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A theoretical description of the vibrational sum frequency generation spectroscopy of interfaces

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A theoretical description of the vibrational sum frequency generation spectroscopy of interfaces
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Perry, Angela S
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Water
Molecular dynamics
Liquid/vapor interface
Nonlinear spectroscopy
SFG
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Abstract:
ABSTRACT: Our work investigates theoretical approximations to the interface specific sum frequency generation (SFG) spectra at aqueous interfaces constructed using time correlation function (TCF) and instantaneous normal mode (INM) methods. Both approaches lead to signals in excellent agreement with experimental measurements. This work demonstrates how TCF and INM methods can be used in a complementary fashion to describe interfacial vibrational spectroscopy. Our approach is to compare TCF spectra with experiment to establish that our molecular dynamics (MD) methods can reliably describe the system of interest. We then employ INM methods to analyze the molecular and dynamical basis for the observed spectroscopy. We have been able to elucidate, on a molecularly detailed basis, a number of interfacial line shapes, most notably the origin of the complex O-H stretching SFG signal, and the identity of several intermolecular modes in the SFG spectra for the water/vapor interface.
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Thesis (Ph.D.)--University of South Florida, 2005.
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Includes bibliographical references.
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by Angela S. Perry.
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ATheoreticalDescriptionoftheVibrationalSumFrequencyGenerationSpectroscopyofInterfacesbyAngelaS.PerryAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofChemistryCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:BrianSpace,Ph.D.RandyLarsen,Ph.D.DavidMerkler,Ph.D.DavidRabson,Ph.D.DateofApproval:July6,2005Keywords:water,moleculardynamics,liquid/vaporinterface,nonlinearspectroscopy,SFGcCopyright2005,AngelaS.Perry

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AcknowledgmentsFirstandforemost,IthankGodforHiscountlessblessingsinmylifeandfortheabilitytosucceedinthiseld.I'dliketoexpressmygratitudetomyhusbandMike,myparents,andmyextendedfamilywhoselove,support,andbeliefinmehasbeenneverending.Fromthebottomofmyheart,thankyou.IgivemythanksandappreciationtoProfessorBrianSpace,whosesupportandencouragementhasbeenunwavering.Hispassionforsciencewasinspiringandmadethisjourneyexciting.ManythanksalsotoProfessorPrestonMoore,whosehelpandadvicehavebeeninvaluable.Iwouldalsoliketothankmycommittee:Dr.RandyLarsen,Dr.DavidMerkler,andDr.DavidRabson.Inaddition,mysincerethanksgoesouttomyfellowgroupmembers:ChristinaRidley,BenRoney,TonyGreen,ChristineNeipert,AbeStern,andespeciallyRussellDeVane,whobeganthisendeavorwithme.Ioermythanksandappreciationforyoursupportandfriendship.

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NotetoReaderNotetoReader:Theoriginalofthisdocumentcontainscolorthatisnecessaryforunderstandingthedata.TheoriginaldissertationisonlewiththeUSFlibraryinTampa,Florida.

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TableofContentsListofFiguresiiiAbstractiv1Introduction12TheoreticalBackgroundofNonlinearPolarization42.1CalculatingthePolarizationinLimitingCases{MonochromaticandIm-pulsiveLight..................................102.2TheMeasuredIntensityIncludingDielectricEectsfromtheInterfacialBoundaries..................................132.3WaveVectorandPhase-MatchingConsiderations.............162.4!1;!2intheDipoleApproximation...................183TimeCorrelationFunctionTheory253.1TheoreticalDevelopment...........................253.2TCFMethodforSFGSpectroscopy.....................274InstantaneousNormalModeTheory304.1TheoreticalDevelopment...........................324.2INMMethodforSFGSpectroscopy.....................335ThePolarizabilityModel34i

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6SFGSpectroscopyoftheWater/VaporInterface386.1SimulationModelsandMethods.......................426.2Results.....................................497Conclusion61References63AbouttheAuthorEndPageii

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ListofFigures2.1CoplanarGeometryoftheBeams......................86.1DierentPotentialsChangeSpectra.....................446.2DierentPolarizabilityModelsChangeSpectra..............476.3SFGTCFandINMSpectrafortheWater/VaporInterface........506.4SFGTCFSpectrafortheO-HStretchingRegion.............516.5SFGTCFSpectrafortheIntermolecularRegion..............536.6Water/VaporInterfaceSnapshot.......................546.7ProbabilityDistributionoftheDirectionCosine..............576.8RealandImaginaryComponentsoftheSpectra..............586.9RealandImaginaryComponentsintheO-HStretchingRegion......59iii

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ATheoreticalDescriptionoftheVibrationalSumFrequencyGenerationSpectroscopyofInterfacesAngelaS.PerryABSTRACTOurworkinvestigatestheoreticalapproximationstotheinterfacespecicsumfre-quencygenerationSFGspectraataqueousinterfacesconstructedusingtimecorrelationfunctionTCFandinstantaneousnormalmodeINMmethods.Bothapproachesleadtosignalsinexcellentagreementwithexperimentalmeasurements.Thisworkdemon-strateshowTCFandINMmethodscanbeusedinacomplementaryfashiontodescribeinterfacialvibrationalspectroscopy.OurapproachistocompareTCFspectrawithexperimenttoestablishthatourmolec-ulardynamicsMDmethodscanreliablydescribethesystemofinterest.WethenemployINMmethodstoanalyzethemolecularanddynamicalbasisfortheobservedspectroscopy.Wehavebeenabletoelucidate,onamolecularlydetailedbasis,anum-berofinterfaciallineshapes,mostnotablytheoriginofthecomplexO-HstretchingSFGsignal,andtheidentityofseveralintermolecularmodesintheSFGspectraforthewater/vaporinterface.ThesuccessofbothapproachessuggeststhattheorycanplayacrucialroleininterpretingSFGspectroscopyatmorecomplexinterfaces.iv

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Chapter1IntroductionLiquidwaterinterfacesareubiquitousandimportantinchemistryandtheenvironment.Thus,withtheadventofinterfacespecicnonlinearopticalspectroscopies,suchinterfaceshavebeenintenselystudied{boththeoretically[1{16]andexperimentally.[17{44]SFGspectroscopyisapowerfulexperimentalmethodforprobingthestructureanddynamicsofinterfaces.SFGisasecondorderpolarizationexperiment,andthemorecommonelectronicallynonresonantexperimentisthefocusoftheworkhere.SFGspectroscopyisdipoleforbiddenincentrosymmetricmedia{suchasliquids.Interfacesservetobreaktheisotropicsymmetry,andproduceadipolarsecondordersignalthatissensitiveonlytotheinterfaceinmostcases.Contributionsfrombulkallowedquadrupolareectshavebeendemonstratedtobenegligibleinsomecases,[45,46]butcanbeincludedifnecessary,[14]and,inthatcase,contributionsfromthebulkandinterfaceareobtainedinthesumfrequencysignal.TheSFGexperimenttypicallyemploysbothavisibleandinfraredIRlasereldoverlappingintimeandspaceattheinterface,andcanbeperformedinthetimeorfrequencydomain.[17,23,47{54]IntheabsenceofanyvibrationalresonanceattheinstantaneousIRlaserfrequency,astructurelesssignalduetothestatichyperpolarizabilityoftheinterfaceisobtained.[5,20,26]WhentheIRlaserfrequencyisintunewithavibrationattheinterface,aresonantlineshapeisobtainedwithacharacteristicshapethatreectsboththestructuralanddynamicalenvironmentattheinterface.[2,15,55]1

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Recentyearshaveseenagreatincreaseinthenumberofexperimentalgroupsper-formingSFGinvestigations.Incontrast,molecularlydetailedtheoreticalsimulationsofSFGspectraarecomparativelyfew,andhaveonlyrecentlybegunmakingasignicantimpact.Likeallvibrationalspectroscopies,thegoalofSFGspectroscopyistoinferstruc-turalanddynamicalpropertiesfromtheobservedspectroscopicsignatures.Incontrasttomoretraditionalvibrationalspectroscopies,SFGlineshapestendtobemorecom-plexreectingtheuniqueenvironmentthatispresentataninterfacialboundary,andarenotnearlyaswellunderstood.Thus,theadventofeectivetheoreticalsimulationtechniquespromisestohelprealizethepotentialofSFGspectroscopytopermitdetailedcharacterizationofinterfacesonparwiththatdoneinthebulk.Further,inanalogywithcondensedphaseexperiments,SFGexperimentshaverecentlybegunbeingperformedus-ingavarietyoftimeandfrequencydomaintechniquestakingadvantageoftheexibilityinherentinmeasuringasecondorderpolarizationsignal.[25,56,57]Chapter2presentsageneraltheoryofnonlinearpolarization,andspecializingtosecondorderprocesses.ThisformalismisneededtotheoreticallydescribecertainSFGex-perimentsespeciallytimedomainmeasurementsthatdonotutilizeeectivelymonochro-maticelds.Section2.1describeshowthegeneralformulassimplifyinidealizedlimits{inwhichmostextantexperimentshavebeenperformedorinterpreted.TheexpressionderivedtherearethosefrequentlypresentedintheSFGliterature.Next,relevantconsiderationsconcerningopticalexperimentsatinterfacesarepre-sentedincludingtheoriginandimportanceofFresnelfactors,andthephenomenologicalexpressionforthemeasuredsecondorderSFGintensityintermsofthesignaleldSection2.2.Therelationshipbetweenthecommonexperimentalpolarizationcondi-tionsoftheexperimentaleldsSSP,SPS,PPP,andPSSandmicroscopicCartesian2

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susceptibilitytensorelementsisalsopresented.Thewavevectorandphasematchingcon-ditionsthatneedtobesatisedforcoherentnonlinearopticalexperimentsarediscussedinSection2.3.Next,Section2.4presentsformalexpressionsfordipolarcontributionstothesumfrequencysignal.ThemicroscopicformulasforthedipolarSFGsusceptibilitytensorarealsopresentedalongwithadiscussionoftherotatingwaveapproximationinthiscontext.Theseexpressionsprovidethetiebetweentheearlierphenomenologicalexpression,andtheformulasneededtorelateasystem'sdynamicstoanSFGsignal.Chapters3and4containthetheoreticalbackgroundfortheTCFandINMmethods,respectively.Section3.2discussesthetimedomainapproachtocalculatingSFGspectra.Section4.2presentsafrequencydomainapproachtocalculatingSFGsignals.Chapter5presentsthepolarizabilitymodelusedthroughoutthiswork.Chapter6discussestheoreticalsimulationsandtheirresultswithafocusonthewater/vaporinterface;comparisonwithexperimentisstressed.Results,includingtheidenticationofnovelspeciesatthewater/vaporinterface,arepresented.Chapter7presentsconclusions,andabriefdiscussionoffuturedirectionsfortheoreticalstudiesofSFGspectroscopy.3

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Chapter2TheoreticalBackgroundofNonlinearPolarizationSFGexperimentsmeasureasecond-orderpolarizationgeneratedcoherentlyinadirectiongivenbytheexperimentalwavevectorandphase-matchingconditions.[25,56,58]Itisoneofseveralsecond-orderprocessesthatarepossiblewhentwoelectriceldsinteractwithamedium.ThefocusofthisworkisonSFGexperiments,althoughsomeoftheformalismpresentedismoregeneral.Suchmeasurementsareinterfacespecicbecauseeven-orderpolarizationgeneratingtermsareforbiddenincentrosymmetricmedia.Thiscanbeunderstoodbyconsideringreversingthedirectionofalltheelectriceldsinanexperimentforanisotropicsystem.Doingsomustchangethesignofthepolarizationbecausealldirectionsareequivalentonaverage.[58]However,evennumbersofeldswillmakethepolarizationequaltoitsnegative{aconditionthatinsiststhepolarizationiszero,i.e.P=)]TJ/F17 11.955 Tf 9.299 0 Td[(P=0.[56]Ataninterface,orincertainnoncentrosymmetricsolids,[25]theisotropyofthesystemisbroken.Thisleadstoasecond-ordersignalwithinthedipoleapproximation,andinthiscase,thesignalisproportionaltotheproductofthesusceptibilityandtheelectriceldsasdescribedbelow.Itshouldalsobenoted,evenincentrosymmetricmedia,bulkquadrupolarcontribu-4

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tionstoSFGsignalsarepossible,buthavebeenshowntobenegligibleinmostcasesinvolvingliquidinterfacesinthecommonSFGreectedgeometry.[45,46,54,59]Whenlightimpingesonaninterface,aSFGsignalisgenerated.Thissignalisbothreectedfromtheboundary,andtransmittedthroughtheinterface.[60]Theycan,however,beimportantforexperimentsperformedinthetransmissiongeometry.[46]Inthecaseofbulkquadrupoles,theircontributiontothesecond-ordersignalisproportionaltoderiva-tivesoftheeldwhichinvalidatestheabovesymmetryargument.Likeallnonlinearopticalexperiments,bothtimeandfrequencydomainapproachestoSFGarepossible.[47,56]Todate,mostSFGexperimentshavebeenperformedinthefrequencydomain,and,eectively,inthelimitofmonochromaticelds.[17,23,51{54]However,thereisgrowinginterestinusingbothtime-domain,mixedtimeandfrequencydomainapproaches,[47{50]andalsolookingatothersecond-orderprocessessuchasdierencefrequencygenerationDFGspectroscopy.[61{64]Thetheoreticalmethodsdiscussedbelowarecapableofdescribinganyofthesesecond-orderprocesses.Thus,beforespecializingthetheoreticalexpressionstothetypicalmonochromaticfrequencydomainexperiment,itishelpfultoexaminetheformaltheoreticalstructureofsecond-ordernonlinearprocesses.Theresultingexpressionswillberequiredincalculatingsignalsfromexperimentsoutsideofthefrequency-monochromaticortime-impulsivelimit,e.g.typicaltime-domainexperiments.Suchexperimentsarebecomingincreasinglycommonbecausetheycanprovide,inprinciple,informationdistinctfromidealfrequency-domainexperiments.[48]First,consideringanNthorderprocessanN+1wave-mixingexperiment,anelectric5

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eldisappliedattimetatpositionr,andcanbewritten:Er;t=NXn)]TJ/F17 11.955 Tf 5.479 -9.684 Td[(Enteiknr+Ente)]TJ/F21 7.97 Tf 6.587 0 Td[(iknr.1InEquation2.1,knisthewavevectorspecifyingtheeld-propagationdirection.Equa-tion2.1ispartitionedintocomponentsthatareslowlyvaryinginspace,andthosethatarespatiallyhighlyoscillatory.[25,56,65]Theslowlyvaryingspatialcomponent,Ent,cangenerallybefurtherdecomposedintotemporallyntandspatiallyEndependentparts.[65]Thissubsequentseparationallowstheeldtoberewrittenintheform:Er;t=NXn)]TJ/F17 11.955 Tf 5.479 -9.683 Td[(Ennteiknr+Ennte)]TJ/F21 7.97 Tf 6.586 0 Td[(iknr.2InEquations2.1and2.2,thesumonnisincludedbecause,inthemostgeneralcase,exacttimeorderingoftheappliedeldscannotbeassumed.[56]Inpractice,experi-mentsinthetimedomaintypicallyuserelativelyshortpulsesthatareseparatedandorderedintimewhilethefrequencydomaintechniquesemploynearlymonochromaticlasereldsthatoverlapintimeandspace{suchconsiderationssimplifytherequiredanalysisconsiderably.Giventheeld,theobservablenonlinearpolarization,PN,withinthedipoleap-proximationwheretheresponsefunctionandsusceptibilitytensorsareindependentofrandktakestheformofamultipletimeintegrationoverthematerialresponsefunction,RN,whichcontainsallthesystemvariablesandinformationtobeprobed.Thema-terialsysteminthetimedomainisdescribedbytheresponsefunction,andistypicallyreferredtoasthesusceptibilityinthefrequencydomain.PNr;t=Z10d1Z10dNRN1;;NjEr;t)]TJ/F20 11.955 Tf 11.956 0 Td[(1Er;t)]TJ/F20 11.955 Tf 11.955 0 Td[(N.36

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InEquation2.3,theverticallinerepresentsNtensorcontractions.InanNorderex-periment,thereareNrelevanttimes{correspondingtothenumberofEr,t'sintheexpressionforPNr,tthatareeachrepresentedbythesuminEquations2.1or2.2.Consequently,whenexacttimeorderingoftheappliedeldscannotbeassumed,andEquation2.1isusedtodescribetheappliedelds,asumofNNtermsdeterminetheNthorderpolarization.Thepolarizationcanthenbewrittenas:PNr;t=2NNNXsPNks;t=2NNNXsPNteiksr.4InEquation2.4,ksisthesumofthewavevectorsassociatedwiththeappliedeldsandrepresentsthedirectioninwhichthegeneratedsignalwillpropagate.AsisshowninFig-ure2.1,consideringasecond-orderexperimentprobinganinterface,afterthenonlinearsignalisgenerated,itwillinteractwiththeinterfaceproducingareectedandtransmit-tedsignalwithmodiedwavevectorsthisissuewillbediscussedinSection2.3.Note,PNks;tisacomplexquantity,anditisoneoftheNNprocessesthatdeterminesthetotalNthorderpolarization,PNr;t.Further,PNr;tisarealquantity,andisthesumofallthePNks;ttermsEquation2.4.However,onceaparticularksischosene.g.,bytheexperimentalgeometrythesignalisacomplexquantity,andtherealandimaginarypartscanbemeasuredseparately{e.g.,inaheterodynedetectedexperiment.[66]Inprinciple,thesumofallNNtermsmustbeevaluatedtocalculatethetotalNthorderpolarization.Consideringsecondorderexperiments,thisleadsto16distinctcontributions.Inpractice,thepolarizationgeneratedforagivenexperimentisassociatedwithaparticularwavevectorandphase-matchingcondition.Thisimplieswhenthetwo7

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Figure2.1:CoplanarGeometryoftheBeams.Coplanargeometryoftheincident,re-ectedandtransmittedbeams.12istheangleofincidencewithrespecttothez-axisofthevisibleIReld.SFGDFGistheanglethegeneratedSFGDFGsignalisradiatedat.k1k2isthewavevectorofthevisibleIReld.ksrksTisthewavevectorofthereectedtransmittedeld,andks=k1+k2.Allincidenteldsareassumedtolieinthesamexzplanewhichisnormaltothesurface.incidentwavevectorsadd,aSFGsignalisgenerated,andwhenthetwoincidentwavevectorsinteractsuchthattheresultingwavevectorisequaltotheirdierence,aDFGsignalisgenerated.Forsurfaceprobingspectroscopies,thedirectionthesignalDFGand/orSFGpropagatesinwillbeguidedbySnell'slinearandnonlinearrefractionandreectionlawsinconjunctionwiththeoriginalpropagationdirectionsofincidentwavevectors,k1andk2.Thus,withtheproperexperimentalsetupinwhichdetectors8

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areplacedattheappropriatephasematchedangle,itistypicaltoonlydetectoneofthepossiblesecond-ordernonlinearprocesses,e.g.SFGorDFG.Phase-matchingcriteriaforsurfaceprobingspectroscopieswillbediscussedmorethroughlyinSection2.3.Inasimilarmanner,informulatingtheoreticaldescriptionsoftheresponsefunction,R,agivenexperimentmayonlybesensitivetoapartoftheresponse,anditisconve-nienttodiscardportionsthatdonotcontributesignicantly.[61]Thisisaccomplishedbyidentifyingtermsintheresponsefunctionthatoscillateintimesoastophasecancelwiththosefromtheappliedelds,andsubsequentlydiscardingtheremainingterms.ThisiscalledtherotatingwaveapproximationRWA.Computationally,thisapproximational-lowsforinclusionofonlyfullyresonantLouivillespacepathwaysanddependsimplicitlyonthereferenceormodelsystembeingconsidered.[56,61,65]TheRWAisacomputa-tionalconvenience;itispossibletoincludetheentireresponsefunction,andperformtheintegrationinEquation2.3orpresentedspecicallyforsecond-orderprocessessuchasSFGinEquation2.5belowexplicitly.[67]Note,aparticularexperimentmeasureseitherthemodulusofthecomplexPNks;thomodynedetectionoritsrealorimaginarypartsheterodynedetection.[66,68]Meth-odsusinginterferenceeectsviahomodynedetectionbetweenbulkandinterfacialcon-tributionstoanSFGsignalhavealsobeenusedtoseparatelymeasuretherealandimaginarycontributionsataqueousquartzinterfaces.[17,69]Therealandimaginarypartsoftheresponsecontaininformationthatisnotobtainableviameasuringthemod-ulusofthesignal.[3,17]Thelimitsofidealfrequencyandimpulsiveeldswillsimplifythecalculationofthepolarizationaswillbeshowninthefollowingsection.9

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2.1CalculatingthePolarizationinLimitingCases{MonochromaticandImpulsiveLightThesecond-ordertimedependentpolarizationisdenedby[25,56,58,70,71]Pks;t=eiksrZ10d1Z10d2R1;2:E1t)]TJ/F20 11.955 Tf 11.955 0 Td[(1E2t)]TJ/F20 11.955 Tf 11.955 0 Td[(2.5InEquation2.5,R1,2isthesystem'ssecond-orderresponsefunctionarealquan-tity,1and2representtimedelaysbetweentherstandsecondelds,andthesecondeldandtimeofsignaldetectionrespectively.Note,thesymbol:"denotescontractionofthetwodimensionaltensorresponseorsusceptibilitywiththeelds.Adirectrela-tionshipbetweenthisquantityandthesecond-orderfrequency-dependentsusceptibilitycanbeestablishedbyrepresentingthetime-dependentcomponentsoftheappliedeldsastheirFouriertransform:Pks;t=eiksr 42Z1d!1Z1d!2Z10d1Z10d2R1;2:E1!1E2!2e)]TJ/F10 6.974 Tf 6.226 0 Td[(i!1t)]TJ/F10 6.974 Tf 6.227 0 Td[(1)]TJ/F10 6.974 Tf 6.227 0 Td[(i!2t)]TJ/F10 6.974 Tf 6.226 0 Td[(2.6ThedoubleFourier-Laplacetransformofthesystem'sresponsefunctionisnowiden-tiedasthesecond-ordersusceptibility,:[56,70]!1;!2=Z10d1Z10d2R1;2ei!11ei!22.7BecausethesusceptibilityresultsfromaFourier-Laplacetransformoftherealresponsefunction,itisacomplexquantitytheFouriertransformoftheresponsefunctionis,however,arealfunction.SubstitutionofEquation2.7intoEquation2.6givesthetime-dependentpolarizationintermsofthebackFouriertransformoftheproductofthe10

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frequencydomaineldsandsusceptibility:Pks;t=eiksr 2Z1d!1Z1d!2!1;!2:E1!1E2!2e)]TJ/F21 7.97 Tf 6.586 0 Td[(it!1+!2.8Fouriertransformingthepolarizationwithrespecttotimegivesthefrequencydependentpolarizationwherethesignalfrequency,s,isthetransformvariableconjugatetot,and!s=!1+!2:Pks;s=eiksr 2Z1d!1Z1d!2!1;!2:E1!1E2!2Z1dte)]TJ/F21 7.97 Tf 6.586 0 Td[(it!s)]TJ/F18 7.97 Tf 6.587 0 Td[(s.9Pks;s=eiksr 2Z1d!1Z1d!2!s)]TJ/F15 11.955 Tf 11.955 0 Td[(s!1;!2:E1!1E2!2.10UnlikeEquation2.7,causalitydoesnotrequireaFourier-Laplacetransformbecausenosystemfunctionisdirectlyinvolvedinthetransform.Integrationofthecomplexex-ponentialovert,thetimeofsignaldetection,resultsinthedeltafunctioninEquation2.10.Inthelimittheappliedeldsaremonochromaticthelimitinwhichmostfre-quencydomainSFGexperimentsareperformed,theymayberepresentedascomplexexponentialsinthetimedomainandwillbedeltafunctionsinthefrequencydomain:E!i=2Ei!i)]TJ/F15 11.955 Tf 12.514 0 Td[(i.Thus,anideal-frequencydomainexperimentdirectlyprobesthesusceptibility:Pks;s=2eiksr1;2:E1E2!s)]TJ/F15 11.955 Tf 11.955 0 Td[(s.11Note,asaresultoftheformof,s=1+2isthesignalfrequency,andisatthesumoftheinputfrequencies;isdescribedinSection2.4.InanSFGexperiment,bothfrequencieswouldbepositive,andasumfrequencydetected.InaDFGexperiment,one11

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oftheinputfrequencieswouldhaveanegativesignassociatedwithit,andadierencefrequencywouldbegenerated.Thelimitofidealfrequencymonochromaticappliedeldsisjustonecommonmeansofsimplifyingtheintegralsnecessarytocalculatethesecond-orderpolarization.Theoppositelimit,theimpulsivelimit,occurswhenthefastestmaterialtimescaleismuchlongerthanthedurationsoftheappliedelds.Thislimitisoftenassumedintheoreticaldevelopmentsforsimplicationpurposes,butgenerallyisnottrulyjustiable{temporalpulsedurationsarenotcurrentlyfasterthaneventheshortestvibrationaltimescale.[56]Inthislimit,theappliedeld'stemporalenvelopesbehaveasdeltafunctions-makingtheevaluationofEquation2.5trivial.Theresultingexpressionforanimpulsiveeldatsometime~isEi0=Ei0)]TJ/F15 11.955 Tf 12.564 0 Td[(~e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!i~.12Pks;t=eiksrRt)]TJ/F15 11.955 Tf 12.563 0 Td[(~1;t)]TJ/F15 11.955 Tf 12.563 0 Td[(~2:E1E2e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!1~1+!2~2.13Note,theappliedeldsinEquation2.12canalsohaveanexponentialphasefactoras-sociatedwiththem.[72]WhileEquation2.12impliestheexperimentprobestheentireresponsefunction,nite,yetshort,pulsesthatdonothavetheinnitefrequencyspec-trumthatadeltafunctionpulsewouldcontainareonlyresonantwithcertainpartsoftheresponsefunction.Tocorrectforthisdeciency,itisusefultowritetheresponsefunctionintermsofLouivillespacepathways.[56]Inthislanguage,oneshouldonlyincludefullyresonantLouivillespacepathwaysofthesusceptibilityresponsetensorinthepolarizationcalculation{thisisequivalenttoinvokingtheRWA.[56,65,73]In12

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practice,thislimitiscommonlyassumedforsimplicityincalculatingthepolarization;itmakesevaluationoftheintegralsinEquation2.5trivial.Inthiscase,itisnecessarytoincludeonlypathwaysthatareexpectedtocontributeforagivensetofeldsandarelevantmodelsystem.[56,72]2.2TheMeasuredIntensityIncludingDielectricEectsfromtheInterfacialBoundariesBecauseinterfacesnecessarilyincludedielectricboundaries,theequationsderivedthusfarneedtobemodiedaccordingly{themeasuredsignalwillincludefactorsduetoin-teractionswiththeboundaries.[74{76]Theeldsintheabovederivationsarelocaltothemedium,andSFGexcitationeldsmusttravelthroughthevacuumbeforeoverlappingattheinterfaceforliquid/vapororgas/solidinterfacesorthroughsomeothermediumwhenconsideringaburiedinterface.Whentheeldscombineattheinterface,asecond-ordernonlinearsignalisproducedthatinteractswiththedielectricboundaries.Hence,theobservedeldsmustberelatedtothelaboratory-generatedeldsthroughFresnelcoecients.[25,60]Inaboundlessmedium,theFresnelcoecientsreducetounity,andthelaboratoryandlocaleldsarethesame.[74]Experimentally,itistheintensitygeneratedatthesumfrequencyofthetwoinputbeamsthatismeasuredintypicalhomodynedetectionexperiments.Equation2.14[25,46,70,74,77]describestherelationshipbetweentheeld,Er;!s,andthemeasuredintensity,I!s,generatedatthesumfrequencyofthetwoinputelds:I!s=cp 1!sjEr;!sj2 2.1413

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Er;!sisfoundthroughtheuseofnonlinearMaxwell'swaveequation{knowingthenonlinearpolarizationonecansolvefortheeld.[25,78,79]Theexactformoftherelationshipbetweenthenonlinearpolarizationandthemeasuredintensitygenerallydependsontheboundaryconditionsofthemedium,thedirectionEr;!spropagatesin,howwellphase-matchingcanbeachieved,whetherornottheslowly-varying-envelopeapproximationismade,andtheformoftheappliedelds{i.e,monochromatic,Gaussian,etc.Ingeneral,thelocaleldsareapproximatedasmonochromatic,andtheslowly-varying-envelopeapproximationisassumedtobevalid.Explicitexpressionsusingvariousapproximationsareavailableintheliterature.[25,46,60,74]Generally,theintensityisfoundtobeproportionaltothesquareofthesumfrequencymultipliedbytheamplitudeofthenonlinearpolarization:I!s/!2sjP!seiksrj2.15TheproportionalitycoecientsincludetheFresnelfactorsthataretypicallycalculatedusinganappropriatemodeloftheexperimentalsetup.[15]Inordertocomparedirectlytheoreticalandexperimentalspectra,itisnecessarytoincludetheFresnelfactors{especiallywhencomparingrelativeintensitiesfromdierentpolarizationconditions.[2,15]AsdemonstratedinSection2.1,inthelimitofmonochromaticelds,thepolarizationdirectlyprobesthesusceptibilityEquation2.11.FollowingfromEquation2.15,inthislimit,themeasuredintensitywillalsodirectlyprobethethesquaredmodulusofthesusceptibilitytensorofthesystem.Intotal,thesurfacesusceptibilitytensorcontains27elements.Considerationofsymmetryconditionsforatypicalazimuthallyisotropic14

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interfacerequiresallbut7elementsofthesusceptibilitytensortovanishbecausetheelementsneedtobeinvariantwithrespecttosymmetryoperationsthatpreservetheazimuthalsymmetry.[63]Additionally,ofthe7nonvanishingcomponents,onlyfourare,ingeneral,uniquexzx=yzy;xxz=yyz;zxx=zyy;zzz.Here,thesubscriptsonaretheCartesiandirectionsinthelaboratoryframe.[77,80]Byutilizingdierentpolarizationconditions,itispossibletoprobedirectlythreeofthefournonvanishingsusceptibility-tensorcomponentsindependently.[25,77,80]Eachofthethreelighteldswithcorrespondingfrequencies!SFG;!vis;!IRinSFGexperimentscanbeeitherSorPpolarized.Spolarizedlighthasapolarizationvectorparalleltotheinterface,andthePpolarizationisatanangletiltedtothesurfaceandliesinaplanethatisperpendiculartotheinterface.Ifthexyplaneistakentobetheinterface,itisusuallydenedthatSpolarizedlighthasasingleCartesianpolarizationvectorcomponentalongtheyaxisb^ywhilePthevectorliesinthexzplanewithcomponentsc^x+d^z.[77,81]DierentcombinationsofSandPpolarizedeldsallowfordirectmeasurementofthefollowingtensorelements:[77,80,82]eff;SSP=sinIRLyy!SFGLyy!visLzz!IRyyz.16eff;SPS=sinvisLyy!SFGLzz!visLyy!IRyzy.17eff;PSS=sinSFGLzz!SFGLyy!visLyy!IRzyy.18eff;PPP=)]TJ/F20 11.955 Tf 9.299 0 Td[(cosSFGcosvissinIRLxx!SFGLxx!visLzz!IRxxz.19)]TJ/F20 11.955 Tf 9.298 0 Td[(cosSFGsinviscosIRLxx!SFGLzz!visLxx!IRxzx15

PAGE 23

+sinSFGcosviscosIRLzz!SFGLxx!visLxx!IRzxx+sinSFGsinvissinIRLzz!SFGLzz!visLzz!IRzzzHere,LrepresentstheFresnelfactorsforthegiveneldslinearFresnelfactorsforthevisibleandIReldsandanonlinearfactorforthesumfrequencyeld[23],and!iistheanglethattheeldatfrequency!imakeswithrespecttothesurfacenormal.Weuseefftodenotetheeectivesusceptibility-unlikeijk,effexplicitlyaccountsfortheFresnelfactors.TheSandPindiciesoneff;denotehowtheelds!SFG,!vis,and!IRrespectivelyarepolarized-i.e.,SorP.Intheaboveexpressions,becauseofthechosenexperimentalgeometry,threeofthepolarizationconditionsSSP,SPS,PSSdirectlyprobesinglesusceptibilitytensorcomponentswhilethePPPconditionhascomponentsofalluniqueallowedCartesiantensorelements.2.3WaveVectorandPhase-MatchingConsiderationsIncoherentnonlinearopticalexperiments,thesignalisgeneratedatawelldenedangleinthelaboratoryframethatisdeterminedbythewavevectoroftheincidentradiation.However,onlycertainexperimentalgeometrieswillgenerateadesiredPNkssignal.Therequiredgeometriesmustsatisfyphase-matchingconditionsthatareaconsequenceoftheinput-wave-vectorchoice.Tounderstandthephase-matchingconditionsthatneedtobemet,consideramonochromaticplanewave,expikjr)]TJ/F20 11.955 Tf 11.955 0 Td[(i!jt,anditsassociatedwavevector,kj.Itsfrequency,!j,andwavevectorarerelatedbythecomplexrefractiveindex,n!j=Refn!jg+iImfn!jg:[58,62]kj=n!j!juj c.2016

PAGE 24

Here,cisthespeedoflightinvacuum,andujisaunitvectorwhichgivesthedirectionofthewavevector.Eachappliedeldthenhasanangleofincidence,j,andadistincttime-dependentphase,j,associatedwithitseeFigure2.1:jt=Refn!ig!j cujr)]TJ/F20 11.955 Tf 11.955 0 Td[(!jt.21Inthecontextofsecond-orderexperiments,whenthetwoincidentwavevectorsoftheappliedeldsataninterfaceadd,k1+k2,aSFGsignalisgenerated.Alternatively,whenks=k1k2aDFGsignalisgenerated.Becausetheincidenteld'swavevectorsareoverlappedatthemedium'sinterface,theeldassociatedwithkswillbetransmittedthroughthemediumandalsoreectedfromthesurfaceofthemediumexceptforthecaseoftotalinternalreection.Intheelectricdipoleapproximationinisotropicmedia,boththereectedandtransmittedsignalsareinterfacespecic.Ifbulkquadrupolarcontributionsareimportant,thetransmittedsignalmaycontainasignicantcontributionfromthebulkthatisnotalwaysseparablefromtheinterfacialsignature.[45,46,83]Snell'sLaw,inconjunctionwithmedium-specicpropertiesandtheincidenteld'swavevectors,mustbeconsideredwhendeterminingthewavevectorofthereected,krs,andtransmitted,kTs,signals.Itshouldbenoted,althoughthetwoincidenteld'swavevectorsinitiallymaycom-binetogivek1+k2and/orks=k1k2,itisnotguaranteedameasurablesignalwillbereected.Thisisduetophase-matchingconditions,aconsequenceofenergyandmomentumconservation,thatmustsatised.ThisconsiderationleadstoSFGandDFGsignalsonlybeingdetectableatanglesthatsatisfythefollowingequationsinatypical17

PAGE 25

SFGexperimentalgeometrydetailedinFigure2.1:[62]sinSFG=!1sin1+!2sin2 !1+!2.22sinDFG=!1sin1)]TJ/F20 11.955 Tf 11.955 0 Td[(!2sin2 !1)]TJ/F20 11.955 Tf 11.955 0 Td[(!2.23Note,althoughEquation2.22alwayshasasolution,Equation2.23doesnot.Specically,forthegivenconditions,thismeansSFGwillalwaysemitasignalwhileDFGwillemitasignalonlywhenEquation2.24issatised:!1sin1)]TJ/F20 11.955 Tf 11.956 0 Td[(!2sin2 !1)]TJ/F20 11.955 Tf 11.955 0 Td[(!221.242.4!1;!2intheDipoleApproximationThesystem'ssusceptibility,,containsalltheopticalinformationaboutthematerialsystem,and,isthus,thefocusoftheoreticalinvestigationsintoSFGinterfacialvibra-tionalspectroscopy.Inordertocalculate,orequivalentlythesystem-responsefunction,R,itisnecessarytodevelopamicroscopicdescriptionofit.Further,itisdesirabletorepresenttheresponsefunctioninaformamenabletocalculation,andonethatcanexploitthepowerofMDinterfacialsimulations.MDiscapableofaccuratelydescribingboththestructureanddynamicsofevencomplexinterfaces.[2{6,10{13,84{86]Speci-cally,itwillbeshowntheSFGresponsefunctionisproportionaltotheimaginarypartoftheone-timecross-correlationfunctionofthesystemdipoleandpolarizability.[1{5,7]Inordertopursuethisgoal,startingfromdensity-matrixtheory,andusingpertur-bativetechniques,aformalexpressionforthesecond-ordersusceptibilityinthedipoleapproximationcanbederived.[5,25,58,87]Usingthismethod,SFGijk!isdenedby18

PAGE 26

asumofsixtermsshownbelow.[6,58]FourofthetermscontributetotheresonantSFGsignalcontainedinR1andR2,andtheremainingcontributetothenonresonantportionofthesignalNR1andNR2.ThetwotermsinR2containtheexpressions!IR+!ng+ing,andmayinitiallyappeartobenonresonant;isanarbitrarycon-vergenceparameterintime[88]thatisfrequentlyinterpretedphysicallyasadipole-dephasingrate[6,58]thatwouldberesponsibleforasinglemode'shomogeneouslinewidthinthefrequencydomain.Inclusionofthesetermsintheresonantsusceptibilityis,however,necessarytodevelopageneraltheory.Neglectingthesecontributionsresultsinanexpressionvalidonlywhen~!kT,wherekisBoltzmann'sconstantandTisthesystemtemperaturethisisequivalenttomakingtheRWAinthiscase.Note,althoughweuse!IRand!vis,thisiseasilygeneralizabletotwoarbitraryappliedelds.Inthefrequencydomain,SFGpqr!takestheformSFGpqr!SFG;!vis;!IR=Xg;n;mgR1+R2+NR1+NR22.25R1=pgnqnm !SFG)]TJ/F20 11.955 Tf 11.955 0 Td[(!ng+ing)]TJ/F20 11.955 Tf 41.666 9.167 Td[(qgnpnm !vis+!ng+ingrmg !IR)]TJ/F20 11.955 Tf 11.955 0 Td[(!mg+imgR2=qnmpmg !SFG+!mg+img)]TJ/F20 11.955 Tf 42.939 9.167 Td[(pnmqmg !vis)]TJ/F20 11.955 Tf 11.955 0 Td[(!mg+imgrgn !IR+!ng+ingNR1=qgnpmgrnm !SFG+!mg+img!vis+!ng+ingNR2=pgnqmgrnm !SFG)]TJ/F20 11.955 Tf 11.955 0 Td[(!ng+ing!vis)]TJ/F20 11.955 Tf 11.955 0 Td[(!mg+imgIntheaboveexpressionsthatdenethesixcomponentsofthesecond-ordersusceptibility,!ngisthefrequencycorrespondingtotheenergydierencebetweenenergylevelsnand19

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g.InEquation2.25,gistheinitial-statethermalpopulation,andthesumisovervibroniclevels.;isadipolematrixelementbetweenstatesandfordipolevectorcomponent.Approximating1/!sfg1=!vis,theresonantcontributionscanbesimpliedbyrewritingthemintermsofpolarizabilitiesanddipoles.Giventhedenitionofpolar-izabilityinEquation2.26,thetworesonantterms,R1andR2,simplifytoEquations2.27and2.28respectively:pq!=Xg;npgnqng )]TJ/F20 11.955 Tf 9.298 0 Td[(!+!ng)]TJ/F20 11.955 Tf 11.955 0 Td[(ing+qgnpng !+!ng+ingg.26R1=)]TJ/F20 11.955 Tf 40.248 9.168 Td[(pqgmrmg !IR)]TJ/F20 11.955 Tf 11.955 0 Td[(!mg+img.27R2=)]TJ/F20 11.955 Tf 40.152 9.168 Td[(rgnpqng !IR+!ng+ing.28LetRespqrdenoteonlythesumoftheresonanttermsR1andR2.Replacingthedenom-inatorsinbothoftheresonanttermswiththeintegralidentitiesR10dte)]TJ/F21 7.97 Tf 6.587 0 Td[(it!)]TJ/F21 7.97 Tf 6.586 0 Td[(!o)]TJ/F21 7.97 Tf 6.587 0 Td[(i=)]TJ/F21 7.97 Tf 6.587 0 Td[(i !)]TJ/F21 7.97 Tf 6.586 0 Td[(!o)]TJ/F21 7.97 Tf 6.587 0 Td[(iandR10dteit!+!o+i=i !+!o+i,andthentakingtheimpliedlimitthatgammagoestozero,givesEquation2.29.Equation2.30followsasanexactrewriteofEquation2.29,andexpressesthesusceptibilityintermsofthecrosscorrelationofthesystemdipoleandpolarizability:Respqr="i ~XgmZ10e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!mgtei!IRtpqgmrmgdt)]TJ/F20 11.955 Tf 14.383 8.087 Td[(i ~XngZ10ei!ngtei!IRtpqngrgndt#g.29Respqr=i ~Z10dteit!IR)]TJ/F20 11.955 Tf 11.727 8.088 Td[(i ~Z10dteit!IR.3020

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InderivingEquation2.29from2.30,pqtisidentiedastheHeisenbergrepresentationofthepolarizabilityoperatorattimet,andasumoverstatesisperformedtoremovearesolutionoftheidentity.[89,90]ExpressingthecorrelationfunctioninEquation2.30explicitlyasthesumofitsrealandimaginarycomponentsreducesEquation2.30toEquation2.31,below.Note,=CRt+iCIt=,andthesubscriptsRandIwillbeusedthroughoutthemanuscripttorepresenttherealandimaginarypartsbothofwhicharethemselvesrealofcomplexquantities.[89]InthefrequencydomaintheTCFisreal,andtakestheformCw=CR!+CI!{whereCRisevenCR!=CR)]TJ/F20 11.955 Tf 9.298 0 Td[(!whileCI!isodd)]TJ/F20 11.955 Tf 9.298 0 Td[(CI!=CI)]TJ/F20 11.955 Tf 9.299 0 Td[(!:[89,91,92]Respqr!IR=2 ~Z10dteit!IRCIt.31Note,RespqrispresentedasanexplicitfunctionoftheIRfrequencybecausetheotheropticalfrequenciesareabsorbedimplicitlyintothepolarizability.Equation2.31isanearlyexactrewriteexactotherthansubstituting1/!sfg1=!visoftheperturbationexpression,andisthecentralresultofthissection;itlinksthesusceptibilitytoaTCFofthesystem'sdipoleandpolarizability.Thequantum-mechanicalTCFisamenabletocalculationusingclassicalMDsupplementedbyasuitablespectroscopicmodelviaquan-tumcorrectingtheclassicalTCF;[3,93,94]whileEquation2.31includestheimaginarypartoftheTCF,classicalTCFmethodscanonlyapproximatetherealpart,butaswillbedetailedbelow,therealandimaginarypartsoftheTCFaresimplyrelatedinthefrequencydomain.[95,96]TheliteraturecontainsexamplesofusinganexpressionsimilartoEquation2.31,but21

PAGE 29

writtenintheRWA.[5,12,14]Itwillnowbedemonstratedthatonlyinthehigh-frequencylimit,andbecauseoftheexactfrequencyrelationshipbetweentherealandimaginarycomponentsofthecorrelationfunction,canR2beexcludedfromtheresonantcomponentofthesusceptibility.Inthisapproximation,theresonantsusceptibilityisgivenbyonlytherstterminEquation2.29{i.e.,theresonantsusceptibilityisthengivenastheFouriertransformofthefullTCF:Respqr=i ~XgmZ10e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!mgtei!IRtpqgmrmgdt=i ~Z10dtei!IR.32Toproceed,thecorrelationfunctionisexpressedasitsrealandimaginarytime-dependentcomponents.Next,boththerealandimaginarycomponentsofthecorrelationfunctionarerepresentedastheircorrespondingFouriertransforms,andtheorderofintegrationisswitched:Respqr=i ~Z10dtei!IRtCRt)]TJ/F20 11.955 Tf 11.955 0 Td[(iCIt.33Respqr=i 2~Z10dtei!IRtZ1d!ei!tCRw+1 2~Z10dtei!IRt)]TJ/F11 10.909 Tf 8.485 0 Td[(iZ1d!ei!tCIw2.34Respqr=i 2~Z1d!CRwZ10dteit!IR+!)]TJ/F11 10.909 Tf 19.205 7.38 Td[(i 2~Z1d!CIwZ10dteit!IR+!.35TheintegrationoverdtinEquation2.35caneasilybeperformedandresultsinadeltafunctionandprincipal-partcontribution:[88]Respqr=i 2~CR!IR+i 2~CI!IR+P 2~Z1d!CI!+CRw !IR+!.3622

PAGE 30

Equation2.36containstheFouriertransformsofboththerealandimaginarypartsoftheTCFincontrasttotheexactresult;Equation2.31afterperformingthetimeintegrationisproportionaltoonlythesumoftheFouriertransformoftheimaginarypartoftheTCFplusaprincipal-partcontribution.However,becausethereexistsanexactdetailed-balancerelationshipbetweentherealcomponentsofthefrequency-domaincorrelationfunction,CIw=tanh~!=2CRw,Equation2.36canberewrittenasRespqr=i 2~[1+coth~!IR]CI!IR+P 2~Z1d!CI!+coth~!=2CIw !IR+!.37Inthehighfrequencylimit,wherecoth~!=2!1,Equation2.37isthecorrectexpres-sionrelatingtheresonantsusceptibilitytotheimaginarycomponentofthecorrelationfunction.Equation2.32hasbeenusedintheliteraturetocalculatetheresonantsusceptibility,and,inthosecases,thefullquantumTCFisapproximatedastheclassicalTCF.[6,12]Equation2.32isaccurateatsucientlyhighfrequency,butwhenadoptingaclassicalapproach,itisbettertolinktheclassicalcorrelationfunctiontothequantumTCFviaquantum-correctionapproaches[2,3,97]{althoughquantumcorrectionaectsthemag-nitudeofthesignalmorethanthelineshape.Further,Equation2.32isnotaccurateatlowerfrequencieswheremanyinterestinginterfacialphenomenaoccur.[98]However,todate,SFGexperimentshavefocusedonhigh-frequencyintramolecularvibrationsduetotechnicallimitations{mosttunableIRlasersarenotyetcapableofprobinglowerfre-quencies.Eventhoughsomenonlinearcrystalse.g.GaSecangenerateinfraredbeamswithstrongenoughintensityinthelowerwavenumbers,theyarenotwidelyapplied,andcurrenttypicalSFGexperimentsemployoptical-parametric-amplierOPAgenerated23

PAGE 31

tunableIRradiationthathasinsucientpowertocreatemeasurableSFGsignalsbelowabout1000cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.[18,53,99]Thereare,however,free-electronlasersourcesthatproducesucientlyintenselightforSFGexperimentsintothefarIR.[98]Computationally,theseregionsoflowerfrequencycanbeanalyzed,[2,3]andhaverevealednovellow-frequencyspeciespresentatthewater/vaporinterface.[2,3,8]24

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Chapter3TimeCorrelationFunctionTheoryMDisoftenusedtocalculatethermodynamicequilibriumproperties.Inaddition,pertur-bationsofequilibriumsystemsmayalsobeexaminedinlinear-responsetheory.Centraltothisideaistheuctuation-dissipationtheorem,whichstatesthattherelaxationofsmallperturbationsofasystemcanberelatedtotheuctuationsthatoccursponta-neouslyinequilibriumsystems.ThisOnsager'shypothesisearnedLarsOnsagerthe1968NobelPrizeinchemistry.[100]3.1TheoreticalDevelopmentFluctuationsinanobservableabouttheequilibriumensembleaveragewheretheanglebracketsindicateensembleaveragecanbewrittenasAt=At)]TJ/F20 11.955 Tf 12.619 0 Td[(.1Now,weknowthat=0,butifwecorrelateAttoaspontaneousuctuationattimezero,Ct==)]TJ/F20 11.955 Tf 12.62 0 Td[(2.225

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wecanshowthatatshorttimesC==.3Whereasatlongtimes,AtbecomesuncorrelatedtoA:Ct!
.4Butweknowthat=0,soatlongtimesCt!0.Thisdecayofcorrelationwithincreasingtimeistheregressionofspontaneousuctuationsandcanberelatedtotherelaxationofasystemfromaperturbation.TheHamiltonianforthissystemisH,andtheequilibriumaverageoftheobservablecanbewrittenasthetraceoftheobservable:=Tre)]TJ/F21 7.97 Tf 6.586 0 Td[(HA Tre)]TJ/F21 7.97 Tf 6.587 0 Td[(H.5Ifweperturbthesystemattimet=0withH=)]TJ/F20 11.955 Tf 9.299 0 Td[(fA,wherefisanappliedeldthatcouplestoA,wecanfollowtherelaxationofAtbackto:A=Tre)]TJ/F21 7.97 Tf 6.586 0 Td[(H+HA Tre)]TJ/F21 7.97 Tf 6.587 0 Td[(H+H.6At=Tre)]TJ/F21 7.97 Tf 6.586 0 Td[(H+HAt Tre)]TJ/F21 7.97 Tf 6.587 0 Td[(H+HIfweexpandAtaboutH,andnotethatclassicallyHandHcommuteweobtainAt=Tr[e)]TJ/F21 7.97 Tf 6.587 0 Td[(H)]TJ/F20 11.955 Tf 11.956 0 Td[(H+:::At] Tr[e)]TJ/F21 7.97 Tf 6.586 0 Td[(H)]TJ/F20 11.955 Tf 11.955 0 Td[(H+:::].7=Tr[e)]TJ/F21 7.97 Tf 6.587 0 Td[(HAt)]TJ/F20 11.955 Tf 11.955 0 Td[(HAt+AtTre)]TJ/F21 7.97 Tf 6.586 0 Td[(HH=Tre)]TJ/F21 7.97 Tf 6.586 0 Td[(H] Tre)]TJ/F21 7.97 Tf 6.586 0 Td[(H+OH2=)]TJ/F20 11.955 Tf 9.298 0 Td[([)]TJ/F20 11.955 Tf 12.619 0 Td[(]+OH2Usingequation3.1andinsertingtheexpressionforH,wehave[100]At=f+Of23.826

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3.2TCFMethodforSFGSpectroscopyThefollowingisaderivationofatime-domainapproach[1{3,5]tocalculatingtheresonantsusceptibility.AtthecoreofthisapproachiscalculatingthecrossTCFofthesystemdipoleandpolarizability,.This,beingtheproductofarstandsecondranktensor,vanishesinisotropicmedia.[101]Theresonantpartofthesusceptibilityisgivenbytheimaginarypartofthisquantum-mechanicalTCFviaEquation2.31.Thegoal,inthiscontext,istorewriteEquation2.31inaformthatisamenabletocalculationusingclassicalTCFtheoryinordertotakeadvantageofthepowerofthemolecularlydetaileddescriptionoeredbymany-bodyclassicalMD.Toproceed,theimaginarypartoftheone-timeTCFisrelatedinfrequencyspaceexactlytotherealpart:CI!=tanh~!=2CR!;wherethesubscriptsdenotetheFouriertransformoftherealandimaginarypartsofthecomplexfunctionCtwhichisarealfunctionoffrequency,i.e.C!=1 2R1dte)]TJ/F21 7.97 Tf 6.587 0 Td[(i!tCRt+iCIt=CR!+CI!.SubstitutingintoEquation2.31givesRes!=2 ~Z10dtei!tCIt3.9CIt=Z1d!0ei!0ttanh~!0=2CR!0Note,CItiswritteninaformthatcanbecalculatedusingtherealpartofthecorrelationfunction{whichisapproximatelyobtainable,afterquantumcorrection,usingclassicalMDandTCFapproaches.Duetocausality,theFourier-LaplacetransformgivesarealandimaginarypartinEquation3.9asthecosineandsinetransformofCItrespectively.Equation3.9canbesimpliedbychangingtheorderofintegration{performingthe27

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frequency-domainintegralrst.DeningtherealandimaginarypartsofRes!=ResR!+iResI!:I!=1 ~tanh~!=2CR!=2 ~Z10sin!tCItdt.10R!=1 ~PZ1tanh~!=2CR!0 !+!0d!0=2 ~Z10cos!tCItdt.11ToobtainEquations3.10-3.11,theidentityR10ei!tdt=iP !+!wasused.Pdesignatestheprincipalpart.[88]ThecurrentfocusofSFGexperimentsisonintramolecularvibrations,andtocalculateobservablesinthisspectralregion,classicalmechanicsisclearlyinvalid.Buildingonourpreviouswork,theclassicalcorrelationfunctionresult,whichisamenabletocalculationusingMDandTCFmethods,isquantumcorrectedusingaharmoniccorrection"factor:CR!=CCl!~! 2coth~!=2.[97,102]Thiscorrectionfactorisexactinrelatingtherealpartoftheclassicalharmoniccoordinatecorrelationfunctiontoitsquantummechanicalcounterpart.ItisworthnotingitisnotuncommoninmodelingvibrationalspectroscopyviaTCFstoseetherealpartofthequantumTCFreplacedbytheclassicalTCFthathasthesameeventimesymmetry,andbecomesequivalentclassicallyatlowfrequencieswhere~!kT.Thisapproachisreasonableindescribingvibrationallineshapesbutdoesnotgiveaccurateintensities.Itisgenerallybettertousetheharmoniccorrectionfactor.SimilarcaveatsapplytoreplacingthefullquantumTCFwithitsclassicalcounterpart,but,inthatcase,oneneglectstheimaginarypartoftheTCFentirelythatmaynotmatterverymuchwhenconsideringhigh-frequencyphenomenaforwhichCI!=CR!becausethehyperbolictangentfunctionisapproachingunity.28

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Usingtheclassicalharmoniccoordinatequantumcorrectionfactordoesnotaccountforthefactthedipoleandpolarizabilitycontainhigherordersofthecoordinates{ex-actcorrectionsforharmonicsystemsofthistypearestillpossible,butunneededthelineardipoleandPlaczekapproximationareadequate.[97]Usingthisresult,theTCFapproximationtotheresonantpartoftheSFGspectrum,Res,takestheformI!=!CCl!.12R!=PZ1CCl!0!0 !+!0d!0.13CClt=.14InEquation3.14,theanglebracketsrepresentaclassicalTCFthatcanbecomputedusingMD,andasuitablespectroscopicmodel.[5,103]Finally,Equations3.12-3.13givetheTCFsignalinaformamenabletoclassicalsimulation.Note,whileiteasiertoevaluateI!usingEquation3.12,R!ismoreeasilycomputedbydoingthecosineintegralasinEquation3.11.[3]Consideringthreepossibleindependentpolarizationconditions,SSP,PPP,andSPS,fortheTCFinEquation3.14,thelastindexinthepolarizationdesignationcorrespondstotherstindexintheTCF.Forexample,theSSPandPPPpolarizationconditionsprobedipolarmotionsnormaltotheinterface,andtheSPScaseissensitivetodipolarchangesparalleltotheinterface.Note,thePPPconditionissensitivetomotionsbothparallelandperpendiculartotheinterface.[77]Further,theSSPandPPP/SPSprobediagonal/o-diagonalpolarizabilitymatrixelementsrespectively.29

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Chapter4InstantaneousNormalModeTheoryINMtheoryhasbeensuccessfulindescribinganumberofliquid-stateproperties.[104{114]Further,INMtheoryisanintuitivelyappealingandeectiveapproachtomodelinginterfacialspectroscopy.[1{3]TheINMapproachtospectroscopycanbeinterpretedasanapproximationtothetrueBorn-OppenheimervibrationaldensityofstatesDOS.[93]Spectroscopicquantitiesmaybeevaluatedbymakingaharmonicoscillatorapproxima-tiontotherelevantfrequency-domaingolden-ruleexpression,leadingtoanapproxima-tiontothespectrumintheformofaweightedDOSwDOS.ForSFGspectroscopythisbecomesadipoleandpolarizabilityderivativeweightedINMDOS,wherethederivativeofthedipoleorpolarizabilityiswithrespecttotheINMs.TheINMwDOScanbeconstructedfromthermodynamicinformation,notrequiringexplicitdynamicalinput,althoughitcontainsshorttimedynamicalinformation.Inseveraldenseliquids,whererotationsaresignicantlyhindered,INMintermolec-ularspectrahavebeenshowntobealmostequivalenttoTCFderivedspectra,[115{121]andthishasledtoaninterpretationoftheintermoleculardynamicsintheslowmod-ulationlimitofKubo'smotionalnarrowingtheory.[55]Intramolecularlineshapescan30

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beinterpretedasbeinginthemotionally-narrowed,fast-modulationlimit,wheretheunderlyingINMspectraldensity"isnarrowedintoaslimmerlineshape.TheKubotheoryofmotionalnarrowingfocusesonthelineshapeassociatedwithatime-dependentharmonicoscillator,anditbecomessimpleinthelimitsofslowandfastmodulation.Intheslow-modulationlimitthelineshapereectsallthefrequencyuctuationsthattheoscillatorexhibits;inthefast-modulationmotionally-narrowedregimethelineshapeisnarrowed.IninterpretinganINMwDOSinthiscontextoneassumesthatINMsarewelldenedforsometime[122,123]andthattheyuctuateinfrequency.ThiskindofinterpretationissuggestedbythetimedependenceoftheINMfrequencies.ItwillbeshownthatconsistentwiththeKubopicture,wehavefoundthattheintramolecularINMbandsarebroaderbuthavethenearlythesamecentralfrequencyastheircorrespondingTCFlineshape,andtheintegratedintensitiesoftheTCFandINMbandsareapproximatelyequal.[2{4]ThesetwoobservationssuggesttheidenticationoftheINMwDOSasanunderlyingspectraldensity,i.e.theintramolecularlineshapewithmotionalnarrowingeectsremoved.AmajorstrengthoftheINMmethodisthatitisafrequency-domainmethod,allowingforthemodesatanygivenfrequencytobeanalyzedinmoleculardetail.INMtheoryprovidesatractablemethodforexaminingtheunderlyingmolecularmotionsresponsibleforgeneratingtheassociatedINMandpresumablyTCFspectroscopicsignature.Thisispossiblebecause,asnotedabove,INMintermolecularspectraareofteninnearagreementwithcorrespondingTCFresults;whenconcernedwithintramolecularlineshapesitisargued,thattheINMspectrumstillrepresentstheunderlyingspectraldensity"thatis31

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motionallynarrowedtogivetheobservedlineshape.4.1TheoreticalDevelopmentInterfacesaremodeledviatypicalMDsimulationsinordertogeneratecongurationalensemblesthatarerepresentativeofthepotentialenergylandscape.Thetotalpotentialenergyofeachcongurationcanbeexpandedasfollows:V=Vr0+NXi=1@V @rir0ri+NXi;j=1@2V @rirjr0rirj.1Thenormalcoordinatesareobtainedbydiagonalizingthematrixofsecondderivativesofthepotentialenergy,theHessianmatrix[104,110,119,122,124{127]:Di;j=@2V @qi@qj.2whereqi=m1=2iriisthemass-weightedcoordinateforatomiwithmassmi,andpositionvectorri.Diagonalizationoftheforceconstantmatrixyieldsasetof3NeigenvalueswhicharethesquaresoftheINMfrequencies,!2.ImaginaryINMsarethoseinwhichtheeigenvalueisnegative,thereforethefrequencyofthemodeisimaginary.Thecorre-spondingeigenvectorsmapthesystemcoordinatesontotheINMcoordinatesQ,whichhavethefollowingform:Qk=Xiuikqi;.3whereuikistheithelementoftheeigenvectork.TheINMapproximationtothevibra-tionaldensityofstatesDOSisahistogramoffrequenciesobtainedoveranensembleaverageofcongurations:!=h3NXk=1!)]TJ/F20 11.955 Tf 11.955 0 Td[(!ki:.432

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4.2INMMethodforSFGSpectroscopyWeconstructedanINMapproximationtoSFGspectra.Todoso,itissucienttoeval-uateEquations3.12-3.14foraharmonicsystem.Todoso,itisconvenienttoinvokeboththePlaczekandlinear-dipoleapproximationtoevaluatetheresultingmatrixelements{althoughhigher-ordercontributionscanbeincluded,andsimpleanalyticexpressionsresultforthesecontributions.AnequivalentapproachistoevaluateCCltforclassicalharmonicoscillators,andquantumcorrecttheresultingexpressionusingtheharmonic-correctionfactor,CR!=CCl!~! 2coth~!=2,torelateCCltandCRt:CCl!=<@i=@Ql@jk=@Ql!)]TJ/F20 11.955 Tf 11.955 0 Td[(!lkT !2>.5InEquation4.5,!listhefrequencyofmodeQl,andtheanglebracketsrepresentaver-agingoverclassicalcongurationsofthesystem.CCl!isthenbacktransformedintothetimedomain,andusedinEquations3.12-3.13inplaceoftheclassicalTCFtoobtainanINMapproximationtothespectroscopy.ItwillbedemonstratedtheTCFapproach,whichdoesnotinvokethePlaczekandlinear-dipoleapproximationsexceptimplicitlyinquantumcorrectingtheresults,givesresultsincloseagreementwiththeINMresultsandEquation4.5isthereforesucient.33

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Chapter5ThePolarizabilityModelInthisworkreferenceismadetothepointatomicpolarizabilityPAPAmodel[128,129].Inordertocorrectlymodelthesystemdipole,includingboththepermanentandinduced-dipolecontributions,polarizationmustbeexplicitlyincludedinthecalculations.Histori-cally,polarizabilityhadbeentreatedasanadditivequantity[130]wherebyvariousatomsorfunctionalgroupswereassignedempiricalpolarizabilitiesandtheircorrespondingsumwasthemolecule'spolarizability.However,thishypothesisisrepeatedlycriticizedbe-causethisapproachneglectstheelectriceldinducedbytheinduceddipolesthemselves.Silberstein[131]overcamethisobjectionbyintroducingapolarizabilitymodelwhichexplicitlyincludedthedipole-dipoleinteraction.ThismethodwasfurtherdevelopedbyApplequist[128,129],avoidingperturbativeapproximations,andistheapproachadoptedinthiswork.ConsiderasystemofNatoms.InourpointchargeandPAPAmodel,[128,129]eachatomihasapointcharge,qiandapointpolarizability,i.Eachatomispolarizedbyanelectriceld,E0i.Thisgivesrisetoaninduced-dipolemoment,i,whichitselfcontributestothetotalelectriceldEi:34

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i=iEi.1Ei=E0i+Xj6=iTijj.2E0i=Xjrriqj rij.3=Xjqjrij r3ij.4whereTijistheinduced-dipoleeldtensorwhichhastheformTij=rrirrj1 rij.5Tij=)]TJ/F15 11.955 Tf 13.837 8.088 Td[(3 r5ij26666664x2xyxzxyy2yzxzyzz237777775)]TJ/F15 11.955 Tf 16.495 8.088 Td[(1 r3ijI.6whererijisthedistancebetweenatomsiandj,andx;y;zarethevectorcomponentsofrij.Iistheidentitymatrix.Expandingequation5.1wegeti=i"E0i+NXj6=iTijj#.7whichcanberewrittenas)]TJ/F18 7.97 Tf 6.586 0 Td[(1ii+NXj=1;j6=iTijj=Ei.8Equation5.8couldbewritteninmatrixformas266666666664)]TJ/F18 7.97 Tf 6.586 0 Td[(11T12T1NT21)]TJ/F18 7.97 Tf 6.586 0 Td[(12T2N............TN1)]TJ/F18 7.97 Tf 6.586 0 Td[(1N37777777777526666666666412...N377777777775=266666666664E1E2...EN377777777775.935

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ormorebrieyA=E.10HereAisan3N3Nreal,symmetricmatrixandandEare3N1columnvectors.IfwecalculatetheinverseofmatrixA,andwritetheinversematrixBusing33elementsBij,thenB=266666666664B11B12B1NB21B22B2N............BN1BNN377777777775.11MultiplyEquation5.10byB=BE.12ThisisasetofNlinearequationsui=NXj=1BijEij:.13Foroursimulationsandanalysis,wemaysolveforthesystem'sinduceddipoleMind,whichisasumoftheindividualinduceddipoles,usingeithermatrixinversion,orinaniterative,self-consistentfashionMind=NXi=1i="NXi=1NXj=1Bij#Ei.14Fromequation5.14,weseethattheinduceddipoleofthesystem,ismerelyafunctionoftheinter-nucleardistances.However,whentheinteratomicdistancerijapproachesij1=6,unphysicallylargevaluesfortheinduced-dipolemomentsresult.Inordertocorrectthisproblem,adampingtermisintroduced.36

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Inordertocalculatethepolarizabilitytensorofthesystem,wemaketheassumptionofaconstantelectriceld,Ei,andutilizeEquations5.1and5.13,todeterminethati=NXj=1Bij.15wherepreviously,iwasthescalar,pointisotropicpolarizability,whichwasdynamicallycoupledthroughchainsofdipolarinteractions,viatherelaytensorBij.iisthe33po-larizabilitytensoronatomi.Nocouplingexistsbetweenthelocalpolarizabilitytensors;theyare,generallyspeaking,morecomplexanisotropicandpossiblyunsymmetricandalreadycontainthecouplingbetweeni's.Fromthesetensors,wemaynowcalculatethesystempolarizabilitytensorA:A=NXi=1NXj=1Bij.16Wenowhavethepolarizabilityasafunctionofinter-nucleardistances.Aswiththesystemdipole,thepolarizabilitytensormaybecalculatedusingeithermatrixinversionoraniterativemethod.Theiterativemethodisderivedfrombreakingupthepolarizabilitytensor,intermsoftheoriginalisotropicpointpolarizabilities,i,andthedipolerelaytensorTij.Foraconstantelectriceld,i=iE0.Substitutionintoequation5.7yieldsi=0iI)]TJ/F25 11.955 Tf 11.955 11.357 Td[(Xj6=iTijj.17wherewehaveused0itodenotethepointisotropicpolarizability.37

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Chapter6SFGSpectroscopyoftheWater/VaporInterfaceInthischapter,classicalMDmethodsareusedtomodelthedynamicsofthewater/vaporinterface.Twocomplementarytheoreticalapproaches{quantum-correctedTCFandINMmethods{usethecongurationsgeneratedbyMDasinputtodescribetheSFGspectrumoftheinterface,andtoascertainthemolecularoriginoftheSFGsignal;bothINMandTCFmethodsrelyonasuitablespectroscopicdipoleandpolarizabilitymodel.Thisdualapproachwasdemonstratedtobehighlyusefulinunderstandingcondensedphasespectroscopyofwater,otherliquids,andinterfaces{classicalmechanics,especiallyinthecontextofquantum-correctedTCFs,hasproventobesurprisinglyeectiveinmodelingintramolecularvibrationalspectroscopy.[1,2]Inparticular,TCFmethodshaveprovidedaquantitativedescriptionoftheO-Hstretchinglineshapeinambientliquidwater,andINMmethodshaveservedtoidentifythemolecularmotionsthatresultintheobservedsignal;thesecomplementarytechniquesareequallyeectiveformodelingwaterinterfacialspectroscopy.[1{6,94,116{119,132,133]AnINMapproximationtoSFGspectroscopyisquantummechanicalbyconstruction,butoersalimiteddynamicaldescription.Asaresult,inbulkwaterandotherliquid38

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stateintramolecularlineshapes,INMintramolecularresonancesarebroaderthantheirTCFcounterparts,buthavethesamecentralfrequencyandintegratedintensity.ThissuggeststheintramolecularINMspectrumrepresentsanunderlyingspectraldensitythatisdynamicallymotionallynarrowedintheactuallineshape.[55]ThisisalsofoundtobethecasehereforSFGspectrainallpolarizationconditions.Thisresultcontrastswithapreviousreportbyus,[1]andevidencefromtheliterature.[5,6]PreviousTCFandINMcalculationsoftheSSPpolarizationSFGO-Hstretchingspectrumofthewater/vaporinterfacewereverynoisy,andsuggestedthespectrahadequalbreadth{thus,suggestingmotionalnarrowingeectswerenotapparentinthespectra.Thesuccessofanapproxi-mate,non-dynamical,frequency-domaintechnique,[6]andthesimilarityofthespectratothoseobtainedusingTCFmethods,[1,5]appearedtobefurtherevidenceofspectrathatcouldbedescribedintheinhomogeneously-broadenedlimit.[122]Thatmethod,[6]however,containsanempiricallyadjustablelinewidththateectivelyaccountsforsomemotionalnarrowingmakingitdiculttodrawconclusions.Becauseofmethodologi-caladvances,itisnowpossibletocalculatewell-averagedTCFandINMspectra,andtheyunambiguouslydemonstrateSFGO-Hstretchinglineshapesatleastatthewa-ter/vaporinterfacearesignicantlymotionallynarrowedtoadegreereminiscentofthebulk.[93,134]Thissuggestsdynamicalmotionalnarrowingeectsareimportantatinterfaces,andthedynamicsarebestdescribedasintermediatebetweenthefastandslowmodulationlimitsofmotionalnarrowing.Intheslow-modulationinhomogeneously-broadenedlimit,allfrequencyuctuationsoftheoscillatorarerepresentedinthelineshape.[55,122,134]39

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ArecentstudyintheShengroupalsosuggestedmotionalaveragingeectsmaywellbesignicantintheSPSgeometry,and,inthatcase,thefreeO-Hstretchingpeakisgreatlydiminished.Althoughthatstudydidnotaddressmotionalnarrowing,thepresenceofmotionalaveragingsuggestsmotionalnarrowingisimportantbecauseitisduetofastreorientationalmotionswithinthevibrationalrelaxationtimeforthemodethatwouldalsobeexpectedtoresultinmotionalnarrowing.InordertoobtainbetterTCFresults,long-timecrosscorrelationsbetweenthesystemdipoleandpolarizabilityneedtobefollowed.BecausemolecularsimulationsofinterfacesinCartesianspacenecessarilyproducetwointerfaces,simulationtimeswerelimitedtothemoleculardiusiontimebetweeninterfacessomoleculescouldnotcon-tributetothesignalatbothinterfacesduringoneMDrun.[1]ThisleadstoTCFswithoutlong-timedecaysthatarediculttoFouriertransformaccurately.Inthiswork,aweakrestrainingpotentialisaddedthatconnesthemoleculesovertimetothehalfofthesim-ulationboxtheystartininthedimensionnormaltotheinterfacewithoutsignicantlyperturbingtherelevantshorttimedynamicsandaveragestructureoftheliquidthatcontributestotheinterfacialspectroscopy;eventhoughthemoleculardiusionconstantnormaltotheinterfaceischanged,themoleculeisonlycontributingtothespectrumwhileresidentattheinterface,andisfreeofanysignicantexternalpotential.ThismodicationpermitsthecalculationofTCFsouttoarbitrarilylongtimes{resultinginsharpspectrathatincludeintermolecularspectrallineshapes.Surprisingly,awell-denedintermolecularmodewasfoundtobeprominentinthespectrum.[2]Itiscenteredat875.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,andiscomparableinintegratedintensitytotherestoftheinter-40

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molecularlineshape{thelineshapealsohasanintensitythatisapproximatelyone-sixthofthemagnitudeoftheintensefreeO-Hstretchingpeakforspectratakeninpolariza-tiongeometriesthataresensitivetodipolederivativesnormaltotheinterfaceSSPandPPP.UsingINMmethods,theresonanceisshowntobeduetoawaggingmodelocalizedonindividualwatermolecules.Watermoleculescontributingtothisresonanceareataslightangletotheinterfacewiththeiroxygenatomsanchoredintheinterface,andthehydrogenatomswaggingnearlynormaltotheinterface.Thepresenceofanotherpopu-lation,asidefromthefreeO-Hstretch,ofinterfacialmoleculeswasrecentlyproposedviaindirectevidence,[8,43,44]andthathypothesisisstronglysupportedbythiswork.Here,wehavedirectlyobservedaspectroscopicallydistinctspecies,andclearlyidentiedthevibrationalmoderesponsibleforthelineshape.Thus,experimentalsetupsthatpermittakingspectraatrelativelylowwavelengthscouldprobethismodeasacomplimenttotheinformationcontainedinthefreeanddonorO-Hstretchingmodes.Atlowerfrequen-cies,well-denedhinderedtranslationalmodesarefoundbothparallelandperpendiculartotheinterface.TheperpendicularmodesareprominentinthepolarizationconditionssensitivetodipolarchangesnormaltotheinterfaceSPPandPPPwhiletheparallelmodesaremorepronouncedintheSPSgeometrywhichissensitivetomotionsalongtheinterface.Here,itisshownbycarefullyexaminingtherealandimaginarypartsoftheSFGsignal,individualmodecontributionstoanobservedlineshapecanbeidentied.Usingthisapproach,thefreeO-Handthenewlydiscoveredmodeswereidentiedasindividualspectroscopicspeciesonetypeofoscillatorattheinterface,andthedonor"O-Hregion41

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consistsofthreedistinctspecies.ThislastconclusionagreeswithresultsfromacarefuldeconvolutionofO-Hstretchingsignalinanearlierexperimentalworkthatalsofoundthreespecies{eachwithapproximatelythesamecentralfrequency.[20]Thus,asapreludetomorecomplexinterfaces,thisjointTCF/INMapproachisap-pliedtothewater/vaporinterfaceproducinggoodagreementwiththeshapeandrelativeamplitudesofSFGmeasurementsforallindependentpolarizationconditions.[15]TheMD,dipole,manybodypolarizationmethods,andassociatedparametersarealsosum-marizedinSection6.1.Thetheoreticalresults,andtheircomparisontoexperiment,arediscussedinSection6.2.6.1SimulationModelsandMethodsMDsimulationswereperformedusingacodedevelopedattheCenterforMolecularModelingattheUniversityofPennsylvania,andusesreversibleintegrationandextendedsystemtechniques.[135]MicrocanonicalMDsimulationswereperformedonambientH2Owithadensityof1.0g/cm3,andanaveragetemperatureof298K.Tocreateaninterface,acubicsimulationboxofequilibratedliquidwaterwasextendeddoubledalongthezaxis,andthesystemwasallowedtoequilibratecreatingtwowater/vaporinterfaces.Theinterfacesweresucientlyfarapartsoastheydidnotinteractstrongly,andEwaldsummationwasincludedinthreedimensions.[11]Thedensityproleofthesystemwasmonitoredtoverifyequilibration.[11]Inallcases,theresultsweretested,andfoundtobesystemsizeindependent.Mostresultsweregeneratedfrom216moleculesimulations,andsmallersystemsizesdownto64moleculesweretried,anddidnotalter42

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theresults.[2]MDsimulationswereconductedusingaexiblesimplepointchargeSPCmodelthatincludedaharmonicbendingpotential,linearcrosstermsandMorseO-Hstretchingpo-tentials,Vr=De)]TJ/F20 11.955 Tf 12.309 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(r2.[93]TheMorseO-Hstretchingpotentialusedherewasslightlysofterthanpreviouswork;ourvalueis2.50185A)]TJ/F18 7.97 Tf 6.586 0 Td[(1insteadof2.566A)]TJ/F18 7.97 Tf 6.586 0 Td[(1.[1,93]ForaMorsepotential,theforceconstant,k,canbeapproximatedask=2De2.As-sumingaharmonicoscillatorwithfrequency!=q k m,thisimpliestheratio1:2isproportionaltotheratio!1:!2.Therefore,a2.5%changeintheexponentialMorseparameterimpliesa2.5%shiftinthespectralfrequencies,andthisbehaviorisdemon-stratedinFigure6.1.ThisanalysisassumestherelevantcoordinatesaresimpleonedimensionalO-Hstretchingmodes.Ifseveraldistinctlydierenttypesofmodeswerepresent,achangeintheshapeofthebroadO-Hstretchingsignalwouldbeexpected.ThisisadditionalevidencethatinterfacialnormalmodesarewellapproximatedassimpleO-Hstretches.[1,2,6]Figure6.1highlightsthespectralchangesresultingfromusingasofterMorsepoten-tial.Theslightlysofterpotentialdoesnotaltertheintermolecularregionofthespectraaswouldbeexpected.Theintermolecularportionofthespectrumhaspolarizabilityanddipolederivativeschangesthataredueprimarilytoreorientation.Thesechangesthendependonthepolarizabilitytensorandthedipolesthemselves,andnottheirderivatives.Ontheotherhand,theintramolecularregionofthespectraissimplyshiftedtothered{thispointwillbereturnedtowhendiscussingthemodalcompositionofthebroadO-Hstretchinglineshape.ThischangeresultedinthefreeO-Hstretchingfrequencyinbetter43

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Figure6.1:DierentPotentialsChangeSpectra.SFGSSPTCFspectraforthewa-ter/vaporinterfacehighlightingthespectralchangesintheuseoftwodierentMorsepotentials{theoriginalMorsepotentialdashedblueline,andasofterMorsepotentialsolidgreenline.Thesofterpotentialresultsinashiftofapproximately100.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1intheO-Hstretchingspectrum.agreementwithexperimentalvalueseventhoughtheMorsepotentialchangeisalmostimperceptibletothenakedeye.ThisimpliesthecenterofthelineshapeisverysensitivetothelocalfrequencyalongtheMorsepotentialastheO-Hstretchingmotion,perturbedbyhydrogenbondingintheliquid,exploresthehighlyanharmonicpotentialsurface.InperformingtheMD,partialpointchargeswereplacedontheatomsthatwerechosentoreproducethecondensedphasedipolemoment.Atthewater/vaporinterface,thetruewaterdipolefallsfromitscondensedphasevalue,about2.9Debeye,tothat44

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inthegasphase,1.8Debeye,overadistanceofonlyafewmolecularlayers.[86]Itwouldseempolarizabledynamicswouldbeessentialtomodelthedynamicsofaqueousinterfaces,buttheuseofnon-polarizableMDseemstoadequatelyrepresentthestructureofthewater/vaporinterface.Apreviousworkusingapolarizablemodelinthiscontextisconsistentwiththisobservation.[5]EvaluatingtheTCFinEquations3.12-3.14presentsaproblemforinterfacialsystems.TheinterfacewasconstructedusingthestandardMDgeometrywithvacuum/vaporaboveandbelowthewater.[6,10]Unfortunately,thisproducestwointerfaceswithaveragenetdipolesinoppositedirections.CalculatingtheSFGspectrumoftheentiresystemwouldleadtopartialcancellationoftheSFGsignal,andmeaninglessresults.Anotherproblemarisesinthatmoleculesatoneinterfacecandiusetotheotherinterfaceovertime.Inthiscase,simulationtimesarelimitedtothemoleculardiusiontimebetweeninterfacessothatmoleculescannotcontributetosignalatbothinterfacesduringoneMDrun.ThisleadstoTCFswithoutlong-timedecaysthatarediculttoFouriertransformaccurately.[1,5]InordertoobtainbetterTCFresults,long-timecrosscorrelationsbetweenthesystemdipoleandpolarizabilityneedtobefollowed.Aweaklaterallyisotropicre-strainingpotentialwasaddedeectivelyconningthemoleculesovertimetothehalfofthesimulationboxtheystartininthedimensionnormaltotheinterfacewithoutsignif-icantlyperturbingtherelevantshorttimedynamics;eventhoughthemoleculardiusionconstantnormaltotheinterfaceischanged,themoleculeisonlycontributingtothespectrumwhileresidentattheinterface,andisfreeofanysignicantexternalpotential.45

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ThismodicationpermitsthecalculationofTCFsouttoarbitrarilylongtimesresultinginsharpspectrathatincludeintermolecularspectrallineshapes.Therestrainingpoten-tialisoftheformV==r9with=2:3K,=2:474A,andr=0isatthecenterofthebox.Therestrainingpotentialbecomesnegligibleneartheinterface,andisonlysignicantwithin2.0Aoftheboxcenter.Theinterfacialdensityprolewasunchanged{demonstratingtherestrainingpotentialuseddidnotperturbtheaveragestructureoftheliquidthatcontributestotheinterfacialspectroscopy.TheMDwasperformedwithoutexplicitpolarizationforces;whentheSFGTCForINMspectrumarecalculated,polarizabilityisincludedinthecalculations{over3million3fstimestepswereincludedincalculatingtheMDandTCFs.Themodelemployedincludesfullmany-bodypolarizationeectsincludedexplicitlyviaaPAPApolarizabilitymodel[128,129,136]withpointpolarizabilitiesontheatomsO=1:1482A3,H=0:3304A3.[137]Thepermanentdipoleswerecalculatedbasedonabinitiodata.[1,5]TheSFGsignalissensitivetobothdipoleandpolarizabilityderivatives.PAPApolarizabilitymodelsnaturallyincorporateparametersthatdeterminethepolarizabilityderivatives.Toimplementthis,itissucienttomakethepointpolarizabilitiesontheatomiccentersO-Hbondlengthdependent.[128,129,136]Thepointpolarizabilitiesthenchangeas:r=0r+0r.risdisplacementfromtheequilibriumbondlength.The0parametersforhydrogenandoxygen0O=2:7A2,0H=)]TJ/F15 11.955 Tf 9.298 0 Td[(1:06A2aresomewhatdierentthaninourpreviousmodel,butstillgivereasonablevaluesforthegasphaseRamanandIRtransitionmoments.[1]Figure6.2chighlightsthedierencesbetweenthepreviousandcurrentmodelfortheSFGSSPTCFspectra.46

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Figure6.2:DierentPolarizabilityModelsChangeSpectra.TheaIRTCFspectraforliquidwater,thebisotropicRamanTCFspectraforliquidwater,andthecSFGSSPTCFspectraforthewater/vaporinterfacehighlightingthespectralchangesintheuseoftwopolarizabilitymodels{previousmodeldashedbluelineandcurrentmodelsolidgreenline.ItisinterestingtonotethenewmodelcapturesthefreeO-Hmodemoreaccuratelywithoutsignicantlyperturbingtheintermolecularregionofthespectra{intramolec-ularspectraaresensitivetodipoleandpolarizabilityderivativesthatdonotsigni-cantlychangethemagnitudeofthedynamicallymoreimportantdipoleandpolarizabil-ity.Note,thesmalldierencesintheintermolecularspectrumarelikelyduetotherelativelypooraveragingthatwasdoneincalculatingthespectrumusingthepreviousmodel.Inthiscaseonlyone-fteenthofthenumberofcongurationswereincludedin47

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thecalculation,andtheSFGTCFswereslowtoconverge.[1,14]Thus,evenrelativelysmallchangestothesederivativescangreatlyeectthespectroscopicobservablewithoutchangingtheessentialphysicsoftheproblem{e.g.,theidentityoftherelevantmodesandmotions.Figure6.2alsopresentstheainfraredandbisotropicRamanTCFspectrarel-evanttotheSSPpolarizationconditionbecauseitprobesdiagonalelementsofthepo-larizabilitymatrixforliquidwaterusingbothmodels.Again,onlytheintramolecularregionofthespectrachanged.FortheO-Hstretchingregion,increasedasymmetryinthelineshapeisapparentforthenewmodelwithashoulderontheblueside.Thisisconsistentwithpreviousworkthatidentiedthisshouldertobeduetoinstancesinwhichahydrogendoesnotformahydrogenbondinthebulk[138,139];thiswouldbeanalo-goustothefreeO-Hstretchfoundininterfacialspectra.Thenewpolarizabilitymodeldoesabetterjobathighlightingthisnon-hydrogenbondedfrequencydistributionforliquidwater,and,consequently,allowsformoreaccurateinterfacialspectra.Thegurealsoclearlydemonstratesthepowerofcalculatingspectroscopicobservablestoanalyzecondensedphaseandinterfacialstructure.Interestingly,theshoulderonthebluesideofthebulkRamanandIRspectrumisatthesamecentralfrequencyasthefreeO-Hmodeattheinterface{stronglysuggestingthepresenceoffree,non-hydrogenbonded,O-Hmodesinbulkwater.48

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6.2ResultsFigure6.3displaysthetheoreticalSFGSSPspectrafortheentirewatervibrationalspec-trumderivedfrombothTCFandINMmethods.BoththeTCFandINMresultsareinabsoluteunits,andnoparameterswereadjustedindisplayingthedata.TheINMandTCFspectrawerefoundtointegratetothesamevalueovertheentire0.-5000.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1range,andseparatelyovertheO-Hstretchingregion2000.-5000.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1forallpolar-izationconditionstheothersarenotshown.ThisbehaviorisstrongevidencefortheinterpretationoftheINMlineshapeasanunderlyingspectraldensitythatismotionallynarrowedintheobservedspectrum.[55]INMapproximationstospectroscopyoeronlyalimiteddynamicaldescription,andcorrespondtoanunderlyingspectraldensitythatistypicallybroaderthantheobservedlineshapewhenconsideringintramolecularmodes.Asanexample,inbulkwaterandotherliquidstateintramolecularlineshapesINMintramolecularresonanceswerefoundtobebroaderthantheirTCFcounterparts,butwiththesamecentralfrequencyandintegratedintensity.OurTCFandINMspectrainFigure6.3unambiguouslydemonstrateSFGO-Hstretchinglineshapesatthewa-ter/vaporinterfacearesignicantlymotionallynarrowedtoadegreereminiscentofthebulk.[93,134]ThisresultalsosuggestsSFGspectraaresensitivetobothstructureanddynamics.TheINMspectrumclearlyexhibitsthesameresonances,butisbroader.Thisimpliestheobservedlineshapesaremotionallynarrowed,anddynamicalcontributionstoSFGsignalsareimportant.[15]Figure6.4presentsTCF-derivedtheoreticaldescriptionsoftheSFGspectraintheO-Hstretchingregionforthewater/vaporinterface.Thethreepossibleindependent49

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Figure6.3:SFGTCFandINMSpectrafortheWater/VaporInterface.SFGSSPspectraforthewater/vaporinterfacefortheentirewatervibrationalspectrumusingTCFsolidgreenlinemethodandINMdashedbluelinemethod.polarizationconditions,SSP,PPP,andSPS,intheelectronicallynonresonantexperi-mentaredisplayed.Thersttwoindicescanbeinterpretedastheelementofthesystempolarizabilitytensor,andthelastindexastheelementofthesystemdipolethatisbeingprobed.Inthedata,forallpolarizations,wehaveincludedtheSSPnonresonantcontri-butionthisisonlystrictlycorrectfortheSSPpolarizationcondition,andservesasanestimateintheothercases,NRes!,whichisasmallnegativeconstant,[5,140]andthefullsignalisgivenby:jSFG!j2/jRes!+NRes!j2.InordertoaccountfortheFresnelcoecientsthatmodifytheexperimentalintensities,wehaveadjustedtherelativeintensitiesofourtheoreticalspectrasotheycanbemoreeasilycomparedwith50

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experimentalresults.[15] Figure6.4:SFGTCFSpectrafortheO-HStretchingRegion.SFGTCFspectraforthewater/vaporinterfaceintheO-Hstretchingregionforthreepolarizations:SSPsolidblackline,PPPdashedredline,andSPSsolidgreenline.Theinsetisexperimentaldata[15]forthesamepolarizationsusingthesamecolorscheme.InFigure6.4,thefreeO-Hpeakisprominentat3700.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1andtherestoftheO-Hstretchingregionhasamorecomplicatedshape.TheinsetofFigure6.4displaysexperi-mentaldatafortheO-Hstretchingregiontakeninthesamepolarizationgeometries.[15]Therelativeintensitiesagreenearlyquantitativelybetweentheoryandexperiment.ThefreeO-HstretchinglineshapeiscapturedveryaccuratelybythetheoryandtherestoftheO-Hregionhasasimilarshape.TheratioofrelativeintensitiesbetweenthefreeO-HandtherestoftheOHstretchingbandareabout2:1forbothexperimentandtheory.51

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Clearlythetheorycapturestheessentialfeaturesofthespectrumanditspolarizationdependence.Toquantitatethepolarizationdependence,theratiooftheSSP:PPPin-tensitiesforthefreeOHstretch,wherethesignaltonoiseisbest,is13:10:1forthetheoreticalexperimentalspectra,andtheSPSisaboutafactorofthreesmallerthanthePPPinbothcases.Theagreementiswellwithintherelativeerrordemonstratingthesuccessoftheoreticalmethods.TheagreementbetweentheTCFandexperimen-talspectrum,includingtherelativeintensitiesofthedierentpolarizationconditions,isexcellent,andwithinthestatisticalerrorovermostofthefrequencyrange.[2]ThespectrumintheSSPgeometrythatcorrelatesthedipolemomentcomponentnormaltotheinterfacewithdiagonalpolarizabilitymatrixelementsintheplaneoftheinterfacee.g,withthezaxistakenasthesurfacenormaldirectionleadstothemostintensespectrumduetoarelativelysizable,andchanging,netnormaldipolemomentattheinterface,andtherelativelylargediagonalpolarizabilityelements;waterhasanearlydiagonalpolarizabilitymatrixwithnearlyequalelementsinboththegasphaseandbulk.Note,thePPPpolarizationconditionissensitivetoacombinationofallallowedsusceptibilitytensorelementsincontrasttoSSPandSPSthatonlyprobeasingletensorelement.[77]Thus,theessentialfeaturesofthespectrum,anditspolarizationdependence,arecapturedverywellbytheTCFtheorywiththecaveatthatabsoluteintensitiesoftheintramolecularmodesarequitesensitivetothechoiceofpolarizabilityparameters.ThepolarizationdependenceofthesignalisdemonstratedinFigure6.4.Forpolar-izationsthataresensitivetodipolederivativesnormaltotheinterface{SSPandPPP52

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Figure6.5:SFGTCFSpectrafortheIntermolecularRegion.SFGTCFspectraforthewater/vaporinterfaceintheintermolecularregionforthreepolarizations:SSPsolidgreenline,PPPdashedblueline,andSPSdottedredline.{thesignalhasanintenselineshape.Incontrast,fortheSPSgeometry,whichissensi-tivetodipolederivativesparalleltotheinterface,onlyahintofasignalisfound.TheSPSpolarizationconditionalsoprobessmallo-diagonalpolarizabilitymatrixelements.TheseresultsalsosuggestbyevaluationofthepolarizationdependenceoftheSFGspec-tra,givenaknowledgeoftheexpectednatureofthepolarizabilityanddipolederivatives,allowsinterfacialmoleculargeometriestobeinferredviathespectra.[17,77,141]Whiletheintermolecularspectrumofbulkwatershowslittlestructure,theinterfacialspectraiscomplexasshowninFigure6.5.ThegurehighlightstheintermolecularSFGTCFspectraforthethreeindependentpolarizationconditions,SSP,PPPandSPS.53

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Figure6.6:Water/VaporInterfaceSnapshot.Asnapshotofawater/vaporinterfacecontaining216.watermoleculesfeaturingINMsfromdierentregionsofthespectra.ThewatermoleculeshowninblueisrepresentativeofafreeO-Hmodeat3694.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Thewatermoleculeshowningreenisrepresentativeofawaggingmotionat858.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Thewatermoleculeshowninyellowhighlightsatranslationperpendiculartotheinterfaceat46.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Thewatermoleculeshowninblackhighlightsatranslationparalleltotheinterfaceat197.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Thepolarizationsthataresensitivetodipolederivativesnormaltotheinterface,SSPandPPP,showawell-denedintermolecularmodeat875.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1thatiscomparableinintensitytotherestoftheintermolecularstructureandapproximatelyone-sixththeintensityoftheintensefreeO-Hstretchingpeak[2].UsingINMmethodslookingatthenatureoftheINMsinthesamespectralregion,theresonanceisshowntobeduetoawaggingmodelocalizedonasinglewatermolecule,ataslightangletotheinterface,54

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withtwohydrogensvibrating/libratingnormaltotheinterface,andtheoxygenanchoredintheinterface.[2]Thehydrogens,pointingintothevaporphase,arehydrogenbondedtoanoxygenatomattheinterface.TheSSPandPPPalsoshowanintenseintermolecularmodeat95.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.UsingINMmethodstheresonanceisfoundtobeduetotranslationsperpendiculartotheinterface.TheSPSspectra,whichissensitivetodipolederivativesparalleltotheinterface,showsanintermolecularmodeat220.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Thismodeisaresultoftranslationsparalleltotheinterface.TheimportanceofpolarizationsensitivityinSFGexperimentsis,thus,highlighted.Further,wehaveobservedspectroscopicallydistinctspecies,andclearlyidentiedthevibrationalmodesresponsibleforthelineshape.Hence,experimentalsetupsthatpermittakingspectraatrelativelylowwavelengthscouldprobethesemodesasacomplimenttotheinformationcontainedinthefreeanddonorO-Hstretchingmodes.Thesethreedistinctpopulationsofwatermoleculesattheinterfacewerepreviouslyundescribed{otherworkshaveinferredtheexistenceofsomethinglikethewaggingmode.[8,43,44]ThismightbeconsideredsurprisinggiventhelargenumbersofMDsimulationsofthewater/vaporinterfacethathavebeenperformedpreviously.Thisobservationhighlightsthepowerofcalculatingspectroscopicobservablesinassessinginterfacialstructureanddynamics.Notonlycantheresultsbedirectlycomparedwithexperiment,thusvalidatingtheMDmodel,thespectroscopiccalculationservesasalterofthedynamicsextractingouttheidentityofcollectivecoordinateswithwell-denedfrequenciesthatpersistattheinterface.Figure6.6highlightsthevibrationalmodesfromtheintermolecularandintramolec-55

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ularregionofthespectra.AtypicalfreeO-Hmode,showninblue,producesthehigh-frequencyfeatureat3700.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Itiscleartheoxygenatomisanchoredintheinterface,andtheO-Hisoscillatingfreelyabovetheinterface.Thewaggingmodegivingrisetothespectralfeatureat875.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1isdisplayedingreenattheoppositeinterface.Here,theoxygenatomisanchoredintheinterface,andthetwohydrogensarevibratingintothevaporphase.Arepresentativeperpendiculartranslationalmodewithlineshapecen-teredat95.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1isshowninyellow,andtheroughlyparalleltranslationalmodewithlineshapecenteredat220.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1isshowninblack.TheseresultsdemonstratehowINMapproachdoesnotrequireaprioriassumptionsaboutthenatureofinterfacialmodesbutdoesrevealtheirphysicalcharacteristics,andhowdierentmolecularmotionscontributetothespectrum.Figure6.7displaysthedistributionofthedirectioncosinefromthesurfacenormalofO-Hvectorspointingintothevapor.Thisresultcompareswellwithprevioustheoreticaldata.[6]Weseeanenhancementinprobabilityatcos1.Wealsondapproximately20.%ofsurfacewatermoleculeshaveafreeO-Hbondpointingoutoftheliquid,andintothevaporwhichisconsistentwithprevioustheoretical[6]andexperimental[26]work.Thisanalysisalsopointsoutitisnecessarytotalkofbroaddistributionsofanglesatthewater/vaporinterfaces,andthatrelativelylesscanbelearnedfromsingleaveragevaluesoforientations.Figure6.8adisplaystherealandimaginarypartsfortheSSPspectrumcalculatedviaequations3.11and3.12.Examiningtherealandimaginarypartsofthespectrumcanoerinsightsunavailablefromthemodulusalone.Therealandimaginarypartscould56

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Figure6.7:ProbabilityDistributionoftheDirectionCosine.TheprobabilitydistributionofthedirectioncosinefromthesurfacenormalofO-Hvectorspointingintothevapor.bemeasuredexperimentallyviaaheterodynedetectionscheme,orbytakingadvantageofinterferenceeectsbetweenbulkandinterfacialcontributionstothespectrum.[17]Toseetheadvantagesofseparatelyexaminingtherealandimaginarycontributions,itisusefultowritetheresonantSFGsignalofasingleharmonicmode,Q,withlineardipoleandpolarizabilityinfrequencyspaceasthefollowing:[6]R!/@i=@Q@jk=@Q!)]TJ/F20 11.955 Tf 11.956 0 Td[(!IR !)]TJ/F20 11.955 Tf 11.955 0 Td[(!IR2+2.1I!/@i=@Q@jk=@Q !)]TJ/F20 11.955 Tf 11.956 0 Td[(!IR2+2.2InEquations6.1-6.2,isamathematicalconvergenceparameterthatphysicallycanbeinterpretedasahomogeneouslinewidth.Thesignalmagnitudeisseentobeproportional57

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Figure6.8:RealandImaginaryComponentsoftheSpectra.RealsolidgreenlineandimaginarydashedbluelinecomponentsoftheaSFGSSPTCFspectraforthewater/vaporinterfaceandforbbulkwatercalculatedastheFourier-Laplacetransform.totheproductofdipoleandpolarizabilityderivatives.Equations6.1-6.2implyasingletypeofmodewillleadtoanimaginarycontributionthatisasymmetricwell-denedpeakLorentzianincharacterwhiletherealpartwillchangesign,dippingbelowzero,atthemaximumoftheimaginaryportion.Ifmorethanonespeciesiscontributingtothesignalinagivenregion,amorecomplexlineshapewillresultfromtheoverlappingsignals.ExaminingtherealandimaginarycontributionsinFigure6.8a,itisclearseveraloftheresonancesareessentiallysinglemodeincharacter:thefreeO-H00.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,thesmallbendingcontributionatthesurface800.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,thewaggingmode75.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,andtranslationalmodes.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1and220.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Thereissomeoverlapin58

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thetranslationalmodes,anditisinstructivethehigherfrequency220.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1mode,thatispronouncedonlyintheSPSmodulusspectrum,alsoshowsupintheSSPrealandimaginaryspectra. Figure6.9:RealandImaginaryComponentsintheO-HStretchingRegion.RealsolidgreenlineandimaginarydashedbluelinecomponentsoftheSFGSSPTCFspectraforthewater/vaporinterfacefortheO-Hstretchingregion.Thearrowshighlightthreeseparatemodescenteredat3195.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,3325.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,and3400.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Figure6.9highlightstheO-Hstretchingregion{fromapproximately3000.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1to3600.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Carefulexaminationofthespectrumrevealsthreeseparatemodesinthisregioncenteredat3195.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,3325.cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,and3400.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Remarkably,thisagreesverywellwithpreviousexperimentalworkthatdeconvolutedthespectruminthisregion.Thatanalysisrevealedthreemodespresentinthesameregioncenteredat3200.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,59

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3310.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1and3420.cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1{nearlythesamefrequencies.[20]Thisisstrongevidencefordistinctpopulationsofwatermoleculesinthisdonor"O-Hregionofthespectrum.FurtherworkisneededtoidentifythenatureofthesedistinctO-Hstretchingspecies.Note,thetheoreticalandexperimentalspectrahaveasomewhatdierentshapeinthisregion,andthismanifestsitselfintherelativeintensitiesofthedierentcontributions.TheexperimentalspectrumismorepronouncedonthebluesidetotheredofthefreeO-Hpeakcomparedtothetheoreticalresult,andthesubpopulationsidentiedonthatsideofthelineshapearerelativelylargeraswell.ThisismostlikelyduetothespectroscopicintensitiesofthesespeciesviaourspectroscopicmodelratherthandierentpopulationsofthesespeciesattheinterfacewithintheMDmodel.However,furtherinvestigationisrequiredtodenitivelydemonstratethis.Itshouldalsobenoted,aspointedoutinanearlierwork,[6]orientationalinformationcanalsobededucedfromtherelativesignsoftheimaginarymodelineshapesgivenknowledgeofthethesignsoftheprefactorsinEquations6.1-6.2thedipoleandpolarizabilityderivatives.Tofurthershowtheutilityoftherealandimaginarymodalanalysis,Figure6.8bdisplaystherealandimaginarypartsofthebulkwaterO-HstretchingregioncalculatedastheFourier-Laplacetransform.WhilealinearIRexperimentdoesnotmeasurethisobservable,thetransformcanstillbeappliedasananalysistool.Figure6.8bisstrongevidencetherearetwodistinctspeciesinthebulk,andthehigherfrequencymoietyarisesfromthebulkfreeO-H,non-hydrogenbondedmolecules.[138,139,142]60

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Chapter7ConclusionSFGexperimentalmeasurementsaregrowinginnumberandimportance;theyareprovid-ingvaluableinformationaboutinterfacialstructureanddynamicsthatwouldbediculttomeasure,orarenotobtainableotherwise.Theoreticalstudiesareonlynowsucientlysophisticatedthattheycanbegintoplaythemajorrolesimulationhasinmodelingandinterpretingcondensedphasespectroscopy.Inprinciple,SFGspectroscopyiscapableofgivingacompletepictureoftheinterface{includingstructureanddynamics.Realizingthispromisedependscriticallyonthespectrabeingreliablyinterpreted,andthemethodsdescribedherearecapableofunambiguouslycharacterizingthenatureofSFGspectra{includinginferringsubpopulationsofmoleculesfromcomplexlineshapes.Still,avigorousinterplaybetweentheoryandexperimentisneededtofurtherdeveloptheinterpretativeandpredictivepoweroftheoreticalstudies.TheinvestigationofmorecomplexinterfacesusingtheimprovedTCFmethods,describedhere,willhelptobothinterpretthelargeandgrowingbodyofexperimentaldata,andtopredictheretoforeunexploredinterfacialvibrationalstructure.Further,experimentaladvancesarelikelytoextendthefrequencyrangeforSFGmeasurementsintothefarIRwheretheorypredictsimportantinterfacial61

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speciesarepresent.Lastly,theoreticalandexperimentalmeasurementsofboththerealandimaginarypartsoftheSFGsignalasopposedtomeasuringthesquaredmodulusasinthetypicalhomodynedetectedexperimentshowgreatpromiseinhelpingunravelcomplexSFGlineshapes.62

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AbouttheAuthorAngelaS.PerryreceivedaBachelor'sofArtsDegreefromtheUniversityofSouthFloridainMayof2000.SheenteredtheDoctoralprogramattheUniversityofSouthFloridaintheFallof2000.ShebeganworkincomputationalchemistrywithProfessorBrianSpaceinDecemberof2000.Inadditiontograduatecoursework,Ms.PerryattendedtheGordon-KenanChemicalPhysicsSummerSchool,heldatRogerWilliamsUniversity.ShehasalsoattendedseveralGordonResearchConferencesandAmericanChemicalSocietyNationalMeetings.Asthegraduatestudentrepresentativein2002,Ms.PerrywastheChairfortheCastleStudentResearchConferenceatUSF.WhileinthePh.D.programattheUniversityofSouthFlorida,Ms.PerrywasarecipientoftheGeorgeBursaDoctoralFellowshipAward.ShewasawardedtwosummerawardsfromtheTharpEndowedScholarshipFundandthreetravelawardsfromtheGordonConferenceChair'sFund.Shehascoauthoredvepublicationsandmadeseveralpresentationsatregionalandnationalmeetings.


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A theoretical description of the vibrational sum frequency generation spectroscopy of interfaces
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Thesis (Ph.D.)--University of South Florida, 2005.
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ABSTRACT: Our work investigates theoretical approximations to the interface specific sum frequency generation (SFG) spectra at aqueous interfaces constructed using time correlation function (TCF) and instantaneous normal mode (INM) methods. Both approaches lead to signals in excellent agreement with experimental measurements. This work demonstrates how TCF and INM methods can be used in a complementary fashion to describe interfacial vibrational spectroscopy. Our approach is to compare TCF spectra with experiment to establish that our molecular dynamics (MD) methods can reliably describe the system of interest. We then employ INM methods to analyze the molecular and dynamical basis for the observed spectroscopy. We have been able to elucidate, on a molecularly detailed basis, a number of interfacial line shapes, most notably the origin of the complex O-H stretching SFG signal, and the identity of several intermolecular modes in the SFG spectra for the water/vapor interface.
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Adviser: Brian Space.
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