Two structures, two spectra

Most symmetry arguments run forwards: a structure is given and its consequences are computed. This essay runs it backwards, which is what a spectroscopist actually does. Two candidate structures for the same formula are proposed, each is asked what spectrum it would give, and the two answers are different enough that a measurement chooses between them.

The remarkable part is how little the measurement has to be. No frequency needs to be assigned. No force constant needs to be known. Nothing needs to be calibrated. The bands need only be counted, and the two structures predict different counts.

Linear against bent #

A triatomic XY₂ has two arrangements available. Linear, with the group D∞h if the two ligands are identical; bent, with C₂ᵥ. The counting runs as follows and every step of it is computed rather than quoted.

The number of modes differs before anything else does. A linear molecule has only two rotations — turning it about its own axis moves no atom — so it has 3N−5 vibrations rather than 3N−6. For N = 3 that is four against three.

Degeneracy then collapses the linear count. Its bending mode occurs twice, in two perpendicular planes, and the two are carried into each other by the molecule’s own symmetry, so they share an energy exactly. Four modes give three distinct frequencies. The bent molecule’s three modes give three frequencies, none degenerate.

Selection rules then split them differently. The linear molecule has a centre of inversion, so no representation of its group carries both a linear and a quadratic function and no mode is active in both experiments: its symmetric stretch is Raman active only, and its antisymmetric stretch and its bend are infrared active only. The bent molecule has no centre, and all three of its modes are active in both.

So the two candidates predict:

Carbon dioxide shows two infrared fundamentals and one Raman fundamental with no coincidence, and has no microwave spectrum. Sulfur dioxide shows three of each, all coincident, and has a microwave spectrum. Both structures are settled by counting.

The coincidence test is the sharp one #

Of the rows in that table, the coincidence count is the one that carries the most weight, and it is worth seeing why.

A centre of inversion divides every representation into gerade and ungerade. The dipole components x, y and z are odd under inversion, so they can only be carried by ungerade representations; products like x², xy and z² are even, so quadratic functions can only be carried by gerade ones. No representation of a centrosymmetric group carries both, so no vibration of a centrosymmetric molecule can be active in both experiments.

That is the rule of mutual exclusion, and it is a theorem about the group rather than a rule of thumb about molecules. It runs in one direction with certainty: a centre of inversion implies no coincidences. The converse — no coincidences implies a centre — is not a theorem, since a molecule could have coincidences forbidden for some other reason, but in practice the inference is safe and it was the standard way of establishing centrosymmetry before diffraction was routine.

The selection rules are one theorem essay derives the exclusion from the vanishing-integral theorem in two lines. What is being used here is its contrapositive as an experimental test, and the test is a count of how many lines two spectra have in common.

Planar against pyramidal #

The XY₃ case is the other classic, and it works by a different mechanism, since neither candidate has a centre of inversion.

The predictions:

The mode counts are identical and the observable counts are not. A planar XY₃ has one frequency — the symmetric stretch — that appears in the Raman spectrum and cannot appear in the infrared, so the two spectra have two coincidences of four frequencies. A pyramidal XY₃ has four of four.

Boron trifluoride shows three infrared fundamentals and ammonia shows four. Boron trifluoride has no dipole moment; ammonia’s is 1.47 debye. Both are decided.

The polarity half of that test needs no spectrum at all, and it is the cheapest of the three. A planar XY₃ has a threefold axis and a horizontal mirror plane, so its group is D3h and symmetry forbids a dipole — no permanent dipole, therefore no pure rotational spectrum however sensitive the instrument. A pyramidal XY₃ is C3v, which permits one along the axis, and ammonia’s 1.47 debye is it.

Three tests, and what each is worth #

The three lines of evidence used above are not equally strong and it is worth grading them.

The count of bands. Strong when the two candidates differ in the count, weak otherwise, and vulnerable in one direction: a band that is allowed may be too weak to see, so an observed count is a lower bound on an allowed count. That asymmetry is the subject of what an absence proves, and it means the inference three bands are seen, therefore three are allowed needs care where four bands are seen, therefore at least four are allowed needs none.

The count of coincidences. Strongest of the three. A coincidence is a positive observation, and observing even one rules out a centre of inversion outright. It is also the test least sensitive to intensity, since it compares two spectra of the same sample rather than comparing an intensity to zero.

The presence of a dipole. Cleanest of all when it applies, because a pure rotational spectrum is either there or it is not and the arithmetic that predicts it is exact — symmetry forbids a dipole settles the question from the group alone. It applies only where the two candidates differ in polarity, which for XY₂ and XY₃ they do.

Used together the three overdetermine the answer, which is what makes structural spectroscopy work: each is individually defeasible and their agreement is not.

A third pair: square planar against tetrahedral #

The XY₄ case is the one where the counting is most decisive, because the two candidates differ in almost every column at once.

The coincidence row is again the sharpest. A square planar XY₄ has a centre of inversion and a tetrahedral one does not, so the first can show no coincidence at all and the second shows four. One observed coincidence settles it, and no assignment or force field is needed to notice one.

There is also an absence in the square planar case that has no analogue in the tetrahedral one: its b₂u mode is silent, active in neither experiment. Seven frequencies, six observable. That is the kind of gap what an absence proves is about, and it is why the band count alone is a weaker test than the coincidence count.

Only one of the two candidates can be drawn as a spectrum here, and the reason is worth stating rather than working around.

There is no matching figure for xenon tetrafluoride because this site holds no force field for it, and the generator refuses rather than inventing one. That is the right refusal and it is also the shape of the argument: the counting above needed no force field at all, which is why it settled the structure and why a spectrum with positions in it is a stronger instrument that is harder to come by. Its atoms sort into orbits without any dynamics — all four fluorines are one orbit, since the fourfold axis carries each onto the next — so a resolving spectrum shows one fluorine environment, and a structure with two above the plane and two below would show two.

What each candidate has to be asked #

Doing this properly means asking each structure the same questions and comparing the answers, rather than asking whether one structure is consistent with the data. Four questions cover it:

How many vibrations? 3N−6, or 3N−5 if linear. This is geometry and is settled before any group is generated.

How many distinct frequencies? The mode count with each degenerate species counted once, which needs the group’s irreducible representations and their dimensions.

How many of them are active in what? Which species carry x, y, z and which carry quadratic functions, read off the same table.

Is there a dipole? Fixed by the group alone, and answerable with no spectrum at all.

Every one of those is computed here from the coordinates of the candidate, and the whole procedure needs no experimental input until the last step, where the two predicted patterns are compared with one measured one. That is what makes the argument strong: the predictions were made before the data were consulted, and they are not adjustable.

Where the argument gets harder #

Two candidates that share a point group cannot be told apart by any of this. The counts are computed from the group, so two structures with the same group predict the same counts whatever their bond lengths and angles — and settling between them needs the positions of the bands rather than their number, which needs a force field, which the force field is not in the spectrum shows a spectrum does not determine.

The other hard case is a molecule flexible enough to be several structures at once. Five sites are not alike is the standing example: phosphorus pentafluoride’s trigonal bipyramid has two inequivalent kinds of fluorine and its nuclear magnetic resonance spectrum shows one, because the molecule exchanges them faster than the measurement can resolve. A vibrational spectrum is fast enough to see the static structure and a magnetic resonance spectrum is not, which is a statement about timescales rather than about symmetry — and it is where a spectrum counts environments, not atoms begins.

It is worth remembering what the counting is a count of. Carbon dioxide’s four modes are drawn out in when a mode becomes a bond stretch: two of them are the same bend in perpendicular planes, the symmetric stretch leaves the carbon still and changes no dipole — which is exactly why it is infrared silent — and the antisymmetric stretch moves the carbon against both oxygens and creates one. Every activity in the figures above is that kind of statement about a displacement, arrived at from the group instead of from the picture.

The historical case this argument settled #

Nitrous oxide is the example worth knowing, because the answer was not the one anybody expected and the counting is what settled it.

Its formula is N₂O and two linear structures are available: the symmetric N–O–N, with a centre of inversion, and the unsymmetrical N–N–O, with none. Both are linear, both have four vibrations, and both give three distinct frequencies — so the mode counts do not separate them at all.

The coincidence count does. A centrosymmetric structure can show no band in both spectra; an unsymmetrical one shows every band in both. Nitrous oxide’s infrared and Raman spectra share their bands, so the molecule has no centre of inversion, so it is N–N–O.

That is a structural conclusion of the strongest available kind, drawn from a positive observation, before diffraction on gases was practical. The same argument run on carbon dioxide gave the opposite answer and established it as symmetric.

The unsymmetrical linear case is worked in a finite group, since C∞v has infinitely many operations — an infinite group, worked in a finite one is how hydrogen cyanide’s analysis is done — and every species that comes out of it carries both a linear and a quadratic function, so every band appears in both spectra. Carbon dioxide’s analysis, in the figures above, splits into two disjoint halves instead. That contrast is the whole of the nitrous oxide argument.

One more property of the method deserves stating: it is cheap. Every count above comes from a point group and a set of coordinates, and both are available for a candidate structure before it has been made. A spectroscopist can work out what each of several proposed structures would show, in an afternoon, and then need only enough spectrum to tell the predictions apart.

That asymmetry — expensive measurement, free prediction — is what made the technique the standard route to molecular geometry for thirty years, and it is why the arithmetic is worth having in a form that runs rather than in a table.

What the arithmetic assumes #

A free molecule at rest. In a crystal or a solution the effective symmetry is the site symmetry rather than the molecule’s own, which is generally lower, and lower symmetry means fewer forbidden bands. A rule of mutual exclusion applied to a solid can fail for that reason without anything being wrong with the theorem.

Fundamentals only. Overtones and combination bands follow the same theorem with a product of species in the middle, which is a weaker restriction, so a real spectrum contains weak features the counts above do not predict. Counting every peak rather than every fundamental will overcount.

No intensities anywhere. What is computed is what is allowed, not how strong anything is, which is the boundary stated in selection rules are one theorem.

The same table for four atoms, and for five #

The linear-against-bent case is the smallest the argument applies to, and the same counting settles the larger shape questions — which is worth setting out, because it turns one worked example into a procedure.

Three ligands round a centre. The candidates are a flat trigonal arrangement and a pyramidal one. The flat one has a mirror plane through the centre perpendicular to its axis, which separates the coordinates from their products; the pyramidal one does not. So the flat arrangement gives three infrared bands, three Raman bands and two coincidences, and the pyramidal one gives four of each with four coincidences.

Counting alone distinguishes them: four bands in each spectrum means pyramidal, three means flat, and the number of coincidences confirms it.

Four ligands round a centre. The candidates are tetrahedral and square planar. The square is centrosymmetric, so mutual exclusion applies and there are no coincidences at all; the tetrahedron has no centre and gives two. The band counts differ as well — a tetrahedron gives two infrared bands and four Raman ones, and a square gives three and three.

Again the counting is decisive, and again no frequency, assignment or force constant enters.

That is the whole procedure and it is worth stating as one:

Write down the candidate shapes. Reduce the vibrations of each in its own group. Count the infrared-active species, the Raman-active ones, and the ones that are both. Compare against the spectrum.

The comparison is between integers, so it either matches or it does not, and the answer does not depend on how well anything was measured.

What it does depend on is the bands being observed, and that is where the method’s failures live. A band too weak to see reduces the count, and a reduced count is the signature of a different shape — so a missed band does not produce an uncertain answer but a confidently wrong one.

That risk is asymmetric and it is worth knowing which way. Missing a band always makes a molecule look more symmetric than it is, because higher symmetry is what reduces counts. A structure assigned as flat on the basis of three bands where four exist is the standard failure, and the standard precaution is to look hard for a fourth before concluding.

There is a second precaution that costs nothing and catches the same error from the other side. The counts have to be consistent with each other. The three numbers — infrared bands, Raman bands, coincidences — are not independent: the coincidence count cannot exceed either of the others, and the total number of distinct frequencies has to match what the reduction predicts for the candidate shape. A set of observed counts that satisfies none of the candidates is a set with a missing band in it, and the inconsistency says so before any shape is assigned.

So the method has an internal check as well as an answer, and the check is the more useful half in practice. A spectrum that matches one candidate exactly is evidence; a spectrum that matches none is a warning that something was not seen — and the second outcome is far more informative than being told the molecule has an unexpected shape.

It is worth adding what the procedure cannot do, since the counting is otherwise so accommodating. It distinguishes candidates that have been written down, and it says nothing about a shape nobody thought of. A molecule with a distortion the candidate list did not contain will produce counts matching none of them, which the comparison catches — but the comparison reports not one of these, and finding what it is instead requires proposing the shape and reducing it.

That is the ordinary limitation of an argument by elimination, and it is why the method is at its strongest where the possibilities are genuinely few: three atoms in a line or bent, four in a plane or a pyramid, five in one of two arrangements. Those are questions with two answers, and counting integers settles a question with two answers completely.

Who settled which #

The XY₂ argument was made for carbon dioxide in the early 1930s and is one of the first structural conclusions drawn from vibrational spectra: the absence of coincidences established the centre of inversion, and therefore the linear symmetric structure, before any diffraction measurement on the gas was possible. The same reasoning settled nitrous oxide as linear but unsymmetrical — N–N–O rather than N–O–N — because it shows coincidences, which a centrosymmetric structure cannot.

Placzek’s 1934 treatment of Raman scattering gave the polarisability selection rules their modern form and made this kind of argument systematic. The technique remained the standard route to molecular geometry for gases and liquids until gas-phase electron diffraction and microwave spectroscopy matured, and it is still the fastest way to establish a centre of inversion.

Still open: what an absence can carry #

The counting arguments here treat a missing band as evidence. What an absence proves asks how much weight an absence can carry, and separates the two quite different reasons a band can fail to appear — forbidden by a theorem, or allowed and weak — with the computed silent modes as the extreme case. Beyond it lies the experiment whose counting is about atoms rather than motions: a spectrum counts environments, not atoms.