RealQM Molecules with Valence Electrons

Standard Quantum Mechanics stdQM teaches that the electrons of an atom successively fill an expanding system of shells around the kernel, with 2 electrons in the 1st shell, 8 in the 2nd, 18 in the 3rd, more generally with (at most) $2m^2$ electrons in shell $m$.

RealQM is classical 3d space continuum mechanics model of atoms and molecules based on non-overlapping electron densities subject to Coulomb interaction. In particular, RealQM gives a rationale for the shell system as a packing problem. For an atom with kernel charge $Z$  electrons successively fill a sequence of non-overlapping spherical shells $S_m$ of radius $r_m\sim \frac{m^3}{Z}$ of thickness $dS_m\sim \frac{m^2}{Z}$ with $\sim m^2$ electrons of width $de_m\sim \frac{m^2}{Z}$ in each shell, for $m=1,2,...,M,$ with $M^3\sim Z$, which gives the sequence $2, 8, 18, 32, 50,..$. Compare with previous post connecting to charge density $\psi^2(r)\sim \frac{Z}{r^2}$ as function of kernel distance $r$. 

Check out by running Atom Simulator giving support to $\psi^2(r)r^2$ approximately constant as  emergent design principle revealed by RealQM, with in particular the total energy in $S_m$ dropping off as $\frac{1}{m+1}$.

In reality outer shells are filled with slower progression and the outermost shell is filled with valence electrons which determine interaction with other atoms as chemistry based on valence bonds. The number $K$ of valence electrons shows to range from 1 to 4 representing different columns in the periodic table,  while noble gasses have 8. 

To reduce computational cost all electrons except the valence electrons can be homogenised into a reduced form of RealQM with the valence electrons interacting with a Z-kernel surrounded by a charge density of total charge $Z-K$, thus effectively involving only $K+1$ electron densities. You can test 2-atom chemistry here: 

• Valence +2+2  (MgO, SO and MgS)
• Valence +1+2  (LiO, NaO)     

with good agreement of atomisation energy. This shows promise that RealQM can simulate complex chemistry in computations involving only valence electrons, thus with computational cost for a molecule with $N$ atoms scaling as $N^2$, way smaller than the exponential cost for multidimensional stdQM.