The coordinate it was already soft along

How far, and along which coordinate finished a sequence of three. 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 a resolution of a distorted structure’s departure from its ideal shape into the symmetry species of the ideal group, with the weights summing to one to a part in a billion and each part idempotent under re-projection.

It ended by saying where that stops being a symmetry problem.

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. 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.

This essay builds that comparison. Most of the work is in establishing that the two halves are talking about the same thing.

Two calculations that share nothing #

The comparison joins two calculations with no common input.

The species decomposition takes a molecule’s coordinates, generates its point group’s operations from them, sorts those into conjugacy classes, and projects a displacement onto each irreducible representation. Masses do not appear. Force constants do not appear. It is a statement about geometry and about a group — the same construction every group a molecule can fall to is built on.

The force field takes a molecule’s internal coordinates, builds a Hessian from a set of fitted constants, mass-weights it and diagonalises. Symmetry does not appear, except as whatever the coordinates happen to have.

Two objects computed from disjoint inputs cannot be compared unless they agree somewhere first, and there is exactly one case where they must: a distortion built from one normal mode is entirely that mode’s species.

They do agree, to better than a part in a thousand of the weight, for every molecule tested. A displacement built from methane’s softest mode comes back as T₂ and nothing else; one built from its stiffest comes back as T₂ as well, which is a complication addressed below; ammonia’s softest is A₁ and its stiffest is E, and the decomposition returns each of them.

What a unit distortion costs #

Agreeing on a label is a weak check. The strong one is the cost.

A distortion of a stated size along a stated coordinate costs a definite amount of energy, and the amount can be computed by resolving the distortion onto the normal coordinates and adding up ω²q². For a distortion built from one mode that reduces to that mode’s own frequency, and the check is that it does.

It does, for every molecule, to a part in a million: the ratio of the stiffest mode’s unit cost to the softest’s is exactly the ratio of their frequencies. Methane 2.328594, ammonia 3.496531, water 2.391394.

That agreement is not automatic and getting it required fixing something. The normal modes are orthogonal in the mass-weighted metric and not in plain Cartesian coordinates, so a projection done without the masses double-counts the overlap between modes. The first version of this calculation returned a ratio of 2.250 for methane where the frequencies say 2.329 — a four per cent discrepancy, entirely plausible as a physical result, and a dropped factor of √m.

A number that is close to the right one is the most expensive kind of wrong, and the only way to catch one is an exact answer to compare it with. Normal modes are not bond stretches is where the mass weighting first matters, and it matters for the same reason.

Two modes, one species #

Methane produces the complication that makes the comparison worth stating carefully. Its softest vibration is of species T₂ at 1362.7 cm⁻¹ and its stiffest is also T₂, at 3173.3.

So a species is not a steepness. A T₂ subspace of a tetrahedral molecule is three-dimensional and there are two of them among methane’s vibrations; a distortion of T₂ symmetry could be either, or any mixture, and its cost is anywhere between the two frequencies.

That limits what the comparison can conclude, and the limit is worth being explicit about. Finding that a distorted structure’s dominant species is the softest mode’s species is evidence that the molecule fell down its own slope, and it is not proof: the same species may contain a stiff mode, and a structure pushed along that one would carry the same label.

The remedy is available and it is a projection rather than a label. Resolving a distortion onto the normal coordinates themselves, rather than onto the species, gives its cost directly and distinguishes the two T₂ sets. That is the quantity the last figure here is built on, and the species decomposition is the coarser reading that a group alone can supply.

The rigid directions, which are not distortions #

Six of the 3N directions move a molecule without deforming it — three translations and three rotations — and a decomposition that assigned them to species would be reporting the orientation of a set of coordinates rather than anything about the structure.

That is not a hypothetical. A measured structure comes with an arbitrary orientation, and comparing it against an ideal one involves subtracting two sets of coordinates whose frames are unrelated, so the difference is dominated by rigid motion before anything is projected.

The species decomposition removes them first and reports how much of the displacement they carried. Here that is checked in the direction that can fail: translating a molecule bodily comes back as more than 99.9 per cent rigid, for all three molecules.

Where the comparison would say something #

With both halves in place, the test the decomposition asked for is a definite procedure: resolve a real structure’s departure from its ideal shape, find the dominant species, and ask whether it is the softest mode’s.

Where the two agree, the molecule is sitting somewhere along its own shallowest direction, and nothing outside it needs to be invoked. Where they disagree — where a structure is distorted along a stiff coordinate — something has paid for it, and the payment is a quantity: the cost of that distortion at that amplitude, which the force field gives.

Two of this collection’s own findings are of the first kind. The vibration that lowers the symmetry is a molecule distorting along a mode that is soft precisely because the distortion is favourable — a first-order Jahn–Teller case, where the softness and the distortion have one cause. Copper is never quite octahedral is the same thing in a d shell.

And one is of the second. A molecule whose crystal structure is distorted along a stiff coordinate has been pushed by its neighbours, and the energy that push supplies is computable from the amplitude and the frequency without knowing anything about the neighbours.

Why two independent halves are worth the trouble #

The decomposition uses only coordinates and a group; the force field uses only masses, force constants and measured frequencies. Neither can see the other’s inputs, so an agreement between them is not an identity dressed up as a result. It is the same logic as checking a numerical integral against a closed form: two routes that share nothing, arriving at one number.

That independence is also what makes the comparison useful afterwards. A decomposition tells a reader which symmetry a distortion broke; a force field tells them what breaking it cost. Neither alone can say whether a molecule was pushed or fell, and the two together can, because falling and being pushed differ in cost rather than in direction.

The same independence sets the limit. Where the force field is poor the cost is poor, and no amount of symmetry will repair it; where the ideal group is misidentified the species are wrong, and no force field will notice. Each half checks the other only where both are sound.

What a distortion does to a level scheme is the other half of why a species matters: a descent in symmetry splits a degenerate set, and which set splits is decided by the species of the distortion rather than by its size.

The three molecules, in their own terms #

The numbers are worth reading as chemistry rather than as a validation, because the ordering they give is not the one a first guess produces.

Ammonia has the widest range: 1020.1 cm⁻¹ for its softest and 3566.7 for its stiffest, a factor of 3.50. The softest is the umbrella inversion coordinate, which is totally symmetric — an A₁ mode — and which is soft because it is the coordinate ammonia goes through when it turns itself inside out. A molecule whose shallowest direction is totally symmetric is a molecule whose distortions cannot be told from a change of size by symmetry alone, which is worth knowing before a decomposition is trusted.

Water runs 1648.6 to 3942.6, a factor of 2.39, and its softest is the bend and its stiffest the antisymmetric stretch. That is the textbook ordering and it is here as a control.

Methane runs 1362.7 to 3173.3, a factor of 2.33, with both ends of species T₂. Its two A₁ and E modes sit between them.

So the softest coordinate is a bend in all three, which is the general fact that bending a molecule is cheaper than stretching it — and the size of the factor is what varies. That factor is the whole of what the force field adds to the decomposition: a species tells a reader which direction, and the ratio tells them what it cost.

The frequencies priced above are ammonia’s computed spectrum against the measured one, and a cost computed from them is as good as that agreement and no better. That is the honest bound on everything here.

Where a species appears twice in one molecule’s list, a distortion of that species has no single cost — which is methane’s complication and the reason the softness above is quoted per species rather than per mode.

A point group depends on a tolerance, with four distorted benzenes admitted or refused by different ones. The whole of the argument since has been an attempt to replace that threshold with a quantity.

Pricing the distortion a crystal imposes #

The agreement between the two halves is a calibration, and what it licenses is the measurement the whole comparison was built for: taking a real distorted structure and saying what its distortion cost.

The procedure is now complete. Project the displacement onto the normal modes; the projection gives an amplitude in each; the force field gives a price per unit amplitude; and the product, summed, is the energy the distortion took. Every ingredient has been checked against the other on the case where both apply.

The natural application is a molecule in a crystal. The same compound in two polymorphs has two slightly different molecular geometries, and the difference between them is a displacement that the packing imposed — so decomposing it says which coordinates the crystal pushed along and how hard.

The answer is nearly always the same in shape and it follows from the arithmetic rather than from any survey. Packing forces are weak — a few kilojoules a mole, against vibrational quanta of tens — so the energy available to distort a molecule is small, and it buys a large amplitude only in a mode whose price is low.

So a molecule in a crystal is distorted almost entirely along its softest modes: torsions, low-frequency bends, ring puckers. Its bond lengths, which are priced at thousands of wavenumbers, barely move at all.

That is a prediction with a testable shape, and it explains something a crystallographer sees constantly. Two polymorphs of one compound differ in their torsion angles by degrees and in their bond lengths by thousandths of an ångström, and the ratio between those two departures is not a fact about crystals — it is the ratio of two force constants, and the decomposition here computes it.

It also identifies which structural differences are worth interpreting. A torsion angle that differs between two crystal forms is a coordinate the molecule was soft along, and the crystal has said nothing about the molecule by moving it. A bond length that differs is expensive, so something has genuinely acted on it, and that is the difference worth explaining.

The same reasoning answers the essay’s own question, and the answer is a comparison rather than a category. A distortion the molecule fell down by itself is one along a coordinate whose own curvature is negative or nearly so — a mode with a very low frequency, or an imaginary one. A distortion something pushed it along is one whose mode has an ordinary frequency and which therefore cost energy that had to come from outside.

So the diagnostic is the frequency of the mode the displacement lies in, and the projection performed here supplies exactly that. A large amplitude in a soft mode is the molecule; a large amplitude in a stiff one is its surroundings — and the two are told apart by a number the force field already contains, with no further calculation and no appeal to what the surroundings happen to be.

What this cannot say #

The force fields are fitted. Every frequency here comes from a valence force field fitted to measured frequencies, so a cost computed from it is as good as that fit and no better. What is exact is the ratio of two costs to the ratio of two frequencies, because both come from the same Hessian.

The tolerance is still in there. Identifying the ideal group at all needs one, and the tolerance is a decision is where what that decision is worth gets measured.

Harmonic only. A distortion large enough to matter chemically is often large enough for the quadratic approximation to be poor, and nothing here has a cubic term.

No anharmonic softening. A mode that is soft because the potential flattens — an inversion coordinate, most obviously ammonia’s — has a harmonic frequency that overstates its stiffness. The ring that cannot hold still is where that shows up in a different subject.

Three molecules. Methane, ammonia and water are the three with force fields and clean groups; extending the comparison to a real distorted structure needs a molecule whose measured geometry differs from its ideal one, and this collection’s set of those is small.

And no cause is identified. The comparison distinguishes “fell down its own slope” from “was pushed”, and it does not say by what. Naming the pusher needs the neighbours, which is a crystal calculation not done here.

What was checked #

A distortion built from one mode is entirely that mode’s species, above 99.9 per cent of the weight, for three molecules — the identity that makes two independently computed objects comparable at all.

One built from the stiffest mode is the stiffest’s species, which is what lets the test say no as well as yes.

The cost of a unit distortion is its own frequency, so the ratio of the stiffest to the softest reproduces the frequency ratio to a part in a million — the check that the pricing is the force field’s.

And the refusal is a rigid motion: translating a molecule bodily must come back as more than 99.9 per cent rigid, or a decomposition would be reporting the orientation of a coordinate set rather than a distortion.

Still open: a measured crystal distortion, and repeated species #

The comparison is built and it has been run on constructed distortions rather than on measured ones, which is the obvious open question and needs data rather than new calculation: a molecule whose crystal geometry departs from its ideal one, resolved and priced. What would come out is an energy — how much the crystal paid to hold the molecule out of shape — and that number is comparable with a lattice energy, which puts a molecular calculation and a solid-state one on one axis. What holds a solid together is the boundary that comparison would cross.

The nearer question is the degeneracy methane exposed. When two modes share a species, a species label does not price a distortion, and the fraction of molecules for which that happens is a property of point groups rather than of chemistry: it is the question of how often an irreducible representation appears more than once in the vibrational reducible representation. That is computable for every point group, and the answer would say how often the coarse test above is enough and how often the projection onto normal coordinates is needed instead.