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Nuclear Wave Packet Dynamics on Folded Effective-Potential Surfaces

Permanent Link: http://ufdc.ufl.edu/UFE0043613/00001

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Title: Nuclear Wave Packet Dynamics on Folded Effective-Potential Surfaces
Physical Description: 1 online resource (113 p.)
Language: english
Creator: Hall, Benjamin T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

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Subjects / Keywords: scattering -- schrodinger -- wavepacket
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A perennial problem in quantum scattering calculations is reducing the 3N degrees of freedom (three for each atom or electron present) to a more computationally manageable number, while still retaining the information present in each degree of freedom. We present a method of extracting a folded, non-adiabatic, effective potential energy surface from Electron Nuclear Dynamics (END) trajectories; we then perform nuclear wave packet dynamics on that surface and calculate differential cross sections for two-center, one (active) electron systems. We further discuss extensions and refinements to this method.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Benjamin T Hall.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sabin, John R.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043613:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043613/00001

Material Information

Title: Nuclear Wave Packet Dynamics on Folded Effective-Potential Surfaces
Physical Description: 1 online resource (113 p.)
Language: english
Creator: Hall, Benjamin T
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: scattering -- schrodinger -- wavepacket
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A perennial problem in quantum scattering calculations is reducing the 3N degrees of freedom (three for each atom or electron present) to a more computationally manageable number, while still retaining the information present in each degree of freedom. We present a method of extracting a folded, non-adiabatic, effective potential energy surface from Electron Nuclear Dynamics (END) trajectories; we then perform nuclear wave packet dynamics on that surface and calculate differential cross sections for two-center, one (active) electron systems. We further discuss extensions and refinements to this method.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Benjamin T Hall.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Sabin, John R.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043613:00001


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NUCLEARWAVEPACKETDYNAMICSONFOLDEDEFFECTIVE-POTENTIALSURFACESByBENJAMINTHOMASHALLADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011BenjaminThomasHall 2

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ACKNOWLEDGMENTS ThanksgoouttoErikDeumensandYngveOhrnforassistanceinthinkingaboutandperformingtheresearchforthiswork.ThisworkwouldnothavebeenpossiblewithoutthecomputationalresourcesoftheUniversityofFloridaHighPerformanceComputingCenter. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 1.1HistoricalOverview ............................... 10 1.2GoalsandMotivationforPresentWork .................... 13 2REVIEWOFALTERNATECOMMONMETHODS ................. 16 2.1GeneralConsiderations ............................ 16 2.2SurfaceHopping ................................ 18 2.3Semi-classicalCoupledChannel ....................... 20 2.4Quantum-NucleiCoupledChannel ...................... 23 2.4.1PerturbedStationaryState(PSS) ................... 25 2.4.2CommonReactionCoordinate(CRC) ................ 25 2.4.3OtherMethods ............................. 27 2.5ElectronNuclearDynamics .......................... 27 2.5.1DerivationofENDEquationsofMotion ................ 28 2.5.2MinimalElectron-NuclearDynamics ................. 29 2.6UnoccupiedNiches ............................... 31 3CONSTRUCTIONOFDYNAMICPOTENTIALSURFACES ........... 33 3.1Movingbeyond(A)diabaticPotentialSurfaces ................ 33 3.1.1ASimpleExample ........................... 36 3.1.2BacktoScattering ........................... 37 3.2ReactionCoordinateDescription ....................... 38 3.3Notation ..................................... 41 3.4PotentialSurfacesforSingly-sampledRegions ............... 42 3.5PotentialSurfacesforMultiply-sampledRegions .............. 43 3.6PrescribingPotentialValuesforClassically-forbiddenRegions ....... 44 3.7NumericalConsiderations ........................... 46 3.8Absorbing(Optical)PotentialsforGridBoundaries ............. 46 4EVOLUTIONALGORITHMFORWAVEPACKETSONENDSURFACES ... 53 4.1FormalDenitionofScatteringProblem ................... 53 4.2SpatialIntegration ............................... 54 4.3Time-EvolutionMethods ............................ 56 4

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4.4AnatomyofaStep ............................... 59 4.5EnforcingContinuityattheFold ........................ 59 5CALCULATINGREACTIONCROSSSECTIONS ................. 61 5.1ClassicalScatteringCrossSections ..................... 62 5.1.1ClassicalDeectionFunctions ..................... 62 5.1.2ClassicalDifferentialCrossSections ................. 64 5.1.3TotalClassicalCrossSections ..................... 66 5.2Semi-ClassicalScatteringCrossSections .................. 67 5.2.1Potential-DependentSchiffApproximation .............. 68 5.2.2Potential-IndependentSchiff-TrujilloApproximation ......... 71 5.3QuantumScatteringCrossSections ..................... 74 5.3.1ScatteringMatrixFormalism ...................... 74 5.3.2PartialWaveExpansions ........................ 77 5.3.3CrossSectionsforTime-DependentCalculations .......... 81 6APPLICATIONOFTHEENDWAVEALGORITHMTOSIMPLESYSTEMS ... 86 6.1GeneralConsiderations ............................ 86 6.2One-DimensionalSystems .......................... 87 6.2.1AnalyticPotentials ........................... 87 6.2.2AsymmetricSquareBarrier ...................... 87 6.3Two-DimensionalSystems ........................... 89 6.3.1AnalyticPotentials ........................... 89 6.3.2H++H(1s) ............................... 90 6.3.3H+He2+ ................................ 92 6.3.4H++H(2s) ............................... 93 6.4DiscussionofResults ............................. 94 7CONCLUSIONS ................................... 104 7.1OverviewofFolded-PotentialWavePacketDynamics(FPWPD) ...... 104 7.2PerformanceofENDwave ........................... 104 7.3FurtherRenements .............................. 105 7.3.1ExtensiontoThreeorMoreSpatialDimensions ........... 105 7.3.2ExtensiontoMolecularTargets .................... 106 7.3.3ExtensiontoMultipleActiveElectrons ................ 106 7.3.4ImplementationImprovements .................... 107 7.4FinalThoughts ................................. 109 REFERENCES ....................................... 110 BIOGRAPHICALSKETCH ................................ 113 5

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LISTOFFIGURES Figure page 3-1One-dimensionalscatteringpotentials ....................... 49 3-2Transmissionandreectionfromsquarebarrier .................. 49 3-3Sketchofscatteringinspace-xedcoordinates .................. 50 3-4Sketchofscatteringinreactioncoordinates .................... 50 3-5Single-trajectorypotentialcurve ........................... 51 3-6He2++Htransformedpotentialsurface ...................... 52 5-1Schematicofclassicalscatteringtrajectories ................... 84 5-2Hard-spheredeectionfunction ........................... 84 5-3Morse-potentialdeectionfunction ......................... 85 5-4Differentialcrosssectionforhardspherescattering ................ 85 6-1Transmissionthroughsquarebarrier(planewave) ................ 95 6-2Transmissionthroughsquarebarrier(wavepacketandcalculated) ....... 95 6-3ProbabilityofTransfer(1DYukawapotentials) ................... 96 6-4ElasticMomentumDistribution(2DYukawapotentials) .............. 96 6-5TransferMomentumDistribution(2DYukawapotentials) ............. 97 6-6ElasticDCS(2DYukawapotential) ......................... 97 6-7EffectivepotentialsurfaceforH++H(1s)(10eVcollisionenergy) ........ 98 6-8Scatteredmomentumdistribution(H++H(1s)) .................. 98 6-9ElasticDCS(H++H(1s)) .............................. 99 6-10EffectivepotentialsurfaceforHe2++H(1s)(10eVcollisionenergy) ....... 99 6-11Scatteredmomentumdistribution(He2++H) ................... 100 6-12ElasticDCS(He2++H(1s)) ............................. 100 6-13TunnellingpotentialsurfaceforH++H(2s) ..................... 101 6-14LowerfoldedpotentialsurfaceforH++H(2s) ................... 101 6-15UpperfoldedpotentialsurfaceforH++H(2s) ................... 102 6

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6-16Elasticscatteredmomentumdistribution(H++H(2s)) .............. 102 6-17Transferredmomentumdistribution(H++H(2s)) ................. 103 6-18DifferentialCrossSection(H++H(2s)) ....................... 103 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNUCLEARWAVEPACKETDYNAMICSONFOLDEDEFFECTIVE-POTENTIALSURFACESByBenjaminThomasHallDecember2011Chair:JohnRSabinMajor:Physics Aperennialprobleminquantumscatteringcalculationsisreducingthe3Ndegreesoffreedom(threeforeachatomorelectronpresent)toamorecomputationallymanageablenumber,whilestillretainingtheinformationpresentineachdegreeoffreedom.Wepresentamethodofextractingafolded,non-adiabatic,effectivepotentialenergysurfacefromElectronNuclearDynamics(END)trajectories;wethenperformnuclearwavepacketdynamicsonthatsurfaceandcalculatedifferentialcrosssectionsfortwo-center,one(active)electronsystems.Wefurtherdiscussextensionsandrenementstothismethod. 8

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CHAPTER1INTRODUCTION Quantumscatteringcalculationsinvolvingheavyprojectileshaveonlyrarelybeencompletedusingthefully-coupledSchrodingerEquation,ineitheritstime-dependent(TDSE)ortime-independent(TISE)form.Instead,almostallapproachesapproximatesomepartoftheproblem.CommonapproximationsincludethosebasedontheBorn-Oppenheimer(BO)separationofslowly-varying(nuclear)andquickly-varying(electronic)degreesoffreedom[ 1 2 3 4 5 ]and,morerecently,theapproachofDeumensandco-workerstocalculatethefully-coupleddynamicsofelectronswithclassicalnuclei.TheBO-basedmethodswillbereferredtocollectivelyasPre-computedPotentialEnergySurface(PPES)methods,whilethelattergoesbythenameofElectronNuclearDynamics(END)andisthestartingpointforthepresentwork. Itmustbestressedatthispointthatalltheapproximationschemes,withintheirregionofvalidity,mustapproachthesamevalues.Athighenoughenergies(usuallybetweenabout100eV/amuandtheionizationthreshold),substantialagreementisfoundbetweenallofthecommonmethodsandimplementations.Atthoseenergies,choiceofmethodislefttoindividualmethodologicalpreference.Atlowenergies(fromafewtensofeV/amudowntothermalenergies)thisbroadagreementislost.Themethodsdivergefromeachother(oftenevenqualitatively);thisdisagreementisexacerbatedbytheabsenceofreliableexperimentaldata,especiallyfordifferentialelectron-transfercrosssections.Again,thisisduetoabreakdowninthefundamentalassumptionsoftheapproximation:theBOseparation(usedbyPPESmethods)canonlybereliablytruncatedtoloworderforhighcollisionenergies[ 6 ],whilethe(necessary)restrictiontoclassicalnucleirestrictstheaccuracyofENDcalculations. Atthecore,alloftheapproximationschemesandmodelsreducetotheneedtoreducethedimensionalityoftheproblem.Fully-coupledTDSErequires3N+1degrees(threeforeachnucleusorelectronplusonetimecoordinate).Transforming 9

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tocenter-of-momentumcoordinatesallowsonetoneglectthethreedegreesoffreedomrelatingtothecenterofmass;incorporationofsymmetrycanfurtherreducethedimensionalityofthenucleardegreesoffreedom.Forthesimplestcollisionsystems,thoseconsistingoftwonucleiandoneelectron(e.g.,H++H),thisreductionstillleaves6degreesoffreedom:2nuclear,3electronic,and1fortime.Morecomplexsystemshavemuchhigherdimensionality.ReductionofdimensionalityisrequiredasthecomputationalcomplexityoftheSchrodingerequationscalesexponentiallywiththedimensionality.Thisreductionof(electronic)dimensionalityisaccomplishedinPPESmethodsbyexpansioninadiabaticordiabaticstates,leadingtocoupledequationswheretheonlygriddedcoordinateisthereactioncoordinate.ForEND,theelectronicdimensionalityishandledviatheexpansionincoherentstates,butatthecostofrequiringclassicalnuclearmotion. 1.1HistoricalOverview Thefoundationsofquantumscatteringtheoryareasoldasquantummechanicsitself;Rutherford'smodeloftheatom,whichleddirectlytoBohr'squantizedmodel,wasbasedonascatteringexperimentinvolving-particlesandgoldfoil.Assuchamatureeld,however,itseemsthatalltheinterestingproblemseitherhavebeensolvedorhavebeenshowntobeunsolvable.Thissuppositionisnotcorrectinthelastfewdecadesmanyproblemsthatwerecomputationallyintractablehavebeenexploredandmanymoreremaintobeinvestigated.Mostoftheexpansionhasbeencourtesyoftheexplosionofcomputationalpoweravailabletoresearchers.Despitethereadyavailabilityofcomputationalresources,thetechniquesusedthroughouttheatomicandmolecularscatteringcommunityremainlargelybasedonthosemethodsdevelopedwhensuchresourcesweresparseornon-existent.Thissectionwillprovideabird's-eyeviewofthemethodsusedinscatteringtheorycurrentlyandattempttotiethembacktotheirhistoricalroots.MoremathematicalandpracticaldetailsaboutthemethodsdescribedherecanbefoundinChapter 2 andthereferencescontainedtherein. 10

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Atitsmostbasiclevel,quantumscatteringtheoryconcernsitselfwithcollisionalchangesinthequantumstatesofatoms,ionsandmolecules.Thesechangescanbeelectronic(electronicexcitations,rearrangements,ejectionorabsorption,etc.)ornuclear(bondbreakingandrearrangement,roto-vibrationalexcitationsorde-excitations,etc.)Sometimesweareinterestedintheangulardistributionofthescatteredparticles;atothertimeswewantprobabilitiesforvariousstatechangestooccur.Thecommonfactorinallofthisvarietyisthatalloftheseeffectshappenoutofequilibrium;boundstatemethodsareoflessapplicability.Thisimposesspecialconsiderationsinthestructuringandsolutionoftheproblems. Mostearlytheoreticalscatteringworkwasintheeldofnon-reactivecollisions(i.e.wherethecollisionpartnersexitinthesamephysicalcongurations,nobondsarebrokennorelectronstransferred),largelywithrigidrotorscollidingwithsurfacesorstructurelesspointparticles[ 7 ].Thesecalculationscouldbedone(forlimitedsetsofavailableroto-vibrationalstatesorotherspecialcases)analyticallybyexpandingtheprojectilewavefunctioninangularmomentumeigenstatesandsolvingthecoupledequationsderivedtherebyfromtheSchrodingerequation.Manyoftheimportanttheoreticaltoolsusednowweredevelopedinthisearlyperiod(between1920and1960),butlayfallowuntiltheadventofscienticcomputing.BornandOppenheimershowedthatitispossibletoseparatefast(electronic)motionfromslow(nuclear)motionin1927[ 8 ];thiswasformalizedandlledoutbyEckartin1935[ 9 ].Bythemid-1950smanyoftheseminaltextsbyauthorssuchasMohrandMassey[ 7 10 ],Landau[ 11 ],andZener[ 12 ]werewrittenandinuse;thecomputationallimitationsofthatera'stechnologyforcedtheadoptionofseveralapproximationsandmethodsthatstillcolormodernscatteringtheory. Adominatingapproximationthathasitsrootsinthisearlyeraistheuseofadiabatic(andnowdiabatic)potentialsurfaces.Theconstructionofpotentialenergysurfaceswaspossiblebyxingthenucleiandonlyconsideringtheelectroniceigenfunctionsateach 11

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discreteconguration.Eacheigenstatecorrespondstoadifferent,independentpotentialenergysurface.Formallythisisnotanapproximationifthecomplete(innite)setofeigenstatesisused;inpracticeonlyatruncatedsetcanbeused(seeChapter 2 formoredetail).Rearrangementcollisionsrequireachangeofstateandthusnon-adiabaticbehaviorthisisintroducedviaoff-diagonalHamiltonianelementscalledcouplingtermswhichmediatetransfersbetweensurfaces.ThisisthestartingpointforallPPESmethods. Withtheuseofadiabaticpotentialenergysurfacescomestheneedforabettercoordinatesysteminwhichthevarioustypesofmotioncouldbeefcientlydecoupled.Thisledtothedevelopmentanduseofinternalcoordinates(andthusangularmomentumeigenstates)foreachcollisionpartner,withtheinter-partnerdistancebeingthesolecontinuouscoordinate.Inthelanguageofchemicalreactions,theinternucleardistanceisthereactioncoordinateforthescatteringprocess.Allotherareexpandedindiscretestates(e.g.,partialwavesfortherotationalcoordinates).Theexpansioninpartialwavesrequiressphericalcentralscatteringpotentials,arestrictionobservedmainlyinthebreach.AsshowninSection 2.3 ,usingtheinternucleardistanceasthereactioncoordinateleadstoinsuperabledifcultiesforelectrontransferproblemswithoutmomentuminthewavefunctionthephysicallocalizationoftheelectronsatasymptoticseparationscannotbemaintained.UseofElectronTranslationFactors(ETFs)andswitchingfunctions(againfordetailsseeSections 2.3 and 2.4 )canalleviatethedifcultiesatthecostofsignicantlycomplexifyingthecoupledequationsforthenuclearwavefunctionsaswellasbeinganadhoc,system-dependentcorrection.Thereactioncoordinatedescriptioncarriesmostoftheintuitivesimplicityoftheclassicone-dimensionaltunnellingmodels,atthecostofhavingasystem-dependentcoordinatesystemwhichrequiresapproximationstotheunderlyingequationsofmotionandaprioriknowledgeofthenaloutcomes.Thisgreatlyrestrictstherangeofapplicabilityofthetheory;extendingittonewsystemsrequiressignicant 12

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trialanderrorandaestheticchoicesbytheresearcher.Comparabilitybetweendifferentcalculationsisthushindered. Unsatisedwiththeadhocnatureofmostscatteringcalculations,Deumens,Ohrnandco-workersbeganinthemid-1990stoderiveanexplicitlytime-dependant,non-potential-surface-basedtheorythatwouldincludethefullinteractionsbetweennucleiandelectronswithoutusingtheBorn-Oppenheimerapproximation.Startingwiththetime-dependentvariationalprincipleleadstoaEND,atheorythatincorporateselectronicmomentum(andsosolvesthelocalizationproblem)andnon-adiabatictransitionsinspace-xedCartesiancoordinates.Nospecialsystem-dependentcoordinatesareneeded,nocouplingelementsneedbecomputed,andthetheoryextendswithoutsubstantialmodicationtoanynon-relativisticcollisionsystem(exceptionization,whichnoneofthecommonmethodshandlewell)[ 13 ]. Electron-NuclearDynamicsdoeshaveonecaveatthough,whichlimitsitsaccuracyatlowcollisionenergies.Tomaintainthefullcouplingbetweennucleiandelectrons,thenucleimustberestrictedtobehavingasclassicalparticles(formally,thezero-widthlimitofGaussianwavepackets).Thetrajectories(unlikethoseofclassical-trajectoryCloseCoupling,describedinSection 2.3 )arecomputeddynamicallyfromtheinstantaneousforcesexertedbytheelectronsandtheothernuclei.Thisallowsunrestrictedfragmentation(withouthavingtospecifythedesiredchannelsaheadoftime),butlimitstheaccuracyoftheangulardistributionsofthescatteredparticlesasnointerferenceispossiblebetweenadjacentclassicaltrajectories. 1.2GoalsandMotivationforPresentWork TheprimarygoalofthecurrentworkistodeneapathtoremovingthelimitationofENDtoclassicalnuclei.Wewanttobringbackinthenon-classicalphysicsseenineventhesimplesttunnellingcalculationsandjoinittotheelegantelectronicbehaviordescribedbyEND.Hereweproposeanalternateviewpointonthequantumscatteringproblemandtestareferenceimplementationofthisframework.Muchistobemade 13

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ofthefactthatthecurrentworkcombinestheintuitiveconceptualideasofthereactioncoordinatedescriptionwiththeeleganceandsimplicityoftheCartesiancalculations.Thenumericalimplementationshouldnotbetakenasnal;norshouldtheresultsbeunderstoodasanythingotherthansuggestiveoftheaccuracypossiblewiththepresentmethod. Whilepotentialenergysurfacesaregenerallytakentobeadiabatic(ordiabaticatmost),itispossibletodeneaneffectivepotentialenergysurface.Thisisthepotentialenergyactuallyexperiencedateachpointinspace,includingtheinstantaneousnon-adiabaticeffects;itisalinearcombinationofthepossibleelectronicenergiesplusthenuclearrepulsion.Thefully-coupledquantum-nuclearversionofthiseffectivepotentialcaninprinciplebeconstructed;inpracticeitisprohibitivelydifcultandamountstosolvingtheentirescatteringprobleminonego.Instead,justasadiabaticpotentialsurfacesarecreatedbytakingxednucleargeometriesandsolvingtheelectroniceigenvalueproblemateachgeometry,onecouldconstructtheeffectivepotentialenergysurfacefromasetofdynamic(time-dependent)trajectoriesofclassicalnucleirespondingtoinstantaneousforces.ENDprovidesthesetrajectoriesandwhiletheelectronicenergyisnotneededasaninputforEND,itcanbecalculatedasanoutputalongeachtrajectory.Theclassicaltrajectoriesthenserveassamplingpointsontheeffectivepotentialenergysurface.Itmustbestressedthatthissurfaceisnotadiabatic.Infact,itcannotbeeasilydecomposedintopurelyadiabatictermsitnaturallyincludestheexcitations.SincethechangeintheENDelectronicstateissmooth,thesurfacesgeneratedtherebywillalsobesmooth.Theextractionandcharacterizationofthesesurfacesistheprimaryfocusofthepresentwork;weshowthattheyintrinsicallyincorporatealloftheinformationofthereaction-coordinatedescriptionbycreatingacomplicatedtopologyinspace-xedrelativecoordinates.Wetalkofthesesurfacesashavingbeenfoldedsothatatvarioustimesinthehistoryofeachtrajectorythesystemenergymayhavedifferentvaluesateachspatialpoint.Thisfoldingisanalogousto 14

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theseparateadiabaticsurfaces,butthefoldedsurfacesareindependentexceptatthefold,insteadofbeingseparatesolutionsthatarecoupledeverywherebyoff-diagonalHamiltonianmatrixelements. Oncethepotentialenergysurfacesareconstructed,weusestandardwavepacketpropagationtoolstoimplementacomputerprogramforevolvinganuclearwavepacketonthefoldedeffectivepotentialsurfacesandtheextractionofdifferentialcrosssections.HereafterwecallthisreferenceimplementationENDwave.WritteninPython[ 14 ]usingtheScipy[ 15 ](includingNumpy)numericallibraries,ENDwaveisarstattemptatimplementingtheideasofthisworkinausableform.WeshowexamplesofboththeeffectivepotentialenergysurfacesandthedifferentialandtotalcrosssectionsoutputfromENDwaveforseveralsimple(two-center,one-effective-electronsystems).Furtherworkisindicatedtoimprovenumericalaccuracy,easeofuse,computationalefciency,andrangeofapplicabilityofENDwave. 15

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CHAPTER2REVIEWOFALTERNATECOMMONMETHODS Note:allequationsarepresentedinatomicunits.WeareinterestedinsolvingtheTime-DependantSchrodingerEquation(TDSE) i@ @t)]TJ /F4 11.955 Tf 14.99 3.02 Td[(^H(~R;t)=0(2) inawaythatisnumerically-tractable.InEquation 2 ,^H=^T+^VisthesystemHamiltonian,composedofthekineticenergyoperatoranda(globallyknown)potentialenergyoperator. Inexplicitform,thekineticenergyoperatorofEquation 2 forasingleparticlecanbewrittenas1 2mr2.Thebehaviorofthecollisionpartnerisassumedtobecontainedinthepotential.Thus,theTDSEisasecond-orderpartialdifferentialequation.Multiplemethodsexistfordealingwitheachofthederivatives;theintegratorchosenforthespatialderivativesis(inprinciple)independentofthemethodchosentotime-evolvethesolution. 2.1GeneralConsiderations Beforeanydiscussionofthedetailsofmethodsandapproximationstothescatteringproblemwillbeuseful,thenatureoftheproblemunderconsiderationandideascommontoallsuchmethodsmustbedened.Thatisthepurposeofthissectionandsubsections.Thefollowingsectionsdiscussmethodsandapproximationscommonlyusedinthesolutionofthescatteringproblem. Thegeneralscatteringproblemisdenedasfollows:Considertwoentities(atoms,ions,ormolecules)AandBinitiallyatasufcientlylargedistancefromeachotherthattheymaybeconsiderednon-interactive(thiswillbecalledtheasymptoticlimitinthefollowing).Beforetheybegininteracting,bothentitiesareinstationarystatesoraknownsuperpositionofstationarystates.ThisistheinitialstateIofthesystem.Oneorbothoftheentitiesisthengivensomeinitialmomentum~kidirectedtowards 16

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theotherpartner.TheresultingkineticenergyisthecollisionenergyE,sincetheinteractionpotentialenergycanbesettozeroasymptotically,andisconservedduringtheinteraction.Theentitiesapproachoneanotherandscatteroffthemutualpotential(orequivalentlyexperienceforcesduetotheother'scharge,etc.).ThisscatteringresultsinadistributionofnalstatesfJg.Ingeneral,allthefollowingmethodsattempttoconstructtheprobabilityamplitudeP(I!J)ofndingthescatteredsysteminaparticularstateJstartingfromaninitialstateI. TheprimarydivisioninmethodsisbetweenthemethodsthatrequireprecomputedelectronicPotentialEnergySurfaces(PES)andthosethatdonot.AllofthemethodshereindescribedrequirePESexceptforElectron-NuclearDynamics(END).ThesesurfacesarecomputedwiththeelectronicHamiltonianforxednucleiatasetofnucleargeometriesusingveryhighlevelmethods(MRSCF,CASSCF,CCSD(T),etc)andtheneitherttoananalyticformorinterpolatedusingsplines.AllmethodsthatrelyonPESareexquisitelysensitivetosmalldeviationsinthepotentialsurfaces,especiallyinregionswheretheelectronicstatesareclose-lying,givingrisetolargeoff-diagonalHamiltonianmatrixelements.Nuclearmotionisaddedbackinafterthetheelectronicbehaviorissolved,leadingtonucleardynamicson(coupled)potentialsurfaces.Thesesystemscanbewritteneitherastime-independentsteady-stateorastime-dependentdynamicsystemsofequations. Non-PESmethodssuchasENDattempttoincludethefullcouplingbetweennucleiandelectronsandthusaresingle-passsolutions.Thenucleimoveintheeldcreatedbythereal-timemotionsoftheelectrons.Unfortunatelythiscalculationrequirestheuseofclassicalnuclei,thatisnucleirepresentedbythezero-widthlimitofGaussianpackets.ThusENDcanbedescribedasdescribingthescatteringofclassicalnucleicoupledtoquantum-mechanicalelectrons.ENDisinherentlydynamic. 17

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2.2SurfaceHopping ExplanationsinthissectionfollowthoseofDoltsinis[ 16 ].ThesimplestPESmethodiscalledsurfacehopping.Inmostimplementations,thetotalwavefunctionisexpandedinatruncatedbasisoftheelectronicstatesj(r;R)ofthequasi-moleculeABateachnucleargeometryR.Thenucleiarepresumedtoactclassically,followingaprescribedtrajectoryR(t).Thistrajectoryiseitherastraightline(theimpulseapproximation),acoulombtrajectory,orsomeothertrajectoryparametrizedforeasysolutions.Thetrajectoriesaregenerallydenedwithoutreferencetothespecicdynamicsofthesystembeingstudied.Theresultinganzatzforthewavefunctionis: (r;t)=Xjaj(t)j(r;R)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(iZEj(Rdt)):(2) TheexpansioncoefcientsajinEquation 2 arecalculatedbyinsertingtheanzatzintotheTime-DependentSchrodingerequation(TDSE),multiplicationbyiandintegrationoverelectroniccoordinates.Thisresultsinthecoupledequations_aj=)]TJ /F9 11.955 Tf 11.29 11.36 Td[(XjajCije)]TJ /F10 7.97 Tf 6.59 0 Td[(iR(Ej)]TJ /F10 7.97 Tf 6.58 0 Td[(Ei)dt (2)Cijhij@ @tjji: (2) Atthispoint,achoicemustbemade.Thesystemcanbepropagated(bysolutionofNewton'sequationsofmotion)onthesurfacedenedbythepopulation-weightedaverageenergy.Thisleadstothemean-eldorEhrenfestapproach.Intheory,forthismethodnoexpansionintoadiabaticstatesisnecessarytheaverageenergycanbecomputedon-the-yduringthenumericalevolutionofNewton'sequationsofmotion(Eq 2 ):MKRK=FK(R) (2)FK(R)=rKEi(R): (2) 18

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Themean-eldapproachhasbothadvantagesanddisadvantages.Advantagesincludesimplicityofimplementation,conceptualsimplicity,andrelativelylowcomputationalcost.Ithascriticalawsthough:First,theon-the-yversionislimitedtosemi-classicalnuclei.Second,themean-eldnaturemeansthatsparsely-occupiedchannelswillbeundersampledandtheaveragepathwillbeverydifferentfromthetruepathforthatchannel.Third,contributionsfromenergeticallyinaccessiblechannelsfrequentlycontaminatethetotalwavefunction(whichisunphysical).Fourth,althoughmoreminorfromourpointofview,apureadiabaticstatethatpassesthrougharegionofstrongcouplingwillexitinamixed(ienon-stationary)state,whichmanyresearchersconsiderundesirable.Lastly,theEhrenfest(mean-eld)approachdoesnotsatisfymicroscopicreversibility. Amoresophisticatedchoiceissurfacehopping.Insteadofevolvingsolelyononeaveragedsurface,theenergy(Ei)usedinEquation 2 isallowedtohopfromonechanneltoanother,basedonthemagnitudeofthecouplingelementsandaprinciplecalledthefewest-switchescriterion.Thisprovidesameaningfortheprobabilitytohopfromsurfaceitosurfacejatanytime:deneijaiaj (2)thenPij(t;t)=2
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thatthemoredetailedmethodsbelowcannothandle.Themajordrawbackisthelackofaccuracywhencomparedtocoupled-channelorothernon-statisticalmethods.Thislackofaccuracyisevenworseastheenergydecreases.Surfacehoppingisdynamicinnature:theequationsofmotionareintegratedwithrespecttotimefromthedistantpasttothedistancefuture. 2.3Semi-classicalCoupledChannel ThetreatmentinthissectionisadaptedfromDelos[ 1 ]Thenextrenementofthebasicmechanismusedbysurfacehoppingisthesemi-classicalcoupledchannelapproach,alsocalledclassical-trajectorycoupledchannel(CTCC).StartingfromaverysimilaranzatztoEquation 2 ,theinstantaneouselectronicstateji(r;R)iandenergiesiarecalculatedalongapre-denedtrajectory,formingasetofenergiesandstates.Thetotalstateisexpandedinthebasisoftheseadiabaticelectronicstates: (r;R)=Xkdk(t)Fkk(r;R)(2) whereFkaretheelectron-translationfactors,givenby(formolecularstatesandrectilineartrajectories)Fk=exp(imvsk(r;R)) (2)sk(r;R)=1 2[fk(r;R)+]r)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 8(1)]TJ /F5 11.955 Tf 11.96 0 Td[(2)R: (2) ThefunctionsfkinEquation 2 areswitchingfunctionswhichvarysmoothlywithelectroniccoordinaterandswitchtheelectronfromonecentertotheother.Noconsensusexistsastothebestformfortheswitchingfunctions. Uponsubstitutingtheanzatz(Equation 2 )intotheTDSE,multiplyingbyjFjandintegratingoverelectroniccoordinates,weextractthecoupleddifferentialequationsforthenuclearcoefcientsdk(t): i_dk=XjSjk)]TJ /F3 7.97 Tf 6.59 0 Td[(1[hjk+v(Pjk+Ajk+jk)]dj;(2) 20

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wherehjk=kjk (2)Sjk=hjjki (2)Pjk=hjj)]TJ /F5 11.955 Tf 17.94 0 Td[(irRjki (2)Ajk==hjj[h;sk]jki (2)=)]TJ /F5 11.955 Tf 11.95 0 Td[(S)]TJ /F3 7.97 Tf 6.58 0 Td[(1h (2)jk==hjj(sk)]TJ /F11 11.955 Tf 11.96 0 Td[(sjhjki (2)jk==hjjsk)]TJ /F11 11.955 Tf 11.96 0 Td[(sjjki: (2) Alltermsappearingintheseequationsareassumedtobetruncatedtolinearpowerinv(asthisisderivedforslowcollisions).Forcollisionsathigherenergies,differentapproximationsarerequired.Essentially,Pcontainsthecouplingsbetweenstates,Acancelsthetranslation-dependentpieces(thatwouldotherwiseresultinspuriousasymptoticcouplings,violationsoftranslationalinvariance,etc),andisacorrectiontotheHamiltonian,accuratetorst-orderinvelocity. Todeterminethescatteringprobabilitiesandcross-sections,Equations. 2 areintegratedusingstandardtechniquesfromt=0;R=)]TJ /F5 11.955 Tf 9.29 0 Td[(R0tot=Tfinal;R=+R0whereR0islocatedwelloutsidetherangeofinteractionofthepotential.Theintegralstate-selectivecross-sectioncanbethenextractedfromtheopticaltheorem (I!J)=4 k=(dJ)j=0:(2) Angular-differentialcross-sectionsareveryrarelyreported,althoughtheycanbederived(e.g.,fromtheSchiffapproximation).Thisisaconsequenceofusingpre-denedtrajectorieswhichingeneralarenotreectiveofthetruetrajectoriesinthesemi-classicallimit. 21

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Afundamentaldifcultyofcoupledchannelcalculations(bothclassical-trajectoryandquantum-nuclearversions)istheselectionofchannelstoincludeintheexpansion(Equation 2 ).Intheory,onecouldincludeallmolecularelectronicstatesfoundbytheelectronicstructurecalculations.Thisisgenerallyimpracticalduetothepoor(_N3)scalingwithtotalchannels.Anothersimpleoptionistoincludeallopen(asymptoticallyaccessible)channelsandnoneoftheclosed(asymptoticallyinaccessible)channels.Thisisproblematicintworespects:First,thischoicefrequentlystillincludesanimpracticalnumberofchannels,includingchannelsthathaveverylowpopulationsasymptotically.Thisneedlesslyincreasescomputationalcost.Second,ithasbeenfrequentlynotedthatforsomemethods(andsystems)someclosedchannelsareimportanttoensureconvergence.Thismeansthatwithoutpriorknowledgeoftheimportantchannelsinthescatteringprocessunderconsideration,channelsmustbechosenbytrial-and-error,increasingthesizeofthebasissetfromlowenergyupwardsuntilsatisfactoryconvergenceisreached.Forsomesystemsandprocesses,asfewasthreestatesareneeded;forothersasmanyasseveralhundredareneeded[ 3 17 ]. SeveralpossibletrajectoriescanbeusedforCTCC,eachofwhichgivesslightlydifferentresults.Forthisreason,oneofthemodelconsiderationsiswhichtrajectorytouse.Themostcommonchoicefornon-Coulomb-interactingsystemsisthestraight-linetrajectory.Athighvaluesofmomentum,thisisareasonableapproximationtothemotionoftheheavycollisionpartner.Forequal-masssystems,oratlowenergies,thetruesemi-classicaltrajectoriesdeviatemarkedlyfromstraightline.Thisleadstoimpropersamplingofthereactionspace,andsotheelectronicdynamicsdeviatefromthecorrectvaluesastheenergydecreases.Thesamearguments,andproblems,occurwiththeCoulombtrajectoriesusedforcharge-chargeinteractions.Morecomplicatedtrajectoriescanbeused,butatthecostofmakingthecoupledequations(Equation 2 )muchmorecomplicatedanddifculttosolve,asthevelocitiesusedintheETFsarenolongerconstantineitherdirectionormagnitude. 22

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Asageneralrule,trajectoriesforlargeimpactparameterarewell-describedbyeitherstraightlinesorCoulombtrajectories.Fortwo-center(ionoratomonionoratom),theintegralcross-sectioncontributionsfromeachimpactparametergoasP(b)b.Thusthesmallimpactparameters,whichdiffergreatlyfromastraightline,areweightedagainstinthecross-sectionintegral.Thus,aslongasthedominantinteractionstayseitherCoulombicornon-Coulombic,xedtrajectorieswillreasonablyapproximatetheintegralcross-section.ThiscanbeseeninRefs.[ 18 19 ],wherethemoreaccuratesemi-classicaltrajectoriesderivedfromtheElectron-NuclearDynamicsformalismyieldresultsthatagreewellwiththexed-trajectoryCTCCcalculations. Thexed-trajectoryapproximationsufferswhenmorecomplicatedsystemsareinvolved.Oneexampleisacollisionofanatomorionwithamolecule(eg.H++CH4).Fortrajectoriesfaroutsidethebond-lengthofthemolecule,thepotentialis(roughly)spherical,sostraight-linetrajectoriesarene.Howevertrajectorieswithimpactparameterssmallerthan(orequalto)thebondlengthwillbesubstantiallydistorted,butstillnotbeweightedoutbytheintegral.Similareffectshappenwithmultiply-chargedionsasprojectiles(eg.C3++H!C2++H+).Here,thedominantinteractionchangesfromacharge-dipole-typepotentialtoaCoulombicpotentialatsomepointwithinthecalculation;thepointoftransitionwilldependonenergy,impactparameter,andthedetailsofthecalculationinawaythatisdifcultifnotimpossibletopredictapriori.Thus,thetruetrajectorieswillswitchfromstraight-linetoCoulombicatmost,ifnotall,impactparameters,badlydistortingP(b)forallb. 2.4Quantum-NucleiCoupledChannel ThenextBorn-Oppenheimer-basedapproximationbeyondCTCCisquantumMolecularOrbitalCoupledChannel(MOCC).Onceasetofpotentialsurfacesisavailableforagivensystem,thetotalwave-functionforthesystemisexpandedinthestationary-statemolecularorbitalsn(r;R)forelectronsandpartialwavesFn(R)fornucleiinthefollowingmanner(whererandRaretheelectronicandnuclearcoordinates 23

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respectively)[ 1 ]: (r;R)=XnFn(R)n(r;R):(2) Oncetheequationsareexpressedintheappropriatecoordinates(oftenaJacobisystemadaptedtotheinitialconditionsofthecollision),theelectroniccoordinatesareintegratedout,leavingmatrixelementswhichcoupleelectronicstates.Theinnerproductsbetweenthemolecularorbitalsn(r)andthesystemHamiltonianH(r),denotedhn(r;R)jH(r;R)jm(r;R)iareelementsoftheaptlynamedcouplingmatrix,andcontainallrequiredelectronicinformationforthecollision.Notethatoncethecouplingmatrixelementsareobtained,nofurtherelectronicinformationisusedinthecalculation. UponsubstitutionofEquation 2 intothetime-independentSchrodingerequation(H)]TJ /F5 11.955 Tf 12.86 0 Td[(E)=0,asetofcoupledintegro-differentialequationsisobtainedforthenuclearfunctionsFn(R).Inmatrixform,theseequationsaresimilartothoseofCTCC(ieEquation 2 ).ThenuclearfunctionsareexpandedinpartialwavesFn=R)]TJ /F3 7.97 Tf 6.59 0 Td[(1fnl(R)Dnlm()andtheangularbehaviorisdealtwithanalytically,leavingasetofcoupledequationsfortheradialfunctionsfnl(R). Theseequationsareintegratedbystandardtechniquesfromsmallnuclearseparationstotheasymptoticregion.AtthispointtheS-matrixandthusthescatteringamplitudecanbeextractedandthedesiredpropertiescalculated. Fortwo-atomsystemswithoneactiveelectron,MOCCmethodsarequiteeffectivetherequisitepotentialsurfacesaresimpleandthenumberofopenchannelsissmall.MOCCmethodsscaleverybadlywithsystemsizeandusuallyrequiretheuseofsemi-empiricalpseudo-potentialsforcoreelectronsamongotherapproximations.Manychannels,includingclosedoneshavingnoasymptoticpopulation,mustbeincludedtoaccuratelydescribetransitionsbetweenstates. AdvantagesoftheMOCCmethodincludeacorrectquantumdescriptionofnuclearbehavior,easily-interpretedphysicalprinciples,andcomparativelysimple-to-implement 24

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numericalprocedures.However,theapproximationsusedcanleadtoun-physicalornumericallyunsatisfactorybehavior,especiallyasthesizeofthesystemincreases.Sinceallpotentialsurfacesareadiabatic,withcouplingallowedbetweensurfaces,stronglynon-adiabaticprocessessuchascoulombexplosionsarenotmodeledwell.Calculationofthepotentialsurfacesformolecularsystemsattherequiredresolutionisalsooftenprohibitive.Inaddition,thecalculationisverysensitivetothechoiceofchannels,basisfunctions,andcoordinateswhichresultsinadistinctlackofagreementbetweenpractitionersofdifferentclose-couplingimplementations.Theexactdescriptionofthenuclearstatedependsonthemethodused:afewcommonmethodsaredescribedinthenextsubsections. 2.4.1PerturbedStationaryState(PSS) Thesimplestdescriptionofthenuclearcoordinates[ 1 ]istousetheinternuclearvectorRasthereactioncoordinate.ThisapproximationisusuallytermedthePerturbedStationaryStateapproximation(PSS).ThisisequivalentinCTCCtoneglectingtheETFs.Atverylowenergies,thisapproximationisdecent,butrapidlydegenerateswithincreasingenergy.ThisimproperbehavioriscausedbythenatureofthemolecularstatesntheyarethestationarystatesofthemoleculeatnucleargeometryR,andassuchhavenowell-describedmomentum.Thismeansthatasthecollisionpartnersseparate,theelectronicprobabilitytriestobridgethedistanceandretainitstwo-centercharacter,insteadofbreakingintoatomiceigenstates.Mathematically,thisshowsupasspuriouselectroniccouplingsatlargedistance,whichinduceselectronictransitionswherenonesuchshouldoccur.Coordinate-dependentcouplingsarealsoacommonresult[ 1 ]. 2.4.2CommonReactionCoordinate(CRC) Becauseofthedifcultiesandnon-physicalbehaviorofthePSSformulationdescribedabove,severalextensionsweredevelopedinthe1980s[ 1 2 4 ].ThemostcommonistheCommonReactionCoordinate(CRC)formulation.Other,more 25

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specializedmethodsaredescribedinthenextsubsection.CRC-MOCCreliesonreplacingthesimpleinternuclearvectorRwithamixedelectronic-nuclearcoordinate,calledthecommonreactioncoordinate.ThismethodstartswiththesameanzatzasthePSSformulation(Equation 2 ),butreplacesReverywherewith.Theexactformofthereactioncoordinatevariesbetweenimplementations,butacommonlyusedformis[ 4 20 21 ]: =R+1 s(R;r)=R+1 f(R;r)r)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2f2(R;r)R(2) wheref(R;r)isthesametypeofswitchingfunctionasinEquation 2 andplaythesameroleofswitchingtheelectroniccoordinatefromonecentertotheother.isthereducedmassofthecollisionsystem. SubstitutingtheanzatzEquation 2 (withRreplacedby)intotheTISE,expandingderivativesininversepowersofthereducedmass(ofwhichweonlykeeptermsproportionalto)]TJ /F3 7.97 Tf 6.59 0 Td[(1),andintegratingoutelectroniccoordinatesyieldsequationsthat,inmatrixform,aresimilartothoseinEquation 2 .Theexactdenitionsofthecouplingmatricesonlydifferslightlyfromthoseabove.Thissimilarityinform(andpurpose)leadstothecommonideathattheCRCformalismisthequantumversionoftheETFsusedinclassical-trajectorymethods. Afterthenewequationshavebeenderived,theexpansioninpartial-wavesandsolutionoftheequationsarethesame(allowingfortheextramatricesneeded)asinthePSSmodel. Theprimarydifcultyhereistheconstructionofthecouplingelements,whichareneededforallmolecularstatesinthebasisateachgeometry.Theaccuracyofthecalculationisprimarilydeterminedbytheaccuracyofthecouplingelementsneartheavoidedcrossingsbetweenpotentialsurfacesofthesamesymmetry.Forsystemswithmultipleactiveelectrons,CRC-MOCCcanbeextendedwithsignicantpenaltiestobothanalyticdifcultyandnumericalcost. 26

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2.4.3OtherMethods TwoothermethodsusedtoavoidthespuriouscouplingproblemsofPSScalculationsaretheHyper-SphericalCoupled-Channel(HSCC)method[ 4 22 ]andtheadvancedadiabaticapproach[ 5 23 24 ]orHiddenCrossingsCoupled-Channel(HCCC). HSCCisparticularlyadaptedtotwo-center,one-activeelectronsystems.Essentially,thismethodconsistsofexpressingtheHamiltonianandassociatedpotentialsandwavefunctionsinmass-weightedhyper-sphericalcoordinates.Thisinherentlycorrectsfortheimproperlong-rangecouplings;noarbitraryoruser-selectedswitchingfunctionsorexpansionsinthereducedmassarenecessary.Generally,thecouplingelementsaresimpleinhyper-sphericalcoordinates.Thedisadvantagesincludemuchmorecomplexequations,lackofsimplephysicaldescription,andmostdamagingly,theabsoluteinabilitytoextendthetheorytomultipleactiveelectronsormolecularparticipants.Hyper-sphericalcoordinatesareonlysuitableforathree-bodyproblem. HCCCisaradicalalternativetostandardmethods.ThecoreconceptofHCCCistheanalyticextensionofthepotentialsurfaceE(R)tothecomplexplane.Thisleadstoasetofhiddencrossings,notunlikethewell-knownavoidedcrossingsofthe(real-valued)Born-Oppenheimersurfaces.Contourintegralsaroundthesehiddencrossingsgiveprobabilitiesfortransitions.FormoredetailsseeRefs.[ 5 ],[ 23 ],and[ 24 ]. 2.5ElectronNuclearDynamics ElectronNuclearDynamics(END)isanapplicationoftheTime-DependentVariationalPrinciple(TDVP)includingexplicitcouplingbetweenelectronsandnucleiatalltimes.Nopotential-energysurfacesareusedandneithertheBorn-Oppenheimernortheadiabaticapproximationisenforced.ThetheoryunderlyingENDanditsapplicationstophysicalsystemshasbeenexplainedindetailelsewhere[ 13 ].Thissectionwillfocus 27

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onthecurrentimplementationoftheENDequations,namelyintheENDynecomputercode.AbasicderivationoftheENDequationsofmotionisalsogiveninSection 2.5.1 2.5.1DerivationofENDEquationsofMotion ThefollowingderivationcloselyfollowsthatfoundinDeumensetal.[ 13 ]aswellasotherpublishedworks. Westartwiththequantummechanicalactiondenedas A=Zt2t1Ldt;(2) whereListheLagrangiandenedas L=hjid dt)]TJ /F4 11.955 Tf 15.46 3.02 Td[(^Hjihji)]TJ /F3 7.97 Tf 6.59 0 Td[(1:(2) Here^HistheHamiltonianoperatorforthesystemandrepresentsthesetofparametersusedtodenethewave-function.VaryingtheactionwithrespecttotheparametersgivestheEuler-Lagrangeequationsintheusualfashion.Applyingtheprincipleofleastactiongivesthecoupledpartial-differentialENDequations: 0B@0)]TJ /F5 11.955 Tf 9.3 0 Td[(iCiC01CA0B@__1CA=0B@@E @@E @1CA;(2) where E(;)=hj^Hji hj0i(2) istheenergyofthesystemandthegeneratorofinnitesimaltimetranslations.Cistheinvertiblemetricmatrixdenedas C=@2lnS @@(2) whichdenesthecouplingsbetweendynamicalvariables.ThequantitySinthedenitionofC,(Equation 2 ),istheoverlapS=hj0i. 28

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Formaximalefciencyandnumericalstability,thesetofparametersshouldbenon-redundantandgenerateacompletesetofwave-functions.ENDusesgeneralizedcoherentstatesintheThoulessrepresentation[ 13 ]toaccomplishtheserequirements.TheparametersusedaretheThoulessparametersz,andsothestatejicanbewrittenasjz(t);R(t);P(t)i,whereRandParethe(classical)positionvectorsandmomentaofthenuclei.TheThoulessparametersareofnecessitycomplex[ 13 ].Withthewave-functionparametrizedincoherentstates,traditionallyproblematicpartsofthewave-functionsuchastheelectronictranslationfactors(ETFs),whicharerequiredinthemolecular-orbitalcoupled-channelapproach,becomenaturaloutgrowthsofthetheoreticalframework.Also,thegeneratedstateisautomaticallyofthepropersymplecticform,havingbothgeneralizedcoordinatesandconjugatemomenta.Whenthewave-functionisexpandedinatomicorbitalsaspertheLinearCombinationofAtomicOrbitals(LCAO)approximation,thedynamicorbitalsarenotgenerallyorthogonal.Thisisanalyticallymorecomplicatedthanusingorthonormalmolecularorbitals,buthasmajorcomputationaladvantages. 2.5.2MinimalElectron-NuclearDynamics Electron-nucleardynamicstheoryiscurrentlyimplementedintheENDynepackagemaintainedanddevelopedbyErikDeumensattheUniversityofFlorida.Itscurrentversion,5.A.4,iscalledaminimalimplementationinthatitrestrictsthewave-functiontoasingleThoulessdeterminant,butamulti-determinantalversioniscurrentlybeingcodedanddebugged.Thissectiondescribesthecurrentproductionversionanditsstrengthsandweaknesses;mentionismadeofthoseweaknessesthatwillbecorrectedbythemulti-referenceversion. MinimalENDhasthreemajordeciencies,oneofwhichiscorrectedbythemulti-referenceversion;anotherthesecondandthirdareintrinsictothemethod. 29

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Therstdeciencyistherestrictionofthewave-functiontoasingle-determinant,spin-unrestrictedform.Theminimalwave-functionfortheelectronstakestheform det(h(rn))(2) with h(r)= h(r)+Xp p(r)zph(2) wherethe k(r);k=p;h,aretheorthonormaltwo-componentspin-orbitalsexpressedinthe(theoreticallycomplete)basisofspin-orbitals.Generallythe karecreatedinthestandardLCAOmannerfromGaussian-typespatialorbitalsmultipliedbyeitheranorspinfunction.TheseGaussian-orbitalsarecenteredonnucleiandarethusreferredtoasatomicorbitals. Restrictingthewave-functiontoasingle-determinantalformisqualitativelyneformostsystems,butintroduceswell-knownerrorsinthequantitativebehaviorofmanysystems.Forexample,apuresingletstateofanopen-shellatomormoleculecannotbeproperlyrepresentedbyasingle-determinantalsystem.Thuseventhecollisionofground-statehydrogenwithground-stateHeliumsuffersfromcontaminationbythe(higher-energy)tripletstate,leadingtoanincorrectestimationofelectron-transferprobability.Thisdeciencyisslatedtobecorrectedbythemulti-determinantalversionofENDyne. Asecond,moreintrinsicproblemwithENDisthetreatmentofthenuclei.Originally,thenucleiweretreatedasthezero-widthlimitofaGaussianwave-packetforconvenience[ 13 ]withtheexpectationthatlaterversionswouldrelaxthatapproximation.Laterunpublishedinvestigationrevealedthatunfreezingthewave-packetsresultedinadegradationoftheaccuracyasitrequiresthecouplingbetweenelectronsandnucleitobeofasimpleproducttype,ratherthantheproperdynamiccoupling.Assuch,neitherminimalnormulti-referenceENDynewillnativelyincorporatequantumnuclei. 30

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ENDnucleartrajectories,whilenotfully-quantalinnature,aredynamicinnature;thepositionsandmomentaofthenucleifullyrespondtotheinstantaneousforcesofbothothernucleiandtheelectronicstructure.Thisdynamicnaturemakesthemacceptableatlargeenergies,wherethedeBrogliewavelengthofthenucleiissmallandthequantumbehaviorissuppressed.Asthecollisionenergyapproachesthechemicalregime(afeweV/amuandbelow),however,thenucleartrajectoriesdeviatemarkedlyfromclassicalbehavior.ThisisespeciallyapparentforlightfragmentssuchasHydrogen;tunnelingandinterferencebetweenadjacenttrajectoriesbecomeapparentandcontributetothestructureofthecross-section.Attheselowerenergies,therestrictiontoclassicaltrajectoriesisaseriousawinENDtheoryandtherearecurrentlynoplannedmethodstocorrectitwithinENDyneitself.AnotherdifcultysharedbetweenENDandthemean-eldsurfacehoppingapproach(Section 2.2 )isthatinassociatingasinglevalueofforceofpotentialwithatrajectoryateachpointinherentlyundersamplesthesmall-probabilitytrajectories.Inthissense,ENDisanextensiontothemean-eldsurface-hoppingscheme. Athirddifculty,presentinbothENDandMOCC,isthatofionization.Neithertheorycancoupletothecontinuumstatesresponsibleforionization,leadingtocontaminationoftheelectron-transfercross-sectionatenergiesabovetheionizationthresholdduetothelackofopenionizationchannels.MOCCpartiallyavoidsthiscontaminationbycalculatingstate-speciccross-sectionsandsummingtogetthetotal;minimalENDcannotaccomplishthis. 2.6UnoccupiedNiches Areviewofcommonmethodsleavesusconvincedthatthereisanicheforanewapproach,oratleastanextensionofanexistingapproachinanewdirection.Manymethodsexistforpotential-surface-mediateddynamics.However,thesemethodsallrequireexpensivesurfacesandcouplingelements,complicatedcoordinatesystems,anddifferentapproximationsandexpansionsasthescaleoftheproblem(eitherinenergy 31

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orinnumberofparticipants)changes.ENDdoesn'trequiresurfacesorcouplingtermsandisrelativelyscale-agnostic;itfails,howevertoaccountforquantumnucleareffectssuchastunnelingorinterferencebetweentrajectories.Inthenextchapter,weproposeatwo-stagecorrectiontothepresentENDcalculationstocorrectfortheneglectofquantum-nuclearinterference. 32

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CHAPTER3CONSTRUCTIONOFDYNAMICPOTENTIALSURFACES 3.1Movingbeyond(A)diabaticPotentialSurfaces Inthissectionwediscussthemotivationsforconstructingeffective-potentialsurfacesanddescribe,usingsimpleone-dimensionalexamples,thecorrespondencebetweenthefoldedeffective-potentialsurfacesandthetime-independenttunnellingmodelforscatteringcalculations.Extensiontohigherdimensionalityisshowntobeastraight-forwardprocedurethatrequiresnoadditionalapproximationsoradhocmanipulations. Acommonpointofviewforchemicalreactionsistoplotthepotentialenergyexperiencedbythecomplexasafunctionofthereactioncoordinate,acoordinatethatischosentobestdescribetheessentialdynamicsofthereaction.Theheightsandwidthsofbarriersbetweenreactantsandproductscontrolsthekineticsofthereaction.Inessence,thispotentialsurfaceformsamapofthereaction,fromthepointofviewofareactionparticipant.Tobringthisintothescatteringframework,wewanttoconsiderwhathappenstoanatomorion(forsimplicitywe'llrestrictthisdiscussiontotwo-centerscattering,butthetheoryextendswithoutfundamentalproblemstolargersystems)asitapproachesanotheratomorion.Sincetheelectronicstateisnotrestrictedtoadiabaticstatesbutcancontinuouslychangeinresponsetotheinstantaneousforces,therecordoftheelectronicpotentialenergy(possiblyplustheinter-nuclearrepulsion)providesacontinuoushistoryoftheforcesexperiencedbythenuclei.Fromtheprojectile'sframeofreference,itmovesstraightaheadandthetargetispushedaway.Thisstraight-lineevolutionisexactlyequivalenttothereaction-coordinatedescription.Itcanbeparametrizedinmanyways(seeSection 3.2 foronepossibleway),buthasthesamecharacteristicsasasimpletunnelling/scatteringsystem.Multi-dimensionaltunnellingmapscanbemadebystackingthevaryingpotentialsexperiencedduringeachtrajectorytogether.Withagoodparametrization, 33

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thetunnellingmapcanbecontinuouslytransformedintoasetofstandardCartesianpotentialsurfaces,suitableforwavepacketscattering. Letusconsiderageneralcaserst.Initially,theprojectileisinaknownstationaryelectronicstate,withpotentialenergyV0.Asitspeedstowardsthetarget,itexperiencesachangeinpotentialduetobothnuclearrepulsionandchangesinelectronicstate.Thishappenssmoothlyasthenuclearseparationsdecrease.Atcertainpointsthechangeinelectronicstateinvolvesatransitiontoanotherstationarystate,possiblyhavingswitchedcenters.Asaresult,theoutgoingprojectilenowhasadifferentasymptoticpotentialenergyV1.Inthetime-of-ightrepresentation,thisisquitesimple;astandardbarrier-causedtunnelling/scatteringpotentialcurve.Inrelativecoordinates,thisisnotsosimple.Take,forexampleahead-oncollision.ItwillretraceitsownCartesiantrajectory,butwithadifferentenergy.Thisleadstoafoldedeffectivepotentialsurface;thecoordinatesofthefoldarewherethetransitionbecameirrevocable.Eachpointinspacewillhaveatleasttwopotentialenergiesassociatedwithit,onefortheincomingprojectileandonefortheoutgoingone.DescribingthesurfacesformedandtheabilityforwavepacketcomponentstotransfercontinuouslyfromonetotheotherisdifcultinCartesiancoordinates(wherethecalculationscanactuallyberun).Thetime-of-ightdescriptionholdsthekey.Thephysicsmustbethesameinthetworepresentations;intunnellingit'sthecontinuityandsmoothnessofthewavefunctionthatmatter,withtheshapeofthebarriercontrollingtransmissionandreection.Sincethetunnellingandscatteringdescriptionscanbemappedontoeachother,continuity,smoothness,andtheshapeofthepotentialsnearthefoldmustcontrolthemulti-surfacescatteringaswell.Thuswecanthinkintermsoftunnelling,butcalculateinscatteringproperties.Wedon'tactuallyneedtocomputethereactioncoordinateandrewritetheSchrodingerequationinthosecoordinateswecanworkinfamiliarcenter-of-massCartesiancoordinateswiththecondencethatourresultscorrespondtophysicalreality. 34

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Toshowthatthiscorrespondenceistrue,wemustexamineboththeone-dimensionaltunnellingcaseandtheone-dimensionalfoldedpotentialscatteringcase(wherethereturnpathisthesameastheincomingtrajectory,butatadifferentenergy).First,wemustexplorethegeneraltunnellingcaseinonespatialdimension.Wethensolveaspeciccase;althoughsimpleinform,thisexampleshowsallthecharacteristicswe'relookingfor.Westartwithapiecewise-continuouspotentialV(x)denedovertheentirerealline:x1.Sincethisisascatteringpotential(asopposedtoabound-stateproblem),thepotentialV(x)mustapproachafree(at)potentialasxapproaches.AnexceptiontothisistheCoulombpotential,specialmentionofwhichwillbemadelater.Wewritethispiecewisepotentialas V(x)=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:V1(x)for
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whereC1+isthecoefcientoftherightwardmovingwaveandC1)]TJ /F1 11.955 Tf 10.41 1.79 Td[(isfortheleftward(reected)wave.Thewavefunctionineachregioncanbewrittensimilarlybysubstituting i(x)forexp(ikx)andCiforC1.Inthelastregionnoreectedwaveispossible,soCn)]TJ /F4 11.955 Tf 10.41 1.79 Td[(=0.Inallotherregionsbothtermsmustoccur. Atthispoint,calculatingthetransmissionandreectioncoefcientsisassimpleasmatching i)]TJ /F3 7.97 Tf 6.58 0 Td[(1(xi)with i(xi)aswellasthederivatives 0i)]TJ /F3 7.97 Tf 6.59 0 Td[(1(xi)and 0i(xi).RepeatingthisprocessyieldsC1)]TJ /F1 11.955 Tf 10.41 1.8 Td[(andCn+asafunctionoftheincomingcoefcientC1+.ThereectionandtransmissioncoefcientsR=jC1)]TJ /F5 11.955 Tf 7.08 1.79 Td[(=C1+j2andT=((V1)]TJ /F5 11.955 Tf 11.97 0 Td[(E)=E)jCn+=C1+j2arethusnormalized:R+T=1.Inmorethanonedimension,thisprocedureismuchmorecomplicatedtoactuallycompute,buttheideasremainthesame:matchthefunctionandthenormalderivativeattheboundarybetweenregionsandcomputethefractionoftheincidentuxthatisreectedtowardsandthefractionthatistransmittedtowards+1. 3.1.1ASimpleExample Tostart,wetakeasimple,althoughnotthesimplestpossible,example.Thepotentialforthiscaseis V(x)=8>>>><>>>>:0forx)]TJ /F5 11.955 Tf 21.92 0 Td[(a=2V2for)]TJ /F5 11.955 Tf 9.3 0 Td[(a=2a=2(3) Wethushavethreeregionsinspace:anasymptoticregionwithzerointeractionpotentialenergy,anotherasymptoticregion(ontheotherside)withpotentialenergyV1,andabarrierofheightV0andwidtha.Figure 3-1 showsthisgraphically;thesamegurealsoshows,withadottedline,asimplephysicalpotentialtowhichthissquarebarriercanbethoughtofasanapproximation.Intermsoftheprojectileenergy(E,therearethreeregionsofinterestaswell:0V2wherethetransmissionshouldbecomesaturatedasEincreases.Thisis 36

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showninFigure 3-2 forincidentplanewaves;forE>0:5hartree,thereection(dotted)isunity,andthetransmission(solid)iszero.Thetransmissionthenrapidlyincreasestonearunity.Thedecayingoscillationsarecausedbyk2apassingthroughn;thisallowsfortotalreectionorpartialreectionevenforenergiesmuchabovethebarrier.Moredetailaboutthissystem,includingtheresultsofrecastingitasanincidentwavepacketandcomparisonstotheCartesiananaloguecanbefoundinSection 6.2.2 Itisimportanttonotethattheonlydataneededtocreatethetransmissionandreectioncoefcientswerethewidthandheight(relativebothtoreactantandproductregions)ofthepotentialsurface.Foranon-squarebarrier,thisreducestothemaximumheightofthebarrier(abovetheentrychannel)andtherelativeslopesofthepotentialnearthepeak.Physically,thismakessense:theslopeofthepotentialis(in2+dimensions)thegradient;thegradientinturndenestheforcesonthepieceofthewavepacketatthatlocation. 3.1.2BacktoScattering Howdoesthesimpleexamplecitedaboverelatetothescatteringproblem?Asweshowbelow,theeffectivescatteringpotential(includinginternuclearrepulsion)incenter-of-massCartesiancoordinatescanbemappedontothetunnellingproblem.InCartesiancoordinatestheprojectilehastheopportunitytoscatterbackoveritsinitialtrajectory,formingmultipleuncoupledsurfaces.Theymaybedegenerate,butthetransitionsbetweenthesesurfacesonlyoccurnearaspecicareainphysicalspace,thefoldbetweenthesurfacesThisfoldisthepoint(inonedimension)or,ingeneral,surfaceofdimensionalityonelessthanthatofthesystematwhichtheevent(suchaselectronicexcitation,transfer,orsoforth)canbesaidtohappen.Inonedimension,itwillalwaysbethesmallestinter-nucleardistanceattained;intwodimensionsittendstofollowtheboundaryoftheenergeticallyforbiddenregionquitetightly.Atthisfold,ourexperiencewiththetunnellingcase(whichmustyieldthesamephysicalresult)tellsusthatensuringthatthewavefunctioniscontinuousandsmoothatallpointsguarantees 37

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thepropertransferofwavefunctiondensityfromonesurfacetothenext.Atallotherpoints,awayfromthefold,thewavefunctioncanevolvefreely.Foldswithhigh(local)gradientsshuntwavefunctiondensityawayfromthefold,eitherpreventingitfromtransferringorkeepingitonthenewsurface.Foldswithlowlocalgradients(forces)allowthewavefunctiontosloshbackandforthmanytimesbeforeleavingthearea,resultinginsignicantreection. 3.2ReactionCoordinateDescription Theforegoingdiscussionisinspace-xedCartesiancoordinates,leadingtothecomplextopologically,butsimplecomputationallypotentialsurfaces.Acharacteristicofspace-xedcoordinatesistheexistenceofforbiddenregions,regionsinspacewherethewave-functionisenergeticallyforbiddenfrompenetrating(asdescribedinSection 3.6 ).Figure 3-3 showsaselectionofarbitrarytrajectoriesapproachingthescatteringcenter(solidcircle).Thedashedparaboliccurveshowstheborderoftheforbiddenregion,andthusthetransitionpointbetweensurfaces.Trajectoriesthatencounterthisboundaryaredeectedbackwards,possiblyatdifferentenergies(dottedlines).ThisistheviewpointusedcomputationallywithinENDwave. Analternate,butcompletelyequivalentdescriptionisfromthepointofviewoftheprojectile.ThisisatransformationofthecoordinatesystemintoonewherethesametrajectoriesshowninFigure 3-3 becomestraightlinesinthenew(non-Cartesian)coordinatesystem.ThisisshowninFigure. 3-4 and 3-5 ;Figure 3-4 showsthesamesetupasFigure 3-3 ,transformedintothisnewcoordinatesystem(iefrom(xrel;zrel)to(;).ThepotentialalongeachtrajectoryisnowdescribedbyformssimilartoFigure 3-5 ,wherethecuspinFigure 3-5 correspondstotheintersectionofthetrajectory(solidline)withthe(transformed)borderoftheforbiddenregion.Inthisrepresentation,theentireprocesscanbedescribedasmulti-dimensionaltunnellingthroughabarrier.Forenergieslessthantheasymptoticseparation(EV0,bothtransmissionandreection 38

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(elasticscatteringintheCartesianpicture)occur,withangle-dependentratiosthatdependonboththeincomingenergyandtherelativegradientsofthepotentialsurfacesnearthecusp(orforbiddenregion),justasinstandardtime-independentwavefunctiontunnelling.Fromadynamicperspective,theincomingwavepacketapproachesthebarrierandsplits,partofitcontinuingeitheroverthebarrierortunnellingthroughit,therestreectingbacktowards=.Thepartofthedensitythattraversesthebarrierbecomestheproductandtherestremainsreactant-like. ThisalternatedescriptionisconceptuallysimpleandstraightforwardlygeneralizestoN>2dimensions,butiscomputationallydifcult.Manycommonmethods(suchastheMOCC-CRCapproachdescribedinSection 2.4 )requiretheoperatortocreate(eitherapriorioradhoc)theappropriatereactioncoordinates;thechoiceofcoordinatesalterstheformofthecoupledequationstobesolved.Oftenill-suitedcoordinatesmustbechosenforcomputationalpurposeswhichthenleadstoproblemssuchasspuriouscouplingelementsatlongrangeororigin-dependenceintheresults.TheCartesianapproachiscomplicatedconceptually,butquitesimplecomputationally,asnocoordinatetransformationsmustbeperformed,nordotheequationshavetobemodied.Wethereforeproposetothinkinthereactioncoordinate,butcalculateintheCartesianframe.Thisisdoablesincethetwoarerelated(atleastintheory)byaone-to-one-and-ontotransformation,implyingthattheresultsmustbethesameineitherframe. Oneparticularlysimpleunfoldedcoordinatesystemfortwoormoredimensionsismotivatedbyatechniqueusedtocontroltheonsetoftime-dependenceinperturbationcalculations(seeShankar[ 25 ]forexamples).InessencewetransformfromCartesiancoordinates(x;z)atoneextremetoimpactparameterandtimecoordinates(b;t)atthe 39

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other.Wedeneanewcoordinatesystem(x0(t);z0(t))bythetransformationx0(t)=(1)]TJ /F5 11.955 Tf 11.96 0 Td[()b+x(t)z0(t)=(1)]TJ /F5 11.955 Tf 11.96 0 Td[()v0t+z(t) (3) whereisacontinuousparameterwithvaluesbetweenzeroandunitythatdescribeshowunfoldedthecoordinatesare,xrepresentsallcoordinatesperpendicularto^k(t=0),baretheimpactparameters(c.f.Section 3.3 )andv0=p 2E=mistheinitialspeedofthe(classical)trajectories.Thesignicanceoftheparameteristocontinuouslyconvertasingle,unfoldedsurfacewithnoforbiddenregions(=0)intoastrictly-Cartesian,multiple-valuedsurface(=1).Atintermediatevaluesof,thesurfacehascharacteristicsofboth,allowingthevisualizationofexactlyhowthesurfacestransformintoeachother. AparticularlystrikingexampleofthevalueofthissimpletransformationisshowninFigure. 3-6 .HereweseethesurfacegeneratedbycollidingHe2+withground-stateatomichydrogenatanenergyof10eV.Thecollisionenergyissuchthatthereisnoelectrontransfer(allstate-to-stateprobabilitiessmallerthan10)]TJ /F3 7.97 Tf 6.59 0 Td[(5).Shownarefourvaluesof:=0:0;0:1;0:32;1:0,chosentoillustratetheevolutionoftheunfoldingprocess.Plottedinblackontopofthesurfacesareverepresentativetrajectorieswithimpactparametersb=0:;1:0;2:5.When=0(Figure 3-6 a)thehead-ontrajectorygoesdownintotheattractiveregion,upovertherepulsivecore,andouttheotherside.Thetrajectoriesathigherimpactparametersenterthewelllater,leadingtotheasymmetryinz0.Athighervaluesof(Figure. 3-6 (b,c)),thehead-ontrajectorybackscatterswhiletheothersscatteratvariousangles.At=1(thenormalCartesiancoordinates,Figure 3-6 d),thetrajectoriesbendaroundthewelltoconserveangularmomentum,leadingtoaforbiddenregion(whitearea)inthemiddleofthegrid. 40

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3.3Notation VectorsRandPrepresentclassicalnuclearcoordinatesandmomentumalongENDtrajectories;vectorsXand^Parethecorrespondingquantalcoordinatesandmomentumoperators. Considertheinteractionbetweentwonuclearcentersasafunctionoftimeandtrajectory.Initially,weconstructabundleofENDinitialconditionsR(t=0)=R020=b;z0.Theseareinspace-xedCartesiancoordinates,andareconsideredtobedenseintheappropriateranges(tobediscussedlater).AftertheseinitialconditionsarepropagatedviatheENDequationsofmotion,theinteractionpotentialEel(R(t;R0);trel)isextractedalongeachtrajectory,andtransformedtorelativeCartesiancoordinates.Therelativetimetrelisdenedasthedifferencebetweensimulationtimetandthetimetd(z0)requiredtoreachtheedgeoftheinteractionregionasdenedbelow.Sincethetrajectoryoutsidetheinteractionisastraightlineatconstantspeed, td=(zinteraction)]TJ /F5 11.955 Tf 11.96 0 Td[(z0)v)]TJ /F3 7.97 Tf 6.58 0 Td[(1:(3) TheinteractionpotentialispresumedtohaveaniterangedenedbyXmaxsuchthatjEel(Xmax)+Vnn(Xmax)j<,whereisanarbitrarly-smallthreshold.TheregionborderedbyXmaxiscalledtheinteractionregionI;AR2)]TJ /F4 11.955 Tf 12.53 0 Td[(Iistheasymptoticregion.Forsimplicity,wecantaketherectangularvolumethatminimallyenclosesIastheworkinginteractionregionwithoutanysignicantloss. Thebundleofinitialconditionsisconstructedasfollows.First,denetheinitialvelocity(commontoalltrajectories)asv0=v0^z.Second,therangeofimpactparametersbmustbelargeenoughtocovertheprojectionoftheinteractionregionontothex-yplane(ieorthogonaltothevelocity).Inthetwo-dimensional,symmetriccase,thismeansb2[)]TJ /F5 11.955 Tf 9.3 0 Td[(bmax;bmax]wherebmaxischosentobelargerthanthemaximumextentoftheinteractionregioninthex-direction.Thebundlemustalsohaveanextentinthetransversedirection(^z).Intheory,thisshouldbeinnite,justliketheselection 41

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ofimpactparameters.Inpractice,itisenoughtohaveenoughtrajectoriessothatthelasttrajectoryreachestheedgeofIatthesametimethatthersttrajectoryisjustexitingI.Thiscaneitherbeexpressedintwocomplementaryways.First,asapositionzlast0=)]TJ /F5 11.955 Tf 9.3 0 Td[(zmax=v0TwherethetimeTisthetimerequiredforthersttrajectory(thatstartedattheedgeofI)toexitI.Second,asadelayintimesothatthelasttrajectorystartsattheedgeofIattimet=td=T. Aprimarypoint(relevantthroughout)isthateachtrajectoryis(bythenatureoftheevolution)independentofallothertrajectories.Inessence,wearepreparingaseparatecollisionsystemforeachvalueofR0.Thus,forxedb,thelasttrajectoryexperiencesthesamepotentials(andsamplesthesamepoints)attimetasthersttrajectorydidattimet+T.Inanalogytoaroller-coasterride,allridersexperiencethesameride,justatdifferenttimes. Ingeneral,theregionaccessedbytheclassicaltrajectoriesisnotequaltothewholeinteractionregion(muchlesstheentirecongurationspace).Thisisduetopotentialbarriershigherthantheincidentkineticenergy.Theseclassically-forbiddenregionscanbeaccessedbythequantumwavepacket,however,throughtunnelling,decayintothebarrier,andothersimilarmechanisms.Thismeansthatanyconsistentevolutionschememustprescribevaluesofthepotentialinsidetheclassically-forbiddenregion,wherenoclassicaltrajectoriesreach.DifferentmethodsandconsiderationsfordoingthisareconsideredinSubsection 3.6 Section 3.4 describesthemostbasiccase,whereforeveryX,thereexistsoneandonlyonepairoftimestandinitialconditionsR0suchthatR(t;R0)=X.Section 3.5 examinesindividuallytheconditionsunderwhichmultipleR(t;R0)aremappedtoasingleX. 3.4PotentialSurfacesforSingly-sampledRegions ForaregionwheretrajectoriesdonotcrossorwhereonlyonlyonetrajectorysamplesagivenvalueofXovertheentiretimeintervalunderconsideration,the 42

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extractionofthepotentialisquitesimple.AtthenaltimeT,foreveryX2I,ndavalueofR0suchthat R(T;R0)=X:(3) Then Eel(X)=Eel(R(T;R0);T):(3) Thisvalueisunique,andthemappingiscomplete. 3.5PotentialSurfacesforMultiply-sampledRegions Duetothenatureofscattering,mostpointsX2Rnwillbesampledmultipletimesduringtheevolutionoftheclassicalbundleoftrajectories.Thiscaneitherhappenwhenthesametrajectory(iecorrespondingtothesameR0)passesthroughthatpointatmultipletimes(t1,t2,uptotn)orifmultipletrajectoriesRi0passthroughXattimesti. AstheelectronicenergyEeldependsonthepasthistoryofthetrajectory,ingeneraltheenergieswillbedifferent.Severalpossiblecasesmayobtain.Take,forexample,thecaseoftwotrajectorieswitharepulsivepotential.Thersttrajectoryhasanimpactparameterofbi=bmax)]TJ /F5 11.955 Tf 12.1 0 Td[((isaninnitesimalpositivequantity);thesecondhasimpactparameterbi+1=bmax+.Botharetobetakeninthelimit!0.Thetrajectorylabelledbiwillbeinnitesimallydeectedoutwardbytherepulsivepotential,whiletrajectorybi+1willnotbedeected(byassumptionitliesoutsidetheinteractionregion).Thus,trajectorybiwillcrossthepathofbi+1,justataslightlylatertime.Theinteractionwill,ingeneral,altertheelectronicenergy:Eel(bi)6=Eel(bi+1)(otherdependencesuppressed).However,sincethesetrajectoriesareinnitesimallyclose,asmoothpotential(whichcanbeassumed)willonlychangetheenergyinnitesimally.Thisholdsgenerallyintheabsenceofelectronicrearrangementsorexcitations,allthescatteredenergieswillbeclosetotheunscatteredones.Electronicrearrangementsorexcitationsleadtoaseparatebandofenergies,allclosetogetheraswell.Ingeneral,asuitablyconstructedbundleoftrajectorieswillsampleallelectronicchannels(thatareclassicallyaccessible).Thisleadstoamultiply-valuedsurface,onwhichthewave-packetevolution 43

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canproceed.NotethatthesesurfacesarenotseparateeigenvaluesofastationaryHamiltonian,andthusarenotcoupledbyoff-diagonalHamiltonianelements. Inpractice,wewillemploybinningtechniquestoproduceaseriesofseparateelectronicsurfaces.Todothis,weemploythesametechniqueasinSection 3.4 ,excepthereproducingasetofRi0suchthatR(T;Ri0)=X:Overmostofthespace,thevaluesoftheelectronicenergycorrespondingtoeachofthesepointswillclusteraroundadiscretesetoflevels.AlltrajectoriesthatyieldenergiesclosetogetherinenergyatXarebundledtogetherasbelongingtoasurface(effectivelybinningtheenergies).ExperienceshowsthatthereisaregionofIinwhichtheenergiesallcollapseontoeachother,becomingarbitrarilycloseasthedensityoftrajectoriesandtime-stepsincreases.Incomingtrajectoriesapproachatoneenergy,whichincreases(ifcomingfromthegroundelectronicstate)andmeetstheoutgoingenergylevel,whichisalsochanging.Thisregionwheresurfacesmeetiscalledthefoldwhereonestategoesoverintoanother.Computersdon'tdealwellwithmultiply-valuedpotentials,sothewavefunctiononeachsurfaceispropagatedseparatelyandcontinuityandrst-orderdifferentiabilityareenforcedinthisjunctionregion.Thisenforcementofsmoothnesscouplesthemultiplebranchestogether. 3.6PrescribingPotentialValuesforClassically-forbiddenRegions Experienceshowsthattherecanbearegionofthescatteringspacethatcannotbesampledbyanyclassicaltrajectory.Purelyrepulsiveinteractionsleadtoashadowofsortsbehindthetarget'sinitiallocation:trajectoriesthatapproachtooclosearestronglyscattered,andthosefarenoughawayareslightlyrepelled.Eveninteractionsthatareattractiveforsomeregionhaveashadowaslongastheclose-rangebehaviorisrepulsive(whichshouldbethecaseforallatomicorionictargets).Theboundaryoftheforbiddenregionalsocloselymatchesthatofthejuncturebetweensurfaces(explainedabove).Thisclassically-forbiddenregionposesaproblemfortheconstructionofaglobalsetofpotentialsurfaces:Thequantumwave-packetisnotintrinsicallyrestrictedbythe 44

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classicallimitationsandcanpenetrateintotheseregions.Thismeansthatapotentialmustbeassignedtoallpointsinthecomputationalspace,includingonesnotsampledbyanytrajectory,nomatterhowdensethebundle(unlesswecandeterminethatthewavefunction(s)atthosepointsareexactlyzero). Acomplicatingconsequenceofthemethodsneededtocouplethevariouspotentialfoldstogetherleadstoaduplicationofthewavepacket;twoidenticalpacketsaregeneratedwithoppositemomenta.Onlyoneofthepacketsisphysical(toconserveparticlenorm).Ifahardwallisplacedattheboundaryoftheforbiddenregion,thespuriouspacketreectsoffthiswallandinterfereswiththephysicalpacket,leadingtofeedbackontothelowersurfaceandpotentialnumericalinstability.Thisisanunderstandableresultphysicallyintheframeofthewavepacket,itdoesn'tchangedirectionsbuttravelsstraightforward.Thisisquitedifferentfromareection,wherewavepacketstopsandturnsaround.Sincethefoldedregionisusuallyco-locatedwiththeforbiddenregion,forcingareectionnon-physicallysuppressestransferofprobabilitydensitytotheothersurface. Tohandlethis,weutilizethefollowingtrick.Thegriddedrepresentationofthepotentialfortheuppersurfaceisinvertedinthedirectionofmotion(iez!)]TJ /F5 11.955 Tf 26.54 0 Td[(z)andthetransferisbetweenthelocationsz0(x)ononegridandLz)]TJ /F5 11.955 Tf 12.33 0 Td[(z0(x)ontheothergrid.Thisprovidestheillusionofachangeindirectionforthewavepacketwithouthavingtoactuallychangethemomentumcomponents.Thetrickamountstopivotingtheuppergridaroundthefoldandattachingtheminthisoverlappingfashion.Pointsbeyondz0(x)onthelowergridandbeforeLz=z0(x)arethentheforbiddenregions.ThepotentialiscontinuedbyalinearextrapolationfromthefoldpointandanabsorbingpotentialisappliedasdescribedinSection 3.8 isappliedtocatchandharmlesslyabsorbtheduplicatewavepacket.Notethatthederivationofthetrickrequirestheinsightgainedfromthepointofviewofthewavepacket.Thisfurtherillustratesthepowerofhavingmultipleinterpretationsofthesameunderlyingphysicalevents. 45

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3.7NumericalConsiderations Numerically,aninnitely-densegridofinitialconditions(bothinbandz0)isimpractical.Infact,ifpossible,wewouldprefertoonlyonesetoftrajectories,differingonlyinb.Thiscanbedone,aslongaswetreateachtime-stepindependently(ieactasifwhenthenumericalbundleoftrajectoriesisattime-stept,thereisanotherbundlethatstartedonetime-steplater,andsoon.Thisallowsustotakethetotalhistoryofthebundleasifitwerethefullsetoftrajectorieswithdifferentinitialdistances. Anotherpairofdifcultiesarethatthegridoftrajectoryvaluesisnotinnitelydenseineithertimeorimpactparameter.ThismeansthattherearearepointsXthatliebetweensamplepoints.Interpolationwillberequiredtollinthegaps.Sincethechangesinpotentialareexpectedtobesmooth,interpolationshouldnotprovidetoomuchextraerror.Specialcarewillhavetobetakentoensurethattrajectoryvaluesbelongingtoseparatesurfacesarenotmixedintheinterpolation. 3.8Absorbing(Optical)PotentialsforGridBoundaries Ideallythepotentialsurfacesusedwouldbeinniteinextent.Thisiscomputationallyimpractical,soasdescribedaboveweonlydocomputationsonasmallniteregionofspace.Thisgivesrisetothequestionofwhattodoattheboundaries.FurthercomplicatingtheissueisthattheuseoftheFourierrepresentationofthekineticenergyoperatorasdescribedinChapter 4 naturallyimposesperiodicboundaryconditionsontheevolution:anywavepacketdensitythatleavesonesideisreplicatedontheother.Thisiscompletelynon-physicalforscatteringproblems.Furthermore,nowavefunctionorderivativevalueconditionscanbespeciedapriori(sincethewholespatialtruncationisnotphysicallyrequired).Manymethodshavebeenproposed(seeRef.[ 26 ]andreferencesthereinforexamples)todealwiththisproblem;theonewehaveadoptedisavariantoftheabsorbingpotentialoropticalpotentialmethod[ 26 ]. Weenlargethecomputationalregionadistancexabsorbinalldirectionsbeyondthatwhichisnecessaryforthescatteringandaddtothenaturally-existingpotential 46

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oneachsurfaceanegativeimaginarytermVimag.Theactionofthispotentialis,asthenamesuggests,tosmoothlyabsorbanywavefunctiondensitythatentersthebufferregion.TheprescriptionforVimagisessentiallyarbitrary,butafewprinciplesandsuggestionsapply.First,thetransitionfromreal-valuedcomputationalpotentialtocomplex-valuedabsorbingpotentialmustbesmooth.Thus,Vimagattheboundaryofthebufferareashouldbezero;inaddition,thederivativeoftheabsorbingpotentialatthatpointshouldalsomatchthatofthephysicalpotential.Second,changesinthegradientoftheabsorbingpotentialshouldbeasgradualaspossible.Thisistoalsominimizethepotentialforreections.Third,themaximumvalueoftheabsorbingpotentialVmaxshouldbeofthesameorderofmagnitudeasthemaximumenergycontainedintheinitialpacket.Alloftheseprescriptionsservetominimizethepossibilityofspuriousreectionsthatwouldpropagatebackintothecomputationalregion. TwofunctionalformsforVimagdominateintheliterature:linearfunctionsandgaussianfunctions.Linearfunctions(Equation 3 )aresimpletoimplementandhaveconstantslope(thussatisfyingthesecondprescriptionabove),butdonotmatchthederivativeattheinneredgeofthebufferregion(sincethat'susuallyzero). Vimag=)]TJ /F5 11.955 Tf 25.22 8.09 Td[(Vmax L)]TJ /F5 11.955 Tf 11.95 0 Td[(xabsorb(x)]TJ /F5 11.955 Tf 11.95 0 Td[(xabsorb)(3) Gaussianfunctions(Equation 3 )don'thaveasbadofamismatchbetweenderivativesattheedge,buthavewidelyvaryingslopesandhavetobetweakedtoensurenodiscontinuityinvalueattheedgeofthebuffer;inthisequationVasymptstandsforthevalueofthe(real)potentialattheedgeofthebufferandisathresholdforequality(usuallylessthan10)]TJ /F3 7.97 Tf 6.59 0 Td[(6Hartree).Vimag=)]TJ /F5 11.955 Tf 9.3 0 Td[(Vmaxexp()]TJ /F5 11.955 Tf 9.3 0 Td[((x)]TJ /F5 11.955 Tf 11.96 0 Td[(L)2)=1 L)]TJ /F5 11.955 Tf 11.96 0 Td[(xabsorblnVmax Vasympt+ (3) 47

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Itseemsvaluabletoustoactuallyimplementathirdfunctionalform,thequadraticfunction(Equation 3 ). Vimag=)]TJ /F5 11.955 Tf 20.86 8.09 Td[(Vmax (xabsorb)2(x)]TJ /F5 11.955 Tf 11.95 0 Td[(L)2(3) Thisfunctionalformhastheadvantagesofboththeotherforms:ithasthecorrectvalueandderivativeattheedgeofthebufferandtheslopechangessmoothlyandgradually.Ourexperienceisthatitdramaticallyoutperformstheothertwobuffersundermostconditionsandforthewidestrangesofincidentenergies.Relatedfunctionalformscanbeusedintheclassically-forbiddenregionstoabsorbspuriousduplicatewavepacketdensity. 48

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Figure3-1. One-dimensionalscatteringpotentials:Realisticexponentialpotential(dotted)andsquarebarrierapproximation(solid) Figure3-2. Transmission(solid)andreection(dotted)offthesquarebarriershowninFigure 3-1 formonochromaticincidentplanewave.Asymptoticpotentialdifferenceis0.5hartree,formingthethresholdfortransmission. 49

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Figure3-3. Cartesianrepresentationofthescatteringprocess:Trajectories(solidlines)approachtheboundaryoftheforbiddenregion(dashedcurve)andcontinue(dottedlines),possiblyonadifferentfoldofthepotentialsurface. Figure3-4. Reactioncoordinaterepresentationofthescattering(tunnelling)process:Trajectories(solidlines)passthroughthebarrier(peakrepresentedbydashedcurve)andareeitherreectedortransmitted. 50

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Figure3-5. PotentialsurfacealongsingletrajectoryinFigure 3-4 ;potentialcuspislocatedalongdashedcurve.V0representsasymptoticseparationbelowwhichnotransferofdensitycanoccur. 51

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(a)=0:0 (b)=0:1 (c)=0:32 (d)=1:0 Figure3-6. END-baseddynamicpotentialsurfaceforHe2++Hunderlineartransformationfrom(b;v0t)coordinates(=0)to(x;z)coordinates(=1) 52

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CHAPTER4EVOLUTIONALGORITHMFORWAVEPACKETSONENDSURFACES Mostofthediscussionhereisbasedonlong-standingmethodsandalgorithmsandcanbefreelyinterchangedwithothersuchmethods.Theuniquepartsarefoundintheconnectionbetweenthesurfaces.Toourknowledge,nopublishedimplementationincludesallthepiecesdenedherein. 4.1FormalDenitionofScatteringProblem FollowingthediscussioninNewton[ 6 ]ch.6,weconsidertheproblemofndingthewavefunctionj(t)iatsometimet=tfwithaneyetowardsextractingthedifferentialcross-section. Formally,thisisequivalenttosolving(inthenon-relativisticregime)theTime-DependantSchrodingerequation. i@ @t)]TJ /F4 11.955 Tf 14.99 3.02 Td[(^Hj(t)i=0(4) where^HistheHamiltonianoperatorforthesystem,commonlyseparatedintothekineticenergyoperator^Tandaninteractionpotential^V.Formultiple-particlesystems(asopposedtoscatteringofasingleparticleoffaxedsurface),^Tcanbewrittenas ^T=1 2XiM)]TJ /F3 7.97 Tf 6.58 0 Td[(1ir2Ri:(4) InEquation 4 wedenotebyMithemassofparticleiandrRithegradientwithrespecttothecoordinatesofthei'thparticle.Thus,Equation 4 isasecond-order(inspace),rst-order(intime)complexpartialdifferentialequation;forarbitrary^V,theTDSEisnotseparable,althoughitisseparableinspeciccases. Evenbeforeboundaryorinitialconditionsareconsidered,severalpropertiescanbederivedfromthebasictheoryofpartialdifferentialequations.Forallcontinuouspotentials,thewavefunctionanditsrstspatialderivativermustbecontinuousin 53

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allrepresentationsandatallpointsinthephase-space.Wealsorequirethatmustbeniteeverywhere. Thetime-dependantscatteringequationsareusuallywrittenintheinitialvalueformulation:(t)=in(t)+Z1dt0G+(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)in(t0) (4)in(t)limt0!iG+(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)(t0) (4) whereG+(t)]TJ /F5 11.955 Tf 12.79 0 Td[(t0)isthepropagatorforthefullHamiltonianandobeysthedifferentialequation i@ @t)]TJ /F4 11.955 Tf 14.99 3.02 Td[(^HG+(t)=(t):(4) Formally,Equation 4 solvestheTDSEwiththeinitialcondition(t!)=in.Assuming(fornow)appropriateconvergence,theintegralequation( 4 )canbesolvediteratively(seeEquation6.15ainRef.[ 6 ]). 4.2SpatialIntegration Ingeneral,thespatialderivativesintheTDSE(Equation 2 )canbehandledeitherbynite-differencing,leadingtotheFDfamilyofmethods,orbyexpansioninauxiliaryfunctions,whosespatialderivativesareanalyticallytractable.IncludedinthesecondfamilyisthesubfamilyofmethodsbasedontheDiscreteFouriertransform.Thediscussioninthissectionwillbeprimarilylimitedto1Dmulti-dimensionalsolvershavemuchthesamefeaturesanddrawbacksastheir1Dcounterparts. Thedifferencesbetweenthefamiliesareentirelyintherepresentationofthekineticoperator^Tthepotentialislocalandcanbeappliedbydirectmultiplicationinbothcases. Finite-Differencemethods(FD)arelocally-polynomialapproximationstothewavefunction;theexactexpressiondependsontheorderpoftheapproximatingpolynomial.Generallyordersbeyondaboutp=4arerarelyusedduetocomputationalcost. 54

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FollowingtheanalysisinKosloff[ 27 ],allFDmethodsconvergeasN)]TJ /F10 7.97 Tf 6.59 0 Td[(pwiththenumberofgridpointsN.Theyalsosufferfromtheirsemi-localcharacter;thekineticenergyoperatorisfullynon-localinposition-space.Sincethesemethodsdon'treproducethespectrumofthekineticoperatorwell,phaseerrorsaccumulaterapidly.Thismeansthatthecommutationrelationshipsbetweenquantizedvariablesareonlyapproximatelyobeyed.Anomalousdispersionisinevitable(sincetheHamiltonianispoorlyrepresented).Allinall,FDmethodsforthespatialderivativestendtorequireverynegridstoachievethesamespatialaccuracyastheexpansionmethods.Theirchiefbenetsarethattheyaresimpletoimplementand,forsufcientlynegrids,canrepresentalmostanyequationrequiredwithnofurtherassumptions. Expansionmethods,alsocalledFiniteElement(FE)methodsexpandthewavefunctioninauxiliaryfunctionswhosederivativesareeitheranalyticallytractableorwherethekineticoperatorhasaparticularlysimpleform.MostcommonlyusedforgeneralpurposecodesistheDiscreteFourierTransform(DFT)[ 28 ]initsfastform(theFastFourierTransform,orFFT).TheDFTexpandsthewavefunctionas (x)=N=2)]TJ /F3 7.97 Tf 6.58 0 Td[(1Xk=)]TJ /F10 7.97 Tf 6.58 0 Td[(N=2)]TJ /F3 7.97 Tf 6.58 0 Td[(1akexpi2kx=L:(4) Thishassomeimmediateadvantages:thecoefcientsakcanbedirectlyinterpretedasthemomentumrepresentationwavefunction,evaluatedatpointpkandamomentum-spacegridcanbeconstructedwithspacingp=2=L.Also,thekineticenergyoperatorhasaparticularlysimplerepresentation: ^T=1 2mr2R=Xkp2k 2m:(4) AlocalrepresentationofthekineticoperatormeansthatitcanbeappliedbydirectmultiplicationontheFourier-transformedwavefunction.Thenon-localityoftheoperatoristakencareofbythetransformitself. 55

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UsingtheFFTdoesrequirecertainassumptionstoensureconvergence,however.Forstrictconvergence,thewavefunctionmustbeband-limited(havenitesupportinmomentumspace),whichnoquantumwavefunctiontrulyis.Fortunatelyaslongasthewavepacketusedfallsofffastenoughthatthevalueatthegridedgescansafelybesettozeroissufcientforpracticalpurposes.AnotherissueisthatthenaturalboundaryconditionsfortheFFTareperiodic(iefN+1=f0)whichisnotnaturalforscatteringproblems.Thismeansthatthecomputationalregionmustbeboundedbyeitherinnitepotentialwallsoranabsorbingboundaryofsometype(asdiscussedinSection 3.8 FFTmethodsconvergeexponentiallywithN,conserveenergyand(forasinglesurface)normautomatically,scalesemi-linearly(Nln(N))andtendtonotaccumulateerrorsnearlyasfastastheFDmethods.ThroughoutthesimulationcommunitytheconsensusisthatrepresentingthekineticoperatorusingFFTisthepreferredspatialintegrationmethod. 4.3Time-EvolutionMethods Wenowturntomethodsforadvancingthesolution(~R;t0)tosomelatertimet0.Justlikewiththespatialderivatives,wecouldapproximatethetime-dependenceasapolynomialandformthenitedifferenceapproximationtothetimederivative.ThisleadstotheSecond-OrderDifferencing(SOD)scheme,aswellasmoreinvolvedversions(implicitdifferentiation,predictor-correctorschemes,etc.).DirectlyimplementingSODyieldsthefollowingrecursionrelationshipbetweenthefunctionatthreepointsintime(subscriptsnlabelthetimestept=nt+t0.): n+1=n)]TJ /F3 7.97 Tf 6.58 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(i2t^Hn:(4) Thismethodhascertainadvantages(especiallywhenusedwithaFDschemefor^H).Itisextremelysimpletoimplement,requireslittlecomputationaleffortperstep,since^Hispredominantlyzero,andissecond-orderaccurateint.Higher-order(int)methodsareavailable,butgenerallyrequiresubstantiallymoreeffortperstep;thusthenetbenet 56

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ofthemethodsisrelativelysmall.AnothermajorproblemisthatexplicitdifferentiationmethodssuchasSODareonlyconditionallystable.Erroraccumulatesinthephaseofthewave-function,leadingtoanomalousdispersionornumericalinstability.ThisisespeciallycommonwithFDapproximationstothespatialderivatives. Asecondclassofevolutionschemesarethespectralorpseudo-spectralmethods.TheprimaryoneinvolvedhereisinterpolationinenergybytheChebyshevpolynomials[ 29 ]Tn.Chebyshevevolutionandrelatedschemesareglobalmethodsthesolutionatt=t0isdirectlyobtainedfromtheinitialguessatt0withoutneedingtocalculatetheintermediatestepsatt=t0+nt.Thesemethodsareextremelyaccurateandefcient;infact,theyaremoreefcientasthetotalsimulationtimeisextendedandcanbemadearbitrarilyaccurate.Theyarealsoabsolutelystable.Theydorequirearescalingoftheeigenvaluestotherange[-1,1];thisofcourserequiresaband-limitedHamiltonian,whichwedonothave(inprinciple).Ourwavepacketcanapproximateone,sothisisnotinsuperable.Wewoulduseoneofthesetypesofmethods,exceptforonemajorproblem.Wecannotseehowtoconnectthefoldsofthepotentialwithoutbreakingtheevolutionintosmallerstepsandenforcingcontinuityanddifferentiabilityacrossthefold.Doingthiseliminatesmostoftheadvantagesofthespectralmethods.Also,ifaCoulombpotentialispresentthesystemcannotbecastintoaband-limitedform,andsothespectralmethodsloosemostoftheirpower.Forthesereasons,wemustreluctantlyabandontheglobalspectralmethodsandturntotheso-calledsplit-operatorschemes. Allsplit-operator(SO)methodsrelyontheformalsolutionforanarbitrarypotential: (~x;t)=e)]TJ /F10 7.97 Tf 6.59 0 Td[(iRtt0^H(~x;t0):(4) If^Vistime-independent(orifthetime-stepdt=t)]TJ /F5 11.955 Tf 11.98 0 Td[(t0issmallenoughthat^Visconstantovertheduration),Equation 4 canbesimpliedto (~x;t)=e)]TJ /F10 7.97 Tf 6.59 0 Td[(idt^H(~x;t0)=e)]TJ /F10 7.97 Tf 6.58 0 Td[(idt(^T+^V)(~x;t0):(4) 57

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TheexponentialofanoperatorisdenedintermsoftheTaylorexpansion: exp(^A)=1+^A+^A2 2!+:::;(4) severalfastalgorithmsexistfortakingexponentialsofsquarematrices.Oneimportantresultisthattheexponentialofadiagonalmatrixisthematrixoftheexponentialsofthediagonalelements.Thus(exp^A)ij=exp^Aijaslongas^Aislocal. ImplementingEquation 4 directlyresultsinarst-orderaccuratealgorithm(duetothenon-commutativityof^Vand^T).Sincethisisunattractiveduetotheprohibitivelysmalltimestepsrequired,mostinvestigatorsre-factortheexponentialas e)]TJ /F10 7.97 Tf 6.59 0 Td[(idt^He)]TJ /F10 7.97 Tf 6.58 0 Td[(idt 2^Te)]TJ /F10 7.97 Tf 6.58 0 Td[(idt^Ve)]TJ /F10 7.97 Tf 6.59 0 Td[(idt 2^T:(4) Thisissecond-orderaccurateindtandcanbeimplementedquitestraightforwardlyusingtheFFT.SinceinthecurrentimplementationVistime-independent,wecanformtheexponentialsofthematricesrequiredbeforetheiterationbegins.Eachbranchofthe(folded)potentialenergysurfaceistreatedseparately,asthereisnoHamiltoniancouplingbetweenthebranches.Thekineticoperatorisdiagonalinmomentum-space,andthepotentialmatrixisdiagonalinposition-space,soexponentiationofthematrixreducestotheexponentialoftheelements. Furthersplittingispossible,leadingtohigher-orderapproximations[ 30 ];goingbeyondthirdorfourthordergenerallyleadstoexcessivecomputationalcostrelativetothetimesavedbythehigherorder.Wethereforeimplementsecond-orderSOasthetime-integratorforENDwave Second-orderSOmethodshaveallthebenetsoftheSODscheme,aswellasbeinganaturalttoFFTspatialschemes.Theyarequiteeasytoimplementandarestableformuchlargertime-stepsthanSOD;whilenotasfastoraccurateasthespectralclassofmethods,usingSOwecanenforcetheconditionsatthefoldinastraightforwardfashionalso,norequirementofspectralrangeoftheHamiltonianisneeded.Sincethe 58

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FFTisusedforthekineticoperator,dynamicgridsinbothpositionandmomentumarerelativelytriviallyavailable,furthersavingcost. 4.4AnatomyofaStep Atthebeginningofeachtimestep,westartwiththeFourier-transformedwavefunction,withvaluesateachgridpoint k(t)where isthevectorofwavefunctionswithcomponentsthevalueatthatpointoneachsurface.Variablesinthemomentumrepresentationwillbedenotedbytildesoverthecorrespondingposition-spacefunctions.Subscriptsklabelthemomentum-spacegridpoints,anditheposition-spacepoints. Tostarteachstep,weactthersthalf-stepkineticoperatortoeachfunctioninthemomentumrepresentation: ~ 0k=exp)]TJ /F5 11.955 Tf 9.3 0 Td[(idt 2k2 2m~ k:(4) Next,weapplytheinverseFFTto~ 0(t),yielding (t).Nowtheactionofthepotentialmatrixissimplemultiplication: 0Ii=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(idtVIi) Ii:(4) Atthispoint,continuityandrst-orderdifferentiabilityareenforcedatthefold(theintersectionofthesurfaces)asexplainedinthefollowingsection.Afterenforcingtheseconditions,thefunctionsareFourier-transformedandasecondkineticexponentialisappliedexactlyasinEquation 4 .Theresultingfunctionsformtheinputtothenextiteration. 4.5EnforcingContinuityattheFold Afterthetheapplicationofthepotentialenergyoperator,theconnectionbetweensurfacesmustbeenforced.IfitwereaHamiltoniancoupling,thiswouldhappenautomaticallybyapplyingthepotentialenergy.Sinceit'smerelyacontinuitycondition,itmustbeenforcednumericallyeachtimestep. Therelevantconditionsconnectingthevariousbranchesofthefoldedpotentialsurfacearecontinuityandrst-orderdifferentiabilityatthebranchcut(inthiscase, 59

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simplytheintersectionbetweenthesurfaces).Ingeneral,thiscutwillnotoccurexactlyatagridpoint;thisisnotaproblem,sincewecantakethetwogridsasoverlappingforasmallregion(5-10gridpoints,0:2au).Werstndtheindexesofthosegridpointswithinthissmallband;thishappensbeforeevolutionbeginsandonlyhastobedoneonceperpairofsurfacesasthegridsdonotchange.Thegridpointswillbelabelledcollectivelyasjj. AftertheFouriertransformationintothepositionrepresentationandtheapplicationofthepotentialenergyoperatorexp(iVdt),thewavefunctioniscopiedtoatemporaryarraytemporaryWaveFunction.Thenthemaincomputationalgridsrepresentingthewavefunction(xWaveFunction)areassignedasfollows(Iistheindexofthelowersurface,Jthatoftheuppersurface):xWaveFunctionIjj=temporaryWaveFunctionJjjxWaveFunctionJjj=temporaryWaveFunctionIjjxWaveFunction=2jj=temporaryWaveFunction=2jj (4) Sincemultiplepointsaretransferredsimultaneously,themomentumanddensityarealsotransferredcorrectly.Wavefunctiondensitythatentersfromthetopsurfacetransferstothebottomsurfaceandviceversa.However,sincethisisequivalenttoapplyinganarrowsquarewindowfunctiontothepositionspacewavefunction,themomentumrepresentationhasspurioussidelobes.Theprimarylobeisatthenegativeofthemomentum,soanon-physicalduplicateoftheactualfunctioniscreated.Thefeedbackbetweenthetwosurfacemakesbothsurfacesexperiencethisdoubling.Asaresult,absorbingpotentialsareneededintheforbiddenregionstoabsorbthisextradensity(seeSection 3.8 forfurtherdetailsonthesepotentials). 60

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CHAPTER5CALCULATINGREACTIONCROSSSECTIONS Onemajorgoalofscatteringcalculationsistheextractionofthereactioncrosssection(i!f)oritsone-dimensionalanaloguethestate-selectivetransitionprobabilityP(i!f).Inbothcases,thenotationi!fdenotesthatsubsectionofprocesseswheretheinitialstateiisconvertedintothedesirednalstatef.Thesecrosssectionsandreactionprobabilitiesareusedinthederivationofrateconstants[ 31 ],stoppingpower,aswellastoevaluatethecontributionofvariousprocessestotheoverallscatteringevent. Crosssectionscanbecalculatedatseverallevelsoftheory,presentedhereinincreasingaccuracyatagivenlevelofcalculationtheory.Generally,calculationsinvolvingpredeterminedorclassicaltrajectoriesfornuclearmotionarerestrictedtoatbestsemi-classicalmethodsforcalculatingcrosssections(section 5.2 ),whereascalculationswithnuclearwavepacketsorotherwisefullyquantumnuclearbehaviorcanapplythemethodsfoundinSection 5.3 .TheclassicalformulasforcalculatingcrosssectionsarepresentedinSection 5.1 forcompletenessandtomotivatethediscussionofthesemi-classicalandquantumformulas. Atalllevelsoftheory,thetotalcrosssection(i!f)isaquantitywithdimensionsofareawhichisdenedastheratioofthetotalincomingparticleuxinstateitothescattereduxinstatef.Thestatelabelsincludeallquantumnumbersandarrangementinformationnecessarytospecifytheinitialornalstate.Spin,electronicexcitations,vibration,rotation,molecularstructureinformation(tospecifynuclearrearrangements)alloftheseareincludedinthestatelabels.Ofcourse,oftentheexperimentortheorycannotmeasurespecicquantities.Thisnecessitatestheformationofstate-averagedcrosssections. Ofspecialinterestamongthestate-averagedcrosssectionsistheintegralcrosssection,formedfromtheangulardifferentialcrosssection.Theangulardifferentialcross 61

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sectioninthelaboratoryframed(;) disdenedas[ 25 ] d(;) dd=numberofparticlesscatteredintod/sec numberincident/sec/area(5) whereisthesolidanglesubtendedbythedetector,(;)arethestandardsphericalanglesrelativetotheinitialmomentumofthebeam,andallotherstatelabelsaresuppressedforreadability.Fromthisangulardistribution,theintegralcrosssectionissimplytheintegralovertheentiresphere: =Z20dZ0dsin()d(;) d(5) Otherstate-averagedcrosssectionscanbeformedastheappropriatesumsorintegralsoverthestate-selectivedifferentialcrosssections.Assuch,theextractionfromthescatteringcalculationofthestate-selectivedifferentialcrosssectionissufcienttodeterminemostoftheinterestingpropertiesofthescatteringsolution,andwillbetheprimaryfocusofthiswork. Inpractice,integralcrosssectionscanbeeasilyextractedfromexperimentaldataorfromtheoreticalcalculationswithoutresorttotheangulardifferentialcrosssection.Infact,generallyitisintegralcrosssectionsthatarereportedintheliterature.Relativelyfewangulardifferentialresultsexistasthesearesignicantlyhardertocalculatefromtheoryormeasureexperimentally. 5.1ClassicalScatteringCrossSections 5.1.1ClassicalDeectionFunctions ThegeometricsetupfortheclassicalapproximationisshowninFigure 5-1 .Theprojectilestartssufcientlyfarfromthe(xed)scatteringcentersothattheinteractionforcesarenegligible(thepotentialisat).Itsinitialmomentumvectorkiisdirectedalongthe+zaxis,anditstartsadistance(calledtheimpactparameter)bperpendiculartothezaxis(hereafterinthexdirection).Forcentralpotentials,alltrajectoriesareconnedtothisplane,andtheresultingdistributionofoutgoingtrajectoriesis 62

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cylindrically-symmetrical.Forsphericalcentralpotentials,theresultingtrajectoriesaresphericallysymmetrical.Forcollisionsbetweentwoparticles,wetransitiontothecenter-of-massreferenceframewherethecenterofmassmovesatconstantspeedvCMandtherelativecoordinatesandmomentafollowEquation. 5 and 5 [ 32 ]whereisthereducedmassmpmt=(mp+mt)andthelabelstandpreferencethetargetandprojectilerespectively.rCM=(rt)]TJ /F11 11.955 Tf 11.95 0 Td[(rp)lab (5)pCM=(vp)]TJ /F11 11.955 Tf 11.96 0 Td[(vt)lab (5) Thescatteringangleinthecenter-of-massframeisrelatedtothescatteringangleoftheprojectile#byEquation 5 where=mp=mt. tan#=sin +cos(5) Afterscattering,the(centerofmass)momentumoftheprojectilekfisdirectedatangle(b)=cos)]TJ /F3 7.97 Tf 6.58 0 Td[(1(kf(b)=ki).Angle(b)iscalledthedeectionfunctionandcontainsallinformationnecessaryforthescatteringcrosssection.ForageneralsphericalpotentialV(r),thedeectionfunctionforaparticleofmassmandenergyEfollowsEquation 5 ,whererministhelargestzerooftheradical[ 6 ]. =)]TJ /F4 11.955 Tf 11.96 0 Td[(2Z1rmindrr)]TJ /F3 7.97 Tf 6.59 0 Td[(2b)]TJ /F3 7.97 Tf 6.59 0 Td[(21)]TJ /F5 11.955 Tf 13.15 8.09 Td[(V E)]TJ /F5 11.955 Tf 11.96 0 Td[(r)]TJ /F3 7.97 Tf 6.58 0 Td[(2)]TJ /F3 7.97 Tf 6.58 0 Td[(1=2(5) .TwoexamplesofclassicaldeectionfunctionsareshowninFigure. 5-2 and 5-3 .ThepotentialV(r)forFigure 5-2 isthatofahardsphereofradiusR=1au: V(r)=8><>:1forrR(5) ForFigure 5-3 thepotentialV(r)istheMorsepotential[ 33 ](usefulforttinginteractionsbetweenneutralmolecules),givenbyEquation 5 ,withwell-depth=4:0hartree, 63

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distancetominimumrm=1:0au,andwidthparametera=1:0au)]TJ /F3 7.97 Tf 6.59 0 Td[(1.ThedeectionfunctioniscalculatedbynumericquadraturefollowingtheapproximationsinRef.[ 34 ],tableIIwith1000quadraturepointsandcollisionenergyE=0:5hartree. V(r)=[exp()]TJ /F4 11.955 Tf 9.3 0 Td[(2a(r)]TJ /F5 11.955 Tf 11.96 0 Td[(rm)))]TJ /F4 11.955 Tf 11.96 0 Td[(2exp()]TJ /F5 11.955 Tf 9.3 0 Td[(a(r)]TJ /F5 11.955 Tf 11.96 0 Td[(rm))](5) Notethatforthehardspherepotential(Equation 5 ),thedeectionfunction(Figure 5-2 )isfortheheadoncollision(implyingtotalbackscattering),decreasessmoothlyuntilb=R=1au,atwhichpointitiszeroforb>R,exactlyasintuitivelyexpected.Thekinkinthescatteringangleatb=Rcomesfromthesharpboundaryinthepotential:trajectorieswithb=R)]TJ /F4 11.955 Tf 13.04 0 Td[((smallandpositive)arescattered,trajectorieswithb=R+arecompletelyunaffectedbythepotential.Realisticpotentialshavesmoothdeectionfunctions. ThedeectionfunctionfortheMorsepotential(Figure 5-3 )ismuchmorecomplicatedand,aswillbeshown,posessignicantproblemsfortheclassicalcrosssection,bothtotalanddifferential.Justasforthehardsphere,head-on(b'0)trajectoriesareback-scatteredbylargeanglesandthescatteringangleapproacheszeroforlargeb.However,forintermediate,energy-dependentangles,thetrajectoriesentertheattractivewellofthepotentialandspendatleastpartofanorbitnearthescatteringcenter(ie<0).Theminimuminthescatteringanglenearb=5aucorrespondstoacausticforthetrajectoriesandiscalledarainbowangle. 5.1.2ClassicalDifferentialCrossSections Thederivationoftheangulardifferentialcrosssectiond=dfromthedeectionfunction(forcylindricalsymmetryandcenter-of-masscoordinates)proceedsfollowingMcDanieletal.[ 32 ].Weconsiderthecrosssectionforscatteringintothesolidangleelementd=2sindbetweentheanglesofand+d.Thosetrajectoriesthatendinsolidangleelementdbyconstructionwereincidentinanannularringofarea2bdbbetweentwocirclesofradiusbandb+db.Therelationshipbetweenband(b) 64

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isgivenbyEquation 5 ;hencethedifferentialcrosssectionisgivenbyEquation. 5 and 5 .jbdbj=jd()sin(b)dj (5))d d=b sindb d: (5) AscanbeimmediatelyseenfromEquations. 5 and 5 ,therearetwogeneralpossibilitiesforsingularitiesintheclassicaldifferentialcrosssection:First,as(b)approachesmthederivativedb=disusuallynite.ThisleadstothedivergenceofthedifferentialcrosssectioncalledthegloryangleorforwardpeakduetothesininthedenominatorofEquation. 5 .Experimentallythisdoesnotposeaseriousproblem,asthedetectorsarenotpositionedat=0soasnottobeswampedbytheun-scatteredbeam.Forinniterangepotentials,thepresenceofagloryanglecausestheintegralcrosssection(denedinthenextsubsection)todivergenon-physically.Asecond,moreserioussingularityoccursforpotentialswithbothattractiveandrepulsiveregionssuchastheMorsepotentialabove(Equation 5 ).AscanbeseeninFigure 5-3 ,thedeectionfunction(b)crossestheaxisnearb=1:4auwithniteslope(causingaglorypeak)andalsohasaminimumnearb=5au.Attheminimum(calledtherainbowangle),thetrajectoriesclustertogetherandcausethecrosssectiontodivergeaswell;mathematicallythisisduetothevanishingofthederivativejd(b)=dbj.Morecomplicatedpotentialsmayhavemultiplesuchrainbowangles.Eachrainbowconstitutesafailureoftheclassicalmodelandtheremovalofrainbow-causeddivergencesisakeyfeatureofthesemi-classicalmodelsdescribedlater. Theabovediscussionisrestrictedtoelasticscattering.ItispossibletomodifytheclassicalmodeltotakeintoaccountreactionprobabilitiesbymultiplyingtheelasticdifferentialcrosssectiondE=dbythereactionprobabilityPi!f(b)(Equation 5 d(i!f) d=Pi!f(b)dE d=Pi!f(b)b sindb d:(5) 65

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Thisisanapproximationofdubiousaccuracyorvalueforseveralreasons.Transitionsbetweendiscretestatesinherentlyinvolvequantumprocesses;thepresenceoftheseprocessesinherentlymeansthattheclassicalapproximationisofuncertainvalidityatbest.Second,interferencephenomenabetweentrajectoriesmustbetakenintoaccountatthelevelofthetransitionamplitudes,notattheleveloftheprobabilities.Third,theinherentinaccuraciesanddivergencesoftheclassicaldifferentialcrosssectionsuggestthatitisbettertomovedirectlytoasemi-classicalorquantummodelwhichincorporatessuchprobabilitiesdirectlywithoutdivergences. Totransformfromcenter-of-masscoordinatestolabcoordinatesforthecaseofastationarytargetandcylindricalsymmetry,wemustmultiplybytheappropriatemass-weightedtransformation[ 32 ](Equation 5 ). d(#) dlab=[(mp=mt)2+2(mp=mt)cos+1]3=2 1+(mp=mt)cosd() dCM(5) 5.1.3TotalClassicalCrossSections Total(orintegral)classicalcrosssectionscanbecalculatedintwobasicways(oneforelasticprocessesandtheotherforreactivepathways).Therstway,suitablefornite-rangeelasticinteractions,istosimplyintegratetheclassicaldifferentialcrosssectionwithrespecttosolidangle(Equation 5 ). =Z20dZ0sinddE d(5) Equation 5 willdivergeforinniterangepotentials(suchastheCoulombinteraction)wherethereisapoleintheforward(=0)crosssection.Itwillalsodivergewheneverthereisarainbowangleinthedifferentialcrosssection.ForinelasticprocessestheclassicalintegralcrosssectioncanbedenedastheintegralofEquation 5 oversolidangles;thissharesallthesamedifcultiesastheelasticintegralcrosssectionaswellasthevalidityproblemsthatwerediscussedalongwiththedenitionofthereactiveclassicaldifferentialcrosssection. 66

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Asecondmethodofcalculatingaclassical(no-interference)integralcrosssectionisvalidforreactivecrosssectionsanddoesnotsharetheconvergenceissuesforinnite-rangeinteractionsorrainbowangles.Westartbycalculatingforeachtrajectory(orimpactparameter)theprobabilityPi!f(b)ofstartinginconguration(orchannel)iandendinginchannelf.Thesechannelscouldinvolvemolecularfragmentation,ionization,electrontransfer,electronicexcitation,orroto-vibrationaltransitions.Theprobabilityfornon-reactivescatteringisformallyPi!i(b)=1)]TJ /F9 11.955 Tf 12.07 8.96 Td[(Pf6=iPi!f(b).Thisformaldenitionisnotveryusefulinpractice,asmostmethodsforcalculatingtheprobabilitiescannotdetermineprobabilitiesforallchannels.Ionizationisparticularlyproblematicformostmethods,aswillbediscussedinChapter 2 .Reactionprobabilitiesshouldbecomenegligibleasb!1,boundingtherangeofimpactparametersneeded.Oncetheprobabilityforthedesiredstateiscalculatedforarangeofimpactparameterssufcienttoextinguishtheprobability,theintegralcrosssectioncanbecalculatedasinEquation 5 i!f=Z10Pi!f(b)bdbZbmax0Pi!f(b)bdb(5) Thisapproachisnotperfect;aswithallclassicalcrosssections,interferenceeffectsbetweenadjacenttrajectoriesarenotconsidered(ascanbeseenin 5 withtheintegralovertheprobability,notovertheprobabilityamplitude(whichincludesthephaseinformation).Forclassicaltrajectoriestheprobabilityamplitudeisnotwelldened,beingapurelyquantumeffect.AllinallEquation 5 formsthebackboneforcalculationsofclassicalreactioncrosssections.Specicexamplesofcalculationsusingthisdevelopedmethodinvolvingthereactivedifferentialcrosssectionsandreactionprobability-basedintegralcrosssections(Equations 5 and 5 )willbediscussedinChapter 6 5.2Semi-ClassicalScatteringCrossSections Manysemi-classicalcorrectionshavebeenproposedtopartiallycorrectforthepathologiesandinaccuraciesoftheclassicalcrosssections(bothdifferentialandintegral),whilenotcompletelyabandoningtheimpactparameterdescriptionof 67

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thescatteringevent.Thisissomewhatofacompromisebetweentheeaseoftheclassicaldifferentialandintegralcrosssectionsandtherigorandaccuracyoffullyquantumscatteringcalculations.ThisreviewwillonlycommentindetailontheSchiffapproximationdevelopedbyL.I.Schiffin1956[ 35 ]andrewritteninapotential-freeformbyKillian[ 36 ]. Thebasic,underlyingideabehindallsemi-classicalformulationsistoexpandtheSchrodingerequationinpowersof~andtakethelimit~!0.Ifalltermslinearorhigherin~areneglected,theformulasshouldreducetotheclassicalresults.TheprocessfordoingthiswasdevelopedrstbyAiry[ 37 ]forscatteringoflightandthensimultaneouslyadaptedtotheSchroodingerequationbyWentzel,KramersandBrillouin,aswellasLiouville,GreenandJeffreys.ThisprocessisvariouslycalledtheWKBapproximation,theLG,ortheWKBJapproximation.AdetailedderivationcanbefoundinShankar[ 38 ]. 5.2.1Potential-DependentSchiffApproximation Schiff'sinsightintothederivationofthesemi-classicalcrosssectionwastoapplytheso-calledmethodofstationaryphase[ 39 ]totheinniteBornseries(Equation 5 )[ 35 ].InthissectionwesketchthederivationofSchiff'soriginalformulasforthescatteringcrosssection;moredetailcanbefoundinRef.[ 35 ]. Bothderivationsstartwiththedesiretondsolutionstothetime-independentSchrodingerequation 5 thathaveasymptoticformsgivenbyEquation 5 .InEquation 5 theincidentkineticenergyis~2k2=2mandthescatteringpotentialenergyisV=~2U=2m;inEquation 5 ,k0andkfrepresentvectorsofmagnitudekalignedalongtheinitialandnalmomentarespectivelyandfisthescatteringamplitude. )]TJ /F2 11.955 Tf 5.48 -9.69 Td[(r2+k2)]TJ /F5 11.955 Tf 11.95 0 Td[(U =0 (5) !exp(ik0r)+r)]TJ /F3 7.97 Tf 6.59 0 Td[(1exp(ikr)f(kf;k0) (5) Inthisnotation,theinniteBornseriesforthescatteringamplitudefcanbewrittenasEquation 5 ,whereGistheoutgoingGreen'sfunctionappropriatetotheSchrodinger 68

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equation.f(kf;k0)=1Xn=1ZZexp()]TJ /F5 11.955 Tf 9.3 0 Td[(ikfrnU(rn)G(rn)]TJ /F11 11.955 Tf 11.95 0 Td[(rn)]TJ /F3 7.97 Tf 6.59 0 Td[(1)U(rn)]TJ /F3 7.97 Tf 6.58 0 Td[(1)G(rn)]TJ /F3 7.97 Tf 6.58 0 Td[(1)]TJ /F11 11.955 Tf 11.96 0 Td[(rn)]TJ /F3 7.97 Tf 6.59 0 Td[(2)U(rn)]TJ /F3 7.97 Tf 6.59 0 Td[(2U(r2)G(r2)]TJ /F11 11.955 Tf 11.96 0 Td[(r1)U(r1)exp(ik0r1)d1dn (5) ThemethodofstationaryphaseapproximatestheBornSeries(Equation 5 )bynotingthefactthattheexponentialexp[i(k)]TJ /F11 11.955 Tf 11.96 0 Td[(k)]oscillatesrapidlyasthedirectionofchanges,exceptwhen^isnearlyparalleltok.Asaresult,themajorcontributionstotheintegralsinEquation 5 comefromeitheraparaboloidregioncenteredaround^kwithradiusoforder(=k)1 2oraspherewithradiusoforderk)]TJ /F3 7.97 Tf 6.58 0 Td[(1centeredattheorigin.Thus,byrewritingtheBornSeriestotakeadvantageofthisforthetwocaseswecanderivemuch-simpliedequationsforthesemi-classicalscatteringamplitude. Largescatteringangles(>(kR))]TJ /F18 5.978 Tf 7.78 3.25 Td[(1 2)resultinndistinctstationary-phaseregions.Asaresult,wecanrewritetheBornSeriesbyintroducingnewvariablesj=rj+1)]TJ /F11 11.955 Tf 12.28 0 Td[(rj(j=1;:::;n)]TJ /F4 11.955 Tf 11.8 0 Td[(1)andq=k0)]TJ /F11 11.955 Tf 11.81 0 Td[(kf.RewritingEquation 5 andtakingtheleadingtermsinleads(aftersubstantialsimplication)tothenalformforthescatteringamplitudeforlargescatteringangles(Equation 5 ).f(kf;k0)=()]TJ /F4 11.955 Tf 9.3 0 Td[(4))]TJ /F3 7.97 Tf 6.58 0 Td[(1Zexp(iqr)U(r)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(i=2k)Z10U(r)]TJ /F4 11.955 Tf 12.28 3.16 Td[(^k0s)ds+Z10U(r+^kfs)dsd (5) NotethatEquation 5 hasreducedtheentirescatteringproblem(atleastforlargeangles)toatrioofintegrationsoverthepotential;theouterintegralistheFouriertransform(modulonormalization)ofthepotentialtimesthesumofthephaseshiftsinducedbytraversingthepotentialalongtheinitialandnalmomentaindividually.Thisisasimpleresultthat,givensuitablyaccuraterepresentationsofthepotentialsurface,canbedirectlyappliedtocalculatetheelasticscatteringamplitude.Itdoesnotapplyto 69

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transitionorinelasticscattering,aswillbediscussedafterthediscussionofthesmallangleresult.Formostproblems,experienceshowsthatthecrosssectionissmallforlargeangles,andsothisresultisonlyofinterestforthesakeofcompleteness. Forsmallscatteringangles(<(kR))]TJ /F18 5.978 Tf 7.78 3.26 Td[(1 2),thendistinctstationary-phaseregionsfoundinthelarge-anglecasecondensetoasinglesphericalregionforeachvalueofnallowingfurthersimplicationoftheresult.Wesetthedirectionoftheinitialmomentumtobe+^z;thenthecomponentofthemomentumtransferinthisdirectionqz1 2k2issmallcomparedto1=2R,soexp(iqzzm)canbereplacedbyunity.ThisalsoallowsustoreplacetheshiftedpotentialintegralsinEquation 5 bythefollowing:Z10U(r)]TJ /F4 11.955 Tf 12.27 3.15 Td[(^k0s)ds=ZzU(x;y;z)dzZ10U(r)]TJ /F4 11.955 Tf 12.28 3.15 Td[(^kfs)ds=Z1zU(x;y;z)dz: Thisprocedure,alongwithsometrivialsubstitutions,allowsustoperformtheallbutonesetofintegralsinEquation 5 ,leavingjustanintegraloverthetransversecoordinatesxandyandasumovern.Thesummationturnsouttobetrivial,resultinginthesymmetry-independentformofthesmall-angleelasticscatteringamplitudefoundinEquation 5 f(kf;k0)=(ik=2m)ZZexp(i[qxx+qyy])1)]TJ /F4 11.955 Tf 11.95 0 Td[(exp()]TJ /F5 11.955 Tf 9.3 0 Td[(i=2k)ZU(x;y;z)dzdxdy(5) TheintegralsinEquation 5 areoverallvaluesoftheirrespectivecoordinates.Thisapproximationisvalidfor(kR))]TJ /F18 5.978 Tf 7.78 3.26 Td[(1 2,1=kR<1,andV=E1(Visthemaximumvalueattainedbythepotential;Eisthecollisionenergy).Itdoesnotrequiresphericalsymmetry(unlikethepartialwaveexpansion),butdoesrequirethatthescatteringbeelastic.Afurthersimplicationcanbemadeforcylindricallysymmetricpotentials(iethoseisotropicaroundthezaxis):wewritethecoordinatesas(b;)wherebisthemagnitudeoftheimpactparameterb=p x2+y2.Performingthetrivialintegrationover 70

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resultsinthestandardformfortheSchiffsmall-anglescatteringamplitude: f(kf;k0)=ikZ10J0(qb)1)]TJ /F4 11.955 Tf 11.95 0 Td[(exp()]TJ /F5 11.955 Tf 9.3 0 Td[(i=2k)ZU(b;z)dzbdb:(5) Forverysmallenergies(VR=~v1)bothEquation 5 andEquation 5 reducetothe(rst-order)Bornamplitude:fB(kf;k0)=()]TJ /F4 11.955 Tf 9.3 0 Td[(4))]TJ /F3 7.97 Tf 6.58 0 Td[(1Zexp(iqr)U(r)dr: Asapracticalrule,theSchiffformsareverynearlyexactwhereverthecoreunderlyingassumptionshold.Thus,forelasticscatteringintosmallangleswithwell-denedpotentialsurfaces,Equation 5 givesresultsidenticaltotheexactresultswithmuchlessworkforrealisticsystems.However,therequiredapproximationsarelethalformostsystemsofinterest.Wewouldprefertouseasingleapproximationatallanglesandforsystemsthatmixelasticandinelasticprocesses(non-resonantchargetransfer,forexample).Inaddition,severalmethodsforsolvingtheelectronicproblem,mostnotablyElectronNuclearDynamics(END)(Section 2.5 ),donotuseornativelygeneratetheelectronicpotentialenergysurfaces;thusforthesemethodstheSchiffapproximation(atleastinstandardform)isuseless. 5.2.2Potential-IndependentSchiff-TrujilloApproximation ToamelioratethesedifcultieswhilenotabandoningthebeautyandsimplicityofSchiff'swork,RemigioCabrera-TrujillocameupwithamethodbasedonastatementinSchiff'soriginalpaper[ 35 ]:UsetherelationshipbetweenthepotentialandtheclassicaldeectionfunctiontoturntheintegraloverthepotentialinEquation 5 orEquation 5 intoanintegraloverthetrajectoriesascomputedbyEND[ 40 ].Thisprocedurehasbeenusedandextended(withcaveats)tocoverinelasticprocessesbyKillian[ 36 ].Theoutlineoftheprocessfollows(thedetailscanbefoundinRefs.[ 40 ]and[ 36 ]). Startingwiththeaxially-symmetricform(Equation 5 ),wenotethatthemagnitudeofthepositionoftheprojectilermustbegivenbyr2=z2+b2,asthe 71

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impactparameterbisperpendiculartothezaxis.ThismeansthatwecantransformtheintegraloverzinEquation 5 intooneoverr: dz=dr [1)]TJ /F4 11.955 Tf 11.96 0 Td[((b=r)]1=2:(5) TheargumentoftheintegraltransformsaccordingtoEquation 5 afteraccountingforsymmetry.1 2kZ1U(b;z)=1 2kZ1U(r) [1)]TJ /F4 11.955 Tf 11.96 0 Td[((b=r)]1=2dr=1 kZ10U(r) [1)]TJ /F4 11.955 Tf 11.95 0 Td[((b=r)]1=2dr (5) Equation 5 istheMassey-Mohrapproximationforthesmall-anglelimitofthesemi-classicalphaseshift(b)[ 11 7 10 ],whichcanberelatedtothedeectionfunctionbytheexpression[ 41 11 42 ] (b)=2 kd(b) db:(5) TogetherwiththeapproximationtotheintegrandsimplicationsinEquation 5 ,therelationshipbetweenthesemi-classicalphaseshiftandthepotentialsufcetoreducetheexpressionfortheSchiffsmall-anglescatteringamplitudetotheSchiff-Trujillopotential-independentform(Equation 5 ,wherethefunctionJ0istheBesselfunction1oftherstkindoforderzero). f(ko;kf)ikZ10J0(qb)[1)]TJ /F4 11.955 Tf 11.96 0 Td[(exp(2i(b))]bdb(5) FormallythisisnomoreapproximatethantheSchiffsmallangleexpression(Equation 5 ),butcanbecalculatedwithoutreferencetopotentialenergysurfaces(whicharenotnativelycalculatedinEND).Thelargeangleexpression(Equation 5 )cannotbe 1BettercalledtheWoodardfunction 72

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similarlysimplied;fortunatelymostphenomenaofinterestoccurforsmallangles[ 36 ]andthisiswheremostexperimentaldifferentialcrosssectionsarereported. ToextendtheexpressionfoundinEquation 5 toelectron-transferprocesses(forwhichitisnotformallyvalid,duetotheinelasticityofmostsuchprocesses),ithasbeencommonpracticetosimplymultiplythedifferentialcrosssectionfoundfromtheSchiff-Trujilloapproximationbytheprobabilityforsuchprocessestooccur.This,itmustbenoted,cannotbetotallyaccurate.BuiltintotheSchiffapproximation(andevenmoresointotheTrujilloextensiontosaidapproximation)istheassumptionthatkf=k0,thatisthattheprocessiselastic.Exceptforresonantelectrontransferprocesses(suchthattheasymptoticelectronicenergiesfortheinitialstateandnalstateareequal),allelectron-transferprocessesareinelastic,somedeeplyso.Thisinelasticitycanbeineitherdirection;ifEf
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theextendedSchiff-Trujillocrosssectionsareaccurate,althoughitdivergesgreatlyastheenergydecreases(due,webelieve,toquantumnucleareffects).Asaresult,wewouldliketogobeyondthesemi-classicalregimetoafullyquantummechanicalcrosssectionusableforelasticandinelasticprocesses. 5.3QuantumScatteringCrossSections Thegoaloffully-quantumcalculationsistoincludeallinterference,tunnelling,boundandcontinuumstatetransitionsforalldegreesoffreedom.Ingeneral,wedenefourpropagators(Green'sfunctions)[ 6 ],differingonlyinwhichHamiltonianisusedinbytheirdeningSchrodingerequation(Equation 5 :i@ @t)]TJ /F4 11.955 Tf 14.99 3.03 Td[(^H0G(t)=(t)i@ @t)]TJ /F4 11.955 Tf 15 3.02 Td[(^HG(t)=(t): (5) HerethetotalHamiltonianHhasbeensplitintotwoparts:^H=^H0+^H0.^H0isthenon-interactingHamiltonianwhichfornon-relativisticcasesisjustthekineticenergyoperator^T=^p2=2m.H0hasalltheinteractivepartsoftheHamiltonian.Thepropagatorshavetheappropriateinitialconditions:G+(t)=G+(t)=0fort<0G)]TJ /F4 11.955 Tf 7.09 -4.94 Td[((t)=G)]TJ /F4 11.955 Tf 7.08 -4.94 Td[((t)=0fort>0: (5) ThusthepropagatorsG+andG+allowustopropagateawavefunctionknownattimettoatimet0>t;similarlythepropagatorsG)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(andG)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(propagatebackwardsintime(iefort0
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fort>t0(usingG+)andwasequalto(t0)att=t0[ 6 ].Ifwelett0approachthedistantpast,wecandenethestatein(t)accordingtoEquation 5 : in(t)limt0!iG+(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)(t0):(5) Thisstatein(t)istheincomingwavefunction;inthedistantpastitwasequaltosome(known)initialstateandevolveswithoutinteracting. UsingG)]TJ /F1 11.955 Tf 10.4 -4.33 Td[(wedenetheoutgoingcounterparttoin: out(t)limt0!1)]TJ /F5 11.955 Tf 9.3 0 Td[(iG)]TJ /F4 11.955 Tf 7.08 -4.94 Td[((t)]TJ /F5 11.955 Tf 11.95 0 Td[(t0)(t0):(5)outisthe(freely-evolving)statethatwill,inthefarfuture,beequaltothecompletestate.Usingthetwofreestatesinandout,wecanformthescatteringstates()(t)accordingtoEquation. 5 .(+)(;t)=in(;t)+Z1dt0G+(t)]TJ /F5 11.955 Tf 11.95 0 Td[(t0)H0(+)(;t)()]TJ /F3 7.97 Tf 6.58 0 Td[()(;t)=out(;t)+Z1dt0G)]TJ /F4 11.955 Tf 7.08 -4.93 Td[((t)]TJ /F5 11.955 Tf 11.95 0 Td[(t0)H0()]TJ /F3 7.97 Tf 6.59 0 Td[()(;t) (5) Thestate(+)(t)isdenedasbeingcontrolled(equaltosomeknownfunctionin(;t)wherethelabelreferstoallvariablesthatcommutewith^Ho)inthedistantpast;similarly()]TJ /F3 7.97 Tf 6.58 0 Td[()willbecontrolledinthedistantfuture[ 6 ].Wehavethechoice:wecaneitherpropagate(+)forwardintime,endingupwithamixtureofthecontrolledstateinandthescatteredstate;orwecanpropagate()]TJ /F3 7.97 Tf 6.58 0 Td[()backwardintime,resultinginasimilarmixtureofcontrollednalstateandscatteredstate.Ofcourse,thedifcultyinallofthisisgeneratingappropriateGreen'sfunctionsaccordingtoEquation 5 .Eitherchoiceofstartingstaterequiresiterativesolutions,astheintegralsinEquations. 5 dependontheunknownfunctions()explicitly. Thereisanotherwaytoapproachtheissuehowever:wecombineboththeinteractingGreen'sfunctionsGandthefreeGreen'sfunctionsGtodenethewave 75

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operator(+):(+)=1)]TJ /F5 11.955 Tf 11.95 0 Td[(iZ1dt0G+(t)]TJ /F5 11.955 Tf 11.96 0 Td[(t0)H0G)]TJ /F4 11.955 Tf 7.09 -4.94 Td[((t0)]TJ /F5 11.955 Tf 11.95 0 Td[(t)=1)]TJ /F5 11.955 Tf 11.95 0 Td[(iZ1dtG+()]TJ /F5 11.955 Tf 9.3 0 Td[(t)H0G)]TJ /F4 11.955 Tf 7.09 -4.93 Td[((t): (5) Theactionofthewaveoperator(+)istoconvertthecontrolledincomingstatein(t)directlyintotheequivalentcompletestate(includingscatteredamplitudes)(t): (t)=(+)in(t):(5) ItmustbenotedherethatthewaveoperatorisunitaryifandonlyifthefullHamiltonian^Hhasnoboundstates.Therangeofthewaveoperatoristhecontinuumof^H;boundstatesareannihilatedbythewaveoperator.Thisisusuallynotaproblemforheavy-ionscatteringcalculationasanyboundstatesarewellseparatedfromthecontinuumscatteringstates.Similarly,anadvancedwaveoperator()]TJ /F3 7.97 Tf 6.58 0 Td[()canbedenedtoconvertoutintobyinterchangingthesuperscriptsintheintegralinEquation 5 [ 6 ]. Usingthewaveoperatorsthusdened,wecannallydenetheprobabilitythatastate,controlledtobein(;t)inthedistantpast,willbeinaknownstate0(;t)inthefarfuture.Sincethewaveoperatorsareinvertible(onthescatteringstatesonly),wecanwritetheoutgoingvectorintermsofitsincomingfunction: out(t)=()]TJ /F3 7.97 Tf 6.59 0 Td[()y(+)in(t):(5) Asaresult,wecanexpresstheprobabilityforstatein(;t)=0(;t)togoovertostate0(;t)as:limt!1h0(;t)j(+)(;t)i=h0(;t)jout(t)i=h0(;t)j()]TJ /F3 7.97 Tf 6.59 0 Td[()y(+)jin(t)i=h0(;t)j^Sj0(;t)i: (5) 76

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Equatio 5 denestheS-matrix:^S()]TJ /F3 7.97 Tf 6.59 0 Td[()y(+).TheS-matrixisaunitarytransformation(aslongasinandoutarecomplete)whichtakeswelldenedstates0denedinthedistantpastanddistantfutureintoeachother.Inthematrixrepresentation,theelementsof^Saretransitionamplitudesbetweenstatesjiandji.Thus,theirsquaresarethestate-to-statecrosssections(althoughusuallysumsoverdegeneratestatesareemployedtosimplifytheoutput). InprincipletheSmatrixcontainsallknowableinformationaboutthescatteringsystemanddenesthefully-quantumcrosssectionfortransitionsbetweenallstates;variousauxiliarydenitionscanbeusedforelasticorinelasticcollisions.EssentiallynoapproximationsaremadeinthederivationoftheSmatrix.Unfortunately,calculationoftheSmatrixisinpracticequitecomplicatedandsubjecttosignicantlimitations.AswasdiscussedinChapter 2 ,useofcompletewavefunctionsisimpracticalnumerically(thisinvolvinganinnitebasis);furthermore,theevolutionofthewavefunctioncanonlybecarriedoutfornitetime(orinthetime-independentversionsmostcommonlyused,nitedistance)whichviolatestheessentialassumption(thatthewavefunctionsarewell-knownindistantpastandfarfuture).Workaroundsarepossible,butquitecomplicated,fortimedependentcalculations;fortime-independentmethodswithsphericalpotentialsthepartial-waveexpansion(Subsection 5.3.2 )allowsthesecondproblemtobealmostcompletelyobviated.Therstproblem(incompletescatteringchannelbasis)ispresentinanynumericalSmatrixcomputation. 5.3.2PartialWaveExpansions Wecanwritetheasymptotictimeindependentsolutionforthegenericscatteringproblemas(settingthedirectionoftheincidentplanewaveas+^z)[ 25 6 ]limr!1 (ki;kf)= in(ki)+ sc(ki;kf):=exp(ikiz)+f(;)exp(ikfr) r (5) 77

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Thetotalwavefunction(asymptotically)isseparatedintoanincident,un-scatteredplanewave in(ki)=exp(ikiz)andascattered,distortedsphericalwave sc(ki;kf)=f(;)exp(ikfr)=r.Thefunctionf(;)isthescatteringamplitudeandcontainsallpossibleinformationaboutthescatteringevent;thedependenceonkisgenerallysuppressed,butimportant.Forgeneralpotentials,thisrelationship(Equation 5 )isformallytrue,butuselessforpracticalcomputations. Forsphericalpotentials(V(r=V(r)),theasymptoticforminEquation 5 canbeappliedrelativelyeasilytothecalculationofdifferentialandintegralcrosssections.Themethodfordoingthisisreferredtoasthepartialwaveexpansion[ 25 ].SinceanyfunctionofcanbeexpandedinLegendrepolynomials Pl(cos)=4 2l+11=2Y0l(5) whereY0listhesphericalharmonicforangularmomentumquantumnumberlwithzeroz-component(m=0),weexpandthescatteringamplitudef(;k)withkdependentcoefcients,asinEquation 5 f(;k)=1Xl=0(2l+1)al(k)Pl(cos)(5) Herethecoefcientsal(k)arecalledthepartialwavecoefcientsforangularmomentumlandrepresentthescatteringinto(conserved)angularmomentum~l.Sincetheincomingplanewavecanbeexpandedas exp(ikiz)=exp(ikircos)=1Xl=0il(2l+1)jl(kir)Pl(cos);(5) wherethefunctionsjl(kir)arethesphericalBesselfunctionsoforderl,eachpartialwavescatterswithnocouplingtoanyotherpartialwaveandcanbecalculatedseparatelyfornon-reactivescattering.ForreactivescatteringamoreinvolvedapproachisnecessaryaswasdiscussedbrieyinSection 2.4 ,butthefundamentalpatternremainsthesame.AlthoughthesuminEquation 5 isinnite,thecontributionsfor 78

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partialwaveswithangularmomentumbeyondlmaxkr0,wherer0istherangeofthepotential,willbenegligible;forshort-rangepotentials(kr0smallinsomesense)onlytherstfewpartialwaveswillcontributesignicantlyandthesumcanbetruncatedafterlmaxwithoutseriouserror[ 25 ]. Asymptoticallyallparticlesmustreducetofree(non-interacting)particles;thismeansthattheirradialwavefunctionsforeachpartialwavemustbeequaltotheradialfreewavefunctionsuptoaphaseshiftcausedbythepotential.Equation 5 showsthisphaseshiftexplicitlyfortheasymptoticwavefunction: l(k;r))401()222()403(!r!11 rAlsin[kr)]TJ /F5 11.955 Tf 11.96 0 Td[(l=2+l(k)]Pl(cos)(5) whereAlisaconstantandl(k)isthephaseshift,thesemiclassicallimitofwhichwasusedinSection 5.2.2 .Comparingthecoefcientsoftheincomingwaveandthefullwavefunction(givenbyasumoverEquation 5 ),whichcoefcientsmustbethesamesincetheinteractiononlyproducesoutgoingwaves,wendthatthecoefcientsAlmustbegivenby Al=2l+1 2ikexp(i[l=2+l])(5) andthusthecompletewavefunction(Equation 5 )reducesto (k;r))401()222()403(!r!1exp(ikz)+"1Xl=0(2l+1)e2il)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2ikPl(cos)#eikr r(5) whichimpliesthatthepartialwavecoefcientsal(k)mustbegivenby al(k)=e2il)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2ik(5) andthepartial-waveSmatrixelementforscatteringintopartialwavelis[ 25 ] Sl(k)=exp(2il(k)):(5) 79

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Notethatimplicitinthisderivationwastheassumptionofelasticityki=kf=k.Thiscanberelaxedbytheinclusionofthefractionkf=kiinthecalculationofthedifferentialandintegralcrosssection. Fromthiswecancalculatecrosssections:theelasticdifferentialcrosssectionisgivenby d=d=jf()j2=1 k1Xl=0(2l+1)eilsin(l)Pl(cos)2(5) andtheintegralcrosssection(afterusingtheorthogonalityconditionforLegendrepolynomials)becomes =4 k21Xl=0(2l+1)sin2l:(5) Asanexample,takethecaseofthehard-spherepotential(Equation 5 .ThewavefunctionobviouslyiszeroforrRandidenticaltothefreewavefunctionforr>R.Thuswesolvebyseparationofvariablesfortheradialportionofthewavefunction:[ 25 ] F(r)=XlFl(r)=Xl(Aljl(kr)+Blnl(kr))(5) wherejl(kr)andnl(kr)arethesphericalBesselfunctionsoftherstandsecondkind,respectively.SinceFl(r=R)mustvanishforalllseparately,weobtain Bl=)]TJ /F5 11.955 Tf 11.56 8.08 Td[(jl(kR) nl(kR)Al:(5) Substitutingintotheasymptoticform(Equation 5 ),weimmediatelyseethattheradialportionmustbegivenby Fl(r))402()222()402(!r!11 kr[Alsin(kr)]TJ /F5 11.955 Tf 11.96 0 Td[(l=2))]TJ /F5 11.955 Tf 11.96 0 Td[(Blcos(kr)]TJ /F5 11.955 Tf 11.96 0 Td[(l=2)]=(A2l+B2l)1=2 krsinkr)]TJ /F5 11.955 Tf 13.15 8.09 Td[(l 2+l(5) whichdeneslas l=tan)]TJ /F3 7.97 Tf 6.58 0 Td[(1Bl Al=tan)]TJ /F3 7.97 Tf 6.59 0 Td[(1jl(kR) nl(kr):(5) 80

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Forthewidthofthepotentialusedabove(R=1:0au)andkeeping100partialwavesforanelectron(m=1)withenergyE=k2=2m=2:0hartree,thedifferentialcrosssectioniscalculatedaccordingtoEquation 5 anddisplayedinFigure 5-4 .Notethatforthiscase,100partialwavesismanymorethanneededsimpletestsshowthattheintegralcrosssectionisconvergedtoarelativeerroroflessthan1.0x10)]TJ /F3 7.97 Tf 6.59 0 Td[(7afteronly5partialwaves.Thedifferentialcrosssectionattensoutasapproacheszerothisisindirectcontrasttotheclassicalcase,wherethecrosssectiondivergesinthislimit.Increasingtheenergy(forthiscase)onlychangesthenumberofpartialwavesneededforconvergencethecrosssections(differentialandintegral)areindependentoftheenergy. 5.3.3CrossSectionsforTime-DependentCalculations Alloftheprecedingmethodsareeitherformallytime-independentorareunderstoodtobetakentothatlimit.Whatisneededisamethodthatcreatespartial(andintegral)differentialcrosssectionsofcomparablequalitytotheSmatrixmethodsbutthatcanbeappliedeasilytotime-dependent,wave-packet-basednumericalcomputations.Thissectiondiscussesonesuchmethod,adaptedforENDwavepropagation. WestartwiththeformalismderivedinSection 5.3 .Inourcase,wehaveawavepacketdenedasymptotically: (x;t=0)=g(x;z)eik0z(5) whereg(x;z)isanenvelopefunction;inENDwavethisisaGaussianofform g(x;z)=Aexp()]TJ /F5 11.955 Tf 9.3 0 Td[(x2=2s2x)exp()]TJ /F4 11.955 Tf 9.3 0 Td[((z)]TJ /F5 11.955 Tf 11.95 0 Td[(z0)2=2s2z):(5) Thewidthparameterssxandszarechosensothatsxsz;thisguaranteesthatthemomentum-spacerepresentationiswellconnedinthexdirectionandhasnitewidthinthezdirection,centeredatkz=k0.Aisthenormalizationconstantsothath (x;t=0)i=1.WewishtopropagateitforwardforanitetimeTsufcienttoremove 81

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theentire(scattered)wavepacketfromtheinteractionregion.Atthispointweneedtoanalyzethewavepacketandextractthescatteringamplitude(similartoEquation 5 ). Forthesakeofthisdiscussion,assumethatwehaveallowedthewavefunctiontoevolvethroughtheinteractionregionandknowthewavefunction (x;t=T)atsometimeT,chosentomake (x2I;t=T)negligible,whereIistheinteractionregion.SinceEquation 5 isonlydenedforr!1andxedk,wetransformtomomentumspaceviaspatialFouriertransform,yieldingthemomentum-spacewavefunction: ~ (k;T)=Zdx (x;T)exp(ikx):(5) Sincetheinitialwavefunctionhadnegligiblewidthinkx,weknowthattheentirewavefunctionfor^k6=^zmusthavebeenscatteredbythepotential.Thus,forallkfnotalongtheinitialdirection,thenuclearscatteringamplitudeandelasticdifferentialcrosssectionbecomef(kf;k0)=kf k01=2~ (kf6=kf^z;T) (5)dE d=jf(kf;k0)j2: (5) Crosssectionsat=0canbecalculatedbyinterpolation;thisisinaccordwiththeexperimentalmethodology,whereparticleuxintheforwarddirectionisnotmeasureddirectlyinthepresenceofascatterer. Forsystemswithelectronicenergeticeffects,thenuclearscatteringamplitudecanbecalculatedasaboveforeachresultingsurface.Alltransitionsbetweensurfacesoccurbyconstructionforx2Isothesurfacesareindependentforx=2I.Calculationofstate-to-statedifferentialcrosssectionsthenrequiresthefollowingprocedure.First,wecalculatefromENDynetrajectoriesthestate-to-statetransitionamplitudesaij=hijjiwherethestatesiaretheorthogonalSCForbitalsfortheisolatedatomsormoleculesandjaretheENDdynamicorbitals(formoredetailsseeRef.[ 43 ]).Thedifferential 82

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crosssectionscanbethenassembled: dij d=2XIaijfI(E;)2(5) whereIlabelssurfaces,iandjelectronicstates,E=k2f=2m,and=tan)]TJ /F3 7.97 Tf 6.59 0 Td[(1(kx=kz).CurrentlyinENDwaveweonlyimplementtherstpart(thecalculationofthenuclearscatteringamplitudesfI(E;).Thetoolsexisttoextractthestate-to-statetransitionamplitudesaij;furtherworkisneededtoimplementandtestthematchingandinterpolationproceduresrequiredtoproperlyperformthesuminEquation 5 .Wedonotexpectthatitwillbedifcult,merelytime-consuming. Extractionofcrosssectionsfromwavepacketstudies(suchasweimplementinENDwave)isstraightforward;theprimarydifcultiesareinproperlyinterpolatingtheresultsandavoidingorremovingnumericalartifactssuchaswindowing(aliasing).TheseconcernsarediscussedinmuchmoredetailinChapter 6 83

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Figure5-1. Schematicdrawingoftrajectoriesforclassicalprojectilescatteringshowingimpactparameterb,distanceofclosestapproachR0,andscatteringangle. Figure5-2. Energy-independentdeectionfunction(b)forhardspherepotentialwithR=1au. 84

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Figure5-3. Deectionfunction(b)forMorsepotential[ 33 ]withE=0.5hartree,=4hartree,a=1:0au)]TJ /F3 7.97 Tf 6.59 0 Td[(1,andrm=1:0au. Figure5-4. Differentialcrosssection(inau2)calculatedbypartialwaveexpansionforelectronscatteringoffofahardspherewithradiusR=1auatenergyE=2hartree. 85

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CHAPTER6APPLICATIONOFTHEENDWAVEALGORITHMTOSIMPLESYSTEMS Atthecurrenttime,thealgorithmandmethodsdescribedabovehavebeenimplementedinPythonandarebeingtestedagainstbothmodelsystemsandsimplerealsystems.Presentedherearesomeofthebenchmarkingandmodel-potentialtestresultsinbothoneandtwospatialdimensions. 6.1GeneralConsiderations ThecurrentimplementationofENDwaveisinfourpartswithauxiliarytools,allwritteninPython( http://www.python.org )usingthenumpyandscipylibraries(availableat http://www.scipy.org ).Twoofthemodulescontaincodecommontobothoneandtwodimensionalscattering;thesetwoarethemainmoduleENDwave.py,containingtheinitializationroutinesanddimension-independentfunctioncallstothecorrectiterators,andthepotentialhandlingroutinereadpotential.py,whichcreatesoneandtwo-dimensionalanalyticsurfacesaswellasreadpotentiallescreatedfromENDtrajectories.Thepotential-handlingmodulealsoimplementstheabsorbingboundaryconditions.Theothertwomodules(run1d.pyandrun2d.py)containdimension-specicroutinesforcreatingtheinitialstate,evolvingthestatefromonetimesteptothenext,connectingthesurfaceacrossthefold,andreadingthenalsavedstateandextractingthedifferentialcrosssection.TheusermustsupplyENDwave.pywithacorrectlyformattedinputlecontainingreferencestothebinaryles(innumpyformat)containingthepotentialsurfaces;ENDwave.pythenhandlesallthecallstotheothermodules.Nographicaluserinterfaceisavailableatthistime. Ascurrentlyimplemented,thenumericalefciencyofENDwaveisnotoptimal;mostofthiscanbecorrectedwithoutchangingthealgorithm.Thetheoreticallowerboundforoperationcountforthesplit-operatoriteratorateachtimestepisthatofthefourFastFourierTransforms(FFTs)requiredeachstep:4Nmlog(N)[ 28 ],whereNisthenumberofgridpointsinanydirectionandmisthedimensionalityofthesystem.Thecostofthe 86

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continuityconditionatthefoldeachtime-stepisN.Foronedimensionothereffectssuchasthestopcondition(theprogramhaltswhenlessthanonepercentofthewavefunctiondensityremainsontheprimarygrid;testingthisisanN2operation)greatlyexceedthetimerequiredfortheFFTitself.IntwodimensionsthescalingoftheFFTissuchthatitisthedominanteffectoncomputationalcomplexity. 6.2One-DimensionalSystems 6.2.1AnalyticPotentials Arsttestshowstheeffectofhavingmultiplesurfaces.Figure 6-3 showstheprobabilityoftheincidentparticlebeingfoundatthegivenenergywiththeindicatedinternalconguration(surface).ThisparticularexamplewascalculatedusingtwoYukawa[ 25 ]typepotentials,offsetby0.1hartree.Asthegureshows,theenergydistributionsareverysimilar,exceptthattheprobabilitiesofbeingfoundonthehighersurfaceareoffsetbythesame0.1hartreeasthepotentials.Thecauseofthesmall('10)]TJ /F3 7.97 Tf 6.59 0 Td[(10)lowenergypeakontherearrangementcurveisunknownatthistimeandmaybeanartifactoftheabsorbingpotential. 6.2.2AsymmetricSquareBarrier AsdiscussedinSection 3.1.1 ,asimpletestcaseforwhichanalyticsolutionsareavailableistheasymmetricsquarebarrier.Forthisexample,thepotentialenergycurveisdenedby V(x)=8>>>><>>>>:0forx0V0for0a(6) whereV0=1:0hartree,V1=0:5hartree,anda=2:0au.TheanalyticformoftransmissionfunctionT(E)foranincidentplanewaveisshowninFigure 6-1 .NotethesharponsetoftransmissionwhenE=V1andthedecayingoscillationsforE!1.Therstisaconsequenceofconservationofenergy:nopropagatingstatesexistforEa.Themaximaoftheoscillationsoccurwhenp 2m(E)]TJ /F5 11.955 Tf 11.96 0 Td[(V0)=nwithn 87

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integer;theresultingperfecttoftheover-barriersegmentofthewavefunctionresultsinperfecttransmission.Allofthisistobecomparedtothedashedlineinthesamegure(Figure 6-1 whichrepresentstheclassicaltransmissionfunctionforthesamepotential.Classically,thetransmissioniszerountilE=V0,andthenisunity;classicalparticlesareeitherfullytransmittedorfullyreected. Planewaves(iemono-energeticincidentfreeparticles)areanapproximationtoaphysicalparticle,especiallyforheavierparticlessuchasatomsormolecules.Toexaminetheeffectsofanincidentparticlewithuncertainmomentum(aswellastocomparebetterwiththeresultsfromENDwave)wecalculatedthetransmissioncoefcientasafunctionoftheaverageenergyofaGaussianwavepacket,givenbyEquation 6 ,wherek=p 2mEisthemomentumoftheincidentplanewavecomponentandk1=p 2m(E)]TJ /F5 11.955 Tf 11.95 0 Td[(V1)isthecorrespondingplanewavecomponentintheexitchannel. T(E)=ZdkAk1 kexp()]TJ /F5 11.955 Tf 9.29 0 Td[((k)]TJ /F4 11.955 Tf 12.28 3.15 Td[(k)2)Tpw(k)(6) Theeffectofthewavepacketsmoothingistoincorporateplanewaveswithbothhigherandlowerenergiesthanthepeak.Thismeansthatforallbutthenarrowestpacketstheonsetoftransmissionisnolongersharp;evenatnearlyzeromeanenergysomeofthepackethasenoughenergytotunnelthroughthebarrier.Sincethereisnolongerasinglewavenumber,theperfectwave-lengthmatchingthatresultedintheoscillatorypeaksiswashedoutbytheothercomponents,resultinginaslowmonotonicrisetocompletetransmissionastheenergyincreasesabovethebarrier.TheseeffectscanbeseeninFigure 6-2 Alsotobeseeninthatsamegure(Figure 6-2 )istheresultsoftwoENDwavecalculations.Therst,signiedbythedashedlinesandpointsisasinglesurfacewavepackettunnellingcalculationwiththesamebarrierasabove;thecomputationalregionis20auwideandweused512gridpoints.Thisservesasacheckthatthepropagation(withoutfoldedsurfaces)isworkingproperly;theaccuracyisquiteacceptable.The 88

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second,shownasdisconnectedpoints,isafolded-surfacerepresentationofthesameproblem.Thepotentialcurvehasbeenhingedaboutthepointx=1:0auandrotatedbackonitselftoforma10-aulongfoldedsurfacewithatransferpointatx=1:0au.Notethatthesedotslieveryclosetothepreviouslycalculatedones,boththeoreticalandcomputed,showingthatthefoldedsurfacedynamicsaccuratelymodelthephysicalones.Theaccuracyiswithintheexpectednumericalaccuracy,giventhenumberofpointsandtimestepchosen. 6.3Two-DimensionalSystems 6.3.1AnalyticPotentials AsarsttestwepresenttheresultsfortwodimensionalYukawapotentials(Equation 6 ). Vi=giexp()]TJ /F5 11.955 Tf 9.3 0 Td[(kiR) R+dVi:(6) Theparametersforthetwopotentialsareg1=2:0hartree,g2=1:0hartree,k1=k2=2:0au)]TJ /F3 7.97 Tf 6.59 0 Td[(1,dV1=0hartree,anddV2=0:1hartree.Thisproducestwopurelyrepulsivepotentialsthatriseas1/RnearR=0;thetwooverlapatR=1:0auandhaveanasymptoticdifferenceof0.1hartree(2.62eV). Figures 6-4 and 6-5 showacolormapoftheelasticandtransfer,respectively,momentumdistributionoftheasymptoticoutgoingwavefunction.AscanbeseeninFigure 6-4 ,twostructuresappear:onecenterednearpz=5au)]TJ /F3 7.97 Tf 6.59 0 Td[(1andtheotherrightnearpz=0.Theforwardpeakisthereectedpulsefromthepotentialbarrier,hardlyscatteredfromitsinitialconguration.ThefainterbandsoneithersidearetheeffectsofFourieraliasingcausedbytoonarrowofawindowusedtoaccumulatetheasymptoticwavefunction.Itispresentedhereasanexampleofthepathologiesthatcanoccurwhentheinitialparametersaresetwithoutcare.Usingagreaternumberofgridpointsandincreasingthewidthoftheasymptoticregionwouldbothpartiallyalleviatethiseffect,butnothingcancompletelyremoveit.Allotherresultsshouldbeinterpretedwiththisinmind.Figure 6-5 showsthetransferredportionofthesamewave 89

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function.Thepeakisthedirecttransfer,butthelowmassmeansthatthewavepackethasspreadenormouslybythetimeitreachestheasymptoticregion.Interferenceandaliasingprovidethecircularbanding.Notethatonlytherstquadrantofthewholemapisshown:thesecondquadrant(negativepxandpositivepz)isthemirrorimageofthisquadrant,andallnegativepzpointscanonlybereachedbyaliasing.Therstquadrantcorrespondstotheangularregion0=2. Figure 6-6 showstheelasticandtransferdoubly-differentialcrosssectionsatthemeanpacketenergyof408eVwithareducedmassof1.Asthebarrierheightisonly27eVinheight,thetransferredcrosssection(dashed)isbetweenveandmanyordersofmagnitudehigherthantheelastic.Itdoesnotfalloffwithanglemuch;partofthisisduetothealiasingproblemdiscussedabove,butpartofitisthatthepotentialhasnoforbiddenregionforR=p x2+z2>1(wherethepotentialsoverlap)Thisleadstoalmostsphericaltransfer.Theelasticcrosssectionfallsoffrapidlywithangleaswemoveawayfromthecentralpeakintotheside-lobesofthealiasing.Thestructuresarepredominantlyaliasing-related.Thissystemshowsthatimproperorinadequatechoiceofgridsleadstoscatteringthatisqualitativelycorrect,butwhosecrosssectionscannotbetrustedduetoaliasingeffects.Ingeneral,thecurrentimplementationofENDwaveshouldbeusedtodothescatteringabetterimplementationofthecross-section-determinationroutinesisrequiredbeforethecrosssectionscanbefullytrusted. 6.3.2H++H(1s) TherstrealisticsystemtestedwithENDwavewaslow-energy(10eVcollisionenergyinthelabframe)protonscollidingwithground-statehydrogenatoms.The1sgroundstateissphericallysymmetric,soonlyoneorientationisneeded.Theelectronicdynamicswecalculatedusingthetriple-zetaaugmentedbasisofDunning[ 44 ];thebasissetreproducesthelowestfewatomicelectronicstatesadequatelyforthistest.Thedegeneratenatureoftheatomicenergylevelsensuresthat,whileelectron 90

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transferisnon-trivial,thetransferwillbedominatedbytheresonanttransferbetweengroundstatesonthetwonuclei.Thispresumptionisbornoutinpracticeasseeninthepotentialsurface(Figure 6-7 )extractedfromtheENDynetrajectories.Theback-scatteredtrajectoriesendupasymptoticallywithenergiesidenticaltothatwhichtheyhadinitially(beforethecollision).Thisresultsinasingle,purelyrepulsive,effective-potentialsurface. LikefortheanalyticYukawapotentials(Subsection 6.3.1 ),wepresentthecolormapoftherstquadrantofthemomentumdistributionoftheasymptoticwavefunction.Thewhiteregionsnearpx=0arehighpeaksthatareoffthescaleusedtoshowtherestofthedistribution.Iftheyareincluded,thetailisnearlyinvisibleatthatscale.Aliasingispresent(asseenbythebanding),butnotnearlytothesamemagnitudeasintheYukawacaseaboveastwiceasmanypointsineachdirectionwereusedandthereducedmassis918.0(thereducedmassoftwoprotons),insteadof1asintheYukawacase.Thepatternisnearlycircularandtightlylimitedtonearthemagnitudeofthemeanvalueofthemomentumfortheincidentpacket. Thenuclearangulardifferentialcrosssectionanenergyof12.0eVascalculatedbyENDwaveisshowninFigure 6-9 .Asthereisonlyonesurface,notransfercrosssectionsareshown.Thisisnottoimplythatnochangeintheelectronicstateoccursjustthatsincetheentiretyofthetransfer(atthisenergy)istothedegenerateprojectile1sstate,regionsofnon-trivialelectrontransferhavethesameasymptoticenergyasstateswithoutanytransfer.Toderivetheproperelectrontransfercrosssections,onewouldhavetoextractthestate-to-statetransitionamplitudes(usingprojectionoperatorsfromthedynamicstateontothestationaryself-consistenteldorSCFstates)andmultiplythecomplexamplitudes(asafunctionofangle)bythecorrespondingmomentumcoefcientsofthenuclearwavefunctionbeforesquaring.Nevertheless,afewinsightscanbegleanedfromexaminingthenucleardifferentialcrosssection. Astobeexpectedatsuchlowenergy,thecrosssectionstaysnearlyconstantouttonearly30degrees,unlikeathighenergywheredifferentialcrosssectionstypicallyhave 91

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fallenoffmultipleordersofmagnitudebythesameangle.Thesharpdropoffabove30degreesisconsistentwiththemaximumforwardscatteringangleexpectedfromtheshoulderinthepotential.Theoscillationsathigherangleseemtobecharacteristicofmanysystemsstudied.Theymaybepurelynoise,butitseemsunlikelythatallsystemsstudied,nomatterthegridspacingorotherdetailswouldshowthesametypeofoscillations.Thesmalleroscillationsontheshelf,betweenoneand10degreesaremuchmorelikelyacombinationnoiseandresidualsfromthenitegridspacingusedtoextractthecrosssections;ascurrentlyimplemented,thealgorithmchoosesallpointswithinauser-selectableenergyincrementofthedesiredenergyandusesthemasthepointsatthatenergyforthecrosssection.Thisleadstosmall-scaleuctuationsinthecrosssection.AllofthistogetherleadsustobelievethatabetterimplementationofthealgorithmsdescribedinSubsection 5.3.3 wouldgreatlyimprovethereliabilityofthecrosssectionsproducedbyENDwave. 6.3.3H+He2+ Anothersinglesurfaceexample,thistimewithbothattractiveandrepulsiveregions,islow-energy(<10eV)collisionsofHe2+(alphaparticle)withH(1s).DespitethestrongCoulombattractionbetweenthealphaparticleandthegroundstateelectron,thewidegapsbetweentheHe+electronicstatesandtheHstatesleadtoverylittletransfertheprobabilityoftransferislessthan10)]TJ /F3 7.97 Tf 6.59 0 Td[(6foralltrajectories.ThepotentialsurfaceusedforENDwaveevolutionisshowninFigure 6-10 .Theblankregionsformtheclassically-forbiddenregionatthisenergy;notrajectoriespenetratedthisregion. Thecolormapoftheasymptoticmomentumdistribution(Figure 6-11 )showssignicantdeviationsfromtheonesshownpreviously.Thesolidlineisalineofconstantenergyequaltothemeanvalueoftheenergy(0.92eV).Notethatthepeakoccursnearpx=0:5au)]TJ /F3 7.97 Tf 6.58 0 Td[(1insteadofnearpx=0.Thisisthesignatureofacaustic(rainbowangle).Abovethisrainbowangle,thewavefunctionfallsoffveryquicklywithangle,ascanbeseeninFigure 6-12 ,whichshowsthedifferentialcrosssectionatthepeakenergyofthe 92

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incidentpacket.Thecrosssectionismarredbysignicantnoiseandaliasingeffects,butshowsthehillnearfourdegrees(theapproximatelocationoftherainbowangle)followedbyasharpdropoff.Again,interpretationofthecrosssectionsmustbedoneinlightoftheconcernsraisedabove. 6.3.4H++H(2s) Anexampleofarelativelycomplicatedfoldedsurfacederivesfromcollisionsofprotonswithhydrogen(asabove,exceptwiththetargetintherstexcitedstate).ENDtrajectorieswerecalculatedfromb=0:0autob=10:0auusingthesamebasisasinSection 6.3.2 .Theinitialstateofthetargethydrogenwas2s,leadingtoaninitiallysphericalelectronicwavefunctionandobviatingtheneedformultipleorientations.Thiswouldnothavebeenpossiblewithanyofthep-typehydrogenfunctions. Theunfoldedpotentialsurface(c.f.Section 3.1.1 )andtwo-surfacedecompositionareshowninFigures 6-13 6-14 ,and 6-15 .Theunfoldedsurface(Figure 6-13 )showsthecomplicatedtopologyofthesurface.Nointeractionhappensuntilv0t>10auwherebeginsaslightattractiveregion.Alargepotentialhillisthenencountered(thenuclearrepulsion);onthefarside,thetrajectoriestakeononeoftwocharacters.Therstisaelectronicde-excitationtothe1sgroundstate(oneithercenter)alongwithasmallincreaseinnuclearkineticenergy;thesecondispartialtransferorde-excitation,leadingtoenergiesintermediatebetweenthegroundstateandtheexcitedstate). Asimpletwo-surfacedecompositionisshowninFigures 6-14 and 6-15 .Notethattheslopeofthehillfortheuppersurfaceisshallowerthanthatfortheincidentsurface;thisisduetothepartialreductioninCoulombforcesduetotheelectrontransfer.Almostallofthe(small)differencesbetweensurfacesareoutsidetheattractiveregionasexpectedthisiswhentheelectroncloudsonthetwocentersseparateandtheinteractionisreducedtojustpolarization-type(vanDerWaals)forces[ 45 ]. Colormapsoftheasymptoticmomentumdistributions(elasticandtransferrespectively)areshowninFigures 6-16 and 6-17 .Theelasticscattering(Figure 6-16 ) 93

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showstheeffectsofaliasingthesameasalltheothersystemsexamined,butotherwiseisconnedtoasmallareaofmomentumspacecenteredattheincidentenergy,asshouldbeexpectedforelasticscattering.Thewidthofthescatteredpacketisthesameasthatoftheincidentwavepacket.Lookingatthewavefunctiononthesecond(non-incident)surface(Figure 6-17 ),weseeamuchdifferentresult.Here,mostofthescatteredpacketisoff-axis(thebeamthatisun-scatteredisalsonottransferredatall)andmuchbroadenedbythetransfer;thecenterofthebandisabout9.0au)]TJ /F3 7.97 Tf 6.59 0 Td[(1,whereasontheelasticsurfaceitiscenterednear8.5au)]TJ /F3 7.97 Tf 6.59 0 Td[(1.Thisconformstoexpectations,asthehighermomentumportionsofthebeamhavegreaterprobabilitiesoftransferring(aswasseeninthediscussioninSection 6.2.2 ).Thestipplingistheeffectsofaliasingandmutualinterference. Notmuchcanbesaid(forreasonsdiscussedintheprevioussections)aboutthedifferentialcrosssections(Figure 6-18 ).Attheselowenergies(4:4eVmeanpacketenergy),notmuchtransferoccurs.Theelasticcrosssectionfallsoffslowlybutconsistentlywithangle;thetransferiseffectivelyconstant(barringnoise)outtoquitehighanglesbeforefallingoffveryrapidly.Theeffectsofaliasingmakeithardtotellwhichifanyofthesmallerstructuresareactualandwhichareartifacts. 6.4DiscussionofResults ApplyingENDwavetothevesystemspresentedhereinleadsustosomegeneralconclusionsaboutthevalidityandaccuracyofthemethod.Extractionofthepotentialsurfacesistrivialforasinglesurface;formorecomplicatedtopologiesitispossiblebutstillrequiressignicantuserintervention.Itwouldbeadvisedtodeviseabettersetoftoolsfordistinguishingthesurfacesfromeachotherautomatically.Thenumericalimplementationofthewavepacketdynamicsisadequateforone-dimensionandforgeneratingthemomentumdistributionsofthescatteredstateintwodimensions.Signicantrenementisneededfortheimplementationofthecross-section-extractionalgorithmsintwodimensions. 94

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Figure6-1. Analytictransmissionofincidentplanewaveonasymmetricsquarebarrier(Equation 6 )showingquantumtunnelling(solid)andclassical(dashed) Figure6-2. TransmissionofGaussianwavepacketthroughasymmetricsquarebarrier(Equation 6 ):ENDwavefoldedsurface(points),ENDwavesingle-surfacetunnelling(dashedwithpoints),andexact(dashedwithoutpoints). 95

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Figure6-3. Energydistributionofprobabilityforelastic(solid)andrearrangement(dashed)reactionsontwoYukawapotentialsasymptoticallyseparatedby0.1hartree Figure6-4. ColormapoftheelasticmomentumdistributionfortransferbetweentwoYukawapotentials;notealiasing(banding)duetoinsufcientmomentumresolution.Forwardpeakistheportionsthatwerejustreectedfromthebarrier;peaknearzeromomentumhasbeentransferredbackfromuppersurface 96

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Figure6-5. ColormapofthetransfermomentumdistributionfortransferbetweentwoYukawapotentials;notethelargespreadinmomentumduetothelightmass(m=1)andthecircularbandingpatterns(aliasing)duetoanarrowtransferwindow. Figure6-6. Elastic(solid)andtransfer(dashed)doubly-differentialcrosssectioninau2fortwoYukawapotentials[ 25 ]asafunctionofangleatthemeanpacketenergyE=15hartree(408eV) 97

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Figure6-7. EffectivepotentialenergysurfaceforH++H(1s)at10eVinthelaboratoryframe.Sinceallelectronictransfersareresonant,onlyonesurfaceresults. Figure6-8. ColormapofthemomentumdistributionofscatteredwavefunctionforH++H(1s).Whiteareanearkx=0;kz=14isduetoclippingthecolorscale;itishigherthan1.2x106. 98

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Figure6-9. ElasticdifferentialcrosssectionforH++H(1s)fromENDwaveforE=0.44hartree=12.0eVinthelabframe. Figure6-10. EffectivepotentialenergysurfaceforHe2++H(1s)at10eVinthelaboratoryframe.Attractiveportionofpotentialleavesinaccessibleholeinthecenter.Noelectronictransfersoccur(atthisenergy). 99

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Figure6-11. ColormapofthemomentumdistributionofscatteredwavefunctionforHe2++H(1s).Whiteareanearkx=0,kz=10isapeakoffthescaleoftheplot. Figure6-12. ElasticdifferentialcrosssectionforHe2++H(1s)fromENDwaveforE=0.034hartree=0.92eV 100

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Figure6-13. Tunnellingpotentialsurface(c.f.section 3.1.1 )forH++H(2s)at10eVlaboratorycollisionenergy.Notetheripplesforv0t>15;thesewillbecometheuppersurfacewhenfolded Figure6-14. LoweroftwofoldedpotentialsurfacesforH++H(2s)at10eVlaboratorycollisionenergy.Notetheforbiddenregionandthesmallattractiveregionaswellasthecomplicatedexitregion(z>0). 101

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Figure6-15. UpperoftwofoldedpotentialsurfacesforH++H(2s)at10eVlaboratorycollisionenergy.Notetheforbiddenregionandthesmallattractiveregionaswellasthecomplicatedexitregion(z>0). Figure6-16. Colormapofthemomentumdistributionoftheelastically-scatteredwavefunctionforH++H(2s). 102

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Figure6-17. Colormapofthemomentumdistributionofthetransferred(inelastic)wavefunctionforH++H(2s). Figure6-18. Elastic(solid)andtransfer(dashed)differentialcrosssectionforH++H(2s)fromENDwaveforE=0.16hartree(4.4eV). 103

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CHAPTER7CONCLUSIONS 7.1OverviewofFolded-PotentialWavePacketDynamics(FPWPD) Afterreviewingtheavailablemethodsforapproachingthereactivescatteringproblem(Sections 2.1 2.5 ,itseemsthatthereisanicheforaquantumcorrectiontothesemi-classical,non-Born-Oppenheimermethods.Inthelatterhalfofthispaper,Ihavelaidoutaproposalandinitialimplementationofatwo-stagecorrectiontoElectronNuclearDynamics(END),whichshouldatleastpartiallyllthisniche,tobecalledFolded-PotentialWavePacketDynamics(FPWPD)andENDwaverespectively.Itsharesacommonideologywiththemean-eldsurfacehoppingsetupdescribedinSection 2.2 ,builtonaframeworkoffoldedpotentialsurfacesderivedfromENDsemi-classicaltrajectorycalculations.Itcanbeextendedtomultiple-center,multiple-electronsystems,andshouldworkatmostenergieswithoutneedforreformulation.Thisapproachaddsbackintunnellingandquantuminterferenceeffectsfordifferentialcross-sections,butshouldnotbeexpectedtosignicantlyincreasetheaccuracyofENDfortotalcross-sections.Italsodoesnotaddresstheaveraged-trajectoryissueinherentintheclassicaltrajectories. 7.2PerformanceofENDwave ExtractionofthepotentialsurfacesfromENDtrajectoriesandconstructionofinterpolatedsurfacessuitableforwavepacketstudiesisstraightforwardandprovidesthehoped-forresults.TheresultingsurfacescanbeunderstoodeitherinthetunnellingorintheCartesianscatteringdescription,aswasshownwithexamples.Theimplementation(ENDwave)works;boththesinglesurfacedynamicsandthetransferbetweensurfacesfollowsthephysicalprinciples(conservationofenergyandmass)asfarastheunderlyingintegratorsallowthesplit-operatormethoddoesnotfullyconserveenergy[ 30 ].AsdescribedinSection 7.3 ,manyrenementsandaugmentationscanandshouldbemadetoENDwavebeforeitisusedinaproductionenvironment. 104

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7.3FurtherRenements Asnotedabove,thisworkpresentsthebasicfoundationforafolded-potentialrepresentationofthescatteringproblem,basedonENDtrajectoriesandelectronicdynamics.Inthissectionwepresentextensionstoboththetheoreticalunderpinningsandtothenumericalimplementation(ENDwave).Itshouldbenotedthatnoneoftheseshouldalterthefundamentalunderstandingspresentedhereinnorrequireareformulationofthetheory;theymerelyextendtherangeofapplicabilityofthetheoryorimprovetheaccuracyoftheimplementation. 7.3.1ExtensiontoThreeorMoreSpatialDimensions Aspresentedinthebodyofthetext,ENDwavewaslimitedtooneortwospatialdimensions(plustime).Thiswasentirelyforconvenienceinexpositionandnumericalcalculations.Nothinginthederivationofthefoldedpotentialsurfaceslimitedthenumberofdimensionsatall.Ofcoursevisualizationofthefoldedsurfacesin3+dimensionsbecomesquitedifcult;visualizingtheunfoldingprocedureisalsoprohibitiveinthreeormoredimensions.Themathematicalandphysicalinsights,ontheotherhand,arenotaffectedbythehigherdimensionalityatall. ToconductanENDwavecalculationinthreeormorespatialdimensionsthefollowingstepswouldhavetobetaken.NotethatextendingENDwavetohigherdimensionalityismerelyaprogrammingtask;thephysicalmodelisindifferenttodimensionality. 1. ConductENDcalculationsfromarepresentativesampleoforientationsoverasufcientlydense(c.f.Section 3.3 )gridofimpactparameters. 2. Extractpotentialvaluesateachsampledpointandconstructpotentialsurfaceincenter-of-massCartesiancoordinates.Thiswillrequirenewscripts,asthecurrentextract-potential.pyscriptiswrittenfortwospatialdimensions. 3. Anewbranchofcodemustbeaddedtoaccommodatehigherdimensionalityindriver.py. 4. Alsoneededwouldbeanewiteratormoduleimplementingallthesamefunctionalityofrun1d.pyandrun2d.py.MostofthechangeswouldneedtobeintheintegrationandFouriertransformmodules. 105

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7.3.2ExtensiontoMolecularTargets ExtendingENDwavetomoleculartargetsissimilarinprincipleto,andinfactrequirestheextensiontohigherdimensionality.Havingsaidthat,thereareadditionaladaptationsneeded.Asitstands,ENDwaveignoresthetargetexceptasasourceofthedynamicpotentialenergysurface.Asallcalculationsaredoneinthecenter-of-massframe,themotionofthetargetistheinverseofthatofthe(calculated)projectile.Obviouslythiseliminatesthepossibility(weresuchpossibleforatomictargetsandprojectiles)offragmentation.Simplycalculationelectroniceffects(transfer,excitation,etc.)withmoleculartargetsrequiresonlythehigher-dimensionalyextensionsdescribedabove.ENDynealreadyhandlesthecaseof(classical)moleculartargets;theresultingfoldedpotentialsurfaceswouldbesignicantlymorecomplicatedthanthoseforatomictargets,buttheENDwavecodealreadyhandlesanarbitrarynumberoffolds.Calculatingfragmentationcrosssections(orroto-vibrationalexcitations,etc.)requiresadifferentapproachandfurtherstudy.Onepossibleapproachwouldbetoinverttheproblem:usethemolecularsystemastheprojectileandinitializeitwiththepropersetof(atomic)wavepackets.Thebestapproachtothisproblemisstillanopenquestionthatrequiressignicantfurtherstudy.Itmustbenoted,however,thatthebasicunderlyingtheoryofthefoldedsurfacesisagnosticastothestructureofthetargetsandprojectiles.Noassumptionsweremade,onlysimplicationsforeaseofpresentation.Thedifcultyisentirelyintheimplementationwhichremainsbeyondthescopeofthiswork. 7.3.3ExtensiontoMultipleActiveElectrons ElectronicdynamicswithinENDwavearecapturedinthefoldedpotentialenergysurfaces(andangularstate-to-statetransitionamplitudes)generatedfromENDynetrajectories.ExtendingENDwavetomultipleactiveelectrons(iecaseswheretwoormoreelectronshavenon-zeroprobabilitiesofexcitationortransfer)requiresalmostnoeffortbeyondthatinvolvedinextendingENDynetobetterhandlesuchsystems.Multiple-activeelectronsystemsfrequentlyinvolveelectronicwavefunctionswhich 106

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cannotbeaccuratelydescribedusingasingleFockdeterminant;multi-referenceENDyneisthusrequired.ENDwavecanhandlethesesystemswithoutsignicantalteration;theincreaseinelectronicactivitywillgeneratemorecomplicatedeffectivepotentialenergysurfacesbutthatcanbehandledascurrentlyimplemented. 7.3.4ImplementationImprovements Thenal(andlargest)areaopenforimprovementistheactualimplementationinsoftwareofthemethodsandalgorithmsdescribedherein.Thissectionpresentsanon-exhaustivelistofpossibleimprovementswithdiscussion.Theserangefromimprovingcalculationefciencybyusingadifferentintegratororrewritinginacompiled(ratherthaninterpretedlanguage)toallowinghigherenergiesbyimplementingdynamicgridsandbeyond. AsdiscussedinSection 6.1 ,thetheoreticalbestoperationcountpercycleisdominatedbytheNmlog(N)performanceoftheFastFourierTransform[ 28 ]wheremisthedimensionalityofthesystem.Thisistheabsolutelimitofperformance.AstheFourierTransformandothermathematicaloperationsaredoneusingtheBasicLinearAlgebraSystem(BLAS)librarieswrappedinPythonfunctions,weexpectthattheperformanceoftherawcomputationsisverynearthetheoreticalmaximum.Improvementsinper-stepcalculationefciencywillthereforecomefromreducingoverheadelsewhere.Overallperformancecanbeincreasedbyusingintegratorsthatallowlargertime-stepsatthesamelevelofaccuracy(reducingthetotalnumberofstepsandthusfunctioncalls)andbyusingdynamicgrids(allowingsmallernumbersofpointsandthusfasterindividualsteps).Itmustbestressedhoweverthatthesearepurelyimplementationimprovements.Theyareindependentofthetheory. Acurrentlimitationisthatthenumberofgridpoints(N)isdirectlyconnectedtothemaximummomentumrepresentableontheFourier-transformedgrid.TheNyquistcondition[ 28 ]statesthatforaspatialsamplingratex=L=(N)]TJ /F4 11.955 Tf 12.69 0 Td[(1),themaximumrepresentablemomentumis(2x))]TJ /F3 7.97 Tf 6.58 0 Td[(1=1 2(N)]TJ /F4 11.955 Tf 12.31 0 Td[(1)=L.Thusthereisatrade-offbetween 107

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therangeofenergyrepresentableandthespeedofthecalculation;largerdynamicrangesrequiremorepoints,whichthenrunslower.Thislimitationisaddressableinseveralways(whichcanbeimplementedsinglyorinconcert).First,onecouldaccepttheincreaseinper-steptimeandinsteaduseanintegratorsuchasanexpansioninChebyshevpolynomials[ 29 ]ortheLanczositerator[ 29 ].Thesemoreaccurateintegratorsrequiremanyfewerstepsforconvergenceandthuswillrunmuchfaster.Useofsuchiteratorsisnotwithoutproblems,however.UseoftheChebyshevmethod,forexample,requiresthattheenergiesbeband-limited,whichscatteringcalculationsarenot,atleastinprinciple.Also,sincetheChebyshevmethodisglobal,theenforcementofcontinuityatthefold(Section 4.5 )wouldhavetobecompletelyreimplemented,anditisnotclearatthistimehowthatshouldbedone. Onecanobservethatduringthecalculations,themomentumrepresentationofthewavefunctionisquitelocalized;mostoftheavailablespaceisunoccupied.Thisleadsustobelievethatasystemofembeddedordynamicgridscoulddrasticallyreducethenumberofgridpointsneededbyconcentratingthepointsintheregionsthewavefunctionoccupies.Sincetheregionofmomentumspaceoccupiedisnotconstant,thegridswouldhavetomovewiththewavepacket.Asearchoftherelevantliteraturesuggeststhatsuchregriddingispossible,butwehavebeenunabletondasuitablereferenceimplementationandcreatingourownwouldhavetakensignicantlymoretimethanwasavailable.Thisisapromisingregionoffurtherresearch,asitwouldsimultaneouslyremoveseverallimitations:therangeofenergieswouldnotbelimitedbythemassorthephysicalsizeofthesystems,andcalculationsonmorecomplexsystemswouldrunmuchfaster. Parallelizationispossible,buthasnotbeenexploredmuch.TheFFTroutinesarealreadymulti-threadedandarethedominatingfactorintherun-timeofthecalculations.Signicantworkwouldberequiredtoparallelizetherestofthecalculation,aseachstep 108

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requirestheinformationfortheentiregridatthepreviousstep.Itisunclearatthistimehowmuchbenettherewouldbetosuchatimeinvestment. 7.4FinalThoughts Manyproblemsbenetfrommultipleviewpoints.Thefully-coupledwavefunctionscatteringproblemforheavyparticipants(nucleiandmolecules)canbedenedexactly,butcannotbesolvedpracticallyexceptforafewarticialcases.Inthisworkwepresentanalternativetothecommonly-usedMolecularOrbitalCloseCoupling(MOCC)method,namelythatofFolded-PotentialWavePacketDynamics(FPWPD),withpotentialsurfacesderivedfromElectronNuclearDynamicsclassicaltrajectories.AlsopresentedisapreliminaryimplementationofFPWPDinPythonwithcalibrationandperformanceresults,calledENDwave.FPWPDisabridgebetweenthestationary-statelanguageofMOCCandthefully-dynamicpresentationnativetothescatteringproblem.Thekeyinsightisthatthedifcultandnon-uniquetransformationtotunnellingcoordinateswherethescatteringprocessiseasytounderstandisunnecessary.Thecalculationcanbecarriedoutinrelativecoordinates(thisimplementationusesCartesiancoordinates,butthatisnotafundamentalrequirement)wherethedifferentialoperatorsaresimpleanddonotrequireapproximation,whilestillretainingtheintuitiveaccessibilityofthemorecomplicatedproduct/reactantcoordinates.Thisavoidstheneedtocalculatecouplingelementsormodifythedifferentialoperatorsbasedontheindividualsystemathand.Everythingisuniquelydened.Thecostforallofthisisthatthetopologyoftheelectronicsurfacesusedforthewavepacketsismademorecomplex;thesurfacesfoldbackonthemselvesandhaveregionswherenotrajectoriespenetrate.Wehavediscussedtoolsandprinciplesfordealingwiththesecomplicationsandfounditstraight-forward.Alltold,FPWPDandENDwavehavemuchpromiseastoolsforexploringquantumscatteringofheavynuclei. 109

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[41] N.F.MottandH.S.W.Massey,TheTheoryofAtomicCollisions,Thirded.(OxfordUniversityPress,1965),p.369ff. [42] E.Mason,J.Vanderslice,andC.Raw,J.ChemPhys40,2153(1964). [43] N.L.Guevaraetal.,Phys.Rev.A83,052709(2011). [44] T.H.DunningJr.,J.Chem.Phys.90,1007(1989). [45] J.E.Lennard-Jones,Onthedeterminationofmolecularelds,inProc.R.Soc.London,,AVol.106,pp.463,1924. 112

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BIOGRAPHICALSKETCH BenjaminHallwasbornin1982inIdahoFalls,Idaho.HeenrolledatBrighamYoungUniversityin2000asaNationalMeritScholar,spenttwoyears(2001-2003)asamissionaryforTheChurchofJesusChristofLatter-daySaintsinLithuania,Latvia,andEstonia,andgraduatedfromBYUwithaBachelorofSciencedegreeinphysicsin2006.AfterbeingacceptedforPh.DstudiesattheUnviersityofFloridaintheDepartmentofPhysics,hehastaughtintroductoryclassesinthedepartmentandconductedresearchintheQuantumTheoryProject. 113