What an absence proves

Two bands are missing from a spectrum. One is missing because a theorem says its intensity is zero. The other is missing because it is weak and the instrument’s noise is not. On the chart they look the same — flat baseline in both places — and the conclusions they support are not remotely the same.

This site treats that distinction as central, and has since its first essays. Exactly zero makes the same point about an overlap integral: where symmetry forbids an interaction the overlap is not small but zero, and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number. The spectroscopic version is the same theorem with a different integrand.

Why a forbidden intensity is exactly zero #

The transition moment is an integral over all space of a product of three things: the final state, the operator, and the initial state. If any operation of the molecule’s group changes the sign of that product, the integral is zero — the operation maps the region of integration onto itself and pairs every contribution with its own negative.

That is the vanishing-integral theorem, and selection rules are one theorem derives it once and applies it five times. What matters here is the character of the conclusion. It is not an estimate. No approximation has been made, no small parameter has been neglected, and improving the calculation cannot change the answer, because there is no calculation — the cancellation is exact and it is a property of the symmetry rather than of the wavefunctions.

Benzene’s lowest singlet is the worked case. Multiply the characters of the ground state, the dipole operator and the excited state class by class — a₁g × e₁u × b₂u in D6h — reduce the product, and it contains no totally symmetric part at all, so the integral is zero. Selection rules are one theorem does that multiplication and prints it. One product per column and a division by the group order settles the question, with no reference to any wavefunction anywhere.

The contrast with an allowed transition is worth stating just as plainly. Symmetry says a transition may be non-zero; it does not say by how much. Allowed is a permission and forbidden is a prohibition, and only one of the two is quantitative.

Silent modes: the extreme case #

A vibration is infrared active when its symmetry species carries x, y or z, and Raman active when the species carries a quadratic function. A species can carry neither, and then the vibration is invisible in both experiments at once.

The computed census puts the number at nine of benzene’s twenty frequencies, one of sulfur hexafluoride’s six, one of xenon tetrafluoride’s seven and one of ethene’s twelve.

The count runs the other way as the symmetry falls, and the two ends of the range are worth setting beside each other, because it is the trend rather than either number that makes an absence structural.

Put six of them in one table and the column of silent frequencies is monotone in the symmetry, with water at the bottom of it holding none.

Silent modes are not unobservable in principle, and it is worth naming the three routes to them, because each works by removing whatever the theorem depended on.

Inelastic neutron scattering has no dipole selection rule at all: a neutron scatters off nuclei rather than coupling to a dipole, so every mode appears. The theorem does not apply because the operator in the integral is different.

Combination bands. The selection rule for a transition to a state with two vibrational quanta involves the product of two species, and a product of two silent species can perfectly well be active. So benzene’s silent modes appear in combination with others, at sums of frequencies, weakly.

Lowering the symmetry. Put the molecule in a crystal and the operations available are those of its site rather than its own; the species that carried nothing may correlate onto a species that carries something. Descent in symmetry computes exactly that correlation for three descents, and the arithmetic here is the same arithmetic.

Each route works by changing the premise. None of them makes the original statement wrong.

The asymmetry, and what it costs #

Because forbidden is exact and allowed is permission, evidence from a spectrum flows more easily in one direction than the other.

Seeing a band is strong evidence. It proves the transition is allowed, which is a statement about the group, which is a statement about the structure. One observed coincidence between an infrared and a Raman spectrum rules out a centre of inversion outright — that is two structures, two spectra’s sharpest test and it rests entirely on a positive observation.

Not seeing a band is weak evidence, and how weak depends on what else is known. It is consistent with the band being forbidden, with its being allowed and weak, with its lying under another band, and with its being outside the range measured. A structural argument that reads “only three bands appear, therefore only three are allowed, therefore the molecule is bent” has the inference running the wrong way through a one-sided bound.

The practical repair is to make the argument depend on presences rather than absences wherever possible, and to state which absences it does depend on. The XY₂ argument in two structures, two spectra survives that test: the linear structure is established by the absence of coincidences, which is an absence — but it is an absence of a whole class of agreements between two independent spectra, and every one of the six bands involved is present and accounted for.

Two ways a real spectrum loses a band that is allowed #

Neither of these is a symmetry effect, and both are common enough that a working spectroscopist meets them before meeting a genuine silent mode.

A small dipole derivative. A mode’s infrared intensity depends on how much the molecular dipole changes as the mode is executed, and there is no rule requiring that to be large. A mode in a nearly non-polar molecule, or one that moves charge in two directions that nearly cancel, can be allowed and negligible. No such derivative is computed here — selection rules are one theorem sets that boundary.

Accidental degeneracy. Two frequencies can coincide without symmetry requiring it, in which case a spectrum shows one band where two are allowed. Symmetry-required degeneracy is exact and stable — degeneracy is a group theorem fixes which degeneracies a group permits — while an accidental one is a coincidence of numbers that isotopic substitution generally destroys. That destruction is the standard test: substitute, and see whether the band splits.

Carbon dioxide is where the exact case can be set against both of those. Two structures, two spectra draws its sticks: the symmetric stretch at 1,354 wavenumbers is Raman active and infrared forbidden, because the molecule’s dipole does not change at all as the two oxygens move outward together and the integral that would give it an infrared intensity vanishes identically. No improvement in an infrared instrument will find it, which is exactly what neither of the two paragraphs above can say.

The two absences of carbon dioxide, which are different absences #

One molecule shows both kinds at once, and separating them is the clearest available demonstration that the distinction is real rather than pedantic.

Carbon dioxide is linear, so its group has infinitely many operations and the analysis is done in a finite subgroup — an infinite group, worked in a finite one is how, and the activities it returns are four exact statements. Its symmetric stretch is Raman active and infrared forbidden; its antisymmetric stretch and its bend are infrared active and Raman forbidden. Between them those four mean the infrared and Raman spectra of this molecule have no band in common at all.

The first absence is the symmetric stretch missing from the infrared spectrum. As the two oxygens move outward together the molecule’s dipole moment stays at zero, by symmetry, throughout the motion — so the derivative that would give the band its intensity is identically zero. Nothing about the size of the molecule, the strength of the bond or the sensitivity of the spectrometer enters.

The second is the antisymmetric stretch missing from the Raman spectrum. Its polarisability derivative vanishes for the corresponding reason, and it vanishes just as exactly.

Both absences are theorems, and the pair of them is the rule of mutual exclusion. What makes carbon dioxide the good example is that it exhibits both, so neither absence can be attributed to sensitivity — the same molecule, the same sample, the same instrument, and a band that is present in one experiment and absent in the other.

The asymmetry, in the form a structural argument uses it #

Because a positive observation is worth more than a negative one, structural arguments should be built to lean on presences. It is worth writing out what that looks like in practice.

Strong: this spectrum shows a coincidence, so the molecule has no centre of inversion. One observed band in each of two spectra at the same frequency, and a theorem does the rest.

Strong: this spectrum shows six fundamentals, so at least six are allowed, so the structure that allows four is refused. Counting upwards from what is present.

Weak: this spectrum shows three bands, so only three are allowed. An allowed band can be too weak to see, so the observed count bounds the allowed count from below and not from above.

Weakest: this band is missing, so the molecule has that symmetry. True only if every other reason for a band to be missing has been excluded, which usually cannot be done from one spectrum.

The XY₂ argument in two structures, two spectra is built the strong way round: the linear structure is established by an absence of coincidences between two spectra, but every one of the six bands involved is present and accounted for, and the absence is of an agreement rather than of a band.

Why the zero is worth insisting on #

There is a temptation to treat “forbidden” as shorthand for “very weak”, partly because forbidden bands do appear in real spectra — a symmetry-forbidden electronic transition of benzene is responsible for its near-ultraviolet absorption, and the band is easily measured.

The resolution is that the observed band is not the forbidden transition. It is a different transition, in which the molecule changes vibrational state at the same time as its electronic state, and the product of species in that integral is not the product in the forbidden one. The vibration that lowers the symmetry works through the argument in the case of a degenerate state; the same arithmetic run for benzene’s forbidden singlet shows that a vibration of species e₂g makes the product contain a dipole component, and benzene has four such vibrations.

So the forbidden transition really does have zero intensity, and something else has the intensity that gets attributed to it. That is not a quibble: the vibronic mechanism has different selection rules, a different temperature dependence and a different polarisation from the transition it stands in for, and treating it as “the forbidden band, slightly allowed” gets all three wrong.

The same pattern runs through a whole group at once. Selection rules are one theorem prints every dipole transition an octahedral molecule allows or forbids, and the block structure of that grid is the Laporte rule with every dash in it an exact zero. It is also why the d–d absorptions of octahedral transition-metal complexes are a hundred times weaker than allowed transitions rather than absent — why a d–d band is weak is the arithmetic — and what is observed there is again a vibronic mechanism, and again a different transition from the forbidden one.

An absence that is not about symmetry at all #

There is a third kind of missing band, and it is worth separating from both of the others because its cause is neither a theorem nor a weak intensity.

A band can be missing because the mode is not there. That sounds circular and is not: two frequencies can coincide accidentally, so a spectrum shows one line where the counting predicts two, and nothing about symmetry has been violated. Degeneracy is a group theorem fixes which degeneracies a group permits; nothing forbids two independent modes from happening to land on the same number.

The two cases are told apart experimentally by substitution. A symmetry-required degeneracy is exact, so an isotopic substitution that keeps the symmetry moves both partners together and they stay coincident. An accidental one is a coincidence of numbers, and almost any substitution destroys it — the line splits.

Two structures, two spectra runs that test on methane, where the degeneracies are genuine: its four distinct frequencies move by four different ratios on deuteration and the degeneracies survive exactly — the t₂ triple stays a triple and the e pair stays a pair, because the substitution keeps the tetrahedral symmetry. An accidental coincidence in the same spectrum would have come apart.

So the full list of reasons a band is absent runs to four, and only the first is a theorem: forbidden by symmetry (exact, and no instrument will find it); allowed but weak (a small dipole derivative, and a better instrument might); overlapped (present, and hidden under something else); and accidentally coincident with another band (present, and indistinguishable from it until the symmetry is perturbed). An argument that treats a flat baseline as the first of those has excluded the other three without saying so.

What this essay does not claim #

Not every computed zero is exact. The overlaps computed in exactly zero come out at 10⁻¹⁶ rather than at 0 because they are evaluated numerically, and the distinction between “zero by symmetry, evaluated to arithmetic noise” and “small” is one to check rather than assume: a quadrature that returned 10⁻⁴ for a forbidden overlap would be reporting a broken quadrature.

The silent-mode counts assume the free molecule. They are exactly the counts for an isolated molecule in its equilibrium geometry, and every real measurement is on something else — a gas at finite pressure, a liquid, a crystal. The counts are still the right starting point and the deviations from them are informative, which is how site-symmetry effects are measured in the first place.

Nothing here is about intensity except negatively. The whole content of the essay is that zero is a special value; every non-zero value is outside what a symmetry argument computes.

How to tell the two absences apart #

The two look identical in one spectrum, and they behave differently under a change of conditions — which turns the distinction from a matter of interpretation into a measurement.

A weak band gets stronger. Increase the concentration, lengthen the path, cool the sample to sharpen the lines, use a more sensitive detector, and a band that was merely faint appears. Every one of those changes multiplies a small intensity by something and multiplies zero by nothing.

A forbidden band does not. Its intensity is zero by a theorem about the molecule’s symmetry, and no amount of sample or signal averaging changes it.

So the first test is simply to try harder, and it is conclusive in one direction: a band that appears was never forbidden.

The sharper test is the reverse and it is the one worth having, because not appearing is a negative result that could always be blamed on sensitivity. Lower the symmetry deliberately and watch what happens.

Substituting one atom with an isotope, dissolving the molecule in a solvent that interacts with it, or freezing it into a crystal at a site of lower symmetry all remove some of the operations the forbiddenness rests on. A genuinely forbidden band appears under such a change, from nothing, with an intensity that grows as the symmetry is lowered further. A band that was merely weak changes by a few per cent, as everything else in the spectrum does.

That is a positive test for a negative result, and it is the only kind that settles the question. The forbidden band’s appearance is evidence that it was forbidden — the intensity came from somewhere, and the somewhere is the symmetry that was removed.

Which gives the practical procedure two steps and an order. Try harder first, because a band that turns up costs nothing to find and settles the matter. Then break the symmetry, because a band that turns up only then was forbidden, and one that never turns up at all was neither forbidden nor weak — it was not there.

The third outcome deserves its own line because it is the one a reader is least prepared for. A band predicted by a reduction and absent under every condition means the assignment is wrong rather than the molecule: the structure assumed in the reduction is not the structure in the sample, and the count that was being tested has been refused.

So the three outcomes of the procedure answer three different questions. A band that appears when the sensitivity improves says the spectrum was incomplete. One that appears when the symmetry is lowered says the theorem was doing the work. One that never appears says the shape was wrong — and only the last is a result about the molecule.

Who drew the distinction #

The vanishing-integral argument is standard group theory by the 1930s and appears in Wigner’s 1931 book in essentially its modern form. The specific care about silent modes belongs to the vibrational literature: Placzek’s 1934 treatment established which species are Raman active, and the systematic tabulation of infrared-inactive and Raman-inactive species followed immediately, because the question “is this band missing or forbidden?” was the practical obstacle to every structural argument being made at the time.

The habit this essay argues against — reading a flat baseline as an absence of the vibration rather than an absence of its permission — is a modern one, and it comes from the same place most confusions about symmetry come from: a rule remembered without the derivation that says what kind of statement it is.

From motions to environments #

The counting so far has been of motions. Counting something else entirely finds the same theorem underneath: a spectrum counts environments, not atoms computes the orbits of a molecule’s atoms under its own group, which is what a magnetic resonance spectrum reports, and finds a molecule whose two environments give one signal for a reason that has nothing to do with symmetry at all.