Chemistry Reference
In-Depth Information
Tab l e 4 . 5
The multiplication table for the C
2v
point group using the
matrix representation of operators for the x and y basis.
C
2v
E
C
2
σ
v
(
XZ
)
σ
v
(
YZ
)
10
01
−
1
10
1
−
01
10
0
10
E
−
0
−
−
1
10
01
−
01
10
1
10
0
10
C
2
−
0
−
10
0
1
−
01
10
01
−
1
10
10
0
σ
v
(
XZ
)
−
−
−
01
10
1
−
1
10
01
10
10
0
σ
v
(
YZ
)
0
−
−
Problem 4.6:
By forming matrix products, check each of the entries in Table 4.5 and
confirm that each product gives the matrix for the operation expected from Table 4.4a.
4.5
Diagonal and Off-Diagonal Matrix Elements
4.5.1 Ammonia, NH
3
,
C
3v
In the H
2
O example the basis is set up to represent the 2p
x
and 2p
y
orbitals on the oxygen
atom. The operations on this basis only ever give a character of
1 for each orbital,
because the orbitals are either unaffected or have the phase pattern reversed. We now move
on to examples in which the basis set is changed in more complex ways.
Figure 4.6 shows the basis of N H bond vectors in the ammonia molecule, which
belongs to the point group
C
3v
. These vectors give the direction of movement for the
H atoms in vibrational modes of the molecule involving N H stretching motion. From the
figure we can see that the
C
3
1
operation causes each basis vector to move to the position of
one of its neighbours. In Figure 4.6, the basis vectors after the transformation have been
marked with a prime, and comparing these to the original set we can write,
+
1or
−
b
3
=
b
1
after
C
3
1
b
1
=
b
2
(4.8)
b
2
=
b
3
C
3
C
3
1
b
1
'
b
2
b
2
'
b
3
b
3
'
b
1
Figure 4.6
The ammonia molecule (point group C
3v
)withabasisofN
H bonds marked.
