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?
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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.
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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? Journal of Chemical Education 75 (1998), 1384.
Scerri also wrote a book on the subject
(The periodic table: its story and its significance, 2007).
Several book reviews are available online:
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 the Americal Scientist,
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.''
For a very recent review on the expert level, see the paper The physics behind chemistry, and the Periodic Table by Pyykkü. 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 Schroedinger equation) for their correct explanation. He treats the periodic system shorter than the title would suggest, but makes up for this in this paper.