Chemistry Reference
In-Depth Information
(i) The product of any two members must also be a member of the group; this means
that the group is 'closed', i.e. a new symmetry operation cannot be generated by
combining the ones in the group.
(ii) There must be an identity operation
E
; this is just a matter of remembering to write
it down, since all objects have the identity element. Any molecule is unchanged
by the identity operation (i.e. doing nothing), and so all molecules have at least
E
.
(iii) Every member of a group must have an inverse, i.e. if you carry out an operation
there must be another member of the group that undoes that operation. This is one
reason for having the identity operator; for example, the inverse to
σ
v
is
σ
v
itself,
since
E
. Reflections and the inversion operation are their own inverses
in this way, but operations involving rotations have more complex inverses, as
detailed in Table 2.3.
σ
v
σ
v
=
2. The symmetry elements of a point group are defined with respect to a global axis system
and so do not move under any of the operations of the group.
3. To check that a group is closed, a multiplication table should be constructed giving all
the products of operations in the group.
4. Symmetry operations need not commute: In general, the order in which two symmetry
operations are applied will affect the result, giving different equivalent single opera-
tions. The conditions under which operations do commute (i.e. the result is the same
irrespective of order) is discussed in Section 2.4.3.
2.7 Completed Multiplication Tables
Tables 2.5 and 2.6 give the completed multiplication tables for H
2
O and NH
3
.
Tab l e 2 . 5
A completed multiplication table for the H
2
O symmetry
operations.
Firstoperation---------------------
Second
operation
E
C
2
σ
v
(
XZ
)
σ
v
(
YZ
)
E
E
C
2
σ
v
(
XZ
)
σ
v
(
YZ
)
C
2
C
2
E
σ
v
(
YZ
)
σ
v
(
XZ
)
σ
v
(
XZ
)
σ
v
(
XZ
)
σ
v
(
YZ
)
E
C
2
σ
v
(
YZ
)
σ
v
(
YZ
)
σ
v
(
XZ
)
C
2
E
Tab l e 2 . 6
A completed multiplication table for the NH
3
symmetry operations.
Firstoperation-------------------------
C
3
1
C
3
2
Second
operation
E
σ
v
A
σ
v
B
σ
v
C
C
3
1
C
3
2
E
E
σ
v
A
σ
v
B
σ
v
C
C
3
1
C
3
1
C
3
2
E
σ
v
C
σ
v
A
σ
v
B
C
3
2
C
3
2
C
3
1
E
σ
v
B
σ
v
C
σ
v
A
σ
v
A
σ
v
A
σ
v
B
σ
v
C
E
C
3
1
C
3
2
C
3
2
C
3
1
σ
v
B
σ
v
B
σ
v
C
σ
v
A
E
C
3
1
C
3
2
σ
v
C
σ
v
C
σ
v
A
σ
v
B
E
