What is Symmetry and Why it is Important?

Some object are "more symmetrical" than others. A sphere is more symmetrical than a cube because it looks the same after rotation through any angle about the diameter. A cube looks the same only if it is rotated through certain angels about specific axes, such as $90^o$, $180^o$, or $270^o$ about an axis passing through the centers of any of its opposite faces, or by $120^o$ or $240^o$ about an axis passing through any of the opposite corners.

Here are also examples of different molecules which remain the same after certain symmetry operations: $NH_3$, $H_2O$, $C_6H_6$, $CBrClF$. In general, an action which leaves the object looking the same after a transformation is called a symmetry operation. Typical symmetry operations include rotations, reflections, and inversions. There is a corresponding symmetry element for each symmetry operation, which is the point, line, or plane with respect to which the symmetry operation is performed. For instance, a rotation is carried out around an axis, a reflection is carried out in a plane, while an inversion is carried our in a point.

We shall see that we can classify molecules that possess the same set of symmetry elements, and grouping together molecules that possess the same set of symmetry elements. This classification is very important, because it allows to make some general conclusions about molecular properties without calculation. Particularly, we will be able to decide if a molecule has a dipole moment, or not and to know in advance the degeneracy of molecular states. We also will be able to identify overlap, or dipole moment integrals which necessary vanish and obtain selection rules for transitions in polyatomic molecules.

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