Modern Spectroscopy

174 6 VIBRATIONAL SPECTROSCOPY
where i refers to a particular vibration. When each vibration is treated in the harmonic
oscillator approximation the vibrational term values are given by
Gðv iÞ¼o i v i þ d i
2
ð6:68Þ
where o i is the classical vibration wavenumber, v i the quantum number and d i the degree of
degeneracy.
6.2.4.1 Infrared spectra of linear molecules
Linear molecules belong to either the D 1h (with an inversion centre) or the C 1v (without an
inversion centre) point group. Using the vibrational selection rule in Equation (6.56) and the
D 1h (Table A.37 in Appendix A) or C 1v (Table A.16 in Appendix A) character table we can
see that the vibrational selection rules for transitions from the zero-point level (S þ g in D 1h,
S þ in C 1v) allow transitions of the type
in D 1h and
S þ u S þ g and P u S þ g ð6:69Þ
S þ S þ and P S þ
ð6:70Þ
in C 1v.
For all types of S vibrational levels the stack of rotational levels associated with them is
given by
F vðJÞ ¼B vJðJ þ 1Þ D vJ 2 ðJ þ 1Þ 2
ð6:71Þ
just as for a diatomic molecule (Equation 5.23). Two such stacks are shown in Figure 6.24
for a S þ u and S þ g vibrational level.
The rotational selection rule is
D J ¼ 1 ð6:72Þ
as for a diatomic molecule, and the resulting spectrum shows a P branch ðD J ¼ 1Þ and an
R branch ðD J ¼þ1Þ with members of each separated by about 2B, where B is an average
rotational constant for the two vibrational states, and a spacing of about 4B between Rð0Þ and
Pð1Þ. Treatment of the wavenumbers of the P and R branches to give the rotational constants
B 1 and B 0, for the upper and lower states, respectively, follows exactly the same method of
combination differences derived for diatomic molecules in Equations (6.27) to (6.33).