100
4 Two Computational Schemes of χ (2)
are included in the time correlation function in Eq. (4.39). Wei and Shen argued
that the SPS spectra are particularly sensitive to the motional effect of molecular
orientation [26]. This is because the χ
(2)
yzy tensor component associated to the
SPS spectra includes α yz and μ y components, and the off-diagonal element of
polarizability α yz is readily reduced by the orientational average.
The realistic situations fall between the two extreme cases of Eqs. (4.34)
and (4.39). We note that the motional effect of molecular orientation is naturally
incorporated in the MD calculation of χ (2) by the time correlation function formula.
4.5 Solutions to Problems
4.5.1 Polarization Ratios
[Problem 4.1] Derive Eqs. (4.14) and (4.15) on the basis of the above assumptions.
The ratios B and C are presented using Eq. (3.41) by
B =
χ
(2)
yyz
χ
(2)
yzy
=
N · α
(2)
yyz
N · α
(2)
yzy
=
α
(2)
yyz
α
(2)
yzy
,
C =
χ
(2)
zzz
χ
(2)
yyz
=
N · α
(2)
zzz
N · α
(2)
yyz
=
α
(2)
zzz
α
(2)
yyz
.
Therefore, we formulate α
(2)
yyz , α
(2)
yzy , and α
(2)
zzz using Eqs. (3.43), (3.47) and (4.13) as
follows.
α
(2)
yyz =
p
q
r
D yp D yq D zr
∂α p q
∂q 1
∂μ r
∂q 1
= D yξ D yξ D zζ Rαμ + D yη D yη D zζ Rαμ + D yζ D yζ D zζ αμ
= (cos ψ sin φ + cos θ cos φ sin ψ) 2 (cos θ) Rαμ
+ (− sin ψ sin φ + cos θ cos φ cos ψ) 2 (cos θ) Rαμ
+ (− sin θ cos φ) 2 (cos θ) αμ
= (cos θ sin
2 φ + cos 3 θ cos 2 φ) Rαμ + cos θ sin
2 θ cos 2 ψ αμ
= (cos θ + cos 3 θ)
Rαμ
2
+ (cos θ − cos 3 θ)
αμ
2
,
(4.40)
α
(2)
yzy =
p
q
r
D yp D zq D yr
∂α p q
∂q 1
∂μ r
∂q 1