description of a slightly distorted benzene is “D6h with a small perturbation” or “D3h” is decided by what is being computed with it, and that is a judgement rather than a measurement.

The length the tolerance ought to be compared against #

A tolerance in ångström is a decision with no scale attached, and there is a physical scale available — one that says what a departure from symmetry has to exceed before the molecule can be said to have it.

The scale is the zero-point amplitude. A molecule at absolute zero is not at the minimum of its potential; its nuclei are spread by their own vibrational ground state, and how far can be computed. In the small molecules where it has been, every hydrogen is spread by more than 0.09 ångström, methane’s by 0.134, and the heavy atoms by an order of magnitude less.

That supplies the comparison a bare tolerance lacks. A structural asymmetry smaller than the atoms’ own zero-point spread is not an asymmetry the molecule has. It is smaller than the distance the atoms move while sitting still, and any symmetry operation is being applied to a set of mean positions that the molecule visits only on average.

The numbers make the point sharply. Benzene bent by a hundredth of an ångström, which the sweep carries across five groups, is bent by a tenth of its own hydrogens’ zero-point amplitude. Asking whether such a structure is D6h is asking a question about a set of coordinates rather than about a molecule, and the five answers the sweep returns are five answers about the coordinates.

So the tolerance is a decision, and it is not a free one. Set it below the zero-point amplitude and the search is resolving distinctions the molecule does not make; set it far above and it is forgiving distortions that are real. A tolerance of the order of a tenth of an ångström for a molecule with hydrogens, and a hundredth for one without, is the range those amplitudes suggest — and it is a range with a physical argument behind it rather than a habit.

What dropping exactness changes #

The standard results about point groups are that a group is decidable from coordinates, that what it settles about a dipole or about chirality follows from the symbol alone, that lowering the symmetry splits labels in a computable way, and that the groups a molecule can fall to form a lattice with a definite size.

All of those assume the coordinates are exact. Dropping the assumption shows that the decision procedure has a parameter, that the parameter changes the answer for distortions far smaller than a bond length, that the sequence of answers is not even a chain of subgroups, and that three pieces of arithmetic which pass every check on exact structures are wrong in ways that only an imperfect structure can show.

The open question is what a near-symmetry buys — whether a quantity that a group forces to vanish exactly is small when the group is only approximate, and by how much — which is the question the last section leaves open and which needs an energy to answer.