The VSEPR theory provides a useful guide to predicting molecular geometries. One implication of the theory is that the orientation of ligands in three-dimensional space is such that the seperation between each pair of them is maximum.
This is analogous to the mathematical problem of symmetrically distributing points on the surface of a sphere. With 2, 3, 4, or 6 points, the solutions are trivial. The required distributions would be linear, triangular, tetrahedral, and octahedral respectively. An issue arises when we attempt to distribute 5 points on the surface of a sphere.
We could try to arrange the points such that they lie on the vertices of a regular polyhedron inscribed in the sphere. However, from my knowledge of Euclidean geometry, I believe that such a shape with 5 vertices simply cannot exist.
Chemists suggest an approximately symmetric distribution such as placing the points on the vertices of a square pyramid. But from a mathematical perspective, there ought to be a better solution.
Question: What is the nearest approximation to a symmetric distribution which we can achieve? Can the approximation be generalized to any other number of points which cannot be distributed at the vertices of a platonic solid?
The definition of a symmetrical distribution here is that any non-labelled point on the sphere is indistinguishable from another such point. A nearly symmetrical distribution implies the least variation in the relative distances between the points (distances measured along the surface).
The ambiguity in the definition of symmetry makes the question more difficult to answer. Any additional insights or numerical methods to solve the problem are also welcome.