Beyond the Electric Dipole Approximation: Electric and Magnetic Multipole Contributions Reveal Biaxial Water Structure from SFG Spectra at the Air-Water Interface

TL;DR

This work provides a first-principles, simulation-based framework to predict the full multipole content of SFG spectra at interfaces, including electric dipole, quadrupole, and magnetic dipole contributions, enabling depth-resolved insights. By applying it to the air–water interface, the authors achieve quantitative agreement with experiments across bending and OH-stretch regions and reveal that higher-order multipoles are essential for accurate interpretation. Subtracting quadrupole and magnetic contributions isolates the second-order electric-dipole susceptibility, which acts as a quantitative probe of interfacial molecular orientational anisotropy and exposes pronounced biaxial ordering of interfacial water. The approach yields detailed, depth-resolved pictures of interfacial structure and shows how local-field and dielectric profiles shape SFG responses, offering a general route to extract interfacial organization from SFG spectra.

Abstract

The interpretation of sum-frequency-generation (SFG) spectra has been severely limited by the absence of quantitative theoretical predictions of higher-order multipole contributions. Magnetic dipole and electric quadrupole contributions are determined by bulk properties but appear in all experimental SFG spectra, obscuring the connection between measured spectra and interfacial structure. We present the simulation-based framework to predict the full set of multipole spectral contributions. This framework also yields depth-resolved spectra, enabling the precise spatial localization of spectroscopic features. Applied to the air-water interface, our approach achieves quantitative agreement with experimental spectra for different polarization combinations in both the bending and stretching regions. Higher-order multipole contributions are crucial for correctly interpreting SFG spectra: in the bending band, the electric dipole and the magnetic dipole contributions have similar intensities, while the electric quadrupole contribution is significantly larger. In the OH-stretch region, the electric quadrupole contribution is found to be in large part responsible for the characteristic shoulder at 3600/cm. Crucially, subtracting the quadrupole and magnetic contributions isolates the second-order electric dipole susceptibility, which is a quantitative probe for interfacial molecular orientational anisotropy. This electric-dipole susceptibility reveals a pronounced biaxial ordering of water at the air-water interface. By resolving a fundamental limitation of the interpretation of SFG spectroscopy, our framework allows for the detailed extraction of interfacial water ordering from SFG spectra.

Figures (6)

• Figure 1: SFG spectra and decomposition into multipole components. (a) & (b): Sketch of the different second-order molecular multipole contributions to the total SFG spectra $\tilde{S}^{(2)\prime \prime}_{yyz}$ and $\tilde{S}^{(2)\prime \prime}_{zzz}$, respectively. Here, $\mu_i^{(2)}$, $Q_{ij}^{(2)}$ and $m_i^{(2)}$ denote the induced molecular second-order electric dipole, electric quadrupole, and magnetic dipole moments, respectively. The blue region represents water and the white region air. (c): Comparison of predicted and experimental imaginary SFG spectra $\tilde{S}^{(2)\prime \prime}_{yyz}$ in the bending-frequency region. The SFG spectrum is decomposed into the pure electric dipole (DD), the electric dipole - electric quadrupole cross (DQ), the electric quadrupole (Q), and the magnetic dipole contribution (M) in (e). (d) & (f): The same analysis for $\tilde{S}^{(2)\prime \prime}_{zzz}$. Results for the OH-stretch frequency region are shown in (g)-(j). Experimental data is taken from Fellows et al.fellowsImportanceLayerDependentMolecular2025b, Yu et al.yuFresnelFactorCorrection2023, and Chiang et al.chiangDielectricFunctionProfile2022a. The predicted absolute spectrum $|\tilde{S}^{(2)}_{yyz}|$ is compared with various published experimental data fellowsImportanceLayerDependentMolecular2025byuFresnelFactorCorrection2023senguptaNeatWaterVapor2018azhangQuantitativeConsistencyIntensity2025a in (k).We red-shift our predictions for the bending and stretch bands by $28cm^{-1}$ and $166cm^{-1}$, respectively, to match the experiments. The boundary between the two shifted regions is set at $2500cm^{-1}$. Grey dashed boxes indicate the regions shown in the insets.
• Figure 2: Depth-dependent analysis of the bending band. The mass density profile is shown in (a), the local field factors in the visual frequency range $f^\mathrm{VIS}_x(z)$ and $f^\mathrm{VIS}_z(z)$ defined in Equation \ref{['eq:def_aver_loc_fac_l']} in (b). Lorentz theory predictions in bulk water are denoted by dashed horizontal lines. The black solid line denotes the vacuum value. For illustrative purposes, a snapshot from the simulation is shown in the background. (c): Comparison of the spatially resolved integrals over the bending band, as defined in Equation \ref{['eq:def_integral_bend']}, between $\tilde{\chi}^{(2,\mathrm{DL})\prime \prime}_{yyz}(z)$ and the prediction based solely on molecular orientation, $\tilde{\chi}^{(2,\mathrm{ORI})\prime \prime}_{yyz}(z)$, defined in Equation \ref{['eq:chi2ori']}. The electric dipole second-order susceptibility $\tilde{\chi}^{(2,\mathrm{DL})\prime \prime}_{yyz}(z)$, defined in Equation \ref{['eq:chi_2DL']}, is shown at selected positions in (d). These positions are marked in (f), where the corresponding two-dimensional profile is presented. The second-order response profile of the SFG signal $\tilde{s}^{(2)\prime \prime}_{yyz}(z)$ is presented in (e) and $\tilde{\chi}^{(2,\mathrm{ORI})\prime \prime}_{yyz}(z)$ in (g). The spectra are red-shifted by $28cm^{-1}$.
• Figure 3: The orientation distribution function $\rho_\mathrm{ORI}(\theta, \psi)$ is shown for the same positions relative to $z_\mathrm{GDS}$ as in Figure \ref{['fig:bending_zres']} in (a)-(c). Three orientational species are distinguished: planar, pointing in, and pointing out, as illustrated at the top of panel (d). These orientations are marked with dots and contours in (a)-(c). Additionally, a snapshot of the simulation box viewed from the air is provided in (d), where molecules are color-coded according to their orientational species. (e) displays the fraction of each orientational species as a function of $z$. The molecular hyperpolarizability tensor component $\tilde{\beta}_{yyz}(\theta,\psi)$, defined in Equation \ref{['eq:def_beta_ijk']}, is shown as a function of the molecular orientation in (f). The color indicates the intensity at bending-band frequencies, obtained by integrating the imaginary part $\tilde{\beta}"_{yyz}(\theta,\psi)$ using the same integration boundaries as in Equation \ref{['eq:def_integral_bend']}. This intensity is decomposed into its uniaxial and biaxial components, defined in Equations \ref{['eq:def_beta_uniax']} and \ref{['eq:def_beta_biax']} and shown in (g) and (h), respectively. The position-resolved profile of the bending band is decomposed into uniaxial and biaxial contributions in (i). The frequency-dependent coefficients $\tilde{\beta}^{lm\prime \prime}_{yyz}$ of $\tilde{\chi}^{(2,\mathrm{ORI})\prime \prime}_{yyz}$ defined in Equation \ref{['eq:def_beta']}-\ref{['eq:def_beta_biax']} are presented in (j).
• Figure 4: Second-order response profile $\tilde{s}^{(2) \prime \prime}_{ijk}\left( z \right)$ of the OH-stretch band as defined in Equation \ref{['eq:s2_ijk']} and the decomposition into its multipole components $\tilde{s}^{(2,\beta)\prime \prime }_{ijk}\left( z \right)$ as defined in Equation \ref{['eq:s2_ijk_beta']}. A snapshot of the simulation box is shown in (a). The mass-density profiles with respect to $z_\mathrm{GDS}$ and $z_\mathrm{WCS}$ are presented in (b). Slices of $\tilde{s}^{(2) \prime \prime}_{ijk}\left( z \right)$ are shown for selected positions in (c) & (d) relative to $z_\mathrm{GDS}$ and in (e) & (f) relative to $z_\mathrm{WCS}$, colored vertical lines in (a), (b), (g)-(j) indicate the positions. The profiles $\tilde{s}^{(2)\prime \prime}_{ijk}\left( z \right)$ are shown relative to $z_\mathrm{WCS}$ in (g) & (h) and relative to $z_\mathrm{GDS}$ in (i) & (j). The profile relative to $z_\mathrm{GDS}$ is dissected into its molecular multipole components $\tilde{s}^{(2,\mathrm{D}) \prime \prime}_{ijk}\left( z \right)$ and $\tilde{s}^{(2,\mathrm{Q}) \prime \prime}_{ijk}\left( z \right)$ in (k) & (l) and (m) & (n), respectively. All spectra are red-shifted by $166cm^{-1}$.
• Figure 5: Linear optical dielectric and IR absorption profiles at the air water interface. (a): Snapshot of the simulation. (b): The water mass density profile is compared to the real parts of the tensorial optical dielectric profiles. (c) & (d): The linear absorption profiles $\omega \tilde{\varepsilon}"_{xx}(z)$ and $\omega \tilde{\varepsilon}"_{zz}(z)$ in the IR and low THz frequency range are shown as a 2D plot. (e)-(j): The imaginary part of the linear absorption profile is shown at the same positions relative to the Gibbs dividing surface as in Figure \ref{['fig:s2ijk']}. The $xx$-component defined in Equation \ref{['eq:def_eps_xx']}, the $zz$-component defined in Equation \ref{['eq:def_eps_zz']}, and the isotropic average defined in \ref{['eq:def_eps_iso']} are presented in (e), (f), and (g), respectively. Additionally, the predicted linear absorption spectrum of bulk water and the experimental spectrum bertieInfraredIntensitiesLiquids1996 are shown in (g). (h)-(j): Results in the stretch region. The bending band in (e)-(g) is redshifted by $28cm^{-1}$ and the stretching band in (h)-(j) is red-shifted by $166cm^{-1}$.