342
6 Molecular Systems
V (φ) =
1
2 V σ (1 − cos σ φ) ,
(6.6.2)
which suffices for most cases. Perhaps the simplest and most widely discussed
examples of molecules falling into this category are ethane, C 2 H 6 , and methyl
chloroform, CH 3 CCl 3 .
6.6.1 Setting the Stage
We shall focus here only upon the simplest type of internal rotational motion,
specifically, the rotational motion of two symmetric top groups about a common
rotation axis. For ethane, for example, this means treating each methyl group as
a rigid symmetric top. In classical mechanical terms, we have a rotational kinetic
energy, T rot , given as
T rot =
1
2 I 1 ω
2
1 +
1
2 I 2 ω
2
2 ,
(6.6.3a)
with I 1 , I 2 the moments-of-inertia of the two tops, and ω 1 and ω 2 their angular
velocities. By analogy with the transformation from space-fixed coordinates and
momenta (see Sect. 6.2.3), we can transform from space-fixed (rotational) angular
velocities ω 1 and ω 2 to centre-of-mass angular velocity and relative angular
velocity ω by employing the total angular momentum J tot = I tot = I 1 ω 1 + I 2 ω 2 ,
in which I tot = I 1 + I 2 is the total moment-of-inertia. The relative angular velocity
is defined as ω ≡ ω 2 − ω 1 , and is associated with the relative rotational motion of
the two tops about the common axis of rotation. We thus have two equations,
=
I 1
I tot
ω 1 +
I 2
I tot
ω 2 ,
(6.6.3b)
ω = ω 2 − ω 1 ,
(6.6.3c)
to be inverted to give ω 1 and ω 2 as
ω 1 = −
I 2
I tot
ω ,
(6.6.4a)
ω 2 = +
I 1
I tot
ω .
(6.6.4b)
the second term in the Fourier series expansion is less than 0.005, respectively, 0.02, times the
leading term.
6 Molecular Systems
V (φ) =
1
2 V σ (1 − cos σ φ) ,
(6.6.2)
which suffices for most cases. Perhaps the simplest and most widely discussed
examples of molecules falling into this category are ethane, C 2 H 6 , and methyl
chloroform, CH 3 CCl 3 .
6.6.1 Setting the Stage
We shall focus here only upon the simplest type of internal rotational motion,
specifically, the rotational motion of two symmetric top groups about a common
rotation axis. For ethane, for example, this means treating each methyl group as
a rigid symmetric top. In classical mechanical terms, we have a rotational kinetic
energy, T rot , given as
T rot =
1
2 I 1 ω
2
1 +
1
2 I 2 ω
2
2 ,
(6.6.3a)
with I 1 , I 2 the moments-of-inertia of the two tops, and ω 1 and ω 2 their angular
velocities. By analogy with the transformation from space-fixed coordinates and
momenta (see Sect. 6.2.3), we can transform from space-fixed (rotational) angular
velocities ω 1 and ω 2 to centre-of-mass angular velocity and relative angular
velocity ω by employing the total angular momentum J tot = I tot = I 1 ω 1 + I 2 ω 2 ,
in which I tot = I 1 + I 2 is the total moment-of-inertia. The relative angular velocity
is defined as ω ≡ ω 2 − ω 1 , and is associated with the relative rotational motion of
the two tops about the common axis of rotation. We thus have two equations,
=
I 1
I tot
ω 1 +
I 2
I tot
ω 2 ,
(6.6.3b)
ω = ω 2 − ω 1 ,
(6.6.3c)
to be inverted to give ω 1 and ω 2 as
ω 1 = −
I 2
I tot
ω ,
(6.6.4a)
ω 2 = +
I 1
I tot
ω .
(6.6.4b)
the second term in the Fourier series expansion is less than 0.005, respectively, 0.02, times the
leading term.