Equidistant Electrons
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 when there are two electrons in an energy shell.
There are two reasons for this. First, the electron would like to minimize the number of paths that it has to travel. Second, 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.
An electron's stringent requirements.
Here are an electron's stringent requirements. The points on a sphere have to:
- keep electrons as far away from each other as possible
- remain equidistant from other points
- have the shortest possible distance between two points
- Minimize the total number of paths available
When the electrons all follow the rules above, none will have a shorter energy path than the other.
Here are a table and a diagram 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) |
|---|---|---|---|---|
one | Great circle | Not applicable | 2 π R | Infinite |
two | Half Great circle | 2 R | π R | Infinite |
Three | Arcs of an Equilateral triangle | √3 R (1.73R) | 2/3 π R | 3 |
Four | Arcs of Tetrahedral edges | 4/√6 R (1.63R) | 0.608 π R | 6 |
R = radius of sphere
1, 2, 3, or 4 points on a sphere
A tetrahedron seems to meet the requirements
As one can visualize from the table and the diagram, 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).
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. As I will describe in the next article, these are the electron's bench points.
Picture credits:
- By adison pangchai. Model of Abstract Atom Structure. Vector illustration. ID: 550452931.
- By alexfan32. Two protons and two electrons of H2 - Molecular Hydrogen in vector, ID: 1594438201.
- Grafixo. Geodesics on a sphere with 1, 2, 3, or 4 points, uploaded Nov 2021.
- By sciencepics. One tetrahedral void showing the geometry, ID: 255673555.