5.1 Local Field Correction Factor
111
and δE(i) is denoted with the superscript “loc” to designate the local field of i-th
molecule. 3 The last expression of Eq. (5.16) means that δμ(i) is also represented as
the local response of i-th molecule to the change in the local field δE loc (i).
Then we deal with a variation of the external field of light, δE ext . This gives rise
to the uniform variation of E 0 , δE 0 (1) = δE 0 (2) = · · · = δE 0 (N) = δE ext , since
the external field of light is considered uniform in the microscopic scale in question.
Accordingly, Eq. (5.16) indicates δμ(i) to be
δμ(i) =
N
j =1
α
eff (ij ) δE
0 (j ) =
⎛
⎝
N
j =1
α
eff (ij )
⎞
⎠ δE
ext .
This formula is written in the component representation using the factor g(ij )
or f (i),
δμ p (i) =
N
j
q
α
eff
pq (ij ) δE
ext
q =
N
j
x∼z
q,r
α pr (i) g rq (ij ) δE
ext
q
=
x∼z
q,r
α pr (i) f rq (i) δE
ext
q =
x∼z
r
α pr (i) δE
loc
r (i).
(5.17)
Equation (5.17) includes the variation of the local field δE loc
r (i) given by
δE
loc
r (i) =
j
x∼z
q
g rq (ij ) δE
ext
q =
x∼z
q
f rq (i) δE
ext
q .
(5.18)
Equation (5.18) clearly indicates that f (i) is the correction factor for the local field
at the i-th molecule with respect to the uniform external field. By addition of the
uniform external field, the dipole moment of the whole system δM changes by
δM =
N
i=1
δμ(i) =
⎛
⎝
N
i=1
N
j =1
α
eff (ij )
⎞
⎠ δE
ext .
This formula indicates the polarizability of the whole system A = ∂M/∂E ext ,
which presents the response of M with respect to the uniform external field E ext , as
A =
N
i=1
N
j =1
α
eff (ij ).
3 In the following we employ E loc with the superscript “loc” for E to distinguish the local field
from other kinds of field.