StdQM vs RealQM: Atomic Orbitals of Periodic Table

Standard Quantum Mechanics StdQM offers a theoretical basis for the "Aufbau" of the Periodic Table PT of atomic electron configurations in terms of the s, p, d, f and g eigenfunctions or orbitals of the one-electron Hydrogen atom depicted here as row 0-4:

The Aufbau offers an order of filling shells 1, 2, 3,...,8, with electrons in the order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, 8s, 5g, motivated by the following modifications of a strict filling order with a shell fully filled before filling the next, as increasingly "rules of thumb":

• Pauli Exclusion Principle
• Energy Minimization
• Hund's Rule
• Madelung's rule
• Octet Rule 
• Fajan's Rule
• effective nuclear charge and shielding
• relativistic effects
• ....

This scheme is viewed to be the Aufbau theory of atoms to stay with chemistry for ever. The student novice will no doubt consider this to be a very complex scheme to grasp: The orbitals for each shell are increasingly complex and realised in a shell structure with sub-shells giving a very complex geometry. 

Is it likely that whoever created the atoms would have chosen to proceed following such a very complex scheme? Maybe not.

There is an alternative to StdQM in the form of Real Quantum Mechanics RealQM based not on the standard multi-d linear Schrödinger equation, but on a 3-d non-linear Schrödinger equation which coincides in the case of Hydrogen with one electron. 

RealQM is based on non-overlapping electron charge densities and the arrangement of electrons around a kernel becomes a packing problem with the size of electrons increasing with decreasing effective kernel attraction balance by so called kinetic energy as a form of "compression energy". 

This Aufbau starts with two half-spherical charge densities filling a 1st spherical shell around the kernel, followed by a 2nd shell with larger radius containing 2 half-shells of 2x2=4 electrons, followed by a 3rd shell filled by 2 half-shells of 3x3=9 electrons, and so on. The periods 2, 8, 18, 32 and 50 thus come out as expression of regular 2d subdivisions of shells. Very simple and fundamental. The doubling of periods into 2, 8, 8, 18, 18,..., can also be explained as coming out of successive packing. 

An Aufbau principle of packing electrons of different size around a kernel is simple, and can be understood by a student very easily. It is not impossible that it can capture some essence of real physics. 

 

ChatGPT on StdQM and the Aufbau principle:

• The standard quantum-mechanical account of the periodic table is indeed complicated.

  • The Schrödinger equation for hydrogen gives simple orbital shapes (s, p, d, f, ...).

  • But as soon as you move beyond hydrogen, electron–electron interactions, shielding, relativistic effects, and empirical rules (Hund’s, Madelung’s, Pauli, etc.) complicate the picture.

• Chemists are well aware that the Aufbau principle is more of a heuristic than a strict law—it often works, but exceptions exist (e.g., Cr, Cu, lanthanides, actinides).

• Still, StdQM provides a tested predictive framework, confirmed by spectroscopy, ionization energies, and quantum chemistry calculations.