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Sum Frequency Generation Spectra From Velocity-Velocity Correlation Functions Remi Khatib, and Marialore Sulpizi J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b00207 • Publication Date (Web): 01 Mar 2017 Downloaded from http://pubs.acs.org on March 4, 2017

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The Journal of Physical Chemistry Letters

Sum Frequency Generation spectra from velocity-velocity correlation functions Rémi Khatib and Marialore Sulpizi∗ Johannes Gutenberg University Mainz, Staudinger Weg 7, 55099 Mainz, Germany E-mail: [email protected]

∗

To whom correspondence should be addressed

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The characterization of liquids at the interface has seen in the last years tremendous progresses thanks to the development of surface selective spectroscopic techniques, such as Sum Frequency Generation (SFG). 1–4 The molecular interpretation of the experimental spectra has also fostered parallel advances in the computational community aiming to an accurate calculation and interpretation of such spectra. Following the pioneer work of Morita 5–8 several groups have contributed to improve the accuracy of the calculated spectra. 9–12 We have recently presented a method to calculate SFG spectra from velocity-velocity correlation functions including interface specific selection rules. 13 Since the velocities are a natural output in molecular dynamics (MD) simulations, they can be directly obtained without the additional cost of the direct calculation of the dipole moments and polarizabilities. 14 The advantage is that accurate Density Functional Theory (DFT) -based MD simulations can be used with no overhead cost, which is particularly useful for description of interfaces where an accurate classical force field is not available. The use of suitable velocity-velocity correlation functions requires the introduction of some approximations, such e.g. in the case of water, the projection of the velocities on the O-H bonds, which restricts the approach to the stretching region only. This is indeed the approximation we introduced in Ref. 13 and which was also similarly adopted in a parallel work in Ref. 9. To lift such a limitation we present here a reworked expression for the SFG signal, which is based on a projection of the atom velocities on the molecular normal modes. This, not only permits to address the stretching region, as the sum of the symmetric and antisymmetric contributions, but also to extend the SFG calculation to the bending region and allows us to discuss the coupling between stretching and bending. As an application of our method we calculate the SFG spectrum at the water-air interface with ab initio MD simulations based on DFT. Although such a system represents, probably, the most investigated one by both experimental and computational SFG approaches, a consensus is still missing on the molecular interpretation, e.g. of the hydrogen bonded region.

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In the first part of this letter we revise the bond-projected approach of Ref. 13, then we introduce the new expression based on the velocity projection on the normal modes. As an application the SFG spectrum of the water/air interface is presented and dissected. We aim to calculate the resonant part of the second order susceptibility tensor χ(2,R) , which according to the classical expression 6 is: (2,R) χζηκ

−i = kB T ωIR

Z

+∞

˙ κ (0)i dt eiωIR t hA˙ ζη (t)M

(1)

0

where the indexes ζηκ refers to the polarization of the SFG, visible and IR beams respectively, ωIR is the frequency of the IR beam, M and A are the dipole moment and the polarizability of the system and h...i stands for a statistical average. The total dipole moment and polarizability derivatives for the system can be decomposed in terms of the molecular and bond contributions:    ˙ =  M

NP mol

P

i=1 ǫ NP mol P

   A˙ =

i=1

ǫ

µ˙ li,ǫ

(2)

α ˙ li,ǫ

where µ˙ li,ǫ (α ˙ li,ǫ ) is the dipole moment (polarizability) of the bond ǫ of the i-th molecule. Therefore, the correlation function in Eq. 1 can be rewritten as: ˙ κ (0)i hA˙ ζη (t)M =

N mol X

X

l α˙ ζη,i,ǫ (t)µ˙ lκ,i,ǫ (0)

ǫ i=1 N mol X X

+

+

i=1 N mol X

ǫ

l (t)µ˙ lκ,i,−ǫ (0) α˙ ζη,i,ǫ

X

l α˙ ζη,i,ǫ (t)µ˙ lκ,j,ζ (0)

i,j=1 ǫ,ζ i6=j

4


(3)

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The first term of the right-hand side is the bond autocorrelation, the second term accounts for the correlation between two bonds in the same water molecule, the third term is for the correlation between bonds in two different water molecules. In the following, we will refer to “intra-” and “inter-molecular” correlation to describe the sum of the first two terms and of the three terms respectively. We restrict the inter-molecular correlation to the sum over the water molecules within a 4 Å distance, namely those belonging to the first solvation shell. Similarly, a cutoff was introduced in Ref. 15 for the calculation of the response function at the lipid-water interface. The key step in our approach is a first order expansion of the dipole moment (µi,ǫ ) and the polarizability (αi,ǫ ) with respect to a change in the O-H bond length (ri,ǫ ) 13.    µ˙ li,ǫ = Dm,i Db,i,ǫ ∂µb r˙i,ǫ ∂r b  T T  α Db,i,ǫ Dm,i r˙ ˙ li,ǫ = Dm,i Db,i,ǫ ∂α ∂r i,ǫ

(4)

where the exponents/indexes l and b refer to the lab and bond frameworks respectively (details on the frameworks definition and the matrix connection between them can be found in the SI section 2). The partial derivatives are parametrized (see SI section 4 for details) and the velocities are obtained from the MD simulations. Therefore, the calculation of these time-derivatives do not require any additional cost with respect to that of simple trajectory accumulation. This is the main advantage over the method previously used by Sulpizi et al.. 14 In Fig. 1, we report the Im χ and Re χ for the water/air interface calculated (from a total of 8 independent trajectories of 40 ps each) with the approximation introduced in Eq. 4. In the Im χ we observe two main features: (i) a positive peak at high frequency usually attributed to free O-H peak, (ii) a negative and broad peak in the 3000-3600 cm-1 region. No re-crossing to positive frequencies is observed around 3100 cm-1 , at odd with the spectrum obtained in Ref. 14. Such a difference is possibly due to the very limited number of configurations sampled in the 3 ps trajectory of Ref. 14. This spectrum is overall in agreement 5



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with those obtained by other groups with different methodologies. 7,9,12 However, the free OH peak does not present a shoulder on the lower frequency side. 16,17,22 The difference with respect to Ref. 17 may be due to some structural difference in the water surface as result of the different water description, which, in our case, includes the full electronic structure at DFT-BLYP D3 level. In Fig. 1, we also compare the simulated spectra with the experimental one published by Nihonyanagi et al.. 18 In the comparison to the experiments, the Fresnel factors and the non-resonant background are also taken into account. The non-resonant term cannot be computed and corresponds to a frequency independent contribution which aligns the computed baseline to the experimental one. We determined that (2,R) (2,NR) χ⊥⊥zl ≈ −0.8 max Im χ⊥⊥zl

(5)

(2,R) where max Im χ⊥⊥zl is the intensity of the free O-H peak and where χ⊥⊥zl stands for the

average of the xl xl zl and yl yl zl elements. This value is slightly more important than those

used by Morita et al. 7 (50% of the maximal intensity). The details of all the contributions (resonant part, non-resonant part, and Fresnel factors) are discussed in the SI section 8. The auto- (intra-) correlation is derived from the first (first two) element(s) in Eq. 3 and is represented by the black (red) line in Fig. 1. Auto- and intra-molecular correlations give comparable results in agreement with the experiment. The inter-molecular correlation introduces a red-shift of the negative band of about 150 cm-1 . Such a shift had been already pointed out by Morita et al. 5 who claimed that it was necessary to consider χ(2,R) as a sum of molecular hyperpolarizabilities and not as the hyperpolarizability of the whole system. As a further step we now consider the velocities projection on the water normal modes, → − identified by the collective variables R j in the molecular framework. The normal modes of a gas phase water molecule include the symmetric stretching (SS), the antisymmetric stretching (AS) and the bending (B).

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order to facilitate the discussion about the decomposition of the SFG signal, we will write “R1 .R2 ” to describe the part associated with the polarizability of the mode R1 correlated with the dipole moment of the mode R2 . The precise description of the collective motions and the way we parametrized ∂µm i ∂Rj

∂αm i ∂Rj

and

can be found in the SI section 4.

In Fig. 2 the ⊥⊥zl spectra obtained with the two different methods (bond projection vs normal modes projection) are compared in the stretching region. The two spectra nicely overlap. The spectrum obtained with the bond projection (red dashed line) can be further analysed in terms of its normal modes decomposition. The main contributions in the stretching region are associated to the SS and the AS (Fig. 2) modes while the B mode does not play a significant role (see Fig. 8 in SI). Interesting the overall spectrum broadening is the result of the “SS.AS” and the “SS.SS” contributions which are peaked at different frequencies (around 3300 and 3470 cm-1 , respectively). No direct coupling between the stretching and the bending is found to contribute to the stretching region, namely the “B.SS” and “SS.B” contributions are both zero (see Fig. 8 in the SI). In this sense we cannot observe here a direct effect of the Fermi resonance to the broadening of the negative peak around 3400 cm-1 as suggested in Refs 19–21. A positive peak at 3500 cm-1 and associated to the AS.AS contribution is also present. However since its intensity is relatively low it does not really give raise to a pronunced shoulder 16,17,22 at 3600 cm-1 in the overall signal (black dashed line in Fig. 2). In addition, we can also spatially localize the water molecules which contribute to the different peaks reported in Fig. 2. For this purpose we only include in the calculation of (2,R)

the χ⊥⊥zl , intra the water molecules within a certain distance (along the zl -axis) from the surface and we observe the evolution of the peaks intensity with an increasing thickness of the included layer. In Fig. 3, we have reported the relative intensities associated with the main three contributions in the stretching region.

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with the experimental one. Similarly, the agreement between the calculated and experimental real parts is very good. As we did for the stretching region, we can associate the simulated spectrum to specific orientation of the water molecules. The evolution of the “B.B” feature according to the distance from the surface, along the zl -axis (Fig. 3) resembles very closely the behaviour associated to the “SS.SS” contribution at 3470 cm-1 . This suggests that the water molecules responsible for these two features are certainly the same. This is fully coherent with the model that we have previously proposed (Fig. 4) since the “B.B” element of molecules A and D are equal to zero while those of the molecule B is positive. In our spectrum we only observe a positive band, as in experiments of Ref. 24, but at odd with other experimental 25 and computational 23,26 results. We do not observe a negative band attributed in Ref. 25 to the water molecules with a free OH. Overall the weak ASAS band at 3600 cm-1 and the absence of a negative peak in the bending could point to a different orientation of the free OH at the interface, which in turn may depend on the water model. A more extensive discuss about this can be found in the SI. We have presented a method to calculate the SFG spectra of water interfaces using suitable velocity-velocity correlations functions. The method is based on the projection of the atomic velocities on the single water normal modes and permits to accurately calculate the spectra using velocities correlation function, with the advantage of no extra cost with respect to the accumulation of an ab initio molecular dynamics trajectory. Our approach permits on one side to decompose the SFG signal in term of contributions with a symmetric, antisymmetric, and bending character, therefore elucidating the molecular origin of the different peaks in the spectra. On the other side, it also permits to describe the bending region, beyond the bond projection approach. The calculated signal in the bending region closely resembles the recent experimental data from phase resolved SFG experiments. Our method is expected to be extremely valuable for all those heterogeneous systems where a force field parametrization is not available or difficult to obtain.

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Acknowledgement Financial support from the Deutsche Forschungsgemeinschaft under grant n◦ SU 752/2 and Computer time from HRLS-Germany (project number 2DSFG) are greatly acknowledged.

Supporting Information Available • SI-Fig. 1: Average density and random snapshot of the water slab. • SI-Fig. 2: Representation of the molecular and the bond frameworks. • SI-Tab. 1: Collective motions associated with the normal modes. • SI-Tab. 2: Average values calculated for the length derivatives of the dipole moment and polarizability. • SI-Fig. 3: Distribution of the length derivatives of the dipole moment and polarizability. • SI-Tab. 3: Average values calculated for the normal modes derivatives of the dipole moments. • SI-Tab. 4: Average values calculated for the normal modes derivatives of the polarizabilities. • SI-Fig. 4: Comparison between the correlation functions and the Gaussian functions used for the apodization. • SI-Fig. 5: Evolution of the inverse Laplace transforms of the correlation functions according to the Gaussian functions used for the apodization. • SI-Fig. 6: SFG spectra associated with the ⊥⊥zl in the bending and in the stretching region. • SI-Fig. 7: Fresnel factors used in the bending and in the stretching region.

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• SI-Fig. 8: Minor contributions of the SFG spectra associated with the ⊥⊥zl . • SI-Fig. 9: Scheme presenting two different models of a water molecule with a free OH at the water surface. This material is available free of charge via the Internet at http://pubs.acs.org/.

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(23) Ni, Y.; Skinner, J. L. IR and SFG vibrational spectroscopy of the water bend in the bulk liquid and at the liquid-vapor interface, respectively. J. Chem. Phys. 2015, 143, 014502. (24) Kundu, A.; Tanaka, S.; Ishiyama, T.; Ahmed, M.; Inoue, K.-I.; Nihonyanagi, S.; Sawai, H.; Yamaguchi, S.; Morita, A.; Tahara, T. Bend Vibration of Surface Water Investigated by Heterodyne-Detected Sum Frequency Generation and Theoretical Study: Dominant Role of Quadrupole. J. Chem. Phys. Letters 2016, 7, 2597–2601. (25) Dutta, C.; Benderskii, A. V. On the Assignment of the Vibrational Spectrum of the Water Bend at the Air/Water Interface. J. Phys. Chem. Lett. 2017, 8, 801–804. (26) Nagata, Y.; Hsieh, C.-S.; Hasegawa, T.; Voll, J.; Backus, E. H. G.; Bonn, M. Water Bending Mode at the Water-Vapor Interface Probed by Sum-Frequency Generation Spectroscopy: A Combined Molecular Dynamics Simulation and Experimental Study. J. Chem. Phys. Letters 2013, 4, 1872–1877.

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