Concept

The rule of mutual exclusion — where it appears

The rule that no vibration of a centrosymmetric molecule is both infrared and Raman active. It follows from parity, and its converse is false: a molecule can obey it without having a centre of inversion.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

6 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

How many frequencies, not how many modes

Benzene has thirty vibrations. It has twenty distinct frequencies, and of those, eleven can be seen — four in the infrared and seven in the Raman, with no band in common. The other nine are invisible to both experiments, exactly.

spectra · Spectrum

Dipole selection rules in Td. For every pair of symmetry species, whether an electric dipole transition between them is allowed and along which polarisation. An entry is allowed exactly when the triple product of representations contains the totally symmetric one.

Selection rules are one theorem

An integral over all space vanishes unless the integrand is totally symmetric. Every selection rule in spectroscopy is that sentence with a different integrand — and the rule of mutual exclusion falls out rather than being remembered.

symmetry · Representation

4 molecules, counted. How many vibrations each molecule has, how many symmetry species they fall into, and how many frequencies are infrared active, Raman active, both, or neither. Every column after the first counts frequencies rather than modes, because a degenerate pair is one line in a spectrum. The molecules with a centre of inversion are the ones with nothing in the both column — mutual exclusion as a computed count. Nothing here uses a force constant.

Two structures, two spectra

A linear XY₂ gives two infrared bands, one Raman band and no band in common. A bent XY₂ gives three of each and three in common. Counting settles the shape, without a force constant, an assignment or a single measured frequency.

spectra · Spectrum

benzene: 30 vibrations. The vibrational representation, obtained by subtracting the translations and rotations from the full set of Cartesian displacements, with the infrared and Raman activity of each species read off the same character table.

What an absence proves

A band that symmetry forbids is not weak. Its intensity is zero, exactly, by a theorem — while a band that is merely too faint to see is absent for reasons no theorem covers. The two look identical in a spectrum and support completely different conclusions.

spectra · Spectrum

Mutual exclusion across every group here. The 20 groups with tables here, each with its highest rotation order, whether it has a centre of inversion, which of its representations carry a coordinate, which carry a product of coordinates, and whether any carries both. Every group with a centre excludes, which is a theorem. 1 group without a centre excludes as well — D5h — so the rule does not run backwards, and the counterexample needs a fivefold axis.

Mutual exclusion does not prove a centre

A centrosymmetric molecule shows no band in both its infrared and its Raman spectrum. The rule is a theorem and its converse is read off as though it were part of it — but ferrocene in the gas phase has no centre of inversion and no coincidence either, and the reason is that a fivefold axis separates the coordinates from their products where a threefold or fourfold axis cannot.

spectra · Spectrum

Three quarters, exactly, for every mode that is not totally symmetric. The depolarisation ratio of every Raman-active mode of five molecules, computed from a bond-polarisability model. 18 of the 26 sit on three quarters to better than a millionth, and they are exactly the modes that are not totally symmetric: the mean polarisability derivative is a trace, a trace is invariant, and an invariant has no derivative along any other species. The polarised ones below the line are the totally symmetric modes, and where they sit is a property of the model rather than of the group.

The one intensity symmetry does fix

Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.

symmetry · Spectrum

Named alongside it

The objects these essays reach for when they reach for this one.

Character table Selection rules Irreducible representations Raman activity Vibrational modes Infrared activity Degeneracy Point group Symmetry-forbidden transitions Dipole moment Parity (g and u)Silent mode

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