108
5 Molecular Theory of Local Field
E =
⎛
⎜
⎜
⎜
⎝
E(1)
E(2)
. . .
E(N)
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
E x (1)
E y (1)
E z (1)
E x (2)
. . .
E z (N)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
E
0
=
⎛
⎜
⎜
⎜
⎝
E 0 (1)
E 0 (2)
. . .
E 0 (N)
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
E 0
x (1)
E 0
y (1)
E 0
z (1)
E 0
x (2)
. . .
E 0
z (N)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(5.5)
where the components μ(i), μ 0 (i), E(i), E 0 (i) for the i-th molecule are threedimensional vectors. On the other hand, α and T are 3N × 3N matrices, whose
components are written in the following form,
α =
⎛
⎜
⎜
⎜
⎝
α(1) 0 · · · 0
0 α(2)
. . .
. . .
0
α(N)
⎞
⎟
⎟
⎟
⎠
, T =
⎛
⎜
⎜
⎜
⎝
0
T (12) · · · T (1N)
T (21)
0
T (2N)
. . .
. . .
. . .
T (N1) T (N2) · · · 0
⎞
⎟
⎟
⎟
⎠
.
(5.6)
In this Eq. (5.6), α is symmetric since α(i) denotes the 3×3 symmetric polarizability
tensor of the i-th molecule. T is also symmetric as T (ij ) is a symmetric 3×3 matrix.
Then Eq. (5.4) is formally solved as follows.
E = E
0
− T μ = E
0
− T
μ
0
+ αE
,
(5.7)
μ = μ
0
+ αE = μ
0
+ α
E
0
− T μ
.
(5.8)
Therefore,
E = [1 + T α]
−1
E
0
− T μ
0
= g
E
0
− T μ
0
,
(5.9)
μ = [1 + αT ]
−1
μ
0
+ αE
0
= g
T
μ
0
+ αE
0
= g
T μ
0 ,
(5.10)
where μ 0 = μ 0 + αE 0 . g is a 3N × 3N matrix defined by
g = [1 + T α]
−1
or g
T
= [1 + αT ]
−1
.
(5.11)
Equations (5.9) and (5.10) manifest the physical meaning of g and g T . Equation (5.9) indicates that g is the factor to modify the electric field from E 0 − T μ 0
to E,
E
0
− T μ
0
g
− → E.