60
3 Microscopic Expressions of Nonlinear Polarization
+
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ r |m
m|μ q |n
n|μ p |g
+ ω ng + ii ng
ω 2 + ω mg + ii mg
−
g|μ r |m
m|μ p |n
n|μ q |g
( − ω nm + ii nm )
ω 2 + ω mg + ii mg
=
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |n
n|μ q |m
− ω ng + ii ng
−
g|μ q |n
n|μ p |m
+ ω nm + ii nm
m|μ r |g
ω 2 − ω mg + ii mg
+
1
¯
h 2
g,n,m
ρ
(0)
m
g|μ q |n
n|μ p |m
+ ω nm + ii nm
−
g|μ p |n
n|μ q |m
− ω ng + ii ng
m|μ r |g
ω 2 − ω mg + ii mg
= −
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)
where the last expression includes the matrix elements of the Raman tensor
g|α pq (()|m
defined by
g|α pq (()|m
= −
1
¯
h
n
g|μ p |n
n|μ q |m
− ω ng + ii ng
−
g|μ q |n
n|μ p |m
+ ω nm + ii nm
.
(3.37)
The matrix element
g|α pq (()|m
is associated to the transition probability of
Raman scattering from the state g to m [2, 8, 10, 11]. The diagonal element at
g = m,
g|α pq (()|g
, is equal to χ
(1)
pq,g (() in Eq. (3.26), the first-order susceptibility
at the state g. We also note that the nonresonant term χ (2),nonres in Eq. (3.35) can be
represented in the same manner to be
χ
(2),nonres
pqr
= −
1
¯
h
g,m
ρ
(0)
g − ρ
(0)
m
g|α pr (()|m
m|μ q |g
ω 1 − ω mg + ii mg
.
(3.38)
The χ (2) expressions in Eqs. (3.36) and (3.38) include product of the transition
dipole moment m|μ|g and the Raman tensor g|α(()|m in the numerator.
Therefore, the SFG process includes both infrared active (m|μ|g = 0) and
Raman active (g|α|m = 0) transitions. This conclusion is in accord with the
basic selection rule of SFG in Chap. 1 that a system having inversion symmetry
has no active vibrational mode to SFG due to the IR-Raman mutual exclusion
theorem.
The principal difference between Eqs. (3.36) and (3.38) is seen in ω 2 (infrared)
or ω 1 (visible) included in the denomenator. In the vibrational SFG spectroscopy,
the vibrational transition could be in resonance with the infrared frequency ω 2 . The
vibrationally resonance term χ (2),res could be interpreted formally as successive
transitions by the infrared absorption and anti-Stokes Raman scattering, as schemat-
3 Microscopic Expressions of Nonlinear Polarization
+
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ r |m
m|μ q |n
n|μ p |g
+ ω ng + ii ng
ω 2 + ω mg + ii mg
−
g|μ r |m
m|μ p |n
n|μ q |g
( − ω nm + ii nm )
ω 2 + ω mg + ii mg
=
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |n
n|μ q |m
− ω ng + ii ng
−
g|μ q |n
n|μ p |m
+ ω nm + ii nm
m|μ r |g
ω 2 − ω mg + ii mg
+
1
¯
h 2
g,n,m
ρ
(0)
m
g|μ q |n
n|μ p |m
+ ω nm + ii nm
−
g|μ p |n
n|μ q |m
− ω ng + ii ng
m|μ r |g
ω 2 − ω mg + ii mg
= −
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)
where the last expression includes the matrix elements of the Raman tensor
g|α pq (()|m
defined by
g|α pq (()|m
= −
1
¯
h
n
g|μ p |n
n|μ q |m
− ω ng + ii ng
−
g|μ q |n
n|μ p |m
+ ω nm + ii nm
.
(3.37)
The matrix element
g|α pq (()|m
is associated to the transition probability of
Raman scattering from the state g to m [2, 8, 10, 11]. The diagonal element at
g = m,
g|α pq (()|g
, is equal to χ
(1)
pq,g (() in Eq. (3.26), the first-order susceptibility
at the state g. We also note that the nonresonant term χ (2),nonres in Eq. (3.35) can be
represented in the same manner to be
χ
(2),nonres
pqr
= −
1
¯
h
g,m
ρ
(0)
g − ρ
(0)
m
g|α pr (()|m
m|μ q |g
ω 1 − ω mg + ii mg
.
(3.38)
The χ (2) expressions in Eqs. (3.36) and (3.38) include product of the transition
dipole moment m|μ|g and the Raman tensor g|α(()|m in the numerator.
Therefore, the SFG process includes both infrared active (m|μ|g = 0) and
Raman active (g|α|m = 0) transitions. This conclusion is in accord with the
basic selection rule of SFG in Chap. 1 that a system having inversion symmetry
has no active vibrational mode to SFG due to the IR-Raman mutual exclusion
theorem.
The principal difference between Eqs. (3.36) and (3.38) is seen in ω 2 (infrared)
or ω 1 (visible) included in the denomenator. In the vibrational SFG spectroscopy,
the vibrational transition could be in resonance with the infrared frequency ω 2 . The
vibrationally resonance term χ (2),res could be interpreted formally as successive
transitions by the infrared absorption and anti-Stokes Raman scattering, as schemat-