Group frequencies, and where they stop

The practical spectroscopy of organic molecules runs on group frequencies. A band near 1,700 wavenumbers is a carbonyl; one near 2,250 is a nitrile; one near 3,300, broad, is a hydroxyl. These assignments work — they have been the working method for seventy years and correlation tables fill the endpapers of textbooks — and they are in tension with everything normal modes are not bond stretches computes.

Both are true, and the boundary between them is calculable. A group frequency exists when a normal mode is nearly all one internal coordinate, and whether it is has an answer in arithmetic rather than in a table.

The criterion, and what it gives across twenty-one frequencies #

Displace a molecule along a normal mode and measure how much each internal coordinate changed. Square them, normalise, and the largest share is the answer to “is this the frequency of anything?” Set the bar at nine tenths — a mode holding 90 per cent of its motion in one coordinate deserves that coordinate’s name.

Applied to every mode of the six molecules with fitted force fields here, the result is stark:

• Water, three modes: the bend is 99.8 per cent in the H–O–H angle. Both stretches are 50.0 per cent in each of the two bonds.
• Sulfur dioxide, three modes: the bend is 99.7 per cent in the angle. The antisymmetric stretch is 50.0 in each bond; the symmetric stretch is 36.3 in each bond and 27.5 in the angle.
• Carbon dioxide, three distinct frequencies: the bend is 100 per cent in the angle. Both stretches are 50.0 per cent in each bond.
• Ammonia, four distinct frequencies: nothing localised. The umbrella is 30.3 per cent in each of three angles and 3.0 in each of three bonds; the stretches are about 26 per cent in each bond and 7 in each angle.
• Methane, four distinct frequencies: nothing localised. Every stretching mode is 25.0 per cent in each of four bonds; every bending mode 16.5 per cent in each of six angles.
• Boron trifluoride, four distinct frequencies: the out-of-plane mode is 100 per cent in its own coordinate. The rest are thirds.

Four of twenty-one. And the four have something in common that has nothing to do with force constants: each is a coordinate that no symmetry operation of the molecule carries into any other coordinate.

Why symmetry decides it #

The argument is short and it is exact, in the sense that it holds before any force constant is chosen.

A normal mode transforms as one irreducible representation of the molecule’s point group — which is not an assumption but a computed fact here, checked against the group-theoretic count for every molecule in this collection. Now suppose some symmetry operation carries coordinate a into coordinate b. Then a mode with all its amplitude in a would be carried by that operation into a quite different motion, with all its amplitude in b, and no single irreducible representation contains both. So there is no such mode.

A coordinate with a symmetry partner cannot hold a mode by itself. Water’s two O–H bonds are exchanged by its twofold axis, so its stretching modes must be the symmetric and antisymmetric combinations, half in each bond. Methane’s four bonds sit in one orbit under a group of twenty-four operations, so its stretching modes are quarters. That is the whole reason, and it is why no amount of care with the force field would produce a localised C–H stretch in methane.

The converse is what makes group frequencies work. A coordinate that is alone in its orbit — the single angle of a triatomic, boron trifluoride’s out-of-plane coordinate, the C=O bond in a ketone, the O–H in an alcohol — has no partner it could be forced to share with, and nothing in symmetry stops it holding a mode. Site symmetry, and what it constrains is where the orbit argument is set out; here it is being used as a predictor of which bands can be named.

The second condition: nothing nearby to mix with #

Being alone in its orbit is necessary and not sufficient. Two coordinates of the same symmetry species mix in proportion to how strongly they are coupled and how close their uncoupled frequencies are, and there is a clean pair of cases showing both outcomes.

Water’s symmetric stretch is 50.0 per cent in each bond and 0.0 per cent in the bend, even though the fitted field couples stretch to bend with a constant of 0.248 mdyn per ångström. Sulfur dioxide’s symmetric stretch is 36.3 per cent in each bond and 27.5 per cent in the bend — a quarter of the mode is a bending motion.

The difference is not the coupling constant, since sulfur dioxide’s fitted field has no stretch–bend constant at all. It is kinematic: at an angle of 119.5 degrees the symmetric stretch and the bend of a heavy triatomic move the central atom in ways that overlap strongly, while at water’s 104.5 degrees with a light pair they hardly overlap. And it is the frequency separation: sulfur dioxide’s two a₁ modes are at 518 and 1,168, water’s at 1,649 and 3,833. Two modes of the same species a factor of 2.3 apart mix a little; the same two with a light central atom hardly mix at all.

So a group frequency needs a coordinate alone in its orbit and no near neighbour of the same species. The carbonyl band satisfies both handsomely: the C=O bond of a ketone is unique, its frequency is far above every C–C stretch and far below every C–H stretch, and the nearest thing of the same symmetry species is hundreds of wavenumbers away.

Breaking the symmetry makes every mode a group frequency #

The prediction that follows from the orbit argument is testable in one step: destroy the symmetry, and the modes should localise. Substituting one of water’s two hydrogens does exactly that.

Three of three, against one of three for H₂O. Nothing about the bonds changed — force constants are properties of an electronic energy surface that does not know which isotope is sitting on it, and the geometry is identical. What changed is that the two O–H coordinates stopped being interchangeable, so nothing forced the modes to share.

This is also why isotopic dilution is a standard technique. A dilute solution of HOD in D₂O gives a genuine, isolated O–H stretch to look at, which H₂O cannot provide at any concentration; and the same trick applied to a C–H bond in a deuterated molecule isolates a genuine C–H oscillator. The technique is usually explained by saying it “decouples” the oscillators, which is right, and the mechanism is that it removes the symmetry that was coupling them.

Boron trifluoride, taken apart at the top of this essay, is the case in miniature. Its out-of-plane mode at 719 wavenumbers is the one motion in the molecule that comes out pure, and it has a name — the umbrella — for exactly the reason given here: there is one out-of-plane coordinate and nothing to share it with. The two degenerate pairs at 480 and 1,454 are mixtures of stretching and bending, in a ratio that no correlation table would guess.

What this means for reading a spectrum #

The working rules that come out of the arithmetic are narrower than the correlation tables and are defensible.

A unique bond gives a band that may be named. Carbonyls, nitriles, hydroxyls, isolated metal–hydrides: one such bond in the molecule, no symmetry partner, no near neighbour of the same species. These are the assignments that hold up.

A symmetric group of bonds gives a set of bands that may be named collectively. A methylene group’s two C–H bonds are exchanged by a mirror plane, so its stretching modes are the symmetric and antisymmetric combinations — which is exactly how they are labelled in practice, and the practice is right. What is not available is one band per bond.

A highly symmetric molecule gives bands that belong to nothing smaller than the molecule. Methane’s spectrum has no C–H stretch in it; it has an a₁ mode and a t₂ mode, each a quarter in each bond. Calling either “the C–H stretch of methane” is a shorthand for a motion of the whole molecule.

Every assignment is a claim about a composition, and the composition can be computed. That is the point worth carrying away. The question is never whether a correlation table has a row for the band; it is whether a mode of that molecule is nine tenths one coordinate, and answering it needs a force field but not a very good one — the compositions here are set by symmetry and kinematics, and are much less sensitive to the fitted constants than the frequencies are.

Two coordinates, and how much they mix #

The second condition can be put as arithmetic rather than as a rule of thumb, and the form it takes explains both the successes and the failures of correlation tables.

Two coordinates of the same symmetry species, with uncoupled frequencies λ₁ and λ₂ and a coupling c between them, produce modes whose mixing angle satisfies

$$\tan 2\theta = \frac{2c}{\lambda_1 - \lambda_2}.$$

Everything about group frequencies is in the denominator. Two coordinates far apart in frequency barely mix whatever their coupling, because the difference dominates; two close together mix completely for any coupling at all, because the difference vanishes and the angle goes to forty-five degrees.

That is why the C=O band is reliable: at 1,700 wavenumbers it sits in a gap, with C–C stretches below and C–H stretches far above, and there is nothing of the same species within hundreds of wavenumbers to mix with. And it is why the C–C region between 800 and 1,300 is the notorious “fingerprint” — dozens of coordinates of similar frequency, all mixing freely, giving a pattern characteristic of the whole molecule and interpretable band by band by nobody.

A fingerprint region is not a failure of the technique. It is what a correlation table looks like when the denominator is small.

What a correlation table is really recording #

Set against the arithmetic, a correlation chart turns out to be a compressed statement about compositions rather than about bonds, and reading it that way explains its shape.

The entries with narrow ranges are the localised modes. A nitrile band quoted as 2,240–2,260 is a claim that the C≡N coordinate holds essentially the whole mode in every molecule anybody has looked at — which follows from its being unique in its orbit and isolated in frequency.

The entries with wide ranges are the partly mixed ones. A C–O stretch quoted as 1,000–1,300 is a claim that the coordinate dominates a mode whose remaining character varies from molecule to molecule, and the width of the range is a measurement of how much it varies.

The entries that do not exist are the delocalised ones. No table has a row for the C–H stretch of methane, because methane’s spectrum has no C–H stretch in it — only an a₁ mode and a t₂ mode, each a quarter in each bond.

So the table is not a list of bond properties that happens to have exceptions. It is a list of modes that happen to be localised, and the criterion computed in this essay is the thing it is implicitly a list of.

What the calculation cannot settle #

The bar of nine tenths is a convention. Nothing distinguishes 89 per cent from 91, and a mode at 70 per cent in one coordinate is genuinely intermediate — its band moves when the rest of the molecule changes, which is the practical signature of imperfect localisation and the reason correlation tables give ranges rather than values.

No molecule here has a carbonyl. The argument about ketones above is made by symmetry rather than by computation, because this site’s attempt at a formaldehyde force field failed for a reason worth recording: a planar three-coordinate centre has three angles that sum to 2π, so its internal coordinate set is redundant and whole combinations of bending constants change nothing observable. Seven free constants reproduced all six frequencies with a C=O constant of 22 mdyn per ångström and the two C–H stretches assigned the wrong way round. The molecule stays in the library for its symmetry, and the frequency argument is made where the arithmetic is sound.

Anharmonicity is absent, and it matters more here than elsewhere. A real X–H stretch is 3 to 5 per cent below its harmonic value, and the shift depends on the mode. Every number in this essay is harmonic and every comparison is between harmonic quantities.

The other reason a group frequency exists #

The computation here finds that the four coordinates behaving as group frequencies are the four that symmetry leaves alone, and that is exact and is not the whole account — because group frequencies work extremely well in molecules with no symmetry at all, where the argument has nothing to say.

The second mechanism is a separation of frequencies rather than of symmetries.

Two oscillators coupled by an off-diagonal force constant mix by an amount that goes as the coupling divided by the difference between their frequencies. Make that difference large and the mixing goes to nothing however real the coupling is: the two modes stay on their own coordinates, not because nothing connects them, but because nothing they are connected to is at the right frequency to respond.

The numbers make it a good approximation rather than a marginal one. Off-diagonal force constants in an organic molecule correspond to couplings of tens of wavenumbers. A carbonyl stretch sits near 1,700 and the nearest skeletal stretches near 1,000, so the separation is several hundred — an order of magnitude larger than the coupling, and the mixing is a per cent or two.

That is why the group frequency tables work at all. Almost none of the molecules they are applied to have a symmetry element anywhere near the group in question, and the isolation is entirely a matter of the group’s frequency lying in a gap.

It also says exactly which groups get tables and which do not, and the pattern in a tabulation confirms it. The reliable group frequencies belong to unusually stiff or unusually light oscillators — O–H and N–H above 3,000, C≡N and C≡C near 2,200, C=O near 1,700 — all of them far above the crowd of skeletal C–C stretches and bends between 800 and 1,400. Inside that crowd there are no group frequencies worth tabulating, and the region is called the fingerprint region precisely because every band in it is a mixture peculiar to the molecule.

So the two mechanisms divide the ground between them. Symmetry isolation is exact and rare; frequency isolation is approximate and common, and the practical usefulness of the whole idea rests on the second — which is why the tables are quoted with a range of twenty wavenumbers rather than as an identity.

Who this is due to #

The systematic use of group frequencies dates from the 1940s and 1950s, when infrared instruments became routine and correlation charts started appearing — Colthup’s chart of 1950 is the standard ancestor. The theoretical account of when they should be expected is Wilson’s, in the same apparatus normal modes are not bond stretches rests on, and the potential-energy distribution — the composition computed here in its simplest honest form — was introduced by Morino and Kuchitsu in 1952 precisely to answer this question quantitatively.

The symmetry half of the argument is older than any of it. That a mode must transform as an irreducible representation is a consequence of the molecule’s group commuting with its Hamiltonian, which is the same theorem behind degeneracy is a group theorem and behind every selection rule.

Where to read on #

The argument that began with a molecule’s modes ends here, with when one of them can be named. The rest of the spectroscopy turns the calculation around and uses spectra as evidence rather than as objects: how many frequencies, not how many modes counts what a spectrum can show at all, two structures, two spectra settles a geometry by counting bands, and the rotational spectrum is a moment of inertia begins the one measurement in this subject that returns a structure outright.