How much symmetry is left

A tolerance sweep showed that the point group of a structure is not a property of the structure alone: four distortions of benzene, none of them larger than six hundredths of an ångström, were reported as six different groups depending on how much error the search was told to forgive. The conclusion was that the tolerance is a decision, and it stopped there.

What it did not do is name the quantity the decision is a threshold on. A threshold is a number compared against something, and the something has never been written down here — not in that essay, not in the search, and not, as far as can be told, in most of the software that reports point groups.

The quantity is available, it is continuous, and computing it needs no search at all.

The measure #

The distance from a structure to the nearest structure that does have a given symmetry, divided by the structure’s own size:

$$S = 100 \times \frac{\sum_i |\mathbf{r}_i - \mathbf{q}_i|^2}{\sum_i |\mathbf{r}_i - \bar{\mathbf{r}}|^2}$$

where $\mathbf{q}$ is that nearest structure. It is zero exactly when the symmetry is exact, it grows smoothly as the structure is pushed away, and it is dimensionless — a molecule twice the size with twice the distortion scores the same. The convention that multiplies by a hundred and calls it a percentage is the one the continuous-symmetry-measure literature uses, and it is kept here so the numbers are comparable with published ones.

Nothing in that expression is a threshold, and nothing in it is a verdict.

The nearest symmetric structure is an average #

The apparent difficulty is finding $\mathbf{q}$, and it is not a difficulty: the nearest symmetric structure can be written down.

Take a group of operations $g$, each with the permutation $P_g$ it induces on the atoms. Then

$$\mathbf{q}_i = \frac{1}{h}\sum_g g^{-1}\, \mathbf{r}_{P_g(i)}$$

is the average of the atom’s whole orbit brought back to one place, and it is symmetric by construction: applying any operation of the group to the set $\mathbf{q}$ reproduces the set exactly. It is also the least-squares nearest such set, which is what makes it the right one to measure against.

Both halves are checked rather than assumed. The folded structure is required to satisfy the operation it was folded onto, to 10⁻⁹; and folding a structure that is already symmetric is required to change it by nothing at all, which is the statement that the operation is a projection.

There is one way for the construction to have no answer, and it is worth stating because it is where the measure stops rather than becomes large. The permutation has to exist: each atom’s image under an operation has to be nearest to one atom, and different atoms have to claim different images. A structure distorted far enough that two atoms compete for one image has no permutation and therefore no measure, and that case is reported rather than patched.

The floor, which is the coordinates rather than the arithmetic #

Before anything is distorted it is worth asking what the measure gives for a structure that is supposed to be exact, because the answer is not zero and the reason is instructive.

Benzene, methane and water measure 10⁻²⁹ against their own groups — that is double-precision arithmetic reporting an exact zero, and it is what an exactly symmetric set of coordinates should give. Ammonia measures 3.5 × 10⁻⁷.

The difference is not in the arithmetic. It is that ammonia’s stored coordinates are quoted to four decimal places and its threefold axis is therefore only threefold to about 10⁻⁴, while benzene’s and methane’s are generated from expressions that are exact in binary. So the measure of an “ideal” structure is a measurement of how well its coordinates were written down.

That sets a floor on what any reported measure can mean. A structure quoted to four decimals cannot be shown to be nearer than about 10⁻⁷ to its group, and a measure quoted more precisely than that is reporting the file it came out of. It also explains an earlier repair: searching ammonia’s group at a tolerance of 10⁻⁶ finds no threefold axis at all, because at that tolerance ammonia as stored has not got one.

What the folding does, on the smallest case #

Water and a twofold axis is small enough to follow all the way through, and it makes the projection concrete.

The group has four operations. Under the identity every atom stays put; under the twofold rotation the oxygen stays put and the two hydrogens exchange; the two mirrors do one or the other of those. So the oxygen’s orbit average is the oxygen’s own position reflected onto itself, which fixes it on the axis, and each hydrogen’s orbit average is the mean of its own position and its partner’s brought back by the rotation.

The folded structure is therefore the one with both O–H distances equal to their mean, both angles to the axis equal to their mean, and the oxygen exactly on the axis — which is what anybody would have drawn by hand, and is now the output of one average rather than of a judgement. What the arithmetic adds is that the same construction works for twenty-one atoms and a group of order twenty, where nobody would draw it by hand, and that it is provably the nearest such structure rather than a plausible one.

Measuring the four distortions #

The four cases from the tolerance sweep can now be asked the other question.

Two results, and they point in opposite directions.

The order agrees. Ranked by the measure, the four go pyramidal, one atom, alternate, stretch. Ranked by the tolerance each needs, they go the same way. That is worth recording as a result rather than as a disappointment: on these four the sweep is not misleading about which is worse. A check that the two orderings disagree was tried first and refused, which is why it is not the claim.

The spacing does not. The alternating structure and the stretched one sit 0.00962 and 0.00980 from D6h — a difference of two per cent, which is nothing — and the search needs 0.08 Å to call the first D6h and 0.15 Å to call the second. And in the other direction: the alternating structure and the one with a single atom moved both first come out as D6h at 0.08 Å, and one of them is 72 per cent further from D6h than the other.

So a tolerance is not a threshold on how symmetric a structure is. It is a threshold on the largest single displacement an operation produces, and that quantity and the measure are related by a factor that depends on the distortion.

Why the two differ, in one sentence #

A maximum reads the worst atom. A measure reads all of them.

For a distortion concentrated on one atom the two nearly coincide, because the worst atom is the only atom that has moved. For a distortion spread over every atom the measure grows with the number of atoms while the largest displacement does not grow at all. A twelve-atom molecule therefore has far more room between the two criteria than a three-atom one.

That is a statement that can be checked across molecules, and it is the wider claim.

The last column is the ratio between the two kinds of distortion for each molecule, and it is not always on the same side of one. For benzene and ferrocene the spread-out distortion is three to six times further from symmetric at the same tolerance; for ammonia and methane the concentrated one is further. So the disagreement between the two criteria is not even a consistent bias that could be corrected by a factor.

The molecules where it matters least, and most #

The measure explains something the tolerance sweep left as an oddity: why small molecules are so much better behaved under a point-group search than large ones.

For a structure with a few atoms the two criteria are nearly the same test, and the choice of tolerance genuinely is the whole decision. For a structure with twenty, a maximum-displacement test is reading one number out of sixty and calling it the symmetry of the molecule.

The extreme case in this collection is ferrocene: twenty-one atoms and a group of order twenty. One badly-placed hydrogen and a fifth of an ångström of thermal spread across the whole structure look identical to a maximum-displacement test, and differ by a factor of five in the measure — which is the difference between a defect and a temperature, reported as one number either way by the test everybody uses.

What the measure does not do #

Four limits, and the first two are the ones that would have to be lifted before these numbers could be compared with published ones.

The group and its orientation are fixed by the ideal structure. The standard treatment minimises over orientations as well: it asks for the nearest structure with some sixfold axis rather than with this sixfold axis. Every structure here is in the same frame as its own ideal, so the difference does not enter — but a measured structure arriving in an arbitrary frame would need that minimisation, and it is a search rather than an average.

The permutation is fixed too. It is found by nearest-image assignment, which is right while the distortion is small and has no answer once two atoms compete for one image. The general problem is a matching over all permutations that respect the elements, which for anything larger than a few atoms is where the cost of the method actually lives.

It is not a physical quantity. No energy appears anywhere: two structures at the same measure can be a hundredth of a kilojoule apart or a hundred, depending on what is stiff. The measure says how far a structure is from a shape, not how far it is from a minimum, and the relation between the two is the force field — which is a different thing entirely and a fitted one.

It says nothing about which operation is lost. The number above is against a whole group, and a structure can lose a mirror while keeping every rotation exactly. Measuring against each operation separately says which, and that decomposition is the natural next thing to compute.

What the measure normalises by, and why that is a decision too #

A continuous measure repairs the tolerance’s worst defect and it does not repair all of them, because turning a displacement into a dimensionless number requires dividing by a length — and which length is a choice of the same kind as the tolerance was.

The construction is a mean squared displacement: how far each atom has to move to reach the nearest structure of the desired symmetry, squared and averaged. That has units of area, so it has to be divided by something with the same units before it can be quoted as a pure number or compared between molecules.

The usual divisor is the mean squared distance of the atoms from the centroid — the molecule’s own size. It is a sensible choice and it is not the only one. Dividing by the largest such distance, or by the mean bond length squared, gives numbers in different ratios, and two molecules that rank one way under one normalisation can rank the other way under another.

So the measure is scale-free by construction, which is exactly what a tolerance in ångström is not, and the price is that a comparison between two molecules of different sizes now depends on how size was defined.

The direction of the effect is predictable and is worth stating. A large molecule is flattered. The same absolute displacement of one atom, in a molecule twice the size, is divided by four times the squared extent and reports a symmetry measure four times smaller. A distortion of a hundredth of an ångström in benzene and the same distortion in a metal cluster of thirty atoms are the same physical departure and produce very different numbers.

That is the tolerance problem inverted rather than solved. A fixed tolerance in ångström admits amounts of asymmetry differing by a factor of fifty-six across a set of molecules; a fixed symmetry measure admits amounts of physical displacement differing by whatever the sizes differ by. Neither is wrong; they answer different questions, and only one of them can be answered at a time.

Which of the two a reader wants depends on what the number is for.

A tolerance is the right instrument for an assignment. The question is this structure D6h is about whether the departure is within the resolution of the measurement or within the molecule’s own motion, and both of those are lengths.

A measure is the right instrument for a comparison. The question which of these two structures is further from octahedral needs a quantity that does not grow with the molecule, and a length cannot supply it.

So the tolerance and the measure are not a problem and its solution. They are two quantities, each with a convention inside it, and the useful practice is to say which is being reported and what it was divided by — which is one clause and is almost never present.

There is one comparison for which both are safe and it is worth naming, because it is the commonest. Comparing one molecule against itself — two polymorphs, two temperatures, a computed structure against a measured one — holds the size fixed, so the normalisation cancels and the tolerance is applied to a constant scale. Every difficulty here arises from comparing different molecules, and a great deal of what a symmetry measure is used for is not that.

Who did this, and when #

Continuous symmetry measures are Avnir’s and Zabrodsky’s, from the early 1990s, and the folding construction above is theirs. The application that made them widely used is in coordination chemistry, where the question is this six-coordinate complex an octahedron or a trigonal prism? has no yes-or-no answer for any real structure and a perfectly good continuous one — a measure against each ideal shape, with the smaller winning.

What is different here is only the framing. The argument here arrives at the measure from the other end: not from wanting to classify a shape, but from finding that a point-group search reports a name whose meaning depends on a number nobody thinks about. The measure is what that number should have been a threshold on.

What this changes about the tolerance sweep #

Nothing in the tolerance sweep was wrong, and one thing in it now reads differently.

That essay found that the raw count of accepted operations rises monotonically with the tolerance while the symbol does not — the search finds more and calls it something erratic. The measure explains the first half completely: an operation is accepted when its largest miss is below the tolerance, and raising the tolerance can only accept more. The second half stays exactly as strange as it was, because a symbol is assembled from counts of perpendicular axes and vertical planes, and a structure part way between two symmetries has some of each.

One half of that picture is at least monotone: what the search finds can only grow as the test weakens, so a group once found is never lost by loosening. That is a guarantee about the procedure and not about the answer, and it is the reason a threshold on the displacement can be defended at all. The measure is the quantity that would let such a threshold be set to mean something — a fixed amount of asymmetry rather than a fixed distance.

So the practical recommendation the measure makes possible, and the sweep alone could not: report the measure beside the symbol. A structure described as D6h with a measure of 0.006 and one described as D6h with a measure of 0.14 are different claims, and at present they are written identically.

Still open: the measure as a sum over species #

The decomposition named above is the obvious open question and it connects two fields. A distortion is a displacement vector on the atoms, and a displacement vector can be resolved into the symmetry species of the ideal group — which is exactly what the vibrational analysis does when it sorts modes into species. The totally symmetric part of a distortion changes nothing about the group; every other part is what the measure is measuring.

That would make the measure a sum over species, and would say not only how far a structure is from its group but along which coordinates it has left — which is the same information a spectroscopist gets from which forbidden bands have appeared, arrived at from the geometry instead.