Physical Chemistry Third Edition

22.4 The Rotation and Vibration of Polyatomic Molecules 933

PROBLEMS

Section 22.3: Nuclear Spins and Wave Function Symmetry

22.19The common isotope of oxygen,^16 O, hasI0. Will

(^16) O 2 take on even values or odd values ofJin the
electronic ground state? Determine the term symbol for
the electronic ground state. How will theJvalues differ
from theJvalues of O 2 with one^16 O nucleus and one
(^17) O nucleus?
22.20.Fluorine has only one isotope in the earth’s crust,^19 F,
withI 1 /2. Refer to Chapter 20 for information about
its electronic ground state. Describe the rotational states
of F 2 in its electronic ground state.

22.4 The Rotation and Vibration of Polyatomic

Molecules

Rotation and vibration are more complicated in polyatomic molecules than in diatomic

molecules, and we consider only an approximation that is equivalent to the harmonic

oscillator-rigid rotor energy level expression of Eq. (22.2-29). To obtain the energy

level expressions, we pretend that the rotating molecule is somehow prevented from

vibrating and that the vibrating molecule is prevented from rotating.

Rotation of Polyatomic Molecules

We now assume that all bond lengths and bond angles of a polyatomic molecule are

locked at their equilibrium values, so that the molecule cannot vibrate and rotates as a

rigid body. The classical rotation of a rigid body is described in terms ofmoments of

inertiataken relative to three mutually perpendicular axes that pass through the center

of mass of the object. For an object consisting ofnmass points, the moment of inertia

about an axis is defined to be

Iaxis

∑n

i 1

mir^2 i(axis) (22.4-1)

wheremiis the mass of theith mass point andri(axis)is the perpendicular distance

from this mass point to the specified axis. The moments of inertia about thex,y, and

zaxes are

Ix

∑n

i 1

mi

(

y^2 i+z^2 i

)

(22.4-2a)

Iy

∑n

i 1

mi

(

x^2 i+z^2 i

)

(22.4-2b)

Iz

∑n

i 1

mi

(

x^2 i+y^2 i

)

(22.4-2c)

There are six additional quantities, which are calledproducts of inertia:

IxyIyx

∑n

i 1

mixiyi (22.4-3)