great circles around tetrahedron
Two electrons use the tetrahedron for their travels
Electrons are always looking for the shortest distance of travel from one stable (or low energy) position towards the next stable position. Electrons are in a stable position when they meet the stringent requirements set out in article 8.
Two electrons traveling from one vertex of a tetrahedron to the next vertex have the best situation. The electrons are always as far from each other as they can possibly be; they are equidistant, and they have a minimum number of paths to travel. Most importantly, the travel paths along the arcs between two vertices of a tetrahedron are shorter geodesic paths.
Two atoms join forces
Suppose two tetrahedra join together such that one of their edges is adjoining.
Adjoining spheres and tetrahedra
Each hydrogen atom has one electron. When the two atoms come together, the two electrons have different options of travel.
- They can both end up on the two vertices that lie on the adjoining edge between the two nuclei. These are bonding positions, as they hold the atoms together. This option is shown in the image above and in the top leftmost image of the image below.
- Or one electron can be in a bonding position, and one in an anti-bonding position (the remaining images below). This leads to a higher energy situation in which the two atoms dissociate
Tetrahedral connections
As was discussed in module II, atoms with 2 electrons have the ability to be in a more stable situation as the two electrons flow on tetrahedral bench points.
Thus, when two hydrogen atoms' form adjoining tetrahedra with one of the edges, two electrons can be shared by both of the atoms simultaneously. Each atom has the best of both worlds. Neither has to deal with a higher energy one-electron situation.
Bonding vertices are frequented more often than non-bonding vertices
As electrons travel in great circles from one vertex to the next, how often do they physically end up on the two "bonding" vertices?
As I showed in article 9 (and the image above), there are 3 great circles for each tetrahedral vertex inscribed within a sphere. As one can visualize, when one of those vertices is shared between two tetrahedra, there will be 6 great circles passing through it.
Thus, an electron traveling around the nucleus - and following tetrahedral geodesics - will travel twice more often to each of the 2 bonding vertices (6 times versus 3 times for the remaining 4 anti-bonding vertices). The image below shows the top bonding vertex on the adjoining edge has 6 great circles passing through it.
What is even more interesting is that electrons preferentially choose those bonding vertices, spending much of their time in the internuclear zone. Thus, bonding vertices are statistically preferred both in terms of space as well as time. The latter will be discussed in the next article.
6 great circles on one vertex
Picture credits:
- By magnetix. Overlap of s-orbitals in hydrogen, ID: 340098287.
- Grafixo. Geodesics along the edges of a tetrahedron. Three great circles shown converging at one vertex, uploaded Nov 2021.
- Grafixo. Two adjoining spheres with inscribed adjoining tetrahedra, uploaded Dec 2021.
- Grafixo. Different positions for electrons along bench points on two adjoining tetrahedra, uploaded Nov 2021.
- Grafixo. Six great circles around one of the bonding vertices on the adjoining edge of two tetrahedra, uploaded Dec 2021.