6.2 Diatomic Molecules
299
versa, while if the nuclei in a homonuclear molecule are bosons, then the nuclear
interchange symmetries of ψ
total and ψ nuclear spin must be the same.
As it happens that a preponderance of homonuclear diatomic molecules have
1 +
g ground electronic states, for which we have seen that ψ el is even, the nuclear
interchange symmetry of ψ
total is determined solely by the symmetry of the
rotational wavefunction. Hence, for a diatomic molecule X 2 ( 1 +
g ) whose nuclei
are fermions, the combined rotational-nuclear spin partition function takes the form
z rot−nuc (T ) = I a (2I a + 1)
j even
(2j + 1)e
−j (j+1)) rot /T
+ (I a + 1)(2I a + 1)
j odd
(2j + 1)e
−j (j+1)) rot /T .
(6.2.108)
Prominent examples of such diatomic molecules are H 2 , F 2 , with nuclear spins I a =
1
2 , Cl 2 (both 35 Cl and 37 Cl have nuclear spin I a =
3
2 ), and I 2 (with I a =
7
2 ).
Similarly, for X 2 ( 1 +
g ) molecules whose nuclei are bosons, we have
z rot−nuc (T ) = (I a + 1)(2I a + 1)
j even
(2j + 1)e
−j (j+1)) rot /T
+ I a (2I a + 1)
j odd
(2j + 1)e
−j (j+1)) rot /T .
(6.2.109)
Prominent examples of such diatomic molecules are D 2 , 14 N 2 , 6 Li 2 (these are the
only nuclei with I a = 1), 12 C 2 , and 28 Si 2 (both the 12 C and 28 Si nuclei have nuclear
spin I a = 0).
Example 6.3 Molecular oxygen.
Molecular oxygen is a special case, as it has a 3 −
g ground electronic term.
From Fig. 6.9, we see that the electronic wavefunction for this electronic term
is antisymmetric to the interchange of the two indistinguishable nuclei of a
homonuclear diatomic molecule. This means that ψ
total (1, 2) for an O 2 molecule
will therefore be symmetric to the nuclear interchange for O 2 molecules that are
in rotational states for which j is odd, and antisymmetric for O 2 molecules that
are in rotational states for which j is even. For an O 2 molecule whose nuclei
are equivalent bosons the requirement that the total wavefunction total (1, 2) be
symmetric to the nuclear interchange results in the odd-j rotational states being
coupled with the symmetric nuclear spin pair-states, and vice versa. The opposite
outcome will be obtained for an O 2 molecule whose nuclei are equivalent fermions.
The most common oxygen isotope by far is 16 O, with nuclear spin I a = 0. In this
case there is only a single, symmetric, nuclear spin pair-state, and it must therefore
be coupled with the odd rotational levels of 16 O 2 . Consequently, 16 O 2 molecules
possess only odd rotational levels, and the combined rotational-nuclear partition
function z rot−nuc (T ) for 16 O 2 is therefore given by
299
versa, while if the nuclei in a homonuclear molecule are bosons, then the nuclear
interchange symmetries of ψ
total and ψ nuclear spin must be the same.
As it happens that a preponderance of homonuclear diatomic molecules have
1 +
g ground electronic states, for which we have seen that ψ el is even, the nuclear
interchange symmetry of ψ
total is determined solely by the symmetry of the
rotational wavefunction. Hence, for a diatomic molecule X 2 ( 1 +
g ) whose nuclei
are fermions, the combined rotational-nuclear spin partition function takes the form
z rot−nuc (T ) = I a (2I a + 1)
j even
(2j + 1)e
−j (j+1)) rot /T
+ (I a + 1)(2I a + 1)
j odd
(2j + 1)e
−j (j+1)) rot /T .
(6.2.108)
Prominent examples of such diatomic molecules are H 2 , F 2 , with nuclear spins I a =
1
2 , Cl 2 (both 35 Cl and 37 Cl have nuclear spin I a =
3
2 ), and I 2 (with I a =
7
2 ).
Similarly, for X 2 ( 1 +
g ) molecules whose nuclei are bosons, we have
z rot−nuc (T ) = (I a + 1)(2I a + 1)
j even
(2j + 1)e
−j (j+1)) rot /T
+ I a (2I a + 1)
j odd
(2j + 1)e
−j (j+1)) rot /T .
(6.2.109)
Prominent examples of such diatomic molecules are D 2 , 14 N 2 , 6 Li 2 (these are the
only nuclei with I a = 1), 12 C 2 , and 28 Si 2 (both the 12 C and 28 Si nuclei have nuclear
spin I a = 0).
Example 6.3 Molecular oxygen.
Molecular oxygen is a special case, as it has a 3 −
g ground electronic term.
From Fig. 6.9, we see that the electronic wavefunction for this electronic term
is antisymmetric to the interchange of the two indistinguishable nuclei of a
homonuclear diatomic molecule. This means that ψ
total (1, 2) for an O 2 molecule
will therefore be symmetric to the nuclear interchange for O 2 molecules that are
in rotational states for which j is odd, and antisymmetric for O 2 molecules that
are in rotational states for which j is even. For an O 2 molecule whose nuclei
are equivalent bosons the requirement that the total wavefunction total (1, 2) be
symmetric to the nuclear interchange results in the odd-j rotational states being
coupled with the symmetric nuclear spin pair-states, and vice versa. The opposite
outcome will be obtained for an O 2 molecule whose nuclei are equivalent fermions.
The most common oxygen isotope by far is 16 O, with nuclear spin I a = 0. In this
case there is only a single, symmetric, nuclear spin pair-state, and it must therefore
be coupled with the odd rotational levels of 16 O 2 . Consequently, 16 O 2 molecules
possess only odd rotational levels, and the combined rotational-nuclear partition
function z rot−nuc (T ) for 16 O 2 is therefore given by