In what follows, we set 𝑚 = 2𝑛 -3 to denote the full dimension of 𝒞 𝑛 . Following Zou, et al. (and independently by Canfield ), some or all of the normal modes 𝒒 𝑎𝑛𝑔(𝑗) , 1 ≤ 𝑗 ≤ 𝑚, can be combined using 𝑚dimensional polar (hyperspherical) coordinate systems akin to those used in the treatment Jahn-Teller analyses . These coordinate systems utilise 𝑚 -1 angular coordinates (𝜑 1 , 𝜑 2 , … , 𝜑 𝑚-1 ) which describe the mixing of the modes, and a linear radial coordinate which indicates the magnitude of the displacements determined by the mixed modes. Given the ABn stereoisomeric configurations lie in 𝑆 2 , it is appropriate to represent this linear radial coordinate as an angular coordinate 𝜃: a 𝐛 𝑖 , 1 ≤ 𝑖 ≤ 𝑛, unit vector rotates about the origin by magnitude 𝜃 in the direction indicated by 𝒒 𝑎𝑛𝑔(𝑗) , 1 ≤ 𝑗 ≤ 𝑚. The appropriate bounds for these 𝑚-dimensional polar (hyperspherical) coordinates are: