Modern Spectroscopy

140 6 VIBRATIONAL SPECTROSCOPY
Figure 6.1 Variation of dipole moment m with internuclear distance r in a heteronuclear diatomic
molecule
6.1.2 Raman spectra
In both heteronuclear and homonuclear diatomic molecules the polarizability (see Section
5.3.2) varies during vibrational motion leading to a vibrational Raman effect. This variation
can be visualized as being due to the polarizability ellipsoid (Figure 5.14) expanding and
contracting as the bond length increases and decreases with vibration. A classical treatment,
analogous to that for rotation in Section 5.3.2, leads to a variation with time of the dipole
moment m, induced by irradiation of the sample with intense monochromatic radiation of
wavenumber ~n, givenby
m ¼ a 0;v A sin 2pc~nt 1
2 a 1;v A cos 2pcð~n þ oÞt ð6:11Þ
þ 1
2 a 1;v A cos 2pcð~n oÞt
In this equation a 0;v is the average polarizability during vibration, a 1;v is the amplitude of the
change of polarizability due to vibration, A is the amplitude of the oscillating electric field of
the incident radiation (Equation 5.44) and o is the vibration wavenumber. Equation (6.11) is
similar to Equation (5.46) for rotation except that the second and third terms in Equation
(6.11) correspond to Raman scattering with a wavenumber of ð~n þ oÞ and ð~n oÞ, the anti-
Stokes and Stokes Raman scattering, respectively. Whereas for rotation a changes at twice
the rotational frequency, during a complete vibrational cycle a goes through only one cycle
and, therefore, changes at the same frequency as the frequency of vibration. This accounts
for the absence of the factor of two in Equation (6.11) compared with Equation (5.46).
As for the change of dipole moment, the change of polarizability with vibrational
displacement x can be expressed as a Taylor series
a ¼ ae þ da
x þ
dx e
1
2!
d 2 a
dx 2
x
e
2 þ ::: ð6:12Þ