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Molecular dynamics and time correlation function theories

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Molecular dynamics and time correlation function theories
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DeVane, Russell H
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Xenon
Water
Nonlinear spectroscopy
Raman
IR
2DIR
CS₂
Fifth-order raman
Dissertations, Academic -- Chemistry -- Doctoral -- USF   ( lcsh )
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Abstract:
ABSTRACT: The research presented in this thesis makes use of theoretical/computational techniques to calculate nonlinear spectroscopic signals and molecular volumes. These techniques have become more practical with advances in computational resources and now are an integral part of research in these areas. Preliminary results allude to the power of these techniques when applied to relevant problems and suggest that much progress can be made in understanding the complex nature of nonlinear spectroscopic signals and molecular volume contributions. The nonlinear spectroscopy work involves writing the quantum mechanical response functions in terms of classical time correlation functions which are amenable to calculation using classical molecular dynamics. The response functions reported in this thesis include the fifth order response function, probed in the fifth order Raman experiment, and the third order response function probed in the two dimensional infrared experiment. The molecular volume calculations make use of modern algorithms used in molecular dynamics simulations to calculate the full thermodynamic volumes of molecules.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2005.
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Includes bibliographical references.
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by Russell H. DeVane.
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.MolecularDynamicsandTimeCorrelationFunctionTheoriesbyRussellH.DeVaneAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofChemistryCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:BrianSpace,Ph.D.TomKeyes,Ph.D.RandyLarsen,Ph.D.JenniferLewis,Ph.D.DavidMerkler,Ph.D.DateofApproval:September22,2005Keywords:xenon,CS2,water,nonlinearspectroscopy,Raman,IR,2DIR,fth-orderRamancCopyright2005,RussellH.DeVane

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AcknowledgmentsFirstandforemost,Ithankmymotherandfatherfortheirneverendingsupportwithallendeavorsinmylife.Thisisforyou.ToProf.BrianSpace,Ioerthemostsincerethanksforbeingawonderfulteacherandfriend.IthasbeenawonderfulexperiencethatIwoulddoalloveragain.ToProf.TomKeyes,Prof.PrestonMooreandProf.RandyLarsen,Iamgratefulforthemanystimulatingcommunications.Yourhelpandadvicehavebeeninvaluable.Iwouldalsoliketothankmycommittee:Prof.TomKeyes,Prof.RandyLarsen,Prof.JenniferLewisandProf.DavidMerkler.Andlastbutnotleast,tomyfellowgroupmembers,AngelaPerry,ChristinaRidley,BenRoney,TonyGreen,ChristineNeipert,AbeStern,andJonBelof,youareanelotandIwilltrulymissyou.

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NotetoReaderNotetoReader:Theoriginalofthisdocumentcontainscolorthatisnecessaryforun-derstandingthedata.TheoriginaldissertationisonlewiththeUSFlibraryinTampa,Florida.

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TableofContentsListofFigures......................................iiiListofTables......................................vAbstract.........................................vi1Introduction....................................12TimeCorrelationFunctionTheoriesofSpectroscopy..............42.1LinearAbsorption............................42.2Non-linearSpectroscopy.........................63ThePolarizabilityModel.............................123.1CalculatingthePolarizability......................143.2RotationalInvariants...........................144TheFifthOrderRamanResponseFunction..................194.1FifthOrderRamanExperiment.....................194.2Polarization................................204.3FifthOrderResponseFunction.....................214.4ConstructingaTCFTheory.......................234.5FODIDvs.MBP.............................375Two-dimensionalInfraredSpectroscopy.....................425.12DIRExperiment.............................435.22DIRResponseFunction.........................44i

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5.3ConstructingaTCFTheory.......................466FifthOrderRamanResults...........................636.1ResultsforAmbientCS2........................636.2ResultsforLiquidxenon.........................7072DIRResults...................................907.1ApplicationstoAmbientWater:ModelsandComputationalDetails.907.2The2DIRSpectrumofAmbientWater.................957.3QuantumCorrections...........................1038MolecularVolumeCalculations.........................1078.1MolecularDynamics...........................1078.2ThePluckMethod............................1099MolecularVolumeResults............................1139.1MethodsandApplicationstoaModelSystem.............1139.2ResultsfromaSimpleModelSystem..................12010Conclusion.....................................128References........................................130AbouttheAuthor...................................EndPageii

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ListofFigures4.1Rotationalinvariants...............................406.1FifthorderTCF..................................646.2jRt1;t2jforCS2................................666.3ThreetimeslicesofRforCS2.........................686.4jRxxxxxxt1;t2j2FODIDxenonresults......................716.5ExactvsTCF...................................726.6Exactvs.TCFslices...............................736.7jRxxxxxxt1;t2j2MBPforxenon.........................746.8MBPvs.FODID.................................756.9jRxxzzxxt1;t2j2forxenon............................796.10jRt1;t2j2semipolarized............................816.11jRzzxxxxt1;t2j2slice...............................826.12jRt1;t2j2depolarized.............................836.13jRt1;t2j2errorslices.............................856.14Fifthordercombinations.............................887.12DIRTCF.....................................947.22DIRTCFslices.................................967.32DIRTCFmixedtime-frequency........................98iii

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7.42DIRspectra...................................997.52DIRspectraslices................................1027.62DIRspectrawithquantumcorrection.....................1059.1Betasheet.....................................1149.2Betasheetvolumeuctuations..........................1159.3Betasheetvolumeuncertainty..........................1179.4Methanevolumeuctuations...........................1229.5Methanevolumeuncertainty...........................1239.6Methaneradialdistributionfunctions......................1249.7Methaneanionandcation............................126iv

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ListofTables4.1FifthordergR=gIrelationships..........................326.1Fifthorderpathways...............................76v

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MolecularDynamicsandTimeCorrelationFunctionsTheoriesRussellH.DeVaneABSTRACTTheresearchpresentedinthisthesismakesuseoftheoretical/computationaltechniquestocalculatenonlinearspectroscopicsignalsandmolecularvolumes.Thesetechniqueshavebecomemorepracticalwithadvancesincomputationalresourcesandnowareanintegralpartofresearchintheseareas.Preliminaryresultsalludetothepowerofthesetechniqueswhenappliedtorelevantproblemsandsuggestthatmuchprogresscanbemadeinunderstandingthecomplexnatureofnonlinearspectroscopicsignalsandmolecularvolumecontributions.Thenonlinearspectroscopyworkinvolveswritingthequantummechanicalresponsefunc-tionsintermsofclassicaltimecorrelationfunctionswhichareamenabletocalculationusingclassicalmoleculardynamics.Theresponsefunctionsreportedinthisthesisincludethefthorderresponsefunction,probedinthefthorderRamanexperiment,andthethirdorderresponsefunctionprobedinthetwodimensionalinfraredexperiment.Themolecularvol-umecalculationsmakeuseofmodernalgorithmsusedinmoleculardynamicssimulationstocalculatethefullthermodynamicvolumesofmolecules.vi

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Chapter1IntroductionTheavailabilityofmodern,highspeedcomputershasmadetheuseofcomputationalmeth-odsforscienticinvestigationapracticalapproach.Althoughlimitedtoapproximationsonsimplemodelsystemsintheearlydays,computationaltechniquesarenowcapableofaccuratelymodelingandinvestigatingrelevant,complexsystems,e:g:biologicallyrelevantsystemssuchasproteinsembeddedinalipidmembrane.Workpresentedinthisthesisinvolvestheapplicationofcomputationalmethodstotwoseparateareasofinvestigation.First,classicalmoleculardynamicsMDmethodsandclassicaltimecorrelationfunctionTCFtheoryformalismwasusedtoinvestigateandpredictnon-linearspectra.Second,classicalMDmethodswereusedtodevelopamethodforcalculatingmolecularvolumesthatisapplicabletobiologicallyrelevantsystems.Advancesinlasertechnologyhaveallowedthedevelopmentofnovelnon-linearspectro-scopictechniquesthatpromisetoprovidenewinsightintothestructureanddynamicsofmaterials.Thesenon-lineareects,notobviousineverydayexperience,arisewhenthema-terialisallowedtointeractwithelectriceldsofsucientintensity,suchasthoseprovidedbylasers.Initially,one-dimensionalexperimentssuchastheopticalKerreectandcoherentanti-StokesRamanscatteringwereexpectedtoprovidenewinformationaboutthesystembeing1

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investigated.However,itwasshownthattheseexperimentsprobedthesameinformationat-tainableinconventionallinearexperiments.Asanalternative,multi-dimensionaltechniqueswereproposedasnecessaryforgainingnewinsight.Inparticular,thetwo-dimensionalDfth-orderRamantechniquewasproposedtohelpdistinguishbetweenthecontributionstospectrallineshapesarisingformhomogeneousandin-homogeneouscontributions.Re-cently,anothermultidimensionaltechniquethathasshownmuchpreliminaryhopeisthetwo-dimensionalinfrared2DIRexperiment.Thistechniquehastheabilitytoprovidetimeresolvedstructureonsub-picosecondtimescales.Whilethesetechniquesprovideawealthofinformation,itcomesattheexpenseofmorecomplexspectra.Tomakeprogressinunravelingtheinformationprovidedinthesespectra,armtheoreticalfoundationisnecessary.Thequantummechanicalnatureoftheproblemmakesitdiculttoapproachtheoreticallyandthusnewtechniquesarerequired.Applicationoftimecorrelationfunctiontheoriesprovidesameansofcalculatingnon-linearspectroscopythatisamenabletotheuseofstandardclassicalcomputationalmethods.Suchapproacheswereshowntobepossibleforone-dimensionalspectroscopyexperiments.Originalresearchpresentedinthisthesisincludesthedevelopmentoftimecorrelationfunctiontheoriesforthefth-orderresponsefunctionprobedinthefth-orderRamanexperimentandthethird-orderresponsefunctionprobedinthe2DIRexperiment.Amicroscopicunderstandingofthecontributionstomolecularvolumesisnecessarytocomplementmodernphotothermalexperimentaltechniquesthatarecapableofmeasuringmolecularvolumechangesofsolvatedmoleculesonnanosecondtimescales.Thesecondareaoforiginalresearchpresentedinthisthesisinvolvesthedevelopmentofatechniqueforcal-culatingtimeresolvedmolecularvolumesusingisothermal-isobaricNPTMDsimulations.Thesocalledpluck"methodallowsthecalculationofthermodynamicvolumesforsolvatedcondensedphasesystems.Thistechniquealsoallowsthecontributionsfromelectrostrictiontobeeasilyevaluated.2

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InChapter2,thetimecorrelationfunctionformalismfortheinteractionoflightandmatterisintroduced.Inaddition,thederivationofatimecorrelationfunctionformofaonedimensionalresponsefunctioninwhichanexactTCFtheoryispossibleisoutlinedtoprovideanideaoftheapproachusedtodevelopthefthorderRamanand2DIRTCFtheories.Chapter3presentsthepolarizabilitymodelusedinthecalculations.Chapters4and5presentthedevelopmentofthetimecorrelationfunctiontheoriesofthefth-orderRamanresponsefunctionandthethird-orderresponsefunctionprobedinthefthorderRamanand2DIRexperimentrespectively.Chapter6and7discussesresultsforthefth-orderresponsefunctionandthe2DIRsignalcalculations.InChapters8and9wepresentthepluck"methodandresultsofitsapplicationrespectively.Finally,Chapter10presentsconclusions,andabriefdiscussionoffuturedirectionsfortheoreticalstudiesoffthorderRamanand2DIRspectroscopy.3

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Chapter2TimeCorrelationFunctionTheoriesofSpectroscopyThischapterpresentstheoriginsoftimecorrelationfunctiontheoriesfordescribingtheinteractionoflightandmatter.WebeginwithFermi'sGoldenRulefromtimedependentperturbationtheoryandshowhowthetimecorrelationfunctionformalismarisesnaturally.Wethenshowthedevelopmentofatimecorrelationfunctiontheoryforasingledimensionalspectoscopictechniquebasedonanon-linearresponsefunctionandhowthatexperimentprobesthesameinformationastheconventionallinearexperiments.TCFtheorieshelptoelucidatethefactthatspectralfeaturesarisefromthedynamicsofthesystemandthatthespectraaremerelyfrequencydomainrepresentationsofsuchdynamics.Forexample,thelinearabsorptionspectraisgivenbythedipolecorrelationfunctionofthesystemwhichisdependentonthedynamicsofthenucleiandelectronsaswillbeshownbelow.2.1LinearAbsorptionForanelectriceld,Et=Eocos!t.1interactingwithasystemofNmoleculesthatareinitiallyinthequantumstatei,amaterialtransitionwilltakeplacetosomenalstatefifthefrequencyoftheelectriceldsatises4

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theBohrrelation,Ef)]TJ/F18 11.955 Tf 12.011 0 Td[(Ei=~.TheGoldenRule,fromtimedependentperturbationtheory,givestheprobabilitythatsuchatransitiontakesplaceandisgivenbyPi!f!=E2o 2~2jMfij2[!fi)]TJ/F18 11.955 Tf 11.955 0 Td[(!+!fi+!]:.2where,Mfi=isthedipolematrixelementand!fi=!f)]TJ/F18 11.955 Tf 11.833 0 Td[(!i.Theperturbation,theinteractionwiththeelectriceld,isgivenbyH0t=)]TJ/F18 11.955 Tf 9.299 0 Td[(MEt.3withMbeingthetotalelectricdipoleoperatorofthesystem.Herewehaveinvokedtheelectricdipoleapproximationsinceweconsiderthemoleculestobemuchsmallerthanthewavelengthoftheelectriceld.[1]Torelatetheprobabilityoftransitionstoaphysicalobservablesuchastheabsorptionlineshape,itisnecessarytoshowaconnectionbetweentheabsorbedradiationandthetheprobabilityofthetransitions.Theamountofenergylostfromtheincidenteldwillshowupasachangeintheintensityoftheincidentradiationuponexitingthematerial.Theexpressionshowingtheamountofenergylostfromtheradiationisgivenby)]TJ/F15 11.955 Tf 13.331 3.022 Td[(_Erad=XiXfi~!fiPi!f:.4whereiistheprobabilitythatthesystemisinstatei.Theabsorptioncrosssection,orcoecient,whichdescribestheamountofradiationabsorbedbythesystem,isgivenbytheratioofenergylostbytheelectriceldandthatincidentonthesystem!=_Erad _Einc:.5where_Einc=c 8E2o.Thecrosssection,!nowtakestheform!=42 ~cn!)]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~!XiXfijMfij2!fi)]TJ/F18 11.955 Tf 11.955 0 Td[(!:.65

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TheabsorptioncrosssectionisrelatedtotheabsorptionlineshapebyI!3~cn!! 42!)]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!:.7Further,byintroducingthedenitionoftheDiracdeltafunction,!=1 2Z1ei!tdt.8wehaveI!=3 2Xi;fiZ1dte[Ef)]TJ/F20 5.978 Tf 5.756 0 Td[(Ei ~)]TJ/F19 7.97 Tf 6.586 0 Td[(!]it.9where,intheHeisenbergrepresentationMt=eiH0t=~Me)]TJ/F19 7.97 Tf 6.587 0 Td[(iH0t=~.10andtheclosurerelationisdenedasXfjf>.12writtenastheFouriertransformofadipole-dipoletimecorrelationfunction.[1,2]2.2Non-linearSpectroscopyTheTCFtheoryapproachwasshowntobesuccessfulfornon-linearresponsefunctionsaswellandleadtotherecognitionofthelackofnovelinformationbeingprovidedbyearlyexperimentsthatdependonthethird-orderresponsefunctionsuchastheOKEtechnique.[3{6]StartingwiththequantummechanicalthirdorderresponsefunctionprobedinthethirdorderRamanexperiment,wehaveRt=i ~<[t;]>.136

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wherethesquarebracketsindicatethecommutatorrelationship,[a;b]=ab)]TJ/F18 11.955 Tf 11.089 0 Td[(ba.Expandingthecommutatorsgives,Rt=i ~[)]TJ/F18 11.955 Tf 12.619 0 Td[(]2.14wheretheanglebracketsindicatethequantummechanicaltrace=1 QTrftg:.15Wenowhavetheresponsefunctionwrittenintermsoftwoquantummechanicaltimecor-relationfunctions.Intheclassicallimit,~!0,thepolarizabilityoperatorscommuteand=.Thus,takingtheclassicallimitofthequantummechanicalTCF'swouldimplynosignalforthethirdorderRamanexperiment.However,sincethisisnotthecase,itisnecessarytowritetheresponsefunctionintermsofoneofthequantumTCF's.Intheclassicallimit,itbecomestheclassicalTCFthatiseasilycalculatedusingclassicalmoleculardynamicstechniques.WewillidentifyeachofthetwoquantumTCF'sintheresponsefunctionasCtandDtwheretheyaredenedas,Ct=Dt=:.16NowwiththeresponsefunctionwrittenasRt=i ~[Ct)]TJ/F18 11.955 Tf 13.02 0 Td[(Dt]wewillidentifyrelationshipsbetweenthetwoTCF's.ThequantummechanicaltracesarewrittenoutwiththeoperatorsinHeisenbergnotationandaregivenasDt=1 QXs.17andCt=1 QXs:.187

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ItiseasytoshowthattheTCF's,CtandDt,arerelatedasCt=DtgivingRt=i ~[Ct)]TJ/F18 11.955 Tf 11.955 0 Td[(Ct]:.19ThisleadtoatimedomainresponsefunctionthatisdependentonlyontheimaginarypartofthequantumTCFgivenbyRt=)]TJ/F15 11.955 Tf 9.299 0 Td[(2 ~[CIt]:.20However,CIthasnovalidclassicallimitthusmakingitimpossibletocalculateusingclassicaltechniques.ItisnecessarytoidentifyarelationshipbetweentherealandimaginarypartsofthequantumTCFsuchthattheresponsefunctioncanbewrittenintermsofarealTCF,whichhasavalidclassicallimit.Toidentifythisrelationship,itishelpfultoworkinthefrequencydomainandconsiderthefrequencydomainformsofCtandCt.TheFouriertransformisdenedasFT[Ct]=C!=1 2Z1dte)]TJ/F19 7.97 Tf 6.586 0 Td[(i!tCt:.21andsimilarlyforCt.C!isgivenbyC!=1 QXsXte)]TJ/F19 7.97 Tf 6.587 0 Td[(Esstts!)]TJ/F18 11.955 Tf 11.955 0 Td[(Ets=~.22whereQisthepartitionfunction,st=andEts=Et)]TJ/F18 11.955 Tf 11.015 0 Td[(Es.TheFouriertransformofCtgivesC)]TJ/F18 11.955 Tf 9.298 0 Td[(!=1 QXsXte)]TJ/F19 7.97 Tf 6.587 0 Td[(Esstts!)]TJ/F18 11.955 Tf 11.955 0 Td[(Est=~2.23ComparingC!andC)]TJ/F18 11.955 Tf 9.298 0 Td[(!weseethatippingtheindicesofC)]TJ/F18 11.955 Tf 9.299 0 Td[(!,s$t,wecaneasilyrewriteC)]TJ/F18 11.955 Tf 9.299 0 Td[(!asC)]TJ/F18 11.955 Tf 9.298 0 Td[(!=1 QXsXte)]TJ/F19 7.97 Tf 6.587 0 Td[(Ettsst!)]TJ/F18 11.955 Tf 11.955 0 Td[(Ets=~:.248

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With~!=Ets,weareabletorelatethetwofrequencydomaincorrelationfunctionsasC!=e~!C)]TJ/F18 11.955 Tf 9.298 0 Td[(!:.25ThisrelationshiprepresentsadetailedbalancerelationshipandapplyingittothefrequencydomainresponsefunctionR!=i ~[C!)]TJ/F18 11.955 Tf 11.955 0 Td[(C)]TJ/F18 11.955 Tf 9.299 0 Td[(!].26wehaveR!=i ~[C!)]TJ/F18 11.955 Tf 11.955 0 Td[(e~!]:.27Intheclassicallimit,weTaylorexpandtheexponentialgivingR!~!0=i ~[C!)]TJ/F15 11.955 Tf 11.955 0 Td[(1+~!]:.28SimplicationleadstothefrequencydomainformoftheresponsefunctionintheclassicallimitR!~!0=i!C!cl:.29Inthetimedomain,theresponsefunctionwilltaketheformthatisamenabletocalcula-tion.ThefrequencydomainresponsefunctionbecomesaderivativeoftheTCFinthetimedomainviatheFouriertransformrelationshipgivenasd dtCt=d dtZ1d!ei!tC!.30whichiseasytoshowisequivalenttod dtCt=iZ1!d!ei!tC!:.31Thus,inthetimedomaintheresponsefunctionisgivenasthederivativeoftheonetimeTCFRt=dCt dt.329

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whereCt=:.33ThisisthecorrelationfunctionthatismeasuredintheconventionalRamanexperiment.ThusthisnonlineartechniqueprovidesnonewinformationthatisnotavailableinthelinearRamanexperiment.Analternativeapproachwhichwillproveusefulinthederivationofthefthorderre-sponsefunctioninvolvestheuseoftanhrelationships.WebeginwiththetimecorrelationfunctionCt=[CRt+CIt]:.34WecanthendenethefouriertransformsfortherealandtheimaginarypartsintimeasfollowsCR!=1 2Z1e)]TJ/F19 7.97 Tf 6.586 0 Td[(i!tdtCRt.35andCI!=1 2Z1e)]TJ/F19 7.97 Tf 6.586 0 Td[(i!tdtCIt:.36Inthefrequencydomain,thecorrelationfunctionsarereal,CR!=CR!2.37CI!=CI!.38whereCI!indicatestheFouriertransformoftheimaginarypartoftheTCF.WecandenetheFouriertransformsoftherealandimaginaryTCF'sasCR!=1 2[C!+C)]TJ/F18 11.955 Tf 9.298 0 Td[(!].3910

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CI!=1 2[C!)]TJ/F18 11.955 Tf 11.955 0 Td[(C)]TJ/F18 11.955 Tf 9.299 0 Td[(!]:.40Then,usingthedenitionofthetanh~!=e~!=2)]TJ/F19 7.97 Tf 6.586 0 Td[(e)]TJ/F20 5.978 Tf 5.756 0 Td[(~!=2 e~!=2+e)]TJ/F20 5.978 Tf 5.756 0 Td[(~!=2,wehaveCI! CR!=1)]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~! 1+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!.41rewrittenasCI! CR!=tanh~! 2:.42ApplyingtanhrelationshipsgivesR!=i ~[tanh~! 2CR!]2.43whereintheclassicallimitwehaveR!~!0=i ~[CR!~!].44whichisthesameasthefrequencydomainresponsefunctiongiveninequation2.29.AsimilarapproachwillbeusedtoderivetheTCFtheoryofthefthorderresponsefunctioninChapter4.ItwillbeshownthatitimpossibletorewritethefthorderresponsefunctionexactlyintermsofasingleTCFaswaspossibleintheabovecase;however,itispossibletomakeapproximationsthatleadtoaneectiveTCFtheory.Finally,theapproachwillbeappliedagaininChapter5tothethirdorderresponsefunctionprobedinthe2DIRexperiment.11

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Chapter3ThePolarizabilityModelInordertocalculatethetimecorrelationfunctionsnecessaryforpredictingspectroscopicsignalsviatimecorrelationfunctions,asuitable,practicalpolarizabilitymodelisnecessary.Forthecalculationsinthiswork,weusedanatomdipoleinteractionmodelthattakesintoaccounttheinteractionofdipoleswiththeeldcreatedbytheotherdipolesinthesystem.ThemodelisknownasapointatompolarizabilityapproximationPAPAmodeliseectiveatpredictingmolecularpolarizabilities.[7]Theinduceddipolemomentofuniti,consistingofthecontributionfromtheelectriceldandallotherinduceddipolesinthesystem,isgivenbyi=i[Ei)]TJ/F19 7.97 Tf 23.375 14.944 Td[(NXj=1;i6=jTijj].1whereEiistheappliedelectriceldatsitei,Tijisthedipoleeldtensorandiisthepolarizabilitytensorofuniti.ThetracelessdipoleeldtensordictatestheinteractionofsitesiandjandisgivenbyTij=)]TJ/F15 11.955 Tf 12.734 8.088 Td[(3 r524x2)]TJ/F15 11.955 Tf 11.955 0 Td[(1=3r2xyxzxyy2)]TJ/F15 11.955 Tf 11.956 0 Td[(1=3r2yzxzyzx2)]TJ/F15 11.955 Tf 11.955 0 Td[(1=3r235.2whereristhedistancebetweenunitsiandjandx,yandzarethecartesiancomponentsofthevectorfromunititoj.[7]12

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Solvingforthemanybody"polarizability,webeginbyrewritingequation3.1as)]TJ/F16 7.97 Tf 6.586 0 Td[(1ii+NXj=1;i6=jTijj=Ei.3whichcanberewrittenintothematrixequationEquation3.4.26664)]TJ/F16 7.97 Tf 6.587 0 Td[(11T12T1NT21)]TJ/F16 7.97 Tf 6.587 0 Td[(12T2N.........T21)]TJ/F16 7.97 Tf 6.587 0 Td[(1N377752666412...N37775=26664E1E2...EN37775.4Ifwedenethe3N3Nmatrixas~Asuchthatequation3.4canbewrittenas~A~=~E.Solvingfor~,wehave~=~B~E.5whereB=A)]TJ/F16 7.97 Tf 6.587 0 Td[(1andisgivenby~B=26664B11B12B1NB21B22B2N............BN1BN2BNN37775.6whereBijare33matrices.Equation3.5iscanberewrittenasNequationsequivalenttoi=[PNj=1Bij]E,wherewehaveconsideredthemoleculetobeinauniformelectriceldsuchthatEi=Eforalli.[7]Thetotalinduceddipolemomentinthemoleculeisthengivenbymol=NXi=1i=[NXi=1NXj=1Bij]E.7withthemolecularpolarizabilitybeinggivenbymol=[NXi=1NXj=1Bij]:.813

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3.1CalculatingthePolarizabilityPolarizationforcesarenotexplicitlyincludedintheMDcalculations.However,incalculatingthesystempolarizabilitytobeusedintheTCF,fullmanybodypolarizationeectswereincludedbysolvingtheequation3.9.Theexpressionfortheeectivepolarizability,~i,forsiteatomiisgivenby~i=i+inXj6=iTrij~j.9whereiistheisotropicpointpolarizabilityforsiteiandTrijisthedipoleeldtensorbetweensitesiandjinasystemwithnsites.[7]Thetotalsystempolarizabilityisgivenbysummingtheeectivepolarizabilitiesforallsites:=nXi=1~i.10Twoformsofthemanybodypolarizabilitymodelwereusedinourcalculations;truncatingEq.3.9totermsrstorderinTrij,therstorderdipoleinduced-dipolemodelFODID,andtheexactsolution,inniteorderdipoleinduced-dipoleMBP.TheMBPmodelrequiresiterativelysolvingEq.3.9oramatrixinversionmatrixinversionwasemployedinthepresentcalculationswhiletheFODIDonlyrequiresasingleiterationofEq.3.9.AlthoughtheFODIDapproximationissucientforsomespectroscopiccalculations[8],itwillbeshownthatthisapproximationneglectslargecontributionstothefthorderRamansignal.3.2RotationalInvariantsAssessingthepolarizationdependenceofaspectroscopicsignalisgreatlyfacilitatedbytheintroductionofrotationalinvariantsofthesystempolarizability.[8,9]Foranisotropicsystem,itispossibletoimprovesamplingofatensorbyallowingthelaboratoryframetoberotatedthroughall4steradiansofthesystem.Byincludingallthesecontributionsitispossible14

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towritecombinationsoftensorsintermsofalimitednumberoftermscalledrotationalinvariants.WecandenearotationmatrixR,thatcanbeusedtotransformatensorfromagivencoordinateframetoanother[9]R;;=0@coscoscos)]TJ/F20 5.978 Tf 7.453 0 Td[(sinsincoscossin+sincos)]TJ/F20 5.978 Tf 5.756 0 Td[(cossin)]TJ/F20 5.978 Tf 5.756 0 Td[(sincoscos)]TJ/F20 5.978 Tf 7.453 0 Td[(cossin)]TJ/F20 5.978 Tf 5.756 0 Td[(sincossin+coscossinsinsincossinsincos1A.11Here,andtakeonvaluesform0to2andtakesvaluesfrom0to.Rotationabouttheoriginalzaxisisdenedbyandtheoriginalyaxisby.Finally,denestherotationaboutthenewzaxis.Rotationofatensorfromagivenframeitoanotherframei0thentakestheformai=Xi0Rii0ai0.12Foratensorsuchasthepolarizabilitytensorijthatweareinterestedin,therotationtakestheformij=Xi0;j0Rii0Rjj0i0j0.13wherethesummationincludesallcombinationsofthecoordinatesx0,y0andz0.Foraproductoftensorswecandenetheorientationalaverageas aijbkl= Xi0;j0Xk0;l0Rii0Rjj0Rkk0Rll0ai0j0bk0l0.14wherewehavetheproductoftwotensorsshownherethatcouldrepresentthepolariz-abilitytensormeasuredattwodierenttimesuchasthatwhichappearsinaTCF,e:g:.Sincetheaveragingofthetensorsisaccomplishedthroughintegrationrotationaboutthe4steradiansofthesystem,wecanrewritetheorientationalaverageas aijbkl=Xi0;j0Xk0;l0 Rii0Rjj0Rkk0Rll0ai0j0bk0l0.1515

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wherethetensorshavebeenremovedfromtheaveragesincetheyhavenoangledepen-dence.[9]Theaverageoftherotationmatricesisthengivenby Rii0Rjj0Rkk0Rll0=Z20dZpi0dZ20dRii0Rjj0Rkk0Rll0sin:.16Fortheproductoftwotensors,thereareonlythreenonzeroaverages,xxxx,xyxy,xxyyandtheirpermutationswithonlytwoofthesebeingunique.Forthepresentcaseweareinterestedintheproductofthreetensors,aswillbeshowninchapter4,wewillbecalculatingTCF'softhefrom.Theorientationalaverageoftheproductofthreetensorelementsisgivenby aijbklcmn=Xi0;j0Xk0;l0Xm0;n0 Rii0Rjj0Rkk0Rll0Rmm0Rnn0ai0j0bk0l0cm0n0.17Forthiscase,nonzeroorientationalaveragesexistforthepolarizationcombinationsxxxxxx,xxxxyy,xxxyxy,xxyyzzandxyxzyzandtheirpermutationswheretheseindicesindicatetheindicesijklmnfor.Fortherotationmatrices,thereareelevennonzeroorientationalaveragesgivenby R6ii0=1 7.18 R4ii0R2ij0=1 35.19 R4ii0R2jj0=3 35.20 R2ii0R2ji0R2ki0=1 105.21 R2ii0R2ji0R2jj0=2 105.22 R2ii0R2ji0R2kj0=1 35.2316

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R2ii0R2jj0R2kk0=8 105.24 R3ii0Rji0Rij0Rjj0=)]TJ/F15 11.955 Tf 13.421 8.087 Td[(1 70.25 R2ii0Rij0Rik0Rjj0Rjk0=)]TJ/F15 11.955 Tf 16.347 8.088 Td[(1 210.26 R2ii0Rjj0Rjk0Rkj0Rkk0=)]TJ/F15 11.955 Tf 13.42 8.088 Td[(1 42.27 Rii0Rij0Rji0Rjk0Rkj0Rkk0=1 105.28wherei,jandkarenotthesameandrepresentx,yandzinthecartesiansystem.[9]Substitutingthesevaluesintoequation3.17andsummingoverallcombinationsofindices,theaveragesarereducedtocombinationofthreerotationalinvariants,thetraceTra=Xiaii.29thepairproductPPa;b=Xi;jaijbij.30andthetripleproductTPa;b;c=Xi;j;kaijbijcij:.31Itisnowpossibletowriteanycombinationpolarizationofproductsofthreetensorsintermsofthesethreeinvariants.Usingtheinvariants,theaveragescanbewrittenas axxbxxcxx=1 105TraTrbTrc+8 105TPa;b;c.32+2 105[TraPPb;c+TrbPPa;c+TrcPPa;b]:17

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axxbxxcyy=1 35TraTrbTrc)]TJ/F15 11.955 Tf 19.004 8.088 Td[(4 105TPa;b;c.33)]TJ/F15 11.955 Tf 16.347 8.088 Td[(1 105[TraPPb;c+TrbPPa;c]+2 35TrcPPa;b: axxbxycxy=)]TJ/F15 11.955 Tf 16.347 8.088 Td[(1 210TraTrbTrc+1 35TPa;b;c.34+1 42TraPPb;c)]TJ/F15 11.955 Tf 19.004 8.088 Td[(1 105[TrbPPa;c+TrcPPa;b]:wherethethreepolarizationconditionsaretheconditionsofinterest;theotherthree,xxyyzz,xxyzyzandxyxzyzwouldfollowinasimilarway.[9]TheorientationalaveragesareusefulforevaluatingtheeectsofusingtheFODIDap-proximationtothepolarizability.Inaddition,itispossibletopredictcontributionstoalineshapeforvariouspolarizationconditions.Bothofthesewillbeaddressedinchapter4.18

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Chapter4TheFifthOrderRamanResponseFunctionInthischapter,aTCFtheoryofthefthorderresponsefunctionispresented.TheworktooktheapproachsimilartothatwhichwassuccessfulinrelatingthirdorderspectroscopiestoclassicalTCF'saswasshowninChapter2.[3{6]WhileitisnotpossibletoobtainasimpleTCFexpressionthatisexactinthiscase,[10]signicantprogressalongtheselinesispossible.4.1FifthOrderRamanExperimentThefthorderRamanexperimentisasixwavemixingtechniquethatinvolveselectronicallynon-resonantRamanexcitationofasystemwithapairoffspulsesthatoccurattime0.Thisleavesthesysteminavibrationalcoherence.Aftersometimedelay,t1,offreeevolution,thesystemisexcitedwithasecondpairoffspulsesthattransfersthesystemtoanewvibrationalcoherenceorapopulationstate.Afterasecondtimedelay,t2,thesystemisprobedwithasinglepulsethatgeneratesascatteringevent.Astrongechosignal,whent2=t1,wouldindicatethattheexcitationsarelonglived,outlivingthereownperiod.Alternatively,thelackofanechowouldsuggestthatdynamicsarebeinginterferedwithonthereowntimescalesandarerelativelyshortlivedhomogeneouslybroadened.[8,11,12]19

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4.2PolarizationTheprincipleobservableinspectroscopyisthematerialpolarizationthatarisesfromtheinteractionofthematerialwiththeappliedradiationeld.ExpansionofthepolarizationintermsofapowerseriesintheradiationeldisgivenbyPr;tPr;t+PNL.1wherePNLPr;t+Pr;t+:::.2withthesuperscriptindicatingthepowerintheelectriceld.[5]Thelinearterm,Pr;t,representsthepolarizationarisingfromaweakappliedradiationeld.Thistermisrespon-sibleforprocessessuchasabsorptionandisprobedinthelinearIRexperiment.ThetermsreferedtoasPNLarethenon-lineartermsthatareprobedinnon-linearexperiments.Inisotropicmedia,evenpoweredpolarizationtermsareidenticallyzero.Therefore,Pr;tistherstsurvivingtermforasystemwithinversionsymmetry.Manynon-linearexperi-mentsarerelatedtoPr;tincludingphotonecho,transientgrating,coherentanti-stokesRamanscatteringandthe2DIRexperiment.[13]ThenextpolarizationtermtosurviveinanisotropicmaterialisPr;twhichisprobedinthefthorderRamanexperiment.Inthischapter,thefthorderpolarizationisofinterest.Thepolarizationofthematerialdependsontheelectriceldsappliedatearliertimesandisgivenbytheintegralofthematerialresponsefunction,R,andappliedelectricelds,Easfollows:PNr;t=Z10d1Z10dNRN1;;NEr;t)]TJ/F18 11.955 Tf 11.956 0 Td[(1Er;t)]TJ/F18 11.955 Tf 11.955 0 Td[(N.3Theresponsefunctionisrealandcausaltensorofrankn+1.InEquation4.3,representsthetimedelaysbetweenpulses.TheTCFtheoriespresentedinthisthesisareformulatedintermsofthematerialresponsefunctions.20

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4.3FifthOrderResponseFunctionThequantummechanicalexpressionfortheelectronicallynon-resonantfthorderpolariza-tionresponseisgivenby:[5,14{16]Rt1;t2=i ~2Trft1+t2[t1;[;]]g.4InEquation4.4,=e)]TJ/F19 7.97 Tf 6.586 0 Td[(H=Q,forasystemwithHamiltonianHandpartitionfunctionQatreciprocaltemperature=1=kT,andkisBoltzmann'sconstant;TrrepresentsthetraceoftheoperatorsandisthesystempolarizabilitytensorandtheGreeksuperscriptsdenotetheelementsbeingconsidered.Thesquarebracketsrepresentthecommutatoroftheoperators.Thetracegivesaclassicallimitthatisanorder~2contributionthatresultsfromacombinationoffourtwotimecorrelationfunctionsthatarethemselvesequivalentclassically.Thisissimilartothethirdorderspectroscopiesresultingfromanorder~contributionofthetraceinthethirdorderresponsefunction,Rt=i ~Trf[t;[;]]g.[5,14,16]Thetracecanbewrittenasthedierencebetweentwoonetimecorrelationfunctionsthatareequivalentclassically,e.g.hti)-244(hti;theanglebracketsarethetraceoftheoperatorsdividedbythepartitionfunctioninthestandardnotation.[1]InthatcasethedierencebetweenthequantummechanicalTCF'sistheimaginarypartoftheTCFthatisexactlyrelatedtotherealpartinfrequencyspace,CI!=tanh~!=2CR!,wherethesubscriptsdenoterealfunctionsthataretheFouriertransformoftherealandimaginarypartsofCt=hti.IntheclassicallimitCR!istheFouriertransformoftheclassicalTCF,[1]makingthethirdorderresponsefunctionandthelinearspectroscopiessimplyrelatedtoconventionalTCF's.ForR,thetraceofthenestedcommutatorsmustbeoforder~2intheclassicallimit,tocanceltheprefactorof~)]TJ/F16 7.97 Tf 6.586 0 Td[(2.InSection4.4itwillbedemonstratedthatamultiplicative21

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factorofleadingorder~canbeobtainedexactlyusingfrequencydomaindetailedbalancerelationshipsbetweenthetwotimequantummechanicalTCF'sthatconstituteRt1;t2.AcombinationofrealandimaginarypartsofatwotimeTCFremain,andwerequiretheirO~contributionfortheclassicallimit.Ifarelationshipbetweentherealandimaginarypartsofthetwotimecorrelationfunctionexisted,theexplicit~-dependencecouldbedeterminedasinthethirdordercase,andtheclassicalfthorderresponsecouldbewrittenassecondderivativesintimeofaclassicalTCF.However,noexactanalyticrelationshipisdiscernible.Toproceedfurther,asimpleapproximaterelationshipisshowntoexistbetweentherealandimaginarypartsofthetwotimeTCFforaharmonicsystemwithnonlinearpolarizability,thatisrequiredtoproduceafthorderresponseforaharmonicsystem.[9,17,18]Inthespiritofquantumcorrection,thisrelationshipisusedtowritetheexactquantummechanicalresponsefunctionapproximatelyintermsofclassicalTCF's.TheresultingTCFexpressionisthencalculatedfromfullyanharmonicmoleculardynamicscalculationssupplementedbyasuitablespectroscopicpolarizabilitymodel.Theapproximateexpressionisdemonstratedtohavecorrectlimitingbehaviorsandleadstoatwodimensionalspectrumforambientcarbondisuldeinexcellentagreementwithexistingexperimentalandtheoreticalwork.Theproposedapproachmakesthecalculationoffthorderresponsefunctionspracticalforawidevarietyofchemicallyinterestingsystems.22

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4.4ConstructingaTCFTheoryExpandingthecommutatorsinthetraceinEquation4.4givesthefthorderresponsefunctionas:Rt1;t2=)]TJ/F15 11.955 Tf 9.299 0 Td[(1=~2[gt1;t2)]TJ/F18 11.955 Tf 11.955 0 Td[(ft1;t2)]TJ/F18 11.955 Tf 11.955 0 Td[(ft1;t2+gt1;t2].5InEquation4.5,ft1;t2=ht1t2i;gt1;t2=ht2t1i;ft1;t2=ht2t1i,andgt1;t2=ht1t2i;thesuperscriptstarindicatesthecomplexconjugate.Thesuperscriptnotationondenotingthepolarizationdirectionsissuppressedandtheresultsapplytoallpossiblepolarizations.TheexpressioninsidethesquarebracketsinEquation4.5isthedierencebetweentherealpartofthefunctionsfandg,gRt1;t2)]TJ/F18 11.955 Tf -458.701 -28.891 Td[(fRt1;t2,withaleadingtermorder~2intheclassicallimit.ThisalsodemonstratesthatRisrealintime,whilethecorrelationfunctionsarecomplexfunctionsoftime.Itisconvenienttoevaluatefandgexplicitlyintheenergyrepresentation.ft1;t2=1 QXihije)]TJ/F19 7.97 Tf 6.586 0 Td[(He)]TJ/F19 7.97 Tf 6.586 0 Td[(iHt1=~eiHt1=~eiHt2=~e)]TJ/F19 7.97 Tf 6.587 0 Td[(iHt2=~jiiInsertingthreecompletesetsofenergyeigenstates,Pjihj,withHji=Eji,operatingandsimplifyinggives:ft1;t2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.587 0 Td[(Eiikkjjieit1Eki=~eit2Ekj=~E=E)]TJ/F18 11.955 Tf 11.955 0 Td[(E=hjjiAsnotedabove,thefunctionsallobey,ft1;t2=f)]TJ/F18 11.955 Tf 9.299 0 Td[(t1;)]TJ/F18 11.955 Tf 9.298 0 Td[(t2,andthisalsoimpliesthatthematrixelementscanbechosenreal,andwillbetakenasreal.23

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Inthefrequencydomain:f!1;!2=1 2Z1dt0e)]TJ/F19 7.97 Tf 6.587 0 Td[(i!1t11 2Z1dte)]TJ/F19 7.97 Tf 6.587 0 Td[(i!2t2ft1;t2f!1;!2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.586 0 Td[(Eiikkjji!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eki=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Ekj=~.6Similarmanipulationsleadtothefrequencydomainfunctionsbelow.Toobtainthemintheformpresented,theidentitythatthedoubleFouriertransformofthecomplexconjugateoftheTCF'sgivesthecomplexconjugateofthefrequencydomainfunctionevaluatedatnegativethefrequencyargument,e.g.FT[ft1;t2]=f)]TJ/F18 11.955 Tf 9.299 0 Td[(w1;)]TJ/F18 11.955 Tf 9.299 0 Td[(w2,isused.FTrepre-sentstheFouriertransformprocessshowninEquation4.6,andthefactthatthefrequencydomainfunctionsareallrealisalsoused.Thefunctionscanbeseentoberealbecause,e.g.FT[ft1;t2]=f!1;!2=FT[f)]TJ/F18 11.955 Tf 9.299 0 Td[(t1;)]TJ/F18 11.955 Tf 9.299 0 Td[(t2]=f!1;!2.ft1;t2=ht2t1i.7f)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.586 0 Td[(Eiikkjji!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eik=~!2)]TJ/F18 11.955 Tf 11.956 0 Td[(Ejk=~gt1;t2=ht2t1i.8g!1;!2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.586 0 Td[(Ekikkjji!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eki=~!2)]TJ/F18 11.955 Tf 11.956 0 Td[(Ekj=~gt1;t2=ht1t2i.9g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.586 0 Td[(Ekikkjji!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eik=~!2)]TJ/F18 11.955 Tf 11.956 0 Td[(Ejk=~Also,anothercorrelationfunctionht1;t2ispresentedthatdoesnotappearinthere-sponsefunction,buthasthesameclassicallimitasfandgandwillproveusefullater;thissetoftwotimecorrelationfunctionsrepresentstheentiresetthatcanbecreatedwiththe24

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relevantoperatorsforthetwotimeTCF's:ht1;t2=ht1t2i.10h!1;!2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.587 0 Td[(Ejikkjki!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eki=~!2)]TJ/F18 11.955 Tf 11.956 0 Td[(Ekj=~ht1;t2=ht2t1i.11h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2=1 QXiXjXke)]TJ/F19 7.97 Tf 6.586 0 Td[(Ejikkjji!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eik=~!2)]TJ/F18 11.955 Tf 11.956 0 Td[(Ejk=~InEquations4.7-4.11theformspresentedarearesultofusingindexswitchingtomaxi-mizethesimilarityofthedeltafunctions.Forexample,anotherformofEquation4.11canbeobtainedbyswitchingtheiandjindicies.Indexswitchingisequivalenttotakingadvantageofcyclicpermutationsofthetracetoobtaindierentexpressions.TheHermiticityofthepolarizabilitywasalsousedtoequate,e.g.ik=ki,whichistrueforrealmatrixelements.ItisclearfromEquations4.6-4.11thatthesetoffunctionsf!1;!2;g!1;!2,andh!1;!2dieronlyintheBoltzmannfactorweightingtheexpressions,e.g.e)]TJ/F19 7.97 Tf 6.587 0 Td[(Eiinf!1;!2ande)]TJ/F19 7.97 Tf 6.587 0 Td[(Eking!1;!2.Thesameappliestothesetoffunctionofnegativefrequencyf)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2,andh)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2.Thisimpliesthefollowingrelationshipsbe-tweenthefrequencydomainfunctions:^!=!1)]TJ/F18 11.955 Tf 11.955 0 Td[(!2.12e~!1g!1;!2=f!1;!2.13e~!2g!1;!2=h!1;!2.14e)]TJ/F19 7.97 Tf 6.587 0 Td[(~^!f!1;!2=h!1;!2.1525

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e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!1f!1;!2=g!1;!2.16e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!1g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2=f)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!24.17e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!2g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2=h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!24.18e~^!f)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2=h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!24.19e~!1f)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2=g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!24.20InEquations4.12-4.20enforcingtherelevantdeltafunctionsallowsthefrequencyfactortobetakenoutsidethesummations.Thereis,however,nodirectrelationshipbetweenthefunctionsofpositiveandnegativefrequency.Itisnowusefultotakesumsanddierencesofthefunctions,andusetherelationshipsinEquations4.12-4.20,e.g.f!1;!2g!1;!2=)]TJ/F18 11.955 Tf 5.479 -9.684 Td[(e~!11g!1;!2.Takingtheratioofthesumanddierencesleadstothefollowingrelationships:f!1;!2)]TJ/F18 11.955 Tf 11.955 0 Td[(h!1;!2=tanh~^!=2[f!1;!2+h!1;!2].21f!1;!2)]TJ/F18 11.955 Tf 11.956 0 Td[(g!1;!2=tanh~!1=2[f!1;!2+g!1;!2].22h!1;!2)]TJ/F18 11.955 Tf 11.955 0 Td[(g!1;!2=tanh~!2=2[h!1;!2+g!1;!2].23f)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2)]TJ/F18 11.955 Tf 11.955 0 Td[(h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2=tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~^!=2[f)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2+h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2].24f)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2)]TJ/F18 11.955 Tf 11.955 0 Td[(g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2=tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~!1=2[f)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2+g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2].25h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2)]TJ/F18 11.955 Tf 11.955 0 Td[(g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2=tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~!2=2[h)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2+g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2].2626

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Thisallowsthefthorderresponsefunctiontoberewrittenas:R!1;!2==~2tanh~!1=2.27[f!1;!2)]TJ/F18 11.955 Tf 11.955 0 Td[(f)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2+g!1;!2)]TJ/F18 11.955 Tf 11.955 0 Td[(g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2]Intheclassicallimit,Equation4.27hasaprefactorof~!1=2,andtoobtainanexpressionintermsofclassicalTCF'satermorder~needstobeobtainedfromthefunctionsinthesquarebrackets.TheexpressioninsidethesquarebracketsinEquation4.27isthedierencebetweentheFouriertransformoftheimaginarypartsofthefunctionsft1;t2andgt1;t2,thatwillbedenotedasgI!1;!2)]TJ/F18 11.955 Tf 12.153 0 Td[(fI!1;!2,wherethesubscriptsdenotetheFouriertransformoftherealRorimaginaryIpartsoftheTCF's,bothofwhicharethemselvesrealfunctionsoffrequency.ItisnowusefultowriteEquation4.27intermsofonlyonefunction,g!1;!2,usingEquations4.13and4.17:R!1;!2==~2tanh~!1=2[+e~!1g!1;!2)]TJ/F15 11.955 Tf 11.956 0 Td[(+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!1g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2].28Toobtaintheclassicallimit,~!kT,theexponentialsareexpandedandtermsorder~areretained,R!1;!2=1 ~2tanh~!1 2.29[+~!1g!1;!2)]TJ/F15 11.955 Tf 11.956 0 Td[()]TJ/F18 11.955 Tf 11.955 0 Td[(~!1g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2]R!1;!2=1 ~2tanh~!1 2.30[2g!1;!2)]TJ/F18 11.955 Tf 11.955 0 Td[(g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2)]TJ/F15 11.955 Tf 11.956 0 Td[(~!1g!1;!2+g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2]27

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R!1;!2=1 ~2~!1 2.31[4gI!1;!2+2~!1gR!1;!2];whereitisunderstoodthattheleadingcontributionsin~totheTCFaretobekept,zero-orderforgRandrst-ordergI.InthetimedomainEquation4.31takestheform:Rt1;t2=)]TJ/F18 11.955 Tf 9.298 0 Td[(2@2=@t21gRt1;t2)]TJ/F15 11.955 Tf 11.956 0 Td[(2i ~@=@t1gIt1;t2.32ItisworthnotingthatEquation4.31canbeobtainedfromthefrequencydomainversionofEquation4.5bywritingR5,atthatpoint,intermsoftheg!1;!2usingthefrequencydomainrelationshipsbetweenfandgandTaylorexpandingtheresultingexponentials.Equation4.28is,however,exactforallfrequenciesandisastartingpointtodevelopanapproximationthatisvalidforintramolecularorhighfrequencyintermolecularspectroscopyforwhich~!kT.Equation4.32isexactintheclassicallimit.When~!kTthefunction,gR!1;!2becomestheFouriertransformoftheclassicalTCF,h)]TJ/F18 11.955 Tf 9.298 0 Td[(t1t2i=ht1t1+t2i,andthetimeshavebeenreorderedbecausetheycommuteintheclassicallimit.Foraonetimecorrelationfunction,Ct,asimplerelationshipexistsbetweentherealandimaginarypartsofthefunction,CI!=tanh~!=2CR!.WhileitisclearthatgImakesacontributionthatisorder~,thereisnogeneralrelationshipbetweentherealandimaginarypartsofg.Therefore,thegRterminEquation4.32canbecalculatedfromclassicalTCF'sbutgIhasnoapparentclassicallimit.NeverthelessanapproximateconnectionbetweengRandgIcanbeestablishedforahar-28

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monicsystemwithanonlinearpolarizability,andthisisthesimplestharmonicsystemthatproducesafthorderresponse.[9,17,18]Thegoalistodeterminetherelationshipbe-tweentherealandimaginarypartsofg!1;!2foraharmonicsystemwithpotentialenergy,V=1=2m2Q2.SucharesultleadstoanapproximatecorrelationfunctionexpressionforR.Toproceed,thepolarizabilityisexpandedtosecondorderintheharmoniccoordinateinordertoevaluatethematrixelementsinEquation4.8:=0+0Q+1=200Q2.33InEquation4.33theprimesrepresentderivativeswithrespecttotheharmoniccoor-dinate,Q.WhenEquation4.8isevaluatedfortheharmonicsystem,thelowestordernon-zerocontributionstoRinvolvetwo'andone";thecorrelationfunctionshavecontributionsfromone0andtwo0,thatcanbedenotedasg011!1;!2,g101!1;!2andg110!1;!2,butthesedonotcontributetotheresponsefunctionwhentreatedexactly;thispointwillberevisitedbelow.Thesuperscriptsongrepresentthepowerofthecoor-dinatethatisusedtoevaluatethepolarizabilitymatrixelementsappearingasikkjjirespectively.Thetotalcontributiontotherstnon-vanishingorder,isthenwrittenas:g!1;!2=g211!1;!2+g121!1;!2+g112!1;!2.Theevaluationinvolvesstandardre-sultsfortheharmonicoscillatorandevaluatingseveralgeometricseries,[19]andgives,inthefrequencydomain:g211!1;!2=00001 Q~ 2m21 2Xke)]TJ/F19 7.97 Tf 6.587 0 Td[(~k.34hk+1k+2!1+2!2++k+1k+1!1!2+29

PAGE 39

+kk+1!1!2)]TJ/F15 11.955 Tf 11.955 0 Td[(+kk)]TJ/F15 11.955 Tf 11.955 0 Td[(1!1)]TJ/F15 11.955 Tf 11.955 0 Td[(2!2)]TJ/F15 11.955 Tf 11.955 0 Td[(iFouriertransformingtothetimedomain:g211t1;t2=0000~ 2m2.35h2 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2[e)]TJ/F19 7.97 Tf 6.587 0 Td[(i2t1e)]TJ/F19 7.97 Tf 6.586 0 Td[(i2t2]+[e)]TJ/F19 7.97 Tf 6.586 0 Td[(~+1 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][e)]TJ/F19 7.97 Tf 6.587 0 Td[(it2]+[e)]TJ/F19 7.97 Tf 6.586 0 Td[(~+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~ )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][e)]TJ/F19 7.97 Tf 6.586 0 Td[(it2]+[e)]TJ/F19 7.97 Tf 6.587 0 Td[(~ )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][ei2t1eiat2]iAddingthecomplexconjugatetoyieldtherealpartinthetime:g211Rt1;t2=0000~ 2m21 4h[+2e)]TJ/F19 7.97 Tf 6.587 0 Td[(~ )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][cost1+t2].36+[e)]TJ/F19 7.97 Tf 6.587 0 Td[(~+6e)]TJ/F19 7.97 Tf 6.586 0 Td[(~+1 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][cost2]iSubtractingthecomplexconjugatetoyieldstheimaginarypartinthetime:g211It1;t2=0000~ 2m21 4h[e)]TJ/F19 7.97 Tf 6.586 0 Td[(~)]TJ/F15 11.955 Tf 11.955 0 Td[(2 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][sint1+t2].37+[e)]TJ/F19 7.97 Tf 6.586 0 Td[(~)]TJ/F15 11.955 Tf 11.956 0 Td[(1 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][sint2]iSimilarmanipulationsleadtothefollowingexpressionsintheothertwocases:g121Rt1;t2=0000~ 2m21 4h[+2e)]TJ/F19 7.97 Tf 6.587 0 Td[(~ )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][cost1+2t2].3830

PAGE 40

+[e)]TJ/F19 7.97 Tf 6.587 0 Td[(~+6e)]TJ/F19 7.97 Tf 6.586 0 Td[(~+1 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][cost1]ig121It1;t2=0000~ 2m21 4h[e)]TJ/F19 7.97 Tf 6.586 0 Td[(~)]TJ/F15 11.955 Tf 11.955 0 Td[(2 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][sint1+2t2].39+[e)]TJ/F19 7.97 Tf 6.586 0 Td[(~)]TJ/F15 11.955 Tf 11.956 0 Td[(1 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][sint1]ig112Rt1;t2=0000~ 2m21 4h[e)]TJ/F19 7.97 Tf 6.587 0 Td[(~ )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2][cost1)]TJ/F15 11.955 Tf 11.955 0 Td[(t2].40+[e)]TJ/F19 7.97 Tf 6.587 0 Td[(~+2e)]TJ/F19 7.97 Tf 6.586 0 Td[(~+3 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][cost1+t2]ig112It1;t2=0000~ 2m21 4h[e)]TJ/F19 7.97 Tf 6.587 0 Td[(~)]TJ/F15 11.955 Tf 11.955 0 Td[(3 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~2][sint1+t2]i.41Thethreefunctionsjustdiscussedarecomposedoftwotermswithdistincttor!de-pendenceforatotalofsixcontributionstog.Eachhasanorder~relationshipbetweentherealandimaginaryparts.However,togetaclassicalresult,therelationshipforallthetermsmustbeofthesameformbecausethedierentcontributionscannotbedis-tinguishedinperformingaclassicalMDandTCFcalculation;thispointwillberevisitedbelow.Whilenoexpressionworksperfectlyforalltheterms,wewillnowdemonstratethat31

PAGE 41

1 2 3 4 5 6 gR gI !1 !2 -~!1=4+!2=2 g211term1 4Eq.4.36 -4~Eq.4.37 2 -~ g211term2 8Eq.4.37 -2~Eq.4.38 0 -1/2~ g121term1 4Eq.4.38 -4~Eq.4.39 2 -5/4~ g121term2 8Eq.4.39 -2~Eq.4.40 0 -1/4~ g112term1 4Eq.4.40 0 -1/4~ g112term2 8Eq.4.41 -6~Eq.4.41 -3/4~ Table4.1:ThesuccessoftherelationshipgI!1;!2=tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~!1=4+!2=2gR!1;!2fortheharmonicsystemisdemonstrated.Intheclassicallimit,fortheproposedrelationshipbetweentherealandimaginarypartsofgtowork,multiplyingthecoecientofgRcolumn2bythevalueof)]TJ/F18 11.955 Tf 9.298 0 Td[(~!1=4+!2=2column6shouldgivethecoecientofgIcolumn3foreachterm.Theproposedrelationshipworksexactlyforthreeofthetermsandverynearlyfortheotherthree.gI!1;!2=tanh)]TJ/F18 11.955 Tf 9.298 0 Td[(~!1=4+!2=2gR!1;!2isanexcellentapproximationintheclassicallimit,andleadstoaTCFapproximationforRthathasreasonablelimitingbehaviors.Intheclassicallimit,toleadingorderin~;tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~!1=4+!2=2=)]TJ/F18 11.955 Tf 9.298 0 Td[(~!1=4+!2=2.Table4.1showsthelimitingformofthefrequencyprefactorsandthevalueof)]TJ/F18 11.955 Tf 9.298 0 Td[(~!1=4+!2=2forthesixtermslistedabove,andeachcontributiontogconsistsoftwoterms.Intheclassicallimit,fortheproposedrelationshipbetweentherealandimag-32

PAGE 42

inarypartsofgtowork,multiplyingthecoecientofgRcolumn2bythevalueof)]TJ/F18 11.955 Tf 9.299 0 Td[(~!1=4+!2=2column6shouldgivethecoecientofgIcolumn3foreachterm.Theproposedrelationshipworksexactlyforthreeofthetermsandverynearlyfortheotherthree.Forthetermsthatarenothandledquitecorrectly,namelytheg211term2andg112term1,thisleadstoerrorsinlteringthedynamics"accordingtotheharmonicsystemdynamics.Whilethepresenttheoryisnotaharmonictheory{itreliesonfullyanharmonicdynamicstocalculatetherelevantTCFderivatives{theharmonicsystemservestoweightthedierentphononprocessesastheywouldcontributeinsuchasystem.[17]Thustheer-rorsassociatedwiththisapproximationaretorelativelyoverweighttheprocessesassociatedwiththestatedterms.Inasubsequentpaperitwillbeshownthatthisapproximationleadstoresultsinliquidxenon,forwhichthefthorderresponsecanbecalculatednumericallyexactly,[20]innearlyquantitativeagreement.[21]Thisanalysisfocusesonthefrequencyprefactorsandthevalueofeachfrequencyargu-ment,!1and!2foreachterm.Thisissucientbecauseoddfunctionsoffrequency,suchastanh~!,areabletoconverttherealpartofsuchafunctiontotheimaginarypartbyconvertingthesumofdeltafunctionstoadierence.ThedeltafunctioncombinationsarethemselvestheresultofFouriertransformingthesinandcosfunctionsassociatedwiththeimaginaryandrealpartsofgrespectivelythatarereferencedexplicitlyinTable4.1.Forexample,inonetimedimension:FT[cost]==2[!)]TJ/F15 11.955 Tf 11.955 0 Td[(+!+].42FT[sint]==2i[!)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F18 11.955 Tf 11.955 0 Td[(!+]4.4333

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Whentanh~!multipliesEquation4.42thefrequencyargument!isreplacedbythevalueoftheharmonicfrequency,.hasthesamemagnitudebutoppositesignforeachdeltafunctioninEquation4.42.Theresultingfunctionoftheharmonicfrequencycanthenbefactoredoutandadierenceofdeltafunctionsremain,i.e.thoseinEquation4.43.ThustherelationshipgI!1;!2=tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~!1=4+!2=2gR!1;!2isnearlyexactfortheharmonicsystemintheclassicallimit.Thesmallerrorsassociatedwiththisapproximationservetoslightlyoverweightthephysicalprocessesassociatedwiththetermsthatarenotaccountedforexactly.Inthespiritofquantumcorrectionschemes,[4,22,23]inwhichanexactresultforarelativelysimplesystemisusedtomakequantummechanicalexpressionsamenabletoclas-sicalcalculation,therelationshipgI!1;!2=tanh)]TJ/F18 11.955 Tf 9.298 0 Td[(~!1=4+!2=2gR!1;!2isusedinEquation4.31togiveintheclassicallimit:R!1;!2==~2~!1=2[)]TJ/F15 11.955 Tf 9.298 0 Td[(2~!2gR!1;!2+~!1gR!1;!2].44Inthetimedomain,Equation4.44takestheform:Rt1;t2=)]TJ/F18 11.955 Tf 9.299 0 Td[(2=2[@2=@t21gRt1;t2)]TJ/F15 11.955 Tf 11.955 0 Td[(2@2=@t1@t2gRt1;t2].45Equation4.45istheprincipleresultoftheTCFtheoryofthefthorderresponsefunctionandrepresentsanapproximationtothefthorderresponsefunctionthatonlyrequirescalculatingaclassicalTCF,albeitasomewhatnovelcorrelationofthreevariableswithtwotimearguments,identifyinggRt1;t2=ht1t1+t2iintheclassicallimitwith,=)]TJ/F18 11.955 Tf 15.383 0 Td[(<>.HerewehaveidentiedgRt1;t2asthecorrelationfunction34

PAGE 44

ofthepolarizabilityuctuations;thisdoesnotaecttheresponsefunctionortheformaldevelopmentoftheTCFtheory,butisanecessarystep.ThereisaremainingproblemwiththeresultgI!1;!2=tanh)]TJ/F18 11.955 Tf 9.298 0 Td[(~!1=4+!2=2gR!1;!2.46inthatitworkswellforthehigherordertermsg211,g121andg112,butiswrongforthelowerterms,g011,g101andg110,thathaveexactrelationshipsbetweentherealandimaginarypartsthatdierfromthechoicemadehere;theexactrelationshipsleadtothecancellationbetweenthetwotermsinEquation4.32andthusnocontributiontothefthorderresponse.Physically,calculatingRt1;t2usingEquation4.45andgRt1;t2=ht1t1+t2ileadstocontaminationfromthelowerordertermsinvolving0,thestaticpolarizability,thatdonotcontributetothefthorderexperiment.Tocorrectthisproblemitissucienttousethecorrelationfunctionofthepolarizabilityuctuations:gRt1;t2=ht1t1+t2i.ToseethatthiseliminatesthesecontributionsconsideraharmonicsystemintheenergyrepresentationwiththepolarizabiltyexpandedviaEquation4.33:<>=0+Pi0hijQjii+1=200PihijQ2jii.Thesecondtermisclearlyzeroandthethirdtermisorder~andwouldnotcontributeclassically.Thuscorrelatingthepolarizabilityuctuationseliminatescontributionsfromthestaticpolarizabilitybyconstructionandthusdoesnotallowthelowerordercontaminating"terms,g011,g101andg110tocontributetothecorrelationfunction.Thisparticulardemonstrationreliesonaharmonicsystembutasimilarargument,basedonordersof~isclearlypossible;alltermsbuttherstareproportionaltosomepowerof~.Further,itiseasytoshowthattheoriginalRt1;t2expression,Equation4.4,is35

PAGE 45

unaectedbythesubstitutionof=)]TJ/F18 11.955 Tf 12.619 0 Td[(<>forandthustheTCFtheoryofthefthorderresponseisestablished.Equation4.45canalsobeobtainedmoredirectlybystartingwiththethefthorderresponsefunctiondirectly,Equation4.47below.Toobtainaclassicallimit,oneofthecommutatorsofthepolarizabilityisreplacedbyaPoissonbracket,itsclassicallimit,[A;B]=i~fA;Bg;thecurlybracketsrepresentthePoissonbracketofoperatorsAandB.[15]TheresultingexpressioncontainsaclassicalTCFpieceandPoissonbracketcontribution,butherethequantumversionisretained.ItisworthnotingthatthepresentTCFtheoryofthefthorderRamanresponsefunctionresultsinaTCFapproximationforthePoissonbracketcontributiontotheclassicalresponsefunction.RewritingEquation4.4gives:Rt1;t2=i ~2h[[t1+t2;t1];]i.47Rt1;t2=)]TJ/F18 11.955 Tf 9.299 0 Td[(2@2=@t21)]TJ/F18 11.955 Tf 11.955 0 Td[(@2=@t1@t2ht1t1+t2i.48+ i~@=@t1ht2)]TJ/F18 11.955 Tf 9.299 0 Td[(t1i)-222(ht2)]TJ/F18 11.955 Tf 9.299 0 Td[(t1iTherstterm,Equation4.48,isalreadyaTCFcontribution.Thesecondtermcanbeidentied,usingEquations4.8and4.11asgt1;t2)]TJ/F18 11.955 Tf 12.653 0 Td[(ht1;t2.UsingEquation4.18thiscanberewritteninfrequencyasg!1;!2)]TJ/F18 11.955 Tf 12.374 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!2g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2.Taylorexpandingtheex-ponentialandkeepingtermsorder~gives2gI!1;!2+~!2gR!1;!2.UsinggI!1;!2=tanh)]TJ/F18 11.955 Tf 9.299 0 Td[(~!1=4+!2=2gR!1;!2givestheresultinEquations4.44and4.45.Thisap-proachdoes,however,immediatelyinvoketheclassicallimit,andthegeneralityofEquation4.27islost.36

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Nextthebehaviorofourtheoryofthefthorderresponsefunction,Equation4.45isassessed.Itiseasytoshowthatatthetimeorigin,t1=t2=0,R;0=0.Thisimpliesthat)]TJ/F18 11.955 Tf 5.48 -9.684 Td[(@2=@t21ht1t1+t2ij0;0.49=2)]TJ/F18 11.955 Tf 5.48 -9.683 Td[(@2=@t1@t2ht1t1+t2ij0;0;andthiscanbeeasilyveried;[1]theverticallinedenotesevaluatingafunctionatthespeciedtimepoints.PerhapsamoredemandingtestisthatRt1;t2=0=0.Thisimpliesthat)]TJ/F18 11.955 Tf 5.479 -9.684 Td[(@2=@t21ht1t1+t2ijt1;0.50=2)]TJ/F18 11.955 Tf 5.479 -9.684 Td[(@2=@t1@t2ht1t1+t2ijt1;0;whichisalsotrue.Itisencouragingthatwendthesecorrectlimitingbehaviors.4.5FODIDvs.MBPPolarizationforcesarenotexplicitlyincludedintheMDcalculationsusedtogeneratecong-urationsforcalculationsoftheTCF's.However,incalculatingthesystempolarizabilitytobeusedintheTCF,fullmanybodypolarizationeectsareincludedbysolvingthedipole-induceddipoleequationsforthePAPAmodelandtheseresultswillbecomparedwithanapproximaterstorderevaluationdescribedinChapter3.[7]Theexactclassicalcalcula-tionofRt1;t2wasperformedonlyintheFODIDapproximationtomakethecalculationfeasible.[8]37

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Assessingthepolarizationdependenceofthesignalisgreatlyfacilitatedbymakinguseofrotationalinvariantsofthesystempolarizability.[9]ThetaskoffullyunderstandingtheimplicationsofusinganFODIDapproximationismadeeasierbyconsideringtheresponsefunctiondirectlyintermsoftherotationalinvariantsdiscussedinChapter3.[8,9]Becausecorrelateductuationsofthetotalsystempolarizabilityarebeingconsideredforanatomicliquidthesingleatompolarizabilityisstaticonlytheinducedpolarizabilitycontributes.IntheFODIDapproximation,inwhichtheonlytermskeptarerstorderinTrijatracelesstensorthisleadstoTr=0.Toseehowthismanifestsitselfintheoverallresponsefunctionwehavetoconsiderhowtheinvariantscombinetogivetheorientationalaverageofaproductofthreetensors.[9] axxbxxcxx=1 105TraTrbTrc+8 105TPa;b;c+2 105[TraPPb;c+TrbPPa;c+TrcPPa;b].51Equation.51presentsthefullypolarizedcaseandthelineoverthetensorproductrepre-sentsisotropicaveraging.TheimportantthingtonotehereisthatPPtermsalwaysappearcoupledtoaTrterm.TheformofEq..51,linearcombinationsofinvariantproducts,holdsnotjustforthefullypolarizedcaseshownherebutforallofthepossiblepolariza-tionconditions.ThevariouscontributionsareweighteddierentlyintheothercasesandthecoecientsareshowninChapter3.[9]ConsideringtheFODIDapproximation,theTrcontributionsarezeroandthiseliminatesanycontributionfromthePPandleavesonlytheTPcontributionmakingallpolarizationconditionsequivalenttowithinaconstant.IntheMBPcasetheTrcontributionissmallbutthePPissucientlylargethattheproduct38

PAGE 48

contributessignicantlytotheoverallsignal.Figure.1presentsasliceoftheinvariantsforliquidxenonalongt1=0fortheMBPmodel.Therelativemagnitudesofthefourdistinctcontributionsalongt1=0whereTrPPt1;t2=Trt1PP;t2areclear.Specically,theplotshowstheTPsolidline,TP;t1;t2;.52theTrPPproductcombinationslinewithcirclesanddashedrespectively,TrPPt1;t2.53Trt2PP;t1.54andTrdottedTrTrt1Trt2:.55Alsoshown,forcomparison,inFig.4.1istheTPusingtheFODIDmodellinewithsquares.EachoftheTrPPtermshaveamplitudesthataresizableandsignicantcomparedtotheTPcontributions.ItisevidentfromFig.4.1thatneglectingthesignicantPPcontributionsintheFODIDcaseseverelyaltersthelineshape.Inaddition,theFODIDapproximationpredictsthatallofthepolarizationconditionswillyieldidenticallineshapeswithvaryingamplitudessinceonlytheTPappearswithdierentweightings.Indeed,itwillbeseeninChapterthatthetwomodelsyieldvastlydierentspectraandthatwithintheMBPmodelthevariouspolarizationsprovidedistinct39

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Figure4.1:RotationalInvariants.TimeslicesforshorttimesoftherotationalinvariantsforRxxxxxxt1;t2fromEq..51.Shownisthetripleproduct,TP;t1;t2,forMBPsolidandFODIDlinewithsquares,thetracepairproductcombinationsforMBP,TrPPt1;t2linewithcircles,Trt2PP;t1dashed,andTrTrt1Trt2dotted.Theslicewastakenalongt1=0whereTrPPt1;t2=Trt1PP;t2.information.ItisworthnotingthatboththeMBPandFODIDmodelspredictsimilarlineshapeswithaslightvariationinmagnitudesfortheTPcontributions.Thisresultimpliesthatthefthorderresponsefunctioncontainsseveralterms,whencalculatedusingaMBPmodel,thatdonotcontributetoaFODIDcalculationallthetermsthatcoupletotheTr,thatiszerowithinaFODIDapproximation.Thisresultisinstarkcontrasttothethirdorderpolarizedspectrum,thatisdominated,fornoblegasliquids,bypairproductcontributionsandthereforedoesnotchangesignicantlywithin40

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aFODIDcalculationalthoughasmallTrcontributionisneglectedwithintheFODIDapproximation.[8]Obtainingathirdorderisotropicspectrumdoes,however,dependonusingaMBPmodelbecauseitvanisheswithinaFODIDtreatment.ItisthereforenotsurprisingthatthefullypolarizedfthordersignalchangessignicantlywhenaFODIDmodelisused;suchatreatmentneglectstermsthatclearlyareimportant.41

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Chapter5Two-dimensionalInfraredSpectroscopyRevealingthethree-dimensionalmolecularstructureoftransientspeciesincondensedphasesisexperimentallychallenging,butnecessarytounderstandthenatureoftime-dependentpro-cesses,suchasproteinfolding.ThecongestedspectraproducedundertheseconditionsoftencannotberesolvedusinglinearinfraredIRspectroscopyorothertraditionaltechniques.X-raydiractionandmultidimensionalnuclearmagneticresonanceNMReectivelyre-vealtime-averagedthree-dimensionalstructures,butfailtoaccuratelydescribeshort-livedspecies,suchaspeptidesinsolution.Recently,2DIRspectroscopyhasbeenthesubjectofex-tensivetheoreticalandexperimentalstudy,hasshownpromiseforprovidingnewinformationaboutthesetimeevolvingstructures.[24{28]WhilemultidimensionalNMRonlycapturesprocessesonamillisecondtimescale,2DIRcanapproachatimescaleontheorderofpicosec-ondsorevenfemtoseconds.[29{33]Recently,2DIRtechniqueshavebeenusedtoinvestigatethecoupledcarbonylstretchesofRhCO2C5H7O2,[30,34{37]thenuclearpotentialenergysurfaceofcoupledmolecularvibrations,[38]vibrationalrelaxation,[39,40]interactionsbe-tweensolventandsolute,[41{43]conformationaluctuationsinpeptides,[29,32,33,44],the42

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three-dimensionalstructureofpeptidesandsmallproteins,[31,45{51],andthecouplingofcytidineandguanosineinDNA.[52]Using2DIR,couplingsandprojectionanglesbetweentwocoupledanharmonicvibrationscanalsobeinferred.[38]AmethodsimilartotheoneusedinChapter4forthefthorderresponsefunctionwillproveusefulheretosimplifytheresponsefunctionprobedinthe2DIRexperimentandobtainanapproximateTCFtheorythatisamenabletoevaluationusingclassicalMDandTCFtechniques.5.12DIRExperiment2DIRusesasequenceoffourtimeorderedfemtosecondlengthIRpulsestocreatevibra-tionalcoherences,couplethem,anddetectthenalpolarizationstateofthesystembeinginvestigated.Intheexperiment,athird-ordernonlinearpolarizationisthusgeneratedandsubsequentlydetected.ThispolarizationPiattimetissimplytheconvolutionofthethirdorderresponsefunctionRwiththethreeinputelectricelds:[5,53,54]Pi1;2;t=Z10Z10Z10Rijk`t1;t2;t3E3jt)]TJ/F18 11.955 Tf 11.955 0 Td[(t3E2kt+2)]TJ/F18 11.955 Tf 11.955 0 Td[(t3)]TJ/F18 11.955 Tf 11.955 0 Td[(t2E1`t+1+2)]TJ/F18 11.955 Tf 11.955 0 Td[(t3)]TJ/F18 11.955 Tf 11.955 0 Td[(t2)]TJ/F18 11.955 Tf 11.956 0 Td[(t1dt1dt2dt3.1TheErepresentthepulsedlaserelds,andtisthetimeelapsedafterthenallaserpulse.Thevariable1isthetimedelaybetweentherstandsecondpulses,and2isthedelaybetweenthesecondandthirdpulses.Inthevibrationalecho2DIRexperiment2=0.[5,36]Alsonotethatintheidealizedlimitofdeltafunctionpulsesthepolarizationbecomesequivalenttotheresponsefunctioninthetimedomain,whichbecomestheexperimentalob-43

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servable.Becausethewaitingtimesarebydenitionpositive,inthefrequencydomaintheFourier-Laplacetransformoftheresponsefunctionisoftenreported,givingarealandimag-inarypartthatcontainsboththeFouriertransformoftheresponsefunctionandprincipalpartintegralsoverthefrequencydomainresponsefunction.WiththetheoreticalexpressionstobedevelopedwehavenodicultytakingthefullFouriertransform,whichcontainsalltime-domaininformation.Thenumericalresultsonwaterpresentedbelowwillfocusonthecasewhere2=t2=0althoughamoregeneraltheoryisdeveloped.Thetnarethetimeintervalsbetweentheeld-matterinteractions,andequalthenifthepulselengthsaresubstantiallyshorterthanthetimescaleofthedynamics.Thethird-orderresponsefunctionRisafourthranktensoranddependsonthepolarizationcomponentsi;j;k;`fXYZgoftheincidentlaserelds.5.22DIRResponseFunctionForaresonant2DIRexperiment,Risgivenby:[36]Rijk`t1;t2;t3=)]TJ/F18 11.955 Tf 9.299 0 Td[(i ~3h[[[it3+t2+t1;jt2+t1];kt1];`]i.2InEquation5.2,therepresentthedipolemomentoperatorsinthesubscriptedlabo-ratoryCartesiandirectionandthetimedependentdipoleistheHeisenbergrepresentationoftheoperator,t=eiHt=~e)]TJ/F19 7.97 Tf 6.586 0 Td[(iHt=~.Thesquareandtheanglebracketsrepresentcommu-tatorsandquantummechanicalaveragesrespectivelyinastandardnotation.[1]EvaluationoftheRexpressioniscomputationallydemandingbecauseitrequirestheevaluationof44

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severalquantummechanicalTCF'sthatresultfromexpandingthecommutators.Evenintheclassicallimit,thecommutatorsbecomePoissonbracketsofthedynamicvariablesatvarioustimesthatareprohibitivelydiculttoevaluatecomputationally.[8,14]TheoreticalmethodsthatapproximatelyrelatequantummechanicalandclassicalTCF'shavesuccessfullybeenappliedtoonedimensionalspectroscopiesaspresentedinChapter2thathaveanexactclassicallimitintermsoftimederivativesofaclassicalTCFandtodescribinghighfrequencyspectroscopyusingquantumcorrectionschemestoapproximatetherelevantquantummechanicalTCF.[4,5,22,55{61]TheRexpressioncontainsatraceofnestedcommutators,whichmustbeimaginaryandhavealeadingcontributionoforder~3tocanceltheprefactorofi=~3intheclassicallimit;thisisausefulwaytoexaminetheexpressionbecauseatheoryintermsofasingleTCFmusthaveaclassicallimitoforder~0.ItshouldbenotedthatweareinterestedinaresultthatisappropriateforhighfrequenciesandexaminingthelowfrequencyexpansionisusedprimarilyasanalytictoolinseekinganexpressionintermsofasingleTCF,althoughlowfrequency2DIRspectroscopyisalsopossible.Expansionofthecommutatorsrevealsfourdistinctquantummechanicaltimecorrelationfunctions.InSection5.3,itwillbedemonstratedthatRcanbewrittenapproximatelyintermsofonlyoneoftheseTCF's.Itwillalsobedemonstratedthatintheclassicallimit,~!!0,acontributionoforder~3,isobtainedfromthelinearcombinationofTCF'sthatcancelstheprefactor.Withoutresortingtoapproximations,one~prefactoriseliminatedexactly,andRcanberewrittenintermsofonlytwoTCF's.Thisisaccomplishedusingfrequencydomaindetailedbalancerelationships45

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betweentwopairsofquantummechanicalTCF'sappearinginR.Theremainingtwo~arenoteliminatedexactly,butcanbebyusingexactresultsforaharmonicsystemwithalinearlyvaryingdipoletorelatetheTCF's.TheresultingexpressionisexactfortheharmonicsystemandcanbeappliedtofullyanharmonicdynamicsderivedusingclassicalMD.Theuseoftheresultsfortheharmonicreferencesystemonlyservestoweightthedierentdynamicaleventsthatcontributetotheresponsefunctionastheywouldbeforaharmonicsystem.Consequently,ourtheoryhastheessentialfeaturethataharmonicoscillatorwithalinearlyvaryingdipoleno"elec-trical"anharmonicitygivesno2DIRsignal;theapproximationservestolterout"thelessinterestingharmonicdynamicsandenhancestheanharmoniccouplingsintheresultingsignal[36].Anharmonicdynamicsandcouplingsareamainfocusof2DIRspectroscopyandtheyarecriticalinextractingphysicalinsight.[36,51]ItispossibletouseadierentreferencesystemtorelatetheTCF'sandacubicandquarticanharmonicoscillatorsystemtreatedpertur-bativelymaybeanothergood,albeitmorecomplicated,choiceforthepresentmethodsandthisavenueisbeingpursued.[62]5.3ConstructingaTCFTheoryThecommutatorsinEquation5.2canbeexpandedtogivefourTCF'sandtheircomplexconjugates.Intermsofthesetimecorrelationfunctions,Rcannowbewritten46

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Rijk`t1;t2;t3=)]TJ/F18 11.955 Tf 9.299 0 Td[(i ~3[At1;t2;t3)]TJ/F18 11.955 Tf 11.955 0 Td[(Bt1;t2;t3)]TJ/F18 11.955 Tf 11.955 0 Td[(Ct1;t2;t3+Dt1;t2;t3)]TJ/F18 11.955 Tf 9.299 0 Td[(Dt1;t2;t3+Ct1;t2;t3+Bt1;t2;t3)]TJ/F18 11.955 Tf 11.955 0 Td[(At1;t2;t3].3InEquation5.3thestarsuperscriptrepresentsthecomplexconjugateofthecomplextimedomainTCFandtheexpressioninsquarebracketscanbeseentobethesumanddierenceoftheimaginarypartsofthefourTCF's,namely2AI)]TJ/F18 11.955 Tf 12.622 0 Td[(BI)]TJ/F18 11.955 Tf 12.622 0 Td[(CI+DI.ThesubscriptsIorRdenotetheimaginaryandrealpartsofatimedomainTCForofitsFouriertransformrespectively.AtthispointevaluationoftheRexpressionrequiresthecalculationoffourdierentquantummechanicaltimecorrelationfunctions,aformidabletask.Tominimizecomputationaleort,itisdesirabletorewriteRintermsofonlyoneTCFthatcanbeapproximatedasitsclassicalcounterpart.ThefourTCF'sthatappearinEquation5.3are:At1;t2;t3=hit3+t2+t1jt2+t1kt1`iBt1;t2;t3=hjt2+t1it3+t2+t1kt1`iCt1;t2;t3=h`jt2+t1it3+t2+t1kt1iDt1;t2;t3=h`it3+t2+t1jt2+t1kt1i.4Becausethedipolemomentoperatorscommuteclassically,thefourTCF'shavethesameclassicallimit.ToproceedandtoverifytheclassicallimitofaTCF,itishelpfultorewriteitintheenergyrepresentation,forexample:47

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At1;t2;t3=hit3+t2+t1jt2+t1kt1`i=1 QXahaje)]TJ/F19 7.97 Tf 6.586 0 Td[(HeiHt3+t2+t1=~ie)]TJ/F19 7.97 Tf 6.587 0 Td[(iHt3+t2+t1=~eiHt2+t1=~je)]TJ/F19 7.97 Tf 6.587 0 Td[(iHt2+t1=~eiHt1=~ke)]TJ/F19 7.97 Tf 6.586 0 Td[(iHt1=~`jai.5InEquation5.5thedipolemomentoperatorsaremultipliedby=e)]TJ/F19 7.97 Tf 6.586 0 Td[(H=Qandthetraceistaken.Qisthepartitionfunctionandat=eiHt=~ae)]TJ/F19 7.97 Tf 6.586 0 Td[(iHt=~.Next,itispossibletoinsertfourcompletesetsofenergyeigenstates,PajaihajwithHjai=Eajai,operate,andsimplify[63]togetexpressionsoftheform:At1;t2;t3=1 QXabcde)]TJ/F19 7.97 Tf 6.586 0 Td[(Eaiadjdckcb`baeiEabt1=~eiEact2=~eiEadt3=~Eab=Ea)]TJ/F18 11.955 Tf 11.955 0 Td[(Ebab=hajjbiNext,considerthecomplexconjugateofAt1;t2;t3;notethatthedipolemomentop-eratormatrixelementsarehermitian,i.e.ij=ji.Thematrixelementscanbechosenandaretakenasreal,itisthenclearthatAt1;t2;t3=A)]TJ/F18 11.955 Tf 9.299 0 Td[(t1;)]TJ/F18 11.955 Tf 9.298 0 Td[(t2;)]TJ/F18 11.955 Tf 9.298 0 Td[(t3.At1;t2;t3=h`kt1jt2+t1it3+t2+t1iAt1;t2;t3=1 QXabcde)]TJ/F19 7.97 Tf 6.587 0 Td[(Eaiadjdckcb`baeiEbat1=~eiEcat2=~eiEdat3=~.6ThetripleFouriertransformofAt1;t2;t3givesthefrequencydomainfunctionA!1;!2;!3.SimilarmanipulationsgivethefrequencydomainfunctionsforB,C,andD.48

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Indexswitchingexchangingsummedoverdummyindicies,whichisequivalenttotakingcyclicpermutationsofthetrace,havebeusedtomaximizethesimilaritybetweenthefourexpressions.[28,63]A!1;!2;!3=1 QXabcde)]TJ/F19 7.97 Tf 6.587 0 Td[(Eaiadjdckcb`ba!1)]TJ/F18 11.955 Tf 11.956 0 Td[(Eab=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F18 11.955 Tf 11.955 0 Td[(Ead=~B!1;!2;!3=1 QXabcde)]TJ/F19 7.97 Tf 6.587 0 Td[(Eaidcjadkcb`ba!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F18 11.955 Tf 11.955 0 Td[(Edc=~C!1;!2;!3=1 QXabcde)]TJ/F19 7.97 Tf 6.586 0 Td[(Ebidcjadkcb`ba!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F18 11.955 Tf 11.955 0 Td[(Edc=~D!1;!2;!3=1 QXabcde)]TJ/F19 7.97 Tf 6.586 0 Td[(Ebiadjdckcb`ba.7!1)]TJ/F18 11.955 Tf 11.956 0 Td[(Eab=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F18 11.955 Tf 11.955 0 Td[(Ead=~UsingthefactthattheFouriertransformsoftheTCF'sarereal,[63]itissimpletondfrequencydomainexpressionsforthecomplexconjugatesoftheTCF's.TakingthetripleFouriertransformofthecomplexconjugatesgivesthesamefrequencydomainfunctionsevaluatedatnegativefrequency,i.e.FT[ft1;t2;t3]=f)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3.A)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3=1 QXabcde)]TJ/F19 7.97 Tf 6.586 0 Td[(Eaidajcdkbc`ab!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F18 11.955 Tf 11.956 0 Td[(Eda=~49

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B)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3=1 QXabcde)]TJ/F19 7.97 Tf 6.586 0 Td[(Eaicdjdakbc`ab!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F18 11.955 Tf 11.955 0 Td[(Ecd=~C)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3=1 QXabcde)]TJ/F19 7.97 Tf 6.587 0 Td[(Ebicdjdakbc`ab!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F18 11.955 Tf 11.955 0 Td[(Ecd=~D)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3=1 QXabcde)]TJ/F19 7.97 Tf 6.587 0 Td[(Ebidajcdkbc`ab.8!1)]TJ/F18 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F18 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F18 11.955 Tf 11.956 0 Td[(Eda=~Atthispoint,ifthefrequencydomainexpressionsonlydieredbytheBoltzmannfactorweightingthem,itwouldbepossibletoderivedetailedbalancerelationshipsbetweenallofthem.ThisisonlythecaseforthepairsofTCF'sA,DandB,C.UsingtheBoltzmannfactorstorelatethemembersofeachpairandthenenforcingthedeltafunctions,fourdetailedbalancerelationshipsarise.SimplerelationshipsbetweenmembersofdierentpairsandbetweenpositiveandnegativefrequencyTCF'sdonotgenerallyexist.[63]ThisistrueeventhoughthetimedomainTCF's,e.g.AandA,havethesamerealpartandtheirimaginarypartsarethesamefunctionofoppositesign;boththerealandimaginaryportionofthetimedomainTCFcontributetothefrequencydomainTCF's.D!1;!2;!3=e~!1A!1;!2;!3D)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3=e)]TJ/F19 7.97 Tf 6.586 0 Td[(~!1A)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3C!1;!2;!3=e~!1B!1;!2;!3C)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3=e)]TJ/F19 7.97 Tf 6.586 0 Td[(~!1B)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3.950

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IftheclassicallimitofeachTCFistakenase)]TJ/F19 7.97 Tf 6.587 0 Td[(E!1,anunexpectedproblemarises.Themembersofeachpaircanclearlybeseentohavethesameclassicallimit,i.e.A,DandB,Cbuttheequivalencebetweenthepairsisnotobvious.Thispointwillbere-examinedlater.AlongwiththerelationshipsshowninEquation5.9,takingtheratiosbetweensumsanddierencesoftimecorrelationfunctionswillprovidefourrelationshipsusefulforsimplifyingtheRexpression.[63]D!1;!2;!3)]TJ/F18 11.955 Tf 11.956 0 Td[(A!1;!2;!3=tanh~!1=2[D!1;!2;!3+A!1;!2;!3]D)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3)]TJ/F18 11.955 Tf 11.955 0 Td[(A)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1=2[D)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3+A)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3]C!1;!2;!3)]TJ/F18 11.955 Tf 11.956 0 Td[(B!1;!2;!3=tanh~!1=2[C!1;!2;!3+B!1;!2;!3]C)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3)]TJ/F18 11.955 Tf 11.955 0 Td[(B)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!35.10=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1=2[C)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3+B)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3]FouriertransformingRintothefrequencydomainanddirectlysubstitutingthere-lationshipsfromEquations5.9and5.10intoit,thethirdorderresponsefunctioncanbewrittenexactlyintermsofonlytwo-timecorrelationfunctions,AandB.R!1;!2;!3=i ~3tanh.11~!1 2[+e~!1B+)]TJ/F18 11.955 Tf 11.955 0 Td[(A+++e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!1B)]TJ/F21 11.955 Tf 9.741 -4.936 Td[()]TJ/F18 11.955 Tf 11.955 0 Td[(A)]TJ/F15 11.955 Tf 7.084 -4.936 Td[(]51

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InEquation5.11thenotationf+=f!1;!2;!3andf)]TJ/F15 11.955 Tf 11.468 -4.339 Td[(=f)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3isintro-ducedforclarityandfrepresentsanyofthefunctionsA,B,C,D.Thisequationisexactforallfrequenciesandimpliesthatthe2DIRsignaliszeroalongthe!1=0frequencyaxis.AtthispointitisinsightfultoexaminetheclassicallimitofR.Theexponentialsareexpandedandtermsoforder~areretained.Theresultisthatone~prefactoriseliminatedexactlyfromRintheclassicallimit.R!1;!2;!3=i ~3~!1 2[+~!1B+)]TJ/F18 11.955 Tf 11.955 0 Td[(A++)]TJ/F18 11.955 Tf 11.955 0 Td[(~!1B)]TJ/F21 11.955 Tf 9.741 -4.936 Td[()]TJ/F18 11.955 Tf 11.955 0 Td[(A)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(].12Itiseasilyproventhatf++f)]TJ/F15 11.955 Tf 10.406 -4.339 Td[(=2fR,wherefR!1;!2;!3denotestheFouriertransformoftherealpartofft1;t2;t3,andf+)]TJ/F18 11.955 Tf 9.662 0 Td[(f)]TJ/F15 11.955 Tf 10.406 -4.338 Td[(=2fI,wherefI!1;!2;!3istheFouriertransformoftheimaginarypartofft1;t2;t3,bothofwhicharethemselvesrealfunctions.TheserelationshipsallowRtobewrittenR!1;!2;!3=i!1 ~2[2BR)]TJ/F18 11.955 Tf 11.955 0 Td[(AR+~!1BI)]TJ/F18 11.955 Tf 11.956 0 Td[(AI].13Equation5.13makesitclearthattheFouriertransformoftheimaginarypartofAandBmusthavealeadingdierenceorder~andtherealpartsorder~2forameaningfulclassicallimittoexist.Atthispoint,backFouriertransformingtothetimedomaingivesRwrittenintermsoftimederivatives.Ingeneralft1;t2;t3=Zei!1t1ei!2t2ei!3t3f!1;!2;!3d!1d!2d!3d dt1ft1;t2;t3=Zi!1ei!1t1ei!2t2ei!3t3f!1;!2;!3d!1d!2d!3d2 dt21ft1;t2;t3=)]TJ/F18 11.955 Tf 9.298 0 Td[(iZ!21ei!1t1ei!2t2ei!3t3f!1;!2;!3d!1d!2d!352

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Then,inthetimedomain,Rcanbewrittenintermsoft1derivativesasshown.Thisexpressionisexactintheclassicallimit.Rt1;t2;t3=2 ~2d dt1BRt1;t2;t3)]TJ/F18 11.955 Tf 17.631 8.087 Td[(d dt1ARt1;t2;t3+i2 ~d2 dt21AIt1;t2;t3)]TJ/F18 11.955 Tf 15.265 8.088 Td[(d2 dt21BIt1;t2;t3.14InordertofurthersimplifyR,itisnecessaryatthispointtomakeapproximations.ByevaluatingtheTCF'sinvokingaharmonicapproximationitispossibletondarelationshipbetweenAandB.AharmonicpotentialV=m2q2=2isassumedwithpartitionfunctionQ=e)]TJ/F20 5.978 Tf 5.756 0 Td[(~=2 1)]TJ/F19 7.97 Tf 6.586 0 Td[(e)]TJ/F20 5.978 Tf 5.756 0 Td[(~.Additionally,thedipolemomentfunctionsappearinginthetimecorrelationfunctionsareexpandedouttorstorderintheharmoniccoordinate,givingtherequiredmatrixelements:[63]ij=0ij+0qijqij=~ 2m1=2[i;j+1j+11=2+i;j)]TJ/F16 7.97 Tf 6.586 0 Td[(1j1=2].15InEquation5.15,theprimesrepresentderivativeswithrespecttotheharmoniccoordi-nateq.InbothAandB,expansionofthefour'sgivesthesumofsixteenterms,eachdistinctinthepowersofcoordinatesusedtoevaluatethedipolemomentmatrixelements.Manyofthesetermscanbeneglectedbecausetheirdeltafunctionsforcethemtoequalzero.Additionally,fromexaminingEquation5.14itisclearthattocontributetoR,aterminthetimedomainmusthaveanonzeroderivativewithrespecttot1althoughEquation5.14onlyconsiderstheclassicallimittheneglectedhigherordertermswouldalsoinvolvehigherordert1derivatives.Undertheseconditions,onlyfournonzerotermsremainforeach53

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TCF:threetermswithtwo0andtwo0denoted0110,1100and1010todenotetheorderofcoordinatethatisusedinevaluatingthedipolematrixelementsintheorderpresentedaboveinEquations5.7and5.8,thenonetermwithfour0denoted1111.The0110,1100,and1010termsareidenticalinTCF'sAandB.BecauseRisnowexpressedintermsofdierencesbetweenTCF'sAandB,thesethreetermsvanish.The1111termistheonlyonelefttoconsider.InAt1;t2;t3the1111termisthesumofsixdistinctparts,labeledathroughf.ThesixtermsfromA1111are:A1111at1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~22e)]TJ/F16 7.97 Tf 6.587 0 Td[(2~eit1+2t2+t3A1111bt1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~22e)]TJ/F19 7.97 Tf 6.587 0 Td[(it1+2t2+t3A1111ct1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2e)]TJ/F19 7.97 Tf 6.587 0 Td[(~+e)]TJ/F16 7.97 Tf 6.586 0 Td[(2~eit1+t3A1111dt1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~e)]TJ/F19 7.97 Tf 6.586 0 Td[(it1+t3A1111et1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~22e)]TJ/F19 7.97 Tf 6.587 0 Td[(~eit3)]TJ/F19 7.97 Tf 6.587 0 Td[(t1A1111ft1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.956 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~22e)]TJ/F19 7.97 Tf 6.586 0 Td[(~e)]TJ/F19 7.97 Tf 6.586 0 Td[(it3)]TJ/F19 7.97 Tf 6.587 0 Td[(t1.16Similarly,forTCFB,the1111termcanbewrittenasthesumoftermsathroughf.TermsaandbareidenticaltothosefoundforTCFA,whiletermscthroughfhavedierentprefactors.ThefourtermsuniqueB1111termsare:B1111ct1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.586 0 Td[(~22e)]TJ/F19 7.97 Tf 6.587 0 Td[(~eit1+t354

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B1111dt1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.955 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~22e)]TJ/F19 7.97 Tf 6.586 0 Td[(~e)]TJ/F19 7.97 Tf 6.587 0 Td[(it1+t3B1111et1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.956 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2e)]TJ/F19 7.97 Tf 6.587 0 Td[(~+1eit3)]TJ/F19 7.97 Tf 6.587 0 Td[(t1B1111ft1;t2;t3=0i0j0k0`~ 2m21 )]TJ/F18 11.955 Tf 11.956 0 Td[(e)]TJ/F19 7.97 Tf 6.587 0 Td[(~2e)]TJ/F16 7.97 Tf 6.587 0 Td[(2~+e)]TJ/F19 7.97 Tf 6.586 0 Td[(~e)]TJ/F19 7.97 Tf 6.587 0 Td[(it3)]TJ/F19 7.97 Tf 6.587 0 Td[(t1.17Consideringonlythe1111terms,itisnowpossibletodeneanexactrelationshipbetweenthetwoTCF'sAandBanduseittosimplifyR.Fortermsathroughf,therelationshipbetweenA1111andB1111iseasilyfoundbycomparingtheircofactors.TheresultisasetofsixrelationshipsbetweenthetwoTCF's.Theserelationshipsholdinboththetimeandfrequencydomain,sincetheyonlydependontheterms'coecients.A1111a=B1111aA1111b=B1111bA1111c=1 2+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~B1111cA1111d=1 2+e~B1111cA1111e=2 1+e~B1111eA1111f=2 1+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~B1111f.18Carefullyconsideringthesixrelationshipsshownabove,itispossibletodeneafunctiong!1;!2;!3anditsnegativefrequencycounterpartg)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3thatrelatesAandBinpositiveandnegativefrequencyintermsofthedynamicalvariables!1,!2,and!3;thisfunctionisnotnecessarilyuniquebutisthesimplestfunctionalformthatwasobtained.55

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ThefrequencydomainTCFdeltafunctionsallowtheharmonicfrequencytobewrittenintermsof!1,!2,and!3intheusualfashion.A!1;!2;!3=g!1;!2;!3B!1;!2;!3g!1;!2;!3=1+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!1+!3)]TJ/F19 7.97 Tf 6.586 0 Td[(!2=2 1+e)]TJ/F19 7.97 Tf 6.586 0 Td[(~!1)]TJ/F19 7.97 Tf 6.587 0 Td[(!3=2A)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3=g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3B)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!35.19g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!3=1+e)]TJ/F19 7.97 Tf 6.587 0 Td[(~!2)]TJ/F19 7.97 Tf 6.586 0 Td[(!1)]TJ/F19 7.97 Tf 6.586 0 Td[(!3=2 1+e)]TJ/F19 7.97 Tf 6.586 0 Td[(~!3)]TJ/F19 7.97 Tf 6.587 0 Td[(!1=2Todemonstratethisresult,theexplicitexpressionsforAandBEquations5.16and5.17canbesubstitutedintoEquation5.18andandthenenforcingthedeltafunctionsimmediatelyleadstotheresultsinEquation5.19{thistypeofmanipulationisdemonstratedindetailinapreviouspaper.[63]Atthispoint,itisinterestingtorevisitthequestionoftheclassicallimitsofTCF'sAandB.Above,itwasnotobviousthatthesetwoTCF'ssharedthesameclassicallimit.Itisworthnotingthatas~!0,thesixrelationshipsin5.18allapproachA=B.Additionally,g!1;!2;!3!1andg)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3!1.ItisclearthatthetwoTCF'shaveidenticalclassicallimitsfortheharmonicsystemwithalineardipole.Iftheclassicallimitistakenbyexpandingtheexponentialsouttorstorderin~,g!1;!2;!3andg)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3canbesimpliedasshownbelow:g!1;!2;!3=1+~!2=4)]TJ/F18 11.955 Tf 11.955 0 Td[(~!3=2g)]TJ/F18 11.955 Tf 9.298 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3=1)]TJ/F18 11.955 Tf 11.955 0 Td[(~!2=4+~!3=2.2056

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Next,g!1;!2;!3andg)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.298 0 Td[(!2;)]TJ/F18 11.955 Tf 9.298 0 Td[(!3canbeincorporatedintoEquation5.12tosim-plifytheRexpressionevenfurther.TCFAhasbeencompletelyeliminatedfromR,leavingbehindonlyTCFB.IntheclassicallimitR!1;!2;!3=i ~3~!1 2~ 4!3)]TJ/F18 11.955 Tf 11.955 0 Td[(!2.21[2~!1BR!1;!2;!3+4BI!1;!2;!3]Twoofthethree~prefactorshavebeeneliminatedcompletelyandRiswrittenintermsofonlyBR!1;!2;!3andBI!1;!2;!3,theFouriertransformsoftherealandimaginarypartsofBt1;t2;t3;inthislimitBRbecomestheclassicalTCFthatcanbecomputedusingMD.FindingarelationshipbetweenBRandBIwillallowtheremovalofthelast~prefactorintheclassicallimitandalsogiveanexpressioninwhichthefrequencyfactorsarenotTaylorexpandedentirelyintermsofBRthatisvalidforallfrequencies{theprimarygoalofthispaper.Consideringaone-timecorrelationfunctionft,fR!andfI!haveasimplefunc-tionalrelationship,fI!=tanh~!=2fR!.[63,64]Ifasimilarrelationshipexistsbe-tweenBI!1;!2;!3andBR!1;!2;!3,RcanthenbewrittenintermsofonlyBR.How-ever,sucharelationshipisnotimmediatelyobvious.Toattempttondone,eachnonzerotermofBt1;t2;t3terms0011,0101,1001andthesix1111terms,wasseparatedintoitsrealandimaginarycomponentsandthetwopartswereFouriertransformedseparately.TheratiobetweenBI!1;!2;!3andBR!1;!2;!3wasthendeterminedforeachterm.Com-paringthenineratios,anexactrelationshipforthetermsconsidered,similarinformtothe57

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aboverelationshipfortheonetimecorrelationfunction,becomesapparent.BI!1;!2;!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1+2!2)]TJ/F18 11.955 Tf 11.955 0 Td[(!3 4BR!1;!2;!3.22Equation5.22isexactfortheharmonicsystem.itshouldbenotedweareonlyconsideringthe1111termsfortheharmonicsysteminconstructingthisrelationship{thelowestordertermsthatcontributetoRwithintheharmonicapproximationcunt.WiththeabilitytorelateBR!1;!2;!3andBI!1;!2;!3,itispossibletomakeonenalsimplication,whichremovesthenalremaining~prefactorintheclassicalRexpression.Intheclassicallimit,thenalexpressionforRisgivenbyR!1;!2;!3=3 8!21!3)]TJ/F18 11.955 Tf 11.955 0 Td[(!21!2)]TJ/F15 11.955 Tf 11.955 0 Td[(5!1!2!3+2!1!22+2!1!23BR!1;!2;!3Rt1;t2;t3=3 8d3BR dt21dt2)]TJ/F15 11.955 Tf 11.955 0 Td[(2d3BR dt1dt22)]TJ/F15 11.955 Tf 11.956 0 Td[(2d3BR dt21dt3)]TJ/F15 11.955 Tf 11.955 0 Td[(2d3BR dt1dt23+5d3BR dt1dt2dt3.23Thederivativesappearinginthetimedomainexpressionsresultfromtakingtimederiva-tivesoftheFouriertransformofB!1;!2;!3.Toexaminehighfrequencyphenomena,takingtheclassicallimitoftheRisnotdesirable.Inthiscase,inthefrequencydomainexpressiontakestheform:R!1;!2;!3=i ~3tanh~!1=2+e~!1)]TJ/F18 11.955 Tf 11.955 0 Td[(g!1;!2;!31)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh[~!1+2!2)]TJ/F18 11.955 Tf 11.956 0 Td[(!3=4]+1+e)]TJ/F19 7.97 Tf 6.586 0 Td[(~!1)]TJ/F18 11.955 Tf 11.955 0 Td[(g)]TJ/F18 11.955 Tf 9.299 0 Td[(!1;)]TJ/F18 11.955 Tf 9.299 0 Td[(!2;)]TJ/F18 11.955 Tf 9.299 0 Td[(!31)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh[~!1+2!2)]TJ/F18 11.955 Tf 11.955 0 Td[(!3=4]58

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BR!1;!2;!3.24Equation5.24isnowinaformthatcanbeevaluatedusingclassicalMDandTCFtech-niques.BRt1;t2;t3isreplacedbyitsclassicalcounterpartandFouriertransformedintothefrequencydomain.Thefrequencyfactorsarethentakenintoaccountandtheresultingfunc-tioncanbebacktransformedtothetimedomaintoproducethedesiredresponsefunctionthatcanthenbeusedinEquation5.1;Equation5.24isthecentralresultofthischapterandwillbeusedtoexaminethe2DIRspectrumofambientwaterinSection7.Asnotedintheintroduction,Equation5.24wouldgivezerosignalforaharmonicsystemwithalineardipole.ThisresultcanbeveriedbysubstitutingtheEquationsin5.17intoEquation5.24.Thustheuseoftheharmonicreferencesystemmakestheexpressionsensitivetoanharmoniccouplingswhilelteringouttheharmoniccontributions.Wehavepresentedafullythree-dimensionaltheory.Whileexpressingtheresponseintermsofaclassicalcorrelationrepresentsanenormoussimplication,athree-timeclassicalcorrelationremainsaformidablechallenge.Becauseofthat,andbecauseoftheavailableexperiments,weaimtodiscussa2Dtheoryindetail.Therearetwoquestions:dotheexperimentsprobea2Dresponsefunction,andisthe2Dresponsedeterminedbya2Dclassicalcorrelation?TheIRechoexperimentnominallyhast2=0.Fornitepulselengthsthisconditionisnotliterallyenforced.However,astheideallimitingcase,andagoodapproximationtotherealexperiment,wewillevaluatethe2Dresponse,specically:R!1;!3.2559

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R!1;t2=0;!3=1 23Zdt1e)]TJ/F19 7.97 Tf 6.586 0 Td[(i!1t1Zdt2e)]TJ/F19 7.97 Tf 6.587 0 Td[(i!2t2Zdt3e)]TJ/F19 7.97 Tf 6.587 0 Td[(i!3t3Rt1;t2;t3t2Equation5.25impliesthatweneedtoevaluatetheresponsefunctiongivenbytheFouriertransformofEquation5.24att2=0.WithinourtheoryitiscomputationallyfarmoreconvenienttoevaluatetheTCF,BRt1;t2;t3usingtheconditiont2=0.ThisiscomplicatedbytheformofEquation5.24becausetheTCFismultipliedbyacomplicatedfunctionoffrequencyandtheproductwouldneedtobebackFouriertransformedtoimplementtheconditionintimeandingeneralthisdoesnotleadtoevaluatingBRt1;t2=0;t3.However,inthelimitthatallthreefrequenciesarehigh,~!1;2;31,anapproximatesimplicationispossibleandthisistherelevantexperimentalfrequencyregime.DeningeverythingontherighthandsideofEquation5.24thatmultipliesBRash!1;!2;!3givesRwithinourtheoryas:Rt1;t2=0;t3=.26Zd!1ei!1t1Zd!2ei!2t2Zd!3ei!3t3h!1;!2;!3BR!1;!2;!3t2=0Equation5.26onlyimpliesthattheTCFonlyneedstobeevaluatedatt2=0ifhisnotafunctionof!2.Inthelimitofhighfrequencythisisthecase,althoughat3000cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1itisstillweakly!2dependentandthelimitingvalueisnotreacheduntil6000cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1when!1!33000cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1at300K.Thus,hhardlychangesoverthewidthofanintramolecularresonanceandtheresultinglineshapeswillbelittlechanged.Therefore,forthepurposesofdemonstratingthetheorywewillassumethehighfrequencylimitandtakehasonly60

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afunctionof!1and!3,h!1;!3.InthiscaseEquation5.26canbeevaluatedtogiveBR!1;t2=0;!3andwhenEquation5.24isevaluatedalarge!2ischosen,inthiscase!2=10000cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1.Totestthechoiceavarietyofsmaller!2valuesweretriedandthelineshapeswereverysimilar.This2Dtheoryinvolvesfurtherapproximationbutisnotalimitationofthegeneraltheorypresentedbutratheracomputationalconveniencetoavoidcalculatingthethree-timeTCFthatisgenerallyrequiredinEquations5.1and5.24.EquatingtheTCFBRt1;t2;t3withitsclassicalcounterpartisafurtherapproximationandthisisfrequentlydoneinthecaseofonedimensionalTCF'sandtendstochangethemagnitudeofspectroscopicsignalsmorethantheactuallineshapesthemselves.[60]AbetterapproachistoquantumcorrecttheFouriertransformoftheclassicalTCFbasedontherelationshipbetweenthequantumandclassicalTCFforanexactlysolvablemodelsystem,e.g.aharmonicsystem.Inthemultidimensionalcasethisissomewhatmoredicult.OneproceedsbyformingtheratioofthetermsintheBfunction,Equation5.17,withtheirclassicallimit.ThatratioisthenusedtoquantumcorrectthefrequencydomainclassicalTCF.Theprobleminthemultidimensionalcaseisthatthe!1,!2,and!3dependenceisunder-determinedandonlytheharmonicfrequencydependenceonisknown;intheonedimensionalcasethereisaonetoonecorrespondencebetweentheharmonicfrequency,andthespectroscopicobservationfrequencyconjugatetotandnoambiguityresults.InChapter7twoquantumcorrectionschemesthatarebothexactfortheharmonicmodelsystemresultsarepresentedanditisdemonstratedthattheresultinglineshapesarenotgreatlychanged.Further,thelackofauniquequantumcorrectionschemeisnotamajor61

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limitationgiventhatchallenging2DIRexperimentsdonotfocusonabsoluteintensitiesandthepresentapproachrepresentsacomputationallytractabletheoryof2DIRspectroscopythatcanbeappliedtochemicallyinterestingsystemssimulatedinatomisticdetail.62

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Chapter6FifthOrderRamanResultsInthischapter,resultsfromtheapplicationoftheTCFtheoryforthefthorderRamanre-sponsefunctionfromchapter4arepresented.TherstsystemtestedwasambientCS2whichisalsocomparedtoavailableexperimentalresults.LiquidxenonwasalsoinvestigatedandtheTCFresultscomparedtoextantexactclassicalcalculations.Althoughtheexperimentaldataavailableissparse,itwillbeseenthattheTCFtheoryreproducescertaincharacteris-ticsofexperimentalresultsaswellasdisplayinglimitingbehavior.Forliquidxenon,itwillbeshownthattheTCFquantitativelyreproducestheexactclassicalcalculations.6.1ResultsforAmbientCS2TotestEquation4.45againstexperimentalandothertheoreticalapproaches,thetwotimeTCFwascalculatedforambientCS2andliquidxenon.ClassicalMDsimulationswereperformedusingacodedevelopedattheCenterforMolecularModelingattheUniversityofPennsylvania,whichusesreversibleintegrationandextendedsystemtechniques.[65,66]MicrocanonicalMDsimulationswereperformedonambientCS2withanappropriatedensity63

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[58,67{70]andanaveragetemperatureof298K.7.0millionfourfemtosecondMDtimestepswereperformed,andTCFdatawascorrelatedevery8.0fsfemtosecondsforatotalof3.5millioncongurations;thefthorderresponsefunctionsderivedfromthesecomputationsweresmoothedbyaveragingeveryvepointstogethertobettervisualizetheoverallshapeoftheresultingfunctions.Figure7.1showsthetwotimeTCF,ht1t1+t2iforthefullypolarizedgeometrywithallofthepolarizabilityelementsarethesameCartesiandirection. Figure6.1:Thetwotimecorrelationfunctionofthepolarizability,ht1t1+t2i.TheTCFdecaystozeroatlongtimeswithonlylimitedstructureatintermediatetimes,incontrasttoitshighlystructuredderivatives.Thefunctiondecaysmostrapidlybetweenzero64

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and500fsandshowssomeoscillatorybehavioralongt2,witht1=0.EvenaftersignicantaveragingtheTCFshowssomenoiseatlongertimes,althoughtheshorttimeinformationbetweenzeroandonepicosecond,thatissignicantincalculatingthefthorderresponsefunction,iswellaveraged.Figures6.2a-bpresentthemagnitudeofthefullypolarizedfthorderresponsefunction,jRt1;t2jcalculatedusingEquation4.45withtwodierentTCFtimessteps.Thelimitingbehaviors,withRt1;t2goingtozeroattheoriginandeverywherealongt2=0areapparent.Further,alongt1=0theresponseexhibitsanintermediate-timeplateau.ThisbehaviorhasbeenobservedexperimentallyforambientCS2.[11,12,14]Theridgealongt1=0hasalsobeenobservedinnumericallyexactcalculationsofRperformedinliquidxenon[8]andformodelsystems.[5,18,71,72]ItisencouragingthatthepresentTCFtheorycapturesthisdistinctivefeatureofthefthorderresponse.Thefthorderresponsefunctiondoesexhibitoscillatorybehaviorwithafrequencycor-respondingtosymmetricstretching658.cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1.ToverifytheoriginofthefeatureaCS2modelwasrunwithafrequency20.cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1largerandtheoscillationsinthesignalappearedwiththenewfrequency.Thesymmetricstretchistheonlyvibrationalmodethathasanon-zerorstpolarizabilityderivativeandallvibrationalcoordinateshavezerosecondpolariz-abilityderivativesinthegasphase.Thephysicalmechanismpermittingtheintramolecularmodesignalappearinginthefthorderresponsefunctioniscurrentlyunderinvestigationandmaybeinteractioninducedwithintermolecularcouplingsleadingtonon-zerosecondpolarizabilityderivativesforaparticularmode.Non-zerosecondpolarizabilityderivatives65

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Figure6.2:TheapplicationofthepresenttheoryofjRt1;t2jisshownforambientCS2.TheTCFusedtogeneratethefthorderresponsefunctioniscalculatedwithatimestepofadt=0.008psandbdt=0.04ps.66

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wouldberequiredforatleastanindependentharmonicmodetocontributeascanbeseeninEquation4.34.OthertheoreticalinvestigationsinCS2haveusedrigidmodels[73,74]ordonotstate/controlthepolarizabilityderivativestomatchexperimentalvalues.[14]Figure6.2bshowstheresponsethatresultsfromcalculatingtheTCFusingalargertimestepthatnaturallyltersoutmostofthehighfrequencyoscillationsandmakestheoverallshapeofthesurfaceeasiertodiscern.Overall,inthefthorderresponse,asinglelargepeakisfoundatshorttimesforbothtimearguments,andthisissimilartotheexperimentalresultsthatareavailable.RecentworkbyJansenet.al.hasalsodescribedthefullypolarizedfthorderresponseofambientCS2usinganon-equilibriumsimulation/niteeldapproachthatcanbeshowntoreproducethethirdorderpolarizationresponsecalculatedforthesamemodelusingTCFmethods.[73,74]Theresultsarediculttocomparebecausethepolarizabiltymodelsusedaredierent.ThepresentmodelwasemployedbecauseitquantitativelyreproducedtheOKEspectrumofliquidCS2fromboilingtofreezingatatmosphericpressure.Inanycase,Jansenobtainedresultsforfourdierentpolarizabiltymodelsallofwhichshowthesignaldecayingoverliketimescalesandexhibitinglimitingbehaviorssimilartotheresultspresentedhere.Tryingourtheorywithdierentpolarizationmodelsisplannedforfutureinvestigations.Figure6.3highlightstimeslicesofR,t1=0upper-left,t2=0upper-right,t1=t2lower-left,andcomparesthetheoreticaltimeslicesonanabsolutescalelower-right.Figure6.3alsoshowrecentlypublishedexperimentaldataforthesametimeslices;theexperimentalmeasurementsarenotinabsoluteunitsandarescaledtomatchthetheoretical67

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Figure6.3:ThreetimeslicesofRforCS2;upper-leftalongt2witht1=0,upper-rightalongt1witht2=0,andlower-leftalongt1=t2areshown.Thetheoreticalresultslineswithsymbolsarecomparedwithrecentexperimentalmeasurementslineswithnosymbols.[11]lower-rightThetheoreticaltimesslicesarecomparedwithanabsolutescalet1=0-lineswithcircles;t1=t2-lineswithtriangles;t2=0-nosymbols.InthisandsubsequentguresthetheoereticalresultsaregiveninthestatedabsoluteunitswithinEquation4.45takenasunity.68

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data.Incontrasttothetheoreticalresults,theexperimentalmeasurements[11]arenotsimplythenuclearcontributiontoR.Theyalsoincludeanelectronicresponseatshorttimesandperhapssomecontaminationfromotherprocesses.[11]Nonetheless,theagreement,especiallytherateandlocationofdecayofthefunctionsisverysimilaroutsideof100fsalthoughitappearssomewhatlongerfort1=0.Theelectroniccontributiontothesignalisdominantintheexperimentalsignalfortimesuptoatleast100fs.Ameasureofthisisthesignalalongt2=0wherethenuclearfthorderresponsevanishesandtheprinciplecontributiontotheexperimentalsignalisfromelectronicresponse.Ourtheoreticalexpression,Equation4.45,alsogiveszeroalongt2=0andthetheoreticaldatademonstratethisresult.Therstfewpointsaresmallnonzerovaluesthatarestatisticallyequivalenttozero;takingderivativesatpointsclosetotheaxisrequiresnumericalalgorithmsthatarecorrecttolowerorder,andthesignalresultsfromthecancellationofpositiveandnegativecontributions.Alongt1=0bothsetsofdatashowarapiddecayandaplateauatabout500fsthatcanbeseenasaridgeinFigure6.3upper,andtheagreementbetweenthetwosetsofdataisexcellent.Thetheoreticaldatashowanoscillationwithaperiodcharacteristicofthesymmetricstretch.Thisoscillation,inphaseandmagnitude,isalsohintedatintheexperiments,butitisunclearhownoisytheexperimentalmeasurementsare.Fouriertransformingboththetheoreticalandexperimentalonedimensionaldatasetsalongt1=0doesgiveapeakofsimilarwidthinthesymmetricstretchingspectralregion.Atlongertimes,afterafewpicoseconds,theresponsefunctionapproacheszeroeverywhere.69

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Alongt1=t2thereisasmallornosignalandnosignicantechothathasbeenthesub-jectofdiscussionconcerningR.[8]Theexperimentaldataalongt1=t2areverysimilartot2=0againreectingthesmallnuclearresponsethatisobserved.Alargecontribu-tionalongt1=t2couldbeinterpretedasanecho,indicatingtheexistenceoflonglivedintermolecularmodesbutthisappearsnottobethecase.Thisabsenceofasignicantre-sponsealongt1=t2suggeststhatintermolecularmodesarehighlydamped.FromEquation4.45,thecontributionalongt=t1=t2canbeshowntoresultfromthedierencebetweentwoTCF's:h_t_ti)-281(h__tti,inwhich_representsatimederivative.EvidentlytheTCF'sareverysimilar.6.2ResultsforLiquidxenonThemostrigoroustestofthistheoryistocomparetheresultingspectratothosefromexactnumericalresultsforthesamemodelsystem.Toevaluatethisapproachexactly,theclassicallimitistakendirectlybyreplacingthecommutatorswithPoissonbrackets.Then,Risseentocontainbracketsofvariablesatdierenttimes,whichrequirestheexceedinglydiculttaskofcalculatingthedependenceofamany-bodydynamicalvariableonitsinitialconditions.CalculatingRinthiswayisonlypracticalforsmallsimplesystemsandresultsforliquidxenonwerereportedpreviously.[8]Tocompareourtheorytothoseresults,microcanonicalMDsimulationswereperformedfortheneatliquidxenonconsistingof108atoms.TheatomsinteractedviaaLennard-Jonespairpotentialandthesystemsreduceddensityandtemperaturewere3=.8andkT==1.0andthesamemodelparameterswereusedasinthe70

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Figure6.4:TCFFODIDresultsforjRxxxxxxt1;t2j2liquidxenon.Contoursareevenlyspacedfrom0.1-0.4.exactcalculation.[8]Resultsforthesquareofthefullypolarizedfthorderresponsefunction,jRxxxxxxt1;t2j2,withintheFODIDapproximation,areshowninFigure6.4.Figure6.5showstheresultusingourTCFtheoryisoverlayedontheexactclassicalcalculationperformedbyMaandStratt[8];eventhoughanexactTCFtheoryisnotpossible,[10]thepresenttheoryisveryeective.Itcapturesthecharacteristicfeaturespresentintheexactcalculationincludingthesamedecaytimesandthelackofanechosignalalongthediagonal{thatimpliesthattheintermolecularmodeshavelifetimesoflessthanafullperiod.Thesignalissharplypeakedaroundt130fs,t2350fs.Decaytimesvaryalongeachaxiswiththesignaldyingoutalongt1by400fs;alongt2thereisalongtimedecaythatcontinuesbeyondthe1psthatisshown.Incontrasttoliquidxenon,theplotofthespectrumofCS2[11,63]isroughlysymmetric,so71

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Figure6.5:ResultsforjRxxxxxxt1;t2j2,usingtheFODIDapproximationforliquidxenon.Theexactclassicalmoleculardynamicscalculationred,Strattet.al.,wasreportedinarbi-traryunitsandisnormalizedheretoourTCFtheorydatagreenatthemaximumpoint.theextremeasymmetryofthexenonsimulation,withthepeakpracticallyonthet2axis,isstriking.[11,63,73]ThecapabilityoftheTCFtheorytocorrectlyyieldsymmetricorasymmetricspectraisfurtherstrongevidenceofitsgeneralapplicability.TohighlighttheeectivenessoftheTCFtheory,Figure6.6showsasliceofjRxxxxxxt1;t2j2alongt2witht1=0.Thedashedlineisfromtheexactclassicalcalculationandthesolidlinemarkedwithsquaresrepresentsthesameresultshiftedforwardintimeby34.7fs.ThesolidlinewithoutsymbolsisfromourTCFtheory.TheTCFresultandthetimeshiftedexactcalculationshowquantitativeagreement.ThedierencebetweentheexactresultandtheTCFtheory,theshiftintime,isverylikelyduetonitesystemsizeeectsontheexactcalculation.MaandStrattwerelimitedtousingonly32atomsintheirsimulation{even72

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Figure6.6:SlicesofjRxxxxxxt1;t2j2alongt2witht1=0.Thedashedlinerepresentstheexactclassicalcalculation,thelinemarkedwithsquaresrepresentsthetimeshiftedexactclassicalcalculationandthesolidlinewithoutsymbolsrepresentstheresultfromtheTCFtheory.withintheFODIDapproximationthatisfarlesscomputationallydemandingthanaMBPevaluationofthepolarizability.Totesttheeectofthesmallsystemsizetheyperformedasimplercalculationofaonedimensionalsliceof@2gRt1;t2 @t1@t2jt2=0for32and108atoms,becauseanon-Poissonbracketpieceofthefthordersignalwiththisformmaybeidentied.[75]TheresultinglineshapehadthesameshapebutwithatimephaseshiftnearlyidenticaltothatapparentinFigure6.6seeFigure4oftheirpaperimplyingthatthepresenttheorymayagreeevenbetterthentheguresuggests.[8]Figure6.7showsjRt1;t2j2forliquidxenonusingtheMBPmodel.[76]TheMBPresultisapredictionoftheexperimentalfthorderresponsefunction{xenonishighlypolarizable73

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Figure6.7:TCFMBPresultsforjRxxxxxxt1;t2j2forliquidxenon.Contoursareevenlyspaced0.1-0.8.andtheFODIDapproximationisnotstrictlyvalid.TheuseoftheFODIDmodeleectivelyactstoremovecontributionsfromthetraceofthepolarizabilitymatrix[8]{oneofthreematrixinvariantsinisotropicmediathatgiverise,indierentcombinations,tothesignalsfordierentpolarizationconditions.[9]Althoughthetracecontributionsaresmallin1Dcorrelationfunctions,[8]in2Dcorrelationfunctionstheinvariantsappearasproductsandazerotraceinvariantkillsoothersignicanttermsthatincludesizableinvariantsmultipliedbythetracecontribution.[9]Thus,theuseoftheFODIDapproximationinthe2Dcorrelationfunctionsactstoremovesignicantcontributionsthatleadtodierentsignals{includingremovingtheechocontribution.Figure6.8comparesresultsforjRxxxxxxt1;t2j2usingtheMBPmodelredandtheFODIDmodelgreen.ThemostnotabledierencesarethattheMBPsignalhasfaster74

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Figure6.8:TCFMBPtheoryresultsforjRxxxxxxt1;t2j2usingMBPmodelred,peaksear-liervsFODIDapproximationgreen.decaytimes,alongboththet1andt2axis,andthepeakisshiftedtoearliertimesalongt2.Thesignalischaracterizedbyastrongpeakaroundt145fs,t2125fsanddiesoutby200fsalongbotht1andt2.AsintheFODIDapproximation,noechosignalappearsalongthediagonalandthesignalappearsfeaturelessbeyond300fsinbotht1andt2directions.WhiletheFODIDmodelcapturesmuchoftheresponse,thedistinctdierencesobservedarearesultoftheexclusionofthepolarizabilitytensorinvariantcontributionsmentionedabove.Theutilityofmultidimensionalspectroscopyistheabilitytoselectoutspecicsysteminformationduetothetwo-dimensionalnatureoftheexperiment.ThiscanbeaccomplishedbyusingthevariouspolarizationconditionstoenhanceordiminishspecicLiouvillepath-75

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waysthatcontributetoRt1;t2.[9]TheuseoftheFODIDapproximationremovesthisutilitybypredictingthesameTCFforallpolarizationconditionsasdiscussedinSec..3;however,theMBPmodelretainsthiscapability.Table6.1showsasummaryoftheanal-ysisperformedbyFourkaset.al.toaidinevaluatingthespectrapresentedhere.[9]TheleftcolumnindicatestheexperimentalpolarizationconditionsconsideredandthetoprowshowsthecontributionstotheresponsefunctionintermsofLiouvillepathways;beloweachpathwayistheeectofeachpolarizationconditiononthecorrelationfunctioncontributingtoit. R R1 R2 R xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx xxxxzz xxxxzz xxxxzz xxxxzz xxxxzz xxzzxx xxzzxx xxzzxx xxzzxx xxzzxx zzxxxx zzxxxx zzxxxx zzxxxx zzxxxx xzxxxz xzxxxz xzxxxz xzxxxz xzxxxz xzxzxx xzxzxx xzxzxx xzxzxx xzxzxx xxxzxz xxxzxz xxxzxz xxxzxz xxxzxz Table6.1:AsummaryoftheanalysisperformedbyFourkaset.al..[9]Thetoprowrepre-sentsthepathwaysthatcorrespondtothoseinFourkas'sanalysisandtheeectofvariouspolarizationconditionsleftcolumnisindicatedineachcolumn.Thexxzzxxcombinationenhancestheechosignal.76

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Ideally,enhancingthepathwayleadingtotheecho-likesignal,R2inTable6.1,and/ordiminishingtheotherswouldleadtointerestinginformation.NotethatthepathwaynotationusedhereisthatofFourkaset.al.anddiersfromotherliteraturereferencesincludingourearlierwork.[15,63,77,78]AmajorreasonfordevelopingfthorderRa-manspectroscopywastoprobeintermoleculardynamicsformodeswithcharacteristicsthatwouldproduceanechosignal.Theutilityofpolarizationconditionsisshownintable6.1;eachpolarizationcondition,exceptthefullypolarized,leadstoauniquecorrelationfunc-tioncontributingforonepathway.Inthefullypolarizedcondition,thedierentLiouvillespacepathwaysareequallyweightedsuggestingthatthisisapoorchoiceforselectingoutspecicinformation;inaddition,thismakesitimpossibletodetermineifasinglepathwayisdominantinthefullypolarizedsignal.Theotherpossiblepolarizationconditionsprovideameansbywhichcertainpathwayscanbeenhancedwhileothersarediminishedandthisanalysisisbasedonaharmonicmodelofthedynamicswherenonlinearityinthepolarizabil-ityprovidesthesignal.[9]Toaidinidentifyingthedominantpathwayforeachpolarizationcondition,Table6.1hasbeenincluded.Forsemipolarizedxxxxzz,xxzzxx,zzxxxxanddepolarizedxzxxxz,xxxzxz,xzxzxxpolarizationconditions,[17]thepathwaywithacor-relationfunctionthathastheodd"indexzzforsemipolarized;xxfordepolarizedonleadstoenhancementofthatpathway.Forexample,forthexxxxzzpolarizationcondition,theoddindex,zz,isoninthecorrelationfunctionxxxxzz.FromTable6.1,thiscorrespondstothepathwayR2.TheenhancementresultsfromanemphasisofthedominantinvariantcombinationTrPP;forthoseforms.Thepolarization77

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conditionsxxzzxxandxzxxxzleadtoacorrelationfunctionfortheecho-likepathwayoftheformxxzzxxandxzxxxzrespectively,andthusenhancethatpathway.Theotherpathwaysareenhancedbyotherpolarizationconditionsinasimilarway.Itshouldbepointedoutthattheleadingordertermsinfthordernon-resonantspectroscopyinvolve,e.g.termsoftheform.[9]Incaseswherethereisdominantanharmonicity,termsoftheformcanmakecontributions.[79]However,thiscontributionisnotexpectedtobelargeinaLennard-Jonesuidundertheseconditions.ThelowerpanelofFigure6.9showsjRxxzzxxt1;t2j2usingtheMBPmodelwiththeinstantaneousnormalmodeINMvibrationalDOSforxenonappearingintheinset.[76,80]Themostnotablefeatureofthisgureistheechopeakthatexistsalongthediagonal.Theechosignalisellipticalwiththelengthalongthediagonalbeingnearlytwiceaslongasthewidthperpendiculartothediagonal.InfthorderRamanspectroscopy,theechopeakimpliesthatanintermolecularmode,excitedattimezero,isstilloscillatingatthetimeofthemeasurementoneperiodlaterandsuggeststhatthespectrumisdominatedbyinhomogeneouscontributions.AsshownisFigure6.9,associatingaperiodwiththeoscillationleadstotherstechosignatureat53cm)]TJ/F16 7.97 Tf 6.587 0 Td[(100fsandtheendofthesignaloccursatafrequencyof12cm)]TJ/F16 7.97 Tf 6.586 0 Td[(150fs.TheechoonsetfrequencyisinthetailoftheINMDOSandthelastechooccursnearthemaximumoftheINMDOSandtheechosignaturespansnearlytheentireINMDOS.TheappearanceofanechoinsuchasimpleliquidimpliesthatintermolecularmodesmaygenerallyliveatleastaperiodinmorecomplicatedliquidsthathavesignicantLennard-Jonesinteractions.ThesignalalongthediagonalofFigure6.978

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Figure6.9:jRxxzzxxt1;t2j2withMBPforliquidxenon.Thesolidlinesparalleltothet2axisrepresentthewidthandlocationofthebeginningandendoftheechopeakalongthediagonal.1ps,0.45ps{usingthediagonaltochoosethelinesleadstothemoverlappingo-diagonalfeaturesinthegure.TheinsetshowstheINMDOSforliquidxenonwithverticallinesplacedtoshowthebreathofthevibrationalperiodsassociatedwiththeechopeak.79

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wasttoasingleexponentialoftheforme)]TJ/F19 7.97 Tf 6.587 0 Td[(t=,wheretisthetimeandisthelifetime,andwasfoundtohavealifetimeofapproximately190fs.Thisservesasanestimateofmodelifetimes.Notethattheechosignalissmallerinmagnitudethantheothercontributionsthathideitinotherpolarizationgeometries.ThepresenceofanechosignalfortheRxxzzxxpolarizationconditionwassuggestedbyFourkaset.al.andonlyappearsusingtheMBPmodel.[9]TheuseoftheFODIDapproximationactstoremovedistinguishingcontributionstodieringpolarizationconditionsbyonlyallowingthetripleproductinvarianttocontributewithvaryingmagnitudes.[8,9]Todate,molecularliquidshavenotshownanechofeaturein2DRamanexperiments.Thisimpliesthattheotherpathwaysaredominatingtheechosignalmakingitdiculttodetect.Thestrongersignalinmolecularliquidsmaybedominatedbyrotationsbutthisremainstobeexamined.However,itmaybethatthepresenceofasinglemoleculepolarizabilityinthatcasewilloverwhelmthelessermagnitudeechocontribution.Figure.10representstheremainingsemipolarizedpolarizationconditions.TheupperpanelshowsjRzzxxxxt1;t2j2withitsstrongridgeatt1=0alongt2.Thesignalisinstantlynonzeroalongt1andbeginstogrowinalongt2ataround200fsandtheridgecontinuesbeyondthe1psshownhere.Inthet1directionthesignaldecaystozerobyt1200fs.Figure6.11showsasliceofthezzxxxxpolarizationsignaltakenalongt2witht1=0wherethedecaytimecanbeseentoextendtoapproximately3ps.Thesignalalsoappearstodecayinphaseswithperiodsofsloworalmostnodecayfollowedbyperiodsofrapiddecay.ThelowerpanelofFigure6.10showsjRxxxxzzt1;t2j2whichnearlyquantitatively80

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Figure6.10:ResultsforsemipolarizedpolarizationconditionsofjRt1;t2j2,usingtheMBPmodelforliquidxenon:zzxxxxtoppanel,xxxxzzlowerpanel.Upperpanel:Threecontourlinesareevenlyspacedfrom0.05-0.15;lowerpanel:Fourcontourlinesareevenlyspacedfrom0.02-0.08.81

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Figure6.11:TimesliceofjRt1;t2j2forthezzxxxxpolarizationalongt2witht1=0.reproducesthefullypolarizedresultinshapewithadierentamplitudeduetothedierentweightingcoecients,jRxxxxxxt1;t2j2.ThissuggeststhatthereisonedominantpathwaythatcontributestothefullypolarizedsignalanditisenhancedinthejRxxxxzzt1;t2j2.Usingtheanalysisfromabove,itwouldbeexpectedthatthepathwayenhancedinthispolarizationwouldbethepathwaylabeledRintable6.1.Figure6.12representsthedepolarizedpolarizationconditions.TheupperpanelshowsjRxzxxxzt1;t2j2whichresemblesthefullypolarizedresult.UsingTable6.1,wecanpredictthatthispolarizationwouldbeexpectedtoenhancetheecho-likepathwaybyobservingthefactthatthecorrelationfunctioncontributingtothatpathwaytakestheformxzxxxz.Thesignalisslightlyelongatedalongthediagonalsuggestingthatthepathwayleadingtotheechoisenhancedalthoughwithlesseciencythanthexxzzxxpolarization.AsdiscussedbyFourkaset.al.,earliertheoreticalanalysissuggestedthatthesignalshouldnotbeinstantly82

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Figure6.12:DepolarizedresultsforjRt1;t2j2,usingtheMBPmodelforliquidxenon:xzxxxztoppanel,xxxzxzmiddlepanelandxzxzxxlowerpanel.Upperandmiddlepanel:vecontourlinesspacedfrom0.02-0.1;lowerpanel:fourcontourlinesspacedfrom0.01-0.04.83

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nonzeroalongeitheraxis,butshouldbuildinalongbotht1andt2.[9]Although,thetheorypredictsasignalthatisrigorouslyzeroalongt2=0,thesignalinFigure6.12showssomenoisealongthataxisduetonumericalerrorintakingtherequiredderivatives.ThetoppanelofFigure6.13showsslicesalongt2=0dotted-dashedlineandt1=t2solidlineforjRxzxxxzj2plottedwitherrorbars0:0081A3=ps2.Theerrorestimatewascalculatedalongthet2=0axisrepresentingthemaximumestimatederrorduetotheneedtocalculatealowerordernumericalderivativeonthezeroaxisandthelackofanyrealsignal.TheresultforjRxxxzxzt1;t2j2Figure6.12middlepanelshowsacombinationofvariouspolarizationcharacteristics.FromTable6.1,wewouldexpectasignalthatenhancesthepathwayR211,thesameenhancedinthezzxxxxpolarization.Indeed,bothxxxzxzandzzxxxxsharethedominantridgealongt2witht1=0.Thesharp,fastrisingpeakneartheoriginresemblesthatseeninthefullypolarizedresultwithsimilardecaytimes.Thesignalisinstantlynonzeroalongt1andgrowsinratherquicklyalongt2.Thesignalthencontinuesinthet2directionwellbeyondthe1pstimeshown.Inthet1directionthesignaldecaystozeroby200fs.NotethattheRpathwayiszeroforaharmonicoraBrownianoscillatorsys-tem.[15,77,78]ThereforetheridgecontributionmustdominatedbyanharmonicdynamicalcontributionsaswassuggestedbyanearlierMDandINManalysis.[8,81]ThelowerpanelshowsjRxzxzxxt1;t2j2.Itwouldbeexpectedthatthispolarizationwouldenhancethesamepathwayenhancedinthexxxxzzpolarization.Bothpolarizationconditionsdobearstrongresemblancetothefullypolarizedwithshorterdecaytimesalong84

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Figure6.13:TimeslicesofjRt1;t2j2forthexzxxxzpolarizationupperalongt2=0dotted-dashedlineandt1=t2solidlineandforxzxzxxalongt1=t2plottedwitherrorbars.Theerrorestimatesforthesliceswere0.0081A3=ps2forthet2=0sliceofxzxxxzand0.0025A3=ps2forthet1=t2sliceofxzxzxx.85

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thediagonalinthexzxzxxsignal.Thisagainagreeswithearliertheoreticalwork.[9].Thesmallbumpappearingaround250fsissuggestiveofanecho.ThelowerpanelofFigure6.13showsthet1=t2sliceofjRxzxzxxj2plottedwitherrorbars0:05A3=ps2.Thissuggeststhatthesmallbumpisnotanartifactofnumericalerrorinthecalculations.Itispossibletoenhancespecicpathwaysmorebycombiningsignalsfrommultiplepo-larizations.TheupperpanelofFigure6.14showsjRxxzzxxt1;t2)]TJ/F18 11.955 Tf 12.583 0 Td[(Rxxxxzzt1;t2j2whichenhancestheechosignalrelativetotheotherfeatures.Figure6.10uppershowsaridgealongt1=0andlowershowsarelativelysharppeakneartheoriginaspike.ThedominantpathwaysforthesetwoguresareRupperandRlower.[9]There-forethesetwopathwayswillbereferredtohereastheridge"andspike"pathwaysrespec-tively.Takingthedierenceofxxzzxxandxxxxzzpolarizationsignalsactstodiminishthespike"pathwaycontributionthatisadominantfeatureinboththexxxxzzFigure6.10andxxzzxxFigure6.9signals.Asshownintable6.1thecorrelationfunctionsthatcontributetotheridge"pathwayarexxxxzzandxxzzxxforxxxxzzandxxzzxxrespectively.Withineachcorrelationfunction,the0scommuteandsothecorrelationfunctionsareiden-ticalforthesetwopolarizationconditions.Thisleadstoperfectcancellationoftheridge"contributionforthiscombination.Note,thatforthezzxxxxpolarizationconditions,thecorrelationfunctionthatcontributestotheridgepathwayiszzxxxx.Thiscorrelationfunctionisnotidenticaltothosefromeitherxxxxzzorxxzzxx,thustakingthedierenceofzzxxxxandxxxxzzorzzxxxxandxxxxzzwouldnotleadtoperfectcancellationoftheridgepathway".Theridge"contributioncanbeseeninboththezzxxxxFigure6.10and86

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xxzzxxFigure6.9signals.ThelowerpanelofFigure6.14showsjRxzxxxzt1;t2)]TJ/F18 11.955 Tf 12.307 0 Td[(Rxzxzxxt1;t2j2inwhichsimilarcancellationtakesplace.Thespike"pathwaycontributioniseectivelyremovedalongwithexactcancellationoftheridge"pathwaycontributionwiththiscombination.NotethesignaltonoiseratioissignicantlylowerinthedepolarizedsignalsasseeninthelowerpanelofFigure6.14.Lastly,whileitispossibletopreferentiallyenhanceapathwayrelativetotheotherpathways,itisnotpossibletoreducethefthorderresponsetoasinglepathway.Thepathwaysareconvolutedinsuchawaythatnocombinationofvariouspolarizationconditionsignalswillyieldaresultthatdependsonasinglepathway.AlthoughtheanalysisbyFourkaset.al.wasappliedtoexistingexperimentalCS2results,itisgeneralandappliesequallywelltoliquidxenon.Thedierencesinthesemipolarizedanddepolarizedspectraisnotatallunexpected.Acombinationoffactorscouldleadtothesedierencesincludingdiminishedabilityofthedepolarizedconditionstoenhance/diminishthepathwaysrelativetooneanother.Ifindeedthereisonedominantpathwaythatleadstothefullypolarizedspectra,thenitwouldbeexpectedthatthedepolarizedresultswouldhavefullypolarizedcharacteristics.Inaddition,theeectofthepathwayRhasnotbeenstressedinouranalysis.Thefactthatthispathwaycontributesequallywithineachofthesetwopolarizationformssemipolarizedanddepolarizedmakesitsomewhatmorediculttoappreciateitscontributionwithoutfurtheranalysis.Itisalsoremarkablethevarietyoflineshapesthatcanbeobtainedforthe2DRamanspectrumofevensimpleliquids.Thissuggestsrichinformationcontentinthesignalthatrequiresbetterunderstandingto87

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Figure6.14:jRxxzzxxt1;t2)]TJ/F18 11.955 Tf 15.065 0 Td[(Rxxxxzzt1;t2j2upperpanelandjRxzxxxzt1;t2)]TJ/F18 11.955 Tf -458.701 -28.892 Td[(Rxzxzxxls)]TJ/F15 11.955 Tf 7.085 3.33 Td[(t1;t2j2lowerpanelforliquidxenonusingtheMBPmodel.Sixcontourlinesareevenlyspacedfromtoppanel0.02-0.12andbottompanel0.01-0.05.88

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extract.89

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Chapter72DIRResultsApplicationoftheTCFtheoryofthethirdorderresponsefunctionprobedinthe2DIRexperimentispresentedinthischapter.Thepresenttheoryofthethirdorderresponsefunctionand2DIRspectroscopywasappliedtoambientliquidwater.RecentexperimentshaveexaminedtheOHstretchingregionfordilutesolutionsofHODinD2O;thereareexperimentalchallengesassociatedwithexaminingneatH2OduetoitsstrongIRabsorbance.[82,83]ApplyingthepresenttheorytoHODinD2Oisthesubjectofanongoinginvestigation.7.1ApplicationstoAmbientWater:ModelsandComputationalDetailsToobtaintimeorderedwatercongurations,classicalMDsimulationswereperformedusingacodedevelopedattheCenterforMolecularModelingattheUniversityofPennsylvania,whichusesreversibleintegrationandextendedsystemtechniques.[66,84]AexibleSPCmodelwasusedtoperformtheMDsimulations.microcanonicalsimulationswereperformedonasystemconsistingof64H2Omoleculeswithadensityof0.99g/cm3andatemperatureof295K;theseconditionsproduceapressureof1.0atmosphereforthiswatermodelas90

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determinedbyconstanttemperatureandpressureNPTsimulations.Previousstudiesdemonstratedasystemsizeof64watermoleculesisadequatetoreproducetheIRspectrumofwater.[85]Thewaterintramolecularpotentialincludesaharmonicbendingpotential,linearcrosstermsandaMorseOHstretchingpotentialandthemodelparametersarepresentedinpreviousworks.[61,64]MDsimulationswithalengthof1.4million1.0fstimestepswereperformedandcongurationswerestoredevery4.0fsproducingatotalof350,000congurationsthataresubsequentlyusedtocalculatetwo-timeTCF's;thistimespacinggivesaNyquistfrequencyof4167cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1.[86]TheMDwasperformedwithoutexplicitpolarizationforces.However,aspectroscopicmodelisusedtocalculatethetimedependentdipoleofliquidwaterthatexplicitlyincludesmanybodypolarizability.Themodelisparametrizedtoproduceaccuratedipolesandpo-larizabilitytensorsandtheircoordinatederivativesinthegasphase.SpecicallyapointatomicpolarizabilityPAPAmodelisusedthathasbeensuccessfullyappliedinquantita-tivelyreproducingthelinearIRspectrumofwater[61,85]andtheSFGspectroscopyofthewater/vaporinterface.[64]ThePAPAmodelaccuratelyaccountsforinduceddipolesandtheirderivativesinthecondensedphaseandthisisevidencedbythesuccessofthemodelinreproducingcondensedphasespectra.Inaddition,apermanentdipolemodel,ttoabinitiocalculationsusedinanotherstudy[87]wasalsoadoptedandcheckedsuccessfullyagainstexperimentalIRgasphaseintensities.[88]InduceddipolederivativesareresponsibleformostoftheobservedliquidstateIRintensityintheOHstretchingregionwhilethebendingintensityislargelydeterminedbythepermanentdipolederivative.91

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Toevaluatethepresenttheoryinthespeciccaseof2DIRspectroscopywithnotimedelaybetweenthesecondandthirdlaserpulsesitisnecessarytocalculatetheclassicaltwo-timeTCF,BRt1;t2;t3=hjt1it3+t1kt1`iandsubsequentlyFouriertransformtheresult.ThisimplicitlyinvolvescalculatingtheTCFatbothpositiveandnegativetimeinperformingtheintegral:BR!1;!3=1 22Z1dt1e)]TJ/F19 7.97 Tf 6.586 0 Td[(i!1t1Z1dt3e)]TJ/F19 7.97 Tf 6.586 0 Td[(i!3t3hjt1it3+t1kt1`i.1Tocalculatethecorrelationfunctionitisconvenienttoonlyworkwithpositivetimesandthusitisnecessarytoconsiderthebehaviorofthetwo-timeTCFinfourdierentcasescorrespondingtoquadrantsinthetwodimensionalplanedeterminedbyt1andt3.ItiseasytoshowthatinthediagonalquadrantsboththequantumandclassicalTCF'sareidentical;thequadrantsaredenotedbelowbyasuperscript++fort1>0t3>0,and\000whent1<0t3<0.Thustheycanbeevaluatedforpositivevaluesofthetimeargumentsas:B++=hjt1it3+t1kt1`i.Intheo-diagonalquadrants+;)]TJ/F15 11.955 Tf 9.299 0 Td[(and)]TJ/F18 11.955 Tf 9.299 0 Td[(;+thetwofunctionsarealsoidenticalandcanbeevaluatedforpositivevaluesofthetimeargumentsas:B+)]TJ/F15 11.955 Tf 10.405 -4.339 Td[(=hjt1+t3it1kt1+t3`t3i,andheretimestationaritypropertieshavebeenusedtorewritetheTCFintermsofpositivetimes;[1]theorderofthedipoleevaluationisirrelevantclassicallyandispresentedinthequantummechanicallycorrectform.ThustwodierentclassicalTCF'sneedtobecalculatedandbothcontributetothe2DIRsignalatagivenpairoffrequencies.NotethatthiscanbeavoidedincalculatingRt1;t2intheclassicallimit,Equation5.23,becausetheresultisgiveninthatcaseby92

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timederivativesthatcanbecalculatedvianitedierenceusingthedatalocalintimeandFouriertransformingtheTCFcanbeavoided.[63]Theschemeemployedhere,wherehighfrequencyvibrationsareofinterest,istoperformaseriesofonedimensionalFFT'stoevaluateEquation7.1andusetheB++datainthediagonalquadrantsandB+)]TJ/F15 11.955 Tf 11.937 -4.339 Td[(dataintheo-diagonalquadrants.ThenatureofthecontributionmadebythetwoTCF'scanbeunderstoodbyrewritingEquation7.1as:BR!1;!3=1 22Z10dt1cos!1t1Z10dt3cos!3t3)]TJ/F18 11.955 Tf 5.479 -9.684 Td[(B+)]TJ/F15 11.955 Tf 9.741 -4.936 Td[(+B+++1 22Z10dt1sin!1t1Z10dt3sin!3t3)]TJ/F18 11.955 Tf 5.479 -9.684 Td[(B+)]TJ/F21 11.955 Tf 9.741 -4.936 Td[()]TJ/F18 11.955 Tf 11.955 0 Td[(B++.2Therefore,thefrequencycontributionstothe2DIRspectrumarisefromaddingthedoublecosinetransformofthesumofthetwoTCF'stothedoublesinetransformoftheirdierence.Therefore,B+)]TJ/F15 11.955 Tf 12.509 -4.338 Td[(andB++werecalculatedforliquidwaterusingthemodeldescribedabove.ThetwodimensionalTCF'sdecayedslowly[89]andwerecalculatedusingamaximumcorrelationtimeof20psineachtimedimension.OverthisdurationeachTCFdecaysapproximately90%fromitsinitialvaluebuthasnotquitereachedtheasymptoticvalueofzeroandthenalvalueat20psisslightlydierentfordistincttimeslices.Therefore,dierentbaselinevaluesaresubtractedoforeachsliceinperformingtheseriesofonedimensionalFouriertransforms.Thisdoesnotsignicantlyaectthelineshape,andthiswascheckedbyperformingtheidenticalprotocolonthelinearIRabsorptionlineshape[85]andtherelevantTCF,,issimilartothe2DIRtimeslicesanditwasunaectedascomparedtoaFouriertransformedTCFthatdecayedfullytozero.Ideally,longercorrelationtimeswouldbeemployed,allowingthesignaltodecaytozero.However,thecomputational93

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Figure7.1:ThemagnitudeoftheFouriertransform,inarbitraryunits,ofthetwo-timecor-relationfunctionofthesystemdipole,jBR!1;!3j,isshownfortheintramolecularstretchingregion.demandsbecomelargeasthetimeisincreasedduetothesmalltimestepneededt=4.0fstoproduceaNyquistfrequencythatallowstheresolutiontheO-Hstretchfrequencyofwater[86]andthetwodimensionalnatureofthedata.Forexample,toallowtheTCFtoget99%ofthewaytothezerobaselinewouldrequireamaximumcorrelationtimeofapproximately50psthatrepresentthewellaveragedportionofacalculationcorrelatedouttoabout100psandthiswouldproduceseveralgigabytesofdatatobestoredandprocessed.94

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7.2The2DIRSpectrumofAmbientWaterFigure7.1showsthemagnitudeoftheFouriertransformedfullypolarizedtwo-timeTCFi.e.thedipolecomponentsareallthesame,jBR!1,!3j,fortheintramolecularvibrations;multiplyingthisresultbytheappropriatefrequencyfactorsinEquation5.24wouldgiveourapproximationtothethirdorderresponsefunction.Forthepurposeofdemonstratingthetheoryonlythefullypolarizedsignalwillbepresented.NotethatBRt1;t2;t3anditsFouriertransformsarerealfunctionswhileRt1;t3isrealanditsFouriertransformispurelyimaginary.Theybothhavepositiveandnegativecontributionsintimeandfrequency.Asexpected,astrongdiagonalsignaldominatesthefrequencydomainlandscapeindicatingstrongself-couplingofvibrationalmodes[51]{dominantpeaksarepresentinthewaterbending1800cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1andstretching3300cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1regions.Also,thereisacontinuousnon-zerosignalalongthediagonalanalogoustothenon-zerosignalpresentinthelinearIRspectruminthisregion.Asidefromthedominantdiagonalsignal,slowlydecayingridgesarepresentthatrunalong!3.Theseridgesarepositionedat!1=1800cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1and3300cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1indicatingcouplingbetweenwaterbendingandstretchinginthecondensedphaserevealedviathetwo-timecorrelationfunction.Also,notethatwhilewaterhasadistinctantisymmetricandsymmetricstretchinthegasphase,thecondensedphasenormalmodesarenearlypurelocalO-Hstretchingmodesandthemixing"ofthetwogasphasevibrationalresonancesintheliquidleadstothebroadO-Habsorption.[22,60,61]OnedimensionalfrequencyslicesofBR!1,!3areshowninFigures7.2a-ctorevealthedetailedlineshapes.Figures7.2aand7.2bareedgeslicesalong!3with!1=0cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1and95

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Figure7.2:ThreeonedimensionalfrequencyslicesofBR!1,!3areshown:aalong!3with!1=0,balong!1with!3=0,andcalong!1=!3Whilethespectraareshowninabsoluteunits,thediagonalsliceisshownsmoothedusingasimplemulti-pointaverageinbothfrequencydirectionstoeliminatetheoscillationsnearandalongthediagonalapparentinFigure7.1andthisleadstoasmallersignalmagnitude.96

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!1with!3=0cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1respectively.Figure7.2arevealsasignalwithadierentphaseforthebendandstretch.Figure7.2bshowsasignalsimilartothelinearIRexperimentalthoughthebendisrelativelymoreintense.Figure7.2cshowsadiagonalslice,where!1=!3,ofBR!1,!3andherethephasesofthebendandstretcharereversedfromFigure7.2aandtheintensityofthebendisdiminishedwithrespecttotheO-Hstretching.TherelativeintensitiesalongthediagonalarereminiscentofthelinearIRexperiment.Seeingdieringphasesfortheresonancesisinterestingandsimilarphaseinformationfromthe2DIRexperimenthasbeusedtoextractstructuralinformationaboutthesystembeingprobed,forexampleindeterminingtherelativeorientationoftransitiondipolesofcoupledvibrationalmodes.[53]Todisplaythedatainadierentformat,Figures7.3a-bdisplayaseriesofsingletimeFouriertransformsofthetheTCF,BRt1,!3andBR!1,t3respectively.Inbothcasesthesignalmagnitudesdecaymonotonicallywithtimeandthewidthsstayaboutthesame.Itremainstobeseenwhatwayofexaminingthespectraprovesmostuseful.Forexample,questionsariseinthismixedtimefrequencyrepresentationsuchas-whatphysicalinsightcanbeobtainedbyunderstandingtherateofdecayofthesignalintimeatagivenfrequency?Toquantitatethesignaldiminution,thedecayoftheOHstretchingbandmaximumamplitudevstimewasttobothsingleanddoubleexponentialsandthetwoexponentialtwasmuchbetteratcapturinganapparentfastandslowerdecayprocess.TheOHstretchingpeakhadafastandslowtimedecayconstantof42fs2:4psrespectively.ThesenumberscomparefavorablytorecentspectroscopicmeasurementsofvibrationalrelaxationofOHstretchinginD2Othatalsondtwodecaytimescales.[53,90,91]Therapid,resonantdecaytimeis97

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Figure7.3:1-DFouriertransformsoftheTCFareshown:,aBRt1,!3,bBR!1,t3.Ineachcasethetimesshownareslicesat0,1,2,3,4,5and6ps.98

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Figure7.4:Theoreticalspectraforambientwaterarepresented.upperThelinearIRspectrumofwaterisshowninthetoppanel.lowerAcontourplotoftheapplicationofthepresenttheoryofthethirdorderresponsefunction,R!1,!3isshowninthebottompanel.fasterinourcasebecausetheOHoscillatorisstronglycoupledtootherOHstretchesviahydrogenbondingandintheircasetheODstretchisonlynon-resonantlycoupled.Theslowerrelaxationtimeisduetostructuralrearrangementsthatwouldbeexpectedtobesimilarforbothpureandisotopicallymixedwaterandtheexperimentsobtainsimilarvaluesforthisrelaxationtime.Displayingthetimedependentfrequencydatahighlightstherichinformationcontentof2DIRspectroscopyandoneofthegoalsofusingshortpulsesintimeistoresolvetimedependentstructuralanddynamicalfeatures.Figure7.4uppershowsthetheoreticallinearIRspectrumthatresultsfromFourier99

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transformingthedipole-dipoleautocorrelationfunctionderivedfromthewatermodelde-scribedabove.[22]Inthegure,thelocationofthemajorvibrationalresonancesisclearanditispresentedtoaidininterpretingthemorecomplicated2DspectrumpresentedinFigure7.4lower.Applicationofthepresenttheoryofthefullypolarizedthirdorderresponsefunctionispresentedasacontourplotofthenon-quantumcorrectedsignalR!1,!3inFigure7.4lower.AsexplainedinSection5.3,Equation5.24isevaluatedusingBR!1,t2=0,!3andsetting!2=10000cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1tomimicthe2DIRexperiment.Rt1;t3becomesequivalenttothethirdorderpolarizationPt1;t2=0;t3inthelimitofdeltafunctionpulsesalthoughthislimitisnotgenerallyexperimentallyfeasibleandinthatcase,Equation5.1needstobeevaluatedtocalculatetheobservedpolarization;thislimitingcaseisagoodapproximationtotherealexperiment.[36]ItshouldbenotedthatthebackFouriertransformofR!1,!3isthetimedomainre-sponsefunctionRt1;t3thatismeasuredintheidealshortpulselimitexperiment.How-ever,afrequencydomaindisplayofexperimentaldatarequiresuseoftheFourier-LaplacetransformofthetimedomainresponsefunctionthatincludesboththefrequencydomainresponsefunctionfullFouriertransformitselfandprincipalpartintegralsoverthefunc-tion.Forsimplicity,weshowonlyR!1,!3.Itcontainsthesameinformationastheexperimentalobservable,andissusceptibletothesamephysicalinterpretation,butdoesnotcorrespondexactlytotherealorimaginarypartsoftheFourier-Laplacetransform.Figure7.4lowerrevealsastrongechosignalalongthediagonalwithmajorpeaksat100

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about1850cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1bendingand3300cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1stretching.Thefrequencyfactorsthatmulti-plythefrequencydomainTCFinEquation5.24signicantlymodifytherelativeintensitiesonandothediagonal.Thediagonalpeaklocatedat3300cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1iselongatedparalleltothediagonalindicatinginhomogeneousbroadening.Itisalsoelongatedatanangletothediagonal,parallelto!3axis,indicatinglifetimebroadening.[34]Theangleoftheelonga-tionofpeaksrelativetothediagonalyieldsinformationonthedegreeofcorrelationofthebroadening.[34]Althoughthemagnitudeofthesignalintheo-diagonalcouplingregionsislowcomparedtothediagonal,signicantsignalispresentandexperimentalpulsesequencescanbecon-structedthatsuppresstheoftenlessinterestingstrongdiagonalfeatures;[51]aprincipalgoalof2DIRspectroscopyistouncovercouplingsbetweenvibrationalmodes.Thebroadandextensivecouplingspresentinthewater2DIRspectrumareconsistentwiththefastvibrationalenergyredistributionknowntotakeplaceinwater.[90,92]Figures7.5a-dshowonedimensionalfrequencyslicesofR!1,!3.Figure7.5adisplaysaslicealong!1with!3=0cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1.Theintensewaterbendingandstretchingpeakslocatedintheregionsof1850cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1and3300cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1respectively,dominatethefrequencyslicelineshape.Figure7.5bshowsthediagonal,!1=!3,sliceofR!1,!3.Theintensediagonalwaterstretchingpeakshowsupinthe3300cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1regionandthelessintensewaterbendingpeakshowsupinthe1850cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1region.Figure7.5cpresentsaslicealong!1with!3=1800cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1.Thewaterbendingpeak,locatedintheregionof!1=1850cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1,correspondstoadiagonalpeak.TheinsetprovidesdetailsoftheweakerwaterOHstretchpeak,withanegative101

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Figure7.5:FourfrequencyslicesofR!1,!3areshown.Insetsprovidedetailedstructureoflineshape.aalong!1with!3=0cm)]TJ/F15 11.955 Tf 7.085 -4.338 Td[(1,balong!1=!3,calong!1with!3=1800cm)]TJ/F15 11.955 Tf 7.085 -4.338 Td[(1,dalong!1with!3=3300cm)]TJ/F15 11.955 Tf 7.084 -4.338 Td[(1.102

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amplitude,locatedinthe!1=3300cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1region.Alsoofinterestaretherelativeintensitiesofthediagonalando-diagonalpeaks.Figure7.5dshowsaslicealong!1with!3=3300cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1.Thedominantpeakislocatedintheregionof!1=3300cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1andrepresentsthedominantpeakofthediagonal.O-diagonalpeaksarealsolocatedapproximatelyintheregionof!1=1850cm)]TJ/F16 7.97 Tf 6.586 0 Td[(1ascanbeseenintheinset.7.3QuantumCorrectionsAswaspointedoutinSection5.3,EquatingtheTCFBRt1;t2;t3withitsclassicalcounter-partisafurtherapproximationandthisisacommonpracticeinthecaseofonedimensionalTCF's.AbetterapproachistoquantumcorrecttheFouriertransformoftheclassicalTCFbasedontherelationshipbetweenthequantumandclassicalTCFforanexactlysolvablemodelsystem,e.g.aharmonicsystem.Inthemultidimensionalcasethisissomewhatmoredicultandthiswillbediscussedbelow.TwoquantumcorrectionschemesforourtheoryofRwereconstructedandtestedforplausibility.Theseschemesprovidetwoalternativesthatrelatetherealpartofthequantummechanicaltwo-timeTCFtothecorrespondingclassicalTCF.ThequantumcorrectionsarechosentoexactlyrelatetherealpartofthesixlowestordercontributingtermsforaharmonicsystemBfunction,Equation5.17,withtheirclassicallimits.ThatratioofthequantumandclassicalTCF'sisthenusedtoquantumcorrectthefrequencydomainclassicalTCF:[63]BQR!1;!2;!3=~!1 8[!1)]TJ/F18 11.955 Tf 11.955 0 Td[(!3+2!2coth2~!1 2.3+!1+!3)]TJ/F15 11.955 Tf 11.955 0 Td[(2!2csch2~!1 2]103

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BC!1;!2;!3:BQR!1;!2;!3=~!3 8[!3)]TJ/F18 11.955 Tf 11.955 0 Td[(!1+2!2coth2~!3 2.4+!1+!3)]TJ/F15 11.955 Tf 11.955 0 Td[(2!2csch2~!3 2]BC!1;!2;!3:BQRrepresentstherealpartofthequantumtimecorrelationfunctionandBCrepresentsitsclassicallimit.NotethatonlytheharmonicfrequencydependenceonisknownfromEquation5.17andnotthe!1,!2,and!3dependence.Itmightbepossibletounambiguouslydeterminethisdependenceforamorecomplexreferencesystem,perhapsbyconsideringastepwisetimedependentharmonicoscillatormodelsystem.Here,thetwopossibilitiesgiveninEquations7.3and7.4werechosenasthesimplestfunctionalformsrelatingBQRandBCexactlyforaharmonicsystem.Equation7.3maybemorereasonablethanEquation7.4becausethelatterequationpredictsnosignalalongthe!1axiswith!3=0,althoughEquation5.24approacheszeroforhighfrequenciesevenuncorrected.Equation7.3givesthethirdorderresponseaszeroalongthe!3axiswith!1=0andthisisconsistentwiththeexactresult,Equation5.11.Figures7.6a-cshowslicesofthequantumcorrectedR!1,!3signalinordertocomparethetwoquantumcorrectionschemesconsidered.Figures7.6a-bshowthe!1with!3=0cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1sliceandthe!1=!3slicerespectivelyusingthequantumcorrectionschemeinEquation7.3.Figure7.6cshows!1=!3forthequantumcorrectionschemeinEquation7.4anditwouldgivenosignalalong!1with!3=0cm)]TJ/F16 7.97 Tf 6.587 0 Td[(1.Thechangestothepresentedsignalsforbothschemesis104

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Figure7.6:ThreefrequencyslicesofthequantumcorrectedR!1,!3signalareshown.Insetsprovidedetailedstructureoflineshape.aalong!1with!3=0withquantumcorrec-tionschemeshowninEquation7.3,balong!1=!3withquantumcorrectionschemeshowninEquation7.3andcalong!1=!3withquantumcorrectionschemeshowninEquation7.4.105

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largelyonlyinmagnitudeandthereisnosignicantdierenceinlineshapewithandwithoutquantumcorrectionasisnotedbycomparingthethree!1=!3slices,Figure7.5bandFigures7.6b-c.Thisresultsisconsistentwithquantumcorrectionschemesforonedimensionalspectroscopiesthatchangethemagnitudeofthesignalmorethanthelineshapebecausequantumcorrectionsarenearlyatfunctionsoverthewidthofavibrationalresonance.[60,93]ThisimpliesthatthepresentclassicalMDbasedTCFtheorymaybeabletocapturetheessentialfeaturesof2DIRspectroscopyevenwithoutquantumcorrection.106

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Chapter8MolecularVolumeCalculationsThischapterpresentsanovelMDmethodfordeterminingboththermodynamicvolumesofsolvatedmoleculesandtimedependentvolumechangesinthecondensedphase.ThemethodutilizescontemporaryMDmethods,[94,95]specically,extendedsystem,NPTMDtodeterminethevolumeofasystemwhilesimultaneouslyexploringthedynamicsofthesystem.8.1MolecularDynamicsTheoriginalresearchpresentedinthisthesiswasallperformedwiththeaidoffullyatomisticclassicalmoleculardynamicssimulations.ThesesimulationsinvolveaccountingforallatomsinthesystemandsolvingNewton'sequationsofmotionforeachatom.Theabilityofclassicalmechanicstosucientlydescribethedynamicalbehaviorofmaterialsisthecaseforawiderangeofmaterial.Infact,quantumeectsareofconcernonlywhenweareconsideringlightatomsormoleculesorhighfrequencyvibrationssuchthatthevibrationalenergyismuchgreaterthanthethermalenergyofthesystem,e:g:h>>KT.107

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Forpedagogicalpurposes,wecanconsideraverysimple,ideologicalsimulationtoun-derstandthebasicfeaturesofasimulation.Wewouldrststartwiththesysteminterestandtheconditionsthatdeneite:g:thenumberofparticles,volumeorthedensity,etc.Initialvelocitiesandpositionsarethenselected.Typically,wecouldstartfromalatticeforsimplesystemssuchasatomicspecies,orformorecomplicatedstructuressuchasproteins,acrystalstructurecouldprovidetheinitialconguration.VelocitiescanberandomlyselectedfromaBoltzmanndistribution.Fromtheinteractionpotential,wecalculatetheforcesonalltheatomsandusetheseforcesintheintegrationoftheequationsofmotion.Theforcecalculationandintegrationstepsarerepeatedforeachtimestep.NumericalalgorithmsareusedtointegratetheequationandarebasedonTaylorexpansionsofthepositionvariable.Ateachtimestep,wecanrecordtheinstantaneousmeasurementsofinterest.CalculatingobservablesfromclassicalMDrequiresbeingabletoexpresstheobservableintermsofthepositionand/ormomentumofthesystemparticles.Simulationswhichinvolvexingthenumberofparticles,N,andthevolume,V,ofasystemandsolvingNewtonsequationsnaturallyhavetheenergy,Easaconstantofmotion.FortheseNVEsimulations,thetimeaveragesfromthesimulationsdocorrespondtoen-sembleaveragesfromthethermodynamicmicro-canonicalNVEensemble.However,NPTMDalgorithmsarenotstrictlyequivalenttomicrocanonicalNVEdynamics,althoughthemethodemployedheresamplestheisothermal-isobaricNPTensembleexactly.[94]Thesemethodscoupletherealsystemvariablestoctitiousvariablesthatregulatethethermody-namicpropertiesofintereste.gthermostatthetemperatureandbarostatthepressuresuch108

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thattheyuctuatearoundthedesired,presetaveragevalues.ThemethodsforcalculatingthermodynamicvolumesisthereforeexactforagivenMDpotentialenergymodel,andtheonlyissueiswhetherdynamicaleventsobservedarephysicallyrelevant.NPTdynamicsdoescloselymimictruemicrocanonicaldynamicse.g.theperturbationofthedynamicsisorder1=p 3NwhereNisthenumberofatomsinthesystem,andhasevenbeensuggestedasthemethodofchoiceforbiophysicalsystems.[96]8.2ThePluckMethodThethermodynamicvolumeofasolutionisobtaineddirectlyfromthevolumecoordinateinNPTMD.Consider,e.g.,asinglesoluteinasolvent.Solutevolumescanbecalculatedbyplucking"thesolvatedspeciesfromthesystem,i.e.simulatingtheremainingsystemintheabsenceofsolute-solventforces.Afterre-equilibration,thevolumeoftheremainingsystem,inthiscasepuresolvent,isdeterminedusingNPTMD.Thedierencebetweenthesolutionandsolventvolumesdeterminesthesolutemolecularvolume.Plucking"themoleculefromanotherwiseequilibratedsystemoftenproducesaninitialconditionforasystemcongurationallynearthenewequilibriumandthusaidsinequilibration;thisisespeciallyimportantinsimulatingcomplexsystems.Also,insimplesolutionsconsistingofasingleyetvariablesolute,thevolumeoftheneatsolventneedonlybedeterminedonce,simplifyingthecomputation.Themethodis,however,notlimitedtosimplesolutionsandcanbeusedinverycomplexbiologicalsimulationstodeterminemolecularvolumesofanyofthecomponentsofanassemblyofbiomoleculesandsolvents.109

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NPTdynamicsproducesauctuatingvolumecoordinateandthethermodynamicvol-umeistheaveragevalueovertime.Thisleadstoanuncertaintyinthethermodynamicvolumecalculationthatmustbeassessed.ItisdemonstratedinSection9.2thatthevolumeuctuationsareGaussianandthusthestandarddeviationofthevolumeuctuationisausefulmeasureoftheuncertainty.However,asaconsequenceofthedynamicalnatureoftheMD,successivevolumevaluesarenotstatisticallyindependent.Followingearlierwork,thecorrelationtimeofthevolumecoordinateiscalculatedandonlyuncorrelatedvaluesaresampled.[97{99]Thisisequivalenttosamplingmorefrequentlyandcorrectingforthecorrelationbetweenvolumes.Fortheaqueoussystemsinvestigatedheresolutevolumeun-certaintiesofabout1.0ml/moleareobtainedfromafewnanosecondsofdynamicsandallthevolumevaluesinthepaperaregivenwithrespecttothechangeinsolutevolume.ThemotivationinusingNPTdynamicstocalculatevolumechangesviathisapproachistomimicphotothermalexperimentsonbiologicalsystemsthatalsodeterminemolecu-larvolumechangesonnanosecondtimescaleswithsimilarprecision.[100{102]Forexample,photothermalexperimentscanidentifyprotein/peptideintermediateswithcharacteristicvol-umesthathavelifetimesofseveralnanoseconds.Thepresenttheoreticalmethodscanbeemployedinthesamefashion.Transientspecieswithcharacteristicvolumescanbeiden-tiedbystatisticallysignicantchangesinthevolumecoordinateovertime,indicativeofmetastableequilibriumbetweenthesoluteandsolvent.MDcanthenprovidemicroscopicresolutiontotheobservationbyidentifyingthestructuresofanyintermediates.Further,whiletheNPTMDisnotdynamicallyexact,itisalwayspossibletoverifycongurational110

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eventsbyrepeatingsimulationsmicrocanicallytoverifytheveracityofthedynamics.Thedynamicinterpretationoftheproposedmethodsisthereforeacomputationalconvenience.Computationaleciencyis,however,clearlydesirablegiventheinherentchallengeofsim-ulatinginterestingbiologicalsystemsforhundredsofnanoseconds,arelativelyshorttimescaleoverwhichtolookforlargeconformationalchangesinbiopolymers.Toaccesslongertimescalesandtosamplevolumeeciently,multipletimescaleintegrationtechniquesareemployedandpermittheuseoflargerMDtimesteps.[95]Itisnotablethatphotother-malmethodscanalsomapoutenthalpyprolesoversimilartimescales,andMDdirectlycomplementsthesemeasurementsbyprovidingamolecularinterpretationoftheenergetics.Othereectivemethodsexisttocalculatemolecularvolumesandrelatedexcesscompress-ibilities[103{106]butthepluck"methodisideallysuitedtomodelingbiologicalsystemsandtheirtimeevolution.Theexibilityindissectingthephysicaloriginofvolumechangesusingthepluck"methodwillalsobedemonstratedbytheexamplesthatwillbepresented.Forexample,byvaryingpotentialenergyinteractionsbetweenasoluteandsolventthevolumecanbedissectedintodierentcontributionsinathermodynamicallyconsistentmanner.[107]Itisimportanttonotethattheprecisionofmeasuringvolumechangeswillbeaectedbythedurationofthesimulationthatiscomputationallypossibleforlargersystems.Forexample,whenmodelinglargesolvatedproteinse.g.greaterthanonehundredresidues,simulationsarecurrentlylimitedtodurationsontheorderofahundrednanoseconds.Itisdemonstratedbelowthat10nsofMDwouldpermitthedeterminationofthevolumetowithinabout4.0ml/mole.Also,theproteinvolumeislargerinthiscaseandtherelativeer-111

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rorinthevolumemeasurementisreduced.Further,volumechangesassociatedwithfoldingmaybegreater,onaperresiduebasis,forlargerproteins.[108]InSection9.1,themolecularvolumeofa-sheetpeptide,thathasbeeninvestigatedexperimentallyusingphotothermalmethods,[100]iscalculatedtodemonstratetheanalysisinvolvedinthepluck"method.Thetechniqueisthenappliedonmodelsystemsinclud-ingneatwater,bothneutralandctitiouschargedaqueousmethaneinSection9.2.Aseriesofmoleculeswaschosentohighlightthemethodsabilitytoprobetherelativevolumechangesassociatedwithelectrostrictionandionicsolvation.Thepresentmethodscanalsobeextendedtocalculateexcesscompressibilitiesbysimulatingatdierentpressuresandcalculatingthecompressibilityvianitedierence.[104{106]112

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Chapter9MolecularVolumeResultsInthischapter,resultsarepresentedfortheapplicationofthepluck"method.Twosystemsareinvestigated.Fortestingthemethod,asimplemodelsystemofsolvatedmethaneistestedandtheeectsofelectrostrictionareinvestigatedbyadjustingthepointchargesontheatoms.Electrostrictioncontributionsareshowntobepotentiallysignicantdependingonthenatureofthesystemleadingtolargevolumechanges.Preliminarydataforabeta-sheetpeptideispresentedhereaswell.9.1MethodsandApplicationstoaModelSystemFigure9.1presentsasnapshotofasolvated-sheetpeptideincludingapanelwiththesolventremovedforbettervisualization.Thepeptidewaschosenbecauseitiscurrentlythesubjectofexperimentalinvestigationusingphotothermalmethods.[100]Chanandco-workerssynthesizedacaged"unfoldedversionofthepeptidethatcanbephotolysedinneatwatersolutiontoinitiatepeptidefolding.[100,109]Usingphotothermalmethodsitisthenpossibletomapoutvolumeandenthalpyprolesduringtheroughly1.0-second113

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Figure9.1:Asnapshotofthe-sheetpeptideisshownrightwithsolventandleftwithwaterremovedforbettervisualizationandinadierentorientationtobetterdisplaythethreedimensionalstructureofthepeptide.Thecolorsrepresenttheatomtypesasfollows:C-green,O-red,N-blueandH-white.foldingprocess.ThefoldedstructurewascreatedbasedonanNMRstructure,[109]andtwoseparatefoldedsystemswerepreparedandtheirequilibriumstructureswerecomparedandfoundtobeessentiallyindistinguishable.Todemonstratethepluck"methodFigure9.2showsatimetraceofthesolvatedpeptidevolumecoordinate.Forcomparison,therelaxationofthesystemvolumeafterthepeptideisplucked"fromsolutionsisalsoshown.TheinsetofFigure9.2showstheshorttimedynamicsofthesystemtohighlightthetransientvolumechange.Thesystemquicklyapproachesanewequilibriumandhasasystemvolumecharacteristicofpurewater50.0psafterremovalofthepeptide.Equilibriumvolumeuctuationsarethenfollowedforseveralnanosecondsinordertodeterminethevolumeof114

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Figure9.2:Thecurvewiththelargeraveragevalueshowsatimetraceofthevolumeuctu-ationsforthefoldedaqueous-sheetpeptide.Thelowercurvedisplaystherelaxationoftheneatwatersystemvolumetoitsequilibriumvalueafterthepeptideisplucked"fromsolution.Theinsetpresentstheshorttimevolumeuctuationstohighlighttherelaxationtoequilib-rium.Notethatthevolumeoftheplucked"systemtransientlyspikestoavaluegreaterthanthepreviousequilibriumuctuationsafterthepeptideisremoved.Thehorizontallinesrepresenttheaveragesystemvolume.thesystemstoadesiredprecision;2.0nsofthedynamicsareshowninFigure9.2.Fluctuationsofobservablequantitiesfromtheirmeans,obtainedfromMDsimulations,aretypicallyGaussian.Theycanthusbecharacterizedbytheirstandarddeviation,.TheinsetofFigure9.3presentsahistogramofthevolumeuctuationsofthesolvatedpeptidesystem.ClearlythedistributioniswelldescribedasGaussian.Ifthesuccessivevaluesof115

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thevolumewereuncorrelated,onecouldcalculatetheuncertaintyofthevolumesimplyas:V==p @.1InEquation9.1@isthetotalnumberofsamples.However,closelyspacedvaluesofanobservablequantityinthiscasethevolumeisofinterestarenotstatisticallyindependentbecausetheyareconnectedtoeachotherimplicitlybythedynamicalequationsofmotion.Itispossibletodeneacorrelationtime,tc=st,duringwhichthevolumesarenotindependent,andsisthestatisticalineciencyornumberofcorrelateddatameasurements.[97,99]Multiplyingsbyt,thetimelengthbetweensuccessivemeasurements,givesthecorrelationtime,tc.ThesvalueiscalculatedbyperformingvolumeaveragesoverblocksoftimeofsuccessivelylongerlengthsendingwiththeentirelengthoftheMDrun.Theparametersisformallydenedby:[98]s=limB!1B2B=22B=1 NBNBXB=1B)]TJ/F18 11.955 Tf 12.62 0 Td[(2.2InEquation9.2,NBisthenumberofblocksoflengthBsuchthattheproduct,NBB=@,thetotalnumberofsamples.TheinherentcorrelationsthenmodifytheuncertaintyinthevolumeasV=p s=@.ThismodiedformulareducestoEquation9.1ifthetimebetweensuccessivevolumesamplesislongerthantc,andinthatcases=1.ForlongMDrunsitiscomputationallyconvenienttosaveaminimalamountofdataandthereforechooseasamplingtimeslightlygreaterthantc,andthisapproachwasadopted;sandtcaredeterminedinitiallyviatrialMDruns.116

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Figure9.3:Theuncertaintyinthevolumeofthe-sheetpeptidesystemasafunctionofthelengthoftheMDsimulation.Volumeuncertaintiesforspecictimesofinterestarealsoiden-tied.Theinsetdisplaysahistogramofthesystemvolumeuctuationsandasuperimposedgaussianfunctionwithastandarddeviationcalculatedfromthevolumeuctuations.117

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Figure9.3alsodisplaysacurvedemonstratinghowtheVvaluesdecreaseovertimeforthesolvated-sheetpeptide.Thegraphdemonstratesthat50.0nsofdynamicsresultsinanuncertaintyof0.68ml/mole.Thisseveralnanosecondtimescalecorrespondstoatypicalphotothermalexperimentaltimeresolutionforidentifyingdynamicalintermediates.Thevolumeresolutionoverthistimescaleissucienttoidentifyrelativelymodestconformationalchangesinapeptide/proteinorotherbiomolecule.Infact,onestudyestimatedthatvolumechangesofabout3.0ml/mole/residuearetobeexpectedforahelixtocoiltransitioninaprotein.[107]Becausetheuncertaintyinthevolumediminishesonlyasthesquarerootofthenumberofvolumemeasurements,50.0nstimesscalesareneededtoreducetheerrortothe0.68ml/molelevel.However,itisencouragingthatmeaningfulvolumedierenceswereobtainedfromonlyananosecondofdynamicswheretheerroris4.2ml/mole.Thegurealsogivesavolumeof1668.+/-2.4ml/moleforthefoldedpeptidefromthe7.5nsofdynamicsthatwereperformedinthisstudy.Themolarvolumeoftheneatwatersystemwasdeterminedoncewithsucientprecisiontonoteectthenetvolumeuncertainty,andthiswillbediscussedfurtherinSection9.2.Forthefoldedpeptide,thevalueofthestatisticalcorrelationtimewasfoundtobe,tc=2.4ps.Preliminaryinvestigationsinvolving5.8nsofMDforanunfoldedcongurationofthepeptideresultedinavolumeof1672.+/-3.1ml/mole.Itisinterestingthatthefoldedandunfoldedstate,towithinthepresentstatisticalcertainty,havethesamevolume.Thisissomewhatsurprisingduetothedierenceinsolvationstructureassociatedwitheachstate.LongerMDsimulationsarerequiredtodistinguishbetweenthevolumesofthese118

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twodistinctconformationalstates.Further,thisresultdoesnotnecessarilyimplythatdynamicalintermediateswithsignicantlydierentvolumesarenotpresentduringfolding.Furtherexperimentalandtheoreticalinvestigationsarerequiredtodeterminethevolumechangesduringfolding.Thedeterminationandstructuralanddynamicalorigin[106]ofanyvolumechangeuponfoldingisthesubjectofongoinginvestigation.Itisnotablethatlargerproteinsmayexhibitlargerperresiduevolumechangesmakingthemamenabletoinvestigationusingthepresenttechnique.[108]ThesimulationswereperformedusingacodeoriginallydevelopedbytheKleingroupattheCenterforMolecularModelingattheUniversityofPennsylvania,andtheSpacegroupiscurrentlyaco-developeranduserofthecode.Itisafastcodethatincludesparallelexecution,extendedsystem,particlemeshEwaldandmultipletimescaleintegrationalgorithms.ThecodehasbeenemployedinanumberofbiologicalMDsimulations.[65,110{112]TheextendedsystemMDmethodsemployedrequireacouplingbetweentherealsystemandextendedsystemvariablesandthiscouldaltertheeectivenessofthevolumesampling.Avarietyofcouplingconstantsrepresentingthecouplingofthebarostattothesystembetweenthevolumecoordinateandthemolecularcoordinatesweretriedinpreliminarysimulations.Physicallyacceptablevaluesofthebarostat"mass[94,95]ledtoonlyaweakdependenceonthesamplingeciencyforthesolvatedpeptidesystem.Inthesimulations,thewatermodelwasaexibleSPCmodeldescribedelsewhere.[22,60,61]Theforceeldincludespartialchargesonthehydrogenandoxygenatomsrepre-sentingthecondensedphasepermanentdipole.TheproteinforceeldwasAmber9.All119

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theaqueousmethanesystemsemployed62watermolecules.Thepeptide[109]hadnonetchargeandwassolvatedwith810watermoleculesusingcubicperiodicboundaryconditions.Anallatommethanemodelwasusedincludingaexibleforceeldttoreproduceex-perimentalinfraredfrequencieswithharmonicC-Hbonds.[113]Lennard-Jonesinteractionswereintroducedonlybetweenthemethanecarbonandwateroxygenwith=3:33Aand=51:0K.TheequilibriumbondlengthfortheC-Hinteractionis1.09A,anditsgeometryistetrahedral.Whensimulatingionicsystemsthenetchargeonthesystemiscanceledbyemployinganeutralizingbackgroundinthestandardfashion.[97]Inallcases,thetemper-aturewas298Kandthepressurewas1.0atmosphere.Themultipletimescaleintegrationmethodsallowedthestableuseof4.0fstimestepsperformingNPTdynamicsand8.0fstimestepswhensimulatingatconstantNVE.Thissystemservestointroducethemethodanditsproperties.UsingNPTMDinthisfashionpermitsboththecalculationofequilibriummolarvolumechangesandidenticationofmetastabledynamicintermediatespeciesexhibitingdistinctmolarvolumes.Themethodwillalsobeideallysuitedtodecomposingvolumechangesintodieringphysicaloriginssuchasattributingavolumechangetopeptideorsolventrearrangementsandassessingtheroleof,e.g.stericsandelectrostatics;thiswillbediscussedfurtherinthenextsection.9.2ResultsfromaSimpleModelSystemInordertotesttheapproachanddemonstrateitsexibilitymolarvolumeswerecalculatedforneatwaterandasolvatedmethanemodel.Thesesystemswerethesubjectofearlierinvestigations,althoughsomewhatdierentpotentialswereusedinthatwork.[104]First,120

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thevolumeoftheexibleSPCwaterwascalculatedtohighprecisiontominimizetheerrorincalculatingvolumesofaqueoussolvatedspecies,andwasfoundtobe18.0+/-.0057ml/mole.Thestatepointconsideredinallthesestudiesisapressureof1.0atmosphereandatemperatureof298K.ToassesstheeectsofelectrostaticforcesinsolvationaqueousmethanewassimulatedforavarietyofmodelswithdieringpartialchargesontheCH4atomsandforanunchargedmodel.Whentherearenopartialchargesonthehydrogensthemethane-waterpotentialenergyinteractionbecomesequivalenttoanunitedatomdescriptionofmethane.Thepartialchargesintherstmodelwerettotheelectrostaticpotentialsurfacecalculatedviaabinitioelectronicstructuremethodsthatreproducetheoctupolemomentofgasphasemethanetheresultingchargesare-.52e)]TJ/F15 11.955 Tf 11.234 -4.338 Td[(onCand+.13e)]TJ/F15 11.955 Tf 11.234 -4.338 Td[(onH.[114]Withthesepartialchargesthemethanevolumeis31.54+/-.41ml/moleandwithoutthemitis31.74+/-.41ml/mole.Thevolumecorrelationtimewasfoundtobetc=1.6psforthesesystems.ThevolumeofamethaneiscalculatedasthedierencebetweenthesystemvolumeoftheaqueousmethaneandaneatwatersystemwiththesamenumberofH2Omoleculesasthesolvatedmethane,i.e.thevolumeoftheoriginalaqueoussystemafterthemethaneisplucked"out.Theseuncertaintieswereobtainedfrom10.0nsofdynamics.Theslightvolumechangeobtainedisnotstatisticallysignicant;theelectrostrictioneectsassociatedwiththesolvatedhighlysymmetricmethane,thatlacksapermanentdipoleandquadrupolemomentbuthasanoctupolemoment,areexpectedtobesmall.Figure9.4showsthetimeevolutionofthesolvatedunchargedmethanesystemvolume.Figure9.5showsthetimedependenterrorestimatesfortheunchargedmethanemodel,andthequadrupolarmethaneerrorestimatecurveisverysimilar.Thesevolumeuncertaintiesaresimilartothatobtainedinthecaseofthesolvated-sheetpeptide,suggestingthattheobservedbehaviorwouldbesimilarinotheraqueoussystems.Figure9.5alsodemonstratesthatslightlyover100nsofdynamicswouldberequiredtoresolveanydierencebetween121

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Figure9.4:Thecurveshowsatimetraceofthevolumeuctuationsfortheaqueousunchargedmethanesystem.Theaveragesystemvolumevalueanditsuncertaintyarealsorepresentedinthegure.122

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Figure9.5:TheuncertaintyintheunchargedaqueousmethanesystemasafunctionofthelengthoftheMDsimulationisshown.Volumeuncertaintiesforspecictimesofinterestarealsoidentied.Theinsetdisplaysahistogramofthesystemvolumeuctuationsandasuperimposedgaussianfunctionwithastandarddeviationcalculatedfromthevolumeuctu-ations.thesemethanemodels.TheinsetofFigure9.5demonstratestheGaussiannatureofthesystemvolumeuctuations.Tofurthertesttheeectsofelectrostaticmomentsonsolvationandtoassesstheasso-ciatedvolumechanges,bothctitiousmonopolarchargedanddipolarmethanewassim-ulated.Thedipolarmethaneconsistedofagainplacingapartialchargeof-.52e)]TJ/F15 11.955 Tf 12.155 -4.338 Td[(onCanda+.52e)]TJ/F15 11.955 Tf 11.362 -4.338 Td[(ononlyoneofthehydrogens,whiletheotherthreewereuncharged.Thesechargesresultinapermanentdipoleof2.7Debye,comparabletothatofliquidwaterthathasanaveragedipoleof2.4Debyeforthemodelusedhere.Thepermanentdipolarmethaneexhibitsamolecularvolumechangefromtheunchargedmethaneviaelectrostrictionwith123

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Figure9.6:leftTheradialdistributionfunctionbetweenthemethanecarbonatomandthewateroxygenatomsisshown.rightTheradialdistributionfunctionbetweenthemethanecarbonatomandthewaterhydrogenatomsisdisplayed.Thesolidlinerepresentstheanion,thedashedlinethecationandthedottedlineistheunchargedmolecule.netvolumedecreaseof1.73+/-1.02ml/molecomparedtotheunchargedmethanemodel.Therelativelysmallvolumechangeassociatedwithdipolarsolvationisconsistentwiththelackofvolumechangeobserveduponsolvationoftheoctupolarmodel.Next,achargeof+e)]TJ/F15 11.955 Tf 10.183 -4.339 Td[(and-e)]TJ/F15 11.955 Tf 10.183 -4.339 Td[(wasplacedonthemethanecarbonandthehydrogenswereleftuncharged.Inbothcasestheaqueouschargedmodelsproduceddramaticelectrostrictioneects,andthesolvatedanionhadthelargestvolumechange.Thevolumechangewas-40.13+/-.48ml/moleforanionicsolvationand-20.96+/-.39ml/moleforcationicsolvation.Theerrorsaretheresultof10.0nsand12.0nsofdynamicsrespectively.Noticethatthisimpliesanegativesolutionvolumefortheanion.Thelargeranionicelectrostrictioneectisduetothenatureofitssolvation.Thisresulthighlightstheimportanceofproperlyaccountingforelectrostaticinteractionsincalculatingmolecularvolumes,andthiseectmaywellbeimportantinvolumechangesassociatedwiththedynamicsofbiomolecules.124

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Figure9.6showstheradialdistributionfunctionbetweenthemethanecarbonatomandFigure9.6leftthewateroxygenatomsandFigure9.6rightthewaterhydrogenatoms.Theradialdistributionfunctionsarepresentedforbothsolvatedionsystemsalongwiththeunchargedmethanesystem.ItisclearthatthesolvatedanionpermitsthewaterhydrogentopenetrateeectivelyintothecarbonvanderWaalssphere,thusmaximizingtheinteractionbetweenthepositivepartialchargeonthehydrogenandthenegativeioniccharge.Thecationisalsotightlysolvatedcomparedtotheneutral,andbothionsdisplayafarmorestructuredsolvationshellthantheneutral.Thecationhydrogenrstneighborpeakisinapproximatelythesamelocationastheneutralbutissharper,indicativeofmoreordering;theanionsecondneighborpeakisshiftedslightlyinwardfromtheneutralsrstpeak.Electrostrictioneectsareessentiallyscreenedoutbyabout5.5A.TheradialdistributionfunctionsinFigure9.6areclearlyconsistentwiththeobservedvolumechangesanddemonstratethephysicalmechanismgivingrisetoelectrostriction.Solvatingthecationdrawstheoxygenatomsinmoretightlytothemethaneandcausessomesolventordering,butdoesnotdramaticallydisruptthesolventstructure.Solvatingtheanioncausesthewatertopreferentiallypointahydrogenintowardthenegativecharge,creatinganewspeciesofcoordinatedhydrogenatomsappearingbetween1.5and2.2AinrightsideofFigure9.6.AstrengthofMDisinprovidingdetailedmolecularmechanismsforobservedstructureanddynamics.TheupperpanelofFigure9.7showsasnapshotofthesolvatedanion.Theanionissolvatedsuchthatthewateralignsitselfwithoneofthehydrogenseectivelypenetratingintowardthenegativecarbonatomwhilethesecondhydrogenisheldatadistance.ThisisconsistentwithFigure9.6whereweseehighorderingwiththerstsharphydrogenpeakrepresentingthehydrogenthateectivelypenetratesandasecondsharppeakrepresentingthesecondhydrogenbeingheldatadistancefromthenegativecarbon.ThelowerpanelofFigure9.7alsodemonstratesthatthewatersolvatesthecationwiththeoxygenapproachingclosetothemethanemoleculeandthehydrogenspushedback;however,125

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Figure9.7:Representativesnapshotofsolvatedmethaneaniontopandcationlower.126

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thebulkieroxygencannotpenetrateaseectively.127

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Chapter10ConclusionReducingtheproblemofcalculatingthefthorderresponsefunctiontothecalculationofatwo-timeTCFrepresentsaconsiderabledecreaseindiculty,eventhoughsuchTCFarenotcurrentlywellunderstood.OnemightalsoconcludethatnovelinformationislesslikelytobefoundinR,althoughthequestionremainsopen.Byinvokingatwo-timeTCFourtheorystands,onascaleofdiculty,inbetweencalculationofatwo-timePoissonbracketandthegeneralizedLangevinequation[72]theory,whichexpressesRwithconventionalTCF's,asdoesthemodecouplingtheory[115]ofDennyandReichman.GiventhepresentcomputationallytractabletheoryforR,examiningthetemperaturedependenceofthesignalforCS2,xenon,waterandotherliquidswillhelpestablishthenatureoftheRmeasurementanditsvariability.Thetheoreticalresultsfortheharmonicsystem,Equations4.33-4.41canbeusedtoformulateaquantummechanicallyderivedINMtheoryofR.[60,116]Constructingsuchatheory,thatwouldincludeonlyharmonicdy-namics,wouldpermitcomparisonwiththepresentTCFtheorythatisahybridofharmonicandfullyanharmonicdynamicalbehaviors.Workalongtheselinesisbeingpursued.ApplicationoftheTCFtheorymethodstoRt1;t2hasalsoledtoapracticalTCFtheoryofthe2DIRexperiment.[117]ItwouldappearthatalthoughexactTCFtheoriesareimpossible,approximateyeteectiveTCFtheoriesofallnonlinearspectroscopymaybe128

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possible.ThesetheoriesbringthepowerofMDsimulationstobearthediculttaskofthe-oreticallymodelingthespectroscopyofcomplexcondensedphasechemicalsystems.WhenmultidimensionalnonlinearopticalexperimentswererstproposeditwasexpectedtheycouldhaveanimpactsimilartointroducinghigherdimensionaltechniquesintoNMR.AmajorimpedimenttodevelopingandinterpretingthesespectroscopieshasbeenthelackoftractableyetaccuratemolecularlydetailedtheoryofthespectroscopyandTCFtheoriesservetollthisvoid.ThepresentcomputationallytractabletheoryforR,the2DIRtechniquecannowopenupthepossibilityofexaminingcomplexmolecularsystemsandotherpolarizationconditionstobetterunderstandtheinformationcontentofthenonlinearspectroscopy.Thepresenttheoryiswellsuitedasbothapredictiveandinterpretivetoolandretainsafullymoleculardetaileddescriptionof2DIRspectroscopy.Wehavepresentedanapproachtocalculatingmolarvolumechangesthatisespeciallyusefulforcomparingwithphotothermalexperimentalresults.Boththeexperimentalandtheoreticalmethodspermitthedeterminationoftimedependentmolarvolumechanges,andtheidenticationofmetastableintermediatespecieswithlifetimeslastingtensofnanosec-onds.Themethodisalsousefulinaccountingforthemolecularlydetailedoriginofobservedmolarvolumechanges,includingdissectingsuchchangesintophysicallymeaningfulpartialcontributionsfromdierentpotentialenergyinteractions.Theobservationthatanionicsol-vationleadstolargermolarvolumechangesthancationicsolvationisinagreementwithexperimentallymeasuredtrendsandtheobservedvolumechangesareonthesameorderofmagnitude.[118]TheMDmodelsemployedhereservetodemonstratethepowerofthemethodspresentedforpredictionofspectroscopicsignalsandmolarvolumechangesassociatedwithsolvationinthepresenceofdierentelectrostaticelds.129

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AbouttheAuthorRussellH.DeVanereceivedaBachelor'sofScienceDegreefromFloridaSouthernCollegeinDecemberof1993.Afteranumberofyearsinindustry,heenteredtheDoctoralprogramattheUniversityofSouthFloridaintheSpringof2001.HebeganworkincomputationalchemistrywithProfessorBrianSpaceimmediately.WhileinthePh.D.programattheUniversityofSouthFlorida,Mr.DeVanewasarecipientoftheGeorgeBursaDoctoralFellowshipAward.HewasawardedtwosummerawardsfromtheTharpEndowedScholarshipFund.Hehassevenrstauthorpublicationsandmadeseveralpresentationsatregionalandnationalmeetings.Finally,Mr.DeVaneappliedforandwasoeredtheNSFBiologicalInformaticsPost-doctoralFellowship.HehasacceptedtheNSFfellowshipandwillcontinuehistrainingattheUniversityofPennsylvanniaunderthedirectionofDr.MikeKlein.


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ABSTRACT: The research presented in this thesis makes use of theoretical/computational techniques to calculate nonlinear spectroscopic signals and molecular volumes. These techniques have become more practical with advances in computational resources and now are an integral part of research in these areas. Preliminary results allude to the power of these techniques when applied to relevant problems and suggest that much progress can be made in understanding the complex nature of nonlinear spectroscopic signals and molecular volume contributions. The nonlinear spectroscopy work involves writing the quantum mechanical response functions in terms of classical time correlation functions which are amenable to calculation using classical molecular dynamics. The response functions reported in this thesis include the fifth order response function, probed in the fifth order Raman experiment, and the third order response function probed in the two dimensional infrared experiment. The molecular volume calculations make use of modern algorithms used in molecular dynamics simulations to calculate the full thermodynamic volumes of molecules.
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