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# On the periods in the periodic table (orWhyisanoblegasstable?)

I've succeeded in making the question entirely unintelligible with all my additions.So I thought I would summarize it in the simplest form I could manage and add it to the title. The question is thereby somewhat narrower in scope than my original query, but I would be happy to have this focused version answered.

The main point is that the loose answer

(A) because the outer shell is filled

frequently heard is definitely wrong. If we take the usual definition of a shell, all the noble gases but Helium have justthe s and p subshells (corresponding to the representations $V_0\otimes S$ and $V_1\otimes S$) filled in the outermost shell, and this is not a full shell once the atom is bigger than Argon.The wikipedia article on noble gases offers an amusing formulation whereby, fora noble gas, 'the outer shell of valence electrons is considered to be "full"' (my emphasis).

All this led to my initial confusion: I thought that the term 'shell' must mean somethingelse for multi-electron systems in a manner adapted to the answer (A). Such an alternative definition does not seem to exist, and it is not at all obvious how to come up withone in a non-tautological way. So I believe the essence of what I am asking is whether or not there is

a natural mathematical explanation for the stabilityof noble gases

that is more or less independent of experimentsconfirmed by difficult computations using approximation schemes.

Anyone interested in the background and details is invited to read the incoherent paragraphs below,or to look up a proper reference like Atkin's book on physical chemistry.

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L-shell: $2s\oplus 2p$

and so on. (In case you're wondering, the representations after $V_3\otimes S=f$ are labeled consecutively in the alphabet.) In The key point is that, in this form, a shell does not necessarily consist of a grouping of subshells of similar energies when more than a few electrons are present. Rather, the conventional description of the phenomena says that more than one shell can be incomplete in an atom. For example, in the fourth row of the periodic table, starting with potassium (K) and up to copper (Cu), they say both the $M$-shell and the $N$-shell are incomplete. If this is confusing to you, I suggest you don't worry about it. I just wanted to point out that my question what is a shell?' has a clear-cut answer in this usage. If we don't want to go against this convention, we need to stick to the version

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Having read some more here and there, I should make some terminological corrections, in case I mislead anyone with my ignorance. I will do so here, and leave the text below as written, in case the temporary confusion is helpful to other interested non-experts.

Firstly, as far as I can tell, the word 'orbital' does seem to refer to a wavefunction, not a representation. So the eigenfunctions in the second occurrence of the representation $V_1\otimes S$ will be called the $3p$-orbitals. That is, the individual wavefunction is an orbital, while the representation itself might be referred to as the so-and-so orbitals, in the plural. The definitive term for any given occurrence of an irreducible representation in $L^2\otimes S$ seems to be subshell.

Secondly, I finally read the wikipedia article on shells. It doesn't help much with my questions, but it does describe the convention regarding the term 'shell'. As far as I can tell, the shells are simply the direct sums of the following form:

K-shell: $1s$

L-shell $2s\oplus 2p$

M-shell: $3s \oplus 3p \oplus 3d$

N-shell: $4s\oplus 4p \oplus 4d\oplus 4f$

O-shell: $5s\oplus 5p \oplus 5d\oplus 5f\oplus 5g$

and so on. (In case you're wondering, the representations after $V_3\otimes S=f$ are labeled consecutively in the alphabet.) In this form, a shell does not necessarily consist of a grouping of subshells of similar energies. Rather, the conventional description says that more than one shell can be incomplete in an atom. For example, in the fourth row of the periodic table, starting with potassium (K) and up to copper (Cu), they say both the $M$-shell and the $N$-shell are incomplete. If this is confusing to you, I suggest you don't worry about it. I just wanted to point out that my question what is a shell?' has a clear-cut answer in this usage. If we don't want to go against this convention, we need to stick to the version

`What determines a period?'

I'm very sorry for all the confusion.