H2 potential energy; electron bench points


A simple explanation for chemical bonds: Electron bench points

By Juman Hijab

Reading time: minutes

Original date: November 4, 2021  

Updated: August 21, 2023

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What are electron bench points?

H2 potential energy; electron bench points

By magnetix. Shutterstock ID: 467129510. energy curve of hydrogen bonding H2.

A chemical bond is a fluid, multiform thing

                                                                                                                                                     Linus Pauling¹

                                                                                                                                                     1901 -  1994

In this post, I will describe the positions in which electrons seem to rest in longer than in others. I call these electron "bench points".

Shortest distance between two points

In a femtosecond time frame, electrons are always moving around with a profusion of positions (aka orbitals) around the nucleus.

However, there are some positions that are more likely (lower energy) than others. Electrons will understandably hang out in those positions for longer periods of time, like a traveler resting on a bench for a bit before getting up again to check alternate paths.


Grafixo. Geodesics, uploaded Nov 2021.

Like other minute particles, electrons follow shortest distance paths as they search for lower energy positions. On a sphere, the shortest distance between two points is that which follows one of the great circles of the sphere (geodesic). 

An electron will move from one geodesic to the next looking for that which has the shortest distance between two points. If it bumps into an arc that is longer, it will revert back to one of the shorter arcs. An equilibrium is reached when none of the geodesics is longer than the next. 

Electron bench points are equidistant from each other

The key here is that points on the sphere have to be equally spaced from each other. This will help the electron have equal geodesics from which to choose. Amazingly, it turns out  that there aren't that many choices that an electron has. 

The reason for this is that the electron would like to minimize the number of paths that it has to travel. Moreover, when there is more than one electron, the electrons have to be spaced as far apart from each other as they can be -  since they repulse each other.

These are the electron's stringent requirements. The points on a sphere have to:

  1. keep electrons as far away from each other as possible
  2. remain equidistant from other points
  3. have the shortest possible distance between two points 
  4. Minimize the total number of paths available

A table that  shows an electron's options

Here is a table showing 1, 2, 3, and 4 points on a sphere that could potentially meet the above requirements: 

# of points
Geodesic Path Structure
How far apart are 2 points
Length of   Geodesic path
# possible Geodesics paths
(arcs of great circles)
Great circle
Not applicable
2 π R
Half Great circle
2 R
π R
Arcs of an Equilateral triangle
√3 R (1.73)
2/3 π R
3* (see below)

Arcs of Tetrahedral edges

4/√6 R (1.63)

0.608 π R


R = radius of the sphere

1, 2, 3, or 4 points on a sphere

Grafixo. Geodesics on a sphere with 1, 2, 3, or 4 points, uploaded Nov 2021.

Three or four points are better than one or two points

As one can visualize from the table and the diagrams, an electron with 1 or 2 points on a sphere can choose from an infinite number of great circle (or half great circle) geodesics. However, when there are 3 or 4 points on a sphere, one has a system where the points meet the criteria listed above (far away from each other, equidistant, shortest distance path, and minimal number of paths).

For 3 points, they would lay on a great circle at 120º from each other. For 4 points, they would lay on the vertices of a tetrahedron, at 109.5º degrees from each other. Unfortunately, though, when there are only 2 electrons, an equilateral triangle option will not fly: one of the geodesics will be twice the length of the other (see the figure above with 3 points). 

A tetrahedron seems to meet the requirements

From the table one can see that the 4 points on a tetrahedron meet the criteria listed above:

  • The points are reasonably far apart (1.63R versus the furthest separation of 2R).
  • All four points are equidistant from each other, since they are on the vertices of a tetrahedron (a Platonic solid).
  • Any two points on a tetrahedron have the shortest geodesic of the four options listed in the table.
  • Finally, there is a finite number of geodesic paths.

When an electron finds points such as the ones on the vertices of a tetrahedron, it will tend to come back to those points again and again. Those points represent positions that allow the electron the shortest distance of travel from one stable (or low energy) position towards the next stable position.

Over time, as electrons repeatedly choose those favorite options, there is superimposition of those points. These will form our electron "bench points". 


By sciencepics. Tetrahedron with 4 vertices (tetrahedral void showing the geometry). Shutterstock, ID: 255673555.T

*Actually, 3 points don't work as well on equilateral triangles as there are 3 great spheres and positions will not be equidistant. When there are 3 points, they choose alternating vertices of an octahedron.

Chemical bonds as a result of mutual bench points

Knowing electron bench points helps predict function and structure. For example, when methane interacts with oxygen (to create the fire on your stove), it is methane’s electrons’ most common resting points that are interacting with oxygen’s most common resting points. Chemical bonds form when the electrons of both atoms meet the stringent requirements noted above - as well as help reduce the overall energy of the system.

Keep in mind that electron positions are fluid and never stationary; the only consistent characteristic is that some positions are longer lasting than others. This is the image that Linus Pauling (1901 - 1994) had in the early 1930s: that bonds between atoms are not rigid structures; his “chemical bond was a fluid, multiform thing” (from Hager, Force of Nature: The life of Linus Pauling).

Pauling's sixth rule

This forms the basis for Pauling’s sixth rule concerning chemical bonds:   "Of two eigenfunctions (composite formulas)…., the one with the smaller mean value of r, that is, the one corresponding to the lower energy level for the atom, will give rise to the stronger bond". In other words, the strongest bond will form where there is the highest probability of electrons (eigenfunction) living in that arrangement.

If you would like to learn more, the course "How do ultrafast electrons create solid H2 bonds?explains how this works for a sigma bond in the Hydrogen (H2) Molecule.


  1. Hager, Thomas. Force of Nature: The Life of Linus Pauling (originally published by Simon & Schuster, 1995). Monroe Press, Ebook edition, 2011.
  2. Chemical bonds - hyperphysics
  3. Chemical bonding. Atkins, Peter W. "Chemical bonding". Encyclopedia Britannica, 20 Dec. 2018, https://www.britannica.com/science/chemical-bonding. Accessed 16 August 2021.


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