6.4 χ (2) Formula with CRK Model
135
and V 0 ,
V
0
ai =
j ( =i)
site
b
Q 0
bj
|R(ai) − R(bj )|
.
(6.39)
Using G and V 0 , the solutions of Eqs. (6.35) and (6.36) are given as follows:
V ai =
j
site
c
[G
−1
] ai,cj
V
0
cj − R(cj ) · E 0
,
(6.40)
Q ai =
j
site
c
[G
T −1
] ai,cj
Q
0
cj −
site
b
K cbj (R(bj ) · E 0 )
=
j
site
c
[G
−1
] cj,ai
Q
0
cj −
site
b
K cbj (R(bj ) · E 0 )
.
(6.41)
Now we can present the dipole moment and polarizability of the whole system
using the above solutions. The dipole moment is represented by
M =
i
site
a
Q ai R(ai)
=
i
j
site
a
site
c
[G
−1
] cj,ai Q
0
cj R(ai)
(E 0 = 0).
(6.42)
The polarizability tensor of the whole system is given by differentiating M with
respect to E 0 and then setting to E 0 = 0,
A =
∂M
∂E 0
=
i
site
a
∂Q ai
∂E 0
R(ai)
= −
i
site
a
site
b
K ab,i R(ai) ⊗
⎡
⎣
j
site
c
[G
−1
] bi,cj R(cj )
⎤
⎦
(E 0 = 0),
(6.43)
where ⊗ stands for tensor product. We further consider the local field correction for
the sum frequency field in the same manner as in Sect. 5.2, and thereby formulate
the effective polarizability A eff as