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.  

  

Atomisation Energy for X2 Molecules Valence Bond

As an important step in reducing computational cost in ab initio molecular dynamics presented in the previous post, we let RealQM compute atomisation energies E of X2 valence bond molecules with the X atom represented by 1 or 2 valence electrons surrounding a +1 or +2 kernel of certain radius R for a range of X and R and compare with experimental observations:

For +2 we get (code):

• R = 0     E = 0.38   compare with postulated E<0 for He2 and discussion below
• R = 0.5  E = 0.29   compare with 0.23 for C2
• R = 1     E= 0.19    compare with 0.18 for O2
• R=1.2    E= 0.14
• R=1.5    E=0.029  compare with 0.004 for Be2

We see increasing E with decreasing R as the full X kernel charge increases. Comparing with observations we can fit a proper R to X to be used in valence bond computations of molecules including X. 

We use these values to compute atomisation energies for 

• HeH     (code)      
• BeH     (code)
• CH       (code)
• OH       (code)

with fair agreement with observation. Recall that HeH does not form from He+H since He+H dissociates into He+ and H. An upcoming post will investigate if HeH can form directly from He and H. So even if HeH has lower energy than He and H separate, it does not guarantee that it will form, since a physical path to lower energy is required.  

 

For +1 we get (code):

• R = 0      E = 0.17    compare with 0.17 for H2
• R = 0.5   E = 0.09    compare with 0.11 for B2
• R = 1      E = 0.05   compare with 0.04 for Li2
• R = 1.5   E = 0.002  compare with 0.003 for Na2

We see the same pattern of increasing E with decreasing R with H2 to be compared to Na2. For 3 valence electrons represented by N see next post.

What stands out is the atomisation energy of He2 to be compared with that of H2 both with R=0 and maximal E. H2 with E=0.17 is the accepted observed standard value, while it is commonly claimed that He2 cannot form because E<0. 

RealQM gives E=0.38 for He2 indicating that He2 can form (maybe under pressure). 

So we have here a neat case to test RealQM: Does He form a He2 molecule, or not?