Finite Element Quantum Mechanics 3: Explaining the Periodicity of the Periodic Table

According to Eric Scerri, the periodic table is not well explained by quantum mechanics, contrary to common text book propaganda, not even the most basic aspect of the periodic table, namely its periodicity:

• Pauli’s explanation for the closing of electron shells is rightly regarded as the high point in the old quantum theory. Many chemistry textbooks take Pauli’s introduction of the fourth quantum number, later associated with spin angular momentum, as the foundation of the modern periodic table. Combining this two-valued quantum number with the ear- lier three quantum numbers and the numerical relationships between them allow one to infer that successive electron shells should contain 2, 8, 18, or $2n^2$ electrons in general, where n denotes the shell number. 
• This explanation may rightly be regarded as being deductive in the sense that it flows directly from the old quantum theory’s view of quantum numbers, Pauli’s additional postulate of a fourth quantum number, and the fact that no two electrons may share the same four quan- tum numbers (Pauli’s exclusion principle). 
• However, Pauli’s Nobel Prize-winning work did not provide a solution to the question which I shall call the “closing of the periods”—that is why the periods end, in the sense of achieving a full-shell configuration, at atomic numbers 2, 10, 18, 36, 54, and so forth. This is a separate question from the closing of the shells. For example, if the shells were to fill sequentially, Pauli’s scheme would predict that the second period should end with element number 28 or nickel, which of course it does not. Now, this feature is important in chemical education since it implies that quantum mechanics can- not strictly predict where chemical properties recur in the periodic table. It would seem that quantum mechanics does not fully explain the single most important aspect of the periodic table as far as general chemistry is concerned. 
• The discrepancy between the two sequences of numbers representing the closing of shells and the closing of periods occurs, as is well known, due to the fact that the shells are not sequentially filled. Instead, the sequence of filling fol- lows the so-called Madelung rule, whereby the lowest sum of the first two quantum numbers, n + l, is preferentially oc- cupied. As the eminent quantum chemist Löwdin (among others) has pointed out, this filling order has never been derived from quantum mechanics. 

On the other hand, in the new approach to atomic physics I am exploring, the periodicity directly connects to a basic partitioning or packing problem, namely how to subdivide the surface of a sphere in equal parts, which gives the sequence $2n^2$ by dividing first into two half spheres and then subdividing each half spherical surface in $n\times n$ pieces,  in a way similar to dividing a square surface into $n\times n$ square pieces.  With increasing shell radius an increasing number of electrons, occupying a certain surface area (scaling with the inverse of the kernel charge), can be contained in a shell. 

In this setting a "full shell" can contain 2, 8, 18, 32,.., electrons, and the observed periodicity 2, 8, 8, 18, 18, 32, 32, with each period ended by a noble gas with atomic numbers 2 (He), 10 (Neon), 18 (Argon), 36 (Krypton), 54 (Xenon),  86 (Radon), 118 (Ununoctium, unkown), with a certain repetition of shell numbers, can be seen as a direct consequence of such a full shell structure, if allowed to be repeated when the radius of a shell is not yet large enough to house a full shell of the next dignity. 

Text book quantum mechanics thus does not explain the periodicity of the periodic table, while the new approach am I pursuing may well do so in a very natural way.   Think of that.