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

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This question is a little bit trying to understand physics or chemistry 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.

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 or in chemistry?** I have read about spins of electrons but am unsure if this fits here or not.

Fiedler Graph of Helium:

[![Fiedler Graph of Helium][1]][1]

Fiedler Graph of Lithium:

[![Fiedler Graph of Lithium][2]][2]

Fiedler Graph of Beryllium:

[![fiedler_graph_of_beryllium][3]][3]

Fiedler Graph of Bor:

[![fiedler_graph_of_bor][4]][4]

Fiedler Graph of Carbon:

[![Fiedler Graph of Carbon][5]][5]

[Here are some pictures of the signed graphs.][6]

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