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4 Rotation of the Polyatomic Molecule
2T R = ω
2
x I xx + ω
2
y I yy + ω
2
z I zz + 2ω x ω y I xy + 2ω y ω z I yz + 2ω x ω z I xz
(4.11)
or in matrix notation
2T R = ˜
ωIω
(4.12)
where ˜
ω =
ω x ω y ω z
is the transpose of the vector ω.
It is always possible to rotate the molecule fixed axis system, so that the offdiagonal terms of I vanish. The new axes are called principal inertial axes, and
the three elements of the diagonal tensor are called principal moments of inertia
(eigenvalues of the inertia matrix). By convention, they are noted I a ≤ I b ≤ I c .
• When the three moments of inertia are identical, the molecule is called spherical
top
• When the three moments of inertia are different, the molecule is called asymmetric
top
• When two moments of inertia are different, the molecule is called symmetric top
– If I b = I c , the molecule is called prolate top
– If I a = I b , the molecule is called oblate top.
A particular case is the linear molecule. If z is the axis of the molecule, x a = y a
= 0 and
I x = I y =
N
a=1
m a z
2
a and I z = 0
(4.13)
In this case, rotations are possible around the x and y axes and there are only two
degrees of freedom.
4.3 Angular Momentum
The angular momentum P may be written
P =
N
a=1
m a r a × v a =
N
a=1
m a r a × (ω × r a )
(4.14)
In the principal inertial axis system, the expression of the angular momentum is
a diagonal 3 × 3 matrix with P a = I a ω a , etc.
For a molecule in free rotation, E = T R = constant and |P| = constant.
The rotation is defined by two equations