6.4 Molecular Spectra: Nuclear Spin Effects
335
We note also that the rotational constant B 0 will differ slightly for these three
isotopologues, so that the characteristic temperatures will also be slightly different
(by less than 1%, however): for the present discussion we may hence employ a
common value of rot for all three isotopologues.
This set of three carbon dioxide isotopologues illustrates quite dramatically the
differences in behaviour that are associated with quantum- statistical effects.
Rotational Spectra for Nonlinear Molecules
In general for molecular spectroscopy, the full quantum mechanical form for
the rotational partition function is required for any interpretation of the intensity
patterns in the rotational fine-structure transitions in infrared spectra. For linear
molecules, we were fortunate enough that they all fell into two categories, described
by the operations of either the C ∞v point symmetry group (noncentrosymmetric
linear molecules) or the D ∞h point symmetry group (centrosymmetric linear
molecules). Nonlinear molecules belong to a whole assortment of point symmetry
groups, ranging from groups corresponding to a very low degree of symmetry in the
molecule to groups corresponding to a very high degree of molecular symmetry. At
the low symmetry end are molecules from possessing no symmetry at all (point
symmetry group C 1 ) through molecules possessing a single rotation axis (point
symmetry group C n ) or a small number of symmetry elements (such as point
symmetry group C 2v ) to molecules possessing tetrahedral (point symmetry group
T d ), octahedral (point symmetry group O h ) or icosahedral (point symmetry group
I h ) symmetry. Molecules having low degrees of symmetry (such as S 2 , C 2v , D 2h )
are often asymmetric top molecules, those with higher degrees of symmetry (such
as C 3v , D 2 , D 2d ) are often symmetric top molecules, and those with the highest
degrees of symmetry (T d , O h , I h ) are spherical top molecules. Both because much
molecular spectroscopy is more concerned with spectral frequency patterns than
with intensity patterns and because it is often very difficult to measure absolute
intensities, the role of the rotational partition function in determining line intensity
patterns is not always of great interest except for centrosymmetric linear molecules
where the intensity patterns are so comparatively dramatic.
Most polyatomic molecules have large moments-of-inertia and correspondingly
small values for their characteristic rotational temperatures. The nuclear spin
contributions to the combined rotational-nuclear spin partition function are largest
for temperatures that do not exceed 10 rot to 20 rot . For this reason, the only
nonlinear molecules for which nuclear spin symmetry effects may be expected to
play a role will be spherical top or light symmetric top hydrides, like CH 4 , CD 4 ,
SiH 4 , CDH 3 , or CHD 3 . We shall therefore comment briefly only upon the behaviour
of this set of molecules.
The relevant expression for z rot−nuc (T ) for a tetrahedral molecule like CH 4 has
been given explicitly [25] as
335
We note also that the rotational constant B 0 will differ slightly for these three
isotopologues, so that the characteristic temperatures will also be slightly different
(by less than 1%, however): for the present discussion we may hence employ a
common value of rot for all three isotopologues.
This set of three carbon dioxide isotopologues illustrates quite dramatically the
differences in behaviour that are associated with quantum- statistical effects.
Rotational Spectra for Nonlinear Molecules
In general for molecular spectroscopy, the full quantum mechanical form for
the rotational partition function is required for any interpretation of the intensity
patterns in the rotational fine-structure transitions in infrared spectra. For linear
molecules, we were fortunate enough that they all fell into two categories, described
by the operations of either the C ∞v point symmetry group (noncentrosymmetric
linear molecules) or the D ∞h point symmetry group (centrosymmetric linear
molecules). Nonlinear molecules belong to a whole assortment of point symmetry
groups, ranging from groups corresponding to a very low degree of symmetry in the
molecule to groups corresponding to a very high degree of molecular symmetry. At
the low symmetry end are molecules from possessing no symmetry at all (point
symmetry group C 1 ) through molecules possessing a single rotation axis (point
symmetry group C n ) or a small number of symmetry elements (such as point
symmetry group C 2v ) to molecules possessing tetrahedral (point symmetry group
T d ), octahedral (point symmetry group O h ) or icosahedral (point symmetry group
I h ) symmetry. Molecules having low degrees of symmetry (such as S 2 , C 2v , D 2h )
are often asymmetric top molecules, those with higher degrees of symmetry (such
as C 3v , D 2 , D 2d ) are often symmetric top molecules, and those with the highest
degrees of symmetry (T d , O h , I h ) are spherical top molecules. Both because much
molecular spectroscopy is more concerned with spectral frequency patterns than
with intensity patterns and because it is often very difficult to measure absolute
intensities, the role of the rotational partition function in determining line intensity
patterns is not always of great interest except for centrosymmetric linear molecules
where the intensity patterns are so comparatively dramatic.
Most polyatomic molecules have large moments-of-inertia and correspondingly
small values for their characteristic rotational temperatures. The nuclear spin
contributions to the combined rotational-nuclear spin partition function are largest
for temperatures that do not exceed 10 rot to 20 rot . For this reason, the only
nonlinear molecules for which nuclear spin symmetry effects may be expected to
play a role will be spherical top or light symmetric top hydrides, like CH 4 , CD 4 ,
SiH 4 , CDH 3 , or CHD 3 . We shall therefore comment briefly only upon the behaviour
of this set of molecules.
The relevant expression for z rot−nuc (T ) for a tetrahedral molecule like CH 4 has
been given explicitly [25] as