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]

Thanks for your help!

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[6]: https://sagecell.sagemath.org/?z=eJzNV21v2zYQ_h4g_0Fzvkg2TZOyHVtFPSD1shTbsjXtMAwThECWaZupRKmU3Cb_fkdSr7bbtEAHTIAt6fjcC-_I08ONTBMrfx-zUAq8ZlGaZGnOC54Ki8OjLKw3yyv0K5OCxfB0frbRChHPnnCUirwIRZFX0B2KkJXtnnIehfF9PYyst0_rFZPb87PzsySNrcUJkN-7-phuw7VMrUrWC3wSKJ3SvNgn2ZMV5pbIzs-uwYrv0zG-DND5mdW6fHeCL9F0gidHI1M8QbMpjFLXxdOjYQ-PEZ1jF9HpGHvIpTM8OwLNAeROATSeKczUA5vjCcFugPzLyxmaTl3kzfAUzdTfRP25nnI5OeFxMvfAyNhzMUWXE4MHoxA7ohSPj_BqvnROpmg8xdSlVE3TG0-1swkdTyF-gMw9D9HxHFPqzdBs7GHXQ_MZBQShQceiTu-abaytDBP7-sF5YYYlK_ZSWElYSP5o-37CBYz6PEDw_xA41iaVFre4sGQotsyOmRp3nEAPPJwYcCpPtwzk8gu-wlVufA2_ky_Scnb7AOumCcEIwbVaTWVqBDyXpoxAuQkbN2JAK2vq4hsrXCzIi26lZPpJmSTBwBZO32-nncU5O42mwcC_ffDDIUUrM7tVy63TsgER4zDLmFjboOucSiTcHMuq68uKu04eSJ0H0sy0tDB0-7ek37eH1Gnrt9TvlHpt08g4UrmliJZplEhEILgjWECMue3AtNovHZcwkO9XZehgCWn1tvt3fCvY-kaG2Q58ohxeF8O6EndlPE04NyAxaKep613lvqksb6e4VZeDxdUZK-t-50OkwUtihWJt6YAgIv3Cf1g8vLAurDDaFwwk25h1tXWEOFyv79karKsZ62wfeCCYvFyoDWFcOS8XIGm5I423Z-2TY_PG6o9ti4PuBNJVsc-_dgYD2l1HNxWuVcaqgP-fwv33tSo9SL7dFV_p4Lliff_61BX6QMgzO73esz5BJGipLrmM9knERNHp8KdsiIPW0PQQGCB1E4mgEeiuIp5pFwpYtwsZrvk-P_rSwMxMIOUUS5NKmH-QhQ1iZ2S7lZxW8uGaFXZjTWH6jUibgstoZZKLwpYwM9tBkqpbN2uSmmwvXTBff_QI-lbuYKnko28kEEaJGMJSkRfDY-B9_nkFYthQyYw0SfK-hCaGYhm2NT2mYm0k0WRNBX3E5bowMHqK73UhJAgQ1GC7ePvTz-3vx7LpBEvIPKRfL3y9yspP1bJejgK1ntufN3GwEJfPfbauum7LQLo2BktcQLPKgXoz2zHLI4tLShLph55k6x7qbSVjAu6reM_g9sTiOP0ED6nudfAQPYWiB1oiTFiuFV-D9LUa-o3D3yv19Ap-S_j9Dr8_FFq1TNG0TIo0_3EaoqOnXm1Ss8R7LGYJ7HQY6SFL-1NszUCulYLPRqpeuiEzZf0aACWJuTBGBGoaQqo2emubYZAkdg8OHKtetYf-UZA6ffByuNM7EX4g3ER3ojV1keor1OuI6nZVSj-m8ZtQhnHM4rRIs6o1VLwZq37Qwr6DM0vMHgF2oNino00YFankYVzP5KM-FDVqRhrphIAUdDKuOglwMjFyMXH62zBJQvUyoPCqBuA-quwpv7KshGB_qQYKheg_9vtuJX3_SwJi7xJP5lZfo8xIHCYrGIj6u5GtUX1KCBnBsc05rBzoIA1BSmlENWX0HBRJJCtwsQFjcHC0xb0-WwooQL4YV8N_32tAscEbXtzrTbBRVa_TGu3SmOXvn2xYDur4V-e4DEMZqCSRqnYEdY7hq5_hhIVCj6PwkecL4iA6YqL23GL8F5-jExcGWlFtZcznAxoMzVO12jUsU2silNA0xmvb1p3IQSUORWmcykUU-2Lo1lqxKnrMBevMQtuKMXsslEs_y2o8SMtAKjfaPOzLwDgigeOYrnaR7xM7ix2c79JPtlP2OSARwPBoiw_BUbvgYs8aVpWrNPhA1aHZBt1jTq7o7QGZKveP4U6atVkvqo2kLhYbxcPzUalnKNEX9D7jr6Q9JxRvTJM9dVrIW7AMNqN9U-WnkV8p8onDfZEmqcx2PE_utzLdZ21MNWcAySTVIEuDTLOJeV7YV86xAl9bKRwWW7bbaleYr-2DtpTrWajDebarZ6lDbzNppAjefRyu4GS5-FPuWTWtfwGVIH9T&lang=sage&interacts=eJyLjgUAARUAuQ==