90
4 Two Computational Schemes of χ (2)
Equations (4.14) and (4.15) are used to the polarization analysis of SFG
measurements. These equations include only three parameters, R, cos θ and cos 3 θ ,
and thus suitable to simple modeling of the polarization analysis. R is estimated
by various means, such as the depolarization ratio of the Raman scattering [14], ab
initio or density functional theory calculations, and simple geometric argument of
molecules. If one further assumes a relation of cos θ and cos 3 θ , one could derive the
molecular orientation θ from the B and/or C ratios. The simplest assumption would
be cos 3 θ ≈ (cos θ) 3 . Then the tilt angle is estimated from B and C in Eqs. (4.14)
and (4.15), respectively, to be
cos θ ≈
B − (1 + R)/(1 − R)
B − 1
1/2
or cos θ ≈
(1 + R)C − 2R
(1 − R)(C + 2)
1/2
.
(4.16)
We note that the assumption cos 3 θ ≈ (cos θ) 3 is valid when the tilt angle θ is
well determined (or the distribution of θ is sufficiently narrow) at the interface. We
further discuss the polarization analysis with the help of MD simulation in Chap. 10.
4.3 Time-Dependent Representation
4.3.1 Time Correlation Function
This section presents an alternative expression of χ (2) based on the time correlation
function, instead of using the perturbation formula in the preceding sections.
According to the theory of statistical mechanics, physical quantities associated to
the response to external perturbation are generally expressed using time correlation
functions [13, 14].
χ (2),res in Eq. (3.36) is converted to an equivalent formula with time correlation
function as follows,
χ
(2),res
pqr ((, ω 1 , ω 2 ) = −
1
¯
h
g,m
ρ
(0)
g − ρ
(0)
m
g|α pq |m
m|μ r |g
ω 2 − ω mg + ii mg
(3.36)
=
i
¯
h
∞
0
dt
g,m
ρ
(0)
g − ρ
(0)
m
g|α pq |m
m|μ r |g exp
i(ω 2 − ω mg + ii mg )t
=
i
¯
h
∞
0
dt
g,m
ρ
(0)
g − ρ
(0)
m
g|α pq (t)|m
m|μ r |g exp (iω 2 t)
=
i
¯
h
∞
0
dt
g,m
ρ
(0)
g
g|α pq (t)|m
m|μ r |g − g|μ r |m
m|α pq (t)|g
exp (iω 2 t)