6.3 Extension to the Ideal Polyatomic Gas
317
to the classical limit, in particular, can be quite different for centrosymmetric
and noncentrosymmetric polyatomic molecules, as has already been noted for
diatomic molecules. We shall consider all three types of correction to the rotational
contributions obtained using the classical limit T rot separately for linear and
nonlinear molecules.
Linear Molecules
Heteronuclear diatomic molecules are the simplest members of the class of noncentrosymmetric linear molecules and homonuclear diatomic molecules are likewise
the simplest centrosymmetric linear molecules. Nuclear spin and rotational degrees
of freedom may always be decoupled for heteronuclear diatomic molecules, as there
are no nuclear interchange effects. Thus, z rot−nuc (T ) may always be written as
z rot (T )z nuc , with z nuc = (2I a + 1)(2I b + 1) and I a , I b the nuclear spins of the
distinguishable nuclei. For homonuclear diatomic molecules, however, z rot−nuc (T )
cannot, in general, be so decoupled due to the definite nuclear interchange symmetry imposed upon the molecular wavefunctions by the Pauli Principle for
indistinguishable particles. We also found [see Eq. (6.2.100)] that in the hightemperature limit T rot , for which the partition function sums over even and
odd rotational quantum numbers are each equal to one-half the partition function
sum over all rotational quantum numbers, z rot−nuc (T ) then becomes z rot (T )z nuc ,
with z nuc = (2I a + 1) 2 . Precisely this same behaviour will be found for general
noncentrosymmetric and centrosymmetric linear molecules, except that z nuc will be
given by
z nuc =
N a
i=1
(2I i + 1) ,
with I i the nuclear spin for nucleus i and N a the number of atoms in the molecule.
Because the moments-of-inertia associated with the end-over-end rotational
motions for such molecules will be large, the corresponding characteristic rotational
temperatures will typically be fractions of a kelvin. Consequently, any temperature
at which the substance is still in gaseous form will correspond to the hightemperature limit, so that there is no longer a need to consider symmetry restrictions
explicitly for the purposes of calculating thermodynamic function values for such
molecules. Note, however, that we must still consider indistinguishable particle
interchange symmetry requirements when dealing with rotational spectroscopy.
Euler–Maclaurin corrections to the classical limit expression for the rotational
partition function, as well as centrifugal distortion and vibration–rotation interaction
corrections can be expressed in the same way as those for diatomic molecules. In
particular, the Euler–Maclaurin corrections to the classical limit for z rot (T ) are given
as [7]
317
to the classical limit, in particular, can be quite different for centrosymmetric
and noncentrosymmetric polyatomic molecules, as has already been noted for
diatomic molecules. We shall consider all three types of correction to the rotational
contributions obtained using the classical limit T rot separately for linear and
nonlinear molecules.
Linear Molecules
Heteronuclear diatomic molecules are the simplest members of the class of noncentrosymmetric linear molecules and homonuclear diatomic molecules are likewise
the simplest centrosymmetric linear molecules. Nuclear spin and rotational degrees
of freedom may always be decoupled for heteronuclear diatomic molecules, as there
are no nuclear interchange effects. Thus, z rot−nuc (T ) may always be written as
z rot (T )z nuc , with z nuc = (2I a + 1)(2I b + 1) and I a , I b the nuclear spins of the
distinguishable nuclei. For homonuclear diatomic molecules, however, z rot−nuc (T )
cannot, in general, be so decoupled due to the definite nuclear interchange symmetry imposed upon the molecular wavefunctions by the Pauli Principle for
indistinguishable particles. We also found [see Eq. (6.2.100)] that in the hightemperature limit T rot , for which the partition function sums over even and
odd rotational quantum numbers are each equal to one-half the partition function
sum over all rotational quantum numbers, z rot−nuc (T ) then becomes z rot (T )z nuc ,
with z nuc = (2I a + 1) 2 . Precisely this same behaviour will be found for general
noncentrosymmetric and centrosymmetric linear molecules, except that z nuc will be
given by
z nuc =
N a
i=1
(2I i + 1) ,
with I i the nuclear spin for nucleus i and N a the number of atoms in the molecule.
Because the moments-of-inertia associated with the end-over-end rotational
motions for such molecules will be large, the corresponding characteristic rotational
temperatures will typically be fractions of a kelvin. Consequently, any temperature
at which the substance is still in gaseous form will correspond to the hightemperature limit, so that there is no longer a need to consider symmetry restrictions
explicitly for the purposes of calculating thermodynamic function values for such
molecules. Note, however, that we must still consider indistinguishable particle
interchange symmetry requirements when dealing with rotational spectroscopy.
Euler–Maclaurin corrections to the classical limit expression for the rotational
partition function, as well as centrifugal distortion and vibration–rotation interaction
corrections can be expressed in the same way as those for diatomic molecules. In
particular, the Euler–Maclaurin corrections to the classical limit for z rot (T ) are given
as [7]