Symmetry-Constrained Properties Behave Differently for 2D or 3D Structures under the Same Point Group

In chemistry and physics, two molecules belonging to the same point group are expected to behave similarly regarding various properties, following their character tables. Here, we show that the derivative of the dipole moment with respect to the normal coordinate of vibration might show different symmetry constraints if the molecule is planar, even if these molecules belong to the same point group. Examples of pairs of molecules featuring these conditions are presented. These findings open a new path toward a much deeper understanding of how 2D materials behave so differently compared to 3D materials featuring the very same atoms and arrangements (graphene and graphite, for example); chemists and physicists dealing with 2D materials could benefit from looking more deeply into pure mathematical relations that might be governing 2D systems in a different way when compared to 3D systems. The aid from mathematicians is welcomed.