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

Authors: Yuan Wang; Liping Huang
Journal: International Journal of Optics
 (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 tensorα′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.


---

## Body

## 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 βi′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 βi′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 βi′j′k′l′ and βi′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:(1)βi′j′k′=−12ωqε0αi′j′′⋅μk′′,(2)βi′j′k′l′=−12ωqε0αi′j′′⋅αk′l′′,(3)βi′j′k′l′m′=−12ωqε0βi′j′k′′⋅αl′m′′.When we get the ratio betweenμ′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 Solvesβ′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 theβ′i′j′k′ of single bond CO with C∞v symmetry in Figure 1.Figure 1
Molecular coordinate system of (C)∞v symmetric molecular group.The reduced massGA1 of single bond CO and the normal coordinate QA1 of symmetric vibration mode are as defined as follows [17]:(4)GA1=1MC+1MO,(5)QA1=ΔrGA1.MC and MO 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(6)Δr=QA1GA1.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(7)Δra=0,(8)Δrb=0,(9)Δrc=Δr.The relationships between the hyper-Raman microscopic polarizability tensor element differentiation of CO are as follows:(10)βζξξ′=βζηη′=r1βζζζ′,βξξζ′=βξζξ′=βηηζ′=βηζη′=r2βζζζ′.Aboveβ′ξηζ = ∂βξηζ/∂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 = ∂βζζζ/∂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:(11)μξ=β0r2EζEξ+r2EξEζΔζ=2β0r2EξEζΔζ,(12)μη=β0r2EζEη+r2EηEζΔζ=2β0r2EηEζΔζ,(13)μζ=β0r1EξEξ+r1EηEη+EζEζΔζ,where Eξ, Eη, and Eζ 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:(14)μa=β0r2EcEa+r2EaEcΔr=2β0r2EaEcΔr,(15)μb=β0r2EcEb+r2EbEcΔr=2β0r2EbEcΔr,(16)μc=β0r1EaEa+r1EbEb+EcEcΔr.We take the expression (6) of |Δr| into expressions (14)–(16), compared with the dipole moment expression μi′=∑j′,k′=a,b,cβi′j′k′Ej′Ek′ in the molecular coordinate system, and the expressions of the hyper-Raman microscopic polarizability tensor βi′j′k′ are as follows:(17)βaac=βaca=βbbc=βbcb=r2β0QA1GA1,(18)βcaa=βcbb=r1β0QA1GA1,(19)βccc=β0QA1GA1,take βi′j′k′ derivative of QA1. The expressions of the hyper-Raman microscopic polarizability tensor differential β′i′j′k′ of CO are as follows:(20)βaac′=βaca′=βbbc′=βbcb′=r2β0GA1,(21)βcca′=βcbb′=r1β0GA1,(22)βccc′=β0GA1,from expressions (20)–(22), the ratios r1HR and r2HR of β′i′j′k′ for the C∞V symmetry molecular group in reference [14] can be simplified as follows:(23)r1HR=βcaa′βccc′=r1,(24)r2HR=βaac′βccc′=r2.


## 3. BAM Solvesβ′i′j′k′ of CO2

Coupling theβ′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 QA1 and QB1 of CO2 are defined as follows [17]:(25)QA1=Δr1+Δr22GA1,(26)QB1=Δr1−Δr22GB1,in which, |Δri| is the modulus of vibration vector of COi, which is along the ζi axis in the bond coordinate system (ξi,ηi,ζi), GA1 and GB1 are the reciprocal of reduced masses, and the expression of GA1 and GB1 are as follows [17]:(27)GA1=1MB,(28)GB1=2MA+1MB.Figure 2
Molecular coordinate system of CO2.It can be seen from expressions (25) and (26) that(29)Δr1=12QA12GA1+QB12GB1,(30)Δr2=12QA12GA1−QB12GB1.From Figure2, the projection components of the vibration vectors Δr1 and Δr2 in the molecular coordinate system are as follows:(31)Δr1a=Δr1,Δr1b=0,Δr1c=0,(32)Δr2a=−Δr2,Δr2b=0,Δr2c=0.According to the definition of molecular group dipole moment, three components of COi (i = 1,2) dipole moment in the bond coordinate system are as follows:(33)μξi=β0r2EζiEξi+r2EξiEζiΔri=2β0r2EξiEζiΔri,(34)μηi=β0r2EζiEηi+r2EηiEζiΔri=2β0r2EηiEζiΔri,(35)μζi=β0r1EξiEξi+r1EηiEηi+EζiEζiΔri.The bonds CO1 and CO2 are identical except for orientation, so β0, r1, and r2 of the two single bonds are the same. Eξi, Eηi, and Eζi represent the projection of the external light field in the bond coordinate system. From Figure 2, the expressions of Eξi, Eηi, and Eζi are as follows:(36)Eξ1=Ec,Eη1=−Eb,Eζ1=Ea,(37)Eξ2=Ec,Eη2=Eb,Eζ2=−Ea.By projecting the dipole moments of two single bonds into the molecular coordinate system and adding them together, the results are as follows:(38)μa=μζ1−μζ2,(39)μb=−μη1+μη2,(40)μc=μξ1+μξ2.By comparing the expressions (38)–(40) with the expressions of dipole moment in the molecular coordinate system μi′=∑j′,k′=a,b,cβi′j′k′Ej′Ek′, the expressions of βi′j′k′ are as follows:(41)βccc=βcaa=βcbb=βaac=βaca=βbbc=βbcb=0,(42)βaaa=β0Δr1−Δr2,(43)βacc=βabb=r1β0Δr1−Δr2,(44)βcca=βcac=βbba=βbab=r2β0Δr1−Δr2.From expression (26), we could derive that(45)Δr1−Δr2=QB12GB1.Substituting expression (45) into expressions (41)–(44), we obtain the βi′j′k′ derivative of QB1, the expressions of β′i′j′k′ for the antisymmetric vibration mode are as follows:(46)β′cccAS=β′caaAS=β′cbbAS=β′aacAS=β′acaAS=β′bbcAS=β′bcbAS=0,(47)β′aaaAS=β02GB1,(48)β′accAS=β′abbAS=r1β02GB1,(49)β′ccaAS=β′cacAS=β′bbaAS=β′babAS=r2β02GB1.From the expressions (46)–(49), the relationship of β′i′j′k′ for the antisymmetric vibration mode are as follows:(50)RAS,a1HR=β′accASβ′aaaAS=RAS,b1HR=β′abbASβ′aaaAS=r1,(51)RAS,a2HR=β′ccaASβ′aaaAS=RAS,b2HR=β′bbaASβ′aaaAS=r2.From the expressions (50)–(51), the ratios RAS,a1HR, RAS,b1HR, RAS,a2HR, and RAS,b2HR, 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 ofr1 and r2

The ratios ofr1 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 IHV and IVV 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 βIJK′2 (I,J,K = x,y,z) is shown in expression (53) [19].Figure 3
The experimental optical path for measuring the hyper-Raman depolarization rate.(52)ρ=IHVIVV=βzyy′2βzzz′2,(53)β′IJK2=∑i,j,k,i′,j′,k′βijk′βi′j′k′′∫02π∫02π∫0πRIiRJjRKkRIi′RJj′RKk′sinθdθdψdϕ/∫02π∫02π∫0πsinθdθdψdϕ.In whichRIi 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 ρSS and antisymmetric vibration mode of the hyper-Raman depolarization rate ρAS, we could be using the values of ρSS and ρAS to determine the value of r1 and r2.For example, for the CO2 which is C2v symmetric type, the bond angle τ = 180°, MC = 12, and MO = 16. The ρSS and ρAS are shown in the expressions (54) and (55), respectively.(54)ρSS=IHV−SSIVV−SS=24+8r1−96r2−16r1r2+108r12+96r2239+6r1+12r2+100r1r2+25r12+100r22,(55)ρAS=IHV−ASIVV−AS=8+40r1−32r2−80r1r2+92r12+32r2255+62r1+116r2+100r1r2+23r12+108r22.The value ofr1 and r2 can be solved by taking the experimental measured ρHRSS and ρHRAS 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 differentialsβ′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 tensorα′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.



---

*Source: 2710506-2022-03-10.xml*