Abstract
Now we provide the computational schemes of SFG spectroscopy on the basis of the microscopic theory described in the preceding Chap. 3. The theme of this chapter is to define two methods of calculating χ (2) spectra, via energy representation and time-dependent representation. These methods can be utilized to calculate the SFG spectra by molecular dynamics (MD) simulation. These methods connect the formal theory of χ (2) to actual spectra of interfaces that consist of molecules, and thus open new routes of SFG analysis with the aid of MD simulation.
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Notes
- 1.
The molecular hyperpolarizability \(\boldsymbol {\alpha }^{(2),\text{res}}_{l}\) in Eq. (4.4) includes the suffix l to allow for distinguishing different species.
- 2.
If the polarizability derivative tensor is approximated with a spherical tensor (R ≈ 1) by \(\partial \alpha _{p^\prime q^\prime }/\partial q_1 \approx c \: \delta _{p^\prime q^\prime }\), the yy element (∂α yy∕∂q 1) is shown to be independent of the transformation,
$$\displaystyle \begin{aligned} \left( \frac{\partial \alpha_{yy}}{\partial q_1} \right)= \sum_{p^\prime}^{\xi \sim \zeta} \sum_{q^\prime}^{\xi \sim \zeta} \mathcal{D}_{y p^\prime} \mathcal{D}_{y q^\prime} \left( \frac{\partial \alpha_{p^\prime q^\prime}}{\partial q_1} \right) \approx \sum_{p^\prime}^{\xi \sim \zeta} \sum_{q^\prime}^{\xi \sim \zeta} \mathcal{D}_{y p^\prime} \mathcal{D}_{y q^\prime} \; c \: \delta_{p^\prime q^\prime} = c \; (> 0). \end{aligned}$$Since the anisotropy is not large (R ≃ 0.8 in Table 4.2), the qualitatively similar trend should hold for the acetonitrile molecule.
- 3.
Therefore, the matrix elements for an arbitrary operator A ∘ (\(=\alpha _{pq}^{\circ }\) or \(\mu _r^{\circ }\)) are represented using energy eigenstates m, n of the entire system as
$$\displaystyle \begin{aligned} \langle m | A^{\circ} | n \rangle = \begin{cases} \langle m | A | n \rangle & (m=n) \\ 0 & (m \ne n) \end{cases} . \end{aligned}$$This is equivalent to the long-time average of 〈m|A|n〉,
$$\displaystyle \begin{aligned} \langle m | A^{\circ} | n \rangle = \lim_{T \to \infty} \frac{1}{T} \int_0^T dt \; \langle m | A(t) | n \rangle = \lim_{T \to \infty} \frac{1}{T} \int_0^T dt \; \exp \left( \frac{i (\mathcal{E}_m - \mathcal{E}_n) t}{\hbar} \right) \langle m | A | n \rangle. \end{aligned}$$Note that A ∘ is commutative with \(\mathcal {H}\), and thus \(A^{\circ } (t) = \exp (i \mathcal {H} t/\hbar ) A^{\circ } \exp (-i\mathcal {H} t/\hbar ) = A^{\circ }\).
- 4.
In the classical mechanics, the distinction between quantum average 〈A〉 and statistical average \(\overline { A }\) in Sect. 3.1 disappears. Accordingly the classical time correlation function 〈A(t)B〉cl is equivalent to \(\overline { A(t) B }\).
- 5.
Here we denote time correlation function by angle bracket 〈A〉 and orientational average by over-bar \(\overline { A }\). In the classical mechanics, both notations 〈A〉 and \(\overline {A}\) indicate the statistical average and thus they are essentially equivalent in this context.
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Morita, A. (2018). Two Computational Schemes of χ (2) . In: Theory of Sum Frequency Generation Spectroscopy. Lecture Notes in Chemistry, vol 97. Springer, Singapore. https://doi.org/10.1007/978-981-13-1607-4_4
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