Qualitative values between two electrons in an atom or how to interpret these values?

Question

This question is a little bit trying to understand physics through geometry of simplex:

Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with order number $Z = i$ and $j$-th electron to be ionized.

As an example, the numbers $13.6 \cdot E_{i,j}$ might be found in the following lower triangular matrix:

$$ \left(\begin{array}{rrrrr} 13.6000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 \\ 24.6000000000000 & 54.4000000000000 & 0.000000000000000 & 0.000000000000000 & 0.000000000000000 \\ 5.40000000000000 & 75.6000000000000 & 122.500000000000 & 0.000000000000000 & 0.000000000000000 \\ 9.30000000000000 & 18.2000000000000 & 153.900000000000 & 217.700000000000 & 0.000000000000000 \\ 8.30000000000000 & 25.2000000000000 & 37.9000000000000 & 259.400000000000 & 340.200000000000 \end{array}\right) $$

The value $13.6$ refers to the ionization energy of hydrogen in $eV$. All values are scaled with $1/13.6$ the value of hydrogen. So hydrogen itself has value $1 = 13.6/13.6$. Helium has for the first (outermost) electron the value $24.6/13.6$ and for the second $54.4/13.6$. I did this scaling to have a kind of independence of the units of energy, but it might be accused to be arbitrary since it arbitrarily chooses Hydrogen as a reference point. The other numbers used in the script to produce the Fiedler graphs / simplices, can be found here as CRC values. Then $|E_{Z,i}-E_{Z,j}|$ can be interpreted in physics as the hypothetical energy of light emitted or absorbed when an electron of an $Z$-atom moves from $i$ to $j$ state.

We can write $|E_{Z,i}-E_{Z,j}|$ as:

$$|E_{Z,i}-E_{Z,j}| = |\psi(Z,i)-\psi(Z,j)|^2$$

where $\left < \psi(Z,i), \psi(Z,j) \right> = \min(E_{Z,i},E_{Z,j})$ is the dot product in some Hilbert space.

In this Hilbert space we have:

$$\text{ squared norm of vector } = |\psi(Z,i)|^2 = \text{ ionization Energy of electron }i$$

and also:

$$\text{ squared Distance } = \text{ Energy }$$

We can look, as in the book of Miroslav Fiedler, at the matrices:

$$M_Z := ( |E_{Z,i}-E_{Z,j}|)_{1\le i,j\le Z}$$

$$M_{Z,0} := \left(\begin{array}{rr} 0 & \mathit{e^T} \\ e & M_Z \end{array}\right)$$

$$Q_{Z,0} := -2 M_{Z,0}^{-1} =: \left(\begin{array}{rr} q_{00} & \mathit{q_0^T} \\ q_0 & Q_Z \end{array}\right)$$

We define an undirected graph on the electrons:

$$q_{ij} <0 \rightarrow \text{ acute angle between faces in simplex} = (-1) = s_{ij}$$ $$q_{ij} =0 \rightarrow \text{ right angle between faces in simplex} = (0)= s_{ij}$$ $$q_{ij} >0 \rightarrow \text{ obtuse angle between faces in simplex} = (+1)= s_{ij}$$

One might then also compute the radius of the circumscribed hypersphere:

$$r_Z = \frac{\sqrt{q_{00}}}{2} = \sqrt{-\frac{\det(M_Z)}{2\det(M_{Z,0})}}$$

Question: Is there an interpretation of the qualitative value $s_{ij} = -1,0,+1$ between two electrons in an atom in terms of known physics? I have read about spins of electrons but am unsure if this fits here or not.

Here are some pictures of the signed graphs, which seem to be path graphs!

Edit: I updated the graphs. There was a numerical issue with $q_{ij}$ and Sagemath. I corrected this now. It seems that the Fiedler graphs are all path graphs. This means that it is possible using the acute graph to do a graph coloring on two colors: (spin up and spin down) for the electrons.

To the question of @CarloBeenakker: Here is a picture of copper (Cu) after doing dimensionality reduction (TruncatedSVD) on the Gram matrix, we can clearly see three clusters:

Thanks for your help!

Another edit: In the following table I have compiled the radius of the circumscribed sphere of the corresponding simplex of electrons and converted it to energies in eV, kJ/mol and wavelength of light in m:

element =  H
radius (energy in eV, energy in kJ/mol, wave length in m)
0.000000000000000 0.000000000000000 +infinity

element =  He
radius (energy in eV, energy in kJ/mol, wave length in m)
7.45000000000000 718.776000000000 1.66430940430255e-7

element =  Li
radius (energy in eV, energy in kJ/mol, wave length in m)
29.2750000000000 2824.45200000000 4.23539028592792e-8

element =  Be
radius (energy in eV, energy in kJ/mol, wave length in m)
52.1000000000000 5026.60800000000 2.37986661459770e-8

element =  B
radius (energy in eV, energy in kJ/mol, wave length in m)
82.9750000000000 8005.42800000000 1.49431817560157e-8

element =  C
radius (energy in eV, energy in kJ/mol, wave length in m)
163.125000000000 15738.3000000000 7.60098394608674e-9

element =  N
radius (energy in eV, energy in kJ/mol, wave length in m)
119.650000000000 11543.8320000000 1.03628124212737e-8

element =  Mg
radius (energy in eV, energy in kJ/mol, wave length in m)
488.754690000000 47155.0524912000 2.53687694783123e-9

element =  Cu
radius (energy in eV, energy in kJ/mol, wave length in m)
2889.97265500000 278824.561754400 4.29038836772454e-10

Answer 1

The key physics that governs the ionisation energies is shell formation; electrons in the same atomic shell have similar ionisation energies; the number of electrons in the $n$-th shell is $2n^2$, so you get the sequence $2, 8, 18, 32$;

For example, for Lithium, with $Z=3$ electrons you get $E_{3,1}\ll E_{3,2}\approx E_{3,3}$, because $E_{3,1}$ refers to an electron in the $n=2$ shell, while $E_{3,2}$ and $E_{3,3}$ both refer to electrons in the $n=1$ shell.

How might the simplex construction capture the $2n^2$ shell formation rule...?

---

The origin of the $2n^2$ rule is the number of eigenvalues $2l+1$ of the angular momentum operator in the $n$-th shell: $\sum_{l=0}^{n-1}(2l+1)= n^2$. The spin up or down doubles this to $2n^2$.

I am not aware of a geometric construction of this. A group theoretic explanation is considered in Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?