water: 3 vibrations

One of the figures on representations: Character tables generated from a molecule's own operations, bases reduced in them, and what a descent in symmetry does to a level.

Ten essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

Hybridisation does not explain

The prediction against the measurement. Two bands with intensities in a 1:3 ratio is what the shape requires; the energies at which they fall — 12.7 and 23 electronvolts — are experimental and are marked as such. The single band that four equivalent bonds would give is drawn dashed, because a prediction with nothing to beat is not a prediction.

Symmetry forbids a dipole

Ethene, which has a centre of inversion, and where no mode is active in both. That is the rule of mutual exclusion, and it is not stated anywhere in the calculation that produced this figure — it comes out because a centrosymmetric group has no representation carrying both a linear and a quadratic function, so the two lists cannot overlap.

A density of states is not a spectrum

The molecular case for contrast: methane’s photoelectron spectrum has two bands because four hydrogen 1s functions span two symmetry species, and the count of bands is settled before any energy is computed. Both statements — how many, and which are allowed — become statements about the shape of a curve once the molecule has a few hundred atoms.

Character tables and reduction

Symmetry says the hydrogen orbitals span two species with degeneracies one and three, so there must be two ionisation energies in a 1:3 intensity ratio. The energies at which the bands fall are experimental and marked as such; the split is not. The single band four equivalent bonds would give is drawn dashed.

How many frequencies, not how many modes

Methane’s nine vibrations sorted into species, which is where the four distinct frequencies come from. One a₁, two e and six t₂ — and the six t₂ modes are two sets of three at two frequencies, so nine modes give four numbers and the arithmetic that produces the four is a reduction rather than a diagonalisation.

Benzene’s thirty vibrations, by species, with the activity of each read off the same character table the reduction used. Ten of the eighteen species present carry neither a linear nor a quadratic function, and the modes in them cannot appear in an infrared or a Raman spectrum at all.

Sulfur hexafluoride, where the pattern is at its sharpest. Fifteen vibrations, six frequencies, two infrared active, three Raman active and one — the t₂u triple — visible to neither, with all three counts fixed by the group before any force constant is mentioned.

Selection rules are one theorem

The same theorem in Oh, applied to vibrations rather than to electronic states. Sulfur hexafluoride’s fifteen modes sort into six species, and not one of them is active in both spectra — because the group has a centre of inversion, the dipole operator is ungerade and every quadratic is gerade, so no species can carry one of each. The electronic version of exactly that argument is the Laporte rule: every transition joining two gerade species or two ungerade ones is forbidden, for the same reason and from the same column of the same table.

Water’s three modes, obtained by subtraction. Two of a₁ symmetry and one of b₂, and all three active in both the infrared and the Raman spectrum — which is what a molecule with no centre of inversion permits.

Methane’s nine modes: a₁ ⊕ e ⊕ 2t₂. Only the two t₂ sets are infrared active — six of the nine modes — while all nine are Raman active. So an infrared spectrum of methane shows two bands where a naive count would expect nine, and the missing seven are missing for a reason that has nothing to do with their being weak.

What an absence proves

Benzene’s thirty vibrations by species, with the activity of each read off the character table. Nine of its twenty distinct frequencies belong to species carrying neither a dipole component nor a quadratic function. Those vibrations exist and cannot appear as fundamentals in either standard experiment, at any sensitivity whatever.

Sulfur hexafluoride’s fifteen vibrations, sorted into species with each species’ activity read off the group rather than off a spectrum. Fifteen modes give six frequencies, and one of the six — the t₂u — carries neither a dipole component nor a quadratic function, so it is invisible in both experiments at once. That is not a weak band. It is a frequency the molecule has and no instrument of this kind can reach.

Ethene at the other extreme of the same table, where the symmetry is low enough that almost nothing is lost: twelve modes, twelve frequencies, D2h with no degenerate representation at all — and still one silent species. The count of silent modes rises with symmetry, and the useful form of that is the converse: a molecule with a silent mode has a symmetry high enough to produce one, which is a structural fact read off an absence.

Degeneracy is a group theorem

The prediction and the measurement side by side. Two bands are required by symmetry and two are observed; their positions are quoted from experiment because nothing here computes an energy. A single band would have refuted the tetrahedral geometry, which is what makes this a test rather than an illustration.

Why a d–d band is weak

The fifteen vibrations of an octahedral ML₆ complex, reduced in its point group: A1g ⊕ Eg ⊕ T2g ⊕ 2T1u ⊕ T2u. Three of the five species are even and two are odd, and the two odd ones are the ones this argument needs.

Fewer bands than electrons

The prediction and the measurement side by side for methane: two bands in a three-to-one intensity ratio, from a reduction with no energy in it, against the observed spectrum. The agreement is a check on the group theory rather than on any calculation of the energies.