How far, and along which coordinate

The argument about symmetry measures ended on a question rather than an answer.

It had found that a point group recovered from coordinates depends on a tolerance nobody thinks about, that the right quantity is a continuous measure of how far a structure is from a shape, and that the measure has a defect: it says how far and not which way. A structure can lose a mirror while keeping every rotation, and one number cannot say so.

The answer is a projection, and this essay builds it.

The construction #

A displacement field on $N$ atoms is a vector in a $3N$-dimensional space, and that space carries a representation of the ideal structure’s group: each operation moves an atom’s arrow to the atom the operation sends that atom to, and rotates the arrow.

Given a representation, the projection operators of the group resolve any vector into parts, one per species:

$$P_\Gamma = \frac{d_\Gamma}{h}\sum_R \chi_\Gamma(R)\, D(R)$$

and the squared lengths of the parts add to the squared length of the whole. So a distortion becomes a list of numbers, one per species, summing to one.

Three things have to be true of that figure or the construction is wrong, and all three are checked rather than assumed. The weights sum to one. Each part, projected again, is entirely itself. And a distortion built to be one species comes back as that species and nothing else.

The part that moves the molecule #

Before any of it, six directions have to be removed.

Three translations and three rotations move the structure without deforming it, and a measured structure expressed in an arbitrary frame carries some of all six — there is nothing in a list of coordinates that fixes the orientation. In the displacement used above they hold 1.14 per cent.

Leaving them in would put weight into whatever species the rotations belong to and read as a distortion. This is the same subtraction the vibrational analysis makes when it takes 3N Cartesian displacements down to 3N − 6 modes, and it is made here for the same reason.

The species that changes nothing #

The most useful thing in the decomposition is the first entry, and it is the one that explains what the measure has been doing all along.

A totally symmetric distortion is invariant under every operation of the group. A structure displaced by one is therefore still in the group: every operation still maps it onto itself. Benzene breathing — every carbon moving radially outwards, every hydrogen following — changes every bond length and no symmetry element.

Applying the totally symmetric part of the arbitrary distortion above, on its own, moves every atom by an average of 0.0200 ångström. The point group recovered from the resulting coordinates is still D6h. The continuous symmetry measure against the ideal structure is 1.3 × 10⁻²⁹ per cent, which is arithmetic noise.

So the measure is blind to the totally symmetric part by construction, and that is not a defect: a measure of how far a structure is from a shape should not count a change of size. But it means the measure is already a sum over the non-symmetric species, and the decomposition is not adding a refinement — it is showing what the measure was made of.

Which subgroup each species leaves #

The projection makes available a table that nothing else produces: apply each species as a distortion on its own, and recover the point group of the result from the coordinates.

Four features of that table are worth reading.

Every species except the first lowers the symmetry, and to a definite subgroup — a property of the species rather than of the amplitude, since scaling a distortion cannot restore an operation it broke.

Two species appear nowhere in the displacement space. B1g and A1u describe no motion of twelve atoms arranged as benzene is; the 3N representation simply does not contain them. That is checkable independently: reduce the Cartesian representation and those two have coefficient zero.

Two different species leave the same subgroup. B1u and B2u both take D6h to D3h, by different distortions — one puckering the ring, the other alternating the bonds. A subgroup does not determine the species that produced it.

And the two-dimensional species leave less symmetry than the one-dimensional ones, because a two-dimensional distortion has a direction within its own subspace and a generic direction is less symmetric than a special one.

This is the correlation table of descent in symmetry arrived at from the opposite end. There, a subgroup is chosen and the species followed down into it. Here, a species is chosen and the subgroup it leaves is found by looking at the structure.

What this makes measurable #

The decomposition turns a vague statement into a measurement, and the statement it turns is one chemists make constantly.

“The complex is distorted from octahedral” becomes a list: so much of the distortion is Eg, so much T2g, so much A1g. The first of those is a Jahn–Teller distortion, the second a trigonal twist, the third a uniform expansion that is not a distortion of shape at all — and the three have different causes and different consequences.

That is exactly the information a spectroscopist gets from the other direction. A forbidden band appears in a spectrum when the molecule’s distortion has a component of the species that makes it allowed, so the appearance of a band is a measurement of one entry in this list. The vibronic argument that makes d–d bands weakly allowed is precisely this, and the decomposition computes from the geometry what the spectrum reports from the intensities.

The question the measure left open #

The tolerance is a decision states the measure’s limit in as many words: it “says nothing about which operation is lost”, and “measuring against each operation separately says which, and that decomposition is the natural next thing to compute”.

What is computed above is not quite what that sentence proposed. Measuring against each operation separately would give a list as long as the group’s order — twenty-four numbers for benzene, most of them redundant, since operations in the same class are related by a change of frame. Measuring against each species gives twelve, they are independent, they sum to the whole, and each names a coordinate rather than an element.

So the question was answerable, and the answer improves on the question.

What the symmetry measure was all along #

Read back through the projection, the continuous symmetry measure is expressible in the same terms.

The measure folds a structure under all the operations of a group and compares the result with the original. The folding is exactly the projection onto the totally symmetric part, extended to the full coordinate vector rather than to a displacement — so the measure is the squared length of everything except the totally symmetric part, normalised.

That is why it is blind to breathing, why it is always non-negative, and why it is zero exactly when the structure has the symmetry. And it is why the decomposition is the natural next step rather than an alternative: the measure is one term of it, aggregated.

The line of questioning starts earlier, with what a point-group search finds as its tolerance is loosened. The count of operations rises monotonically and the symbol does not, because a symbol is a name for a set rather than a number — which is the defect a measure resolved into species was invented to repair.

What a crystallographer already does with this #

The construction is not new to everybody. Two fields use it routinely under names that do not obviously mean the same thing.

Structural chemistry calls it a symmetry-mode analysis, and uses it to describe a distorted crystal structure as a parent structure plus a small number of symmetry-adapted displacements. A phase transition then becomes a statement about which mode’s amplitude grew, and the amplitude is an order parameter.

Coordination chemistry calls a version of it a shape measure, and uses it to say whether a six-coordinate complex is nearer an octahedron or a trigonal prism — the question the continuous measure was built for, with a continuous answer where a symbol has none.

Both are doing the projection above, and neither usually says that the first component — the totally symmetric one — is invisible to the measure that goes with it. That omission is harmless where the comparison is between two structures of the same size and matters where it is not, because a structure that has merely expanded scores as undistorted while its bonds have all changed.

The ligand-field arguments this site makes elsewhere are downstream of exactly that: a splitting is a symmetry statement, and which splitting occurs is decided by which species the distortion is in.

Where the model stops #

The ideal structure has to be chosen. Everything here is a displacement from a reference, and the reference is a structure with the symmetry in question — which for a real molecule means deciding what shape it is nearly, and the continuous measure exists precisely because that decision has no yes-or-no answer.

The permutation is fixed by nearest image. Each atom’s displacement is measured to the ideal atom nearest it, which is right while the distortion is small and ambiguous once two atoms compete.

Nothing here is an energy. A distortion of one species may cost a great deal or nothing at all, and the decomposition says how much of it there is, not what it is worth. The relation between the two is a force field, which is a fitted thing.

And the subgroup found for each species is the generic one. A distortion of a two-dimensional species has a direction within its subspace, and special directions leave more symmetry than generic ones — the table above reports what a generic direction leaves, which is the least it can be.

The projectors are built from a table generated from the molecule’s own operations: twelve species, twelve classes, and the orthogonality relations checked rather than assumed. Every number in this essay is a sum over that table, so an error in it would show up as a decomposition whose weights did not add to the whole.

One number, twelve numbers, and what each is for #

It is worth ending the construction with the three descriptions side by side, because each is right for a different question.

A symbol — D6h, C2v — is a classification and answers what kind of thing is this. It is discrete, it is what every table is indexed by, and it depends on a tolerance.

A measure is one number and answers how far from that kind. It is continuous, it needs no threshold, and it is blind to size.

A decomposition is a list and answers along which coordinates. It is what connects a geometry to a spectrum, to a mode, and to a mechanism.

A structure reported with all three is reported completely, and the cost of the third is a projection that takes a fraction of a second. That is the practical recommendation, and it is smaller than the argument that produced it.

A measure of shape and not of size #

The totally symmetric part moving every atom by two hundredths of an ångström and lowering the symmetry measure by nothing is stated here as an observation about one distortion. It is exact and general, and the general form is the more useful statement.

A symmetry measure asks how far a structure is from having a symmetry. A totally symmetric displacement preserves every operation of the group — that is what totally symmetric means — so a structure displaced along one is still exactly as symmetric as it was, and the measure cannot move.

So the measure is blind to a whole coordinate, and it is blind by construction rather than by insensitivity. The blindness is not approximate: the contribution is zero.

Which makes the measure a description of shape and not of size. Take a molecule and expand it uniformly by ten per cent; every bond lengthens, every atom moves, and the symmetry measure is unchanged to the last digit, because a uniform expansion is the most totally symmetric displacement there is.

That has a practical consequence which runs in the measure’s favour, and it is worth extracting because it is not obvious. The commonest systematic error in a computed structure is totally symmetric. A method that gets bond lengths slightly too long gets them all slightly too long, which is an expansion — so a symmetry measure computed on a calculated geometry is insensitive to exactly the error that geometry most reliably has.

Two conclusions follow for using the two quantities together.

A symmetry measure and a bond length are independent reports. Neither constrains the other, and a structure can be reproduced perfectly in its symmetry measure and be wrong by a per cent in every distance.

And a distortion should be decomposed before it is priced. The totally symmetric part of a displacement is invisible to the symmetry measure and entirely visible in the geometry; every other part is the reverse. Reporting the total displacement mixes the two, which is why the projection is worth more than the single number it started from.

The blindness also settles a question about the projection’s arithmetic that would otherwise look like a coincidence. The twelve species’ contributions sum to one, and the totally symmetric one takes 3.4 per cent of that total while contributing nothing to the symmetry measure — so the projection and the measure are not two ways of writing the same quantity. The projection accounts for the whole displacement; the measure accounts for the part of it that lowers the symmetry. Their difference is exactly the totally symmetric share, which is why a structure can be a long way from its ideal geometry and no distance at all from its ideal symmetry.

What is quoted, and what is computed #

Nothing is quoted. The operations are generated from the coordinates by closure, the classes by conjugation, the characters matched against the tabulated table for the group — which is itself checked for orthogonality — and the projectors assembled from them. The distorted structures’ point groups are recovered by the same coordinate search that finds any molecule’s point group.

The distortion resolved in the first figure is a stated deterministic function of the atom index, so that the percentages are reproducible rather than typical.

What was checked #

The weights sum to one, to a part in a billion, which is the statement that the projectors resolve the identity.

An arbitrary distortion has components of several species, or the decomposition would have nothing to do.

Each part, resolved again, is entirely itself, to a part in a hundred million — idempotence, which fails for any error in the representation matrices.

A breathing distortion is entirely totally symmetric and has no component of anything else, which is the refusal: a decomposition that spread a pure distortion over several species would be reporting its own arithmetic.

A structure displaced by the totally symmetric part alone is still in the full group, with a symmetry measure of zero although every atom has moved by 0.02 ångström.

And a part of any other species takes the structure out of the group.

Still open: which cause drove the distortion #

The last three arguments about symmetry have been about one thing: the gap between a symbol and a structure. A point group is a label, the label depends on a tolerance, the tolerance should have been a measure, and the measure should have been a list.

The list is where it stops being a symmetry problem and becomes a chemistry one. Each entry is a coordinate along which the structure has left its ideal shape, and each has a cause — a Jahn–Teller instability, a packing force in the crystal, a substituent’s steric demand, a Peierls distortion in an extended structure. The decomposition says which coordinates are involved and in what proportion; what it cannot say is which cause is responsible.

Deciding that needs the energies, and the natural test is a comparison: the species a molecule is distorted along, against the species its softest vibration belongs to. Where they agree, the distortion is the molecule falling down its own shallowest slope. Where they do not, something outside the molecule pushed it — and that comparison is computable from the geometry and the force field together.