82
4 Representations
Fig. 4.6 T 1u distortion space for UF 6 with coordinates as defined in Eq. (4.103); T z is the translation 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 σ 2 |σ 1 is −10.5 ◦
for the three T 1u z-components, which we shall abbreviate as follows:
Q U =
√
MMZ 0
Q σ =
√
m
√
2
((Z 1 + Z 6 )
Q π =
√
m
2
((Z 2 + Z 3 + Z 4 + Z 5 )
(4.103)
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 = MMZ 0 + m((Z 1 + Z 2 + Z 3 + Z 4 + Z 5 + Z 6 )
=
√
MQ U +
√
2mQ σ +
√
4mQ π
(4.104)
We can remove this degree of freedom from the function space by a standard orthogonalization procedure. One option is to construct first a pure stretching mode, which
does not involve the Q π coordinate. This mode is denoted by |σ 1 . The remainder
of the function space, which is orthogonal both to the translation and to this pure
stretching mode, is then denoted by |π 1 . Normalizing these modes with respect to
mass-weighted coordinates yields:
|σ 1 =
−
√
2mQ U +
√
MQ σ
√
M + 2m
|π 1 =
−
√
4mMQ U − m
√
8Q σ + (M + 2m)Q π
√
(M + 2m)(M + 6m)
(4.105)