Implementation of the Incremental Scheme for Highly Efficient Correlation Methods Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathemathisch-Naturwissenschaftlichen Fakultät der Universität zu Köln vorgelegt von Joachim Friedrich aus Leideneck Köln, 08.05.2007 ii Berichterstatter: Prof. Dr. M. Dolg Prof. Dr. U. Deiters Tag der mündlichen Prüfung: 25.06.2007 Contents 1 Introduction 1 2 General Theory 5 2.1 TheMany-ParticleProbleminQuantumMechanics . ... 5 2.2 Hartree-FockTheory .......................... 6 2.3 Multi-ConfigurationHartree-FockTheory . ... 8 2.4 The Concept of Size-Extensivity/Size-Consistency . ....... 8 2.5 ConfigurationInteractionTheory . 9 2.5.1 Multi-Reference Configuration Interaction Theory . .... 10 2.6 CoupledClusterTheory. 10 2.7 Localization............................... 11 2.8 PeriodicSystems ............................ 15 2.8.1 LatticeTranslationVectors . 15 2.9 Finite-ClusterApproach . 16 2.10 GraphTheory .............................. 17 2.10.1 METISGraph-Partitioning . 17 3 The Incremental Scheme 19 3.1 TheIncrementalExpansionforaMolecule. .. 19 3.1.1 ExactnessoftheIncrementalExpansion . 21 3.1.2 ApproximationstotheIncrementalScheme . 23 3.1.3 EnergyScreening.. .. .. .. .. .. .. 26 3.1.4 FormalScaling ......................... 27 3.2 DerivationoftheCorrelationEnergyforaSolid . ..... 28 3.3 Incremental Correlation Energy for Small Domains . ..... 29 3.3.1 Expansion of the Fragmental Energies in a Further Expansion 30 3.3.2 Separate Treatment of the Translational Symmetry . ... 31 3.4 IncrementalExpansionfornon-DisjointCells . ..... 33 3.4.1 ThePrefactorsforDisjointCells . 34 3.4.2 ThePrefactorsfornon-DisjointCells . 35 iii iv CONTENTS 3.5 Incremental Expansion for a Multi-reference Case . ...... 36 3.5.1 TruncationoftheVirtualSpace . 37 3.6 TreatmentofSymmetry. 37 3.6.1 SymmetricOne-SiteDomains . 38 3.7 ErrorAnalysisintheIncrementalExpansion . .... 39 4 Automatization of the Incremental Scheme 43 4.1 ObtainingGroupsofOccupiedOrbitals . 43 4.1.1 ExcitationSpacesforOne-SiteDomains. 44 4.1.2 Construction of the n-SiteDomains . 45 4.1.3 ObtainingCorrelationEnergies . 46 4.2 TruncationbyDistance . 46 5 Implementation 47 5.1 Interfaces ................................ 47 5.2 Foster-BoysLocalization . 47 5.3 Parallelization.............................. 48 5.3.1 TheServer ........................... 48 5.3.2 TheClient ........................... 49 5.3.3 TheWrapper .......................... 49 5.4 VisualizationoftheDomains . 52 5.5 SymmetryforMolecules . 53 5.5.1 GeneratingtheOperatorsofthePointGroup . 54 5.5.2 SymmetryAdaptedOne-SiteDomains. 55 5.5.3 Symmetry Classified n-SiteDomains . 56 5.6 SymmetryforPeriodicSystems . 57 5.7 AlgorithmtoObtaintheVirtualSpace . 58 6 Applications 59 6.1 Molecules................................ 59 6.1.1 HydrocarbonCompounds . 59 6.1.2 TransitionMetal/ActinideCompounds. 59 6.2 IntermolecularInteractions . 68 6.2.1 WaterClusters ......................... 68 6.2.2 π–π/CH–π–Interactions .................... 72 6.2.3 TheAurophilicAttraction . 74 6.2.4 DNABasePair ......................... 76 6.2.5 Reaction Pathways of the 4-exo/5-endo Cyclization . 76 6.3 SymmetricSystems........................... 81 CONTENTS v 6.3.1 AWaterCluster......................... 81 6.3.2 Circular Cis, Trans-CyclotriazineCluster . 86 6.4 ClusterCompounds........................... 88 6.4.1 BoronHydrides......................... 88 6.4.2 MercuryClusters .. .. .. .. .. .. .. 89 6.5 PotentialEnergySurfaces. 91 6.5.1 Octane ............................. 91 6.5.2 Hexayne ............................ 93 6.6 Polymers ................................ 96 6.7 ExcitedStates.............................. 97 7 Symmetric Localization 101 8 Summary and Outlook 105 8.1 Summary ................................ 105 8.2 Outlook ................................. 106 Literature 107 A List of Abbreviations 117 B The Incremental Code 119 B.1 RequiredLibraries . .. .. .. .. .. .. .. 119 B.2 QuantumChemistryPackages . 120 B.3 MolproInputExample . 120 B.4 MOLCAS6.4Dependencies . 120 B.5 DaltonDependencies . 120 C Molecular Data 125 C.1 ExemplaryListsofIncrements . 125 C.2 ListsofIncrementsforCI-basedMethods . 127 C.3 4-exo/5-endo Cyclization ........................ 131 vi CONTENTS List of Figures 2.1 Parallelepiped spanned by the fundamental lattice vectors ~a1, ~a2, ~a3.. 15 2.2 Exampleofagraphwith8vertices. 17 2.3 The various phases of the multilevelgraph bisection. ....... 18 3.1 Asystemdividedintofoursubsystems. 19 3.2 The four one-site increments in a system of 4 spatial parts. ...... 20 3.3 All possible two-site increments within a space of four one-site incre- ments................................... 20 3.4 A test system with 3 spatial parts 1, 2, 3................. 22 3.5 Pictorialviewof theinvolvedsets, ifthesetof domains X is partitioned into two sets A and B........................... 24 3.6 A 3 3 supercell built from a unit cell with atoms/centers of charge at × thecorners. ............................... 35 3.7 Error distribution of the incremental series for 10, 15 and20domains. 40 3.8 Error distribution of the incremental series with 20 domains using ar- −10 O tificial order-dependent energies of (εX = X 10 10 ). ..... 42 · · 5.1 A sketch of the required input data files for the incremental calculation 49 5.2 Dataflowwithintheserver/clientstructure. ..... 50 5.3 Data flow for an incremental calculation within the MOLPRO envi- ronment. ................................ 51 5.4 Visualization of the concept of a wrapper class as data container for thenetworktransfer. .. .. .. .. .. .. .. 51 5.5 Centers of charge for naphthalene colored by their membership to the one-sitedomains. ............................ 52 5.6 Centers of charge for naphthalene colored by their membership to the one-sitedomains. ............................ 53 6.1 RI-BP86/SVP optimized structures of some hydrocarbon molecules. 60 3+ 6.2 Optimized structures of TiCp2Cl2, MoCl6, MoF6, Nb2Cl10, [Pu(H2O)8] 62 6.3 Centers of charge of MoF6 ....................... 64 vii viii LIST OF FIGURES 6.4 Centers of charge of MoCl6 ....................... 64 6.5 RI-BP86/SVP optimized structure of Au4Cl4 11 (D2d). ........ 66 6.6 RI-BP86/SVP optimized structure of a set of eight water molecules. 68 6.7 (H2O)11 clustertakenfromBulusuetal. 69 6.8 Sandwich structure of the benzene dimer at a benzene-benzene dis- tanceof3.75Å ............................. 73 6.9 IndolemethanecomplexoptimizedbyRingeretal. .... 73 6.10 MP2 optimized structure of Au2(PH-C2H2-S)2 ............. 75 6.11 RI-BP86/SVP optimized guanine-cytosine base pair. ....... 76 6.12 Intermediates of the 4-exo/5-endo cyclization. 78 6.13 Convergence of the incremental correlation energies for intermediates of the 4-exo and 5-endo cyclization. .................. 79 6.14 RI-BP86/SVP optimized structure of (H2O)6 (C3)............ 81 6.15 RI-BP86/SVP optimized structure of (N3H3)5 (C5). .......... 86 6.16 RI-BP86/SVP optimized structures of B5H11 17 (C1), Hg13 18 (Ih), and Hg20 19 (Td)............................. 88 6.17 RI-BP86/SVP optimized structure of n-octane with a fixed C4-C5 dis- tanceof2.05Å.............................. 91 6.18 Potential energy scan along the C4-C5 distance of n-octane...... 92 6.19 Errors of the incremental correlation energy for n-octane........ 92 6.20 RI-BP86/SVP optimized structure of dodeca-hexayne with a fixed C5- C6distanceof1.55Å. ......................... 93 6.21 Potential energy scan along the C5-C6 distance of dodeca-hexayne. 94 6.22 Errors of the incremental expansion of the CCSD correlation energy ofhexayne. ............................... 95 6.23 Structure of the used hydrocarbon chain and the symmetric cell of (C2H2)n.................................. 96 6.24 H28 testsystem ............................. 99 B.1 File dependencies for CI-based calculations within the MOLCAS 6.4 andDIESELframework. 122 B.2 File dependencies for the incremental CCSD calculations within the DALTONframework.. 123 List of Tables 6.1 Comparison of the incremental energies with the full CCSD calcula- tionsforthehydrocarbonsinfigure6.1. 61 6.2 Comparison of the incremental energies with the full CCSD calcula- tionsforaseriesofcomplexes . 65 6.3 Comparison of the incremental energies with the full CCSD calcula- tions for Au4Cl4 ............................. 66 3+ 6.4 Incremental correlation Energy of [Pu(H2O)8] ............ 66 6.5 Comparison of the convergence of the incremental energy of TiCp2Cl2 with respect to different dsp and tcon values .............. 67 6.6 Comparison of the incremental energies for the (H2O)8-cluster . 69 6.7 Convergence behavior of the incremental scheme with respect to the density parameter tdens ......................... 70 6.8 Performance of the approximation of the incremental scheme with re- specttoadynamicdistancethreshold . 71 6.9 Performanceoftheapproximationschemeeqn. 3.26 . .... 72 6.10 Performance of the dynamic distance threshold R ( ) ....... 72 min Oi 6.11 Comparison of the incremental energies with the full CCSD calcula- tions for the benzene dimer and the methane indole complex . ... 74 6.12 Comparison of the incremental energies with the full CCSD calcula- tions for Au2(PH-C2H2-S)2 ....................... 75 6.13 Comparison of the incremental energies with the full CCSD calcula- tionsfortheguanine-cytosinedimer . 77 6.14 Performance of different quantum chemical methods for the 4-exo/5- endo cyclization. ............................ 80 6.15 Comparison of the error introduced by the incremental expansion and savingwithrespecttosymmetry . 82