Investigating pure vibrational dephasing of I 3 − in solution:Temperature dependenceof T 2 * for the fundamental and first harmonic of ν 1Erez Gershgoren, Zhaohui Wang, Sanford Ruhman, Jiri Vala, and Ronnie Kosloff Citation: The Journal of Chemical Physics 118, 3660 (2003); doi: 10.1063/1.1539844 View online: http://dx.doi.org/10.1063/1.1539844 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/118/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Temperature and solvent dependence of vibrational relaxation of tri-iodide: A simulation study J. Chem. Phys. 119, 6119 (2003); 10.1063/1.1603718 Br − - H 2 and I − - H 2 anion complexes: Infrared spectra and radial intermolecular potential energy curves J. Chem. Phys. 117, 3256 (2002); 10.1063/1.1486435 Femtosecond photoelectron spectroscopy of I 2 − ( Ar ) n clusters (n=6,9,12,16,20) J. Chem. Phys. 111, 10566 (1999); 10.1063/1.480433 Vibrational relaxation of I 2 in complexing solvents: The role of solvent–solute attractive forces J. Chem. Phys. 109, 9494 (1998); 10.1063/1.477611 Structure and dynamics of molecular ions in clusters: I 2 − in flexible CO 2 J. Chem. Phys. 107, 9326 (1997); 10.1063/1.475229

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Investigating pure vibrational dephasing of I 3À in solution: Temperature

dependence of T2* for the fundamental and first harmonic of n1

Erez Gershgoren, Zhaohui Wang, and Sanford Ruhmana)

Department of Physical Chemistry and the Farkas Center for Light Induced Processes,The Hebrew University, Jerusalem 91904, Israel

Jiri Vala and Ronnie Kosloffa)

Department of Physical Chemistry, and The Fritz Haber Center for Chemical Dynamics, The HebrewUniversity, Jerusalem 91904, Israel University, 91904, Israel

~Received 2 August 2002; accepted 27 November 2002!

Pure n1 vibrational dephasing of triiodide is recorded in ethanol and methyl-tetrahydrofuranesolutions from 300 to 100 K, for the vibrational fundamental and its first overtone. Using impulsiveRaman spectroscopy, dephasing is demonstrated to be homogeneous throughout the temperaturerange studied. Independent measures ofT1 prove that population relaxation contributes negligiblyto the dephasing rates. The reduction in temperature gradually leads to a;2-fold decrease in the rateof pure dephasing. With cooling the ratio ofT2(n51)* /T2(n52)* reduces slightly but remains in therange of 2.7 to 2. These results are discussed in terms of Kubo lineshape and Poisson dephasingtheories. Neither of these consistently explains the experimental observations assuming reasonableintensities and rates of intermolecular encounters in the solutions. ©2003 American Institute ofPhysics. @DOI: 10.1063/1.1539844#

I. INTRODUCTION

Vibrational relaxation is a subject of both practical andconceptual importance. It is the key to vital stages of chemi-cal reactivity, such as thermal activation of transitions be-tween stable ground state molecular structures, and the sub-sequent dispersal of excess energy and stabilization of thereactants.1 Conceptually, it provides a critical testing groundfor theoretical models for intermolecular interactions in con-densed phases.2 Accordingly, IR and Raman lineshapes,3 aswell as ultrafast laser spectroscopy,4,5 have been applied torecord vibrational energy and phase relaxation of molecularsolutes. Even in the limited context of relaxation involvingthe lowest rungs of a vibrational ladder, analysis of experi-mental results can prove challenging. To begin with, energyand phase relaxation are intertwined, through the well-known relation

1/T251/2T111/T2* , ~1!

where T2 and T2* represent observed and pure dephasingtimes, respectively, andT1 is the inverse rate of populationdecay. It is therefore almost always necessary to investigateboth processes in order to delineate the various contributionsto the linewidth or equivalently to the dephasing time whenmeasured directly.

In liquids, pure dephasing often dominates the decay ofvibrational coherence. It is almost universally described interms of a stochastic model, assuming that solvent-inducedshifts in the vibrational frequencyvv cause the oscillators tolose step with each other. The distribution of local solventarrangements is envisioned to generate a Gaussian spread of

instantaneous frequencies, characterized by a rms width ofD.The solvent motions induce spectral diffusion that effectivelyrandomizes the oscillator frequency on a characteristic aver-age time scaletc . After assuming that the spectral modula-tion is a Gaussian process, the results of the analysis are asfollows: When the spectral diffusion is much slower than 1/D(Dtc@1), the coherence decay is Gaussian, dictated bystatic inhomogeneous broadening according to Eq.~2!,

^Q~ t !Q~0!&;exp~2t2D2/2! ~2!

~whereQ is the displacement coordinate!. At the other ex-treme, whenDtc!1, extensive spectral diffusion is going onbefore the phase is washed out, the vibrational coherencewill be erased exponentially~Lorentzian lineshape!, accord-ing to

^Q~ t !Q~0!&;exp~2D2t tc!. ~3!

This is the ‘‘motional narrowing,’’ or ‘‘fast modulation’’ limitof the model, originally formulated by Kubo6 to explainmagnetic resonance data, and later adapted to vibrational re-laxation by Oxtoby.2 In the midrange between the two, ahybrid dynamics of dephasing is predicted starting outGaussian, and reverting to an exponential tail at longerdelays.7 When the autocorrelation of the frequency is as-sumed to decay exponentially (Cv;e2t/tc), this range canbe described analytically:

^Q~ t !Q~ t !&5exp$2D2@tc2~e2t/tc21!1ttc#%, ~4!

consistently breaching the limits of inhomogeneous and ho-mogeneous broadening.

a!Authors to whom correspondence should be addressed. Electronic mail:sandy@fh.huji.ac.il

JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 8 22 FEBRUARY 2003

36600021-9606/2003/118(8)/3660/8/$20.00 © 2003 American Institute of Physics

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The model described above has proven most valuable inrationalizing the microscopic mechanisms that underliesolvent-induced pure dephasing. Within its context predic-tions have been made concerning the changes ofT2* withshifts in temperature, density, or viscosity.8–10 For specificmolecular systems the values ofD and oftc have been de-termined and compared with theoretical predictions. Oneprediction of this model that has not been as thoroughlytested deals with the relative pure dephasing rates of vibra-tional harmonics. The modulation mechanism must be inde-pendent of the observed harmonic and thereforetc is identi-cal for coherence at all harmonics ofvn . The spread offrequencies, however, is a different story, and does dependon n, the order of the measured coherence:

D~nth harmonic![Dn;n•D. ~5!

Thus, in the fast modulation limit,T2* (n)51/Dn2 tc

51/D2n2 tc , and depends quadratically onn. In the staticlimit, T2* (n);1/Dn51/nD, leading to a linear dependence.

The exposition above predicts a quadratic dependence ofthe rate of dephasing on the harmonic order of the coherencein the fast modulation limit. Subquadratic behavior can alsobe expected, but only when the phase decay is clearly non-exponential. In that case, when neither of the limiting casesdominates the dynamics, no clear definition of a ‘‘dephasingtime’’ exists. Strangely enough, in the majority of casestested, a subquadratic dependence ofT2* on n has beenobtained,11–13 even when the conditions for fast modulationwere demonstrated to hold, and exponential phase decay wasmeasured at all harmonics.14 In a recent paper, Gayathriet al. have tried to provide a mechanistic explanation withinthe Kubo lineshape model to subquadraticn dependence ofmolecular nitrogen in various thermodynamic states, sug-gesting that coupling of vibration and rotation provides anadditional inhomogeneous contribution to the broadening,not taken into consideration in the usual application of thetheory.15 A multiphased decay of ^v(0)v(t)&, thefrequency—frequency time autocorrelation function, overseparable and very disparate phases in time, was also dis-cussed as a mechanism for the nonideal relative dephasingrates.

An alternative interpretation of the observations de-scribed above is presented in a theoretical study byYamaguchi.16 A model is formulated where collisions in theliquid can induce sudden phase jumps in the vibrations, andrather than being the results of many slight modulations, jus-tifying the Gaussian approximation, the phase relaxation isthe result of uncorrelated and large changes of vibrationalphase. Mathematically this leads to Poisson dephasing dy-namics. This model is also characterized by two parameters,an average rate of phase changing interactions (g51/t i),and the typical size of the jump in phase (df0). Two addi-tional assumptions are applied, as presented in Eq.~6!:

H P~df!5e2df/df0/df0 , for df>0 ~6a!

and P~df!50, for df,0 ~6b!

Pi~m,t !51

m! S t

t iDe2t/t i

.

The exponential distribution of phase jumps reflects theirthermally activated nature, and Eq.~6b! describes the prob-ability of our solute experiencingm collisions within timespant.

The resulting dephasing dynamics differ markedly fromthe Kubo model. First, phase relaxation is always exponen-tial. Second, the dephasing rates of harmonics range fromquadraticn dependence when the jumps are small~back tothe Gaussian regime!, to zeron dependence when the jumpsare large. At this limit, the collisions are so potent that eachcan totally eliminate the solute molecule from contributing toany of the harmonics. Thus, the dephasing takes place at allharmonics at a rate that is identical to the rate of these cata-strophic collisions. In terms of the parameters above, thepure dephasing time resulting from the Poisson model is

T2* ~n!5t i b11n2~df0!2c

n2~df0!2 ~7!

leading to a gradual transition between the limiting dephas-ing behaviors as the average phase jump increases.

The molecular system under study here is the linear tri-iodide ion I 3

2 , which has been the subject of combined ex-perimental and theoretical study in our groups for sometime.17–19 Using impulsive photolysis,I 3

2 dissociation andrecombination dynamics were recorded in various polar mo-lecular solvents. The solvent was found to influence reactiv-ity not only by dephasing nuclear motion and dissipatingexcess photon energy, but also by determining the initial ge-ometry and degree of symmetry of the reactant.20–22 Theultrafast photoexcitation that served to vertically promotemolecules onto the reactive surface, simultaneously gener-ated vibrational coherences in the ground state through reso-nant impulsive stimulated Raman scattering~RISRS!,23–25

giving rise to spectral modulations observed in the UV ab-sorption spectrum. Their decay provided a direct measure ofthe ensuing vibrational dephasing between low vibrationallevels of the ion.

In view of the central role played by the solvent in thephotochemistry of this ion, a specific effort was made to gaindetailed insight into static and dynamic aspects of triiodidesolvation in polar liquids—based upon the RISRS observ-able. Starting with the study of the RISRS process itself, andlater applying this knowledge to unraveling triiodide vibra-tional dynamics, the following discoveries were made.

~1! RISRS proved to be a facile method for characterizingground state vibrational dephasing at the bottom of thevibrational ladder.

~2! Fortuitous tuning of pump and probe frequencies, or ap-plying properly timed multiple excitation pulses allowedselective measurement of fundamental and overtonedephasing times with high precision. Comparing resultswith those reported from resonance Raman furtherproved RISRS to be the only method capable of reliableT2 measurements for this system!26

~3! Independent measures ofT1 demonstrate that puredephasing dominates dephasing inI 3

2 in all solventsstudies.27

3661J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Vibrational dephasing of I32 in solution

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~4! When studied in glass forming liquids, from 300–100 K,homogeneous fundamental dephasing was observedthroughout. The dephasing times increased gradually bya factor of;2 over the range, despite the drastic changesto viscosity.28

~5! T2 (n51)* /T2 (n52)* '2.5 for both ethanol and acetonitrileat 300 K.26

In the present study we investigate the mechanisms forthe deviation of fundamental and harmonic pure dephasingtimes from the predictions of Kubo lineshape theory. In par-ticular, T2 (n51)* /T2 (n52)* was measured as a function of tem-perature, throughout the range from room temperature to thatof glass formation. The next section is dedicated to present-ing the two theories outlined above on an equal footing, inorder to expose the molecular consequences of the math-ematical assumptions and to explain the motivation of ourexperiments. Following the experimental section, our resultswill be presented and analyzed in terms of both models. Theapparent inadequacy of both in explaining our results will bepresented and discussion.

II. THEORY

While the general approach represented by Yamaguchi’sstochastic model is adopted, it will first be reformulated, as aconsistent quantum dynamical theory. The advantages ofsuch an approach are~a! that from it all the physical observ-ables of the system can be evaluated, and~b! the simulationof inherently quantum mechanical variables, such as changesin time of molecular absorption and emission are possible.This requires abandoning a weak coupling approach, since itnecessarily leads to Gaussian dynamics. One such approach,inspired by the exponential vibrational dephasing, is to de-scribe the open system by the Quantum Master equation. Inthis case the state of the system is characterized by the den-sity operatorr of the primary system subject to the equationof motion:

]r

]t52

i

\@H,r#1LD~r!, ~8!

whereH represents the Hamiltonian of the systems andLD

the dissipative bath contribution, which in Lindblads semi-group form becomes29

LD~r!5(i

FirFi121/2$FiFi

1 ,r%, ~9!

whereFi are a operators from the Hilbert space of the pri-mary system representing the dissipative force due to thebath. The distinction between the Hamiltonian dynamics andthe dissipative part is not unique, which reflects the fact thatthe boundary between the system and the bath is arbitrary.The Hilbert space of the system has to include all the opera-tors representing observables. It is customary to include partof the bath explicitly. This allows one to describe effectssuch as inhomogeneous broadening and spectral diffusion.The approach is in analogy with the Brownian oscillator,where a collective over a damped bath coordinate is includedexplicitly.30

In light of the experimental observations, the paved road of aperturbation expansion assuming a weak coupling betweenthe system and the bath is abandoned to a strong couplingdescription. This requirement eliminates the approach basedon a classical calculation of bath correlation functions. Forthe quantum case only two strong coupling limiting caseshave been worked out. The analysis focuses on elastic sys-tem bath encounters for whichLD* (H)5019 ~whereLD* is thedissipative superoperator in the Heisenberg representation!.At this point we should try to classify the origin of thesesystem bath elastic encounters. One possibility is that thedephasing is the result of the accumulation of rare uncorre-lated events, i.e., a Poisson model. Each of these events canbe described by a scattering operatorS such that after asingle event the state of the system is changed toSrS1. Thetotal dissipative rate of change under a flow of uncorrelatedscattering events with a rateg51/t i is then

rD5g~SrS12r!. ~10!

The Poisson model fits Lindblads form if we identify theoperator F in Eq. ~9! with AgS.29,31 The simplest puredephasing scattering operator that does not alter the energyof the system is the phase shift operatorS5e2 i /hHf, wheref is the elastic phase shift characterizing the scattering event.The accumulation of these phase shifts generates the dephas-ing process leading to the equation of motion:

]r

]t52

i

\@H,r#1g~e2 i /hf@H,* #2I !r, ~11!

~where@H,* #A5@H,A#), which in the energy representationof the density operator becomes

]rnm

]t52 ivnmrnm1g~e2 ivnmf21!rnm , ~12!

wherevnm5(En2Em)/\ is the Bohr frequency. In the har-monic casevnm5(n2m)n. It is obvious in Eq.~12! thatonly the off-diagonal terms of the density operator decay.The simple Poisson model can be augmented by assuming adistribution of scattering events weighted byP(f), the prob-ability density of a phase shiftf. The linearity of the Semi-group equations of motion allow averaging, leading to

]r

]t52

i

\@H,r#1gS E P~f!df~e2 i /\f@H,* #2I ! D r.

~13!

We employ a symmetrized version of phase jump probabilitydistribution of the exponential distribution of Yamaguchi,Eq. ~6!, i.e.,

P~f!5df0e2ufu/df0. ~14!

The exponential distribution of phase jump fits the assump-tion that the process is activated, i.e., the phase jump is de-termined by the energyEc of the collision eventdf0

}Ec /kBT. The symmetric version eliminates the frequencyshift associated with the process. A different viewpoint is toassume that the rate of eventsg is activated and that thephase jump distribution is normal:

3662 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Gershgoren et al.

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P~f!5A 1

2pse2f2/2s2

. ~15!

For the normal distribution, the averaging in Eq.~13! can becarried out explicitly, leading to

]r

]t52

i

\@H,r#1g~e~2s2/2\2!†H,@H,* #‡2I !r, ~16!

which in the energy representation becomes

]rnm

]t52 ivnmrnm1g~e~2s2/2!vnm

221!rnm . ~17!

For the harmonic dephasing case with a frequencyn, Eq.~17! leads to

T2~n!5t i

~e~2s2/2!n2n221!

. ~18!

The exponential distribution, Eq.~15!, leads to a differentdecay pattern:

]rnm

]t52 ivnmrnm2gS vnm

2

df021vnm

2 D rnm , ~19!

which leads to theT2(n) relation of Yamaguchi, Eq.~7!.An alternative strong coupling dephasing model is based

on the concept of the Hamiltonian of the primary systemcontinuously modulated by the bath:

H5H f ~ t !, ~20!

where f (t) is the bath modulation function. A Guassianmodel is defined when the modulation function can be de-scribed by a Guassian stochastic process. For the harmoniccase f (t) is interpreted as the temporary frequency of theoscillator. The dephasing is therefore a result of frequencymodulation. In the fast modulation limit,

^ f ~ t !&B5v0

and

^ f ~ t ! f ~ t8!&B5Gd~ t2t8!; ~21!

Eq. ~21! leads to the following Lindblads semigroup form:32

LD~r!52G2†H,@H,r#‡, ~22!

which in the energy representation becomes

]rnm

]t52 ivnmrnm12G2vnm

2 rnm . ~23!

Equation~23! recovers the result that the dephasing rate ofthe nth harmonic is quadratic inn, i.e., T2(n)51/n2n2G2.This result will also be obtained in the weak coupling limit.Comparing the Gaussian dephasing model~Eq. 23! to thePoisson dephasing model@Eqs. ~7! and ~18!#, we find thatwhen the phase jumps are small, i.e.,s or df0

21, the dephas-ing becomes the result of the accumulation of many smallphase jumps naturally, leading to a Gaussian dephasingmodel, i.e., G25g(s2/2) for the normal distribution andG25g/df0

2 for the exponential distribution. For a largevalue of s or df0

21 each event leads to complete random-ization of phase, leading to

]rnm

]t52 ivnmrnm2grnm , ~24!

i.e., T2(n)51/g is independent ofn, and is proportional tothe rate of collision eventsg.

Equations~20! and ~24! represent the two limits of thePoisson model. For small jumps the dephasing rate will scalequadratically with the energy gapvnm and for large phasejumps the scaling becomes independent of the energy differ-ence. In addition, if the phase jump is activated we expectthat the process will become more Gaussian at lower tem-peratures. If only the rate of eventsg is activated we expecta slowdown of the dephasing at lower temperature whilemaintaining the Poisson character.

III. EXPERIMENT

The light pulses were generated in a 1 kHz homemademultipass amplified Ti sapphire laser system that is describedin detail elsewhere.26 The output of the 15 fs oscillator isstretched to 20 ps, and injected into a eight pass amplifiercavity, which is pumped with;10 mJ pulses from an intra-cavity doubled Q-switched Nd:YLF laser. The ultimate out-put after compression consists of a train of 28 fs pulses,centered at 790 nm, containing;0.6 mJ of energy per pulse.Pump, push, and probe pulses were all derived by doublingportions of the amplified pulses in BBO crystals that were, atmost, 100m thick, and were demonstrated to introduce neg-ligible pulse broadening.

The samples were prepared as previously described,26 inethanol and in 1-methyl THF. Ethanol samples were stop-pered in 1 mm quartz cuvettes, which were pressed againstthe cold finger of a closed cycle helium cryostat~RMC!.Methyl-THF samples were glove-box loaded into a cell con-sisting of two 2 mm thick sapphire windows~crystran!, and a0.5 mm teflon spacer. The internal volume was accessedthrough holes drilled in one sapphire flat, and both vacuumtightness around the holes, and thermal contact, were pro-vided by indium wire pressed between the aluminum cellholder and the windows. The sample temperature was read atthe base of the cell, and actual solution temperatures wereestimated to be within 2° of these readings. The temperaturewas regulated by a Lake Shore 330 controller coupled to aDT-400 series diode sensor. During the irradiation the samplewas translated alternately inY andZ directions~X being thedirection of irradiation! via a stepper motor actuated lineartranslation stage on a precision lab jack~microcontrol!. Ratesof translation were demonstrated to eliminate any local heat-ing or sample bleaching effects. Methods of data collectionand automation were as described previously.

The integrity of the cooled samples was monitored byabsorption spectroscopy. A fiber bundle halogen lampilluminator–open electrode deuterium lamp combinationproduced a stable and continuous irradiation source over therange of the triiodide UV absorption. Some scattering thatbuilt up at the lower edge of the temperature range interferedwith a quantitative determination of sample concentrations.The positions and widths of the two intense near-UV bands

3663J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Vibrational dephasing of I32 in solution

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were obtained by fitting the measured spectra to a superpo-sition of three Gaussians bands plotted against photon en-ergy.

IV. RESULTS

The center of the I32 absorption peaks red shifted, and

the bands narrowed systematically upon cooling. In order touncover the mechanism for the spectral narrowing observedin both solvents, a theoretical simulation of the spectra wasconducted using model potentials that were described previ-ously and used to simulate I3

2 photochemistry.18,19 Figure 1presents the widths predicted by the theory together withthose obtained from the fitting. An example of a measuredspectrum along with a nonlinear least squares fit is presentedin the insert. The theoretical predictions lie slightly outsidethe error limits of the measured widths at room temperature,but the trends of change with temperature agree well with theexperimental changes. Another noteworthy result is that thewidths obtained in ethanol and in MTHF are identical withinerror. This is remarkable in view of the charged nature of thechromophore, and the marked difference in polarity andcharge distribution for the two solvents. While ethanol is avery polar solvent, especially adept at stabilizing localizednegative molecular charge, THF is a sparingly polar solvent,which is better at stabilizing cations. In both solvents thewidth of the upper and lower energy transitions are essen-tially the same at the various measured temperatures.

Another unusual aspect of the temperature-dependentspectra involves the spectral shifting of these peaks withtemperature. Figure 2 depicts the peak positions as a functionof the temperature in both solvents, using the designation ofthe transitions used in the insert to Fig. 1. The transitionenergies shift to the red as temperature is reduced, and thelower of the two main absorption bands~;27 000 cm21!shifts almost two times more (Dn;450 cm21) over the

same temperature range~300–100 K! than that above it~;34 000 cm21!. Furthermore, the effect of cooling is againequal for both solvents, indicating perhaps that the essentialeffects of cooling on the interaction of the solvent with theion is not influenced by the differences between these twosolvents.

Dephasing dynamics were recovered from RISRS spec-tral modulations, as previously described.26 Degenerate twoand three pulse experiments employing 390 nm pulses wereused to record the decay of spectral modulations reflectingcoherent ground state vibrations induced by the pump se-quence. The results were first fit to a sum of complex expo-nentials using the ‘‘filter diagonalization method.’’33 Thisproduced a perfect fit to the data, but generated a large ma-jority of terms that were devoid of direct significance to thevibrational dynamics. Knowing the vibrational frequenciesof the ions, a minimal subset of diagonalization terms thatwere deemed relevant were used as a first guess for fittingthe data using a Marquardt–Levenberg least squares algo-rithm. Two pulse data in MTHF at three temperatures is pre-sented in Fig. 3, along with the products of the describedanalysis. The first panel shows the raw data. The secondshows the modulations once subtracted from the dc back-ground, along with the fit provided by the filter diagonaliza-tion method shown as a solid black line. Finally, in the lastpanel, all of the terms oscillating near the doubled frequencyare isolated from all others in the diagonalization series, andfit to an exponentially decaying sinusoid using least squaresroutines. The motivation of this step in the analysis was firstto estimate the range of confidence to assign the final param-eters of the fitting, such as frequencies, phases, and dephas-ing times. Another goal was to determine whether deviationsfrom exponential decay could be discerned in the spectralmodulations. Within the Kubo lineshape picture, once out-side the fast modulation limit, the pure dephasing no longermakes a Lorentzian contribution to the lineshape, but ratherone following a Voight function as in Eq.~4!. To test thedephasing dynamics of individual harmonics, all the terms inthe filter diagonalization result aside from those exhibiting a

FIG. 1. A comparison of experimental and simulated peak widths at varioustemperatures. An experimental spectrum is presented in the inset to demon-strate the quality of the fit provided by the sum of Gaussians, and to desig-nate the peaks.

FIG. 2. Absorption peak positions as a function of temperature for triiodidein ethanol and MTHF solutions. See the text for details.

3664 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Gershgoren et al.

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frequency very near that of the harmonic of interest weresubtracted from the data. The remainder was fit to an expo-nentially decaying sinusoid, to a Gaussian decaying one, andalso to Eq.~4! multiplied by a sinusoid.

In both solvents, despite the extreme variations ofsample viscosity, no significant deviations from exponentialdecay of the dephasing in either the fundamental or first har-monic were detected. This is in accord with an earlier reportof the fundamental dephasing dynamics in ethanol alone.28 Itraises the question of whether our method of measurementcaptures the dynamics from an early enough stage to allowthe observation of an initial fleeting Gaussian stage of relax-ation in an intermediate dephasing regime. The answer tothis question is probably negative, due to the presence ofrapid spectral changes that accompany bond fission in I3

2 ,allowing us to characterize the decay of the coherent vibra-tions only after a delay of;300 fs. However, the examina-tion of Voight line functions shows that even in the case ofintermediate modulation, the decay constants fit to the expo-nential tails that characterize later stages of phase relaxationwill maintain the quadratic dependence onn.34 Accordingly,all dephasing times reported in this paper were retrieved byfitting the data to exponential decays.

Results of the dephasing measurements are presentedgraphically in Figs. 4 and 5. The exponential phase relax-ation times of the fundamental and the first harmonic aredepicted in picoseconds along with the pure dephasing timeratios as a function of the sample temperatures. The error indeterminingT2* usually reflects the scatter of values obtainedfrom the analysis of multiple transient transmission measure-ments. In cases where the three pulse method provided morereproducible numbers for the harmonic dephasing times, theerror reflects the deviation of values obtained from thisscheme alone. The first obvious trend is the mild increase inT2 (n51)* and in T2 (n52)* , roughly by a factor of 2 as thetemperature is reduced from 300 to 100 K in both liquids.But no less important is the overall similar trend observed in

the T2 (n51)* /T2 (n52)* ratio, which for both solvents remainsbelow 3 at all temperatures, reducing mildly from;2.6 tonearly 2 at 100 K. In particular, we wish to draw attention tothe fact that throughout the temperature range studied, thevalue of 4 predicted in the Kubo theory lies outside the gen-erous and very realistic range of error associated with theexperimental dephasing time ratio.

V. DISCUSSION

The subquadratic ratio ofT2 (n51)* /T2 (n52)* observed inthe symmetric stretch vibration of I3

2 in two solvents, studiedover a wide temperature range, is in conflict with establishedtheories of vibrational dephasing in liquids. Before an alter-native explanation can be considered it is important to criti-cally evaluate the experimental procedures. The real timepump–probe RISRS method is based on impulsively induc-ing a coherent vibrational motion on the ground electronicsurface and observing the subsequent dynamics by a weakprobe pulse. It has been verified that the weak probe pulse isequivalent to a position measurement of the vibrationalcoordinate.26 As a result, the measurement is phase sensitive;

FIG. 3. Transient absorption scans taken in an MTHF solution of I32 at 400

nm, following excitation at the same frequency. A three-panel chronologyrepresenting the phases of signal analysis, extracting the dephasing timesfrom the raw spectral modulations. See the text for details.

FIG. 4. Pure dephasing times of the fundamental and first harmonic sym-metric stretching vibrations of triiodide in ethanol as a function of tempera-ture. The ratio of these times is also shown in the graph.

FIG. 5. The same as Fig. 4 for triiodide in MTHF.

3665J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Vibrational dephasing of I32 in solution

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therefore the decay of modulations with the correspondingfrequency is a direct measurement of dephasing. For the cur-rent measurements of I3

2 in solution, the duration of thepump and probe pulses were an order of magnitude shorterthan the fundamental vibrational period. With these tools athand the evidence favoring exponential decay of the modu-lations vs other functional forms such as Gaussian is verystrong.

The rate of dephasing also has a contribution from vi-brational relaxation. Moreover, for this contribution the ratioT2 (n51)* /T2 (n52)* is two, meaning that if the contribution ofvibrational relaxation to dephasing is significant, the ob-served exponential decay and subquadratic ratio can be ex-plained. Independent measurements based on the rate of therecovery of the bleach originating from the recombination ofI221I, have shown vibrational relaxation times of 3–4 ps in

ethanol in the same temperature range of the currentexperiments.27 In THF this process is even slower and ischaracterized by a decay time nearly twice that determined inethanol. In addition, the experimentally determinedT1 ofdiiodide that has the same frequency, reduced mass, andcharge as the symmetric stretch of I3

2 is also in the range ofa few picoseconds. Thus, the contribution of vibrational re-laxation to dephasingT252T1 is an order of magnitudeslower than the observed phase decay rates, leading to theconclusion that the dominant contribution to the decay ofmodulations is pure dephasing.

The established view of vibrational pure dephasing inliquids assumes that the vibrational mode is weakly coupledto the solvent bath. The dephasing is then the result of anaccumulation of many solvent fluctuations that modulate thefrequency. Such a picture is consistent with a Gaussiandephasing model and the quadratic decay ratio for a har-monic sequence of frequencies. Anharmonicities couldchange the decay ratios. For most vibrational modes the am-plitude of motion increases more than linearly with the vi-brational quantum number. This would lead to ahigher thanquadradicdecay ratio thus overruling anharmonicity as thesource of the effect.

In the gas phase, dephasing is caused by an accumula-tion of uncorrelated binary collisions. If the collisions aresoft the total result is again caused by an accumulation ofmany individual events, leading to a Gaussian model. If hardcollisions prevail each elastic collision leads to large reshuf-fling of phase leading to the Poisson dephasing model. Cansuch a Poisson description be used as a model for observeddephasing of I3

2 in polar solvents? The first step is to fit thelarge body of experimental observations to this model. Thiscan be done using either Eq.~18! for the normal distributionof phase jumps or Eq.~7! for the exponential distribution ofphase jumps. The two parameters to be fitted are the rate ofeventsg and the average phase jump, eithers or df0 .

As can be seen in Figs. 6 and 7, the Poisson model fitsthe data. Typical fitted average phase jumps are of the orderof 50°–60°. From these figures it is clear that only a weaktemperature dependence is observed. Also, the average phasejump increases when the temperature decreases. This obser-vation rules out the possibility that the phase jump is acti-vated, which favors the normal Poisson phase jump model

over the exponential one. In order to be able to distinguishexperimentally between the two models, the dephasing rateof the third harmonic and higher should have been measured.Although in some traces the third harmonic can be identifiedthe error bars are too large to be able to have any definiteexperimental discrimination between the models. Figure 6shows the rate of eventsg in the Poisson model as a functionof temperature. At room temperature the average time be-tween events is 300 fs. At 100 K the rate of events decreasesto one every 600 fs.

The success of the Poisson phase jump model to fit thedata motivates the search for a microscopic mechanism thatcould justify it. In the gas phase where collisions are rare, thePoisson model is a natural consequence. In condensed mediathe origins of these events are not clear. It is usually arguedthat the high density of the liquid phase excludes hard binary

FIG. 6. Average phase jumps as a function of temperature to fit the experi-mental data of I3

2 in both solvents. Solid lines refer to a normal distributionmodel, and the dashed lines to an exponential distribution of jump ampli-tudes.

FIG. 7. Rate of phase jumps to fit the experimental data in both solvents.The solid and dashed curves refer to the normal and exponential distributionof phase jump amplitudes, respectively.

3666 J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Gershgoren et al.

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collisions. Triiodides ionic character suggests long-range in-teractions with the surrounding polar solvent molecules. Thispicture would favor a Gaussian dephasing model accumu-lated from many small contributions. In metals scattering ofpolarization waves have been identified as a source of Pois-son dephasing.35 A possibly more appropriate model for I3

2

could be a rare solvent rearrangement process around thesolute. In any case, the mechanism responsible for phasedecay in then1 of I3

2 must also explain the negligible tem-perature dependence of bothT2* and the ratio T2* (n51)/T2* (n52). We note in passing that in earlier studies,vibrational dephasing in glass-forming liquids is rapidlyoverwhelmed by inhomogeneous contributions, as soon asthe solvent viscosity increases significantly.36 The lack of asimilar effect here may be due to the brevity of the stage ofdephasing here at all temperatures.

But, the search for a molecular mechanism for explain-ing our findings is further complicated by realizing just howwidespread deviation from quadratic harmonic pure dephas-ing times actually is. A wide variety of molecular vibrationsin solution have shown similar behavior, suggesting that theobserved phenomenon of deviations from Kubo lineshapepredictions is quite general.14,15This fact alone suggests thatthe underlying mechanism might be common to all the citedmolecular systems. However, they vary so drastically both inthe nature of the solvent–solute interaction potentials, in thestructure of the vibrators, and in the frequencies of the sys-tem and the bath, that at present a general mechanism eludesus. We can, however, rule out mechanisms proposed in theliterature, which are based on the gradual transition from fastto slow modulation while climbing the harmonic ladder. Thiscan be based on the perfect exponential character of thedephasing dynamics observed throughout.

The consequence of the discussion above is that thecommonly used picture of vibrational dephasing based onthe weak coupling between system and bath is expected to bedeficient. This means that established computational proce-dures based on a combination of molecular dynamics forobtaining the bath correlation functions, and perturbativetreatments for calculating spectroscopic observables, can bemisleading.37,38A consistent theoretical model should there-fore explain the minor frequency shift of the symmetricstretch of I3

2 as well as the strong collision characteristics ofthe dephasing. More work both experimental and theoreticalhas to be performed before such a theory can be established.

VI. CONCLUSIONS

The pure dephasing times of I32 in two glass-forming

liquids have been measured in the time domain from 300down to 100 K, both at the fundamental frequency and itsfirst harmonic. The exponential phase decay was found toincrease gradually from;1 to ;2 ps over this range for bothsolvents. The ratio of fundamental to harmonic pure dephas-ing times remained in the range 2–2.7 at all temperatures,and showed only a slight dependence on temperature. Thewidely used Kubo lineshape model which predicts a ratio of4 for these dephasing times is incapable of explaining these

results. Furthermore, similar deviations have been encoun-tered in a growing number of small molecular systems insolution, indicating the generality of this finding. A schemeassuming Poisson collision statistics consistently matches thedata.

ACKNOWLEDGMENTS

Enlightening discussions with E. Geva, R. Lynden-Bell,J. Skinner, and B. Bagchi are gladly acknowledged. Researchwas supported by the Israel Science Foundation. The Farkasand Fritz Haber research centers are supported by the Min-erva Gesellschaft, GmbH, Munich, Germany.

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3667J. Chem. Phys., Vol. 118, No. 8, 22 February 2003 Vibrational dephasing of I32 in solution

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