7.4 Invariance to Molecular Origin
185
Using Eq. (7.120), the χ IQ ((, ω 1 , ω 2 ) term in Eq. (7.29) is transformed to
χ
IQ
pqr
((, ω 1 , ω 2 ) = χ
IQ
pqr ((, ω 1 , ω 2 )
−
∞
z b
dz
⎛
⎝
molecule
l
z l α D0
l,pqr ((, ω 1 , ω 2 )δ(z − z l )
⎞
⎠ ∂
∂z
f p (z, ,)f q (z, ω 1 )f r (z, ω 2 )
+ · · · ,
(7.121)
where we neglected the terms of higher order derivatives. We also get the transformed expression of χ
IQB
pqr ((, ω 1 , ω 2 ) from Eqs. (7.30) and (7.119) as follows,
χ
IQB
pqr
((, ω 1 , ω 2 ) = χ
IQB
pqr ((, ω 1 , ω 2 )
−
⎛
⎝
molecule
l
z l α D0
l,pqr ((, ω 1 , ω 2 )δ(z b − z l )
⎞
⎠ f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 )
+ · · · .
(7.122)
Equations (7.114), (7.121) and (7.122) indicate that the sum of these terms is
invariant against the shift of the molecular origin,
χ
ID
pqr ((, ω 1 , ω 2 ) + χ
IQ
pqr ((, ω 1 , ω 2 ) + χ
IQB
pqr ((, ω 1 , ω 2 )
= χ
ID
pqr
((, ω 1 , ω 2 ) + χ
IQ
pqr
((, ω 1 , ω 2 ) + χ
IQB
pqr
((, ω 1 , ω 2 ).
(7.123)
On the other hand, the bulk contribution χ B0 is invariant in itself by the change
of the molecular origin. This is evident because the factors ξ
Q2,β
1
, ξ
Q1,β
2
, ξ
Q1,β
3
and
ξ
Q2,β
3
involved in Eq. (7.41) are invariant. These four factors are given in Eqs. (7.42),
(7.43), (7.44), (7.45) as the difference of two susceptibility terms, e.g. ξ
Q2,β
1
=
χ
D2,β
1
−χ
Q,β
1
in Eq. (7.42), and thus the additional terms originating from the shift of
molecular origin in Eq. (7.120) cancel each other. The alternative expression of the
bulk contribution, χ B , given in Sect. 7.2.4 by Eqs. (7.54) and (7.55), (7.56), (7.57),
(7.58), (7.59), (7.60) is also invariant for the same reason.
In conclusion, the effective nonlinear susceptibility χ
(2)
eff,G in Eq. (7.47) is well
defined irrespective of the choice of the molecular origin. This invariance is requisite
in order that χ
(2)
eff,G is related to experimental observables. Moreover, the sum of
the interface contributions, χ ID +χ IQ +χ IQB , and the bulk contribution, χ B0 or
χ B , are respectively invariant. Consequently, the interface and bulk contributions
in the SFG/SHG signals are respectively well defined, though each of three terms
consisting of the interface contribution, χ ID , χ IQ , χ IQB , is affected by the definition
of the molecular origin.
185
Using Eq. (7.120), the χ IQ ((, ω 1 , ω 2 ) term in Eq. (7.29) is transformed to
χ
IQ
pqr
((, ω 1 , ω 2 ) = χ
IQ
pqr ((, ω 1 , ω 2 )
−
∞
z b
dz
⎛
⎝
molecule
l
z l α D0
l,pqr ((, ω 1 , ω 2 )δ(z − z l )
⎞
⎠ ∂
∂z
f p (z, ,)f q (z, ω 1 )f r (z, ω 2 )
+ · · · ,
(7.121)
where we neglected the terms of higher order derivatives. We also get the transformed expression of χ
IQB
pqr ((, ω 1 , ω 2 ) from Eqs. (7.30) and (7.119) as follows,
χ
IQB
pqr
((, ω 1 , ω 2 ) = χ
IQB
pqr ((, ω 1 , ω 2 )
−
⎛
⎝
molecule
l
z l α D0
l,pqr ((, ω 1 , ω 2 )δ(z b − z l )
⎞
⎠ f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 )
+ · · · .
(7.122)
Equations (7.114), (7.121) and (7.122) indicate that the sum of these terms is
invariant against the shift of the molecular origin,
χ
ID
pqr ((, ω 1 , ω 2 ) + χ
IQ
pqr ((, ω 1 , ω 2 ) + χ
IQB
pqr ((, ω 1 , ω 2 )
= χ
ID
pqr
((, ω 1 , ω 2 ) + χ
IQ
pqr
((, ω 1 , ω 2 ) + χ
IQB
pqr
((, ω 1 , ω 2 ).
(7.123)
On the other hand, the bulk contribution χ B0 is invariant in itself by the change
of the molecular origin. This is evident because the factors ξ
Q2,β
1
, ξ
Q1,β
2
, ξ
Q1,β
3
and
ξ
Q2,β
3
involved in Eq. (7.41) are invariant. These four factors are given in Eqs. (7.42),
(7.43), (7.44), (7.45) as the difference of two susceptibility terms, e.g. ξ
Q2,β
1
=
χ
D2,β
1
−χ
Q,β
1
in Eq. (7.42), and thus the additional terms originating from the shift of
molecular origin in Eq. (7.120) cancel each other. The alternative expression of the
bulk contribution, χ B , given in Sect. 7.2.4 by Eqs. (7.54) and (7.55), (7.56), (7.57),
(7.58), (7.59), (7.60) is also invariant for the same reason.
In conclusion, the effective nonlinear susceptibility χ
(2)
eff,G in Eq. (7.47) is well
defined irrespective of the choice of the molecular origin. This invariance is requisite
in order that χ
(2)
eff,G is related to experimental observables. Moreover, the sum of
the interface contributions, χ ID +χ IQ +χ IQB , and the bulk contribution, χ B0 or
χ B , are respectively invariant. Consequently, the interface and bulk contributions
in the SFG/SHG signals are respectively well defined, though each of three terms
consisting of the interface contribution, χ ID , χ IQ , χ IQB , is affected by the definition
of the molecular origin.