4.1 Energy Representation
83
χ (2),res . The instantaneous value of χ (2),res at each time step is statistically averaged
over ensemble of MD snapshots. The averaged value of χ (2),res is thereby obtained,
χ
(2),res
pqr
=
molecule
l
ξ ∼ζ
p
ξ ∼ζ
q
ξ ∼ζ
r
D l,pp D l,qq D l,rr α
(2),res
l,p q r .
(4.4)
Equations (4.3) and (4.4) provide a route to evaluate the resonant term χ (2),res
through quantum chemical and molecular dynamics calculations. 1 On the other
hand, the nonresonant term χ (2),nonres is regarded as a constant over the ω 2
frequency range of SFG measurements. In practical analysis of SFG spectra, it
could be treated as an empirical parameter to fit the experimental spectra. Then
χ (2) = χ (2),res +χ (2),nonres is obtained by Eq. (3.33). The above procedure illustrates
the essential idea to calculate χ (2) on the basis of the energy representation.
Advantages and disadvantages The above analysis method of SFG spectra based
on the energy representation of χ (2) modeling has been proposed by Morita
and Hynes [15], and first applied to the water-vapor interface. This method has
been subsequently applied and developed by other researchers [4, 21, 22, 27].
The advantages of the energy representation of χ (2) modeling are summarized as
follows.
1. The model of molecular hyperpolarizability is based on quantum chemical
calculations of constituent molecules of the interface. Chemical specificity of
the molecules is readily reflected in the calculation of SFG spectra.
2. The calculated χ (2) in Eq. (4.4) is straightforwardly decomposed into and
assigned to molecular species, since Eq. (4.4) represents χ (2) as the summation
of molecular contributions. The effect of molecular orientation on the χ (2) tensor
is also readily analyzed [8, 9].
3. Computational cost of MD simulation is relatively modest, compared to the timedependent representation of χ (2) modeling in Sect. 4.3. The MD simulation is
used for sampling the molecular orientations at interfaces, and the molecular
orientation is readily investigated using conventional force fields.
The above features indicate that the energy representation of χ (2) modeling provides
a convenient tool to interpret experimental spectra from molecular properties and
orientation with modest cost of computation.
On the other hand, this method involves some difficulties as follows.
1. Accurate molecular modeling is hard to be generalized in condensed environment. This is particularly the case for hydrogen-bonding systems, because
the perturbation on the vibrational frequency and transition matrix elements in
Eq. (4.3) is substantial.
1 The molecular hyperpolarizability α
(2),res
l
in Eq. (4.4) includes the suffix l to allow for distinguishing different species.
83
χ (2),res . The instantaneous value of χ (2),res at each time step is statistically averaged
over ensemble of MD snapshots. The averaged value of χ (2),res is thereby obtained,
χ
(2),res
pqr
=
molecule
l
ξ ∼ζ
p
ξ ∼ζ
q
ξ ∼ζ
r
D l,pp D l,qq D l,rr α
(2),res
l,p q r .
(4.4)
Equations (4.3) and (4.4) provide a route to evaluate the resonant term χ (2),res
through quantum chemical and molecular dynamics calculations. 1 On the other
hand, the nonresonant term χ (2),nonres is regarded as a constant over the ω 2
frequency range of SFG measurements. In practical analysis of SFG spectra, it
could be treated as an empirical parameter to fit the experimental spectra. Then
χ (2) = χ (2),res +χ (2),nonres is obtained by Eq. (3.33). The above procedure illustrates
the essential idea to calculate χ (2) on the basis of the energy representation.
Advantages and disadvantages The above analysis method of SFG spectra based
on the energy representation of χ (2) modeling has been proposed by Morita
and Hynes [15], and first applied to the water-vapor interface. This method has
been subsequently applied and developed by other researchers [4, 21, 22, 27].
The advantages of the energy representation of χ (2) modeling are summarized as
follows.
1. The model of molecular hyperpolarizability is based on quantum chemical
calculations of constituent molecules of the interface. Chemical specificity of
the molecules is readily reflected in the calculation of SFG spectra.
2. The calculated χ (2) in Eq. (4.4) is straightforwardly decomposed into and
assigned to molecular species, since Eq. (4.4) represents χ (2) as the summation
of molecular contributions. The effect of molecular orientation on the χ (2) tensor
is also readily analyzed [8, 9].
3. Computational cost of MD simulation is relatively modest, compared to the timedependent representation of χ (2) modeling in Sect. 4.3. The MD simulation is
used for sampling the molecular orientations at interfaces, and the molecular
orientation is readily investigated using conventional force fields.
The above features indicate that the energy representation of χ (2) modeling provides
a convenient tool to interpret experimental spectra from molecular properties and
orientation with modest cost of computation.
On the other hand, this method involves some difficulties as follows.
1. Accurate molecular modeling is hard to be generalized in condensed environment. This is particularly the case for hydrogen-bonding systems, because
the perturbation on the vibrational frequency and transition matrix elements in
Eq. (4.3) is substantial.
1 The molecular hyperpolarizability α
(2),res
l
in Eq. (4.4) includes the suffix l to allow for distinguishing different species.