Chemistry Reference
In-Depth Information
Z
C
6
,
C
3
,
C
2
6
5
4
z
v
Y
3
u
1
2
X
Figure 6.24
A basis of 3N vectors for the D
6h
molecule benzene.
freedom of each atom. The
X
and
Y
directions are taken so that C H
1
is aligned with
Y
;
this means that the
u
vectors of the atoms C
1
,H
1
,C
4
and H
4
are also along
Y
, but the other
u
basis vectors are not.
There are 36 basis vectors, and the effect of each operation of the
D
6h
group gives the
reducible character set shown in Table 6.15. These characters are obtained by considering
the effect of an example operation from each class in the point group on the basis.
Table 6.15
The reducible representation obtained for benzene with a basis of three vectors
per atom.
D
6h
3
C
2
3
C
2
E
2
C
6
2
C
3
C
2
i
2
S
3
2
S
6
σ
h
3
σ
d
3
σ
v
36
0
0
0
−
4
0
0
0
0
12
0
4
As usual, the identity operator
E
leaves all basis vectors unchanged and so simply counts
the number of basis vectors.
The element for the rotations 2
C
6
,2
C
3
and 2
C
2
is the vertical axis through the centre of
the molecule. Any rotation around this axis interchanges the atoms, and so all basis vectors
are changed completely.
The 3
C
2
axes, through opposite atoms, interchange all atoms except the four on the axis
employed. For these four atoms the
z
and
v
are reversed but
u
is unaltered, giving character
−
4.
The operations in the classes 3
C
2
,
i
,2
S
3
and 2
S
6
all interchange atoms so that none
remains in its original position, giving a character of 0.
The
2
+
1
=−
1 per atom on the axis and so a total of
−
σ
h
operation is a reflection in the plane of the molecule; this leaves all atoms in
place with all
u
and
v
basis vectors unaltered and each
z
is reversed, giving a character of
1 per atom, making a total of 12.
The mirror planes for the 3
σ
d
class contain the principal axis and dissect opposite bonds
in the ring. Each
σ
d
operation swaps atoms with their images; hence the character is 0.
σ
v
planes each contain the principal and a
C
2
axis; that is, on reflection, four
atoms are left in place with
u
and
z
unaltered but their
v
reversed, giving a character of 1
per unaltered atom and so a total of 4.
The 3
