Is the Mendeleev table explained in quantum mechanics? Does anybody know if there exists a mathematical explanation of Mendeleev table in quantum mechanics? In some textbooks (for example in F.A.Berezin, M.A.Shubin. The Schrödinger Equation) the authors present quantum mechanics as an axiomatic system, so one could expect that there is a deduction from the axioms to the main results of the discipline. I wonder if there is a mathematical proof of the Mendeleev table?
 A: Yes, quantum mechanics – even non-relativistic quantum mechanics for several electrons orbiting nuclei – fully, quantitatively, and comprehensively explains all of chemistry (including biochemistry and, in fact, biology). This fact has been known since the late 1920s.
To understand the periodic character of the properties of the elements, one must realize that already the Hydrogen atom has energy eigenstates given by quantum numbers $(n,l,m)$ as well as the binary $s_z$. Energy as well as degeneracy increases as a function of $n$. When many electrons are allowed (to neutralize the positive electric charge of the nucleus), the Pauli principle (coming from the antisymmetry of the electrons' wave functions, a fact that may be deduced from quantum field theory but may be assumed as another axiom of the simplified quantum mechanical model) says that the electrons will gradually fill the states with the ever higher values of $n$. Every time one fills all states with $n<n_0$ up to some $n_0$, one gets inert gases. When one more electron is added to the new shell, we get highly reactive elements (because they include one loosely bound electron in the outer shell), and so on.
The only variation one has to add to make the calculation of the atomic energy levels exact are the electron-electron interactions (if there are at least two electrons). They slightly reorder the shells that are being filled, $1s, 2s, 2p, 3s, 3p, 4s, 3d$, and so on... The problem (aside from the basic Hydrogen problem) obviously can't be solved analytically but there exist lots of numerical techniques to find the right results and everything that has been calculated - and some of the calculations were very precise - agrees with the observations. The calculations become more complex for larger atoms (or molecules), of course. But when the size is large enough, one may use new simplifying assumptions or approximations so it's not necessary the case that it's always harder to understand/calculate larger objects.
A: While quantum mechanics explains the gross features of the periodic system, many fine details of the periodic table of elements are computable numerically from various approximations to QED, but are conceptually ill understood. See, e.g., 

Eric R. Scerri, How Good Is the Quantum Mechanical Explanation of the Periodic System? J. Chem. Educ. 75, 1384 (1998).

Scerri also wrote a book on the subject (The periodic table: its story and its significance, 2007). Several book reviews are available online:


*

*
Werner Kutzelnigg writes in his review: ''I am personally skeptical 
whether a genuine PS [periodic system] of the elements that 
incorporates all chemical properties of the elements (e.g., their 
tendency to form covalent, ionic, semipolar, multicenter, or 
hypervalent bonds) will ever be formulated. Another issue about which 
one would like to learn more is whether the periodic system has a 
chance to survive in the realm of superheavy elements.''

*
Michael Laing describes (for the Platinum Metals review) in his 
review anomalies of platinum.


It is difficult to derive from the periodic table (or from quantum 
mechanics) precise, generally valid laws about chemical elements.
In a 2008 paper for American Scientist (96 no. 1, 52-58 (2008)), The past and future of the periodic table, 
Scerri writes about the predictive power of the periodic system, 

if one considers all of Mendeleev's many predictions 
  of new elements, his powers of prophecy appear somewhat less 
  impressive, even to the point of being a little worrying. In all 
  Mendeleev predicted a total of 18 elements, of which only nine were 
  subsequently isolated. [...] the Davy medal, which predates the Nobel 
  Prize as the highest accolade in chemistry, was jointly awarded to 
  Mendeleev and Julius Lothar Meyer, his leading competitor, who did not 
  make any predictions. Indeed, there is not even a mention of 
  Mendeleev's predictions in the published speech that accompanied the 
  joint award of the Davy prize. It therefore seems that this prize was 
  awarded for the manner in which the two chemists has successfully 
  accommodated the then-known elements into their respective periodic 
  systems rather than for any foretelling.
it is possible to predict that subsequent main shells of the atom 
  can contain a maximum of 2, 8, 18 or 32 electrons. This is in perfect 
  agreement with the lengths of periods in the chemist's periodic table. 
  The simple quantum mechanical theory does not, however, account for the 
  repetition of all period lengths except for the first one. Indeed, 
  this problem has continued to elude theoretical physicists until quite 
  recently. Appropriately enough, it was a Russian physicist, the late 
  Valentin Ostrovsky, who recently published a theory to explain this 
  feature, although it is not yet generally accepted. Although the theory 
  is too mathematically complicated to explain here, Ostrovsky's work 
  and some other competing accounts demonstrate that the periodic table 
  continues to be an area of active research by physicists as well as 
  chemists even though it has existed for nearly 140 years.''

For a very recent review on the expert level, see the paper

The physics behind chemistry, and the Periodic Table by Pekka Pyykkü (Chem. Rev. 112, 371 (2012)).
He mentions that a number of important effects (such as the color of 
gold, the liquidity of mercury, or the voltage of a lead-acid battery) 
need QED (more precisely the Dirac-Coulomb-Breit approximation to QED 
rather than the textbook nonrelativistic Schrödinger equation) for 
their correct explanation. He treats the periodic system shorter than 
the title would suggest, but makes up for this in

this paper (Phys. Chem. Chem. Phys. 13, 161 (2011)).
Of interest may also be papers by

Bonchev
and

Kibler; the latter relates the periodic system to the dynamical 
symmetry group $\rm SO(4,2)$ of the hydrogen atom.
A: From what I learned here and at MathOverflow, where this question was originally posted, I deduce that the answer to it is “no”. I.e. the mathematical proof of the Mendeleev table was  not found by now. I am sorry to inform you, dear physicists. And this has nothing in common with the nuances you are talking about, in particular with the problem of “superheavy elements”. Even the simpliest properties of elements are formally not explained from the mathematical point of view, since no axiomatic theory was constructed for this (in contrast to some other physical disciplines, like classical mechanics). Those of you, who don’t agree can join the discussion at MathOverflow for explaining your opinion to mathematicians.
