Tutorials in vibrational sum frequency generation spectroscopy. III. Collecting, processing, and analyzing vibrational sum frequency generation spectra

At this point in this Tutorial series, we have understood the foundations of BB-VSFG, seen how a spectrometer is constructed, and discussed data collection. The final step of the journey is to understand how we can read and analyze a BB-VSFG spectrum. While this is too broad a topic to give a comprehensive treatment to in this article (and what you interpret from your VSFG spectra is the science you do), we aim to give a general introduction—discussing things such as qualitative interpretation, spectral fitting, and briefly encountering orientational analysis. As we will shortly see, the overriding message is that VSFG spectra can be very complex to interpret, and we advise caution in interpretation.

A. Qualitative interpretation: Reading your VSFG spectrum

Let us initially consider a qualitative interpretation—reading the spectrum by eye. Reading a VSFG spectrum is, initially, much like reading an IR or a Raman spectrum. We have the same axes (energy and intensity), and peaks at certain frequencies show (essentially) the presence of chemical bonds vibrating at that frequency in your system. However, VSFG spectra can show dramatically different information than the IR or Raman spectra of the same substance in the same spectral region. This is most classically illustrated by comparing the IR spectrum of water with the VSFG spectrum of water in the same region.

Figure 4 [panel (a1)] shows a transmission IR spectrum of water, between 3000 and 3800 cm−1. The broad band centered at 3400 cm−1 is characteristic of the hydrogen bonded O–H stretches in bulk water. In contrast, Fig. 4 [panel (a2)] shows a VSFG spectrum of water—which looks markedly different to the IR absorption spectrum. The broad band at 3400 cm−1 is still present (reflecting the O–H stretches of hydrogen bonded water molecules at the interface), but there is now a sharp peak present at around 3700 cm−1. This peak corresponds to the free O–H, or “dangling O–H,” which is the O–H bond that sticks out of the water surface into the air. This bond is not hydrogen bonded to other molecules, and, thus, the vibrational frequency is not red-shifted due to hydrogen bonding. This peak is not visible in the transmission IR spectrum due to the much larger number density of bulk water molecules that contribute to the spectrum—but in the VSFG spectrum, the contribution from these bulk water molecules is zero due to the symmetry of the χ(2) tensor. A subtle but important point is that the spectrum shown in panel (a2) is best thought of as a VSFG spectrum of the entire air-water system (which only shows contributions from the interface due to symmetry), than a spectrum purely of the air-water interface. This distinction becomes important when either of the two bulk phases are not centrosymmetric (e.g., chiral liquids), when there are quadrupolar contributions,1212. S. Sun, J. Schaefer, E. H. G. Backus, and M. Bonn, J. Chem. Phys. 151, 230901 (2019). https://doi.org/10.1063/1.5129108 or when there are ions present in the water phase that generate a strong local electric field at the interface, which can produce a contribution to the VSFG spectrum through the χ(3) tensor.xii

xiiThis will be a four-wave mixing process–the input fields being the IR, VIS, and the local electric field, and the output being the SF output modulated by addition of the local field from the interface.

,1313. P. E. Ohno, H.-F. Wang, and F. M. Geiger, Nat. Commun. 8, 1032 (2017). https://doi.org/10.1038/s41467-017-01088-0

We see from the preceding example that VSFG spectra can be interpreted in a similar way to any IR or Raman spectrum. However, we stated in Paper I in this series that there are several phenomena that can complicate the analysis of VSFG spectra. Here, we discuss these further and provide some guidance for dealing with them.

That VSFG is a coherent spectroscopy has a significant impact on the appearance and interpretation of spectra, and we refer the reader to Sec. II D of Ref. 22. J. D. Pickering, M. Bregnhøj, A. S. Chatterley, M. H. Rasmussen, S. J. Roeters, K. Strunge, and T. Weidner, Biointerphases 17, 011202 (2022). https://doi.org/10.1116/6.0001403 for an initial discussion of the fundamentals of how coherence manifests in spectra. Coherence causes oscillators that are close in energy to interfere with one another, and this can drastically alter lineshapes and apparent peak positions. This can mean that the position of a peak in the spectrum does not necessarily coincide with the central frequency of the underlying oscillator. In a similar vein, the presence of a strong nonresonant backgroundxiii

xiiiOften nonresonant backgrounds are small on dielectric substrates like water that we consider here but can be very substantial on metal substrates.

can cause line shape distortion and peak shifting due to the interference of the resonant peaks with the nonresonant background. All of this raises a clear question—if, for example, a peak in a VSFG spectrum is shifted away from the corresponding peak in a bulk IR spectrum is the shift due to:1.

Different chemical environments/structures at the interface, like the appearance of the dangling O–H in water spectra?

2.

Interference between neighboring peaks that cause the apparent position to shift?

3.

Interference between the resonant peak and a nonresonant background that cause the apparent position to shift?

There is no universal answer to this question, and it depends entirely on the interface under study. Clearly, option 1 is most interesting to us as surface scientists, but we must discount options 2 and 3 if we are to make this conclusion. Do you expect that there are nearby oscillators with a similar frequency to your observed peak that can interfere with each other? Or do you expect that the chemical environment of the oscillator at the interface is different to the bulk? It is often necessary to have other information, besides the VSFG spectrum, if you want to fully understand your interface. VSFG alone can provide information, but it is much more powerful if used in conjunction with other techniques. A glance at the literature will show that VSFG is often paired with techniques, such as ellipsometry,1414. T. A. Dramstad, Z. Wu, and A. M. Massari, J. Chem. Phys. 156, 110901 (2022). https://doi.org/10.1063/5.0076252 surface tension measurements,1515. Y. Rao, X. Li, X. Lei, S. Jockusch, M. W. George, N. J. Turro, and K. B. Eisenthal, J. Phys. Chem. C 115, 12064 (2011). https://doi.org/10.1021/jp201799z Brewster angle microscopy,16,1716. E. H. G. Backus, D. Bonn, S. Cantin, S. Roke, and M. Bonn, J. Phys. Chem. B 116, 2703 (2012). https://doi.org/10.1021/jp207454517. E. M. Adams, A. M. Champagne, J. B. Williams, and H. C. Allen, Chem. Phys. Lipids 208, 1 (2017). https://doi.org/10.1016/j.chemphyslip.2017.08.002 NMR,1818. T. Weidner, N. F. Breen, K. Li, G. P. Drobny, and D. G. Castner, Proc. Natl. Acad. Sci. U.S.A. 107, 13288 (2010). https://doi.org/10.1073/pnas.1003832107 FT-IR,1919. B. Ding, J. Jasensky, Y. Li, and Z. Chen, Acc. Chem. Res. 49, 1149 (2016). https://doi.org/10.1021/acs.accounts.6b00091 near edge x-ray absorption fine structure spectroscopy,2020. T. Weidner, J. S. Apte, L. J. Gamble, and D. G. Castner, Langmuir 26, 3433 (2010). https://doi.org/10.1021/la903267x and x-ray photoelectron spectroscopy,21,2221. T. Weidner and D. G. Castner, Annu. Rev. Anal. Chem. 14, 389 (2021). https://doi.org/10.1146/annurev-anchem-091520-01020622. T. Weidner and D. G. Castner, Phys. Chem. Chem. Phys. 15, 12516 (2013). https://doi.org/10.1039/c3cp50880c among others. Our advise would be to be wary of interpreting too much from a VSFG spectrum without other information to help guide your conclusions.

The preceding discussion may give the impression that we are very limited in what we can interpret from VSFG spectra on their own—this is not quite true, although we do advise prudence. We stated in the first article in this series that the advantages of VSFG come from being able to access and measure the full χ(2) tensor. To do this requires some more quantitative interpretation—and in a homodyne experiment, this requires that we fit the VSFG spectra to obtain the underlying parameters.

B. Quantitative interpretation: Fitting VSFG spectra

Our fundamental aim in VSFG is to determine χ(2) for our system and, thus, learn something about the structure of our interface. However, in a homodyne BB-VSFG experiment, we know that we do not directly measure χ(2), but the square modulus: |χ(2)|2. This difference is significant because χ(2) is a complex quantity and so taking the square modulus inherently discards information about the phase of χ(2). Recall from Paper I in this series that χ(2)(ω)=χNR(2)+χR(2)(ω)=|χNR(2)|eiϕ+|χR(2)(ω)|eiθ(ω),(1)where χNR(2) refers to the nonresonant contribution to the VSFG signal (which has no intrinsic frequency dependence), and χR(2)(ω) refers to the resonant contribution, which is much more strongly frequency dependent and contains the information about the structure of our interface. Recall also that χR(2)(ω)=|χR(2)(ω)|eiθ(ω)=Aω−ω0+iΓ,(2)where A is the strength of the vibrational mode being studied, ω0 is the central frequency of the resonance, and Γ is the linewidth of the resonance. Taking the modulus squared of Eq. (2) results in a Lorentzian function with half-width at half-maximum linewidth Γ and peak height A2Γ2, as discussed previously. It is also clear that taking the modulus squared of Eq. (2) will remove the term in θ(ω). However, taking the modulus squared of Eq. (1) will not completely remove the phase terms, as the presence of the nonresonant term means that there will be an interference between the resonant (θ) and nonresonant (ϕ) phases, such that |χ(2)(ω)|2=|χNR(2)|2+|χR(2)(ω)|2+2|χNR(2)||χR(2)(ω)|cos[θ(ω)−ϕ].(3)Equation (3) is true for a single resonance, but in practice, there may be several resonances in one spectrum such that χ(2) can be expressed as χ(2)(ω)=χNR(2)+∑jχR,j(2)(ω).(4)For a spectrum with j resonances. Each of these resonant terms has a phase θj, which will interfere with the nonresonant phase ϕ, and with the other resonant phases of the resonances near resonance j. This leads to a complex expression for the total χ(2) with many interference terms that depend on both ϕ−θj and θi−θj, where i≠j. The fundamental point here is that while we cannot resolve the individual phases in a homodyne VSFG experiment (we cannot uniquely measure θ and ϕ, only their relative differences), we are sensitive to the differences between phases due to these interference terms in χ(2). Thus, the phase of each resonance cannot simply be ignored in homodyne VSFG, even though it cannot be measured. It is important to note that the phase θj is the intrinsic phase of the resonance χR(2) (the intrinsic phase of the Lorentzianxiv

xivGoing through the algebra to find θ will reveal that it is determined solely by the amplitude A and linewidth Γ of the underlying Lorentzian—it is not an extra free parameter by itself.

). Additional phase factors (eiψ) are sometimes added to χR(2) to simulate the effect of different oscillators “pointing” in different directions relative to each other—this will be discussed further later.The best way to determine the full χ(2) tensor is to perform a heterodyned VSFG measurement, where the phase is directly determined by comparison to a local oscillator with a known phase.23–2523. M. Thämer, R. Kramer Campen, and M. Wolf, Phys. Chem. Chem. Phys. 20, 25875 (2018). https://doi.org/10.1039/C8CP04239J24. S. Yamaguchi and T. Otosu, Phys. Chem. Chem. Phys. 23, 18253 (2021). https://doi.org/10.1039/D1CP01994E25. Y. R. Shen, Annu. Rev. Phys. Chem. 64, 129 (2013). https://doi.org/10.1146/annurev-physchem-040412-110110 However, heterodyne VSFG spectrometers are more complex to build (and the data slightly more complex to process) than homodyne VSFG experiments, so homodyne experiments are still more widely used. A common approach to the problem of determining χ(2) from a homodyne VSFG spectrum is to fit the VSFG spectrum using Eq. (4), and an appropriate nonlinear least squares fitting algorithm. We have fitted our spectrum in panel (b2) in Fig. 2, with three resonant peaks to illustrate how a typical fit looks. These peaks correspond to the symmetric stretch of the CH2 group (2840 cm−1—shoulder), the symmetric stretch of the CH3 group (2880 cm−1), and a Fermi resonance of the CH3 group (2940 cm−1). Fitting parameters are tabulated in the Appendix (see Table I in Appendix). As would be expected from a dielectric substrate (pure water in this instance) and relatively strongly ordered system that gives a strong resonant response (stearic acid), the amplitude of the nonresonant background was ∼1000 times lower than the resonant amplitudes and, thus, does not impact the spectrum severely in this case. More detail on the specifics of fitting VSFG spectra is provided in Lambert’s excellent tutorial review2626. A. G. Lambert, P. B. Davies, and D. J. Neivandt, Appl. Spectrosc. Rev. 40, 103 (2005). https://doi.org/10.1081/ASR-200038326 and is not reproduced here, but we will make some general points regarding the reliability and interpretation of fitted homodyne VSFG spectra.Table iconTABLE I. Fitting parameters for the fitted spectrum shown in Fig. 2 panel (b2).ParameterResonance 1Resonance 2Resonance 3ω0 (cm−1)2837 (11.5)2883 (1.41)2942 (2.77)Γ (cm−1)19.9 (11.9)9.7 (1.29)9.9 (2.65)A (a.u.)−1.02 (0.59)1.11 (0.19)0.68 (0.16)Nonresonant A (a.u.)−0.009 (0.004)Nonresonant phase2.35 (1.28)The first thing to decide upon when fitting a VSFG spectrum is the number of resonances to use in the fit—this is not necessarily as simple as counting the number of visible peaks in the spectrum due to interference effects and the possibility of closely spaced resonances to be unresolved due to the resolution of the upconversion pulse. When this has been decided, each resonance has three parameters that can be varied—the amplitude (A), linewidth (Γ), and central frequency (ω0). Additionally, the nonresonant background is fitted with an amplitude and phase.xv

xvThough the amplitude may be very small for typical experiments on dielectric substrates, such as the air-water interface.

This means that there are at least five parameters that can be varied, and a spectrum containing three peaks will have 11 parameters that can be independently varied to perform the fit. This results in a very large parameter space to explore, and it can be challenging to be sure that the global minimum has been found2727. I. Sgura and B. Bozzini, Int. J. Non-Linear Mech. 40, 557 (2005). https://doi.org/10.1016/j.ijnonlinmec.2004.03.004—especially if spectra are noisy and peaks start to overlap (with interference effects becoming significant). As such, it is often the observation of people doing VSFG analysis that there are several sets of parameters that are observed to “fit” the experimental spectrum and that the initial guesses fed to the algorithm can have a significant bearing on the final result.2828. B. Busson and A. Tadjeddine, J. Phys. Chem. C 113, 21895 (2009). https://doi.org/10.1021/jp908240d Indeed, this observation is backed up by the observation by Le Rille2929. A. Le Rille and A. Tadjeddine, J. Electroanal. Chem. 467, 238 (1999). https://doi.org/10.1016/S0022-0728(99)00047-9 that in a homodyne experiment, there can be more than one χ(2) that will give rise to the same |χ(2)|2. The non-uniqueness of the fitting parameters is established in the literature, and Busson and Tadjeddine propose an alternative method for extraction of χ(2), by considering all sets of parameters that could give rise to a given spectrum, and then using prior knowledge of the system chemistry to find the set that is most likely the “true” set.2828. B. Busson and A. Tadjeddine, J. Phys. Chem. C 113, 21895 (2009). https://doi.org/10.1021/jp908240d Here, we arrive at a familiar message—the homodyne BB-VSFG spectrum alone is often not sufficient to fully characterize the chemistry of the interface, and complementary information is required to build a full picture.

The large parameter space that needs to be explored raises another question—how should we define initial guesses for various parameters when doing the fit? And should we fix certain parameters to narrow down the parameter space? This is a common weapon in the VSFG fitters’ arsenal. For example, if it is known that the spectral linewidths are all limited by the bandwidth of the VIS pulse, then fixing the linewidth to this value is a sensible course of action. Similarly, if it is known that the resonant frequencies of the observed peaks are the same as those in the bulk, then they can be fixed to a value obtained from, e.g., a high-resolution FT-IR spectrum. For measurements taken on the same substrate on the same spectrometer, then the nonresonant phase can also be fixed to allow a more insightful comparison between different spectra. It is also often observed that the sign of the resonant amplitude is fixed for certain resonances in certain regions—the argument being that the relative directions the transition dipole moments of different resonances point in are fixed relative to the surface coordinates (however, a look into the literature will reveal that there is no real agreement as to how these should be fixed). In any case, it is clearly often useful to have more information about your spectrum that can be used to guide the fitting process.

A particular issue that arises in VSFG of species on water subphases is what to do when trying to fit the spectrum of the water itself. How best to fit water spectra is still an active area of research, and we do not propose to enter into a lengthy discussion here. There can be a strong χ(3) response from water, which will contain effects from salts and charged species at the interface, which will then be added into Eq. (4), with its own phase term. Clearly, uniquely fitting this in addition to all the χ(2) terms in a homodyne spectrum is not possible. Our view here is that fitting the water spectra is best avoided unless you are very clear on what you are doing, and on what you hope to get out of the analysis, as meaningfully interpreting a fitted water spectrum is not as straightforward as interpreting other VSFG spectra.Finally, we wish to discuss the use of additional phase factors (eiψ) in modeling VSFG spectra. The purported use of this is to simulate how different resonances can have different relative phases (i.e., in addition to θ, the intrinsic phase of the Lorentzian). These are then included in the description of χR(2),The idea being that if two resonances, χR,1(2) and χR,2(2) are, for example, 180° out of phase, then ψ1−ψ2=π. This would be equivalent to generalizing the definition of A so that it can now be a complex number. Inclusion of ψ adds another fitting parameter and another phase factor that will interfere with the nonresonant phase and other resonances to produce the final spectrum. Many of the line shapes that can be produced by the addition of this extra phase factor can also be produced by varying other parameters—and we would advise extreme caution in using this for spectral fitting, as at some point the parameter space expands so much that the reliability of any individual set of parameters is questionable.xvi

xviOur experience is that is often all too easy in VSFG fitting to include additional peaks, factors, and constants (often with tenuous or very limited physical justification) to make the fits look better. More parameters do not automatically mean a fit is less reliable—but it is important to ensure that fitting parameters used are consistent with the signal—to—noise and resolution in the experiment and can be justified physically.

Be very sure that if you choose to do this that you are able to justify it physically.

It may seem from the preceding discussion that we are being rather negative about how useful VSFG fitting is, but this is not our intention. While it can be challenging to determine accurate and detailed chemical information by fitting a single VSFG spectrum, often a much more fruitful and reliable strategy is to compare several VSFG spectra when physical parameters (such as pH, temperature, concentrations, etc.) are changed. In this way, you can look for systematic changes in the VSFG spectra as the interfacial chemistry changes and are less susceptible to being tricked by the fitting process. For instance, if you measure a VSFG spectrum as a function of temperature, and see that one fitting parameter is consistently changing as you cool down the sample, while all others remain constant, then you can infer much more reliable chemical information, than you could from the value of that one parameter from a single spectrum. Interpreting relative changes between spectra recorded on the same spectrometer is much more reliable than comparing absolute fitting parameters between your spectra and other published spectra.

An essential exercise for anyone starting to fit VSFG spectra is, in our view, to turn the process around and try to generate simulated spectra using Eq. (1). In this way, you can learn how different resonances interfere with one another, and what effect varying different parameters will have on the final spectrum. This will give a clearer picture of how reliable fitting results are. It is also very useful to use a fitting algorithm that will show the uncertainty (and other statistics) associated with each fitted parameter—we have found the Python library LMFIT (Ref. 3030. M. Newville, T. Stensitzki, D. B. Allen, and A. Ingargiola, “LMFIT: Non-linear least-square minimization and curve-fitting for python” (2014).) to be very good in this regard. Having the uncertainties allows you to both get a feel for how precise the parameters are and propagate the uncertainties through further analysis and obtain an accurate measure of the uncertainty associated with a calculated property (such as the tilt angle of a species from the interface) from the fit parameters.

C. Further interpretation of VSFG spectra

When values for χ(2) have been extracted from spectra (via fitting or otherwise), then there are other analyses that can be performed to discover more about the chemistry of the interface. A very common analysis on biological interfaces is analyzing the tilt angles of species at the interface by considering ratios of χ(2) components from different polarization combinations31,3231. G. Ma and H. C. Allen, Langmuir 22, 5341 (2006). https://doi.org/10.1021/la053522732. H.-F. Wang, W. Gan, R. Lu, Y. Rao, and B.-H. Wu, Int. Rev. Phys. Chem. 24, 191 (2005). https://doi.org/10.1080/01442350500225894 or by considering amplitude ratios of specific peaks within a single spectrum (this is commonly done using CH2 and CH3 peaks to determine alkyl chain ordering—see Ref. 2626. A. G. Lambert, P. B. Davies, and D. J. Neivandt, Appl. Spectrosc. Rev. 40, 103 (2005). https://doi.org/10.1081/ASR-200038326 for an instructive tutorial review). These methods were first developed for self-assembled monolayers of alkane thiols on gold surfaces but have been widely used for analyzing orientations of fatty acids at water surfaces and proteins at membrane surfaces.21,33–3521. T. Weidner and D. G. Castner, Annu. Rev. Anal. Chem. 14, 389 (2021). https://doi.org/10.1146/annurev-anchem-091520-01020633. T. W. Golbek, M. Padmanarayana, S. J. Roeters, T. Weidner, C. P. Johnson, and J. E. Baio, Biophys. J. 117, 1820 (2019). https://doi.org/10.1016/j.bpj.2019.09.01034. S. Hosseinpour, S. J. Roeters, M. Bonn, W. Peukert, S. Woutersen, and T. Weidner, Chem. Rev. 120, 3420 (2020). https://doi.org/10.1021/acs.chemrev.9b0041035. Z. Chen, Biointerphases, 17, 031202 (2022). https://doi.org/10.1116/6.0001859 In using a method like this with a fitted χ(2) tensor, it is essential to have a grasp on the uncertainties associated with the fit—and to propagate these through the orientation analysis such that the uncertainty of the final tilt angle(s) can be ascertained. It is also highly advisable to understand the assumptions associated with the underlying theory and to decide whether or not these assumptions are appropriate for your specific system—someone else having previously done the analysis on a similar system does not necessarily mean it is appropriate. A more thorough review of these concepts is provided by Wang et al.3232. H.-F. Wang, W. Gan, R. Lu, Y. Rao, and B.-H. Wu, Int. Rev. Phys. Chem. 24, 191 (2005). https://doi.org/10.1080/01442350500225894

Thus far, we have described ways to analyze VSFG spectra that rely solely on manipulation of experimental data to extract parameters that can be related to the underlying chemistry of the interface. An alternative approach to extract useful chemical information from VSFG spectra is to calculate VSFG spectra. The fundamental idea is common across spectroscopy—calculate a spectrum based on an interfacial molecular simulation and compare this with your experimental spectrum. Once an acceptable match between the experimental and calculated spectra is reached, then the assumption is that the underlying simulated interfacial structure is correct, and then information such as the phase of different oscillators and molecular tilt angles can be readily accessed from the simulation. A variety of approaches to the spectral calculation for VSFG have been developed, the applicability of which depends on the exact interface being studied, and, here, we briefly recount some common methods and provide suggestions for further reading. We focus primarily on the calculation of spectra of proteins at interfaces as an example, but many methods are more widely applicable.

One common method in the calculation of VSFG spectra of proteins at interfaces is to map the protein structure to a VSFG spectrum, where the vibrational frequencies of the protein at the interface are based on the gas-phase frequencies modulated by effects such as hydrogen bonding and inter-residue coupling.36,3736. J. K. Carr, L. Wang, S. Roy, and J. L. Skinner, J. Phys. Chem. B 119, 8969 (2015). https://doi.org/10.1021/jp507861t37. S. J. Roeters, C. N. van Dijk, A. Torres-Knoop, E. H. G. Backus, R. K. Campen, M. Bonn, and S. Woutersen, J. Phys. Chem. A 117, 6311 (2013). https://doi.org/10.1021/jp401159r This “frequency mapping” approach was derived from techniques developed for simulated IR spectra38–4338. C. R. Baiz et al., Chem. Rev. 120, 7152 (2020). https://doi.org/10.1021/acs.chemrev.9b0081339. Y. S. Lin, J. M. Shorb, P. Mukherjee, M. T. Zanni, and J. L. Skinner, J. Phys. Chem. B 113, 592 (2009). https://doi.org/10.1021/jp807528q40. T. C. Jansen and J. Knoester, J. Chem. Phys. 124, 044502 (2006). https://doi.org/10.1063/1.214840941. M. Reppert and A. Tokmakoff, J. Chem. Phys. 138, 134116 (2013). https://doi.org/10.1063/1.479893842. T. Hayashi and S. Mukamel, J. Phys. Chem. B 111, 11032 (2007). https://doi.org/10.1021/jp070369b43. J.-H. Choi and M. Cho, J. Chem. Phys. 134, 154513 (2011). https://doi.org/10.1063/1.3580776 and is very efficient—it can easily be averaged over a varied interfacial system. The drawback is that it is difficult to develop, and the results can be difficult to validate. A fundamentally different approach is being developed, where the vibrational frequencies are obtained from dynamical simulations (such as classical molecular dynamics, ab initio dynamics, quantum mechanics/molecular mechanics, or ring polymer dynamics). The vibrations are then monitored as a function of time to generate a time-correlation function, the Fourier transform of which results in a calculated VSFG spectrum.44,4544. G. R. Medders and F. Paesani, J. Am. Chem. Soc. 138, 3912 (2016). https://doi.org/10.1021/jacs.6b0089345. F. Tang, T. Ohto, S. Sun, J. R. Rouxel, S. Imoto, E. H. G. Backus, S. Mukamel, M. Bonn, and Y. Nagata, Chem. Rev. 120, 3633 (2020). https://doi.org/10.1021/acs.chemrev.9b00512 However, a key point here is that it is not always necessary to calculate a VSFG spectrum to gain useful information about your system. For example, in the case of MD simulations, it can be quite challenging to generate an accurate VSFG spectrum from the simulation. However, other useful information (such as molecular orientations and density profiles at the interface) can be extracted without calculating a full VSFG spectrum. Such information can be very helpful in interpreting spectra and more often readily obtained without the care needed to produce an accurate spectral simulation.

Computational approaches to describe interfaces (in combination with experimental VSFG spectra) have gained a lot of traction recently, as modern computational power and algorithms allow the description of large interfacial systems while maintaining molecular-level resolution. The development of computational techniques to calculate VSFG spectra is very active and in the future will likely allow even more information to be extracted from the experimental VSFG spectra.

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