Ultrafast Vibrational Echo Spectroscopy of Liquid Water from First

Article pubs.acs.org/JPCB

Ultrafast Vibrational Echo Spectroscopy of Liquid Water from FirstPrinciples Simulations Deepak Ojha and Amalendu Chandra* Department of Chemistry, Indian Institute of Technology, Kanpur, India 208016 ABSTRACT: Vibrational echo spectroscopy has become a powerful technique to study vibrational spectral diffusion in water and aqueous solutions. The dynamics of vibrational spectral diffusion is intimately related to the hydrogen bond fluctuations in liquid water and other hydrogen bonded liquids. Earlier theoretical calculations of vibrational echo spectroscopy of aqueous systems were based on classical molecular dynamics simulations involving empirical force fields of water. In the current work, we have employed the method of ab initio molecular dynamics simulation to calculate the spectral observables of vibrational echo and two-dimensional infrared (2D-IR) spectroscopy of liquid water at room temperature under Condon and cumulant approximations. The time scales extracted from the temporal decay of the frequency−time correlation function (FTCF), short-time slope of three pulse photon echo (SP3E), dynamic line width (DLW), and the slope of nodal line of 2D-IR spectra are found to be in reasonably close agreement with each other which reinforces the assertion that signatures of FTCF can be captured using three pulse photon echo and 2D-IR spectroscopy.

1. INTRODUCTION The study of liquid water has remained a subject of profound interest in several branches of science ranging from chemistry, biology, physics, geology, to oceanography.1−4 The ubiquitous nature of water is the prime issue of concern in chemistry where it acts as solvent and reaction medium for a wide range of chemical reactions. It is equally a subject of scientific curiosity for those who look to unravel the mechanistic path of various chemical reactions and biological processes in aqueous media. Liquid water is also known to exhibit anomalous behavior at ambient conditions like density maximum at 4 °C. With further lowering of temperature, several macroscopic properties like density, thermal expansion coefficient, and heat capacity seem to deviate significantly from those of other simple liquids of comparable molecular mass and structure. The reason for the anomalies shown at the macroscopic level is believed to be deeply rooted in the hydrogen bonded structure and dynamics at the microscopic level. There have been a plethora of experimental and theoretical studies on water which have revealed a lot of information about the microscopic behavior of water.5−17 While the X-ray and neutron scattering experiments18,19 are able to determine the structure factors which provide the radial distribution functions on Fourier inversion, nuclear magnetic resonance dipolar relaxation experiments provide the reorientational time scale of water molecules.20 The anomalous behavior of liquid water is believed to be strongly related to its extensive three-dimensional network of hydrogen bonds which fluctuates in ultrafast time scales. The environmental changes due to hydrogen bond fluctuations also alter the vibrational frequencies of water molecules. Thus, the dynamics of hydrogen bond breaking and reformation can be correlated with the vibrational frequency © 2015 American Chemical Society

relaxation of a local chromophore such as the OH or OD stretch of a water molecule in liquid water.6−14 It is well-known that the infrared (IR) spectrum corresponding to the OH vibrational stretch of liquid water exhibits a broad peak around 3400 cm−121 which envelops specific signature peaks corresponding to the hydrogen bond dynamics. The broad peak is primarily due to the static distribution of the chromophore in different local solvent environments which is also known as the inhomogeneous broadening. With the advent of ultrafast lasers, several new experimental techniques like transient hole burning, three pulse photon echo (3PEPS), and two-dimensional infrared (2D-IR) spectroscopy have been devised which involve multiple matter−radiation interactions in a time dependent manner, thus providing a short time window in which the dynamics of the system can be observed in the spectrum.7−12,22−37 The inhomogeneous broadening is eliminated as the system is not allowed to encompass all possible local solvent environments in the narrow time window of spectral resolution. A majority of the existing experiments have considered the system of isotopically diluted solution of HOD in H2O or D2O and have probed the spectral diffusion of OD or OH stretch of the HOD solute.7−12,22−33 This allows spectral isolation of the OH/OD stretch mode of interest and avoids any strong coupling with the stretch modes of solvent H2O or D2O molecules which otherwise can complicate the dynamics due to many fast competing dynamical processes arising from various resonantly coupled intra- and intermoSpecial Issue: Biman Bagchi Festschrift Received: March 31, 2015 Revised: July 1, 2015 Published: July 10, 2015 11215

DOI: 10.1021/acs.jpcb.5b03109 J. Phys. Chem. B 2015, 119, 11215−11228

Article

The Journal of Physical Chemistry B

used Condon or cumulant approximations. Thus, the effects of non-Gaussian dynamics and transition dipole changes are automatically included in these calculations. It may, however, be noted that the existing theoretical studies of vibrational echo spectroscopy and 2D-IR of liquid water are based on empirical force field models for water molecules, and expectedly, the dynamics was found to vary considerably with force fields chosen for water.43 Recently, Mallik et al.59,60 calculated the vibrational hole dynamics and frequency correlation functions of deuterated liquid water and aqueous ionic solutions by using ab initio molecular dynamics simulations77,78 without involving any empirical interaction potentials. A very recent study looked at the mechanism of spectral diffusion of OH stretch mode in liquid water by generating frequency-resolved nonequilibrium states by means of a reverse nonequilibrium ab initio molecular dynamics algorithm.61 The method of ab initio molecular dynamics simulation has also been used to calculate the THz spectrum of liquid water.79 However, to the best of our knowledge, a theoretical study of the ultrafast vibrational echo and 2D-IR spectroscopy of liquid water from ab initio simulations without using any empirical potential models is not yet available in the literature. In the current work, we present such a study and calculate observables of vibrational echo and 2D-IR spectroscopy like the integrated echo intensity, initial slope of three-pulse echo, dynamic line width, and the slope of nodal line from first-principles simulations. While the dynamics of water molecules is determined through calculations of many-body potentials and forces from quantum density functional theory within Kohn− Sham formulation,80,81 the frequency fluctuations are calculated through a time series analysis of the simulated trajectory using the wavelet method.59,60,82,83 The effects of anharmonicity are included in calculation of the transition frequencies. The time dependence of various spectroscopic observables of ultrafast vibrational echo and 2D-IR spectroscopy are obtained through calculations of the relevant third order response functions first within the second order cumulant approximation and then within the Condon approximation where the relevant integrals involving the frequency fluctuations are calculated directly without approximating them by two-point frequency correlations.38−41,62,63 Since our work does not invoke any empirical intermolecular interaction potential of water for determining the instantaneous dynamics at the microscopic level, it makes a significant contribution toward theoretical studies of the dynamics of nonlinear spectroscopic observables from a microscopic point of view. Our study also widens the scope of theoretically linking many other dynamical phenomena such as the hydrogen bond switching mechanism, anisotropy decay, and the kinetics of hydrogen bond formation to various vibrational echo spectroscopic observables from first-principles simulations. The framework of our current theoretical calculations also holds good for studying reactive situations such as proton transfer reactions in acidic and basic solutions and ligand-exchange reactions in aqueous media since the atoms in a first-principles simulation of the type employed here move in many-body quantum potentials which dynamically allow bond formation and breaking events with associated rearrangement of the electronic structure in a seamless manner. The rest of the paper is organized as follows. In section 2, we discuss the basic theoretical formulation of linear absorption and time dependent nonlinear photon echo infrared spectroscopy at different levels of approximations. The computational details of ab initio simulations are presented in section 3. The

lecular modes. Experiments on such isotopically diluted HOD in H2O or D2O systems using 3PEPS and 2D-IR spectroscopy have generally reported a short time scale of 100−200 fs and a longer time scale of 1−2 ps for the dynamics of vibrational spectral diffusion occurring in liquid water at room temperature. Some of the recent experiments34−37 also probed the vibrational dynamics of OH stretch modes in pure liquid H2O instead of using isotopically diluted HOD in H2O or D2O and found a significantly faster relaxation (∼50 fs) for the loss of structural correlation due to fast population relaxation from the excited state and energy transfer to resonantly and also nonresonantly coupled intra- and intermolecular vibrational modes. There have also been a number of theoretical studies on spectral diffusion of local OH/OD stretch modes in liquid water and also of resonantly coupled OH/OD modes in water which helped to interpret the findings of these experiments and also provided further insight into the underlying molecular dynamics.8−10,13,14,38−61 In the present work, however, we are primarily concerned with the spectral diffusion of local OD stretch modes in liquid water without the complications of fast population relaxation and resonant energy transfer to other modes. Regarding the existing theoretical studies of photon echo spectroscopy of a local OH/OD mode in water, we note the work of Skinner and co-workers who developed a robust electronic structure/molecular dynamics (ES/MD) based method40−43 involving an empirical mapping between the electric field and frequency fluctuation and employing such a map in classical molecular dynamics simulations. A similar empirical relation was also developed by Cho and coworkers45−47 between the vibrational frequency and electrostatic potential in an ES/MD setup to study the spectral dynamics in water and an aqueous solution of N-methylacetamide. There have also been studies48−51 which invoked perturbation theory to extract the frequency fluctuations from the forces projected along the OH bonds where the forces and the dynamics were again calculated using classical molecular dynamics with empirical potential models of water. All these studies have provided useful insights into the dynamics of vibrational spectral diffusion, particularly into the dynamics of frequency fluctuations, and the presence of strong correlations between the peak shift dynamics and frequency−time correlation function of stretch modes of water. All these existing theoretical studies of vibrational echo and 2D-IR spectroscopy first expressed the echo and 2D-IR intensity in terms of several third order nonlinear response functions which depend on frequency and transition dipole fluctuations62−64 and then calculated these response functions at different levels of approximations such as second order cumulant and Condon approximations. The second order cumulant approximation is exact for a Gaussian process. However, it has been generally shown that the frequency fluctuations in water have substantial non-Gaussian character;10,65−71 hence, one needs to move beyond the second order cumulant approximation for a reliable treatment of the spectral dynamics. Calculations without involving any of the two above-mentioned approximations, the so-called non-Condon level calculations, have also been carried out in recent years.41−43 We also note in this context that some of the recent work14,55,72−76 has directly obtained the nonlinear response functions through nonequilibrium molecular dynamics simulations on liquid water or by numerically solving the Schrodinger equations of stretch vibrations upon application of an external field without invoking the commonly 11216


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The Journal of Physical Chemistry B results of our calculations of various nonlinear spectroscopic observables are presented in section 4. A brief discussion comparing the current results with those of some of the available experimental and other theoretical work is presented in section 5. Finally, our conclusions are briefly summarized in section 6.

principle a stationary Gaussian process, to further reduce eq 4 to the following simplified form41,62

2. THEORETICAL FORMULATION In this section, we present the basic equations of linear absorption and nonlinear time dependent echo spectroscopy by following the formulation of Mukamel.62 The basic equations and their reductions for various approximations have also been nicely presented by Skinner and co-workers41 and we will closely follow their notations for different dynamical quantities in the present work. 2.1. Linear Spectroscopy. Linear response of a system to the external electric field of an electromagnetic radiation generates the linear absorption spectrum which can be expressed mathematically as the Fourier transform of the dipole time correlation function

where g1(t) is given by

× ⟨exp( −g1(t ))⟩

g1(t ) =

∫−∞ dt exp(−iωt )⟨μ(0)μ(t )⟩

(1)

where μ is the dipole operator of the concerned vibrational mode of a chromophore. For the fundamental transition from the ground to the first vibrational level, v = 0 → 1 where v stands for the vibrational quantum number, the above equation within semiclassical approximation can be expressed as41,62,84 ∞

I(ω) ∼

∫−∞ dt exp(−iωt )

× ⟨μ10 (0)·μ10 (t )exp[i

∫0

t

dτω10(τ )]⟩

(2)

where μ10 is the 1−0 matrix element of the dipole operator between states v = 0 and v = 1. ω10 is the instantaneous transition frequency between the ground and first excited states of the vibrational mode of interest. We note that eq 2 now contains equilibrium classical statistical mechanical averages, and also it does not include the lifetime effects of the excited state. Ignoring the orientational dynamical effects of μ10 and defining δω10 = ω10 − ⟨ ω10 ⟩ (⟨ω10 ⟩ is the average 0−1 transition frequency), we rewrite the above equation in the following form

∫0

I(ω) ∼

× ⟨exp[i

(6)

(7)

× ⟨μ10 (0)μ10 (t1)μ21(t1 + t 2)μ21(t1 + t 2 + t3) × ψRP(t3 , t 2 , t1)⟩ (3)

(8)

R 4(t3 , t 2 , t1) = R 5(t3 , t 2 , t1) = exp(− i⟨ω10⟩t3 − i⟨ω10⟩t1) × ⟨μ10 (0)μ10 (t1)μ10 (t1 + t 2)μ10 (t1 + t 2 + t3) × φNP(t3 , t 2 , t1)⟩

(9)

R 6(t3 , t 2 , t1) = − exp(− i⟨ω21⟩t3 − i⟨ω10⟩t1) × ⟨μ10 (0)μ10 (t1)μ21(t1 + t 2)μ21(t1 + t 2 + t3)

∫−∞ dt exp(−i(ω − ⟨ω10⟩)t )

× ψNP(t3 , t 2 , t1)⟩

t

dτδω10(τ )]⟩

dτ″C(τ″)

R3(t3 , t 2 , t1) = − exp(− i⟨ω21⟩t3 + i⟨ω10⟩t1)

∞

∫0

τ′

× φRP(t3 , t 2 , t1)⟩

In order to further simplify the above equation, we make the well-known Condon approximation where the instantaneous transition dipole moment is assumed to be independent of nuclear degrees of freedom, thus treating μ10 as a constant and moving it out of the ensemble averaging. Within this Condon approximation, eq 3 becomes41,62 μ102

∫0

× ⟨μ10 (0)μ10 (t1)μ10 (t1 + t 2)μ10 (t1 + t 2 + t3)

t

dτδω10(τ )]⟩

dτ ′

= exp(− i⟨ω10⟩t3 + i⟨ω10⟩t1)

∫−∞ dt exp(−i(ω − ⟨ω10⟩)t )

× ⟨μ10 (0)μ10 (t )exp[i

t

R1(t3 , t 2 , t1) = R 2(t3 , t 2 , t1)

∞

I(ω) ∼

∫0

(5)

In eq 6, C(t) represents the frequency−time correlation function (FTCF) given by C(t) = ⟨δω10(0)δω10(t)⟩. Following earlier work,41 we will denote the formulas of eqs 4 and 5 as Condon and cumulant approximations of linear absorption spectrum, respectively. 2.2. Third Order Nonlinear Echo Spectroscopy. The third order nonlinear echo spectroscopic observables of time dependent infrared spectroscopy can be formally expressed as convolution of response functions with the applied electric field.62−64 These third order response functions can be further simplified under different levels of approximations. For vibrational echo and 2D-IR experiments, three ultrafast laser pulses with wave vectors k1, k2, and k3 and adjustable time delays are sequentially targeted on the system. Subsequently, an echo is produced due to constructive interference in the rephasing direction kRP = −k1 + k2 + k3 accompanied by free induction decay (FID) in the nonrephasing direction kNP = k1 − k2 + k3. We denote the delay time between the first and second pulses as t1, that between the second and third pulses as t2 while the delay time between the third pulse and the measurement of the echo signal is denoted as t3. Again, ignoring the excited state lifetime and transition dipole rotational effects, the three pulse photon echo response functions can be expressed as41,62−64

∞

I(ω) ∼

∞

∫−∞ dt exp(−i(ω − ⟨ω10⟩)t )

I(ω) ∼ μ102

(10)

where (4)

φRP(t3 , t 2 , t1) = exp[i

We next make the commonly used second order cumulant expansion, which implies that the frequency fluctuation is in

∫0

t1

dτδω10(τ ) − i

t1+ t 2 + t3

∫t +t 1

dτδω10(τ )]

2

(11) 11217


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The Journal of Physical Chemistry B φNP(t3 , t 2 , t1) = exp[− i

∫0

t1


t1+ t 2 + t3

∫t +t 1

R 4(t3 , t 2 , t1) = R 5(t3 , t 2 , t1)

dτδω10(τ )]

2

= μ104 exp( −i⟨ω10⟩t3 − i⟨ω10⟩t1)

(12)

ψRP(t3 , t 2 , t1) = exp[i

∫0

t1


t1+ t 2 + t3

∫t +t 1

× exp( −g1(t1) − g1(t 2) − g1(t3) dτδω21(τ )]

+ g1(t 2 + t1) + g1(t 2 + t3)

2

(13)

ψNP(t3 , t 2 , t1) = exp[− i

∫0

t1


t1+ t 2 + t3

∫t +t 1

− g1(t1 + t 2 + t3))

dτδω21(τ )]

R 6(t3 , t 2 , t1) = −μ104 κ 2exp(iΔt3) exp( −i⟨ω21⟩t3 − i⟨ω10⟩t1)

2

(14)

× exp( −g1(t1) − g2(t 2) − g3(t3)

We note that the response functions R3 and R6 include contributions from 1−2 excitations as the second excited vibrational level (v = 2) also plays a relevant role in three pulse photon echo experiments. μ21(t) is the 1−2 transition dipole, and ω21(t) is the time dependent 1−2 transition frequency. The fluctuation δω21(t) = ω21(t) −⟨ω21 ⟩, where ⟨ω21 ⟩ is the average 1−2 transition frequency. Next, we will make two approximations similar to that made for the linear spectrum. The first one is the Condon approximation where it is assumed that the instantaneous transition dipoles are independent of nuclear coordinates, and hence can be treated as constants. Under this Condon approximation, the response functions of eqs 7−10 can be rewritten as41,62−64

+ g2(t 2 + t1) + g2(t 2 + t3) − g2(t1 + t 2 + t3))

=

g 2 (t ) =

(15)

R 4(t3 , t 2 , t1) = R 5(t3 , t 2 , t1) = μ104 exp(− i⟨ω10⟩t3 − i⟨ω10⟩t1)⟨φNP(t3 , t 2 , t1)⟩ (16) R3(t3 , t 2 , t1) (17)

R 6(t3 , t 2 , t1)

∫0

dτ″⟨δω21(0)δω10(τ″)⟩

(23)

∫0

t

dτ″⟨δω21(0)δω21(τ″)⟩

(24)

τ′

i=1

(25)

where Si(ω1 , t 2 , ω3)

+ i⟨ω10⟩t1)

∫0

∞

dt1

∫0

∞

dt3 exp(iω3t3 − iω1t1)R i(t1 , t 2 , t3)

(26)

dt3 exp(iω3t3 + iω1t1)R i(t1 , t 2 , t3)

(27)

for i = 1−3 and

× exp( −g1(t1) + g1(t 2) − g1(t3)

Si(ω1 , t 2 , ω3)

− g1(t 2 + t1) − g1(t 2 + t3)

= (19)

∫0

∞

dt1

∫0

∞

for i = 4−6, respectively. In integrated three pulse echo peak shift (3PEPS) experiments, the intensity along the rephasing direction integrated over t3 is measured. The expression of the integrated intensity is given by

R3(t3 , t 2 , t1) = −μ104 κ 2 exp(iΔt3) exp( −i⟨ω21⟩t3 + i⟨ω10⟩t1) × exp( −g1(t1) + g2(t 2) − g3(t3) − g2(t 2 + t1) − g2(t 2 + t3) + g2(t1 + t 2 + t3))

dτ ′

τ′

6

=

+ g1(t1 + t 2 + t3))

∫0

S(ω3 , t 2 , ω1) = Re[∑ Si(ω3 , t 2 , ω1)]

R1(t3 , t 2 , t1) = R 2(t3 , t 2 , t1) =

dτ ′

(18)

Next, we make the well-known second order cumulant approximation which is strictly valid when the underlying processes are Gaussian. Under this second order cumulant approximation, the expressions of the response functions can be further simplified as41,62−64

μ104 exp( −i⟨ω10⟩t3

t

In eqs 20 and 22, Δ = ⟨ω21 ⟩ − ⟨ω10 ⟩, and κ is equal to μ21/μ10. It may be noted that Δ is zero for a harmonic oscillator; hence, its nonzero value measures the extent of anharmonicity of the system. We note that, in experimental studies of the nonlinear infrared spectroscopy, the polarization is obtained as the convolution of the applied pulses with the response functions. In our present calculations, we assume delta function pulses so that the polarization is proportional to the total response function. For the 2D correlation spectroscopy, we calculate all the six response functions and then perform two-dimensional Fourier transformation to get the spectrum in frequency domain as expressed below41,62−64,85

exp(− i⟨ω10⟩t3 + i⟨ω10⟩t1)⟨φRP(t3 , t 2 , t1)⟩

= − μ102 μ212 exp(− i⟨ω21⟩t3 − i⟨ω10⟩t1)⟨ψNP(t3 , t 2 , t1)⟩

∫0

and g 3( t ) =

= − μ102 μ212 exp(− i⟨ω21⟩t3 + i⟨ω10⟩t1)⟨ψRP(t3 , t 2 , t1)⟩

(22)

where gn(t) (n = 1−3) is related to the time correlations of fluctuating transition frequencies. g1(t) is already defined by eq 6, and the expressions for g2(t) and g3(t) are given by

R1(t3 , t 2 , t1) = R 2(t3 , t 2 , t1) μ104

(21)

I(t1 , t 2) =

(20) 11218

∫0

∞

3

dt3 |∑ R i(t1 , t 2 , t3)|2 i=1

(28)


Article

The Journal of Physical Chemistry B The peak shift t*1 (t2) for a given t2 is defined as the value of t1 for which the echo intensity exhibits a maximum. An alternative to the extraction of frequency correlation function from the integrated echo is the calculation of initial slope of the integrated three-pulse echo intensity, known as the short-time slope of 3-pulse echo or S3PE, which can be mathematically expressed as S (t 2 ) =

∂I(t1 , t 2) |t1= 0 ∂t1

treatment of such large systems appears to be a highly challenging task at present. By simulating a neat liquid of 108 D2O molecules, we have 216 local OD modes of vibration rather than possibly just one if a dilute solution of HOD in H 2 O was taken. This clearly enhances the statistical convergence of our calculated results. We note that similar simulations of neat water with local mode approximation of the OH/OD stretch were also done in many of the earlier theoretical studies of spectral diffusion in water.17,40−43,50,59,60 From an experimental point of view, since the effects of vibrational population relaxation and resonant excitonic couplings34−37 are not incorporated in our current calculations, the present work can be better considered as an approximation of the spectral diffusion of HOD dissolved in H2O. We note, however, that earlier experiments9,28,31 have revealed comparable spectral diffusion dynamics of HOD in H2O and D2O systems which, in turn, can be linked to very similar hydrogen bond dynamics of H2O and D2O liquids at room temperature. Hence, the normalized dynamics of spectral diffusion of the current study can also be considered as approximate results for HOD in liquid D2O. Since the transition dipole moments are assumed to be constants within the Condon approximation employed here (hence, also for the cumulant approximation), we chose the value of the 0−1 transition dipole, μ10, to be unity in our calculations. We could safely use the above value since any precise numerical value of the transition dipole moment constant would contribute to the overall intensity of the calculated spectrum but would not affect the spectral features associated with the dynamics. The relative value of the higher order transition dipole moment of the chromophore, namely μ21, was determined within an anharmonic approximation of the OD stretch. As described in more detail in section 4.2, the higher order anharmonic states are determined by invoking the well-known Morse oscillator approximation for the OD stretch modes. Under this anharmonic description, the ratio of the 0−1 and 1−2 transition dipole moments, i.e., κ of eqs 20 and 22, is calculated to be 1.34. We note in passing that the value of the same quantity (κ) under harmonic approximation is 1.414.39

(29)

The normalized frequency−time correlation function is then given by86−88 C(t 2) =

S(t 2 ) S(0)

(30)

3. AB INITIO SIMULATIONS The ab initio molecular dynamics simulations have been performed by using the Car−Parrinello method77,78 and the CPMD code.89 The system consists of 108 D2O molecules in a cubic box of edge length 14.6 Å corresponding to the density of liquid water at 300 K. The electronic structure of the system was represented by Kohn−Sham formulation80 of density functional theory within a plane wave basis. The normconserving atomic pseudopotentials of Troullier−Martins90 were employed for the inner core electrons, and the plane wave expansion of the orbitals was truncated at the kinetic energy cutoff of 80 Ry. A fictitious mass of 800 au was assigned to the electronic orbital degrees of freedom, and the coupled equations of motion were integrated using a time step of 5 au In our simulations, the BLYP functional91,92 with dispersion corrections,93,94 the so-called BLYP-D functional, was chosen for electronic structure calculations. Although the BLYP functional predicts the structure of liquid water reasonably well, it does not properly account for atom−atom dispersion interactions which can be significant in many systems such as those having complex hydrogen bonded network. In fact, many recent studies95−105 have shown that inclusion of dispersion corrections provides better phase equilibria and also other equilibrium and dynamical properties of liquid water. We, therefore, incorporated the dispersion interactions as per the scheme suggested by Grimme.93,94 The initial configuration was generated using classical molecular dynamics simulation employing the SPC/E106 potential of water molecules. The classically equilibrated system was further equilibrated through ab initio molecular dynamics simulation in canonical ensemble for 15 ps at 300 K by employing the Nose−Hoover thermostat.107,108 For calculations of the dynamical and spectral properties, the system was then further simulated for another 55 ps in the microcanonical ensemble. After the simulation trajectory was generated, we carried out a time series analysis of all the OD modes to calculate their fluctuating frequencies. The details of the frequency calculations are provided in the next section. Unlike many of the previous experimental studies which considered the vibrational spectral diffusion of dilute solutions of isotopically labeled HOD in D2O or H2O,7−12,22−33 we have considered neat D2O where each OD mode is considered as a local chromophore for calculations of frequency fluctuations. Simulations of dilute HOD/H2O or HOD/D2O solutions necessarily require consideration of very large systems so as to have a sufficient number of HOD molecules, but ab initio

4. RESULTS 4.1. Frequency Fluctuations in the Ground State. The time dependent fluctuations in the frequency of OD modes due to environmental fluctuations are determined through a time series analysis of the trajectory obtained from ab initio molecular dynamics simulation. Details of such time series calculations of OD frequencies, where each OD mode is treated as a local oscillator, have been described in our earlier work;59,60 hence, we present only the key features here. We have used the method of wavelet analysis82 which uses a time dependent function that is expressed in terms of the translations and dilations of a mother wavelet ⎛t − b⎞ ⎟ ψa , b(t ) = a−1/2ψ ⎜ ⎝ a ⎠

(31)

The coefficients of the wavelet expansion are given by the wavelet transform of f(t) L ψ f (a , b) = a−1/2

+∞

∫−∞

⎛t − b⎞ ⎟ dt f (t )ψ ̅ ⎜ ⎝ a ⎠

(32)

where a and b are real quantities with a > 0. We have used the so-called Morlet−Grossman form109 for the mother wavelet. 11219


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three-pulse echo28 and 2D-IR27 experiments which are discussed in details in the next section. Theoretical assignments of the time scales of frequency correlation to different dynamical features have been done48,49,59,60,110 on the basis of the power spectrum of atomic velocity correlations and population correlation function formalism of hydrogen bond dynamics. When compared with earlier ab initio BLYP results,59,60 it is found that the inclusion of dispersion interaction does not alter the qualitative nature of the dynamics of frequency correlation. However, at quantitative level, the dispersion interactions are found to make the dynamics slightly faster. We have also calculated the first moment ⟨ω(t)⟩ for the two subensembles along the red and blue sides of the mean frequency in order to look at the frequency dependence of vibrational spectral diffusion, and the results are shown in Figure 1b. It is seen that the dynamics of the chromophores on the two different sides of the frequency distribution are not indistinguishable unlike that expected for a stationary Gaussian process that can be approximated within the second order cumulant approximation. Those on the red side seem to decay at a steeper rate at short times and are also accompanied by a distinct oscillatory pattern at sub-100 fs regime arising from underdamped motion with stronger hydrogen bonds. Such underdamped motion also marks its presence in the frequency−time correlation function in similar time domain. Thus, we can infer that the oscillatory kink in the frequency correlation function primarily originates from the distinct dynamics of water molecules on the red side of frequency distribution. In order to identify the pattern of frequency relaxation, we have calculated the two-dimensional joint probability distribution of fluctuating frequencies for different waiting times. The results of the joint probability distribution are shown in Figure 2. It is seen that, at short waiting times, the distribution is

The inverse of the scale factor a is proportional to the frequency, and thus the wavelet transform at each b gives the frequency content of f(t) over a time window about b. Following our earlier work,59,60 we consider the function f(t) for a local OD oscillator at time t to be a complex function with its real and imaginary parts corresponding to the instantaneous fluctuations in OD distance and the associated momentum. The stretch frequency of this bond at time t = b is then calculated from the scale a that gives the maximum modulus of the corresponding wavelet transform. The frequency calculations are then performed for all the OD oscillators present in the current simulation system. The decay of the time correlation function of OD frequency fluctuations is shown in Figure 1a. The correlation function

Figure 1. Time dependence of the (a) correlation function of frequency fluctuations of OD stretch modes in liquid water and (b) first moment of frequency along the red and blue sides of the vibrational spectrum of OD stretch modes of liquid water. The results of this figure and also of subsequent figures are obtained by treating each OD mode as a local mode of vibration in our calculations.

shows a biphasic decay. The associated time scales are determined by making a biexponential fit. Although the slight kink seen in the short time decay of the correlation function suggests that a damped triexponential fit could also have been used,48,49,59,60 the biexponential fit is preferred here because later, for the 2D-IR and 3PEPS metrics where no clearly noticeable kink is found, a biexponential fit is used to extract the time scales of spectral diffusion. The fitting function used is f (t ) = a0e−t / τ0 + (1 − a0)e−t / τ1

(33)

Figure 2. Joint probability distributions of observing frequencies ω1 and ω2 at a time gap of t2 for a given OD oscillator. The results are averaged over all the OD oscillators in the system.

where τ0 and τ1 are the two relaxation times having weights a0 and (1 − a0), respectively. The shorter time scale is found to be 90 fs and can be attributed to intact intermolecular stretching and bending motion of water molecules while the longer time scale of 1.91 ps is identified with the hydrogen bond lifetime of water molecules. These results generally agree with those of frequency correlations reported in a recent study for the same functional but for different system size and simulation protocol.105 It also generally agrees with the frequency correlation function extracted from experiments based on

almost linear indicating that the chromophore has not yet completely sampled all the possible local solvent environments. The distribution gets broadened with increase in waiting time, more so at the lower frequency part, and gradually for the waiting time of around 1800 fs, it becomes essentially spherical indicating that the spectral diffusion and inhomogeneous 11220


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the anharmonicity constant corresponding to a Morse oscillator at time t. With the dissociation energy (De) and anharmonicity parameter (xe) known, we then calculated the time dependent energy gaps corresponding to the 0 → 1 and 1 → 2 excitations. The results of such calculations for a selected part of the simulation trajectory are shown in Figure 3. The fluctuations of

broadening are complete. It may be noted that, for small waiting times, the probability distribution displays substantial asymmetric pattern with nonuniform elongation with respect to the red and blue sides of the mean frequency. This implies a breakdown of the assumption of stationary Gaussian statistics which is made under the cumulant approximation. In fact, this non-Gaussian behavior has been one of the motivations behind our going beyond the cumulant approximation and carrying out calculations of spectral observables at the Condon level which does not assume any Gaussian statistics for the underlying frequency fluctuations. 4.2. Fluctuations in Transition Frequencies. For the evaluation of the fluctuations in the transition frequencies for excitations to the first and second excited states, it is noted that the OD stretch vibration in liquid water has substantial anharmonic character. A good description of such an anharmonic oscillator can be made through the well-known Morse potential which is given by111 V (r ) = De[1 − e−a(r − re)]2

(34)

where De is the bond dissociation energy, re is the equilibrium bond length, and a is the anharmonic parameter that determines the curvature of the potential ⎛ μν 2 ⎞1/2 a=⎜ 2 ⎟ ⎝ 8π De ⎠

(35)

where ν is the harmonic oscillation frequency which is related to the curvature of the potential at its minimum by the following equation 1 ⎛ d2V (r ) ⎞ ν2 = ⎜ ⎟ 4π 2μ ⎝ dr 2 ⎠r = r

e

Figure 3. (a) Fluctuations in 0 → 1 and 1 → 2 vibrational transition frequencies of an OD oscillator over a selected period of the trajectory. (b) Instantaneous fluctuations in the anharmonicity Δ over the entire simulation trajectory. (c) Correlation plot of ω21 with ω10 for a given OD oscillator. The blue line in part c shows the hypothetical case of a harmonic oscillator where ω21 = ω10.

(36)

The main advantage of describing the OD anharmonic motion by a Morse oscillator is that its vibrational energy levels are known analytically which are given by111 Ev = (v + 1/2)hν − (v + 1/2)2 xehν

the calculated anharmonicity Δ (=ω10 − ω21) are also shown in this figure. The average value of Δ is calculated to be 157 cm−1 which is in very good agreement with the experimental value of 162 cm−1 for the anharmonicity of OD stretch vibration in liquid water.27 4.3. Echo Intensity and S3PE. Experimentally, heterodyned three pulse echo peak shift (3PEPS) studies constitute a powerful nonlinear spectroscopic technique which is employed to extract the time dependence of frequency correlation function and underlying dynamics at molecular level. However, an intrinsic disadvantage of the spectroscopic technique involving 3PEPS is that the extraction of slow time scales is often difficult on account of several parameters involved and intricate numerical fitting.86,112,113 Thus, in lieu of 3PEPS, several alternative observables have also been introduced to extract the frequency correlation function like short-time slope of three pulse echo (S3PE).87 The S3PE is defined as the slope of echo intensity at time t = 0. It has also been reported that, within impulsive limit, S3PE is better suited for the determination of frequency correlation function than the peak shift function. However, when the approximation of impulsive limit is relaxed, it behaves on similar lines as 3PEPS but still remains a useful alternative.88 Since we are working within impulsive limits, we resort to the S3PE metric for comparison with the dynamics of frequency−time correlation function.

(37)

where v is the vibrational quantum number and the anharmonicity constant xe = ν/4De. The frequency gaps ω10 and ω21, which appeared in the response functions R3(t3,t2,t1) and R6(t3,t2,t1), can be obtained from the energy levels of eq 37 by using the following relations: ω10 = (E1 − E0)/hc and ω21 = (E2 − E1)/hc. In the current work, we adopted a methodology wherein an implicit assumption was made that the hydrogen bond fluctuations in the surrounding environment alters the force constant or the curvature of the associated anharmonic potential, hence its fundamental oscillation frequency, but the dissociation energy of the covalent bond is assumed to remain essentially unaffected by environmental fluctuations. This appears to be a reasonable assumption since the dissociation energy of the OD covalent bond is known to be more than 2 orders of magnitude greater than typical frequency fluctuations caused by environmental fluctuations; hence, any change in its value due to thermal fluctuations in the surrounding environment is expected to be relatively very small. Thus, using the fundamental harmonic frequency obtained from the time series analysis at a given time as described in section 4.1 and a fixed value of the dissociation energy of the OD oscillator, which is 478 kJ mol−1 as obtained for a single water molecule through the same level of quantum chemical calculations, we estimated 11221


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The Journal of Physical Chemistry B We have also calculated the integrated echo intensity along the rephasing direction as a function of t2 or the population time for both cumulant and Condon approximations. The results are shown in Figure 4. It is found that initially, with the

Figure 5. Time dependence of the S3PE of vibrational echo spectroscopy of OD stretch in liquid water calculated within second order cumulant and Condon approximations.

inhomogeneous broadening. This can be seen from Figure 6 where the calculated linear IR spectrum of OD stretch modes is shown for both cumulant and Condon approximations. The corresponding experimental results of the absorption spectrum of OD stretch in liquid water21 are also shown in this figure. The frequency in this figure is expressed in terms of deviation

Figure 4. Time dependence of the integrated echo intensity of vibrational echo spectroscopy of OD stretch in liquid water calculated within (a) second order cumulant and (b) Condon approximations.

increase in population time, the intensity shows a surge which is popularly called vibrational echo. However, for waiting time of about 1 ps, the inhomogeneous broadening is found to have taken over the vibrational dynamics, and the intensity now essentially corresponds to normal free induction decay (FID) signal. The results of S3PE are shown in Figure 5. In order to obtain time scales from normalized S3PE, a numerical biexponential fit was performed. The biexponential fit to the normalized S3PE gives a short time scale of 76 fs and a longer time scale of around 1.95 ps. The S3PE calculated within the Condon approximation shows a faster relaxation on short as well as long time scales but stands in qualitative agreement with S3PE derived from the second order cumulant approximation. For the Condon approximation, the longer time scale is found to be 1.77 ps. These numbers both for Condon and cumulant approximations are in reasonable agreement with those extracted from FTCF discussed in section 4.1. It may be noted that a similar faster dynamics of spectral diffusion for the Condon approximation compared to that for the cumulant approximation was also found earlier in the work of ref 41 where an empirical potential was used to model liquid water. We note that the normalized S3PE captures the relaxation pattern of peak shifts with minimal deviations, and the time scales known from experiments for peak shifts28,30 are reasonably close to our findings for S3PE. These results reinforce the conclusion that peak shift or its analogues can be utilized experimentally to extract the time scales of vibrational spectral diffusion in aqueous systems.9,10,26−29 4.4. Linear and 2D-IR Spectra. The linear spectrum is sensitive to the distribution of eigenstates of a complex system. However, in water the intrinsic dynamics and its signature peaks in the spectrum get overshadowed under the

Figure 6. Linear absorption spectrum of OD stretch in liquid water calculated within the second order cumulant and Condon approximations. The experimental absorption spectrum21 is also shown. The values of the average frequency ⟨ω10⟩ are given in the text for the calculated cumulant, Condon, and experimental absorption spectrum. 11222


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The Journal of Physical Chemistry B from the average frequency ⟨ω10⟩, where the values of ⟨ω10⟩ are 2430, 2418, and 2500 cm−1 for the calculated cumulant, Condon, and experimental absorption spectrum, respectively. The wide inhomogeneously broadened spectrum does not allow identification of any intrinsic dynamical time scales of the system. The linear spectrum within cumulant approximation is a Gaussian centered at the mean OD stretch frequency. The calculated full width at half-maximum (fwhm) is found to be 162 (160) cm−1 for the cumulant (Condon) approximation which can be compared with the corresponding experimental result of 170 cm−1 as obtained from the absorption spectrum of ref 21. The calculated linear spectrum under Condon approximation is slightly asymmetric on the higher frequency side. There is a broad shoulder on the blue side of the spectrum which corresponds to the dangling and weakly hydrogen bonded OD modes in liquid water. As described before, the width of the spectrum is not informative about the different agents of broadening which manifest in the linear spectrum. A plausible way to overcome the difficulty is to perform four wave mixing experiments.5,114 The experiments provide a time window in which the system is allowed to relax dynamically, and by adjusting the length of the time window, we can determine the time scale at which the inhomogeneous broadening overtakes the natural relaxation dynamics of the system. The calculated 2D-IR spectra are shown in Figures 7

Figure 8. Two-dimensional infrared spectra of OD stretch in liquid water at different time delays calculated within Condon approximation.

scale of vibrational spectral relaxation in liquid water. The results of DLW are shown in Figure 9a for both the cumulant

Figure 7. Two-dimensional infrared spectra of OD stretch in liquid water at different time delays calculated within second order cumulant approximation. Note that the results of this figure and also of other figures of this work are obtained by treating each OD mode of deuterated water as a local mode of vibration in our calculations.

Figure 9. (a) Time dependence of the dynamic line width (DLW) of 2D-IR spectra of OD stretch in liquid water calculated within second order cumulant and Condon approximations. (b) Slope of the nodal line (SNL) of 2D-IR spectra of OD stretch in liquid water calculated within second order cumulant and Condon approximations.

and 8 for the cumulant and Condon approximations, respectively. The plots of Figures 6−8 show the limitations of the linear spectrum and the ability of the 2D-IR spectra to dynamically resolve the linear spectrum by eliminating the inhomogeneous broadening. The 2D-IR spectra reveal the time scales on which the system relaxes. In order to quantify the precise time scales of the underlying dynamics, several observables have been proposed such as the dynamic line width,27 nodal line,30 and central line slope.115 The dynamic line width (DLW) is defined as the full width at half-maximum (fwhm) of the linear projection of the 2D-IR spectrum corresponding to its maximum intensity at a given time. The time scale at which the DLW of the spectrum converges to the constant value can be attributed to the time

and Condon approximations. A biexponential fit gives the short time scale of 270 fs and a longer time scale of 2.4 ps for the cumulant approximation. For the Condon approximation, the DLW relaxes to its final value at a faster rate. The longer time scale for the latter case is found to be 2.1 ps. Similarly, the slope of the nodal line is defined as the slope of the line dividing the two contours corresponding to 0 → 1 and 1 → 2 vibrational excitations. The decay of the nodal line for different waiting times also captures the time scale of vibrational spectral diffusion time. The results of the nodal line are also shown in 11223


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The Journal of Physical Chemistry B Figure 9b. Unlike DLW which is more of a spectroscopic metric that directly captures the vibrational spectral relaxation as modification in fwhm, the nodal line is more of a geometric metric. Yet, the metric changes with time due to spectral diffusion. A biexponential fit to the normalized nodal slope gives the short time scale of 145 fs while the longer one is found to be around 1.92 ps for the cumulant approximation. The slope of nodal line under the Condon approximation gives the time scale of 1.58 ps for the slower relaxation constant. These results also seem to be in reasonably good agreement with the time scales predicted from frequency−time correlation function discussed earlier in section 4.1. Details of these time scales and associated weights are included in Table 1 for all the observable metrics calculated in this work. Table 1. Time Scales of Vibrational Spectral Diffusion Extracted from Various Metrics in the Present Worka metric

τ0 (fs)

τ1 (ps)

a0

FTCF S3PE DLW SNL

79 81 (69) 240 (260) 146 (94)

1.91 1.92 (1.77) 2.4 (2.1) 1.92 (1.58)

0.71 0.66 (0.70) 0.64 (0.81) 0.39 (0.55)

Figure 10. Comparison of the current theoretical results of (a) frequency−time correlation function (FTCF), (b) short-time slope of 3-pulse echo (S3PE), (c) dynamic line width (DLW), and (d) slope of nodal line (SNL) with the known experimental results. For FTCF, the experimental results are taken from ref 28 for experiment a and ref 27 for experiment b. The experimental results of peak shift, DLW, and Nodal slope are taken from refs 28, 27, and 30, respectively.

a

Time scales in parentheses correspond to the calculations done under Condon approximation while those out of parentheses are obtained within the cumulant approximation.

5. DISCUSSION In this section, we discuss our current results in the context of some of the recent experimental and other theoretical results of vibrational spectral diffusion in liquid water. Since each OD mode of liquid D2O of the current study has been considered as a local independent mode of vibration, the current results of frequency fluctuations can be meaningfully compared with the experimental results of OD modes of HOD in liquid H2O at high dilution where each such OD mode acts essentially as an independent local oscillator. Also, since the frequency fluctuations occur due to hydrogen bond fluctuations in the surrounding solvent, the normalized results of the current study can also be compared with those of OH modes of HOD in liquid D2O. Some of these comparisons with available experiments are shown in Figure 10. We note that isotopic mixtures of HOD in liquid D2O or H2O have been studied rather extensively using different experimental techniques like pump−probe, photon echo, and 2D-IR spectroscopy.10,22−33 Experiments based on pump−probe rely on the time dependent change in the absorption as a function of pump and probe frequencies which is then used to numerically extract the frequency correlation function and spectral diffusion time scale. Bakker and co-workers25 carried out several such experiments using mid-infrared beams to investigate the vibrational dephasing of HOD in liquid H2O. They estimated the vibrational spectral diffusion time of OD stretch to be 1.8 ps which can be compared with the time scale of 1.9 ps found for the decay of FTCF in the current work. The experimental results of FTCF as extracted from peak shift28 and 2D-IR experiments27 are included in Figure 10a. As can be seen, the overall experimental biphasic decay with a short time scale of about 100 fs and a longer time scale of 1.4 ps appears to be in general agreement with results of the current study. The differences in the decay pattern of the two sets of experimental results and the current theoretical results mainly arise from

differences in weight of the short-time component of FTCF. The time scales, however, are found to be rather similar. Next, we discuss the results of peak shift experiments on HOD in liquid D2O.28 In the current study, we have calculated the decay of S3PE which can be readily compared with the dynamics of peak shift28 as shown in Figure 10b. Again, the overall decay pattern predicted by the current theoretical study is found to be in good agreement with experimental results. The agreement appears to be even better for the Condon approximation. The experiments of 2D-IR spectroscopy usually report results of spectral diffusion through various metrics like dynamic line width (DLW), slope of nodal line (SNL), etc. Such 2D-IR studies of OD stretch in liquid H2O have been carried out by Fayer and co-workers,27 and their results are compared with the current ones in Figure 10c. The corresponding results of the slope of nodal line (SNL)30 are shown in Figure 10d. Again, the current results are in qualitative agreement with those of experiments. The experimental results of SNL are found to fall between the cumulant and Condon results of the current work. Apart from experiments, many theoretical studies have also been carried out in recent years to investigate the dynamics of vibrational spectral diffusion in liquid water. These simulations have investigated the relation between vibrational frequency fluctuations and rapidly evolving hydrogen bond network, vibrational hole dynamics, and nonlinear 3-pulse echo and 2DIR spectroscopic observables. All the existing studies of vibrational echo and 2D-ID spectroscopy involved dynamical simulations involving empirical potential models. For the purpose of comparison, we consider the simulation results of two water models, namely, SPCE106 and SPC-FQ116 which have been studied by Skinner and co-workers43 and another study by Jeon and Cho47 on OD stretch of HOD in H2O using the flexible SPC/Fw model117 of solvent water and a semiempirical quantum mechanical potential for the HOD 11224


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the second order cumulant and Condon approximations.41,62 It would be worthwhile to relax the existing levels of approximations and perform calculations at the non-Condon level41,43 and incorporate the excited state lifetime effects along with finite width of the laser pulses43 while the dynamics is generated by ab initio molecular dynamics without involving any empirical potential models. It would also be interesting to calculate ultrafast two-dimensional infrared anisotropy55,118−121 of water and aqueous solutions from first-principles simulations. Work in these directions is in progress.

solute. The simulations of ref 43 using the SPC/E model predicted a time scale of 0.98 ps which is faster than the corresponding results of both experiments and our current study. The SPC-FQ model produced a time scale of 1.45 ps which agrees very well with experiments and also closer to the current experimental results. The work of ref 47 reported a time scale of 1.6 ps for the decay of SNL which is in good agreement with the current result of 1.58 ps found for the Condon approximation.

■

6. SUMMARY AND CONCLUSIONS We have presented a theoretical study of vibrational echo and two-dimensional infrared spectroscopy of liquid water from first-principles simulations without using any empirical potentials for water−water interactions. Spectral calculations are done at two different levels of theoretical framework, namely, the second order cumulant and Condon approximations, by considering OD modes in liquid D2O as local oscillators. The time scales of spectral diffusion through different photon echo and 2D-IR observables like frequency− time correlation function (FTCF), short time slope of three pulse photon echo (S3PE), and 2D-IR observables such as slope of the nodal line (SNL) and dynamic line width (DLW) are calculated and are found to be in reasonably good agreement with the experiments and also with some of the earlier theoretical studies based on empirical potential models of water.10,25−29,31,41,43,47 However, slight variations in the time scales of these quantities are also found depending on the nature of the decay pattern of the respective observables. The long time decay of all the observables, which is around 1.5−2 ps, is related to inhomogeneous broadening and spectral diffusion in liquid water. This long time decay is determined by the hydrogen bond lifetime in the complex microscopic environment of liquid water. It is found that the Condon approximation leads to a faster dynamics as seen from the time scales derived from FTCF, S3PE, DLW, and SNL. Unlike the second order cumulant approximation which is strictly based on the assumption that the dynamics is a Gaussian process, Condon approximation directly incorporates the instantaneous frequency fluctuations along the time dependent frequency trajectory. Thus, the condon approximation includes the effects of non-Gaussian dynamics that are known to be present in aqueous systems.10,41,43,65−67 It is found that the time scales of spectral diffusion improve and get closer to the known experimental and other theoretical results as one moves from the second order cumulant to the Condon approximation. The two-dimensional joint probability plots also give indicative time scales which are in better agreement with those predicted from the correlation spectra calculated at Condon level without the assumption of Gaussian dynamics. This is expected since the results of the joint probability distributions clearly reveal the presence of substantial non-Gaussian features in the distributions. The loss of memory is found to be complete by about 2 ps as determined by the sphericity of both the 2D-IR spectrum and joint probability distributions. The current work can be extended to many other systems of chemical interest. For example, it would be interesting to perform similar calculations of vibrational echo spectroscopy of aqueous solutions of ionic and molecular solutes, including those containing protonic defects such as an excess proton or a hydroxide ion, from first-principles simulations. Also, the response functions in the current work are calculated within

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support from Department of Science and Technology (DST) and Council of Scientific and Industrial Research (CSIR), Government of India.

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Ultrafast Vibrational Echo Spectroscopy of Liquid Water from First

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