Physical Chemistry , 1st ed.

that affects the number of possible wavefunctions (that is, the degeneracies),
and thus the partition function. Ultimately, we will see effects on the thermo-
dynamic properties of homonuclear diatomic gases that are caused directly by
restrictions of the Pauli principle.
Recall that we defined the overall molecular partition function as
Qqtransqelectqvibqrotqnuc

We also separate the overall wavefunction of a molecule into similar parts:

totaltranselectvibrotnuc

Of the five parts of,transand vibcan always be considered symmetric as
far as the Pauli principle is concerned (no matter which type of particle—a
fermion or a boson—the nucleus is). The electronic wavefunction electis al-
most always symmetric. For homonuclear diatomic molecules, there is usually
a superscript on the term symbol of the ground electronic state that implies
symmetric behavior; however, some diatomic molecules—O 2 is the notewor-
thy one—have a superscript minus () in their term symbol, indicating that
the ground electronic state is actually antisymmetric. Ignoring these rare ex-
ceptions (but see the end-of-chapter exercises), ultimately the rotand the
nucpartition functions combine to determine the overall symmetry ofQfor
the molecule.
If the nuclei of the atoms in the diatomic molecule have integer spins, they
are bosons and the complete wavefunction (translational-nuclear-vibrational-
electronic-rotational) mustbe symmetric with respect to exchange. If the nu-
clei of the atoms in the diatomic molecule have half-integer spins, then they
are fermions and the complete wavefunction mustbe antisymmetric upon ex-
change. Given that transand vibare symmetric and electis almost always
symmetric for homonuclear diatomic molecules, the overall wavefunction
symmetry behavior dictates what combinations ofrotand nucare allowed.
What we find ultimately is that the combinations ofrotand nuchave dif-
ferent degeneracies, so that the populations of molecules in various rotational
states are skewed from normal expectations.
Nuclei are fermions or bosons depending on whether their spins are half-
integers or integers. For a nucleus that has a spin of magnitude I, there are
2 I 1 possible spin states (just as there are 2J 1 possible rotational states).
For twonuclei, there are (2I 1)(2I 1) (2I 1)^2 possible combinations.
Some of these combinations will be symmetric, and some will be antisym-
metric. (This is analogous to the consideration of symmetric and antisym-
metric electron spin states for the He atom in Chapter 12.) It turns out that
for all nuclei, boson or fermion, there are 2I^2 Iantisymmetric spin states,
and the remaining states [out of the (2I 1)^2 states] are symmetric.
Finally, we note that rotis symmetric for even values of the Jrotational
quantum number, and is antisymmetric for odd values of the Jrotational
quantum number.
We therefore have the following two scenarios:

Scenario 1: boson nuclei (that is, integer spin nuclei)

tottrans
vib
elect
nuc
rot
sym sym sym sym* sym sym,Jeven
degen: (I 1)(2I 1) degen: 2J 1
or
antisym antisym,Jodd
degen: 2I^2 I degen: 2J 1

18.5 Diatomic Molecules: Rotations 631