330
6 Molecular Systems
6.4.2 Polyatomic Molecular Spectra
We have seen in Sect. 6.4.1 that the full quantum mechanical expression for the
rotational partition function is needed to explain the intensity patterns both of
pure rotational spectra and of the rotational fine structure (typically in the form
of P - and R-branches) observed in high-resolution vibrational spectra of diatomic
molecules. Moreover, we have seen that there are quite dramatic effects in the
form of alternating intensities for adjacent rotational lines and, in some cases, even
missing lines in the spectra of homonuclear diatomic molecules. These effects could
all be explained via nuclear spin symmetry arguments arising from application of
the Pauli Principle for indistinguishable particles.
The arguments utilized in our discussion of rotational spectral intensity patterns
for diatomic molecules translate straightforwardly to rotational spectroscopic transitions in linear polyatomic molecules. However, for nonlinear polyatomic molecules,
the application of symmetry arguments is considerably more complicated than it is
for linear polyatomic molecules. For this reason, we shall split our discussion into
two parts, one dealing with linear molecules, the other with nonlinear molecules.
Rotational Spectra for Linear Polyatomic Molecules
It should not be too surprising that we may literally apply all of the arguments
presented in Sect. 6.4.1 to linear molecules in general, given that diatomic molecules
themselves form a subset of linear molecules, with heteronuclear diatomic
molecules the smallest noncentrosymmetric linear molecules (point symmetry
group C ∞v ) and homonuclear diatomic molecules the smallest centrosymmetric
linear molecules (point symmetry group D ∞h ).
As the concepts presented in Sect. 6.4.1 apply directly to linear molecules, we
shall simply consider three examples: a simple noncentrosymmetric linear molecule,
N 2 O (structure NNO) and two centrosymmetric linear molecules, C 2 H 2 (structure
HCCH), and CO 2 (structure OCO).
Example 6.10 Nitrous oxide, N 2 O.
The nitrous oxide molecule, N 2 O, has a pure rotational electric-dipole-allowed
absorption spectrum in which transitions involving j = 0 to j = 11 lie in the
microwave (MW) region, while those involving j ≥ 12 lie in the far infrared
region of the electromagnetic spectrum. The spectral lines shown in Fig. 6.18 arise
from transitions between lower and upper rotational states that both belong to the
ground vibrational level of the N 2 O molecule: the spectrum consists of a series of
transitions with adjacent spectral lines approximately equidistant from one another.
Transitions for which both j and j lie in the same vibrational level are necessarily
such that j − j > 0, so that pure rotational spectra always consist solely of an
6 Molecular Systems
6.4.2 Polyatomic Molecular Spectra
We have seen in Sect. 6.4.1 that the full quantum mechanical expression for the
rotational partition function is needed to explain the intensity patterns both of
pure rotational spectra and of the rotational fine structure (typically in the form
of P - and R-branches) observed in high-resolution vibrational spectra of diatomic
molecules. Moreover, we have seen that there are quite dramatic effects in the
form of alternating intensities for adjacent rotational lines and, in some cases, even
missing lines in the spectra of homonuclear diatomic molecules. These effects could
all be explained via nuclear spin symmetry arguments arising from application of
the Pauli Principle for indistinguishable particles.
The arguments utilized in our discussion of rotational spectral intensity patterns
for diatomic molecules translate straightforwardly to rotational spectroscopic transitions in linear polyatomic molecules. However, for nonlinear polyatomic molecules,
the application of symmetry arguments is considerably more complicated than it is
for linear polyatomic molecules. For this reason, we shall split our discussion into
two parts, one dealing with linear molecules, the other with nonlinear molecules.
Rotational Spectra for Linear Polyatomic Molecules
It should not be too surprising that we may literally apply all of the arguments
presented in Sect. 6.4.1 to linear molecules in general, given that diatomic molecules
themselves form a subset of linear molecules, with heteronuclear diatomic
molecules the smallest noncentrosymmetric linear molecules (point symmetry
group C ∞v ) and homonuclear diatomic molecules the smallest centrosymmetric
linear molecules (point symmetry group D ∞h ).
As the concepts presented in Sect. 6.4.1 apply directly to linear molecules, we
shall simply consider three examples: a simple noncentrosymmetric linear molecule,
N 2 O (structure NNO) and two centrosymmetric linear molecules, C 2 H 2 (structure
HCCH), and CO 2 (structure OCO).
Example 6.10 Nitrous oxide, N 2 O.
The nitrous oxide molecule, N 2 O, has a pure rotational electric-dipole-allowed
absorption spectrum in which transitions involving j = 0 to j = 11 lie in the
microwave (MW) region, while those involving j ≥ 12 lie in the far infrared
region of the electromagnetic spectrum. The spectral lines shown in Fig. 6.18 arise
from transitions between lower and upper rotational states that both belong to the
ground vibrational level of the N 2 O molecule: the spectrum consists of a series of
transitions with adjacent spectral lines approximately equidistant from one another.
Transitions for which both j and j lie in the same vibrational level are necessarily
such that j − j > 0, so that pure rotational spectra always consist solely of an