Physical Chemistry , 1st ed.

Scenario 2: fermion nuclei (that is, half-integer spin nuclei)

tottrans

vib

elect

nuc 

rot

antisym sym sym sym* sym sym,Jodd

degen: (I 1)(2I 1) degen: 2J    1

or

antisym sym,Jeven

degen: 2I^2 I degen: 2J     1

The * on electserves as a reminder that in some cases the electronic state may

be antisymmetric. Although the degeneracies of the symmetric and antisym-

metric nuclear wavefunctions are the same for fermions and bosons, they are

not equal to each other. This means that a different number of possible total

wavefunctions are available for the homonuclear diatomic molecules, and this

will affect the number of molecules occupying each rotational state! Understand

what the above scenarios imply. If a molecule has a fermion as a nucleus, then

symmetric nuclear states will exist only for odd values of the Jrotational quan-

tum number. Similarly, if the nuclei are in an antisymmetric spin state, the

molecules will exist only with symmetric rotational states, that is, even values

of the Jrotational quantum number.

What this does is skew the “normal” population of measured rotational

states for homonuclear diatomic molecules, because the degeneracies are dif-

ferent. In fact, in spectra of homonuclear diatomic molecules—and also any

linear molecule that is symmetric with respect to a plane perpendicular to the

molecular axis, like C 2 H 2 —there is a profound intensity alternation because of

the different populations of odd and even rotational states, based on the above

analysis. Figure 18.2 shows a spectrum in which the intensity pattern shows

such behavior. This is one of the more spectacular experimental verifications

of the Pauli principle.

Example 18.7

Diatomic hydrogen has nuclei with a spin of^12 . What would be the expected

ratio of molecules in odd rotational states to molecules in even rotational

states?

Solution

With a spin of^12 , hydrogen’s nuclei are fermions. Therefore, according to

scenario 2 above, there will be (I 1)(2I 1) (^12  1)(2 ^12  1)  3

632 CHAPTER 18 More Statistical Thermodynamics

0.4

0.0

0

 (um)

1.55

Absorbance

1.51

0.3

0.2

0.1

1.52 1.53 1.54

Figure 18.2 This vibrational spectrum of
acetylene, C 2 H 2 , shows intensity variations that
are due to the effect of the nuclear wavefunction’s
symmetry on the degeneracies of the overall wave-
function of the molecule. This is one of the few
direct consequences ofnuclearwavefunctions in
chemistry.Source:L. W. Richards,J. Chem. Ed.,
1996, 43: 645.