Chemistry Reference
In-Depth Information
Table 12.4
The multiplication table of the point group C
3v
C
3
C
3
C
3v
I
s
1
s
2
s
3
C
3
C
3
I
I
s
1
s
2
s
3
C
3
C
3
C
3
I
s
2
s
3
s
1
C
3
C
3
C
3
s
3
s
1
s
2
I
C
3
s
1
s
1
s
3
s
2
C
3
I
C
3
s
2
s
2
s
1
s
3
C
3
I
C
3
s
3
s
3
s
2
s
1
C
3
I
language of group theory, the irreducible representations (in short, irreps)
of that group. For each irrep, symmetry operations are given in the formof
matrices like (12.21) or (12.23), whose order equals the dimensionality of
the irrep.
6
The characters are simply the trace of suchmatrices, and are the
only numbers given in textbooks. While C
2v
has only one-dimensional
irreps, the higher symmetry C
3v
group has two one-dimensional irreps (A
1
and A
2
) and one two-dimensional irrep (E).
7
Tables 12.2 and 12.4 give themultiplication tables of the two groups. In
constructing a multiplication table, we recall that R
k
¼
R
i
R
j
is the result
of the intersection of the column headed by R
i
and the row headed by R
j
,
and that the operation on the right must be done first. It is seen that each
symmetry operation occurs once in each row.
We now give a few further definitions and fundamental theorems.
12.2.1 Isomorphism
Two groups G and G
0
of the same order h are isomorphic if (i) there is a
one-to-one correspondence between each element G
r
of G and G
0
r
of
G
0
ð
r
¼
1
; ...;
h
Þ
and (ii) the symbolic multiplication rule is preserved,
namely, if G
r
G
s
¼
G
t
in G
()
G
0
r
G
0
s
¼
G
0
t
in G
0
. If only (ii) is true then
the groups are said to be homomorphic.
;
2
12.2.2 Conjugation and Classes
Any two elements A and B of a groupG are said to be conjugate if they are
related by a similarity transformation with one other element X of the
p
3
6
In C
3v
, c
¼
1
2.
7
Cubic groups have also three-dimensional irreps (T).
=
2, s
¼
=
