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Time-Resolved Sum Frequency Generation Spectroscopy: A Quantitative Comparison Between Intensity and Phase-Resolved Spectroscopy Ellen H.G. Backus, Jenée D Cyran, Maksim Grechko, Yuki Nagata, and Mischa Bonn J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b12303 • Publication Date (Web): 12 Feb 2018 Downloaded from http://pubs.acs.org on February 13, 2018

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The Journal of Physical Chemistry

Time-Resolved Sum Frequency Generation Spectroscopy: a Quantitative Comparison Between Intensity and Phase-Resolved Spectroscopy

Ellen H.G. Backus*, Jenée D. Cyran, Maksim Grechko, Yuki Nagata, and Mischa Bonn* Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany *email: [email protected]. [email protected]

Abstract Time-resolved and two-dimensional sum frequency generation (TR-SFG and 2D-SFG) spectroscopies are promising tools in the experimental study of molecular dynamics, specifically at interfaces. Most implementations of TR/2D-SFG spectroscopy rely on a pump-probe scheme, where an excitation pulse excites a fraction of interfacial molecules into the first excited state of a specific vibrational mode, and a subsequent SFG probe pair detects the time-dependent changes of the surface vibrational response. In steady state SFG spectroscopy, phase-resolved detection (also known as heterodyne-detection), as opposed to SFG intensity measurements, has been shown to be useful in unraveling the steady-state response of interfacial vibrations. Here, we explore the merits of phaseresolved TR/2D-SFG spectroscopy. This purely theoretical and numerical study reveals that for a typical response from aqueous interfaces, the intensity 2D-SFG measurements contain the same information content as phase-resolved 2D-SFG measurements. We specifically analyze the frequency-dependence of the bleach lifetime (reflecting vibrational relaxation), and the timedependent slope of the on-diagonal features observed in a 2D spectra. We show that for different systems, the intensity-based and phase-resolved 2D-SFG measurements provide the same information and are quantitatively very similar. We investigate the effect of different lineshapes, anharmonicity, and non-resonant signal contributions, and show that none of these effects substantially change the conclusion that intensity-based and phase-resolved 2D-SFG measurements provide equivalent information.

Introduction Infrared-visible surface sum-frequency generation (SFG) spectroscopy has been instrumental in advancing our understanding of molecular arrangement and molecular phenomena at surfaces and interfaces. The success of SFG stems from the ability to provide specifically the vibrational response of molecules at the interface. More precisely, SFG reports on the region for which inversion symmetry is broken. For bulk water exposed to air, i.e. the water-air interface, this is a relatively thin layer of a few Angstroms;1 for water at charged interfaces, the range over which the symmetry is broken can amount to many nanometers.2–5 Generally, the frequency-dependent SFG intensity is proportional to the square of the effective second-order non-linear response, : 1 ACS Paragon Plus Environment

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∝

1

The overall effective response can contain, in addition to resonant terms, i.e. , =

∑ , , also a non-resonant term ( , ), as well as possible contributions from the third-

order nonlinear optical response . A detailed discussion about the different contributions can be found in Ref. 2,3 and is outside the scope of the present manuscript. In the vicinity of a resonance,

the response from a single oscillator n is a complex quantity, typically approximated by a Lorentzian lineshape:

, =

2

with amplitude , center frequency and linewidth Γ for oscillator n. Access to the separate real

and imaginary parts of , or, equivalently, the amplitude and phase of the overall response, is relevant for determining the precise amplitude and sign of , and clarifying the effect of possible interference between different modes on the intensity spectrum. The sign of the peak reflects the absolute orientation of the transition dipole moment of the studied vibrational transition, and can thus be used to determine absolute molecular orientation (‘up’ or ‘down’) at the interface. The imaginary part of the overall , , Im , , is equivalent to the bulk absorption

coefficient, and is additive in terms of its different contributions, as opposed to , which also contains the cross terms of different resonant contributions and of the resonant and nonresonant contributions. As such, several approaches have been developed to determine the complex , rather

than . The phrases ‘phase-resolved’ and ‘heterodyne-detected’ SFG have been used to indicate approaches that provide both the amplitude and phase, or, equivalently, the real and imaginary parts, of . The term ‘heterodyne-detected’ SFG has been introduced specifically to describe broadband SFG spectroscopy; for scanning frequency SFG spectroscopy, historically the term ‘phase-resolved’ is used. Both ‘phase-resolved’ and broadband ‘heterodyne-detected’ SFG rely on interfering the surface SFG signal with a signal at the same frequency of known phase, so that the phase of the surface SFG signal can be determined. Although determining only the sign of Im for dominating spectral features using phase-resolved detection is relatively straightforward, it has proven more challenging to elucidate details of the spectral shape of Im in an unambiguous manner. This problem is related to the experimental challenges associated with determining the reference phase reliably, and is well illustrated by the recent discussion of the presence of – or lack thereof– a positive peak in the SFG response of the O-H stretch mode at the water-air interface at low frequencies.6–11 Furthermore, the spectral shape of Im can appear quite broad and featureless. In this case, reliably determining the exact phase of the SFG response with a small signalto-noise ratio is very challenging. This complication presents itself for the H-O-H bending SFG spectra.12–15 More recently, time-resolved (TR)16–18 and two-dimensional (2D)19–22 versions of SFG spectroscopy have been developed, including phase-resolved 2D-SFG spectroscopy.21–26 In these TR spectroscopies, an additional pulse is added to the pulse scheme to excite a significant fraction of oscillators to their first excited vibrational state. Both the spectral width of the response (reflecting the (in-)homogeneous linewidth) as well as the spectral relaxation time (reflecting spectral diffusion 2 ACS Paragon Plus Environment

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and vibrational relaxation) can in principle be obtained from these experiments. Such data can provide important new physical insights into interfacial water. The excitation-frequency dependent lifetime, for instance, can report directly on aqueous heterogeneity at different water interfaces.27 For TR and 2D-SFG spectroscopies, it has been claimed that only phase-resolved or, heterodyne-detected SFG spectroscopy can provide the ‘true’ rate of spectral diffusion and/or the ‘true’ vibrational dynamics.22,24,28,29 Along these lines, a previous theoretical study demonstrated that intensity 2D-SFG spectra can indeed be complicated by the interference of different contributions to the overall SFG spectral line shape.30 Nonetheless, phase-resolved 2D-SFG data on systems that had been previously studied with intensity 2D-SFG measurements – specifically aqueous interfaces – have not provided new physical insights in these systems that could not be obtained from the intensity 2DSFG-spectra. Here, we show, using simple numerical simulations, that for a typical response from aqueous interfaces, the intensity 2D-SFG measurements contain the same information content as phaseresolved 2D-SFG measurements. We specifically analyze the frequency-dependence of the bleach lifetime (reflecting vibrational relaxation), and the time-dependent slope of the on-diagonal features observed in the 2D spectra. We show that for different systems, the intensity-based and phaseresolved 2D-SFG measurements provide the same information and are quantitatively very comparable. We investigate the effect of different lineshapes, anharmonicity, and non-resonant signal contributions on the 2D-SFG spectra, and show that none of these effects substantially change the conclusion that intensity-based and phase-resolved 2D-SFG measurements provide equivalent, and quantitatively similar information. Moreover, we also illustrate the limitations of intensity 2DSFG experiments. For a system with a second order nonlinear response containing several narrow, overlapping resonances with amplitudes of opposite sign, phase-resolved experiments might be the better choice, as artificial cross peaks might appear in the intensity spectrum.

Theory General Expression of Differential Spectra The steady state SFG intensity is given by equation 1. For an intensity-based differential measurement, the TR signal reads:

∆ ! ∝ "#$% !" −

(3)

where ∆ is the time-dependent change in intensity of the SFG signal, #$% ! = + ∆ ! is

the second order nonlinear susceptibility with the excitation pulse and is the second order nonlinear susceptibility in absence of the excitation pulse; ∆ ! is the time-dependent excitationpulse-induced change in the second order nonlinear susceptibility. As mentioned above, phase-resolved detection enables the measurement of the imaginary

part of , Im[ ], which provides information on the absolute orientation of interfacial molecules. The TR phase-resolved spectrum is obtained from the difference between the spectra in presence and absence of the excitation pulse: 3 ACS Paragon Plus Environment


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∆*[ !] = Im +#$% !, − Im = Im∆ !.

(4)

It is important to note that the intensity detection and phase-resolved detection techniques can result in significantly different shapes of the observed 2D spectra. The intensity of the spectral features varies based on the detection method. In order to understand the discrepancies in intensity, we will use below the response function formalism to present the polarizations for the different signal contributions following ref 31. We omit the time dependence for simplicity. For the SFG signal with no excitation, the polarization is

./01/ 233 ∝ −401 4vis 49:; ? , 4EF , and 4BCD are the transition dipole moments associated with the IR excitation between ground (0) and the first vibrational excited state (1), visible pulse excitation between electronic ground state and virtual state, and the SFG emission from the virtual state to the ground state, respectively. Since combined 4GHI and 4JKL go through the electronic excitation, 4GHI 4JKL can be written as the transition polarizability which is based only on the vibrational energy level transition32, resulting in the widely used expression for the SFG response:1,33,34

./01/ 233 ∝ −4>? M>? ? 〈1|2〉 + w1|ŷ |2{ + ⋯

(13)

Since 〈0|1〉 = 〈1|2〉 = 0, we obtain M? = √2M>? approximately. By inserting these relations into the

above equation, we obtain ∆Im[ ] = 0 for the case of the harmonic potential. However, in reality, the ESA has a different frequency than the bleach/SE due to anharmonicity. Thus, two signals are

observed in a pump-probe spectrum. Eq. (11) and the relations 4? = √24>? and M? = √2M>? (i.e. as is commonly done, assuming the harmonic ratio between the transition dipole moment even in the case of the anharmonic oscillator) show that the ESA signal has the same intensity, but opposite sign as the bleach/SE signal. 5 ACS Paragon Plus Environment


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In a SFG intensity experiment, we obtain the following expression for the signal under pump excitation: V

V

V

ijki l ∝ ".mn#o%p + .Bh + .hB + .ijki lqq " ,

(14)

which results in many cross terms. If we assume that the 0-1 transition and the 1-2 transition are well separated and do not overlap in frequency, the cross terms with the two different transitions will be V

zero, like for example the term .hB .SFG,pump off . By substituting equation (6), (7), and (8), the difference between pump on and pump off is V

V

V

ΔBCD ∝ .ESA + 4.bleach + 4.bleach .pump off =

−4>? ? ijki : 4>? ijki − 1

(16)

Also here, the ESA and bleach/SE have opposite signs as the term 4>? ijki is typically much smaller than 1. In this case, the ESA will also be much smaller than the bleach/SE signal. Assume for example that 4>? ijki = 0.1, then the ESA signal is 0.1 while the bleach/SE is 0.9.

A more intuitive picture for the ratio of the bleach/SE and ESA signal could be obtained with the following consideration

mn#o%p/Bh ∝ > − ? ; hB ∝ ? −

(17)

where $ is the population of vibrational level x. If 10% of the population is excited from the ground state to the first excited vibrational state, then ? =0.1, while > =0.9. The intensity is equal for the bleach/stimulated emission and the excited state absorption peaks in the difference spectra for the phase-resolved detection regime since the transition dipole moment associated with the 1-2 transition is as mentioned above a factor of ~√2 larger than that of the 0-1 transition and this transition occurs twice due to the two interactions with the pump pulse. In contrast, in the SFG intensity measurements, the signal is given by the square of . Thus, the 0-1 transition intensity, proportional to > − ? , is reduced from 1 to 0.64 resulting in a difference signal of 0.36, while the 1-2 transition intensity is significantly lower, at [2? ] = 0.04. The ratio between bleach/SE and ESA is thus 9:1 and as such, the peaks corresponding to the decrease of the 0-1 transition dominate the intensity spectrum.

Time Dependence of Differential Spectra Beyond the intensity differences, one expects the vibrational dynamics inferred from a homodyne and phase-resolved experiments to be very similar. This becomes evident when equation (3) is expanded to: 6 ACS Paragon Plus Environment

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∗

∗

∆ ! = ∆ ! + ∆ ! + Δ ! ,

(18)

with the asterisks indicating complex conjugates. Since the last term Δ ! is negligibly small, one can recast equation (20) as: ∆ ! = 2 ∆ ! + 2* * ∆ !.

(19)

This notation shows that the intensity difference spectrum is a simple sum of the real part of the static nonlinear susceptibility multiplied with the real part of the time dependent change and the imaginary part of the static susceptibility multiplied with the imaginary time dependent change. As the real and imaginary parts are related by the Kramers-Kronig relation, the time evolution of the real and imaginary parts are governed by the same vibrational dynamics. Therefore, for the intensity difference measurement, the differential signal is just a ‘heterodyning’ of the dynamics obtained in a phase resolved measurement (eq. 4) with the static, i.e. ground state, spectrum. In other words, the dynamics obtained from an intensity difference spectrum should be similar to the dynamics in a phase-resolved experiment. Therefore, to obtain vibrational lifetimes and spectral diffusion timescales, an intensity experiment will suffice in many cases. Of course, if one is interested in the real and imaginary spectral changes upon excitation, a phase-resolved experiment is the better choice. Even if the real and complex components of the static spectrum are known, extracting

the complex Δ from the differential intensity experiments is ambiguous as equation 19 shows.

Time Dependence for Lorentzian Lineshape Model For the limiting case of one single Lorentzian resonance for mode n on top of a NR signal, neglecting excited state absorption and approximating ∆ ! by ∆ / / − + YΓ , with ∆ ≪ , eq. 3 and 4 turn into: ∆ ! = ∆

/ +

# ∆

/ +

# Ak ∆*[ !].

∆*[

∆

∆ /

!] = *∆ ! =

/ −

Ak ∆

/ = (20) (21)

From equations 20 and 21, it is clear that the dynamics governing the time-variations of the intensity and the phase resolved spectrum are exactly identical, in the limit of low excitation density. The only difference is a time-independent multiplication factor.

Protocol for Model Calculations For more realistic, more complex line shapes, the analytical expressions discussed above are less straightforward. Here, we use a numerical procedure to allow for the direct comparison of the results from intensity measurements with results from phase-resolved measurements. To ensure that we model a realistic, complex situation, we employed dynamics from the model that has been proposed 7 ACS Paragon Plus Environment


recently36 to describe the response of interfacial water at the water (H2O)-air interface as well as the D2O-lipid interface, with positively charged lipids. For the resonant part of the second-order susceptibility, we assume inhomogeneous broadening, with lineshapes consisting of several complex Lorentzians of which the intensity of each vibrational chromophore follows a Gaussian distribution. The imaginary SFG spectrum is calculated from the sum of all Lorentzians and in certain cases a nonresonant contribution described by an amplitude and a phase, while the square of the sum is taken in case of the SFG intensity spectrum. As explained above, for obtaining the 2D SFG response, the spectra in presence and absence of the excitation pulse are subtracted. The spectra in presence of the pump are calculated as follows. For time zero, the overlap integral between each Lorentzian line and an 80 cm−1 FWHM Gaussian excitation pulse at each pump frequency, which are 20 cm-1 spaced, are calculated. To match typical experiments, the overlap integral is normalized to a maximum bleach of 10%. The frequency dependence of the transition dipole moment is taken into account,37,38 where we assume the same frequency dependence for an OD stretch oscillator as for an OH stretch oscillator by scaling the frequency axis for the mass difference. As this frequency dependence is not well defined at high frequencies, we assume a constant transition dipole moment above 2621 cm-1, following Ref 39, and again scaling the frequency by the mass difference. The anharmonicity between the ground state and first excited state is infinite or set to 150 cm−1, which is in agreement with experimental obtained values.40–42 The 1 to 2 transition is in the first case outside our probe region, but will appear at 150 cm-1 lower frequency than the 0 to 1 transition in the last case.

Figure 1: Model used to calculate 2D-SFG spectra. For a given sub-ensemble of OH/OD groups (consider, for instance, the sub ensemble corresponding to the green Lorentzian line), near-resonant energy transfer can occur out of (green arrows) and into (red arrows) that sub ensemble, where the transfer rate is determined by the degree of spectral overlap with other subensembles (shaded pink areas). Vibrational relaxation (i.e. depopulation of the state) occurs through overlap with the overtone of the bending state, the spectrum indicated as the purple Gaussian line shape. The rate of vibrational relaxation is proportional to the overlap between the green Lorentzian and the purple Gaussian line shapes (shaded in purple).

The time-dependent 2D spectra are calculated using the model schematically shown in figure 1. In brief, the model incorporates spectral diffusion through near-resonant vibrational energy transfer as well as intramolecular vibrational relaxation (IVR) by coupling of the O-H (O-D) stretch 8 ACS Paragon Plus Environment

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mode to the bending overtone. Spectral diffusion is modeled by allowing intensity transfer between the different Lorentzian oscillators with a rate proportional to the spectral overlap of different Lorentzian oscillators. To mimic experimentally observed spectral diffusion times, a proportionality factor of 1.7×10-5 is used. The concept is schematically depicted in figure 1 by the pink shaded areas for the overlap of the Lorentzian marked in green and neighboring oscillators. In the model, IVR occurs exclusively by coupling to the D2O bend overtone. The spectral response of the bend overtone of D2O is modeled by a Gaussian centered at 2361 cm−1 with a full width at half maximum (FWHM) of 58 cm−1.43 The rate of vibrational relaxation is proportional to the spectral overlap between the bend overtone Gaussian line shape and the Lorentzian describing a certain oscillator. The spectral overlap for the Lorentzian marked in green is represented by the purple shaded area in figure 1. Experimental relaxation times are obtained by scaling the overlap with a factor 0.018. The frequency dependence of the transition dipole was taken into account throughout the whole calculations to convert spectral intensity into population and in the overlap integrals as explained in ref 39. Following ref 44, the system of coupled differential equations has been solved, resulting in the time-dependent populations at selected time points. With these time-dependent populations, C(t) with the pump excitation on, the different contributions to the 2D spectra can be calculated for the bleach, stimulated emission, and excited state absorption: , mn#o%p = ∑ ,

Bh

= ∑

F ∗

(22)

F ∗

(23)

∗F ∗

(24)

, hB = ∑

in which Ωnn is the anharmonicity of the potential for the vibrational mode n and sign(An) is + or -

depending on the orientation of the molecules. The total response with the excitation on is a sum of these three terms and the nonresonant response, which is assumed to be the same as for the non-pumped case. In case of cross peaks, we obtain one additional term in which the anharmonic coupling Ωnm accounts for cross peak anharmonicity:

, %lFF i#o = ∑

(25)

A more common way to model vibrational dynamics is using a frequency-frequency correlation function (FFCF) formalism.31 The FFCF describes the ‘memory loss’ of the frequency of oscillators in an ensemble. Within the description of the Kubo lineshape model, the FFCF is described by an exponential decaying function. When the vibrational chromophores are governed by multiple characteristic dynamics, the FFCF is characterized by multi-exponential (or non-exponential) decay dynamics. For example, the FFCF for bulk water has been described using three exponential functions.45,46 Subsequently, the FFCF is used to calculate a lineshape function, which is used as input in the response function. The modelling from the current paper is in principal equivalent to the FFCF procedure. The energy transfer is also taking place in a ‘random walk’-type manner, by hopping between discretely spaced Lorentzian lines. The model employed here is characterized by two time constants for the frequency-frequency correlation function. The 106 cm-1 (FWHM) homogeneous broadening of each Lorentzian is determined by the fastest time constant, which represents the frequency memory decay due to the librational motion. The individual Lorentzians can be associated 9 ACS Paragon Plus Environment


with water in specific hydrogen bonding motifs, giving rise to a slow component in the decay of the FFCF. As such, the frequency migration from one Lorentzian to the other in this model can be connected with the frequency change due to the hydrogen bond exchange.

Results and Discussion Starting with the simplest system that can be modeled using our approach, we calculate the intensity and phase-resolved responses for an inhomogeneously broadened resonance, consisting of 41 Lorentzians lines spaced by 25 cm-1 with a FWHM for each of 106 cm-1, distributed over a Gaussian line shape with a FWHM of 166 cm-1, resulting in an overall linewidth of about 240 cm-1 FWHM. These parameters were chosen to mimic the water O-D stretch response. The Lorentzians constituting the overall line shape are assumed to have negative amplitude. The corresponding intensity, Im[ ] , and Re[ ]spectra are shown in figure 2a-c.

Figure 2: Modeled (a) SFG intensity, (b) Im[ ], and (c) Re[ ] spectrum assuming for the resonant part a -1 -1 Gaussian distribution (FWHM: 166 cm ) of 41 Lorentzian lineshapes each with a FWHM of 106 cm . The dotted lines show the same response including a nonresonant response. The nonresonant response also has a finite (frequency-independent) imaginary component, as typically observed experimentally for water.

The corresponding intensity, Im[ ] and Re[ ] 2D spectra, for three representative delay times for this inhomogeneously broadened resonance assuming an infinite anharmonicity, i.e. no excited state absorption in the nearby frequency range, are shown in figure 3a-c calculated as described above. At t = 0 ps, there is a positive response in the imaginary spectrum that is elongated 10 ACS Paragon Plus Environment

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along the diagonal and can be assigned to the bleach and stimulated emission. As the imaginary part is negative to start with, the difference spectrum is positive. However, for the intensity spectrum, the difference between the pump and un-pumped signal is negative. Similar to the imaginary spectrum, illustrated in figure 3b, there is only one elongated peak along the diagonal in the intensity spectrum in figure 3a. As time progresses, the signal decays due to coupling with the bending mode and the signal becomes elongated along the horizontal due to vibrational energy transfer. The differential response for the real part of the spectrum is shown in figure 3c.

Figure 3: Modeled (a) 2D-SFG intensity, (b) Im[ ] and (c) Re[ ] spectra assuming the steady-state spectra shown in figure 2, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. For this example, the anharmonicity of the vibrational transition was chosen to be infinitely large.

The equivalent 2D spectra calculated for resonances with a finite anharmonicity, set to 150 cm (a typical value for D2O), are shown in figure 4. In the finite anharmonicity scenario, the imaginary part has an additional negative response at lower probe frequency due to excited state absorption. The positive and negative signals have roughly equal intensity. As explained in the theory -1

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section, for the intensity spectrum the signal from the excited state absorption is weak and appearing below the diagonal. With increasing time, all signals decay and change from being elongated along the diagonal to being elongated along the horizontal. Figure 5 shows the equivalent 2D spectra for the case with finite anharmonicity of 150 cm-1 and a small, largely real, nonresonant contribution (not affected by excitation). The static spectra of this case are plotted in Fig. 2 as dotted lines. The 2D spectra (figure 5) look very similar to the case without the nonresonant signal (figure 4). The 2D spectra shown in figures 3, 4, and 5 were analyzed as one would analyze experimental 2D spectra: for one, by inferring the relaxation time of specific sub-ensembles within the inhomogeneous band from the time evolution of on-diagonal elements, and fitting a single exponential to the relaxation dynamics. In detail, the relaxation times for the imaginary differential spectra are obtained by integrating all positive signals at a specific pump frequency and specific time point, i.e. in a vertical slice in the 2D spectrum. For each pump frequency, the integrals are plotted as a function of time and fitted with a single exponential decay. The inset of figure 6b shows the integral as function of delay time with an exponential fit for the case assuming 150 cm-1 anharmonicity, pumped at 2460 cm-1. The relaxation times observed from the exponential fits are depicted as a function of pump frequency as a grey line in panels 6a, 6b, and 6c for the three different cases: infinite anharmonicity, 150 cm-1 anharmonicity, and 150 cm-1 anharmonicity with a nonresonant signal included. The curves for the intensity spectrum are obtained for a given pump frequency by integrating all points from the diagonal upwards, i.e. for 2500 cm-1 pumping integrated from 2500 cm-1 to 3220 cm-1. Subsequently, the relaxation time is obtained by fitting the integrals as a function of delay time with a single exponential fit. The results are depicted as black lines in the left column of Fig. 6 for the three different cases. This analysis shows that the lifetimes obtained from the intensity spectra, at least for the infinite anharmonicity, are less dispersive than the „true“ ones obtained from the imaginary two-dimensional response. This is not a result of the finite (10%) bleach used in the calculation; the same discrepancy is observed for extremely small bleach. Upon adding anharmonicity, the difference above 2450 cm-1 remains the same as for infinite anharmonicity. However, below 2450 cm-1, the intensity curve is now below the curve for the imaginary data. Clearly, the interference between the 0-1 bleach and the 1-2 transition speeds up the apparent relaxation time artificially. Adding a nonresonant signal does not change the curves significantly. The second typical analysis of 2D spectra is to quantify the central line slope (CLS) as a function of delay time,47,48 shown as dots and fitted lines in figures 3, 4 and 5. For the 2D intensity spectra, the slope is obtained by determining the frequency at which the 2D spectrum shows a minimum in the vertical slices at a given pump frequency, illustrated with the open circles in the 2D plots, and subsequently fitting a line, solid line in the 2D plots, through the obtained frequency as function of pump frequency. The result of this analysis at each time point is depicted as a black line in figure 6d-f for the three different cases. For the imaginary difference 2D spectrum with infinite anharmonicity, the same procedure has been followed. For the normal anharmonicity case and the case with nonresonant contribution, the nodal frequency is determined for each pump frequency and each time delay. See for example the open circles in figure 4b. Subsequently, a straight line is fitted through the symbols. The grey curves in figure 6d-f show the CLS as a function of delay time for the 2D imaginary difference spectra. The CLS at zero delay time is a direct measure for the heterogeneity, and the decay of the CLS reflects spectral diffusion, i.e. the decay of the FFCF. The results in figures 6d, 6e, and 6f show that the timescale inferred for the intensity and phase resolved 12 ACS Paragon Plus Environment

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measurements are very similar. A single exponential fit results in a timescale of ~200 fs for both scenarios. However, the initial slope values are not the same. At infinite anharmonicity the intensity spectrum underestimates the slope, while at normal anharmonicity the intensity spectrum slightly overestimates the slope. The presence of a nonresonant signal does not change this conclusion. For a small anharmonicity, i.e. 50 cm-1, the initial slope of the intensity spectrum is even 25 % higher than the slope from the imaginary spectrum. The under- and overestimations of the slope originates from subtle interferences in the intensity spectra between both ground state and excited state signals with the ‘background‘ signal from the remaining unexcited oscillators. The difference in the initial slope illustrates that one has to be careful to conclude directly from the initial slope value information about the amount of homogeneous broadening and fast spectral diffusion beyond the time resolution of the experiment. However, the dynamics are indistinguishable between the intensity and phase-resolved results.

Figure 4: Modeled 2D SFG intensity (a), Im[ ] (b) and Re[ ] spectra assuming the steady-state spectra shown in figure 2, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. For -1 this example, the anharmonicity of the vibrational transition was chosen to be 150 cm .



Figure 5: Modeled 2D (a) SFG intensity and (b) Im[ ] spectra assuming the steady-state spectra shown in figure 2 as dotted line, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. -1 For this example, the anharmonicity of the vibrational transition was chosen to be 150 cm and a small mainly real nonresonant contribution was added.


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Figure 6. Results of the analysis of simulations presented in figures 3, 4, and 5, for the cases with infinite and -1 normal (150 cm ) anharmonicity, and normal anharmonicity including a non-resonant contribution, respectively. Left: lifetime as a function of excitation frequency for infinite (a) and normal (b) anharmonicity and (c) normal anharmonicity with a nonresonant contribution. The curve for the imaginary spectrum is obtained by integrating over the positive signal, while the curve for the intensity spectrum is obtained by integrating over all points from the diagonal upwards for each time point, and fitting an exponential function to the decay. The inset in panel (b) shows the integral as function of delay time for the imaginary case with -1 pumping at 2460 cm . The line is a single exponential fit through the data. Right: slope from the 2D plot between 2300 and 2600 cm-1 as a function of delay time for infinite (e) and normal (e) anharmonicity and (f) normal anharmonicity with a nonresonant contribution; for the intensity case the curves are obtained by fitting a linear curve through the minimum determined by a Gaussian fit through the vertical slices. For the normal anharmonicity cases, the curve for the imaginary spectrum is obtained by fitting a linear curve through the zero crossing, while the curve for the infinite anharmonicity is obtained by fitting a linear curve through the maximum determined by a Gaussian fit through the vertical slices.

To model a more realistic system, for which different modes could interfere constructively or destructively, we next present results for the static response corresponding to the water air interface. The intensity spectrum, along with the imaginary part of the response, is shown in figure 7. The spectral features between 2300 and 2600 cm-1 reflect the response of O-H groups of water molecules hydrogen-bonded to a neighbor. The narrow residence at high frequency reflects the response of water OD groups sticking out into the air and not having a hydrogen bond with another molecule. To model the hydrogen-bonded part of this spectrum, 41 Lorentzians with a FWHM linewidth of 106 cm-1, a frequency spacing of 6.25 cm-1, and an intensity given by a double Gaussian distribution with each a FWHM width of 91 cm-1 and centered at 2370 and 2500 cm-1 are used. The free OD part is modeled with 5 Lorentzian lineshapes spaced by 2 cm-1 with equal amplitude and a FWHM of 30 cm-1. The nonresonant signal has a phase of -3 radians.



Figure 7. Modelled SFG intensity (a) and Im[ ] (b) spectrum mimicking the SFG spectrum recorded from the D2O water-air interface. The three peaks are inhomogeneously broadened to different degrees, to best approximate the experimentally observed spectra. See the text for the details of the spectral parameters used to obtain these spectra.

The corresponding intensity and phase-resolved 2D spectra are shown in figure 8. The spectra have been calculated at zero delay time between pump and probe, but for varying values of the cross-anharmonicity, i.e. the anharmonic coupling, between the free O-H resonance around 2750 cm-1 and the hydrogen-bonded vibrations, centered around 2350 and 2500 cm-1. It is apparent from this figure that the intensity and phase-resolved spectra look qualitatively quite similar, apart from the absence of the clear hot band (excited state SFG signal) in the intensity spectra. For both positive and negative cross-anharmonicity, the coupling is apparent for both the intensity and phase-resolved response at ijki = 2300 − 2600 cm-1 and ilm# = 2750 cm-1. The surprising observation is the appearance of a seeming cross peak in the intensity spectrum at ijki = 2750 cm-1 and ilm# = 2600 cm-1 in the absence of any cross anharmonicity (black circle in lower left panel). Similar „cross“ peaks have been previously reported in simulations of 2D SFG intensity data.30


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Figure 8: Modeled 2D (a) SFG intensity and (b) Im[ ] spectra assuming the steady-state spectra shown in figure 7, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. The spectra are shown at zero delay time, with varying coupling between the free O-H resonance around 2750 cm-1 and the hydrogen-bonded vibrations, centered around 2350 and 2500 cm-1, respectively. The black circle in the bottom left panel marks the artificial cross peak.

Subsequently, we model the time dependence of the H-bonded water molecules. The cross peak anharmonicity between the hydrogen bonded OD and the free OD group is set to be zero in this case. Moreover, energy transfer between these two water ensembles is also switched off. Basically, energy transfer only occurs within the hydrogen bonded water molecules. The frequency dependent lifetime and the slope dynamics are plotted in Figure 9. As in the model cases explained above, also for this realistic example of the water-air interface, the dynamical results for the imaginary and intensity spectrum are similar. However, if one is interested in the free OH/OD region, the heterodyne experiments might be more appropriate due to the appearance of an artificial cross-peak mentioned above and shown in figure 8.



Figure 9. (a) Lifetime as a function of excitation frequency for the modeled water-air interface; the lifetime curve for the imaginary spectrum is obtained by integrating over the positive signal, while the curve for the intensity spectrum is obtained by integrating over all points from the diagonal upwards for each time point, -1 and fitting an exponential function to the decay. (b) Slope from 2D plots between 2290 and 2560 cm as a function of delay time; the curve for the imaginary spectrum is obtained by fitting a linear curve through the zero crossing, while the curve for the intensity spectrum is obtained by fitting a linear curve through the minimum determined by a Gaussian fit through the vertical slice.

Conclusions We have studied the information content in intensity and phase-resolved 2D-SFG spectra for spectral responses similar to that of interfacial water, and find that apart from subtle differences –in particular in certain cases the absolute value of the centerline slopes– the results are very similar. We find that the time-dependent decay of the centerline slope, one of the key observables in 2D spectroscopies, is indistinguishable between the intensity and phase-resolved 2D SFG spectra for the model systems studied here, closely mimicking the response of aqueous interfaces. Given the experimental challenges associated with accurate determination of the phase of the SFG signal, especially over the long time the data are collected, it seems advisable to perform 2D SFG spectroscopy in the intensity domain. However, when the second order nonlinear response of the system under study contains several narrow, overlapping resonances with amplitudes of opposite sign,30 phase resolved experiments might be the better choice. In this case, the intensity spectrum might contain artificial cross peaks. A similar conclusion has been drawn for solution phase experiments.31,49 In cases where the oscillators are well separated, frequency resolved photon echo spectroscopy could be used, while for overlapping bands the heterodyned photon echo technique, i.e. 2D-IR spectroscopy, should be the appropriate method.


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Acknowledgements This work was partially funded by an ERC Starting Grant (Grant No. 336679). J.D.C thanks the Alexander von Humboldt foundation for financial support.

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(18) Bonn, M.; Bakker, H. J.; Ghosh, A.; Yamamoto, S.; Sovago, M.; Campen, R. K. Structural Inhomogeneity of Interfacial Water at Lipid Monolayers Revealed by Surface-Specific Vibrational Pump−Probe Spectroscopy. J. Am. Chem. Soc. 2010, 132 (42), 14971–14978. (19) Zhang, Z.; Piatkowski, L.; Bakker, H. J.; Bonn, M. Ultrafast Vibrational Energy Transfer at the Water/Air Interface Revealed by Two-Dimensional Surface Vibrational Spectroscopy. Nat. Chem. 2011, 3 (11), 888–893. (20) Zhang, Z.; Piatkowski, L.; Bakker, H. J.; Bonn, M. Communication: Interfacial Water Structure Revealed by Ultrafast Two-Dimensional Surface Vibrational Spectroscopy. J. Chem. Phys. 2011, 135 (2), 021101. (21) Xiong, W.; Laaser, J. E.; Mehlenbacher, R. D.; Zanni, M. T. Adding a Dimension to the Infrared Spectra of Interfaces Using Heterodyne Detected 2D Sum-Frequency Generation (HD 2D SFG) Spectroscopy. Proc. Natl. Acad. Sci. 2011, 108 (52), 20902–20907. (22) Inoue, K.; Nihonyanagi, S.; Singh, P. C.; Yamaguchi, S.; Tahara, T. 2D Heterodyne-Detected Sum Frequency Generation Study on the Ultrafast Vibrational Dynamics of H2O and HOD Water at Charged Interfaces. J. Chem. Phys. 2015, 142 (21), 212431. (23) Hsieh, C.-S.; Okuno, M.; Hunger, J.; Backus, E. H. G.; Nagata, Y.; Bonn, M. Aqueous Heterogeneity at the Air/Water Interface Revealed by 2D-HD-SFG Spectroscopy. Angew. Chem. Int. Ed. 2014, 53 (31), 8146–8149. (24) Singh, P. C.; Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Ultrafast Vibrational Dynamics of Water at a Charged Interface Revealed by Two-Dimensional Heterodyne-Detected Vibrational Sum Frequency Generation. J. Chem. Phys. 2012, 137 (9), 094706. (25) Inoue, K.; Singh, P. C.; Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Cooperative Hydrogen-Bond Dynamics at a Zwitterionic Lipid/Water Interface Revealed by 2D HD-VSFG Spectroscopy. J. Phys. Chem. Lett. 2017, 8 (20), 5160–5165. (26) Laaser, J. E.; Xiong, W.; Zanni, M. T. Time-Domain SFG Spectroscopy Using Mid-IR Pulse Shaping: Practical and Intrinsic Advantages. J. Phys. Chem. B 2011, 115 (11), 2536–2546. (27) Cyran, J. D.; Backus, E. H. G.; Nagata, Y.; Bonn, M. Structure from Dynamics: Vibrational Dynamics of Interfacial Water as Probe of Aqueous Heterogeneity. J Phys Chem B submitted. (28) Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Ultrafast Dynamics at Water Interfaces Studied by Vibrational Sum Frequency Generation Spectroscopy. Chem. Rev. 2017. (29) Singh, P. C.; Inoue, K.; Nihonyanagi, S.; Yamaguchi, S.; Tahara, T. Femtosecond Hydrogen Bond Dynamics of Bulk-like and Bound Water at Positively and Negatively Charged Lipid Interfaces Revealed by 2D HD-VSFG Spectroscopy. Angew. Chem. Int. Ed. 2016, 55 (36), 10621–10625. (30) Bredenbeck, J.; Ghosh, A.; Nienhuys, H.-K.; Bonn, M. Interface-Specific Ultrafast TwoDimensional Vibrational Spectroscopy. Acc. Chem. Res. 2009, 42 (9), 1332–1342. (31) Hamm, P.; Zanni, M. T. Concepts and Methods of 2D Infrared Spectroscopy; Cambridge, 2011. (32) Auer, B. M.; Skinner, J. L. IR and Raman Spectra of Liquid Water: Theory and Interpretation. J. Chem. Phys. 2008, 128 (22), 224511. (33) Du, Q.; Superfine, R.; Freysz, E.; Shen, Y. R. Vibrational Spectroscopy of Water at the Vapor/Water Interface. Phys. Rev. Lett. 1993, 70 (15), 2313–2316. (34) Bonn, M.; Nagata, Y.; Backus, E. H. G. Molecular Structure and Dynamics of Water at the Water–Air Interface Studied with Surface-Specific Vibrational Spectroscopy. Angew. Chem. Int. Ed. 2015, 54 (19), 5560–5576. (35) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press, 1999. (36) Livingstone, R. A.; Zhang, Z.; Piatkowski, L.; Bakker, H. J.; Hunger, J.; Bonn, M.; Backus, E. H. G. Water in Contact with a Cationic Lipid Exhibits Bulklike Vibrational Dynamics. J. Phys. Chem. B 2016, 120 (38), 10069–10078. (37) Loparo, J. J.; Roberts, S. T.; Nicodemus, R. A.; Tokmakoff, A. Variation of the Transition Dipole Moment across the OH Stretching Band of Water. Chem. Phys. 2007, 341 (1–3), 218–229. (38) Loparo, J. J.; Roberts, S. T.; Nicodemus, R. A.; Tokmakoff, A. Erratum to “Variation of the Transition Dipole across the OH Stretching Band of Water” [Chem. Phys. 341 (2007) 218–229]. Chem. Phys. 2009, 361 (3), 185. 20 ACS Paragon Plus Environment

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TOC Graphic


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∆𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝐹𝑖𝑛 − 𝐹𝑜𝑢𝑡 − 𝑁𝐷2𝑂 ∗ 𝜏𝑏𝑒𝑛𝑑 Förster Transfer; 𝐹𝑖𝑛 , 𝐹𝑜𝑢𝑡

Vibrational Relaxation (D2O) via Fermi Resonance; 𝜏𝑏𝑒𝑛𝑑 : ACS Paragon Plus Environment


Modeled (a) SFG intensity, (b) Im[χ^((2) )], and (c) Re[χ^((2) )] spectrum assuming for the resonant part a Gaussian distribution (FWHM: 166 cm-1) of 41 Lorentzian lineshapes each with a FWHM of 106 cm-1. The dotted lines show the same response including a nonresonant response. The nonresonant response also has a finite (frequency-independent) imaginary component, as typically observed experimentally for water. 81x116mm (300 x 300 DPI)

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Modeled (a) 2D-SFG intensity, (b) Im[χ^((2) )] and (c) Re[χ^((2) )] spectra assuming the steady-state spectra shown in figure 2, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. For this example, the anharmonicity of the vibrational transition was chosen to be infinitely large. 175x175mm (300 x 300 DPI)



Modeled 2D SFG intensity (a), Im[χ^((2) )] (b) and Re[χ^((2) )] spectra assuming the steady-state spectra shown in figure 2, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. For this example, the anharmonicity of the vibrational transition was chosen to be 150 cm-1. 82x117mm (300 x 300 DPI)


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Modeled 2D (a) SFG intensity and (b) Im[χ^((2) )] spectra assuming the steady-state spectra shown in figure 2 as dotted line, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. For this example, the anharmonicity of the vibrational transition was chosen to be 150 cm-1 and a small mainly real nonresonant contribution was added. 82x118mm (300 x 300 DPI)



Results of the analysis of simulations presented in figures 3, 4, and 5, for the cases with infinite and normal (150 cm-1) anharmonicity, and normal anharmonicity including a non-resonant contribution, respectively. Left: lifetime as a function of excitation frequency for infinite (a) and normal (b) anharmonicity and (c) normal anharmonicity with a nonresonant contribution. The curve for the imaginary spectrum is obtained by integrating over the positive signal, while the curve for the intensity spectrum is obtained by integrating over all points from the diagonal upwards for each time point, and fitting an exponential function to the decay. The inset in panel (b) shows the integral as function of delay time for the imaginary case with pumping at 2460 cm 1. The line is a single exponential fit through the data. Right: slope from the 2D plot between 2300 and 2600 cm-1 as a function of delay time for infinite (e) and normal (e) anharmonicity and (f) normal anharmonicity with a nonresonant contribution; for the intensity case the curves are obtained by fitting a linear curve through the minimum determined by a Gaussian fit through the vertical slices. For the normal anharmonicity cases, the curve for the imaginary spectrum is obtained by fitting a linear curve through the zero crossing, while the curve for the infinite anharmonicity is obtained by fitting a linear curve through the maximum determined by a Gaussian fit through the vertical slices. 82x89mm (300 x 300 DPI)


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Modelled SFG intensity (a) and Im[χ^((2) )] (b) spectrum mimicking the SFG spectrum recorded from the D2O water-air interface. The three peaks are inhomogeneously broadened to different degrees, to best approximate the experimentally observed spectra. See the text for the details of the spectral parameters used to obtain these spectra. 81x76mm (300 x 300 DPI)


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Modeled 2D (a) SFG intensity and (b) Im[χ^((2) )] spectra assuming the steady-state spectra shown in figure 7, with spectral diffusion and vibrational relaxation in the model as summarized in figure 1. The spectra are shown at zero delay time, with varying coupling between the free O-H resonance around 2750 cm-1 and the hydrogen-bonded vibrations, centered around 2350 and 2500 cm-1, respectively. The black circle in the bottom left panel marks the artificial cross peak. 82x123mm (300 x 300 DPI)



(a) Lifetime as a function of excitation frequency for the modeled water-air interface; the lifetime curve for the imaginary spectrum is obtained by integrating over the positive signal, while the curve for the intensity spectrum is obtained by integrating over all points from the diagonal upwards for each time point, and fitting an exponential function to the decay. (b) Slope from 2D plots between 2290 and 2560 cm-1 as a function of delay time; the curve for the imaginary spectrum is obtained by fitting a linear curve through the zero crossing, while the curve for the intensity spectrum is obtained by fitting a linear curve through the minimum determined by a Gaussian fit through the vertical slice. 81x87mm (300 x 300 DPI)


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TOC figure 82x43mm (300 x 300 DPI)


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