Descent in symmetry

A molecule almost never stays in the group it starts in. Replace one hydrogen of methane and Td becomes C₃ᵥ. Stretch an octahedral complex along one axis and Oh becomes D₄ₕ. Let benzene’s bonds alternate and D₆ₕ becomes D₂ₕ.

In each case nothing has yet happened to the electrons. What has happened is that some of the operations that justified the labels have gone, and the labels have to be reassigned.

What restricting a character means #

An irreducible representation of a group assigns a character — a number — to each class of operations. That is all a row of a character table is.

When the symmetry drops, the subgroup’s operations are a subset of the parent’s. Every one of them is still an operation of the parent, so the parent’s representation still assigns it a character, and reading off only those entries gives a character of the subgroup.

That character will not usually be irreducible in the subgroup. Reducing it — the same sum over classes, the same division by the group order — says which of the subgroup’s representations it decomposes into, and that decomposition is the correlation.

So a correlation table is not reference data. It is one reduction per representation, and every one of them can be computed in a line.

The choice that has to be made #

There is one thing in the construction that is not computed, and it is worth being exact about it because it is where the physics enters.

The subgroup’s operations have to be identified with particular operations of the parent. Descending from Td to C₃ᵥ means keeping one of the four threefold axes, and which one is a choice about which substituent was changed. Descending from Oh to D₄ₕ means choosing which of the three fourfold axes survives, which is a choice about which way the octahedron was stretched.

That choice is what “descent in symmetry” means, and stating it is the honest form of the construction. Everything after it is arithmetic. The correspondence is declared here as a map from each class of the subgroup to the class of the parent its operations came from — for Td to C₃ᵥ, the identity to the identity, the two C₃ rotations to the eight, the three σᵥ to the six σd.

A missing entry in that map is not a small omission. It would leave a character restricted using an undefined index, and the reduction would return something. So an incomplete map throws.

What it decides #

Read the figure above and the prediction is immediate: methane’s threefold bonding level splits into a pair and a single the moment one hydrogen is replaced, whatever it is replaced by and however small the difference.

That is not an approximation. There is no representation of C₃ᵥ that can hold three functions, so a threefold level cannot survive into it. The magnitude of the splitting depends on the substituent; the fact of it does not.

That is the same kind of statement as degeneracy is a group theorem makes about the parent group, run in reverse: there, the group said which degeneracies were available; here, the loss of operations says which of them can no longer be.

Chloromethane’s photoelectron spectrum duly shows the split. So does methanol’s, and fluoromethane’s, and every other monosubstituted methane. A model that had predicted three orbitals staying together would have been refuted by any of them.

The same argument in the octahedral case is the whole of ligand-field theory’s starting point. The five d orbitals of a free atom span a single fivefold level — a degeneracy no point group can hold, since none of them has a five-dimensional representation. In Oh they split into $t_{2g}$ and $e_g$, a three and a two. Stretch the octahedron and $t_{2g}$ splits again into $b_{2g}$ and $e_g$, because D₄ₕ has nothing three-dimensional either. Three levels where there was one, decided entirely by which representations exist.

Reading a descent backwards #

The construction is also useful in the other direction, and the reverse reading is the one a spectroscopist uses.

Suppose a measurement finds three levels where a symmetric structure would give two. That count is a constraint on the geometry: it rules out every group whose correlation gives two, and it is consistent with the ones that give three. The reduction is exact, so the elimination is exact, and no calculation has been done.

This is how a great deal of structural inference actually works. Counting infrared and Raman bands and asking which point group could produce that many of each is a standard determination, and the rule of mutual exclusion — a molecule with a centre of inversion has no band appearing in both spectra — is a special case of it. Selection rules are one theorem is where that method is set out.

The important discipline is that a count is evidence and not a proof. Two structures of different symmetry can happen to give the same number of bands, and a band can be too weak to see. What the symmetry gives is a clean sufficient condition for exclusion, which is a great deal more than most inferences in structural chemistry get.

The Jahn–Teller argument, in full #

The theorem is usually stated as: a non-linear molecule in a degenerate electronic state distorts to remove the degeneracy. The symmetry half of the argument is what has been built above.

Take a molecule with a partly filled degenerate level. Distorting it lowers the symmetry; the lower group cannot hold the degeneracy; the level splits; and the electrons occupy the lower piece. The energy gained is first order in the distortion and the elastic cost is second order, so for a small enough distortion the gain wins and the symmetric geometry is not a minimum.

Every step of that is decided by the group except the last comparison, which is a matter of energies. The same division appeared in which angles are symmetry and which are the model, where the arrangement of four domains is pinned by symmetry and the arrangement of seven is decided by a force law nobody states. Symmetry says whether a distortion can split the level and into what; it does not say how far the molecule moves, and it does not say which of several possible distortions is chosen.

Cyclobutadiene is this site’s worked case. Square, at D₄ₕ, its four π electrons put two into a degenerate non-bonding pair. Rectangular, at D₂ₕ, that pair must split, because every representation of D₂ₕ is one-dimensional. Delocalisation is stabilising, and other things that are false in general computes what the splitting is worth; the statement that it must happen is the one made here.

Cyclobutadiene is the standing example: a filled bottom orbital, then two degenerate non-bonding levels with one electron each. The degeneracy is required by the square ring’s symmetry and the half-filling is what makes it unstable, so the molecule leaves the group rather than keep the degeneracy — which is a descent forced by the electrons rather than chosen by an experimenter.

The descent that is not a distortion #

Not every drop in symmetry involves moving an atom, and the cases where nothing moves are the ones most easily missed.

A molecule in a crystal sits at a lattice position whose own symmetry is generally lower than the molecule’s. Its labels are then the labels of the site group, not of the free molecule, and degeneracies the free molecule has are split by the environment alone — the crystal-field splitting that started this whole construction in Bethe’s hands. Site symmetry, and what it constrains works through what a local group decides.

An applied field does the same thing without a crystal. An electric field along one axis of a tetrahedral molecule reduces Td to C₃ᵥ as surely as a substituent does, and the Stark splitting that follows is a correlation table read at low field strength.

And a molecule can descend in symmetry by being watched. A measurement fast compared with a fluxional rearrangement sees a distorted structure; one slow compared with it sees the average, which may have a higher symmetry than any instantaneous geometry. Which group is the right one is then a question about the experiment rather than about the molecule.

Every claim in this essay begins with a group recovered from coordinates rather than assigned, and methane’s is Td by that route. The descent begins by deciding which of its twenty-four operations survive a substitution, and that decision is a fact about the coordinates rather than about the label.

Ammonia’s three hydrogen 1s functions reduce in C₃ᵥ to a₁ ⊕ e, and that is exactly what methane’s a₁ ⊕ t₂ correlates into once a fourth substituent is made different: a single stays a single, and a triple splits into a single and a pair. The correlation table says so before any reduction is performed, and the reduction agrees.

What was computed, and how #

Three descents are built here and every row of every one of them is a reduction rather than an entry.

For each class of the subgroup, the declared map names the parent class its operations came from, and that name is checked against the parent’s own class list — a class the parent does not have is refused rather than resolving to nothing. Each parent representation’s characters are then read off at those classes and reduced in the subgroup, which brings with it the checks every reduction carries: whole-number multiplicities to within a millionth, and a rebuild of the character the multiplicities came from.

Two further claims are checked for each descent, and both can fail.

Dimension is conserved. Whatever a representation becomes, the dimensions of the pieces must add to the dimension it started with. A correlation that turned a threefold species into two singles would pass every reduction check and be wrong, and this is what catches it.

What cannot be held must split. If a parent representation’s dimension exceeds the largest dimension available in the subgroup, the correlation is required to return more than one piece. That is the Jahn–Teller precondition stated as an assertion, and it is the reason the three descents chosen here are the three that matter: each has a parent with a representation the child cannot hold.

The character tables are the same thirteen used throughout, each verified against four internal relations before any reduction touches it.

Where the model stops #

The correspondence between classes is declared, not derived. It could be derived — by generating both groups from coordinates and matching operations by their axes — and that is a heavier construction than the argument needs. What is stated instead is the choice, in the descent’s own definition, along with what physical change it corresponds to.

Only three descents are built. Td to C₃ᵥ, Oh to D₄ₕ, D₆ₕ to D₂ₕ. Those cover the substitution, the tetragonal distortion and the Peierls case, which are the three argued about here. A general correlation between any group and any subgroup is a bigger calculation.

Symmetry cannot say which distortion. The Jahn–Teller argument gives a mode of distortion whose symmetry is contained in the product of the degenerate representation with itself, and there is usually more than one candidate. Which one is chosen, and how far, is an energy question.

A distortion that lowers symmetry is not always available. Jahn and Teller’s proof is exhaustive precisely because the existence of a suitable mode has to be checked case by case, and linear molecules are the exception where none exists.

Nothing here is an energy at all. These figures contain no calculation of any level’s position, and a correlation diagram drawn with lines sloping down the page — as they usually are — is claiming an ordering the arithmetic here does not supply.

The generalisation #

The recurring shape is that a symmetry argument answers whether and how many, and never how much.

How many bands a spectrum can show. Whether a molecule can be polar or chiral. Whether an integral must vanish. How degenerate a level may be. And now: whether a level must split when a symmetry is removed. Each is exact, each is available before any calculation, and each is silent about magnitudes.

That silence is the point rather than a limitation. A statement that survives every improvement to a model is worth more than a number that does not, and the two are complementary: symmetry frames the question and a calculation fills it in. Where they conflict, the calculation is wrong.

The habit generalises past chemistry. Asking what a change of symmetry permits, before asking what a system does, is the same move whether the object is a molecule, a crystal or an equation — and it is the reason point groups from coordinates comes first rather than as a piece of notation near the end.

The descent a molecule undergoes by being put somewhere #

The arithmetic here is presented as a calculation to be done when a molecule distorts, and there is a second and commoner occasion for it that involves no distortion at all: putting the molecule into a solid.

A molecule in a crystal sits at a site, and that position has its own symmetry — the operations of the crystal that leave that position fixed. That site symmetry is almost always lower than the free molecule’s, sometimes drastically, and the molecule’s labels descend to it exactly as this arithmetic describes.

The consequence is a set of bands that should not be there.

A centrosymmetric molecule in the gas phase obeys mutual exclusion: no vibration is active in both the infrared and the Raman spectrum. Put the same molecule at a crystal site with no centre of inversion and the exclusion is gone — the descent has merged the g and u labels into ones that carry no parity, every mode is potentially active in both, and bands appear in each spectrum that the free molecule forbids.

Their appearance is not a failure of the rule. It is the rule applied to the right group, and the intensity of the newly allowed bands measures how far the site departs from the molecule’s own symmetry.

The full analysis has three groups in it and the descent runs through all of them: the molecular group, the site group it descends to, and the factor group of the whole unit cell, which can split a single molecular mode into several if the cell contains more than one molecule. Each step is a correlation of the kind computed here, and the three together predict how many bands a crystalline sample shows where the gas showed one.

That gives the arithmetic a use beyond distorted molecules. A solid-state spectrum is a gas-phase spectrum after two descents, and reading it means running the correlation twice — which is why a compound’s infrared spectrum in a crystal has more bands than its own point group allows, and why the extra ones are informative about the packing rather than about the molecule.

It also gives a reason for a practice that otherwise looks like fussiness. A vibrational spectrum quoted for structural purposes is measured in the gas phase, or in a dilute solution, or in an inert matrix at low temperature — and never on a neat solid if it can be avoided. The reason is not that solids are inconvenient; it is that a solid’s spectrum is the molecule’s after a descent to a site group nobody chose, and the band count it produces is a count for a different group.

A matrix isolation experiment is the extreme form of that care: the molecule is frozen into a solid of argon, where the site is nearly spherical and the descent is nearly nothing, and what comes back is the free molecule’s spectrum with the free molecule’s band count.

Who found it, and when #

Bethe’s 1929 paper on crystal fields is where the descent construction first appears in physics: he needed to know what happened to atomic levels when an atom was put into a lattice with lower symmetry, and the answer was a restriction of representations.

Jahn and Teller’s theorem is from 1937, and its proof is an exhaustive one — they checked every point group and every degenerate representation for the existence of a suitable distortion mode, and found that only linear molecules escape. It is one of the few theorems in chemistry proved by going through all the cases.

Correlation tables were compiled and published through the 1950s and 1960s, and are in the standard appendices of every group-theory textbook, which is where the impression that they are reference data comes from. They are a page of arithmetic, and computing them makes both the choice and the checks visible in a way that reading them does not.

Still open: where the electrons are #

Four essays go from finding a molecule’s group in its coordinates to deciding what that group permits, forbids, and does when it is taken away. What none of them needs is an electron, which is the point — and also the boundary. Everything past here is about where the electrons actually are.