The modification of the scattered photons results from the incident photons either gaining energy from or losing energy to the vibrational and rotational motion of the molecule. J. Chem. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. Complementarity of Raman and Infrared Spectroscopy for Structural B 125, 67176726 (2021). J. Chem. Although the vibrations in a molecule obey the laws of quantum mechanics, molecular systems can be analyzed using classical mechanics to ascertain the nature of the vibrational motion. Walrafen, G. E. & Pugh, E. Raman combinations and stretching overtones from water, heavy water, and NaCl in water at shifts to ca. The permanent polarizability of each molecule i, ii(riO,riH1,riH2), was computed according to ref. Discover the techniques used to assess the authenticity of paintings, Basic features of electromagnetic radiation, Types of electromagnetic-radiation sources, Techniques for obtaining Doppler-free spectra, Total orbital angular momentum and total spin angular momentum, Coherent anti-Stokes Raman spectroscopy (CARS), Laser magnetic resonance and Stark spectroscopies. Theory Comput. The two-dimensional spectra are computed through a double sine transform42,55. Shalit, A., Perakis, F. & Hamm, P. Communication: disorder-suppressed vibrational relaxation in vapor-deposited high-density amorphous ice. Phys. Raman spectroscopy is an alternative way to get information about the infrared transitions within a molecule. J. Phys. 67. 123, 044513 (2005). J. Chem. Both Raman and IR spectra result due to changes in vibration modes of molecules. Ciardi, G., Berger, A., Hamm, P. & Shalit, A. Signatures of intra- and intermolecular vibrational coupling in halogenated liquids revealed by two-dimensional Raman-terahertz spectroscopy. J. Phys. Acc. J. Chem. Even so, due to limited availability of efficient THz emitter materials, not all frequencies have been covered by the reported techniques. Test your knowledge of FTIR and Raman spectroscopy with these 10 questions Difference between IR Spectroscopy and Raman Spectroscopy Kapil, V. et al. Following Hamm44, each water molecule was amended with permanent anisotropic polarizability, which was used for the evaluation of the induced contributions to the dipoles and polarizabilities. The result agrees with the proposed interpretation that the nodal shape of the spectrum results from mechanical anharmonic coupling pathways. The grey squares highlight the peak around (250cm1, 3600cm1). 158, 134201 (2023). So, in many cases, red or near-IR lasers (660-830 nm) are still used for fluorescence suppression in Raman spectroscopy, even if the Raman-scattering efficiency is much weaker in the near-IR. 26, we then establish a structure-spectrum relationship by separating the spectral contributions from molecules exhibiting low or high tetrahedral coordination. Lett. 5, 935940 (2013). http://creativecommons.org/licenses/by/4.0/. Yet, conventional, steady-state spectroscopy of liquids typically produces broad, unresolved spectral features and misses dynamical information. The structure of the first coordination shell in liquid water. Castner, E. W., Chang, Y. J., Chu, Y. C. & Walrafen, G. E. The intermolecular dynamics of liquid water. Benson, R. L., Trenins, G. & Althorpe, S. C. Which quantum statistics-classical dynamics method is best for water? . Unfortunately, the basic discoveries made during this period were not followed up . Namely, the red-shifted O-H stretch band in the TRPMD steady-state spectra reflects in a red shift along 2 of the spectral features in the TIRV spectrum (Fig. Phys. J. Phys. Phys. Natl Acad. 12, 20602064 (2021). Google Scholar. Some nuclear quantum effects can be easily explained in terms of the differences between one-dimensional spectra (Supplementary Fig. IR absorption and anisotropic Raman spectra simulated with TRPMD are presented in Fig. Solids, films, and coatings can be examined by means of several techniques that employ the reflection of radiation from the sample. Magdu, I. These transitions result in the emission of photons with various wavelengths. Each TRPMD equilibrium trajectory was propagated for 50ps (200000 steps) and nonequilibrium trajectories was launched every 1000 steps. J. Phys. This can be quantified by the local tetrahedral order parameter Q81,82,83, which measures how the arrangement of the four neighboring water molecules deviates from the ideal tetrahedral arrangement around the central one. We note, however, that the features in this congested spectral region could be affected by the short times available from our simulations and by artificial broadening. Google Scholar. Impact of nuclear quantum effects on the structural inhomogeneity of liquid water. The classical thermal average of Eq. Phys. In the mid-infrared region a variety of optical-grade crystals, such as calcium fluoride (CaF2), zinc selenide (ZnSe), cesium iodide (CsI), or potassium bromide (KBr), coated with silicon or germanium are employed. 7. Finally, we report the first TRPMD simulation of the two-dimensional IIR spectrum of liquid water and compare it with the MD simulation in Fig. This result is consistent with other experimental observations and theoretical models. i-pi 2.0: A universal force engine for advanced molecular simulations. Chem. Though comparatively new in cement and concrete, IR study is gaining much importance with the advent of user-friendly equipment and continuing research on identification and characterization of reaction products, new materials, etc. J. Chem. Additional details about the model and MD simulations can be found in the Methods section. In the meantime, to ensure continued support, we are displaying the site without styles Nature Communications thanks the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Naserifar, S. & Goddard, W. A. Article Heyden, M. et al. 11, 1689 (2020). Two-dimensional infrared-Raman spectroscopy as a probe of waters tetrahedrality. Lin, H.-W., Mead, G. & Blake, G. A. Mapping LiNbO3 phonon-polariton nonlinearities with 2D THz-THz-Raman Spectroscopy. T.B. J. Chem. I. vibrational dynamics in two-dimensional IR line shapes. The structure-spectrum relationship can explain the temperature dependence of the spectral features corresponding to the anharmonic coupling between low-frequency intermolecular and high-frequency intramolecular vibrational modes of water. Because of the anharmonicity of the molecular vibrations, transitions corresponding to multiples (2i, 3i, etc., known as overtones) and combinations (1 + 2, 23 + 4, etc.) Sci. Google Scholar. Second, following ref. Proc. Cite this article. Importantly, this change in intensity could not be simply predicted from the one-dimensional spectra, which show little change in the intensity of the 250cm1 band. (Each band in a gas phase spectrum of free CO 2 molecules would exhibit a series of closely space, sharp lines due to rotational transitions. 147, 144501 (2017). https://doi.org/10.1038/s41467-023-37667-7, DOI: https://doi.org/10.1038/s41467-023-37667-7. Analytical Vibrational Spectroscopy - NIR, IR, and Raman Appl. Infrared and Raman Spectroscopy of Liquid Water through "First The beam splitter functions to divide the source signal into two parts for the formation of an interference pattern. Polarizability refers to the ease of distorting electrons from their original position. Two dashed, grey lines specify the region excluded from subsequent two-dimensional IIR spectra simulations (0.62Raman vs IR | Process Instruments Inc J. Chem. Mead, G., Lin, H. W., Magdu, I. Phys. Their development was enabled by the recent advances in the generation of strong THz pulses that are needed to induce a nonlinear light-matter interaction48. J. Phys. In the following, we focus on the two frequency regions indicated by the grey rectangles. J. Chem. Struct. Nat. Phys. The following experimental parameters were used for all collected spectra: 1s acquisition time, 495 mW power. Gaiduk, A. P., Gustafson, J., Gygi, F. & Galli, G. First-principles simulations of liquid water using a dielectric-dependent hybrid functional. Chem. Proc. Furthermore, the libration-stretch peak shifts from 1700cm1 in the MD spectrum to 1800cm1 in TRPMD, which aligns with the differences between MD and TRPMD IR absorption spectra. Seki, T. et al. J. Chem. 11, 88628874 (2020). 16,27,91. Res. Two-dimensional TIRV spectra of liquid water at different temperatures, as calculated using Eq. Phys. J. Chem. Phys. Infrared spectroscopy historically has been divided into three regions: the near infrared (4,00012,500 inverse centimetres [cm1]), the mid-infrared (4004,000 cm1), and the far infrared (10400 cm1). To verify this interpretation, we would ideally construct approximate dipole and polarizability models that depend linearly on atomic coordinates, which would allow us to study the mechanical anharmonic coupling pathways directly. USA 116, 24582463 (2019). Then, the total dipole moment and polarizability are given by. The technique involves shining a monochromatic light source (i.e. Publishers note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Lett. J. Chem. Phys. J. Phys. Solution Chem. 7a, b). Reply to Head-Gordon and Paesani: Liquid water, a branched polymer with~100-fs short-lived heterogeneous hydrogen bonds. Sci. Phys. Article Ojha, D., Karhan, K. & Khne, T. D. On the Hydrogen Bond Strength and Vibrational Spectroscopy of Liquid Water. Similarly, THz-THz-Raman spectroscopy recently revealed signatures of anharmonic coupling between phonons of ionic solid LiNbO356. A., Welsch, R., Allodi, M. A., Miller, T. F. & Blake, G. A. A 123, 72787287 (2019). Guide to Raman Spectroscopy | Bruker Sci. Hasegawa, T. & Tanimura, Y. For polyatomic molecules the situation is compounded by the simultaneous motion of many nuclei. Correspondence to Commun. Infrared and Raman spectroscopy - Mineral Physics 6d, e) appears at lower 1 frequencies for low Q. Cho, M. Theoretical description of two-dimensional vibrational spectroscopy by infrared-infrared-visible sum frequency generation. Proc. You are using a browser version with limited support for CSS. J. Therefore, an accurate determination of the full response function will be needed to experimentally measure the spectral features appearing in the sine-transform TIRV spectra. Infrared and Raman Spectroscopy | ScienceDirect Furthermore, the two-dimensional IIR spectrum at 280K contains peaks close to diagonal, around 1=2=250cm1, due to anharmonicity of the hydrogen-bond stretching modes. J. Phys. The authors thank Haw-Wei Lin, Roman Korol, and Vignesh C. Bhethanabotla for helpful discussions. J. Chem. J. Chem. In the Fourier-transform spectrometers, the entire optical path can be evacuated to prevent interference from extraneous materials such as water and carbon dioxide in the air. In particular, such studies could provide additional information on the combination bands appearing in IR absorption and Raman scattering spectra that have challenged physical chemists for decades. Chem. 148, 244105 (2018). J. Phys. Sci. The containment of liquid and gas samples must be in a cell that has infrared-transmitting windows such as sodium chloride, potassium bromide, or cesium iodide. 33, 8197 (2004). Overall, the temperature dependence of the IIR spectra can be almost exclusively assigned to the changes in the distribution of the local tetrahedral order parameter and the effect of the local structure on the spectral features. Perspective: Echoes in 2D-Raman-THz spectroscopy. 18, 1842418430 (2016). 46,62,63, and the response function was computed for 250fs in both t1 and t2. Two-dimensional IIR spectra of liquid water simulated with classical MD at different temperatures. Raman spectroscopy - Wikipedia Grey squares indicate the regions of the spectrum that we studied in more detail, namely the region of TIRV spectroscopy and the low-frequency region that covers intermolecular modes and the intramolecular bending mode. Am. IR absorption (a) and anisotropic Raman (b) spectra of liquid water at 300K simulated with the classical MD and TRPMD, compared with the experiments of refs. 136, 094516 (2012). 1), even though the corresponding two-dimensional spectral feature clearly appears in the IIR spectrum. Paolantoni, M., Lago, N. F., Albert, M. & Lagan, A. Tetrahedral ordering in water: raman profiles and their temperature dependence. Therefore, the combination bands in the one-dimensional spectra can be orders of magnitude weaker than the fundamentals and still exhibit strong off-diagonal peaks in the two-dimensional spectra. Raman and infrared (IR) are two complementary vibrational spectroscopy techniques that enable label-free, noninvasive, and nondestructive structural characterization of analyzed specimens. Phys. Natl Acad. Coherent two-dimensional terahertz-terahertz-Raman spectroscopy. Donaldson, P. M. Photon echoes and two dimensional spectra of the amide I band of proteins measured by femtosecond IR-Raman spectroscopy. Morawietz, T. et al. The figure on the right compares NIR gasoline data and data obtained with our Raman analyzer. 128, 224511 (2008). Comprehensive analysis of cellulose content, crystallinity, and lateral PubMed Tran, H. et al. Recently, we have developed a new ring-polymer MD (RPMD) approach65, which can simulate, at least approximately, the nuclear quantum effects on the two-dimensional IIR spectra. IR Spectroscopy - an overview | ScienceDirect Topics Chem. T.B. (11), which approximates the mechanical anharmonic coupling contribution to the TIRV spectrum. (4) and time-dependent response functions R(t1,t2) equal to those used in Fig. Indeed, we see that the diagonal peak at around (1650cm1,1650cm1) is much lower in intensity than the off-diagonal libration-bending peak, implying that the anharmonicity within the bending mode is weaker than its anharmonic coupling to the intermolecular librations. b Low-frequency part of the spectrum. STEP 2: Break modes into its component irreducible representations. This technique covers the region of the electromagnetic spectrum between the visible (wavelength of 800 nanometres) and the short-wavelength microwave (0.3 millimetre). J. Chem. Reddy, S. K., Moberg, D. R., Straight, S. C. & Paesani, F. Temperature-dependent vibrational spectra and structure of liquid water from classical and quantum simulations with the MB-pol potential energy function. J. Chem. The following article discusses the differences between IR spectroscopy and Raman spectroscopy. 9, 851857 (2018). Two-dimensional IIR spectra of liquid water at 280K, simulated with equilibrium-nonequilibrium MD (a, c) and TRPMD (b, d). Internet Explorer). A., Welsch, R., Allodi, M. A., Miller, T. F. & Blake, G. A. Appl. Phys. However, due to the presence of light hydrogen atoms, their effect might not be negligible66. In our simulations, we used N=32, which is sufficiently large for liquid water in the studied temperature range25. 150, 100901 (2019). This discrepancy can be explained by the fact that the fundamentals of the one-dimensional spectra follow harmonic selection rules, while the peaks in the two-dimensional spectra appear solely due to the anharmonic excitation pathways. Kananenka, A. Specifically, the electrical anharmonicity dominates at higher temperatures, where the tetrahedral order parameter is low, whereas the mechanical coupling, due to the anharmonic terms in the potential energy surface, is pronounced at lower temperature and higher tetrahedrality. Phys. A polarizable water model for intramolecular and intermolecular vibrational spectroscopies. 2) in a simulation with the doubled number of initial conditions. Zhao, W. & Wright, J. C. Doubly vibrationally enhanced four wave mixing: the optical analog to 2D NMR. Begui, T., Tao, X., Blake, G. A. Auer, B. M. & Skinner, J. L. IR and Raman spectra of liquid water: theory and interpretation. Instrumentation Spectral Analysis Literature Figure 18.1.1 shows a portion of the Raman spectrum for carbon tetrachloride and illustrates several important features. The spectra observed in this region are primarily associated with the internal vibrational motion of molecules, but a few light molecules will have rotational transitions lying in the region. Rev. PubMed In this contribution, we present an integrated theoretical and computational framework (named many-body molecular dynamics or MB-MD) that, by systematically removing uncertainties associated with existing approaches, enables a rigorous modeling of vibrational spectra of water from quantum dynamical simulations. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 117, 17081713 (2002). Hasegawa, T. & Tanimura, Y. Finally, we observe that the 400cm1<1<1000cm1 region of the IIR spectrum, corresponding to the librations of water molecules, carries rich information about liquid water and should be further explored experimentally. To target the low-frequency modes directly, a number of hybrid spectroscopic techniques have been proposed, involving different sequences of THz, IR, and visible pulses, such as the THz-THz-Raman38,39,40,41, THz-Raman-THz, Raman-THz-THz42,43,44,45, and THz-IR-Raman (also called THz-IR-visible46,47 or TIRV). 151, 194507 (2019). 1). In addition, the shape of the spectral peaks changes drastically. J. Chem. The bend+libration combination band is an intrinsic, collective, and strongly solute-dependent reporter on the hydrogen bonding network of liquid water. USA 103, 79737977 (2006). (2) of the main text, which is comparable to the numbers used in Refs. J. Phys. After an introduction to the basic principles of IR and Raman spectroscopy, new developments in applications of these . Lett. In total, this resulted in 1.024105 initial conditions. With the development of Fourier-transform spectrometers, this distinction of areas has blurred and the more sophisticated instruments can cover from 10 to 25,000 cm1 by an interchange of source, beam splitter, detector, and sample cell. Raman Spectroscopy - an overview | ScienceDirect Topics Google Scholar. Richardson, J. O. et al. 153, 124112 (2020). B. Vibrational frequencies can also be obtains by single level fluorescence (like vibronic features in the absorption spectrum). Analogously, there will be a rotational Raman spectrum for molecules with no permanent dipole moment that consequently have no pure rotational spectra. Most of the results presented below rely on the validity of the classical MD approach, which neglects the quantum-mechanical properties of atomic nuclei. Chem. Proc. Then, we simulated the spectra (Fig. Since RPMD is known to suffer from the spurious resonance issue, where unphysical peaks due to artificial harmonic springs appear in the spectra, we employed its thermostatted version (TRPMD)73,74. To model water, we used a flexible, point-charge qTIP4P/F force field66, which has been well studied for spectroscopic simulations25 and benchmarked against a number of experimental thermodynamic properties of water, including the radial distribution functions, dielectric constant, density, and melting point. Different spectral regions have been probed to investigate intermolecular hydrogen-bond bending and stretching modes (up to 300cm1), frustrated rotational (librational) modes (4001000cm1), intramolecular bending (1650cm1) and stretching (30003800cm1) modes, as well as combination bands between intermolecular and intramolecular modes. Located at the Chenoweth Laboratory, our facility provides resourceful advanced spectroscopic analysis of versatile organic and inorganic samples, including agricultural, environmental, food, and biomedical materials, as well as polymers and heavy metals. Raman spectroscopy is an important tool in the field of vibrational spectroscopy and is complementary to infrared absorption spectroscopy, the latter being the more common vibrational spectroscopy. NIR has very broad bands arising from overtones and harmonics that contain only subtle differences between different gasoline blends whereas Raman spectroscopy provides excellent spectral resolution of fundamental vibrations allowing for . Raman and Infrared spectroscopy. 49, 18601866 (2018). The most prominent change in the spectrum is the disappearance of the negative peak around (250cm1, 3600cm1). Chem. 8, 46404644 (2017). This generated 2.56105 samples for Eq. The absorption of infrared radiation is due to the vibrational motion of a molecule. PDF Understanding Raman Spectroscopy In this research, Raman and mid-infrared (MIR) spectroscopy in conjunction with selected chemometrics approaches were used to investigate the effects of ageing on the protein structure in Cheddar cheese. Infrared spectroscopy and Raman spectroscopy, when applied to food samples, have the unique ability to interrogate their chemical structure. Although this relaxation has been largely assigned to the coupling with the intramolecular bending overtone, the intermolecular, hydrogen-bond modes are also believed to play an important role34,35,36,88. Grechko, M. et al. To conclude, we have presented MD and TRPMD simulations of the two-dimensional IIR spectra of liquid water. The authors gratefully acknowledge support from the National Science Foundation Chemical Structure, Dynamics and Mechanisms program (grant CHE-1665467). PubMed A more detailed analysis is possible based on the work of Auer and Skinner85, who studied the dependence of the intramolecular stretching frequency and the corresponding dipole moment derivative on the electric field E generated by the surrounding water molecules at the hydrogen atom of the central molecule and projected along the OH bond. & Tokmakoff, A. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. 9, 31093117 (2013). Two-dimensional infrared-Raman spectroscopy as a probe of waters tetrahedrality, $$R({t}_{1},{t}_{2})=-\frac{1}{{\hslash }^{2}}{{{{{{{\rm{Tr}}}}}}}}\{[[\hat{{{\Pi }}}({t}_{1}+{t}_{2}),\, \hat{\mu }({t}_{1})],\, \hat{\mu }(0)]\hat{\rho }\}$$, $${R}^{{{{{{{{\rm{MD}}}}}}}}}({t}_{1},{t}_{2})=\frac{\beta }{\varepsilon }\langle [{{\Pi }}({q}_{+\!,\, {t}_{2}})-{{\Pi }}({q}_{-,\, {t}_{2}})]\dot{\mu }({q}_{-{t}_{1}})\rangle,$$, \(\hat{\mu }=\mu (\hat{{{{{{{{\bf{q}}}}}}}}})\), \(\hat{{{\Pi }}}={{\Pi }}(\hat{{{{{{{{\bf{q}}}}}}}}})\), \(({{{{{{{{\bf{q}}}}}}}}}_{\pm,0}={{{{{{{{\bf{q}}}}}}}}}_{0},{{{{{{{{\bf{p}}}}}}}}}_{\pm,0}={{{{{{{{\bf{p}}}}}}}}}_{0}\pm \varepsilon {{{{{{{\boldsymbol{{\mu }}}}}}}^{{\prime} }}}({{{{{{{{\bf{q}}}}}}}}}_{0})/2)\), $$R({\omega }_{1},{\omega }_{2})=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R({t}_{1},{t}_{2}){{{{\rm{sin}}}}} ({\omega }_{1}{t}_{1}){{{{\rm{sin}}}}} ({\omega }_{2}{t}_{2})d{t}_{1}d{t}_{2}.$$, $$\tilde{S}({\omega }_{1},{\omega }_{2})=| [S({\omega }_{1},{\omega }_{2})+S({\omega }_{2}-{\omega }_{1},{\omega }_{2})]{E}_{{{{{{{{\rm{THz}}}}}}}}}({\omega }_{1}){E}_{{{{{{{{\rm{IR}}}}}}}}}({\omega }_{2}-{\omega }_{1}) |,$$, $$S({\omega }_{1},{\omega }_{2})=\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }R({t}_{1},{t}_{2}){e}^{-i{\omega }_{1}{t}_{1}}{e}^{-i{\omega }_{2}{t}_{2}}d{t}_{1}d{t}_{2},$$, $${\omega }_{{{{{{{{\rm{OH}}}}}}}}}={\omega }_{{{{{{{{\rm{OH}}}}}}}}}^{(0)}-{\omega }_{{{{{{{{\rm{OH}}}}}}}}}^{(1)}E-{\omega }_{{{{{{{{\rm{OH}}}}}}}}}^{(2)}{E}^{2},$$, $${\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} }={\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} (0)}+{\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} (1)}E,$$, \({\omega }_{{{{{{{{\rm{OH}}}}}}}}}^{(\alpha )}\), \({\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} (\alpha )}\), \({\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} }\), \(\partial {\omega }_{{{{{{{{\rm{OH}}}}}}}}}/\partial {q}_{{{{{{{{\rm{HB}}}}}}}}}=-({\omega }_{{{{{{{{\rm{OH}}}}}}}}}^{(1)}+2{\omega }_{{{{{{{{\rm{OH}}}}}}}}}^{(2)}E)\partial E/\partial {q}_{{{{{{{{\rm{HB}}}}}}}}}\), \(\partial {\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} }/\partial {q}_{{{{{{{{\rm{HB}}}}}}}}}={\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} (1)}\partial E/\partial {q}_{{{{{{{{\rm{HB}}}}}}}}}\), \(|\partial {\mu }_{{{{{{{{\rm{OH}}}}}}}}}^{{\prime} }/\partial {q}_{{{{{{{{\rm{HB}}}}}}}}}|\), $${{{{{{{{\boldsymbol{\mu }}}}}}}}}^{{{{{{{{\rm{ind}}}}}}}}}=\mathop{\sum}\limits_{\begin{array}{c}i,\, j\\ i\ne j\end{array}}{{{{{{{{\boldsymbol{\Pi }}}}}}}}}_{i}\cdot {{{{{{{{\bf{E}}}}}}}}}_{ij},$$, $${{{{{{{{\boldsymbol{\Pi }}}}}}}}}^{{{{{{{{\rm{ind}}}}}}}}}=-\mathop{\sum}\limits_{\begin{array}{c}i,\, j\\ i\ne j\end{array}}{{{{{{{{\boldsymbol{\Pi }}}}}}}}}_{i}\cdot {{{{{{{{\bf{T}}}}}}}}}_{ij}\cdot {{{{{{{{\boldsymbol{\Pi }}}}}}}}}_{j},$$, \({{{{{{{{\bf{E}}}}}}}}}_{ij}={\sum }_{a\in \{{{{{{{{\rm{M,H1,H2}}}}}}}}\}}{Q}_{ja}{{{{{{{{\bf{r}}}}}}}}}_{ja,i{{{{{{{\rm{O}}}}}}}}}/{r}_{ja,i{{{{{{{\rm{O}}}}}}}}}^{3}\), $${{{{{{{\boldsymbol{\mu }}}}}}}}=\mathop{\sum}\limits_{i}{{{{{{{{\boldsymbol{\mu }}}}}}}}}_{i}+{{{{{{{{\boldsymbol{\mu }}}}}}}}}^{{{{{{{{\rm{ind}}}}}}}}},$$, $${{{{{{{\boldsymbol{\Pi }}}}}}}}=\mathop{\sum}\limits_{i}{{{{{{{{\boldsymbol{\Pi }}}}}}}}}_{i}+{{{{{{{{\boldsymbol{\Pi }}}}}}}}}^{{{{{{{{\rm{ind}}}}}}}}}.$$, $${I}^{{{{{{{{\rm{IR}}}}}}}}}(\omega )\propto -\omega {{{{{{{\rm{Im}}}}}}}}\int\nolimits_{0}^{\infty }\langle {{{{{{{\boldsymbol{\mu }}}}}}}}(t)\cdot \dot{{{{{{{{\boldsymbol{\mu }}}}}}}}}(0)\rangle {e}^{-{t}^{2}/2{\sigma }_{t}^{2}}{e}^{-i\omega t}dt,$$, $${I}^{{{{{{{{\rm{Raman}}}}}}}}}(\omega )\propto -A(\omega ){{{{{{{\rm{Im}}}}}}}}\int\nolimits_{0}^{\infty }\langle {{{{{{{\rm{Tr}}}}}}}}[{{{{{{{\boldsymbol{\beta }}}}}}}}(t)\cdot \dot{{{{{{{{\boldsymbol{\beta }}}}}}}}}(0)]\rangle {e}^{-{t}^{2}/2{\sigma }_{t}^{2}}{e}^{-i\omega t}dt,$$, \({{{{{{{\boldsymbol{\beta }}}}}}}}(t)={{{{{{{\boldsymbol{\Pi }}}}}}}}(t)-{{{{{{{\bf{I}}}}}}}}{{{{{{{\rm{Tr}}}}}}}}[{{{{{{{\boldsymbol{\Pi }}}}}}}}(t)]/3\), \(A(\omega )={[1-\exp (-\beta \hslash \omega )]}^{-1}\), \(f({t}_{1},{t}_{2})=\exp (-{({t}_{1}+{t}_{2})}^{12}/{\tau }_{12})\), \({{{{{{{{\rm{[length]}}}}}}}}}^{3}\times {{{{{{{{\rm{[dipole]}}}}}}}}}^{2}/{{{{{{{{\rm{[energy]}}}}}}}}}^{2}\equiv {e}^{2}{a}_{0}^{5}/{E}_{{{{{{{{\rm{h}}}}}}}}}^{2}\), https://doi.org/10.1038/s41467-023-37667-7.
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