110
5 Molecular Theory of Local Field
In the last expression of Eq. (5.13), f (i) is a 3 × 3 matrix for each i as,
f pq (i) =
N
j =1
g pq (ij ).
(5.14)
f (i) indicates the correction factor for the dipole moment of the i-th molecule.
Polarizability A The polarizability of the whole system A is defined as the
derivative of the dipole moment M with respect to the external field. Here we
introduce the generalized effective polarizability of 3N × 3N tensor, α eff =
∂μ/∂E 0 , as the derivative of μ with respect to the electric field E 0 . It is derived
by differentiating Eq. (5.10) as
α
eff
=
∂μ
∂E 0 = g
T α = [1 + αT ]
−1 α = α[1 + T α]
−1
= αg.
(5.15)
α eff is written by
α
eff
=
⎛
⎜
⎜
⎜
⎝
α eff (11) α eff (12) · · · α eff (1N)
α eff (21) α eff (22)
. . .
. . .
. . .
α eff (N1)
· · · α eff (NN)
⎞
⎟
⎟
⎟
⎠
,
using the 3 × 3 component matrices α eff (ij ) = ∂μ(i)/∂E 0 (j ). α eff (ij ) is the
derivative of the dipole moment of the i-th molecule with respect to the electric
field at the j -th molecule. Note that α eff (ij ) may have finite values for i = j due to
the interaction between induced polarizations, in contrast to α in Eq. (5.6).
Suppose that the external field at j -th molecule varies by δE 0 (j ), it changes
the dipole moment of the same j -th molecule. As a consequence, the modified
dipole changes the polarization of a neighboring i-th molecule by δμ(i) through
the electrostatic intermolecular coupling. The first-order variation of δμ(i) is given
with the variation of δE 0 (j ) by
δμ(i) =
N
j =1
α
eff (ij ) δE
0 (j )
=
N
j =1
α(i) g(ij ) δE
0 (j ) = α(i) δE
loc (i).
(5.16)
The last expression of Eq. (5.16) is obtained by using the differential relation of
Eq. (5.9) with respect to E 0 ,
δE(i) =
j
g(ij ) δE
0 (j ) ≡ δE
loc (i),