Quasiclassical and semiclassical formulation of the electron nuclear dynamics theory : a method for molecular dynamical processes

289 Pages·1997·10.6 MB·English

by  Morales, Jorge Alberto

QUASICLASSICALANDSEMICLASSICALFORMULATION
OFTHEELECTRONNUCLEARDYNAMICSTHEORY:A
METHODFORMOLECULARDYNAMICALPROCESSES
By
JORGEALBERTO MORALES
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1997
Tomymother, MarthaB.L.SamperiodeMorales.
ACKNOWLEDGMENTS
IwouldliketothankDr. YngveOhrnandDr. ErikDeumensfortheirwiseguidance,
warmencouragement,andadamantsupportduringtheseexhilaratingyearsintheirgroup.
Bothhavecreatedafertileandfriendlyatmospheretodevelopmyprofessionalskillsand
tosustain myinterests inscience.
Thisgratitudeisalsoextendedtotheothermembersofmyadvisorycommittee.
IwouldliketothankDr. JohnKlauderforhisthoughtfulcommentsaboutcoherent
state theory.
IwouldliketothankDr. DavidA.MichaandDr. KeithRungefortheirsupport
andhospitalityduringtheyearandhalfIspentintheirgroup.
IwouldliketothankpastandpresentmembersofDr. Ohrn'sgroupfortheirhelp
and friendship: Dr. Agustin Diz, Mr. Juan Oreiro, Mr. Denis Jacquemin and Mr.
MagnusHedstrom. Inthisregard, IwouldspeciallyliketothankmydearfellowDr.
Benny Mogensen forhis cordial friendship during all these years and formaking it
possibleformetovisithisbeautifulcountry,Denmark. IwouldalsoliketothankMr.
MauricioCoutinho-Netoforthemanygoodmomentswespenttogetherauditingcourses
and attendingconferences.
Thecompletionofthisthesisworkinexorablycalledforthehelpandsupportfrom
manypeopleintheQuantumTheoryProjectandoutside. Unlikesomeofmyungrateful
colleges, Idonotfeartoacknowledgealltheinvaluableassistancefrommydifferent
benefactors. IwouldliketothankforsodiversereasonsDr. DarioBeksic,Dr. Herbet
iii
Dacosta,ProfessorJoaquimDelphinoDaMotta-Neto,Mr. PiotrRozyczkoandhiswife
Ewa("Piotra"),Dr. MarcelNooijen,Ms. LynnSalemi,Dr. RichardJohannesAlphonse
vonMathar, Dr. HenkMonkhorst, Ms. JudyParkerandherfamily,Dr. JeffKrause,
Dr. RodBartlett,Dr. FrankHarris,Dr. JamesDeyrup,Dr. JamesHorvath,Mr. Karl
Zachary,Dr. FabioZuluaga,Dr. PedroRuesta,Ms. NormaBarahona,Ms. GerrySams,
Ms. CeciliaColetti,Mr. SergioCabadoandhiswifeMonika, Ms. GloriaAtkinsand
Mr. Larry Monroe.
MymostspecialgratitudeistomytwodearestfriendsintheQuantumTheoryProject:
Dr. S.AjithPereraandDr. Raymond"ThePig"Sadeghi. Whatcouldhavebeenofmy
harshlifeiftheothertwomusketeershadnotbeenaroundhere? Iwouldliketothank
bothofthemfortheirloyalfriendshipandtheircontinuoushelpinallaspectsoflife.
IwouldliketothankallthepeopleinArgentinawhoalsomadethisthesispossible:
mymother,mysisterMariaMartha,myrecentlydeceasedgrandmotherAngela,andmy
friendsFernandoMarcer,RobertoMarcer,Dr. JorgeZobenica,Dr. CarlosDiaz,Lie.
MarceloRadicioni, andDr. MarceloFernandez.
Finally,IwouldliketothankmydearestAnaRosaSegarraforallthelove,caring,
encouragementandsupportshegavemethisyear. Gracias,linda!
MaythereaderforgivemeforwhomIamunconsciouslyforgettingandagreewithme
forwhomIamconsciouslyomitting. "Elolvidoeselmejorperdonylapeorvenganza."
iv
TABLE OFCONTENTS
ACKNOWLEDGMENTS
iii
LISTOFTABLES viii
LISTOFFIGURES ix
ABSTRACT xiii
CHAPTERS
1. INTRODUCTION 1
2. ANOVERVIEWOFTHESCATTERINGTHEORYMETHODS 4
ExperimentsandTheoryforScatteringProcesses 4
ScatteringBeamExperiments 4
ObservablesinScatteringBeamProcesses: CrossSections 6
TheTheoryforScatteringProcesses 9
QuantumMechanicalScatteringTheory 11
FormalScatteringTheory 11
Time-IndependentScatteringTheory 13
Time-DependentScatteringTheory 34
SemiclassicalScatteringTheory 48
TheSemiclassicalTheoryinGeneral 48
TheJeffreys-Wentzel-Kramers-Brioullin(JWKB)Approximation 51
TheBohr-SommerfeldQuantizationRule 56
TheEinstein-Broullin-Keller(EBK)QuantizationRule 58
Miller-MarcusSemiclassicalS-matrix 59
HellerSemiclassicalWavePacketDynamics 65
Non-AdiabaticSemiclassicalMethods: TransitionSurfaceHopping
Model(TSHM) 65
3. THEENDTHEORYFORTIME-DEPENDENTDYNAMICS 68
GeneralOutlineoftheENDTheory 68
TheQCSDENDWaveFunction 72
TheQCSDDynamicalEquations 78
GeneralCharacteroftheQCSDENDDynamicalEquations 82
OtherENDModels: TheDoubleWavePacketENDTheory 84
4. THECOHERENTSTATETHEORYINTHEENDCONTEXT 87
TheGeneralTheoryofCoherentandStates 87
DefinitionofCoherentStates 87
Quasi-ClassicalCoherentStates 92
TheCoherentStateTheoryandtheQCSDENDTheory 94
AProposedRotationalCoherentStateforTheENDTheory 96
PreviousRotationalCoherentStates 96
RotationalHamiltonianandRelatedOperators 98
GroupRelationships 100
CoherentStateConstruction 102
OperatorAveragesinTheCoherentState 105
ParameterizationinPhysicalTerms 106
TimeEvolutionofTheCoherentState 108
5. THECROSSSECTIONSINTHEDIFFERENTTHEORIES 110
QuantumCrossSections 110
GeneralDefinition 110
Atom-AtomScattering 113
Atom-DiatomScattering 115
ClassicalandSemiclassicalCrossSections 118
CoordinateTransformations: CenterofMassandLaboratoryFrames . 118
ClassicalTotalCrossSections 123
ClassicalPartialCrossSections 140
SemiclassicalCrossSections 146
6. THEENDCROSSSECTIONS 149
TheENDS-Matrix 149
GeneralOverview 149
TheENDCrossSectionsI:Atom-AtomScattering 151
TheAtom-AtomENDWaveFunction 151
UncouplingoftheAngularandLinearCoordinates 155
TheAtom-AtomENDS-Matrix 158
ENDCrossSectionsfortheAtom-AtomScattering 164
TheENDCrossSectionsII:Atom-DiatomScattering 169
TheAtom-DiatomENDWaveFunction 169
TheAtom-DiatomENDS-Matrix 172
TheENDCrossSectionsfortheAtom-DiatomScattering 175
OtherENDScatteringProperties 178
TheENDAverageEnergyLoss 178
7. CALCULATIONRESULTS 180
GeneralConsiderations 180
TheProton-HydrogenMoleculeSystem 181
ExperimentsandPreviousTheory 181
InitialConditionsandFinalStateAnalysis 183
ENDResults 186
TheProton-MethaneSystem 204
InitialConditionsandFinalStateAnalysis 204
ENDResults 209
TheProton-WaterSystem 227
InitialConditionsandFinalStateAnalysis 227
ENDResults 229
vi
8. CONCLUSIONANDFUTUREWORK 232
APPENDIXA. THEDIRACDELTAFUNCTION 236
APPENDIXB. THET*vTENSOROPERATORS 243
APPENDIXC. THEROTATIONALCOHERENTSTATEMEASURE 247
APPENDIXD. AVERAGESINTHEROTATIONALCOHERENTSTATE ... 250
APPENDIXE. AVERAGEOVERINITIALANGULARVARIABLES 259
REFERENCES 261
BIOGRAPHICALSKETCH 271
vii
LIST OFTABLES
Table Page
7.1: Toryipeentoatfiopnroacnesdspprorjoedcuticleedibmypavcatripoaursamientiteira.lcDondmietainonssdaisssgociivaetniobny,Rtarget
rearrangement,andNT&CTnontransferinelasticandchargetransfer
scattering,respectively 188
7.2: TheintegralchargetransfercrosssectioninA2calculatedinthree
differentways,IOSA,TSHM andEND,comparedtoaninterpolated
experimental, value 204
7.3: Integrationgridfortheorientationsofthetargetmethanemolecule. The
sixbasicorientationsareshowninFigure 1 206
7.4: Typesofprocessproducedbydifferentinitialconditions(orientationof
targetandimpactparameterofprojectile) 211
7.5: Chargetransferintegralcrosssectionsandpercentageofintegralcross
sectioncomingfromtheD2,D3,E,andFD2channelsforthedifferent
targetorientations 217
7.6: Classicalandsemiclassicalrainbowangle(degrees)perorientationinthe
laboratoryframe. ThecorrespondingimpactparameterinBohrisalso
listed 218
7.7: Analysisofthefragmentationofmethane. Comparisonbetween
experimentandtheory. Allvaluesin% 223
7.8: Selectedtargetorientationthatcreate36gridpointsforrotational
averaging 229
viii
LIST OFFIGURES
Figures Page
7.1: InitialconditionsfortheH++HosystemfortheENDcalculations, aand
Barethepolarandtheazimuthalangles,respectively,ofthebondvector
inthedepictedLabframe 185
7.2: xvs. zpositionforthethreecenters(nuclei)inaENDH++H2
trajectorybelongingtotheorientation[90°,0°]forwhichthedynamics
remainsinthexzplane. Thetotalevolutiontimeis2000a.u. Theprocess
shownisforadissociation(D)atimpactparameterb=0.3a.u. The
dissociationoftheoriginaltargetisclearlydepicted 190
7.3: ThesameasinFig. 7.2butforarearrangementprocess(R)atb=1.0a.u. . . . 190
7.4: ThesameasinFig. 7.2butforanontransfer/chargetransfer(NT/CT)atb
= 1.6a.u 191
7.5: Mullikenpopulationofaspinelectronsvs.timeforthethreecenters(nuclei)
inaENDH++H-2trajectorybelongingtotheorientation[90°,0°]. The
processshownisforadissociation(D)atimpactparameterb=0.3a.u 191
7.6: ThesameasinFig.7.5butforarearrangementprocess(R)atb=1.0a.u. ... 192
7.7: ThesameasinFig. 7.5butforanontransfer/chargetransfer(NT/CT)atb
= 1.6a.u 192
7.8: Relativenucleardistancesvs. timefortheorientation[90°,0°]andimpact
parameterb= 1.6a.u. Thebond"dilution"oftheH2astheprojectile
approaches,theinitialvibrationalexcitation(correlatedtotheCTprocess,
seeFig. 7.7),andthevibrationalexcitationafterthecollisioncanreadily
bediscerned 193
ix
7.9: Laboratoryscatteringangle0\abvs. impactparameterbforsome
representativeorientations[a,(3).Thecasedepictedcorrespondonlytothe
NT/CTscattering(nodissociationofrearrangement)forsakeofsimplicity.
Theprimaryrainbowanglecanbeseenfor2.0a.u.<b<3.0a.u.inall
cases. Intheorientationswithfi=0°,a(zero)gloryanglecanbeobserved
atimpactparameterslowerthanthatoftheprimaryrainbow. Intheother
orientations,a(nonzero)secondaryrainbowanglecanbeseen 195
7.10: Orientation-averagedweightedprobabilityvs. impactparameter. Allthe
channelsshownalongwithbj^j^{b)=bastheupperstraightline. CT
isthetotalchargetransferviaeitherdissociationorrearrangementorpure
scattering,Disnontransferdissociation,andVf=0. 1. 2, 3. 4,final
vibrationalstateoftheH2moleculeintheNTscatteringcase. TheCT
probabilityisacontinuousfunctionoftheimpactparameterbutislowat
highimpactparameter. Observethepredominanceofthedissociationand
chargetransferprocessesatlowimpactparameters,andthehigher
vibrationalexcitationattheimpactparametersoftherainbow 196
7.11: H2NTvibrationalenergytransferAEvjbvslaboratoryscatteringangle
^lab- Orientation-averagedresultsfromEND,IOSA,andTHSM
calculationsalongwithexperimentaldata 196
7.12: Totalnontransferdifferentialcrosssectionsvslaboratoryscattering
angle, orientationaveragedresultsfromEND,IOSA,andTHSM along
withexperimentaldata. ThelatterhavebeennormalizedtotheEND
resultsbymatchingtheexperimentaltotalNTdifferentialcrosssectionat
theexperimentalrainbowangle 198
7.13: ThesameasinFig. 7.12butforthefinalvibrationalstateVf=0 198
7.14: ThesameasinFig. 7.12butforthefinalvibrationalstateVf= 1 199
7.15: ThesameasinFig. 7.12butforthefinalvibrationalstatevf=2 199
7.16: ThesameasinFig. 7.12butforthefinalvibrationalstatevf=3 200
7.17: ThesameasinFig. 7.12butforthefinalvibrationalstatevf=4 200
7.18: ThesameasinFig. 7.12butforthefinalvibrationalstatevf=5 201