Three chemical systems -SiF4, XeF4, and a chiral silaneare implemented to demonstrate practical application of this rigorous approach and the corresponding potential energy surfaces (PESs) are presented. These reveal that, while configuration spaces are well-defined topological manifolds (continuous, smooth, and differentiable), the corresponding PES for a specific embodiment (specific atoms are assigned to AB4) will contain discontinuities idiosyncratic of the system.