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Reply| March 11 2024

Response to “Comment on ‘Sum-frequency vibrational spectroscopy of centrosymmetric molecule at interfaces’” [J. Chem. Phys. 160, 107101 (2024)]

Ren-Hui Zheng

0000-0002-0236-9675

;

Ren-Hui Zheng ^a)

¹

Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences

, Zhongguancun, Beijing 100190,

People’s Republic of China

a)Authors to whom correspondence should be addressed: [email protected] and [email protected]

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Wen-Mei Wei

0000-0001-9072-6232

;

Wen-Mei Wei ^a)

²

Department of Chemistry, College of Basic Medicine, Anhui Medical University

, Hefei, Anhui 230032,

People’s Republic of China

a)Authors to whom correspondence should be addressed: [email protected] and [email protected]

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Shuo-Cang Zhang

0000-0003-1582-0645

Shuo-Cang Zhang

¹

Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences

, Zhongguancun, Beijing 100190,

People’s Republic of China

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a)Authors to whom correspondence should be addressed: [email protected] and [email protected]

J. Chem. Phys. 160, 107102 (2024)

https://doi.org/10.1063/5.0177277

The first-order electric dipole hyperpolarizability of the centrosymmetric benzene molecule is zero, but benzene shows very strong sum-frequency vibrational spectroscopy (SFVS) at interfaces in experiment. In Ref. 1 entitled “Sum-frequency vibrational spectroscopy of centrosymmetric molecule at interfaces,” we perform a theoretical study on SFVS of benzene, which is in good agreement with experiment. Its strong SFVS mainly comes from the interfacial electric quadrupole (EQ) hyperpolarizability rather than the symmetry-breaking electric dipole, bulk EQ, and interfacial and bulk magnetic dipole hyperpolarizabilities. Hirano et al.² have commented on our results of Ref. 1.

We would like to address five points mentioned in the preceding Comment.² The first point is the local field effect. In the preceding Comment, Hirano et al. pointed out that they fully described the local field effects of benzene by calculating them explicitly with the self-consistent polarizable molecular dynamics (MD) simulation in Ref. 3, which is more rigorous than a phenomenological dielectric model in both surface and bulk regions. As we are aware, polarizability, dielectric constant, and local field effects arise from the interactions between molecules and optical fields. These interactions depend on factors such as the orientation, structure, and properties of molecules, as well as the frequencies of the optical fields. Therefore, they are all frequency-dependent. For example, the dielectric constant of water is 78.5 in the electrostatic field; however, it is about 1.77 at 532 nm.

In Ref. 4, there are detailed formulas for calculating the local field correction considering the influences of molecular polarizabilities and intermolecular couplings that are obtained using electric dipole–dipole interactions. That method has its advantages: molecular polarizability and electric dipole moments are both calculated in real-time during the polarizable MD simulation, and they can promptly respond to the changes of molecular environment, where the effects of orientation and vibration are well incorporated in the flexible and polarizable MD simulation. In general, this method may suffice for computing the local field for molecules at interfaces, particularly in non-resonant cases.

However, that method has a fatal flaw: the calculation process neglects the influence of optical frequencies and uses static polarizability instead of frequency-dependent polarizability. In Ref. 3, Kawaguchi et al. applied that method to calculate the local field corrections using the static polarizability for the IR, visible, and sum-frequency (SF) light beams, and they said that the electronic polarizability of benzene can be well regarded as constant over the frequency range of IR, visible, and SF under electronically non-resonant conditions, and thus, they can describe the electronic polarization with the static polarizability to consider the polarization couplings.

Benzene is a very special molecule, whose local field corrections at interfaces are very sensitive to the optical frequencies even in non-resonant cases; for example, the XX components L_XX of the Lorentz local field correction factors of parallel displaced (PD) benzene dimer at 3252, 532, and 457 nm are 7.3623, 11.4892, and 14.9484 at interfaces (see Table SI in Ref. 1), respectively. The frequency-sensitive and robust nature of local field corrections significantly amplify the interfacial EQ contribution to benzene’s SFVS, constituting the primary factor behind its anomalous SFVS behavior.

In a previous study,⁵ we also use this approach to calculate the local field corrections for methanol at interfaces, where L_XX at 3534, 532, and 462 nm are 1.638, 1.667, and 1.678, respectively, which is insensitive to the light frequencies in non-resonant cases.

The strong frequency dependence of the local field correction factors even in non-resonant cases found by us does not refer to the bulk local field but to the interfacial one, which results from the benzene special orientation and properties at interfaces. The local field correction at interfaces is more complex than that in bulk.

The Lorentz model in the bulk used by Kawaguchi et al. is valid enough,³ and we also apply this model to compute the local field corrections in the bulk.¹

The second point is on the symmetry breaking of benzene. Hirano et al. commented that we actually did not examine the symmetry breaking in our argument and picked up some dimer configurations and discussed possible contribution of EQ from such dimers.² They also said that since the orientational distribution of liquid benzene is quite random at the surface, it is hard to prove that the “typical” dimer configurations dominate the population or the SFG signal.²

Benzene molecule possesses conjugated large π bonds, and its structure is relatively stable, so weaker interactions, such as van der Waals forces, are expected to exert a relatively minor impact on the overall structure. However, in the case of benzene dimers, the π–π interactions are relatively strong, leading to a significant disruption of benzene’s symmetry. This is why we focus solely on the consideration of three dimmers. These relatively strong π–π interactions can substantially alter the arrangement and symmetry of the molecules, which is crucial for studying the properties and behavior of benzene molecules.¹

Just as pointed out by Hirano et al., benzene has six equivalent C–H bonds, which form normal modes in the isotropic environment. These modes are sensitive to the external perturbation and readily mixed to form local modes in the inhomogeneous environment. We think that van der Waals and electrostatic interactions from the environment influence the C–H bonds. We posit that π–π interactions exert a considerable strength and wield a predominant influence on C–H normal modes. This assertion is grounded in the remarkably short distance characterizing π–π interactions and their relatively high intensity. The proximity of the interacting aromatic rings contributes to the potency of these interactions, thereby establishing their dominance in shaping the vibrational behavior of C–H bonds. The external perturbation stems mainly from the π–π interactions.

The third point is on the field variations at interfaces. In the preceding Comment, Hirano et al. pointed out that our fundamental definition of the interfacial and bulk susceptibilities [

{\overset{\leftrightarrow}{χ}}_{S}^{(2)}

and

{\overset{\leftrightarrow}{χ}}_{B}^{(2)}

in Eq. (3) of Ref. 1] is questionable in many respects, and our formulas do not include the field variations in the interface region to take account of the EQ contributions. In Ref. 1, we present the formulas for the interfacial second-order EQ susceptibility

χ_{S q}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2})

without considering the field variations and for the bulk second-order EQ susceptibility

χ_{B q}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2})

⁠. They can be expressed as¹

\begin{gathered} \begin{aligned} χ_{S q}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) & = - ({\vec{k}}_{1} + {\vec{k}}_{2}) \cdot {\vec{χ}}_{Q, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) \\ + {\vec{k}}_{1} \cdot {\vec{χ}}_{Q 1, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) + {\vec{k}}_{2} \cdot \end{aligned} \\ {\vec{χ}}_{Q 2, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}), I, J, K, L = X, Y, Z, \end{gathered}

(1)

\begin{align} χ_{B q}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) & = - i ({\vec{k}}_{1} + {\vec{k}}_{2}) \cdot {\vec{χ}}_{Q, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) \\ + i {\vec{k}}_{1} \cdot {\vec{χ}}_{Q 1, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) + i {\vec{k}}_{2} \cdot {\vec{χ}}_{Q 2, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}), \end{align}

(2)

where

{\vec{k}}_{1}

and

{\vec{k}}_{2}

are the wave vectors of the Vis and IR beams, respectively, and

{\vec{χ}}_{Q, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2})

⁠,

{\vec{χ}}_{Q 1, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2})

⁠, and

{\vec{χ}}_{Q 2, I J K L}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2})

are the EQ susceptibilities interacting with the SF, Vis, and IR beams, respectively.

In Ref. 6, Sun et al. presented the formulas for calculating the electric dipole susceptibility

χ_{S d}^{(2)}

and the interfacial EQ susceptibility

χ_{S q}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) = χ_{SqZ}^{(2)} + χ_{q Z}^{(2)}

(⁠

χ_{SqZ}^{(2)}

is the interfacial EQ term and

χ_{q Z}^{(2)}

is the bulk EQ term), considering the field variations in the interface region,

χ_{S d}^{(2)} = \int_{interface} f_{I} (ω, Z) χ_{d, IJK}^{(2)} f_{J} (ω_{1}, Z) f_{K} (ω_{2}, Z) dZ,

(3)

\begin{align} χ_{SqZ}^{(2)} & = \int_{interface} [- \frac{\partial f_{I} (ω, z)}{\partial Z} χ_{Q, IJKZ}^{(2)} f_{J} (ω_{1}, Z) f_{K} (ω_{2}, Z) \\ + f_{I} (ω, Z) χ_{Q 1, IJKZ}^{(2)} \frac{\partial f_{J} (ω_{1}, Z)}{\partial Z} (ω_{2}, Z) \\ + f_{I} (ω, Z) χ_{Q 2, IJKZ}^{(2)} f_{J} (ω_{1}, Z) \frac{\partial f_{K} (ω_{2}, Z)}{\partial Z}] dZ, \end{align}

(4)

χ_{q Z}^{(2)} = - χ_{Q Z, I J K Z}^{(2)},

(5)

where ω, ω₁, and ω₂ are the light frequencies of the SF, Vis, and IR beams, respectively,

χ_{d, IJK}^{(2)}

is the electric dipole susceptibility,

χ_{Q Z, I J K Z}^{(2)}

is the bulk EQ susceptibility interacting with the SF light, and f is the ratio of the interfacial dielectric constant to the bulk one.⁶ Neglecting the field variations, we rewrite Eq. as

\begin{align} χ_{SqZ}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) & = - ({\vec{k}}_{1 Z} + {\vec{k}}_{2 Z}) \cdot {\vec{χ}}_{Q Z, I J K Z}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) + {\vec{k}}_{1 Z} \cdot {\vec{χ}}_{Q 1 Z,IJKZ}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}) \\ + {\vec{k}}_{2 Z} \cdot {\vec{χ}}_{Q 2 Z,IJKZ}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2}), \end{align}

(6)

where

{\vec{k}}_{1 Z}

and

{\vec{k}}_{2 Z}

are the Z component of the wave vectors of the Vis and IR beams, respectively. For SSP SFVS, the interfacial EQ term in Eq. can be given by¹

\begin{gathered} χ_{Q Z, S S P}^{(2)} = L_{QY Z} (ω) L_{Y Y} (ω_{1}) L_{Z Z} (ω_{2}) \cos β \sin β_{2} χ_{Q, Y Z Y Z}, \\ χ_{Q1Z,SSP}^{(2)} = - L_{Y Y} (ω) L_{QY Z} (ω_{1}) L_{Z Z} (ω_{2}) \cos β_{1} \sin β_{2} χ_{Q 1, Y Y Z Z}, \\ χ_{Q2Z,SSP}^{(2)} = - L_{Y Y} (ω) L_{Y Y} (ω_{1}) L_{QZZ} (ω_{2}) \sin β_{2} \cos β_{2} χ_{Q 2, Y Y Z Z}, \end{gathered}

(7)

where L is the Lorentz field correction; β is the exit angle of the SF light; β₁ and β₂ are the incident angles of the Vis and IR beams, respectively; and χ_Q,YZYZ, χ_Q1,YYZZ, and χ_Q2,YYZZ are the macroscopic EQ hyperpolarizabilities interacting with the SF, Vis, and IR beams, respectively. In addition, the bulk EQ term in Eq. can be expressed as¹

χ_{q Z, S S P}^{(2)} = - L_{QY Z} (ω) L_{Y Y} (ω_{1}) L_{Z Z} (ω_{2}) \cos β \sin β_{2} χ_{Q, Y Z Y Z} .

(8)

In Fig. 1, we plot the imaginary parts of SSP SFVS from the contributions of the interfacial EQ term, the bulk EQ term, and the sum of the two terms. We find that the interfacial EQ term dominates, and the bulk EQ term has some influence. SFVS calculated by Eq. (1) is similar to that obtained by Eqs. (6)–(8), which indicates that the interfacial EQ susceptibility in Eq. (1) is correct when we do not consider the field variations at interfaces.

FIG. 1.

(a) Computational SSP SFVS of ImχSq(2) (m2/V) from the contributions of the interfacial EQ susceptibility [black, see Eq. (1)], the interfacial EQ term [blue, see Eq. (7)], the bulk EQ term [red, see Eq. (8)], and the sum of the two terms (green) of parallel displaced (PD) dimer. (b) Experimental SFVS of ImχSS,YYZ(2) (m2/V) at the air/benzene interface.6

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(a) Computational SSP SFVS of $Im χ_{S q}^{(2)}$ (m²/V) from the contributions of the interfacial EQ susceptibility [black, see Eq. (1)], the interfacial EQ term [blue, see Eq. (7)], the bulk EQ term [red, see Eq. (8)], and the sum of the two terms (green) of parallel displaced (PD) dimer. (b) Experimental SFVS of $Im χ_{S S, Y Y Z}^{(2)}$ (m²/V) at the air/benzene interface.⁶

Now, we discuss something about Eqs. (1) and (2). Without the field variations, the interfacial [Eq. (1)] and bulk [Eq. (2)] EQ susceptibilities appear identical at first glance. However, the molecular orientation averaging required for them are different: The surface EQ susceptibility requires an averaging of molecular orientations at interfaces, while the bulk one requires an isotropic averaging in bulk.

For example, at interfaces, the relationships between the macroscopic and molecular EQ hyperpolarizabilities β_σρκl are obtained through the Euler rotational matrix,

χ_{Q, I J K L}^{(2)} = N_{s} \sum_{σ ρ κ l} ⟨ R_{I σ} R_{J ρ} R_{K κ} R_{L l} ⟩ β_{Q, σ ρ κ l},

(9)

where N_s is interfacial molecular density and R_Iσ, R_Jρ, R_Kκ, and R_Ll are the elements of the Euler rotational matrix. However, after isotropic averaging, the macroscopic EQ hyperpolarizabilities

χ_{Q 1}^{(2)}

arising from the interactions between the Vis light and molecular EQ in bulk can be expressed as⁷^,

\begin{gathered} \begin{aligned} χ_{Q 1, Y Z Y Z}^{(2)} & = N_{B} [\frac{1}{10} (β_{Q 1, y x y x} + β_{Q 1, x x y y} + β_{Q 1, y y z z} + β_{Q 1, z y z y} + β_{Q 1, z x z x} + β_{Q 1, x x z z}) + \frac{1}{15} (β_{Q 1, x x x x} + β_{Q 1, y y y y} + β_{Q 1, z z z z}) \\ - \frac{1}{30} (β_{Q 1, y x x y} + β_{Q 1, x y y x} + β_{Q 1, y z z y} + β_{Q 1, z y y z} + β_{Q 1, z x x z} + β_{Q 1, x z z x})], \end{aligned} \\ \begin{aligned} χ_{Q 1, Z X X Z}^{(2)} & = N_{B} [- \frac{1}{15} (β_{Q 1, y x y x} + β_{Q 1, x x y y} + β_{Q 1, y y z z} + β_{Q 1, z y z y} + β_{Q 1, z x z x} + β_{Q 1, x x z z}) + \frac{1}{15} (β_{Q 1, x x x x} + β_{Q 1, y y y y} + β_{Q 1, z z z z}) \\ + \frac{2}{15} (β_{Q 1, y x x y} + β_{Q 1, x y y x} + β_{Q 1, y z z y} + β_{Q 1, z y y z} + β_{Q 1, z x x z} + β_{Q 1, x z z x})], \end{aligned} \\ \begin{aligned} χ_{Q 1, Z Z Z Z}^{(2)} & = N_{B} [\frac{2}{15} (β_{Q 1, y x y x} + β_{Q 1, x x y y} + β_{Q 1, y y z z} + β_{Q 1, z y z y} + β_{Q 1, z x z x} + β_{Q 1, x x z z}) + \frac{1}{5} (β_{Q 1, x x x x} + β_{Q 1, y y y y} + β_{Q 1, z z z z}) \\ + \frac{1}{15} (β_{Q 1, y x x y} + β_{Q 1, x y y x} + β_{Q 1, y z z y} + β_{Q 1, z y y z} + β_{Q 1, z x x z} + β_{Q 1, x z z x})], \end{aligned} \end{gathered}

(10)

and the relationships between

χ_{Q}^{(2)}

and

χ_{Q 1}^{(2)}

are as follows:⁷,

\begin{gathered} χ_{Q, Z Y Y Z}^{(2)} = χ_{Q 1, Y Z Y Z}^{(2)}, \\ χ_{Q, X X Z Z}^{(2)} = χ_{Q 1, Z X X Z}^{(2)}, \\ χ_{Q, Z Z Z Z}^{(2)} = χ_{Q 1, Z Z Z Z}^{(2)}, \end{gathered}

(11)

where N_B is the molecular density in bulk, β_Q1 is the molecular EQ hyperpolarizability,

χ_{Q}^{(2)}

is macroscopic EQ hyperpolarizabilities arising from the interactions between the SF light and molecular EQ in bulk, and

({\vec{k}}_{1}, {\vec{k}}_{2})

is omitted in the above formulas. Indeed, the surface EQ susceptibility is defined per unit area, while the bulk EQ susceptibility is defined per unit volume. They do not have the same form. Finally,

{(χ_{SSq}^{(2)})}_{(t, r)} = χ_{S q}^{(2)} - \frac{χ_{B q}^{(2)} ({\vec{k}}_{1}, {\vec{k}}_{2})}{i Δ k_{(t, r) Z}}

is used to calculate the EQ SFVS. The detailed definitions of the above symbols can be found in Refs. 1 and 7.

For simplicity, we discuss the electric vector of an optical light propagating along the Z-direction, expressed as $\overset{⇀}{E} = \vec{ε} e^{i (kZ - 2 π ν t)}$ ⁠, where $\overset{⇀}{ε}$ is the amplitude vector of the electric field, k is the Z component of wave vector, ν is the optical frequency, and t is time. The electric vector derivative with respect to Z can be given by $\nabla_{Z} \overset{⇀}{E} = \nabla_{Z} (\overset{⇀}{ε} e^{i (kZ - 2 π vt)}) = (\nabla_{Z} \overset{⇀}{ε} + ik \overset{⇀}{ε}) e^{i (kZ - 2 π vt)} \neq ik \overset{⇀}{E}$ because there may be the following relationship at interfaces: $| \nabla_{Z} \overset{⇀}{ε} | ≫ | k \overset{⇀}{ε} |$ ⁠. In Ref. 8, Ye and Shen developed a method for computing the local field correction of the optical field for molecules adsorbed on surface, considering the influences of the electric field and its gradient [see Eqs. (6)–(8) in Ref. 8]. We apply this method to calculate the local field correction at interfaces.

The fourth point is on Fermi resonance. In the preceding Comment, Hirano et al. said that we also ignored the bulk EQ (χ^IBQ) term, which is relevant to the Fermi resonance bands of liquid benzene. The Hirano et al.’s statement does not reflect objective facts. $χ^{IBQ} = χ_{q Z}^{(2)}$ [see Eq. (5)], which is included in the interfacial EQ susceptibility in Eq. (1) (see our discussions of the third point on the field variations at interfaces). Furthermore, we calculate the bulk EQ susceptibilities in the reflection and transmission arrangements, respectively (see Figs. 5 and S15 in Ref. 1).

In SFVS, Fermi resonance may be very important. We have computed Fermi resonance for SFVS of methanol at interfaces and elucidated the corresponding vibrational peaks of the CH₃ group.⁵

Indeed, we do not directly compute Fermi resonance for benzene at interfaces. However, we have studied SFVS under the influences of the C–H bending overtones (see Fig. 4 in Ref. 1), where the second-order derivatives of the electric dipole moments and polarizabilities are computed. The corresponding calculated imaginary part of SSP SFVS is small with a value of −3.97 × 10⁻²² m²/V (see Fig. 4 in Ref. 1). SFVS from the C–H bending combinations is also small. The wavefunction due to Fermi resonance is a linear combination of the overtone and combination wavefunctions of the C–H bending vibrations and the fundamental wavefunctions of the C–H stretching vibrations.⁵ Thus, the contribution of Fermi resonance is small.

Our conclusion is that the influence of Fermi resonance on SFVS of benzene at interfaces is relatively small. We do not claim that Fermi resonance can be neglected, which is not contradictory to experimental observations by Hommel and Allen.⁹ However, Fermi resonance is relatively unimportant compared with the interfacial EQ contribution.

In Ref. 3, Kawaguchi et al. underestimated the interfacial EQ SFVS, and they found that the contributions from the bulk EQ and symmetry breaking dominated. Hence, Fermi resonance is not negligibly small when compared with these two minor contributions.

The fifth point concerns the absolute amplitude and line shape of SFVS for benzene. We plot Fig. 2, which includes experimental SFVS,⁶ our computational SSP SFVS from the EQ contribution of the PD dimer at interfaces,¹ and Hirano et al.’s SFVS at the gas/benzene interface from the symmetry-breaking surface term, bulk EQ term, and the sum of the two terms.² Our results are in good agreement with experiment in terms of vibrational frequencies, amplitudes, and line shapes. Comparing Hirano et al.’s theoretical SFVS with the experiment, we observe that their vibrational frequencies are significantly higher than those observed in the experiment, and there are notable differences in the spectral line shapes.

FIG. 2.

(a) Experimental SFVS of ImχSS,YYZ(2) (m2/V) at the air/benzene interface,6 (b) computational SSP SFVS of ImχSq,SSP(2) (m2/V) from the EQ contribution of the PD dimer at interfaces,1 and (c) the calculated SFVS at the gas/benzene interface from the symmetry-breaking surface term (black), bulk EQ term (blue), and the sum of the two terms (red).2

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(a) Experimental SFVS of $Im χ_{S S, Y Y Z}^{(2)}$ (m²/V) at the air/benzene interface,⁶ (b) computational SSP SFVS of $Im χ_{S q, S S P}^{(2)}$ (m²/V) from the EQ contribution of the PD dimer at interfaces,¹ and (c) the calculated SFVS at the gas/benzene interface from the symmetry-breaking surface term (black), bulk EQ term (blue), and the sum of the two terms (red).²

Furthermore, a crucial point is that they did not calculate the absolute amplitude of SFVS. The experiment provided the absolute spectral amplitude,⁶ while Hirano et al. only computed the relative amplitude.² Due to the absence of absolute amplitude calculations, Hirano et al.’s research lacks credibility and persuasiveness. We strongly recommend that Hirano et al. consider calculating the absolute amplitude.

In conclusion, we respond to the five points raised by Hirano et al. regarding their comments and research results. We calculate the local field correction at the benzene gas/liquid interface using the method developed by Ye and Shen⁸ and observe that the interfacial local field correction is highly sensitive to optical frequencies even in non-resonant cases. This correction is substantial, significantly enhancing the interfacial EQ contribution to SFVS, which is the dominant term. However, Kawaguchi et al. neglected the influence of optical frequencies and uses static polarizability to calculate the local field correction for the IR, visible, and SF beams, leading to an underestimation of the local field correction at interfaces. Our computational SFVS aligns well with experimental results in terms of vibrational frequencies, amplitude, and line shapes. However, their vibrational frequencies are much higher than those observed in experiment, and their spectral line shapes differ from the experimental results. Significantly, they omit the calculation of the absolute SFVS amplitude. SFVS without absolute amplitude is deemed unreliable and lacks persuasive strength.

This work was supported by the National Natural Science Foundation of China (NNSF) (Grant Nos. 91856122 and 22373109) and Beijing Municipal Science & Technology Commission (Grant No. Z191100007219009).

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Author(s)