A Simplified Method of Microscopic Polarizability Tensor Differential of Hyper-Raman Spectroscopy Based on the Bond Additivity Model

Wang, Yuan; Huang, Liping

doi:https://doi.org/10.1155/2022/2710506

International Journal of Optics

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Research Article | Open Access

Volume 2022 | Article ID 2710506 | https://doi.org/10.1155/2022/2710506

A Simplified Method of Microscopic Polarizability Tensor Differential of Hyper-Raman Spectroscopy Based on the Bond Additivity Model

Yuan Wang¹and Liping Huang²

Academic Editor: Wonho Jhe

Received12 Oct 2021

Revised07 Feb 2022

Accepted15 Feb 2022

Published10 Mar 2022

Abstract

Coherent anti-Stokes Raman spectroscopy (CARS) and Ccherent anti-Stokes hyper-Raman spectroscopy (CAHRS), as other high-order nonlinear spectroscopy techniques, are widely exploited in many research fields, such as dynamic processes, gene expression spectrum screening, high-resolution spectroscopy, and nonlinear high-resolution imaging. However, it is difficult to make a quantitative analysis of the spectral signals that involve a large number of high-order micropolarizability tensors. It is reported that the CARS and CAHRS microscopic hyperpolarizability tensor elements can be decomposed into the product of the differentiation of Raman microscopic polarizability tensor α′_i′j′ and hyper-Raman microscopic polarizability tensor β′_i′j′k′ so that the high-order spectra can be simplified to the analysis of low-order spectra. In this paper, we use the bond additivity model (BAM) combined with experimental corrections to address the carbon dioxide (CO₂) molecule and present the simplified scheme for differentiation of hyper-Raman microscopic polarizability tensor elements β′_i′j′k′. Taking advantage of this approach, combined with the experimental correction, the differentiation of Hyper-Raman microscopic polarizability tensor elements β′_i′j′k′ of the CO₂ is obtained and the expressions of β′_i′j′k′ for antisymmetric vibrations of CO₂ are deduced. Finally, substituting the differentiation of Raman microscopic polarizability tensor elements α′_i′j′ reported in the literature into the ratio above can obtain the proportional relationship between the microscopic polarizability tensor elements of CARS and CAHRS of the CO₂. This method can provide the basis for the quantitative analysis of high-order nonlinear spectral profiles.

1. Introduction

In the past ten years, as a high-order nonlinear spectroscopy technique, coherent anti-Stokes Raman spectroscopy (CARS) and its surface enhancement methods [1–6] and coherent anti-Stokes hyper-Raman spectroscopy (CAHRS) [7–9] have gained important applications in kinetic processes, gene expression profile screening, high-resolution spectroscopy, nonlinear high-resolution imaging technology, and so on. However, the abovementioned research only carried out qualitative analysis of spectral signals and lacked quantitative analysis of spectral signals. This limits its application to some extent. As a high-order nonlinear coherent optical process, the spectral signals of CARS and CAHRS are contributed by the third and fourth-order microscopic polarizability tensor elements of molecular groups [10]. For n-order nonlinear coherence spectroscopy, the number of molecular microscopic polarizability tensor elements is 3ⁿ⁺¹. For example, CARS involves 81 molecular microscopic polarizability tensor elements. These large numbers of molecular microscopic polarizability tensor elements will lead to greater difficulties in the quantitative analysis of high-order nonlinear coherent optical processes [10]. In addition, in view of the nonlinearity of CARS and CAHRS, the complexity of such approaches results increased. In order to facilitate the analysis of the results, the most commonly used experimental configuration is the vertical incidence type [2]. However, with the in-depth study of the theory of high-order nonlinear optical processes and the development of high-order nonlinear optical technologies, experimental techniques and experimental methods will gradually be improved. In the case of noncollinear experimental configurations with non-normal incidence, the acquisition of the high-order microscopic polarizability tensor element is particularly important for the quantitative analysis of spectral signals.

Based on the second-order nonlinear optical process of the sum frequency generation vibrational spectroscopy (SFG-VS) quantitative analysis method [11–13], we propose a method to simplify the high-order microscopic polarizability tensor elements of CARS and CAHRS [10, 14]. By analyzing the symmetry of molecular groups, the number of independent nonzero high-order microscopic polarizability tensor elements are reduced. Thus, CARS microscopic polarizability tensor element β_{i′j′k′l′} is expressed as the product of two Raman microscopic polarizability tensor elements differential α′_i′j′ and α′_k′l′. The microscopic polarizability tensor element of CAHRS β_{i′j′k′l′m′} is expressed as the product of the hyper-Raman microscopic polarizability tensor element differential β′_i′j′k′ and the Raman microscopic polarizability tensor element differential α′_l′m′. The ratio between β_{i′j′k′l′} and β_{i′j′k′l′m′} could be simplified by using the ratio between α′_i′j′ and β′_i′j′k′. Therefore, the problem of solving the ratio of high-order microscopic polarizability tensor elements is simplified to the problem of solving the ratio of two low-order microscopic polarizability tensor elements differential. Finally, the problem comes down to how to find the ratio of Raman and hyper-Raman microscopic polarizability tensor elements differential. In the literature [11, 14], the 2-order, 3-order, and 4-order microscopic polarizability tensor elements expressed by the hyper-Raman microscopic polarizability tensor element differential β′_i′j′k′, the Raman microscopic polarizability tensor element differential α′_i′j′, and the dipole moment tensor element differentials μ′_i′ are as follows:

When we get the ratio between μ′_i′, α′_i′j′, and β′_i′j′k′, we can obtain the proportional relationship between the microscopic polarizability tensor elements of CARS and CAHRS of CO₂ through the expressions (3)–(5).

Bond additivity model (BAM) is usually used to solve the ratio of Raman microscopic polarizability tensor elements differential in literature [11, 15, 16]. Notably, when Wang group [17] discussed the SFG-VS signal at the air/methanol interface, it was found that the ratio between the differential α′_i′j′ of the Raman microscopic polarizability tensor derived from the traditional BAM was brought into the SFG-VS quantitative analysis method, and the results were inconsistent with the experimental measurement. Therefore, the results of BAM are corrected by the method of experimental correction, so that the results are in good agreement with the measured interface sum-frequency vibration spectra. In this paper, by using the BAM and its experimental modification, a simplified scheme is proposed to calculate the hyper-Raman microscopic polarizability tensor differential β′i′j′k′ of the carbon dioxide (CO₂) molecule. This approach is quite general, and it can be extended to other nonlinear coherent Raman spectroscopic techniques to simplify the relationship between the microscopic polarizability tensor elements and quantitative analysis of the experimental signal, such as time-domain Raman spectroscopy [18, 19], stimulated Raman scattering [20, 21], and nonlinear high-resolution microscopy [5, 6]. In order to carry out this work, the anharmonic mode couplings and higher-order corrections to the molecular polarizabilities have been neglected in this paper.

2. BAM Solves β′_i′j′k′ of Single Bond

The BAM method is based on the assumption of the local mode of a molecular group and the assumption of the local mode of a single bond. In this model, the hyper-Raman microscopic polarizability corresponding to the stretching vibration of a single bond in the group is coupled by symmetry analysis, and the expression of the hyper-Raman microscopic polarizability tensor differential corresponding to each normal vibration mode of the molecular group is obtained. In the BAM method, the discussion of a single bond is the basis of the whole molecular group discussion. Therefore, we first discuss the β′_i′j′k′ of single bond CO with C_∞v symmetry in Figure 1.

The reduced mass G_A1 of single bond CO and the normal coordinate Q_A1 of symmetric vibration mode are as defined as follows [17]:

M_C and M_O are the masses of carbon and oxygen atom, and _Δr is the vibration vector of CO along the ζ-axis in the bond coordinate system (ξ,η,ζ). It can be seen from expressions (5) that

For CO, the molecular coordinate system (a,b,c) coincides with the bond coordinate system (ξ,η,ζ), so the three projection components of the vibration vector _Δr in the molecular coordinate system (a,b,c) are

The relationships between the hyper-Raman microscopic polarizability tensor element differentiation of CO are as follows:

Above β′_ξηζ = ∂β_ξηζ/∂Δζ, Δζ = |_Δr| is the mode of the vibration vector _Δr along the ζ-axis. r₁ and r₂ are the ratios of the differential of the hyper-Raman microscopic polarizability tensor of CO. For the convenience of calculation, the β₀ = ∂β_ζζζ/∂Δζ. According to the definition of molecular group dipole moment, three components of single bond AB dipole moment in the bond coordinate system are as follows:where E_ξ, E_η, and E_ζ represent the projection of the external light field in the bond coordinate system. The expressions (11)–(13) are transformed into the molecular coordinate system, and the results are as follows:

We take the expression (6) of |_Δr| into expressions (14)–(16), compared with the dipole moment expression in the molecular coordinate system, and the expressions of the hyper-Raman microscopic polarizability tensor β_i′j′k′ are as follows:take β_i′j′k′ derivative of Q_A1. The expressions of the hyper-Raman microscopic polarizability tensor differential β′_i′j′k′ of CO are as follows:from expressions (20)–(22), the ratios and of β′_i′j′k′ for the C_∞V symmetry molecular group in reference [14] can be simplified as follows:

3. BAM Solves β′_i′j′k′ of CO₂

Coupling the β′_i′j′k′ of two single bond CO, the expression of β′_i′j′k′ of CO₂ can be obtained.

For CO₂ shown in Figure 2, the normal coordinates Q_A1 and Q_B1 of CO₂ are defined as follows [17]:in which, |_{Δr i}| is the modulus of vibration vector of CO_i, which is along the ζ_i axis in the bond coordinate system (ξ_i,η_i,ζ_i), G_A1 and G_B1 are the reciprocal of reduced masses, and the expression of G_A1 and G_B1 are as follows [17]:

It can be seen from expressions (25) and (26) that

From Figure 2, the projection components of the vibration vectors _Δr1 and _Δr2 in the molecular coordinate system are as follows:

According to the definition of molecular group dipole moment, three components of CO_i (i = 1,2) dipole moment in the bond coordinate system are as follows:

The bonds CO₁ and CO₂ are identical except for orientation, so β₀, r₁, and r₂ of the two single bonds are the same. E_ξi, E_ηi, and E_ζi represent the projection of the external light field in the bond coordinate system. From Figure 2, the expressions of E_ξi, E_ηi, and E_ζi are as follows:

By projecting the dipole moments of two single bonds into the molecular coordinate system and adding them together, the results are as follows:

By comparing the expressions (38)–(40) with the expressions of dipole moment in the molecular coordinate system , the expressions of β_i′j′k′ are as follows:

From expression (26), we could derive that

Substituting expression (45) into expressions (41)–(44), we obtain the β_i′j′k′ derivative of Q_B1, the expressions of β′_i′j′k′ for the antisymmetric vibration mode are as follows:

From the expressions (46)–(49), the relationship of β′_i′j′k′ for the antisymmetric vibration mode are as follows:

From the expressions (50)–(51), the ratios , , , and , are only related to r₁ and r₂ of single bond CO. And r₁ and r₂ can be obtained by measuring the hyper-Raman depolarization rate [17].

4. Experimental Correction of r₁ and r₂

The ratios of r₁ and r₂ can be obtained by experimentally measuring the hyper-Raman depolarization rate [22]. The experimental optical path for measuring the hyper-Raman depolarization rate is shown in Figure 3. (x,y,z) represents the laboratory coordinate system. The direction of the incident light is parallel (H) to or perpendicular (V) to the horizontal plane (xy plane). The outgoing light is detected in a direction at a 90° angle to the incident light, and the detected polarization direction is the vertical (V) direction. By selecting the polarization of the incident and outgoing light, the signals I_HV and I_VV of the two polarization combinations can be obtained. The expression of the hyper-Raman depolarization rate of a certain vibrational mode of the studied molecule is shown in expression (52), and (I,J,K = x,y,z) is shown in expression (53) [19].

In which R_Ii is the matrix element of the Euler transformation from the molecular coordinate system to the laboratory coordinate system, β′_ijk is the hyper-Raman microscopic polarizability tensor differentials. The expressions (46)–(49) are introduced into expression (53) to obtain the relationship between the hyper-Raman depolarization rate and the r₁, r₂, and τ. If for a certain molecular group, can be measured to the symmetric vibration mode of the hyper-Raman depolarization rate ρ_SS and antisymmetric vibration mode of the hyper-Raman depolarization rate ρ_AS, we could be using the values of ρ_SS and ρ_AS to determine the value of r₁ and r₂.

For example, for the CO₂ which is C_2v symmetric type, the bond angle τ = 180°, M_C = 12, and M_O = 16. The ρ_SS and ρ_AS are shown in the expressions (54) and (55), respectively.

The value of r₁ and r₂ can be solved by taking the experimental measured and into expressions (54) and (55).

Using the experimentally modified BAM method, we can get the ratio of the hyper-Raman microscopic polarizability tensor element differentials that is consistent with the experimental results. Combining the ratio between the Raman microscopic polarizability tensor differentials, and bringing the above relation into the expressions (50) and (51), we can greatly simplify the relationship between CARS and CAHRS microscopic polarizability tensors.

Through the presented analysis, we obtained the ratios between the hyper-Raman microscopic polarizability tensor element differentials β′_i′j′k′. Combining the ratios between Raman microscopic polarizability tensor element differentials α′_i′j′ in the literature, we obtain the proportional relationship between the microscopic polarizability tensor element of CARS and CAHRS.

5. Conclusions

In this paper, a simplified method of microscopic polarizability tensor differential of hyper-Raman spectroscopy based on BAM is investigated. In the first, we use the differentiation of Raman microscopic polarizability tensor α′_i′j′ and the hyper-Raman microscopic polarizability tensor β′_i′j′k′ to mean the CARS and CAHRS microscopic hyperpolarizability tensor elements, so the high-order spectra can be simplified to the analysis of low-order spectra. Then, taking the CO₂ molecule as an example, we use the bond additivity model method to analyze the normal vibration mode of the single bond stretching vibration to obtain the hyper-Raman microscopic polarizability tensor element differential β′_i′j′k′ of the single bond CO. On that basis, by analyzing the symmetry of the CO₂ molecule, the β′_i′j′k′ of the two single bonds are vector-coupled to obtain the expressions of the hyper-Raman microscopic polarizability tensor element differential β′_i′j′k′ of the two normal vibration modes of the CO₂ molecule. Finally, combining the ratios between Raman microscopic polarizability tensor element differentials α′_i’j, we derived the relationship between the microscopic polarizability tensor element of CARS and CAHRS. The above analysis provides a theoretical basis for the quantitative analysis of high-order nonlinear spectral signals in symmetric molecular systems with 2 or 3 atoms.

This approach is quite general, and it can be extended to other nonlinear coherent Raman spectroscopic techniques to simplify the relationship between the microscopic polarizability tensor elements and quantitative analysis of the experimental signal, such as time-domain Raman spectroscopy, stimulated Raman scattering, and nonlinear high-resolution microscopy.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Yuan Wang and Liping Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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International Journal of Optics

A Simplified Method of Microscopic Polarizability Tensor Differential of Hyper-Raman Spectroscopy Based on the Bond Additivity Model

Abstract

1. Introduction

2. BAM Solves β′i′j′k′ of Single Bond

3. BAM Solves β′i′j′k′ of CO2

4. Experimental Correction of r1 and r2

5. Conclusions

Data Availability

Conflicts of Interest

References

Copyright

2. BAM Solves β′_i′j′k′ of Single Bond

3. BAM Solves β′_i′j′k′ of CO₂

4. Experimental Correction of r₁ and r₂