Many-body effects in the stimulated Raman response of binary mixtures: A comparisonbetween theory and experimentThomas l. C. Jansen, Audrius Pugzlys, Gheorghe Dan Crınguş, Jaap G. Snijders, and Koos Duppen Citation: The Journal of Chemical Physics 116, 9383 (2002); doi: 10.1063/1.1475763 View online: http://dx.doi.org/10.1063/1.1475763 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/116/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Stimulated Raman scattering in the evanescent field of liquid immersed tapered nanofibers Appl. Phys. Lett. 102, 201110 (2013); 10.1063/1.4807170 Coarse-grained models for fluids and their mixtures: Comparison of Monte Carlo studies of their phase behaviorwith perturbation theory and experiment J. Chem. Phys. 130, 044101 (2009); 10.1063/1.3050353 Thermal diffusion measurements and simulations of binary mixtures of spherical molecules J. Chem. Phys. 127, 014502 (2007); 10.1063/1.2746327 Mesoscale modeling of complex binary fluid mixtures: Towards an atomistic foundation of effective potentials J. Chem. Phys. 124, 074105 (2006); 10.1063/1.2161207 Comparison of low-order multireference many-body perturbation theories J. Chem. Phys. 122, 134105 (2005); 10.1063/1.1863912

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Many-body effects in the stimulated Raman response of binary mixtures:A comparison between theory and experiment

Thomas l. C. JansenTheoretical Chemistry, Materials Science Center, Rijksuniversiteit Groningen (RuG), Nijenborgh 4,9747 AG Groningen, The Netherlands

Audrius Pugzlys and Gheorghe Dan CrıngusUltrafast Laser Laboratory, Materials Science Center, Rijksuniversiteit Groningen (RuG), Nijenborgh 4,9747 AG Groningen, The Netherlands

Jaap G. SnijdersTheoretical Chemistry, Materials Science Center, Rijksuniversiteit Groningen (RuG), Nijenborgh 4,9747 AG Groningen, The Netherlands

Koos DuppenUltrafast Laser Laboratory, Materials Science Centre, Rijksuniversiteit Groningen (RuG), Nijenborgh 4,9747 AG Groningen, The Netherlands

~Received 26 November 2001; accepted 12 March 2002!

The subpicosecond dynamics of binary mixtures of carbon disulfide and alkane have been studiedusing third-order time-resolved Raman techniques. Both the anisotropic and the isotropic responseswere investigated. These depend differently on many-body contributions to the first-ordersusceptibility and probe different modes in the liquid. The anisotropic response is dominated bysingle molecule effects, whereas the isotropic response is completely determined by many-bodycontributions since the single molecule response vanishes. To interpret the experimental results,molecular dynamics simulations were performed on model mixtures. The effect of dilution on thesubpicosecond response cannot be explained by many-body effects in the first-order susceptibilityalone. Aggregation due to permanent quadrupole moments on the carbon disulfide molecules anddensity changes upon dilution are also inadequate explanations for the observed effect. Apparentlythe character of the many-body dynamics itself is modified by the change of the molecular forcefields, when carbon disulfide molecules are replaced by alkanes. ©2002 American Institute ofPhysics. @DOI: 10.1063/1.1475763#

I. INTRODUCTION

Femtosecond laser spectroscopy techniques are powerfulmethods to study the ultrafast dynamics in liquids. Experi-ments such as the~heterodyned! optical Kerr effect1–3 andtransient grating scattering4,5 allow the observation of in-duced motions in real time rather than as resonances. For amolecule to be Raman active the polarizability has to becoordinate dependent. In anisotropic molecules the polariz-ability depends on the orientational coordinate of the mol-ecules and hence a rotational Raman response is observed. Inthe liquid phase this response will be highly-influenced bythe many-body interaction between the individual rotatingmolecule and the surrounding molecules. The many-body in-teractions result in line broadening of the response in thefrequency domain, corresponding to an, often exponentially,decaying tail in the time domain.

The molecular polarizability itself is also affected by thepresence of other polarizable molecules in the neighborhood,due to local field effects. In liquids, the Raman response istherefore also determined by the coordinate dependence ofthe many-body~macroscopic! counterpart to the polarizabil-ity, i.e., the first-order susceptibility. Now, not only the indi-vidual single molecule coordinates but also the intermolecu-lar coordinates become important. The many-body effects in

the first-order susceptibility give rise to response due to dy-namics in the local structure, allowing observations of colli-sions, collective movements and structural effects.

The influence of many-body interactions on the first-order susceptibility can be investigated through dilution stud-ies. Such studies provide information not only on the many-body dynamics and the local field effects in the liquid, butalso on the structure of the diluted liquid. The formation ofclusters of molecules in the mixture will tend to preserve theresponse from many-body effects in the first-order suscepti-bility, whereas a solvent effectively isolating the polarizablemolecules from each other will suppress this part of the re-sponse to some extent.

Various liquid mixtures have been investigated experi-mentally using third-order nonlinear Raman response tech-niques to probe the ultrafast dynamics.1–3,6–12Diluted carbondisulfide belongs to one of the most studied systems becauseof the intense anisotropic response of this molecule. Studiesof carbon disulfide have been done in mixtures withalkanes,2,3,6,7 chlorine substituted methane1,10,11 and variousalcohols.11 Also molecular dynamics~MD! simulations wereperformed previously on mixtures of carbon disulfide andcarbon tetrachloride,13,14 but to our knowledge no calcula-

JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 21 1 JUNE 2002

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tions have been reported of the third-order nonlinear Ramanresponse of liquid mixtures.

The anisotropic third-order nonlinear Raman responsecan be roughly separated into three main features, i.e., thediffusive, the interaction-induced and the librational re-sponse. The diffusive response is caused by the diffusiverealignment of single molecules. Only this contribution canbe clearly recognized because of its distinct and slow expo-nential decay. The interaction-induced response is originat-ing from intermolecular motions getting intensity from thedipole-induced dipole effects~local field effects! and highermultipole and collision effects.15 The librational responsecomes from single molecules moving in the local potential ofneighboring molecules.

The isotropic third-order nonlinear Raman response is ingeneral much weaker than the anisotropic response. It issolely due to the interaction induced effects. Very few inves-tigations have been done on the isotropic response,16–19 de-spite of the fact that it provides an opportunity to investigatemany-body effects without the disturbing influence of thesingle molecule response. The obvious reason for avoidingthe isotropic response has been its very low intensity.

All investigations on mixtures until now have been in-terpreted in terms of analytical models at a macroscopic levelof theory, assigning the nondiffusional response either to li-brational degrees of freedom1,2,6 or interaction inducedresponse.11 These approaches were based on an interpretationin phenomenological terms, such as homogeneous or inho-mogeneous broadening of the Raman response or an atomiccollision model, originally proposed for low density media.20

Recently a macroscopic model was presented21 that de-scribes the diffusional and nondiffusional response in termsof abstract oscillators in which the microscopic many-bodydynamics is summarized.

Getting an understanding of the optical response on amicroscopic~molecular! scale in terms of molecular proper-ties such as force fields, atomic masses and polarizabilitieswould be preferable. Using MD simulations one can performstudies on real systems to get insight into the effects of dilu-tion on a microscopic scale. This is particular valuable whenthese results are compared to experimental investigations ofhow changes in the local environment of molecules affect themany-body part of the Raman signals.

In this paper the effect of dilution is investigated forbinary mixtures of carbon disulfide with various alkanes. Theanisotropic and the isotropic responses are examined bothexperimentally and theoretically. In Sec. II the theory used toanalyze the data is developed and in Sec. III the experimentsare described. The experimental results are compared withtheory and molecular dynamics simulations of an idealizedmixture in Sec. IV, together with a discussion of possibleexplanations of the deviations. The conclusions are presentedin Sec. V.

II. THEORY

The change of the nonlinear Raman spectrum upon dilu-tion will be considered for the simple case, where the dy-namics of the system does not change upon dilution. Further-more, it will be assumed that only one of the two

components in the mixture contributes to the Raman re-sponse. The interaction induced contribution to the suscepti-bility will be treated within the first-order approximation tothe dipole-induced dipole model. In this model the effectivemolecular polarizabilities, when local fields are present, aregiven by22–25

Pi(1)5a i1a i(

j Þ iT i j a j1

4p^x (1)&3

a i . ~1!

Herea i is the single molecule polarizability tensor,T i j is thedipole tensor, andx (1)& is the constant average susceptibil-ity. The first-order~linear! approximation is used here to pro-vide insight into the physics of interaction-induced opticalresponse. When performing actual calculations of the re-sponse~see Sec. IV!, the full DID effect will be taken intoaccount.

The first-order dipole-induced dipole polarizability is theresponse to the macroscopic electric field inside a dielectricmedium and not to the external electric field, eliminatingsample shape dependent effects. This means that the mol-ecules feel the local field generated by a dielectric medium inthe total space around them. This is the first term of Eq.~1!.The dipole-induced dipole coupling in the local surroundingof a molecule will be taken into account explicitly throughthe second term of Eq.~1!. The volume in which this is donewe will call the cavity. Since the coupling is calculated ex-plicitly in this volume, a term has to be subtracted that con-tains the effect of the dielectric medium inside the cavity.This is the third term in Eq.~1!.

The full dipole-induced dipole effect is accounted for ifin the second term of Eq.~1! the polarizabilitya j of themolecules generating the local fields on the considered mol-ecule i is replaced by the effective polarizabilityP j . Thistakes into account that the dipole on moleculej is also in-duced by a local field. The set of equations will then have tobe solved self-consistently.22–25As mentioned above, in theMD calculations that will be reported in Sec. IV, this self-consistency was fully taken into account.

To shorten the notation the induced polarizabilitya iT i j a j will be abbreviated withDi j . The instantaneous sus-ceptibility in an ensemble is given by the ensemble averageof the effective polarizabilities

x (1)51

V (i

S a i1(j Þ i

Di j 14p^x (1)&

3a i D , ~2!

whereV is the ensemble volume.The third-order Raman response is given by the time

correlation function24,26–28

xabcd(3) ~ t !52

1

2kT^xab

(1)~ t !xcd(1)~0!&. ~3!

Substituting the instantaneous susceptibility Eq.~2! intothis equation reveals six types of terms, which will be de-noted RA, DI, CA, C1, C2, and C3, respectively. These wewill now describe one at a time, omitting the proportion-ality factor 2(1/2kTV2) and the indices for the polarizationdirections.

9384 J. Chem. Phys., Vol. 116, No. 21, 1 June 2002 Jansen et al.

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The single molecule realignment term~RA! is propor-tional to

RRA~ t !5(i , j

^a i~ t !a j~0!&

5(i

^a i~ t !a i~0!&1(i

(j Þ i

^a i~ t !a j~0!&

5(i

^a i~ t !a i~0!&. ~4!

This term only depends on the rotational motion of singlemolecules, since the derivative of the correlation between themolecular polarizabilities of two randomly chosen differentmolecules, in the second term on the second line, vanishes.

The dipole interaction terms~DI! are proportional to

RDI~ t !

5(i ,k

(j Þ ilÞk

^D i j ~ t !Dkl~0!&

5(i

(j Þ i

~^D i j ~ t !Di j ~0!&1^D i j ~ t !D ji ~0!&!

1(i

(j Þ i

(kÞ i , j

S ^D i j ~ t !Dik~0!&1^D i j ~ t !Dki~0!&

1^D i j ~ t !Dk j~0!&1^D i j ~ t !D jk~0!&D

1(i

(j Þ i

(kÞ i , j

(lÞ i , j ,k

^D i j ~ t !Dkl~0!& . ~5!

The last term vanishes, because the correlation between thedipole interaction on two independent pairs of moleculesdoes not vary in time.

The cross terms between the single molecule realign-ment and the dipole interaction~C1! are proportional to

RC1~ t !5(i ,k

(j Þ i

~^ak~ t !Di j ~0!&1^D i j ~ t !ak~0!&!

5(i

(j Þ i

~^a i~ t !Di j ~0!&1^a i~ t !D ji ~0!&

1^D i j ~ t !a i~0!&1^D j i ~ t !a i~0!&!

1(i

(j Þ i

(kÞ i , j

~^a i~ t !D jk~0!&1^D jk~ t !a i~0!&!

5(i

(j Þ i

~^a i~ t !Di j ~0!&1^a i~ t !D ji ~0!&

1^D i j ~ t !a i~0!&1^D j i ~ t !a i~0!&!. ~6!

Here the last equality is based on the fact that the correlationbetween the polarizability (a i) on one molecule and the po-larizability on a second one induced by yet another molecule(D jk) is constant in time.

The remaining contributions to the third-order Ramanresponse arise from correlations with the last term of Eq.~2!.This term eliminates the effect from a dielectric medium in-side the cavity since the dipole-induced dipole coupling is

explicitly taken into account in that volume. All of thesecontributions can be expressed in terms of the single mol-ecule polarizabilities in Eqs.~4! and~6!, scaled with a factor4p^x (1)&/3 or (4p^x (1)&/3)2. The cavity correction term~CA! is given by

RCA~ t !5S 4p^x (1)&3 D 2

(i

^a i~ t !a i~0!&. ~7!

The single molecule realignment-cavity correction crossterm ~C2! is given by

RC2~ t !524p^x (1)&

3 (i

^a i~ t !a i~0!&. ~8!

And finally the cross term between the dipole interaction andthe cavity correction terms~C3! is given by

RC3~ t !

54p^x (1)&

3 (i

(j Þ i

S ^a i~ t !Di j ~0!&1^a i~ t !D ji ~0!&

1^D i j ~ t !a i~0!&1^D j i ~ t !a i~0!&D .

~9!

Dilution can be looked upon as replacing Raman activemolecules with molecules that do not contribute to the Ra-man response. Removing active chromophores from the so-lution is equivalent to reducing the summations in Eqs.~4!–~9!. From the single summation in the last line of Eq.~4! itcan be seen that the single molecule realignment part of theresponse scales linearly with the concentration. The C1 crossterm that depends on the correlation between the rotationalmotion of a molecule and a dipole interaction involving thesame molecule consist of a double summation and is grow-ing quadratic with the concentration. The dipole interactionterm ~DI! involves both a double and a triple summation,giving rise to quadratic and cubic growth, respectively.

In the terms involving the cavity correction~CA, C2,and C3!, as a first approximation the constant average sus-ceptibility ^x (1)& can be taken to be proportional to the con-centration. Consequently these terms scale quadratic~C2!and cubic~C3 and CA!, with the concentration, respectively.These three last contributions should be seen as correctionsto the terms involving the dipole interaction, since the cavitycorrection ensures that the dielectric medium inside the cav-ity, where the interactions are taken explicitly into accountthrough the dipole interaction scheme, is not counted twice.

This analysis suggests that in general terms scaling lin-ear, quadratic and cubic in the concentration upon dilutioncan be found. It should be mentioned that by the use of thefirst-order DID model instead of the full self-consistent DIDmodel terms are omitted that also scale with quadratic andhigher powers, but only the single molecule realignmentterm scales linearly.

Some of the terms found can sometimes be excluded byusing symmetry arguments. For instance, the terms includingsingle molecule polarizabilities are vanishing in the isotropicresponse since the trace of the single molecule polarizabilityis constant as long as intramolecular vibrational motion can

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be neglected. The isotropic response is then dominated bythe terms in Eq.~5!, scaling quadratic and cubic in the con-centration.

In a real mixture, the ideal conditions considered till nowwill never be found. Structural, dynamical and chemicalchanges of the liquid can take place upon dilution. Differ-ences in molecular shapes and sizes of the two componentswill change the structure and dynamics of the liquid. Further-more, the differences in the weak forces, binding the mol-ecules together in the liquid, might change the structure anddynamics as well. For instance, in mixtures of water andethanol the structural changes due to redistribution of hydro-gen bonds result in a considerable decrease of the molarvolume. In liquids with weaker bonding types, similar effectswill take place but on a smaller scale. In some cases, specificintermolecular forces can also have drastic effects. Dimers,clusters, micelles, and molecular aggregates are examples ofsystems where individual molecules associate with eachother. Such molecular structures can be expected to be rela-tively stable upon dilution and they will often give rise todistinctively different optical responses than unassociatedsingle molecules. Chemical reactions in the mixture, givingrise to breaking or formation of covalent bonds, will ofcourse also change the response considerably.

In general the optical signals are strongly dependent onthe concentration of Raman active molecules. The growthrate depends on how many individual molecules are neededto produce the response. Only one Raman active molecule isneeded in the single molecule reorientational response~RA!,while at least two molecules are involved in the interactioninduced parts of the response leading to quadratic or higherorder dependence in the concentration.

To get a picture of the terms that are determining theRaman response, molecular dynamics simulations on dilutedsystems using the full DID model,22–25 were performed.Thus, the information content of the experiments concerningthe structural dynamics of liquids can be evaluated. In suchsimulations a fraction of the molecules can be made invisibleand excluded from the susceptibility calculations. This isidentical to reducing the sums in Eqs.~4!–~9!. In this way anideal solution is obtained, where the structure and dynamicsof the system is maintained during the process of dilution.

The ideal mixture will thus be a mixture of two almostidentical molecules. Experimental deviation from ideal be-havior will provide valuable information on the differentproperties of the two types of molecules and about their re-spective interactions. Investigations on a wider range of mix-tures can provide valuable information on molecular interac-tions in general.

III. EXPERIMENT

Kerr effect and transient grating scattering experimentswere employed to study the third-order time resolved Ramanresponse of carbon disulfide and carbon disulfide/alkanebinary mixtures.

The OHD-Kerr experiments were performed as proposedby McMorrow et al.1 Briefly, we used a Ti:sapphire oscilla-tor ~Mai Tai, Spectra-Physics! delivering;70 fs pulses cen-tered around 800 nm at an 82 MHz repetition rate. After

pre-compression in a fused silica prism pair, the pulses withan energy of 7 nJ/pulse were split into pump and probebeams with a ratio 10:1, respectively. The probe pulse wasvariably delayed by a computer controlled delay stage. Thepump and probe beams, polarized at 45° with respect to eachother, were focussed into a sample by using a spherical mir-ror of r 525 cm. The necessary pump and probe polarizationorientations were set by 3 mm thick Glan–Taylor polarizersand al/2 plate in the pump beam. A 90° out-of-phase localoscillator field for the signal was generated by insertion of al/4 plate in the probe beam and detuning of the probe polar-izer by ;1.5°. By measuring the cross-correlation functionof the pump and probe beams in a 20mm BBO crystal andapplying a deconvolution procedure in frequencydomain,29,30 the distortions introduced by the instantaneouselectronic response were separated from the~delayed! re-sponse due to the induced nuclear dynamics in the liquid.

The transient grating experiments in BOXCAR geom-etry were performed by using a 1 kHz Ti:sapphire laser sys-tem ~Hurricane, Spectra-Physics! and optical parametric am-plifier ~OPA!. The laser system produces 120 fs, 800mJpulses at 1 kHz centered at 800 nm. About;300 mJ pulseswere split off to pump a traveling wave optical parametricamplifier ~TOPAS, Light Conversion, Ltd.!. The sample wasexcited with pulses centered at a wavelength of 700 nm~sec-ond harmonics of the signal wave of the TOPAS at 1400nm!. Before splitting into two pump and a probe beam, theoutput was compressed to;50 fs in a double-pass compres-sor based on two fused silica prisms. In addition to pulseshortening, the compressor allows spatial separation of thedifferent spectral components of the parametric light~signaland idler beams!. The pulse shape was determined by fre-quency resolved optical gating~FROG!.31,32The experimentswere performed with pulses, attenuated to an energy of,15nJ per pulse. After setting the polarization of the beams byl/2 plates and 3 mm thick Glan–Taylor polarizers, they werefocussed into the sample with a spherical mirror ofr 525cm. Different tensor elements of the third-order nonlinearoptical response functionx i jkl

(3) can be determined by varyingthe polarizations of the interacting beams as well as selectinga polarization direction in the detection. The signal was fil-tered by a Glan–Taylor polarizer, detected by a silicon pho-todiode, processed by lock-in amplifier, digitized and storedin a computer for further analysis.

The samples consisted of binary mixtures of CS2 andalkane solvents~pentane, heptane, and decane! placed in a 1mm standing quartz cell. To avoid heating effects the samplewas stirred with a glass-coated metal stirrer placed inside thecell and a rotating magnet. CS2 as well as pentane, heptane,and decane~all spectroscopic grade! were obtained fromMerck and Lab-Scan and used without further purification.In order to remove dust particles the solvents were filtered byusing 0.2mm pore size filters directly before injection intothe cell.

IV. RESULTS AND DISCUSSION

The third-order Raman response was calculated usingMD for idealized diluted carbon disulfide. These mixtures, as

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already described in Sec II, consist of chromophore carbondisulfide molecules that contribute to the optical response,and so-called ghost carbon disulfide molecules. These latterones do not contribute to the Raman response but fill out thevoid between the chromophores and preserve the dynamicbehavior of the liquid. We choose to calculate such an ideal-ized diluted sample rather than a specific mixture in order toobtain general information on the effects of dilution uponmany-body effects. The deviations from this model, that areobserved in experimental results, can then be directly relatedto specific physical effects that provide information on themicroscopic structure and dynamics of the liquid.

The time correlation function method24,26 was employedto calculate the response function from the dynamics, whichwas simulated using the conditions described in our earlierpaper.28A common literature force field33 of atomic Lennard-Jones type was used,

VLJ54eF S s

r D 12

2S s

r D 6G . ~10!

Heree is the depth of the potential well ands a character-istic distance. The simulations were done with 64 and 256molecule simulation boxes in 5 ns trajectories. The degree ofdilution was set by varying the relative number of chro-mophores with polarizabilities and ghost molecules withoutpolarizabilities.

The experiments were performed on mixtures of carbondisulfide and the alkanes pentane, heptane, and decane. Someof the important physical properties of the pure liquids arelisted in Table I. The mixtures have been prepared with vol-ume fractions of carbon disulfide of 30%, 50%, 70%, and100% for the anisotropic OHD-Kerr experiments and from 0to 100% with 10% intervals for the isotropic transient gratingexperiments.

In the OHD-Kerr experiment, used to measure the aniso-tropic response, the sample is excited by two pump fieldsoriginating from the same laser pulse. After a time delay thesignal is measured as the induced rotation of the polarization.The observed signal for pure CS2 is shown in Fig. 1. At zerotime delay an instantaneous response of electronic origin isobserved, followed by a rising nuclear signal. After reachingits maximum, the signal starts to decay, first nonexponen-tially and later in an exponential way. The electronic part ofthe response can be eliminated together with the pulse shapedependence by deconvolution of the signal,29,30 so that thepure impulsive nuclear anisotropic response is obtained.

In the transient grating experiments, two different pumpbeams are used to induce a grating in the sample from whicha delayed prope pulse is Bragg scattered. For parallel polar-

ized pump pulses and prope and detection polarizations atmagic angle, the isotropic response is measured. The signalmeasured for pure CS2 is shown in Fig. 1. An instant elec-tronic response is observed as well as a weak isotropicnuclear response with finite rise time. In principle the elec-tronic response can be removed using a deconvolutionprocedure,34 but if the nuclear signal is very weak, noiseintroduced by this procedure will severely contaminate thesignal. Therefore the deconvolution procedure has not beenapplied. Instead, the isotropic response was only examined atlong time delays, where the electronic response vanishes.

In the anisotropic OHD-Kerr experiment, the relaxationtime in pure CS2 is found to be 1.72 ps. Others have reportedvalues of 1.70 ps,10,11 1.65 ps,2 and 1.61 ps1,6, which are allcomparable to the decay rate measured here. The MD simu-lation yielded a decay rate of 1.44 ps. The difference be-tween the experimental and calculated realignment relax-ation times is due to a slightly lower diffusion rate in themodeled liquid35 than in the experiment, and to some extentto the noise in the tail of the calculated response.

The observed diffusional decay times at delays largerthan 2 ps are listed in Table II. It is seen that the reorienta-tional relaxation time of the carbon disulfide molecules isdecreasing upon dilution with pentane, relatively constantupon dilution with heptane and increasing when carbon di-sulfide is diluted with decane. This behavior correlates wellwith the relative values of the viscosity, listed in Table I.According to the Stokes–Einstein–Debye relation9 the reori-entational relaxation time is

TABLE I. Density, viscosity, and refractive index of the pure liquids at20 °C ~Ref. 40!. The concentration of CS2 molecules in the mixture withalkanes is given at the volume percentage.

Liquid r ~g/ml! h ~cP! nD

Pentane 0.6262 0.240 1.3575Heptane 0.6837 0.409 1.3878Decane 0.7300 0.92 1.4102CS2 1.2632 0.363 1.6319

FIG. 1. The measured response of pure CS2. The OHD-Kerr signal~a! isshown in full line together with the nuclear response~long-dashed! and thepure electronic response~dashed line!. This is the anisotropic part of thestimulated Raman response. The transient grating signal~b! was measuredemploying a probe pulse with polarization at magic angle compared to thepump pulses. The electronic response dominates the weak nuclear responsethat is also shown with a magnification of a factor of 100. This is theisotropic part of the stimulated Raman response.

9387J. Chem. Phys., Vol. 116, No. 21, 1 June 2002 Stimulated Raman response of binary mixtures

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tD5ch

kT1t0 , ~11!

wherec is a positive constant. In the MD simulations of theideal mixtures the relaxation time should be constant. Valuesbetween 1.41 and 1.53 ps are found when only consideringthe data after 1.5 ps. The relative big spread is an indicationof the uncertainty due to the fact that the Raman signal in-tensity in the tail becomes small quite fast.

Many authors have chosen to fit the observed responseto phenomenological models of the dynamics. For instance, asum of diffusional and librational response2,3,6 was consid-ered, and also a sum of diffusional and interaction inducedresponse.11 Theoretical investigations27 show that both libra-tional and interaction induced response are of importanceand any model based fit should include both effects to beable to give a correct description of the generation of theRaman signals. Since the same response can be fitted tomodels ignoring either the librational or the interaction in-duced response, the physical interpretation of such fits isquestionable. When interpreted in terms of the correlationfunctions given in Eqs.~4!–~9!, the diffusional decay and thelibrational response both belong to the single molecule termEq. ~4!, while the interaction induced response is covered byEqs.~5!–~9!.

Experimentally, the relative importance of the singlemolecule and many-body contributions can, in principle, beinferred from the concentration dependence of the intensityof the Raman response. When the structural, dynamic andchemical effects upon dilution are sufficiently small, only thesingle molecule term depends linearly on the concentration~see Sec. II!. So, fitting the response to a third-order polyno-mial in the concentration at every delay gives the relativeimportance of the various terms. In the MD simulations thesingle molecule term can be found in the same way, but alsoby simply excluding the DID terms. Comparing the calcu-lated single molecule term found in both ways gives a checkon the fitting procedure. In Fig. 2 the linear scaling parts ofthe third-order responses are shown for the dilution with thethree alkanes, together with the linear scaling part of theideal dilution calculation and the calculated third-order re-sponse excluding interaction induced effects.

The linear scaling responses of the experimental solu-tions are all very similar to each other. The linearly scalingresponse of the ideal dilution calculations resembles thesingle molecule response calculated excluding interaction in-duced effects. However, there is a striking difference be-tween the experimental and the theoretical linear scalingcurves of Fig. 2. Examining the normalized Raman re-sponses at various concentrations of CS2 in Fig. 3 shows

little difference between the experiments, whereas the inten-sity of the peak decreases rapidly compared to the tail upondilution in the simulated response.

Steffen et al.2 suggested that the almost constant ratiobetween the initial and the diffusive response upon dilutionmight be due to the formation of CS2 aggregates bound to-gether by the big quadrupole moments in CS2. Larger do-mains of CS2 would indeed explain the observed discrepancybetween the experiment and calculations. This theory can betested by adding charges on the chromophore molecules andnot on the ghost molecules in order to simulate a mixture ofmolecules with large quadrupole moments and moleculeswith no quadrupole moments.

Charges of20.308e on the carbon atoms and 0.154e onthe sulfur atoms were placed on the chromophores in a cal-culation with 32 chromophore molecules and 32 ghost mol-ecules. The resulting carbon–carbon radial distribution func-tions are shown in Fig. 4. Only small differences areobserved between the chromophore–chromophore,chromophore–ghost, and ghost–ghost radial distributionfunctions, so aggregation apparently does not occur. Thestimulated anisotropic Raman response of this mixture isshown in Fig. 5, showing no difference compared to the re-sult without quadrupole moments.

Another possible explanation for the discrepancy be-tween the simulation of the idealized mixture and the experi-ment may be found in the fact that all alkanes have densitieshalf the size of the CS2 density~see Table I!. If fluctuationsof the density play a role in the initial response, one couldimagine that surrounding the chromophores with moleculeswith lower density enhances the initial response. This couldthen explain why the initial response is not suppressed ex-perimentally, while it is in the simulation of the idealizedmixture, where the density is constant. This idea can be in-vestigated by lowering the mass of the ghost molecules.

Reducing the mass of the ghost molecules from 76.143g/mol to 58.0716 g/mol and thereby the density of the liquidfrom 1.26 to 1.09 g/ml when an equal number of chro-mophore and ghost molecules are present, shows only minorchanges in the Raman response as can be seen in Fig. 5. So,the change of the density upon dilution apparently also can-not explain the observed discrepancy between simulation andexperiment.

TABLE II. The time constant in ps for the diffusive decay extracted fromdata after 2 ps~1.5 ps for the MD!.

Solvent 100% 70% 50% 30%

Pentane 1.72 1.56 1.48 1.40Heptane 1.72 1.67 1.68 1.78Decane 1.72 1.74 1.85 2.03MD 1.44 1.53 1.41 1.48

FIG. 2. The linearly scaling part of the anisotropic response found in thethree alkanes, from the ideal mixture calculation and from a calculationexcluding multibody effects in the polarizability.

9388 J. Chem. Phys., Vol. 116, No. 21, 1 June 2002 Jansen et al.

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The van der Waals interactions between alkane mol-ecules are in general weaker than the intermolecular interac-tions found in highly polarizable molecules as CS2. In theCS2–alkane mixtures the CS2 molecules can be expected toexperience more shallow potentials than in the pure liquid. Itis possible that this enhances the librational and interactioninduced responses at early times. Varying the depth of the LJpotential on the ghost molecules, can give a clue on theimportance of such effects. The presence of more shallowpotentials was used previously as an argument to explain thedecrease of the maximum in the frequency domain~Fouriertransformed! spectrum of the Raman response upondilution.2,3,6,7

Examining the dependence on the force field is ratherdifficult, since changing the force field will most likely alsochange the volume of the liquid considerably and hence thevolume fraction of CS2. Furthermore, properties such as theviscosity and the density will also be affected, as is of coursealso the case in the real experiment. A simulation was per-formed with a potential depthe of 120 K instead of 183 K onthe sulfur atoms in the ghost molecules. The combinationrule e i j 5(e i i e j j )

1/2 for interactions of these sulfur atomswith the other types of atoms was applied. This mimicks themore shallow potentials expected in alkane mixtures. The

FIG. 3. The normalized CS2–alkane anisotropic Raman response, measured by OHD Kerr experiments at CS2 volume fractions of 30%, 50%, 70%, and100%.~a! Pentane,~b! heptane,~c! decane, and~d! ideal mixture simulation.

FIG. 4. Radial distribution functions~center of mass! for calculations withchromophores~C! with quadrupole moments and ghost molecules~X! with-out quadrupole moments.

FIG. 5. The calculated anisotropic response of mixtures of 32 chromophoresand 32 ghost molecules. The response of an ideal mixture is compared withcalculations, where the chromophores have quadrupole moments~Q-pol!,where the molecular weight of the ghost molecules is lowered~low density!and where the depth of the LJ potential is reduced from 183 K to 120 K onthe ghost molecule sulfur atoms.

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volume grows with approximately 10% giving a decrease ofthe effective volume fraction of CS2 from 0.5 to 0.457. Thesimulated response is shown in Fig. 5, where it has beenscaled to give approximately the same diffusive tail as theother responses. Now a clear difference is observed com-pared to the idealized solution calculation. The initial re-sponse is stronger even though the volume fraction of CS2 islower. This clearly shows that the solvent force field is cru-cial for the Raman response. Typical potential depths used todescribe interactions of CH3 and CH2 groups in liquid al-kanes are well below 100 K,36,37 but these could not be ap-plied in the simulations since the ghost molecules would thenbe similar to the small alkanes ethane and propane. These areboth gaseous at room temperature because of their small size.

A general explanation for the deviations between thesimulations on the idealized solutions and the experimentscould be that the model potentials are not good enough todescribe the response. The force field is an atomic LJmodel,33 which has been used for virtually all studies oncarbon disulfide. In the present study the molecules are keptrigid and no charges are distributed on the atoms. Otherstudies38,39have included intramolecular motion and charges,but that does not cause any significant change of the lowfrequency spectrum considered here. Carbon disulfide is avery anisotropic molecule and a high anisotropy can there-fore be expected in the van der Waals forces. As also statedin the paper where the LJ model, used here, was firstdescribed,33 a model with isotropic atomic forces cannot de-scribe the anisotropic interactions of a molecule properly atall distances. It will have an isotropic asymptotic behavior inthe long distance limit instead of the correct anisotropicasymptotic behavior. However the substantial differences ob-served between theory and experiment at low concentrationsare unlikely to arise from small errors in a model that gives agood description of the pure liquid.

So far, we discussed the concentration dependence of theanisotropic Raman response, i.e.,xxzxz

(3) . The isotropic re-sponse is about one order of magnitude weaker than the an-

isotropic one. Since the isotropic response was measured us-ing transient grating scattering, i.e., a homodyne detectiontechnique, the square of the response function is measuredand the intensity of the isotropic signal is therefore about twoorders of magnitude weaker than the anisotropic signal mea-sured with the same technique. This unfortunately leads tofar bigger uncertainties in the experimental results.

The isotropic response contains no contributions fromsingle molecule effects. So, if the mixture is behaving as inthe idealized simulations, the isotropic response containsquadratic and cubic scaling terms due to interaction inducedcontributions, but no linearly scaling terms. The power de-pendence in the volume fraction of the measured signal isshown in Fig. 6 together with the measured signal for pureCS2. At short delay times, where the electronic responsedominates, the intensity is growing with the second power inthe volume fraction. Since this is a homodyne experiment, inwhich the square of the response function is measured, thisindicates that the electronic response is predominantly asingle molecule property, depending on the number of CS2

molecules present in the mixture. At longer delay times(.200 fs!, where the nuclear response is expected to domi-nate the response, the power dependence of the measuredsignal is clearly larger than 2; it reaches a value around 3. Inthe theory section it was shown that the interaction-inducednuclear response is expected to grow at least quadratically inthe volume fraction and the measured intensity therefore hasto show at least fourth and higher-order power dependence.The fact that this is not seen here shows that a mixture ofdifferent power dependences on the concentration is present.This means that not only two-body interactions in the first-order susceptibility are important but also other many-bodyinteractions. Fitting to one power is therefore too simple.

In Fig. 7 the concentration dependence of the nuclearisotropic response~square root of the signal intensity! isshown at 300 fs, where the electronic response is expected tohave vanished completely. Within the precision of the experi-ment the behavior of the different CS2 alkane mixtures arethe same. The behavior of the simulated response seems tooverestimate the response at high CS2 fractions. If CS2 ag-gregates were formed in the liquid one would expect theopposite effect, i.e., slower disappearance of the multibodyisotropic response than predicted by simulations on the idealmixture.

FIG. 6. The power dependence~dots! of the isotropic signal intensity in amixture of CS2 and pentane. The electronic signal scales quadratically. Aslightly higher power dependence is found in the nuclear part of the re-sponse. The response for pure CS2 is shown as the dotted line, with thenuclear tail also displayed 20 times amplified as the solid line.

FIG. 7. The concentration dependence isotropic response at 300 fs in thethree CS2–alkane mixtures and in the simulation of an ideal mixture.

9390 J. Chem. Phys., Vol. 116, No. 21, 1 June 2002 Jansen et al.

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V. CONCLUSIONS

In this study the anisotropic and isotropic third-order Ra-man responses have been measured for mixtures of carbondisulfide with a series of alkanes. The experimental resultswere compared with molecular dynamics simulations of dif-ferent model mixtures. Significant differences were foundbetween the simplest model and the experiments in the sub-picosecond regime. The initial response does not disappearas fast as predicted by the theoretical calculations. Usingmolecular dynamics we examined some of the possible mi-croscopic explanations for the peculiar deviations. We suc-ceeded in ruling out both aggregation due to large quadru-pole moments on carbon disulfide and density fluctuations inthe inhomogeneous medium as sources of the discrepancies.In contrast, the shallowness of the potentials found in al-kanes could provide a convincing explanation, as shown bythe model calculations. The shallow potentials are probablyleading to an increase of the librational response at a rate thatcompensates for the decrease of the interaction-induced re-sponse, when diluting. Intramolecular solvent vibrations andthe lack of a true anisotropic potential for carbon disulfidecannot be ruled out completely as partial explanations of thedifferences, since only rigid molecules have been used in thesimulations. Further investigations with systematic variationsof solvent potentials and vibrational degrees of freedom willbe needed to get a more detailed microscopic picture of theresponse.

We showed that the isotropic response is measurable andreveals important information about the interaction inducedresponse. Also in the isotropic response deviations were ob-served between the theory and experiments. In contrast to theinitial anisotropic response, the isotropic response seems todisappear faster upon dilution in the high concentration limitthan predicted by the simulations, which again providesstrong evidence against the formation of aggregates. Furtherstudies of the isotropic response should provide more infor-mation about the nature of the intermolecular motion andinteraction-induced effects in the third-order Raman re-sponse.

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