292
6 Molecular Systems
odd rotational states of H 2 will have a statistical weighting which is three times
that of the even rotational states. This weighting has a profound effect both on the
spectroscopy and low temperature thermodynamics of bulk H 2 gas.
We have seen from our discussion following Eq. (6.2.81) that ψ
total has four
contributing factors, namely, ψ tr , ψ vib , ψ el , and ψ rot , of which ψ tr and ψ vib are
always symmetric to the nuclear interchange (as they are unaffected by the interchange process), so that the interchange symmetry of ψ
total depends entirely upon
the interchange symmetries of ψ el and ψ rot . Moreover, we have seen that the nuclear
interchange symmetry of ψ el can be determined from the electronic term symbol
associated with the wavefunction for that electronic state. Electronic wavefunctions
ψ el that are symmetric (even) or antisymmetric (odd) under both inversion, ı, of
the electronic coordinate system and reflection across a vertical mirror plane, σ v ,
are symmetric to the nuclear interchange, while electronic wavefunctions that have
opposite symmetry under these two operations are antisymmetric to the nuclear
interchange. We have also determined that the nuclear interchange symmetry of
the rotational wavefunction, ψ rot ≡ ψ jm , (j , m are rotational angular momentum
quantum numbers) is given by (−1) j .
A key point arising from our consideration of the effect of symmetry upon the
behaviour of homonuclear diatomic molecules in a (g, +) electronic term is that
the rotational and nuclear spin partition functions cannot be trivially separated into
the product of two independent partition functions z rot (T ) and z nuc (T ), but must in
general be considered together as a single partition function z rot−nuc (T ).
For H 2 we thus see that the combined nuclear spin-rotational partition function
z rot−nuc has in general the structure
z rot−nuc (T ) = z ortho (T ) + z para (T ) ,
(6.2.93)
in which the individual partition functions for the ortho and para modifications are
given by
z ortho (T ) = 3
j =1,3,5,...
(2j + 1)e
−j (j+1)) rot /T ,
(6.2.94)
and
z para (T ) =
j =0,2,4,...
(2j + 1)e
−j (j+1)) rot /T .
(6.2.95)
We thus see that the restrictions imposed by the Pauli Principle cause z rot−nuc for
hydrogen not to factor in general into the product of rotational and nuclear spin
partition functions.
Although the discussion thus far has been focussed upon molecular hydrogen, it
will be clear from what has been said that much the same result, i.e., an inability to
represent the rotational and nuclear spin contributions to the partition function for a
homonuclear hydrogen molecule in general as the product of rotational and nuclear
6 Molecular Systems
odd rotational states of H 2 will have a statistical weighting which is three times
that of the even rotational states. This weighting has a profound effect both on the
spectroscopy and low temperature thermodynamics of bulk H 2 gas.
We have seen from our discussion following Eq. (6.2.81) that ψ
total has four
contributing factors, namely, ψ tr , ψ vib , ψ el , and ψ rot , of which ψ tr and ψ vib are
always symmetric to the nuclear interchange (as they are unaffected by the interchange process), so that the interchange symmetry of ψ
total depends entirely upon
the interchange symmetries of ψ el and ψ rot . Moreover, we have seen that the nuclear
interchange symmetry of ψ el can be determined from the electronic term symbol
associated with the wavefunction for that electronic state. Electronic wavefunctions
ψ el that are symmetric (even) or antisymmetric (odd) under both inversion, ı, of
the electronic coordinate system and reflection across a vertical mirror plane, σ v ,
are symmetric to the nuclear interchange, while electronic wavefunctions that have
opposite symmetry under these two operations are antisymmetric to the nuclear
interchange. We have also determined that the nuclear interchange symmetry of
the rotational wavefunction, ψ rot ≡ ψ jm , (j , m are rotational angular momentum
quantum numbers) is given by (−1) j .
A key point arising from our consideration of the effect of symmetry upon the
behaviour of homonuclear diatomic molecules in a (g, +) electronic term is that
the rotational and nuclear spin partition functions cannot be trivially separated into
the product of two independent partition functions z rot (T ) and z nuc (T ), but must in
general be considered together as a single partition function z rot−nuc (T ).
For H 2 we thus see that the combined nuclear spin-rotational partition function
z rot−nuc has in general the structure
z rot−nuc (T ) = z ortho (T ) + z para (T ) ,
(6.2.93)
in which the individual partition functions for the ortho and para modifications are
given by
z ortho (T ) = 3
j =1,3,5,...
(2j + 1)e
−j (j+1)) rot /T ,
(6.2.94)
and
z para (T ) =
j =0,2,4,...
(2j + 1)e
−j (j+1)) rot /T .
(6.2.95)
We thus see that the restrictions imposed by the Pauli Principle cause z rot−nuc for
hydrogen not to factor in general into the product of rotational and nuclear spin
partition functions.
Although the discussion thus far has been focussed upon molecular hydrogen, it
will be clear from what has been said that much the same result, i.e., an inability to
represent the rotational and nuclear spin contributions to the partition function for a
homonuclear hydrogen molecule in general as the product of rotational and nuclear