| It can readily be seen that the set of operations
defined for the water molecule satisfy the requirements of a mathematical
group: |
|
Rule I
|
The combination (multiplication)
of two symmetry operations does indeed result in a symmetry operation
that is also a member of the group. |
|
Rule 2
|
Clearly the identity element
is equivalent to the symmetry operation that leaves the molecule
unchanged, namely E. |
|
Rule 3
|
Associative multiplication
can be shown to hold by the following examples: |
| |
C2.{  v( xz) .v( yz)}
= C2.C2 = E
{ C2.v( xz)} .v( yz)
= v( yz) .v( yz)
= E
|
|
Rule 4
|
For the water molecule,
the symmetry operations C2, v(xz)
and v(yz)
are their own inverses: i.e. |
| |
C2.C2 = E
v( yz) .v( yz)
= E
v( xz) .v( xz)
= E
|
|
| It should be noted, however,
that it is not generally true that symmetry operations are
their own inverses. |
The complete set of
symmetry operations of a molecule always form a mathematical group, termed
in fact a point group because all
the symmetry elements intersect at a point within the molecule which is
not shifted by any of the symmetry operations. The point group of the
water molecule is denoted C2v, according to the Schoenflies
notation which is commonly used.
|
| Using similar reasoning,
the symmetry elements and operations of other point groups can be derived,
and the appropriate Schoenflies
symbol assigned. In Table 1 overleaf, the essential symmetry elements
for the various point groups are listed. The word "essential" is used
since some of the symmetry elements listed in this table for a given point
group necessarily imply the existence of orders which are not listed.
In Table 2 an exhaustive list of the symmetry elements for each
point group is given. |