A label that prices nothing

A comparison between the distortions a molecule is asked to make and the ones it is already soft along prices each distortion by its symmetry species: a displacement of species Γ is accommodated by the vibrations of species Γ, so if a molecule has one such vibration the label is a complete answer and the price is that mode’s force constant.

Methane showed where that breaks. Two of its nine vibrations share a species, so the label picks a two-dimensional space and does not say which member of it the distortion goes into — and the essay recorded the question rather than answering it: how often does that happen?

It is a question about point groups and about how a molecule’s coordinates fall into orbits, not about chemistry, and every molecule with known coordinates can be asked. Degeneracy is a group theorem and so is this: nothing about the answer depends on a force constant.

The census #

For each molecule, the point group is found from the coordinates, the Cartesian representation is reduced, the translations and rotations are subtracted, and what is left is the vibrational representation. A species appearing with a coefficient above one is a species that prices nothing.

Every molecule in the census has a repeated species, seventeen of seventeen, and the mean share of vibrations affected is 71.4 per cent.

That is a much stronger answer than the question anticipated. It asked how often the coarse test fails; the answer is that it fails somewhere in every molecule, and for most of their distortions.

The two ends of the range #

The two octahedral molecules are the best case at 40 per cent, and the reason is structural: an octahedron’s fifteen vibrations spread over five distinct species and only $T_{1u}$ repeats. The two molecules with no symmetry at all are the worst case at 100 per cent, trivially — with one species everything is in it.

Between them the pattern is not simply more symmetry is better. Ammonia, a fairly symmetric molecule, is at 100 per cent; xenon tetrafluoride, less symmetric than an octahedron, is at 44. What decides it is how many orbits of coordinates a molecule has: an orbit of equivalent bonds contributes one totally symmetric combination, so a molecule with two kinds of internal coordinate has A₁ twice and one with one kind has it once.

Methane is the instructive case here, and it connects to something measured elsewhere. Methane’s A₁ appears once, not twice, although it plainly has two kinds of internal coordinate — four bonds and six angles. The reason is that the totally symmetric combination of the six angles is the redundancy: it describes no displacement of any atom, so it is not a vibration and does not appear in the count.

The redundancy that costs methane’s force field ten unmeasurable directions is what saves its A₁ from being ambiguous. The same object, doing opposite things to two questions — which is worth holding onto, because a redundancy is normally described as a nuisance and here it is removing an ambiguity that every other molecule of its size carries.

Nine of them are not forced #

The obvious reading of a repeated species is a pigeonhole: a molecule has $3N - 6$ vibrations and its group has a limited number of species, so a large enough molecule must repeat.

That is true and it is not what is happening. A group’s species can accommodate $\sum \dim(\Gamma)$ modes before one has to appear twice — the sum of the dimensions, since a two-dimensional species holds two modes at one coefficient. Comparing:

Nine of the seventeen have room to spare and repeat anyway. For those, the repetition is a fact about how the molecule’s coordinates fall into orbits and not about how many boxes its group has — which is why the count cannot be predicted from the group alone, and has to be done molecule by molecule.

What this does to pricing by species #

Pricing by species is not wrong; it is coarse in a way that can now be quantified.

Where a species appears once, the label is a complete answer and the price is a force constant. That is 28.6 per cent of the vibrations in this census, averaged over the molecules — about two distortions in seven.

Where it appears more than once, the label picks a subspace and the projection onto normal coordinates is needed. That is the other five in seven, and it is the named repair, without any statement of how often it would be needed. It is needed most of the time.

That is the same shape of caution a mode’s percentage composition needs: a label that reads like a property of the molecule turns out to be a property of a convention, and the honest version has a number attached. The practical consequence is a rule with a number in it: a species argument about a distortion should be checked against the count of that species in the vibrational representation before it is trusted, and the check costs one reduction. It succeeds about twice in seven, and a selection rule is a different kind of statement that survives all of this intact: whether a mode is active does not depend on which member of a repeated species it is.

Methane is the worked example and it is worth carrying through. Its representation is A₁ + E + 2T₂, so a distortion of species A₁ is priced completely — there is one totally symmetric vibration and it is the symmetric stretch. A distortion of E is priced completely too. A distortion of T₂ is not: there are two T₂ vibrations, the antisymmetric stretch near 3019 cm⁻¹ and the deformation near 1306 cm⁻¹, they differ in frequency by a factor of 2.3, and the label does not say which of them a given displacement goes into. Six of methane’s nine vibrations are in that pair.

So for methane the species test is right about a third of the distortions and silent about the rest, and the two members of the ambiguous pair differ in stiffness by more than twice — which is the size of the error a species argument can make when it guesses.

What the totally symmetric species is counting #

The census keeps pointing at one mechanism and it is worth stating properly, because it explains the whole spread from 40 per cent to 100.

A molecule’s internal coordinates fall into orbits: sets of coordinates the symmetry operations permute among themselves. Methane’s four C–H bonds are one orbit; its six H–C–H angles are another. Each orbit contributes exactly one totally symmetric combination — the sum of its members, which every operation leaves alone.

So the multiplicity of A₁ in the vibrational representation is the number of orbits, less one for every redundancy, less one more if the molecule is linear, which none of these is. Methane: two orbits, one redundancy, one A₁. Water: two orbits (two bonds, one angle — the angle is its own orbit), no redundancy, two A₁. Ammonia: two orbits, no redundancy, two A₁.

That accounts for the smallest molecules immediately. A three-atom molecule has two orbits and therefore two totally symmetric vibrations out of three, which is 67 per cent ambiguous before anything else is considered — and there is no arrangement of three atoms that avoids it.

The ambiguity is not a feature of complicated molecules. It starts at water. What large molecules add is more of it, and what high symmetry buys is species enough to spread the rest of the modes over — which is why the octahedra do best without doing well.

The one thing that removes an A₁ is a redundancy, and redundancies are rare: they need a molecule whose coordinates are more numerous than its motions, which among these seventeen is methane and boron trifluoride and nothing else.

Why the fraction is a fair measure #

Two other measures were available and are worse.

Counting species rather than modes would say methane has one ambiguous species out of three, which is 33 per cent and understates it: the ambiguous species is $T_2$, which is three-fold degenerate and appears twice, so it accounts for six of the nine vibrations. What a reader cares about is how many distortions the label fails on, and that is a count of modes.

Counting molecules gives 100 per cent for every census and says nothing. It is the right answer to does this ever happen and the wrong one to how much of the time.

So the measure is the share of a molecule’s $3N - 6$ vibrations that belong to a species appearing more than once, and the check that keeps it honest is that the modes summed over species with their degeneracies come back as $3N - 6$ exactly.

What is quoted, and what is computed #

Nothing is quoted. No frequency, no force constant, no energy of any kind appears here. The molecules are coordinate sets, the point groups are found by searching those coordinates for operations and closing them under multiplication, and every reduction is the reduction formula applied to characters computed from the found operations.

The capacities are sums of the dimensions of the irreducible representations, read off the character tables this collection generates rather than from a book — and the tables are verified against four internal relations before use, as they have been since the essay that built them.

The repetition is what lets modes mix #

A species appearing more than once is presented here as a failure of a labelling scheme, and it is the precondition for something measured from the other side — because two vibrations can mix if and only if they belong to the same species.

Symmetry forbids mixing between species. Two modes of different species have exactly zero coupling, whatever the force constants are, and each is confined to its own coordinate as far as the group can arrange. Two modes of the same species have no such protection: the force field couples them, they mix, and what comes out is a pair of combinations rather than two separate motions.

So the 71.4 per cent measured here is not merely the fraction of modes whose label fails to identify them. It is the fraction of modes that are mixtures, and the remaining 28.6 per cent — the ones whose species appears once — are the modes that are pure by symmetry.

That is exactly the division the count of group frequencies found independently. The coordinates that behave as group frequencies, belonging to one internal coordinate and nothing else, are the ones symmetry leaves alone; and leaves alone means is the only member of its species. Four of twenty-one distinct frequencies, in that count; a quarter or so, in this one.

Two conclusions follow that neither count reaches alone.

A unique species is a guarantee and a repeated one is a licence. A mode alone in its species cannot mix with anything, ever, at any force field. A mode sharing its species mixes by an amount the force constants decide — which may be large or small, and which symmetry cannot bound.

And the repetition is not a defect to be designed out. A molecule with enough symmetry that every species appears once would have every mode pure and every frequency attributable to one coordinate, which would make vibrational spectroscopy far simpler and is achieved by essentially no molecule with more than a handful of atoms. The count of species grows with the group and the count of modes grows with the atoms, and the second grows faster.

So this finding and the group-frequency count are one statement counted two ways: most vibrations belong to a species that something else also belongs to, and are therefore mixtures rather than motions of one bond — which is why a label prices nothing, and why a frequency is not a bond stretch.

The nine molecules whose groups have room to spare and repeat anyway deserve a separate note, because they are the cases where the repetition is not forced. For those, the species could in principle have been distributed one per mode and are not — which says the repetition is a property of what the coordinates are rather than of how many species are available. A molecule’s internal coordinates come in orbits: sets of equivalent bonds or angles that the group permutes among themselves, and each orbit contributes a whole batch of species at once. Two orbits contributing the same species is how a repeat arises with room to spare, and two orbits is what a molecule with two kinds of bond has.

So the repetition tracks chemical variety rather than symmetry poverty. A molecule with several kinds of bond will repeat species however symmetric it is, and only a molecule with one orbit of coordinates could avoid it — which means essentially only the simplest ones.

What this cannot say #

Nothing here is about whether the modes are degenerate. A species appearing twice is two separate sets of modes with the same symmetry and generally different frequencies; a species of dimension two or three is one set of modes with the same frequency. Degeneracy is a group theorem and repetition is not — the first is fixed by the character table alone and the second needs the molecule.

A repeated species is not a failure of symmetry. It is the ordinary situation, and the reduction is exactly right: what fails is the shortcut of treating a species label as a mode label. Symmetry has not stopped settling anything; it settles which subspace, and a subspace of dimension above one needs one more step.

The census is the molecules these essays use. They were chosen, across many essays, for other reasons and are not a sample of chemistry — they are small, mostly highly symmetric, and skewed towards the cases a character table is easy to apply to. A survey over a real database would almost certainly find a higher ambiguous share, since larger molecules have more modes and no more species.

And two of the collection’s molecules are missing. Carbon dioxide, hydrogen cyanide, carbonyl sulfide and xenon difluoride are linear, their groups have infinitely many operations, and the reduction formula divides by the order — so they are handled in a finite subgroup elsewhere and are not in this count.

What was checked #

Every molecule in the census has a species that appears more than once — checked as a share of exactly one, so a single clean molecule would fail the check rather than being averaged away.

The mean ambiguous share is above a half, which is the statement that the coarse test fails on most distortions rather than on a minority of them.

For most of them the repetition is not forced by the group’s capacity, checked as a majority — the finding that separates a fact about orbits from a fact about pigeonholes.

And the species add up. Every molecule’s modes summed over species with their degeneracies must equal the number of vibrations the reduction reports, or every number above is about something else.

The capacities come from the character tables this collection generates rather than from a reference: a group’s capacity is the sum of its species’ dimensions, and it is larger for the larger groups without being proportional to their order. Two groups of the same order can have different capacities, which is one more thing a label does not price.

Still open: pricing the repair with a force field #

The obvious open question is the repair, priced. Where a species appears more than once the projection onto normal coordinates is what settles which mode a distortion goes into, and that projection needs a force field — which exists here for six molecules and not for the rest. Running the coarse test and the projection side by side on those six would say how far apart their answers are, which is the number that decides whether the shortcut is a good approximation or merely a cheap one.

The nearer question is the orbit structure the census keeps pointing at. A molecule’s totally symmetric species appears once per orbit of internal coordinates, less one for each redundancy — which is a formula, not an observation, and it is used here as an explanation without being checked. Computing the orbit count and the redundancy count for each molecule and comparing their difference against the multiplicity of A₁ would turn the explanation into an identity, and methane is already the worked example: two orbits, one redundancy, one A₁.