What a group forbids to happen

What a group does to a function turns a group of motions into a set of matrices acting on everything defined over the points it moves, and how large a degeneracy may be reads the dimensions of the irreducible pieces as the degeneracies symmetry can force. Both are statements about the levels a crystal has.

The question every spectroscopy actually asks is about the transitions between them, and it has an answer of the same kind — shorter, and with a sharper edge.

One integral, and why it is a symmetry question #

The rate at which light drives a system from one state to another is fixed by an integral over the whole of space: the final state, the operator that does the driving, and the initial state, multiplied together and integrated.

Space is carried onto itself by every operation of the crystal’s point group. So the integral is unchanged by every one of them — which means it equals the integral of the group average of the same product, and a function whose average over the group is zero integrates to zero.

That is the whole argument. The integral vanishes unless the product of the three representations contains the identity, and how many times it does is one line:

m = (1/|G|) Σ χ_f(g)* χ_op(g) χ_i(g)

A zero is a fact about the group and not about the material. No choice of composition, no temperature, no intensity of illumination makes a forbidden transition happen, because nothing in the argument mentioned any of them. That is the same asymmetry Neumann’s principle has in the counting of property tensors, and it is why symmetry arguments in this subject are so much better at saying no than at saying yes.

The operator is the vector #

The operator here is the electric dipole, which is what light at ordinary wavelengths couples to, and its representation is the one the group is already written in: the natural representation, the group’s own matrices acting on a vector.

That makes the computation completely self-contained. Nothing has to be looked up: the character of the operator is the trace of each matrix of the group, which is the quantity that names an operation, and the characters of the levels come from the table built rather than quoted.

A second operator comes for nothing. The symmetric square of the vector representation is what a two-photon process or a Raman scattering couples to, and its character is (χ(g)² + χ(g²))/2 — a formula with the group in it twice and no new input.

Laporte’s rule, derived #

The most famous selection rule in spectroscopy falls straight out of the sum, and it is worth watching it arrive rather than quoting it.

A group containing the inversion — in the plane, the half-turn, which acts as minus the identity — sorts its representations into two kinds: those whose character at the inversion is plus the dimension, and those where it is minus. Call them even and odd. The vector operator is odd, always, because inversion reverses a vector.

The character sum then contains the factor χ_f(−1)·(−1)·χ_i(−1), and if the two states have the same parity that factor is negative one times a positive number at the inversion and positive everywhere it pairs with the identity — the sum cancels exactly and the multiplicity is zero.

So a transition between two states of the same parity is forbidden, in every centrosymmetric group, for a dipole. That is Laporte’s rule, and it is the reason the d–d transitions of an octahedral transition-metal complex are weak: they are parity-forbidden, and the colour of such a compound comes from the small amount of parity mixing that vibrations supply.

The machinery here derives it rather than carrying it as a special case: in class 2 every same-parity pair comes out with multiplicity zero, and the check is over the pairs rather than over a rule.

How much each class forbids #

Running the sum over all ten plane classes gives a census, and it reads the way symmetry censuses on this site usually read.

Two features of that table are worth pausing on.

The trivial class forbids nothing, which is the control every census here needs. A group with one operation has one representation, one possible transition, and no way of forbidding it; a machine that reported a prohibition there would be reporting something about itself.

The two operators differ on most classes. In the classes with a centre they are complementary in the strict sense — what the dipole permits the quadrupole forbids and the other way about — and in the classes without one they overlap partially. That complementarity is why infrared and Raman spectroscopy are done on the same sample: between them they see transitions neither sees alone, and in a centrosymmetric crystal a band appearing in both is evidence that the centre is not there.

The second computation, and what it caught #

A character sum is one line, and one line is exactly the length at which a conjugate in the wrong place or a forgotten class weight produces a plausible wrong table. So every prohibition is computed a second way, with no characters in it.

Basis functions for each representation are projected out of monomials, their products with each dipole component are formed as polynomials, and the product’s average over the group is computed in exact whole numbers. A prohibition says that average must be zero — for every choice of functions, since the argument mentioned none. Forty-eight prohibitions across seven classes were checked and none is violated.

The permissions are a different matter, and the check found out why. Of thirty-six permitted transitions, thirty-two show a non-zero average at the lowest degree, and four do not: two in 4mm and two in 3m, each pairing a one-dimensional representation with the two-dimensional one.

Those four are not errors. The multiplicity counts invariant tensors in the abstract product of three representations, and whether a particular realisation as polynomials produces a non-zero product depends on the realisation — multiplying polynomials is not the tensor product, and it can lose exactly the component the count was about. Going to higher-degree functions recovers them.

That is permitted is not present one level down, and the sharpest version of it this collection has met. A permission is not a prediction that anything happens; it is a statement that nothing in the symmetry prevents it, and here even the functions can decline to show it while the symmetry permits it.

Reading one table #

The 4mm table repays going through cell by cell, because every zero in it has a reason that can be stated in a sentence.

The class has five representations: four one-dimensional and one of dimension two. Call the one-dimensional ones by what they do to the two mirrors and the four-fold, and the two-dimensional one E — it is the representation the vector itself carries, so a level of that kind is a doubly degenerate pair that a rotation mixes.

Every transition between two one-dimensional levels is forbidden. The product of two one-dimensional representations with the two-dimensional operator is two-dimensional, and a two-dimensional representation contains no copy of the identity: an irreducible representation contains the identity only if it is the identity. That single observation kills sixteen of the twenty-five cells at a stroke, and it is why the table is so empty.

Everything permitted goes through E, and that is the general shape of a dipole rule in a class with a degenerate level. A one-dimensional level connects to the degenerate one and to nothing else; the degenerate one connects to itself and to all four of the others. So the spectrum of such a crystal is a spectrum of transitions into and out of one kind of level.

And the diagonal is not all zero. E to E is permitted, because the product of a two-dimensional representation with itself contains every representation the group has, the identity among them. A transition from a degenerate level to itself sounds like nothing happening; what it means is that the two components of a degenerate pair can be driven into one another, which is exactly what a circularly polarised beam does.

Class 2 makes an instructive contrast, being the smallest class with a centre: two one-dimensional representations, the diagonal forbidden by parity and the off-diagonal allowed. That table is Laporte’s rule with nothing else in it.

What the rule cannot say #

The list of things a selection rule does not settle is longer than the list of things it does, and it is the same list what symmetry decides about a material sets out for property tensors.

It says nothing about how strong an allowed transition is. The multiplicity is a count of independent invariant tensors, not an amplitude; the amplitude requires the actual wavefunctions, which requires the material.

It says nothing about whether the levels exist. The classification is of the representations a group has, and a crystal need not have a state carrying every one of them at any accessible energy.

And it assumes the symmetry is exact. A crystal at a finite temperature is vibrating, and a vibration that lowers the symmetry momentarily permits what the static group forbids — which is precisely the mechanism giving parity-forbidden transitions their small but non-zero intensity. The rule remains true of the idealised crystal and the idealised crystal is not what is in the beam.

The same sum, three times over #

It is worth noticing that this essay’s computation is one this collection has now run in three different subjects with three different names, and that the sameness is not a coincidence.

Counting property components. Neumann’s principle asks how many independent components a tensor may have, and answers by averaging a character over the group — the multiplicity of the identity in the representation the tensor carries.

Counting invariant polynomials. How many invariants of each degree asks how many polynomials of a given degree the group leaves alone, and answers with the same average applied to the representation on polynomials.

Counting permitted transitions. This essay asks whether an integral can be non-zero, and answers with the same average applied to a product of three representations.

All three are the multiplicity of the trivial representation, and all three are computed by the same sum over the group. What differs is only which representation is fed to it — a tensor’s, a polynomial space’s, or a triple product’s — and that is what makes the machinery reusable rather than three machineries that happen to look alike. The site’s own code says so: the character table is built once per class and the three computations are three callers.

The reason to point at it is that the interpretations are not obviously related. One is about which numbers a material may have, one is about which functions a symmetry leaves alone, and one is about whether light can move a crystal from one state to another. Underneath, all three are asking how much of something survives being averaged.

Where the exactness stops #

Computed here: the character table of each plane class, built rather than quoted; the vector and quadrupole characters from the group’s own matrices; the multiplicity of the identity in every triple product, for every pair of representations in every class; and, for the seven classes whose characters are whole numbers, an independent check of every prohibition by exact integer arithmetic on polynomials.

Not computed: any intensity, any energy, and any statement about a real material. The three cyclic classes are excluded from the second check because their one-dimensional characters are complex, and rounding them would make the check a check of something else.

And the plane is not space. The thirty-two crystal classes have selection rules of exactly this form and there are more of them, with the extra structure that a vector in three dimensions has components that transform differently under a uniaxial group — which is what makes a transition polarised, and is the part of the subject the plane has no room for.

What a forbidden transition looks like in an experiment #

A prohibition computed on an idealised crystal meets a real spectrometer, and what happens then is worth stating because it is the difference between a rule that is useful and a rule that is merely true.

A forbidden band is not absent; it is weak. Every mechanism that breaks the idealisation gives it a small intensity: a vibration that momentarily lowers the symmetry, a defect that destroys it locally, a magnetic dipole or electric quadrupole contribution that the electric-dipole argument said nothing about. Intensities four to six orders of magnitude below an allowed band are routine, and they are measurable.

So the rule is read as a ratio rather than as a switch. An allowed transition and a forbidden one differ by a factor large enough that no ambiguity survives, and the useful statement is that a band’s strength identifies which kind it is. That is how a spectrum is assigned in practice: the strong features are matched to the permitted entries of a table like the one above, and what is left over is evidence about the mechanism that broke the symmetry.

And the exceptions are informative. A band that should be forbidden and is not weak is a statement that the crystal does not have the symmetry it was assigned — the same evidence a Bijvoet difference gives about a centre, arriving from spectroscopy instead of from diffraction. Selection rules are used as often to test an assignment as to predict a spectrum, and for the same reason a prohibition is worth more than a permission: only the prohibition can be contradicted.

Who found it, and when #

Otto Laporte stated the parity rule in 1924, before quantum mechanics had the form that explains it, as an empirical regularity in the spectrum of iron. Eugene Wigner gave the general argument in 1927 — the integral is invariant, so it vanishes unless the product contains the identity — and that argument has not changed since.

The application to crystals came with Hans Bethe’s work of 1929 on how a free ion’s levels split in a crystal field, which is the paper that made the character tables of the crystallographic point groups a working tool rather than an algebraic curiosity. Everything above is in it.

Where the ladder goes next #

Back, to the action the whole subject rests on: what a group does to a function, and the shell that splits into kinds for the cheapest character in the subject.

Sideways, to the counting that has the same shape and a different object: Neumann’s principle, as one sum counts the independent components of a property tensor by averaging exactly this kind of character, and twenty of the twenty-one is a case where the count comes out zero for a class nobody expects.

And to what happens to all of this at a wavevector rather than at a point: where two levels must meet is the same representation theory done in the little group of a point in reciprocal space, where the selection rules acquire a momentum condition on top of the symmetry one.

The group the rule is about is the static one #

A zero in these tables is absolute, and forbidden transitions are nevertheless observed all the time — weakly, but observed. The two statements are compatible, and reconciling them says exactly what the sum is a statement about.

The sum was taken over the group of the crystal at rest. Every operation in it is an operation of the equilibrium arrangement. A real crystal is never at its equilibrium arrangement: its atoms are displaced, and a displacement is a distortion that generally belongs to none of the group’s operations.

A vibration can therefore lend a transition what the group denies it. Consider a centrosymmetric crystal and a transition forbidden by parity. Excite a vibration of odd parity and, for as long as it is excited, the arrangement has no centre — so the argument that produced the zero does not apply to the distorted arrangement. The transition acquires an intensity proportional to how much of that odd vibration is present, which is why such bands are weak, why they strengthen with temperature, and why they carry vibrational structure the purely electronic transition would not have.

The bookkeeping is a product of three characters instead of two. The state is now labelled by its electronic representation times the representation of the vibration, and the sum is run on that product. A transition is vibronically allowed when some vibration the crystal actually has makes the product contain the operator’s representation.

So the rule survives intact, with its scope named. A zero forbids the transition between those states of that arrangement. It does not forbid a spectrum from showing a line there, and the intensity ratio between an allowed line and a vibronically borrowed one — typically two or three orders of magnitude — is itself a measurement of how far the crystal departs from the arrangement whose group was used.