I’ve been thinking about chemistry lately. I’m always amazed by how far we can get in the study of multielectron atoms using ideas from the hydrogen atom, which has just one electron.
If we ignore the electron’s spin and special relativity, and use just Schrödinger’s equation, the hydrogen atom is exactly solvable—and a key part of any thorough introduction to quantum mechanics. So I’ll zip through that material now, saying just enough to introduce the ‘Madelung rules’, which are two incredibly important rules of thumb for how electrons behave in the ground states of multielectron atoms.
If take the Hilbert space of bound states of the hydrogen atom, ignoring electron spin, we can decompose it as a direct sum of subspaces:
where the energy equals in the subspace
Since the hydrogen atom has rotational symmetry, each subspace further decomposes into irreducible representations of the rotation group Any such representation is classified by a natural number Physically this number describes angular momentum; mathematically it describes the dimension of the representation. The angular momentum is while the dimension is Concretely, we can think of this representation as the space of homogeneous polynomials of degree with vanishing Laplacian. I’ll say a bit more about this next time.
The subspace decomposes as a direct sum of irreducible representations with like this:
So, in the subspace the electron has energy and angular momentum
I’ve been ignoring the electron’s spin, but we shouldn’t ignore it completely, because it’s very important for understanding the periodic table and chemistry in general. To take it into account, all I will do is tensor with to account for the electron’s two spin states. So, the true Hilbert space of bound states of a hydrogen atom is
By what I’ve said, we can decompose this Hilbert space into subspaces
called shells, getting this:
We can then decompose the shells further into subspaces
called subshells, obtaining our final result:
Since
the dimensions of the subshells are
or in other words, twice the odd numbers:
This lets us calculate the dimensions of the shells:
since the sum of the first bunch of odd numbers is perfect square. So, the dimensions of the shells are twice the perfect squares:
Okay, so much for hydrogen!
Now, the ‘Aufbau principle’ says that as we keep going through the periodic table, looking at elements with more and more electrons, these electrons will act approximately as if each one occupies a subshell. Thanks to the Pauli exclusion principle, we can’t put more electrons into some subshell than the dimension of that subshell. So the big question is: which subshell does each electron go into?
This is the question that the Madelung rules answer! Here’s what they say:
 Electrons are assigned to subshells in order of increasing values of .

For subshells with the same value of , electrons are assigned first to the subshell with lower —or equivalently, higher
They aren’t always right, but they’re damned good, given that in reality we’ve got a bunch of electrons interacting with each other—and not even described by separate wavefunctions, but really one big fat ‘entangled’ wavefunction.
Here’s what the Madelung rules predict:
So, subshells get filled in order of increasing and for any choice of we start with the biggest possible and work down.
This chart uses some oldfashioned but still very popular notation for the subshells. We say the number but instead of saying the number we use a letter:
• : s
• : p
• : d
• : f
The reasons for these letters would make for a long and thrilling story, but not today.
The history of the Madelung rules
At this point I should go through the periodic table and show you how well the Madelung rules predict what’s going on. Basically, as we fill a particular subshell we get a bunch of related elements, and then as we go on up to the next subshell we get another bunch—and we can understand a lot about their properties! But there are also plenty of deviations from the Madelung rules, which are also interesting. I’ll talk about all these things next time.
I should also say a bit about why the Madelung rules work as well as they do! For example, what’s the importance of
But that’s not what I have lined up for today. Sorry! Instead, I want to talk about something much less important: the historical origin of the Madelung rules. According to Wikipedia they have many names:
• the Madelung rule (after Erwin Madelung)
• the Janet rule (after Charles Janet)
• the Klechkowsky rule (after Vsevolod Klechkovsky)
• Wiswesser’s rule (after William Wiswesser)
• the Aufbau approximation
• the diagonal rule, and
• the Uncle Wiggly path.
Seriously! Uncle Wiggly!
Understanding the history of these rules is going to be difficult. You can read a lot about their prehistory here:
• Wikipedia, Aufbau principle.
Bohr and Sommerfeld played a big role in setting up the theory of shells and subshells, and perhaps the whole idea of ‘Aufbau’: German for ‘building up’ atoms one electron at a time.
But I was confused about whether Madelung discovered both rules or just one, especially because a lot of people say ‘Madelung rule’ in the singular, when there are really two. So I asked on History of Science and Mathematics Stackexchange:
There are two widely used rules of thumb to determine which subshells are filled in a neutral atom in its ground state:
• Electrons are assigned to subshells in order of increasing value of .
• For subshells with the same value of , electrons are assigned first to the subshell with lower .
These rules don’t always hold, but that’s not my concern here. My question is: which of these rules did Erwin Madelung discover? (Both? Just one?)
Wikipedia seems to say Charles Janet discovered the first in 1928:
A periodic table in which each row corresponds to one value of (where the values of and correspond to the principal and azimuthal quantum numbers respectively) was suggested by Charles Janet in 1928, and in 1930 he made explicit the quantum basis of this pattern, based on knowledge of atomic ground states determined by the analysis of atomic spectra. This table came to be referred to as the leftstep table. Janet “adjusted” some of the actual values of the elements, since they did not accord with his energy ordering rule, and he considered that the discrepancies involved must have arisen from measurement errors. In the event, the actual values were correct and the energy ordering rule turned out to be an approximation rather than a perfect fit, although for all elements that are exceptions the regularised configuration is a lowenergy excited state, well within reach of chemical bond energies.
It then goes on to say:
In 1936, the German physicist Erwin Madelung proposed this as an empirical rule for the order of filling atomic subshells, and most Englishlanguage sources therefore refer to the Madelung rule. Madelung may have been aware of this pattern as early as 1926.
Is “this” still rule 1?
It then goes on to say:
In 1945 William Wiswesser proposed that the subshells are filled in order of increasing values of the function
This is equivalent to the combination of rules 1 and 2. So, this article seems to suggest that rule 2 was discovered by Wiswesser. But I have my doubts. Goudsmit and Richards write:
Madelung discovered a simple empirical rule for neutral atoms. It consists of two parts.
and then they list both 1 and 2.
I have not yet managed to get ahold of Erwin Madelung’s work, e.g.
• E. Madelung, Die mathematischen Hilfsmittel der Physikers, Springer, Berlin, 1936
I got a helpful reply from M. Farooq:
The only relevant section in E. Madelung’s edited book, Die mathematischen Hilfsmittel der Physikers, Springer: Berlin, 1936 is a small paragraph. It seems Madelung just called this for electron bookkeeping and he had no justification for his proposition. He calls it a lexicographic order (lexikographische Ordnung).
I was searching for it some other reasons several months ago: Question in SE Physics. The original and the (machine but edited) translations is shared.
15 Atomic structure (electron catalog) (to p. 301).
The eigenfunction of an atom, consisting of electrons and times positively charged nucleus, can be constructed in the case of removed degeneracy in first approximation as a product of hydrogen eigenfunctions (cf. p. 356), each of which is defined by four quantum numbers defined by . According to Pauli’s principle, no two of these functions may coincide in all four quantum numbers. According to the Bohr principle of structure, an atom with electrons is formed from an atom with electrons by adding another one (and increasing the nuclear charge by 1) without changing the quantum numbers of the already existing electrons. Therefore a catalog can be set up, from whose in each case first positions the atom is built up in the basic state (cf. the table p. 360).
The ordering principle of this catalog is a lexicographic order according to the numbers A theoretical justification of just this arrangement is not yet available. One reads from it:
1. The periodic system of the elements. Two atoms are homologous if in each case their “last electron” in the coincides.
2. The spectroscopic character of the basic term, entered in column 10 . There is namely the character and the multiplicity.
3. The possibilities for excited states (possible terms), where not all electrons are in the first positions of the catalog.
The catalog is the representation form of an empirical rule. It idealizes the experience, because in some cases deviations are observed.
So, it’s clear that Madelung knew and efficiently stated both rules in 1936! By the way, ‘lexicographic order’ is the standard mathematical term for how we’re ordering the pairs in the Madelung rules.
References
I took the handsome chart illustrating the Madelung rules from here:
• Aufbau Principle Chemistry God, 18 February 2020.
This blog article is definitely worth reading!