Does Textbook QM Explain the Periodic Table?

The rows of the periodic table have lengths

2, 8, 8, 18, 18, 32, 32, 50, …

A schoolchild can ask why. Every textbook answers in a paragraph, and the paragraph says: quantum mechanics. Does it?

What the textbook answer actually contains

Two halves, with very different standing.

The capacities are derived. A central field is rotationally symmetric, the ℓ-th spherical-harmonic representation has dimension 2ℓ+1, and spin-half with the Pauli principle doubles it. So 2(2ℓ+1) is a theorem, not a pattern, and 2, 6, 10, 14 are genuinely obtained.

The order is not. The Madelung (n+ℓ) rule is no theorem of the Schrödinger equation. Löwdin set the problem out in 1969 and it has not been closed. Demkov and Ostrovsky did derive the rule exactly — but for a specially chosen potential interpolating between Coulomb and short range, which shows it consistent with some Hamiltonian rather than following from the atomic one. Self-consistent field calculation reproduces most configurations, with about twenty anomalies falling exactly where Madelung fails. And penetration with screening explains the ordering in direction — low-ℓ orbitals reach into the core and are bound harder — but not in the quantitative crossings that make the sequence 2, 8, 8, 18, 18.

There is one further thing about those capacities, sharper than the usual complaint. 2(2ℓ+1) is interaction-independent. It is a rotation-group dimension times spin, so it would be identical in a universe with inverse-cube forces, or any other central law — including laws under which there is no chemistry at all. The orbital capacities carry no information about the Coulomb interaction whatever.

The authority, and how his answer changed

Eric Scerri has spent a career on exactly this question — his research programme is stated as the extent to which chemistry reduces to quantum mechanics — and Chapter 9 of The Periodic Table: Its Story and Its Significance (Oxford, 2007; 2nd edition 2019) is given over to how far modern quantum mechanics can explain the periodic system from first principles. The verdict there is negative: the chemistry of the table has not been reduced to a purely quantum-mechanical explanation, and periodicity and the properties of the elements are not entirely reducible to quantum mechanics, at least not yet.

Two qualifications of his own must travel with that, because they are load-bearing and they are the most misread part of his position. He is not saying the calculations fail. He grants that ab initio computation reproduces configurations and energies; his claim is that reproducing an answer numerically is not deriving the rule. And at least not yet claims no impossibility — the thing has not been done, which is weaker and far more defensible than what is often attributed to him.

And then he moved. Anyone citing Scerri for a flat denial is citing the book and not the man. Writing in Inference in April 2017, he put it this way: “The reduction of the periodic table to quantum mechanics is far more successful than some chemists … have been willing to admit.” On the specific gap he had pressed for twenty years, he allowed that “whether or not the Madelung rule can be derived from quantum mechanics may be irrelevant.” And asked directly whether chemistry reduces to physics, his answer was “Yes, almost but not quite.”

The reason for the shift is technical, and it is the most interesting thing in the dispute: “the building-up of atoms with electrons does not in fact follow this time-honored rule.” The Madelung rule is not a statement about which orbitals fill inside a given atom. It tracks the differentiating electron along the sequence of elements, which is a different object. So a demand for its derivation from the Schrödinger equation may be a demand for the derivation of the wrong thing — and that is Scerri's own ground for setting the demand aside, not an opponent's.

His six questions

Stripped to questions, his critique is a list, and the list is worth more than any verdict:

1.  Is the table DERIVED from quantum mechanics, or only accommodated by it?
2.  Where does the Madelung (n+l) filling rule come from?  (Loewdin, 1969)
3.  Why does each capacity serve TWO periods -- 8,8 then 18,18 then 32,32?
4.  Are electron configurations physical, or artefacts of a model?
    (about twenty elements have anomalous ones)
5.  Does chemistry reduce to physics at all?
6.  Where do the periods actually CLOSE -- and so where do H and He belong?

And his answers

1. Derived, or accommodated? His answer moved. The book: not derived from first principles, the reduction incomplete. By 2017: far more successful than some chemists have been willing to admit. Net position — not fully derived, but considerably more reduced than chemists concede. He attacks the overclaim in both directions, which is why he is hard to recruit.

2. Where does Madelung come from? Book: nowhere — undrived, Löwdin's problem open. 2017: the question may be irrelevant, because the rule does not describe orbital filling in the first place. He withdrew his own demand, which is the strongest form the point could take.

3. Why does each capacity serve two periods? On his account this is not a separate question. The (n+ℓ) grouping yields the period lengths 2, 8, 8, 18, 18, 32, 32 directly, so the doubling is not an extra fact needing explanation — it is the Madelung rule, and it inherits whatever standing the rule has. After 2017 that means he regards the whole locus as possibly misconceived rather than as an outstanding debt.

4. Are electron configurations physical? No — they are models, and this he has pressed hardest of all. Orbitals are not observables, and his well-known critique of the 1999 claim to have imaged d-orbitals in copper oxide turned on exactly that. The twenty-odd anomalies are his standing evidence that configurations are assigned rather than derived.

5. Does chemistry reduce to physics? Verbatim: “Yes, almost but not quite.”

6. Where do the periods close, and where do H and He go? Here he has a firm positive answer rather than a critique: the left-step (Janet) table — periods closing at the alkaline earths, with helium placed over beryllium rather than over neon. He argues for it precisely because it follows the n+ℓ ordering, so that the table matches the rule instead of having a different placement laid over it. (On group 3, separately, he favours Sc–Y–Lu–Lr.)

Five of the six are critiques — positions about what has not been established. Question 6 is the only one where he stakes a positive commitment of his own, and it is therefore the only one on which he can be disagreed with rather than merely answered.

So what is the state of the question

Textbook quantum mechanics derives the capacities and not the sequence, and the doubling inherits the standing of the weaker half. The leading authority on the periodic table says in print that the table has not been reduced to quantum mechanics — and then says the reduction is more successful than chemists admit, and that the missing derivation may not have been the thing to want.

That is not a contradiction. It is a position with a shape: an approximate explanation whose unexplained residue sits in a rule that may have been the wrong demand all along. What it is not is the claim the textbooks make.

A different answer exists

There is an account of the same sequence that does not route through the Madelung rule at all — obtaining the row capacities from how charge packs around a kernel, and reaching the noble-gas closures 2, 10, 18, 36, 54, 86, 118 directly rather than by placing a different arrangement over the rule. It is set out in RealQM for Atoms and the Periodic Table.

Whether it answers Scerri's six any better is a separate question from the one asked here, and it deserves to be asked separately. 

I have asked Scerri for comment, in particular if his stand-point is correctly presented in the post.