The degeneracy no group predicts

Degeneracy is a group theorem is one of the sharper results in the subject. How many orbitals can share an energy is decided before any energy is computed: the dimensions of a group’s irreducible representations are the only degeneracies it permits, and for the rotation group of a central potential those are 1, 3, 5, 7 — one for each value of the angular momentum.

Hydrogen’s second shell has four orbitals at one energy. Its third has nine. Neither number is on the list.

That is not a small discrepancy in a rule. It is a case where the rule’s whole content — the list of numbers a degeneracy is allowed to be — does not contain the number observed, in the one atom whose spectrum is exactly solvable and known best.

The degeneracy, computed rather than quoted #

The first thing to establish is that the degeneracy is real and not a statement about a formula. The energies here are computed from the radial functions themselves.

For a potential going as 1/r the virial theorem gives the total energy as half the potential energy, so E = −Z⟨1/r⟩/2, and ⟨1/r⟩ is an integral over the radial function. Doing that integral for the 2s and for the 2p separately gives 0.250000 for both, and hence −0.125000 hartree for both, with a spread across the shell of 1.2 × 10⁻⁷ — which is the quadrature and not the physics.

The two columns of that table are the whole essay. One is the same across the shell and the other is not, and the first is the one that decides the energy.

What the group actually says #

It is worth being careful about what the group’s claim is, because it is a permission rather than a prohibition.

The group says: any set of orbitals sharing an energy must form a representation, and a representation that is irreducible has a dimension from the list. A set of four orbitals at one energy is a perfectly good representation — it is the s representation plus the p one — and there is nothing forbidden about it. What the group says is that it has no reason to be so, and that a small change to the Hamiltonian which respects the group will generally split it.

That distinction is the standard one between a symmetry-required degeneracy and an accidental one, and the word accidental is the one this essay is about. An accident is something that could easily have been otherwise. This one could not.

Any other potential splits it #

Take the Hamiltonian and add a small correction that respects the rotational symmetry — anything of the form δV®, a function of the distance alone. The group is unchanged, so a group-theoretical argument says nothing about whether the shell splits.

The first-order shift is the expectation value of δV in each orbital, and for a correction going as 1/r² those expectations are the second column of the table above: 0.2500 for the 2s and 0.0833 for the 2p. Three to one. The shell splits immediately, in first order, by an amount proportional to the correction.

So the degeneracy is not robust in the way a symmetry-required one is. A symmetry-required degeneracy survives every perturbation that respects the symmetry; this one survives none of them except those that go exactly as 1/r.

It is worth noting which quantity did the work. The energy is fixed by ⟨1/r⟩, which is Z/n² — a function of n alone. Every other inverse moment depends on l as well. So the degeneracy is the statement that the one moment the energy happens to need is the one that does not know about the angular momentum, and that coincidence holds for the 1/r potential and for nothing else.

Which is a statement about the potential, and the periodic table is its evidence #

The consequence is not hypothetical. Every atom but hydrogen has more than one electron, and the potential an electron feels in such an atom is not 1/r: it is the nucleus screened by the other electrons, which is steeper near the nucleus and shallower far out.

So the 2s and the 2p separate in every atom from lithium onwards, and they separate in the direction the moments predict — the s below the p, because the s has the larger ⟨1/r²⟩ and the correction is attractive where the screening is incomplete.

Slater’s rules assign one effective charge to a whole shell, so they give the 2s and the 2p exactly the same one at every element — which means the model has the accidental degeneracy built into it and cannot be used as evidence for it.

That is the whole of the structure of the periodic table’s short periods, arriving from a symmetry argument. What an electron actually feels is the essay about the screened charge itself, and what the screening model cannot see is about a model crude enough to put the 2s and the 2p in one group and therefore to predict no splitting at all — which is a failure of exactly the kind this essay explains.

The s is the one that moves furthest because its radial function has an inner lobe close to the nucleus, where the screening is least complete and the potential departs most from 1/r. A p has no such lobe and a d has less still, which is the ordering every inverse moment records.

The size of the departure, where it can be measured #

The argument so far gives a direction and a mechanism. The magnitude is available too, from the moments, and it is worth carrying through because it turns a qualitative account into a number.

The first-order shift is the correction times the moment, so the ratio of the shifts is the ratio of the moments and does not depend on the correction’s size: three to one across n = 2 and five to one across n = 3, whatever the perturbation is, provided it goes as 1/r². For a correction of a different shape the ratio changes and the ordering does not — every correction that is more strongly peaked than 1/r near the nucleus shifts the s more.

Measured against real atoms, the direction is right everywhere and the ratio is not, which is the expected result: screening is not a 1/r² correction, it is a complicated function of r that this calculation has not been given. What survives is the qualitative structure — s below p below d within a shell, in every atom, which is what penetration means — and that is a consequence of the moments’ ordering, which is the same for every reasonable correction because ⟨1/r^k⟩ falls with l for every k above one.

The missing symmetry, seen classically #

A degeneracy larger than the group requires is evidence of a symmetry that has not been found, and for this one the missing symmetry has a classical shadow that can be drawn.

An orbit in an inverse-square field is an ellipse that closes: after one turn the particle is back where it started, moving as it was. That is not true of orbits in general, and it is a stronger statement than the conservation of angular momentum — which fixes the plane and the rate of sweeping out area but says nothing about whether the long axis of the orbit stays put.

The classical version is sharp: an orbit under an inverse-square force returns to the same long axis to within three thousandths of a degree over three turns, and an exponent of 2.1 moves it measurably. Two is not a value at which the precession is merely small — it is the value at which it is exactly zero.

An exponent of 1.9 moves the axis 16.9 degrees a turn in the opposite direction, which is what makes the vanishing at exactly two a statement about a conserved quantity rather than about a small number. That conserved vector is the extra symmetry the point group does not contain.

The direction of the long axis is a conserved quantity, and it is conserved for the inverse-square force and for no other. That extra conservation law is the extra symmetry, and it is what makes the quantum degeneracy larger than the rotation group requires. Selection rules are one theorem is the standing account of how a symmetry becomes a statement about integrals; this is the same relation running the other way, from a spectrum back to a symmetry. The two facts — a closed classical orbit and a shell degenerate in l — are the same statement in two languages.

Why this is the right way round #

It would be possible to present this as a coincidence with an explanation attached, and it is worth saying why that reading is backwards.

The reasoning that runs the shell is degenerate, therefore look for a symmetry is a general and productive one, and it is how the extra conservation law was found. A degeneracy is a fact about a spectrum; a symmetry is a fact about a Hamiltonian; and the first is evidence for the second in a way that has repeatedly turned out to be reliable across physics.

So the useful statement is not “hydrogen has an accidental degeneracy”. It is:

A degeneracy the known group does not require is a signal that the group is not the whole symmetry, and the correct response is to look for the rest of it.

That reading also explains why the degeneracy is so fragile. The extra symmetry belongs to the 1/r potential specifically, so any change to the potential destroys it — while the rotational symmetry belongs to any central potential and survives everything.

The word “accidental” and what it costs #

The standard name for this is an accidental degeneracy, and the name has done damage worth pointing at.

It suggests that two levels have coincided by chance, the way two unrelated frequencies in a spectrum might. If that were the situation, the right response would be to note the coincidence and move on, because nothing follows from a coincidence.

What is actually the case is that a symmetry has not been identified. The right response is to look for it, and looking for it in this case produced a conserved quantity, a larger group, and an account of the whole hydrogenic spectrum that never solves a differential equation.

The word survives because the degeneracy was known long before the symmetry, so it was named for what it looked like. That is worth remembering when a spectrum shows a coincidence that a group does not require: the historical answer in the standing example was that the group was incomplete.

What this does not explain #

The filling order of the periodic table is a separate question and it is worth saying so explicitly, because the two are easy to run together.

The aufbau order is not a property of the atom is the essay about the 4s and the 3d: iron’s 3d is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy, and the 4s fills first anyway. That is not the splitting discussed here. It is a many-electron effect involving the repulsion between the electrons, and no argument about a one-electron potential reaches it.

What this essay explains is why the 2s and the 2p are split at all in a many-electron atom, and in which direction. How large the splitting is, and how it competes with the shell spacing to decide a filling order, is a different and harder question.

Other degeneracies that were not required #

The general point has other instances, and gathering them makes the pattern easier to recognise.

One table, three groups is a case of the reverse relation: three different groups share a character table entry for entry, so a degeneracy pattern does not identify the group that produced it. A degeneracy is weaker evidence about a symmetry than the symmetry is about the degeneracy, and both directions have their traps.

Why a character table stops where it stops is the accounting that makes the permitted list finite: the number of representations equals the number of classes, and the squares of their dimensions sum to the order of the group. That is what forbids a fourth dimension in a group whose dimensions are one and three — and hydrogen’s four-fold level is not a violation of it, because the four orbitals are not one irreducible representation.

The thing to hold onto is that the list of permitted dimensions is a list of what a single irreducible representation can be. Any sum of them is permitted too, and a spectrum that shows one is showing a coincidence — until a bigger group is found that makes it a single representation again, which is exactly what happens here.

The other potential with a hidden symmetry #

Hydrogen’s extra degeneracy is called accidental and is not, and the strongest evidence that it is not is that there is exactly one other potential with the same kind of surplus — and the two are singled out by a theorem that has nothing to do with quantum mechanics.

The other is the isotropic harmonic oscillator: a particle in a potential proportional to the square of its distance from a centre. Its levels are degenerate far beyond what rotations permit — the first excited level holds three states, the second holds six, and rotations account for at most five of the six. It too has a hidden symmetry, of a different kind from hydrogen’s, and it too fails the moment the potential is changed.

Two potentials, two surpluses, and no others. The theorem that says so is classical and is about orbits: the only central potentials in which every bound orbit closes on itself are $1/r$ and $r^2$. Every other central force gives an orbit that precesses, never returning to where it began.

That is a statement about planets rather than about atoms, and it is the same statement. A closed orbit is one whose orientation is conserved — there is a constant of the motion fixing where the orbit points — and a conserved quantity is a symmetry. The two potentials with closed orbits are the two with an extra conserved quantity, and the extra conserved quantity is what produces the extra degeneracy when the same potential is treated quantum mechanically.

So the degeneracy is not accidental in either case, and the word survives only because the symmetry responsible is not a rotation and has no picture attached.

It also says why nothing else in chemistry has such a surplus. A many-electron atom’s potential is not $1/r$, a molecule’s is not central at all, and a harmonic well is an approximation that fails as soon as a bond can break. The two exact cases are the hydrogen atom and the harmonic oscillator — which are, not coincidentally, the two problems every course solves exactly and the two whose degeneracies every student is told to memorise.

What a chemist should take from it #

Two things, and both are about reading a level diagram rather than about hydrogen.

A level diagram with two orbitals of different symmetry drawn at the same height is making a claim, and the claim is either that a group requires it or that something else does. If neither, the two levels are not at the same height and the diagram is a sketch. Hydrogen’s is the one case where “something else does” is the answer, and it is the case every other diagram is unconsciously modelled on.

And a degeneracy that a group requires cannot be lifted by any perturbation respecting the group, while one it does not require is lifted by nearly all of them. So the two kinds behave completely differently under any change to a molecule — a substituent, a field, a distortion — and telling them apart is the difference between predicting that a splitting will appear and predicting that it cannot.

What is left #

The classical calculation above is a demonstration rather than a derivation. It shows that the exponent two is special and that the precession changes sign through it; it does not derive the conserved quantity, and the quantum version — an operator that commutes with the Hamiltonian and connects orbitals of different l — is not constructed here at all.

Constructing it would be the natural next step and would put the argument on the same footing as the rest of this field, where a group is generated from a molecule’s own operations and its representations are reduced rather than looked up. The group in question is not a group of rotations of ordinary space, which is why no point-group search reaches it, and constructing it is a larger job than one essay.

The other absence is relativity. The l-degeneracy of a real hydrogen atom is not exact: the spin–orbit interaction and the other relativistic corrections split the 2s from the 2p by a small amount, and part of that splitting is measurable to extraordinary precision. Every correction of that kind goes as some power of r other than −1, so every one of them splits the shell — which is this essay’s argument applied once more, in a place where the numbers are known to twelve figures.

A correction that falls off quickly does its work in the inner lobes of an s function, close to the nucleus — the region a 3p reaches less and a 3d hardly at all. Every inverse moment in this essay is dominated by exactly that region, which is why they order the shell the way they do.