Potential Energy Hypersurfaces for Polyatomic Molecules

For molecules with more than three atoms (or even with just three atoms if the bond angle and both bond lengths are considered independently), it is evident that the potential energy surface cannot be plotted in a way in which we can visualise it, i.e. in three dimensions. As the number of coordinates increases, it becomes impossible to constrain all the necessary coordinates to enable a 3D plot and maintain accuracy. A potential energy surface with more than two coordinates is often termed a potential energy hypersurface. Despite the word surface still being used, it is generally impossible to visualise these hypersurfaces, although they act in the same way as one- or two-coordinate potential energy surfaces. Terms such as ‘saddle point’ as visualisation aids become meaningless on a hypersurface. Often, cuts are taken through the hypersurface to plot the area of interest as a 3D potential energy surface; implying that, in the region of interest, it is possible to constrain some of the coordinates.
Minima on the surface can, however, still be located mathematically for relatively small molecules.