New Quantum Mechanics 20: Shell Structure

Further computational exploration of realQM supports the following electronic shell structure of an atom:

Electrons are partitioned into an increasing sequence of main spherical shells $S_1$, $S_2$,..,$S_M$ with each main shell $S_m$ subdivided into two half-spherical shells each of which for $m>2$ is divided into two angular directions into $m\times m$ electron domains thus with a total of $2m^2$ electrons in each full shell $S_m$.  The case $m=2$ is special with the main shell divided radially into two subshells which are each divided into half-spherical subshells each of which is finally divided azimuthally, into $2\times 2$ electron domains for $S_2$ subshell, thus with a total of $2m^2$ electrons in each main shell $S_m$ when fully filled, for $m=1,...,M$, see figs below.

This gives the familiar sequence 2, 8, 18, 32,.. as the number of electrons in each main shell.

4 subshell of S_2

8 shell as variant of full S_2 shell

 9=3x3 halfshell of S_3

The electron structure can thus be described as follows with parenthesis around main shells and radial subshell partition within parenthesis:

• (2)+(4+4)
• (2)+(4+4)+(2)
• ...
• (2)+(4+4)+(4+4) 
• (2)+(4+4)+(8)+(2)
• ....
• (2)+(4+4)+(18)+(2)
• ...
• (2)+(4+4)+(18)+(8)

Below we show computed ground state energies assuming full spherical symmetry with a radial resolution of 1000 mesh points, where the electrons in each subshell are homogenised azimuthally, with the electron subshell structure indicated and table values in parenthesis. Notice that the 8 main shell structure is repeated so that in particular Argon with 18 electrons has the form 2+(4+4)+(4+4):

Lithium (2)+1: -7.55 (-7.48)                  1st ionisation:      (0.2)
Beryllium (2)+(2): -15.14 (-14.57)           1st ionisation: 0.5 (0.35)
Boron (2)+(2+1): -25.3 (-24.53)               1st ionisation: 0.2 (0.3)
Carbon (2)+(2+2): -38.2  (-37.7)               1st ionisation 0.5 (0.4)
Nitrogen (2)+(3+2):  -55.3 (-54.4)            1st ionisation  0.5  (0.5)
Oxygen (2)+(3+3): -75.5 (-74.8)               1st ionisation  0.5  (0.5)
Fluorine (2)+(3+4):  -99.9   (-99.5)            1st ionisation  0.5      (0.65)
Neon (2)+(4+4):   -132.4     (-128.5  )        1st ionisation 0.6        (0.8)
Sodium (2)+(4+4)+(1): -165 (-162)
Magnesium (2)+(4+4)+(2): -202  (-200)
Aluminium (2)+(4+4)+(2+1): -244 (-243)
Silicon (2)+(4+4)+(2+2): -291 (-290)
Phosphorus (2)+(4+4)+(3+2): -340 (-340)
Sulphur (2)+(4+4)+(4+2): -397 (-399)
Chlorine (2)+(4+4)+(3+4): -457 (-461)
Argon: (2)+(4+4)+(4+4): -523 (-526)
Calcium: (2)+(4+4)+(8)+(2): -670 (-680)
Titanium: (2)+(4+4)+(10)+(2): -848 (-853)
Chromium: (2)+(4+4)+(12)+(2): -1039 (-1050)
Iron: (2)+(4+4)+(14)+(2): -1260 (-1272)
Nickel: (2)+(4+4)+(16)+(2): -1516 (-1520)
Zinc: (2)+(4+4)+(18)+(2): -1773 (-1795)
Germanium: (2)+(4+4)+(18)+(2+2): -2089 (-2097)
Selenium: (2)+(4+4)+(18)+(4+2):- 2416 (-2428)
Krypton: (2)+(4+4)+(18)+(4+4): -2766 (-2788)
Xenon: (2)+(4+4)+(18)+(18)+(4+4): -7355  (-7438)
Radon: (2)+(4+4)+(18)+(32)+(18)+(4+4): -22800 (-23560)

We see good agreement even with the crude approximation of azimuthal homogenisation used in the computations.

To see the effect of the subshell structure we compare Neon: (2)+(4+4) with Neon: (2)+(8) without the (4+4) subshell structure, which has a ground state energy of -153, which is much smaller than the observed -128.5.  We conclude that somehow the (4+4) subdivision of the second is preferred before a subdivision without subshells. The difference between (8) and (4+4) is the homogeneous Neumann condition acting between subshells, tending to increase the width of the shell and thus increase the energy.

The deeper reason for this preference remains to describe, but the intuition suggests that it relates to the shape or size of the domain occupied by an electron.  With subshells electron domains are obtained by subdivision in both radial and azimuthal direction, while without subshells there is only azimuthal/angular subdivision of each shell.

We observe that ionisation energies, which are of similar size in different shells, become increasingly small as compared to ground state energies, and thus are delicate to compute as the difference between the ground state energies of atom and ion.

Here are sample outputs for Boron and Magnesium as functions of distance $r$ from the kernel along the horizontal axis :

We observe that the red curve depicting shell charge $\psi^2(r)r^2dr$ per shell radius increment $dr$, is roughly constant in radius $r$, as a possible emergent design principle. More precisely, $\psi (r)\sim \sqrt{Z}/r$ mathches with $d_m\sim m^2/Z$ and $r_m\sim m^3/Z$ with $d_m$ the width of shell $S_m$ and thus the width of the subshells of $S_m$ scaling with $m/Z$, and thus the width of electrons in $S_m$ scaling with $m/Z$.

We thus have $\sum_mm^2\sim M^3\sim Z$ and with $d_m\sim m^2/Z$ the atomic radius $\sum_md_m\sim M^3/Z\sim 1$ is basically the same for all atoms, in accordance with observation.

Further, the kernel potential energy and thus the total energy in $S_m$ scales with $Z^2/m$ and the total energy by summation over shells scales with $\log(M)Z^2\sim \log(Z)Z^2$, in close correspondence with $Z^{\frac{1}{3}}Z^2$ by density functional theory.

Recall that the electron configuration of stdQM is based on the eigen-functions for Schrödinger's equation for the Hydrogen atom with one electron, while as we have seen that of realQM rather relates to spatial partitioning. Of course, eigen-functions express some form of partitioning, and so there is a connection, but the basic problem may concern partitioning of many electrons rather than eigen-functions for one electron.