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p = α · E
(1)
Here, the α is the polarization tensor, depending on the vibration of the atoms in
molecules [4], while E is the electric field of the incident radiation at time t having
a frequency v, expressed as,
E = E 0 cos(2πvt)
(2)
Further, during vibration of the molecule, the atomic position of the nuclei changes
from their equilibrium results in a variation of the polarizability, which can be
expressed as the Taylor series in the vicinity of the position of the vibration,
α = α 0 +
i
∂α
∂ Q i
Q i +
1
2
i, j
∂
2
α
∂ Q i ∂ Q j
Q i Q j + · · ·
(3)
Here, α 0 stands for the polarizability in the equilibrium position, while Q i is associated with normal coordinates and having dependences on the vibrational frequencies
of molecules. In the above equation, neglecting the higher-order terms, the change
in polarizability can be written as follows:
α = α 0 + α 1 Q i
(4)
where α 1 =
∂α
∂ Q i
.
Furthermore, in ideal condition, the vibration of molecules can be considered as
simple harmonic oscillations, therefore, Q i is expressed as
Q i = Q 0 cos(2πv i t)
(5)
Here, Q 0 is the amplitude of the Q i . Substituting the Q i in Eq. (4), and then combining
Eqs. (2) and (1),
α = α 0 + α 1 Q 0 cos(2πv i t)
(6)
p = α 0 E 0 cos(2πvt) + α 1 Q 0 E 0 cos(2πv i t) cos(2πvt)
(7)
This further can be simplified as the following expression,
p = α 0 E 0 cos(2πvt) +
1
2
α 1 Q 0 E 0 [cos 2π (v − v i )t + cos 2π (v + v i )t]
(8)