Analytically Derived Symmetric VSEPR Geometries from Classical Energy Minimization on a Sphere

The basic valence shell electron-pair repulsion molecular geometries are derived by treating electron domains as identical repelling charges constrained to a sphere. In order to minimize the total Coulombic repulsion under the constraint, Lagrange multipliers and the Gram matrix are used to show that the minimum-energy configurations for N = 2, 3, 4 are linear, trigonal planar, and tetrahedral. Furthermore, we demonstrate that this analytical approach is intractable for N > 4 due to the multi-angular nature of higher coordinate systems. This approach serves as an excellent enrichment for upper-level undergraduates in electromagnetism or mathematical chemistry and advanced linear algebra, also providing an intuitive path that motivates the shift from manual analytical derivations to computational and numerical methods.