Degeneracy is a group theorem

A calculation on methane returns four bonding orbitals, one at one energy and three sharing another. It is natural to read that as a result of the calculation, and to suppose that a better calculation might have split the three.

It could not. The threefold degeneracy was fixed before any energy was computed, by twenty-four operations that permute four hydrogen atoms.

What a degeneracy is, in group terms #

Two orbitals are degenerate when they have the same energy, and the useful question is what forces that.

Suppose $\psi$ is an eigenfunction of the Hamiltonian with energy $E$, and $R$ is a symmetry operation of the molecule. Because $R$ commutes with the Hamiltonian — that is what being a symmetry operation means — the function $R\psi$ is also an eigenfunction, with the same energy $E$.

So the set of functions generated by applying every operation to $\psi$ all share one energy. They span a space, that space is invariant under the group, and the dimension of the smallest such space containing $\psi$ is the degeneracy.

Spaces of that kind have a name: they are the irreducible representations of the group. Their dimensions are listed in the first column of the character table, and they are the only degeneracies available.

That is the whole theorem, and it is exact. It follows from the symmetry of the exact Hamiltonian rather than from any orbital approximation, so unlike almost everything else here it may be stated without hedging.

What each group permits #

The consequences are immediate and are worth reading off directly.

Water, C₂ᵥ. Four representations, all of dimension one. A water molecule cannot have a degenerate orbital level. Not “does not happen to”; cannot, unless something splits by accident.

Ammonia, C₃ᵥ. Dimensions 1, 1, 2. Degenerate pairs are permitted and no triple is.

Methane, Td. Dimensions 1, 1, 2, 3, 3. Triples are permitted, which is why methane has one.

Benzene, D₆ₕ. Dimensions 1 and 2 only. Its π levels come out as a single, two degenerate pairs and a single — which is exactly the pattern aromaticity as a computed shell closure needs for the 4n+2 counting to work, and the pattern is the group’s rather than Hückel’s.

An octahedral complex, Oh. Dimensions 1, 1, 2, 3, 3 twice over — the g and u sets. The famous splitting of five d orbitals into a set of three and a set of two is a statement that the d functions span $t_{2g} \oplus e_g$, and the 3 and the 2 are dimensions on that list.

Methane, and the spectrum that confirms it #

The four hydrogen 1s orbitals of methane span $a_1 \oplus t_2$. So the four bonding orbitals fall into two symmetry species in a one-to-three ratio, and there must be two ionisation energies rather than one.

The photoelectron spectrum shows bands at 12.7 and 23 electron volts, in roughly that ratio of intensity. That is the confirmation of a prediction, not the source of it, and the distinction is the whole reason hybridisation does not explain can say what it says: four equivalent sp³ orbitals would give one band, and the group says four equivalent orbitals cannot be eigenfunctions in the first place.

The energies are not predicted. Symmetry says how many distinct levels there are and how degenerate each is; it says nothing whatever about where they lie. That division of labour is exact and is worth stating in both directions, because the second half is as important as the first.

Counting the levels before computing them #

The same argument that limits degeneracies also fixes how many distinct levels there can be, and the counting is worth doing explicitly because it is the part that gets used.

Take any set of functions on a molecule — the hydrogen 1s orbitals, the three p orbitals of a central atom, the displacement vectors of every nucleus — and work out its character under each class of operations. The reduction formula then says how many times each irreducible representation appears in it, and each appearance is one level.

For methane’s four hydrogen functions the answer is $a_1 \oplus t_2$: two levels, of degeneracy one and three. For the same molecule’s fifteen nuclear displacements the answer contains the translations and rotations as well, and removing them leaves $a_1 \oplus e \oplus 2t_2$ — four vibrational levels, of degeneracies 1, 2, 3, 3, adding to nine, which is $3N - 6$ for five atoms. Both counts come from a table and a sum, and neither needs a force constant.

Two things about that are worth noticing. The count is exact, and it is a strong constraint: a spectrum showing five distinct methane vibrations would refute the tetrahedral geometry outright. And the count is silent about the positions, which is why a symmetry analysis is a frame for a measurement rather than a substitute for one.

Degeneracies the group does not permit #

A level can be degenerate without the group requiring it. Two representations of dimension one can happen to lie at the same energy, and that is an accidental degeneracy.

The distinction is not academic, because the two behave completely differently.

A degeneracy the group requires is stable. Any perturbation that preserves the symmetry leaves it intact, whatever its size or its physical origin: lengthening all four bonds of methane equally, applying a uniform pressure, improving the basis set. The three $t_2$ orbitals stay together because the operations that make them equivalent are still there.

An accidental degeneracy is fragile. Any perturbation at all splits it, and the splitting is first order in the perturbation.

So the useful test is not whether two levels coincide but whether they are required to. In a Hückel calculation on benzene the two levels at $\alpha + \beta$ are required, by the ring’s sixfold symmetry, and they survive any change to $\beta$. The pair at $\alpha - \beta$ likewise. Nothing about those degeneracies is an artefact of a matrix of ones and zeroes.

The Jahn–Teller consequence #

Once degeneracy is a property of the group, an obvious question follows: what happens to a molecule whose degenerate level is only partly filled?

Its energy can be lowered by distorting to a lower symmetry, because the lower group has no representation large enough to hold the level and the level must split — and half the split goes down. That is the Jahn–Teller theorem, and the part of it decided by symmetry is decided completely: which distortions split which levels, and by how many pieces.

Cyclobutadiene is the familiar case. Its four π electrons put two into a degenerate non-bonding pair, so Hückel predicts a triplet, and the molecule escapes by distorting to a rectangle where the pair is split. Delocalisation is stabilising, and other things that are false in general computes the energies; the symmetry statement is that at D4h the pair must be degenerate and at D2h it cannot be.

Working out which representation lands where is a correlation, and it is the subject of descent in symmetry. The same correlation decides what happens to a molecule’s own labels when a substituent is changed, which is how site symmetry, and what it constrains approaches the same problem from the direction of the atoms rather than the orbitals.

Why a bigger basis does not help #

A recurring hope is that a level which looks degenerate in a crude model might separate in a better one, and it is worth seeing exactly why symmetry blocks that.

Improving a calculation means enlarging the space of functions it works in. Every function in that space still transforms somehow under the group’s operations, and every eigenfunction of the Hamiltonian still belongs to some irreducible representation. Adding functions can change which levels appear and where they lie; it cannot invent a degeneracy the group forbids, and it cannot break one the group requires.

That has a practical corollary worth stating plainly. If a numerical calculation on a tetrahedral molecule returns a nearly-but-not-quite threefold level, the residual splitting is numerical error, not physics — the basis is not symmetric, or the grid is not, or the geometry is slightly off. This is one of the standard diagnostics in computational chemistry, and it works because a symmetry statement is exact and a calculation is not.

The same logic runs the other way for a calculation. Hückel results show benzene’s two pairs at exactly the same eigenvalue, to the last bit of a double, because the matrix is exactly symmetric. A version of the eigensolver that split them by a millionth would be reporting its own convergence, and the closed-form check against $2\cos(2\pi k/n)$ is what would catch it.

What was computed, and how #

The character tables here are not typed in as data and read off. Thirteen of them are stored, and each is verified before use against four internal relations: the rows are orthogonal, the columns are orthogonal, the sum of the squared dimensions equals the group order, and the number of representations equals the number of classes. A table that fails any of them throws.

The groups themselves are generated. Given a molecule’s coordinates, the operations that permute its atoms are found, closed under multiplication, and sorted into conjugacy classes by conjugation — so the twenty-four operations of methane’s group are produced from four hydrogen positions rather than looked up. The order of the generated group is then checked against the table’s, which is the check that caught an earlier bug in which benzene’s twenty-four operations silently collapsed to twelve.

The reductions run the standard formula and check two things about every answer: that each multiplicity is a whole number to within a millionth, and that the multiplicities rebuild the character they were computed from. A basis whose reduction returned 2.5 of something would be refused, which is the correct response — a fractional multiplicity means the character was wrong.

The degeneracy claim itself is checkable from the same data: the dimensions available to a group are the first column of its table, and a level of any other degeneracy in a calculation on a molecule of that group is either an error or an accident.

Where the model stops #

Symmetry says which degeneracies are permitted, never which occur. Td permits triples and a particular tetrahedral molecule may have none in a given energy range. The theorem is a constraint, not a prediction.

Spin is not in any of this. The degeneracies discussed here are spatial. A spin-degenerate pair is a different thing, and spin–orbit coupling — which is outside this site’s scope entirely — mixes the two and changes which group is the relevant one.

Time-reversal symmetry adds degeneracies the point group does not. Kramers’ theorem gives every odd-electron system a twofold degeneracy that no spatial group accounts for, and no amount of lowering the spatial symmetry will remove it — only a magnetic field will.

The group has to be the right one. Every statement here is about the point group of a fixed nuclear framework. A molecule that is fluxional on the timescale of the measurement may effectively have a higher symmetry than any of its instantaneous geometries, and a molecule in a crystal has a site symmetry lower than its own.

Accidental degeneracies are not always accidental. Some coincidences that look accidental in the point group are required by a larger hidden symmetry — the classic case being the hydrogen atom, where the equality of the 2s and 2p energies follows from a symmetry of the Coulomb problem that is invisible in the rotation group. A degeneracy the point group does not explain is a reason to look for a bigger group, not a reason to call it a coincidence.

The generalisation #

The pattern this belongs to is the one running through the whole of symmetry: a class of questions that can be answered exactly, before and independently of any calculation, by asking what the group permits.

Whether a molecule can have a dipole moment. Whether it can be chiral. Whether a transition is allowed. How many bands a vibrational spectrum can have. And now: how degenerate a level can be. Each is settled by the group alone, each is exact, and each is available in advance.

What none of them gives is a number with an energy in it. The division is clean and it is worth keeping clean, because the two kinds of statement have completely different standing — selection rules are one theorem makes the same point about integrals, and the theorem there is the same theorem here in a different costume.

Read backwards, a degeneracy is a measurement of symmetry #

The theorem settles what degeneracies a group permits, and the useful direction for somebody holding a spectrum rather than a structure is the other one: an observed degeneracy places a lower bound on the symmetry.

The bound is sharp and it is easy to state. A representation of dimension greater than one requires an axis of order three or higher. Groups built from twofold axes, mirror planes and a centre of inversion — and that is a great many of them — have only one-dimensional representations, so a molecule in such a group cannot have two levels at the same energy for any reason the symmetry supplies.

So observing a genuine degeneracy is observing an axis of order at least three, without measuring a single coordinate.

The qualifier genuine is doing real work and it is the only difficulty in the argument. Two levels can be close together for reasons that have nothing to do with symmetry, and telling an accidental near-coincidence from a symmetry-required degeneracy is the practical problem. Two tests separate them.

A symmetry-required degeneracy is exact. It does not split when the temperature changes, the solvent changes or the molecule is put in a different crystal — unless the change lowers the symmetry, in which case it splits and the splitting appears.

And it splits in a predictable way. Lowering the symmetry deliberately — by isotopic substitution, by a field, by a chemical change to one position — makes a required degeneracy come apart into the pieces the descent predicts, and an accidental one behave arbitrarily.

That is the practical procedure and it is worth its cost. Perturb the molecule and watch whether the coincidence survives. A degeneracy that splits under a perturbation that lowers the symmetry, into the number of components the correlation predicts, is a degeneracy that was symmetry-required — and therefore evidence of a threefold or higher axis, obtained from a spectrum rather than from a structure.

The bound runs only one way, and saying so keeps the inference honest. A degeneracy implies a high-order axis; the absence of one implies nothing. A molecule with a threefold axis may have no degenerate level occupied, or none in the range being measured, or two levels that are degenerate and unresolved. So a spectrum with no observed degeneracy is consistent with any symmetry at all, and only the positive observation carries information — which is the usual asymmetry between what a symmetry argument forbids and what it permits, arriving here in the form of what a measurement can conclude.

Who found it, and when #

The connection between degeneracy and group representations is Wigner’s, from 1927, in the first of the papers that brought representation theory into quantum mechanics. Bethe’s 1929 paper on crystal fields applied it to the splitting of atomic levels in a lattice and is where the $t_{2g}$ and $e_g$ labels come from.

The mathematics predates the physics by decades: Frobenius developed character theory in the 1890s, entirely as pure algebra, and Schur’s lemma — which is what actually forces the degeneracy — is from 1905.

The reception was famously grudging. The phrase Gruppenpest, the group-theory plague, circulated among physicists who resented having to learn it; Slater took pride in avoiding it. The resentment has faded and the labels have not, and a modern chemist who has never seen a character table still says $t_{2g}$.

Still open: what happens when the symmetry is taken away #

This essay says which degeneracies a symmetry permits. What happens to them when the symmetry is taken away is the question every distortion, every substitution and every crystal field is an instance of.