Symmetry, group theory and applications
Lecture 1
Symmetry operation: operation which, when applied to an object, leaves it indistinguishable
from the original object
Symmetry element: geometric entity, e.g., the point, axis or plane with respect to which the
operation is carried out
There are 5 types of molecules symmetry operation:
Identity, proper rotation, reflection, inversion, improper rotation
Identity (E) = do nothing, exists for every object
Proper rotation (Cn)
N = order of the axis, defined by the rotation angle 360/n
If there are several rotation axes within a molecule, the axis with the largest n, if unique, is the
principle axis and is conventionally orientated vertically along the z-axis of the coordinate
system
E.g., H2O has a C2 axis that bisects the H-O-H bond angle
E.g., NH3 has a C3 axis
Can have molecules with several rotation axes, and these can be either collinear or non-
collinear
Reflection (σ)
A vertical mirror plane is a plane that contains the principle rotation axis
A horizontal mirror plane is a plane that is perpendicular to the principle rotation axis, or it is the
plane of the molecule if there is no principle rotation axis
A dihedral mirror plane is a type of vertical mirror plane, which occurs when the mirror plane
bisects two C2 axes that usually lie perpendicular to the principle axis
If you reflect in the xy plane then the x, y coordinates of that point stay the same
E.g., benzene has 7 mirror planes: one horizontal, 3 vertical and 3 dihedral
Lecture 1
Symmetry operation: operation which, when applied to an object, leaves it indistinguishable
from the original object
Symmetry element: geometric entity, e.g., the point, axis or plane with respect to which the
operation is carried out
There are 5 types of molecules symmetry operation:
Identity, proper rotation, reflection, inversion, improper rotation
Identity (E) = do nothing, exists for every object
Proper rotation (Cn)
N = order of the axis, defined by the rotation angle 360/n
If there are several rotation axes within a molecule, the axis with the largest n, if unique, is the
principle axis and is conventionally orientated vertically along the z-axis of the coordinate
system
E.g., H2O has a C2 axis that bisects the H-O-H bond angle
E.g., NH3 has a C3 axis
Can have molecules with several rotation axes, and these can be either collinear or non-
collinear
Reflection (σ)
A vertical mirror plane is a plane that contains the principle rotation axis
A horizontal mirror plane is a plane that is perpendicular to the principle rotation axis, or it is the
plane of the molecule if there is no principle rotation axis
A dihedral mirror plane is a type of vertical mirror plane, which occurs when the mirror plane
bisects two C2 axes that usually lie perpendicular to the principle axis
If you reflect in the xy plane then the x, y coordinates of that point stay the same
E.g., benzene has 7 mirror planes: one horizontal, 3 vertical and 3 dihedral
