Chemistry Reference
In-Depth Information
Fig. 4.6
T
1
u
distortion space for UF
6
with coordinates as defined in Eq. (
4.103
);
T
z
is the transla-
tion of mass. The circle, perpendicular to this direction, is the space of vibrational stretching and
bending, with coordinates defined in Eqs. (
4.104
)and(
4.105
). The angle
10
.
5
◦
σ
2
|
σ
1
is
−
for the three
T
1
u
z
-components, which we shall abbreviate as follows:
√
MZ
0
Q
U
=
√
m
√
2
(Z
1
+
Z
6
)
Q
σ
=
(4.103)
√
m
2
Q
π
=
(Z
2
+
Z
3
+
Z
4
+
Z
5
)
This space is still reducible since it includes the translation in the
z
-direction. The
translation coordinate corresponds to the displacement of the center of mass in the
z
-direction. It is given by
i
m
i
Z
i
, which can be expressed as follows:
T
z
=
MZ
0
+
m(Z
1
+
Z
2
+
Z
3
+
Z
4
+
Z
5
+
Z
6
)
√
MQ
U
+
√
2
mQ
σ
+
√
4
mQ
π
=
(4.104)
We can remove this degree of freedom from the function space by a standard orthog-
onalization procedure. One option is to construct first a pure stretching mode, which
does not involve the
Q
π
coordinate. This mode is denoted by
. The remainder
of the function space, which is orthogonal both to the translation and to this pure
stretching mode, is then denoted by
|
σ
1
|
π
1
. Normalizing these modes with respect to
mass-weighted coordinates yields:
σ
1
=
−
√
2
mQ
U
+
√
MQ
σ
|
√
M
+
2
m
(4.105)
π
1
=
−
√
4
mMQ
U
−
m
√
8
Q
σ
+
(M
+
2
m)Q
π
|
√
(M
+
2
m)(M
+
6
m)
