Quantum Mechanics Contradictions 20: Addendum

The smoothed particle model of atomic physics to be explored computationally in coming posts leads to the following simple model for the ground state energy of an atom with Z + 2  electrons (at most 10 to start with) with 2 electrons in a first inner shell of radius and Z electrons in a second outer shell of radius R, the same for all Z.

We assume the ground state energy is half of the kernel potential energy (in correspondence with the virial theorem) and motivate the constant radius of the outer shell from the idea that the "width" of an electron in the second shell scales like 1/Z and thus Z electrons could fill up a shell of constant radius, in accordance with the observation that atomic radii vary surprisingly little with the number (beyond one or two) of electrons in the outer shell.

The energy contribution e_Z from the second shell will then be Z x Z / D with D = 2 x R, which gives the following contributions with R = 1 with exact values in parenthesis:

• Lithium: e_1 = - 0.5 ( - 0.2)
• Beryllium: e_2 = - 2 ( - 1)
• Boron: e_3 = - 4.5 ( - 1.9)
• Carbon: e_4 = - 8 (- 5.25) 
• Nitrogen: e_5 = - 12.5 (- 9.8)
• Oxygen: e_6 = - 18 (- 15.91)
• Fluorine: e_7 = - 24.5 ( - 24.2)
• Neon: e_8 = - 32 (- 34)

We see a fairly good agreement with respect to the simplicity of the model (complemented by the energies for the inner shell) , which gives good promise to the coming computational exploration.

PS Continuing with a third shell of radius R = 2 and D = 2 x R = 4, we get with
e_Z = (Z -10) x (Z - 10) /D:

• Sodium: e_11 = - 0.25 (- 0.18)
• Magnesium: e_12 = - 1  (- 0.80)
• Aluminium: e_13 = - 2.25 (- 1.9)
• Silicon: e_14 = - 4 (- 3.65)
• Phosphorus: e_15 = - 6.25 (- 6.2)
• Sulfur: e_16 = - 9 (- 9.82)
• Chlorine: e_17 = - 12.25 (- 14.5)
• Argon: e_18 = - 16 (- 20.5) 

Again a fairly good agreement. Taking into account that the atomic radius decreases with increasing number of electrons within the band, improves the fit between model and observation,