Mathematically, symmetry is expressed by group theory. The cases where symmetric groups are continuous groups and discrete groups are called continuous symmetry and discrete symmetry respectively. German mathematician Herman Weil was the first to apply this mathematical method to physics and realized the importance of gauge symmetry.
When a molecule has a center of symmetry, extending a second line from any atom in the molecule to the center of symmetry, you can find another same atom on the other side equidistant from the center of symmetry, that is, every point is symmetrical about the center. Symmetry operation according to the center of symmetry is inversion operation, and it is inversion according to the center of symmetry, which is denoted as I; In=E when n is even, and in=i when n is odd.
A mirror is a plane that divides molecules into two. In the molecule, except for the atoms on the warped surface, the others are arranged in pairs on both sides of the mirror, which can be recovered by reflection operation. The reflection operation is that every point is symmetrical about the mirror surface, and it is recorded as σ; When n is even, σn=E, and when n is odd, σn=σ. The mirror perpendicular to the principal axis is denoted by σh; The mirror passing through the principal axis is denoted by σv; Through the principal axis, the mirror bisecting the included angle of the secondary axis is denoted by σ d.
The basic operation of the anti-axis In is to rotate 360/n around the axis, and then the inversion is carried out according to the center point on the axis, followed by the joint operation of C 1n and I: I1n = IC1n; Rotate 360/n around the In axis, and then reverse according to the center.
The basic operation of the reflection axis Sn is to rotate 360/n around the axis, and then reflect according to the plane perpendicular to the axis, followed by the joint operation of C 1n and σ: s1n = σ c1n; Rotate 360/n around the Sn axis, and then reflect according to the plane perpendicular to the axis.