5.3 Applications 61
with the moment of inertia
Θ=m 1 d 12 +m 2 d 22 =m 12 d^2. (5.19)Herem 12 =m 1 m 2 /(m 1 +m 2 )is the reduced mass.
Notice that the tensorδμν−uμuν, when multiplied with a angular velocity vector
wν, projects onto the directions perpendicular tou. Thus the angular momentum
Lμ=Θμνwνis perpendicular tou, andL·u=0. The rotational motion of a linear
molecule, in 3D, has just 2 and not 3 degrees of freedom. This also holds true, when
the rotational motion is treated quantum mechanically.
Examples for linear molecules are the homo-nuclear molecules (m 1 =m 2 )ofhy-
drogen and nitrogen, viz.: H 2 and N 2. Hetero-nuclear molecules (m 1 =m 2 )are,e.g.
hydrogen-deuterium HD and hydrogen chloride HCl. Also some tri-atomic mole-
cules are linear, e.g. carbon-dioxide CO 2. In this case, the center of mass coincides
with the central C-atom and the moment of inertia is determined by the masses and
distances of the O-atoms.
(ii) Symmetric Top Molecules
The moment of inertia tensor of molecules with a symmetry axis parallel to the unit
vectoruis of the form
Θμν=Θ‖uμuν+Θ⊥(δμν−uμuν), (5.20)whereΘ‖andΘ⊥are the moments of inertia for the angular velocity parallel and
perpendicular tou, respectively. Examples for symmetric top molecules are CH 3 Cl
and CHCl 3 ,orC 6 H 6 .Forprolate, i.e. elongated, particles, one hasΘ‖ >Θ⊥.
Particles withΘ‖<Θ⊥arereferredtoasoblateor disc-like.
The linear molecules discussed above correspond toΘ‖=0.
(iii) Spherical Top Molecules
The special caseΘ‖=Θ⊥applies tospherical topmolecules which have an isotropic
moment of inertia tensor
Θμν=Θδμν, (5.21)
with the moment of inertiaΘ. The regular tetrahedral molecules, CH 4 and CF 4 ,as
well as the regular octahedral molecules CF 6 and SF 6 , are spherical top molecules.
Notice that physical properties described by a second rank tensor, like the moment
of inertia tensor, are isotropic not only for spheres, but also for regular tetrahedra,
cubes and regular octahedra. Physical properties associated with higher rank tensors
are needed to distinguish between the different symmetries.
(iv) Asymmetric Top Molecules
In general, the moment of inertia tensor is characterized by three different principal
valuesΘ(^1 ),Θ(^2 )andΘ(^3 ). By definition, one hasΘ(i)≥0fori= 1 , 2 ,3.