How far does quantum mechanics explain the periodic table?

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.,

E.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:
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 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.''
''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 recent review on the expert level, see the 2011 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.

Of interest may also be papers by Bonchev and Kibler; the latter relates the periodic system to the dynamical symmetry group SO(4,2) of the hydrogen atom.