Selection rules are one theorem

Which transitions are allowed, which vibrations are infrared active, which are Raman active, which orbitals may combine, which molecules may be polar. Five rules, five chapters in most textbooks, and one theorem.

An integral over all space vanishes unless its integrand is totally symmetric under every operation of the group. If any operation changes the sign of the integrand, the integral is zero — the operation maps the region of integration onto itself and pairs each contribution with its negative.

That is the whole of it. What follows is the same sentence applied five times.

The dipole case, worked #

A transition between two states has an intensity proportional to the square of the transition moment,

$$\mu_{fi} = \int \psi_f^{*}\, \hat{\mu}\, \psi_i \, d\tau,$$

an integral of a product of three things: the final state, the dipole operator, and the initial state.

The theorem says this vanishes unless the product representation contains the totally symmetric one. The dipole operator transforms as a vector — as $x$, $y$ and $z$ — and every character table records which representation carries each. So the test is: multiply the characters of the initial state, the dipole component and the final state class by class, and see whether the result reduces to include the totally symmetric representation.

That is one multiplication per column and a division by the group order. Applied to every pair of species at once, it produces the grid above.

Two things about those grids repay attention. More than half of every table is forbidden, which is what makes a spectrum interpretable at all — an unrestricted spectrum would be a continuum of lines. And nothing in producing them mentioned an orbital, an energy or a wavefunction; the input was a set of coordinates.

The Laporte rule is worth watching emerge. It is normally stated as a fact about centrosymmetric molecules and it is a two-line consequence: $x$, $y$ and $z$ are odd under inversion, so a product $g \times u \times g$ is odd and integrates to nothing, and so does $u \times u \times u$. The rule has no content beyond that parity argument, and seeing it appear as a diagonal block of dashes makes the parity visible.

The vibrational case, which needs a count first #

For vibrations the same theorem applies to the same integral, and the work is in finding out what the vibrations are.

The atoms have $3N$ ways to move. Three of those ways translate the whole molecule and three rotate it, and both sets span representations. Subtracting leaves the vibrations.

The count must come out at $3N - 6$, and it is asserted rather than assumed: a subtraction producing anything else means the characters were wrong.

Activity then follows from the same table. A mode absorbs infrared light when its representation carries one of $x$, $y$ or $z$, because those are what the dipole operator transforms as. It appears in a Raman spectrum when its representation carries a quadratic function, because Raman scattering involves the polarisability, which is a second-rank tensor.

Both conditions are read off columns that were already in the table for other reasons. Nothing extra is needed.

Mutual exclusion, which is not a separate rule #

The best demonstration that these are all one theorem is a result usually taught as a fact to memorise.

If a molecule has a centre of inversion, no vibration is active in both the infrared and the Raman spectrum.

The proof is two lines. Under inversion, $x$, $y$ and $z$ are odd; every quadratic function is even. A representation is either gerade or ungerade, so it cannot carry both a linear and a quadratic function. Therefore no mode can be both.

The converse is asserted too, and it has to be for the result to be worth anything: a molecule without a centre of inversion must have at least one mode active in both. Both directions are checked, on every molecule this site draws.

The practical use is the direction that runs backwards. A molecule whose infrared and Raman spectra share no bands has a centre of inversion, and that is a structural conclusion reached without solving a structure. It is how the linear geometry of carbon dioxide and the centrosymmetry of many square-planar complexes were settled long before diffraction was routine.

The overlap case, which is the same integral again #

The third application is the one this site met first, and setting it beside the other two makes the family resemblance unmistakable.

Whether two orbitals may combine is a question about the integral $\int \psi_a \psi_b \, d\tau$ — a product of two things rather than three, and otherwise identical. It vanishes unless the product representation contains the totally symmetric one, which for two irreducible representations happens only when they are the same one.

So the rule “only orbitals of the same symmetry species may mix” is not an extra principle. It is the theorem with two factors instead of three.

The exactly-zero essay computes the corresponding overlap integrals numerically and finds around $10^{-17}$ where this argument says zero, which is the pair of results worth having together: the symmetry argument says whether, and the quadrature says how nearly — and twelve orders of magnitude between a forbidden and an allowed case is the evidence that the two agree.

What was computed, and how #

Everything above rests on a group as a list of matrices, and that list is generated rather than looked up.

The operations found by searching the coordinates are multiplied together to closure, sorted into conjugacy classes by conjugating each by every other, and matched against the tabulated columns by kind, order and size — with geometric rules deciding the two cases where those three are not enough. Only then are characters used, and the table’s four internal consistency relations are asserted first.

The vibrational characters are counts of unmoved atoms weighted by the operation’s trace. The subtraction of translations and rotations is a subtraction of multiplicities, legitimate because the Cartesian displacement space genuinely contains both as subspaces, and checked by the $3N-6$ requirement.

Three assertions are fed deliberate errors in the site’s gate. A forbidden transition declared allowed is refused. An allowed one declared forbidden is refused. And a character constructed not to reduce to whole numbers is refused. An assertion that has never rejected anything proves nothing, and the mutual-exclusion check in particular would be worthless if it only ever confirmed.

The surprise: the rules are about the operator, not the light #

The framing that makes all of this click is not the one usually given.

A selection rule is normally introduced as a statement about photons — about angular momentum being conserved, or about the light having to “grab” the molecule somewhere. That framing is not wrong and it explains nothing about why the rules differ between infrared and Raman.

The symmetry framing explains it in one clause: the rules differ because the operators differ. Infrared absorption goes through the dipole operator, which transforms as a vector. Raman scattering goes through the polarisability, which transforms as a symmetric second-rank tensor. Two operators with different symmetry pick out different representations, and every difference between the two techniques follows.

The same observation extends without effort. A magnetic dipole transition goes through an operator transforming as a rotation — the same axial-vector character that decides whether a molecule may be chiral — which is why its selection rules invert the Laporte parity. A two-photon transition goes through a product of two vectors. Each is a different row of the same character table.

That is a genuine unification rather than a rhetorical one, and it is why “selection rules” is a single topic rather than a list.

Reading a spectrum backwards #

The rules are stated forwards — given a structure, here is what may be seen — and are used backwards, which is where they earn their place.

A vibrational spectrum is a list of frequencies with intensities, taken on a molecule whose structure is in question. The count of bands is the evidence, and the symmetry treatment — the group generated from the coordinates — supplies what each candidate structure predicts.

The classic case is a four-coordinate complex, which may be tetrahedral or square planar. Tetrahedral gives Td: nine modes as a₁ ⊕ e ⊕ 2t₂, of which the two t₂ sets are infrared active and all four species are Raman active. Square planar gives D₄ₕ, which is centrosymmetric — so mutual exclusion applies, and no band appears in both spectra.

That single distinction settles it, and settles it from counting rather than from measuring anything. Two spectra sharing bands means no centre of inversion means not square planar.

The same reasoning distinguishes the two ways a ligand can attach, the two isomers of a disubstituted octahedron — cis is C₂ᵥ and shows more infrared bands, trans is D₄ₕ and shows fewer, because higher symmetry forbids more — and, historically, whether a molecule is bent or linear. Each is a count against a count.

What makes it work is precisely that the rules deliver so little. A prediction of “these many bands, of these species, active here and not there” is an integer statement, and an integer statement can be checked against a spectrum by someone who cannot compute a single frequency.

What it costs #

Generating a group and reducing a representation costs milliseconds. What the rules cost is best stated as what they cannot deliver, and the list is short and important.

No intensities. Symmetry says an integral is zero or that it is not. A permitted transition may be arbitrarily weak, and most of a real spectrum’s interest is in relative intensities that no group can supply.

No frequencies. How many modes a molecule has is a count fixed by the group; where they fall requires force constants and masses, neither of which is a symmetry quantity.

No assignment without a model. Knowing that methane has two infrared-active species does not say which observed band is which, and matching computed species to observed bands is where the real work of vibrational spectroscopy lies. That is the same division of labour the hypervalency argument runs on: symmetry identifies the quantity the question turns on, and something else has to measure it.

So the rules are a filter rather than a prediction. They halve or better the space of possibilities and leave everything quantitative to somebody else.

Where the model stops #

Three limits, and the first is the one that makes real spectra messier than these tables.

Forbidden lines appear. A vibrating molecule is instantaneously less symmetric than its equilibrium geometry, so a mode that breaks the symmetry can lend intensity to a formally forbidden electronic transition. That is vibronic coupling, and it is why the d–d bands of octahedral complexes are visible at all despite being Laporte-forbidden — they are weak, by a factor of a hundred or so, which is exactly what “forbidden but not zero once the symmetry is broken” should look like.

The equilibrium symmetry may not be the relevant one. A flexible molecule visits conformations of different symmetry, and what a spectrum shows is an average.

The linear groups are refused here. C∞v and D∞h have infinitely many operations and the reduction formula divides by the group order, so carbon dioxide — the textbook mutual-exclusion example — cannot be treated by this machinery at all. Its selection rules are worked out by other means, and a site that quietly truncated its rotation axis at some large order would be producing plausible answers for a group the molecule does not have.

Who found it, and when #

The vanishing-integral theorem in this form belongs to Wigner, in the late 1920s, as part of the general application of group representation theory to quantum mechanics.

The individual rules are older and were empirical. Laporte’s parity rule dates from 1925 and was a spectroscopic regularity before it had an explanation. The mutual exclusion rule was stated in the 1930s from accumulated infrared and Raman data.

What group theory did was not discover them but collapse them. Rules that had been separate empirical findings in separate literatures turned out to be one theorem with different operators substituted, and the collapse is why a modern treatment can state all of them in a paragraph.

The collapse also predicted rules nobody had looked for, which is the usual sign that a unification is real rather than tidy. The selection rules for two-photon transitions were worked out from the symmetry of the operator decades before lasers made such transitions observable.

Where the ladder goes next #

The machinery is character tables and reduction.

The same theorem applied to overlap is exactly zero.

The same theorem applied to a permanent moment is symmetry forbids a dipole.

And what a position’s own symmetry constrains, rather than the molecule’s, is site symmetry.

One last consequence of collapsing five rules into one. A reader who has learned the theorem rather than the rules can derive a rule for a case nobody has tabulated — a new operator, an unusual group, a transition between states whose symmetry has just been worked out — by writing down a product of characters and dividing by an order. That is the practical value of a unification, and it is worth more than the tidiness.

What the pictures here cannot show. No figure on this page is a spectrum. The vibrational tables list how many modes there are and which may be seen; they contain no frequency and no intensity, because neither is a symmetry quantity. A reader looking for what a spectrum would actually look like is looking for a calculation this site does not perform.