How many frequencies, not how many modes

“A non-linear molecule has 3N−6 vibrations” is the first thing anybody learns about vibrational spectroscopy, and it is true. It is then used as though it were a prediction about how many bands a spectrum will contain, and between the two statements sit two quite separate reductions, each of them large.

Benzene is the case that makes both visible at once. Twelve atoms, so thirty vibrations. Twenty distinct frequencies, because ten of its symmetry species are two-dimensional and each of those is one line rather than two. Eleven observable bands — four in the infrared, seven in the Raman, and not one frequency in common between them. Nine of benzene’s twenty frequencies appear in neither experiment.

The first reduction: degeneracy #

A symmetry species of dimension greater than one describes a set of modes that are related by the group’s own operations and therefore share an energy exactly. Degeneracy is a group theorem establishes the point in general: which degeneracies exist is fixed by the dimensions in the first column of the character table, before any energy is computed.

The consequence for counting is direct. Methane’s nine vibrations reduce as a₁ ⊕ e ⊕ 2t₂ — a singlet, a doubly degenerate pair and two triply degenerate triples — so nine modes give four frequencies. Sulfur hexafluoride’s fifteen give six. Benzene’s thirty give twenty. Only in a molecule with no degenerate species at all, like ethene, are the two counts the same, and ethene’s twelve modes do give twelve frequencies.

Two things are worth noticing about that collapse. It is exact: the three t₂ modes of methane are not nearly degenerate but degenerate, because a symmetry operation carries each into the others. And it is removable: substituting one hydrogen for deuterium lowers the group from Td to C₃ᵥ, the t₂ species splits into a₁ ⊕ e, and what was one band becomes two. Isotopic substitution is the standard way of confirming that a band is degenerate, and it works by removing the symmetry that made it so.

The second reduction: activity #

A mode is infrared active when its species carries one of x, y or z, and Raman active when its species carries a quadratic function — both consequences of the same theorem, which selection rules are one theorem derives once and applies five times. Nothing requires a species to carry either.

For benzene the arithmetic runs like this. Of twenty frequencies, four belong to species carrying x, y or z — one a₂u and three e₁u — and are infrared active. Seven belong to species carrying quadratic functions — two a₁g, one e₁g and four e₂g — and are Raman active. The two sets do not overlap at all, because benzene has a centre of inversion and no representation of a centrosymmetric group carries both kinds of function. That is the rule of mutual exclusion, appearing as a zero in the census rather than as a sentence.

Nine frequencies are left: one a₂g, two b₂g, two b₁u, two b₂u and two e₂u. They are silent — vibrations the molecule genuinely has, at energies that are perfectly well defined, which neither experiment can reach.

Silent modes are not rare #

The census makes the point across a set of molecules. Sulfur hexafluoride has one silent frequency of six, in species t₂u. Xenon tetrafluoride has one of seven, b₂u. Ethene has one of twelve, au. Benzene has nine of twenty.

The pattern is that silence is a symptom of symmetry. A group with many operations has many representations, most of which carry nothing; a molecule of low symmetry has few, and its species tend to carry both kinds of function, which is why water’s three frequencies are all three infrared and Raman active and none is silent.

Silent modes are not unobservable in an absolute sense. They appear in inelastic neutron scattering, which has no dipole selection rule at all; they appear in combination bands, where the product of two species can be active even when neither is; and they appear as soon as the symmetry is lowered by putting the molecule in a crystal, in solution, or next to another molecule. What they cannot do is appear as a fundamental in either of the two standard experiments — and that is a symmetry statement rather than an intensity one, so no improvement in the instrument changes it.

Where this bites #

The practical consequence is that counting bands does not count vibrations, and any argument that treats the two as interchangeable will go wrong on a symmetric molecule. That includes an argument this site makes elsewhere and has to make carefully: two structures, two spectra settles a geometry by counting bands, and it works precisely because it counts what each candidate structure predicts will be visible rather than what each has.

It also explains why vibrational spectroscopy uses two experiments rather than one. In a centrosymmetric molecule the infrared and the Raman see disjoint halves of the spectrum, so neither alone is close to a complete picture, and running both is not redundancy but necessity. For benzene the two together reach eleven frequencies of twenty, which is the most any pair of these experiments can do.

The count for a molecule with no symmetry at all #

The pattern is easiest to see from its opposite end. A molecule whose group is C₁ has one operation, one irreducible representation, and every mode in it — so the three counts coincide and every vibration is active in everything.

That is the trade symmetry makes, stated as sharply as it can be. A symmetric molecule has a simpler spectrum and the simplicity is a loss of information: fewer lines, but fewer than the molecule has, with the missing ones missing by theorem. An unsymmetrical molecule shows everything and the spectrum is correspondingly harder to read.

It also explains a standard experimental move. Putting a molecule into a low-symmetry environment — a crystal site, a solvent cage, a matrix — lowers its effective group and lets forbidden bands appear weakly. What is being bought is exactly the information symmetry removed, and the price is that every band shifts a little.

Counting the count: where 3N−6 itself comes from #

The first of the three numbers deserves its own paragraph, because it is the only one that is pure geometry.

A molecule of N atoms has 3N Cartesian coordinates. Six combinations of them — three translations and three rotations — move the molecule without deforming it, and the rest deform it. That subtraction is done here as a subtraction of representations rather than of numbers, which is what makes the species come out as well as the count.

For a linear molecule the count is 3N−5, and the missing rotation is worth naming precisely: it is rotation about the molecular axis, which moves no atom at all, so it was never a motion of the molecule. An infinite group, worked in a finite one computes that case, where the subtraction has to be done in a finite subgroup and the count checked against 3N−5 there.

The independent check on all of this is the one normal modes are not bond stretches uses: diagonalise a mass-weighted Hessian and count the eigenvalues that come out at zero. Six for a non-linear molecule, five for a linear one, at about one part in ten million of the largest — a completely different route to the same number.

Three counts, and which one a number refers to #

A number of vibrations, quoted without qualification, may mean any of three things, and this site distinguishes them everywhere:

3N−6 modes. The dimension of the vibrational space. Exact, geometric, and equal to the number of independent ways the molecule can deform.

Distinct frequencies. How many different energies those modes have, which is the mode count with each degenerate set counted once. Smaller than 3N−6 for any molecule with a degenerate species.

Observable bands. How many of those frequencies appear as a fundamental in a given experiment. Smaller again, and different for the infrared and the Raman.

For water all three are three. For methane they are 9, 4 and 2 in the infrared. For benzene they are 30, 20 and 4. The gap grows with symmetry, which is the opposite of the intuition that a symmetric molecule ought to have a simpler spectrum in a way that makes it easier to analyse. It has a simpler spectrum and the simplicity is a loss of information.

Counted across six molecules, the two numbers separate for two independent reasons and the separation is not a fixed fraction of either. That is the practical point: a spectrum’s band count cannot be predicted from 3N−6 without the group, and with the group it needs no force field at all.

Where the three counts are used #

The distinction is not academic bookkeeping; each of the three numbers is the right one for a different question, and using the wrong one is a real error rather than an imprecision.

3N−6 is the right count for a thermodynamic sum. A molecule’s vibrational contribution to its heat capacity, entropy or zero-point energy is a sum over modes, and a degenerate pair contributes twice. Using the frequency count there underestimates methane’s zero-point energy by more than a third.

The frequency count is the right one for a force-field fit. Fitting a model to a spectrum compares computed frequencies with observed ones, and the observed list has one entry per line. The force field is not in the spectrum counts constants against measurements, and the measurements are lines.

The observable count is the right one for a structural argument. Comparing two candidate structures means comparing what each predicts will be seen, which is the third number and not the first — and the two candidates may differ in the third while agreeing in the first, as planar and pyramidal XY₃ do.

Getting these confused produces the standard error of expecting a symmetric molecule’s spectrum to be busy. Sulfur hexafluoride has fifteen vibrations and shows two infrared bands. Nothing has gone wrong: thirteen of the fifteen are either degenerate partners of the two or forbidden.

Two more consequences worth naming before the limits.

A spectrum with more lines than 3N−6 is not a contradiction. Overtones, combination bands and hot bands all appear, and their selection rules are the same theorem applied to products of species — which is a weaker restriction, so they can appear where fundamentals cannot.

A spectrum with fewer lines than expected is the normal case for a symmetric molecule, and the shortfall is computable rather than mysterious. Benzene at eleven of thirty is the extreme here.

What this count cannot tell #

It says nothing about intensity. An allowed band can be too weak to see, and frequently is: the count here is of what symmetry permits, and a permitted band with a small dipole derivative is absent from a real spectrum for reasons this counting does not model. That asymmetry — forbidden is exact, allowed is permission — is the subject of what an absence proves.

It counts fundamentals only. A real spectrum contains overtones and combination bands, whose selection rules are the same theorem applied to a product of species and which are therefore less restrictive: benzene’s silent b₂g modes appear in combination with other modes, at the sum of two frequencies. Counting every band in a real spectrum against 3N−6 will overcount for that reason as often as it undercounts for the reasons above.

It assumes the free molecule. A molecule in a crystal has the site symmetry of its position rather than its own, which is generally lower, and modes forbidden in the free molecule become allowed. That is a standard technique for reaching silent modes and it is the same orbit-and-stabiliser arithmetic as site symmetry, and what it constrains.

The technique with no selection rules #

Nine modes invisible to both experiments is exact and it is not the end of the matter, because the invisibility is a property of how those two experiments couple to a molecule — and there is a third that couples differently and sees everything.

Infrared absorption requires the dipole moment to change during the vibration; Raman scattering requires the polarisability to. Both are properties of the electrons, both transform as something under the molecule’s group, and both therefore vanish for modes of the wrong species. That is where every selection rule here comes from.

Inelastic neutron scattering has neither requirement. A neutron scatters off nuclei, by the nuclear force, and the intensity of a vibrational feature depends on how far the nuclei move in that mode and on how strongly each nucleus scatters. There is no dipole and no polarisability anywhere in it, so there is no symmetry species to be wrong.

Every mode appears. The nine of benzene’s twenty distinct frequencies that neither optical technique can reach are ordinary features of a neutron spectrum, with intensities proportional to the amplitude of nuclear motion rather than to a derivative of an electronic property.

Two further properties of the technique follow from the same mechanism and are worth knowing.

Hydrogen dominates. The neutron scattering cross-section of a hydrogen nucleus is more than an order of magnitude larger than any other atom’s, so a neutron spectrum is essentially a spectrum of the hydrogen motions — which makes it complementary in a second way, since the modes hardest to see optically are often the ones where hydrogens move most.

And the intensities are computable. With no electronic property in the expression, the intensity of a neutron band follows from the mode’s displacement vectors and the nuclei’s cross-sections — quantities a normal-mode calculation supplies directly. That is the reverse of the optical case, where intensities are the part a calculation gets worst.

What it costs is a reactor or a spallation source, which is why the technique is not the default. But the standing statement here — that nine frequencies are invisible — is a statement about two experiments rather than about the molecule, and the modes exist, have frequencies, and have been measured.

That distinction is worth keeping separate from the counting itself, because the counting is exact and the invisibility is contingent. Thirty modes and twenty distinct frequencies is a property of benzene, fixed by its symmetry, and no instrument changes it. Eleven observable is a property of infrared and Raman spectroscopy, and a different experiment has a different number.

The habit that follows is to say which count is being quoted. A molecule has a number of modes; it shows a number of bands; and the second depends on what it is being shown to. Most of the confusion this essay is written against comes from a single word doing both jobs.

There is one more count between the two and it belongs to neither. The number of frequencies a molecule has is not the number of numbers a spectroscopist has to determine, because a mode of a degenerate species has one frequency and several partners, and the spectrum reports the frequency once. Benzene’s thirty modes collapsing to twenty frequencies is that collapse, and it is why a force field for a symmetric molecule has fewer independent constants than its atom count suggests — a point the normal-mode calculation makes from the other end.

Who counted first #

The reduction of the Cartesian displacements and the subtraction of translations and rotations is standard by the 1930s; Placzek’s 1934 treatment of Raman scattering supplied the polarisability half of the activity rules and is where the systematic comparison of infrared and Raman counts begins. The interest at the time was structural: the count of coincidences between the two spectra was one of the few ways available of deciding whether a molecule had a centre of inversion, and it settled the geometry of several molecules before diffraction could.

That use is still the sharpest thing this arithmetic does, and it is the use worth following next.

Where to read on #

Counting is only interesting if the count discriminates. Two structures, two spectra puts two candidate geometries for the same formula side by side and shows that they predict different numbers of bands, different numbers of coincidences and different polarisation behaviour — so a spectrum settles the structure by counting alone, with no force field and no assignment required. After it, what an absence proves takes the asymmetry between forbidden and merely-absent seriously, and a spectrum counts environments, not atoms applies the same counting to a quite different experiment.