Added 22, November:
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 just
the 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, for
a 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 something
else 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 with
one 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 stability
of noble gases
that is more or less independent of experiments
confirmed 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.
Added, 21 November:
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.) The key point is that, in this form, a shell does not 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
`What determines a period?'
I'm very sorry for all the confusion.
Added, 19 November:
After receiving the nice answers from Jeff and Antoine, I realized that I should sharpen the question
somewhat more. I like Neil's formulation, but I think I am asking something much more naive.
Allow me to start by providing more background for mathematicians whose knowledge is as hazy
as mine. As mentioned by Jeff and Antoine, the Hamiltonian for a system of $N$ electrons
moving around a nucleus of charge $Z$ looks like
$$H=-\frac{\hbar^2}{2m}\sum_{i=1}^N \Delta_i-\sum_{i=1}^N \frac{Ze^2}{r_i}+ \sum \frac{e^2}{r_{ij}}.$$
In principle, one would like to understand the structure of
discrete spectrum solutions to the eigenvalue equation
$$H\psi=E\psi.$$
One characterizes solutions using labels called 'quantum numbers'. In general, there is an objective label often
denoted $l$, the angular momentum quantum number. This comes from the fact that there is
an $SO(3)\times SU(2)$ symmetry, breaking up solutions into the irreducible representations $V_l\otimes S$, mentioned
earlier. All the vectors in a given irreducible representation must have the same energy level $E$, because
a basis for the space can be obtained from a highest weight vector by applying elements of $LieSO(3)$.
Any given $V_l\otimes S$ is called a subshell or an orbital (I'm a bit unclear about this because the term
'orbital' is also used for the individual eigenfunctions) and is described using the (historical) labels
$$V_0\otimes S \leftrightarrow s$$
$$V_1\otimes S \leftrightarrow p$$
$$V_2\otimes S \leftrightarrow d$$
$$V_3\otimes S \leftrightarrow f$$
The principal quantum number is determined as follows. We order all the orbitals by energy levels,
and the $n$-th time $V_0 \otimes S$ occurs, we call that subspace $ns$. However, the $n$-th time that
$V_1\otimes S$ occurs, we call it $(n+1)p$. In general, the $n$-th time $V_l\otimes S$ occurs,
the label is
$(n+l+1)$(whatever letter $l$ corresponds to).
For example, the orbital $3d$ is the first occurrence
of the representation $V_2\otimes S$. When there is just one electron in a $1/r$ potential, these $n$ really label
the order of the energy levels, and $d$ really occurs the first time in the third energy level. This is the reason for
the shift in labeling, even though this correspondence with energy levels breaks down for multi-electron systems.
In fact, in neutral atoms with $N=Z$, the energy levels are ordered like
$$1s<2s<2p<3s<3p<4s<3d <4p<5s<4d<5p<6s< \ldots$$
(It's not supposed to be obvious how to fill in the $\ldots$. However, I've been told that knowing $4s<3d$ is essential to passing A-level chemistry in England.)
I hope it is clear from this that the orbitals (or the subshells) are completely well-defined and have a labeling
scheme that is a bit odd, but makes sense when the obvious translation is combined with history.
So the real question is
What determines a shell?
Some energy-non-decreasing consecutive
sequence of subshells is a shell. The shells then determine the rows of the periodic table (if we ignore the added complication
that $Z$ is also increasing as we move right and down).
Now, it is tempting to conclude that the standard
convention is a bit arbitrary. After all, the chemical similarity of the columns could possibly be explained simply by the
fact that
they have the same representation of $SO(3)\times SU(2)$ at the uppermost energy level, and the
same number of electrons in this representation
without any reference to an outermost shell at all. This pattern can be easily seen by the arrangement into blocks shown here. And then, unlike the $N=1$ case,
the energy levels in one shell are not even the same, just rather close to each other.
Unfortunately, the grouping into rows represents real phenomena.
This comes out very clearly, for example, in the graph of ionization energies, with the noticeable peaks
at the end of the rows immediately followed by precipitous drops. So I believe a bit of thought reduces a good deal of my original question to two parts:
-
Is there some reason for a big gap in ionization energy when one moves to an $s$-orbital?
-
Can one show that two orbitals of the same type are necessarily separated by a huge energy
gap, so that it is highly unlikely for them to occur in the same shell?
1 and 2 together make it natural that the dimensions we see in each shell will be of the form
2, 2+6, 2+6+10, 2+6+10+14, etc,
even in the general case. Of course, this doesn't say anything about how many times each combination is
likely to occur.
In any case, I hope there is a mathematical answer to these two questions that doesn't involve a full-blown programme in
hard analysis.
At the more speculative end, one might analyze a family of operators like
$$H_{\epsilon}=-\frac{\hbar^2}{2m}\sum_{i=1}^N \Delta_i-\sum_{i=1}^N\frac{Ze^2}{r_i}+\epsilon \sum \frac{e^2}{r_{ij}}.$$
Is there a way to see the numbers we see occurring via the spectral flow of this family as we go from $\epsilon =0 $ to
$\epsilon=1$? But maybe this is just as difficult as giving a full account of the structures using analysis.
By the way, I certainly wouldn't like to complain, but it is a bit puzzling to me why some people regard this
question as inappropriate for the site. Since I don't keep too well in touch with cultural trends in the
mathematical world, maybe I am unaware of how much things have changed since I was a
Ph.D. student. In those days, a programme like
Prove the stability of matter, based only on the Schroedinger equation
was regarded as an example of an important
mathematical problem motivated by atomic structure, tackled by people like Fefferman and Lieb. The questions I ask here are hopefully much easier, but still research-level
mathematics to my mind.
Original question:
This question 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 lengths 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 Aufbau principle, 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.
Best Answer
You are completely correct in your analysis of the structure one obtains by considering the Schrodinger equation for Z electrons in a central potential due to a nucleus of charge Ze when the Coulomb interaction between electrons as well as relativistic effects such as spin-orbit coupling are ignored. In such a model the periods would indeed be 2, 8, 18 etc. for exactly the reasons you have described. In physics jargon the energy in this model depends only on the principal quantum number $n \in {\mathbb Z}, n>0$ and the allowed $\ell$ values are $\ell\le n-1$. Orbitals with $\ell=0,1,2,3,4, \cdots$ are labelled by $s,p,d,f \cdots$ for historical reasons. Thus at $n=1$ one can have one or two states in the $(1s)$ orbital (accounting for spin), at $n=2$ one has the $(2s)$ and $(2p)$ orbitals with $2(1+3)=8$ states. And at $n=3$ one should have $18$ states by filling the $(3s),(3p),(3d)$ orbitals. But in the real world this is not what happens. This simple model does not give a correct description of atoms in the real world once you get past Argon. I believe the main effect leading to the breakdown is the Coulomb interaction between the electrons.
So there is no simple mathematical model based say just on the representation theory of $SU(2)$ and simple solutions to the Schrodinger equation which will account for the structure of the periodic table past Argon. However one could ask whether including Coulomb interactions between electrons does gives a model which correctly reproduces the next few rows of the periodic table past Argon. I am not an expert on this, but since I doubt there are physical chemists on MO I'll just give my rough sense of things.
To approach this problem with some level of rigor probably requires difficult numerical work and my impression is that this is beyond the current state of the art. However there are rough models which try to approximate what is going on by assuming that the interactions between electrons can be replaced by a spherically symmetric potential which is no longer of the $1/r$ form. This leaves the shell structure as is, but can change the ordering of which shells are filled first. In such a model instead of filling the $(3d)$ shell after Argon one starts to fill the $(4s)$ and $(3d)$ shells in a somewhat complicated order. Eventually one fills the $(4s),(3d),(4p)$ shells and this leads to the line of the periodic table starting at K and ending at Krypton.
Added note: There is one nice piece of mathematics associated with this problem that I should have mentioned, even if it doesn't by itself explain the detailed structure of the periodic table. When Coulomb interactions between electrons and relativistic effects are ignored the energy levels of the Schrodinger equation with a central $1/r$ potential depend only the quantum number $n$, but not on the quantum number $\ell$ which determines the representation $V_\ell$ of $SO(3)$ referred to above. When screening is included this is no longer the case and the energies depend on both $n$ and $\ell$. Why is this? With a $1/r$ central potential there is an additional vector $\vec D$ which commutes with the Hamiltonian. Classically this vector is the Runge-Lenz vector and its conservation explains why the perihelion of elliptical orbits in a $1/r$ potential do not precess. Quantum mechanically the commutation relations of the operators $\vec D$ along with the angular momentum operators $\vec L$ are those of the Lie algebra of $SO(4)$ (for bound states with negative energy). There are two Casimir invariants, one vanishes and the other is proportional to the energy. As a result the energy spectrum depends only on $n$ and can be computed using group theory without ever solving the Schrodinger equation explicitly. Perturbations due to screening, that is from some averaged effect of the Coulomb interactions between electrons, change the $1/r$ potential to some more general function of $r$ and break the symmetry generated by the Runge-Lenz vector $\vec D$. As a result the energy levels depend on both $n$ and $\ell$.