92
4 Two Computational Schemes of χ (2)
This relation elucidates Eq. (4.18) by setting A = α pq . We note again that the above
discussion about the time evolution is based on the Hamiltonian H 0 , not including
the perturbation. The diagonal elements (g = m) in Eq. (4.18) do not contribute to
Eq. (4.17), due to the vanishing factor (ρ
(0)
g − ρ
(0)
m ) = 0 for g = m.
In summary, Eq. (4.17) gives an equivalent expression of χ (2),res to Eq. (3.36).
It indicates that the vibrational resonant term of the nonlinear susceptibility χ (2),res
can be represented by the Fourier-Laplace transformation of the time correlation
function between the Raman tensor α pq and the dipole moment μ r for the interface
system. It is a rigorous expression of χ (2),res on the basis of quantum mechanics.
4.3.2 Classical Analogue
The time correlation formula of Eq. (4.17) allows for an alternative computational
scheme of χ (2),res . The time correlation functions are utilized to evaluate various
properties of statistical mechanics [13], and amenable to be computed by molecular
dynamics (MD) simulation [1, 6]. In order to use this formula with MD simulation,
however, we should obtain a classical version of Eq. (4.17) since usual MD
simulations are carried out on the basis of classical mechanics. Deriving the classical
expression is the theme of this subsection.
χ (2),res in Eq. (4.17) is expressed by the Fourier-Laplace transformation of the
time correlation function F(t),
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
∞
0
dt exp (iω 2 t) F(t),
(4.21)
where
F(t) =
i
¯
h
α pq (t)μ r − μ r α pq (t)
=
i
¯
h
δα pq (t)δμ r − δμ r δα pq (t)
(4.22)
In this expression δα pq (t) = α pq (t) −
α pq
and δμ r = μ r − μ r denote
the displacements from the average values. Equation (4.22) is apparently a
quantum mechanical expression, since it includes ¯
h and a commutation relation
[α pq (t), μ r ] = α pq (t)μ r − μ r α pq (t) = 0. This form is not amenable to the
classical limit, since both ¯
h and the commutation relation would go to zero and thus
F(t) → 0/0. To obtain a classical analogue, we change Eq. (4.21) to an equivalent
form with the canonical time correlation function [13].
A related function G(t) is introduced using the canonical time correlation
function,
G(t) =
α pq (t) − α
◦
pq ; μ r − μ
◦
r
(4.23)
=
1
β
β
0
dλ
exp (λH)
α pq (t) − α ◦
pq
exp (−λH)
μ r − μ ◦
r
,
4 Two Computational Schemes of χ (2)
This relation elucidates Eq. (4.18) by setting A = α pq . We note again that the above
discussion about the time evolution is based on the Hamiltonian H 0 , not including
the perturbation. The diagonal elements (g = m) in Eq. (4.18) do not contribute to
Eq. (4.17), due to the vanishing factor (ρ
(0)
g − ρ
(0)
m ) = 0 for g = m.
In summary, Eq. (4.17) gives an equivalent expression of χ (2),res to Eq. (3.36).
It indicates that the vibrational resonant term of the nonlinear susceptibility χ (2),res
can be represented by the Fourier-Laplace transformation of the time correlation
function between the Raman tensor α pq and the dipole moment μ r for the interface
system. It is a rigorous expression of χ (2),res on the basis of quantum mechanics.
4.3.2 Classical Analogue
The time correlation formula of Eq. (4.17) allows for an alternative computational
scheme of χ (2),res . The time correlation functions are utilized to evaluate various
properties of statistical mechanics [13], and amenable to be computed by molecular
dynamics (MD) simulation [1, 6]. In order to use this formula with MD simulation,
however, we should obtain a classical version of Eq. (4.17) since usual MD
simulations are carried out on the basis of classical mechanics. Deriving the classical
expression is the theme of this subsection.
χ (2),res in Eq. (4.17) is expressed by the Fourier-Laplace transformation of the
time correlation function F(t),
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
∞
0
dt exp (iω 2 t) F(t),
(4.21)
where
F(t) =
i
¯
h
α pq (t)μ r − μ r α pq (t)
=
i
¯
h
δα pq (t)δμ r − δμ r δα pq (t)
(4.22)
In this expression δα pq (t) = α pq (t) −
α pq
and δμ r = μ r − μ r denote
the displacements from the average values. Equation (4.22) is apparently a
quantum mechanical expression, since it includes ¯
h and a commutation relation
[α pq (t), μ r ] = α pq (t)μ r − μ r α pq (t) = 0. This form is not amenable to the
classical limit, since both ¯
h and the commutation relation would go to zero and thus
F(t) → 0/0. To obtain a classical analogue, we change Eq. (4.21) to an equivalent
form with the canonical time correlation function [13].
A related function G(t) is introduced using the canonical time correlation
function,
G(t) =
α pq (t) − α
◦
pq ; μ r − μ
◦
r
(4.23)
=
1
β
β
0
dλ
exp (λH)
α pq (t) − α ◦
pq
exp (−λH)
μ r − μ ◦
r
,