Half the distance from a coin

Pricing a distortion by its symmetry species works when the species appears once, and it appears more than once for 71.4 per cent of the vibrations of seventeen molecules. That essay measured the ambiguity and every essay since has carried it as a sentence: where a species is not unique, only a projection onto the normal coordinates says which vibration a distortion goes into, and a projection needs a force field.

Five molecules have one. With every coordinate set now either complete or incomplete for a reason that has been named, nothing in the counting underneath either half of the comparison is still in doubt, and the comparison can be made.

It is worth about half the distance from a coin to a decision.

What the two halves are #

Each of a molecule’s internal coordinates is used as a distortion. The B matrix’s row for a coordinate is the Cartesian direction that changes it fastest, so stretch this bond becomes a definite displacement field, normalised to unit length.

Two answers are then computed for that field, from different objects.

The coarse one decomposes it into the symmetry species of the molecule’s group. It needs the geometry and the group and nothing else — no masses, no force constants.

The projection takes its overlap with each normal mode, in the mass-weighted metric where the modes are orthogonal. It needs the force field.

The measurement is how much the first determines the second. Having named a species that appears twice, the projection distributes the distortion between that species’ two occurrences, and the coarse test cannot choose. So the number to report is the larger of those two shares as a fraction of their sum: one means the label picks a vibration, one half means it has said nothing.

The check that the two halves are commensurable is a distortion built from a single normal mode. Its species is that mode’s species, and the projection must put the whole of that species’ share on that mode’s own occurrence — so the concentration has to be exactly one. It is, on every mode of ammonia, which is what says the two calculations are describing the same object. The same identity is the one the earlier species-resolution work rests on: the two halves are computed from different things, and a case where they must agree is the only way to know they are aligned.

Degeneracy is not ambiguity #

One distinction has to be made before any number means anything, and getting it wrong changes the answer by a fifth.

A species appearing twice is an ambiguity: there are two different vibrations and the label cannot say which. A species that is degenerate is not: its partners are indistinguishable, and a distortion spread evenly across a degenerate pair has not been left undecided about anything — it has picked that set.

So the shares are summed within each occurrence first, and only then compared between occurrences. Methane’s $T_2$ appears twice with three modes each, so reading the six modes individually gives a chance level of one sixth and a concentration of 0.522; reading the two occurrences gives a chance level of a half and a concentration of 0.746. The second is the one that answers the question, and the first reads as a much worse result than the model deserves.

Every ambiguous case here has exactly two occurrences, so the chance level is one half throughout and the numbers are directly comparable. That is a property of these five molecules rather than of the method: a group with more vibrations than species has to repeat some of them more than twice, and none of the five is large enough to. It is convenient and it is also the sample’s main limitation, since a two-way choice is the easiest one for a label to get half right by accident.

The measurement #

Twenty-four of the twenty-nine internal coordinates across the five molecules fall in a species that appears more than once — so the caveat applies to nearly everything the coarse test is used on.

Across those twenty-four the projection puts an average of 0.761 of the named species’ share on one of its two occurrences. Six of them exceed 0.9 and are settled. Ten are below 0.6 and are not.

That is a more useful answer than either of the extremes the caveat allowed. The label is not decorative: 0.761 against 0.500 is half the available distance, and a reader told this distortion is E′ has learned something about which vibration it excites. It is also not a resolution: on ten of twenty-four the two candidates are within a fifth of each other, and on three of those they are within three per cent.

Which of those two readings matters depends on what the label is being used for, and this argument has used it for two different things. Resolving a measured distortion into species to say which coordinate a molecule fell down by needs the label to identify a direction, and half the distance to a decision is not enough. Pricing a distortion to compare two of them needs only that the label pick the right cost, and since the two occurrences of a species have different frequencies, a 0.76 split gives a cost that is mostly the right one. The same number is adequate for the second use and not for the first.

The floor is worth naming because it is not exotic. Ammonia’s three H–N–H angles all sit at 0.516 — a coin to two decimals — and a bond angle is the commonest distortion there is. So the case where the label says nothing is not a contrived one; it is what happens when a molecule’s two E vibrations of the same species mix a stretch and a bend in comparable amounts.

Which molecules the label works on #

Water’s single ambiguous distortion sits at 0.876 and sulfur dioxide’s at 0.770. Boron trifluoride reaches 0.774 and ammonia 0.751, each settling three of their six. Methane reaches 0.746 and settles none of its ten.

The pattern is not about molecular size. It is about how many species a group has to spread its vibrations over: water has three vibrations across two species, so only one distortion can be ambiguous at all; methane has nine across four, so every species it has repeats and every distortion lands in one.

And methane reaching the same average as ammonia without settling anything is the reading to keep. A mean concentration of three quarters describes two quite different situations — a set of distortions most of which are decided and a few of which are not, and a set every one of which is three-quarters decided. Only the first is a usable instrument, and the mean does not distinguish them.

What it looks like when the label works #

The best case is ammonia’s N–H stretch at 0.986, which the projection assigns to the 3567 cm⁻¹ vibration. Boron trifluoride’s B–F stretch is 0.949 at 1454 cm⁻¹.

Both are stretches, and the condition they satisfy is the one to take away: the two occurrences of their species are far apart in frequency, so one of them is much more nearly a pure stretch than the other and the distortion has somewhere obvious to go. Where the two occurrences are close in frequency they mix, each becomes part stretch and part bend, and a distortion built from one internal coordinate divides between them.

So the coarse test is informative exactly when the force field would have been unnecessary — when the vibrations of a species are well separated and their composition is obvious. That is an unsatisfying result and it is the honest one: the label is reliable where nothing was in doubt.

It also explains the molecule ordering without reference to size. Water’s two $A_1$ vibrations are a bend at 1649 and a stretch at 3832 wavenumbers, which is more than a factor of two apart and about as unmixed as two vibrations of one species can be; methane’s two $T_2$ sets are a bend and a stretch separated by less, and both carry some of the other’s character. The concentration is a measure of how far a molecule is from having pure stretching and pure bending vibrations, and that mixing is what a mode composition reports — so the quantity being measured here has a name in the collection already, approached from the other side.

What the number would have to be to be useful #

It is worth asking what concentration a symmetry label would need to carry before quoting it as an assignment, because the answer is not close to 0.761.

An assignment is a claim that a distortion excites one vibration rather than another, and a claim of that kind is worth making when the alternative is unlikely rather than merely less likely. At a concentration of 0.9 the other occurrence carries a tenth, which is about the level at which a spectroscopist would ignore it; at 0.76 it carries a quarter, which is not ignorable in an intensity or in a frequency shift.

By that standard six of the twenty-four labels here are assignments and eighteen are not. The useful statistic is therefore the count and not the mean, and the count is a quarter.

There is a second reading that is kinder and is also true. A label that splits a distortion 0.76 to 0.24 has not identified a vibration and has identified a pair, with weights — and a pair with weights is exactly what a projection gives, only less precisely. So the coarse test is not a failed assignment; it is a low-resolution projection, and its resolution is now measured. Whether that is good enough is a question about the use rather than about the method, which is why this essay reports a distribution rather than a verdict.

What the comparison rests on #

Five molecules, and the sixth is linear. Carbon dioxide has a fitted field and no bond angles that this construction can use, so it is absent. Five molecules and twenty-nine coordinates is a small sample, and the finding is stated as a range across them rather than as a single number for the method.

The ambiguity counted is the one a label leaves, not the one a spectrum leaves. Two vibrations of the same species may also be too close in frequency for an instrument to separate, which is a different problem with the same symptom, and nothing here mixes them: every concentration above is computed from exact modes at their computed frequencies, however close those are.

Internal coordinates as the distortions. The alternative would be random directions, which would sample the space evenly and would not correspond to anything anybody does. A coordinate is what a chemist means by a distortion — lengthen this bond, open this angle — so the sample is the one the question is about and is not uniform over the space. It is also not independent: the three H–N–H angles of ammonia are related by its threefold axis and give the same concentration to four decimals, so twenty-four distortions are fewer than twenty-four independent measurements.

The mass-weighted metric, and it is not a detail. The normal modes are orthogonal in the metric where each Cartesian displacement is weighted by the square root of its atom’s mass, and not in plain Cartesian coordinates. A projection done without the masses double-counts the overlap between two modes and returns a number close to the right one — which is exactly the kind of error that reads as a result.

The force fields are fitted to observed frequencies, each to its own molecule, and every projection here inherits whatever they get wrong. What the comparison needs from them is the composition of the modes rather than their frequencies, and composition is the part of a force field least constrained by a fit — an earlier essay measured the directions a fit leaves undetermined and they are not small. So the concentrations carry an uncertainty this essay does not quantify.

And the species decomposition is of a distortion, not of a mode. A distortion generally has weight in several species, and the concentration above is computed inside the dominant one. How dominant it is varies: the figure of the four best cases reports it, and where the dominant species carries only half the distortion the label has already lost half the answer before the ambiguity is reached.

A caveat with a number in it is a different object #

The habit: a limitation carried in words for several essays is a measurement nobody made.

The label cannot say which vibration was true, was repeated four times, and was compatible with the label being worth nothing and with its being worth almost everything. Both readings appear in this argument’s own prose at different points, because a sentence has no number in it and the writer picks the emphasis. One comparison settles it at 0.761, which is neither reading.

The corollary is about what made the measurement possible, and it was not the projection — that apparatus has existed since the force fields did. It was knowing that both halves were counted right: the coordinate sets had to be complete, or incomplete for a stated reason, before a comparison between what they span and what the modes are could mean anything. Three essays of counting coordinates was the prerequisite, and the comparison itself is thirty lines.

What the five molecules do not cover #

The sample is five molecules and it is worth naming what kind of molecule is missing from it, because the answer bears on whether 0.761 travels.

Every one of the five has a single central atom with equivalent ligands. That is what gives them high symmetry, few species and therefore species that repeat — and it is also what makes their vibrations cleanly divisible into stretches and bends, which is the condition under which a label can be informative at all. A molecule with two different kinds of bond, or with a low-symmetry group, has more species and fewer repeats, so its labels are less often ambiguous and more often decisive when they are not.

So the five are simultaneously the hardest cases for a label and the ones where the quantity is best defined. A molecule with no symmetry has no repeated species and needs no projection; a molecule with a great deal has repeats everywhere and vibrations mixed enough that the label cannot resolve them. The interesting range is in between, and none of the five is in it.

That is the honest limit on the number. It is measured on the cases where the caveat bites hardest, and the fleet’s own habit is that a quantity measured at one end of a range should say so rather than be quoted as a property of the method.

Who compared what, and when #

Resolving a displacement into symmetry species is elementary group theory; projecting it onto normal coordinates is Wilson’s GF apparatus of the 1940s. The observation that a symmetry label cannot resolve a repeated species is as old as either and is usually left as a remark. The force fields are the fits to observed frequencies, and the internal coordinates are generated from the geometry rather than named.

What is computed here is the two resolutions for every internal coordinate of every molecule with a force field, the concentration of the projection across the occurrences of the coarsely named species, and the separation of that ambiguity from degeneracy.

The numbers worth carrying are 0.761 and 0.500 — a symmetry label’s answer and a coin’s, on the same twenty-four distortions.

Still open: the composition, and a molecule with three occurrences #

The obvious open question is whether the concentration is predictable without the force field. The pattern in the six best and six worst cases is that the label works when the two occurrences of its species are far apart in frequency, and a frequency separation is a force field’s output — but the composition of the two modes is what actually decides it, and there may be a coarser proxy. The candidate is the internal-coordinate content of each occurrence, which is already computed here as a mode composition: if the concentration tracks how differently the two occurrences are composed, then a table of compositions would predict when a symmetry label is worth quoting, and that table is cheaper than a projection.

The nearer question is a species with three occurrences. Every ambiguous case here has exactly two, so the chance level is a half throughout and the measure is a two-way split. A larger molecule — benzene, for which no force field is fitted here, or a substituted methane — would have species appearing three and four times, where a label leaves a three-way choice and the natural measure is an entropy rather than a largest share. Whether the label’s margin over chance holds up as the choice widens is the question that decides whether 0.761 is a property of the method or of five small molecules.