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Multidimensional Quantum Dynamics Edited by Hans-Dieter Meyer, Fabien Gatti, and Graham A. Worth RelatedTitles Fried,JR Rode,B.M.,Hofer,T.,Kugler,M. Computational Chemistry The Basics of Theoretical and Molecular Simulation and Computational 2009 Chemistry ISBN:978-0-471-46244-6 2007 ISBN:978-3-527-31773-8 Höltje,H.-D.,Sippl,W.,Rognan,D., Folkers,G. Dronskowski,R. Molecular Modeling Computational Chemistry of BasicPrinciplesandApplications Solid State Materials Third,RevisedandExpanded AGuideforMaterialsScientists, Edition Chemists,Physicistsandothers 2008 2005 ISBN:978-3-527-31568-0 ISBN:978-3-527-31410-2 Matta,C.F.,Boyd,R.J.(eds.) Cramer,C.J. The Quantum Theory of Essentials of Computational Atoms in Molecules Chemistry FromSolidStatetoDNAandDrug TheoriesandModels Design 2004 2007 ISBN:978-0-470-09182-1 ISBN:978-3-527-30748-7 Multidimensional Quantum Dynamics MCTDH Theory and Applications Edited by Hans-Dieter Meyer, Fabien Gatti, and Graham A. Worth WILEY-VCH Verlag GmbH & Co. KGaA The Editors All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information Prof. Hans-Dieter Meyer contained in these books, including this book, to Ruprecht-Karls-Universität be free of errors. Readers are advised to keep in Physikalisch-Chemisches Institut mind that statements, data, illustrations, procedural details or other items may Theoretische Chemie inadvertently be inaccurate. Im Neuenheimer Feld 229 69120 Heidelberg Library of Congress Card No.: Germany applied for Dr. Fabien Gatti British Library Cataloguing-in-Publication Université Montpellier II Data CTMM, Institut Charles Gerhardt A catalogue record for this book is available from the British Library. UMR 5253, CC 1501 34095 Montpellier Cedex 05 Bibliographic information published by France the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this Dr. Graham A. Worth publication in the Deutsche Nationalbibliografie; University of Birmingham detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. School of Chemistry Birmingham B15 2TT (cid:164) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, United Kingdom Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting Uwe Krieg, Berlin Printing Strauss GmbH, Mörlenbach Binding Litges & Dopf GmbH, Heppenheim Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-32018-9 V Contents Preface XV ListofContributors XVII ListofSymbols XXI 1 Introduction 1 Hans-DieterMeyer,FabienGattiandGrahamA.Worth Part1 Theory 9 2 TheRoadtoMCTDH 11 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 2.1 TheStandardMethod 12 2.2 Time-DependentHartree 13 3 BasicMCTDHTheory 17 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 3.1 WavefunctionAnsatzandEquationsofMotion 17 3.2 TheConstraintOperator 20 3.3 EfficiencyandMemoryRequirements 22 3.4 MultistateCalculations 27 3.5 ParametrizedBasisFunctions: G-MCTDH 28 4 IntegrationSchemes 31 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 4.1 TheVariableMean-Field(VMF)IntegrationScheme 31 4.2 ASimpleConstantMean-Field(CMF)IntegrationScheme 32 4.3 WhyCMFWorks 33 4.4 Second-OrderCMFScheme 34 VI Contents 5 PreparationoftheInitialWavepacket 37 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 5.1 InitialWavepacketasHartreeProduct 37 5.2 EigenstatesandOperatedWavefunctions 38 6 AnalysisofthePropagatedWavepacket 41 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 6.1 RuntimeAnalysisofAccuracy 41 6.2 Spectra 43 6.2.1 PhotoabsorptionSpectra 43 6.2.2 EigenvaluesandFilterDiagonalization 46 6.2.3 Time-ResolvedSpectra 48 6.3 OptimalControl 50 6.4 StatePopulations 50 6.5 ReactionProbabilities 52 7 MCTDHforDensityOperator 57 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 7.1 WavefunctionsandDensityOperators 57 7.2 TypeIDensityOperators 58 7.3 TypeIIDensityOperators 60 7.4 PropertiesofMCTDHDensityOperatorPropagation 61 8 ComputingEigenstatesbyRelaxationandImprovedRelaxation 63 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 8.1 Relaxation 63 8.2 ImprovedRelaxation 63 8.3 TechnicalDetails 66 9 IterativeDiagonalizationofOperators 69 FermínHuarte-LarrañagaandUweManthe 9.1 OperatorsDefinedbyPropagation 69 9.2 AModifiedLanczosScheme 70 9.3 TheState-AveragedMCTDHApproach 71 10 CorrelationDiscreteVariableRepresentation 73 FermínHuarte-LarrañagaandUweManthe 10.1 Introduction 73 10.2 Time-DependentDiscreteVariableRepresentation 74 10.3 CorrelationDiscreteVariableRepresentation 76 10.4 Symmetry-AdaptedCorrelationDiscreteVariable Representation 78 10.5 MultidimensionalCorrelationDiscreteVariableRepresentation 78 Contents VII 11 PotentialRepresentations(potfit) 81 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 11.1 ExpansioninProductBasisSets 81 11.2 OptimizingtheCoefficients 82 11.3 OptimizingtheBasis 83 11.4 ThepotfitAlgorithm 84 11.5 ContractionOverOneParticle 86 11.6 SeparableWeights 86 11.7 Non-SeparableWeights 87 11.8 ComputationalEffortandMemoryRequest 88 12 KineticEnergyOperators 91 Hans-DieterMeyer,FabienGattiandGrahamA.Worth 12.1 Introduction 91 12.2 VectorParametrizationandPropertiesofAngularMomenta 92 12.2.1 Examples 93 12.2.2 GeneralFormulation 95 − 12.2.2.1 DefiningaSetofN 1VectorsandtheCorrespondingClassical KineticEnergy 95 12.2.2.2 IntroductionoftheBody-FixedFrameandQuantization 96 12.2.2.3 IntroductionoftheBody-FixedProjectionsoftheAngularMomenta − AssociatedWiththeN 1Vectors 97 12.3 GeneralExpressionofKEOinStandardPolyspherical Coordinates 99 12.3.1 GeneralExpression 99 12.3.1.1 DefinitionoftheBFframe: Figure12.3 99 12.3.1.2 PolysphericalParametrization 100 12.3.1.3 PropertiesoftheBFProjectionsoftheAngularMomenta 101 12.3.1.4 GeneralExpressionoftheKEOinPolysphericalCoordinates 104 12.3.1.5 IntroductionofaPrimitiveBasisSetofSphericalHarmonics 105 12.4 Examples 106 12.4.1 ScatteringSystems: H2+H2 107 12.4.2 Semi-RigidMolecules: HFCO 108 12.5 Extensions 109 12.5.1 SeparationIntoSubsystems 109 12.5.2 ConstrainedOperators 110 Part2 ExtensiontoNewAreas 111 13 DirectDynamicsWithQuantumNuclei 113 BenjaminLasorneandGrahamA.Worth 13.1 Introduction 113 VIII Contents 13.2 VariationalMulticonfigurationGaussianWavepackets 115 13.2.1 GaussianWavepacketAnsatz 115 13.2.2 EquationsofMotion 117 13.2.3 IntegrationScheme 120 13.2.4 InitialWavepacket 121 13.2.5 DirectDynamicsImplementation 122 13.3 Applications 124 13.4 Conclusions 128 14 MultilayerFormulationoftheMulticonfigurationTime-Dependent HartreeTheory 131 HaobinWangandMichaelThoss 14.1 Introduction 131 14.2 FromConventionalWavepacketPropagationtoML-MCTDH Theory:AVariationalPerspective 132 14.2.1 ConventionalApproachBasedonTime-Independent Configurations 132 14.2.2 TheMulticonfigurationTime-DependentHartreeMethod 134 14.2.3 TheMultilayerFormulationoftheMCTDHTheory 138 14.3 ConcludingRemarks 145 15 SharedMemoryParallelizationoftheMulticonfiguration Time-DependentHartreeMethod 149 MichaelBrillandHans-DieterMeyer 15.1 Motivation 149 15.2 SharedMemoryParallelizationofMCTDH 149 15.2.1 EquationsofMotionandRuntimeDistribution 150 15.2.2 ParallelizationoftheMCTDHCoefficientsPropagation 151 15.2.3 ParallelizationoftheMean-FieldComputation 152 15.2.4 ParallelizationoftheSPFsPropagation 153 15.2.5 ParallelizationScheme 153 15.2.6 LoadBalancingandMemoryRequirements 154 15.3 ResultsandConclusion 156 15.3.1 BenchmarkSystems 156 15.3.2 Amdahl’sLaw 157 15.3.3 Results 157 15.3.4 ConclusionandOutlook 159 16 StronglyDrivenFew-FermionSystems–MCTDHF 161 GeraldJordanandArminScrinzi 16.1 EquationsofMotionforIndistinguishableParticles 161 16.1.1 ModelSystem: Laser-DrivenFew-ElectronSystems 162 16.1.2 Spin 163 Contents IX 16.2 ComputationofOperators 164 K 16.2.1 andMean-FieldOperators 164 16.2.2 SpatialDiscretization 165 16.2.3 One-ParticleOperators 168 16.2.4 Two-ParticleOperators 169 16.2.4.1 RepresentationofHonaCoarseGrid 169 16.2.4.2 H-MatrixRepresentation 170 16.3 Parallelization 171 16.3.1 ApplicationoftheInverseOverlapMatrixS−1 172 16.3.2 ParallelComputationofMeanFields 173 16.3.3 DynamicLoadBalancing 174 16.4 ObservablesandTransformations 174 16.4.1 OrbitalTransformations 174 16.4.2 ProjectionsOntoMultiparticleStates 175 16.4.3 One-andTwo-ParticleExpectationValues 175 16.4.4 All-ParticleObservables 176 16.4.5 Spectra 177 16.5 Applications 178 16.5.1 IonizationofLinearMolecules 178 16.5.1.1 High-HarmonicSpectraofMolecules 179 16.5.2 ColdFermionicAtoms 182 17 TheMulticonfigurationalTime-DependentHartreeMethodfor IdenticalParticlesandMixturesThereof 185 OfirE.Alon,AlexejI.StreltsovandLorenzS.Cederbaum 17.1 PreliminaryRemarks 185 17.2 BosonsorFermions? –UnifyingMCTDHBandMCTDHF 186 17.2.1 BasicIngredients 186 17.2.2 EquationsofMotionwithReducedDensityMatrices 189 17.3 Bose–Bose,Fermi–FermiandBose–FermiMixtures 192 17.3.1 IngredientsforMixtures 192 17.3.2 EquationsofMotionWithIntra-andInter-SpeciesReducedDensity Matrices 194 17.4 Higher-OrderForcesandReducedDensityMatrices 196 17.4.1 IngredientsforThree-BodyInteractions 197 17.4.2 EquationsofMotionWithThree-BodyReducedDensityMatrix 198 17.5 IllustrativeNumericalExamplesforBosons: MCTDHB 199 17.6 DiscussionandPerspectives 204