84
4 Two Computational Schemes of χ (2)
2. Equation (4.4) constructs χ (2) with a simple sum of molecular hyperpolarizabilities. This feature has a merit for decomposition analysis as mentioned in
Advantage 2 above, though it is hard to incorporate the effects of intermolecular
couplings. For example, the vibrational modes Q a are largely perturbed via
intra- and inter-molecular couplings, and may depend on the environment.
Vibrational correlation among neighboring molecules can have a substantial
effect on χ (2) in strongly interacting system [11, 12]. The local field effect arises
from intermolecular dielectric couplings.
3. The above χ (2) model is not capable of determining the damping term a in
Eq. (4.3). It arises from dephasing of the vibrations, and has to be treated as
empirical parameters. The motional effect of molecular orientation on χ (2) is
omitted [26], as discussed in Sect. 4.4.
In summary, the χ (2) model of Eqs. (4.3) and (4.4) is useful to interpret experimental
spectra in a qualitatively plain manner, though it is rather difficult to make this
modeling method a predictive tool of experimental spectra. The χ (2) model of
Eqs. (4.3) and (4.4) may not be convenient to accurately reflect the molecular
behaviors in condensed phase, including the perturbation of molecular properties,
vibrational couplings and dephasing. For example, the perturbation on the molecule
hyperpolarizability α (2),res in Eq. (4.3), generally called “solvent effect” in quantum
chemistry, is critical in the SFG spectrum of water. The first paper by Morita and
Hynes [15] made elaborated modeling of the solvent effect of water, to represent the
substantial perturbation on the transition dipole (∂μ/∂Q a ), transition polarizability
(∂α/∂Q a ), and the frequency ω a of water [18]. However, such elaborated modeling
hinders from applying it to general interfaces beside the pure water. It is therefore
desirable to develop an alternative method which allows for more rigorous and
general modeling of χ (2) and SFG spectroscopy. The need of extending the χ (2)
modeling spurred us to develop another method based on the time-dependent
representation [16, 17]. That method is described in Sect. 4.3 below.
4.2 Examples of χ (2) Tensor and Orientation
The above χ (2) model in Sect. 4.1 is actually quite useful to discuss the qualitative
relation between χ (2) tensor and molecular orientation for specific molecular
species. Here we discuss the relation in some typical examples of O-H and C-H
stretching vibrations, the two most commonly measured vibrational bands by the
SFG spectroscopy. Further details of computational analysis will be discussed in
the application chapters of 9 and 10.
4 Two Computational Schemes of χ (2)
2. Equation (4.4) constructs χ (2) with a simple sum of molecular hyperpolarizabilities. This feature has a merit for decomposition analysis as mentioned in
Advantage 2 above, though it is hard to incorporate the effects of intermolecular
couplings. For example, the vibrational modes Q a are largely perturbed via
intra- and inter-molecular couplings, and may depend on the environment.
Vibrational correlation among neighboring molecules can have a substantial
effect on χ (2) in strongly interacting system [11, 12]. The local field effect arises
from intermolecular dielectric couplings.
3. The above χ (2) model is not capable of determining the damping term a in
Eq. (4.3). It arises from dephasing of the vibrations, and has to be treated as
empirical parameters. The motional effect of molecular orientation on χ (2) is
omitted [26], as discussed in Sect. 4.4.
In summary, the χ (2) model of Eqs. (4.3) and (4.4) is useful to interpret experimental
spectra in a qualitatively plain manner, though it is rather difficult to make this
modeling method a predictive tool of experimental spectra. The χ (2) model of
Eqs. (4.3) and (4.4) may not be convenient to accurately reflect the molecular
behaviors in condensed phase, including the perturbation of molecular properties,
vibrational couplings and dephasing. For example, the perturbation on the molecule
hyperpolarizability α (2),res in Eq. (4.3), generally called “solvent effect” in quantum
chemistry, is critical in the SFG spectrum of water. The first paper by Morita and
Hynes [15] made elaborated modeling of the solvent effect of water, to represent the
substantial perturbation on the transition dipole (∂μ/∂Q a ), transition polarizability
(∂α/∂Q a ), and the frequency ω a of water [18]. However, such elaborated modeling
hinders from applying it to general interfaces beside the pure water. It is therefore
desirable to develop an alternative method which allows for more rigorous and
general modeling of χ (2) and SFG spectroscopy. The need of extending the χ (2)
modeling spurred us to develop another method based on the time-dependent
representation [16, 17]. That method is described in Sect. 4.3 below.
4.2 Examples of χ (2) Tensor and Orientation
The above χ (2) model in Sect. 4.1 is actually quite useful to discuss the qualitative
relation between χ (2) tensor and molecular orientation for specific molecular
species. Here we discuss the relation in some typical examples of O-H and C-H
stretching vibrations, the two most commonly measured vibrational bands by the
SFG spectroscopy. Further details of computational analysis will be discussed in
the application chapters of 9 and 10.