Physical Chemistry Third Edition

936 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

If the molecule is not oriented on thezaxis, thexandycomponents are similar, so that

Ieμr^2 e

The possible values ofLzfor a diatomic or linear molecule are given by Eq. (17.2-31).

LzhM ̄ (M0,±1,±2,...,±J) (22.4-8)

The energy does not depend on the quantum numberM, so the energy level for a

particular value ofJhas a degeneracy of 2J+1.

For a spherical top,IAIBIC. For this case

Eclassical

1

2 IA

(

L^2 A+L^2 B+L^2 C

)



1

2 IA

L^2 (22.4-9)

The quantum mechanical energy is

EqmEJ

h ̄^2

2 IA

J(J+1) (22.4-10)

This formula is the same as that for the energy of a diatomic or linear molecule, but the

degeneracy is not the same. In both cases, there is a quantum numberMthat specifies

the projection of the angular momentum on thezaxis (an axis external to the molecule).

When there are three independent variables in the expression for the classical kinetic

energy, such as the three components of the angular momentum, there are three quantum

numbers. In this case, the third quantum number is for the projection of the angular

momentum on one of the principal axes, say theAaxis. This projection can take on

the values

LAhK ̄ (K0,±1,±2,...,±J) (22.4-11)

The quantum numberKhas the same range of values asM. For a given value ofJ,

there is one state for each value ofMand for each value ofK, so that the degeneracy is

gJ(2J+1)^2 (spherical top) (22.4-12)

The energy levels of symmetric tops and asymmetric tops can depend on the values of

all three quantum numbersJ,M, andK.^5 Problems 22.37 and 22.38 display the energy

levels of oblate and prolate symmetric tops.

In Section 22.3, we found that only half of the values of the rotational quantum num-

berJoccurred for a homonuclear diatomic molecule because of the indistinguishability

of the nuclei. In the case of polyatomic molecules the effect of the indistinguishability

of identical nuclei is more complicated. We assert without proof that the fraction of

the conceivable rotational states that can occur is 1/σ, whereσis called thesymmetry

numberof the molecule. The symmetry number is defined as the number of equivalent

orientations of the molecule in its equilibrium conformation, which means the number

of orientations in which the molecule can be placed and have each nuclear location

occupied by a nucleus of the same kind as in the first orientation.

The symmetry number of any homonuclear diatomic molecule equals 2, correspond-

ing to the result that only half of the conceivable values ofJcan occur. The symmetry

number of a heteronuclear diatomic molecule equals unity, as does that of some poly-

atomic molecules, so that all values of the rotational quantum numbers can occur. The

(^5) G. Herzberg,Infrared and Raman Spectra, Van Nostrand Reinhold, New York, 1945, p. 42ff.