Isotropic and anisotropic Raman scattering from molecular liquids measured by spatially masked optical Kerr effect spectroscopy.pdf

Isotropic and anisotropic Raman scattering from molecular liquids measured byspatially masked optical Kerr effect spectroscopyC. J. Fecko, J. D. Eaves, and A. Tokmakoff

Citation: The Journal of Chemical Physics 117, 1139 (2002); doi: 10.1063/1.1485070 View online: http://dx.doi.org/10.1063/1.1485070 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/117/3?ver=pdfcov Published by the AIP Publishing

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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:143.106.26.202 On: Wed, 25 Jun 2014 13:18:20

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JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 3 15 JULY 2002

This alecular structure and the time-dependent evolution of thesestructures. For room-temperature molecular liquids, dynam-ics that occur on the 10215 to 10211 second time scale havea wide variety of molecular motions and interactions associ-ated with them, including molecular translation and reorien-tation, Coulombic interactions, collision-induced effects, andhydrogen bonding. Numerous femtosecond optical and infra-red spectroscopies have been used to study these dynamics inequilibrium neat liquids or as the response of a solvent tocharge redistribution. The persistent complication is that ex-periments are not in themselves effective at resolving thenature and time scale of one process from another, or de-scribing the correlation of one motion with another.

Among the most widely used time-domain spec-troscopies employed to study collective liquid motions aremethods based on impulsive Raman excitation with nonreso-nant femtosecond pulses. These experiments have been usedto probe the polarizability relaxation dynamics of a variety ofpolar and nonpolar liquids and solutions, supercooled liq-uids, polymers, and liquid crystals.14 Nonresonant third-order nonlinear experiments are straightforward to perform,but they observe collective motions through the many-bodypolarizability, thus requiring copious amounts of modeling

sity for most systems is nearly featureless, making it difficultto extract conclusive information about the motions that con-tribute. More selective methods of probing would help toconstrain the modeling of these systems.

One advantage of using Raman experiments to study in-termolecular dynamics is that the response function is a ten-sor quantity. Since the many-body polarizability tensor canbe separated into isotropic and anisotropic components, Ra-man spectroscopy with polarized light can distinguish be-tween contributions to the molecular dynamics based onsymmetry arguments.6,7 Whereas the anisotropic componentis dominated by molecular reorientation or collective mo-tions with reorientational character, the isotropic response isselective to motions that are totally symmetric in the en-semble average. The isotropic response is entirely due tointermolecular interactions, and may give the best indicationof the frequency dependence of translational density fluctua-tions. It should be important for testing the predictions ofmode coupling theory, which describes density fluctuations.8In the past, the isotropic response has received little atten-tion, since most experimental configurations ~such as OHD-RIKES transient birefringence1! probe only the anisotropicresponse. However, new methods with polarization controlof all of the optical fields are being developed for measuringdifferent tensor elements of the third-order response.915a!Electronic mail: [email protected] and anisotropic Raman scatmeasured by spatially masked optical

C. J. Fecko, J. D. Eaves, and A. Tokmakoffa)Department of Chemistry and George R. Harrison Spectroof Technology, Cambridge, Massachusetts 02139~Received 28 January 2002; accepted 19 April 2002!

Spatially masked optical Kerr effect ~SM-OKE! spectprobe technique capable of measuring isotropic comolecular liquids. In conjunction with traditional opspectroscopy, polarization-selective SM-OKE measureanisotropic and isotropic third-order nonlinear responThese two responses, which allow the intermoleculara complete and independent basis for describingthird-order experiments. The Fourier transform spectfor each liquid and are interpreted in terms of the moto the many-body polarizability. The molecular coresponse for all liquids, while the line shape in the inwith the liquid. For the non-hydrogen-bonding liquisuppressed! as compared with that of the anisotropicisotropic spectrum exhibits completely new featuresresponse is especially notable, since it is exceedingly f 2002 American Institute of Physics. @DOI: 10.106

I. INTRODUCTION

Understanding the dynamics associated with local andlong-range structures within molecular liquids continues tobe an experimental challenge, because only limited methodsexist which are simultaneously sensitive to collective mo-1130021-9606/2002/117(3)/1139/16/$19.00rticle is copyrighted as indicated in the article. Reuse of AIP content is sub

143.106.26.202 On: Wed,ring from molecular liquidsKerr effect spectroscopy

opy Laboratory, Massachusetts Institute

scopy is a nonresonant femtosecond pumpributions to the transient birefringence ofcal-heterodyne-detected optical Kerr effectents are used to experimentally measure theof CS2 , acetonitrile, methanol, and water.

ynamics to be separated by symmetry, formpolarization dependence of nonresonant

l densities of these responses are presentedcular and interaction-induced contributionstributions are suppressed in the isotropicaction-induced portion of the spectra varies, the isotropic line shape is similar ~albeitectrum, but the high-frequency wing of the

n methanol and water. The isotropic watert and distinct from the anisotropic response./1.1485070#

and simulation to relate observables to molecular quantities.5The most common model is the interaction representation ofthe polarizability, in which the polarizability is calculatedusing Coulombic interactions between individual molecules.However, the observed low-frequency Raman spectral den-9 2002 American Institute of Physicsject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 25 Jun 2014 13:18:20

This aIn this paper, we present a simple method for directlyobserving the isotropic nonresonant response from molecularliquids. The spatially masked optical Kerr effect ~SM-OKE!is a two-pulse experiment that measures the isotropic re-sponse and other tensor elements that include isotropic dy-namics. By treating the probe pulse as a Gaussian beam withspatially varying phase, we demonstrate that a partiallyclosed iris introduces an in-phase local oscillator for hetero-dyne detection. This allows experiments with parallel polar-izations to measure a nonresonant ~birefringent! response inthe form of a nonlinear SM-OKE signal.

We apply this method to the study of four model mo-lecular liquids: carbon disulfide, acetonitrile, methanol, andwater. These liquids have been well studied by experimentand simulation. Furthermore, we can compare the isotropicand anisotropic response from weakly and strongly interact-ing liquids, from polar and nonpolar liquids, and from proticand aprotic polar solvents.

To begin our description of the experiment, and the se-lective nature of the isotropic and anisotropic responses, Sec.II presents the different representations of the many-bodypolarizability through which we observe the system. The em-phasis is on the interaction representation of the polarizabil-ity. Section III describes the theory and implementation ofthe SM-OKE experiment. The experimental results for allliquids are then presented and discussed in Sec. IV. We findthat the isotropic response effectively suppresses molecularreorientation in all liquids, and is quite distinct from the an-isotropic response in the hydrogen-bonding liquids. The iso-tropic response from water is particularly curious, since it isexceedingly fast and completely different from the well-characterized anisotropic response.

II. THEORYA. Many-body polarizability tensor

The experimental observable of nonresonant opticalspectroscopies that encodes information about the structureand dynamics of liquids is the many-body polarizability,P(t). To obtain a molecular picture of liquids, this labora-tory frame quantity must be decomposed into contributionsfrom microscopic quantities. Because the polarizability is afunction of all 3N nuclear coordinates, and intermolecularinteractions within the liquid state are extremely complex,the best choice for a physically meaningful microscopic rep-resentation is not readily apparent. One approach is to ex-pand the polarizability in a Taylor series about a set of equi-librium nuclear coordinates.16,17 This implies that the liquidbehaves as a collection of damped oscillators over short timeintervals and therefore lends itself to a normal mode treat-ment of the liquid.1822 A second approach involves a quasi-perturbative site-basis construction of the many-body polar-izability by the addition of intermolecular interactions to acollection of gas phase molecular polarizabilities.2329 Toprovide additional physical insight, the polarizability can befurther decomposed into contributions from rotational andtranslational degrees of freedom.20,23,28 The relationship be-

1140 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002tween the normal mode basis and the site basis representa-tion of the polarizability is nontrivial.16 The interaction rep-

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,resentation of the polarizability provides a more intuitiveframework for a qualitative interpretation of our low-frequency spectra, and it will therefore be discussed in thissection.

1. Interaction representation of the polarizabilityThe polarizability of an isolated molecule is described

by the molecular ~M! polarizability tensor

aM~ t !5aI1b~ t !, ~1!

where a5Tr(aM(t))/3 is the isotropic component, I is theunit matrix, and the anisotropic part is the product of the gasphase anisotropy with the traceless molecular orientation ten-sor, b(t)5gQ(t). In the case of molecules with axially sym-metric charge distributions, the orientation tensor is the sec-ond Legendre polynomial of the unit vector along themolecular axis, Q(t)5 12(u(t)u(t)2I). For isolated mol-ecules, the polarizability varies with the displacement of in-tramolecular degrees of freedom, which are often neglectedfor intermolecular polarizabilities.

In condensed phase systems, the total polarizability canbe decomposed into a contribution from the individual mol-ecules ~M! and interaction-induced ~II! contributions that aredue to interactions between molecules23

P~ t !5PM~ t !1PII~ t !, ~2!

PM~ t !5 (n51

N

anM~ t !. ~3!

The molecular term is merely the sum of the isolated mo-lecular polarizabilities.

In order to treat the II polarizability, a model is neededfor intermolecular interactions within the fluid. For weaklyinteracting molecular liquids, a standard assumption is thatelectronic overlap and frame distortions may be neglected,and that intermolecular interactions take place via long-rangemultipolar forces.5,30,31 Molecular dynamics ~MD! simula-tions have established that truncation of the interactions afterthe dipole-induced-dipole ~DID! term adequately describesdepolarized Raman scattering in many simple fluids.23,24,32 Inthe DID approximation, the II term takes the form

PII~ t !> (n51

N

(mn

anM~ t !"T~2 !~rnm~ t !!"am

M~ t !, ~4!

where T(2)(rmn(t))5ij(rmn(t))21 is the time-dependentdipole interaction tensor, and am

M(t) is the effective polariz-ability of molecule m that accounts for the local field from allother molecules

amM~ t !5am

M~ t !1 (p51

N

(pm

amM~ t !"T~2 !~rmp~ t !!"ap

M~ t !1fl .~5!

Intermolecular interactions act as a time-dependent modifi-cation to the isolated molecular polarizability, since the totalmolecular polarizability is a function of intermolecular dis-tances and orientations. From a computational point of view,

Fecko, Eaves, and Tokmakoffthe polarizability of a single molecule can be calculated to allorders of the DID approximation by using an iterative

ject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 25 Jun 2014 13:18:20

This aalgorithm,28 by matrix inversion,24 or through the use of arenormalized interaction tensor.33 The first-order DID ap-proximation, in which am

M(t) is replaced by the isolated mol-ecule polarizability in Eq. ~4!, is used below. The range ofvalidity of this approximation is questionable; recent calcu-lations point to the importance of cancellation effects be-tween two-, three-, and four-body interactions, especially atlong times.24,25 Other studies have suggested that the inclu-sion of higher order multipole terms or electronic overlapterms is crucial for a calculation of the many-bodypolarizability.31,3436

Partitioning the polarizability according to Eq. ~2! pro-vides a physical basis for its interpretation. For rigid mol-ecules, the only contribution a single molecule can make tothe time-dependent correlation function is through rotation.Because a closely packed environment hinders molecular ro-tation in liquids, molecular reorientation contributes on pico-second time scales and is often treated diffusively.32 The IIterm can be modulated by the change in orientation of onemolecule with respect to another, or by a change in the in-termolecular distance between the two. Thus, it is a measureof local orientational and density fluctuations within the liq-uid. The II polarizability is also sensitive to changes in elec-tronic distributions, so it can contain components that aremodulated at frequencies higher than the single moleculeterm. Because both molecular and II polarizabilities are sen-sitive to orientational fluctuations, it is possible that crosscorrelations between the two make significant contributionsto the dynamics.

In order to determine which polarizability terms contrib-ute to the isotropic and anisotropic response, Eq. ~1! can beused to expand Eq. ~2!, which yields23,25

P5PM ,a1PM ,b1PII ,aa1PII ,bb1PII ,ab1ba. ~6!

The explicit time dependence has been left out. The a and bnotation indicates contributions from the aM isotropic or an-isotropic terms, respectively. For a rigid molecule, the firstterm is constant in time and is not observed in the intermo-lecular Raman signal. The second term, corresponding to ro-tational diffusion, is traceless and can only contribute to theanisotropic polarizability. Likewise, the third term is alsotraceless since it consists of the product of two scalars withthe dipole interaction tensor.30 The two remaining termscan contribute to both the isotropic and anisotropicpolarizability.23 Thus, the many-body polarizability can bepartitioned

P5Piso1Paniso, ~7!

where the isotropic ~scalar! and anisotropic ~traceless! partsof the polarizability contain

Piso5a isoPII ,ab1ba1b isoPII ,bb, ~8!

Paniso5PM ,b1PII ,aa1aanisoPII ,ab1ba1banisoPII ,bb.

~9!For the partitioning constants a iso1aaniso51 and b iso1baniso51; expressions for these variables are given byFrenkel and McTague.23 It is important to note that the iso-

J. Chem. Phys., Vol. 117, No. 3, 15 July 2002tropic polarizability measures purely interaction-induced ef-fects, and that only terms dependent on the anisotropic part

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,of aM contribute to the total isotropic polarizability. Murry,Fourkas, and Keyes therefore conclude that the strength ofthe isotropic response is an indication of the amount of mi-croscopic asymmetry within solvent cavities.17 Last, we notethat if the polarizability is expanded beyond first-order DIDto include three ~or more!-body interactions, the isotropicpolarizability also contains induced contributions from theisotropic single molecule polarizability.

2. Translational and rotational contributionsBecause the correlation between the molecular polariz-

ability and the II component depends on orientation, a moremicroscopically descriptive way to divide the polarizabilityis in terms of rotational and translational contributions20,23,28

P~ t !5Prot~ t !1Ptrans~ t !. ~10!

In brief, each component of PII(0) is projected alongPM(0) to obtain the fraction of the interaction-induced po-larizability that is correlated with molecular orientations,Dmn . Then, the rotational polarizability can be written asP i j

rot(t)5(11D i j)P i jM(t) and the translational contributionas P i j

trans(t)5P i jII(t)2D i jP i jM(t).The relationship between the translational/rotational par-

titioning and the anisotropic and isotropic polarizability isnot trivial, but the rotationtranslation cross terms are ofparticular interest. Rotational and translational terms are or-thogonal for any instant in time by construction; however,they are not necessarily uncorrelated if measured at two dif-ferent times. Because of the transform relationship betweentime and frequency, this requires that the rotationtranslationcross term must vanish when integrated over all frequencies,although it can have nonzero spectral intensity for a givenfrequency. It has been found in simulations that the crossterm is often negative for low frequencies, as discussedabove, but passes through zero and is positive at higherfrequencies.23 This has the effect of shifting the spectral cen-ter of gravity of liquids to higher frequencies.

B. Response functionInformation about the microscopic liquid dynamics as

observed through the polarizability is contained in the mate-rial response function. Within the BornOppenheimer ap-proximation, the third-order response function may be sepa-rated into contributions from electronic and nuclear degreesof freedom R (3)5R (3)el1R (3)nuc. For electronically nonreso-nant transitions, the electronic response function is effec-tively instantaneous in time37

Ri jkl~3 !el~t!5^g i jkl&d~t!, ~11!

and therefore contains no information about molecular dy-namics. The nuclear response function is proportional to thecorrelation function of the polarizability operator

Ri jkl~3 !nuc~t!5

i\

^@P i j~t!,Pkl~0 !#&. ~12!

1141Raman scattering from molecular liquidsHere i, j, k, and l indicate the components of P in the labo-ratory frame ~x,y,z!. When describing intermolecular dynam-


This aics, the quantum-mechanical response function is often re-placed with a classical impulse response function using thefluctuation-dissipation theorem38

Ri jkl~3 !nuc~t!5

1kT

ddt ^P i j~t!Pkl~0 !&. ~13!

The response function can also be represented in the fre-quency domain through a Fourier transform

x~v!5E0

R~t!eivt dt , ~14!

where Im@x(v)# or equivalently x9 is the susceptibility, aspectral density for polarizability relaxation.

Although the response function is a tensor quantity,which in principle consists of 64 elements, symmetry con-straints limit the number of independent elements.39 For anisotropic medium, symmetry with respect to inversion androtations limits the number of independent elements to three,which are related by Rzzzz

(3) 5Rzzyy(3) 1Rzyzy

(3) 1Rzyyz(3)

. Klein-mann symmetry further allows the interchange of indicesthat are time coincident. Thus, the entire electronic responsecan be described by a single tensor component, Rzzzz

(3)el/35Rzzyy

(3)el5Rzyzy(3)el5Rzyyz

(3)el. Two elements are required to com-

pletely describe the third-order nuclear response, sinceRzyzy

(3)nuc5Rzyyz(3)nuc

. For the remainder of this discussion, R willrefer to the third-order nuclear response unless indicated oth-erwise.

The tensor elements of the third-order response can beused to separate the isotropic and anisotropic dynamics ofthe many-body polarizability. Orientational averaging for anisotropic medium allows the response function tensor ele-ments discussed above to be related to the isotropic and an-isotropic polarizability response functions7,40

Raniso~t!512@Rzzzz~t!2Rzzyy~t!#

5 12@Rzyzy~t!1Rzyyz~t!#

1

15kTddt ^P

aniso~t!Paniso~0 !&, ~15!

R iso~t!513@Rzzzz~t!12Rzzyy~t!#

1

kTddt ^P

iso~t!P iso~0 !&. ~16!

Because the anisotropic polarizability transforms as asecond-rank tensor and the isotropic as a zero-rank tensor, nocorrelation can exist between the two23 so that Raniso(t) andR iso(t) represent a complete and independent basis for de-scribing the third-order nuclear response. The isotropic re-sponse has also been referred to as the Rzzmm tensor element,where the index m indicates a projection of the polarizabilityonto the laboratory frame axis oriented at 54.7 relative tothe z axis, i.e., the magic angle.7

The anisotropic response contains only depolarized Ra-man bands in the intramolecular Raman spectra.13 It isstrongly influenced by orientational motion, especially in theintermolecular region of the spectrum where diffusive reori-

1142 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002entation contributes significantly. Raniso can be measured di-rectly in the time domain using optical Kerr effect ~or tran-

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,sient birefringence! spectroscopy.3 In the frequency domain,anisotropic dynamics is observed using depolarized lightscattering or Raman spectroscopy in the IVH geometry. Thelight scattering spectrum is related to the susceptibility by3

SLS~v!52\2 Im@xaniso9 ~v!#/@12exp~2\v/kT !# . ~17!

Note that the strong Raleigh line in light scattering is a con-sequence of the occupation factor. It does not exist in thetime-domain measurement, allowing a more accurate deter-mination of the lowest frequency range of the spectrum.

For intramolecular vibrations, the isotropic responseonly contains resonances of totally symmetric vibrations.13However, the selective nature of the isotropic response forintermolecular motions is not as obvious, since the symmetryaround molecules in the liquid is not well defined. The reori-entational dynamics of individual molecules cannot directlycontribute to R iso because the isotropic polarizability dis-cussed in Eq. ~8! does not contain the molecular term. There-fore, the isotropic response is entirely due to interaction-induced effects and thereby provides information aboutfluctuations of relative orientation and distance between mol-ecules. MD simulations indicate that the isotropic responsehas a larger contribution from density fluctuations than theanisotropic response; however, orientational motions arisingfrom intermolecular interactions may still be dominant.41 R isois measured in the frequency domain by a difference of po-larized and depolarized spectra, IISO5IVV24IVH/3.40 Calcu-lating the difference of two spectra leads to difficulties inaccurately matching the experimental conditions for bothspectra, particularly with a strong Rayleigh wing.42 However,the isotropic response can be directly observed in the timedomain using a probing field that is polarized at the magicangle relative to the excitation fields.7 Alternatively, the an-isotropic and/or isotropic response can be reconstructed bymeasuring Rzzzz and Rzzyy , and using Eqs. ~15! and ~16!. Aswith the frequency-domain measurement, an accurate deter-mination requires that the two measurements be taken underexactly the same experimental conditions.

III. EXPERIMENTA. Heterodyne-detected polarization

The SM-OKE experiment, an electronically nonresonantthird-order nonlinear spectroscopy, utilizes a two-beam near-collinear pumpprobe geometry in which a weak probepulse Epr(kpr ,tpr) is delayed by a time t with respect to astrong pump pulse Epu(kpu ,tpu). The fields are incident attime tn , where n5pu or pr, with an incident wave vectorkn . Also, the fields are linearly polarized along the unit vec-tor n , so that the polarization component of these fieldsalong the jP$x ,y ,z% laboratory axis can be expressed as

Ejn~kn ,tn!5~n!En~ tn!eiknr2ivntn1c.c. ~18!

Here, En represents the time-dependent amplitude of thefield.

The pump beam impulsively excites vibrational and col-lective motion through a Raman process, and the probe

Fecko, Eaves, and Tokmakoffstimulates radiation of a coherent optical field from thesample. The experimental signal can be expressed in terms of


This athe third-order nonlinear polarization P(3), which is gener-ated by two field interactions with the pump and one fromthe probe. For the nonresonant experiment, it is generallyassumed that the lifetime in Raman virtual states is ex-tremely short, and that the Raman excitation interaction andthe stimulation of the signal by the probe field involve twotime-coincident fields. In this limit, the time evolution of P(3)is given by a convolution of the incident fields with the third-order material response function37,38

Pi jkl~3 ! ~ t ,t!

5E jpr~ t2t!E

0

dt8 Ri jkl~3 ! ~t8!Ek

pu*~ t2t8!Elpu~ t2t8!,

~19!where t85t2tpu . Pi jkl represents the component of P(3) inthe i direction of the laboratory frame induced by the j, k,and l components of the incident fields. While the experi-ment is taken to be nonresonant, P(3) can generally be ex-pressed in terms of components that oscillate in-phase andin-quadrature with respect to the probe field

Pi jkl~3 ! ~ t ,t!52 Re@Pi jkl

~3 ! ~ t ,t!#cos~vt !

12 Im@Pi jkl~3 ! ~ t ,t!#sin~vt !. ~20!

The polarization coherently radiates a signal field in thewave vector-matched direction ksig56kpu7kpu1kpr ,which is along the direction of the transmitted probe beam.The phase of the radiated signal is shifted by p/2 with re-spect to the polarization

Ei jklsig ~ t ,t!}iPi jkl

~3 ! ~ t ,t! ~21a!

52 Re@Pi jkl~3 ! ~ t ,t!#sin~vt !

12 Im@Pi jkl~3 ! ~ t ,t!#cos~vt ! ~21b!

5Ei jklbir ~ t ,t!sin~vt !1Ei jkl

dic ~ t ,t!cos~vt !. ~21c!

Here, the signal field components that oscillate in-phase(wsig2wpr50) and in-quadrature (wsig2wpr5p/2) with re-spect to the probe field are commonly known as the dichroicand birefringent parts of the nuclear response, respectively.

The dichroic and birefringent parts of the response arederived predominantly from the evolution of the density ma-trix in the excited and ground states, respectively.43 There-fore, the relative magnitude of the two is dependent on de-tuning of the excitation frequency from electronic resonance.In a nonresonant pumpprobe experiment, the birefringentresponse is typically 10100 times stronger than the dichroicresponse.

In order to detect a signal that is linear in the radiatedfield, a local oscillator ~LO! is introduced to achieve opticalheterodyning, and the interference between the signal andLO is observed.38 If the phase of the LO is well defined andable to be varied with respect to that of the probe field, thedichroic and birefringent contributions can be measuredseparately.38 In general, the heterodyne detected signal is

LO bir LO dic LO

J. Chem. Phys., Vol. 117, No. 3, 15 July 2002Si jkl~t!}uEi u~Ei jkl~t!sin~w !1Ei jkl~t!cos~w !!.~22!

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,Therefore, a LO that is in- or out of phase (wLO50,p) withthe probe selects the dichroic part of the response (Si jkldic ),while the birefringent response (Si jklbir ) is measured with anin-quadrature LO (wLO5p/2,3p/2). The latter is preferablefor a nonresonant experiment, since the birefringent responsedominates.

A local oscillator can be introduced either by mixing thesignal with an external LO field or by intrinsic heterodynedetection, in which the LO is derived from the copropagatingtransmitted probe field. Traditional optical Kerr effect ~OHD-OKE! experiments involve intrinsic heterodyne detection byusing a 45/0/90 polarization geometry for the pump,probe, and analyzer polarizers, and a quarter-wave plate inthe probe beam.1,3 Phase control of the LO is achieved by asmall rotation of the polarizer before (wLO2wpr5p/2) orafter (wLO2wpr50) the sample.44 The OHD-OKE experi-ment is limited by the restriction that only the anisotropiccomponent of the response function can be measured. Toachieve direct control over each field, several experimentalapproaches based on heterodyne detected transient gratingshave been reported.45 An actively stabilized MachZehnderinterferometer has been used for transient grating measure-ments of several independent tensor elements.10 Diffractiveoptics have also been used to generate two pulse pairs thatare used for a nonresonant transient grating experiment withan additional field that acts as a passively phase-locked ex-ternal LO.12,14,46 In these experiments, the phase of the LO isadjusted by a pulse-envelope delay introduced tilting a glassplate in the LO beam or with a half-wave plate.

The full tensorial nonresonant response is completelydetermined by any two independent components, and OHD-OKE experiments are a reliable and straightforward ap-proach to linearly detecting Raniso . Thus, simplified methodsof measuring other elements are of interest. In particular, twobeam ~pumpprobe! methods in which the probe acts as anintrinsic LO would be the simplest way to directly measureRzzzz , Rzzyy , and R iso . The complication is that measure-ment of the dominant birefringent response requires a LOthat is in quadrature relative to the probe and with the samepolarization.

These requirements make it impossible to make transientbirefringence measurements in a pumpprobe geometry ifthe incident and radiated fields are treated as plane waves.However, various nonresonant nonlinear phenomena havebeen described for Gaussian beams in which a radially vary-ing phase applies for the probe ~or LO! beam. Such experi-ments are typically described as transient lens experimentsbased on radial gradients in temperature or the nonlinear in-dex of refraction.47,48 For example, in the Z-scan method, anintense focused optical field induces a nonlinear lens in asample. The effective focal length of this lens is determinedby measuring the intensity of the field transmitted through apartially closed iris after the sample as the position of thesample is moved relative to the waist of the beam. This mea-sures the time-integrated nonresonant response.48

Characterizing the time dependence of the nonlineartransient lens phenomena allows isotropic elements of the

1143Raman scattering from molecular liquidsnonlinear response to be determined. In analogy to transientthermal lens experiments, a two-beam technique was devel-


This alapped probe and signal or using a cutoff filter to collect onlycertain frequency components.15

B. SM-OKE spectroscopyWe have introduced an additional technique that is ca-

pable of observing multiple components of the responsefunction by using an iris to spatially mask the transmittedprobe beam as a way of introducing a LO field with an in-quadrature component.13 It utilizes the principle that, for fo-cused Gaussian beams, the difference in Guoy phase shiftand radius of curvature between the signal and LO fields areboth nonzero at finite differences from the sample. Thus,symmetrically aperaturing the axial part of the probe allowsfor heterodyne detection of the birefringent response. Spa-tially masked optical Kerr effect ~SM-OKE! spectroscopyhas been demonstrated using the intramolecular modes ofCCl4 by experimentally discriminating against the symmetricand asymmetric vibrations in the anisotropic and isotropicresponses, respectively. Recently, Ziegler and co-workershave elaborated on this type of method, discussing in detailthe similarity between position-sensitive Kerr lens spectros-copy and Z-scan experiments.49

The SM-OKE experiment can be explained phenomeno-logically as a nonlinear lensing effect. In the presence ofintense light, a materials complex index of refraction is writ-ten n5n01n2I1fl , where I is the intensity of the radia-tion. For a Gaussian beam, the radially varying intensity pro-file induces a radially varying index of refraction, which inturn acts to focus or defocus the probe. Therefore, the SM-OKE experiment can also be described as a pump-inducedlens that modulates the amount of probe transmitted throughan aperture in the far field. This is fundamentally the same asthe third-order polarization description presented in the pre-vious section, since n2 is directly proportional to thesusceptibility.11,50

In a SM-OKE experiment, the focused pump and probebeams are crossed in a thin sample at a distance z from thewaist of the beams ~see Fig. 1!. The incident probe beam isassumed to have a Gaussian profile

Epr5Epr~ tpr!w exp@2if~z !#

wpr~z !

3expF2 r2wpr

2 ~z !2i

kr2

2Rpr~z !G , ~23!where w is the waist spot size, k52p/l , and z05kw2/2 isthe Rayleigh range. The position-dependent spot size of thebeam is wpr

2 (z)5w2(11z2/z02) and the radius of curvature is

OHD 2w00w10kL ra 2 1 1Si jkl ~ra ,z ,t!5 w0w1@11z2/z02#2

E0

expF2r Sw0

2 1 w12D G

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,Rpr(z)5z(11z2/z02). In Eq. ~23!, f(z) is the linear phase,which is radially symmetric. An iris is centered on the probebeam at a distance d from the sample, and is closed to adiameter of 2ra . As the probe beam passes through thesample, it stimulates the emission of the coherent nonlinearfield in the direction of the transmitted probe. However, thespatial amplitude and phase characteristics of the radiatedsignal differ from that of the probe beam. The waist of theradiated signal field is roughly ) smaller than the radius ofthe incident fields at the sample because the third-order po-larization is created by the overlap of three incident lightfields at the sample. As the probe and signal propagate awayfrom the sample, spatial variations in the relative phase existwhich allow the probe beam to serve as a local oscillator forboth the birefringent and dichroic responses. The physicalorigin of the relative phase shift at finite distances from thesample is the spot-size difference, which leads to a shorterRayleigh range for the third-order signal. Therefore, the axialphase shift and radius of curvature of the probe and third-order signal differ at the iris position, creating a radiallyvarying phase shift between the two ~Fig. 1!. By spatiallyselecting a portion of the overlapping beams with the iris,one can detect the parts of the beams that have the largestintegral phase shift. When the iris is removed the integratedradial phase shift between Ebir and ELO is p/2, as expected inthe plane-wave limit.

The relationship between probe and signal beams maybe expressed quantitatively by utilizing a Gaussian decom-position of the input beam.48,51 In this treatment, the nonlin-ear signal is treated as a spatially varying perturbation of theinput beam. The total complex electric field at the sample isexpressed as a Taylor series expansion of the radially varyingnonlinear phase, so that each term can be easily propagatedthrough space to the iris position. The probe is represented asthe zero-order term (m50) and the third-order radiated sig-nal as the first-order correction term (m51). The expressionfor the measured interference signal after passing through theiris has been given by Gardecki et al.49

bir dic

FIG. 1. Schematic representation of the SM-OKE experimental arrangement~beam propagation is from left to right!. The distances z, d, and ra aredefined in the text.oped by Simon and co-workers in which the induced polar-ization acts as a lens to change the focusing of the probebeam.9,11 Another two-beam technique, implemented byZiegler and co-workers, involves either dispersing the over-

1144 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002 Fecko, Eaves, and Tokmakoff$sin uLO~r ,z !Si jkl~t!1cos uLO~r ,z !Si jkl~t!%r dr , ~24!


This awhere wm02 5wpr

2 (z)/(2m11) is the focused spot size of themth-order expansion term, wm

2 5wm02 @g21d2/dm

2 # is thebeam radius at the iris, dm5kwm0

2 /2 and g511d/Rpr(z).The spatially varying LO phase is

uLO~r ,z !5~u12u0!2kr2

2 S 1R12 1R0D , ~25!where the axial phase of the probe and signal are given by

um5tan21F dgdmG , ~26!

and the axially varying radii of curvature are

Rm5dF12 gg21d2/dm2 G21

. ~27!

Therefore, the phase of the local oscillator depends on thesum of contributions from a radially uniform term, u12u0 ,which is a scaled Guoy phase shift, and a term of oppositesign which does not contribute on axis but increases in mag-nitude with radial distance. The total signal magnitude isgreatest on axis (r50) and decreases radially, as one wouldexpect from a Gaussian beam.

The dependence of Eq. ~24! on ra and z has been dis-cussed by Gardecki et al. for both dichroic and birefriengentresponses. Because birefringence is much larger than dichro-ism for excitation that is far from electronic resonance, welimit this discussion to the birefringent part of the response.The dependence of the signal on the size of the iris openingis determined by both the radial amplitude profile of thebeams and the radially varying phase difference. A plot ofthe phase difference as a function of radius exhibits an ex-tremum on axis but changes sign when the second term inuLO dominates the phase. Therefore, the birefringent signal islargest when the iris is closed just far enough to block theportion of the beam for which the second term in the phase issignificant, but opened far enough to collect a measurablefraction of the beam. The signal vanishes for large ra , sincethe unapertured probe pulse cannot serve as a local oscillatorfor a birefringent response.

The dependence of the signal on the axial position of thesample relative to the beam waist, z, is determined mostly bythe relative amount of Guoy phase shift acquired by eachbeam as it propagates from the sample to the far field. A plotof the birefringent signal as a function of z is antisymmetricwith respect to the beam waist, containing a maximum andminimum on either side of the origin and tending towardszero for large uzu. Thus, there is an optimal distance a sampleshould be placed from the beam waist in order to achieve themaximal SM-OKE signal, which was found to be ;0.8 z0 byGardecki et al.

The final result is that by placing an iris in the probebeam path and moving the sample away from the beamwaist, the self-heterodyned pumpprobe signal may be de-tected. Note that the only restriction placed on the light fieldpolarizations in this method is that both pump field interac-tions are from the same beam and that the signal must have

J. Chem. Phys., Vol. 117, No. 3, 15 July 2002a polarization component along the probe polarization,which allows detection of Rzzzz , Rzzyy , and R iso .

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,C. MethodsThe experimental setup used for the SM-OKE experi-

ments is a modified version of the standard pumpprobeinterferometer, in which a single pulse is split into two rep-licas that travel nearly equal path lengths and are crossed at asmall angle ~56! in the sample. The relative delay be-tween pump and probe is adjusted via a motorized steppertranslation stage ~Newport UTMPP0.1!. For the Z-scan ex-periments, mica zero-order half-wave plates ~CVI! and Glan-Laser polarizers ~CVI! are placed in each arm of the inter-ferometer to allow independent rotation of the linearpolarization of each beam. An additional polarizer centeredon the probe beam immediately after the sample is adjustedparallel to the incident probe polarization, so that the Rzzzz ,Rzzyy , and R iso (5Rzzmm) tensor components can be mea-sured by rotation of the pump beam polarizer only. Thepump and probe are focused to a spot of 2w545 mm by a 10cm achromatic doublet ~CVI!, and crossed ;1/2 Rayleighrange ~1 mm! from the focus. An iris centered on the probebeam is typically placed ;20 cm from the sample andopened to a radius which maximizes the experimental signal.While the distance of the sample from the beam waist andiris opening varied between samples, they were always keptconstant between measurements of a set of tensor compo-nents.

Direct measurements of the anisotropic response func-tion are recorded in a typical OHD-OKE experiment. Themain difference between the OHD-OKE and SM-OKE set-ups is that the half-wave plate in the pump arm is replacedby a quarter-wave plate placed after the polarizer. The probeand analyzing polarizers are crossed, while the pump polar-ization is 45 with respect to the probe. An in-quadraturelocal oscillator is introduced by rotating the probe polarizerby a small angle ~Dw!, and the homodyne contribution isremoved by the subtraction of scans recorded with oppositepolarizer rotations ~6Dw!.44

The experiments are performed using the output of ahome-built cavity-dumped Ti:sapphire oscillator producing atrain of pulses at 500 kHz with a center wavelength of 800nm. Following precompensation in a pair of fused silicaprisms and propagation through the experimental interferom-eter, second harmonic cross correlation in a 10 mm BBOcrystal typically yields pulsewidths of 1618 fs FWHM.Pulse energies are typically 10 nJ for the pump and 1 nJ forthe probe, which is well within the linear detection regime.Signal intensity is detected using a photomultiplier tube, andrecorded using a lock-in amplifier referenced to the 2 kHzfrequency of a chopper placed in the pump arm of the inter-ferometer.

Acetonitrile, carbon disulfide, and methanol were ob-tained from Sigma ~all with at least 99.9% purity!, deionizedwater with resistivity .1.83106 V/cm was obtained fromRICCA, and all liquids were used without further purifica-tion. Because of significant signal contribution from thefused quartz cell walls to the isotropic spectrum, a 0.5 mmthick flowing jet was used to obtain measurements formethanol and water. For these samples, the noise floor is

1145Raman scattering from molecular liquidsdictated by the surface quality of the jet, so great care isneeded in adjusting the flow rate and in centering the beam


This aon the smoothest portion of the jet. It was not possible toobtain an optical quality jet for acetonitrile; however, it waspossible to greatly reduce the cell wall contribution by usinga 1 cm path length fused silica cell, and compressing thepulse using the electronic signal contribution. A jet of CS2was not attempted due to safety considerations; because thesignal was distorted in a long path length cell, a 1 mm pathlength cell was used for CS2 .

D. Data analysisSmall variations in the relative amplitudes and zero de-

lay times (t50) of the signals measured in different polar-ization geometries with the SM-OKE and OHD-OKE experi-ments are corrected using the R (3) symmetry relationships.After recording the time-domain data in all four polarizationconfigurations, a self-consistent reconstruction of the data isperformed by varying the amplitude and delay times of eachtensor component, holding one amplitude and delay timeconstant, and numerically minimizing the least-squares func-tion

u 12@AzzzzSzzzz8 2AzzyySzzyy8 #2AanisoSaniso8 u

1u 13@AzzzzSzzzz8 12AzzyySzzyy8 #2A isoS iso8 u. ~28!

The signal components in Eq. ~28! are experimentally mea-sured nuclear signals, denoted by a prime to differentiatethem from the resulting self-consistently fit data. The fit isperformed for t.100 fs to minimize contributions from thenonresonant electronic response. Because Szzzz8 , Szzyy8 , andS iso8 are all measured using the SM-OKE method, it is ex-pected that their scaling amplitudes should be approximatelyequal. The experimentally observed scaling amplitudes aretypically within 10%15% of each other, indicating that therelative signal amplitude varies slightly between measure-ments due to displacements of pump beam pointing when thepolarizer is rotated. Because the strength of the local oscil-lator in the OHD-OKE setup is different from that in theSM-OKE experiment, the amplitude of the Saniso8 component

FIG. 2. Comparison of the directly measured anisotropic ~upper panel! andisotropic ~lower panel! acetonitrile signals with their respective recon-structed signals. In each panel, directly measured data are displayed with apositive offset with respect to reconstructed data. The relative phase andamplitude of each scan was determined by a self-consistent fit of the dataaccording to Eq. ~28!. In the case of acetonitrile, Azzzz50.98 and Azzyy50.96, where A51 for both under perfect experimental conditions.

1146 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002relative to the others is treated as an unknown and is deter-mined completely by the self-consistent fit. Figure 2 provides

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,an example of the utility of this method, where the aniso-tropic and isotropic signals reconstructed from Szzzz8 andSzzyy8 are compared with the scaled Saniso and experimentallymeasured S iso of acetonitrile. ~A iso is constrained to unity.!The reconstructed quantities are nearly indistinguishablefrom those measured directly, as is the case for carbon disul-fide and methanol as well. It should be stressed that experi-mental amplitude and timing uncertainty is greatly reducedby independently measuring all four tensor components, al-lowing an accurate determination of the two independent re-sponse function components.

The time-domain data are also presented as a spectraldensity x9, the imaginary part of the Fourier transform ~sinetransform! of the time-domain response. As described above,this quantity is closely related to the traditional Raman spec-trum. However, due to convolution of the material responsewith the finite duration of the laser pulses, the amplitude ofeach Fourier component is scaled by the pulse spectral band-width. For a heterodyne-detected experiment, the impulsivelimit is recovered by deconvolution with the second har-monic autocorrelation, which amounts to division of the ex-perimental signal by the autocorrelation in the Fourierdomain.44 In addition, the contribution of the nonresonantelectronic response may be separated from the nuclear por-tion by considering only the imaginary part of the transform.

IV. RESULTS AND DISCUSSIONA. Acetonitrile, carbon disulfide, and methanol

The heterodyne-detected time-domain signals for aceto-nitrile are plotted in Fig. 3. The inset shows the nonresonantelectronic responses, which follow the predicted intensity ra-tios, Rzzzz :Rzzyy :Raniso :R iso>3:1:1:5/3. Each signal con-tains high-frequency underdamped oscillations attributed tointramolecular vibrations superimposed on slowly varyingdamped responses characteristic of the collective liquid re-sponse. The amplitude of the polarized (Rzzzz) response isgreatest and most similar to Raniso , while R iso is muchweaker. Rzzyy has many of the same features as Rzzzz , butwith roughly half the amplitude and opposite sign. It is there-fore clear that the collective response is predominantly an-

FIG. 3. Self-consistent, polarization-selective SM-OKE signals for acetoni-trile. In the main figure, from top to bottom are plotted Rzzzz , Raniso , R iso ,and Rzzyy . The inset shows the relative amplitudes of the electronic re-sponse Rzzzz :R iso :Rzzyy :Raniso51:0.53:0.31:0.31.

Fecko, Eaves, and Tokmakoffisotropic. Two Raman active intramolecular modes lie withinthe laser bandwidth, a symmetric CC stretch at 918 cm21


This aand an antisymmetric C CwN bend at 380 cm21. Bothmodes contribute significantly to the parallel and perpendicu-lar response; however, the anisotropic and isotropic signalseach decay with a single underdamped oscillation.

We demonstrate the polarization selectivity of the mea-surement in the deconvolved Fourier transform spectra ofacetonitrile plotted in Fig. 4. Only the antisymmetric modecontributes to the anisotropic signal and only the symmetricmode appears in the isotropic response. Such polarizationselectivity for intramolecular modes has been observed forCCl4 , and explained in terms of the projection of the lab-frame polarizations onto the molecular symmetry axes.13 Thespectral region below 200 cm21 is due to the intermolecularresponse of the liquid. The spectra indicate that the intermo-lecular modes are largely depolarized, since xZZZZ9 is almostindistinguishable from xaniso9 in this region, and the amplitudeof xaniso9 @x iso9 .

The deconvolved low-frequency anisotropic and isotro-pic Raman spectral density for acetonitrile, CS2 , and metha-nol are compared in Fig. 5. For all liquids, the anisotropicresponse is much stronger than the isotropic. The overallspectral profiles for the isotropic and anisotropic response aregenerally similar, except at frequencies ,10 cm21 and, formethanol, at frequencies .300 cm21. Noise is apparent inthe isotropic spectrum of methanol, due to small jitters of theflowing jet combined with a weak nuclear response. In addi-tion, the isotropic CS2 has a constant nonzero backgroundthat extends to 500 cm21, which we attribute to the wall ofthe fused silica cell. We have found the cell wall response tobe strongly polarized and fairly flat over the region 100500cm21. Thus, it is not possible to quantitatively compare theisotropic and anisotropic intermolecular spectral profile onthe high-frequency edge of CS2 .

The anisotropic spectra of acetonitrile and CS2 agreewell with previously reported OHD-OKE spectra.3,4,44,52Whereas the spectrum appears to show only two distinct fea-tures, a sharp peak at low frequency ~

This ature at frequencies higher than 350 cm21 due to librations ofhydrogen bonded molecules about the CO axis, which isabsent in our spectrum and in frequency domain depolarizedRaman spectra.

In Fig. 6, we compare the low-frequency anisotropic andisotropic spectra for acetonitrile, CS2 , and methanol on a logfrequency plot. For all three liquids, the amplitude of theintermolecular anisotropic response is much larger than theisotropic response, by a factor of .10 in CS2 and methanoland .20 in acetonitrile. The relative isotropic to anisotropicamplitude ratio agrees well with the 1:15 integrated intensityratio28 and 1:12 intensity ratio at the peak of the time-domainresponse29 found in recent MD simulations of CS2 .

Equally important to the magnitude is the frequency de-pendence of each response. In order to make a better com-parison, the isotropic response is scaled to compare the high-frequency edge. Qualitatively, the high-frequency wings ofthe isotropic and anisotropic spectra for acetonitrile and CS2are very similar, but the isotropic response for both liquidsdiffers from that of the anisotropic at frequencies below ;50cm21, where the anisotropic response is dominated by con-

FIG. 6. A comparison of the low-frequency anisotropic and isotropic lineshapes for acetonitrile, CS2 , and methanol. Isotropic spectra are scaled forthe purpose of comparison. Time-domain data were zero padded to approxi-mately ten times their original length to demonstrate how each curve slopestoward zero. The isotropic acetonitrile spectrum is the average of threeseparate scans. Intensities are consistent within each panel but are scaleddifferently for each liquid.

1148 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002tributions from rotational diffusion. For both liquids thesame frequency components are present in the isotropic and

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,anisotropic spectra, yet the amplitudes of the lowest fre-quency contributions are suppressed in the isotropic. Sup-pression of the lowest frequency components is even moredramatic in the isotropic response of methanol, where theresponse is zero within error for the region below 10 cm21.We therefore conclude that molecular rotational diffusion isgreatly suppressed, or perhaps eliminated entirely in the iso-tropic signal.

Suppression of the long-time decay in the isotropic re-sponse has been noted in other experimental investigations.Khalil14 and Constantine15 have both observed that the iso-tropic dichroic response of CS2 is dominated by processesthat decay with intermediate time scales and contain almostno long time decay components. However, Chang11 reportsdepolarization ratios for benzene, tolulene, and benzonitrile,which are mostly depolarized but constant for frequenciesless than 100 cm21, indicating that while the isotropic re-sponse is greatly suppressed with respect to the anisotropicresponse, the lowest frequency components are not preferen-tially eliminated in the isotropic spectrum.

A weak, low-frequency isotropic component has beenobserved in computational investigations of CS2 .17,29 Basedon a decomposition of the polarization according to M and IIterms, it may appear surprising that the isotropic responsecontains dynamics on the time scale of molecular diffusion.However, it has been demonstrated that the effective isotro-pic polarizability is larger for molecules that are aligned thanfor those that are randomly oriented, allowing for aninteraction-induced contribution on a time scale correspond-ing to rotationally diffusive motion.29,41 This phenomenonmay be alternatively described as an MII cross term, asdiscussed by Keyes, Kivelson, and McTague, who postulatedthat the time scale for cross correlations should resemble thatof rotational diffusion.57 Their argument was based on theassumption that preferential molecular alignment within theliquid would act to reduce the gas phase anisotropy, makingthe many-body polarizability more isotropic. Thus, while thecross term is negative for the anisotropic dynamics, it is posi-tive in the isotropic response.

The behavior of the isotropic spectra at frequenciesabove those corresponding to rotational diffusion is differentfor methanol, as compared with CS2 and acetonitrile. Whilethe latter two exhibit isotropic line shapes that are similar totheir anisotropic counterparts, the isotropic spectrum ofmethanol is significantly different from the depolarized spec-trum. After peaking near 25 cm21, it maintains a constantslope towards baseline with nonzero intensity beyond 500cm21. It is possible that the line shapes of the anistropic andisotropic are similar at low frequencies; however, it is clearthat the two differ greatly at the higher frequencies. As dis-cussed above, simulations have predicted that librationsabout the CO bond occur at frequencies as high as 600cm21,22,56 but such motions should be predominantly mo-lecular in nature and should be preferentially eliminatedfrom the isotropic spectrum. Even if librations were to con-tribute through the PII ,bb term, they should also lend inten-sity to the anisotropic spectrum at high frequencies.

Fecko, Eaves, and TokmakoffThe differences observed for methanol may be related tothe Raman noncoincidence effect, which is a frequency dif-


This aference in the first and second moments of the isotropic andanisotropic components of an intramolecular vibration. Theeffect has been explained as a manifestation of transitiondipole coupling and is often associated with local orderingwithin a liquid.58 The intramolecular modes of methanol areknown to have a large Raman noncoincidence,59 and the re-sults presented here bring into question the applicability ofthe effect for intermolecular vibrations. There are clear dif-ferences, since the usual effect involves a perturbation of theobserved frequencies for a localized vibration, while the in-termolecular spectrum is derived from the motions of lessordered liquid structures. Nonetheless, noncoincidence re-quires that the strong intermolecular coupling is observeddifferently in the isotropic and anisotropic spectra, which islikely to have a large impact on line shapes in the intermo-lecular part of the spectrum, as has been measured for metha-nol. While we note that Raman noncoincidence may be im-portant for methanol, we do not exclude other possibleexplanations, such as those discussed below.

Although numerous theoretical investigations have at-tempted to dissect the third-order responses of CS2 , acetoni-trile, and methanol, little has been said about the isotropicspectrum. Some insight into possible contributions to the iso-tropic spectrum can be gained by discussing contributions tothe anisotropic spectrum. As has been stated, II effects havebeen found to contribute to all portions of the spectrum for agiven liquid, although hindered rotations generally dominatethe anisotropic spectrum and contribute at higher frequenciesthan translations. The frequency dependence of the interac-tion induced terms in Eq. ~9! is not straightforward, as hasbeen demonstrated in a recent simulation of CS2 ,25 in whichcontributions were separated according to the number andorder of a and b interactions. While the response arisingfrom P II ,bb interactions contributes at the highest frequen-cies, those arising from purely translational fluctuations inthe P II ,aa tensor are not necessarily found at the lowestfrequencies. It has been argued that cancellation effects be-tween second-, third-, and higher-order DID interactions aremore important in dictating the time scale of a particularcontribution than the translational or rotational symmetry ofthe motions that give rise to them. This result may be used toexplain why we do not observe differences between the non-diffusive components of the isotropic and anisotropic spectrafor CS2 and acetonitrile: Even though the isotropic spectrumdoes not contain contributions from all of the terms in Eq.~6!, the II terms have overlapping time scales so that weak ormissing contributions do not change the line shape apprecia-bly ~as compared to the anisotropic spectrum!. It is not clearwhether such an argument would apply for methanol due tothe presence of hydrogen bonds, and it is also possible thatthe isotropic spectrum of methanol is dominated by a mecha-nism other than DID ~as discussed below!.

MD simulations are generally able to reproduce experi-mental anisotropic intermolecular line shapes with a highdegree of accuracy; however, it is interesting to note areaswhere difficulty has been encountered. In particular, severalstudies have had trouble reproducing the high-frequency

J. Chem. Phys., Vol. 117, No. 3, 15 July 2002wing of the intermolecular feature, even in CS2 andacetonitrile.24,60 The authors have suggested that poor fits

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,may be due to the importance of higher-order terms in themultipole expansion or short-range interactions, both ofwhich were neglected in these simulations. Similar difficul-ties have been encountered in light scattering studies of othersimple systems,35,36 in which it was found that the same con-siderations are even more important in reproducing the iso-tropic line shape than the anisotropic spectrum. We believethat both higher-order multipole terms and short-range inter-actions will be important for simulating the isotropic spectrafor liquids as simple as CS2 , acetonitrile, and especiallymethanol.

B. Water

It has been well established that the intermolecular Ra-man spectrum of liquid water is different from that of otherliquids. The primary difference is that the intermolecular in-teractions are known to contribute to the spectral density atfrequencies as high as 800 cm21. The anisotropic response ofwater has been characterized using both frequency and time-domain techniques.6163 The depolarized spectrum containsthree prominent features ~in addition to low-frequency rota-tional diffusion!, which are often referred to by assignmentsbased on their similarity with the spectrum of ice.61,62 Theintermolecular region consists of distinct peaks centered at60 and 175 cm21; the former is assigned to a transverse, orbending mode and the latter to a longitudinal, or translationalmode. X-ray scattering experiments on liquid water haveidentified these modes as second- and fourth-nearest neigh-bor interactions, respectively, emphasizing the collective na-ture of the motions.64 In contrast, the region between 4001000 cm21 contains an extremely broad band that arisesbecause of hindered rotations of single water molecules. Thebreadth of the band has been attributed to librations aboutdifferent molecular axes,62 with small amplitude motions ofthe light hydrogen atoms undergoing oscillations in less than50 fs. It is because of these extremely fast librations thatwater is able to accommodate changes in the electrostaticcharge distributions of solutes more quickly and efficientlythan any other liquid.

The 16 fs pulses utilized in our experiment are sufficientto excite the entire intermolecular band; however, data analy-sis requires extra precautions to ensure quantitative agree-ment between the fast and slow time scales. As for the otherliquids, all four tensor components were measured experi-mentally. However, instead of performing a self-consistentleast-squares fit using Eq. ~28! in the time domain, each dataset was first Fourier transformed and then fit in the frequencydomain. This procedure is needed because the dynamics onthe time scale of the experiment ~16 fs! makes it unreliable tofit to nuclear dynamics at delay times longer than 100 fs.Because the nonresonant electronic response does not con-tribute to the imaginary part of the transform, the frequency-domain fit is equivalent to the time-domain fit, as long as thephase of one component can be determined accurately. Thedepolarized Raman spectrum of water, for which the phase isnot an experimental variable, has previously been recorded

64

1149Raman scattering from molecular liquidsby Walrafen and has been used here to phase the experi-mental anisotropic response. This phase choice has conse-


This aquences with respect to the intensity of the isotropic responseat high frequency, as will be discussed below.

The experimental time-domain anisotropic and isotropicresponses are plotted in Fig. 7. The anisotropic time-domainresponse is plotted for both the flowing jet and static cell~inset! samples. While the cell data are unreliable at shorttimes due to signal from the walls, the long-time diffusivepeak is entirely due to water and can be measured moreaccurately with less noise. The jet data contain a sharp peakat 65 fs and a slightly broader peak near 200 fs. The formeris predominantly due to the critically damped oscillation ofthe intermolecular longitudinal mode, and the latter to that ofthe transverse mode. Contributions from the librations ap-pear as an asymmetry in the nonresonant electronic spike,which is not easily discernible from the time-domain data.

The self-consistent Fourier transform spectra for Saniso ,S iso , Szzzz , and Szzyy are shown in Fig. 8. The frequency-domain anisotropic spectrum agrees quite well with previ-ously reported spectra in both the frequency and relative in-tensity of all features. It contains all three intermolecularmodes discussed above, with bands peaked at 45, 160, and

21

FIG. 7. Self-consistent anisotropic and isotropic SM-OKE signals for water.The anisotropic signal shown in the inset was taken using a static cell andmay contain unwanted signal from the cell walls at short times, while theanisotropic and isotropic signals in the main panels were taken in the jet andare valid for all times.

1150 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002;450 cm . The transverse mode peak is shifted to lowerfrequencies due to overlap with the rotational diffusion re-

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,sponse. The spectra plotted in Fig. 8 show the direct Fouriertransform of the time-domain data. The spectra are still con-volved with the pulse bandwidth, which is also shown as adotted curve in the top panel. However, the behavior of thedeconvolved spectrum is clear, and it can be seen that theanisotropic signal decreases to baseline in the region be-tween 8001000 cm21, while the isotropic spectrum still hassignificant amplitude.

The isotropic response of water is entirely different fromthe anisotropic response. It contains almost no low-frequency components, and therefore appears in the time do-main predominantly as an asymmetry of the nonresonantelectronic spike. Only a hint of a low-frequency decay can bedetected at the positive time base of the electronic spike inthe isotropic response of Fig. 7. The data can be observedmuch better in the frequency domain, where the most notablefeature is the broad peak extending from 250 cm21 to higherfrequencies. The high-frequency tail of the broad peak fol-lows the intensity of the pulse spectrum, indicating that thefeature extends beyond the experimental bandwidth. Thehigh-frequency isotropic response is also observed as abroad, positive contribution to the Szzzz and Szzyy spectra.

The magnitude of the isotropic response of water indi-cates that intermolecular interactions are unusually large inwater. Its amplitude is substantially larger than would bepredicted based on a first-order DID model @Eq. ~8!#, sincethe polarizability of an isolated water molecule is nearlyisotropic.65 A similar conclusion has been reached for theanisotropic response in MD simulations employing first- andhigher-order DID methods: the zero-order molecular polariz-ability must have significantly more anisotropy than the gasphase value in order to reproduce experimental results.27,66,67

FIG. 8. Fourier transform anisotropic, isotropic, parallel, and crossed con-volved Raman spectra of water. The pulse bandwidth is represented as adotted line, and has the same zero level as the anisotropic spectrum. Theisotropic spectrum represents the average of ten independent datasets, toreduce noise and ensure accuracy; 95% confidence limits are typically630%.

Fecko, Eaves, and TokmakoffThis is one indication that the DID approximation is inad-equate to describe water, since modifying the isolated polar-


This aizability violates the perturbative nature of the approach.At this point we note the consequence of phasing our

data with respect to that of Walrafen. After subtracting back-ground, Walrafens anisotropic water spectrum drops to near-zero intensity at high frequencies ~8001000 cm21!; how-ever, there have been more recent reports which indicate thatthe spectral intensity is nonzero up until the bending region~;1600 cm21!.31,68,69 The consequence of this discrepancy isthat our data represent a lower bounds for the intensity ofboth the anisotropic and isotropic data in the high-frequencyregion. However, because of the self-consistent data collec-tion method, the phases of the two spectra are not indepen-dent; phasing the anisotropic data so that the spectral inten-sity at high frequencies is nonzero would require acorresponding rephasing of the isotropic data to increase itsspectral intensity at high frequencies. In any case, it is clearthat the intensity of the isotropic component is greater thanthat of the anisotropic component at high ~.800 cm21! fre-quencies.

The presence of a broad feature at high frequencies inthe isotropic water spectrum was reported previously byMoskovits et al.,68 and by De Santis, et al.,31 who recon-structed the spectrum from frequency domain IVV and IVHmeasurements. These studies observe a broad band similar toFig. 8 that merges with a peak at the OH bending frequency.The frequency of the maximum intensity and width of theband agree well with the Fourier transform isotropic Ramanspectrum plotted in Fig. 8. Since all three experiments utilizedifferent laser frequencies ~488, 514, and 800 nm! yet ob-serve the same feature, and since all are far from electronicresonance, it is highly unlikely that the band is due to adichroic response. Benassi and co-workers also report that asthe temperature is increased, the integrated intensity of thehigh-frequency isotropic feature decreases and the maximumintensity shifts to lower frequencies.69

There has been debate regarding the depolarization ratioin the translational region ~,250 cm21! of the spectrum.Walrafen originally reported that this region is entirely depo-larized (r50.75), but later found r,0.75 and used this asevidence for the existence of C2v symmetry clusters withinliquid water.61,62 Moscovitz and Michaelian68 later measuredcomplete depolarization, which was again disputed by De-Santis and co-workers,31 who found constant depolarizationratio of r;0.72 below 250 cm21. None of the studies indi-cated structure to the depolarization ratio in the translationalregion. We find that the depolarization ratio remains between0.70 and 0.75 from 10250 cm21, and that around 60 cm21it is less than 0.75.

A comparison of the anisotropic and isotropic spectrareveals few similarities, as opposed to the other liquids ob-served in this investigation. At low frequencies, the only re-gion where the response is nonzero is near 60 cm21, corre-sponding to the transverse mode of the anisotropic spectrum.However, the intensity is so low in comparison to the aniso-tropic intensity that the mode is still almost entirely depolar-ized. At higher frequencies, the shape of the isotropic band isdistinct from the libration band in the anisotropic spectrum.

21

J. Chem. Phys., Vol. 117, No. 3, 15 July 2002While librations are peaked near 450 cm and have zerointensity by about 1000 cm21, the isotropic feature rises to at

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,least 750 cm21, and may become more intense or plateau atfrequencies higher than the observable bandwidth of the ex-periment. Additionally, if the anisotropic band is properlyassigned to arise from pure hindered rotations, one wouldexpect these motions to be suppressed or eliminated entirelyfrom the isotropic spectrum based on symmetry arguments.Therefore, the microscopic origin of the isotropic spectrumof water in the intermolecular region must be different fromthat which gives rise to the anisotropic spectrum.

In order to gain insight into the relevant time scales themolecular- and interaction-induced terms contribute to theRaman spectrum of water, Fig. 9 compares our nuclear re-sponse functions to the anisotropic response calculated bySaito and Ohmine in a MD simulation.27 The experimentalnuclear response functions were obtained by back transform-ing the nuclear spectral densities of Fig. 8. Saito and Ohminehave decomposed the response into contributions from themolecular- and interaction-induced parts of the polarizabilityand from the cross term. The molecular contribution is domi-nated by librations, and undergoes a critically damped oscil-lation on an 80 fs time scale, while the II term contributes ona time scale approximately twice as long. It can be seen thatthe total calculated response matches the experimental aniso-tropic response quite well. The maxima of the experimentalresponse occur on a slightly faster time scale and with adifferent height ratio, both of which can be attributed toslight differences in the librational frequency and intensity;however, the overall agreement is remarkable. On the otherhand, it is nearly impossible to attribute the oscillation of theisotropic nuclear response, shown in Fig. 9~c!, to any of thecomponents of the MD simulation. The isotropic response isclearly faster than even the molecular part of the anisotropicresponse.

C. Isotropic spectra of hydrogen bonding liquidsAn interpretation of the high-frequency features found in

the isotropic Raman spectra of water and methanol is not

FIG. 9. ~a! Anisotropic nuclear response of water simulated by Saito andOhmine ~Ref. 27!. Molecular, interaction-induced, and cross terms havebeen projected from the total response. ~b! Experimental anisotropic nuclearresponse of water obtained by inverse Fourier transform of the spectrum. ~c!Experimental isotropic nuclear response of water.

1151Raman scattering from molecular liquidsstraightforward, since there has been little attention paid tothe isotropic response in previous molecular dynamics simu-


This alations of liquids. Based on what is available in the literature,we exclude several possibilities and further suggest plausibleassignments. The explanations we offer focus on proton ef-fects that may be either quantum-mechanical in nature orhave to do with a localized proton motion.

It has been suggested that isotropic spectral intensity inthe 8001000 cm21 region of water is the overtone of thewater librational band. A similar band has been observed inthe IR spectrum of water and assigned as an overtone band.70Although it is possible for a depolarized intermolecular vi-bration to have a partially polarized overtone, the intensity ofthe fundamental relative to that at higher frequencies is notconsistent with an assignment as an overtone band, even fora highly anharmonic mode. In addition, the temperature de-pendence of the high-frequency intensity is opposite thatwhich would be predicted for an overtone band.69

Another explanation for a polarized intermolecular bandat high frequencies is that higher-order terms in the multipoleexpansion are significant and contribute at high frequencies.The importance of higher-order terms has been suggested bySampoli and co-workers for both H2S and for water.31,69,71MD simulations using a TIP4P potential were performed us-ing an induced polarizability that included interactions be-tween the permanent dipole moment and the first hyperpo-larizability, and interactions of the molecular polarizabilitywith the dipolequadrupole polarizability.69 While these ef-fects were found to be important to achieve a nonzero iso-tropic spectral intensity in the 300700 cm21 region, thelong-range induced polarizability model found no spectraldensity at frequencies greater than 700 cm21, so it is unlikelythat these terms contribute to the observed isotropic re-sponse.

Results similar to those presented here have been foundfor the case of the isotropic component of Raman scatteringby atomic gasses35,36 and liquids,72 ionic metal melts,73 andother simple molecules.74 In general, these studies have mea-sured a depolarization ratio which is a maximum for thelowest frequency components measured, but which dropswith increasing frequency, eventually tending towards zerofor the highest frequencies measured. For example, a studyof He2 revealed that scattering that is completely polarizedbeyond ;500 cm21.36 Numerous models have been used toreproduce the depolarization ratios; while a multipole expan-sion may reproduce the data for low or intermediate frequen-cies, it consistently fails on the high-frequency wing. Theauthors of these studies have concluded that it is important totake into consideration short-range interactions, such as ex-change, induction, and dispersion, which may generally con-tribute to both the anisotropic and isotropic response. It hasbeen suggested that such effects may be taken into accountusing an analytical model that utilizes nonlocal polarizabilitydensity,34 or by means of ab initio calculations that can ac-count for these quantum-mechanical properties directly.75 Byusing ab initio methods, it has been found that these calcu-lations differ substantially from classical models, especiallyfor light atoms such as He where quantum effects are impor-tant.

1152 J. Chem. Phys., Vol. 117, No. 3, 15 July 2002It would be a logical extension to assume that the lighthydrogen atoms of water and methanol are greatly affected

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,by these same quantum-mechanical interactions and there-fore need to be treated as quantum particles. Such a treat-ment is almost certainly not needed at low frequencies, forwhich classical simulations adequately approximate thecenter-of-mass intermolecular motions, as has been provenby numerous simulations. However, quantum-mechanical ef-fects would be expected to contribute significantly for spec-tral contributions at frequencies higher than kT, as arepresent in both hydrogen bonding liquids.

Another possible assignment of the isotropic spectrum ofwater and methanol, which is not necessarily exclusive fromthe quantum-mechanical effects, is a modulation of the po-larizability due to low amplitude motions of individual hy-drogen atoms, possibly even involving proton mobility. Arelated idea has been suggested in the form of hydrogenbond dynamics by Nardone and co-workers, although fewspecifics were proposed.69 It is hard to ignore the fact that thebroad, isotropic band in water is continuous from frequenciescorresponding to the anisotropic librations around 400500cm21 to the intramolecular bends near 1600 cm21. A similarbroad background has been observed in the spectroscopy ofacidic media and attributed to the strong electronic polariza-tion of collective motions coupled to the excess proton.76Simulations and normal mode calculations have been used toassign features in the 9501200 cm21 range to collectivemotions involved in a Zundel-like O H1 O stretch.77 Theassignment of our isotropic band would not necessarily haveto include excess protons, but could involve a motion local-ized to hydrogen atoms, which are known to have an unusu-ally large polarizability when involved in a hydrogenbond.78

Simulation of this type of motion requires a flexible mo-lecular potential, and calculation of the Raman spectrum de-pends heavily on the model used to calculate the polarizabil-ity. Most MD simulations of water thus far have utilizedrigid potentials such as the MCY,79 SPC,66,80 TIPS2,19,27TIP4P,81 and other rigid models,82 most in conjunction withpoint molecular polarizabilities. The fastest dynamics whichcan be captured by these simulations are librations, which isthe reason that the spectral density for each drops to zero by;800 cm21. However, simulations of the IR spectrum ofneutral water21,83 and of the hydrated proton77 in which aflexible potential ~flexible SPC! is employed contain a regionof nonzero intensity between the librations and intermolecu-lar bends ~10001600 cm21!. Although the Raman spectrumis not presented, the results indicate that there exists dynam-ics occurring on a time scale between librations and bends,and it seems likely that a distributed polarizability modelwould contain Raman spectral density in this region. A CarrParinello MD ~CPMD! calculation of the IR spectrum ofwater also contains spectral intensity in this region.84 TheCPMD simulation also treats electrons quantum mechani-cally, so the induction, exchange, and dispersion effects dis-cussed above would also be included therein.

Further calculations are needed to investigate the possi-bility that short-range electronic properties and/or mechani-cal motions contribute to the high-frequency feature we ob-

Fecko, Eaves, and Tokmakoffserve in the isotropic spectra. Such a determination shouldnow be possible based on current classical and CPMD simu-


This alation techniques. One additional possibility includes thequantum effects of proton motion; however, it is not clearthat such a detailed study is needed until the possibilitieslisted above are explored.

V. CONCLUSIONS

We have measured the isotropic and anisotropic responsefunctions for several neat liquids in the frequency range01000 cm21. These results demonstrate the utility of theSM-OKE method for the determination of the low-frequencyisotropic Raman spectrum, and for the accurate scaling of thespectrum with respect to the depolarized spectrum by a self-consistent fit of multiple data sets. Because the nuclear por-tion of the third-order susceptibility has only two indepen-dent components, our measurements contain all of theinformation that can be derived from a one-dimensional Ra-man experiment. We believe the isotropic/anisotropic repre-sentation is the clearest separation of the Raman active inter-molecular dynamics that can be measured directly.

The isotropic response is less intense than the aniso-tropic response by about an order of magnitude for CS2 ,acetonitrile, and methanol; however, the isotropic responseof water is unusually strong. In all cases, the lowest fre-quency components, attributed largely to the rotational diffu-sion of single molecules, are preferentially suppressed oreliminated from the isotropic spectrum, as would be ex-pected from a decomposition of the system polarizabilityinto molecular and induced contributions. At higher frequen-cies, the isotropic line shapes for CS2 and acetonitrile aresimilar to their anisotropic counterparts; however, this is notthe case for methanol and water. The high-frequency wing ofthe isotropic spectrum for both of the hydrogen bonding liq-uids extends farther than the depolarized spectrum, exceed-ing our instrument sensitivity ~;1000 cm21! in the case ofwater.

It is not surprising that the scattering characteristics inthe intermolecular frequency range of the spectrum for thenonpolar (CS2) and polar aprotic ~acetonitrile! liquids differfrom the strongly associating hydrogen-bonding liquids~methanol and water!. However, the particular mechanismthat can account for the differences in isotropic scattering isnot entirely clear. While a DID interaction mechanism hasbeen widely used to account for anisotropic scattering insimple liquids, we do not believe it will be sufficient to re-produce the entire isotropic response, even for the nonasso-ciating liquids. Short-range interactions will be needed tomodel the interaction-induced effects, especially for metha-nol and water. It is obvious that quantum effects should beimportant for these liquids since the bulk of the isotropicscattering occurs above kT, and because light atoms such ashydrogen are especially susceptible to induction and ex-change processes. In addition, it is also possible that the ob-served spectra correspond to motion localized on individualhydrogen atoms. Simulations of liquids that take into accountthese mechanisms will be essential to gain an understanding

J. Chem. Phys., Vol. 117, No. 3, 15 July 2002of the interactions which give rise to the isotropic spectrapresented here.

rticle is copyrighted as indicated in the article. Reuse of AIP content is sub143.106.26.202 On: Wed,ACKNOWLEDGMENTSThe authors have benefited greatly from conversations

with many people, including Dr. Oleg Golonzka, ProfessorLarry Ziegler, Professor John Fourkas, and Professor TomKeyes. This work was supported by the Office of Basic En-ergy Sciences, U.S. Department of Energy ~DE-FG02-99ER14988!. Support for work at the G. R. Harrison Spec-troscopy Laboratory came from the National ScienceFoundation through the MIT Laser Research Facility ~CHE9708265!. A.T. thanks the David and Lucile Packard Foun-dation for a Fellowship.

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Isotropic and anisotropic Raman scattering from molecular liquids measured by spatially masked optical Kerr effect spectroscopy.pdf