<a href="http://mathoverflow.net/questions/80146/is-the-mendeleev-table-explained-in-quantum-mechanics">This question</a> prompts me to ask something more specific about the periodic table. As far as I know, the main significance of the periodic table is that *The elements in the same column have similar chemical properties.* For example, the noble gases at the far right are all pretty stable on their own. The explanation for this is that they have (in the neutral state) a full outermost shell. Now my question is *Is there an explanation, at some reasonable level of mathematical rigor, of when a new shell starts?* That is, where do the length of the periods 2, 8, 8, 18, 18, 32, 32 come from? I confess I've been puzzled by this ever since my university physics course. Allow me to pinpoint my confusion a bit more. I understand that there are some numbers that are important in atomic structure, and these are 2, 6, 10, 14, and so on. This is because of the occurrence of the representations $$V_l\otimes S$$ inside the Hilbert space for a single particle moving in a central potential. These are the *orbitals* one hears about in physical chemistry courses. Here, the $V_l$ are the representations of $SO(3)$ of odd-dimension $2l+l$, while $S$ is the standard two-dim representation of $SU(2)$. Since all the states in a single orbital have the same energy, it is natural that the breaks will occur after some collection of orbitals are all filled. Thus, for a hydrogen atom, the successive shells have dimensions $2n^2$ for $n=1, 2, 3, \ldots$ because the representations $V_l\otimes S$ for $l=1, 2,\ldots, n-1$ occur each with multiplicity one inside the $n$-th shell. So if the periods in the table were of lengths 2, 8, 18, 32, I would have vaguely assumed that the pattern of shells even in general looks like that of the hydrogen atom. But of course, among the known elements, each period length that occurs in the hydrogen atom is repeated twice, except for the first one. So a more precise question is *Is there a reasonable mathematical explanation of this `multiplicity two' of the periods?* The little I recall of discussions in standard textbooks were quite unclear. There are various rules by the name of Hund's rule, the <a href="http://en.wikipedia.org/wiki/Aufbau_principle"> Aufbau principle</a>, and so on, but I couldn't gather from any of them *Where the breaks should occur. * What I do see is that periods end when the orbitals 1p, 2p, 3p, etc. get filled following the Aufbau principle. (Here, p is the chemist's label for $V_1\otimes S$.) So perhaps another version of the question is *Is there some reason that the p-orbitals mark the end of the periods?* To be honest, I've never understood the Aufbau principle either, because I don't know the rationale behind the principal quantum number for the larger atoms. That is, the number 4 in the orbital 4p refers to the 4-th energy level in the case of the hydrogen atom. But for larger nuclei, the $n $ in orbital '$np$' does not refer to the energy level. (This discrepancy is in fact implied by the Aufbau principle.) So what is the significance of the $n$ in general that enables them to play some role in a physical principle? I realize this question is becoming incoherent already. Nevertheless, I would very much appreciate clarification on any sensible version of it at a level of mathematical rigor of your choice. (I am not asking for any axiomatics.) Pointers to an accessible reference would be equally welcome. ------------------------------------------------------------------- As with the earlier question, a word of explanation is in order on the decision to post this on Math Overflow. I will draw upon an analogy I read long ago in an article of George Mackey's that went something like this: Say my mother tongue is Korean. If I would really like to use English fluently, it is probably best eventually to learn from native speakers of English. On the other hand, if I would like a *good translation* into Korean of English literature, it is better to consult an educated Korean who knows a lot about English. Of course answers from real physicists will be very gratefully received, especially if they bear in mind that the query comes from someone who struggles against a serious language handicap.