Symmetry

 Symmetry in nature:

Symmetry element and Operation:

        Symmetry element is a geometric object(a plane, line/axis or point) about which symmetry operation is performed.

Symmetry Operation:

 An operation is a movement (reflection, rotation , inversion) carried out with respect to the symmetry element.

An symmetry operation moves object into an indistinguishable orientation.

     Element                                                                             Operation

1) Mirror plane(sigma)                                                       1) Reflection in the plane
2) Proper axis(Cn, n=360 / angle)                                      2) Rotation about the axis
3) Improper axis (Sn)                                                         3)Rotation followed by reflection in a plane                                                                                                   perpendicular to the axis.

4) Center of Inversion(i)                                                     4) Inversion of all atoms through center.

Five Symmetry operation: Reflection(sigma), Proper rotation(Cn),Center of Inversion(i) ,Improper  Rotation(Sn)

                                  IDENTITY OPERATION

                           

Identity operation (einheit = identity, E) 

• Do nothing operation (simplest operation)
•  Mathematically meaning multiplying by 1
•  All objects have E

                                                 PROPER ROTATION

  Counter clockwise rotation of (360/n) about an axis

                                              INVERSION  OF A  SYMMETRY OPERATION

 

If A & B are two symmetry operations such that 

                                     AB = BA = E 

                          then A is called the inverse of B and vice-versa. 

                                                  PROPER ROTATION

•   The rotation axis with largest n is called principal axis (the C6 axis in the case of benzene molecule)  Objects may have rotations that are perpendicular to the principle axis
•   An object with Cn axis must have zero or ‘n’ perpendicular C2 axes
•   The benzene molecule has C6 , C3 , and C2 axis plus six perpendicular C2 axes.

                                          REFLECTION PLANE OF SYMMETRY (σ)

•   An imaginary mirror plane (internal reflection) within an object 
•  Horizontal mirror plane (σh ): perpendicular to the principal axis
•  Vertical mirror plane (σv ) and dihedral mirror plane (σd ): parallel to the principal axis 

                                                      REFLECTIONS 

Plane of symmetry (σ) 

•  An imaginary mirror plane (internal reflection) within an object 
•  Horizontal mirror plane (σh ): perpendicular to the principal axis 
•  Vertical mirror plane (σv ) and dihedral mirror plane (σd ):
•  parallel to the principal axis 

                                                               INVERSION

 CENTER OF SYMMETRY (i)

• Each point is moved along a straight line through the center of the object to a point an equal distance from the center 
•  Mathematically: i (x ,y, z) = (-x,-y,-z) for all points 
•  An object can have zero or one inversion center. 
•  Benzene molecule has one inversion center

                                               IMPROPER ROTATION

 

Improper Rotation Operation (Sn ) 

 Rotation followed by a perpendicular reflection 

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