CHAPTER 6: Question 4

2D explanations

All molecules contain the identity element. The diagram below shows the location of the C3 axis and three planes of symmetry for ammonia. The point group can be determined as C3v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. These four symmetry elements give rise to the following symmetry operations: E, C31, C32, s, s', s" and it these six symmetry operations which will be used to construct the group multiplication table.

To construct the multiplication table, first place the six symmetry operations along both the horizontal and vertical axis as shown in the shaded boxes in the table below. Since EX = XE = X where X is any symmetry operation, it is easy to fill in the rows and columns of the table corresponding to multiplication by E. It should
not be difficult to see that C31.C31 = C32; C32.C32 = C31; and C31.C32 = C32.C31 = E. In addition, since s2 = E three other boxes of the table can be filled. This should give the table shown below.
 
C3v E C31 C32 s s' s"
E
E
C31
C32
s
s'
s"
C31
C31
C32
E
 
 
 
C32
C32
E
C31
 
 
 
s
s
 
 
E
 
 
s'
s'
 
 
 
E
 
s"
s"
 
 
 
 
E
 

To complete the rest of the table, it is necessary to label the three hydrogen atoms of ammonia and then carry out each pair of symmetry operations in turn and determine which single operation that pair is equivalent to. For example, the diagram below shows that carrying out the symmetry operations C31 then s is equivalent to the single symmetry operation s"

The complete multiplication table is given below, and from this it can be seen that for the point group C3v, the relationship PQ = QP is only true if one of the operations is the identical operation, or if P and Q are C31 or C32. The inverse of a symmetry operation is the member of the group which when multiplied by the symmetry operation produces the identical operation. ie, the inverse of operation P is Q where PQ = E. From the group multiplication table it can be seen that E, s, s', and s" are there own inverses, that the inverse of C31 is C32, and that the inverse of C32 is C31.
 
C3v E C31 C32 s s' s"
E
E
C31
C32
s
s'
s"
C31
C31
C32
E
s"
s
s'
C32
C32
E
C31
s'
s"
s
s
s
s'
s"
E
C31
C32
s'
s'
s"
s
C32
E
C31
s"
s"
s
s'
C31
C32
E
 

 
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