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:

enter image description here

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
  • $\begingroup$ also asked here: physicsoverflow.org/45760/… $\endgroup$ Commented Oct 16, 2023 at 16:42
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    $\begingroup$ "Let $E_{i,j}$ be the ionization energy [...]" is not a mathematical definition (even if such an interpretation might be useful for some users here). Can you state your question in mathematical terms (and then perhaps add a link to a physical interpretation)? $\endgroup$ Commented Oct 16, 2023 at 18:01
  • $\begingroup$ @IosifPinelis: en.wikipedia.org/wiki/… $\endgroup$ Commented Oct 16, 2023 at 18:11
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    $\begingroup$ @mathoverflowUser I am interested in your questions too, so is it possible to do a translation into mathematics for them? So that a person with a much less physics background can still help out. $\endgroup$ Commented Oct 16, 2023 at 21:24
  • $\begingroup$ @ThomasKojar I updated the question with the details you asked. $\endgroup$ Commented Oct 17, 2023 at 0:55

1 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?

  • $\begingroup$ The definition of the shell, is based on the visual impression one has when looking at the ionisation energies. This suggests that the right tool is to do cluster analysis on the vectors of the simplex. I have done this in the updated question for copper and one can clearly see three clusters. However at the moment I have no idea how to "prove" the $2n^2$ rule using the simplex geometry.... $\endgroup$ Commented Oct 17, 2023 at 2:04

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