214
8 Other Topics
and
χ
(2),nonres
chiral
=
1
6
x∼z
p,q,r
ε pqr χ
(2),nonres
pqr
= −
1
6 ¯
h
x∼z
p,q,r
ε pqr
g,m
ρ
(0)
g − ρ
(0)
m
g|α pr (()|m
m|μ q |g
ω 1 − ω mg + ii mg
=
1
6 ¯
h
g,m
ρ
(0)
g − ρ
(0)
m
1
ω 1 − ω mg + ii mg
g|
α yz (() − α zy (()
|mm|μ x |g
++g| {α zx (()−α xz (()} |mm|μ y |g++g|
α xy (()−α yx (()
|mm|μ z |g
.
(8.25)
The above formulas of Eqs. (8.24) and (8.25) include the antisymmetric components
of the Raman tensor
α pq − α qp
. Therefore, the isotropic average χ
(2)
chiral in
Eq. (8.21) should vanish if the Raman tensor is symmetric. The antisymmetric
components of the Raman tensor is generally minor in electronically off-resonant
conditions, as discussed in Sect. A.3. This is the main reason why the chiral SFG has
not been fully exploited for a long time, though it is allowed for symmetry reason.
We also notice in passing that Eqs. (8.24) and (8.25) lead to χ
(2),res
chiral = −χ
(2),nonres
chiral
for ω 1 = ω 2 . This indicates that SHG yield no signal from isotropic bulk materials. 4
However, the intensity of chiral SFG signals is remarkably augmented when
becomes close to the electronic resonant condition [4, 6]. This is because the
antisymmetric components of the Raman tensor become signficantly strong in
electronically resonant or near-resonant conditions. Figure 8.5 displays an example
of chiral SFG spectra from R-1,1’-bi-2-naphtol with varying sum-frequency photon
energy ¯
hh [4]. These spectra demonstrate that the intensity of chiral SFG signal is
augmented by 10 5 in the near resonance condition with an electronic excited state.
Such remarkable enhancement makes the chiral SFG a practically useful tool to
detect chiral vibrational spectra with monolayer sensitivity.
8.2.3 Future Development
The vibrational SFG spectroscopy of chiral probe has many applications, particularly to biological molecules. It could attain monolayer sensitivity to detect chiral
species. Yan et al. argued that the chiral SFG could have interface selectivity in
the electronically off-resonant conditions that the chiral signal from isotropic bulk
vanishes [26].
4 Note that the SHG from chiral interface is allowed [17, 24].
8 Other Topics
and
χ
(2),nonres
chiral
=
1
6
x∼z
p,q,r
ε pqr χ
(2),nonres
pqr
= −
1
6 ¯
h
x∼z
p,q,r
ε pqr
g,m
ρ
(0)
g − ρ
(0)
m
g|α pr (()|m
m|μ q |g
ω 1 − ω mg + ii mg
=
1
6 ¯
h
g,m
ρ
(0)
g − ρ
(0)
m
1
ω 1 − ω mg + ii mg
g|
α yz (() − α zy (()
|mm|μ x |g
++g| {α zx (()−α xz (()} |mm|μ y |g++g|
α xy (()−α yx (()
|mm|μ z |g
.
(8.25)
The above formulas of Eqs. (8.24) and (8.25) include the antisymmetric components
of the Raman tensor
α pq − α qp
. Therefore, the isotropic average χ
(2)
chiral in
Eq. (8.21) should vanish if the Raman tensor is symmetric. The antisymmetric
components of the Raman tensor is generally minor in electronically off-resonant
conditions, as discussed in Sect. A.3. This is the main reason why the chiral SFG has
not been fully exploited for a long time, though it is allowed for symmetry reason.
We also notice in passing that Eqs. (8.24) and (8.25) lead to χ
(2),res
chiral = −χ
(2),nonres
chiral
for ω 1 = ω 2 . This indicates that SHG yield no signal from isotropic bulk materials. 4
However, the intensity of chiral SFG signals is remarkably augmented when
becomes close to the electronic resonant condition [4, 6]. This is because the
antisymmetric components of the Raman tensor become signficantly strong in
electronically resonant or near-resonant conditions. Figure 8.5 displays an example
of chiral SFG spectra from R-1,1’-bi-2-naphtol with varying sum-frequency photon
energy ¯
hh [4]. These spectra demonstrate that the intensity of chiral SFG signal is
augmented by 10 5 in the near resonance condition with an electronic excited state.
Such remarkable enhancement makes the chiral SFG a practically useful tool to
detect chiral vibrational spectra with monolayer sensitivity.
8.2.3 Future Development
The vibrational SFG spectroscopy of chiral probe has many applications, particularly to biological molecules. It could attain monolayer sensitivity to detect chiral
species. Yan et al. argued that the chiral SFG could have interface selectivity in
the electronically off-resonant conditions that the chiral signal from isotropic bulk
vanishes [26].
4 Note that the SHG from chiral interface is allowed [17, 24].