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Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo
Chemical Physics Letters 369 (2003) 107–113 www.elsevier.com/locate/cplett
Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo Sergey Yeremenko 1, Maxim S. Pshenichnikov *, Douwe A. Wiersma Ultrafast Laser and Spectroscopy Laboratory, Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received 2 August 2002; in final form 12 December 2002
Abstract Results of heterodyne-detected photon echo experiments on the OH stretching mode of water are reported and discussed. Two vibrational dynamical processes with time constants of 130 and 900 fs were identified. The former is attributed to bond breaking dynamics of a single hydrogen bond, the latter to rearrangement of the hydrogen-bond network. Ó 2003 Elsevier Science B.V. All rights reserved.
The distinctive feature of water compared to other fluids is its extended three-dimensional hydrogen-bond network, which determines to a large extent its peculiar physical, chemical, and biological properties [1]. In the liquid phase, hydrogen bonds are formed and broken continuously. Therefore, a grasp of liquid water dynamics is key to understanding its dynamical structure. Despite the fact that hydrogen-bond dynamics in water has been a subject of extensive research, there is still much to be learned [2]. Much of what we know on the structural dynamics in water comes from past studies of line
*
Corresponding author. Fax: +31-50-3634441. E-mail addresses: [email protected] (S. Yeremenko), [email protected] (M.S. Pshenichnikov). 1 Also corresponding author.
shapes of intra- and intermolecular vibrational transitions in the far- and mid-IR part of the optical spectrum of water [3–5]. Work of the past 25 years has shown that nonlinear spectroscopic experiments can greatly add to our understanding of condensed-phase dynamics beyond that what can be learned from linear spectroscopy. The application of nonlinear optical methods to the study of vibrational dynamics has been greatly stimulated recently by major advances in infrared femtosecond pulse technology. Because the OH-stretch vibration is directly involved in hydrogen bonding most dynamical studies are focused on this mode. To simplify the problem and exclude collective effects, quite often the OH-stretch of an HDO molecule diluted in heavy water is studied. The infrared (IR) absorption spectrum of HDO in D2 O depicted in Fig. 1, shows all the characteristics of strong hydrogen bonding: a prominent red shift, a complex line
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(02)02001-8
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Fig. 1. Absorption spectrum of O–H stretch vibration of the HDO molecule dissolved in heavy water (solid dots). Thick and thin solid curves present the results of simulations according to models described in the text. The filled peaks designate the positions and the relative intensities of the spectral lines corresponding to OH bonds involved (3400 cm1 ) and not involved ( 3570 cm1 ) in hydrogen bonding [6,7]. The spectrum of excitation pulses is shown by the shaded contour for comparison while the inset depicts its temporal intensity.
shape, and a large line width compared to the gas phase [6,7]. The spectral contour of this local mode can be excellently described, when a Gaussian distribution is assumed of intermolecular oxygen– oxygen (O–O) distances with a mean value of 2.83 ˚ with an rms deviation of 0.4 A (thin line in Fig. A 1). Note that in the simulation we have included the fact that the OH-frequency is not a linear function of the (O–O) distance [8]. Of course, this static description does not capture the underlying hydrogen-bond dynamics, where solvent motions sweep the oscillator frequencies across the whole width of the vibrational transition. To unveil such spectral dynamics, femtosecond nonlinear spectroscopy is needed. Pump–probe spectroscopy has been used extensively to explore OH-bond dynamics in water. For instance, spectral dynamics on a time scale of several picoseconds, obtained in early hole-burning experiments [9,10] were recently pushed into the sub-picosecond time domain [11]. In another pump–probe experiment, a vibrational relaxation time of 700 fs [12] was reported. Furthermore it was concluded that OH-vibrational oscillators undergo a sub-picosecond Stokes shift [13]. It was also
suggested that the hydrogen bond is the dominant accepting mode in vibrational relaxation [14]. A more powerful tool for the exploration of liquid state dynamics is photon echo. In this method a dynamic hologram in frequency space is created, whose fading with time contains information on spectral dynamics [15]. In particular femtosecond photon echo allows detection of fast dynamical processes that are hidden in the spectral contour determined by slower ones. Recently, the first time-integrated photon echo experiment on the OH-stretch vibration of an HDO molecule was reported [16,17]. The observed integrated twopulse echo signals were analyzed in terms of infinitely fast and infinitely slow spectral dynamics, corresponding to homogeneous and inhomogeneous line broadening processes. From the data a homogeneous dephasing time of T2 90 fs was extracted and assigned to a fluctuating solvent force pounding on an anharmonic vibrational oscillator. Although these pioneering echo experiments gave us a glimpse of the ongoing dynamics, the available time resolution of 130 fs and the lack of information on the temporal shape of the echo signal defeated detection of the ultrafast hydrogen-bond dynamics that is expected to occur at 100-fs timescale [18]. In this Letter we report on heterodyne-detected photon echo experiments on the OH stretching mode of HDO molecules, diluted in heavy water. Combining extremely short 70-fs excitation pulses with heterodyne detection of the transient nonlinear polarization enabled us to obtain a more detailed and distinctly different from [16] picture of the OH-vibrational dynamics. The experiments are well described in the framework of a stochastic modulation model by a bimodal vibrational correlation function that exhibits a fast decay of 130 fs and a slower one of 1 ps. Clearly, there is a separation of time scales, but not in terms of infinitely fast and infinitely slow modulation processes as concluded by Stenger et al. [16,17]. We assign the faster time scale to dynamics of a single hydrogen bond while the slower is attributed to reorganization of the hydrogen-bond network. IR pulses of 70 fs at a central wavelength of 3 lm with P10 lJ energy are generated at a 1-kHz repetition rate in a home-built 3-stage optical
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parametrical amplifier (OPA). The pulses are tuned such that their spectral content is positioned at the blue wing of the OH absorption spectrum (Fig. 1). This is done to minimize the effect of excited state absorption, which is red-shifted by 270 cm1 [9]. The amplitude and phase of IR pulses (Fig. 1, inset) are carefully characterized by frequency-resolved pump–probe measurements [19]. The OPA output is split into three parts. Two beams are focused into the sample with a 10-cm mirror to generate the photon echo signal at two phase-matched directions 2k2 k1 and 2k1 k2 . The third beam is aligned colinearly with the echo signal and serves as a local oscillator (LO) to heterodyne the photon echo [20]. Zero delay between all three pulses is determined with an 10-fs accuracy by simultaneous detection of two echoes and a pump–probe signal between the first pulse and the LO. The sample, a 0.6 M solution of HDO in D2 O at room temperature (with a maximum OD 0.5), is pumped through a 100-lm sapphire nozzle to form a free-standing jet. The integrated two-photon echo signal, measured with the LO blocked, is presented in Fig. 2 (solid circles). The nonresonant contribution to the signal – from pure D2 O – is also shown (open circles). The latter can be perfectly described by the third-order correlation function of the IR pulses (dashed line) confirming the instantaneous nature
Fig. 2. Integrated two-pulse photon echo signal. Solid and open dots represent the experimental data points obtained from HDO in D2 O and neat D2 O, respectively. The solid and dashed curves show the results of the simulations.
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of the D2 O response. In contrast, the echo signal is clearly asymmetric and shifted by about 35 fs from zero delay. A contour plot of the measured timeresolved photon echo signal is presented in Fig. 3a. This plot is generated as follows: for a fixed delay t12 between the excitation pulses, the delay of the LO is scanned and the heterodyned echo field recorded by an InSb detector. The resulting interference pattern is then numerically processed using a Fourier bandpass filter to extract the echo transient profile. The white line in Fig. 3a shows that with increasing delay t12 , the echo signal peaks at longer times, consistent with echo-formation. The position of the echo maximum considerably deviates from the one expected for an inhomogeneously broadened system (the dashed curve in Fig. 3a). This is a clear signature of nonMarkovian dynamics. To simulate the experimental data, a stochastic modulation model is used. In this model the transition frequency of the molecular vibration xðtÞ is modulated by bath fluctuations as xðtÞ ¼ xo þ dxðtÞ. Since the spectral width of excitation pulses (230 cm1 ) is comparable with the frequency difference of the 0–1 and 1–2 vibrational transitions (270 cm1 ), both transitions have to be taken into account. Hence, the system–field interaction is described by eight double-side Feynman diagrams, three of which are responsible for rephasing, three for nonrephasing, and two for two-photon processes [21,22]. The nonlinear response functions were adapted to account for a partial correlation in frequency fluctuations of the lower and upper transitions [23]. For each delay between the excitation pulses, the third-order polarization was calculated as a threefold integral comprising the relevant combinations of the experimentally measured electric field and response functions [23]. Because of a high sample OD, the excitation pulse amplitude and phase change substantially on propagation, which has to be accounted for in the simulations. To do this, the split-step Fourier algorithm was used to solve a parabolic wave equation with the third-order polarization as a right-side source term. Since the background (solvent) contribution amounts to as much as a one-fourth of the total integrated echo intensity (Fig. 2, open circles), it is necessary
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Fig. 3. The experimental (left panel) and simulated (right panel) time-resolved two-pulse photon echo profiles for HDO in D2 O. The white curve depicts the positions of the echo maxima. The dotted line is drawn if the overwhelming inhomogeneous broadening is assumed.
to incorporate this effect into the simulations as well. Previous experiments involving nonMarkovian optical dynamics [22,24] suggest as Ansatz a biexponential correlation function: MðtÞ ¼ hdxðtÞdxð0Þi ¼ D2fast expðKfast tÞ þ D2slow expðKslow tÞ: ð1Þ On basis of this correlation function a global fit to the available experimental data (time-resolved transients, integrated two-pulse photon echo and absorption spectrum) was made. The following parameters were obtained: 1=Kfast ffi130 30 fs, Dfast ffi75 cm1 (FWHMffi130 cm1 ), 1=Kslow ffi 900 250 fs, Dslow ffi65 cm1 (FWHM ffi 115 cm1 ). Figs. 1, 2b and 3 show that the correspondence between the model calculations and the experimental data is very good, except for the low-frequency tail of the absorption spectrum (vide infra). The solid line in Fig. 4 shows the correlation function as generated by a global fit of the data. Note that since the fastest dynamics occur on a time scale comparable to the pulse width, knowledge of the pulse phase and intensity is indispensable for a trustworthy analysis. The accuracy in determination of the slow time is hampered by the fact that the signal only lasts to 250 fs. Several comments have to be made on a simulation of the absorption spectrum. First, as Fig. 1 shows, there is a weak shoulder on the high-fre-
Fig. 4. The model correlation function as given by Eq. (1) (solid line) and with the fast part obtained by Fourier-transformation of the spectral density (solid dots). The inset shows spectral densities of anisotropic Raman polarizability [33] (solid dots) and of the fast part of the correlation function (solid line).
quency side of the spectrum. Its position (3570 cm1 ) coincides perfectly with the O–H stretch frequency in dilute solutions of HDO in acetonitrile (AN). While HDO is hydrogen-bonded to AN molecules there is no water cluster formation in this situation [25]. Therefore, the high-frequency shoulder can be interpreted as absorption from OH bonds that are not directly involved in tetrahedral water structures [6]. Assuming the dynamics of these molecules to be identical to the hydrogen bonded ones we calculate the number of such bonds to be about 9% (Fig. 1, thick solid line). This number is reasonably close
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to values obtained in MD simulations (12%) [6] and thermodynamical calculations (15%) [26] owing to the fact that symmetric lineshapes were used. Second, the noticeable discrepancy between simulation and experiment at the low-frequency side needs to be addressed. As pointed out earlier, the frequency of the OH mode is not a linear function of the O–O distance [8]. Yet the stochastic approach used assumes a linear coupling to exist between the vibrational oscillator and the bath. A better fit of this low-frequency wing waits for a stochastic theory that includes quadratic coupling. The departure from linearity, however, does not exceed 10% at FWHM of the spectral width. Therefore, the contribution of anharmonic coupling to the dynamics is expected to be minor and should not affect the correlation function derived [27]. Support for this belief comes from computer simulations that yield a vibrational dephasing time of the echo amplitude of T2 1 ps due to this effect, which exceeds the population lifetime (T1 0:7 ps). Finally, in our experiments we do not excite the spectral region of the OH-band where anharmonic coupling is most important. Therefore, we believe that the effect of nonlinear dependence of the O–H frequency on the O–O distance is minimal in the photon-echo experiment and can be safely neglected. From extensive numerical simulations we have also found that the effect of the 1–2 vibrational transition on the results is negligibly small due to substantial detuning of the excitation spectrum from the 1–2 transition absorption line. With the correlation function being established, the question is: what is the physics behind it. It follows from our analysis that the dynamics of the OH-stretch vibration in water occurs on two different time scales with approximately equal amplitudes. The slow part of the correlation function, at a picosecond time scale, has been observed before experimentally [11,13,17,28,29] and in a molecular dynamics simulations of water [6,7,30,31]. There is a general consensus (although with some reservations [32]) that this time scale is related to structural relaxation of the hydrogen-bond network, e.g., the time during which an initially bonded pair survives regardless to the fact whether
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the pair has been broken or not within this interval [6,7,31]. The fast part of the dynamics, however, has never been directly observed before. To elucidate the molecular dynamics at this time scale, it is instructive to calculate the spectral density generated by the experimental correlation function. Fig. 4 depicts this spectral density as a solid line, which, as the figure shows, strongly resembles the Raman spectrum of heavy water [33]. Its spectral components have been identified as librations, collective oscillations due to the hydrogen-bond network, and intermolecular collisions of water molecules. The HDO molecule, unlike any other probe species [22], is chemically identical to bulk D2 O and, therefore, the OH-stretch vibration is expected to be fully involved in all molecular interactions. For this reason, the spectral density as measured in the Raman experiment is an excellent candidate for the spectral density of bath fluctuations. When the Raman spectral density of heavy water is used to calculate the fast part of the correlation function, while keeping the slow part unchanged we find a correlation function (Fig. 4, solid circles) that is virtually indistinguishable from the one derived experimentally. Not surprisingly the simulated two-pulse photon echo, time-resolved transients, and absorption spectrum are practically identical to those presented in Figs. 1, 2b and 3. Recently, the fast relaxation process was inferred from frequency-domain spectroscopy with THz pulses [34]. The time scale of 150 fs with an amplitude of 3% was deduced assuming a Debye relaxation model. While the former number corroborates our direct measurements, the later one is lower by an order of magnitude, most probably, due to the limited spectral range (2 THz) of THz pulses. In order to demonstrate that for OH-bonds the connection between the correlation function and the spectral density of the anisotropic Raman polarizability is not a mere coincidence, we performed heterodyne-detected photon-echo experiments on HDO in AN. We found that the results can be described well by a single-exponential correlation function with a decay time of 300 fs fully consistent with the anisotropic Raman data on AN. The measured time scale of 130 fs is similar to the so-called average hydrogen-bond lifetime
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(an interval during which an initially bonded pair gets broken) obtained in a number of MD simulations [6,7,31]. Therefore, on the basis of MD simulations and our experiments, the following picture of liquid water dynamics emerges. The correlation time of 130 fs corresponds to the time scale at which a single hydrogen bond of a four-bonds-connected molecule is disturbed by the local environment. With a frequency excursion of 130 cm1 associated with this time scale, 20% of water molecules have frequencies identical to those of the free molecule (3570 cm1 ). For some of these molecules the current bond is broken with either subsequent new bond formation or restoration of the old one. In either case, phase coherence needed for echo formation is unlikely to survive such perturbations, which results in nonMarkovian dynamics of the echo formation (Fig. 3). However, although a local breakdown of the hydrogen-bond network occurs at a 130-fs time scale (the average hydrogen-bond lifetime), its global structure remains intact. It takes 1 ps (the structural relaxation time) before the initial backbone loses its integrity as a consequence of several hydrogen bonds being broken. Summarizing, we have presented a detailed photon-echo study of the OH-stretch vibration dynamics in liquid water. We have shown that water dynamics occur at time scales of 130 and 900 fs. The fast time scale has been assigned to the lifetime of a single hydrogen bond. The slow time scale is assigned to a structural relaxation time of the hydrogen-bond network surrounding a particular molecule. In current experiments, a diluted sample of HDO in heavy water is used. This allows separation of single-molecule from collective dynamics. In neat water, phenomena like F€ orster energy transfer [35] or exciton formation are expected to play a role. To obtain more detailed information on such phenomena, heterodyne-detected photon echo experiments need to be done, where next to the amplitude also the phase of the nonlinear optical polarization is measured [15]. The use of diffractive optics in photon echo experiments seems very promising because of inherently locked phases of excitation pulses.
Acknowledgements We acknowledge financial support from FOM (Fundamenteel Onderzoek der Materie). We thank P. Zemtsov for valuable advice on computer code optimization.
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Hydrogen-bond dynamics in water explored by heterodyne-detected photon echo Sergey Yeremenko 1, Maxim S. Pshenichnikov *, Douwe A. Wiersma Ultrafast Laser and Spectroscopy Laboratory, Materials Science Centre, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received 2 August 2002; in final form 12 December 2002
Abstract Results of heterodyne-detected photon echo experiments on the OH stretching mode of water are reported and discussed. Two vibrational dynamical processes with time constants of 130 and 900 fs were identified. The former is attributed to bond breaking dynamics of a single hydrogen bond, the latter to rearrangement of the hydrogen-bond network. Ó 2003 Elsevier Science B.V. All rights reserved.
The distinctive feature of water compared to other fluids is its extended three-dimensional hydrogen-bond network, which determines to a large extent its peculiar physical, chemical, and biological properties [1]. In the liquid phase, hydrogen bonds are formed and broken continuously. Therefore, a grasp of liquid water dynamics is key to understanding its dynamical structure. Despite the fact that hydrogen-bond dynamics in water has been a subject of extensive research, there is still much to be learned [2]. Much of what we know on the structural dynamics in water comes from past studies of line
*
Corresponding author. Fax: +31-50-3634441. E-mail addresses: [email protected] (S. Yeremenko), [email protected] (M.S. Pshenichnikov). 1 Also corresponding author.
shapes of intra- and intermolecular vibrational transitions in the far- and mid-IR part of the optical spectrum of water [3–5]. Work of the past 25 years has shown that nonlinear spectroscopic experiments can greatly add to our understanding of condensed-phase dynamics beyond that what can be learned from linear spectroscopy. The application of nonlinear optical methods to the study of vibrational dynamics has been greatly stimulated recently by major advances in infrared femtosecond pulse technology. Because the OH-stretch vibration is directly involved in hydrogen bonding most dynamical studies are focused on this mode. To simplify the problem and exclude collective effects, quite often the OH-stretch of an HDO molecule diluted in heavy water is studied. The infrared (IR) absorption spectrum of HDO in D2 O depicted in Fig. 1, shows all the characteristics of strong hydrogen bonding: a prominent red shift, a complex line
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(02)02001-8
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Fig. 1. Absorption spectrum of O–H stretch vibration of the HDO molecule dissolved in heavy water (solid dots). Thick and thin solid curves present the results of simulations according to models described in the text. The filled peaks designate the positions and the relative intensities of the spectral lines corresponding to OH bonds involved (3400 cm1 ) and not involved ( 3570 cm1 ) in hydrogen bonding [6,7]. The spectrum of excitation pulses is shown by the shaded contour for comparison while the inset depicts its temporal intensity.
shape, and a large line width compared to the gas phase [6,7]. The spectral contour of this local mode can be excellently described, when a Gaussian distribution is assumed of intermolecular oxygen– oxygen (O–O) distances with a mean value of 2.83 ˚ with an rms deviation of 0.4 A (thin line in Fig. A 1). Note that in the simulation we have included the fact that the OH-frequency is not a linear function of the (O–O) distance [8]. Of course, this static description does not capture the underlying hydrogen-bond dynamics, where solvent motions sweep the oscillator frequencies across the whole width of the vibrational transition. To unveil such spectral dynamics, femtosecond nonlinear spectroscopy is needed. Pump–probe spectroscopy has been used extensively to explore OH-bond dynamics in water. For instance, spectral dynamics on a time scale of several picoseconds, obtained in early hole-burning experiments [9,10] were recently pushed into the sub-picosecond time domain [11]. In another pump–probe experiment, a vibrational relaxation time of 700 fs [12] was reported. Furthermore it was concluded that OH-vibrational oscillators undergo a sub-picosecond Stokes shift [13]. It was also
suggested that the hydrogen bond is the dominant accepting mode in vibrational relaxation [14]. A more powerful tool for the exploration of liquid state dynamics is photon echo. In this method a dynamic hologram in frequency space is created, whose fading with time contains information on spectral dynamics [15]. In particular femtosecond photon echo allows detection of fast dynamical processes that are hidden in the spectral contour determined by slower ones. Recently, the first time-integrated photon echo experiment on the OH-stretch vibration of an HDO molecule was reported [16,17]. The observed integrated twopulse echo signals were analyzed in terms of infinitely fast and infinitely slow spectral dynamics, corresponding to homogeneous and inhomogeneous line broadening processes. From the data a homogeneous dephasing time of T2 90 fs was extracted and assigned to a fluctuating solvent force pounding on an anharmonic vibrational oscillator. Although these pioneering echo experiments gave us a glimpse of the ongoing dynamics, the available time resolution of 130 fs and the lack of information on the temporal shape of the echo signal defeated detection of the ultrafast hydrogen-bond dynamics that is expected to occur at 100-fs timescale [18]. In this Letter we report on heterodyne-detected photon echo experiments on the OH stretching mode of HDO molecules, diluted in heavy water. Combining extremely short 70-fs excitation pulses with heterodyne detection of the transient nonlinear polarization enabled us to obtain a more detailed and distinctly different from [16] picture of the OH-vibrational dynamics. The experiments are well described in the framework of a stochastic modulation model by a bimodal vibrational correlation function that exhibits a fast decay of 130 fs and a slower one of 1 ps. Clearly, there is a separation of time scales, but not in terms of infinitely fast and infinitely slow modulation processes as concluded by Stenger et al. [16,17]. We assign the faster time scale to dynamics of a single hydrogen bond while the slower is attributed to reorganization of the hydrogen-bond network. IR pulses of 70 fs at a central wavelength of 3 lm with P10 lJ energy are generated at a 1-kHz repetition rate in a home-built 3-stage optical
S. Yeremenko et al. / Chemical Physics Letters 369 (2003) 107–113
parametrical amplifier (OPA). The pulses are tuned such that their spectral content is positioned at the blue wing of the OH absorption spectrum (Fig. 1). This is done to minimize the effect of excited state absorption, which is red-shifted by 270 cm1 [9]. The amplitude and phase of IR pulses (Fig. 1, inset) are carefully characterized by frequency-resolved pump–probe measurements [19]. The OPA output is split into three parts. Two beams are focused into the sample with a 10-cm mirror to generate the photon echo signal at two phase-matched directions 2k2 k1 and 2k1 k2 . The third beam is aligned colinearly with the echo signal and serves as a local oscillator (LO) to heterodyne the photon echo [20]. Zero delay between all three pulses is determined with an 10-fs accuracy by simultaneous detection of two echoes and a pump–probe signal between the first pulse and the LO. The sample, a 0.6 M solution of HDO in D2 O at room temperature (with a maximum OD 0.5), is pumped through a 100-lm sapphire nozzle to form a free-standing jet. The integrated two-photon echo signal, measured with the LO blocked, is presented in Fig. 2 (solid circles). The nonresonant contribution to the signal – from pure D2 O – is also shown (open circles). The latter can be perfectly described by the third-order correlation function of the IR pulses (dashed line) confirming the instantaneous nature
Fig. 2. Integrated two-pulse photon echo signal. Solid and open dots represent the experimental data points obtained from HDO in D2 O and neat D2 O, respectively. The solid and dashed curves show the results of the simulations.
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of the D2 O response. In contrast, the echo signal is clearly asymmetric and shifted by about 35 fs from zero delay. A contour plot of the measured timeresolved photon echo signal is presented in Fig. 3a. This plot is generated as follows: for a fixed delay t12 between the excitation pulses, the delay of the LO is scanned and the heterodyned echo field recorded by an InSb detector. The resulting interference pattern is then numerically processed using a Fourier bandpass filter to extract the echo transient profile. The white line in Fig. 3a shows that with increasing delay t12 , the echo signal peaks at longer times, consistent with echo-formation. The position of the echo maximum considerably deviates from the one expected for an inhomogeneously broadened system (the dashed curve in Fig. 3a). This is a clear signature of nonMarkovian dynamics. To simulate the experimental data, a stochastic modulation model is used. In this model the transition frequency of the molecular vibration xðtÞ is modulated by bath fluctuations as xðtÞ ¼ xo þ dxðtÞ. Since the spectral width of excitation pulses (230 cm1 ) is comparable with the frequency difference of the 0–1 and 1–2 vibrational transitions (270 cm1 ), both transitions have to be taken into account. Hence, the system–field interaction is described by eight double-side Feynman diagrams, three of which are responsible for rephasing, three for nonrephasing, and two for two-photon processes [21,22]. The nonlinear response functions were adapted to account for a partial correlation in frequency fluctuations of the lower and upper transitions [23]. For each delay between the excitation pulses, the third-order polarization was calculated as a threefold integral comprising the relevant combinations of the experimentally measured electric field and response functions [23]. Because of a high sample OD, the excitation pulse amplitude and phase change substantially on propagation, which has to be accounted for in the simulations. To do this, the split-step Fourier algorithm was used to solve a parabolic wave equation with the third-order polarization as a right-side source term. Since the background (solvent) contribution amounts to as much as a one-fourth of the total integrated echo intensity (Fig. 2, open circles), it is necessary
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Fig. 3. The experimental (left panel) and simulated (right panel) time-resolved two-pulse photon echo profiles for HDO in D2 O. The white curve depicts the positions of the echo maxima. The dotted line is drawn if the overwhelming inhomogeneous broadening is assumed.
to incorporate this effect into the simulations as well. Previous experiments involving nonMarkovian optical dynamics [22,24] suggest as Ansatz a biexponential correlation function: MðtÞ ¼ hdxðtÞdxð0Þi ¼ D2fast expðKfast tÞ þ D2slow expðKslow tÞ: ð1Þ On basis of this correlation function a global fit to the available experimental data (time-resolved transients, integrated two-pulse photon echo and absorption spectrum) was made. The following parameters were obtained: 1=Kfast ffi130 30 fs, Dfast ffi75 cm1 (FWHMffi130 cm1 ), 1=Kslow ffi 900 250 fs, Dslow ffi65 cm1 (FWHM ffi 115 cm1 ). Figs. 1, 2b and 3 show that the correspondence between the model calculations and the experimental data is very good, except for the low-frequency tail of the absorption spectrum (vide infra). The solid line in Fig. 4 shows the correlation function as generated by a global fit of the data. Note that since the fastest dynamics occur on a time scale comparable to the pulse width, knowledge of the pulse phase and intensity is indispensable for a trustworthy analysis. The accuracy in determination of the slow time is hampered by the fact that the signal only lasts to 250 fs. Several comments have to be made on a simulation of the absorption spectrum. First, as Fig. 1 shows, there is a weak shoulder on the high-fre-
Fig. 4. The model correlation function as given by Eq. (1) (solid line) and with the fast part obtained by Fourier-transformation of the spectral density (solid dots). The inset shows spectral densities of anisotropic Raman polarizability [33] (solid dots) and of the fast part of the correlation function (solid line).
quency side of the spectrum. Its position (3570 cm1 ) coincides perfectly with the O–H stretch frequency in dilute solutions of HDO in acetonitrile (AN). While HDO is hydrogen-bonded to AN molecules there is no water cluster formation in this situation [25]. Therefore, the high-frequency shoulder can be interpreted as absorption from OH bonds that are not directly involved in tetrahedral water structures [6]. Assuming the dynamics of these molecules to be identical to the hydrogen bonded ones we calculate the number of such bonds to be about 9% (Fig. 1, thick solid line). This number is reasonably close
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to values obtained in MD simulations (12%) [6] and thermodynamical calculations (15%) [26] owing to the fact that symmetric lineshapes were used. Second, the noticeable discrepancy between simulation and experiment at the low-frequency side needs to be addressed. As pointed out earlier, the frequency of the OH mode is not a linear function of the O–O distance [8]. Yet the stochastic approach used assumes a linear coupling to exist between the vibrational oscillator and the bath. A better fit of this low-frequency wing waits for a stochastic theory that includes quadratic coupling. The departure from linearity, however, does not exceed 10% at FWHM of the spectral width. Therefore, the contribution of anharmonic coupling to the dynamics is expected to be minor and should not affect the correlation function derived [27]. Support for this belief comes from computer simulations that yield a vibrational dephasing time of the echo amplitude of T2 1 ps due to this effect, which exceeds the population lifetime (T1 0:7 ps). Finally, in our experiments we do not excite the spectral region of the OH-band where anharmonic coupling is most important. Therefore, we believe that the effect of nonlinear dependence of the O–H frequency on the O–O distance is minimal in the photon-echo experiment and can be safely neglected. From extensive numerical simulations we have also found that the effect of the 1–2 vibrational transition on the results is negligibly small due to substantial detuning of the excitation spectrum from the 1–2 transition absorption line. With the correlation function being established, the question is: what is the physics behind it. It follows from our analysis that the dynamics of the OH-stretch vibration in water occurs on two different time scales with approximately equal amplitudes. The slow part of the correlation function, at a picosecond time scale, has been observed before experimentally [11,13,17,28,29] and in a molecular dynamics simulations of water [6,7,30,31]. There is a general consensus (although with some reservations [32]) that this time scale is related to structural relaxation of the hydrogen-bond network, e.g., the time during which an initially bonded pair survives regardless to the fact whether
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the pair has been broken or not within this interval [6,7,31]. The fast part of the dynamics, however, has never been directly observed before. To elucidate the molecular dynamics at this time scale, it is instructive to calculate the spectral density generated by the experimental correlation function. Fig. 4 depicts this spectral density as a solid line, which, as the figure shows, strongly resembles the Raman spectrum of heavy water [33]. Its spectral components have been identified as librations, collective oscillations due to the hydrogen-bond network, and intermolecular collisions of water molecules. The HDO molecule, unlike any other probe species [22], is chemically identical to bulk D2 O and, therefore, the OH-stretch vibration is expected to be fully involved in all molecular interactions. For this reason, the spectral density as measured in the Raman experiment is an excellent candidate for the spectral density of bath fluctuations. When the Raman spectral density of heavy water is used to calculate the fast part of the correlation function, while keeping the slow part unchanged we find a correlation function (Fig. 4, solid circles) that is virtually indistinguishable from the one derived experimentally. Not surprisingly the simulated two-pulse photon echo, time-resolved transients, and absorption spectrum are practically identical to those presented in Figs. 1, 2b and 3. Recently, the fast relaxation process was inferred from frequency-domain spectroscopy with THz pulses [34]. The time scale of 150 fs with an amplitude of 3% was deduced assuming a Debye relaxation model. While the former number corroborates our direct measurements, the later one is lower by an order of magnitude, most probably, due to the limited spectral range (2 THz) of THz pulses. In order to demonstrate that for OH-bonds the connection between the correlation function and the spectral density of the anisotropic Raman polarizability is not a mere coincidence, we performed heterodyne-detected photon-echo experiments on HDO in AN. We found that the results can be described well by a single-exponential correlation function with a decay time of 300 fs fully consistent with the anisotropic Raman data on AN. The measured time scale of 130 fs is similar to the so-called average hydrogen-bond lifetime
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(an interval during which an initially bonded pair gets broken) obtained in a number of MD simulations [6,7,31]. Therefore, on the basis of MD simulations and our experiments, the following picture of liquid water dynamics emerges. The correlation time of 130 fs corresponds to the time scale at which a single hydrogen bond of a four-bonds-connected molecule is disturbed by the local environment. With a frequency excursion of 130 cm1 associated with this time scale, 20% of water molecules have frequencies identical to those of the free molecule (3570 cm1 ). For some of these molecules the current bond is broken with either subsequent new bond formation or restoration of the old one. In either case, phase coherence needed for echo formation is unlikely to survive such perturbations, which results in nonMarkovian dynamics of the echo formation (Fig. 3). However, although a local breakdown of the hydrogen-bond network occurs at a 130-fs time scale (the average hydrogen-bond lifetime), its global structure remains intact. It takes 1 ps (the structural relaxation time) before the initial backbone loses its integrity as a consequence of several hydrogen bonds being broken. Summarizing, we have presented a detailed photon-echo study of the OH-stretch vibration dynamics in liquid water. We have shown that water dynamics occur at time scales of 130 and 900 fs. The fast time scale has been assigned to the lifetime of a single hydrogen bond. The slow time scale is assigned to a structural relaxation time of the hydrogen-bond network surrounding a particular molecule. In current experiments, a diluted sample of HDO in heavy water is used. This allows separation of single-molecule from collective dynamics. In neat water, phenomena like F€ orster energy transfer [35] or exciton formation are expected to play a role. To obtain more detailed information on such phenomena, heterodyne-detected photon echo experiments need to be done, where next to the amplitude also the phase of the nonlinear optical polarization is measured [15]. The use of diffractive optics in photon echo experiments seems very promising because of inherently locked phases of excitation pulses.
Acknowledgements We acknowledge financial support from FOM (Fundamenteel Onderzoek der Materie). We thank P. Zemtsov for valuable advice on computer code optimization.
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