CHEM106: Assignment 10
Point Group for Molecule
1. Understanding symmetry elements and symmetry operations is essential to learning
group theory of molecules.
A. What is a symmetry element? What is its relation with the symmetry operation?
A symmetry element is a line, point, or plane with respect to which one or more
symmetry operations may be performed.
A symmetry operation is a transformation of an object about a symmetry element
such that the object's orientation and position before and after the transformation
are indistinguishable. A symmetry operation carries every point in the object into
an equivalent point or the identical point.
B. For all the symmetry elements listed in the following table, define the
corresponding symmetry operation and its notation.
Symmetry Element
Identity
Symmetry plane
Inversion center
Proper rotation axis
Improper rotation
axis
Symmetry Operation
leave the object unchanged
Reflection in the plane
Inversion of a point (x, y, z) to (x, y, z)
Rotation by (360/n)o
Rotation by (360/n)o, followed by reflection
in a plane perpendicular to the rotation axis
Symbol
E
σ
i
Cn
Sn
2. The C2h group contains a C2 axis that transforms the (x, y, z) coordinate to
(x, y, z). Define the three-dimensional matrix representation for the C2 operation.
The C2 operation transforms the (x, y, z) coordinate to (x, –y, –z), so the
corresponding matrix representation for the operation is
  1 0 0


 0  1 0
 0 0 1


3. Identify all the symmetry elements in the chloroform (CHCl3) molecule, and assign its
point group.
As shown in the figure below, there is a c3 axis of rotation that passes through the
C-H bond. It is also the principal axis of rotation. There are three σv mirror planes
(each defined by the H-C-Cl bond plane).
Symmetry elements: E, C3, v, v', v''
Point Group: C3v
4. Determine the symmetry group for the following molecules.
A. NH3
C3V
B. CCl4
Td
C. Planar BF3
D2h
D. H2C=CBr2
C2V
E. C6H6
D6h