6.4 Molecular Spectra: Nuclear Spin Effects
333
structure transitions. C 2 H 2 has a rotational constant [22] B 0 = 1.17692 cm −1 ,
giving a characteristic rotational temperature rot 1.69 K, and a j max value
of approximately 9 for temperature T = 298 K. The P - and R-branches of this
overtone band in C 2 H 2 at temperature T = 298 K shown in Fig. 6.19 both clearly
display 3:1 intensity ratios.
Example 6.12 Carbon dioxide, CO 2 .
It is instructive in the case of CO 2 to compare the expected forms for the
rotational fine structures of the vibration–rotation bands for three different CO 2
isotopologues. We shall examine the nature of the fine-structure bands for the most
abundant isotopologue, 12 C 16 O 2 , and compare our results for it with the structures
of the corresponding vibration–rotation bands for two less common isotopologues,
specifically, 12 C 17 O 2 and 18 O 12 C 16 O. As we have seen in previous examples, both
the 12 C and 16 O nuclei are bosons with spin I a = 0: the 18 O nucleus likewise has
spin I a = 0, but the 17 O nucleus is a fermion with spin I a =
5
2 .
Other than producing an overall factor 2I a + 1 = 1 in the combined rotationalnuclear spin partition function z rot−nuc (T ), the nuclear spin of 12 C is irrelevant to
our considerations, as the C atom is located at the centre of the CO 2 molecule. Of
course, any nucleus located at the centre of a molecule will play no role in a spin
symmetry argument.
The 12 C 16 O 2 isotopologue of CO 2 is a centrosymmetric linear molecule with two
equivalent bosonic nuclei, so that if we are considering the fundamental vibrational
transition of the ν 1 (symmetric stretch) vibrational mode of CO 2 (which carries σ +
g
symmetry [22]) for which ψ vib is symmetric to the interchange of the two O nuclei,
we may employ Eq. (6.2.109) directly to obtain the relevant form for z rot−nuc (T ),
namely,
z rot−nuc (T ) =
j =even
(2j + 1)e
−j (j+1)) rot /T .
We thus see from this expression that the rotational fine-structure lines of the ν 1
vibrational transitions in this isotopologue of CO 2 will be associated only with
rotational levels for which the quantum number j is an even integer: this means
that adjacent lines in all ν 1 infrared spectra will be separated by 4B 0 rather than by
2B 0 because of missing lines that correspond to odd values of j . There will thus be
no alternation of intensities in this spectrum, as can be seen from the pure rotational
Raman spectrum of the predominant carbon dioxide isotopologue 12 C 16 O 2 shown
in Fig. 6.20.
However, were we dealing with the rotational fine structure of the CO 2 asymmetric stretch, which has (group) symmetry σ +
u , and is hence characterized by
a vibrational wavefunction that is antisymmetric to the interchange of the two O
nuclei, just the opposite conclusion would be reached. Thus, for example, rotational
fine-structure lines in the ν 3 fundamental vibrational spectrum will involve only
rotational levels for which j is odd. This behaviour has indeed been confirmed
experimentally [22] long ago.
333
structure transitions. C 2 H 2 has a rotational constant [22] B 0 = 1.17692 cm −1 ,
giving a characteristic rotational temperature rot 1.69 K, and a j max value
of approximately 9 for temperature T = 298 K. The P - and R-branches of this
overtone band in C 2 H 2 at temperature T = 298 K shown in Fig. 6.19 both clearly
display 3:1 intensity ratios.
Example 6.12 Carbon dioxide, CO 2 .
It is instructive in the case of CO 2 to compare the expected forms for the
rotational fine structures of the vibration–rotation bands for three different CO 2
isotopologues. We shall examine the nature of the fine-structure bands for the most
abundant isotopologue, 12 C 16 O 2 , and compare our results for it with the structures
of the corresponding vibration–rotation bands for two less common isotopologues,
specifically, 12 C 17 O 2 and 18 O 12 C 16 O. As we have seen in previous examples, both
the 12 C and 16 O nuclei are bosons with spin I a = 0: the 18 O nucleus likewise has
spin I a = 0, but the 17 O nucleus is a fermion with spin I a =
5
2 .
Other than producing an overall factor 2I a + 1 = 1 in the combined rotationalnuclear spin partition function z rot−nuc (T ), the nuclear spin of 12 C is irrelevant to
our considerations, as the C atom is located at the centre of the CO 2 molecule. Of
course, any nucleus located at the centre of a molecule will play no role in a spin
symmetry argument.
The 12 C 16 O 2 isotopologue of CO 2 is a centrosymmetric linear molecule with two
equivalent bosonic nuclei, so that if we are considering the fundamental vibrational
transition of the ν 1 (symmetric stretch) vibrational mode of CO 2 (which carries σ +
g
symmetry [22]) for which ψ vib is symmetric to the interchange of the two O nuclei,
we may employ Eq. (6.2.109) directly to obtain the relevant form for z rot−nuc (T ),
namely,
z rot−nuc (T ) =
j =even
(2j + 1)e
−j (j+1)) rot /T .
We thus see from this expression that the rotational fine-structure lines of the ν 1
vibrational transitions in this isotopologue of CO 2 will be associated only with
rotational levels for which the quantum number j is an even integer: this means
that adjacent lines in all ν 1 infrared spectra will be separated by 4B 0 rather than by
2B 0 because of missing lines that correspond to odd values of j . There will thus be
no alternation of intensities in this spectrum, as can be seen from the pure rotational
Raman spectrum of the predominant carbon dioxide isotopologue 12 C 16 O 2 shown
in Fig. 6.20.
However, were we dealing with the rotational fine structure of the CO 2 asymmetric stretch, which has (group) symmetry σ +
u , and is hence characterized by
a vibrational wavefunction that is antisymmetric to the interchange of the two O
nuclei, just the opposite conclusion would be reached. Thus, for example, rotational
fine-structure lines in the ν 3 fundamental vibrational spectrum will involve only
rotational levels for which j is odd. This behaviour has indeed been confirmed
experimentally [22] long ago.