112
5 Molecular Theory of Local Field
The components of A are written by
A pq =
N
i,j
α
eff
pq (ij ) =
N
i,j
x∼z
r
α pr (i)g rq (ij ) =
N
i
x∼z
r
α pr (i)f rq (i).
(5.19)
Equation (5.19) shows that A is not simply the sum of α(i), but involves the local
field correction factor f (i).
5.2 Local Field Correction for χ (2)
Next, we irradiate the interface system with oscillating electric fields of visible light
E ext (ω 1 ) and infrared light E ext (ω 2 ). The superscript “ext” designates the external
fields of perturbation. 4 Then the nonlinear susceptibility of interface generates the
sum-frequency polarization at the i-th molecule, namely μ (2),0 (i, ,). Here we
formulate the sum-frequency polarization generated at the interface from the sumfrequency polarization at each molecule.
Let us recall the microscopic derivation of nonlinear polarization from the
second-order perturbation on the density matrix in Chap. 3. When we have defined
the perturbation Hamiltonian ˆ
H = − ˆ
μ · E ext (t) in Eq. (3.18) in the semiclassical
theory, ˆ
μ designates the dipole operator of the whole material system (including
the interface) and E ext (t) is the external electric field as the perturbation. The dipole
operator for the whole material system ˆ
μ is given by the sum of dipole operators
of constituent molecules ˆ
μ(j ), i.e. ˆ
μ =
j
ˆ
μ(j ). The second-order polarization in
Eq. (3.29) has been represented with the dipole operator ˆ
μ and the second-order
density matrix ρ (2) by
P
(2),0
p
(t) = Tr
ˆ
μ p ρ
(2) (t)
= Tr
⎡
⎣
⎛
⎝
N
j
ˆ
μ p (j )
⎞
⎠ ρ
(2) (t)
⎤
⎦ =
N
j
μ
(2),0
p
(j, t).
(5.20)
We put the superscript 0 in the left-hand side to emphasize that Eq. (5.20) is the
simple sum of the bare nonlinear polarizations of the molecules. This relation
is converted to the frequency domain by taking the Fourier transformation and
extracting the exp(−iit) component,
P
(2),0
p
(() =
N
j
μ
(2),0
p
(j, ,).
(5.21)
4 We note the distinction between E ext (ω) and the incident field E I (ω) in Chap. 2, as detailed in
Sect. 5.3.
5 Molecular Theory of Local Field
The components of A are written by
A pq =
N
i,j
α
eff
pq (ij ) =
N
i,j
x∼z
r
α pr (i)g rq (ij ) =
N
i
x∼z
r
α pr (i)f rq (i).
(5.19)
Equation (5.19) shows that A is not simply the sum of α(i), but involves the local
field correction factor f (i).
5.2 Local Field Correction for χ (2)
Next, we irradiate the interface system with oscillating electric fields of visible light
E ext (ω 1 ) and infrared light E ext (ω 2 ). The superscript “ext” designates the external
fields of perturbation. 4 Then the nonlinear susceptibility of interface generates the
sum-frequency polarization at the i-th molecule, namely μ (2),0 (i, ,). Here we
formulate the sum-frequency polarization generated at the interface from the sumfrequency polarization at each molecule.
Let us recall the microscopic derivation of nonlinear polarization from the
second-order perturbation on the density matrix in Chap. 3. When we have defined
the perturbation Hamiltonian ˆ
H = − ˆ
μ · E ext (t) in Eq. (3.18) in the semiclassical
theory, ˆ
μ designates the dipole operator of the whole material system (including
the interface) and E ext (t) is the external electric field as the perturbation. The dipole
operator for the whole material system ˆ
μ is given by the sum of dipole operators
of constituent molecules ˆ
μ(j ), i.e. ˆ
μ =
j
ˆ
μ(j ). The second-order polarization in
Eq. (3.29) has been represented with the dipole operator ˆ
μ and the second-order
density matrix ρ (2) by
P
(2),0
p
(t) = Tr
ˆ
μ p ρ
(2) (t)
= Tr
⎡
⎣
⎛
⎝
N
j
ˆ
μ p (j )
⎞
⎠ ρ
(2) (t)
⎤
⎦ =
N
j
μ
(2),0
p
(j, t).
(5.20)
We put the superscript 0 in the left-hand side to emphasize that Eq. (5.20) is the
simple sum of the bare nonlinear polarizations of the molecules. This relation
is converted to the frequency domain by taking the Fourier transformation and
extracting the exp(−iit) component,
P
(2),0
p
(() =
N
j
μ
(2),0
p
(j, ,).
(5.21)
4 We note the distinction between E ext (ω) and the incident field E I (ω) in Chap. 2, as detailed in
Sect. 5.3.