Chemistry Reference
In-Depth Information
By this elimination procedure, the reducible representations are found to be:
u
+
+
g
+
(
z
)
=
2
and
(
x
,
y
)
=
2
u
+
g
(6.12)
A full basis of three vectors per atom has been used, and so some of these irreducible
representations will be for simple motion of the whole molecule and its rotation. The
character table shows that movement in
x
and
y
belongs to
u
. This is a doubly degenerate
representation, so that only one instance need be removed from
(
x
,
y
).
g
. Note that
R
Z
, the symbol
for rotation around
Z
, has not appeared, for any linear molecule rotation around the molec-
ular axis has no effect on the atom coordinates and so is not a degree of freedom for the
molecule.
Motion in the
Z
-direction, on the other hand, is a degree of freedom and has the
representation
Similarly, rotations about
X
and
Y
axes are degenerate in
u
+
. So we are left with the vibrations
=
u
+
+
g
+
=
u
(
z
)
and
(
x
,
y
)
(6.13)
u
+
are the
symmetric and antisymmetric stretch modes, similar to those found for H
2
O. There are
also two bending modes, which are degenerate with one another, bending in the
XZ
plane
and bending in the
YZ
plane. The character table indicates that the antisymmetric stretch
and the bending modes are IR active, and so CO
2
can absorb IR radiation from the surface
of the Earth.
In this analysis of the linear triatomic CO
2
we have found one more vibration than was
obtained for the nonlinear H
2
O case. This is due to the loss of the rotational degree of
freedom around the molecular axis. Instead of the rotation and single bending mode for
H
2
O, we find two degenerate bending modes for the linear triatomic case. For a linear
molecule containing
N
atoms there will always be 3
N
g
+
and
So, for this linear molecule, we have found four vibrational modes.
−
5 vibrational modes, one more
than the 3
N
−
6 for the nonlinear case.
Problem 6.5:
The other greenhouse gas mentioned above is CH
4
, which belongs to
the
T
d
point group. Using the basis shown in Figure 6.8, demonstrate that the C
H
stretching modes of this molecule have the reducible representation shown:
H
1
σ
d
E
8
C
3
3
C
2
6
S
4
6
T
d
Γ
4
1
0
0
2
H
2
H
3
H
4
Figure 6.8
A basis of the four C H bonds for obtaining the reducible representation for the
stretching modes of CH
4
. The corresponding reducible representation is shown to the right.
