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Practical applications of molecular dynamics techniques and time correlation function theories

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Practical applications of molecular dynamics techniques and time correlation function theories
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Kasprzyk, Christina Ridley
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Cyclohexanedione
Water
Nonlinear spectroscopy
2DIR
Raman
Azobenzene
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ABSTRACT: The original research outlined in this dissertation involves the use of novel theoretical and computational methods in the calculation of molecular volume changes and non-linear spectroscopic signals, specifically two-dimensional infrared (2D-IR) spectroscopy. These techniques were designed and implemented to be computationally affordable, while still providing a reliable picture of the phenomena of interest. The computational results presented demonstrate the potential of these methods to accurately describe chemically interesting systems on a molecular level. Extended system isobaric-isothermal (NPT) molecular dynamics techniques were employed to calculate the thermodynamic volumes of several simple model systems, as well as the volume change associated with the trans-cis isomerization of azobenzene, an event that has been explored experimentally using photoacoustic calorimetry (PAC). The calculated volume change was found to be in excellent agreement with the experimental result. In developing a tractable theory of two-dimensional infrared spectroscopy, the third-order response function contributing to the 2D-IR signal was derived in terms of classical time correlation functions (TCFs), entities amenable to calculation via classical molecular dynamics techniques. The application of frequency-domain detailed balance relationships, as well as harmonic and anharmonic oscillator approximations, to the third-order response function made it possible to calculate it from classical molecular dynamics trajectories. The finished theory of two-dimensional infrared spectroscopy was applied to two simple model systems, neat water and 1,3-cyclohexanedione solvated in deuterated chloroform, with encouraging preliminary results.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
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by Christina Ridley Kasprzyk.
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PracticalApplicationsofMolecularDynamicsTechniquesandTimeCorrelationFunctionTheoriesbyChristinaRidleyKasprzykAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofChemistryCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:BrianSpace,Ph.D.RandyLarsen,Ph.D.DavidMerkler,Ph.D.VenkatBhethanabotla,Ph.D.DateofApproval:June26,2006Keywords:cyclohexanedione,water,nonlinearspectroscopy,2DIR,Raman,azobenzenecCopyright2006,ChristinaRidleyKasprzyk

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AcknowledgmentsFirst,IwouldliketothankmyhusbandBruceKasprzykandmyparentsBrianandLynnRidleyfortheirunwaveringloveandsupport.Yourencouragementandloyaltymeantheworldtome.ManythankstoProfessorBrianSpaceforservingasanincrediblementorandleader.ThankyouforyourguidanceandpatienceasIworkedtowardsthisend.IappreciatetheamazingopportunitiesthatyouhaveoeredmeduringmytimeattheUniversityofSouthFlorida.IexpressmygratitudetomycommitteemembersProfessorRandyLarsen,Profes-sorDavidMerkler,andProfessorVenkatBhethanabotlaforyourtime,energy,andhonestadvice.IalsothankProfessorDavidRabsonforservingasmydissertationcommitteechair.Ihaveenjoyedworkingwithallofyou.Iamgratefulforthesupportofmyfellowgroupmembers,ChristineNeipert,BenRoney,AbeStern,TonyGreen,andJonBelof,whohaveaccompaniedmeonthisjourney.Thankyoufortheassistanceyouhavegivenmeand,mostofall,foryourfriendship.Iwishyouwell,andIwillmissyou.Finally,anoteofthankstotheUniversityofSouthFloridaforoeringmethePresidentialDoctoralFellowship,whichallowedmetodevotemyselfsingle-mindedlytomyacademicendeavorsduringtheseveyears.GodhasblessedmerichlywiththeknowledgeIhaveacquiredandthepeopleIhaveencounteredduringthecourseofmygraduatestudies.IamthankfulforeverythingHehasgivenme.

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NotetoReaderNotetoReader:Theoriginalofthisdocumentcontainscolorthatisnecessaryforunderstandingthedata.TheoriginaldissertationisonlewiththeUSFlibraryinTampa,Florida.

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TableofContentsListofFiguresivListofTablesviAbstractvii1Introduction12CalculatingMolecularVolume:MolecularDynamicsTechniques42.1Motivation................................52.2MolecularDynamicsinCalculatingMolecularVolume........52.3CalculatingUncertainty........................73CalculatingMolecularVolume:ModelSystems123.1VolumeofaWaterMolecule......................133.2VolumeofaSimplePeptide......................133.3VolumeofaMethaneMoleculeandElectrostaticEects......164CalculatingMolecularVolume:Azobenzene'sIsomerization224.1AzobenzeneExperimentalDetails...................224.2AzobenzeneSimulationDetails....................244.3ComputationalResultsandDiscussion................265TimeCorrelationFunctionFormalism30i

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5.1LinearAbsorptionofRadiation....................306TCFTheory:One-TimeCorrelationFunction356.1TheOne-TimeCorrelationFunction..................356.2ObtainingRinTermsofCt....................386.3RelatingtheRealandImaginaryPartsofC!...........406.4TransformingtotheTimeDomain..................427TCFTheory:Fifth-OrderRamanSpectroscopy437.1TheFifth-OrderResponseFunction..................447.2RelatingTCFsfandg.........................467.3ClassicalLimitofR.........................497.4ApplyingaHarmonicApproximation.................507.5RemovinggIfromtheRExpression................558TCFTheory:2D-IRSpectroscopyExact598.1TheExperiment.............................618.2Introductiontothe2D-IRTCFTheory................628.3ExpansionoftheRExpression...................648.4TheEnergyRepresentation......................658.5FrequencyDomainTCFs........................688.6Detailed-BalanceRelationships....................708.7TheClassicalLimitofR.......................749TCFTheory:2D-IRSpectroscopyHarmonicApproximation779.1TheHarmonicApproximationwithLinearlyVaryingDipole....789.2ApplyingtheApproximationtoR.................799.3ExpansionoftheDipoleMomentMatrixOperators.........799.4Frequency-DomainRelationshipbetweenTCFsAandB......83ii

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9.5EliminatingtheImaginaryPartofTCFB..............859.6TheFinalRExpression.......................879.7LimitationoftheHarmonicApproximation..............8810TCFTheory:2D-IRSpectroscopyAnharmonicApproximation9210.1TheAnharmonicApproximationwithLinearlyVaryingDipole...9210.2SimplifyingtheApproximation....................9510.3RelatingtheAnharmonicTCFsAandB...............9710.4RelatingtheRealandImaginaryPartsofAnharmonicTCFB...10510.5TheFinalRExpression.......................10811TCFTheory:2D-IRSpectroscopyComputationandResults10911.1TheStepsinCalculatinga2D-IRSpectrum.............11011.2ConsideringaConstantt2Delay....................11311.3FourierTransformingBRt1;t2;t3...................11511.4ImplementingaQuantumCorrectionScheme.............11711.5CalculatingPolarization........................11811.6AmbientWater.............................12011.71,3-Cyclohexanedione..........................13012Conclusions136References139AboutTheAuthorEndPageiii

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ListofFigures2.1GaussianVolumeFluctuations.......................82.2SimulationLengthandVolumeUncertainty...............113.1Synthetic-sheetPeptide.........................143.2Time-DependentVolumeofSolvated-Sheet...............153.3SolvationofAnionicandCationicMethane................183.4MethaneCarbon-HydrogenRadialDistributionFunction........193.5MethaneCarbon-OxygenRadialDistributionFunction.........204.1AzobenzeneSimulationSnapshots.....................254.2AzobenzeneMolecularVolume.......................264.3AzobenzeneNitrogen-HydrogenRadialDistributionFunction......2811.1jBR!1;t2=0;!3jofNeatWater.....................12311.2BR!1;!3ofNeatWater:DiagonalSlice.................12411.3HarmonicThird-OrderResponseFunctionofNeatWater........12511.42DIRO-DiagonalCouplingsinNeatWater...............12611.52DIRQuantumCorrectionScheme....................12711.62DIRSpectrumofWaterwithVariousWaitingTimes..........12911.7StretchingModesof1,3-Cyclohexanedione................13011.8LinearIRSpectrumof1,3-Cyclohexanedione...............13111.92D-IRSpectrumof1,3-Cyclohexanedione.................132iv

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11.10DiagonalSliceof1,3-Cyclohexanedione's2D-IRSpectrum........13311.11O-DiagonalCouplingsin1,3-Cyclohexanedione.............135v

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ListofTables4.1AzobenzeneSimulationResults......................277.1RelatingtheRealandImaginaryPartsofTCFg............579.1ValuesofOmegaforTCFAandB1111Terms.............8410.1AnharmonicConstants...........................9410.2AnharmonicTCFATerms.........................9810.3AnharmonicTCFBTerms.........................9910.4AnharmonicTCFsinTermsofHarmonicTransitions..........10010.5RelatingTCFsAandBundertheAnharmonicApproximation....10110.6RelatingBRandBIundertheAnharmonicApproximation.......107vi

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PracticalApplicationsofMolecularDynamicsTechniquesandTimeCorrelationFunctionTheoriesChristinaRidleyKasprzykABSTRACTTheoriginalresearchoutlinedinthisdissertationinvolvestheuseofnoveltheoret-icalandcomputationalmethodsinthecalculationofmolecularvolumechangesandnon-linearspectroscopicsignals,specicallytwo-dimensionalinfraredD-IRspectra.Thesetechniquesweredesignedandimplementedtobecomputationallyaordable,whilestillprovidingareliablepictureofthephemonemaofinterest.Thecomputationalresultspresenteddemonstratethepotentialofthesemethodstoaccuratelydescribechemicallyinterestingsystemsonamolecularlevel.Extendedsystemisobaric-isothermalNPTmoleculardynamicstechniqueswereemployedtocalculatethethermodynamicvolumesofseveralsimplemodelsystems,aswellasthevolumechangeassociatedwiththetrans-cisisomerizationofazobenzene,aneventthathasbeenexploredexperimentallyusingphotoacousticcalorimetryPAC.Thecalculatedvolumechangewasfoundtobeinexcellentagreementwiththeexperimentalresult.Indevelopingatractabletheoryoftwo-dimensionalinfraredspectroscopy,thethird-orderresponsefunctioncontributingtothe2D-IRsignalwasderivedintermsofclassicaltimecorrelationfunctionsTCFs,entitiesamenabletocalculationviaclassicalmoleculardynamicstechniques.Theapplicationoffrequency-domaindetailedbalancevii

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relationships,aswellasharmonicandanharmonicoscillatorapproximations,tothethird-orderresponsefunctionmadeitpossibletocalculateitfromclassicalmoleculardynamicstrajectories.Thenishedtheoryoftwo-dimensionalinfraredspectroscopywasappliedtotwosimplemodelsystems,neatwaterand1,3-cyclohexanedionesolvatedindeuteratedchloroform,withencouragingpreliminaryresults.viii

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Chapter1IntroductionRecentadvancesintheoreticaltechniquesandcomputationaltechnologyhaveal-lowedthescienticcommunitytogainmeaningfulinsightintointerestingchemicalphe-nomena,evenmattersascomplexasthefoldingpathwayofpeptidesormultidimensionalspectroscopyofcondensedphases.Thesynergyofinnovativeexperimentalprocedureswithcomputationalandtheoreticalinvestigationsyieldsamicroscopicunderstandingofthestructureanddynamicsthatarediculttointerpretbasedonexperimentalresultsalone.Inthisdissertation,theoreticalapproachestotworelevantproblems,themea-surementoftime-dependentmolecularvolumesandthecalculationandinterpretationoftwo-dimensionalinfraredD-IRspectra,arepresented.Modernphotothermalexperiments,includingphotoacousticcalorimetryPAC,arecapableofmeasuringmolecularvolumechangesassociatedwithpeptidefoldingandunfolding,isomerizations,andotherprocessesonapicosecondtimescale.Inthisthesis,theapplicationofmoleculardynamicstechniquestothisproblemispresented.Themethodwasdevelopedwiththehopethatitwouldcomplementexperimentalresultsandprovidedetailedstructuralinformationexplainingmolecularvolumechanges.Isobaric-isothermalNPTmoleculardynamicswasusedtocalculatethevolumesofseveralmodelsystems,includingawatermolecule,amethanemolecule,andatwenty-residue-sheetpeptide,inordertoverifytheutilityofthemethodanddemonstratetheinuenceofelectrostaticinteractionsonmolecularvolume.Finally,themolecularvolumechange1

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associatedwiththetrans)]TJ/F20 11.955 Tf 12.048 0 Td[(cisisomerizationofazobenzene,asimpleorganicmolecule,wascalculatedandfoundtobeinexcellentagreementwithexistingexperimentalresults.Nonlinearspectroscopictechniquesprovideinsightintostructureanddynamicsthattraditionallinearmethodsareunabletoprobe.Whileearlynonlinearexperiments,namelytheopticalKerreectOKEandanti-StokesRamanscattering,didnotoerinformationthatlinearexperimentscouldnotprovide,multidimensionaltechniques,in-cluding2D-Ramanand2D-IRspectroscopy,promisedtorevealstructuralanddynamicaldetailsofcomplicatedsystems.In2D-IRspectroscopy,threetime-orderedelectricaleldsinteractwithasub-stancedescribedbyitsthird-orderresponsefunctionRtogenerateathird-orderpo-larizationPresponsibleforthesignal.Thisspectroscopyworksonsub-picosecondtimescales,allowingittoprovidetime-resolvedstructuresoftransientspecies,incon-trasttoestablishedmultidimensionalNMRandX-rayscatteringtechniques,whichtypi-callyyieldedtime-averagedresults.Inrecentyears,2D-IRspectroscopyhassuccessfullybeenemployedtoinvestigatemanyintricateproblems,includingthehydrogenbondingnetworkofwater,thethree-dimensionalstructureofpeptides,andorganicmolecules.Sincethecomplicatednatureoftheresultingspectraoftenmaketheirinterpretationproblematic,theapplicationoftheoreticalmethodsto2D-IRspectroscopytoextractmeaningfromthespectraiscalledfor.UsingatimecorrelationfunctionTCFformal-ism,thethird-orderresponsefunctionresponsiblefor2D-IRsignal,initiallyacomplicatedquantum-mechanicalexpression,wasderivedintermsofasingleclassicaltimecorrela-tionfunction,anentitywhichiseasilycalculatedviaclassicalmoleculardynamicsMDsimulations.Theresultingtheorywasusedtocomputetheoretical2D-IRspectraoftwomodelsystems,neatwaterand1,3-cyclohexanedionesolvatedindeuteratedchloroform.InChapter2,themoleculardynamicstechniquesusedtocalculatemolecularvolumesareintroduced.Chapter3outlinesthecalculationofthemolecularvolumesofwater,the-sheetpeptide,andmethaneandalsodescribestheroleofelectrostatic2

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interactionsinthesecalculations.Chapter4includestheresultsoftheexperimentalandtheoreticaldeterminationofthevolumechangeassociatedwithazobenzene'strans)]TJ/F20 11.955 Tf 11.147 0 Td[(cisisomerization.Thetimecorrelationfunctionformalism,asitarisesfromFermi'sGoldenrule,isintroducedinChapter5.Chapters6and7,whichcontainthedevelopmentsofTCFthe-oriesforthelinearresponsefunctionandthefth-orderresponsefunctionassociatedwith2D-Ramanexperiments,respectively,areprovidedasbackgroundforthetwo-dimensionalinfraredspectroscopictheorypresentedinsubsequentchapters.Chapter8introducesthe2DIRexperimentandthetheory.Inthischapter,an-alyticalmanipulationsareutilizedtosimplifythethird-orderresponsefunctionexactly.Furthersimplications,accomplishedusingharmonicandanharmonicoscillatorapprox-imationsarediscussedinChapters9and10,respectively.Finally,thecomputationalimplementationofthe2D-IRTCFtheoryisdiscussedandtheoreticalspectraofneatwaterand1,3-cyclohexanedionearedisplayedinChapter11.Chapter12concludesthisworkandreectsonpotentialfutureapplicationsofthesetheoreticaltechniques.3

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Chapter2CalculatingMolecularVolume:MolecularDynamicsTechniquesPhotothermalmethods,includingphotoacousticcalorimetryPACandphotother-malbeamdeectionPBD,permitthemeasurementofmolecularvolumechangesofsolvatedmoleculesonnanosecondtimescales.Photothermalexperimentsareusefulforinvestigatingthethermodynamicprolesassociatedwithinterestingphenomenasuchasthefoldingofapeptide.UsingmoleculardynamicsMDtechniquestomimicexper-imentalmeasurementsprovidesmicroscopicunderstandingofthethermodynamicmea-surements.Tocalculatetime-dependentthermodynamicvolumes,isothermal-isobaricNPTmoleculardynamicssimulationsareperformedonthesystemofinterest.NPTmoleculardynamicsallowsthevolumeofthesystemtouctuateovertimeandresultsinastatisticaluncertaintyintheaveragevolumescalculated.Itwasdiscoveredthatsimulationslastingafewnanosecondswerecapableofdiscerningvolumechangesofapproximately1.0mL/mol,aprecisioncomparabletowhatcanbeachievedinthelaboratory.Inthischapter,themoleculardynamicstechniquesemployedincalculatingmolec-ularvolumesareintroduced.InChapter3,theapplicationofmoleculardynamicstoseveralsimplemodelsystemisdiscussed,andinChapter4,itisdemonstratedthatthesetheoreticalmethodsandphotoacousticcalorimetrypredictthesamevolumechangeforthetrans-cisisomerizationofazobenzene.4

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2.1MotivationAproductiveuseofmoleculardynamicsistosimulatetheprocessesexaminedinphotothermalexperiments,whichdeterminemolecularvolumechangesonnanosecondtimescales1{3andgainmicroscopicinsightintotheexperimentalresults.Suchexper-imentsarecapableofidentifyingproteinandpeptideintermediateswithcharacteristicvolumesthathavelifetimesofseveralnanoseconds.Statisticallysignicantchangesinthevolumecoordinateovertimeindicatethepossiblepresenceoftransientspecies,signaledbymetastableequilibriumbetweenthesoluteandsolvent.TheMDmethodsemployedinthisresearchallowtheidenticationofintermediatestructuresonthemicroscopiclevel.Photothermalexperimentsalsocanmapoutenthalpyprolesoversimilartimescales,andMDsimulationsmaybeusedtoprovidemolecularinterpretationsoftheseenergetics.2.2MolecularDynamicsinCalculatingMolecularVolumeClassicalextended-systemisothermal-isobaricNPT4,5moleculardynamicssim-ulationsplayanessentialroleincomputationallydeterminingmolecularvolumechanges.Thethermodynamicvolumeofasystem,oftenconsistingofasolutemoleculeandsol-vent,canbeextracteddirectlyfromanNPTMDsimulation'svolumecoordinate.Toobtainthevolumeofthesolutemoleculealone,itisstraightforwardtoobtainthevolumeofthesolventaloneandsubtractitfromthetotalsystemvolume.Onepossiblemethodofcomputingthesolventvolumeisto"pluck"thesolvatedspeciesfromthesystemandre-equilibratethesolventintheabsenceofsolvent-soluteforces."Plucking"themoleculefromwhatwasanequilibratedsystemprovidesaninitialconditioncongurationallynearanewequilibrium,andthenewequilibriumisquicklyachieved.Uponre-equilibration,thevolumeofthesolventiseasilydeterminedfromthevolumecoordinate.5

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Thismethodisusefulforsimplesolutions,butalsomayprovehelpfulinexam-iningcomplexbiologicalsystemscomposedofintricateassembliesofbiomoleculesandsolvents.Whileothereectivemethodsofcalculatingmolecularvolumes6{9exist,themethodproposedisidealformodelingthetimeevolutionofbiologicalsystems.NPTmoleculardynamicssimulationsgiverisetoauctuatingvolumecoordinate.Thesys-tem'sthermodynamicvolumeistakensimplyastheaveragevolumeoverthecourseofthesimulation.Upondeterminingtheaveragevolume,itbecomesnecessarytoassesstheuncertaintyassociatedwiththeaverage.InSection2.3,itisdemonstratedthatthevolumeuctuationsassociatedwithNPTmoleculardynamicsareGaussianinnature.Consequently,thestandarddeviationinvolumeisausefulmeasureoftheuncertainty.OneimportantconsiderationinusingNPTmoleculardynamicstocalculatevol-umeaveragesisthat,asaresultofthedynamicalnatureofMDsimulations,successivevolumevaluesarenotstatisticallyindependent.Toavoidaveragingnon-independentvolumemeasurements,itisnecessarytocalculatethecorrelationtimeofthevolumecoordinateandsampledatapointswhichareuncorrelated.10{12Anotherconcernworthnotingisthat,althoughthemethodemployedinoursimu-lationssamplestheNPTensembleexactly,4NPTMDalgorithmsarenotstrictlyequiva-lenttomicrocanonicaldynamics.NPTmethodscouplerealsystemvariablestoctitiousvariablesthatregulatethermodynamicproperties,e.g.thermostatsfortemperatureandbarostatsforpressure,insuchawaythattheyuctuatearoundpre-determinedaveragevalues.Themethodsforcalculatingthermodynamicvolumesareexactforagivenpoten-tialenergymodel,butitisuncertainwhetherdynamicaleventsobservedarephysically6

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relevant.NPTdynamicsareonlyslightlyperturbedfromtrueNewtonianmotionontheorderof1=p 3N,whereNisthenumberofatomsinthesystemrelativetomicrocanoni-calNVEdynamics.Thus,theNPTensembleisoftenrecommendedasoneofthemorereliablemeansofsimulatingbiologicalsystems.13Ifthereliabilityofisothermal-isobaricdynamicsisaconcern,therepetitionofsimulationsinthemicrocanonicalensemblemayservetoverifytheresults.2.3CalculatingUncertaintyAsstatedearlier,uctationsofobservablequantities,suchasvolume,fromtheirmeansduringthecourseofanNPTMDsimulation,aretypicallyGaussianandcharacter-izedbytheirstandarddeviation=p N.Figure2.1,ahistogramofthevolumesmeasuredinmL/molduringamoleculardynamicssimulationofaqueouscis-azobenzene,demon-stratestheGaussiannatureofvolumeuctuationsinNPTmoleculardynamics.Ifsuccessivemeasurementsofmolecularvolumewereuncorrelated,theuncertaintyinvolumewouldbesimplybecalculatedasthestandarddeviationassociatedwiththesetofmeasurements:V==p N.1InEquation2.1,Nisthetotalnumberofsamples.Whenmeasurements,suchasinstantaneousvolumes,arecloselyspaced,theycannotbeconsideredstatisticallyindependent.Theyareinherentlyconnectedbythedynamicalequationsofmotionthatdrivemoleculardynamicssimulations.Toremedythisproblem,acorrelationtimedened7

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Figure2.1:AhistogramofthevolumesinmL/molmeasuredinamoleculardynamicssimulationofaqueouscis-azobenzeneisdisplayed.ThisplotdemonstratestheGaussiannatureofvolumeuctuationsinisothermal-isobaricNPTmoleculardynamics.8

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bytc=stbeyondwhichmeasurementsareconsideredindependent,isdened.Thetimestepbetweensuccessivemeasurementstismultipliedbythestatisticalineciencys,whichindicatesthenumberofcorrelateddatameasurements,10,12toobtaincorrelationtimetc.ThestatisticalineciencyisdeterminedbyperformingvolumeaveragesoverblocksoftimeofincreasingdurationendingwiththelengthoftheentireMDrun.ThisparametersisformallydenedbyFriedbergandCameron11:s=limB!1B2B=2.22B=1 NBNBXB=1hViB)-222(hVi.3InEquation2.2,NBrepresentsthenumberofblockscontainingBmeasurementssuchthattheproductBNB=N,thetotalnumberofobservations.Basedonthisinformation,thecorrelationsbetweensuccessivevolumemeasurementsyieldsamodiedexpressionforvolumeuncertainty:V=p s=N.4Ifthetimebetweenmeasurementsexceedstc,thevalueofsapproachesunity,andEquation2.4reducestoEquation2.1.WhenperforminglongMDsimulations,itisconvenienttospacemeasurementsattslightlygreaterthantcinordertominimizetheamountofdatastoredandtoallowtheuncertaintytobecalculatedsimplyasthe9

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standarddeviation.CorrelationtimetcforagivensystemcanbedeterminedinitiallyfromtheresultsofapreliminaryMDsimulation.Asthelengthofasimulationincreases,theuncertaintydecreases,allowingsmallerandsmallermolecularvolumechangestoberesolved.Todemonstratethisphenomenon,theuncertaintyassociatedwithcis-azobenzene'svolumeisconsidered.Basedonamea-suredcorrelationtimeof1.4ps,thestandarddeviationofcis-azobenzene'svolumeiscomputedasafunctionofincreasingsimulationtime.TheresultisshowninFigure2.2.Theuncertaintyofthevolumedecreasesasthesquarerootofthenumberofvolumemeasurements.Bythetimethesimulationlengthreaches50ns,atimescalerelevanttophotothermalexperiments,theuncertaintyinvolumefallstolessthan0.5mL/mol.Atthispoint,itshouldcertainlybepossibletorecognizerelativelymodestchangesinvolumeassociatedwithchangesinthismolecule'sconformation.Examinationoftheuncertaintiesinothersystems'volumesyieldssimilarresults,indicatingthat50nsofdynamicscangenerallyprovideusefulinformationaboutmolecularvolumes.Forexam-ple,ahelix-to-coiltransitioninapeptideisestimatedtobringaboutavolumechangeofapproximately3.0mL/mol/residue,14achangewhichshouldeasilybediscernedwith50nsofdynamics.AllmoleculardynamicssimulationsusedinmolecularvolumecalculationswerecarriedoutusingacodedevelopedbytheKleinresearchgroupattheCenterforMolecu-larModelingattheUniversityofPennsylvania.Thecodewasimplementedwithparallelexecution,extendedsystemparticlemeshEwaldsummation,andmultipletimescalein-tegrationalgorithms.15,1610

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Figure2.2:Thevolumeuncertaintyofaqueouscis-azobenzeneisdisplayedasafunctionofsimulationlength.By50ns,atimescalerelevanttophotothermalexperiments,theuncertaintyinvolumefallstolessthan0.5mL/mol.11

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Chapter3CalculatingMolecularVolume:ModelSystemsInthischapter,theuseofmoleculardynamicstechniquesinthecalculationofthermodynamicsvolumesisdemonstratedforseveralmodelsystems,includingawatermolecule,asmallaqueouspeptide,andamethanemolecule.Thewatermolecule'scal-culatedvolumeisinexcellentagreementwithacceptedvalues,conrmingthemodel'sabilitytocapturemolecularvolumescorrectly.Thesimulationofthepeptide,althoughyieldinginconclusiveresultsaboutthedierenceinvolumebetweenitsfoldedandun-foldedstates,hintsatthepotentialofthemethodtodiscernvolumechangesinlargerproteins.Intheanalysisofmethane,theeectofelectrostaticsonthemolecule'seectivevolumeisexaminedbymanipulatingtheatomicchargesonthemethanemolecule.Theeectofelectrostaticsoncalculatedmolecularvolume,especiallyinthecaseofanionicandcationicmethane,isdramaticandconformstoexperimentallydeterminedtrends.Theinformationobtainedfromthesemodelsystemswillaidinsettingupandexecutingthesimulationofothersystemsofinterest.12

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3.1VolumeofaWaterMoleculeAsaninitialtestofthisapproachtomeasuringmolecularvolume,themolarvolumeofneatwater,asubjectofearliercomputationalinvestigations,8wasdetermined.Thevolumeofaexiblesinglepointcharge17{19watermoleculewascalculatedtohighprecisionatatemperatureof298Kandapressureof1.0atmospheres.Theresultwas18.00.0057mL/mol,20avaluewhichcorroboratesexistingresults.Thisprecisevalueofawatermolecule'svolumewasusedtodeterminethesolventvolumesinseveralsystemsinvestigatedusingthesemoleculardynamicsmethods.3.2VolumeofaSimplePeptideSinceonepotentialuseofthismethodistocalculatevolumechangesassociatedwiththefoldingofpeptides,itwasappliedtoatwenty-residue-sheetpeptidewhichhasrecentlybeenunderinvestigationwithphotothermalmethods.1ThepeptideisdepictedinFigure3.1.Acaged"formoftheunfolded-sheet,whichcanbephotolyzedinneatwatertoinitiatefolding,wassynthesizedbyChanandco-workers.1,21Usingphotothermalmethods,itispossibletoconstructvolumeandenthalpyprolesassociatedwiththe1.0-sfoldingprocess.Forthepurposesofthecomputationalinvestigation,theinitialfolded-sheetpeptidewasconstructedfromtheNMRstructure.21Thesequenceoftheproteinisgivenbelow:13

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Figure3.1:Animageofthe-sheetpeptideisshowninbothpanels.Intherightpanel,itissolvatedwithwater,andintheleft,itisdisplayedwithoutsolventforeasiervisual-ization.Thecolorsrepresentatomtypes:Cgreen,Ored,Nblue,HwhiteACE)]TJ/F20 11.955 Tf 11.955 0 Td[(VAL)]TJ/F20 11.955 Tf 11.955 0 Td[(PHE)]TJ/F20 11.955 Tf 11.955 0 Td[(ILE)]TJ/F20 11.955 Tf 11.955 0 Td[(THR)]TJ/F20 11.955 Tf 11.956 0 Td[(SER)]TJ/F20 11.955 Tf 11.955 0 Td[(PRO)]TJ/F20 11.955 Tf 11.955 0 Td[(GLY)]TJ/F20 11.955 Tf 11.955 0 Td[(LYS)]TJ/F20 11.955 Tf 11.955 0 Td[(THR)]TJ/F20 11.955 Tf 9.298 0 Td[(TYR)]TJ/F20 11.955 Tf 11.955 0 Td[(THR)]TJ/F20 11.955 Tf 11.956 0 Td[(GLU)]TJ/F20 11.955 Tf 11.956 0 Td[(VAL)]TJ/F20 11.955 Tf 11.955 0 Td[(PRO)]TJ/F20 11.955 Tf 11.956 0 Td[(GLY)]TJ/F20 11.955 Tf 11.956 0 Td[(LYS)]TJ/F20 11.955 Tf 11.955 0 Td[(ILE)]TJ/F20 11.955 Tf 11.955 0 Td[(LEU)]TJ/F20 11.955 Tf 11.955 0 Td[(GLN.1Additionally,anunfoldedcongurationofthepeptidewasbuiltandsimulatedtoprovideabasisforcomparingthevolumesofthefoldedandunfoldedstates.Bothpeptidesystemsweresolvatedwith810exibleSPCwatermolecules.TheAMBERf99forceeld22wasusedtodescribethebonds,bends,torsions,VanderWaalsinteractions,andnon-bondedinteractionsbetweenatomsseparatedbythreebondsknownasone-fourinteractionsinthepeptide,andthepeptidewasconguredtohavenonetcharge.Figure3.2displaysthevolumesofthesolvatedfoldedpeptideandthewatersolventover14

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Figure3.2:Theredcurvedisplaysthevolumeuctuationsofthefoldedaqueous-sheetpeptide.Thebluecurvedisplaysthevolumeuctuationsofthewatersolventafterthepeptideisplucked"out.Theinsetdemonstratesthatthewaterre-equilibratesanditsvolumestabilizesquickly,within0.05ns,oncethepeptideisremoved.2.0nsofsimulationtime.TheinsetinFigure3.2demonstratesthat,afterthepeptideisplucked"outofthewatersolvent,thewaterre-equilibratesanditsvolumestabilizesquickly,within0.05ns.7.5nsofdynamicsonthefolded-sheetgaveavolumeof1668.02.4mL/mol,while5.8nsontheunfoldedcongurationgave1672.03.1mL/mol.20Thesolutionvolumeofthepeptideswerecomputedbysubtractingfromthetotalsystemvolumetheprecisevolumesofthewatermolecules,asdeterminedinSection3.1.Itissome-whatsurprisingthatthefoldedandunfoldedstates,withintheuncertaintiesstated,15

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haveessentiallythesamevolumesincethesolvationstructuresassociatedwitheacharemarkedlydierent.Thissurprisingresultdoesnotnecessarilysuggestthatdynamicalintermediateswithsignicantlydierentvolumesarenotpresentduringthefoldingpro-cess.However,longersimulationsofbothstatesshouldbeattemptedbeforedrawinganyrmconclusionsaboutthevolumechangeassociatedwiththe-sheet'sfoldingprocess.Althoughtheinvestigationofthesmall-sheetwasinconclusive,largerproteins,whichoftenexhibitlargerper-residuevolumechangesduringfolding,maybeidealsub-jectsforthismethodofcalculatingmolecularvolumechanges.3.3VolumeofaMethaneMoleculeandElectrostaticEectsAsanaltestofthemethod,thevolumeofasinglemethanemoleculesolvatedby62watermoleculeswasmeasuredatambientconditionstemperatureof298Kandpressureof1.0atmospheres.Anall-atommethanemotel,includingaexibleforce-eldt,wasusedtoreproduceexperimentalinfraredfrequencieswithharmoniccarbon-hydrogenbonds.23Lennard-Jonesinteractionswereappliedonlybetweenthemethanecarbonandwateroxygenswithparameters=3:33Angstromsand=51:0K.Theequilibriumcarbon-hydrogenbondlengthwassetat1.09Angstroms,andthemoleculewasassumedtohavetetrahedralgeometry.Tomeasuretheeectsofelectrostaticforcesinsolvation,aqueousmethanewassimulatedusingavarietyofmodels,eachplacingdierentpartialchargesonmethane'scarbonandhydrogenatoms.Thepartialchargesintherstmodelof-0.52e)]TJ/F15 11.955 Tf 12.284 -4.338 Td[(oncarbonand+0.13e)]TJ/F15 11.955 Tf 12.284 -4.338 Td[(oneachofthehydrogenatomswerettotheelectrostaticpotentialsurfacecalculated16

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usingabinitioelectronicstructuremethodsthatreproducetheoctupolemomentofgasphasemethane.24Applyingtheserealisticpartialchargesresultedinacalculatedvolumeof31.540.41mL/mol.Incontrast,amethanemoleculewithallpartialchargesremovedexhibitedavolumeof31.740.41mL/mol.20Bothresultswereobtainedfrom10.0nsofdynamics.Thevolumedierencebetweenthetwoversionsofmethaneisstatisticallyinsignicant.Thisresultisnotsurprising,giventhattheelectrostrictioneectsassociatedwiththehighlysymmetricmethanemolecule,whichlacksapermanentdipole,quadrupole,andoctupolemoment,areconsiderednegligible.Tomorecloselyexaminetheeectsofelectrostaticmomentsonsolvationandthesolutionvolumeofthemethanemoleculedipolarmethane,whichdoesnotrepresentarealisticmethanemolecule,wassimulated.Thedipolarmethanewasconstructedbyplacingapartialchargeof+0.52e)]TJ/F15 11.955 Tf 12.267 -4.338 Td[(onthecarbonatomand-0.52e)]TJ/F15 11.955 Tf 12.267 -4.338 Td[(ononeofthehydrogenatoms.Theotherthreeatomsweretakenasuncharged.Theresultingdipolemomentonthemethanemoleculewas2.7Debye,slightlylargerthanwater'sdipolemomentof2.4Debye.Incomparisontotheunchargedmethanemoleculedescribedearlierinthissection,thedipolarmethaneexhibitedavolumeconstrictionof1.731.02mL/mol.Thisrelativelysmallvolumechangeisconsistentwiththenegligiblevolumechangeoccurringwhenoctupolarmethaneissolvatedinwater.Whiledipolarandoctupolarmethanedonotexhibitsignicantvolumedecreasesduetoelectrostriction,thesimulationofmonopolarchargedmethanemoleculesyieldedstrikingresults.Themethaneanionandcationareconstructedbyplacingchargesof+e)]TJ/F15 11.955 Tf -442.915 -33.23 Td[(and-e)]TJ/F15 11.955 Tf 7.085 -4.339 Td[(,respectively,onthecarbonatom.Thefourhydrogenatomsareleftuncharged.17

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Figure3.3:Thesesnapshotsdepictthesolvatedmethanecationleftpanelandanionrightpanel.Theorderingofwatermoleculesaroundthecarbonthegreenatomofmethaneisapparentineachsnapshot.Comparedtotheunchargedmethanemolecule,thecationexhibitedavolumechangeof-20.960.39mL/mol,basedon10.0nsofdynamics.Theanionexperiencedanevenmoredramaticchangeof-40.130.48mL/mol,basedon12.0nsofdynamics,avolumechangewhichgivesthemoleculeanegativevolumewhensolvated.Bothresultsdemonstratethesignicanteectthatelectrostaticscanhaveoncalculatedmolecularvolumesandhighlighttheimportanceofcarefullyaccountingforelectrostaticinteractionsinanysimulation.Thedierenceseeninthevolumesoftheanionandcationcanbeattributedtothenatureofthemethanemolecule'ssolvation.Figure3.3depictsthesolvatedanionicandcationicmethanemolecules.Intherstpanel,whichdisplaysthesolvatedcationicmethane,thewater'selectronegativeoxygenatomsarealignedtobeasclosetotheposi-tivelychargedcarbonatom.Inthesecondpanel,whichshowsthesolvationoftheanion,18

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Figure3.4:Theradialdistributionfunctionbetweenthemethanecarbonandwaterhydro-genatoms.Thesolidredlinerepresentsanionicmethane,thedashedgreenlinecationicmethane,andthedottedbluelineneutralmethane.Thecarbon-hydrogenrstneighborpeakoftheanionicmethaneissharplyshiftedtotheleftrelativetotheothertwoformsofmethane,indicatingthatthehydrogenspenetratethevanderWaalssphereofanionicmethane'scarbon.thewatermoleculesalignthemselveswithhydrogenspointingtowardsthenegativelychargedcarbonatom.Becausethehydrogensarelessbulkythantheoxygenatominwa-ter,thewatermoleculeseectivelymoveclosertothemethaneanionthantothecation,allowingforgreaterelectrostrictionofthesolvent.ExaminationoftheradialdistributionfunctionsbetweenthemethanecarbonandwaterhydrogenFigure3.4andoxygenFigure3.5atomsconrmsthearrangements19

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Figure3.5:Theradialdistributionfunctionbetweenthemethanecarbonandwateroxygenatoms.Thesolidredlinerepresentsanionicmethane,thedashedgreenlinecationicmethane,andthedottedbluelineneutralmethane.Thesharprstneighborpeaksforcationicandanionicmethanesuggestthatthesolventismorehighlyorderedaroundthemethanecarbonthanitisfortheneutralformofmethane.20

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ofatomssuggestedinthesimulationsnapshots.Theanionicmethaneallowswater'shydrogenatomstopenetrateintothecarbonatom'svanderWaalssphereatadistanceof1.5-2.2Angstroms.Thus,asclearlydisplayedinitscarbon-hydrogenrstneighborpeak,whichisshifteddramaticallytotheleftoftheneutralform's,anionicmethaneismuchmoretightlysolvatedthanneutralmethane,indicatedbythebluedottedlines.Theclosecoordinationofthecarbonandhydrogenatomsinanionicmethane'ssolvationmaximizestheinteractionbetweenthenegativelychargedcarbonatomandpartiallypositivelychargedhydrogenatoms.Cationicmethane,indicatedbygreendashedlines,ismoretightlysolvatedthantheneutralform,asdemonstratedbyitscarbon-oxygenrstneighborpeak.Bothanionicandcationicmethanepossessasharpcarbon-hydrogenrstneighborpeak,whichsuggestthepresenceofastructuredsolvationshell.Thesimulationsnapshotsandradialdistributionfunctionsofanionic,cationic,andneutralmethanedemonstratethatmoleculardynamicscanprovideaneectivemi-croscopicpictureoftheelectrostaticinteractionsthatdrivemolecularvolumechanges.Theconclusionthatanionicsolvationyieldslargervolumecontractionsthancationicsolvationisconsistentwithexperimentallymeasuredtrends.25Theapplicationofthismoleculardynamicsmethodtothemodelsystemsofwater,the-sheet,andmethanedemonstrateitspowerfulabilitytoassessmolecularvolumechangesassociatedwithsolvationundervaryingelectrostaticconditions.InChapter4,themethodwillbeusedtoassessthevolumechangeassociatedwithazobenzene'strans-cisisomerization,avolumechangewhichhasbeenmeasuredexperimentallyusingphotoacousticcalorimetry.2621

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Chapter4CalculatingMolecularVolume:Azobenzene'sIsomerizationInthischapter,themoleculardynamicsmethodfordeterminingmolecularvolumechanges,outlinedinChapter2andappliedtoseveralmodelsystemsinChapter3,isusedtomeasurethemolecularvolumechangeassociatedwiththetrans-cisisomerizationofthesimpleorganicmoleculeazobenzene.Theresultsofthesimulationarefoundtobeinexcellentagreementwiththeexperimentalvolumechange,measuredbyProfessorRandyLarsen'slaboratoryattheUniversityofSouthFlorida,determinedusingphotoacousticcalorimetryPAC.4.1AzobenzeneExperimentalDetailsInthePACexperiment,laserpulsesareusedtophotoisomerizeasampleofaque-oustransazobenzenetothecisform.Excessenergynotusedintheisomerizationprocessgeneratesanacousticwave,whichisdetectedbyamicrophoneandmeasuredwithanos-cilloscope.Theamplitudeoftheacousticsignalisproportionaltothemolecularvolumechangeassociatedwithazobenzene'sisomerization.Toisolatethetransisomerofazobenzene,5mgofsolidazobenzenewasdissolvedin2mLofabsoluteethanol.Thesolutionwasilluminatedwithahalogenlampand22

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dilutedwithvevolumesofdeionizedwater.Thetransisomer,whichisonlysparinglysolubleinwater,wasthenlteredout.ThesampletobeusedinthePACmeasurementswaspreparedbysaturatingawatersolutionwiththesolidtransazobenzene.Intheabsenceoflight,boththecisandtransisomersofazobenzenecanremainasmetastableaqueousisomersforseveralhours.Uponillumination,arapidphotoisomer-izationoccurs,andamolecularvolumechange,whichismeasuredbyPAC,accompaniestheconformationaltransition.Thesampleandcalorimetricreferenceacoustictraceswereobtainedasfunctionsoftemperature,andtheratiooftheamplitudesoftheacousticsignalsS=Rwasplottedversus1==Cp.S=REh=Eh=[Q+Vcon==Cp].1InEquation4.1,isthequantumyield,whichtookonavalueof0.26inthisexperiment.26Qistheheatreleasedtothesolvent,isthecoecientofthermalexpan-sionofthesolventK)]TJ/F18 7.97 Tf 6.587 0 Td[(1,Cpistheheatcapacitycalg)]TJ/F18 7.97 Tf 6.586 0 Td[(1K)]TJ/F18 7.97 Tf 6.587 0 Td[(1,isthedensityg/mL,andVcondenotestheconformationalandelectrostrictioncontributionstothesolutionvolumechangeoftheazobenzenemolecule.AplotofEhisexpectedtogiveastraightlinewithaslopeofVcon.SubtractingQfromEhalsoyieldstheenthalpychangeHassociatedwithprocessesfasterthanthetimescaleoftheinstrument,approximately50ns.Aplotoftheexperimentaldatarevealedthatphotoisomerizationoftrans-azobenzenetothecisformyieldedavolumechangeof-41mL/mol.Theobservedvolumecon-23

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tractionwassimilartothatobservedin80:20ethanol:water,aswellasthevolumechangeassociatedwithaqueouscarboxyl-azobenzene.264.2AzobenzeneSimulationDetailsInitialstructuresforcisandtransazobenzenewerebuiltandoptimizedusingtheGAMESSpackage27anda6-31Gbasisset.Theresultingstructureswerecomparedwithestablishedcrystalstructuresandweredeterminedtobesuperimposableandvirtuallyindistinguishable.TheAmberf9922forceeldprovidedbond,bend,torsion,one-four,andvanderWaalsinteractionparameters,andthepartialchargesonazobenzene'satomswerettotheelectrostaticpotentialsurfaceusingtheConnollymethodintheGAMESSpackage.27Azobenzene'stransisomerisaplanarmoleculewhichhasanetdipoleofzeroduetoitssymmetry,whilethecisisomerisanonplanarstructurewithalargegasphasedipoleof3.45Debye.Allsimulationsincluded108explicitexibleSPCwatermolecules.Figure4.1depictsthegas-phaseandsolvatedcisandtransazobenzenemolecules.ThezerovolumereferencewasprovidedbyanNPTsimulationofaboxof108exibleSPCwatermolecules.Thedierencebetweenasolvatedazobenzeneisomerandthevolumeoftheneatwaterwastakenasthemolecularvolumeofeachisomer.Thedierencebetweenthemolecularvolumesofthetransandcisazobenzeneisomerswastakenasthemolecularvolumeassociatedwithazobenzene'strans)]TJ/F20 11.955 Tf 11.955 0 Td[(cisisomerization.Thecorrelationtimeassociatedwithazobenzene'svolumewasdeterminedtobe1.4ps.Basedonthisinformation,volumemeasurementswererecordedevery2.0pstoensurethatsuccessivemeasurementswerestatisticallyindependent.Theuncertaintyin24

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Figure4.1:Equilibriumsolvatedstructuresofcisleftandtransrightazobenzenearedepicted.Thetoppanelsshowthegas-phasestructures,andthebottompanelsshowthemoleculessolvatedwith108watermolecules.Themoleculetypesarerepresentedasfollows:redO,whiteH,greenC,blueN.25

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Figure4.2:Volumeuctuationsforbothcisredandtransgreenazobenzene,aswellasneatwaterblueduring100nsofdynamicsaredisplayed.Theaveragevaluesofthevolumesarerepresentedasstraightdashedlines.eachisomer'smolecularvolumewastakenasthestandarddeviationassociatedwitheachsetofmeasurements.4.3ComputationalResultsandDiscussionTracesofthesolvatedazobenzenevolumeuctuationsfor100psofdynamicsaredisplayedinFigure4.2.Thetoptwotracesrepresentthevolumesofcisandtransazobenzene,whilethebottomtracerepresentsthesolventvolume.Theresultsofazobenzene'ssimulationaresummarizedinTable4.3.Basedon72nsofdynamics,thevolumesofcisandtrans-azobenzeneweredeterminedtobe148.226

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System cis trans V Charged 148.20.3 151.80.3 -3.560.6 Uncharged 152.60.4 148.80.4 3.80.8 Table4.1:Thistabledisplaysthemolecularvolumescalculatedforcisandtransazoben-zene,boththechargedandunchargedsystems.Inthenalcolumn,thevolumechangeassociatedwiththetrans-cisisomerizationisshown.AllresultsaregiveninmL/mol.0.3mL/moland151.80.3mL/mol,respectively.Thevolumechangeassociatedwithisomerizationis-3.560.6mL/mol,aresultwhichcomparesfavorablywiththeexperimentalmolecularvolumechange.26Theabsolutevolumesalsoconformcloselytothevolumesofthecrystalstructures:149-150mL/molforcis-azobenzeneand148-149mL/molfortrans-azobenzene.28,29Whilethesenumbersarenotstrictlycomparableduetothedierentchemicalenvironmentsassociatedwithaqueousandcrystallineazoben-zene,theagreementisstriking.Examinationoftheradialdistributionfunctionofazobenzene'snitrogenatomswiththewaterhydrogenatoms,displayedinFigure4.3,conrmstheseresults.Theradialdistributionfunctionassociatedwithcis-azobenzeneexhibitsamarkedrstneighborpeakshiftedslightlytotheleftoftrans-azobenzene'sbroadrstneighborpeak.Thispeaksuggestsorderingofthesolventandcloserproximityofwater'shydrogenatomstocis-azobenzene'snitrogenatoms.Thetightersolvationofcis-azobenzene,relativetothetransform,resultsinareducedmolecularvolume.27

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Figure4.3:Theradialdistributionfunctionofazobenzene'snitrogenatomswithwater'shydrogenatomsisdisplayed.Thebluelineindicatescis-azobenzene,andtheredlinetrans-azobenzene.Themarkedrstneighborpeakforthecisformindicatesorderingofthesolventandcloserproximityofthesolventmoleculestocis-azobenzene'snitrogenatoms.28

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MolecularvolumesareoftenconsideredintermsofVanderWaalsradiiandthemannerinwhichtheoverlappingspheresllspace.Whilestericinteractionsplayamajorroleindeterminingmolecularvolumes,asdemonstratedinChapter3,electrostaticinteractionsalsomustbeexamined,especiallyinmeasuringdynamicalvolumechanges.Toassesstheroleofelectrostrictioninazobenzene'svolumechange,bothaqueousazobenzenemoleculesweresimulatedwiththeatomicchargesremoved.Inthiscase,thevolumesofcisandtrans-azobenzeneweredeterminedtobe152.60.4mL/moland148.80.4mL/mol,respectively,basedon40.5nsofdynamics.Thevolumeofthecisisomerisnowsignicantlylargerthanthetrans,aresultconsistentwithitslargercrystalstructureandbulkierthree-dimensionalstructure.Thisresultalsosuggeststhatcis-azobenzene'slargedipoleisresponsibleforanexcesselectrostrictionof8mL/mol,whichmakesthestericallylargercisisomerhaveasmallersolvatedvolumethanthetransisomer.Thisstudyofazobenzenedemonstratestheabilityofthiscombinedtheoreticalandexperimentalapproachtoprovideatomisticresolutionoftheoriginofmolecularvolumechanges.Whilethemethodisappliedtoavolumechangebetweentwoequilibriumstatesinthiscase,itmayalsobeusedtoaccuratelydescribethechangesinshapeandvolumeassociatedwithintermediatesalongareactionpathway.29

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Chapter5TimeCorrelationFunctionFormalismTimecorrelationfunctionTCFtheoriesareusefulforlinkingthenatureofasys-tem'sdynamicstospectrageneratedusinginnovativeexperimentaltechniques,suchassumfrequencygeneration,two-dimensionalRaman,andtwo-dimensionalinfraredspec-troscopy.Inthischapter,thetimecorrelationfunctionformalism'snaturalconnectiontoFermi'sGoldenRuleandtime-dependentpertubationtheoryisoutlined.InChapters6and7,thederivationsofTCFtheoriesforone-dimensionalnonlinearspectroscopyandthefth-orderresponsefunctionRarediscussedasbackgroundforthedevelopmentofaTCFtheoryoftwo-dimensionalinfraredspectroscopy,describedinChapters8-10.5.1LinearAbsorptionofRadiationThisanalysisbeginswithasystemofNinteractingmoleculesininitialquantumstatei.ThesystemisdescribedbyitsHamiltonianH0,whereH0j=Ejj.Thesysteminteractswithamonochromaticelectriceldoffrequency!,describedbyEquation5.1.Et=E0cos!t=E0"ei!t+e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!t.130

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E0istheamplitudeoftheeldandisaunitvectorindicatingthedirectionoftheeld.Theeldisconsideredtobespatiallyuniformandthewavelengthsignicantlygreaterthanthesizeoftheinteractingmolecules.Usingtheseassumptions,theinterac-tionbetweentheeldandthemolecules,whichactsasapertubationtoH0,iswrittenasshown.Ht=)]TJ/F20 11.955 Tf 9.298 0 Td[(MEt.2Misthetotalelectricdipolemomentoperatorofthesystem.AccordingtotheFermiGoldenruleoftime-dependentpertubationtheory,theprobabilityperunittimethatatransitionfrominitialstateitonalstatefwilloccurisgivenbyEquation5.3.Pi!f!=E20 2~2jhfj"Mjiij2[!fi)]TJ/F20 11.955 Tf 11.956 0 Td[(!+!fi+!].3MultiplyingEquation5.3by~!fi,theenergydierencebetweenthenalandinitialstates,givestherateofenergylostfromtheradiationinthetransitionfromstateitof.Summingoverallstatesfyieldstheenergylostintransitioningfromstateitoanyotherstate.Finally,multiplyingbytheprobabilityiofbeinginstateiandsummingoveralligives_Erad,therateofenergylossfromtheradiationtothesystemduringthetransition.31

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)]TJ/F15 11.955 Tf 13.332 3.022 Td[(_Erad=XiXfi~!fiPi!f)]TJ/F15 11.955 Tf 13.331 3.022 Td[(_Erad=E20 2~XfXi!fiijhfj"Mjiij2[!fi)]TJ/F20 11.955 Tf 11.955 0 Td[(!+!fi+!].4Itisreasonabletoswitchtheiandfindicesinthesumovertheseconddeltafunction,sinceiandfincorporateallquantumstatesofthesystem.ThisallowsthesimplicationofEquation5.4.)]TJ/F15 11.955 Tf 13.332 3.022 Td[(_Erad=E20 2~XfXi!fii)]TJ/F20 11.955 Tf 11.955 0 Td[(fjhfj"Mjiij2!fi)]TJ/F20 11.955 Tf 11.955 0 Td[(!5.5Ifthesystemisinitiallyinequilibrium,f=e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!fii.Also,enforcingthedeltafunctioninEquation5.5allowsall!fitobewrittensimplyas!andpulledoutofthesum.)]TJ/F15 11.955 Tf 13.332 3.022 Td[(_Erad=E20 2~!)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!XfXiijhfj"Mjiij2!fi)]TJ/F20 11.955 Tf 11.955 0 Td[(!.6Theabsorptivecross-section!isdenedastheratiooftheradiationloss_EradtotheincidentuxofradiationS,shownbelowinEquation5.7.S=c 8nE20.7crepresentsthespeedoflightinavacuumandnistheindexofrefractionofthemedium.UsingEquations5.6and5.7,theabsorptivecross-sectioncanbederived.32

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!=_Erad S42 ~cn!)]TJ/F20 11.955 Tf 11.956 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!XfXiijhfj"Mjiij2!fi)]TJ/F20 11.955 Tf 11.955 0 Td[(!.8Equation5.8maybeusedtodeneanabsorptionlineshapeI!,whichisdenedbelowinEquation5.9.I!=3hcn! 42!)]TJ/F20 11.955 Tf 11.956 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!I!=3XiXfijhfj"Mjiij2!fi)]TJ/F20 11.955 Tf 11.956 0 Td[(!5.9Toderiveatimecorrelationfunctionofspectroscopy,Equation5.9shouldbewrittenintheHeisenbergrepresentation.ThisisaccomplishedbyintroducingtheDiracdeltafunction,denedbelow.!=1 2Z1ei!tdt.10IncorporatingtheDiracdeltafunctionintothelineshapeexpressiongivesEqua-tion5.11.I!=3 2XiXfihij"MjfihfjMjiiZ1e[Ef)]TJ/F21 7.97 Tf 6.586 0 Td[(Ei=~)]TJ/F21 7.97 Tf 6.587 0 Td[(!]it.1133

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Statesiandjareeigenstatesoftheunperturbedsystem,implyingthate)]TJ/F21 7.97 Tf 6.587 0 Td[(iEit=~jii)]TJ/F20 11.955 Tf -436.001 -28.892 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(iH0t=~jiiandhfje)]TJ/F21 7.97 Tf 6.586 0 Td[(iEft=~=hfjeiH0t=~.Usingthisinformation,Equation5.11canbesimpliedevenfurther.I!=3 2Z1dte)]TJ/F21 7.97 Tf 6.586 0 Td[(i!tXiXfihij"MjfihfjMtjiiMt=eiH0t=~Me)]TJ/F21 7.97 Tf 6.587 0 Td[(iH0t=~.12Theclosurerelationship,Pfjfihfj,allowsfortheremovalofthesumoverffromtheequation.Thesumoverinitialstatesigivesanequilibriumensembleaverage,whichcanberepresentedusinganglebrackets.Finally,forisotropicuids,averagingoveralldirectionsgivesthesimpliedequationshownbelowinEquation5.13.30I!=1 2Z1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!thMMtidt.13ThelineshapefunctionI!hasnowbeenwrittenasthetimecorrelationfunctionoftheabsorbingmolecules'dipolemomentoperatorintheabsenceoftheelectriceldpertubation.Equation5.13allowsaconnectiontobeformedbetweenanobservable,givenbyI!andthemotionsofthemolecules'dipolemomentsovertime.34

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Chapter6TCFTheory:One-TimeCorrelationFunctionThetimecorrelationfunctiontheory,besidesbeingrelevanttolinearspectroscopy,hasbeenappliedsuccessfullytononlinearspectroscopies.Ithasalsobeenusedtodemon-stratethelackofnewinformationprovidedbyearlynonlinearexperiments,suchasOKEspectroscopy.31{34Toprovideanintroductiontothetechniquesusedinthedevelopmentofthetwo-dimensionalinfraredspectroscopyTCFtheory,outlinesoflower-ordertheoriesareprovided.Inthischapter,thedevelopmentofthetheoryoftheone-timecorrelationfunctionassociatedwithOKEandRamanexperimentsisdiscussed.Someofthebasicproceduresused,includingthederivationoffrequency-domaindetailed-balancerelation-shipstosimplifyresponsefunctions,willappearinlaterchapters.6.1TheOne-TimeCorrelationFunctionThisdevelopmentbeginswiththethird-orderresponsefunctionprobedinthethird-orderRamanexperiment.3435

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Rt=i ~h[t;]i.1Thesquarebracketsdenotethecommutatoroftand,andtheanglebrack-etsindicatethequantummechanicaltrace.Theisthepolarizabilityofthesystem.ExpansionofthecommutatorbracketsgivesRt=i ~[hti)-222(hti].2Theresponsefunctionisnowdisplayedintermsoftwoone-timequantumme-chanicaltimecorrelationfunctions.Classically,theoperatorstandcommutesuggestingthat,inthislimit,thereisnothird-orderRamansignal.TodevelopameansofcalculatingRusingclassicalmoleculardynamicstechniques,itisnecessarytorewriteEquation6.2intermsofaclassicaltimecorrelationfunction,butinsuchawaythatthesignalisnonzero.ThisisaccomplishedbyderivingEquation6.2intermsoftherealpartofasinglequantummechanicalTCF,whichisessentiallyequivalenttoaclassicaltimecorrelationfunction.Equation6.2actuallyisthedierencebetweenonequantummechanicalTCFCtanditscomplexconjugateCt.Toprovethis,thequantummechanicaltracesaretakenandtheoperatorsarewrittenintheHeisenbergnotation,wheret=e)]TJ/F21 7.97 Tf 6.586 0 Td[(iHt=~e)]TJ/F21 7.97 Tf 6.587 0 Td[(iHt=~.36

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Ct=hti=1 QXahaje)]TJ/F21 7.97 Tf 6.587 0 Td[(HeiHt=~e)]TJ/F21 7.97 Tf 6.587 0 Td[(iHt=~jaiCt=hti=1 QXahaje)]TJ/F21 7.97 Tf 6.587 0 Td[(He)]TJ/F21 7.97 Tf 6.587 0 Td[(iHt=~eiHt=~jai.3ThenacompletesetofstatesPbjbihbjisinsertedtogivetheTCFsinanotherform.BelowinEquation6.4itisclearthathtiandhtiarecomplexconjugates.Ct=1 QXaXbe)]TJ/F21 7.97 Tf 6.586 0 Td[(EaabbaeiEabt=~Ct=1 QXaXbe)]TJ/F21 7.97 Tf 6.586 0 Td[(EaabbaeiEbat=~.4InEquation6.4thenotationEijindicatestheenergydierenceEi)]TJ/F20 11.955 Tf 11.281 0 Td[(Ejandijamatrixelementhijjji.Now,thethird-orderresponsefunctioniswrittenagainintermsofCtandCt.Rt=i ~[Ct)]TJ/F20 11.955 Tf 11.955 0 Td[(Ct].5SinceCt=CRt+iCItandCt=CRt)]TJ/F20 11.955 Tf 12.019 0 Td[(iCIt,whereCRtistherealpartofCtandCItistheimaginarypart,Equation6.5canclearlybewrittenintermsofCItonly.Rt=)]TJ/F15 11.955 Tf 10.797 8.088 Td[(2 ~CIt.637

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6.2ObtainingRinTermsofCtUnfortunately,CIthasnoclassicallimitandcannotbecalculatedviaclassicaltechniques.However,itispossibletoidentifyananalyticalrelationshipbetweentherealandimaginarypartsofCt,allowingRtobewrittenasafunctionofarealTCFwhichhasavalidclassicallimit.Toderivesucharelationship,itishelpfultoFouriertransformCtandCttoobtaintheminthefrequencydomain.C!=FT[Ct]=1 2Z1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!tCtdt.7C)]TJ/F20 11.955 Tf 9.298 0 Td[(!=FT[Ct]=1 2Z1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!tCt6.8Thefrequency-domainTCFsareshownbelow.Theexponentialsappearinginthetime-domainTCFsofEquation6.4arereplacedbydeltafunctions.C!=1 QXaXbe)]TJ/F21 7.97 Tf 6.587 0 Td[(Eaabba!)]TJ/F20 11.955 Tf 11.955 0 Td[(Eab=~C)]TJ/F20 11.955 Tf 9.299 0 Td[(!=1 QXaXbe)]TJ/F21 7.97 Tf 6.587 0 Td[(Eaabba!)]TJ/F20 11.955 Tf 11.955 0 Td[(Eba=~.9TorelatethetwoTCFs,itisusefultorewriteC)]TJ/F20 11.955 Tf 9.299 0 Td[(!byippingtheindicesaandb.Thisactionisequivalenttotakingacyclicpermutationofthetrace.C)]TJ/F20 11.955 Tf 9.298 0 Td[(!=1 QXaXbe)]TJ/F21 7.97 Tf 6.587 0 Td[(Ebabba!)]TJ/F20 11.955 Tf 11.956 0 Td[(Eab=~.1038

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TheonlydierencebetweenC!andC)]TJ/F20 11.955 Tf 9.298 0 Td[(!isnowtheBoltzmannfactor,therstiteminsidethesum.ThisimpliesthatC!=e)]TJ/F21 7.97 Tf 6.587 0 Td[(EaeEbC)]TJ/F20 11.955 Tf 9.299 0 Td[(!=eEbaC)]TJ/F20 11.955 Tf 9.299 0 Td[(!.11ThedeltafunctioninEquation6.10requires!=Eab,thereforeEquation6.11becomesC!=e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!C)]TJ/F20 11.955 Tf 9.298 0 Td[(!.12Thisdetailed-balancerelationshipbetweenC!andC)]TJ/F20 11.955 Tf 9.298 0 Td[(!maybeappliedtothefrequency-domainresponsefunctiontosimplifyit.R!=i ~[C!)]TJ/F20 11.955 Tf 11.955 0 Td[(C)]TJ/F20 11.955 Tf 9.299 0 Td[(!]=i ~)]TJ/F20 11.955 Tf 11.955 0 Td[(e~!C!.13Intheclassicallimit,theexponentialcanbeexpandedouttorstordertogivethefollowingresult.R!=i ~~!C!=i!C!.1439

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6.3RelatingtheRealandImaginaryPartsofC!AnalternativemeansofderivingEquation6.14istoderiveananalyticalrela-tionshipbetweentherealandimaginarypartsofthefrequency-domainTCFC!.Thistechnique,whileseeminglyredundantinthecaseoftheone-timecorrelationfunction,willproveextremelyusefulintheanalysisofhigher-orderTCFs.Inthetimedomain,theTCFCtisthesumofitsrealandimaginaryparts.Ct=CRt+iCIt.15ThetwopartsareFouriertransformedseparatelytoobtainCR!andCI!,bothofwhicharereal.CR!=1 2Z1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!tCRtdtCI!=1 2Z1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!tiCItdt.16ItiseasytodeneCR!andCI!intermsofC!andC)]TJ/F20 11.955 Tf 9.298 0 Td[(!,giventhatFT[Ct]=C!andFT[Ct]=C)]TJ/F20 11.955 Tf 9.299 0 Td[(!.CR!=[C!+C)]TJ/F20 11.955 Tf 9.298 0 Td[(!]=2CI!=[C!)]TJ/F20 11.955 Tf 11.955 0 Td[(C)]TJ/F20 11.955 Tf 9.298 0 Td[(!]=2.17Usingthesimpledetailed-balancerelationshipdenedinEquation6.12,aratiobetweenCI!andCR!iswritten.40

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CI!=[1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!]C! [1+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!]C!CR!.18Multiplicationoftheright-handsideofEquation6.18bye~!=2=e~!=2resultsinatanhrelationshipbetweenCI!andCR!.CI!=tanh~!=2CR!.19Inthefrequencydomain,thelinearresponsefunctioniswrittenasthedierenceofC!andC)]TJ/F20 11.955 Tf 9.298 0 Td[(!,whichisjusttwiceCI!.R!=i ~[C!)]TJ/F20 11.955 Tf 11.955 0 Td[(C)]TJ/F20 11.955 Tf 9.299 0 Td[(!]=i ~[2CI!]6.20Usingthenewlyderivedtanhrelationship,theresultobtained,whentheclassicallimitistaken,isthesameasEquation6.14.R!=i ~[2tanh~!2CR!]R!~!0=i!CR!.2141

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6.4TransformingtotheTimeDomainSincecalculationswillbeperformedinthetimedomain,itisusefultoobtainthisresponsefunctioninthetimedomain.ItiseasilyproventhatEquation6.14actuallyrepresentsatimederivativeofCt.dCt dt=d dtZ1ei!tC!d!.22InEquation6.22,theright-handsidedisplaysthetimederivativeofthereverseFouriertransformofC!,whichgivesCt.CarryingoutthisderivativeyieldsdCt dt=iZ1!ei!tC!d!.23InEquation6.23,theright-handsiderepresentsthereverseFouriertransformof!C!,whichappearsintheresponsefunction.Thus,inthetimedomain,theresponsefunctioncanbewrittenasatimederivativeasshown.Rt=d dthti.24Equation6.24iswrittenintermsofCt,thesameTCFthatisassociatedwiththelinearexperiment,asshowninEquation5.13.ThisresultsupportstheideathatthenonlinearRamanexperimentprovidesnonewinformationbeyondwhatthelinearexperimentdoes.42

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Chapter7TCFTheory:Fifth-OrderRamanSpectroscopyInthischapter,thedevelopmentofthetimecorrelationfunctiontheoryoffth-orderRamanspectroscopyispresented.Whilelittleattentionwillbegiventotheexper-imentaldetailsorcomputationalresults,theprocedureusedinderivingthetheorywillbediscussedthoroughlyasameansofprovidinganintroductiontoseveraltechniquesusedindevelopingthetheoryoftwo-dimensionalinfraredspectroscopy.Manyoftheproceduresusedtoderivethefth-orderRamanTCFtheoryaresimilarinnaturetothetechniquesusedinChapter6toexaminetheone-timeTCFsas-sociatedwithR.Exactfrequency-domaindetailed-balancerelationshipsareemployedtowriteRintermsoftherealandimaginarypartofasinglequantummechanicalTCF.However,unliketheone-timecase,itisnotpossibletoconstructanexactRexpressionthatcanbecalculatedintermsofclassicalTCFs35.Therefore,theharmonicoscillatorwithnonlinearpolarizability,thelowestorderreferencesystemthatgivesasignal,36{38isintroducedasameansofapproximatelyeliminatingtheimaginarypartoftheremainingTCF,therebymakingthefth-orderresponsefunction'scalculationamenabletoclassicalmoleculardynamicsmethods.43

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7.1TheFifth-OrderResponseFunctionInthefth-orderRamanexperiment,asixwavemixingtechnique,apairoffspulsesattimezeroexciteasystem,leavingitinavibrationalcoherence.Afteratimedelayt1,duringwhichthesystemundergoesfreeevolution,thesystemisexcitedbyasecondpairofpulsesthattransfersittoanewvibrationalcoherenceorpopulationstate.Afteranothertimedelayt2,asinglepulseprobesthesystem.Theobservableofinterestinthefth-orderRamanexperimentisthefth-orderpolarizationPt,whichisrepresentedastheconvolutionoftheincidentelectriceldswiththefth-orderresponsefunctionRt1;t2.Thequantummechanicalexpressionfortheelectronicallynonresonantfthorderpolarizationresponseisgivenby34,39{41Rt1;t2=)]TJ/F25 11.955 Tf 11.291 16.857 Td[(1 ~2Trft1+t2[t1;[;]]g.1InEquation7.1,=e)]TJ/F21 7.97 Tf 6.587 0 Td[(H=QforasystemdescribedbyHamiltonianHandpartitionfunctionQ.Thevariablerepresentsthereciprocaltemperature1=kT,wherekisBoltzmann'sconstant.representsthesystem'spolarizabilitytensorandtheGreeksuperscriptsaretheelementsunderconsideration.ExpansionofthecommutatorsrevealsthatRappearsintermsoftwoquantummechanicalTCFs,denedasft1;t2andgt1;t2,andtheircomplexconjugatesft1;t2andgt1;t2.Theresponsefunctionitselfisrealintime,whiletheindividualTCFsarecomplex.44

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Rt1;t2=)]TJ/F25 11.955 Tf 11.291 16.857 Td[(1 ~2[gt1;t2)]TJ/F20 11.955 Tf 11.955 0 Td[(ft1;t2)]TJ/F20 11.955 Tf 11.955 0 Td[(ft1;t2+gt1;t2]7.2ThetwoTCFsandtheircomplexconjugatesaredenedbyft1;t2=ht1t2ift1;t2=ht2t1igt1;t2=ht2t1igt1;t2=ht1t2i.3Noticethatintheclassicallimit,theoperatorscommute,makingTCFsfandgequivalent,andtheentireRexpression,aswritten,becomeszero.InEquation7.2,thecontentsofthesquarebracketsrepresentthedierencebetweentherealpartsofTCFsgt1;t2andft1;t2,i.e.gRt1;t2)]TJ/F20 11.955 Tf 11.955 0 Td[(fRt1;t2.Intheclassicallimit,thetraceinEquation7.1musthavealeading~2prefactortocancelthe1=~2shownintheequation.Usingexactfrequency-domaindetailed-balancerelationshipsbetweenthetwoquantummechanicalTCFs,itwillbepossibletoextractoneofthese~.Afterthisstep,theexpressionremainsintermsoftherealandimaginarypartsofoneTCF.Itisnotpossibletodiscernanexactrelationshipbetweenthesetwoparts,thusbringingoutthenalneededfactorof~.However,asimpleapproximaterelationshipbetweenthetwopartswasdiscoveredbyapproximatingaharmonicoscillatorwithnonlinearpolarizability.45

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7.2RelatingTCFsfandgAsdemonstratedearlierwiththeone-timeTCFs,itisusefultowriteft1;t2andgt1;t2outintheirenergyrepresentationsasameansofndingarelationshipbetweenthem.CarryingoutthissteptheninsertingthreecompletesetsofstatesPijiihijwithHjii=Eijiigivesthefollowingresults.Manipulatingtheindicesi,j,andk,equivalenttotakingcyclicpermutationsofthetrace,allowsthesimilaritiesbetweenTCFsfandgtobemaximized.TheonlydierencebetweenthemistheBoltzmannfactore)]TJ/F21 7.97 Tf 6.587 0 Td[(H.ft1;t2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.587 0 Td[(Eiikkjjieit1Eki=~eit2Ekj=~gt1;t2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.586 0 Td[(Ekikkjjieit1Eki=~eit2Ekj=~.4Similarly,thecomplexconjugatescanalsobewrittenintheenergyrepresentation.Ifthepolarizabilitymatrixelementsarechosenrealsothatij=ij,itisclearthatft1;t2=f)]TJ/F20 11.955 Tf 9.298 0 Td[(t1;)]TJ/F20 11.955 Tf 9.298 0 Td[(t2andgt1;t2=)]TJ/F20 11.955 Tf 9.299 0 Td[(g)]TJ/F20 11.955 Tf 9.299 0 Td[(t1;)]TJ/F20 11.955 Tf 9.299 0 Td[(t2.ft1;t2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.587 0 Td[(Eiikkjjieit1Eik=~eit2Ejk=~gt1;t2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.586 0 Td[(Ekikkjjieit1Eik=~eit2Ejk=~.5AdoubleFouriertransformconvertstheTCFstothefrequencydomain,giv-ingf!1;!2andg!1;!2.ItiseasytoprovethatFT[ft1;t2]=f)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2andFT[gt1;t2]=g)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2.Theproofisoutlinedbelow.Thenalstepoftheproofarisessincethefrequency-domainTCFsarereal,implyingthatf!1;!2=f!1;!2.46

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f!1;!2=1 22Z1Z1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!2t2ft1;t2dt1dt2f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=1 22Z1Z1ei!1t1ei!2t2ft1;t2dt1dt2f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=1 22Z1Z1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!2t2ft1;t2dt1dt2.6Inthefrequencydomain,thefourTCFscanbewrittenintermsofsimpledeltafunctions.f!1;!2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.586 0 Td[(Eiikkjji!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eki=~!2)]TJ/F20 11.955 Tf 11.956 0 Td[(Ekj=~f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.586 0 Td[(Eiikkjji!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eik=~!2)]TJ/F20 11.955 Tf 11.956 0 Td[(Ejk=~g!1;!2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.586 0 Td[(Ekikkjji!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eki=~!2)]TJ/F20 11.955 Tf 11.956 0 Td[(Ekj=~g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=1 QXiXjXke)]TJ/F21 7.97 Tf 6.586 0 Td[(Ekikkjji!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eik=~!2)]TJ/F20 11.955 Tf 11.956 0 Td[(Ejk=~.7SincetheTCFsfandgdieronlyintheBoltzmannfactorsweightingthem,itisstraightforwardtoderivefrequency-domainrelationshipsbetweenthem.f!1;!2=eEke)]TJ/F21 7.97 Tf 6.586 0 Td[(Eig!1;!2=eEkig!1;!2f)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2=eEie)]TJ/F21 7.97 Tf 6.587 0 Td[(Ekg)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=eEikg!1;!27.847

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EnforcingthedeltafunctionsinEquation7.7requiresthat!1=Eki=~inpositivefrequencyand!1=Eik=~innegativefrequency.ThisactionallowstheBoltzmannfactorstobetakenoutsidethesummationsandusedtorelatethepairofTCFs.Thisresultgivesthedetailed-balanceequationswhichwillbeusedtorelateTCFsfandginpositiveandnegativefrequency.f!1;!2=e~!1g!1;!2f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2.9AnotherusefulstepistoconsiderthewayinwhichthesumsanddierencesofTCFsfandgarerelated.Inpositivefrequency,f!1;!2)]TJ/F20 11.955 Tf 11.955 0 Td[(g!1;!2 f!1;!2+g!1;!2=e~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(1 e~!1+1g!1;!2 g!1;!2.10Multiplyingtheratiobye)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1=2=e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1=2givesasimpletanhrelationshipbetweenthedierenceandthesumoffandg.Therelationshipfornegativefrequency,derivedinthesamemanner,isalsoshown.f!1;!2)]TJ/F20 11.955 Tf 11.955 0 Td[(g!1;!2=tanh~!1=2[f!1;!2+g!1;!2]f)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2)]TJ/F20 11.955 Tf 11.955 0 Td[(g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1=2[f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2+g)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2].11UsingEquation7.11,thefrequency-domainfthorderresponsefunctioncanbewrittenasshown.48

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R!1;!2=1 ~2tanh~!1=2[f!1;!2+g!1;!2)]TJ/F20 11.955 Tf -132.967 -31.88 Td[(f)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2)]TJ/F20 11.955 Tf 11.955 0 Td[(g)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2].12Now,usingEquation7.9,itispossibletoexactlyeliminateTCFffromtheRexpression.Havingthefth-orderRamanexpressionintermsofonequantummechanicalTCF,ratherthantwo,willultimatelymakeitscomputationlessexpensive.R!1;!2=1 ~2tanh~!1=2[+e~!1g!1;!2)]TJ/F15 11.955 Tf -115.377 -31.88 Td[(+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2].137.3ClassicalLimitofRIntheclassicallimit,where~!<
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g!1;!2+g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=2gR!1;!2andg!1;!2)]TJ/F20 11.955 Tf 12.325 0 Td[(g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2=2gI!1;!2.TheclassicallimitRexpressionthenbecomesR!1;!2=1 ~2~!1=2[4gI!1;!2+2~!1gR!1;!2].15Inthetimedomain,theRexpressionintheclassicallimitcanbewrittenintermsoftimederivativesofgt1;t2.Rt1;t2=)]TJ/F20 11.955 Tf 9.298 0 Td[(2d2 dt21gRt1;t2)]TJ/F15 11.955 Tf 13.151 8.088 Td[(2i ~d dt1gIt1;t2.167.4ApplyingaHarmonicApproximationInEquation7.15,both~prefactorsarecancelledoutingR,butingI,one1=~stillremains.Consequently,gRcanbecalculatedintheclassicallimitastheclassicalTCFh)]TJ/F20 11.955 Tf 9.299 0 Td[(t1t2i,butgIhasnoapparentclassicallimit.Fortheentireexpressiontobevalidintheclassicallimit,itisusefultondarelationshipbetweengRandgIinordertoremovethis~.Asseeninthepreviouschapter,theone-timecorrelationfunctionCtexhibitsasimplefrequency-domainrelationshipbetweentheFouriertransformsoftherealandimaginaryparts,CI!=tanh~!=2CR!.Tobeginderivingasimilarrelationshipforgt1;t2,anapproximationmustbeconsidered.AnapproximateconnectionbetweengRandgIcanbeestablishedusingaharmonicreferencesystemwithanonlinearpolarizability,thelowestordersystemthatproducesafthorderresponse.50

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TheharmonicsystemisrepresentedbyaharmonicpotentialoftheformV=m2q2=2,whereisafundamentalharmonicfrequencyandqistheharmoniccoordinate.Harmonicstatesareassumed,andenergiesareharmonic,i.e.E=~+1=2.Thepolarizabilitymatrixelementsinthefrequency-domainTCFs,asshowninEquation7.7,areexpandedouttosecondorderintheharmoniccoordinate.ij=0ij+0hijqjji+1 200hijq2jji.17InEquation7.17,theprimesrepresentderivativeswithrespecttotheharmoniccoordinate.0isthestaticpolarizability.Thematrixelementsintermsoftheharmoniccoordinatearewrittenasfollows.42hijqjji=r ~ 2mi;j+1i+11=2+i;j)]TJ/F18 7.97 Tf 6.587 0 Td[(1j1=2hijq2jji=~ 2m[i;j+2[j+1j+2]1=2+2ijj+1 2+i;j)]TJ/F18 7.97 Tf 6.587 0 Td[(2[jj)]TJ/F15 11.955 Tf 11.955 0 Td[(1]1=2].18WhenthepolarizabilitymatrixelementsinTCFgareexpandedaccordingtoEquation7.17,TCFgissplitintoseveralterms,whichwillbedesignatedgabc,wherea,b,andcrepresenttheordertowhicheachpolarizabilityelementistaken.Forexample,g000iswrittenintermsofthree0elements.Thersttermtoconsider,g000,iswrittenasshownbelow.51

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g!1;!2000=1 Q03Xke)]TJ/F21 7.97 Tf 6.587 0 Td[(Ek.19Inthetimedomain,thederivativedg000=dt1=0,implyingbyEquation7.16thattheg000termdoesnotcontributetoR.Next,thetermsg001,g010,andg100,eachofwhichhaveone0andtwo0elements,areconsidered.Writingoutthematrixelementsandenforcingtheincludeddeltafunctionsdemonstratesthatallthreeofthesetermsarezero,andcanthereforedonotneedtobeincludedintheexpansionofg!1;!2.Bythesamelogic,theg111isexcludedfromtheRexpression.Thenexttermstoconsideraretheonescontainingtwo0andone0,namelyg011,g101,andg110.Thesetermsareshownbelow.g011!1;!2=000~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~[e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!2)]TJ/F15 11.955 Tf 11.955 0 Td[(+!2+]g101!1;!2=000~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~[e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(+!1+]g110!1;!2=000~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1+!2++!1+!2+.2052

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Fortermg011thet1derivativesarezero,preventingitfromcontributingtothenalexpression.TheothertwotermsareexcludedbecauseRiscalculatedintermsofpolarizabilityuctuations,meaningthatcontributionsfromstaticpolarizabilityareeliminatedbyconstruction.Therefore,theonlytermslefttoconsideraretheonescontainingtwo00andone0,whichareg211,g121,andg112.Thetotalcontributiontog!1;!2oftherstnonvanishingorderisg!1;!2=g211!1;!2+g121!1;!2+g112!1;!2.21GivenEquation7.21,itisusefultoderiverelationshipsbetweentherealandimaginarypartsofallthreetermsandattempttogeneralizetherelationshiptotheentireTCF.Toderiveeachterm,thematrixelementsarewrittenout,thedeltafunctionsareenforced,andthegeometricseriesgivenbythesumsevaluated.42Asanillustration,termg211isshownbelowinthefrequencydomain.g211!1;!2=00001 Q~ 2m21 2Xke)]TJ/F21 7.97 Tf 6.586 0 Td[(~k[k+1k+2!1+2!2++k+1k+1!1!2++.22kk+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[(].23Torelateitsrealandimaginaryparts,itisnecessarytoreverseFouriertransformg211tothetimedomain.53

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g211t1;t2=0000~ 2m21 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~2[2e)]TJ/F18 7.97 Tf 6.586 0 Td[(2it1e)]TJ/F21 7.97 Tf 6.587 0 Td[(it2+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~+1e)]TJ/F21 7.97 Tf 6.586 0 Td[(it2+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~e)]TJ/F21 7.97 Tf 6.586 0 Td[(it2+2e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~e2it1eit2].24ToderivetherealpartB211Rt1;t2,thecomplexconjugateisaddedtoEquation7.24,andtoderivetheimaginarypartB211It1;t2,thecomplexconjugateissubtracted.Thecomplexconjugateisdeterminedbychangingallito)]TJ/F20 11.955 Tf 9.298 0 Td[(i.g211Rt1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2+2e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~cost1+t2+e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~+6e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+1cost2g211It1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~)]TJ/F15 11.955 Tf 11.955 0 Td[(2sint1+t2+e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~)]TJ/F15 11.955 Tf 11.955 0 Td[(1sint2.25Similarmanipulationscanbeusedtoderivetherealandimaginarypartsoftheg121andg112terms.g121Rt1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2+2e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~cost1+2t2+e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~+6e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+1cost154

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g121It1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~)]TJ/F15 11.955 Tf 11.956 0 Td[(2sint1+2t2+e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~)]TJ/F15 11.955 Tf 11.956 0 Td[(1sint1g112Rt1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~24e)]TJ/F21 7.97 Tf 6.586 0 Td[(~cost1)]TJ/F15 11.955 Tf 11.955 0 Td[(t2+e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~+2e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+3cost1+t2g112It1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~)]TJ/F15 11.955 Tf 11.955 0 Td[(3sint1+t2.267.5RemovinggIfromtheRExpressionThethreecontributionstoTCFgcaneachbesplitintotwotermswithdistincttimeorfrequencydependence,yieldingatotalofsixcontributionstog.Eachshouldoeranorder~relationshipbetweenitsrealandimaginaryparts.Toobtainaclassicalresult,therelationshipforallsixtermsmustbeofthesameform,asthedistinctcontributionscannotbedistinguishedduringthecourseofaclassicalmoleculardynamicssimulation.Whilenosingleexpressionworksperfectlyforallsixterms,ithasbeendeterminedthatthefollowingtanhrelationshipisasatisfactoryapproximationintheclassicallimitandleadstoaTCFtheoryforRthathasreasonablelimitingbehavior.gI!1;!2=tanh)]TJ/F20 11.955 Tf 9.299 0 Td[(~!1=4+!2=2gR!1;!2.2755

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Intheclassicallimit,withexponentialsexpandedouttoorder~,thetanhshowninEquation7.27becomes)]TJ/F20 11.955 Tf 9.298 0 Td[(~!1=4+!2=4.Table7.5displaysthelimitingformofallfrequencyprefactorsandthevalueof)]TJ/F20 11.955 Tf 9.298 0 Td[(~!1=4+!2=4forallterms,withg211,g121,andg112eachsplitintoitstwoparts.Intheclassicallimit,fortheproposedrelationshipinEquation7.27tobevalid,multiplyingthecoecientofgRby)]TJ/F20 11.955 Tf 9.299 0 Td[(~!1=4+!2=4shouldyieldthecoecientofgI.Forthreeterms,theproposedtanhrelationshipgivestheexactdesiredresults,anditcomesveryclosefortheotherthree.Toillustratehowthecoecientsaredened,coecientsC1andC2fortermg211Rarewrittenbelow.g211Rt1;t2=0000~ 2m21 41 )]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2[C1cost1+t2+C2cost2]C1=2+2e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~C2=e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~+6e)]TJ/F21 7.97 Tf 6.586 0 Td[(~+17.28Forthethreetermswhicharenotaccuratelydescribedbythetanhrelationship,errorsinlteringthedynamics"areintroducedintothecalculations.Inspiteofthisinaccuracy,thisapproximationhasbeenusedtoproduceresultsforliquidxenonwhichareinnearlyquantitativeagreementwithexactnumericalcalculations.43{45Intheclassicallimit,thetanhrelationshipisappliedtoEquation7.14toobtainthefth-orderresponsefunctionintermsofgR!1;!2alone.R!1;!2=1 ~2~!1 2[~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(2~!2]gR!1;!2.2956

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Term gR gI !1 !2 )]TJ/F20 11.955 Tf 9.298 0 Td[(~!1=4+!2=2 g211Term1 4 )]TJ/F15 11.955 Tf 9.298 0 Td[(4~ 2 )]TJ/F20 11.955 Tf 9.299 0 Td[(~ g211Term2 8 )]TJ/F15 11.955 Tf 9.298 0 Td[(2~ 0 )]TJ/F20 11.955 Tf 9.299 0 Td[(~=2 g121Term1 4 )]TJ/F15 11.955 Tf 9.298 0 Td[(4~ 2 )]TJ/F15 11.955 Tf 9.298 0 Td[(5~=4 g121Term2 8 )]TJ/F15 11.955 Tf 9.298 0 Td[(2~ 0 )]TJ/F20 11.955 Tf 9.299 0 Td[(~=4 g112Term1 4 0 )]TJ/F15 11.955 Tf 9.298 0 Td[( )]TJ/F20 11.955 Tf 9.299 0 Td[(~=4 g112Term2 8 )]TJ/F15 11.955 Tf 9.298 0 Td[(6~ )]TJ/F15 11.955 Tf 9.298 0 Td[(3~=4 Table7.1:TheusefulnessoftherelationshipgI!1;!2=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh[~!1=4+!2=2]gR!1;!2foraharmonicreferencesystemisdemonstrated.ThesixtermsoftheTCFg!1;!2aredisplayed.Intheclassicallimit,thistanhrelationshipdictatesthatthegRcoecientcolumn2multipliedby)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh[~!1=4+!2=2]column6shouldgivethegIcoecientcolumn3.Thisrelationshipholdsexactlyforg111Term1,g121Term2,andg112Term2,andverynearlyfortheremainingthreeterms.57

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Inthetimedomain,thisexpressioniswrittenoutintermsoftimederivativesofgRt1;t2,aTCFwhichcan,intheclassicallimit,becalculatedusingmoleculardynamicstechniques.Rt1;t2=)]TJ/F20 11.955 Tf 10.494 8.088 Td[(2 2d2 dt21gRt1;t2)]TJ/F15 11.955 Tf 11.955 0 Td[(2d2 dt22gRt1;t2.3058

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Chapter8TCFTheory:2D-IRSpectroscopyExactProbingthethree-dimensionalstructureofmolecules,particularlycomplexpro-teins,peptides,andotherbiomolecules,isanimportantobjectiveinmanyspectroscopicinvestigations.Inthecaseofproteins,thethree-dimensionalstructureisadirectreec-tionofthemolecule'sfunction.Experimentallydeterminingthethree-dimensionalstruc-tureoftransientspeciesincondensedphasesisdicultbutessentialtounderstandingthenatureofmanyinterestingtime-dependentprocesses,suchasproteinfolding.Tra-ditionalexperimentaltechniquesincludingmultidimensionalnuclearmagneticresonanceNMRspectroscopy46{49andX-Raydiraction50,51workonmillisecondtimescalesandeectivelyrevealtime-averagedthree-dimensionalstructures.However,sincemanysys-temsaretransient,suchasproteins,whoseconformationsoftenuctuateseveraltimesonthistimescale,thesetechniquesoftenfailtoaccuratelydescribeshort-livedspecies.Two-dimensionalinfraredD-IRspectroscopy,atechniquecurrentlyunderex-tensivetheoreticalandexperimentalinvestigation,showsgreatpotentialforprovidingnewinformationabouttimeevolvingstructures.52{56UnlikemultidimensionalNMRandX-Raydiraction,2D-IRcanapproachatimescaleontheorderofpicosecondsorevenfemtoseconds.Additionally,2D-IRiscapableofresolvingthestructuresofcomplex59

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molecules,whoselinearspectraareoftenextremelycongestedanddiculttointerpret.Couplings,revealedaspairsofcross-peaksin2D-IRspectra,aresuggestiveoftherela-tiveorientationsofpairsofcoupledanharmonicoscillatorsandconsequentlyhintatthesystem'sthree-dimensionalstructure.Thescienticcommunityhasrecognized2D-IR'sadaptabilitytomanyinterest-ingapplications.2D-IRtechniqueshavealreadyprovenusefulinexaminingthecoupledcarbonylstretchesofRhCO2C5H7O2,57{64thenuclearpotentialenergysurfaceofcou-pledmolecularvibrations,65vibrationalrelaxation,66,67interactionsbetweensolventandsolute,68{70conforationaluctuationsinpeptides,71{74thethree-dimensionalstructureofpeptidesandsmallproteins,75{87,thehydrogenbondnetworkinwater,88,89andthecou-plingofcytidineandguanosineinDNA.902D-IRspectroscopycanbeusedtodeterminethecouplingandprojectionanglebetweentwoanharmonicvibrations.65Inthischapter,atimecorrelationfunctionTCFtheoryoftwo-dimensionalinfraredspectroscopyisdeveloped.Thistheoryallowsthecalculationofasystem'sthirdordernonlinearresponsefunctionRt1;t2;t3associatedwith2D-IRfromclas-sicalmoleculardynamicsMDtrajectories.ItispossibletocalculateasingleclassicalTCFBRt1;t2;t3=hjt2+t1it3+t2+t1kt1`i,Fouriertransform,multiplybyasetofafrequencyfactors,thenbacktransformtheresulttoobtainthetime-domainresponsefunction.Finally,Rt1;t2;t3maybeconvolutedwithaseriesofelectricinputeldstoobtainthethird-orderpolarizationPwhichgeneratesthe2D-IRsignal.60

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8.1TheExperimentAtwo-dimensionalinfraredD-IRexperimentmeasuresthethird-ordernon-linearpolarizationofachemicalsystem.Inresonant2D-IRexperiments,threeinputelectriceldsE,E,andE,associatedwithwavevectorsk,k,andk,crossthematerialofinterest,describedbyitsthird-orderresponsefunctionR,togenerateathird-orderpolarizationP.Theresultingeldradiatesinthephase-matcheddirectionks=)]TJ/F20 11.955 Tf 9.298 0 Td[(k+k+k.ThetimeorderingofeldsE,E,andEmaybevaried,allow-ingkstooriginatefromoneofthreepossiblewavematchingconditions:)]TJ/F20 11.955 Tf 9.299 0 Td[(k1+k2+k3,k1)]TJ/F20 11.955 Tf 12.302 0 Td[(k2+k3,ork1+k2)]TJ/F20 11.955 Tf 12.303 0 Td[(k3.ThepolarizationPiattimetistheconvolutionofthethird-orderresponsefunctionRwiththethreeinputelds,shownbelowasE1`,E2k,andE3j.34,91,92Pi1;2;t=Z10Z10Z10Rijk`t1;t2;t3E3jt)]TJ/F20 11.955 Tf 11.956 0 Td[(t3E2kt+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2E1`t+1+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2)]TJ/F20 11.955 Tf 11.956 0 Td[(t1dt1dt2dt3.1InEquation8.1,tisthetimeelapsedafterthenallaserpulse,1isthetimedelaybetweentherstandsecondpulses,and2isthetimedelaybetweenthesecondandthirdpulses.Thetnrepresenttimeintervalsbetweentheeld-matterinteractionsandequalthenifthepulsedurationsaresignicantlyshorterthanthetimescaleofthedynamics.ThesubscriptsindicatelaboratoryCartesiandirectionsx,y,z.Inavibrationalecho2D-IRexperiment,thedelayt2istypicallysettozero.34,63R,shownbelowinEquation8.2,isthethird-orderresponsefunctionassociatedwitharesonant61

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2D-IRexperiment.63ThethirdorderresponsefunctionRisafourthranktensoranddependsonthepolarizationcomponentsi,j,k,`oftheincidentelds.Rt1;t2;t3=)]TJ/F20 11.955 Tf 9.298 0 Td[(i ~3h[[[it3+t2+t1;jt2+t1];kt1];`]i.2TherepresentthedipolemomentoperatorsinthesubscriptedCartesiandirec-tion.Thetime-dependentdipolemaybewrittenastheHeisenbergrepresentationoftheoperator,t=eiHt=~e)]TJ/F21 7.97 Tf 6.586 0 Td[(iHt=~.Thesquareandanglebracketsindicatecommutatorsandquantummechanicalaveragesrespectivelyinastandardnotation.30Evaluationofthethird-orderresponsefunctionaswrittenisdicultduetotheprohibitivecomputationalexpenseofcalculatingfourdistinctfour-pointquantumme-chanicalTCFs.Evenintheclassicallimit,thecommutatorsbecomePoissonbracketsofthedynamicalvariables,whicharealsoimpracticaltoevaluatecomputationally.41,43TomakeRt1;t2;t3amenabletoclassicalmoleculardynamicscomputationalmethods,itisdesirabletorewriteitintermsoftherealpartofasingleTCF.8.2Introductiontothe2D-IRTCFTheoryAsdiscussedinChapters6and7,theoreticalmethodsthatapproximatelyrelatequantummechanicalandclassicaltimecorrelationfunctionshavebeenusedsuccessfullyforbothone-time17,19,31{34,93{95andtwo-time44,45,96correlationfunctions.AsimilarmethodwillbeemployedtosimplifytheRexpressionandobtainanapproximateTCFtheoryofthethirdorderresponsefunctionandtwo-dimensionalinfraredspectroscopy.62

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ExpandingthenestedcommutatorsinEquation8.2yieldsfourdistinctquantummechanicalTCFsandtheircomplexconjugates.OneimportantobjectiveindevelopingtheTCFtheoryistosimplifytheexpressiontobewrittenintermsofonlyoneoftheseTCFs.Additionally,atheorywrittenintermsofasingletimecorrelationfunctionmusthaveaclassicallimitoforder~0.Thus,anotherobjectiveistoeliminatethei=~3prefactorwhilerelatingandeliminatingTCFs.Itispossibletoeliminateone~prefactorandrewritethethird-orderresponsefunctionintermsoftwoTCFs,At1;t2;t3andBt1;t2;t3,byemployingexactfrequency-domaindetailed-balancerelationshipsbetweenpairsofquantummechanicalTCFsap-pearingintheRexpression.Unfortunately,noexactmeansofeliminatingthenaltwo~andanadditionalTCFhasbeenfound.However,thesesimplicationsmaybeaccomplishedbyresortingtoapproximations.Oneapproximation,whichwasusedwithsuccessindevelopingtheRTCFtheory,isbasedonamodelsystemofaharmonicoscillatorwithalinearlyvaryingdipole.Oncetheapproximationisapplied,ageneralrelationshipbetweentheremainingtwoTCFsbecomesapparent,andtheexpressionisderivedintermsoftherealandimaginarypartsofasingleTCF,Bt1;t2;t3.Finally,arelationshipbetweentherealandimaginarypartsofBt1;t2;t3isemployedtoobtainanRexpressionwhichmaybecalculatedusingclassicalmoleculardynamics.TheresultingexpressionisexactforaharmonicsystemandcanbeappliedtofullyanharmonicdynamicsusingclassicalMDtechniques.Itisimportanttonotethataharmonicoscillatorwithalinearlyvaryingdipolegivesnotwo-dimensionalinfrared63

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signal.Thus,thisversionoftheTCFtheoryservestolterout"harmonicdynamicsandenhancetheanharmoniccouplingsinthesignal.63Sinceanharmonicdynamicsandcouplingsareessentialtotheinterpretationof2D-IRspectra,63,82thedevelopmentoftheRTCFtheoryusingadierentreferencesystemtorelatetheTCFsisappropriate.Ananharmonicreferencesystem,representedasaharmonicsystemtreatedwithacubicpertubation,97wasusedindevelopinganotherversionoftheRtheory.8.3ExpansionoftheRExpressionTheRexpressioninEquation8.2containsatraceofnestedcommutators,whichmustbeimaginaryandhavealeadingcontributionoforder~3inordertocancelthei=~3prefactor.ExpandingthecommutatorsshowninEquation8.2yieldsanexpressionintermsoffourfour-pointdipoletimecorrelationfunctionsandtheircomplexconjugates,asdemonstratedbelowinEquation8.3.Becausethedipolemomentoperatorscommuteclassically,thefourTCFsdenedinEquation8.4areexpectedhavethesameclassicallimit.Rijk`t1;t2;t3=)]TJ/F20 11.955 Tf 9.299 0 Td[(i ~3[At1;t2;t3)]TJ/F20 11.955 Tf 11.955 0 Td[(Bt1;t2;t3)]TJ/F20 11.955 Tf 11.955 0 Td[(Ct1;t2;t3+Dt1;t2;t3)]TJ/F20 11.955 Tf 9.299 0 Td[(Dt1;t2;t3+Ct1;t2;t3+Bt1;t2;t3)]TJ/F20 11.955 Tf 11.955 0 Td[(At1;t2;t3].364

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At1;t2;t3=hit1+t2+t3jt2+t1kt1`iBt1;t2;t3=hjt2+t1it1+t2+t3kt1`iCt1;t2;t3=h`jt2+t1it3+t2+t1kt1iDt1;t2;t3=h`it3+t2+t1jt2+t1kt1i.4InEquation8.3,thestarsuperscriptsrepresentthecomplexconjugatesofthecomplextimedomainTCFs.ThecontentsofthesquarebracketsmaybeseenassumsanddierencesoftheimaginarypartsofthefourTCFs,namely2AI)]TJ/F20 11.955 Tf 11.965 0 Td[(BI)]TJ/F20 11.955 Tf 11.965 0 Td[(CI+DI,sinceforacomplexquantityC,C)]TJ/F20 11.955 Tf 12.493 0 Td[(C=CI.ThesubscriptsRorIdenotetherealandimaginarypartsofatimedomainTCFortheFouriertransformsoftherealandimaginaryparts,respectively.Afteridentifyingthefourtimecorrelationfunctions,thenextstepistoformrelationshipsbetweentheminordertosimplifytheRexpression.8.4TheEnergyRepresentationToproceedwiththesimplicationofRandalsotoverifytheclassicallimitoftheTCFsA,B,C,andD,itishelpfultorewritethemintheenergyrepresentation.Forexample,TCFAt1;t2;t3writtenintheenergyrepresentationisshownbelowinEquation8.5.65

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At1;t2;t3=hit3+t2+t1jt2+t1kt1`iAt1;t2;t3=1 QXahaje)]TJ/F21 7.97 Tf 6.587 0 Td[(HeiHt1+t2+t3=~e)]TJ/F21 7.97 Tf 6.586 0 Td[(iHt1+t2+t3=~eiHt1+t2=~je)]TJ/F21 7.97 Tf 6.586 0 Td[(iHt1+t2=~eiHt1=~ke)]TJ/F21 7.97 Tf 6.586 0 Td[(iHt1=~`jai.5Intheenergyrepresentation,thedipolemomentoperatorsaremultipliedby=e)]TJ/F21 7.97 Tf 6.587 0 Td[(H=Q,whereQisthepartitionfunction,andthetraceistaken.Thetime-dependentdipolesarewrittenintheHeisenbergrepresentation,i.e.at=eiHt=~e)]TJ/F21 7.97 Tf 6.587 0 Td[(iHt=~.Next,insertingfourcompletesetsofenergyeigenstatesoftheformPajaihajwithHjai=Eajai,operating,andsimplifyingyieldstime-domainexpressionsofthefollowingform:At1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Eaiadjdckcb`baeiEabt1=~eiEact2=~eiEadt3=~.6InEquation8.6,Eab=Ea)]TJ/F20 11.955 Tf 12.79 0 Td[(Ebistheenergydierencebetweenstatesaandb.Thedipolemomentoperatormatrixelements,ab=hajjbi,areHermitian,i.e.ij=ji.Ifthedipolemomentoperatormatrixelementsaretakenasreal,itclearthatAt1;t2;t3=A)]TJ/F20 11.955 Tf 9.299 0 Td[(t1;)]TJ/F20 11.955 Tf 9.299 0 Td[(t2;)]TJ/F20 11.955 Tf 9.299 0 Td[(t3.ComparisonofEquations8.6and8.7veriesthisclaim.66

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At1;t2;t3=h`kt1;jt2+t1it3+t2+t1iAt1;t2;t3=A)]TJ/F20 11.955 Tf 9.299 0 Td[(t1;)]TJ/F20 11.955 Tf 9.299 0 Td[(t2;)]TJ/F20 11.955 Tf 9.299 0 Td[(t3=1 QXabcde)]TJ/F21 7.97 Tf 6.587 0 Td[(Eaiadjdckcb`baeiEbat1=~eiEcat2=~eiEdat3=~.7TheremainingthreeTCFsBt1;t2;t3,Ct1;t2;t3,andDt1;t2;t3canbewritteninasimilarmannerintheenergyrepresentation.Indexswitching,equivalenttotakingcyclicpermutationsofthetrace,52,96isusedtomaximizethesimilaritiesbetweentheTCFs.InEquation8.8,thesethreeTCFsarewritteninthetimedomain.NotethatthepairofTCFsAt1;t2;t3andDt1;t2;t3,aswellasthepairBt1;t2;t3andCt1;t2;t3dieronlybytheBoltzmannfactore)]TJ/F21 7.97 Tf 6.586 0 Td[(H.Bt1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.587 0 Td[(Eaidcjadkcb`baeiEabt1=~eiEact2=~eiEdct3=~Bt1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.587 0 Td[(Eaidcjadkcb`baeiEbat1=~eiEcat2=~eiEcdt3=~Ct1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Ebidcjadkcb`baeiEabt1=~eiEact2=~eiEdct3=~Ct1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Ebidcjadkcb`baeiEbat1=~eiEcat2=~eiEcdt3=~Dt1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Ebiadjdckcb`baeiEabt1=~eiEact2=~eiEadt3=~Dt1;t2;t3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Ebiadjdckcb`baeiEbat1=~eiEcat2=~eiEdat3=~.867

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8.5FrequencyDomainTCFsToderiveanalyticalrelationshipsbetweentheTCFs,itisbenecialtoperformthetripleFouriertransformtoobtainfrequencydomainfunctionsoftheformA!1;!2;!3.A!1;!2;!3=1 22Z1dt1Z1dt2Z1dt3e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!2t2e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!3t3At1;t2;t3.9Thefourfrequency-domainTCFsA!1;!2;!3,B!1;!2;!3,C!1;!2;!3,andD!1;!2;!3areshownbelowinEquation8.10.A!1;!2;!3=1 QXabcde)]TJ/F21 7.97 Tf 6.587 0 Td[(Eaiadjdckcb`ba!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Ead=~B!1;!2;!3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Eaidcjadkcb`ba!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Edc=~C!1;!2;!3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Ebidcjadkcb`ba!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Edc=~D!1;!2;!3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Ebiadjdckcb`ba!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eac=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Ead=~.10ItisstraightforwardtoprovethattheFouriertransformsofthecomplexconju-gatesofthefourTCFsgivetheTCFsinnegativefrequency,i.e.FT[ft1;t2;t3]=f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3.Ingeneral,thetripleFouriertransformofatime-domainfunctionft1;t2;t3isdenedasfollows:68

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FT[ft1;t2;t3]=f!1;!2;!3f!1;!2;!3=1 23Z1dt1Z1dt2Z1dt3e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!2t2e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!3t3ft1;t2;t3.11AllofthefrequencyargumentsappearinginEquation8.11aremadenegativetoobtainanexpressionforf)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3.f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3=1 23Z1dt1Z1dt2Z1dt3ei!1t1ei!2t2ei!3t3ft1;t2;t3.12Tonishtheproof,thecomplexconjugateofbothsidesistaken.TherightsideofEquation8.12thenbecomesthetripleFouriertransformofft1;t2;t3.Consequently,FT[ft1;t2;t3]=f)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3.TakingintoaccountthatthefrequencydomainTCFsarereal,96thenalresultisobtained:FT[ft1;t2;t3]=f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3=1 23Z1dt1Z1dt2Z1dt3e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!2t2e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!3t3ft1;t2;t3.1369

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TheFouriertransformsofthecomplexconjugatesofTCFsA,B,C,andDareshownbelowinEquation8.14.A)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Eaidajcdkbc`ab!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Eda=~B)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3=1 QXabcde)]TJ/F21 7.97 Tf 6.586 0 Td[(Eaicdjdakbc`ab!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Ecd=~C)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3=1 QXabcde)]TJ/F21 7.97 Tf 6.587 0 Td[(Ebicdjdakbc`ab!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Ecd=~D)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3=1 QXabcde)]TJ/F21 7.97 Tf 6.587 0 Td[(Ebidajcdkbc`ab!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eba=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(Eca=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Eda=~.148.6Detailed-BalanceRelationshipsByvisualinspectionofEquation8.10,itiseasilyobservedthatTCFsA!1;!2;!3andD!1;!2;!3areexactlythesame,exceptfortheBoltzmannfactorse)]TJ/F21 7.97 Tf 6.586 0 Td[(Eaande)]TJ/F21 7.97 Tf 6.587 0 Td[(Eb.SomesimplemultiplicationcanbeusedtoconvertoneofthetwoTCFsintotheother.D!1;!2;!3=eEae)]TJ/F21 7.97 Tf 6.587 0 Td[(EbA!1;!2;!3.15Enforcingthedeltafunction!1)]TJ/F20 11.955 Tf 9.706 0 Td[(Eab=~forcingEa)]TJ/F20 11.955 Tf 9.705 0 Td[(Eb=~!1allowsEquation8.15toberewrittenintermsof!1.70

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D!1;!2;!3=e~!1A!1;!2;!38.16Innegativefrequency,ananalogousrelationshipisderived.D)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3=e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1A)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!38.17Inasimilarmanner,thefrequencydomainTCFsB!1;!2;!3andC!1;!2;!3canbepairedupandexactdetailed-balancerelationshipscanbederivedbetweenthemandtheirnegativefrequencycounterparts.C!1;!2;!3=e~!1B!1;!2;!3C)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3=e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1B)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3.18Usingthepreviouslyderiveddetailed-balancerelationshipsbetweenTCFpairs,itispossibletoderivesimpletanhrelationshipsbetweensumsanddierencesoftheTCFs.Forexample,considerTCFsA!1;!2;!3andD!1;!2;!3.D!1;!2;!3+A!1;!2;!3=e~!1+1A!1;!2;!3D!1;!2;!3+A!1;!2;!3=e~!1)]TJ/F15 11.955 Tf 11.956 0 Td[(1A!1;!2;!3.19TheratioofD!1;!2;!3)]TJ/F20 11.955 Tf 11.127 0 Td[(A!1;!2;!3toD!1;!2;!3+A!1;!2;!3isshownbelowinEquation8.20.Multiplyingtheratiothroughbye)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1=2=e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1=2revealsatanhrelationshipbetweenthesumanddierenceofA!1;!2;!3andD!1;!2;!3.71

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D!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A!1;!2;!3 D!1;!2;!3+A!1;!2;!3=e~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(1 e~!1+1=e~!1=2)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1=2 e~!1=2+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1=2=tanh~!1=28.20Inananalogousmanner,threeotherusefultanhrelationshipscanbederived.Thefourtanhrelationshipsaresummarizedbelowinequation8.21.D!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A!1;!2;!3=tanh~!1=2[D!1;!2;!3+A!1;!2;!3]D)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1[D)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3+A)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3]C!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(B!1;!2;!3=tanh~!1=2[C!1;!2;!3+D!1;!2;!3]C)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(B)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1[C)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3+B)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3].21Usingthedetailed-balanceandtanhrelationships,itispossibletoeliminateone~prefactor,aswellastheTCFsCandD,fromtheRexpression.Thefrequencydomainthird-orderresponseiswrittenbelowinitsoriginalform.72

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R!1;!2;!3=)]TJ/F20 11.955 Tf 9.299 0 Td[(i ~3[A!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(D!1;!2;!3+C!1;!2;!3)]TJ/F20 11.955 Tf 11.956 0 Td[(B!1;!2;!3+D)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3+B)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(C)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3].22ThetanhrelationshipsarerstemployedtorewritethedierencesbetweenTCFpairsassums.R!1;!2;!3=i ~3tanh~!1=2[A!1;!2;!3+D!1;!2;!3+B!1;!2;!3+C!1;!2;!3)]TJ/F20 11.955 Tf -203.194 -31.88 Td[(A)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(D)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3)]TJ/F20 11.955 Tf -191.744 -31.88 Td[(B)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(C)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3].23Next,thedetailed-balancerelationshipsshowninEquations8.16,8.17,and8.18areusedtorewritetheresponsefunctionintermsofTCFsAandBonly.R!1;!2;!3=i ~3tanh~!1=2f+e~!1[B!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A!1;!2;!3]++e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1[B)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3)]TJ/F20 11.955 Tf 11.956 0 Td[(A)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3]g.2473

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Thethird-orderresponseisnowwrittenintermsoftwoTCFs.NoapproximationshavebeeninvokedtoobtainEquation8.24.ThisequationindicatesthatthevalueofRiszeroalongthe!1=0axis.8.7TheClassicalLimitofRTodemonstratehowone~prefactorcanbeeliminatedfromthethird-orderre-sponsefunction,theclassicallimit,where~!1becomessmall,isconsidered.Inthislimit,thetanhfunctionsandexponentialscanbeexpanded,retainingtermsoforder~.e~!1!1+~!1e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1!1)]TJ/F20 11.955 Tf 11.955 0 Td[(~!1tanh~!1=2!~!1=2.25UsingEquation8.25,theRexpressiondisplayedinEquation8.24canberewrit-ten,asshownbelow.R!1;!2;!3=i ~3~!1 2f+~!1[B!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A!1;!2;!3]+)]TJ/F20 11.955 Tf 11.955 0 Td[(~!1[B)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3]g.26Itiseasilyproventhatf++f)]TJ/F15 11.955 Tf 10.897 -4.338 Td[(=2fRandf+)]TJ/F20 11.955 Tf 12.152 0 Td[(f)]TJ/F15 11.955 Tf 10.896 -4.338 Td[(=2fI,wheref+denotesaTCFinpositivefrequency,f)]TJ/F15 11.955 Tf 10.808 -4.338 Td[(aTCFinnegativefrequency,fRtheFouriertransformoffRt1;t2;t3,andfItheFouriertransformoffIt1;t2;t3.BothfRandfIarethemselves74

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realfunctions.Thesesimplerelationshipsallowthethird-orderresponsefunctiontoberewrittenagain.R!1;!2;!3=i!1 ~2BR)]TJ/F15 11.955 Tf 11.956 0 Td[(2AR+~!1BI)]TJ/F20 11.955 Tf 11.955 0 Td[(~!1AI.27ExaminationofEquation8.27makesitclearthatBI)]TJ/F20 11.955 Tf 11.421 0 Td[(AImusthaveadierenceoforder~andBR)]TJ/F20 11.955 Tf 12.307 0 Td[(AIorder~2fortheRexpressiontohaveameaningfulclassicallimit.Now,performingthereverseFouriertransformbacktothetimedomainwillgiveRintermsoftimederivatives.Ingeneral,thefollowingrelationshipholdstrue.ft1;t2;t3=Z1ei!1t1ei!2t2ei!3t3f!1;!2;!3d!1d!2d!3d dt1ft1;t2;t3=Z1i!1ei!1t1ei!2t2ei!3t3f!1;!2;!3d!1d!2d!3d2 dt21ft1;t2;t3=)]TJ/F25 11.955 Tf 11.291 16.272 Td[(Z1!21ei!1t1ei!2t2ei!3t3f!1;!2;!3d!1d!2d!3.28UsingEquation8.28,thethird-orderresponsefunctioncanberewrittenonemoretimeintermsoft1derivatives.Rt1;t2;t3=2 ~2d dt1BRt1;t2;t3)]TJ/F20 11.955 Tf 17.631 8.088 Td[(d dt1ARt1;t2;t3+i2 ~d2 dt21AIt1;t2;t3)]TJ/F20 11.955 Tf 15.265 8.087 Td[(d2 dt21BIt1;t2;t3.29Equation8.29demonstratesclearlythatone~canbeeliminatedexactlyfromthethird-orderresponsefunctionwhentheclassicallimitisinvoked.Atthispoint,itis75

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necessarytoconsiderapproximationstosimplifyRsucientlytomakeitamenabletocomputationviaclassicalmoleculardynamicstechniques.76

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Chapter9TCFTheory:2D-IRSpectroscopyHarmonicApproximationInthischapter,thedevelopmentoftheTCFtheoryoftwo-dimensionalinfraredspectroscopyD-IRiscontinued.Inthepreviouschapter,itwasdemonstratedthatthethird-orderresponsefunctionresponsibleforthe2D-IRsignalcouldbewrittenexactlyintermsoftwofour-pointdipoletimecorrelationfunctionsAt1;t2;t3andBt1;t2;t3.Furthersimplicationofthethird-orderresponsefunctionisdesirablefortworeasons.First,theresultsofclassicalmoleculardynamicsMDcalculationswillallowthecalculationoffRt1;t2;t3,therealpartofaTCF,butnotfIt1;t2;t3,theimaginarypart.ItisnecessarytoeliminateAIt1;t2;t3andBIt1;t2;t3tomaketheuseofthistheorywithclassicalMDtechniquesfeasible.Second,sincethecalculationofafour-timeTCFiscomputationallydemanding,itmakessensetorewritetheexpressionintermsofjustoneoftheTCFs.Indevelopingatheoryoffth-orderRamanspectroscopy,aharmonicapproxima-tionwassuccessfullyinvokedtoallowcomputationofthefth-orderresponsefunctionR.Basedonthesuccessofthisapproach,thisapproximationwasappliedtothe2D-IRTCFtheory.Theapproximationmadeitpossibletoderivearelationshipbetweenthefrequency-domainTCFsA!1;!2;!3andB!1;!2;!3,thuseliminatingTCFA77

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fromtheexpression.Additionally,itallowedtheexpressiontoberewrittenintermsofBR!1;!2;!3.Thesesimplicationscreatedameansofcalculatinga2D-IRsignalusingclassicalMD.BRt1;t2;t3couldbecalculated,Fouriertransformed,thenmultipliedbyasetoffrequencyfactorstoobtainafrequency-domain2D-IRsignal.9.1TheHarmonicApproximationwithLinearlyVaryingDipoleInusingtheharmonicapproximation,aharmonicpotentialV=m2q2=2.1isassumed.indicatesafundamentalharmonicfrequency.HarmonicenergiesoftheformEa=~a+1=2arealsoassumed.ThepartitionfunctionappearinginalltheTCFstakesthefollowingform.Q=e)]TJ/F21 7.97 Tf 6.586 0 Td[(~=2 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~.2Additionally,thedipolemomentmatrixelementsintheTCFsareexpandedouttorstorderintheharmoniccoordinateqasshown.ij=0ij+0qij78

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Theqijmatrixelementsaregivenbytheexpression42qij=~ 2m1=2[i;j+1j+11=2+i;j)]TJ/F18 7.97 Tf 6.586 0 Td[(1j1=2].3InEquation9.3theprimesdenotederivativeswithrespecttotheharmoniccoor-dinateq.9.2ApplyingtheApproximationtoRExpansionofthefourdipolemomentmatrixelementsinTCFsAandByieldsasumofsixteenterms.Eachtermisuniqueinthepowersofcoordinatesusedtoevaluatethedipolemomentmatrixelements.Manyofthesetermscanbeneglectedbecausetheirdelta-functionsforcethemtoequalzero.Also,examinationofEquation8.29suggeststhat,inordertocontributetothethird-orderresponsefunction,atime-domaintermmusthaveanonzeroderivativewithrespecttot1.EventhoughEquation8.29assumestheclassicallimit,theneglectedhigherordertermswouldinvolvehigherordert1derivatives.9.3ExpansionoftheDipoleMomentMatrixOperatorsAsdemonstratedearlier,frequency-domainTCFA!1;!2;!3iswrittenbelow.A!1;!2;!3=Xabcdiadjdckcb`ba!1)]TJ/F20 11.955 Tf 11.955 0 Td[(Eab=~!2)]TJ/F20 11.955 Tf 11.956 0 Td[(Eac=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(Ead=~.4Thefourdipolemomentmatrixelementsiad,jdc,kcd,and`baareexpandedaccordingtoEquation9.3,yieldingasumofsixteendistinctterms.Severaloftheterms79

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thatarisefromthisexpansioncanclearlybeeliminatedThetermsA0001,A0010,A0100,A1000,writtenintermsofone0andthree0,aswellasA1110,A1101,A1011,andA0111,intermsofthree0andone0comeouttozerowhenthedeltafunctionsbuiltintothedipolemomentmatrixelementsareenforced.Thedeltafunctionsgivenonzerovaluestotheeightremainingterms.However,sevenofthesenonzerotermscanalsobeeliminated.TherstofthesetermsisA0000,whichiswritten:A0000!1;!2;!3=1 Q0i0j0k0`Xae)]TJ/F21 7.97 Tf 6.586 0 Td[(Ea!1!2!3.5Thesumoverindexaisasumoverallpossiblestatesa,andisthereforeconsideredaninnitesum.IttakesonthesamevalueasthepartitionfunctionQ.1Xa=0e)]TJ/F21 7.97 Tf 6.587 0 Td[(Ea=e)]TJ/F21 7.97 Tf 6.587 0 Td[(~=21Xa=0e)]TJ/F21 7.97 Tf 6.586 0 Td[(Ea=e)]TJ/F21 7.97 Tf 6.587 0 Td[(~=2 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~=Q.6Afterevaluatingthesum,enforcingthedeltafunctions,andbackFouriertrans-formingtheresult,A0000isfoundtobetheproductoffour0elements.Thistermiseliminatedfromthethird-orderresponsefunctionsinceitsderivativewithrespecttot1iszero.A0000t1;t2;t3=0i0j0k0`.7Thenextsixnonzerotermscontaintwo0andtwo0matrixelements.Allofthetermsareevaluatedthesameway,byenforcingthedeltafunctionsandassessingthe80

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valueoftheinnitesumsoverindexa.Theresultsareshowninthetimedomain.ThreeofthesetermsdonotcontributetoRbecausetheirderivativeswithrespecttot1arealsozero.A0110t1;t2;t3=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~[e)]TJ/F21 7.97 Tf 6.586 0 Td[(~eit2+e)]TJ/F21 7.97 Tf 6.586 0 Td[(it2]A1100t1;t2;t3=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~[e)]TJ/F21 7.97 Tf 6.586 0 Td[(~eit3+e)]TJ/F21 7.97 Tf 6.586 0 Td[(it3]A1010t1;t2;t3=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~[e)]TJ/F21 7.97 Tf 6.586 0 Td[(~eit2+t3+e)]TJ/F21 7.97 Tf 6.586 0 Td[(it2+t3].8TheremainingthreetermsappeartocontributetoRbecausetheirdeltafunc-tionsgivethemnonzerovaluesandtheyalsopossessnonzerot1derivatives.A0011t1;t2;t3=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~[e)]TJ/F21 7.97 Tf 6.587 0 Td[(~eit1+e)]TJ/F21 7.97 Tf 6.587 0 Td[(it1]A0101t1;t2;t3=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~[e)]TJ/F21 7.97 Tf 6.587 0 Td[(~eit1+t2+e)]TJ/F21 7.97 Tf 6.587 0 Td[(it1+t2]A1001t1;t2;t3=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~[e)]TJ/F21 7.97 Tf 6.587 0 Td[(~eit1+t2+t3+e)]TJ/F21 7.97 Tf 6.586 0 Td[(it1+t2+t3].9ThenalnonzeroterminTCFAisA1111,inwhichallthedipolematrixelementsappearas0.Foreaseofmanipulation,A1111issplitintoasumofsixdistinctterms,indexedathroughf.A1111at1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~22e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~eit1+2t2+t3A1111bt1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~22e)]TJ/F21 7.97 Tf 6.586 0 Td[(it1+2t2+t381

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A1111ct1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2e)]TJ/F21 7.97 Tf 6.586 0 Td[(~+e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~eit1+t3A1111dt1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~2+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~e)]TJ/F21 7.97 Tf 6.586 0 Td[(it1+t+3A1111et1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~22e)]TJ/F21 7.97 Tf 6.587 0 Td[(~eit3)]TJ/F21 7.97 Tf 6.587 0 Td[(t1A1111ft1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.956 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~22e)]TJ/F21 7.97 Tf 6.587 0 Td[(~e)]TJ/F21 7.97 Tf 6.587 0 Td[(it3)]TJ/F21 7.97 Tf 6.587 0 Td[(t1.10Notethatinthefrequencydomain,theexponentialsoftheformeitalphabecomedeltafunctionsoftheform!)]TJ/F15 11.955 Tf 10.765 0 Td[(.Thispropertyofthefrequency-domain1111termswillproveusefulinderivingageneralrelationshipbetweenTCFsAandB.SimilaranalysisisperformedforTCFB!1;!2;!3.Enforcingthedeltafunctionseliminateseightterms,andtheB0000,B0110,B1100,andB1010termsarefoundtohavezerot1derivatives.ThetermsB0011,B0101,andB1001termsareidenticaltotheonesforTCFA.SincetheRexpression,asshowninEquation8.24,iswrittenintermsofdierencesbetweenTCFsAandB,thecommontermsbetweentheTCFscancelouttozeroanddonotneedtobeconsidered.TheonlynonzerotermremainingforTCFBistheB1111term,whichisalsowrittenasasumofsixterms,indexedathroughf.NotethattheB1111aandB1111btermsareidenticaltothosefoundforTCFa,whiletermsB1111cthroughB1111dareunique.B1111at1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~22e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~eit1+2t2+t3B1111bt1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~22e)]TJ/F21 7.97 Tf 6.586 0 Td[(it1+2t2+t382

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B1111ct1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~22e)]TJ/F21 7.97 Tf 6.586 0 Td[(~eit1+t3B1111dt1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~22e)]TJ/F21 7.97 Tf 6.587 0 Td[(~e)]TJ/F21 7.97 Tf 6.586 0 Td[(it1+t3B1111et1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.956 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+1eit3)]TJ/F21 7.97 Tf 6.586 0 Td[(t1B1111ft1;t2;t3=0i0j0k0`~ 2m21 1)]TJ/F20 11.955 Tf 11.956 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~2e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~e)]TJ/F21 7.97 Tf 6.586 0 Td[(it3)]TJ/F21 7.97 Tf 6.586 0 Td[(t1.11ConsideringonlytheA1111andB1111terms,itisnowpossibletodeneanexactrelationshipbetweenthetwoTCFsanduseittofurthersimplifyR.9.4Frequency-DomainRelationshipbetweenTCFsAandBTodiscernarelationshipbetweenTCFsA!1;!2;!3andB!1;!2;!3,itishelp-fultoexaminehowthesixindividual1111termsofthetwoTCFsarerelated.A1111a=B1111aA1111b=B1111bA1111c=1 2+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~B1111cA1111d=1 2+e~B1111dA1111e=2 1+e~B1111eA1111f=2 1+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~B1111fUsingthisinformation,theageneralrelationship,indicatedbythefrequency-domainfunctiong!1;!2;!3,betweenA!1;!2;!3andB!1;!2;!3isderived.83

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Term !1 !2 !3 a 2 b )]TJ/F15 11.955 Tf 9.299 0 Td[( )]TJ/F15 11.955 Tf 9.299 0 Td[(2 )]TJ/F15 11.955 Tf 9.299 0 Td[( c 0 d )]TJ/F15 11.955 Tf 9.299 0 Td[( 0 )]TJ/F15 11.955 Tf 9.299 0 Td[( e )]TJ/F15 11.955 Tf 9.299 0 Td[( 0 f 0 )]TJ/F15 11.955 Tf 9.299 0 Td[( Table9.1:Thevaluesthatfrequencies!1,!2,and!3takeonasdictatedbythedeltafunctionsofthesix1111termsoftheTCFsAandBA!1;!2;!3=g!1;!2;!3B!1;!2;!3g!1;!2;!3=1+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1+!3)]TJ/F21 7.97 Tf 6.586 0 Td[(!2=2 1+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1)]TJ/F21 7.97 Tf 6.586 0 Td[(!3=2.12ThisrelationshipcanalsobeappliedtotheTCFsinnegativefrequency.Thisisaccomplishedbyreplacingallthe!by)]TJ/F20 11.955 Tf 9.298 0 Td[(!.A)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3=g)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3B)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3g)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3=1+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!2)]TJ/F21 7.97 Tf 6.586 0 Td[(!1)]TJ/F21 7.97 Tf 6.586 0 Td[(!3=2 1+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!3)]TJ/F21 7.97 Tf 6.587 0 Td[(!1=2.1384

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MakinguseofEquations9.12and9.13,thefrequency-domainRexpressioncannowbewrittenintermsofB!1;!2;!3alone.R!1;!2;!3=i ~3tanh~!1=2[+e~!1)]TJ/F20 11.955 Tf 11.956 0 Td[(g!1;!2;!3B!1;!2;!3++e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(g)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.298 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3B)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3]9.5EliminatingtheImaginaryPartofTCFBNext,considertheclassicallimitofthesimpliedRexpression.Twoofthethree~prefactorshavebeeneliminatedcompletelyfromtheexpressionandRiswrittenintermsofBR!1;!2;!3andBI!1;!2;!3.R!1;!2;!3=i ~3~!1 2~ 4!3)]TJ/F20 11.955 Tf 11.955 0 Td[(!22[~!1BR!1;!2;!3+4BI!1;!2;!3]9.14Inthislimit,BR!1;!2;!3isequivalenttotheclassicaltimecorrelationfunctionthatcanbecalculatedusingmoleculardynamicsmethods.FindingageneralrelationshipbetweenBR!1;!2;!3andBI!1;!2;!3willallowthenal~prefactortoberemovedandgiveanexpressionsolelyintermsofBR!1;!2;!3thatisvalidforallfrequencies.Aone-timecorrelationfunctionfthasasimplefunctionrelationshipoftheformfR!=tanh~!=2fI!betweentheFouriertransformsofitsrealandimaginaryparts.85

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ArelationshipofthisnaturebetweenBR!1;!2;!3andBI!1;!2;!3isnotimmediatelyobvious.Toattempttondone,eachtermofBt1;t2;t3contributingtothethird-orderresponse,namelyB1111,wasseparatedintoitsrealandimaginaryparts,andthetwopartswereFouriertransformedseparately.Foreachterm,theratiobetweenBR!1;!2;!3andBI!1;!2;!3wasdetermined.Inthisanalysis,theB1111termsweregroupedasB1111ab=B1111a+B1111b,B1111cd=B1111c+B1111d,andB1111ef=B1111e+B1111f.B1111abI!1;!2;!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh[~!1+!2+!3=4]B1111abR!1;!2;!3B1111cdI!1;!2;!3=0B1111cdR!1+!2+!3.15B1111efI!1;!2;!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh[~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(!3=4]B1111efR!1;!2;!3.16ComparisonofthethreeresultsledtothederivationofageneralrelationshipbetweenBR!1;!2;!3andBI!1;!2;!3.Thegeneralrelationshiptakesintoaccountthevaluesintermsofthatthefrequencies!1,!2,and!3takeonbasedoneachterm'sdeltafunctions.BI!1;!2;!3=)]TJ/F15 11.955 Tf 11.291 0 Td[(tanh~!1+2!2)]TJ/F20 11.955 Tf 11.955 0 Td[(!3 4BR!1;!2;!3.17Equation9.17isexactfortheharmonicsystem.Withthisrelationship,itisstraightforwardtomakeonenalsimplicationtotheRexpression,whichremovesthenalfactorof~intheclassicallimitexpression.86

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9.6TheFinalRExpressionIntheclassicallimit,thethird-orderresponsefunctioncannowbewrittenasBR!1;!2;!3multipliedbyasetoffrequencyfactors.ApplyingtheclassicallimitofEquation9.17toEquation9.14givesthefollowingresult.R!1;!2;!3=i ~3~!1 2~ 4!3)]TJ/F20 11.955 Tf 11.956 0 Td[(!2[2~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(~!1+2!2)]TJ/F20 11.955 Tf 11.955 0 Td[(!3]BR!1;!2;!3R!1;!2;!3=i3 8!21!3)]TJ/F20 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!22+2!1!23BR!1;!2;!3.18TakingtimederivativesofthebackFouriertransformofBR!1;!2;!3allowthisexpressiontoberewritteninthetimedomain.Rt1;t2;t3=3 8d3 dt21dt2)]TJ/F15 11.955 Tf 11.956 0 Td[(2d3 dt1dt22)]TJ/F15 11.955 Tf 11.955 0 Td[(2d3 dt21dt22)]TJ/F15 11.955 Tf 11.955 0 Td[(2d3 dt1dt23+5d3 dt1dt2dt3BRt1;t2;t39.19Toexaminehigh-frequencydynamics,takingtheclassicallimitoftheRex-pressionisnotrealistic.Thethird-orderresponsefunctioniswrittenwithouttakingtheclassicallimitasfollows:87

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R!1;!2;!3=i ~3tanh~!1=2f+e~!1[1)]TJ/F20 11.955 Tf 11.955 0 Td[(f!1;!2;!3])]TJ/F15 11.955 Tf 11.955 0 Td[(tanh[~!1+2!2)]TJ/F20 11.955 Tf 11.955 0 Td[(!3=4]++e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1[1)]TJ/F20 11.955 Tf 11.955 0 Td[(f)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3]+tanh[~!1+2!2)]TJ/F20 11.955 Tf 11.955 0 Td[(!3=4]gBR!1;!2;!3.20Equation9.20representsthethird-orderresponsefunctioninaformthatcanbeevaluatedusingclassicalMDandTCFcomputationaltechniques.IncalculatingR,thetimedomainTCFBRt1;t2;t3iscalculatedasitsclassicalcounterparthjt2+t1it3+t2+t1kt1`iandFouriertransformedintothefrequencydomain.ItmaythenbemultipliedbythenecessaryfrequencyfactorsthenbacktransformedtoobtainRt1;t2;t3.TheresultingresponsefunctionmaythenappliedtoEquation8.1toobtaina2D-IRsignal.9.7LimitationoftheHarmonicApproximationItiswell-establishedknowledgethataharmonicoscillatormodelsystemwithalinearlyvaryingdipoleyieldszerosignalina2D-IRexperiment.Thetheorypresentedundertheharmonicapproximationthusservesasameansoflteringharmonicdynamicsoutofsimulationresultsandhighlightinganharmoniccouplings.ItissimpletoprovethatR=0undertheseconditionsusingPoissonbrack-ets,whichareclassicallyequivalenttocommutators.Fortwofunctionsuandvofthevariablespmomentumandqposition,88

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fu;vgq;p=du dqidv dpi)]TJ/F20 11.955 Tf 14.448 8.088 Td[(du dpidv dqi.21Foraharmonicsystem,thefollowingtwoexpressionsareusedtodescribethepositionqanddipole.qt=qft+vgt=0+0q.22ToprovethatR=0undertheseconditions,theleftmostPoissonbracketintheRexpression,betweenit3+t2+t1andjt2+t1,isevaluatedrst,takingintoaccounttheinformationprovidedinEquation9.22.it1;t2;t3=0+0[qft1+t2+t3+vgt1+t2+t3]di dq=0ft1+t2+t3di dv=0gt1+t2+t3jt2+t1=0+0[qft1+t2+vgt1+t2]dj dq=0ft1+t2di dv=0gt1+t2.23ThePoissonbracketcomesouttofit3+t2+t1;jt1+t2g=02[ft1+t2+t3gt1+t2)]TJ/F20 11.955 Tf -110.637 -31.881 Td[(ft1+t2gt1+t2+t3].2489

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Thisresulthasnovelocityorpositiondependence,sotakingthenextPoissonbracketwillgivederivativesofzero,makingtheentireRexpressionzero.Itisinterestingtoconsiderexactlyhowtheharmonictheoryof2D-IRleadstozerosignal.TheindividualTCFsAandBareindividuallynonzerounderthisapproximation.Thus,eithertherelationshipgbetweenthenorthetanhrelationshipbetweenBRandBIcausestheexpressiontorigorouslyequalzero.BeginningwithEquation8.24,itispossibletouseallthe1111termsofTCFsAandB,thetermsthatcontributetotheRexpression,todeterminethedierencesB!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A!1;!2;!3andB)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3.B!1;!2;!3)]TJ/F20 11.955 Tf 11.955 0 Td[(A!1;!2;!3=B1111c)]TJ/F20 11.955 Tf 11.955 0 Td[(A1111c+B1111d)]TJ/F20 11.955 Tf 11.955 0 Td[(A1111d+B1111e)]TJ/F20 11.955 Tf 11.955 0 Td[(A1111e+B1111f)]TJ/F20 11.955 Tf 11.955 0 Td[(A1111fB!1;!2;!3=K[e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!3)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F20 11.955 Tf 11.956 0 Td[(!1+!3++!1+!3)]TJ/F15 11.955 Tf 11.956 0 Td[()]TJ/F20 11.955 Tf 11.955 0 Td[(!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!3+]K=0i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~.25ThedierenceB)]TJ/F20 11.955 Tf 9.299 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.298 0 Td[(!3)]TJ/F20 11.955 Tf 12.119 0 Td[(A)]TJ/F20 11.955 Tf 9.298 0 Td[(!1;)]TJ/F20 11.955 Tf 9.299 0 Td[(!2;)]TJ/F20 11.955 Tf 9.299 0 Td[(!3isidentical,butwith)]TJ/F20 11.955 Tf 9.298 0 Td[(!1replacing!1and)]TJ/F20 11.955 Tf 9.299 0 Td[(!3replacing!3.Next,thesetwodierencesareplacedintoEquation8.24.Theresultingexpressionissplitintofourdistinctterms,eachofwhichisequaltozerowhenitsdeltafunctionsareenforced.90

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R!1;!2;!3=i ~3~!1 20i0j0k0`~ 2m1 1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~e)]TJ/F21 7.97 Tf 6.586 0 Td[(~+e~!1e)]TJ/F21 7.97 Tf 6.587 0 Td[(~)]TJ/F15 11.955 Tf 11.955 0 Td[(1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1!1)]TJ/F15 11.955 Tf 11.956 0 Td[(!3)]TJ/F15 11.955 Tf 11.955 0 Td[(+)]TJ/F15 11.955 Tf 9.298 0 Td[(1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1+e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1+!3+++e~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1e)]TJ/F21 7.97 Tf 6.586 0 Td[(~!1+!3)]TJ/F15 11.955 Tf 11.955 0 Td[(++e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.586 0 Td[(~)]TJ/F20 11.955 Tf 11.956 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1+!3+.26Indevelopingthistheoryof2D-IRspectroscopy,itisusefultondthelowestorderreferencesystemthatwillgiveasignal.Thisreferencesystemisclearlynottheharmonicoscillator,asdemonstratedinthischapter.91

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Chapter10TCFTheory:2D-IRSpectroscopyAnharmonicApproximationAspreviouslystated,thethird-orderresponsefunctionfortwo-dimensionalin-fraredspectroscopygivesnosignalwithareferencesystemofaharmonicoscillatorandlinearlyvaryingdipole.Toobtainanon-zerosignal,thereferencesystemofananhar-monicoscillator,alsowithalinearlyvaryingdipole,isexamined.Theanharmonicityisbuiltintothetheorybyusingacubicpertubationoftheformq3totheharmonicpotential.Anharmonicstatesarerepresentedasasuperpositionofharmonicstates.Theresultofthisdevelopmentisadierentexpressionforthethird-orderresponsefunctionwhichdoesgivea2D-IRsignal.10.1TheAnharmonicApproximationwithLinearlyVaryingDipoleOnewaytorepresentananharmonicoscillatoristhroughtheuseoftheMorsepotential,butthismethodwouldrequiretheanharmonicstatestobedenedintermsofcomplicatedBesselfunctions.Forthesakeofsimplicity,theanharmonicapproximationwasinvokedbyapplyingacubicpertubationtotheharmonicpotentialshowninEquation9.1.92

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V=m2q2=2+q30.1Theconstantindicatesthedegreeofanharmonicityorthestrengthoftheper-tubation.Accordingtorst-orderpertubationtheory,theanharmonicstatescanberepresentedassumsofharmonicstates.jiiANH=jiiHARM+Pi6=khkjq3jii Ei)]TJ/F20 11.955 Tf 11.955 0 Td[(Ekjki0.2NotethatEquation10.2isvalidonlyforsmallpertubations,i.e.2~ m3!5>>1.Thematrixelementshkjq3jiiarenonzeroinonlyafewcases:k=i+1,k=i)]TJ/F15 11.955 Tf 11.858 0 Td[(1,k=i+3,andk=i)]TJ/F15 11.955 Tf 10.147 0 Td[(3.Eachanharmonicstateiscompletelydescribed,withinthisapproximation,asasuperpositionofveharmonicstatesweightedbyconstants.jiianh=jiiharm+ci+1ji+1i+ci)]TJ/F18 7.97 Tf 6.587 0 Td[(1ji)]TJ/F15 11.955 Tf 11.955 0 Td[(1i+ci+3ji+3i+ci)]TJ/F18 7.97 Tf 6.586 0 Td[(3ji)]TJ/F15 11.955 Tf 11.955 0 Td[(3i0.3ThepertubationconstantscaneasilybederivedbysolvingforthematrixelementinEquation10.2.Theconstantsgivezeroresultsfortransitionstonegativestatessuchasi=)]TJ/F15 11.955 Tf 9.299 0 Td[(1.Intheharmoniclimit,allpertubationconstantsgotoone.ci+=)]TJ/F20 11.955 Tf 9.299 0 Td[(bi+13=2 5=2ci)]TJ/F18 7.97 Tf 6.586 0 Td[(=bi3=2 5=293

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Constant Terms k c1+ k0 1+c0+ k1 1+c1)]TJET1 0 0 1 392.739 614.092 cmq[]0 d0 J0.398 w0.199 0 m0.199 28.892 lSQ1 0 0 1 -137.677 -0.398 cmq[]0 d0 J0.398 w0 0.199 m137.877 0.199 lSQ1 0 0 1 -0.2 -28.892 cmq[]0 d0 J0.398 w0.199 0 m0.199 28.892 lSQ1 0 0 1 -254.862 -584.802 cmBT/F20 11.955 Tf 278.911 593.47 Td[(k2 1+c2)]TJ/F15 11.955 Tf 9.742 1.794 Td[(+c0+ k3 1+c3)]TJ/F15 11.955 Tf 9.742 1.793 Td[(+c1+ k4 1+c4)]TJ/F15 11.955 Tf 9.742 1.793 Td[(+c2+ k5 1+c5)]TJ/F15 11.955 Tf 9.742 1.793 Td[(+c3+ Table10.1:Denitionsofseveralanharmonicconstantswhichwillappearintheexpres-sionsforTCFsAandBci+3=)]TJ/F20 11.955 Tf 9.298 0 Td[(bp i+1i+2i+3 95=2ci)]TJ/F18 7.97 Tf 6.587 0 Td[(3=bp ii)]TJ/F15 11.955 Tf 11.955 0 Td[(1i)]TJ/F15 11.955 Tf 11.955 0 Td[(2 95=2b=r 9~ 8m3=Ei+1)]TJ/F20 11.955 Tf 11.955 0 Td[(Ei=~0.4InTable10.1,severalconstantscomposedofsumsofci+andc1)]TJ/F15 11.955 Tf 11.228 1.794 Td[(aredenedforconvenienceinwritingoutexpressionsforTCFsAandBlaterinthechapter.94

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10.2SimplifyingtheApproximationSincethetimecorrelationfunctionsthatresultfromexpansionofthethird-orderresponsefunction'scommutatorscanbeanalyticallyrelatedtoTCFsAandB,thefunctionoftheanharmonicapproximation,aswiththeharmonic,istodeterminearelationshipbetweenTCFsAandB,thenBRandBIwiththeendresultofcalculatingRasBRmultipliedbyasetoffrequencyfactors.A!1;!2;!3=1 Q1XABCDe)]TJ/F21 7.97 Tf 6.586 0 Td[(EAiADjDCkCB`BA!1)]TJ/F20 11.955 Tf 11.955 0 Td[(EAB=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(EAC=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(EAD=~B!1;!2;!3=1 Q1XABCDe)]TJ/F21 7.97 Tf 6.586 0 Td[(EAiDCjADkCB`BA!1)]TJ/F20 11.955 Tf 11.955 0 Td[(EAB=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(EAC=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(EDC=~10.5InEquation10.5,theindicesA,B,C,andDnowrepresentanharmonicstates.InTCFsAandBthedipolematrixelementsareexpandedouttorstorder,i.e.hjji=0+0hjqji.Statesandarenowanharmonicstates,eachoneisasumofveharmonicstatesweightedbyconstants.Consequently,thenalresultisthatforeachTCF,theinnitesumof254termsmustbeevaluated.Itisobviousthatsimplicationisrequiredtomaketheapplicationofthisapproxmationfeasible.OnerealisticsimplicationisassumethatanharmonicstatejAiwillbeground-statedominated,asindicatedbytheBoltzmannfactorinbothTCFs.Formally,thisideaisrepresentedinthetheorybyassumingthatthei+3andi)]TJ/F15 11.955 Tf 12.125 0 Td[(3transitionsmake95

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negligiblecontributions,andthereforemaybeneglected.RestrictionofstatejAiinturnrestrictsthevaluesthatstatesjBi,jCi,andjDicanassume,sinceonlyhnjjn;n1imatrixelementsarenonzero.Anharmonicstateaisrestrictedasasumfromzerotoone.Consequently,thesumsinEquation10.5arerestrictedasshown.A!1;!2;!3=1 Q1XA=02XD=02XB=03XC=0e)]TJ/F21 7.97 Tf 6.587 0 Td[(EAiADjDCkCB`BA!1)]TJ/F20 11.955 Tf 11.955 0 Td[(EAB=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(EAC=~!3)]TJ/F20 11.955 Tf 11.956 0 Td[(EAD=~B!1;!2;!3=1 Q1XA=02XD=02XB=03XC=0e)]TJ/F21 7.97 Tf 6.587 0 Td[(EAiDCjADkCB`BA!1)]TJ/F20 11.955 Tf 11.955 0 Td[(EAB=~!2)]TJ/F20 11.955 Tf 11.955 0 Td[(EAC=~!3)]TJ/F20 11.955 Tf 11.955 0 Td[(EDC=~0.6Additionalsimplicationscanbemadebydecomposingthesumsoveranharmonicstatesintosumsoverharmonicstates.Asaresult,thesumsthatmakeupTCFsAandBbecomeasumofthreeharmonicsums,eachcorrespondingtoaharmonicstatea=0,1,or2,asshownbelow.1XA=02XD=02XB=03XC=0=0Xa=01Xd=01Xb=02Xc=0+1Xa=1+2Xd=02Xb=03Xc=0+2Xa=23Xd=13Xb=14Xc=00.7Theresultingsumscanbedecomposedevenfurtherbyeliminatingtermsthatgivezeroasaresultofimproperoverlapofstates.Afterthesesimplicationshavebeenmade,onlyforty-vetermscontributetoeachTCF.ExpandingthesumsandevaluatingeachonewithalinearlyvaryingdipolesimpliestheTCFsevenfurtherdowntosums96

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ofthirteennonzeroterms.ThisprocedureisthesameforbothTCFsAandB.Tables10.2and10.3displayallthetermscontributingtoTCFsAandB.Atermiswrittenoutasfollows:~ 2m)]TJ/F20 11.955 Tf 11.955 0 Td[(e)]TJ/F21 7.97 Tf 6.587 0 Td[(~0i0j0k0`K0.8Kindicatesthecombinationofpertubationconstantsincludedintheterm.TheseconstantsarethesameforTCFsAandB.indicatestheterm'sdeltafunctions,whicharetypicallydierentforTCFsAandB.10.3RelatingtheAnharmonicTCFsAandBInChapter9'sanalysisinvolvingtheharmonicapproximation,TCFsAandBwererelatedbytakingratiosbetweentermswithidenticaldeltafunctionsandusingtheresultstogeneralizetherelationship.Intheanharmonicapproximation,thereisnoobviousmeansofdeterminingsucharelationship.Instead,allpossibleharmonictransitionsappearingintheanharmonictermsofTCFsAandBareconsidered.TheanalysisissummarizedinTable10.4.Forsimplicity,thepertubationconstantsarenotincludedintheanalysis.First,ageneralformulaisdevelopedfortheharmoniccase,thenthepertubationconstantswillbere-incorporatedintotheexpression.Next,theprefactorsforTCFsAandBarecomparedintermsthathavethesamesetsofdeltafunctions.Wheni<2,someofthecoecientsgotozero.Heavysidestepfunctionsoftheform!areimposedtodealwithcaseswheni=0.Inthesecases,97

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IndicesADBC K TCFA 0110 k21k22 !1+!2!3+ 0121 2k1k22k3 !1+!2+2!3+ 1001 k0k21k2 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 1021 2k0k1k2k3 !1+!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 1201 2k0k1k2k3 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3+ 1212 4k0k2k23 !1+!2!3+ 1232 6k0k23k4 !1+!2+2!3+ 2110 2kk1k22 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2)]TJ/F15 11.955 Tf 11.955 0 Td[(2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 2112 4kk22k3 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 2132 6kk2k3k4 !1+!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 2312 6kk2k3k4 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3+ 2332 9kk3k24 !1+!2!3+ 2334 12kk42k5 !1+!2+2!3+ Table10.2:ThistabledescribesthethirteentermsthatmakeupTCFAintheanharmonicapproximation.TherstcolumnindicatestheindicesofA,B,C,andDcorrespondingtoeachterm,thesecondcolumnthepertubationconstantsK,andthethirdcolumnthedeltafunctionsforTCFA.98

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IndicesADBC K TCFB 0110 k21k22 !1+!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 0121 2k1k22k3 !1+1!2+2!3+ 1001 k0k21k2 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3+ 1021 2k0k1k2k3 !1+!2!3+ 1201 2k0k1k2k3 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 1212 4k0k2k23 !1+!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 1232 6k0k23k4 !1+!2+2!3+ 2110 2kk1k22 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2)]TJ/F15 11.955 Tf 11.955 0 Td[(2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 2112 4kk22k3 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3+ 2132 6kk2k3k4 !1+!2!3+ 2312 6kk2k3k4 !1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 2332 9kk3k24 !1+!2!3)]TJ/F15 11.955 Tf 11.956 0 Td[( 2334 12kk42k5 !1+!2+2!3+ Table10.3:ThistabledescribesthethirteentermsthatmakeupTCFBintheanharmonicapproximation.TherstcolumnindicatestheindicesofA,B,C,andDcorrespondingtoeachterm,thesecondcolumnthepertubationconstantsK,andthethirdcolumnthedeltafunctionsforTCFB.99

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A B C D Value !1,!2,!3TCFA !1,!2,!3TCFB i i+1 i-1 i ii+1 ,0,)]TJ/F15 11.955 Tf 9.298 0 Td[( ,0, i i+1 i+1 i i+12 )]TJ/F15 11.955 Tf 9.298 0 Td[(,0,)]TJ/F15 11.955 Tf 9.298 0 Td[( )]TJ/F15 11.955 Tf 9.299 0 Td[(,0 i i+1 i+1 i+2 i+1i+2 )]TJ/F15 11.955 Tf 9.299 0 Td[(,)]TJ/F15 11.955 Tf 9.298 0 Td[(2,)]TJ/F15 11.955 Tf 9.298 0 Td[( )]TJ/F15 11.955 Tf 9.299 0 Td[(,)]TJ/F15 11.955 Tf 9.298 0 Td[(2,)]TJ/F15 11.955 Tf 9.299 0 Td[( i i-1 i-1 i-2 ii-1 ,2, ,2, i i-1 i-1 i i2 ,0, ,0,)]TJ/F15 11.955 Tf 9.299 0 Td[( i i-1 i+1 i ii+1 )]TJ/F15 11.955 Tf 9.299 0 Td[(,0, )]TJ/F15 11.955 Tf 9.299 0 Td[(,0,)]TJ/F15 11.955 Tf 9.299 0 Td[( Table10.4:AnalysisoftheanharmonictermsofTCFsAandBintermsofharmonictransitions.ColumnsonethroughfourindicatethepatternsthatindicesA,D,B,andCintermsofageneralindexi.Columnveindicatestheprefactorsappearinginfrontofthepertubationconstants.Columnssixandsevenindicatesthevaluesthatfrequencies!1,!2,and!3takeon,asenforcedbyeachterm'sdeltafunctions.ThisinformationwillbeusedtoworktowardsageneralrelationshipbetweenanharmonicTCFsAandB.100

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!1 !2 !3 A=B 2 1 )]TJ/F15 11.955 Tf 9.299 0 Td[( )]TJ/F15 11.955 Tf 9.299 0 Td[(2 1 0 i2 ii+1=[i+)]TJ/F21 7.97 Tf 6.587 0 Td[(!1]2 ii+1 )]TJ/F15 11.955 Tf 9.299 0 Td[( 0 )]TJ/F15 11.955 Tf 9.299 0 Td[( i+12 ii+1=[i+)]TJ/F21 7.97 Tf 6.587 0 Td[(!1]2 ii+1 )]TJ/F15 11.955 Tf 9.299 0 Td[( 0 ii+1 i+12=ii+1 [i+)]TJ/F21 7.97 Tf 6.587 0 Td[(!1]2 0 )]TJ/F15 11.955 Tf 9.299 0 Td[( i2 i+12=ii+1 [i+)]TJ/F21 7.97 Tf 6.587 0 Td[(!1]2 Table10.5:Basedontheanharmonicoscillatorapproximation,ratiosbetweenTCFsAandBarepresentedforeachpossiblesetofdeltafunctionsgivingfrequencies!1,!2,!3theirvalues.TherstthreecolumnsgivethevaluesofthefrequenciesandthefourthgivestheA=Bratios.nosetofdeltafunctionswith!1=cancontributesincethesewouldcorrespondtoforbiddentransitionstonegativestates.Table10.5summarizestheratiosbetweenTCFsAandBforallpossiblesetsofdeltafunctions.ThecontentsofTable10.5wereusedtoderiveageneralrelationshipbetweenTCFsAandBfortheharmoniccase.Pi=0[i+)]TJ/F20 11.955 Tf 9.299 0 Td[(!1]2e)]TJ/F21 7.97 Tf 6.587 0 Td[(i~p !21+!23=2 Pi=0ii+1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i~p !21+!23=2!!3=!1)]TJ/F21 7.97 Tf 6.587 0 Td[(!2=2!10.9InEquation10.9,denotesthemaximumvaluethatstateicanattain.Sincethisexpressionisexactforaharmonicsystem,maybearbitrarilylarge.MakingthesuminniteandevaluatingtheresultinggeometricseriesyieldsthesameanalyticalA=Bratio101

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derivedinChapter9.TheanharmonicpertubationconstantscaneasilybeincorporatedintoEquation10.9toderiveananharmonicexpression.Pi=0[i+)]TJ/F20 11.955 Tf 9.298 0 Td[(!1]2ikb+1kc+1kd+1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i~p !21+!23=2 Pi=0ii+10ik0b+1k0c+1k0d+1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i~p !21+!23=2!!3=!1)]TJ/F21 7.97 Tf 6.586 0 Td[(!2=2!10.10InEquation10.10,thekconstantsaredescribedbyTable10.1andtheiconstantsaredenedbelowinEquation10.11.Theprimesonthedenominator'skandconstantsdenotetheirassociationwithTCFB,whiletheonesinthedenominatorareassociatedwithTCFA.Theb,c,anddindicescorrespondingtoaspecicivalueofindexainEquation10.10determinethevaluesofthekandconstants.i==1<1i==c)]TJ/F18 7.97 Tf 6.586 0 Td[(1+1i<=1+ci+1)]TJ/F15 11.955 Tf 9.741 1.86 Td[(+ci)]TJ/F18 7.97 Tf 6.587 0 Td[(1+0.11Asanexample,iftheupperlimitontheanharmonicsumsis=3,thentheconstantsare0=k1,1=k2,2=1+c1+,and3=c2+.Supercially,Equations10.9and10.10producecorrectresults,butfurtheranal-ysisoftheseexpressionsrevealtheirfatalaws.Bothofthemareill-behavedatzerofrequency,aregionwhichplaysavitalroleinexperimentsandcannotbeoverlooked.Additionally,intheanharmonicexpression,thepertubationconstantsaredenedbythevaluesofb,c,andd,whichinturndependonthevalueofindexi.Theydonotmaintain102

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aconstantvalueforallpossiblesetsofdeltafunctions.Whilethisdependencedoesnotformallyposeanyproblem,itdoesnotallowtheexpressiontobeincorporatedintotheTCFcalculationcode.Therefore,analternativeexpressiontorelateTCFsAandB,whichremediestheseissues,isconsidered.AANH!1;!2;!3=Xi=0[i+)]TJ/F20 11.955 Tf 9.298 0 Td[(!1][i+)]TJ/F20 11.955 Tf 9.299 0 Td[(!3)]TJ/F20 11.955 Tf 11.956 0 Td[(!2+)]TJ/F20 11.955 Tf 9.298 0 Td[(!2]e)]TJ/F21 7.97 Tf 6.587 0 Td[(~iTaBANH!1;!2;!3=Xi=0[i+)]TJ/F20 11.955 Tf 9.298 0 Td[(!1][i+!3)]TJ/F15 11.955 Tf 11.955 0 Td[(2!2+2)]TJ/F20 11.955 Tf 9.299 0 Td[(!2]e)]TJ/F21 7.97 Tf 6.586 0 Td[(~iTb=q !21+!23=2Ta=)]TJ/F20 11.955 Tf 9.298 0 Td[(!1!3kiki+1ki+2+!1)]TJ/F20 11.955 Tf 9.298 0 Td[(!3k2iki+1+)]TJ/F20 11.955 Tf 9.298 0 Td[(!1!3k2i+2ki+1)]TJ/F15 11.955 Tf 11.956 0 Td[(tanh2~!2k2iki+1+1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh2~!1Tb=!1!3kiki+1ki+2+!1!3k2iki+1+)]TJ/F20 11.955 Tf 9.299 0 Td[(!1)]TJ/F20 11.955 Tf 9.299 0 Td[(!3k2i+2ki+1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh2~!2kiki+1ki+2+1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh2~!10.12Foreaseofcalculation,itisdesirabletoremoveallspecialfunctionsfromtheexpressionrelatingTCFsAandB.Inthehighfrequencylimit,theregionofinterestinmosttypicalexperimentsandtheoreticalcalculations,tanh~!=!)]TJ/F20 11.955 Tf 10.666 0 Td[()]TJ/F20 11.955 Tf 9.299 0 Td[(!.Thus,theHeavysidestepfunctionscanbewrittenintermsoftanhfunctionsaccordingtothefollowingrelationships.103

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!=[tanh2~!+tanh~!]=2)]TJ/F20 11.955 Tf 9.299 0 Td[(!=[tanh2~!)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!]=2.13UsingthesedenitionstoremovetheHeavysidestepfunctionsfromEquation10.12,theexpressionrelatingTCFsAandBundertheanharmonicapproximationisthennalized,andisdisplayedinEquation10.14below.Asbefore,takesonavalueofp !21+!23=2.AANH!1;!2;!3=Xi=0e)]TJ/F21 7.97 Tf 6.587 0 Td[(~iiTai+tanh2~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!1 2i+tanh2~!3)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!3 2)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!2BANH!1;!2;!3=Xi=0e)]TJ/F21 7.97 Tf 6.586 0 Td[(~iiTbi+tanh2~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!1 2i+tanh2~!3)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!3 2)]TJ/F15 11.955 Tf 11.955 0 Td[(2tanh~!2Ta=tanh2[~!1)]TJ/F20 11.955 Tf 11.956 0 Td[(!3]kiki+1ki+2+1 4[tanh2~!1+tanh~!1][tanh2~!3+tanh~!3]k2iki+1i+1 4[tanh2~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!1][tanh2~!3)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!3]k2i+2ki+1i)]TJ/F15 11.955 Tf -190.561 -31.881 Td[(tanh2~!2k2iki+1i+1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh2~!1104

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Tb=tanh2[~!1+!3]kiki+1ki+2+1 4[tanh2~!1+tanh~!1][tanh2~!3)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!3]k2iki+1i+1 4[tanh2~!1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!1][tanh2~!3)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh~!3]k2i+2ki+1i)]TJ/F15 11.955 Tf -209.099 -31.881 Td[(tanh2~!2kiki+1ki+2i+1)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh2~!1.1410.4RelatingtheRealandImaginaryPartsofAnharmonicTCFBSinceonlytherealpartofthetime-domainTCFBcanbecalculatedusingclassi-caldynamics,itisnecessarytodevisearelationshipbetweenTCFB'srealandimaginaryparts.ThisisaccomplishedbywritingoutallthetermsofTCFBinthetimedomain,separatingthemintotheirrealandimaginaryparts,andFouriertransformingtherealandimaginarypartsseparately.TheFouriertransformsaregroupedbylikedeltafunc-tionsintotermsBjR!1;!2;!3andBjI!1;!2;!3.RdenotestheFouriertransformofarealpartandIdenotestheFouriertransformofanimaginarypart.Frequency-domainrelationshipsbetweenindividualBjRandBjItermsaredevelopedandthengeneralizedtobecorrectforallterms.TherearethreepossiblegroupingsofdeltafunctionsassociatedwithBRandBI.B1R:!1+!3)]TJ/F15 11.955 Tf 11.955 0 Td[(+!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!3+B1I:!1+!3)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F20 11.955 Tf 11.955 0 Td[(!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!3+B2R:!1)]TJ/F15 11.955 Tf 11.956 0 Td[(!3)]TJ/F15 11.955 Tf 11.955 0 Td[(+!1+!3+105

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B2I:!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!3)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F20 11.955 Tf 11.956 0 Td[(!1+!3+B3R:!1+!2+2!3++!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2)]TJ/F15 11.955 Tf 11.955 0 Td[(2!3)]TJ/F15 11.955 Tf 11.955 0 Td[(B3I:!1+!2+2!3+)]TJ/F20 11.955 Tf 11.955 0 Td[(!1)]TJ/F15 11.955 Tf 11.955 0 Td[(!2)]TJ/F15 11.955 Tf 11.955 0 Td[(2!3)]TJ/F15 11.955 Tf 11.955 0 Td[(0.15TheconstantsassociatedwitheachoftheBjRandBjIarelistedbelow.ThesearethequantitiesusedtodeviserelationshipsbetweentherealandimaginarypartsofTCFB'sterms.B1R:1 2k21k22+2k0k2k23e)]TJ/F21 7.97 Tf 6.586 0 Td[(~+1 2k0k21k2e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+2kk2k23e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~+9 2kk3k24e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~B1I:1 2k21k22+2k0k2k23e)]TJ/F21 7.97 Tf 6.587 0 Td[(~)]TJ/F15 11.955 Tf 13.15 8.087 Td[(1 2k0k21k2e)]TJ/F21 7.97 Tf 6.587 0 Td[(~)]TJ/F15 11.955 Tf 11.955 0 Td[(2kk2k23e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~+9 2kk3k24e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~B2R:[6kk2k3k4e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~+2k0k1k2k3e)]TJ/F21 7.97 Tf 6.586 0 Td[(~B2I:0B3R:k1k22k3+3k0k23k4e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+6kk24k5e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~+kk1k22e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~B3I:k1k22k3+3k0k23k4e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+6kk24k5e)]TJ/F18 7.97 Tf 6.587 0 Td[(2~)]TJ/F20 11.955 Tf 11.955 0 Td[(kk1k22e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~0.16Table10.6summarizestherelationshipsbetweentherealandimaginarypartsofTCFBforthethreedierentgroupingsofdeltafunctions.Becausethesystemisassumedtobeinthehighfrequencylimits,thetanhfunctions,whichappearintheharmonicexpressionEquation9.17andgoto1inthislimit,areintroducedintotherelationships.Theanharmonicconstantsa,b,c,anddappearingintherelationshipsdopossess-dependence,butthisisnotaconcernsincethedeltafunctionsgoverningthevalues106

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Term f,whereBI=fBR 1 )]TJ/F25 11.955 Tf 11.291 9.684 Td[()]TJ/F21 7.97 Tf 6.675 -4.976 Td[(a)]TJ/F21 7.97 Tf 6.587 0 Td[(b a+btanh[~!1)]TJ/F20 11.955 Tf 11.956 0 Td[(!3+2!2=4] 2 )]TJ/F15 11.955 Tf 11.291 0 Td[(tanh[~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(!3+2!2=4] 3 )]TJ/F25 11.955 Tf 11.291 9.684 Td[()]TJ/F21 7.97 Tf 6.675 -4.977 Td[(c)]TJ/F21 7.97 Tf 6.586 0 Td[(d c+dtanh[~!1)]TJ/F20 11.955 Tf 11.955 0 Td[(!3+2!2=4] Constant Denition a 1 2k21k22+2k0k2k23e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+9 2kk3k24e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~ b 1 2k0k21k2e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+2kk2k23e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~ c k1k22k3+3k0k23k4e)]TJ/F21 7.97 Tf 6.587 0 Td[(~+6kk24k5e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~ d 2kk2k23e)]TJ/F18 7.97 Tf 6.586 0 Td[(2~ Table10.6:Basedontheanharmonicoscillatorapproximation,ratiosbetweentheFouriertransformsofTCFB'srealandimaginarypartsarederived.Theupperportionsum-marizestherelationshipsforthethreegroupingsofdeltafunctionsandthelowerportiondenesseveralconstantsusedintherelationships.The-dependenceoftheconstantsisnotaconcernsincethedeltafunctionsgoverningthevaluesofthefrequenciesarethesameforboththerealandimaginarypartsofTCFB.107

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ofthefrequencies!1,!2,and!3arethesameforboththerealandimaginarypartsofTCFB.Takingtheharmoniclimitofthefourconstantsandremovingtherestrictionofremaininginthehighfrequencylimit,thusallowingthesumsoverstatestobecomeinnite,reproducestheharmonicrelationshiplaidoutinEquation9.17.10.5TheFinalRExpressionHavingderivedexpressionstorelateTCFsAandB,aswellastherealandimaginarypartsofTCFB,itisnowpossibletoderiveanalexpressionforthethird-orderresponsefunction.BeginningwithEquation8.24,anexactanalyticalexpressionforRintermsoftherealandimaginarypartsofTCFsAandB,theanharmonicthird-orderresponsefunctionisconstructed.R!1;!2;!3=i ~3tanh~!1=2f+e~!1[1)]TJ/F20 11.955 Tf 11.955 0 Td[(f+ANH!1;!2;!3)]TJ/F15 11.955 Tf 11.955 0 Td[(tanh[~!1+2!2)]TJ/F20 11.955 Tf 11.956 0 Td[(!3=4]++e)]TJ/F21 7.97 Tf 6.587 0 Td[(~!1[1)]TJ/F20 11.955 Tf 11.955 0 Td[(f)]TJ/F21 7.97 Tf -1.277 -8.189 Td[(ANH!1;!2;!3+tanh[~!1+2!2)]TJ/F20 11.955 Tf 11.955 0 Td[(!3=4]gBR!1;!2;!30.17Thisexpression,whichisexpectedtoyieldnonzerosignalforthereferencesys-temcollected,maybeusedtocalculatethethird-orderresponsefunctionusingclassicalmoleculardynamicstechniques,sinceitissimplycomposedofthefrequency-domainTCFBR!1;!2;!3multipliedbyasetoffrequencyfactors.108

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Chapter11TCFTheory:2D-IRSpectroscopyComputationandResultsInChapters8,9,and10,atimecorrelationfunctiontheoryoftwo-dimensionalinfraredspectroscopywasdeveloped.Specically,thesethreechaptersoutlinedthedevel-opmentofatheorywhichallowedthethird-orderresponsefunctionRassociatedwith2D-IRexperimentstobecalculatedfromaclassicalTCF,approximatedasBRt1;t2;t3.Inthischapter,thepracticalaspectsofobtainingR!1;!2;!3fromBRt1;t2;t3,includingaccountingforaconstanttimedelayt2,Fouriertransformingthetime-domainTCF,andincorporatingareliablequantumcorrectionscheme,willbediscussed.Addi-tionally,theissueofusingRtocomputethirdpolarizationP,theobservableina2D-IRexperiment,isexplored.Finally,theapplicationoftheTCFtheorytoneatwater88,98and1,3-cyclohexanedioneindeuteratedchloroform99,twosystemswhichhavebeenprobedexperimentally,willbepresented.TheresultingspectraforbothsystemsareinagreementwithexperimentalresultsanddemonstratethepromiseofthisTCFtheorytoaccuratelyreproduce2D-IRspectraofcondensedphasesystems.109

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11.1TheStepsinCalculatinga2D-IRSpectrumThecomputationofatwo-dimensionalinfraredspectrumfromclassicalmoleculardynamicsisaccomplishedinseveralsteps.First,toobtainaseriesoftime-orderedpositioncongurationsforthesystemofinterest,microcanonicalNVEclassicalmoleculardynamicssimulationsareper-formedusingacodedevelopedattheCenterforMolecularModelingattheUniver-sityofPennsylvania.15,16Thecodeemploysreversibleintegrationandextendedsystemtechniques.Positioncongurationsarestoredfrequentlyenoughtoresolvetherelevantfrequencies.Forexample,incomputingthespectrumofneatwater,congurationswerestoredevery4.0fs,givingaNyquistfrequencyof4167cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.98,100Oncethecongurationsareobtained,itisnecessarytocalculatethedipoleateverystepofthesimulationandusetheresultstoobtainthefour-pointdipolecorrelationfunctionhjt1+t2it1+t2+t3kt1`i.Inthesecalculations,apointatompolarizabilityapproximationPAPAmodel,101whichaccountsfortheinteractionofdipoleswiththeeldcreatedbyotherdipolesinthesystem,isemployed.Theinduceddipoleassociatedwithatomiconsistsofcontributionsfromtheexternalelectriceldandalltheotherinduceddipolesjinthesystem.InasystemofNatoms,atomi'sinduceddipoleisgivenbelow.i=i"Ei)]TJ/F21 7.97 Tf 23.375 14.944 Td[(NXj=1;i6=jTijj#1.1110

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InEquation11.1,Eiistheappliedelectricaleldatpointi,iisthepolarizabilitytensorofi,andTijisthedipoleeldtensor.Thedipoleeldtensoristracelessanddescribestheinteractionofatomsiandj,asshownbelow.Tij=)]TJ/F15 11.955 Tf 12.734 8.088 Td[(3 r50BBBB@x2)]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 11.956 0 Td[(1=3r21CCCCA1.2InEquation11.2,rrepresentsthedistancebetweenatomsiandjandx,y,andzaretheCartesiancomponentsofthevectorfrompointitopointj.Atthispoint,Equation11.1isrewrittentosolveforthemanybodypolarizability.)]TJ/F18 7.97 Tf 6.586 0 Td[(1ii+NXj=1;i6=jTijj=Ei1.3Equation11.3canberecastasamatrixequation.0BBBBBBBB@)]TJ/F18 7.97 Tf 6.587 0 Td[(11T12:::T1NT21)]TJ/F18 7.97 Tf 6.586 0 Td[(12:::T2N.........T21::::::)]TJ/F18 7.97 Tf 6.586 0 Td[(1N1CCCCCCCCA0BBBBBBBB@12...N1CCCCCCCCA=0BBBBBBBB@E1E2...EN1CCCCCCCCA1.4A3NX3NmatrixAisdenedsuchthatEquation11.4becomesA=E.Solvingforyieldstheequation=BE,whereBissimplytheinverseofmatrixA.BismadeupofN3x3matricesBij.111

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B=0BBBBBBBB@B11B12:::B1NB21B22:::B2N............BN1BN2:::BNN1CCCCCCCCA1.5ThematrixEquation11.4cannowbewrittenasasetofNequationsfortheindividualatoms'induceddipoles.i="NXj=1Bij#E1.6Equation11.6assumesthattheelectricaleldEisuniformforallpointsi.Finally,thisequationcanbeusedtodeterminethesystem'stotalinduceddipole.system=NXi=1i="NXi=1NXj=1Bij#E1.7ThePAPAmodelhasbeensuccessfullyusedtoquantitativelyreproducethelinearinfraredspectrumofwater19,102andtheSFGspectrumofthewater/vaporinterface103andthereforeappearstobeasuitablemodelforthesecalculations.OncetheTCFhjt1+t2it1+t2+t3kt1`ihasbeenobtained,itisFouriertransformedtoobtainBR!1;!2;!3.TheresultismultipliedbyasetoffrequencyfactorsasdescribedinEquation9.20fortheharmonicreferencesystemorEquation10.17fortheanharmonicsystemtoobtainR!1;!2;!3.Theresponsefunctionisthenreverse112

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FouriertransformedtothetimedomainandcanthenbeappliedtoEquation8.1toobtainthethirdorderpolarizationPt.Finally,aFourierLaplacetransformofthepolarizationgivesthe2D-IRspectruminthefrequencydomain.11.2ConsideringaConstantt2DelayThetheorypresentedinthepreviouschaptersisafullythree-dimensionaltheory.Expressingthethird-orderresponsefunctionasasingleclassicalTCFmultipliedbyfrequencyfactorsisatremendoussimplication,butthecomputationofathree-timecorrelationisademandingtask.Atthispoint,itisworthconsideringthepossibilityofimplementingatwo-dimensionaltheory,sincemany2D-IRexperimentsxthetimedelayt2atsetvalues,oftenzero.Inassessingthevalidityofsuchatheory,itisimportanttoconsideriftheseexperimentsdo,indeed,probeatwo-dimensionalresponsefunctionandifsucharesponsefunctionisdeterminedbyatwo-dimensionalclassicalcorrelationfunction.Inthe2D-IRechoexperiment,thetimedelayt2issettozero.Fornitepulselengths,thisequalityisnotliterallyenforced,butasanapproximationtorealisticex-perimentalconditions,thevalueoft2isrestrictedusingadeltafunction.Undersuchconditions,theFouriertransformofthetime-domainresponsefunctionRt1;t2=0;t3isgivenbyR!1;!3=R!1;t2=0;!31 23Z1Z1Z1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!2t2e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!3t3Rt1;t2;t3t2dt1dt2dt31.8113

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Equation11.8suggeststhatameansofcalculatingRt1;t3witht2=0isrequiredtoobtainatwo-dimensionalresponsefunction.However,determiningthereverseFouriertransformofEquation9.20or10.17,BRmultipliedbyacomplicatedsetoffrequencyfactors,thencalculatingitsvaluefort2=0isanextremelydiculttask.Usingtheproposedmethods,themostconvenientwayofimposingtherestrictionont2istocalculatethetwo-dimensionalcorrelationfunctionBRt1;t2=0;t3.TheTCFisthenFouriertransformedtoobtainBR!1;t2=0;!3andmultipliedbyasetoffrequencyfactorstoobtainR!1;!2;!3.ReverseFouriertransforminggivesthenalresult,Rt1;t2=0;t3.Rt1;t2=0;t3=Z1Z1Z1ei!1t1ei!2t2ei!3t3h!1;!2;!3BR!1;!2;!3d!1d!2d!3jt2=01.9Onecomplicationwiththisprocedureisthat,ifh!1;!2;!3hasany!2depen-dence,itwillnotnecessarilyfollowthatthenaltime-domainresultisevaluatedforBRt1;t2=0;t3.However,for~!1;2;3>>1,thefrequencyrangeofinterestinmany2D-IRexperiments,itcanbeshownthatthe!2dependenceofh!1;!2;!3weakenssignicantly.Whenallfrequenciesareapproximately3000cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,the!2dependenceisweak,andwhen!2isincreasedto6000cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,itvirtuallydisappears.98Thevalueofh!1;!2;!3changesminimallyoverthewidthofanintramolecularresonanceandhaslittleeectontheresultinglineshapes.Thus,inthehighfrequencylimit,thefunctionhcanbeessentiallybewrittenash!1;!3.Consequently,theselectionofalarge!2114

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andevaluationofEquation11.9usingBR!1;t2=0;!3areappropriateactionsandgivereliableresults.Intheresultsdisplayedinthischapter,BRiscalculatedandFouriertransformedasatwo-dimensionalcorrelationfunctionwithaconstantvalueoft2.Whilethisapprox-imationgivescredibleresults,thecalculationofathree-dimensionalcorrelationfunctionisstillconsideredthemostaccuratewaytoobtaina2DIRsignalfromclassicalMDtrajectories.Thedevelopmentofcodestohandlethefullthree-dimensionaltheoryisunderway.11.3FourierTransformingBRt1;t2;t3TocorrectlytransformtheTCFBRt1;t2;t3tothefrequencydomain,itisnec-essarytoperformafullFouriertransformtoaccountforallpositiveandnegativevaluesofthetimes.BR!1;!2;!3=1 22Z1Z1Z1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!1t1e)]TJ/F21 7.97 Tf 6.587 0 Td[(i!2t2e)]TJ/F21 7.97 Tf 6.586 0 Td[(i!3t3hjt1+t2it1+t2+t3kt1`it2)]TJ/F20 11.955 Tf 11.955 0 Td[(211.10ThedeltafunctioninEquation11.10holdst2constantatdelay2,asdescribedintheprevioussection.ToimplementthisFouriertransformcorrectly,itisnecessarytocalculateBRt1;2;t3infourquadrantsinordertoincludeallpositiveandnegativevaluesoft1andt3.CalculatingallfourquadrantsessentiallyamountstocalculatingfourclassicalTCFsexpressedintermsofpositiveandnegativevaluesoft1andt3.115

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B++=hjt1+2it1+t3+2kt1`iB+)]TJ/F15 11.955 Tf 10.405 -4.936 Td[(=hjt1+2it1)]TJ/F20 11.955 Tf 11.955 0 Td[(t3+2kt1`iB)]TJ/F18 7.97 Tf 6.586 0 Td[(+=hj)]TJ/F20 11.955 Tf 9.298 0 Td[(t1+2i)]TJ/F20 11.955 Tf 9.298 0 Td[(t1+t3+2k)]TJ/F20 11.955 Tf 9.298 0 Td[(t1`iB\000=hj)]TJ/F20 11.955 Tf 9.299 0 Td[(t1+2i)]TJ/F20 11.955 Tf 9.298 0 Td[(t1)]TJ/F20 11.955 Tf 11.955 0 Td[(t3+2k)]TJ/F20 11.955 Tf 9.298 0 Td[(t1`i1.11TimestationaritypropertiesoftheTCFscanbeusedtorewritethefourquad-rants'TCFsintermsofonlypositivetimes.34,98B++=hjt1+2it1+t3+2kt1`iB+)]TJ/F15 11.955 Tf 10.406 -4.936 Td[(=hjt1+t3+2it1+2kt1+t3`t3iB)]TJ/F18 7.97 Tf 6.586 0 Td[(+=hj2it3+2k`t3iB\000=hjt3+2i2kt3`t1+t3i1.12For2=0,theTCFscorrespondingtothediagonalquadrantsB++andB\000areequal,andsoaretheonescorrespondingtotheo-diagonalquadrants,B)]TJ/F18 7.97 Tf 6.586 0 Td[(+andB+)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(.Thisisprovenusingtimereversal,i.e.makingallthetimeargumentsnegative,whichdoesnotchangetheTCF.Thus,having2=0requiresthecalculationofonlytwoTCFs,B++andB+)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(.Fornonzero2,noneofthefourTCFsareequal,makingitnecessarytocalculateallofthemseparately.OncetheappropriateTCFshavebeencalculated,thefullFouriertransformmaythenbeperformedtoobtainBR!1;t2=2;!3.116

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Althoughcalculationswillgenerallybeperformedinthehighfrequencylimit,itisinterestingtonotethat,intheclassicallimit,thetime-domainresponsefunctionissimplywrittenasasumoftimederivativesofBRt1;t2;t3.Inthiscase,itispossibletoexamineonlythepositivequadrantB++sincetheFouriertransformofBRt1;t2;t3isnotrequiredtoobtaintheresponsefunction.11.4ImplementingaQuantumCorrectionSchemeEquatingtherealpartofBt1;t2;t3withtheclassicalTCFhjt1+t2it1+t2+t3kt1`isanapproximation,commonlyimplementedinthecaseofonetimecorrelations.TomaketheclassicalTCFcloserinformtoitsquantummechanicalcoun-terpart,aquantumcorrectionschememaybeapplied.SuchaschemesimplyinvolvesmultiplyingtheFouriertransformoftheclassicalTCFbyasetoffrequencyfactors,whichisderivedbydividingthequantummechanicalBR!1;!2;!3byitsclassicallimit.Undertheharmonicreferencesystem,twodistinctquantumcorrectionschemesweredevelopedforthetheoryofRandtestedforplausibility.ThequantumcorrectionschemeswereconstructedusingthesixlowestordertermscontributingtoTCFB,i.e.theB1111termsdisplayedinEquation9.11.BQR!1;!2;!3=~!1 8[!1)]TJ/F20 11.955 Tf 11.955 0 Td[(!3+2!2coth2~!1=2+!1+!3)]TJ/F15 11.955 Tf 11.955 0 Td[(2!2csch2~!1=2]BC!1;!2;!3.13117

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BQR!1;!2;!3=~!1 8[!3)]TJ/F20 11.955 Tf 11.956 0 Td[(!1+2!2coth~!3=2+!1+!3)]TJ/F15 11.955 Tf 11.956 0 Td[(2!2csch2~!3=2]BC!1;!2;!3.14InEquations11.13and11.14,BQRrepresentstheFouriertransformoftherealpartofthequantumtimecorrelationfunctionBandBCrepresentsitsclassicallimit.ThetwoproposedquantumcorrectionschemesrepresentthesimplestfunctionalformsthatcanexactlyrelateBQRandBCfortheharmonicreferencesystem.Equation11.13maybeconsideredthemorereasonableofthetwoquantumcor-rectionschemesbecauseitpredictszerosignalalongthe!3axiswith!1=0.ThisbehavioriscongruentwithEquation9.20,whosetanh~!1=2prefactorsuggeststhattheresponsefunctioniszerowhen!1=0intheclassicallimit.Incontrast,theschemedescribedbyEquation11.14givesnosignalalongthe!1axiswith!3=0,behaviorthatisnotpredictedbyEquation9.20.11.5CalculatingPolarizationOncetheresponsefunctionRhasbeencomputed,thenextstepinobtainingthe2D-IRsignalistocalculatethethird-orderpolarizationPtasinEquation8.1.ToobtainamoreexplicitexpressionforP,thetimeenvelopesoftheappliedelectriceldsE1`,E2k,andE3jmustbespecied.Thesimplesttimeenvelopeisthe-function,whichresultsinPproportionaltoR.Using-functiontimeenvelopeseliminatestheneedtoperformtheintegrationsre-quiredinEquation8.1,buthastheunfortunateconsequenceofremovingthedependence118

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ofPontheexperimentalwave-matchingconditions.58,59Itisalsonecessarytoconsiderthatinafrequency-domainexperiment,itispossibletogenerateanearlymonochromaticpulse,butinthetime-domainexperimentsofinterestinthisresearch,-functiontimeenvelopesarenotrealistic.Fortheoreticalstudiesperformedintheimpulsivelimit,itiscommontoinvoketherotatingwaveapproximation,whichpreservesonlyresonantterms,inwhichtheopticalfrequencyiscancelledbyamaterialfrequencyofoppositesign,oftheresponsefunction,andtherebytakesintoaccountphase-matchingconditions.Sincethistheoryoftwo-dimensionalinfraredspectroscopyinvolvesthefullthird-orderresponsefunction,itisincorrecttoderivePfromRusing-functionpulses.34Theinclusionofwave-matchingconditionsisrequiredtohighlighttheLiouvillepathwaysofinterestforaparticularexperiment.Onepossibleapproachtothisproblemistoselectaeldtimeenvelopefunctionconsistentwiththoseusedinexperiments,suchasaGaussianfunction,whichretainsthewavevectorassociatedwitheachelectriceld.AGaussiantimeenvelopetakesonthefollowingform,whereisaparameterindicatingitswidth.Et=cos[!t]e)]TJ/F21 7.97 Tf 6.586 0 Td[(t2=1.15Becauseintheexperimentsthetimedelay2istypicallysettoaconstant0,a-functionenvelopeisusedforthesecondelectriceldE2jandanother-functionisusedtoenforcethetime'sconstantvalue.Ifthedynamicsofinterestoccuronamuchlongertimescalethanthepulses,itisacceptabletoassumethatn=tn.63119

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P1;2=0;t=Z10Z10dt1dt2dt3Rt1;t2;t3E3t)]TJ/F20 11.955 Tf 11.955 0 Td[(t3E2t+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2E1t+1+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2)]TJ/F20 11.955 Tf 11.956 0 Td[(t12)]TJ/F20 11.955 Tf 11.955 0 Td[(0E1t+1+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2)]TJ/F20 11.955 Tf 11.955 0 Td[(t1=cos[!1t+1+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2)]TJ/F20 11.955 Tf 11.956 0 Td[(t1]et+2+1)]TJ/F21 7.97 Tf 6.587 0 Td[(t3)]TJ/F21 7.97 Tf 6.586 0 Td[(t2)]TJ/F21 7.97 Tf 6.587 0 Td[(t12=E2t+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2=t+2)]TJ/F20 11.955 Tf 11.955 0 Td[(t3)]TJ/F20 11.955 Tf 11.955 0 Td[(t2E3t)]TJ/F20 11.955 Tf 11.956 0 Td[(t3=cos[!3t)]TJ/F20 11.955 Tf 11.955 0 Td[(t3]et)]TJ/F21 7.97 Tf 6.587 0 Td[(t32=1.16The-functionsinEquation11.16canbeenforcedandthenalresultcanbeadaptedtonumericalcomputationalmethodsinordertoobtainthethird-orderpolar-izationfromthecalculatedresponsefunctionR.11.6AmbientWaterThesuccessfulapplicationofthisTCFtheorytoambientwaterdemonstratesitspotentialtocorrectlycapturethe2D-IRspectraofcondensedphases.98SincewaterisdiculttoinvestigateexperimentallyduetothestronginfraredabsorbanceoftheOHoscillator,mostexperimentalstudieshavefocusedondilutesolutionsofHODinliquidD2O.104,105However,the2DIRspectrumofneatwaterhasrecentlybeendeterminedusingatransientgratingexperiment.88The2DIRspectrumofwater,particularlyintheOHstretchingregion,canprovideinsightintothedistributionofhydrogenbondsinthecondensedphase.Thepresenceofo-diagonalpeaksinthespectrummightsug-120

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gestcouplingofoscillatorsthroughhydrogenbondsormechanismssuchasdipole-dipolecoupling,whichwouldfacilitatethetransferofvibrationalenergybetweenmolecules.88TheresultsofcalculationsonneatwatermightserveasabenchmarkforassessingthevalidityoftheTCFtheory.Beginningwithasystemofambientwater,thetwo-timecorrelationfunctionofthesystemdipoleBRt1;t3wascalculatedforseveralxedvaluesoftimedelay2thenusedtodeterminetheresponsefunctionR!1;!2.Togeneratethetime-orderedpositioncongurationsrequiredforTCFcalcula-tions,microcanonicalclassicalmoleculardynamicssimulations15werecarriedoutonasystemof64exibleSPCwatermolecules.Inapreviousstudy,itwasdemonstratedthatasystemsizeof64moleculeswassucienttoreproducethelinearinfraredspec-trumofwater,19andwasthereforeconsideredreasonableforthesecalculations.Thewaterwassimulatedatadensityof0.99g/cm3andanambienttemperatureof295K,conditionsproducingapressureof1.0atm,asdeterminedbyisobaric-isothermalNPTmoleculardynamics.Thewaterintramolecularpotentialincludedaharmonicbendingpotential,linearcrossterms,andaMorseOHstretchingpotential.19,103Simulationswerecomposedof1.4million1.0fstimestepsandpositioncongurationswerestoredevery4.0fsallowingforresolutionoftheNyquistfrequency,yieldingatotalof350,000con-gurations.Themoleculardynamicswasperformedwithoutexplicitpolarizationforces,butthepointatomicpolarizabilityPAPAmodel,whichexplicitlyincludesmany-bodypolarizability,wasusedtocalculatethetimedependentdipoleoftheliquidwater.Addi-tionally,apermanentdipolemodel,ttoabinitiocalculations,106wasadoptedinthesesimulations.Induceddipolederivatives,accountedforbythePAPAmodel,arerespon-121

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sibleformostoftheobservedliquidstateinfraredintensityintheOHstretchingregion,whilethebendingintensityismostlydeterminedbythepermanentdipolederivative.Forthetimedelay2=0thetwoTCFsB++andB+)]TJ/F15 11.955 Tf 11.084 -4.338 Td[(showninEquation11.11werecalculated.TheTCFsdecayedslowly107andwerecalculatedusingamaximumcorrelationtimeof20psinthet1andt3directions.Overthecourseof20ps,eachTCFdecayedtoapproximately10percentofitsinitialvalue,butdidnotreachtheasymptoticvalueofzero.98Ideally,longercorrelationtimeswouldallowtheTCFstodecayfullytozero,butthecomputationaldemandsbecametoogreatforsuchcorrelationtimestobepractical.Acorrelationtimeof50ps,whichwouldhaveallowedtheTCFtodecaytoonepercentofitsinitialvalue,wouldhaverequiredthestorageandmanipulationofseveralgigabytesofdata.ToeectivelyFouriertransformtheTCFdata,baselinevaluesforindividualtimeslicesweresubtractedfromthedataandaseriesofone-dimensionalFouriertransformswasperformed.Thisproceduredidnotaectthelineshape,ascomparedtotheFouriertransformofaTCFthatfullydecayedtozero.Figure11.1displaysthemagnitudeoftheFouriertransformofthefullypolarizedcorrelationfunctionBRt1;t2=0;t3.Multiplyingthisresultbyeithertheharmonicoranharmonicfrequencyfactorswouldgivethethird-orderresponsefunction.Astrongdiagonalsignal,whichissimilarinappearancetowater'slinearIRspectrumandindicatesstrongself-couplingofvibrationalmodes,82ispresent,asshowninFigure11.2.Dominantpeaksarepresentnear1800cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1and3300cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,liquidwater'sinfraredbendingandstretchingregions,respectively.Slowlydecayingridgesarepresentalongthe!3axis122

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Figure11.1:ThemagnitudeoftheFouriertransformofthetwotimecorrelationfunctionofthesystemdipolejBR!1;t2=0;!3jforambientwater.Theintramolecularstretchingregionisdisplayed.123

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Figure11.2:Aone-dimensionalsliceofthefrequency-domainTCFBR!1;!3corre-spondingtothediagonal,!1=!3isdisplayed.Thissliceissmoothedusingasinglemultipointaverageinbothfrequencydirectionstoeliminateoscillationsnearandalongthediagonal.at!1=1800cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1and3300cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Theseridgessuggestcouplingbetweenwater'sbendingandstretchingmodes.Althoughwaterhasdistinctantisymmetricandsymmetricstretchesinthegasphase,themixing"oftheseresonancesinthecondensedphaseresultsinthebroadinfraredabsorption17{19thatisclearlyshowninthebehaviorofthiscorrelationfunction.One-dimensionalfrequencyslicesofBR!1;!2revealthedetailedlineshapesassociatedwiththisTCF,aswellasthenatureoftheo-diagonalcouplings.InFigure11.2adiagonalsliceofBR!1;!3,whichisreminiscentofthelinearIRexperiment,isdisplayed.Theoppositephaseofthebendingandstretchingmodesshouldbenoted,and124

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Figure11.3:Theoreticalspectraforambientwateraredisplayed.TheupperpanelshowsthelinearIRspectrumofwater,andthebottompanelisacontourplotofthethird-orderresponsefunctioncalculatedundertheharmonicapproximation.maybeusedtoextractstructuralinformationaboutthesystembeingprobed,suchastherelativeorientationsoftransitiondipolesofcoupledvibrationalmodes.62Figure11.3displaysthethird-orderresponsefunctioncalculatedunderthehar-monicapproximation.Thecontourplotrevealsastrongechosignalalongthediagonalwithpeaksat1850and3300cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Thediagonalpeakat3300cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1iselongatedparalleltothediagonal,indicativeofinhomogeneousbroadening,andalsoatanangletothediagonalandparalleltothe!3axis,suggestinglifetimebroadening.60Themagnitudeoftheo-diagonalsignalissmallcomparedtothediagonal,butsignicantsignalisstillvisibleintheseregions.Thebroadandextensivecouplingsthat125

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Figure11.4:Twofrequencyslicesoftheharmonicthird-orderresponsefunctionforneatwateraredisplayed.Theslicesaretakenalong!1with!3=1850wavenumbersleftpaneland!3=3300wavenumbersrightpanel.Theinsetshighlighttheo-diagonalcouplingsbetweenwater'sbendingandstretchingmodes.appearinwater'sthird-orderresponsefunctionareconsistentwiththefastvibrationalenergydistributionthatisfrequentlyobservedinwater'sbehavior.108,109Figure11.4demonstratestheprominenceofo-diagonalcouplingsinwater's2D-IRspectrum.Theleftpanelpresentsasliceofwater'sharmonicthird-orderresponsefunctionalong!1with!3=1850cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Thewaterbendingpeak,adiagonal,iscenteredat!1=1850cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,andtheinsethighlightsitsweakercouplingwiththeOHstretchataround3300cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Therightpaneldisplaysaslicealong!1with!3=3300cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Thedominantdiagonalpeakat!1=3300cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1representstheOHstretch,andaweakcouplingwiththebendingmodeat!1=1850cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1appearsintheinset.Figure11.5displaystheuseofthequantumcorrectionschemedescribedbyEqua-tion11.13.Adiagonal!1=!3sliceofthequantumcorrectedthird-orderresponse126

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Figure11.5:Adiagonalsliceofthequantumcorrectedthird-orderresponsefunctionofneatwaterisdisplayed.Theinsethighlightsthedetailedlineshape.Thequantumcorrec-tionaectsthemagnitudeofthesignal,butdoesnotchangethelineshapesignicantly.127

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functionindicatesthatthequantumcorrectionschemeonlysignicantlyaectsthemag-nitudeofthesignal.ThelineshaperemainsessentiallyunchangedfromitsappearanceinFigure11.4.Thisresultisconsistentwithobservationsofone-dimensionalspectroscopies,forwhichquantumcorrectionsaretypicallyatfunctionsoverthewidthofvibrationalresonances17,32andonlyslightlyaectthelineshapeofthesignal.Theminimaleectofapplyingaquantumcorrectionschemetothethird-orderresponsefunctionimpliesthattheTCFtheoryiscapableofcapturingtheessentialfeaturesof2D-IRspectroscopywithoutquantumcorrection.Water'sthird-orderresponsefunctionwasalsocalculatedforseveraldierentpop-ulationtimes2,andtheresultswerefoundtobeconsistentwiththebehaviorofexistingexperimentalspectra.88AsdisplayedinFigure11.6,theo-diagonalcouplingsbecomemoreprominentaspopulationrelaxationanddephasingtakeplace.110Lookingspeci-callyattheOHstretchingregion,thespectrumcorrespondingto2=0exhibitsinho-mogeneousbroadeningintheelongationofthediagonalpeak.Whenthewaitingtimeisincreasedto48and100fs,thepeakbecomeslesselongated,suggestingrapidrelaxationandmemorylossinthesystem.Ithasbeensuggestedthatthisrelaxationcanbeat-tributedtofastlibrationalmotionshavingaperiodbetween30and90fsinthehydrogenbondnetworkofliquidwater.88InapplyingtheTCFtheoryof2D-IRspectroscopytoneatwater,asystemwhosepropertiesarewellestablishedthroughnumeroustheoreticalandexperimentalstudies,thepotentialofthetheorytocorrectlycapture2D-IRsignalandcouplingswasclearlydemonstrated.128

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Figure11.6:ThewaterrephasingsignalintheOHstretchingregionfor2=0fstop,48fsmiddleand100fsbottomisdisplayed.Theelongatedpeakfor2=0suggeststhepresenceofinhomogeneousbroadening,whichislostby50fs.129

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Figure11.7:Thesesimulationsnapshotsillustratethestretchingmodes,indicatedbyyellowvectors,of1,3-cyclohexanedione.Thetwomodesofinterestarethesymmetricstretchleftpanelat1711wavenumbersandtheantisymmetricstretchrightpanelat1735wavenumbers.11.71,3-Cyclohexanedione1,3-cyclohexanehastwocarbonylgroupswhichgivesymmetricandantisymmetricstretchingbandsat1711and1735cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1inthelinearinfraredspectrum.99Thenatureof1,3-cyclohexanedione'sbendingandstretchingmodesisdisplayedinFigure11.7.Arecent2D-IRexperimentrevealedcrosspeaksdescribingthecouplingofthesetwostretchingmodes.Thesimplicityofthismoleculeandtheavailabilityofexperi-mentalresultsmakes1,3-cyclohexanedioneanidealmodelsystemfortestingtheTCFtheoryof2D-IRspectroscopy.Moleculardynamicscalculationson1,3-cyclohexanedionesolvatedby64CDCl3moleculeswereperformedtogenerateaseriesoftime-orderedpo-sitioncongurations.Thecongurationswerestoredevery8fsinordertoresolvethefrequenciesofinterest.AmodelproposedbyDietzandHeinzinger111andApplequist130

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Figure11.8:Acalculatedlinearinfraredspectrumofsolvated1,3-cyclohexanedioneisdisplayed.Theinsethighlightsthecarbonylstretchingregion.Thesymmetricandanti-symmetricstretchingbandsappearat1705and1760wavenumbers,respectively.polarizabilities112wereemployedtomodelthesolvent,andAmber94parameters22tomodelthe1,3-cyclohexanedione.Atheoreticallinearinfraredspectrum,displayedinFigure11.8ofthesystemshowedtheexpectedsymmetricandantisymmetricstretchingbandsaround1705and1760cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Next,thetwo-dimensionalinfraredspectrumof1,3-cyclohexanedione,displayedinFigure11.9,wascalculatedwith2=0.TheFourier-Laplacetransformofthesystem'sthird-orderpolarization,calculatedundertheanharmonicapproximation,exhibitedastrongdiagonalsignalwithpeaksnear1690and1740cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,asexpectedfromexaminationofthesystem'slinearIRspectrum,andabroado-diagonalridgealong!1centeredat!3=1670cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Theappearanceofthisridgesuggeststhepresenceofcouplingbetween131

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Figure11.9:TheFourier-Laplacetransformofthethird-orderpolarizationof1,3-cyclohexanedione,calculatedundertheanharmonicapproximation,isdisplayed.Astrongdiagonalsignalwithpeaksat1690and1740wavenumbersandano-diagonalridgealong!1with!3=1670wavenumbersarepresent.thestretchingmodesandanothermodewithfrequencyof1670cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1.Afrequencysliceofthediagonal!1=!3,showninFigure11.10conrmstheexistenceofpeaksat1690and1740cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Asintheexperimentalspectrum,bothpeakshaveapositivephase.99Takingfrequencyslicesofthe2D-IRsignalhighlightso-diagonalcouplingsbe-tweenthesymmetricandasymmetricstretchingmodes.Figure11.11showsslicesalong!1with!3=1682and1723cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1.Theplotalong!1with!3=1682cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1hasastrongpositivepeakcenteredat!1=1682cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1correspondingtothesymmetricstretchingmode'sdiagonalsignal,aswellasanegativepeakatcenteredat!1=1723cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1,which132

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Figure11.10:TheFourier-Laplacetransformofthethird-orderpolarizationof1,3-cyclohexanedioneisdisplayedfor!1=!3.Thediagonalexhibitstwostrongpositivepeaksat1690and1740wavenumbers,asdoestheexperimentalspectrum.133

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suggestscouplingwiththeantisymmetricstretchingmode.Theplotalong!1with!3=1723cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1,similarly,exhibitsalargepositivediagonalpeakat!1=1723cm)]TJ/F18 7.97 Tf 6.587 0 Td[(1corre-spondingtotheantisymmetricstretchandasmallernegativepeakat!1=1682cm)]TJ/F18 7.97 Tf 6.586 0 Td[(1toindicatecouplingwiththesymmetricstretchingmode.Asintheexperimentalspectrumof1,3-cyclohexanedione,thereisachangeinphasebetweenthediagonalando-diagonalpeaks,whichindicatesthattheanglebetweenthetransitiondipolesofthesymmetricandantisymmetricstretchingmodesisninetydegrees,avaluelargerthanthemagicangleof54.7degrees.99,113,114Thecalculatedspectrumof1,3-cyclohexanedioneisinstrongagreementwiththeexperimentalspectrumand,justliketheneatwatersystem,supportsthereliabilityofthetimecorrelationfunctiontheoryoftwo-dimensionalinfraredspectroscopy.134

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Figure11.11:Twofrequencyslicesof1,3-cyclohexanedione'sthird-orderpolarizationaredisplayed.Theslicesaretakenalong!1with!3=1682toppaneland1723bottompanelwavenumbers.Thesliceshighlighttheo-diagonalcouplingsbetweenthesymmet-ricandantisymmetricstretchingmodes.135

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Chapter12ConclusionsInthisdissertation,thestrongpotentialofcomputationallytractablemoleculardynamicsandtimecorrelationfunctiontechniquestoprovidemicroscopicunderstandingofexperimentalresultshasbeenclearlydemonstratedviatheresultsoftwoseparateinvestigations,thecalculationofmolecularvolumesusingmoleculardynamicstechniquesandthederivationofatimecorrelationfunctiontheoryoftwo-dimensionalinfraredspectroscopy.Isobaric-isothermalNPTmoleculardynamicssimulationscaneectivelybeusedtocalculatetime-dependentmolecularvolumesinconjunctionwithphotothermalexper-iments,suchasphotoacousticcalorimetryPAC,asdemonstratedintehcaseofthetrans)]TJ/F20 11.955 Tf 13.246 0 Td[(cisisomerizationofazobenzene26andthesolvationofanionicmethane20,25.Thedeterminationofastatisticallysignicantvolumechangeusingthesecomputationalmethodscanpotentiallyidentifythepresenceofshort-livedmetastableintermediatesas-sociatedwithchemicalprocessessuchaspeptidefolding.Computationaltoolsincludingsimulationsnapshotsandradialdistributionfunctionsimpartdetailedmicroscopicknowl-edgeofthestructuralchangesanddynamicsaccompanyingmolecularvolumechanges.ThetheoreticalinvestigationsoutlinedinChapters3and4corroborateexperimental136

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resultsandadditionallyprovideanatomisticlevelofunderstandingoftheforcesandstructuralfactorsthatdrivemolecularvolumechanges.Thissynergisticapproachtode-terminingandinterpretingmolecularvolumechangesmayeasilybeappliedtocomplexbiomoleculesandotherinterestingsystems.Thedevelopmentofatimecorrelationfunctiontheoryoftwo-dimensionalinfraredD-IRspectroscopyservestodecreaseacomplicatedquantummechanicalproblemtothecalculationofasingleclassicaltimecorrelationfunction,aquantitythatcanbecomputedusingmoleculardynamicstechniques.Whileanexactexpressionforthethird-orderresponsefunction,whichdetermines2D-IRsignal,couldnotbederived,theinvo-cationofharmonicandanharmonicapproximationsyieldedcredibletheoretical2D-IRspectraofneatwaterand1,3-cyclohexanedioneindeuteratedchloroform.Preliminarycomputationalresultshintatthepromiseofthistimecorrelationfunctiontheorytore-vealanharmoniccouplingsandtime-dependentthree-dimensionalstructuresoforganicmoleculesandpeptides,astheexperimentsdo.Accurateprocessingofcalculatedtimecorrelationfunctionsandtheapplicationofthetheoryinthreedimensionsarecontinuingareasofresearchwhichareexpectedtodramaticallyimprovethereliabilityandclarityofcalculatedspectra.Onceasolidtheoreticalfoundationfordetermining2D-IRspec-traisestablished,itwillbeahelpfultoolforreproducingandexaminingtheresultsofcurrent2D-IRexperimentsonamicroscopiclevel.Theabilityofmoleculardynamicscoupledwiththistimecorrelationfunctiontheoryof2D-IRspectroscopytomodelin-tricatecondensedphasesystemsmayultimatelyhelpthescienticcommunityovercomethechallengeofinterpretingthedynamicsthatdetermine2D-IRspectra.137

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Theendeavorsofcalculatingmolecularvolumeanddetermining2D-IRspectraoutlinedinthisworkindicatethestrongsynergybetweenexperimentalandtheoreticalwork.Whensolidtheoreticalmethodsaredevelopedandcarefullyrenedusingsimplemodelsystems,theymayservetopredictthebehaviorofinterestingchemicalsystemsindierentenvironments.Usingtheoryandexperimenttogether,aswiththesetworesearchprojects,isapowerfulmethodofprovidingdetailed,atomisticinterpretationsofavarietyofchemicalphenomena.138

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References[1]K.Hansen,R.Rock,R.Larsen,andS.Chan,J.Am.Chem.Soc.122,1156700.[2]R.W.LarsenandT.Langley,J.Am.Chem.Soc.121,449599.[3]R.Larsen,J.Osborne,T.Langley,andR.Gennis,J.Am.Chem.Soc.120,8887998.[4]G.J.Martyna,M.E.Tuckerman,D.J.Tobias,andM.L.Klein,Mol.Phys.87,1117996.[5]M.E.TuckermanandG.J.Martyna,J.Phys.Chem.B104,15900.[6]N.MatubayasiandR.M.Levy,J.Phys.Chem.B104,4210000.[7]L.Lockwood,P.Rossky,andR.Levy,J.Phys.Chem.B104,4210000.[8]L.LockwoodandP.Rossky,J.Am.Chem.Soc.103,198299.[9]V.DadarlatandC.Post,J.Chem.Phys.B105,7152001.[10]M.P.AllenandD.J.Tildesley,ComputerSimulationofLiquidsClarendonPress,Oxford,1989.139

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[11]R.FriedbergandJ.E.Cameron,J.Chem.Phys.52,6049970.[12]G.JacucciandA.Rahman,NuovoCimentoD4,3411984.[13]T.Hansson,C.Oostenbrink,andW.vanGunsteren,Curr.Op.Struct.Biol.12,19002.[14]T.Imai,Y.Harano,A.Kovalenko,andF.Hirata,Biopolymers59,5122001.[15]P.MooreandM.Klein,ImplementationofaGeneralIntegrationforExtendedSystemMolecularDynamics,1997.[16]M.Tuckerman,B.J.Berne,andG.J.Martyna,J.Chem.Phys.97,199092.[17]P.Moore,H.Ahlborn,andB.Space,inLiquidDynamicsExperiment,SimulationandTheory,editedbyM.D.Fayer.andJ.T.FourkasACSSymposiumSeries,NewYork,2002.[18]H.Ahlborn,X.Ji,B.Space,andP.B.Moore,J.Chem.Phys.111,10622999.[19]H.Ahlborn,X.Ji,B.Space,andP.B.Moore,J.Chem.Phys.112,8083000.[20]R.DeVane,C.R..R.W.Larsen,B.Space,P.B.Moore,andS.I.Chan,Biophys.J.85,2801003.[21]P.Chen,C.Lin,H.Jan,andS.Chan,Prot.Sci.10,179401.[22]W.D.Cornell,P.Cieplak,C.I.Bayly,I.R.Gould,J.KennethM.Merz,D.M.Ferguson,D.C.Spellmeyer,T.Fox,J.w.Caldwell,andP.A.Kollman,J.Am.Chem.Soc.117,517995.140

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[23]G.Herzberg,InfraredandRamanSpectraofPolyatomicMoleculesD.VanNos-trandCompany,Inc.,NewYork,1946.[24]E.Sigfridssonn,J.Comp.Chem.19,3771998.[25]R.vanEldik,Chem.Rev.89,5491989.[26]C.Ridley,A.C.Stern,T.Green,R.DeVane,B.Space,J.Miksosvska,andR.W.Larsen,Chem.Phys.Lett.418,1372006.[27]M.Schmidt,K.Baldridge,J.Boatz,S.Elbert,M.Gordon,J.Jensen,S.Koseki,N.Matsunaga,K.Nguyen,S.Su,T.Windus,M.Dupuis,andJ.Montgomery,J.Comp.Chem.14,1347993.[28]A.MostandandC.Romming,ActaChem.Scand.25,3561971.[29]J.Bouwstra,A.Schouten,andJ.Kroon,J.ActaCrystallogr.Sect.CCryst.Struct.Comm.39,1121983.[30]D.A.McQuarrie,StatisticalMechanicsHarperandRow,NY,NY,1976.[31]B.J.Berne,J.Jortner,andR.Gordon,J.Chem.Phys.47,160067.[32]J.Borysow,M.Moraldi,andL.Frommhold,Mol.Phys.56,9131985.[33]E.W.CastnerJr.,Y.J.Chang,Y.C.Chu.,andG.E.Walrafen,J.Chem.Phys.102,65395.[34]S.Mukamel,PrinciplesofNonlinearOpticalSpectroscopyOxfordUniversityPress,Oxford,1995.141

PAGE 153

[35]S.Mukamel,V.Khidekel,andV.Chernyak,Phys.Rev.E.53,R1996.[36]R.L.MurryandJ.T.Fourkas,J.Chem.Phys107,9726997.[37]R.L.Murry,J.T.Fourkas,andT.Keyes,J.Chem.Phys.109,281499.[38]K.OkumuraandY.Tanimura,J.Chem.Phys.107,2267997.[39]S.SaitoandI.Ohmine,J.Chem.Phys.108,2401998.[40]S.SaitoandI.Ohmine,J.Chem.Phys106,4889997.[41]S.SaitoandI.Ohmine,Phys.Rev.Lett.88,20740102.[42]J.E.B.Wilson,J.Decius,andP.Cross,MolecularVibrationsDoverPublications,Inc.,NewYork,1955.[43]A.MaandR.M.Stratt,J.Chem.Phys.116,496202.[44]R.Devane,C.Ridley,B.Space,andT.Keyes,Phys.Rev.E70,501012004.[45]R.DeVane,C.Ridley,T.Keyes,andB.Space,J.Chem.Phys.123,194507005.[46]T.KuhnandH.Schwalbe,J.AmChem.Soc.122,616900.[47]J.Balbach,J.Am.Chem.Soc.122,5887000.[48]M.Blackledge,R.Bruschweiler,J.Griesinger,J.Schmidt,P.Xu,andR.Ernst,Biochemistry32,109601993.[49]A.BaxandS.Grzesiek,Acc.Chem.Res.26,1311993.142

PAGE 154

[50]D.J.Segal,A.Bachmann,J.Hofrichter,K.O.Hodgson,S.Doniach,andT.Kiefhaber,J.Mol.Bio.288,4891999.[51]S.AraiandM.Hirai,Biophys.J.76,219299.[52]M.Cho,J.Chem.Phys.115,4424001.[53]N.-H.Ge,M.T.Zanni,andR.M.Hochstrasser,J.Phys.Chem.A106,96202.[54]S.GnanakaranandR.Hochstrasser,J.Am.Chem.Soc.123,12886001.[55]K.Kwac,H.Lee,andM.Cho,J.Chem.Phys.120,1477004.[56]C.Scheurer,A.Piryatinski,andS.Mukamel,J.Amer.Chem.Soc.123,3114001.[57]M.Khalil,N.Demirdoven,andA.Tokmako,Phys.Rev.Lett.90,047401003.[58]M.KhalilandA.Tokmako,Chem.Phys.266,2132001.[59]M.Khalil,N.Demirdoven,andA.Tokmako,J.Phys.Chem.A.107,5258003.[60]N.Demirdoven,M.Khalil,andA.Tokmako,Phys.Rev.Lett.89,237401002.[61]N.Demirdoven,M.Khalil,O.Golonzka,andA.Tokmako,J.Phys.Chem.A105,8025001.[62]O.GolonzkaandA.Tokmako,J.Chem.Phys.115,2972001.[63]O.Golonzka,M.Khalil,N.Demirdoven,andA.Tokmako,J.Chem.Phys.115,1081401.143

PAGE 155

[64]K.A.Merchant,D.E.Thompson,andM.D.Fayer,Phys.Rev.Lett.86,3899001.[65]O.Golonzka,M.Khalil,N.Demirdoven,andA.Tokmako,Phys.Rev.Lett.86,2154001.[66]P.Hamm,M.Lim,andR.M.Hochstrasser,J.Phys.Chem.B102,6123998.[67]I.V.RubtsovandR.M.Hochstrasser,J.Phys.Chem.B106,9165002.[68]I.V.Rubtsov,J.Wang,andR.M.Hochstrasser,Proc.Nat.Acad.Sci.U.S.A.100,5601003.[69]D.E.Thompson,K.A.Merchant,andM.D.Fayer,J.Chem.Phys.115,317001.[70]D.E.Thompson,K.A.Merchant,andM.D.Fayer,Chem.Phys.Lett.340,267001.[71]J.Bredenbeck,J.Helbing,R.Behrendt,C.Renner,L.Moroder,J.Wachtveitl,andP.Hamm,J.Phys.Chem.B107,8654003.[72]S.Woutersen,Y.Mu,G.Stock,andP.Hamm,Proc.Nat.Acad.Sci.U.S.A.98,1125401.[73]M.T.ZanniandR.M.Hochstrasser,Curr.Op.Struct.Bio.11,5162001.[74]S.WoutersenandP.Hamm,J.Chem.Phys.115,773701.[75]S.WoutersenandP.Hamm,J.Phys.Chem.B.104,11316000.144

PAGE 156

[76]M.Asplund,M.Zanni,andR.Hochstrasser,Proc.Natl.Acad.Sci.USA97,8219000.[77]J.BredenbeckandP.Hamm,J.Chem.Phys.119,1569003.[78]P.Hamm,M.Lim,W.F.DeGrado,andR.M.Hochstrasser,J.Chem.Phys.112,1907000.[79]P.Hamm,M.Lim,W.F.DeGrado,andR.M.Hochstrasser,Proc.Nat.Acad.Sci.U.S.A.96,2036999.[80]S.WoutersenandP.Hamm,J.Chem.Phys.114,272701.[81]M.T.Zanni,M.C.Asplund,andR.M.Hochstrasser,J.Chem.Phys.114,4579001.[82]M.T.Zanni,N.-H.Ge,Y.S.Kim,andR.M.Hochstrasser,Proc.Nat.Acad.Sci.U.S.A.98,112652001.[83]C.Fang,J.Wang,Y.Kim,A.Charnley,W.Barber-Armstrong,A.S.III,S.De-catur,andR.Hochstrasser,J.Phys.Chem.B108,10415004.[84]C.M.Cheatum,A.Tokmako,andJ.Knoester,J.Chem.Phys.120,8201004.[85]A.M.Moran,S.-M.Park,andS.Mukamel,J.Chem.Phys.118,9971003.[86]A.M.Moran,S.-M.Park,J.Dreyer,andS.Mukamel,J.Chem.Phys.118,3651002.145

PAGE 157

[87]S.Woutersen,R.Pster,P.Hamm,Y.Mu,D.S.Kosov,andG.Stock,J.Chem.Phys.117,683302.[88]M.Cowan,B.Bruner,N.Huse,J.Dwyer,B.Chugh,E.Nibbering,T.Elsaesser,andR.Miller,Nature434,19905.[89]J.B.Asbury,T.Steinel,K.Kwak,S.Corcelli,C.Lawrence,J.Skinner,andM.Fayer,J.Chem.Phys.121,124312004.[90]A.T.Krummel,P.Mukherjee,andM.T.Zanni,J.Phys.Chem.B.107,9165003.[91]O.GolonzkaandA.Tokmako,J.Chem.Phys.115,2972001.[92]A.Tokmako,J.Chem.Phys.105,196.[93]H.KimandP.J.Rossky,J.Phys.Chem.B106,8240002.[94]X.Ji,H.Ahlborn,,P.Moore,andB.Space,J.Chem.Phys.113,869300.[95]X.Ji,H.Ahlborn,B.Space,P.Moore,Y.Zhou,S.Constantine,andL.D.Ziegler,J.Chem.Phys.112,4186000.[96]R.DeVane,C.Ridley,B.Space,andT.Keyes,J.Chem.Phys.119,6073003.[97]G.Herzberg,MolecularSpectraandMolecularStructure:VolumeI-SpectraofDiatomicMoleculesD.VanNostrandCompany,Inc.,NewYork,1946.[98]R.DeVane,B.Space,A.Perry,C.Neipert,andC.Ridley,J.Chem.Phys.121,3688004.146

PAGE 158

[99]M.T.Zanni,N.-H.Ge,Y.S.Kim,andR.M.Hochstrasser,Proc.Natl.Acad.Sci.USA98,112652001.[100]W.H.Press,B.P.Flannery,S.A.Teukolsky,andW.T.Vetterling,NumericalRecipesCambridgeUniversityPress,Cambridge,1989.[101]J.Applequist,J.R.Carl,andK.-K.Fung,J.Am.Chem.Soc.94,295272.[102]H.Ahlborn,B.Space,andP.B.Moore,J.Chem.Phys.112,808300.[103]A.Perry,H.Ahlborn,P.Moore,andB.Space,J.Chem.Phys.118,8411003.[104]C.P.LawrenceandJ.L.Skinner,J.Chem.Phys.118,2642003.[105]A.Piryatinski,C.P.Lawrence,andJ.L.Skinner,J.Chem.Phys.118,96722003.[106]A.MoritaandJ.T.Hynes,J.Phys.Chem.B.106,6732002.[107]R.vanZonandJ.Schoeld,Phys.Rev.E.65,01110602.[108]A.Pakoulev,Z.Wang,Y.Pang,andD.D.Dlott,Chem.Phys.Lett380,404003.[109]Z.H.Wang,A.Pakoulev,Y.Pang,andD.D.Dlott,Chem.Phys.Lett378,281003.[110]M.Khalil,N.Demirdoven,andA.Tokmako,J.Chem.Phys.121,3622004.[111]W.DietzandK.Heinzinger,Ber.BunsengesPhys.Chem.89,96885.[112]K.BodeandJ.Applequist,J.Phys.Chem.100,178201996.147

PAGE 159

[113]M.Zanni,S.Gnanakaran,J.Stenger,andR.Hochstrasser,J.Phys.Chem.B.105,6520001.[114]R.Hochstrasser,Chem.Phys.110,501166.148

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AbouttheAuthorChristinaR.KasprzykreceivedaBachelor'sofArtsDegreefromtheUniversityofSouthFloridainMayof2000.InFall2001,sheenteredtheDoctoralprogramattheUniversityofSouthFloridaandimmediatelybeganworkincomputationalandtheoreticalchemistrywithProfessorBrianSpace.WhileinthePh.D.programattheUniversityofSouthFlorida,Ms.KasprzykwasarecipientofthePresidentialDoctoralFellowshipandtheTheodoreandVenetteAskounesAshfordDoctoralFellowshipinChemistry.Ms.Kasprzykalsoattendedthe2002Gordon-KenanChemicalPhysicsSummerSchoolandthe2005GordonResearchConferenceonLiquids,aswellasseveralnationalmeetingsoftheAmericanChemicalSociety.Shehasco-authoredeightpublications.Ms.KasprzykwillcontinuehereducationatDukeLawSchoolinFall2006.


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Practical applications of molecular dynamics techniques and time correlation function theories
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ABSTRACT: The original research outlined in this dissertation involves the use of novel theoretical and computational methods in the calculation of molecular volume changes and non-linear spectroscopic signals, specifically two-dimensional infrared (2D-IR) spectroscopy. These techniques were designed and implemented to be computationally affordable, while still providing a reliable picture of the phenomena of interest. The computational results presented demonstrate the potential of these methods to accurately describe chemically interesting systems on a molecular level. Extended system isobaric-isothermal (NPT) molecular dynamics techniques were employed to calculate the thermodynamic volumes of several simple model systems, as well as the volume change associated with the trans-cis isomerization of azobenzene, an event that has been explored experimentally using photoacoustic calorimetry (PAC). The calculated volume change was found to be in excellent agreement with the experimental result. In developing a tractable theory of two-dimensional infrared spectroscopy, the third-order response function contributing to the 2D-IR signal was derived in terms of classical time correlation functions (TCFs), entities amenable to calculation via classical molecular dynamics techniques. The application of frequency-domain detailed balance relationships, as well as harmonic and anharmonic oscillator approximations, to the third-order response function made it possible to calculate it from classical molecular dynamics trajectories. The finished theory of two-dimensional infrared spectroscopy was applied to two simple model systems, neat water and 1,3-cyclohexanedione solvated in deuterated chloroform, with encouraging preliminary results.
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Cyclohexanedione.
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