Quantum
Numbers to Periodic Tables:

The above diagram is developed from one on the HyperPhysics website but converted to metric units. 
The second problem with the conventional images above is that the negatively charged electrons "associate" with the positively charged nucleus as three dimensional resonant standing waves:
The electrons move about the point positive charge
in a beautiful & subtle quantum mechanical dance
The modes of resonance for single electron systems such as the hydrogen atom are described by the Schrödinger wave equation. Very briefly Schrödinger:
Chemists recognise s, p, d and forbitals. The topologies of these orbitals: the shape, phase & electron occupancy are described by four quantum numbers:
n

The principal quantum number 
l

The subsidiary or azimuthal or angular momentum or orbital shape quantum number 
ml

The magnetic quantum number 
ms

The electron spin quantum number 
From Wikipedia:
These quantum numbers conspire to give spherical sorbitals, dumbbell shaped porbitals that come in sets of three, double dumbbell dorbitals that come in sets of five, etc.
Electrons enter and fill orbitals according to four rules:
Pauli
Exclusion Principal

Orbitals can contain a maximum of two electrons which must be of opposite spin. 
Aufbau
or Buildup Principle

Electrons enter and fill lower energy orbitals before higher energy orbitals. 
Hund's
Rule

When there there are degenerate (equal energy) orbitals available, electrons will enter the orbitals oneatatime to maximise degeneracy, and only when all the orbitals are half filled will pairingup occur. This is the rule of maximum multiplicity. 
Madelung's
Rule

Orbitals fill with electrons as n + l, where n is the principal quantum number and l is the subsidiary quantum number. This rule 'explains' why the 4s orbital has a lower energy than the 3d orbital, and it gives the periodic table its characteristic appearance. 
Certain 'magic' numbers of electrons of electrons exhibit energetic stability: 2, 10, 18, 36, 54, 86 and, one assumes, 118, are associated with the Group 18 inert or noble gases: He, Ne, Ar, Kr, Xe, Rn & Uuo.
The pattern of orbital structure is very rich and can be mapped onto the two dimensions of paper in many different ways.
Some mappings emphasize how the orbitals are ordered and filled with electrons, others stress how the chemical elements and their orbitals are ordered with respect to atomic number Z. Each tells us something different about atomic orbital structure and/or elemental periodicity.
Madelung's Rule says the orbitals fill in the order n + l (lowest n first). This gives the sequence:
 (n = 1) + (l = 0) = 1 1s
 (n = 2) + (l = 0) = 2 2s
 (n = 2) + (l = 1) = 3 2p
 (n = 3) + (l = 0) = 3 3s
 (n = 3) + (l = 1) = 4 3p
 (n = 4) + (l = 0) = 4 4s
 (n = 3) + (l = 2) = 5 3d
 (n = 4) + (l = 1) = 5 4p
 (n = 5) + (l = 0) = 5 5s
 and so on...
A nice electron shell representation from the Encyclopedia Britannica:
Thus, the orbital filling sequence is, from the bottom of this diagram, upwards because the lowest energies fill first:
As electrons are added, the quantum numbers build up the orbitals. Read this diagram, from the top downwards:
The electronic structure can be illustrated adding electrons to boxes (to represent orbitals). This representation shows the Pauli exclusion principle, the aufbau principle and Hund's rule in action.
There are some subtle effects with the d block elements chromium, Cr, and copper, Cu. Hund's rule of maximum multiplicity lowers the energy of the 3d orbital below that of the the 4s orbital, due to the stabilisation achieved with a complete and spherically symmetric set of five 3d orbitals containing five or ten electrons. Thus,
A periodic table – of sorts – can be constructed by listing the elements by n and l quantum number:
The problem with this mapping is that the generated sequence is not contiguous with respect to atomic number atomic number, Z, and so is NOT a periodic table. Try counting the numbers past the red lines  on this representation:
Named after a French chemist who first published in the formulation in 1929, the Janet or LeftStep Periodic Table uses a slightly different mapping that is contiguous with atomic number:
While the Janet periodic table is very logical and clear, it does not separate metals from nonmetals as well as the Mendeleev version, and helium is problem chemically.
However, it is a simple 'mapping' to go from the Janet or LeftStep periodic table to a modern formulation of Mendeleev's periodic table. There are a couple of steps:
Including the movement of He to Group 18:
Drop the fblock down, and move the sblock and d and pblocks into the space produced, and this gives the commonly used medium form periodic table:
Note, this is the 'correct' medium form, with Group 3 as: Sc, Y, Lu & Lr and with the fblock as LaYb and AcNo.
Many medium form PTs incorrectly leave a gap where Lu and Lr are situated, and then add these two elements to the end of the fblock:
The Devil In The DetailMatters are considerably more involved than implied above. Wavefunctions are called orbitals by chemists, but this ignores the existence of real and imaginary portions of the complex wave function – where the term 'complex' is being used in the mathematical sense. Orbitals should not be considered as being physically real, and in principle no experimental technique can directly observe orbitals.
Solutions for the isolated hydrogen atom must be physically spherically symmetric (because the isolated hydrogen atom force field is spherically symmetric) and this gives rise to the 1s, 2s, 3s.. 'shells'. But mathematically one also gets 2p, 3p... solutions (lobes) that are not spherically symmetric because theta and phi are defined. These angles presuppose x, y, z axes that exist mathematically and allow nonsymmetrical solutions. Physically these lobes don't exist for any isolated atom, but chemists talk about them when atoms are closely interacting, i.e. they exist in nonuniform fields, where axes and lobes can make physical sense. The Schrödinger wave equation can only be solved analytically for one electron systems. For multielectron systems it is necessary to use approximations, and the more electrons are involved, the more severe are the approximations. That said, mathematically manipulated orbitals are the foundation of much computational quantum chemistry software. We chemists can fool ourselves into thinking that we understand s and porbitals, because they can be superimposed upon (mapped to) the easily understandable three dimensional x, y, z Cartesian coordinate system. But dorbitals project into 5dimensional space, and forbitals into 7dimensional space. dOrbitals can be mapped to the 3dimensional Cartesian coordinate system, but a mathematical artifact is the 'strange looking' dz^{2} orbital. Real, physical chemical structure and reactivity systems also exist in three dimensional Cartesian coordinate space. Mother Nature must also solve the 5dimensional to Cartesian dimensional reduction when forming transition metal complexes, such as [Cu(H_{2}O)_{6}]^{2+}, that use dorbitals. Understanding comes from crystal field theory that is able to predict what happens when a Cartesian array of charges interacts with dorbitals that are degenerate in 5dimensional space. The geometry causes the 5 dorbitals to split in energy to give three lower energy orbitals and two higher energy orbitals. Thanks to DD, an exphysicist/philosopher of science, for his input into this section.

Segré Chart 
The
Periodic Table: What Is It Showing?

© Mark R. Leach 1999
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