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Developments in Coupled-Cluster Theory Gradients and Potential Energy Surfaces

Permanent Link: http://ufdc.ufl.edu/UFE0022513/00001

Material Information

Title: Developments in Coupled-Cluster Theory Gradients and Potential Energy Surfaces
Physical Description: 1 online resource (236 p.)
Language: english
Creator: Taube, Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: cluster, coupled, derivatives, frozen, hermitian, linearized, natural, orbitals, perturbative, triples
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Despite the successes of coupled-cluster theory (CC) to predict the properties of small- and medium-sized molecules of chemical interest, there are limitations to the conventional approach. The computational cost of the 'gold standard' CCSD(T) method scales as O(N^7) in the number of electrons, limiting the size of systems that can be calculated. Also, CCSD(T) performs relatively poorly away from equilibrium. This study proposes and evaluates partial solutions to those and related problems of coupled-cluster theory. The frozen natural orbital (FNO) coupled-cluster method increases the speed of coupled-cluster calculations by an order of magnitude with no consequential error along a potential energy surface. This method allows the virtual space of a correlated calculation to be reduced by about half, significantly reducing the time spent performing the coupled-cluster calculation. The derivation of both the energy and gradient for FNO-CC and applications to energetic material are presented. The failure of CCSD(T) away from equilibrium is shown to arise from two separate effects. For spin-restricted references, near degeneracy in the orbital space leads to problems with the (T) perturbative correction. For spin-unrestricted references, it is the slow convergence of perturbation theory due to spin-contamination that is the problem. The first of these problems is addressed by using Lambda CCSD(T), a method of the same computational scaling as CCSD(T), that, by using information from CC gradient calculations, improves the description of stretched bonds. To efficiently derive the gradient expression for Lambda CCSD(T) a more general form of the coupled-cluster energy is functional is introduced, allowing the method to be formulated in a stationary manner. The spin-symmetry breaking problem is addressed by using Brueckner orbitals to make the spin-restricted solution more stable across a potential energy surface. This choice then naturally leads to Brueckner LambdaCCSD(T), which is shown to improve the behavior of bond-breaking beyond Lambda CCSD(T) itself. To more fundamentally address the problem of potential energy surfaces, hermitian coupled-cluster theories deriving from expectation-value CC (XCC) are explored. Linearized CC is shown to be an economical method that, using a simple numerical regularization procedure, generates well-behaved potential energy surface. The capabilities of various other modifications of XCC and its truncations are explored. These purely theoretical investigations suggest that XCC-based methods have the potential to improve the capabilites of CC.
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 Andrew Taube.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Bartlett, Rodney J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022513:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022513/00001

Material Information

Title: Developments in Coupled-Cluster Theory Gradients and Potential Energy Surfaces
Physical Description: 1 online resource (236 p.)
Language: english
Creator: Taube, Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: cluster, coupled, derivatives, frozen, hermitian, linearized, natural, orbitals, perturbative, triples
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Despite the successes of coupled-cluster theory (CC) to predict the properties of small- and medium-sized molecules of chemical interest, there are limitations to the conventional approach. The computational cost of the 'gold standard' CCSD(T) method scales as O(N^7) in the number of electrons, limiting the size of systems that can be calculated. Also, CCSD(T) performs relatively poorly away from equilibrium. This study proposes and evaluates partial solutions to those and related problems of coupled-cluster theory. The frozen natural orbital (FNO) coupled-cluster method increases the speed of coupled-cluster calculations by an order of magnitude with no consequential error along a potential energy surface. This method allows the virtual space of a correlated calculation to be reduced by about half, significantly reducing the time spent performing the coupled-cluster calculation. The derivation of both the energy and gradient for FNO-CC and applications to energetic material are presented. The failure of CCSD(T) away from equilibrium is shown to arise from two separate effects. For spin-restricted references, near degeneracy in the orbital space leads to problems with the (T) perturbative correction. For spin-unrestricted references, it is the slow convergence of perturbation theory due to spin-contamination that is the problem. The first of these problems is addressed by using Lambda CCSD(T), a method of the same computational scaling as CCSD(T), that, by using information from CC gradient calculations, improves the description of stretched bonds. To efficiently derive the gradient expression for Lambda CCSD(T) a more general form of the coupled-cluster energy is functional is introduced, allowing the method to be formulated in a stationary manner. The spin-symmetry breaking problem is addressed by using Brueckner orbitals to make the spin-restricted solution more stable across a potential energy surface. This choice then naturally leads to Brueckner LambdaCCSD(T), which is shown to improve the behavior of bond-breaking beyond Lambda CCSD(T) itself. To more fundamentally address the problem of potential energy surfaces, hermitian coupled-cluster theories deriving from expectation-value CC (XCC) are explored. Linearized CC is shown to be an economical method that, using a simple numerical regularization procedure, generates well-behaved potential energy surface. The capabilities of various other modifications of XCC and its truncations are explored. These purely theoretical investigations suggest that XCC-based methods have the potential to improve the capabilites of CC.
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 Andrew Taube.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Bartlett, Rodney J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022513:00001


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DEVELOPMENTSINCOUPLED-CLUSTERTHEORYGRADIENTSAND POTENTIALENERGYSURFACES By ANDREWG.TAUBE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2008 1

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c 2008AndrewG.Taube 2

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Tomyparents 3

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ACKNOWLEDGMENTS MyworkattheUniversityofFloridawouldhavebeenimpossiblewithoutthe guidance,helpandfriendshipofmanypeople.First,andforemost,myadvisorand mentor,RodneyJ.Bartlett,helpedpushmetoreachmypotential,gavemethefreedom toexperiment,andkeptmeexcitedaboutmyworkalongtheway.Heandhiswife BeverlyhavebeentremendouslykindtomeandIcannotthankthemenough. AjithPererahasbeenaninvaluablefriend,asheisforallmembersoftheBartlett group,andIamluckytohavebeenabletoworksocloselywithhim.WhenIrstarrived inthegroup,theeldermembersofthegroup:Ariana,TomHenderson,Carlos,and Anthonymadethelearningcurvealittlelessintimidating.Myclosecollaboratorsand friendsinthegroup,Josh,Igor,andTomHughes,havemadethedailygrindofresearch moreproductiveandalotmorefun.Iwouldalsoliketoacknowledgetheassistanceand manyconversationsIhavehadoverthelastveyearswithMonika,Norbert,Victor, Tomacz,Ann,Denis,Prakash,andTomWatson. IhavehadthelucktohaveaccesstotherestoftheQuantumTheoryProject faculty,whoactedasdefactoadvisors,andalwaysmademefeelwelcome.Iwouldlike toespeciallythankN.Yngve Ohrnforthedozensofconversationsaboutscienceand otherwiseduringmytimehere.AdrianRoitberg,ErikDeumensandSoHiratahavebeen trulygraciouswiththeirhelpandadvice.ThankyoutoKevinIngersentandMikeScott forputtingintheirtimeandeortaspartofmycommittee.LetmealsothanktheQTP sta,especiallyJudyandCoralu,forputtingupwithme. ThemostsurprisingpartofmyPh.D.experiencehasbeenthegreatgroupoffriends thatIhavehadinQTP.Inadditiontothepeoplementionedabove,thelastfewyears wouldnothavebeenthesamewithoutChristina,Lena,SeonAh,Alessandro,Georgios, Joey,Julio,Martin,Kevin,andDan. Finally,Iwouldliketothankmyfamily,especiallymybrothers,MatthewandJames, andmygrandmother,Mrs.HarryAnnetteTaube,foralltheirsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................9 NOTEONUNITS.....................................11 ABSTRACT........................................12 CHAPTER 1INTRODUCTION..................................14 1.1 AbInitio QuantumChemistry.........................14 1.2TheCorrelationProblem............................22 1.2.1IndependentParticleModels......................22 1.2.2SecondQuantization..........................28 1.2.3CongurationInteraction........................32 1.3Coupled-ClusterTheory............................35 1.3.1DerivationandFunctional.......................36 1.3.2TripleExcitations............................42 1.3.3PropertiesandExcitedStates.....................46 1.4Regularization..................................55 1.5OutlineofthisStudy..............................59 2FROZENNATURALORBITALCOUPLED-CLUSTERTHEORY.......60 2.1Introduction...................................60 2.2Theory......................................63 2.2.1Energetics................................63 2.2.2Gradients................................66 2.2.3SmoothnessofthePotentialEnergySurface.............81 2.3Implementation.................................81 2.4ResultsandDiscussion.............................82 2.4.1Calibration................................82 2.4.2EnergeticMaterials...........................93 2.5Conclusion....................................104 3GENERALIZEDCOUPLED-CLUSTERSINGLES,DOUBLESAND PERTURBATIVETRIPLESMETHODS.....................108 3.1Bond-Breaking.................................108 3.1.1Introduction...............................108 3.1.2Theory..................................110 5

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3.1.3Implementation.............................116 3.1.4ResultsandDiscussion.........................117 3.1.5Conclusion................................136 3.2Forces......................................138 3.2.1Introduction...............................138 3.2.2Theory..................................140 3.2.3Implementation.............................145 3.2.4ResultsandDiscussion.........................146 3.2.5Conclusion................................154 3.3BruecknerCCSDT.............................155 3.3.1Introduction...............................155 3.3.2Theory..................................157 3.3.3Implementation.............................161 3.3.4ResultsandDiscussion.........................164 3.3.5Conclusions...............................167 4HERMITIANCOUPLED-CLUSTERTHEORIES.................171 4.1MotivationandIntroduction..........................171 4.2HermitianCoupled-ClusterTheory......................172 4.2.1Expectation-ValueCoupled-ClusterTheory..............173 4.2.2UnitaryCoupled-ClusterTheory....................175 4.2.3Strongly-ConnectedXCC........................176 4.2.4Truncations...............................179 4.2.5BeyondtheGroundState........................181 4.2.6Conclusion................................189 4.3LinearizedCoupled-ClusterTheory......................190 4.3.1Introduction...............................190 4.3.2Theory..................................192 4.3.3Implementation.............................198 4.3.4Results..................................198 4.3.5Conclusion................................204 5CONCLUSIONS...................................207 APPENDIX AORBITALFUNCTIONALSFORDIFFERENTREFERENCEFUNCTIONS.210 BEQUATIONSFORANALYTICALDERIVATIVESOFCCSDT.......215 CEQUATIONSFORFORBCCD.........................219 DFACTORIZATIONSOFCOUPLED-CLUSTERFUNCTIONALS........220 REFERENCES.......................................225 BIOGRAPHICALSKETCH................................236 6

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LISTOFTABLES Table page 2-1IntermediatequantitiesforFNOgradients.....................73 2-2Elementsofthenon-canonicalCPFNO A matrix.................74 2-3ElementsofthecanonicalCPFNO A matrix....................78 2-4PercentageofcorrelationenergyforaseriesofFNOtruncations.........84 2-5BondlengthsinvalencebasissetsusingFNOCCSDTandCCSDT....87 2-6Bondlengthsincore-valencebasissetsusingFNOCCSDTandCCSDT..88 2-7BondanglesinvalencebasissetsusingFNOCCSDTandCCSDT.....89 2-8Bondanglesincore-valencebasissetsusingFNOCCSDTandCCSDT..90 2-9Harmonicvibrationalfrequenciesforclosed-shellmoleculesusingFNO CCSDTandCCSDT..............................95 2-10Harmonicvibrationalfrequenciesforopen-shellmoleculesusingFNOCCSDT andCCSDT....................................96 2-11EnergydierencesforDMNAconformersusingFNOCCSDT.........96 2-12InteractionenergiesforDMNAdimerconformationsusingFNOCCSDT...97 2-13ChairandboatconformersofRDXcalculatedusingFNOCCSDT.......98 2-14Stationarypointsofnitroethanedecomposition..................105 3-1CarbonmonoxidedissociationenergycalculatedwithCCSDTandCCSDT126 3-2AsymptoticbehaviorofN 2 bond-breakingwithvarioustriplesmethods.....127 3-3Mostsignicant T 3 amplitudesforF 2 CCSDT..................130 3-4Bond-breakingcurveforNH 2 calculatedwithROHFandUHFperturbative triplesmethods....................................131 3-5Methylradicalinthe 2 A 00 2 statebond-breakingusingROHFandUHF perturbativetriplesmethods.............................132 3-6EnergiesofH 2 O + stateswithperturbativetriplesmethods............133 3-7BarrierheightswithCCSDTandCCSDT..................135 3-8ComparisonofcriticalpointenergiesforRDXconcertedreactionpathway...136 3-9OptimizedbondlengthsforCCSDTandCCSDT..............150 7

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3-10OptimizedbondanglesforCCSDTandCCSDT...............150 3-11Rootmeansquaredistancesbetweenthetransitionstatestructuresfrom QCISD,CCSDTandCCSDT.........................151 3-12Optimizedtransitionstatefornitromethanetomethylnitrite...........153 3-13Spin-symmetrybreakinginstabilityforHartree-FockandBruecknerreferences.164 4-1LinCCSDoptimizedbondlengths..........................203 4-2LinCCSDoptimizedbondangles..........................203 4-3BarrierheightswithLinCCSD............................205 8

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LISTOFFIGURES Figure page 1-1ConvergenceofcorrelationenergyforCI,CCandMBPTmethods........36 1-2CalibrationcurvesforMBPT,CCSD,andCCSDTenergies.........44 1-3CalibrationcurvesforMBPT,CCSD,andCCSDTgeometries.......45 2-1StepsinanFNOderivativecalculation.......................80 2-2CorrelationenergyasafunctionofcalculationtimeforFNOCCSDT.....85 2-3PotentialenergycurveforHFusingCCSDwithdierentFNOtruncations...86 2-4BondlengtherrorsinvalencebasissetsusingFNOCCSDTandCCSDT.91 2-5Bondlengtherrorsincore-valencebasissetsusingFNOCCSDTand CCSDT......................................92 2-6BondangleerrorsinvalencebasissetsusingFNOCCSDTandCCSDT..93 2-7Bondangleerrorsincore-valencebasissetsusingFNOCCSDTandCCSDT94 2-8Harmonicvibrationalfrequencyerrorsforclosed-shellmoleculesusingFNO CCSDTandCCSDT..............................97 2-9Harmonicvibrationalfrequencyerrorsforopen-shellmoleculesusingFNO CCSDTandCCSDT..............................98 2-10DirectssionandHONOpathwaysfordecompositionofnitroethane......100 2-11Isomerizationtoethylnitritepathwaysfordecompositionofnitroethane.....101 2-12Isomerizationtoethylhydroxynitroxidepathwaysfordecompositionof nitroethane......................................102 2-13Mostimportantpathwaysfordecompositionofnitroethane............103 3-1Hydrogenuoridebond-breakingforCCSDTandCCSDT.........119 3-2Fluorinebond-breakingusingvariousperturbativetriplesmethods........120 3-3Dicarbidebond-breakingusingvariousperturbativetriplesmethods.......121 3-4Waterbond-breakingusingvariousperturbativetriplesmethods.........122 3-5Ethylenebond-breakingwithCCSDTandCCSDT.............123 3-6EthylenetorsionwithCCSDTandCCSDT..................124 3-7Carbonmonoxidebond-breakingwithCCSD,CCSDTandCCSDT....125 9

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3-8Nitrogenbond-breakingusingvariousperturbativetriplesmethods.......127 3-9Weightednon-parallelityerrorsforCCSD,CCSDT,andCCSDT......128 3-10TheRDXconcerteddissociationtriplewhammy"transitionstate........134 3-11ReactionprolecomparisonforRDXconcertedtransitionstate.........137 3-12ForcecurveforHFbond-breakingwithCCSDTandCCSDT........147 3-13ForcecurveforCObond-breakingwithCCSDTandCCSDT........148 3-14BondlengtherrorsforCCSDTandCCSDT.................149 3-15Transitionstatefortherearrangementofnitromethanetomethylnitrite.....152 3-16NitromethanetoMethylnitritereactionpathcomputedbyCCSDTand CCSDT......................................154 3-17Hydrogenuoridebond-breakingwithHartree-FockandBruecknerreferences.165 3-18Fluorinebond-breakingwithHartree-FockandBruecknerreferences.......167 3-19Waterbond-breakingwithHartree-FockandBruecknerreferences........168 3-20Nitrogenbond-breakingwithHartree-FockandBruecknerreferences.......169 4-1RegularizedandunregularizedLinCCSDbond-breakingofHF..........199 4-2RegularizedandunregularizedLinCCSDbond-breakingofH 2 OandCO....201 4-3RegularizedandunregularizedRHFLinCCSDbond-breakingofHF.......202 10

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NOTEONUNITS Hartreeatomicunitsa.u.areassumedthroughouttheequations.More experimentallymeaningfulunitsareusedforresults,andareindicatedinthetext.In Hartreea.u.,Planck'sconstant ~ ,theabsolutevalueoftheelectroncharge e ,the electronmass m e ,andCoulomb'sconstant = 4 0 areallsetto1.Then,energiesare measuredinunitsofHartrees,1E h =27 : 211eV=627 : 5kcal/mol,andlengthsare measuredinBohr,1 a 0 =0 : 52918 A=52 : 918pm. 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DEVELOPMENTSINCOUPLED-CLUSTERTHEORYGRADIENTSAND POTENTIALENERGYSURFACES By AndrewG.Taube August2008 Chair:RodneyJ.Bartlett Major:Chemistry Despitethesuccessesofcoupled-clustertheoryCCtopredictthepropertiesof small-andmedium-sizedmoleculesofchemicalinterest,therearelimitationstothe conventionalapproach.Thecomputationalcostofthegoldstandard"CCSDTmethod scalesas O N 7 inthenumberofelectrons,limitingthesizeofsystemsthatcanbe calculated.Also,CCSDTperformsrelativelypoorlyawayfromequilibrium.Thisstudy proposesandevaluatespartialsolutionstothoseandrelatedproblemsofcoupled-cluster theory. ThefrozennaturalorbitalFNOcoupled-clustermethodincreasesthespeedof coupled-clustercalculationsbyanorderofmagnitudewithnoconsequentialerror alongapotentialenergysurface.Thismethodallowsthevirtualspaceofacorrelated calculationtobereducedbyabouthalf,signicantlyreducingthetimespentperforming thecoupled-clustercalculation.Thederivationofboththeenergyandgradientfor FNO-CCandapplicationstoenergeticmaterialarepresented. ThefailureofCCSDTawayfromequilibriumisshowntoarisefromtwoseparate eects.Forspin-restrictedreferences,neardegeneracyintheorbitalspaceleadsto problemswiththeTperturbativecorrection.Forspin-unrestrictedreferences,it istheslowconvergenceofperturbationtheoryduetospin-contaminationthatisthe problem.TherstoftheseproblemsisaddressedbyusingCCSDT,amethodofthe samecomputationalscalingasCCSDT,that,byusinginformationfromCCgradient 12

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calculations,improvesthedescriptionofstretchedbonds.Toecientlyderivethegradient expressionforCCSDTamoregeneralformofthecoupled-clusterenergyisfunctional isintroduced,allowingthemethodtobeformulatedinastationarymanner. Thespin-symmetrybreakingproblemisaddressedbyusingBruecknerorbitalsto makethespin-restrictedsolutionmorestableacrossapotentialenergysurface.Thischoice thennaturallyleadstoBruecknerCCSDT,whichisshowntoimprovethebehaviorof bond-breakingbeyondCCSDTitself. Tomorefundamentallyaddresstheproblemofpotentialenergysurfaces,hermitian coupled-clustertheoriesderivingfromexpectation-valueCCXCCareexplored. LinearizedCCisshowntobeaneconomicalmethodthat,usingasimplenumerical regularizationprocedure,generateswell-behavedpotentialenergysurface.Thecapabilities ofvariousothermodicationsofXCCanditstruncationsareexplored.Thesepurely theoreticalinvestigationssuggestthatXCC-basedmethodshavethepotentialtoimprove thecapabilitesofCC. 13

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CHAPTER1 INTRODUCTION 1.1 AbInitio QuantumChemistry Chemistryisfundamentallyaboutelectrons.Bonding,reactions,excitations,and propertiessuchasdipolemomentsaredeterminedbythedistributionofelectronsina moleculeandhowthatdistributionchangesintimeorduetoanexternalperturbation. Knowledgeofthedistributionofelectronsanditsresponsepropertiesallowsforthe predictionofawidevarietyofexperimentalresults.Oneofthegoalsofcomputational chemistryistocalculatethesequantities.Theseresultscanbeusedtointerpret experimentalresults,tomakepredictionsaboutmoleculesthatareyettobesynthesized, ortosubstituteforexperimentsthatarediculttoperform.Forexample,thenatureof thereactionsofenergeticmaterials,suchasjetfuelsandexplosives,makesperforming tabletopexperimentsvirtuallyimpossible[1]. Electronsaretrulyquantummechanicalparticles:non-quantummechanicalclassical electronswouldnotbindmolecules[2].Onemustdealwithquantummechanicsdirectly whenattemptingtocalculateelectronicproperties.Theequationsthatdeterminethe propertiesofamoleculeareknown;unfortunately,theyarenotanalyticallysolvable formorethanoneelectron.Instead,onemustconstructcomputationallyfeasible approximationstotheexactequations.Thesearchforbetterapproximationsandtheir applicationtoproblemsofchemicalinterestdenestheeldofquantumchemistry.Often, onewouldliketoperformcalculationswithoutanyinputfromexperiment,usingonly thefundamentalequationsofquantummechanics;thisisknownas abinitio fromthe beginning"quantumchemistry. TheevolutionofquantummechanicalparticlesisgovernedbytheSchrodinger equation[3], )]TJ/F21 11.9552 Tf 11.955 0 Td [(i @ @t j x ;t i = H t j x ;t i ; {1 where H t isthepotentiallytime-dependentHamiltonianoperatorthatdescribesthe 14

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interactionsamongtheparticlesaswellastheirkineticenergy,andisthe wavefunction oftheparticles,whosecoordinatesareindicatedbythevector x .Thewavefunctionisa quantitywhosemagnitudeistheprobabilitydistributionofthequantumparticles[3].The valueofagiventime-independentproperty A isdenedbytheexpectationvalueofthe correspondingoperator, h A i t = h j A j i h j i : {2 HereIuseDiracbra-ket"notation[3],wheretheket" j i isavectorintherelevant Hilbertspaceoftheproblem,thebra" h j isalinearfunctionalinthedual-spaceofthe Hilbertspace,andthebracket"isdenedbytheinnerproductinthespace.Theinner productcanbewrittenasanintegraloveralldegreesoffreedom, h j i t = Z d x x ;t x ;t : {3 Then,assuminganormalizedwavefunction h j i =1,theexpectationvalueis h j A j i t = h j A i t = Z d x x ;t A x x ;t : {4 AcceptablewavefunctionssatisfytheSchrodingerequationEquation1{1subject totheboundaryconditionsoftheproblem.Whilemanychoicesofboundaryconditions arepossible,ofinterestinthisstudyaretheboundstatesolutionstotheSchrodinger equation,thosethatareasymptoticallyvanishing.Foratime-independentHamiltonian whentherearenoexternaleldsorpotentialsonecansimplifytheequationsby recognizingthatacompletesetofboundstatesolutionscanbedeterminedbywriting thewavefunctionintheseparableform j x ;t i = j x ij t i ; {5 wheretheproduct j A ij B i isunderstoodtomeanthetensorproduct j A ij B i .Using separationofvariables,thisproductstructureimpliesthatthetime-independent wavefunction j x i mustsatisfythetime-independentSchrodingerequationTISE, 15

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where,forboundstates, E isarealnumberidentiedasthetotalenergyofthesystem, H j x i = E j x i : {6 Equation1{6isaneigenvalueequationwithmultiplesolutions.Eachsolutionwill correspondtoadierentboundstatewhosetimeevolutionisthephasefactor j t i = e )]TJ/F22 7.9701 Tf 6.587 0 Td [(iEt : {7 Insertingthesesolutionsintoanexpectationvalue, h j A j i t = Z d x x e iEt A x x e )]TJ/F22 7.9701 Tf 6.586 0 Td [(iEt = Z d x x A x x = h j A j i ; {8 whichistime-independent.Sincethispropertyholdsforarbitraryobservables,these boundstatesolutionsofTISEareknownasstationarysolutions.Inparticular,theenergy canbedeterminedastheexpectationvalueoftheHamiltonian E = h x j H j x i : {9 Anarbitrarytime-dependentboundstatewavefunctioncanbewrittenasalinear combinationofthesolutionstoEquation1{6,andthen,usingthetime-dependent Schrodingerequation,onecanfollowtheevolutionofthelinearcombinationasafunction oftime.Inmanyenvironments,chemicalsystemsarefoundtobewell-describedbybeing inthelowestenergystationarystate,thegroundstate.Therefore,itisthisboundstate thatisofmostinterest. Ausefulpropertyofthetime-independentSchrodingerequationistheRayleigh-Ritz variationalprinciple,whichstatesthatthegroundstateenergyforagivenHamiltonianis theminimumexpectationvalueoftheHamiltonianoverallwavefunctionsintheHilbert space.Mathematically, E 0 =min j i h j H j i h j i : {10 Aneasyproofofthisformulacanbemadeifweassumethatthewavefunctionis 16

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normalized.Then,assumingthegroundstateisnon-degenerateandusingthehermicityof H ,theoperator H )]TJ/F21 11.9552 Tf 11.955 0 Td [(E 0 ispositivesemi-denite.Therefore,foranyarbitrary wavefunction, h j H )]TJ/F21 11.9552 Tf 11.955 0 Td [(E 0 j i 0{11 whichisapropertyofapositivesemi-deniteoperator[4]andisequivalentto Equation1{10.Thefactthat E 0 isaminimuminthespaceofall j i indicatesthatthe rstderivativeof E 0 withrespectto j i vanishes,leadingtotheconclusionthatrst-order errorsinthewavefunctionaroundtheminimuminduceonlysecond-ordererrorsinthe energy.Forasmallparameter E [ j i + j i ]= E [ j i ]+ O 2 : {12 Specifyingtheproblemtochemistry,onecanassumethatthesystemcontainsa nitenumberofelectrons, N e ,andanitenumberofnuclei, N n .Further,focusingonthe structureandpropertiesofmolecules,anyexternaleldsareperturbationsandignored atthislevelofapproximation.Requiringthattheatomsofinterestberelativelylight rstthroughthirdrowsoftheperiodictableimpliesthatrelativisticandnuclearsize eectscanbeassumedsmall,andignored.Then,becausethenucleiactaspointparticles, anucleus A iscompletelyspeciedbyitsatomicnumber Z A ,mass M A andspin.Given theseparameters,theHamiltonianis H = )]TJ/F22 7.9701 Tf 14.555 14.944 Td [(N e X i =1 1 2 r 2 i )]TJ/F22 7.9701 Tf 14.68 14.944 Td [(N n X A =1 1 2 M A r 2 A )]TJ/F22 7.9701 Tf 15.219 14.944 Td [(N e X i =1 N n X A =1 Z A j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(r A j + N e X i =1 N e X j>i 1 j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(r j j + N n X A =1 N n X B>A Z A Z B j r A )]TJ/F36 11.9552 Tf 11.955 0 Td [(r B j {13 ThetermsintheHamiltonianare,inorder,theelectronkineticenergy,thenuclear kineticenergy,theelectron-nucleusattraction,theelectron-electronrepulsion,andthe nucleus-nucleusrepulsion.Thewavefunctionis j x ; X i ,where x isthevectorofelectron coordinates, x i = r i ; i r i thepositionofelectron i and i itsspin, X isthevectorof nuclearcoordinates, X A = r A ; A .Thewavefunctionisacomplicatedfunctionthat 17

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dependsonboththeelectronsandthenuclei;however,inmanychemicalprocessesthere isaneectivedecouplingoftheelectronsandthenuclei.Therearelimitationstothis assumption,butformuchofroom-temperaturechemistryitholdssurprisinglywell.This decouplingcanbegivenquasi-rigorousjustication[5,6],butitissimplertoproposean ansatzforthewavefunction, j x ; X i = j x ; X ij X i : {14 Theelectronicwavefunction, j x ; X i ,depends parametrically onthenuclearcoordinates, i.e.foreachchoiceofnuclearcoordinates X j x i isdierent,thoughthenuclear coordinatesdonotappearin j i itself.Thenuclearwavefunctionisdenedas j X i ThisformofthewavefunctionisnotexactandisknownastheBorn-Oppenheimer approximation[7].Aswellasthecomputationaladvantages,thereisphysicalcontent intheBorn-Oppenheimerapproximation:inchemistryoneoftenassumestheexistence ofapotentialenergysurfacePESwithoutwhichsuchchemicallyimportantconcepts astransitionstates,isomers,andspectroscopicassignmentsare,atbest,ill-dened[8]. Whethertheseconceptscanbeconsistentlygeneralizedfornon-Born-Oppenheimercasesis stillanopenquestion. InsertingtheBorn-OppenheimerseparatedwavefunctionintotheTISE [ H e + H n + H ne ] j x ij X i = E j x ij X i {15 where H e = )]TJ/F22 7.9701 Tf 14.555 14.944 Td [(N e X i =1 1 2 r 2 i + N e X i =1 N e X j>i 1 j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(r j j ; {16 H n = )]TJ/F22 7.9701 Tf 14.015 14.944 Td [(N n X A =1 1 2 M A r 2 A + N n X A =1 N n X B>A Z A Z B j r A )]TJ/F36 11.9552 Tf 11.955 0 Td [(r B j ; {17 and H ne = )]TJ/F22 7.9701 Tf 14.555 14.944 Td [(N e X i =1 N n X A =1 Z A j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(r A j : {18 18

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Left-projectingby h X j ,andusingthenormalization h X j X i =1 [ H e + h j H n j i + h j H ne j i ] j x i = E j x i : {19 Approximatingthenucleiasdistinguishableclassicalparticles,locatedatpositions R andignoringnuclearkineticenergyandthecouplingbetweenthenuclearandelectron momenta,then h j H n j i = N n X A =1 N n X B>A Z A Z B j R A )]TJ/F36 11.9552 Tf 11.956 0 Td [(R B j = E nn {20 and h j H ne j i = )]TJ/F22 7.9701 Tf 14.556 14.944 Td [(N e X i =1 N n X A =1 Z A j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(R A j : {21 Deningtheelectronicenergyas E ee = E )]TJ/F21 11.9552 Tf 11.955 0 Td [(E nn ,thentheelectronicTISEis )]TJ/F22 7.9701 Tf 14.556 14.944 Td [(N e X i =1 1 2 r 2 i )]TJ/F22 7.9701 Tf 15.22 14.944 Td [(N e X i =1 N n X A =1 Z A j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(R A j + N e X i =1 N e X j>i 1 j r i )]TJ/F36 11.9552 Tf 11.955 0 Td [(r j j # j x i = E ee j x i : {22 NotethatEquation1{22clearlyshowstheparametricdependenceonnuclearcoordinates oftheelectronicwavefunction,i.e.foreachchoiceof R A ,thereisadierentequation tosolve,thoughthe R A arenotvariablesintheequation.Findingsolutionsto Equation1{22isknownasthe electronicstructureproblem .Thersttwotermsof theelectronicHamiltonianactononlyasingleelectronatatime,whilethelastterm, electron-electronrepulsion,actsontwoelectronsatatime. TheexactHamiltonianoperator H commuteswithseveralsymmetryoperations suchasthepointgroupsymmetryofthemolecule,thetotalspinoperator S 2 ,andthe projectionofspinontothe z -direction S z .IftheHamiltoniancommuteswithanother operatorthentheexactwavefunctioncanalwaysbechosentobeaneigenfunctionofthat otheroperator,e.g.theexactwavefunctioninchemistrycanalwaysbechosentobean spin S 2 eigenfunction.Inquantumchemistry,typicallyonechoosesfunctionsthatare, byconstruction,eigenfunctionsof S z .Theothersymmetriesofthemoleculearemore diculttohandleinpractice. 19

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TheelectronicwavefunctionisavectorintheHilbertspacebuiltfromthevector space R 3 N e [ SU ] N e ,where R iscartesianspace,and SU isthespinspace associatedwitheachelectron.BecausetheHamiltonianinnon-relativisticelectron structuretheoryisspin-independent,onecanchoosetorequireapreferredelectron magneticdirectionandsimplifythetreatmentofspintothetwo-valuedsubspacethat Idenote S 1 = 2 ,whoseonlytwoelementsarethesingle-electroneigenfunctionsof S z = f ; g .Acompletebasisfor R 3 istheset f p r g ,wherethesingleelectronfunctions p r areknownas orbitals .Themany-electronbasisfunctionsfor R 3 N e S N 1 = 2 arethen j e m i = N e O i =1 j p i r i ij i i = j N e Y i =1 p i r i i i ; {23 forallchoicesof p i and i .Iwilldenethevectorlabel x i = r i ; i ,so j e m i = j N e Y i =1 p i x i i : {24 Thentheexactwavefunctioncanbeexpressedas j x i = X m c m j e m i : {25 Thespin-statisticstheoremofquantumeldtheoryprovesthatforfermions,suchas electrons,thewavefunctionmustbeantisymmetricwithrespecttointerchangeofanytwo particles[9].Thatis,theexactwavefunctionisavectorinthesubspace ^ A h R 3 N e S N 1 = 2 i where ^ A istheantisymmetrizer,"whichpermuteselectronswiththepropersignchange. Thisrequirementcanbesatisedbychoosingantisymmetricbasisvectors, j e 0 m i = ^ Aj e m i = 1 p N e N e ^ i =1 j p i r i i i {26 usingtheantisymmetrictensorproduct ^ .Thisdenitionisequivalenttoadeterminant, j e 0 m i = 1 p N e det 0 B B B B @ p 1 r 1 p 2 r 1 p 1 r 3 p 1 r 2 p 2 r 2 p 1 r 3 . . . 1 C C C C A : {27 20

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Becausethesemany-electronbasisvectorscanbeviewedasdeterminants,theyare typicallycalledsingleorSlaterdeterminants.Forsimplicity,thenotation j pqr i = 1 p N e det 0 B B B B @ p r 1 q r 1 r r 3 p r 2 q r 2 r r 3 . . . 1 C C C C A : {28 willbeused. Usingacompletebasisfortheset f p g iscomputationallyinfeasible.Instead,a truncatedbasisfor R 3 mustbeused.Thereareseveralwaysofchoosingsuchabasis:a real-spacegride.g.[10],planewaveseectively,agridinmomentumspace,e.g.[11], waveletse.g.[12],andmanyothers,butthemostcommonand,inmanyways,best choicefornitemoleculesistoexpandinatomicorbitals. AtomicorbitalsAOsarefunctionsthatareassociatedwitheachnucleusinthe molecule,andaresolutionstothehydrogenicSchrodingerequation.Thesefunctionshave thegeneralform,foranAOcenteredatnucleusAinsphericalcoordinates, nlm l r = Y lm l ; f n r e )]TJ/F22 7.9701 Tf 6.587 0 Td [( n j r )]TJ/F37 7.9701 Tf 6.586 0 Td [(R A j ; {29 where Y lm l isasphericalharmonic,whichcarriestheangularmomentumofthe orbital,and f n r isapolynomialradialfunction.Thesetypesoforbitalsareknown asexponentialorSlaterorbitals.Forsimplicity,onetypicallychoosestousereal,cartesian sphericalharmonicsratherthanthetruecomplexsphericalharmonics.Becauseof technicaldicultiesinperformingtwo-electronsintegralsovermultiplecenters,rather thanusingtheexponentialfunctions,gaussianfunctionsareused,notingthattoarbitrary precisionitispossibletoreplaceanexponentialfunctionbyasumovergaussians e )]TJ/F22 7.9701 Tf 6.587 0 Td [( n j r )]TJ/F37 7.9701 Tf 6.586 0 Td [(R A j X d n e )]TJ/F22 7.9701 Tf 6.587 0 Td [( n j r )]TJ/F37 7.9701 Tf 6.586 0 Td [(R A j 2 : {30 ForreasonablyshortexpansionsthesenewfunctionsdonotbehaveastheSlaterorbitals 21

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doeitherveryclosetothenucleuswheretheSlaterorbitalsexhibitanimportantcusp [13],oratlongrangewheregaussianfunctionsfallotoorapidly.However,at intermediatedistancesawayfromthenucleus,wherethevalenceelectronsare concentrated,gaussianfunctionscanbettocloselymatchSlaterorbitals. Deningaparticularlinearcombinationofcartesianprimitive"gaussianstoforma contracted"gaussian, j i = r = X d x m y n z p e )]TJ/F22 7.9701 Tf 6.586 0 Td [( j r )]TJ/F37 7.9701 Tf 6.586 0 Td [(R A j 2 {31 theneachorbitalintheentire R 3 spaceisexpandedasalinearcombinationofthese atomicorbitals j i i = X c i j i {32 undertheconstraintthattheyarenormalized h i j i i =1 X c i c i h j i = X c i c i S {33 Setsofatomicorbitalscalledbasissets"havebeenconstructedwitharangeof propertiesincludingthenumberoffunctionsperelectron,thenumberofshellsofangular momentum,andthenumberofprimitivefunctionsthatmakeupthecontractedgaussians. Examplesofcommonlyusedbasissetscanbefoundin[14{16].Inmostcalculations, oneusesabasissetthatislikelytosuittheproblemandtheavailablecomputational resources.Ifpossible,onewouldliketoconvergethecalculationwithrespecttothebasis suchthatthecalculationaccuratelyreproducesthefullHilbertspace;thisisknownas thecompletebasissetCBSlimit.Inallofthefollowing,Iassumeanite-dimensional atomicorbitalbasisset. 1.2TheCorrelationProblem 1.2.1IndependentParticleModels Thesimplestapproximationtotheexactwavefunctionisasinglemany-electron basisstatedeterminant.Thesestatesareoftenreferredtoasindependentparticle 22

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statesIPSsbecausetheyareantisymmetricproductsofsingle-particlefunctions,sothe electronsarestatisticallyuncorrelatedotherthantheantisymmetry.Therearemany choicesofIPSsandonecanconstructvariousapproximatemodelsthatwillyielddierent IPSsastheirsolution.Thetwomostcommonchoicesofindependentparticlemodelare Hartree-Focktheoryanddensityfunctionaltheory. Hartree-Fock :Hartree-Focktheoryisthechoicetodeterminethesinglebasis stateastheminimumenergydeterminantfortheproblem.Beginningwithanarbitrary normalizeddeterminant j i ,composedoforthonormalorbitals i r ,where i;j = 1 ; ;N e ,theenergyofthatstateis E = h j H i = N e X i =1 h ii + 1 2 N e X i;j =1 h ij jj ij i {34 where h ij = h i r 1 j)]TJ/F15 11.9552 Tf 19.128 8.088 Td [(1 2 r 2 1 )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X A Z A j r 1 )]TJ/F36 11.9552 Tf 11.955 0 Td [(R A j j j r 1 i {35a h ij jj kl i = h ij j kl i)-222(h ij j lk i {35b h ij j kl i = h i r 1 i r 2 j 1 j r 1 )]TJ/F36 11.9552 Tf 11.955 0 Td [(r 2 j j k r 1 l r 2 i : {35c Inthetwo-electronintegralEquation1{35b,thersttermisthecoulombicrepulsion betweentwochargedistributions,andthesecondtermistheexchange,"apurely quantummechanicaleectthatenforcestheantisymmetryofthedeterminant.Requiring thisenergytobeaminimumundervariationsoftheorbitalsthatconserveorthonormality leadstotheHartree-Fockequationforeachorbital j i i )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 r 2 )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X A Z A j r )]TJ/F36 11.9552 Tf 11.956 0 Td [(R A j + J r )]TJ/F21 11.9552 Tf 11.955 0 Td [(K r # j i i = X j ij j j i {36 where J isthecoulomboperatorand K istheexchangeoperator, J r j i i = N e X j =1 Z d r 0 j r 0 j r 0 i r j r )]TJ/F36 11.9552 Tf 11.955 0 Td [(r 0 j {37a 23

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K r j i i = N e X j =1 Z d r 0 j r 0 i r 0 j r j r )]TJ/F36 11.9552 Tf 11.955 0 Td [(r 0 j : {37b Notetheinterchangeofthe r and r 0 in K .Onecandenetheoperatorontheleft-hand sideofEquation1{36astheFockoperator, f r = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 r 2 )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X A Z A j r )]TJ/F36 11.9552 Tf 11.955 0 Td [(R A j + J r )]TJ/F21 11.9552 Tf 11.956 0 Td [(K r {38 whosematrixelementsare f pq = h p j f j q i = h pq + N e X j =1 h pj jj qj i : {39 Thisdenitionshowsthat f y = f ,i.e.thattheFockoperatorishermitian.The N e lowestenergysolutionstoEquation1{36denetheHartree-Fockorbitals.Equation1{36 mustbesolvediterativelybecauseboth J and K dependontheorbitalsthemselves. Atconvergence,theHForbitalscanbeviewedasthedistributionofelectronsinthe averageeldgeneratedtheotherelectrons.Inanitebasisofdimension M ,thereare anadditional M )]TJ/F21 11.9552 Tf 12.732 0 Td [(N e solutionstotheHFequations.Theseunoccupied,orvirtual," solutionsdonothaveadirectphysicalmeaning,butspanthespaceorthogonaltothe Hartree-Fockdeterminantinthebasis.Therefore,theyareaconvenientsetoffunctionsin whichtorepresentcorrectionstotheHFwavefunction.Iwillusethenotationthat i;j;k correspondtooccupiedorbitals, a;b;c correspondtovirtualorbitalsand p;q;r correspond toeither.Giventhisnotation,theHartree-Fockequationsarecompletelyequivalenttothe Brillouincondition[7], f ia = f ai =0 ; 8 i;a: {40 ThisconditionandtheenergyexpressionEquation1{34showthattheHartree-Fock equationsareindependentoftheactualformoftheoccupiedandvirtualorbitals:onecan performanyunitaryrotationwithintheoccupiedorbitalsorwithinthevirtualorbitals withoutchangingtheenergyortheBrillouincondition.Onecanfurtherchoosetoimpose theconditionthattheoccupiedorbitalsandthevirtualorbitalsdiagonalizetheFock 24

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operatorintheirsubspaces,i.e. f ij = f ii ij ;f ab = f aa ab : {41 TheorbitalsthatsatisfythisadditionalconditionarecanonicalHartree-Fock"orbitals. CanonicalHForbitalsareoftencalledmolecularorbitals." BecausetheFockoperatorcompletelydeterminestheHForbitals,itsproperties determinethepropertiesoftheorbitals.TheFockoperatoractsasaneective Hamiltonianfortheeectivesingle-electronSchrodingerequationofthecanonicalHF orbitals, f j i i = i j i i : {42 UnlikethefullHamiltonian,theFockoperatordoes not commutewiththespin operatoringeneral,[ f;S 2 ] 6 =0.ThelowestenergyHartree-Fockwavefunctionwill notnecessarilybeaneigenfunctionof S 2 .Onecanchoosetoimposetheadditional conditionthattheHartree-Fockwavefunctionmustsatisfythesymmetriesofthesystem; thischoiceiscalledrestrictedHartree-FockRHF.Ifinsteadonedoesnotimposethat conditionthenitispossibletohaveasymmetry-broken,lowerenergystate,knownas theunrestrictedHartree-FocksolutionUHF.Inquantumchemistry,UHFisusually takentomeantheHartree-FockwavefunctionthatbreaksspinS 2 andspatial-point groupsymmetry.Whetherthelowerenergysymmetry-brokenUHFsolutionorthe higher-energysymmetry-conservingRHFsolutionispreferredisamatterofdebate.In theliteraturethisproblemisknownasthesymmetrydilemma"[17,18].Formany closed-shellmoleculesnearequilibriumtheUHFsolution is theRHFsolutionandthereis nosymmetrydilemma.TheRHFsolutionandtheUHFsolutionwillsplitfromeachother atcertaindistanceawayfromequilibriumwhentheRHFsolutionwillbecomeunstable withrespecttosymmetry-breakingchangesintheorbitals.Thelocationofthisinstability isdeterminedbytheHartree-Fockinstabilityconditions[19{22]. 25

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Foranopen-shellmolecule,suchasaradical,thereisnoRHFsolution;ifthe spin-symmetryisconservedthenthewavefunctioncannotsatisfytheHartree-Fock equations.Onecanchoosetousethesymmetry-brokenUHFchoice,oronecanchooseto useasymmetry-restrictednon-Hartree-Fockreference f ia 6 =0.Themostcommonchoice, ifsymmetryisimportant,isrestrictedopen-shellHartree-FockROHF.Thisnameisa misnomer,becauseanROHFwavefunctiondoesnotsatisfytheHartree-Fockequations; ROHFdoessatisfyaspin-summedsetofequations f i a + f i a =0 f i s = f s a =0 ; 8 i;a;s {43 where s runsoverallsingly-occupiedorbitals.BecausetheROHFwavefunctionisnota Hartree-Fockwavefunctionitis always unstablewithrespecttotheUHFsolution. DensityFunctionalTheory :AlthoughdensityfunctionaltheoryDFT[23]will notbedevelopedinthisstudy,itservesasabasisformanycomparisons,andistoo importantacomputationalchemistrytechniquetoignore.TheHohenberg-Kohntheorems [24]provedthattheexactground-stateenergyofamoleculeisanunknownfunctionalof theelectronicdensity : E 0 = E DFT [ ] : {44 OnegoalofDFTistodevelopapproximateformsfortheexactdensityfunctional E DFT ItwasshownbyGilbert[25]andbyHarriman[26]thatallphysicaldensitiescanbe writtenintheform r = N e X i =1 j i r j 2 {45 forsingleparticlefunctions i r .ItisnaturaltoreformulateDFTintermsofthese orbitals.ThisapproachwasformalizedbyKohnandSham[27,28],leadingtotheenergy expression E KS [ f i g ]= T s [ f i g ]+ E coul [ ]+ E ext [ ]+ E xc [ ]{46 where T s isthenon-interactingkineticenergy, E coul istheclassicalcoulombicrepulsion, 26

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E ext isthenuclear-electronattraction,and E xc istheunknownexchange-correlation" functionalofthedensitythatincludesallothereects.Explicitly,thesetermsare T s [ f i g ]= N e X i =1 )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 h i jr 2 j i i {47a E coul [ ]= Z d r d r 0 r 0 r j r )]TJ/F36 11.9552 Tf 11.955 0 Td [(r 0 j = N e X i;j =1 h ij j ij i {47b E ext [ ]= Z d r v ext r r = )]TJ/F22 7.9701 Tf 14.555 14.944 Td [(N e X i =1 h i j X A Z A j r )]TJ/F36 11.9552 Tf 11.955 0 Td [(R A j j i i : {47c MakingtheKohn-Shamenergystationarywithrespecttoorthonormalvariationsofthe orbitals,yields )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 r 2 + v ext r + J r + v xc [ ] r i r = N e X j =1 ij j r ; {48 where v xc r = E xc isanunknownlocaldensity-dependentexchangeandcorrelation" xcpotential.Itisnotknownwhetherallphysicaldensitiescorrespondtoasolutionof theseequation;thisisthenon-interacting v -representabilityproblem[23]. ThesimilaritybetweenEquation1{48andEquation1{36isstriking;yet,whileHFis explicitlyanapproximatemethods,KS-DFTis inprinciple exact.Unfortunately,without amethodtogeneratetheexact E xc ,onecannotdeterminetheexactenergy.Commonly, basedonanumberofformalargumentsandnumericalparameterizations,oneproposesan exchange-correlationfunctional,which,inturn,denesanexchange-correlationpotential. AmajordicultyinconstructingaDFTfunctionalistheself-repulsionpresentinthe coulombictermoftheenergythatmustbecancelledbythexcterm.Theimpactofthis termcanbereducedbyreplacingthelocal v xc byalinearcombinationofthenon-localHF exchangeandalocalpotential,hindicatessuchahybrid" v xc,h r = K r + )]TJ/F21 11.9552 Tf 11.955 0 Td [( v xc r : {49 AnexampleisthecommonlyusedB3LYPfunctional[29,30]. 27

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1.2.2SecondQuantization Asingledeterminantisnotgenerallygoingtobetheexactwavefunction.An arbitrarymany-bodystatewillbealinearcombinationofmultipledeterminants.In chemistry,theindependentparticlestatesaresurprisinglygood;inSection1.2.3,Iwill discussthisinmoredetail,butsuceittosayonecanviewtheexactsolutionassome relativelysmallcorrectiontoanIPS.Itishelpfultoorganizethesecorrectionsinsucha wayastominimizethereproductionofeort.Aconvenientwayofdoingthisistouse secondquantizationandnormal-ordering[31]. Secondquantizationintroducesoperatorsthataddorremoveanelectronfroma determinant.Onecandenethecreationoperator p y suchthatactingonadeterminant thatdoesnotcontainanelectroninorbital p ji isthezero-electronket, p y ji = j p i : {50 Itshermitianconjugate,theannihilationoperator p ,removessaidelectronfromorbital p p j p i = ji : {51 Becauseoftherequiredantisymmetryofthewavefunction,onecannothavetwo electronsinthesamestate.Thatisequivalenttorequiringnilpotencyofthecreation orannihilationoperators )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p y 2 = p 2 =0 : {52 Theseoperatorsobeythecanonicalanticommutationrelations[9], [ p;q ] + =0=[ p y ;q y ] + [ p y ;q ] + = pq =[ p;q y ] + ; {53 where[ A;B ] + = AB + BA .Theseanticommutationrelationsbuildinthefermionic symmetryofthewavefunction,sothatanarbitrarydeterminant j 0 i of N e electronscanbe writtenasaproductofcreationoperatorsonthezero-electronstate, 28

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j 0 i = p y q y r y s y | {z } N e operators ji : {54 Intermsoftheseoperators,theHamiltoniancanbewritten H = X pq h pq p y q + 1 4 X pqrs h pq jj rs i p y q y sr: {55 BecausethereareasmanycreationoperatorsasannihilationoperatorsinHamiltonian, thenumberofelectronsisconservedwhenactingwith H Onewouldliketoreformulatetheoperatorsinsuchawaythatthepiecesthatare alreadycontainedinthereferencewavefunctionareremoved.Theprocessofdoingthis isknownasnormal-ordering"andisdenotedbycurlybrakets fg aroundtheoperator. Anoperatorinnormal-orderedformhasavanishingexpectationvalueinthereference wavefunction, j 0 i h 0 jf A gj 0 i =0 : {56 Anarbitrarynormal-orderedoperator A N canbewritten A N = f A g = A )-222(h 0 j A j 0 i : {57 Deningtheorbitalsoccupiedinthereferencefunctionby i;j;k holestatesand thoseunoccupiedby a;b;c particlestates,thenanormal-orderedsecond-quantized operatorhasall i y 'sand a 'stotherightofthe i and a y operators.The i y and a operators arezerowhenactingonthereferencedeterminant, i y j 0 i = a j 0 i =0 : Onenormal-ordersan operatorbycommutingthevarioussecond-quantizedoperatorsusingEquations1{53. Theseanticommutationrelationscreateagreatnumberofterms;tohandlethe proliferationoftermsawidevarietyoftechniqueshavebeenused,includingcomputer algebrasystems[32,33]anddiagrammatictechniques[7,9,31].Thediagrammatic techniqueshavebeenpervasiveandthelanguagedescribingnormal-orderedoperators isoftenbasedonthehowvariousdiagramslook.Forexample,theonlytermsthat 29

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willsurviveinanexpectationvaluearethosethathavenoremainingsecond-quantized operators,i.e.onlydeltafunctionsremain;theseoperatorsarecalledclosed"operators. Inparticular,thenormal-orderedHamiltonianis H N = H )-222(h 0 j H j 0 i = X pq f pq f p y q g + 1 4 X pqrs h pq jj rs if p y q y sr g : {58 Asmentionedabove,anormal-orderedoperatorwithoutanysecond-quantized operatorsremainingisknownasaclosedoperator,converselyanoperatorthathas normal-orderedcreationandannihilationoperatorsremainingisanopen"operator.A connected"operatorisonethatcannotbewrittenastheproductoftwooperators,for example A pq f p y q g ,connected, X q A pq A 0 qr f p y r g ,connected, A pq A 0 rs f p y qr y s g ,disconnected. {59 Thecommutatoroftwoconnectedoperatorsisitselfconnected.Asubscript C will indicatekeepingonlytheconnectedpiecesofanoperator.Alinked"operatorisone thatiseitherclosedandconnectedortheproductofoneormoreopenoperators,withno closedoperatorsintheproduct.Asubscript L willindicatekeepingonlythelinkedpieces ofanoperator.Thelinked-diagramtheorem[34{38]provesthatonlylinkedoperators enterphysicalquantities.Forexample,theenergycanbewrittenasthesumofclosed, connectedoperatorsandthewavefunctioncanbegeneratedbyanopen,linkedoperator. Iftheenergycontainsunlinkeddiagramsclosedanddisconnectedthenitwillnot scaleproperlywiththenumberofparticles.Ifoneincreasesthenumberofparticlesina systematconstantdensitythethermodynamiclimitthentheenergyperparticleshould approachaconstant, lim N !1 N=V =constant E N = e: {60 30

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Thisscalingpropertyisknownassize-extensivity"[39].Theenergyfromunlinked operatorswillhaveadierentdependenceonthenumberofparticles.Ifthedependenceis higherthanlinearin N ,thentheenergyperparticlewilldivergeandifthedependenceis lowerthanlinearin N ,thentheenergyperparticlewillvanish. Onemightthinkthatthethermodynamiclimitisfarfromchemicalsystemsof interest,however,thelackofsize-extensivityimplies,amongotherthings,thaterrorper electron increases withthenumberofelectrons[39,40].Forasize-extensivemethodthe errorperelectronisconstantwiththenumberofelectrons.Therefore,whencomparing twosystemsofdierentnumbersofelectronsusingasize-extensivemethodthereisan inherentcancellationoferrorsthatisnotpresentinamethodthatisnotsize-extensive. Ifamethodincludesunlinkedterms,thentheenergyoftwonon-interactingsystems calculatedbythemethodisnotthesumoftheenergiesoftheindividualsystems,i.e. E A + B 6 = E A + E B : {61 Therefore,chemicalquantitiessuchasheatsofformationandheatsofreactionare ill-denedwhenusinganon-size-extensivemethod. OnceareferencefunctionisdenedtherestoftheHilbertspacecanbewritten asexcitations"ofthereferencedeterminant.Theoperationof f a y i g onthereference determinant j 0 i constructsanewdeterminant j a i i ,wheretheorbital i hasbeenreplaced bytheorbital a .Thisdeterminantisknownasasingleexcitation.Similarly,theactionof f a y ib y j g formsthedeterminant j ab ij i ,adoubleexcitation,andsoon.Becausetheunionof theoccupiedandvirtualorbitalsspanthesingleparticlespace,thesetofallexcitations fromoccupiedorbitalstovirtualorbitalsspansthemany-electronspace.Therefore,an arbitrarywavefunctioninthemany-electronspacecanbewritten j i = c 0 j 0 i + X ia c a i j a i i + 1 4 X ijab c ab ij j ab ij i + {62 wherethefractionsinfrontofthesumsaccountfordoublecountingintheexpansion. 31

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1.2.3CongurationInteraction Usingsecondquantization,theexpansioninEquation1{62canbewrittenasan operatoractingonthereferencedeterminant, j i = C j 0 i ;C = N e X n =0 C n {63 and C 0 = c 0 C 1 = X ia c a i f a y i g {64a C 2 = 1 2! 2 X ijab c ab ij f a y ib y j g C 3 = 1 3! 2 X ijk abc c abc ijk f a y ib y jc y k g {64b andsoonforhigherexcitations.Sincetheoverallnormalizationofawavefunctionwill becancelledinanyexpectationvalue,thenassumingthatthereferencedeterminantis notorthogonaltotheexactwavefunction,onecanchoose c 0 =1andremoveitfromthe operator C ,soEquations1{63arereplacedby j i =+ C j 0 i ;C = N e X n =1 C n : {65 Thischoiceisknownasintermediatenormalization."Foranitesystem,aslongasthe referencefunctionhasthesamesymmetryastheexactgroundstate,thisnormalizationis acceptable;whileforaninnitesystemthisassumptionislikelyto fail [41].Thislackof overlapcomplicatesthedevelopmentofcorrelatedmethodsforsolids. Sinceanarbitrarywavefunctioncanbewrittenintermsofthisexpansion,onecan applytheRayleigh-Ritzprincipletodeterminethegroundstateenergyandwavefunction, E 0 = h 0 j H j 0 i +min C h 0 j )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ C y H N + C j 0 i h 0 j + C y + C j 0 i ; {66 whereIhavefactoredouttheenergyofthereferencedeterminant.Forcomparisonwith othermethods,itishelpfultodeneanenergyfunctional, 32

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E CI = h 0 j )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ C y H N + C j 0 i h 0 j + C y + C j 0 i ; {67 whichappliesforanarbitrarywavefunctionofthe+ C j 0 i form,suchthatthe minimumofthisfunctionalis E = E 0 )-222(h 0 j H j 0 i =min C E CI : {68 Assumingeach c K tobereal,thenonecanmakethisexpressionstationarywithrespect tovariationsin c y K ,where K isanexcitation,yieldingthecongurationinteractionCI equations hKj H N + C j 0 i = Ec K : {69 Equation1{69isaneigenvalueequationforthecorrectiontothereferenceenergyofthe groundstatewavefunctioninthebasis.Ifoneincludesall C n for n =1 ; ;N e ,thenthis isknownasthefullcongurationinteractionFCI.FCIistheexactanswerinagiven basissetandisthetargetforallothermethods.IfthereferenceischosentobeanRHF function,then E canbecalledthecorrelationenergy," E corr = E FCI )]TJ/F21 11.9552 Tf 11.955 0 Td [(E RHF : {70 Ingeneral,thecorrelationenergyisonly1%ofthetotalenergy.However,bondenergies arealsoroughly1%ofthetotalenergy,therefore,thecorrelationenergyiscrucialfor describingchemistry.Toapproachtheexperimentalvalue,onemustbothconvergewith respecttothebasisCBSlimitandconvergewithrespecttothelevelofexcitationFCI limit. GiventhatFCIistheexactanswer,whyuseanyothermethod?Thenumberof coecientsinaFCIexpansionscales O 2 M ,for M basisfunctions,whichisequivalentto exponentialgrowthinthesizeoftheeigenvalueproblem.Therefore,theFCIwillquickly becomecomputationallyinfeasible.Instead,onecantruncatethelevelofexcitation includedintheoperator C .Ifoneincludesonlysingleanddoubleexcitations,themethod 33

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isknownasCIwithsinglesanddoublesCISD,includingtripleexcitationsyieldsCISDT, etc.ThescalingofCISDis O n 2 N 4 ,where n = N e and N = M )]TJ/F21 11.9552 Tf 12.562 0 Td [(N e ,thenumberof virtualorbitals.Thisscalingisonlypolynomialinthesizeofthesystem,whichisfeasible. TruncatedCImethodshavecrucialdierenceswhencomparedtoFCI.InFCI,the referencefunctionisentirelyarbitrary;aslongasitwasnotorthogonaltotheground state,theFCIenergywillbethesamewithanyreferencefunction.Thesameisnottrue foratruncatedCI.Becauseonlysomeoftheexcitationsareincluded,ifthereference functionisagoodchoice,relativelyfewexcitationswillbenecessarytoaccuratelymodel thefullwavefunction.Ifthereferencefunctionisbad,thenmanymoreexcitationsmay benecessary.AmorefundamentalfailureoftruncatedCIisthelackofsize-extensivity. Oneoftheconsequencesofsizeextensivityisnon-interactingseparability;thattheenergy oftwonon-interactingsystemsisthesumoftheirenergies.ToprovethatCIisnot size-extensive,IwillshowthattruncatedCIdoesnotobeynon-interactingseparability. Assumethattherearetwoclosed-shellsystems,aheliumatomandahydrogen molecule,innitelyfarapart.TheCISDequationfortheheliumwhichistheFCIis H He+ C 1 He+ C 2 He j 0He i = E He+ C 1 He+ C 2 He j 0He i : {71 Similarly,theCISDequationforhydrogenistheFCIandis H H 2 + C 1 H 2 + C 2 H 2 j 0H 2 i = E H 2 + C 1 H 2 + C 2 H 2 j 0H 2 i : {72 Becausetheseareclosedshellsystems,theFCIforthehelium-hydrogensystemisan antisymmetrizedproductwavefunctionoftheindividualFCIs, j full i = j HeH 2 i =[1+ C 1 He+ C 2 He][1+ C 1 H 2 + C 2 H 2 ] j 0HeH 2 i =[1+ C 1 He+ C 1 H 2 + C 2 He+ C 2 H 2 + C 1 He C 1 H 2 + C 1 He C 2 H 2 + C 2 He C 1 H 2 + C 2 He C 2 H 2 ] j 0HeH 2 i : {73 34

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The C 1 C 2 generatesanoveralltripleexcitationasingleexcitationonhelium,forexample, andadoubleexcitationonhydrogenand C 2 C 2 generatesaquadrupleexcitationadouble excitationonheliumandhydrogen.However,ifoneusesaCISDwavefunctiontodescribe theoverallsystem,therearenotripleorquadrupleexcitationsincluded.Therefore,the CISDrepresentationoftheoverallsystemis not theFCI,eventhoughCISDwastheFCI forbothofthefragmentsindividually.Thisargumentwasnotarigorousproof,butshows ahallmarkofinextensivemethods,theaccuracyfortheproductoftwonon-interacting systemsislowerthancalculatingeachsystemseparately.Withoutfocusingonthe detailsoftheproofofthisfact,notethatontheright-handsideofEquation1{69,the CIcoecient c K ,whichiseectivelyanopenoperator,ismultiplying E ,whichmust beaclosedoperator.Therefore,theirproductisunlinked.Asdiscussedabove,unlinked productsbreaksize-extensivity. Finally,asshowninFigure1-1,comparedtoothermethodsthatwillbediscussed inlatersections,CIconvergesslowlywithrespecttothecomputationalcost.Thelackof size-extensivitymeansthatthisslowconvergencewillbeevenworsewhenincreasingthe sizeofthesystem. 1.3Coupled-ClusterTheory Theimportanceofextensivityhaslongbeenrecognizedwithinthephysicscommunity andtherehavebeenseveraltechniquesdevelopedthatbuildinthisproperty.Twoofthe mostprominentmethodsaremany-bodyperturbationtheoryMBPTandtheuseof propagatorsGreen'sfunctionsandtheDysonequation.Inbothcases,oneincludesall contributionsthroughsomeorderinperturbationtheory.Insituationssuchashard-core nuclearpotentialsandinthehomogenouselectrongas,onemustsumcertaincontributions toinniteorderintheperturbation.Whilethereareaninnitenumberofchoicesofthese summations,ithasbeenfoundthatcoupled-clustertheory,discussedhere,isaparticularly powerfultechniqueofinnitesummation. 35

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Figure1-1.ConvergenceofthemeanpercentageofcorrelationenergyforCI,CCand MBPTwhencomparedtoFCIforBH,HF,H 2 O,SiH 2 ,CH 2 ,N 2 ,andC 2 at equilibrium[42]. 1.3.1DerivationandFunctional Therequirementofextensivitycanbesatisedbyusinganexponential parameterizationofthewavefunction[43{46],whereanewexcitationoperator T is denedvia j i = e T j 0 i ;T = N e X n =1 T n {74 withthe n -excitationterm, T n ,givenby T n = 1 n 2 X ijk abc t abc ijk a y ib y jc y k : {75 36

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Thisparameterizationofthewavefunctionisknownasthecoupled-clustertheoryCC [42].ThekeydierencebetweentheCCparameterizationofthewavefunctionandaCI parameterization,isthepresenceofproductsof T operators.Truncating T atsingleand doubleexcitations T = T 1 + T 2 e T j 0 i = 1+ T 1 + T 2 + 1 2 T 2 1 + T 1 T 2 + 1 2 T 2 2 + 1 2 T 2 1 T 2 + 1 3! T 3 1 + 1 4! T 4 1 j 0 i : {76 Theproductterms, T 1 T 2 T 2 2 ,etc.,arefullyincludedinthewavefunction.Therefore,it shouldbeclearthatthehelium-hydrogenexample,Equation1{73,willbeproperlytreated usingcoupled-clustertheory. OnecancomparethefullCCFCCtotheFCItoshowthatCCispotentiallyan exacttheory.IntheFCIlimit,thefollowingoperatoridenticationcanberecursively made, T 1 = C 1 T 2 = C 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 T 2 1 {77a T 3 = C 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 1 T 2 )]TJ/F15 11.9552 Tf 14.776 8.088 Td [(1 3! T 3 1 T 4 = C 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 1 T 3 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 T 2 1 T 2 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 T 2 2 )]TJ/F15 11.9552 Tf 14.776 8.088 Td [(1 4! T 4 1 ; {77b continuingforhigherexcitations.Mathematically,thisdenesthe T operatorsasthe cumulantsofthe C operators.Thereareexactlyenoughundeterminedcoecientsonthe left-handsideofthissetofequationstomakethisidenticationexact.Atanytruncated levelofapproximation,however,theCCansatzindirectlyincludesapproximationsto higherexcitationlevelsthanareincludedinthetruncation.Forexample,forCCsingles anddoublesCCSD,onehasthetildesindicatethattheactualvaluesofthevarious operatorsisnotthesameasfromtheCIcalculation ~ C 1 = T 1 ~ C 2 = T 2 + 1 2 T 2 1 {78a ~ C 3 = T 1 T 2 + 1 3! T 3 1 ~ C 4 = 1 2 T 2 1 T 2 + 1 2 T 2 2 + 1 4! T 4 1 : {78b Assumingthat,forexample, T 4 1 = 2 T 2 2 then ~ C 4 C 4 .Therefore,CCSDcanincludethe dominantimpactofquadrupleexcitations,andhigher,withoutneedingquantitieslarger 37

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thansingleanddoubleexcitations.ThedominantcorrectionneededtoaCCSD wavefunctionisthatfrom T 3 Insertingthiswavefunctionansatzintotheexpectationvalueandremovingthe referenceenergy, E = h j H j i h j i = h 0 j e T y H N e T j 0 i h 0 j e T y e T j 0 i ; {79 yieldsanenergyexpressionthatiscomputationallyunsatisfying.Theexponential structureofthewavefunctionleadstoaninniteseriesinproductsof T and T y operators. Theexpansionof e T isnotactuallyinniteforanitenumberofelectrons.Inthatcase, anytermthatgeneratesanexcitationhigherthan N e willbezero,truncatingtheseries. However,theratioofthenumeratoranddenominatorwillremainaninniteseriesin T Truncatingthedenominatorandnumeratorseparatelybreaksextensivity,oneof themajorreasonsforthecoupled-clusteransatz,sothatwouldbeapoormethodof approximation.WhiletherearealternativesthatIwilldiscussinChapter4,hereI willdiscusstheconventionalapproach.Onecaninserttheidentity ^ I = e T e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T into Equation1{79, E = h 0 j e T y e T e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T H N e T j 0 i h 0 j e T y e T j 0 i {80 andonecandenethecoupled-clustereectiveHamiltonian, H as H N = e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T H N e T : {81 TheeectiveHamiltonianisasimilaritytransformationofthebareHamiltonian. Similaritytransformationsdonotchangetheeigenvaluesofamatrix,soafull diagonalizationof H N wouldyieldtheexactenergiesinthebasis,thoughthis approachwouldbeequivalenttoFCIandnotcomputationallyfeasible.Thesimilarity transformationalsobreaksthehermicityoperator; H y N 6 = H N .Thetermsintheenergy not intheeectiveHamiltonianaregivenbythefollowingrearrangement,wherethe summationover K isaresolutionoftheidentityoverallexcitations 38

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h 0 j e T y e T h 0 j e T y e T j 0 i = X K h 0 j e T y e T jKi h 0 j e T y e T j 0 i hKj = h 0 j + X K6 =0 h 0 j e T y e T jKi h 0 j e T y e T j 0 i hKj = h 0 j + X K6 =0 K hKj = h 0 j + : {82 InthelaststepIhaveabstractedthesummationoverdeterminantsintoanoperator, whichisade-excitationoperatordenedby = N e X n =1 n ; n = 1 n 2 X ijk abc ijk abc i y aj y bk y c : {83 Usingthesedenitions,anitecoupled-clusterfunctionalcanbedened E CC = h 0 j + H N j 0 i : {84 Itmaynotseemobviouswhythisformofthefunctionalisanymorenitethanwas Equation1{79.ThefullBaker-Campbell-HausdorBCHexpansionof H is H N = e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T H N e T = H N +[ H N ;T ]+ 1 2! [[ H N ;T ] ;T ] + 1 3! [[[ H N ;T ] ;T ] ;T ]+ 1 4! [[[[ H N ;T ] ;T ] ;T ] ;T ] : {85 TheBCHexpansionterminatesforanynumberofelectronsatfourcommutatorsbecause theHamiltonianisatwo-particleoperator.Theleft-handoperatorislinear,leadingto theconclusionthatEquation1{84containsonlyanitenumberofterms.Ifonetruncates T andatsomelevelofexcitation,forexamplesingleanddoubleexcitations,thenthe CCfunctionalwillhavethesamenumberoftermsforanynumberofelectrons.Because H N canbewrittenasaseriesofcommutators,itcanalsobeidentiedastheconnected productof H and e T H N = )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(H N e T C ,iftheoperator T isconnected,thensois H N 39

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WhileonenowknowstheCCfunctional,Equation1{84,theequationstodetermine T andhavenotbeendetermined.OnewouldliketobeabletoapplytheRayleigh-Ritz variationalprincipleanddeterminetheCCenergyastheminimumofthefunctional. Unfortunately,Rayleigh-Ritzdoesnotapply.Usingtheidenticationthatwhen untruncated h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T = h 0 j e T y h 0 j e T y e T j 0 i = h ~ j ; {86 denesaleft-wavefunction,whichis,inthetruncatedcase,distinctfromthehermitian conjugateoftheright-wavefunction j i = e T j 0 i .Thetwowavefunctionsare biorthonormal,i.e. h ~ j i =1.Then,onecandenetheCCfunctionalas h ~ j H N j i Becausetheleft-andright-wavefunctionsarenotidentical,thereisnolongeranupper boundingprincipleastherewasintheRayleigh-Ritzformula.Thereasonthatthe variationalprincipledoesnotholdcanbeexplainedbyfollowingthesamederivationas beforefortheRayleigh-Ritzprinciple,aroundEquation1{11.Theoperator H N )]TJ/F15 11.9552 Tf 12.306 0 Td [( E is stillpositivesemi-denite,butbecause h ~ j6 = j i ,theboundingprincipledoesnothold. Despitethefactthatthereisnovariationalprinciple,therecanbea stationary principle,whichwillmaketheenergyfunctionallocallystablewithrespectto variationsinitsparameters.Oneshouldrecognizethathavingdierentleftand rightparameterizationsofthewavefunctionisequivalenttoviewingthehermitian operator H asanon-hermitianoperator.Itisrelativelystraightforwardtoshowthat foranon-hermitianoperator,thattheerrorsintheenergywillbesecond-orderinthe wavefunctionparametersifandonlyiftheenergyisstationarywithrespecttovariations ofitsparameters[47,48].Therefore,theCCequivalentoftheRayleigh-Ritzprincipleis E 0 =stat T; h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T H N e T j 0 i : {87 Becausetheenergyislinearin,ifitisstationarywithrespecttovariationsofthen willdisappearfromtheequations,i.e. 40

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stat h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T H N e T j 0 i = h 0 j e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T H N e T j 0 i : {88 Becauseofthislackofdependenceontheactualvalueofitisunnecessarytoimpose stationaritywithrespectto T todeterminetheenergy.Iftheoperator H N isconnected, thenthisexpectationvaluewillbelinked,andtheenergywillbesize-extensive.Aswas mentionedearlier, H N islinkedifandonlyif T islinked.Todeterminetheequationsfor T ,onevariestheCCfunctionalwithrespecttothemultipliers, @ E CC @ K = hKj H N j 0 i =0 : {89 Thisequationisthenanonlinearequationfor T .Assumingthattheinitialguessfor T is connectedforexample,theMBPTamplitudes,thensolvingthisequationwillnever introduceadisconnectedterm.Therefore, T iseasilychosentobeconnectedandthenso is H N .ThisoutlinesthereasonwhyCCissize-extensive. Thecoupled-clusterdoublesCCDequationsarethesimplestnon-trivialexample ofCC,andserveasagoodexampleofthestructureoftheequations.WithinCCD,the doubleexcitationamplitudeequationsare ab ij t ab ij = h ab jj ij i + 1 2 X cd h ab jj cd i t cd ij + 1 2 X kl h ij jj kl i t ab kl )]TJ/F21 11.9552 Tf 11.955 0 Td [(P ij j ab X kc h ak jj ci i t bc jk + P ij X k 6 = i;j f ik t ab jk )]TJ/F21 11.9552 Tf 11.955 0 Td [(P ab X c 6 = a;b f ac t bc ij + 1 4 X klcd h kl jj cd i t cd ij t ab kl + 1 2 X klcd h kl jj cd i P ij )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(t dc jk t ab li + t ac ik t bd jl )]TJ/F21 11.9552 Tf 11.956 0 Td [(P ab )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(t ac lk t db ij + t ac jk t bd il {90 where P qr =1 )-222(P qr P qr j st = P qr P st {91 and P qr permutesthelabels q and r .Thenonlineardependenceof t ab ij onalloftheother T 2 amplitudesisclearfromthisequation.Atthestationarypoint,theCCDenergyis 41

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E CCD = 1 4 X ijab h ab jj ij i t ab ij : {92 InCCSDandallhigherexcitations,theenergy, E CCSD = 1 4 X ijab h ab jj ij i t ab ij + X ia f ai t a i + 1 2 X ijab t a i t b j : {93 Tripleexcitationsandhigherdonotdirectlyentertheenergy,butratheraecttheenergy bymodifyingthevalueof T 1 and T 2 Atagivenlevelofexcitation,thenonlinearCCequationsaremorecostlythanthe eigenvalueCIequations,e.g.CISDisfasterthanCCSD.The scaling ofbothmethodsis thesame.Theslightlyworsecomputationalcostisworthit,asisclearfromFigure1-1. TheCCresultsconvergefarmorerapidlythantheCIresultsdotowardtheFCI.The size-extensivityofCCmeansthatthisconvergencewillbethesameforanarbitrarilysized molecule.Size-extensivityrequiresthattheexactwavefunctionmustbeanexponential. 1.3.2TripleExcitations Tobepredictive,amethodmusthavetighterrorbarsforenergies,geometries, frequencies,etc.,andmustbeapplicabletorelativelyarbitrarymolecularstructures[42]. Atypicalgoalischemicalaccuracy,"totalenergieswithin1kcal/moloftheexactresult. Formoleculesofchemicalinterest,theprimaryrestrictionontheaccuracyofcalculations isthecomputationalcost.Withincoupled-clustertheory,thisstatementiscloselyrelated tothelevelofexcitationsthatareallowedinthecalculation.Formoleculesoftensof atoms,calculationsarefeasibleusingCCSD,whichscalesas O n 2 N 4 ,where n = N e thenumberofelectrons,and N isthenumberofunoccupiedorbitalsintheproblem.In reliablecalculations, N n Unfortunately,CCSDisnotaccurateenoughforpredictivecalculations.Asis illustratedinFigure1-2,theerrordistributionsforCCSDforatomizationenergiesand forreactionenthalpiesdonotcomeclosetoreachingchemicalaccuracy.Inthecase ofatomizationenergiesFigure1-2Aerrorsareontheorderof10kcal/molandthe 42

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distributionisbroad.ForreactionenthalpiesFigure1-2Bthemeanerrorisreasonable, butthebroadnessofthedistributionindicatesthatformanymoleculeserrorswillbe largerthandesired.Theresultsareevenmorestrikingforgeometricproperties,where inFigure1-3ACCSDequilibriumbondlengthsareingreatererrorthanthosefrom many-bodyperturbationtheorythroughsecond-order[MBPT]. Tripleexcitationsarenecessarytoreachthedesiredlevelofaccuracy.CCSDT,which scalesas O n 3 N 5 ,ismuchmorecomputationallyexpensivethanCCSD.Additionally, CCSDTrequiresthestorageofquantitiesthathavedimension O n 3 N 3 ,toolargetostore onmostcomputers.SomeapproximationtoCCSDTmustbeusedtoaddressmolecules ofmorethanafewatoms.Therearetwomaincategoriesofapproximation:iterativeand non-iterative,alsoknownasperturbative,methods.Inbothcasesadditionalstepsthat scaleas O n 3 N 4 arerequired.Whilethisscalingissignicantlybetterthanthe O n 3 N 5 ofCCSDT,itisstillexpensive.Therefore,perturbativemethods,whichrequireonlya single O n 3 N 4 steparepreferredincalculations. Formoleculesnearequilibria,thesequalitiesarebestrepresentedwithin coupled-clustersingles,doubleandperturbativetriples[CCSDT][51,52].CCSDT wasdevelopedinthreestages:thenumericallymostimportantterm,sometimescalled [T],wasintroducedin1985byUrbanetal.[53].Thisapproximationprovidedtheonly fourth-ordertriplescorrectionthatoccurswhenaHartree-Fockreferencefunctionisused inCCtheory.ThenextdevelopmentwasbyRaghavacharietal.[51]whoaddedasingle excitationterm,which,thoughmuchsmallerinsize 5%ofthe[T]correction,acted intherightdirectiontoimproveupon[T].FromtheHFperturbationviewpoint,there wasnoformalreasontoaddthisonefth-ordertermwhenmanyotherfth-orderterms werealsopresent,butifonechoosestoinsteadcountperturbativecorrectionsrelative tonon-HFcaseswhere T 1 occursinrst-order,thenthistermisaproperfourth-order one,recommendingitsinclusion.However,onceweadmitnon-HFcases,whichare importantinprovidingthewealthofapplicationsofCCtheorytoday,anotherterm 43

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A B Figure1-2.NormaldistributionsoferrorsofMBPT,CCSDandCCSDTina cc-pCVQZbasisfortwoenergeticproperties.Themeanerrorsandstandard deviationswereusedtotagaussiandistribution.Thehorizontalaxisisthe errorinkcal/mol,theverticalaxisisameasureofprobabilityofagivenerror. AAtomizationenergyerrorsascomparedtoexperimentforthe16small moleculesfrom[49].BReactionenthalpyerrorsascomparedtoexperiment for13isogyricreactionsofthose16smallmolecules[49]. 44

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A B Figure1-3.NormaldistributionsoferrorsofMBPT,CCSDandCCSDTina cc-pCVQZbasisfortwogeometricproperties.Themeanerrorsandstandard deviationswereusedtotagaussiandistribution.Thehorizontalaxisisthe error,theverticalaxisisameasureofprobabilityofagivenerror.A Equilibriumbondlengtherrorspmascomparedtoexperimentforthe19 smallmoleculesfrom[50].BEquilibriumbondangleserrorsascomparedto experimentfor9smallmoleculesfrom[50]. 45

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occursinfourth-order[52]thatneedstobeadded.Finally,tomakethisgeneralized CCSDTcomputationallyfeasible,itisnecessarytoexploitthefreedomthattheenergy ofCCSDTisinvariantunderasemi-canonicalorbitaltransformationtoretainthe easeofapplicationfornon-HFreferencecases.Together,thesestepsdeneCCSDT, whichcanbeusedforanyreferencedeterminant:restrictedRHF,restrictedopen-shell ROHF,unrestrictedUHF,orquasi-restrictedQRHFHartree-Fock,Kohn-ShamKS, BruecknerB,therstnaturaldeterminantN,etc.ACESII[54,55]providesallas referencefunctions,toexploitthefullexibilityofsingle-referenceCCtheory. CCSDTfulllsmostofthepossibledesiredpropertiesforanenergymodelnear equilibrium:itisaccurate,extensive,orbitallyinvariantforrotationsamongoccupied andvirtualorbitals,applicabletoarbitrarysingle-determinantreferencefunctions,and analyticalderivativesarereadilyavailable[42].ThecomputationalscalingofCCSDTis iteratively O n 2 N 4 withasinglenon-iterative O n 3 N 4 step.Thiscostislowenoughto allowCCSDTtobeusedforawidevarietyofmolecules. 1.3.3PropertiesandExcitedStates GeneralizedHellman-FeymanTheorem :Onecanviewthecalculationofa property,suchasaforce,adipolemoment,oranyotherproperty,asaderivativeofthe energy.Deningaperturbation-dependentHamiltonian H H = H + A {94 where A isthepropertythatoneisattemptingtodetermine.Then,onecanapplythe derivativetotheexpectationvalueexpression @ @ [ h j H j i = E ] =0 @E @ = h @ @ j H j i + h j @H @ j i + h j H j @ @ i : {95 Hellman[56]andFeynman[57]independentlyprovedthatifthewavefunctionisatits variationalminimumthenalltermsinvolving @ @ vanish,andthepropertyisdenedby theperturbationindependentexpression, 46

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@E @ = h j A j i ; {96 whichisequivalenttotheexpectationvaluedenitionofaproperty.Unfortunately,this argumentonlyholdsforvariationalwavefunctions,wheretherstderivativeoftheenergy withrespecttoallparametersinthewavefunctionvanishes.Forcongurationinteraction methods,theenergyisvariationalwithrespecttothecorrelationoperators,thoughnot necessarilywithrespecttothemolecularorbitalsortheatomicorbitals.Thesetermsare dealtwithseparately,yieldingageneralizedHellman-Feynmantheoremwithadditional referencefunctionterms[58].Foranon-variationalwavefunction,likecoupled-cluster theory,thegeneralizedHellman-Feynmantheoremdoesnotapply.Totakeadvantageofa Hellman-Feynman-liketheorem,itisnecessarytosupplementtheenergywithadditional variables,suchthattheenergyis,ifnotvariational,insteadstationarywithrespectto theparametersofthewavefunction.Inthatcase,therstderivativesofthewavefunction vanishaswell. Forcoupled-clustertheorywehaveencounteredsuchanobjectbefore,the coupled-clusterfunctional, E CC [ T; ]= h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T H N e T j 0 i + h 0 j H j 0 i {97 Recognizingthatonecanconsiderthisasamatrixelementbetweentwowavefunctions h ~ j = h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T j i = e T j 0 i {98 thenthefunctionalis E CC [ h ~ j ; j i ]= h ~ j H N j i + h 0 j H j 0 i : {99 Ifboth j i and j ~ i aredeterminedusingastationaryconditionthenageneralized Hellman-Feynmantheoremwillapply[59],leadingto @E @ = h ~ j @H N @ j i + h 0 j @H N @ j 0 i ; {100 47

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wherethereferencefunctioniskeptxed.ItwasshowninEquation1{88thattheground stateenergyisindependentof.However,propertieswillnotbe.Therefore,onemust determineusingastationaryprocedure,bydierentiatingtheenergywithrespectto T introducingtheequations, @ E CC @t K = h 0 j + H N ;a K j 0 i =0 : {101 Then,anarbitrarypropertycanbedeterminedviathecoupled-clusterexpectationvalue h A i CC = h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T Ae T j 0 i : {102 ThisdenitionofaCCexpectationvaluedenesdensitymatricesaswell.The one-particledensitymatrixone-matrixisdenedby pq = h ~ j p y q j i + h 0 j p y q j 0 i = h 0 j + \010 p y q e T C j 0 i + h 0 j p y q j 0 i : {103 Becausetheleft-andright-wavefunctionsarenotthesame,theone-matrixisnot necessarilyhermitian.Inpractice,oneusesthesymmetrizeddensitymatrix 0 pq = 1 2 h ~ j p y q j i + h ~ j q y p j i + h 0 j p y q j 0 i = 1 2 h 0 j + p y q + q y p e T C j 0 i + h 0 j p y q j 0 i : {104 Asimilarexpressionholdsforthetwo-particledensitymatrix. ReferenceFunctions :Formanyproperties,thereferencedeterminantcontributes tothepropertyaswellasthecorrelation.Toformamoregeneralexpectationvaluethat willincludetheseeects,itisnecessarytodetermineasuitablestationaryfunctionalfor thereferenceproblem[60,61].Thespaceofallsingle-determinantscanbeparameterized byaunitaryoperatorthatmixesoccupiedorbitalswithvirtualorbitals[19].Anyunitary operatorcanbewrittenastheexponentialofananti-hermitianoperator.Then,the followingparameterizationcanbemadeforallsingledeterminants 48

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j i = U j 0 i = e j 0 i : {105 isanorbitalrotationanti-hermitianoperatorthatcanbewritten,assumingrealorbitals andnormal-orderingwithrespectto j 0 i = X ia a i \010 a y i )]TJ/F26 11.9552 Tf 11.955 9.684 Td [( i y a : {106 TheHartree-FockequationsareequivalenttotheBrillouincondition, f ia =0,whichcan berewrittenasthestationarypointoftheHFfunctional E HF = h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e j 0 i + h 0 j H j 0 i : {107 wheretheLagrangianmultipliersaredenedby = X ia i a i y a : {108 ThereareothersimplerchoicesfortheHartree-Fockfunctional,buttheydonotlend themselvestobecombinedwiththecoupledclusterfunctional.Thechoiceofwhetherthe referenceisRHForUHFisdeterminedbywhethertheoperatorsthatcompose and arethemselvesspin-restrictedornot.CombiningEquation1{107withtheCCfunctional leadstothefullfunctional, E HF/CC = h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T j 0 i + h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e j 0 i + h 0 j H j 0 i {109 Notethatinthecoupled-clusterfunctional,theunitaryoperatorsexp hadtobe includedtoallowfortheorbitalrotationstoaectthecoupled-clusterresult.Assuming nothingabouttheformalreferencestate j 0 i exceptthatitiscomposedofanorthogonal setoforbitals,astationarypointof E HF/CC correspondstoasolutionoftheCCequations onaHartree-Fockreference.OnecanchoosethattheHFsolutioncorrespondto =0, i.e. j 0 i = j HF i .Imposingstationaritywithrespecttovariationsofgeneratesthe HFequations;stationaritywithrespecttogeneratesthe T equations;stationarity withrespectto T generatestheequations.Thenewcontributionisthatstationarity 49

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withrespectto inducesacouplingbetweentheCCfunctionalandtheHFfunctional todetermine.Thiscouplingisexactlythatdevelopedusingthenon-stationary developmentusingtheinterchangetheorem[62]. TheformofthefunctionalinEquation1{109naturallydenesaresponsedensity matrixarisingpurelyfromcorrelationandarelaxeddensitymatrixarisingfromboth correlationandthechangeintheunderlyingMOs. resp pq = h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T e )]TJ/F22 7.9701 Tf 6.587 0 Td [( f p y q g e e T j 0 i {110a relax pq = h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [( f p y q g e j 0 i + h 0 j p y q j 0 i {110b full pq = resp pq + relax pq : {110c Inpractice,onesymmetrizesthesematrices,asinEquation1{104. Forperturbativetriplesmethods,suchasCCSDT,itiscomputationally advantageoustorequirecanonicalorbitals.Sincecanonicalorbitalssatisfytheequations f ij = i ij and f ab = a ab thesecanbereformulatedintotermsthataugmenttheHF functional, E HF-canon = X i>j ij h 0 j i y e )]TJ/F22 7.9701 Tf 6.586 0 Td [( 0 H N e 0 j j 0 i + X a>b ab h 0 j ae )]TJ/F22 7.9701 Tf 6.587 0 Td [( 0 H N e 0 b y j 0 i {111 wheretheneworbitalrotationparameter 0 includesoccupied-occupiedandvirtual-virtual orbitalrotations,againassumingrealorbitalsandnormal-orderingwithrespectto j 0 i 0 = X p
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H = X pq h p j r j q i p y q {113 wherethesuperscript denotestheperturbedHamiltonianinthebasisdenedby j 0 i However,becausetheatomicorbitalbasisiscenteredonnuclei,aperturbationthataects nuclearpositionwillintroducecontributionsfromtheatomicorbitalsthemselvesintothe derivativeexpression.Similarly,whenusingamagneticeld,itisusefultointroducea eld-dependentcontributionintotheorbitalsthemselvesleadingtogauge-includingatomic orbitalsGIAOs,butthenanymagneticcontainingperturbationwillalsoleadtoan atomicorbitalcontribution. IftheatomicorbitalsdocontributethentheperturbedHamiltonianismore complicated[58,61].Applyingthechainrule, H = H )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 S ;H + {114 where H isthesameasiftheAOswereindependentoftheperturbationand S is thechangeintheoverlapbetweentheAOsduetotheperturbation.Elaboratingonthe overlapterm, S = h @ @ =0 j i + h j @ @ =0 i ; {115 whereIhaveindicatedthattheAOs and dependexplicitlyon .Theperturbed overlaptermsareneededtoforcetheperturbedmolecularorbitalstoremainorthogonal, astheunperturbedorbitalswere. ExcitedStates :Coupled-clustertheoryisapplicabletoexcitedstatesaswellas groundstates,viatheequation-of-motioncoupled-clusterEOM-CCformalism[63{68]or thelinear-responsecoupled-clusterLR-CCapproach[69,70].Iwillshowthatthesetwo approachesyieldthesameequationfortheexcitationenergies. UsingEOMtheparameterizationforanexcitedstate j m i j m i = R m e T j 0 i {116 51

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where R m isalinearCI-likeexcitationoperator, R m = N e X n =0 R m n R m n = 1 n 2 X ijk abc r m abc ijk a y ib y jc y k : {117 Unlikeinthecaseof T r m 0 isincludedintheexpansion.Thistermisneededifthe excitedstateandgroundstatearethesamesymmetrytoorthogonalizetheexcitedstate tothegroundstate.Insertingthisstateintotheexpectationvalue,theenergyis E m = h 0 j e T y R m y H N R m e T j 0 i h 0 j e T y R m y R m e T j 0 i + h 0 j H j 0 i : {118 FollowingthederivationforthegroundstateinSection1.3.1,onecanrewritetheenergy inthenon-hermitianform, E m = h 0 j L m H N R m j 0 i h 0 j L m R m j 0 i + h 0 j H j 0 i ; {119 whereintheFCClimit, h 0 j L m = h 0 j R m y e T y e T h 0 j e T y R m y R m e T j 0 i : {120 Thegroundstateenergycanbewritten E 0 = h 0 j H N j 0 i + h 0 j H j 0 i = h 0 j L m R m j 0 ih 0 j H N j 0 i h 0 j L m R m j 0 i + h 0 j H j 0 i : {121 Assumingthatthecoupled-clusterequationshavebeensolvedinthesamespaceasthe excitationsincludedin L m and R m i.e.if R m includesthroughdoubleexcitations, thentheCCSDgroundstateequationsweresolved,thisisequalto E 0 = h 0 j L m R m H N j 0 i h 0 j L m R m j 0 i + h 0 j H j 0 i : {122 Subtractingthegroundstateenergyfrom E m m = E m )]TJ/F21 11.9552 Tf 11.955 0 Td [(E 0 = h 0 j L m H N R m j 0 i)-222(h 0 j L m R m H N j 0 i h 0 j L m R m j 0 i = h 0 j L m H N ;R m j 0 i h 0 j L m R m j 0 i : {123 52

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Makingthisequationstationarywithrespecttothemultipliers L m yields hKj H N ;R m j 0 i = m r m K ; {124 theequation-of-motionCCeigenvalueequation.DerivingEquation1{124fromanother perspective,onerecognizesthat R m and L m aretheright-andleft-eigenvectorsof thenon-hermitianeectiveHamiltonian.Todetermineexcitationenergiesonly R m isneeded,butforexcitedstateproperties,theleft-eigenvector L m isalsorequired,in directanalogywiththegroundstateproblem,whereisonlyneededforproperties. Inresponsetheory,onecandenetheactionfunctionalas[71,72] A CC = Z t 1 t 0 d E CC {125 wherethetime-dependentenergyfunctionalis, E CC = h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T H N )]TJ/F21 11.9552 Tf 11.956 0 Td [(i @ @ e T j 0 i + h 0 j H j 0 i = h 0 j + H N j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j T j 0 i + h 0 j H j 0 i : {126 where T isthetime-derivativeof T .Iamfreetochoose j 0 i tobetime-independent, leadingtoatime-independentnormal-ordering;thischoicemaybepoorinatrue dynamicalsimulationratherthanacalculationofresponseproperties. TheproperformofthestationaryconditionsfortheactionaretheEuler-Lagrange equations,whichinthiscaseare @ E CC @t K )]TJ/F21 11.9552 Tf 16.36 8.088 Td [(d d @ E CC @ t K =0 @ E CC @ K )]TJ/F21 11.9552 Tf 16.36 8.088 Td [(d d @ E CC @ K =0 : {127 Assumethat H iscomposedofatime-independentHamiltonian H anda time-dependentperturbation V .Tosimplifythenotation,Iwillsuppresstheexplicit dependence.Then,throughzerothorderintheperturbation,theenergyfunctionalis E CC = h 0 j )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ H N j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j T j 0 i + h 0 j H j 0 i : {128 53

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Makingthisequationstationarywithrespecttothezeroth-orderparametersusing Equation1{127, hKj H N j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i hKj T j 0 i =0 ; {129 whichcanbesolvedbychoosing T tosatisfythetime-independentstationarycondition and T =0.Similarly, canbechosentosatisfythetime-independentstationary conditionwith =0. Therst-orderenergyfunctionalis,with V N = e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T V N e T E CC = h 0 j H N j 0 i + h 0 j )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ h H N ;T i j 0 i + h 0 j )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ V N j 0 i )]TJ/F21 11.9552 Tf 11.956 0 Td [(i h 0 j T j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j T j 0 i + h 0 j V j 0 i : {130 Usingthesolutiontothezeroth-orderstationaryconditions,thisfunctionalsimpliesto E CC = h 0 j )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ V N j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j T j 0 i + h 0 j V j 0 i : {131 Fouriertransformingthisexpressionfromtimetofrequency,yields E CC = h 0 j )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ V N j 0 i + h 0 j T j 0 i + h 0 j V j 0 i : {132 Onecanseethatthesecondterminthatequationcanberewritten h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T T e T j 0 i ; {133 whichvanishesduetothebiorthogonalityoftheperturbedrst-orderright-wavefunction andthezeroth-orderleft-wavefunction.Therstorderenergyfunctionalisthen E CC = h 0 j )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ V N j 0 i + h 0 j V j 0 i ; {134 whichisindependentofanyfreeparameters. Thesecond-orderenergyfunctionalis E CC = h 0 j H N j 0 i +2 h 0 j h H N ;T i j 0 i + h 0 j )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ h H N ;T 2 + T i j 0 i 54

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+2 h 0 j V N j 0 i +2 h 0 j )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1+ V N ;T j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j T j 0 i )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 i h 0 j T j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j T j 0 i : {135 Usingthelower-orderstationaryconditionsandorthogonality, E CC =2 h 0 j h H N ;T i j 0 i + h 0 j )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ h H N ;T 2 i j 0 i +2 h 0 j V N j 0 i +2 h 0 j )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ V N ;T j 0 i)]TJ/F15 11.9552 Tf 19.261 0 Td [(2 i h 0 j T j 0 i : {136 Imposingstationaritywithrespecttovariationsin 2 hKj h H N ;T i j 0 i +2 hKj V N j 0 i)]TJ/F15 11.9552 Tf 19.261 0 Td [(2 i hKj T j 0 i =0 : {137 Then,Fouriertransformingandlookingforthehomogeneoussolutionstothisequation thosethatcorrespondtopolesintheresponsefunction, hKj h H N ;T i j 0 i = hKj T j 0 i : {138 Identifying T with R m ,thisequationisidenticaltoEquation1{124,whichprovesthe equationsarethesamefortheexcitationenergies.Todetermineproperties,onemustsolve the equations,whicharenotidenticaltotheequationsfor L m .Thisdierenceisthe sourceoftheextensivityproblemincoupled-clustertheorysecond-orderproperties[73]. 1.4Regularization Thecoupled-clusterequationsareeitherlinearornon-lineariterativeequationsand theirconvergencebehaviorcanbesensitivetothenumericalprocedureusedtosolvethem. TypicalnumericalapproachesareJacobiiterations[4]orpreconditionedconjugateresidual PCRmethods[74].WhilethePCRmethodsarequitestabletheyarenotalways successful.SometimesitisfoundthattheCCequationsareill-posedorill-conditioned. Ill-posedproblemsareunderdetermined,meaningthereisaninnitesetofsolutions,and ill-conditionedproblemsarehighlysensitivetosmallperturbationsintheinputdatato theequations.Becausetheprocessofsolvinganequationinniteprecisionarithmetic 55

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inherentlyintroducessomeerrorintotheproblem[4],ill-conditionedequationsmayhave largenumericalerrors.IntheCCcase,theseerrorscanleadtoeitherbeingunableto convergetheCCequationsorconvergingtothewrongsolutionoftheequations. Astandardnumericaltechniqueforbothill-posedandill-conditionedproblemsis touse regularization [4,75].Inthecaseofanill-posedproblem,regularizationisthe additionofaconstraintthatreducesthenumberofpossiblesolutionsfrominnityto one,unique,solution.Inanill-conditionedproblem,theconstrainttradesosolvingthe unstableequationexactlyforsolvingaslightlymodiedstableequation.Whilethereare manydierentregularizationtechniques,Iwillintroducetwoofthesimplesthere.Asan example,considerthelinearequation Ax = b {139 where A and b areknown.Thesimplesolutiontothisequationis x = A )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 b : {140 Ifanyofthesingularvaluesof A arezero,thentheinverseisundened,andthisproblem isill-posed.Ifthereareafewsmallsingularvaluesrelativetotherest,thenumerical processofinvertingthematrixwillcausetheinverseofthesmallsingularelementsto contaminatetheprecisionoftheinverseofthelargersingularvalues,leadingtolarge errorsintheinverse.Thatwouldbetoanill-conditionedproblem. SingularvaluedecompositionSVDregularizationisbasedonthefollowing observation:anarbitrarycomplex-valuedmatrixcanbewritteninSVDformas A = U T V {141 where U and V areunitarymatrices,and isadiagonalmatrix,withthesingular valuesalongthediagonal.Foramatrixthathaseigenvalues,thesingularvaluesare themagnitudesoftheeigenvalues.Ifthematrix A issingular,thenatleastoneofthe 56

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singularvalueswillbezero.Thishintsatawaytosolvethesingularlinearequation.For anonsingularmatrix A )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = V T )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 U {142 usingtheunitarityofthematrices U and V .Ofcourse,forasingularmatrix )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 isundened.However,theinverseiswell-denedforallsingularvaluesthatare non-zero,soonecanimagineinvertingthematrixwithinthatsubspace.Thisdenes theMoore-Penrosepseudoinverse[4], + ii = 8 > > < > > : ii )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ii 6 =0 0 ii =0 : {143 Then,thepseudoinverseofthematrix A isdenedas A + = V T + U {144 andthesolutiontothelinearequationis x = A + b : {145 Forexactlysingularmatrices,thisprovidesonewayofsolvingthelinearequation. Itcorrespondstoaleastsquarestofthenon-singularvectorstothesolution.For linearequationsthatarenotexactlysingularbutratherill-conditioned,asisthe casenumericallyasoneapproachesasingularity,theideaofapseudoinversecanbe generalized,sothatitregularizes"thenearsingularity, e + ii = 8 > > < > > : ii )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ii > 0 ii {146 forsomesmallconstant > 0.Thentheregularizedmatrixis e A + = V T e + U {147 57

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andthesolutionis x = e A + b : {148 SVDregularizationcanworkwellformanyproblems.WhenattemptingtouseSVD regularizationinacontinuousmodel,suchasmovingalongapotentialenergysurface,the sharpcutocanleadtoproblemswithcalculatingderivatives[75]. Anothersimpleregularizationapproach,whichavoidsasharpcuto,isTikhonov regularization.OnecanreformulateEquation1{139astheminimumoftheleastsquares problem W = k Ax )]TJ/F36 11.9552 Tf 11.955 0 Td [(b k 2 : {149 Whenthematrix A issingularthisequationcanhavemultiple x vectorsthatachievethe minimization.Therefore,onemustaugmentthisequationtopickoutasinglesolution. Thiscanbedonebeaddingtothefunctional W 0 = k Ax )]TJ/F36 11.9552 Tf 11.955 0 Td [(b k 2 + 2 k x k 2 {150 where )]TJ/F15 11.9552 Tf 11.983 0 Td [(isanarbitrarymatrixand isreal,so 2 > 0.Themostcommonchoiceof)]TJ -279.133 -23.908 Td [(istousetheidentitymatrix.Inthatcase,onecanviewtheaugmentedfunctionalasa leastsquarestofthesolutionvectortotheinputdatawithapenaltyifthenormof x becomeslarge.OnecanshowthattheTikhonovapproachisequivalenttothefollowing scalingofthesingularvalues, e )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ii = ii 2 ii + 2 : {151 Thedenominatorcannevervanishinthisscalingbecauseboth 2 ii and 2 arepositive numbers.Thevalueof 2 istheminimumdenominatorpossible,andsetsascaleforthe solution.When ii then e )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ii = ii 2 ii + 2 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ii : {152 TheTikhonovapproachsmoothlyregularizesthesingularvalues,butchangesthesmall, 58

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troublesome,singularvaluesleavingthelargersingularvaluesrelativelyunaected. Choosingtheoptimalvalueof isanon-trivialtask,unfortunately,andthemostcommon approachisnumericalexperiment[75]. 1.5OutlineofthisStudy Inthenextthreechapters,IwilladdressthreegroupsoftechniquesthatIhave workedontoxproblemswithincoupled-clustertheory.InChapter2,Idescribethe developmentandapplicationoffrozennaturalorbitalcoupled-clusterFNO-CCtheory energiesandgradients.FNO-CCisamethoddesignedtoreducethecomputationalcost ofcoupled-clustertheory.Toaddressbond-breakingproblemsofCCSDT,inChapter3, IintroducethemethodCCSDTmethod,withgradients,anditsvarianttheBrueckner CCSDTmethod.Inadeparturefromthestandardformofcoupled-clustertheory, Idevelophermitiancoupled-clustertheoriesinChapter4,bothingeneralandina particulartruncatedlinearizedversionthatisahighlysimpliedbutstillanaccurate coupled-clustermethod. 59

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CHAPTER2 FROZENNATURALORBITALCOUPLED-CLUSTERTHEORY PortionsofthischapterareexcerptedandadaptedwithpermissionfromA.G. TaubeandR.J.Bartlett,Collect.Czech.Chem.Commun. 70 ,837[76] c 2005, InstituteofOrganicChemistryandBiochemistryoftheAcademyofSciencesoftheCzech RepublicandA.G.TaubeandR.J.Bartlett,J.Chem.Phys., 128 ,16410108 [77] c 2008,AmericanInstituteofPhysics. 2.1Introduction Predictionofstructuresofequilibriaandtransitionstatesisamongthemost importanttasksforcomputationalchemistry.Coupled-clustertheoryhasshownitself tobewell-suitedfordeterminingequilibriumstructures,especiallyinthecoupled-cluster singles,doubles,andperturbativetriples[CCSDT]form[42,51,52].Asignicant drawbackofthecoupled-clusterapproachisthehighcomputationalscalingwithrespect tothesizeofthesystem.ForCCSDT,forexample,themostexpensivestepis O n 3 N 4 where n isthenumberofoccupiedorbitalsand N isthenumberofunoccupiedorbitals and M = N + n isthetotalnumberofbasisfunctions.Itiswell-known,however,that standardbasissetsarenotoptimal;onecanreducethesizeofthevirtualunoccupied spacewithoutadverselyaectingthenumericalresults.Inparticular,inlargebasis setstherearecombinationsofvirtualorbitalsthatdonotcontributesignicantly totheCCenergy.Onecan,therefore,reducethecomputationalcostbyidentifying andremovingtheseirrelevantfunctionsfromthebasisset.Thereisalonghistoryof tryingtogeneratesuchspacesforcongurationinteractionCI[78{85]andmany-body perturbationtheoryMBPT[86{88]and,morerecentlyforcoupled-clustertheory [76,89{95].Perhapsthemostpowerfulmethodofdoingsoistousefrozennaturalorbitals FNOs[76,92,96{101].Theseorbitalsuseinformationfromanapproximateone-particle reduceddensitymatrixtochoosethebestsubsetofone-particleorbitalswithinwhich toperformacorrelatedcalculation.WhenusingFNOsbasedontheMBPTdensity 60

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matrix[102,103],thistruncatedorbitalsethasbeenshowntobesurprisinglyeective attruncatinglargerbasissets,allowing 50%ofamodiedunoccupiedorbitalsetto beremovedwithoutsignicantchangestogroundstateCCenergiesanddensitymatrices [76,92,93].AnestimateoftheerrorintheFNOprocedurecanbemadebycomparing theMBPTenergyinthefullbasistothatinthetruncatedbasisset,yieldingasimple correctionthatimprovestheenergeticresultsfurther[76].TheFNOprocedurecouldbe combinedwithfurtherreductionsintheunderlyingcontractedGaussianbasis[104{106] forfurthersavings. Thefrozennaturalorbitalsareabletoachievethisspeed-upbytailoringthebasis settoamoleculeataparticulargeometry.Therefore,whenonewantstocalculate forces,thereisanon-trivialcomponentduetothechangesintheunderlyingstructure ofthefrozennaturalorbitals.Toaccountforthesechanges,onemustintroduceaset ofcoupled-perturbedfrozennaturalorbitalequations,similartothecoupled-perturbed Hartree-Fockequations,andre-arrangethetermsinacomputationallyecientmanner,to avoidcalculatingmanyperturbation-dependentquantities. Gasphasechemistryofenergeticmaterialsisanareathatbothtestsand takesadvantageofcomputationalchemistrytechniques.Thesereactionsare especiallysusceptibletosmallerrorsinthetreatmentofcorrelationbothin energeticsandstructures[1].Tobepredictive,onemustusehigh-levelsofcorrelation andlargebasissets,whichtaxescomputationalresources.Dimethylnitramine DMNAisamodelcompoundforcombustionprocessesthatoccurinmore complicatednitramines,suchas1,3,5-trinitrohexahydro-1,3,5-triazineRDXand 1,3,5,7-tetranitrooctahydro-1,3,5,7-tetrazocineHMX.Accordingtoexperimentaldata frombothsolid-phaseX-raycrystallography[107]andgas-phaseelectrondiraction [108]measurements,theequilibriumgeometryofDMNAshouldbeplanar{ C 2v However,single-moleculetheoreticalstudieshaveconsistentlypredictedaground-state geometrywithonly C s symmetry[109{112].Ithasbeensuggestedthatthesetheoretical 61

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calculationsmaybemorereliablethantheexperimentalresults[110{112].Calculations havebeenperformedusingavarietyofmethods:restrictedHartree-FockRHF, Many-BodyPerturbationTheoryMBPT,multipleDensityFunctionalmethods B3LYP,BLYP,PW91andQuadraticCongurationInteractionwithsinglesanddoubles QCISDinmoderatelysizedbases,e.g.6-31G*[16]orcc-pVDZ[113].Tohavecondence inthetheoreticalpredictions,especiallywithcontradictingexperimentaldata,itis oftennecessarytoincludeatleastthroughtripleexcitationspreferablywithinthe coupled-clustertheoryframeworkandworkinlargebasissets. WhenDMNAorRDXorHMXburns,itisnotsolelyinthegas-phase;ignition occursinthecrystallinesolidphase[114].Therefore,itisimportanttotakeintoaccount theeectofintermolecularinteractiononthesemolecules.ThedimerofDMNAhasbeen calculatedatxedmonomer C 2v geometrywithsymmetry-adaptedperturbation theoryinacc-pVDZbasisset[115].Thisstudyidentiedvelocalminimaforthe interactionsbetweenthexedmonomersusingagridsearchacrossbothanglesand relativeseparationsbetweenthetwomonomersofDMNA,byincludingtheeects ofdispersion,exchange,inductionandelectrostaticinteractionsbetweenmonomers perturbatively. Inthesolidphase,RDXhasoneoftwoforms: -RDX,whichhas C s symmetry, alsocalledtheaxial-axial-axialAAAboatform,and -RDX,whichhas C 3v symmetry, alsocalledtheAAAchairform,andistheexperimentalstructureofthegas-phaseRDX molecule.ResultsatMBPTandvariousdensityfunctionalsappeartoagree,although allresultsareinmoderate-to-smallbases[116,117].Atleastonestudyhasquestioned whetherthe -RDXstructureisaminimumofthelonemoleculeatall,suggestinginstead atwistednon-symmetricconformationasthesecondgas-phaseminimum[118]. Nitroethaneisaprototypeforthedecompositionofthenitroalkaneclassofenergetic materials[119,120].Whencomparedtothebetterstudiednitromethane[121{123], thedecompositionofnitroethaneintroducesoneimportantadditionalpathway:the 62

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eliminationofHONO[124,125].Thispathwayisapparentlythekineticallyfavoredone forthermaldecompositionofnitroethane,andisimportantinthedecompositionofmore complicatedmaterials,suchasRDX.Theenergeticsofdecompositionfornitroethane havebeenstudiedusingDFTusingtheB3LYPfunctional[126]buttobeableto understandtherelativeimportanceofthevariouspathwayswithcondence,high-level correlatedcalculationsshouldbeperformedattheappropriatelyoptimizedstationary points.Nitroethanehasveheavyatomsand,tousealargeenoughbasissettobe denitive,oneneedsseveralhundredbasisfunctions.Toperformmultipleoptimizations andtransitionstatesearcheswithCCSDTinthatsizeofbasisisacomputational challenge;instead,weapplytheFNOprocedure,withanalyticalgradientstocalculatethe potentialenergysurfacefornitroethanedecomposition.WeusebothCCSDTandthe morerecentCCSDT[123,127{129],which,byusinginformationfromtheleft-hand CCSDeigenvector,improvesthedescriptionofbondbreaking. 2.2Theory 2.2.1Energetics TheFNO-CCmethodhasbeensummarizedbeforein[76].Below, i;j;k indicate occupiedorbitals, a;b;c indicatevirtualorbitals,and p;q;r indicatearbitraryorbitals. Asetofimprovedvirtualorbitalsisgeneratedthroughaseriesofrelativelysimple operations.First,thecanonicalHartree-FockHFequationsaresolvedandthenthe MBPTdensitymatrixiscomputedintheresultingvirtualorbitalspace.Thisdensity matrixisdenedas D ab = 1 2 X cij h cb jj ij ih ij jj ca i cb ij ca ij {1 wherethedenominator, ab ij = f ii + f jj )]TJ/F21 11.9552 Tf 12.391 0 Td [(f aa )]TJ/F21 11.9552 Tf 12.391 0 Td [(f bb ; iscomposedofdiagonalFockmatrix elements, f pp .Thedensitymatrixisdiagonalizedyieldingasetofnaturalorbitalswhose occupiedspaceisfrozentotheoriginalHartree-Fockorbitalshencethenamefrozen naturalorbitals".Associatedwitheachnaturalorbitalisanapproximateoccupation number,whichisameasureofimportanceoftheorbital.Then,basedonkeepingthese 63

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orbitalswithhighestoccupation,thevirtualspaceispartitionedintotwosubspaces:a setofkeptorbitalsandasetofdroppedorbitals.TheFockmatrixisformedinthesenew orbitals,andisseparatelydiagonalizedineachofthetwosubspaces.Therefore,attheend oftheprocess,onehasthreesetsoforbitals:canonicalHartree-Fockoccupiedorbitals, akeptsetofHartree-Fockvirtualorbitalsthatarecanonicalamongthemselves,anda droppedsetofHartree-Fockvirtualorbitalsthatarealsocanonicalamongthemselves. OrbitalsintheoriginalHartree-Fockbasiswillbeuncapitalized,whileorbitalsafterthe FNOtransformationwillbedenotedbycapitals.Keptvirtualsareindicatedby A 0 ;B 0 ;C 0 droppedvirtualsare A 00 ;B 00 ;C 00 ,andarbitraryvirtualstheunionofthekeptanddropped orbitalsare A;B;C Hartree-FockorbitalsinagivenbasissetarecompletelydenedbytheBrillouin condition f IA 0 = f IA 00 =0 : {2 Similarly,attheendoftheFNOprocedure,theadditionalBrillouin"conditionis D A 0 A 00 =0 : {3 ThisequationissimilartotheHartree-Fockconditionfornon-canonicalorbitalsasthe o-diagonalelements D A 0 B 0 and D A 00 B 00 canbenon-zero.ThecombinationoftheBrillouin conditionandtheequivalentFNOconditionwillbeusedtocalculategradients. TheoverallsetofFNOmolecularorbitalcoecientsareexpressedinthefollowing equalities.Theelements V P aretheoveralltransformations,while C p aretheoriginal HFtransformationsand U aB aretheadditionalFNOtransformationsinthevirtualspace, where ;;::: areatomicorbitals, V I = C i V A = X b C b U bA : {4 Thenalmolecularorbitalsarenon-canonicalHartree-Fockorbitals,andthereforewill satisfyalloftheHFconditions.Theseorbitalsaredenedby, 64

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j P i = X V P j i : {5 Theoveralltransformations, V P willsatisfyalltheconditionsoftheconventional non-canonicalHartree-Fockorbitals.Ontheotherhand,theFNOtransformations, U aB operatesinaorthonormalbasisbyconstruction. TheFNOprocedurecanbegeneralizedtonon-Hartree-Fockorbitalchoices,aswell. Inanon-HFcase,thedensitymatrixhasanadditionalterm,sothat D ab = X i f bi f ia a i b i + 1 2 X cij h cb jj ij ih ij jj ca i cb ij ca ij : {6 Inmostcases,theFockmatrixwillnotbediagonalineithertheoccupiedspaceorthe virtualspace,i.e. f ab 6 = f aa ab ,therefore,itisnecessarytosemi-canonicalizetheorbitals bydiagonalizingwithineachofthosesubspacesseparately.Thenthedensitymatrix inEquation2{6willbecorrectwithdiagonalFockelementsinthedenominator.For restrictedopen-shellHartree-FockROHForbitals,onewouldliketotruncatetheorbital spaceinsuchawaythatthespin-eigenfunctionpropertyoftheROHForbitalsisnot broken.BecauseROHF-MBPTdoesnotnecessarilyproduceadensitythatisspin-pure, itisnecessarytomanipulatethedensitymatrixfurther.Afterconstructingthedensity matrixinsemi-canonicalorbitals,onemustback-transformthematrixtotheoriginal ROHForbitals.Thenthedensitymatrixcanbespin-summedandnallydiagonalizedand truncated.Thiswillgenerateanewsetofstandardnon-canonicalROHForbitalsinthe truncatedvirtualorbitalspace.Finally,theseorbitalscanbesemi-canonicalized,andthe calculationcanproceed. OncetheFNOshavebeengenerated,aconventionalcoupled-clustercalculationis performed.BoththeCCequationsandtheenergyexpressiondependsolelyonthekept orbitals{thereisnoinuencefromthedroppedorbitals.Theenergycanbewritten E CC = X IA 0 f IA 0 t A 0 I + 1 4 X IJA 0 B 0 h IJ jj A 0 B 0 i t A 0 B 0 IJ + t A 0 I t B 0 J )]TJ/F21 11.9552 Tf 11.955 0 Td [(t B 0 I t A 0 J : {7 65

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DuringtheprocessofgeneratingtheMBPTdensitymatrixfortheFNOtruncationthe MBPTenergyinthefullbasisiscalculated.Also,astherstiterationoftheCCSD equations,theMBPT2energyinthetruncatedbasisisavailable.Anestimateofthe correlationenergythatismissedbyperformingtheFNOtruncationcanbemadeby comparingthosetwoenergies,anddeningacorrectionterm, FNO-CCSD FNO-MBPT= E [Full] )]TJ/F21 11.9552 Tf 11.956 0 Td [(E [Truncated] : {8 Usingthisestimate,acorrectedtotalenergyis E CC Corrected= E CC +FNO-MBPT : {9 2.2.2Gradients AgeneralexpressionforCCgradientsis[130{132] @E CC @ = X P 0 Q 0 P 0 Q 0 @f P 0 Q 0 @ + X P 0 Q 0 R 0 S 0 )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(P 0 Q 0 ;R 0 S 0 @ h P 0 Q 0 jj R 0 S 0 i @ {10 where P 0 ;Q 0 ;R 0 ;S 0 runoverallcorrelatedorbitals. P 0 Q 0 and)]TJ/F22 7.9701 Tf 30.075 -1.793 Td [(P 0 Q 0 ;R 0 S 0 aretheone-and two-particlecoupled-clusterresponsedensitymatricesintheactivespace.Thederivatives @f P 0 Q 0 =@ and @ h P 0 Q 0 jj R 0 S 0 i =@ aretotalderivativesofthemolecularorbitalFock operatorandtwo-electronintegrals.Thesetotalderivativescanbeseparatedintoa pieceduetotheatomicorbitalsandapieceduetothemolecularorbitalcoecients;to calculatethegradienteciently,itisnecessarytodistinguishbetweenthesetwo. ThederivativeofanactiveFNOorbitalwithrespecttoanexternalperturbationis @ j P 0 i @ = X @V P 0 @ j i + X V P 0 @ j i @ : {11 Focusingontherstmolecularorbitaltermonecanparameterizetheresponse @V P 0 @ = X Q V Q V QP 0 : {12 Thecoecient V QP 0 istheequivalentofacoupled-perturbedHartree-Fockcoecientfor 66

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theFNOs,whichwewillrefertoasacoupled-perturbedFNOCPFNOcoecient.Itis importanttonotethattheCPFNOcoecientshavecontributionsfrom all orbitals, includingthosedroppedduringtheFNOprocedure. TheatomicorbitalpiecefromEquation2{11contributestoseveralperturbed integralsthataretransformedintotheFNObasishereandbelowweareassumingreal orbitals: S PQ = X V P V Q @S @ = X V P V Q @ h j i @ ; {13a h PQ = X V P V Q @h @ = X V P V Q @ h j h j i @ ; {13b h PQ jj RS i = X V P V Q V R V S @ h jj i @ : {13c WheretheAOpartoftheFockmatrixderivativeis f PQ = h PQ + X i h PI jj QI i : {14 ThenthefullpartialderivativesoftheFockmatrixandthetwo-electronintegralscanbe written @f PQ @ = f PQ + X R f RQ V RP + f RP V RQ + X RI [ h PR jj QI i + h PI jj QR i ] V RI {15a @ h PQ jj RS i @ = h PQ jj RS i + X U V UP h UQ jj RS i + V UQ h PU jj RS i + V UR h PQ jj US i + V US h PQ jj RU i ] : {15b SubstitutingthesedenitionsintoEquation2{10, @E CC @ = X P 0 Q 0 P 0 Q 0 f P 0 Q 0 + X P 0 Q 0 R 0 S 0 )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(P 0 Q 0 ;R 0 S 0 h P 0 Q 0 jj R 0 S 0 i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 X PP 0 I 00 PP 0 V PP 0 : {16 Theintermediatematrix I 00 PP 0 includesalloftheindirectcontributionstothegradient fromtheorbitalrelaxationandisdenedwith P 0 ; occ =0if P 0 isnotanoccupiedorbital and=1if P 0 isoccupied, 67

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I 00 PP 0 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 X Q 0 P 0 Q 0 + Q 0 P 0 f PQ 0 + X Q 0 R 0 S 0 h PQ 0 jj R 0 S 0 i )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(P 0 Q 0 ;R 0 S 0 + h Q 0 P jj R 0 S 0 i )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(Q 0 P 0 ;R 0 S 0 + h Q 0 R 0 jj PS 0 i )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(Q 0 R 0 ;P 0 S 0 + h Q 0 R 0 jj S 0 P i )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(Q 0 R 0 ;S 0 P 0 + X Q 0 R 0 h Q 0 P jj R 0 P 0 i + h Q 0 P 0 jj R 0 P i Q 0 R 0 P 0 ; occ # : {17 Unliketheenergy,thederivativedependsonorbitalsthataredroppedbytheFNO procedureduetothepresenceoftermssuchas f PQ 0 and h PQ 0 jj R 0 S 0 i Byrequiringorthonormalityoftheperturbedorbitals,theCPFNOcoecientssatisfy V QP + S PQ + V PQ =0 : {18 Theperturbedintegral S PQ isknown,soonecansolvefor V QP V QP = )]TJ/F21 11.9552 Tf 9.298 0 Td [(S PQ )]TJ/F21 11.9552 Tf 11.955 0 Td [(V PQ V PP = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 S PP : {19 Therefore,thereareonlyindependentequationsfor P>Q .Expandingthelasttermof Equation2{16, )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 X PP 0 I 00 PP 0 V PP 0 =2 X P>P 0 X PP 0 V PP 0 + X PP 0 I 0 PP 0 S PP 0 {20 where X PP 0 = I 0 P 0 P )]TJ/F21 11.9552 Tf 11.956 0 Td [(I 0 PP 0 {21 I 0 PP 0 = 8 > > < > > : I 00 PP 0 for P P 0 I 00 P 0 P for P>P 0 : {22 Now,wemustaddresshowtocalculatetheCPFNOcoecients.Asisthecase forCPHF[133],inCCtheorytheirdirectcalculationcanbeavoidedbyusingthe Dalgarno-Stewartinterchangetheorem[62,134],sometimescalledthez-vectormethod forCPHF[135,136].ThegoverningequationsoftheFNOsarethoseexpressedin 68

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Equations2{2and2{3.Dierentiatingtheseequations,onehastherequirementsthat @f A 0 I @ = @f A 00 I @ =0{23a @D A 00 A 0 @ =0 : {23b Takingintoaccountthattheoverallmolecularorbitalmatrix V isacompositeof theHartree-Fockmolecularorbitalmatrix C andtheFNOtransformationmatrix U dierentiatingEquation2{4, @V I @ = @C i @ {24a @V A @ = X b @C b @ U bA + C b @U bA @ : {24b Onecanuseasimilarparameterizationoftheresponsesofthe C and U matricesaswas usedfor V @C p @ = X q C q C qp {25a @U aB @ = X C U aC U CB : {25b Notethat,byconstruction,thematrix U canbeexpandedpurelywithinthevirtualspace. RelatingtheCPFNOcoecients V PQ to C pq and U AB V IJ = C ij {26a V AI = X b U bA C bi {26b V IA = X b C ib U bA {26c V AB = U AB + X cd U cB C cd U dA : {26d Thecoecients C pq areCPHFcoecients,whosesolutionswillbeimplicit;theyobeyan orthonormalitycondition.TheperturbedFNOtransformationobeysaslightlydierent 69

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conditionbecause U actsintheorthonormalHartree-Fockbasis, C qp + S pq + C pq =0{27a U BA + U AB =0 : {27b TherelationsexpressedinEquations2{27canberearrangedtoreducethenumberof independentvariables, C qp = )]TJ/F21 11.9552 Tf 9.299 0 Td [(S pq )]TJ/F21 11.9552 Tf 11.956 0 Td [(C pq C pp = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 S pp {28a U BA = )]TJ/F21 11.9552 Tf 9.299 0 Td [(U AB U AA =0 : {28b ForallchoicesofperturbedFNOorbitals,oneshouldchoosetheunderlyingCPHF coecientsinthevirtualspacetobenon-canonical,todene C ab = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 S ab : {29 ThischoicethencanbeinsertedintoEquation2{26d,yielding V AB = U AB )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 S AB : {30 Thedenitionof C ij willbedependentonchoosingcanonicalornon-canonicalperturbed orbitals.SatisfyingtheHartree-FockperturbedBrillouinconditionsisunchangeddue totheFNOprocedure,becausetheFNOsarestillanon-canonicalsetofHartree-Fock orbitals.BecausetheBrillouinconditionisstillsatised,wedonotneedtoincludesingle excitationcontributionstotheMBPTdensitymatrix.Ontheotherhand,focusingon theperturbeddensitymatrixrevealssomeadditionalcomplexities. TheformofthedensitymatrixillustratedinEquation2{1onlyholdsforcanonical HForbitals.Todirectlyusethatequationtoderiveaperturbeddensitymatrix,asis neededtoimposetheCPFNOconditionEquation2{23b,oneneedstorequirecanonical perturbedunderlyingHForbitals.ThisrestrictionwouldnecessitatesolvingtheCPHF equations foreachperturbation .Weinsteadconstructtheresponsewithoutsolvingthe 70

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CPHFcoecientsbyrstworkingintheoriginalHFbasisthatdenedbythecoecients C .TheformoftheMBPTdensitymatrixfornon-canonicalHForbitalsis D ab = 1 2 X ijc t cb ij t ca ij {31 wheretherst-order T 2 amplitudessatisfytheequation, ab ij t ab ij = h ab jj ij i + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab X c 6 = a f ac t cb ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij X k 6 = i f ik t ab kj : {32 with P pq =1 P pq where P pq interchangesorbitals p and q .Inthecasethatthe Hartree-Fockorbitalsarecanonical,thelasttwotermsvanish,allowingthesolution, t ab ij = h ab jj ij i ab ij ; {33 which,wheninsertedintothegeneralexpressionforthedensitymatrixreturnstheoriginal resultfromEquation2{1.Introducinganexternalperturbationyields @D ab @ = 1 2 X ijc @t cb ij @ t ca ij + t cb ij @t ca ij @ # = 1 2 P + ab X ijc @t cb ij @ t ca ij {34 wheretheperturbed T amplitudesaredenedbytheperturbedamplitudeequation[137], @ ab ij @ t ab ij + ab ij @t ab ij @ = @ h ab jj ij i @ + P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab X c 6 = a @f ac @ t cb ij + f ac @t cb ij @ )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij X k 6 = i @f ik @ t ab kj + f ik @t ab kj @ : {35 Usingthefactthattheunderlying unperturbed orbitalsarecanonicalevenifthe perturbedorbitalsarenot,onecansimplifytheamplitudeequationto, @ ab ij @ t ab ij + ab ij @t ab ij @ = @ h ab jj ij i @ + P )]TJ/F15 11.9552 Tf 7.084 1.794 Td [( ab X c 6 = a @f ac @ t cb ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( ij X k 6 = i @f ik @ t ab kj : {36 Theunperturbed T amplitudesareknownsincetheycorrespondtocanonicalorbitals. Onecanwritetheperturbed T amplitudeequationintermsofknownquantities, 71

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ab ij @t ab ij @ = @ h ab jj ij i @ + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab X c @f ac @ h cb jj ij i cb ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij X k @f ik @ h ab jj kj i ab kj : {37 Thissetofequationsdependsontheoccupied-virtualblockofCPHFcoecientsthrough theperturbedtwo-electronintegralsandtheperturbedFockoperator,ascanbeseenby expandingthefullperturbeddensitymatrix, @D ab @ = P + ab 1 2 X ijc @ h cb jj ij i @ h ij jj ca i cb ij ca ij )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X ijkc @f ik @ h cb jj kj ih ij jj ca i cb kj cb ij ca ij + 1 2 X ijcd @f cd @ h db jj ij ih ij jj ca i cb ij db ij ca ij + @f bd @ h cd jj ij ih ij jj ca i cb ij cd ij ca ij !# : {38 Expandingtheintegralderivatives, @D ab @ = P + ab X cd Y ab;cd S cd + M ab;cd f cd + X i>j Y ab;ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y ab;ji C ij + X ij M ab;ij f ij )]TJ/F26 11.9552 Tf 11.956 11.358 Td [(X i>j Y ab;ji S ij + X ic Y ab;ci C ci + M ab;ci S ci # + D ; ab {39 wheretheintermediatequantities M and Y aredenedinTable2-1andtheperturbed densitymatrix D ; ab isdenedby D ; ab = 1 2 P + ab X ijc h cb jj ij i h ij jj ca i cb ij ca ij : {40 Unliketheothertermsdevelopedsofar,theperturbeddensitymatrixdoesnothavea clearcounterpartintheCCgradientexpression. TheseequationshavebeenderivedintheoriginalHartree-Fockbasis,buttheCPFNO equationisintheFNObasis.Totransformtheresults,oneusestherelation D A 00 A 0 = X bc U bA 00 D bc U cA 0 : {41 Dierentiatingthisexpression, 72

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Table2-1.IntermediatequantitiesforFNOgradients M ab;ij = )]TJ/F26 11.9552 Tf 11.291 8.967 Td [(P kc h cb jj jk ih ik jj ca i cb jk cb ik ca ik M ab;ci = )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 2 P jk h ib jj jk ih jk jj ca i cb jk ca jk )]TJ/F20 7.9701 Tf 13.151 4.707 Td [(1 2 bc P jkd h di jj jk ih jk jj da i db jk da jk M ab;cd = 1 2 P ij h db jj ij ih ij jj ca i db ij cb ij ca ij + 1 2 bc P ije h ed jj ij ih ij jj ea i eb ij ed ij ea ij Y ab;ij = i + j M ab;ij +2 P kl M ab;kl h ki jj lj i +2 P cd M ab;cd h ci jj dj i Y ab;cd = )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 4 P ij h db jj ij ih ij jj ca i cb ij ca ij )]TJ/F20 7.9701 Tf 13.151 4.707 Td [(1 2 c + d M ab;cd Y ab;ci = M ab;ci + P jk M ab;jk h jc jj ki i + h ji jj kc i + P de M ab;de h dc jj ei i + h di jj ec i @D A 00 A 0 @ = X bc @U bA 00 @ D bc U cA 0 + U bA 00 @D bc @ U cA 0 + U bA 00 D bc @U cA 0 @ # : {42 TheperturbedFNOdensitycanthenbewritten,using D B 00 B 0 =0andtherelationship expressedinEquation2{30, @D A 00 A 0 @ = )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X B 0 D A 0 B 0 V A 00 B 0 + X B 00 D A 00 B 00 V B 00 A 0 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 X B 0 D A 0 B 0 S A 00 B 0 + 1 2 X B 00 D A 00 B 00 S B 00 A 0 + P + A 00 A 0 X IB Y A 00 A 0 ;BI V BI + X I>J Y A 00 A 0 ;IJ )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y A 00 A 0 ;JI V IJ + X BC Y A 00 A 0 ;BC S BC + M A 00 A 0 ;BC f BC + X IJ M A 00 A 0 ;IJ f IJ )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X I>J Y A 00 A 0 ;JI S IJ + X IB M A 00 A 0 ;BI S BI # + D ; A 00 A 0 {43 wherethequantities M and Y havebeentransformedtotheFNObasis.Theperturbed quantitieshavethereforebeencompletelyseparatedfromtheCPFNOcoecients,which allowsforaperturbation-independentsolutionoftheCPFNOequations.Togofurther, onemustchoosebetweennon-canonicalandcanonicalperturbedFNOs. Non-canonicalperturbedorbitals :Forthechoiceofnon-canonicalperturbed orbitals,wearefreetodene 73

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Table2-2.Elementsofthenon-canonicalCPFNO A matrix A AI;BJ = h AB jj IJ i + h AJ jj IB i)]TJ/F21 11.9552 Tf 19.261 0 Td [( IJ f AB + IJ AB f II A A 00 A 0 ;BJ = Y A 00 A 0 ;BJ + Y A 0 A 00 ;BJ A A 00 A 0 ;B 00 B 0 = A 00 B 00 D A 0 B 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [( A 0 B 0 D A 00 B 00 V IJ = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 S IJ {44a V A 0 B 0 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 S A 0 B 0 {44b V A 00 B 00 = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 S A 00 B 00 : {44c Theonlytermsthatneedtobesolvedare V AI and V A 00 B 0 ThenCPFNOequationsinmatrixformare 2 6 4 A vo,vo 0 A v 00 v 0 ,vo A v 00 v 0 ,v 00 v 0 3 7 5 2 6 4 V vo V v 00 v 0 3 7 5 = 2 6 4 B vo B v 00 v 0 3 7 5 {45 where B AI = )]TJ/F21 11.9552 Tf 11.956 0 Td [(f AI + S AI f II + X JK S JK h AJ jj IK i {46 B A 00 A 0 = P + A 00 A 0 X BC Y A 00 A 0 ;BC S BC + M A 00 A 0 ;BC f BC + X IJ M A 00 A 0 ;IJ f IJ )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 Y A 00 A 0 ;JI S IJ + X IB M A 00 A 0 ;BI S BI # )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 X B 0 S A 00 B 0 D B 0 A 0 + 1 2 X B 00 D A 00 B 00 S B 00 A 0 + D ; A 00 A 0 {47 andtheelementsofmatrix A areinTable2-2. Theinterchangetheoremcanbewritten X P>Q X PQ V PQ = X T V = X T A )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 B )]TJ/F36 11.9552 Tf 21.918 0 Td [(D or T B = )]TJ/F26 11.9552 Tf 12.552 11.357 Td [(X P>Q D or PQ B PQ : {48 74

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Onecansolve,insteadofEquation2{45,theperturbation-independentequation, 2 6 4 A T vo,vo A T vo,v 00 v 0 0 A T v 00 v 0 ,v 00 v 0 3 7 5 2 6 4 D or vo D or v 00 v 0 3 7 5 = 2 6 4 )]TJ/F21 11.9552 Tf 9.298 0 Td [(X vo )]TJ/F21 11.9552 Tf 9.298 0 Td [(X v 00 v 0 3 7 5 : {49 Thesecondoftheseequations,whichdeterminestheorbitalresponseoftheuncorrelated molecularorbitals,canbesolvedindependentlyoftherst,usingastandardlinear equationsolver.Substitutingthisresultintotheequationforthevirtual-occupiedblockof theorbitalresponsecontributiontothedensitymatrix, X JB D or BJ [ h BA jj JI i + h BI jj JA i + f II AB )]TJ/F21 11.9552 Tf 11.955 0 Td [(f AB IJ ]= )]TJ/F15 11.9552 Tf 12.675 3.022 Td [(~ X AI {50 ~ X AI = X AI + X B 00 B 0 D or B 00 B 0 [ Y B 00 B 0 ;AI + Y B 0 B 00 ;AI ] : {51 Thismodiedz-vectorequationcanthenbesolvedbythestandardmethod[135].Using theseorbitalrelaxationcomponents,onecanformthenaldensitymatrices, D AI = AI + D or AI {52 and D IJ = IJ )]TJ/F26 11.9552 Tf 13.442 11.357 Td [(X A 00 A 0 M A 00 A 0 ;IJ + M A 0 A 00 ;IJ D or A 00 A 0 {53 D AB = BC )]TJ/F26 11.9552 Tf 13.697 11.357 Td [(X C 00 C 0 M C 00 C 0 ;AB + M C 0 C 00 ;AB D or C 00 C 0 : {54 Thenalintermediatematricesare I IJ = I 0 IJ )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X AK D or AK h AI jj KJ i + 1 2 X A 00 A 0 Y A 00 A 0 ;JI + Y A 0 A 00 ;JI D or A 00 A 0 {55a I AI = I 0 AI )]TJ/F21 11.9552 Tf 11.955 0 Td [(f II D or AI )]TJ/F26 11.9552 Tf 13.879 11.358 Td [(X B 00 B 0 M B 00 B 0 ;AI )]TJ/F21 11.9552 Tf 11.955 0 Td [(M B 0 B 00 ;AI D or B 00 B 0 {55b I A 0 B 0 = I 0 A 0 B 0 )]TJ/F26 11.9552 Tf 13.697 11.357 Td [(X C 00 C 0 Y C 00 C 0 ;A 0 B 0 + Y C 0 C 00 ;A 0 B 0 D or C 00 C 0 {55c 75

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I A 00 A 0 = I 0 A 00 A 0 + 1 2 X B 0 D or A 00 B 0 D B 0 A 0 )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 X B 00 D A 00 B 00 D or B 00 A 0 )]TJ/F26 11.9552 Tf 13.878 11.357 Td [(X B 00 B 0 Y B 00 B 0 ;A 00 A 0 + Y B 0 B 00 ;A 00 A 0 D or B 00 B 0 {55d I A 00 B 00 = )]TJ/F26 11.9552 Tf 13.697 11.357 Td [(X C 00 C 0 Y C 00 C 0 ;A 00 B 00 + Y C 0 C 00 ;A 00 B 00 D or C 00 C 0 : {55e Thenaltermlefttoaddressis D ; A 00 A 0 from B A 00 A 0 .Thistermcanonlybesimply expressedasinEquation2{40whentheorbitalsarecanonicalHF.Therefore,to completelyseparatetheperturbedorbitalcontributionrequiresadditionalintegral transforms.First,onemustback-transform D or A 00 A 0 totheHFbasis, D or ab = X C 00 C 0 U aC 00 D or C 00 C 0 U bC 0 : {56 Thisobjectcannowbecontractedwiththeperturbationindependentpiecesof D a 00 A 0 G ab;ij = X c D or cb h ij jj ac i bc ij ac ij ; {57 leadingtotheexpressionfornon-canonicalperturbedorbitalgradients @E CC @ = X PQ D PQ f PQ + X PQRS )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(PQ;RS h PQ jj RS i + X PQ I PQ S PQ + X abij G ab;ij h ab jj ij i : {58 Becauseoftheextra G term,aseparateback-transformationisnecessarytowritethe G ab;ij termintheatomicorbitalbasisbeforecontractionwiththederivativeintegrals. Canonicalgradients :InthecaseofCCSDT,itishighlyadvantageoustoimpose theconditionthattheperturbedorbitalsremainsemi-canonical[52,138].Whenfrozen occupiedorvirtualsareused,thederivativeiscalculatedusingcanonicalperturbed orbitalsaswell[132].Thisconditionisactuallymorestringentthanstrictlynecessary; aslongasmixingoccursonlywithinthefrozenandactivesubsetsoforbitals,theydo notneedtobemaintainedcanonical.Therefore,onemustformulatetheFNOorbital relaxationtermsinsemi-canonicalorbitals.SupplementingtheconventionalBrillouin conditionandtheFNOconditionaretherequirementsthat 76

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@f IJ @ =0= @f A 0 B 0 @ : {59 Thereisnoneedtoimposecanonicalityontheuncorrelatedorbitals f A 00 g becausethe computationaladvantageliesindeterminingtheCCcontributiontothedensitymatrices, whichdoesnotinvolvetheuncorrelatedorbitals.Byimposingthisrequirementonecan nolongerchoose V IJ = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 2 S IJ and V A 0 B 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 2 S A 0 B 0 .However,one can choose V A 00 B 00 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 2 S A 00 B 00 ,sincethedroppedvirtualscanbenon-canonical.Therefore,no iterativeequationshavetobesolvedintheuncorrelated-uncorrelatedsector. ThenewCPHFequationscanbewritteninmatrixform 2 6 6 6 6 6 6 6 4 A oo,oo A oo,vo 00 0 A vo,vo 00 0 A v 0 v 0 ,vo A v 0 v 0 ,v 0 v 0 A v 0 v 0 ,v 00 v 0 A v 00 v 0 ,oo A v 00 v 0 ,vo 0 A v 00 v 0 ,v 00 v 0 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 V oo V vo V v 0 v 0 V v 00 v 0 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 B oo B vo B v 0 v 0 B v 00 v 0 3 7 7 7 7 7 7 7 5 : {60 Theright-handsideofEquation2{60isgivenbytherelations B P 0 Q 0 = )]TJ/F21 11.9552 Tf 11.955 0 Td [(f P 0 Q 0 + S P 0 Q 0 f Q 0 Q 0 + X IJ S IJ h P 0 I jj Q 0 J i {61 B A 00 A 0 = P + A 00 A 0 X BC Y A 00 A 0 ;BC S BC + M A 00 A 0 ;BC f BC + X IJ M A 00 A 0 ;IJ f IJ )]TJ/F26 11.9552 Tf 11.291 11.357 Td [(X I
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Table2-3.ElementsofthecanonicalCPFNO A matrix A IJ;KL = IK JL f II )]TJ/F21 11.9552 Tf 11.955 0 Td [(f JJ A IJ;AK = h IA jj JK i + h IK jj JA i A AI;BJ = IJ f AB )]TJ/F21 11.9552 Tf 11.955 0 Td [( IJ AB f II + h AB jj IJ i + h AJ jj IB i A A 0 B 0 ;CI = h A 0 C jj B 0 I i + h A 0 I jj B 0 C i A A 0 B 0 ;C 0 D 0 = A 0 C 0 B 0 D 0 f A 0 A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f B 0 B 0 A A 0 B 0 ;A 00 C 0 = B 0 C 0 f A 0 A 00 + A 0 C 0 f A 00 B 0 A A 00 A 0 ;IJ = Y A 00 A 0 ;IJ + Y A 00 A 0 ;IJ )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y A 0 A 00 ;JI )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y A 0 A 00 ;JI A A 00 A 0 ;BJ = Y A 00 A 0 ;BJ + Y A 0 A 00 ;BJ A A 00 A 0 ;B 00 B 0 = A 00 B 00 D A 0 B 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [( A 0 B 0 D A 00 B 00 yieldstheorbitalresponsecontributiontotheoveralldensitymatrix.Inthisform,itis obviousthatthesolutionfortheactivevirtual-activevirtualblockdoesnotcoupletothe otherblocks,yielding D or A 0 B 0 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(X A 0 B 0 f A 0 A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f B 0 B 0 : {64 TheFNOblockcanthenbedeterminedbyinsertingtheneworbitalrelaxationtermsfrom theactivevirtualblock X B 00 B 0 D or B 00 B 0 h A 00 B 00 D A 0 B 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [( A 0 B 0 D A 00 B 00 i = )]TJ/F15 11.9552 Tf 12.675 3.022 Td [(~ X A 00 A 0 {65 ~ X A 00 A 0 = X A 00 A 0 + X B 0 f A 00 B 0 D or B 0 A 0 : {66 Furthermore,aftersolvingfortheinactive-activevirtualblock,theorbitalresponseforthe occupied-occupiedblockcanbesolved, X K>L D or KL [ IK JL f II )]TJ/F21 11.9552 Tf 11.955 0 Td [(f JJ ]= )]TJ/F15 11.9552 Tf 12.675 3.022 Td [(~ X IJ D or IJ = )]TJ/F15 11.9552 Tf 12.675 3.022 Td [(~ X IJ f II )]TJ/F21 11.9552 Tf 11.955 0 Td [(f JJ {67 ~ X IJ = X IJ + P )]TJ/F15 11.9552 Tf 7.084 1.794 Td [( IJ X A 00 A 0 Y A 00 A 0 ;IJ + Y A 0 A 00 ;IJ D or A 00 A 0 : {68 78

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Finally,theresponseoftheoccupied-occupiedblockandthevirtual-virtualblockscan beinsertedintotheequationfortheoccupied-virtualblock X JB D or BJ [ AB IJ f AA )]TJ/F21 11.9552 Tf 11.955 0 Td [(f II + h BA jj JI i + h BI jj JA i ]= )]TJ/F15 11.9552 Tf 12.675 3.022 Td [(~ X AI {69 ~ X AI = X AI + 1 2 X P 0 Q 0 D or P 0 Q 0 [ h AP 0 jj IQ 0 i + h AQ 0 jj IP 0 i ] + X B 00 B 0 D or B 00 B 0 [ Y B 00 B 0 ;AI + Y B 0 B 00 ;AI ] : {70 Thisequationnowtsthestandardformofthez-vectorequations. Aftersolvingforalloftheorbitalresponsecomponentsofthedensitymatrices,one candenethefull,relaxed,densitymatricesvia, D IJ = IJ + D or IJ )]TJ/F26 11.9552 Tf 13.441 11.358 Td [(X A 00 A 0 M A 00 A 0 ;IJ + M A 0 A 00 ;IJ D or A 00 A 0 {71a D AI = AI + D or AI {71b D AB = BC + D or AB )]TJ/F26 11.9552 Tf 13.697 11.358 Td [(X C 00 C 0 M C 00 C 0 ;AB + M C 0 C 00 ;AB D or C 00 C 0 {71c andthenalintermediatematrices, I IJ = I 0 IJ )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X PQ h IP jj JQ i D or PQ )]TJ/F21 11.9552 Tf 11.956 0 Td [(f JJ D or IJ {72a I AI = I 0 AI )]TJ/F21 11.9552 Tf 11.955 0 Td [(f II D or AI )]TJ/F26 11.9552 Tf 13.879 11.358 Td [(X B 00 B 0 M B 00 B 0 ;AI )]TJ/F21 11.9552 Tf 11.955 0 Td [(M B 0 B 00 ;AI D or B 00 B 0 {72b I A 0 B 0 = I 0 A 0 B 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f B 0 B 0 D or A 0 B 0 )]TJ/F26 11.9552 Tf 13.697 11.358 Td [(X C 00 C 0 Y C 00 C 0 ;A 0 B 0 + Y C 0 C 00 ;A 0 B 0 D or C 00 C 0 {72c I A 00 A 0 = I 0 A 00 A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f A 0 A 0 D or A 00 A 0 + 1 2 X B 0 D or A 00 B 0 D b 0 A 0 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 X B 00 D A 00 B 00 D or B 00 A 0 )]TJ/F26 11.9552 Tf 13.879 11.358 Td [(X B 00 B 0 Y B 00 B 0 ;A 00 A 0 + Y B 0 B 00 ;A 00 A 0 D or B 00 B 0 {72d I A 00 B 00 = )]TJ/F26 11.9552 Tf 13.033 11.358 Td [(X C 00 C 0 Y C 00 C 0 ;A 00 B 00 + Y C 0 C 00 ;A 00 B 00 D or C 00 C 0 : {72e Again,itisnecessarytoincludethe D ; A 00 A 0 termseparatelyviaaback-transformation,as 79

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1.SolveSCFequations 2.FNOprocedure: aConstructMBPTdensitymatrix bFormFNObasis cConstruct M and Y matricesinFNObasisTable2-1 dStore h ij jj ab i andHForbitaleigenvaluesforconstructionof G ab;ij [Equation2{57] 3.SolvecorrelatedcalculationintruncatedFNObasis 4.Formresponsedensitymatrices and)]TJ0 g 0 G -190.862 -14.446 Td [(5.SolveFNOZ-vectorequationsfororbitalresponse, D or [Equation2{49or Equation2{63] 6.Formrelaxeddensitymatrices D [Equations2{52to2{54orEquations2{71]and G [Equation2{57]andintermediatematrix I [Equations2{55orEquations2{72] 7.Back-transformalldensitymatricestotheAObasis 8.Contractagainstintegralderivatives. Figure2-1.StepsinanFNOderivativecalculation isexpressedinEquations2{56to2{58.Asummaryofthestepsnecessarytocalculatea derivativeusingtheFNOsisshowninFigure2-1. Unfortunately,thesimplecorrectionthatwaspossiblefortruncatedFNOenergies, Equation2{9,isnotaccessibleforgradients.Theequivalentexpressionwouldbe @E CC Corrected @ = @E CC @ + @E [Full] @ )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(@E [Truncated] @ : {73 Unlikethecaseforenergies,thederivativesoftheMBPTenergiesarenotcalculated asabyproductoftheFNOprocedure.Forenergies,thiscorrectioncanbesubstantial, whichraisesthequestionoftheaccuracyofFNOgradientstoreproducetheuntruncated gradients.However,areasonableassumptionisthattheincrementbetweenthetwo MBPTresultsislocallyconstant{i.e.that @ FNO-MBPT =@ 0.Ifthis statementwerenottobethecasethentheeectofthetruncatedorbitalswouldneed tovarysubstantiallypointwisealongapotentialenergysurface.Asisdiscussedin Section2.2.3,suchvariationintroducesmorefundamentalproblemsthantheinabilityto useanMBPTcorrection. 80

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2.2.3SmoothnessofthePotentialEnergySurface TheFNOproceduredevelopedherewillnotnecessarilyyieldrigorouslysmooth potentialenergysurfacesPES.NotethattheFNOtruncationisperformedpointby pointonthePES,withoutconsiderationoftheconnectionbetweenthatpointandother pointsonthepotentialenergysurface.Therefore,ifthestructureorsizeofthespace spannedbythecorrelatedsetofvirtualorbitalschangesasafunctionofthegeometry,it ispossiblethattheenergycouldchangediscontinuously. Tominimizetheimpactofdiscontinuities,thecoderecognizesorbitalsthatareclose inoccupationtothecorrelatedorbitals.Thosewithinacertaintoleranceofthecuto occupationareconsideredtobequasi-degenerateandareretained.Assumingthatthe geometrystepsarenottoolarge,thisprocedureshouldsmoothchangesintheFNO structure.Itshouldbeclearthatthisproblemisnotuniquetothefrozennaturalorbital truncationprocedure,butexistsforallproceduressuchaslocalizedorbitalsmethods thattruncatethecorrelationspaceinageometrydependentway[139{141]. 2.3Implementation TheFNO-CCenergiesandgradientshavebeenimplementedwithintheACESII programsystem[54,55].Ittakesadvantageofrealabelianpointgroupsymmetry,andall equationsarefullyspin-summed.Forenergies,FNO-CCisavailableforsingledeterminant RHF,spin-polarizedUHF,spin-restrictedopen-shellROHF,andquasi-restricted Hartree-FockQRHF[142]references.Forgradients,onlyRHFandUHFreferencesare available. InanFNOenergycalculation,apartialintegraltransformationisperformedbefore theFNOtruncation,andthenafullintegraltransformationisperformedintheresultant truncatedbasis.Thiscomputationaladvantageisunachievableforgradients;instead,one mustperformafullintegraltransformationforboththetruncatedandfullbasissets, requiringthestorageofmoreintegrals.Thecorrelatedcalculationsarethenperformed withinthetruncatedbasis.Formationofthedensitymatricesproceedsintwoparts:rst, 81

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thecorrelatedcontributionsareformedwithinthetruncatedbasis,thenthesedensity matricesareexpandedtothefullbasis,andtheorbitalrelaxationtermsarecalculated andincluded.Onecalculatesthe M and Y intheHFbasisandthenstorestheminthe truncatedFNObasis,sothattheycanbeaddedtotheorbitalrelaxationequations. Theback-transformationoftheFNOdensitymatrixandthe G ab;ij termareperformed separately.Thesetermsarethensummedbeforecontractionwithderivativeintegrals. FNOenergycalculationsonlyrequiretheadditionalcostofanMBPTdensity calculationincludingapartialintegraltransformation;formanymoleculesthespeedup achievedcloselymatcheswhatisexpectedfromthecoupled-clusterscalingexpressions. Ontheotherhand,comparedtogradientcalculationsthatdonotuseFNOs,thelargest addedexpenseisthenecessityofcalculatingandstoringseveralnewintermediatesofa dimensionsimilartothatofthetwo-electronintegrals.Thecomputationalcostisfarless thanthecostoftheCCprocedure,though,anddoesnotchangetheoverallscalingof thecoupled-cluster,but,insteadreducesitscostinapplications.However,theadditional storagecostscouldbeproblematicforsomecombinationsofcomputerandmolecule. Accuracyofthegradientimplementationwasveriedbycomparinganalytical gradientresultstothoseobtainedbynumericaldierentiationoftheenergy. 2.4ResultsandDiscussion 2.4.1Calibration Energies :Thequalityoffrozennaturalorbitalsforequilibriumpropertiesand forthedipolemomentincoupled-clustertheoryhasbeeninvestigatedbefore[92,93]. However,iftheFNOsaretobeusefulforcalculatingenergydierencesbetweendierent conformersofmoleculesandalongreactionpathways,thequalityoftheFNOsmustbe maintainedacrossthepotentialenergysurface.Togaugethiselementofthequality, aseriesofsixsmallmoleculeswereexaminedacrosstheirpotentialenergysurfacesat theCCSDandCCSDTlevelintheDZP[14],cc-pVTZ,andcc-pVQZ[113]basissets retaining20%,40%,60%,80%and100%ofthevirtualorbitals.Theseplotsshowthe 82

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dependenceofpropertiesonthepercentageofthevirtualspaceretainedinthetruncated calculations.Insomeways,thiswayofchoosingatruncationisunsatisfying;itwould bebetterifonewereabletoexaminetheMBPToccupationnumbers,andthenchose propercutosbasedonthesevalues.However,whilewehavelookedintothisissue,we havenotbeenabletodetermineanyconsistenttruncationcriterion:theoccupation numbersgosmoothlyfromhightolowoccupation,withoutanysharpchangesthatwould naturallyindicateaplacetotruncate.Becausethegoalofthemethodistoreducethe computationalcostofthecalculation,atthispointitseemsbettertouseatruncation schemewherethespeedupcanbepredicted,evenifitislesssatisfyingtheoretically. AverageresultsandtimingsarecompiledinTable2-4.Thebestbasissetchoicefor anycalculationistheonethatcangiveyouthegreatestpercentageofthecorrelation energywiththequickestcalculation.Figure2-2isaguideforthatdecisionbasedonthe FNOorbitalscheme. Theupper-lefthandcornerofFigure2-2representsthebestpossiblebasisset.As canbeseen,choosingalargebasissetandthentruncatingitusingtheFNOframework providessuperiorresultscomparedtousingasmallerbasisset.Forexample,forthe moleculesconsidered,usinga40%cc-pVQZbasissetwouldprovidebothagreater percentageofthetotalcorrelationenergy and beaquickercalculationthanusingthefull cc-pVTZbasisset.AnillustrationofthiseectisshowninFigure2-3A,whichshows fourpotentialenergycurvesforthedissociationofhydrogenuorideattheCCSDlevelin dierentbases,eachofwhichhasthesamenumberofactiveorbitals. Thelowestcurveisthefull100%cc-pVQZbasisresult,theothersaretruncations fromthreedierentbases.ThequalitativeformofthePESisidenticalinallfourcases, indicatingthattheFNOsprovideabalancedtreatmentofcorrelationacrossthepotential energysurface.Thebestresults,afterthefullcc-pVQZbasis,arethosefromaretention of20%ofthecc-pVQZbasis.Thistruncatedlargebasissignicantlyoutperformsthe equivalent-sizedfullDZPbasis. 83

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Table2-4.AveragepercentageofcorrelationenergyforaseriesofFNOtruncationsforsix smallmolecules Fractionof% E corr [100%cc-pVQZ] VirtualOrbitalsDZPcc-pVTZcc-pVQZ 20%29 : 661 : 181 : 2 40%47 : 376 : 094 : 0 60%58 : 782 : 298 : 0 80%63 : 185 : 099 : 2 100%66 : 085 : 6100 : 0 Shiftingtheenergiesofthesecurvessothattheirequilibriacoincide,showsthatnot onlydoesusinganFNOtruncatedbasisleadtoconvergencewithtotalenergies,italso leadstoconvergenceindissociationenergies.Figure2-3Bshowsthedissociativetailsof theseshiftedpotentialenergysurfaces;ascanbeseen,the20%cc-pVQZbasiscomes closesttoaccuratelydescribingthedissociation.ThefullDZPbasisunderestimatesthe dissociationenergybymorethan10kcal/mol,whilethe20%cc-pVQZbasisiswithin2 kcal/molofthefullcc-pVQZdissociationenergy. Gradients :TodeterminethecapabilityofFNOtruncatedgradientstoreproduce structures,weappliedFNOCCSDTandCCSDTtothesetofwell-characterized moleculesfromBak,etal.,[50].ComparativestatisticsareshowninTables2-5to2-8. OneimmediateconclusionisthattheFNOconvergencebehaviorisidenticalforboth CCSDTandCCSDT.Meanabsoluteerrorsprobablythebestsinglemeasureofthe resultsarealmostidentical,especiallyforlargerbasissets.Theconvergencewithrespect totruncationoftheFNOgeometriesisnotmonotonic;whilethereisgenerallyatrend thatlesstruncationleadstobetter abs ,thereareexceptions.Evenmoredramaticare themaximumerrors,whichdonotshowaclearconvergencebehavior.Theseresultsare notnecessarilysurprising.Especiallyforthedouble )]TJ/F21 11.9552 Tf 9.298 0 Td [( basissets,thetruncatedbasissets canbecomesosmallthatonecannotconsiderthemmeaningfulpointsforextrapolationof theconvergencebehavior.Unliketheconvergenceoftheenergy,convergenceofgeometric propertieswilltendtobelessclear-cut:optimizedgeometriesaredependentnotjuston 84

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Figure2-2.PercentageofCCSDTcorrelationenergycomparedto100%cc-pVQZresult recoveredasafunctionofthecoupled-clustertimerelativeto20%DZP calculationandasafunctionofbasisset theenergyatapoint,butrathertherelativeenergyatapointtothepointsaroundit. Thereis,therefore,adelicatebalancetothebestchoiceofbasisandmethodforgeometry prediction,leadingtomorecomplicatedconvergencebehavior. ExaminingthetablesofbondanglesTables2-7and2-8,itisclearthatboth methodsunderestimatebondangles,evenwithallthedierenttruncations.Importantly, theFNOtruncationsdonotsignicantlyaectthestandarddeviationsofthegeometries. TobetterunderstandtheconvergencebehavioroftheFNOs,themeanabsoluteerrors fromexperimentforthisdatasetareplottedinFigures2-4to2-7.Eachpointrepresents retaining20%,40%,60%,80%or100%ofthevirtualspaceofthecorrespondingbasis.On thehorizontalaxisisameasureoftherelativesizeofthetruncatedbasissetascompared tothelargestbasisinthecalculation%quadruple )]TJ/F21 11.9552 Tf 9.299 0 Td [( .Forbondlengths,allchoices 85

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A B Figure2-3.AHydrogenuorideRHFCCSDpotentialenergycurvefordierentFNO basistruncationswiththesamenumberofvirtualorbitals.BHydrogen uorideRHFCCSDdissociationenergycurvesfordierentFNObasis truncationswiththesamenumberofvirtualorbitals 86

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Table2-5.Comparisonofoptimizedequilibriumbondlengthsfordierent correlation-consistentbasissets[15,113]formultipleFNOtruncationsfor CCSDTandCCSDT.Thepercentageindicateswhatpercentofthevirtual spaceofeachmoleculewasactive.Fortruncatedbasissets% )]TJ/F15 11.9552 Tf 11.956 0 Td [(80%errors arerelativetotheuntruncatedbasissetresult;for100%,errorsarerelativeto experiment.Averageswerecalculatedoverthesetofmoleculesfrom[50].Only valenceelectronswerecorrelated. isthesignedmeanerror, abs isthe meanabsoluteerror, max isthemaximumabsoluteerror,and std isthe standarddeviation.Allnumbersareinunitsofpm. BasisSet CCSDTCCSDT abs max std abs max std cc-pVDZ: 20%0 : 020 : 662 : 300 : 800 : 050 : 631 : 930 : 74 40% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 170 : 541 : 730 : 70 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 140 : 541 : 850 : 71 60% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 530 : 542 : 430 : 63 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 510 : 522 : 30 : 60 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 260 : 303 : 450 : 51 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 240 : 293 : 220 : 65 100% a 1 : 721 : 724 : 510 : 821 : 691 : 694 : 120 : 76 cc-pVTZ: 20%0 : 300 : 645 : 501 : 210 : 320 : 635 : 471 : 20 40%0 : 150 : 210 : 870 : 250 : 160 : 220 : 880 : 24 60% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030 : 140 : 570 : 19 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 020 : 140 : 530 : 19 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030 : 090 : 450 : 13 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030 : 090 : 410 : 13 100% a 0 : 050 : 220 : 900 : 290 : 020 : 220 : 710 : 27 cc-pVQZ: 20%0 : 050 : 180 : 680 : 240 : 070 : 180 : 770 : 23 40% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 110 : 181 : 110 : 27 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 110 : 171 : 010 : 26 60%0 : 000 : 050 : 260 : 070 : 020 : 050 : 260 : 07 80%0 : 000 : 030 : 100 : 040 : 000 : 020 : 100 : 03 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 060 : 130 : 710 : 19 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 100 : 140 : 710 : 19 a Relativetoexperiment. ofbasissets60%triple )]TJ/F21 11.9552 Tf 9.299 0 Td [( orlargerperformsimilarly.Thepictureinthebondangleplots ismoremixed,withfullconvergencenotachieveduntil40%ofthequadruple )]TJ/F21 11.9552 Tf 9.298 0 Td [( basis, thoughthe60%triple )]TJ/F21 11.9552 Tf 9.299 0 Td [( basisperformsquitewell.Double )]TJ/F21 11.9552 Tf 9.298 0 Td [( basissetsareinadequateat everytruncation.Theseplotsareaguideforthechoiceofanoptimalbasissetofagiven size.Forexample,20%ofacc-pVQZbasisor40%ofacc-pVTZbasisyieldresultsthat areapproximatelythesameforbondlengthsasshowninFigure2-4andhavethesame costastheinferioruntruncatedcc-pVDZbasisset. 87

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Table2-6.Comparisonofoptimizedequilibriumbondlengthsfordierent correlation-consistentbasissets[15,113]formultipleFNOtruncationsfor CCSDTandCCSDT.Thepercentageindicateswhatpercentofthevirtual spaceofeachmoleculewasactive.Fortruncatedbasissets% )]TJ/F15 11.9552 Tf 11.956 0 Td [(80%errors arerelativetotheuntruncatedbasissetresult;for100%,errorsarerelativeto experiment.Averageswerecalculatedoverthesetofmoleculesfrom[50].All electronswerecorrelated. isthesignedmeanerror, abs isthemean absoluteerror, max isthemaximumabsoluteerror,and std isthestandard deviation.Allnumbersareinunitsofpm. BasisSet CCSDTCCSDT abs max std abs max std cc-pCVDZ: 20% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 090 : 662 : 210 : 81 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 050 : 621 : 840 : 74 40% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 580 : 632 : 960 : 73 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 560 : 602 : 860 : 69 60% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 300 : 332 : 340 : 49 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 290 : 312 : 140 : 45 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 140 : 170 : 990 : 24 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 140 : 170 : 980 : 24 100% a 1 : 661 : 664 : 420 : 801 : 631 : 634 : 030 : 74 cc-pCVTZ: 20% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 110 : 262 : 370 : 50 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 090 : 252 : 150 : 47 40% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 240 : 320 : 920 : 34 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 230 : 310 : 910 : 33 60% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 290 : 331 : 340 : 43 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 280 : 321 : 340 : 42 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 090 : 110 : 330 : 12 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 090 : 110 : 330 : 12 100% a 0 : 190 : 261 : 040 : 280 : 150 : 250 : 860 : 27 cc-pCVQZ: 20% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 210 : 250 : 710 : 23 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 210 : 250 : 700 : 22 40% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 060 : 070 : 330 : 10 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 060 : 080 : 300 : 09 60% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 050 : 090 : 520 : 13 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 040 : 090 : 490 : 12 80%0 : 000 : 010 : 050 : 020 : 000 : 010 : 050 : 02 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 010 : 090 : 630 : 17 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 060 : 100 : 640 : 18 a Relativetoexperiment. Thetablesprovidetherequiredinformationtore-zerotheresultstothoseinthe completebasis,toisolatetheFNOeectfromtheexperimentalvalues.Suchplotswould reachzerodeviationmuchmorerapidlyasthetablesshow. Evenmoresensitivetoelectronstructuremethodthangeometriesarevibrational frequencies.Tables2-9and2-10showthedataforvibrationalfrequenciesascomparedto theuntruncatedbasissetresultsforaug-cc-pVDZandaug-cc-pVTZbasissets.Forthe closed-shellmoleculesusedtocalculatetheaveragesinTable2-9,themeanabsoluteerrors 88

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Table2-7.Comparisonofoptimizedequilibriumbondanglesfordierent correlation-consistentbasissets[15,113]formultipleFNOtruncationsfor CCSDTandCCSDT.Thepercentageindicateswhatpercentofthevirtual spaceofeachmoleculewasactive.Fortruncatedbasissets% )]TJ/F15 11.9552 Tf 11.956 0 Td [(80%errors arerelativetotheuntruncatedbasissetresult;for100%,errorsarerelativeto experiment.Averageswerecalculatedoverthesetofmoleculesfrom[50].Only valenceelectronswerecorrelated. isthesignedmeanerror, abs isthe meanabsoluteerror, max isthemaximumabsoluteerror,and std isthe standarddeviation.Allnumbersareinunitsofdegrees. BasisSet CCSDTCCSDT abs max std abs max std cc-pVDZ: 20%0 : 180 : 340 : 660 : 360 : 200 : 290 : 640 : 30 40%0 : 640 : 771 : 360 : 610 : 620 : 761 : 360 : 63 60%0 : 530 : 601 : 000 : 460 : 510 : 580 : 960 : 46 80%0 : 130 : 170 : 490 : 210 : 120 : 160 : 470 : 21 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 991 : 994 : 971 : 53 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 971 : 974 : 921 : 52 cc-pVTZ: 20%0 : 100 : 360 : 890 : 460 : 090 : 340 : 850 : 45 40%0 : 010 : 190 : 470 : 270 : 010 : 180 : 470 : 27 60%0 : 120 : 190 : 410 : 220 : 120 : 190 : 410 : 22 80%0 : 070 : 100 : 270 : 110 : 070 : 100 : 280 : 11 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 900 : 914 : 261 : 31 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 880 : 894 : 201 : 29 cc-pVQZ: 20% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030 : 310 : 680 : 40 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030 : 300 : 650 : 40 40%0 : 080 : 180 : 400 : 230 : 080 : 180 : 410 : 23 60%0 : 040 : 070 : 140 : 080 : 040 : 070 : 130 : 08 80%0 : 010 : 020 : 090 : 030 : 010 : 020 : 090 : 03 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 690 : 693 : 901 : 23 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 680 : 683 : 831 : 21 a Relativetoexperiment. areacceptableforbasissettruncationsof40%ormore,withmeanerrorsof10cm )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 or less.Thisstandsinstarkcontrasttotheopen-shellresultsinTable2-10,where80%of thegivenbasissetsarerequiredtoreproducetheuntruncatedresults.Fortheopen-shell molecules,weuseUHFreferencefunctionsbecausewehavenotyetimplementedFNO gradientsforROHFreferencefunctions.Forcyanideradical,itisknownthatanROHF referencefunctionprovidessignicantlybetterresultsthanUHF[143]forperturbation theory,whichmaybeskewingtheaverages.However,eventheNH 2 resultsshowmore 89

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Table2-8.Comparisonofoptimizedequilibriumbondanglesfordierent correlation-consistentbasissets[15,113]formultipleFNOtruncationsfor CCSDTandCCSDT.Thepercentageindicateswhatpercentofthevirtual spaceofeachmoleculewasactive.Fortruncatedbasissets% )]TJ/F15 11.9552 Tf 11.956 0 Td [(80%errors arerelativetotheuntruncatedbasissetresult;for100%,errorsarerelativeto experiment.Averageswerecalculatedoverthesetofmoleculesfrom[50].All electronswerecorrelated. isthesignedmeanerror, abs isthemean absoluteerror, max isthemaximumabsoluteerror,and std isthestandard deviation.Allnumbersareinunitsofdegrees. BasisSet CCSDTCCSDT abs max std abs max std cc-pCVDZ: 20%0 : 390 : 401 : 060 : 410 : 380 : 381 : 010 : 39 40%0 : 430 : 611 : 260 : 650 : 420 : 601 : 250 : 65 60%0 : 370 : 370 : 650 : 180 : 370 : 370 : 660 : 18 80%0 : 100 : 240 : 990 : 410 : 100 : 240 : 990 : 41 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 991 : 994 : 971 : 53 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 981 : 984 : 931 : 52 cc-pCVTZ: 20%0 : 170 : 290 : 720 : 340 : 160 : 290 : 730 : 34 40%0 : 380 : 491 : 020 : 460 : 380 : 501 : 020 : 47 60%0 : 320 : 371 : 280 : 440 : 320 : 371 : 280 : 44 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030 : 200 : 530 : 26 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 020 : 200 : 520 : 26 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 011 : 014 : 261 : 28 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 001 : 004 : 201 : 26 cc-pCVQZ: 20%0 : 020 : 280 : 630 : 370 : 010 : 270 : 620 : 36 40%0 : 090 : 160 : 390 : 210 : 090 : 160 : 390 : 21 60%0 : 070 : 110 : 280 : 120 : 070 : 100 : 270 : 12 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 010 : 010 : 040 : 02 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 020 : 030 : 100 : 04 100% a )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 700 : 703 : 921 : 23 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 690 : 693 : 841 : 21 a Relativetoexperiment. dependencethantheclosed-shellmolecules.ItispossiblethattheUHFreferencefunction isthesourceofthisdiscrepancy,butfurtherworkisnecessarytopositivelyidentifythe sourceoftheproblem. DeviationsfromexperimentforthesesetsofmoleculesareshowninFigures2-8 and2-9.Moresothanthegeometries,thedeviationsinthevibrationalfrequenciesare non-uniform,withdierentpercentagesexhibitingradicallydierentagreementswith experiment.Intheopen-shellset,whatisimmediatelyclearisthattheresultsagree 90

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Figure2-4.Meandeviationfromexperimentforbondlengthsinpmfortheequilibrium geometriesofthesetofmoleculesfrom[50]asafunctionof correlation-consistentvalencebasissetandFNOtruncationforCCSDTand CCSDT.Thehorizontalaxisistheaveragenumberofvirtualbasis functionsasapercentageofthevirtualspaceofthelargestbasis,cc-pVQZ. Onlyvalenceorbitalswerecorrelated. muchmorepoorlyatallbasissetssizeswithexperimentthantheclosed-shellset. Onesurprisingfeatureoftheopen-shellgureisthattheaugmenteddoublebasisset resultsaresignicantlybetterthanthetripleresults.ThisbehaviorholdsforallFNO truncationsmaintainingmorethan40%ofthebasisset. Theseresultsaremorediculttointerpretthanthoseforgeometriesandenergies. Especiallywhencomparedtoexperiment,theresultsaremuchlessuniformandshow moredependenceonthedegreeofFNOtruncationthanotherproperties.Thisfactshould notbesurprising;ahessiandependsmorestronglyontheenergydierencesaroundthe equilibriumstructurethandoesarstderivative.Anoteofcaution:itispossiblefor 91

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Figure2-5.Meandeviationfromexperimentforbondlengthsinpmfortheequilibrium geometriesofthesetofmoleculesfrom[50]asafunctionof correlation-consistentcore-valencebasissetandFNOtruncationforCCSDT andCCSDT.Thehorizontalaxisistheaveragenumberofvirtualbasis functionsasapercentageofthevirtualspaceofthelargestbasis,cc-pCVQZ. Allelectronswerecorrelated. theFNOproceduretoshowdiscontinuitiesinthevibrationalfrequencies.Wedidnot seethisapppearintheresultsforthesetofmoleculesusedhere,butinothercasessmall changesinthetruncationlevelcanleadtolargerchangesinthevibrationalfrequencies. Thisdependenceillustratestheproblemoflocalsmoothnessaroundanygivenpointonce theFNOprocedurehasbeenapplied.Wearecurrentlylookingmorecloselyatthese issuesinanattempttoprovidelesstruncationdependentvibrationalfrequenciesforall molecules.AtomicnaturalorbitalANObasissets[152,153]havebeenproposedtobe betterforvibrationalfrequenciesthancorrelation-consistentones;thisissuewillneedto beaddressedinfurtherwork. 92

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Figure2-6.Meandeviationfromexperimentforbondanglesindegreesforthe equilibriumgeometriesofthesetofmoleculesfrom[50]asafunctionof correlation-consistentvalencebasissetandFNOtruncationforCCSDTand CCSDT.Thehorizontalaxisistheaveragenumberofvirtualbasis functionsasapercentageofthevirtualspaceofthelargestbasis,cc-pVQZ. Onlyvalenceorbitalswerecorrelated. 2.4.2EnergeticMaterials Threeenergeticmaterialswereaddressedinthisstudy:dimethylnitramineboththe monomeranddimerRDX,andnitroethane. Dimethylnitramine :TheFNOmethodwasusedtomeasuretherelativeenergy ofthetwoDMNAisomersthathavebeenpredicted/observed.Thecalculationwas performedinbothaDZPandcc-pVTZbasisset,with60%ofthevirtualspaceretained. Forcomparisonandverication,calculationswerealsodoneinthefullDZPandcc-pVTZ basissets.TheresultsaresummarizedinTable2-11. 93

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Figure2-7.Meandeviationfromexperimentforbondanglesindegreesforthe equilibriumgeometriesofthesetofmoleculesfrom[50]asafunctionof correlation-consistentcore-valencebasissetandFNOtruncationforCCSDT andCCSDT.Thehorizontalaxisistheaveragenumberofvirtualbasis functionsasapercentageofthevirtualspaceofthelargestbasis,cc-pCVQZ. Allelectronswerecorrelated. Theagreementbetweenthefullcc-pVTZandtruncatedcc-pVTZenergydierences isexcellent.Also,notethedierenceinresultsattheDZPlevelbetweenincluding andnotincludingtheMBPTcorrectiondenedinthecaption.Webelievethat becausetherearesomanyfewervirtualorbitalsinaDZPbasisthaninacc-pVTZbasisa greaterpercentageofthosetruncatedcontainasignicantoccupation.Becauseofthis,a correctionduetothetruncationlevelisnecessaryforasmallbasisset,butnotforalarger basisset. WealsoappliedtheFNOstoinvestigatethedimerofDMNA.Priorworkhad determined,usingsymmetry-adaptedperturbationtheory,veminimastructuresforrigid 94

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Table2-9.Comparisonofharmonicvibrationalfrequenciesfortheselectedclosed-shell moleculesH 2 O[144],NH 3 [145],H 2 CO[146,147],andC 2 H 4 [148]at equilibriumwithdierentaugmentedcorrelation-consistentbasissets [15,113,149]formultipleFNOtruncationsforCCSDTandCCSDT.The percentageindicateswhatpercentofthevirtualspaceofeachmoleculewas active.Fortruncatedbasissets0% )]TJ/F15 11.9552 Tf 11.955 0 Td [(80%errorsarerelativetothe untruncatedbasissetresult;for100%,errorsarerelativetoexperiment.Only valenceelectronswerecorrelated. isthesignedmeanerror, abs isthe meanabsoluteerror, max isthemaximumabsoluteerror,and std isthe standarddeviation.Allnumbersareinunitsofcm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 BasisSet CCSDTCCSDT abs max std abs max std aug-cc-pVDZ: 20% )]TJ/F15 11.9552 Tf 9.299 0 Td [(11278637 )]TJ/F15 11.9552 Tf 9.299 0 Td [(10278637 40% )]TJ/F15 11.9552 Tf 9.299 0 Td [(8103710 )]TJ/F15 11.9552 Tf 9.299 0 Td [(8103610 60% )]TJ/F15 11.9552 Tf 9.299 0 Td [(67348 )]TJ/F15 11.9552 Tf 9.299 0 Td [(57348 80% )]TJ/F15 11.9552 Tf 9.298 0 Td [(1141 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1141 100% a 1826702717257027 aug-cc-pVTZ: 20% )]TJ/F15 11.9552 Tf 9.299 0 Td [(11185422 )]TJ/F15 11.9552 Tf 9.299 0 Td [(10185322 40% )]TJ/F15 11.9552 Tf 9.299 0 Td [(68208 )]TJ/F15 11.9552 Tf 9.299 0 Td [(68207 60% )]TJ/F15 11.9552 Tf 9.299 0 Td [(494815 )]TJ/F15 11.9552 Tf 9.299 0 Td [(494815 80% 0151 0151 100% a 41849253174825 a Relativetoexperiment. geometry C 2v monomers[115].Usingtheirgeometries,wecalculatedinteractionenergies attheCCSDTleveloftheory,comparisontotheSAPTresultsareshowninTable2-12. WhilethedierenceininteractionenergiesbetweenSAPTandCCSDTisonlyon theorderofafewkcal/mol,thereisaqualitativedierencebetweenthetworesults.The energiesofminimaM2andM3areintheoppositeorderintheCCSDTresultcompared tothesymmetry-adaptedperturbationtheorySAPTresults.Determiningthecorrect energeticorderingwillrequirevalidationoftheFNOmethodforinteractionenergies,full geometryoptimizationofthedimersattheFNOlevel,largerbasissets,considerationof basissetsuperpositionerror;andmayrequireinclusionofiterativetripleexcitations. 95

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Table2-10.Comparisonofharmonicvibrationalfrequenciesfortheselectedopen-shell radicalsCN[150]andNH 2 [151]atequilibriumwithdierentaugmented correlation-consistentbasissets[15,113,149]formultipleFNOtruncationsfor CCSDTandCCSDT.Thepercentageindicateswhatpercentofthe virtualspaceofeachmoleculewasactive.Fortruncatedbasissets % )]TJ/F15 11.9552 Tf 11.955 0 Td [(80%errorsarerelativetotheuntruncatedbasissetresult;for100%, errorsarerelativetoexperiment.Onlyvalenceelectronswerecorrelated. is thesignedmeanerror, abs isthemeanabsoluteerror, max isthe maximumabsoluteerror,and std isthestandarddeviation.Allnumbersare inunitsofcm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 BasisSet CCSDTCCSDT abs max std abs max std aug-cc-pVDZ: 20% )]TJ/F15 11.9552 Tf 9.299 0 Td [(9999280122 )]TJ/F15 11.9552 Tf 9.299 0 Td [(105105309137 40% )]TJ/F15 11.9552 Tf 9.299 0 Td [(303010148 )]TJ/F15 11.9552 Tf 9.299 0 Td [(27279143 60% )]TJ/F15 11.9552 Tf 9.299 0 Td [(16163110 )]TJ/F15 11.9552 Tf 9.299 0 Td [(17173412 80% )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(383 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 )]TJ/F15 11.9552 Tf 9.298 0 Td [(121 100% a )]TJ/F15 11.9552 Tf 9.299 0 Td [(696912152 )]TJ/F15 11.9552 Tf 9.299 0 Td [(767612244 aug-cc-pVTZ: 20% )]TJ/F15 11.9552 Tf 9.299 0 Td [(364712663 )]TJ/F15 11.9552 Tf 9.299 0 Td [(22347641 40% )]TJ/F15 11.9552 Tf 9.299 0 Td [(17174520 )]TJ/F15 11.9552 Tf 9.299 0 Td [(19205827 60% )]TJ/F15 11.9552 Tf 9.299 0 Td [(12123012 )]TJ/F15 11.9552 Tf 9.299 0 Td [(13133315 80% 0364 )]TJ/F15 11.9552 Tf 9.299 0 Td [(883015 100% a )]TJ/F15 11.9552 Tf 9.299 0 Td [(10510517152 )]TJ/F15 11.9552 Tf 9.299 0 Td [(11411417351 a Relativetoexperiment. Table2-11.EnergydierencesbetweenC 2v andC s conformersofDMNAusingRHF CCSDTinvariousFNOtruncatedbasissets.MBPTisthedierence betweentheuntruncatedandFNOtruncatedMBPTcalculations.Only valenceelectronsarecorrelated.TotalenergiesareinE h andrelativeenergies areinkcal/mol. BasisSet TotalEnergy C 2v )]TJ/F21 11.9552 Tf 11.955 0 Td [(C s Timehr C 2v C s 100%DZP )]TJ/F15 11.9552 Tf 9.299 0 Td [(338 : 816 )]TJ/F15 11.9552 Tf 9.298 0 Td [(338 : 8309 : 091 : 61 60%DZP )]TJ/F15 11.9552 Tf 9.299 0 Td [(338 : 755 )]TJ/F15 11.9552 Tf 9.298 0 Td [(338 : 7677 : 660 : 25 60%DZP+MBPT )]TJ/F15 11.9552 Tf 9.299 0 Td [(338 : 808 )]TJ/F15 11.9552 Tf 9.298 0 Td [(338 : 8228 : 900 : 25 100%cc-pVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(339 : 125 )]TJ/F15 11.9552 Tf 9.298 0 Td [(339 : 1409 : 2452 : 6 a 60%cc-pVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(339 : 103 )]TJ/F15 11.9552 Tf 9.298 0 Td [(339 : 1189 : 298 : 80 60%cc-pVTZ+MBPT )]TJ/F15 11.9552 Tf 9.299 0 Td [(339 : 128 )]TJ/F15 11.9552 Tf 9.298 0 Td [(339 : 1439 : 298 : 80 a Estimated. 96

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Figure2-8.Meandeviationfromexperimentforharmonicvibrationalfrequenciesin cm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 fortheequilibriumgeometriesoftheclosed-shellmoleculesH 2 O,NH 3 H 2 CO,andC 2 H 4 asafunctionofaugmentedcorrelation-consistentvalence basissetandFNOtruncationforCCSDTandCCSDT.Thehorizontal axisistheaveragenumberofvirtualbasisfunctionsasapercentageofthe virtualspaceofthelargestbasis,aug-cc-pVTZ.Onlyvalenceorbitalsare correlated. Table2-12.Interactionenergieskcal/molforDMNAdimerconformationsusingFNO RHFCCSDTatSAPTgeometries[115].Allcalculationswereperformedin aDZPbasiswith60%ofthevirtualorbitalskept.Onlyvalenceelectronswere correlated. Minimum a SAPTCCSDT M1 )]TJ/F15 11.9552 Tf 9.298 0 Td [(11 : 056 )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 : 355 M2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 934 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 821 M3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 169 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 602 M4 )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 : 855 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 379 a Geometriesfrom[115] 97

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Figure2-9.Meandeviationfromexperimentforharmonicvibrationalfrequenciesin cm )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 fortheequilibriumgeometriesoftheopen-shellmoleculesCNandNH 2 asafunctionofaugmentedcorrelation-consistentvalencebasissetandFNO truncationforCCSDTandCCSDT.Thehorizontalaxisistheaverage numberofvirtualbasisfunctionsasapercentageofthevirtualspaceofthe largestbasis,aug-cc-pVTZ.Onlyvalenceorbitalsarecorrelated. Table2-13.EnergydierencebetweenchairandboatconformersofRDXusingfrozencore RHFCCSDTwithanFNO60%DZPbasis.Geometriesarefrom[118]. TotalenergiesareinE h ,relativeenergiesareinkcal/mol. GeometryCCSDT CCSDTTimehr +MBPTFNObasisFullBasis a AAAChair )]TJ/F15 11.9552 Tf 9.298 0 Td [(895 : 1732 )]TJ/F15 11.9552 Tf 9.298 0 Td [(895 : 331130200 AAABoat )]TJ/F15 11.9552 Tf 9.298 0 Td [(895 : 1728 )]TJ/F15 11.9552 Tf 9.298 0 Td [(895 : 32891451000 Chair-Boatkcal/mol )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 25 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 41751200 a Estimated. 98

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1,3,5-Trinitrohexahydro-1,3,5-triazineRDX :Chakraborty,etal.[118]gave twominimastructuresforRDX:theAAAchairandtheAAAboat.Theyperformed B3LYPcalculationsontheseminima,andfoundthatatthatleveloftheory,theAAA chairwas0 : 75kcal/mollowerinenergythantheAAAboat.Theyalsofoundby performingavibrationalanalysisthattheAAAboatwasnotaminimumforB3LYP. WeusedtheirgeometriesforbothconformersandperformedafrozencoreRHFCCSDT calculationina60%DZPFNObasis.Table2-13liststheresults. Theenergydierencebetweenthesetwogeometriesdeterminedby60%DZP CCSDTis )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 4kcal/mol,onthesameorderastheB3LYPresult )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 75kcal/mol. Furthercalculationsinatruncatedtriplezetabasisandwithfullgeometryoptimization arenecessarytodenitivelydeterminewhichconformationislowerinenergy. Nitroethane :Thedecompositionofnitroethanecanoccurviaseveraldierent pathways[126].SchematicsofthepossiblereactionpathsareshowninFigures2-10 to2-12.Thenumberingoftransitionstatesandintermediatescorrespondstothatused in[126].Tosortouttherelativeimportanceofeachoftheindividualpathways,allofthe relevantspeciesareoptimizedusingFNOCCSDTandFNOCCSDT.Inthemain pathways,thereisonereactantnitroethane,veintermediates,tentransitionstates, andatotaloftwelveproducts.Eachofthese28criticalpointsarefullyoptimizedina cc-pVTZbasissetwith60%ofthevirtualspacekeptusingFNOs.Thisbasissetboth performswellinthecalibrationtestsandissmallenoughtoallowthecalculationstobe completedusingourcomputationalresources.Calculationswereperformedbothlocally, onourSGIAltix,aswellasatDepartmentofDefenseMajorSharedResourceCenters. Thecoreoccupiedorbitalsandcorrespondingcorevirtualorbitalsaredroppedaswell. Fornitroethaneanditsisomers,thisyieldsatotalof15activeoccupiedorbitalsand 117activevirtualorbitals.Theexpectedsavingspergeometryoptimizationstepofeach criticalpoint,ascomparedtoafullbasissetcalculation,isapproximately75%.RHF referencesareusedforclosed-shellspecies,foropen-shellspeciesUHFreferencesareused. 99

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Figure2-10.Schematicoftheone-stepHONOeliminationanddirectssionpathwaysfor decompositionofnitroethane.TheverticalaxismeasurestheZPEcorrected energiesinkcal/molrelativetonitroethanefromCCSDTcalculations with60%ofthevirtualspaceofthecc-pVTZbasissetretainedviaFNOs. Attheoptimizedcriticalpoints,nite-dierencehessiansarecalculatedtoverifythatthe geometriesdo,infact,correspondtoeitherminimaorrst-ordertransitionstates,aswell astodeterminethevibrationalfrequencies,allowingzero-pointenergycorrectionstobe included. Thedecompositionofnitroethanecanbebrokenintofourmainclassesofpathways: directssionofnitroethanetoformethylradicalandnitrogendioxideFigure2-10, single-stepeliminationofHONOFigure2-10,isomerizationtoethylnitritedenoted INT3andsubsequentdecompositionFigure2-11,andisomerizationtoethylhydroxy nitroxide[CH3CHNOHO]denotedINT5andthenfurtherdecompositionFigure2-12. 100

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Figure2-11.Schematicofthedecompositionpathwayofnitroethanethroughisomerization toethylnitrite.TheverticalaxismeasurestheZPEcorrectedenergiesin kcal/molrelativetonitroethanefromCCSDTcalculationswith60%of thevirtualspaceofthecc-pVTZbasissetretainedviaFNOs. Intheguresmentioned,wehaveusedthenotationfrom[126]fortheintermediates, transitionstatesandsomeproductsP4andP8aretwocyclicisomersofnitroethane. Whencomparedtonitromethane,analogiesofeachofthesepathwaysexists{except fortheHONOelimination.Forthesetofpathwaysbeginningwithisomerization toethylnitrite,wefocusonthemechanismthatyieldsthelowestenergyproducts CH 3 CHO+HNO.Forisomerizationthroughethylhydroxynitroxide,wechoosetofocuson thethermodynamicallyminimumsetofproducts,CH 3 CNO+H 2 O{eliminationofwater. Toprovideanestimateoftheimportanceofthesedierentpaths,inFigure2-13weplota qualitativepictureoftheirrelativeenergies. 101

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Figure2-12.Schematicofthedecompositionpathwayofnitroethanethroughisomerization toethylhydroxynitroxide.TheverticalaxismeasurestheZPEcorrected energiesinkcal/molrelativetonitroethanefromCCSDTcalculations with60%ofthevirtualspaceofthecc-pVTZbasissetretainedviaFNOs. Table2-14comparestheresultsfromB3LYPina6-311+Gdf,2pbasisandthe 60%FNOcalculationswithCCSDTandCCSDTattheirrespectiveoptimized geometries.FocusingrstontheB3LYPresultsfrom[126],theenergydierencesbetween thedierentpathwaysarerelativelysmall.Toappropriatelymodelthekineticsofthe decompositionofthesereactions,itisimportantthatthestationarypointenergiesare convergedwithrespecttoelectronicstructure{smallchangesinbarrierheightscanlead tolargedierencesinkinetics. Beforeconsideringthedierencesbetweenthecoupled-clusterresultsandthosefrom DFT,notethattheresultsforCCSDTandCCSDTagreeclosely,withminimal 102

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Figure2-13.Schematicofthemostimportantpathwaysforeachpossibleisomerizationfor thedecompositionofnitroethane.TheverticalaxismeasurestheZPE correctedenergiesinkcal/molrelativetonitroethanefromCCSDT calculationswith60%ofthevirtualspaceofthecc-pVTZbasissetretained viaFNOs. changesinenergyorderingofthedierentspecies,despitethefactthatCCSDTdoes muchbetterforRHF-basedCCbondbreaking.Becauseofthissimilarity,wewillsimply refertotheCCresultswhencomparingagainstB3LYPratherthanchoosingoneor other.Qualitatively,theresultsfromCCandDFTseemtoagreequitewell;products andintermediatesareorderedthesameinCCandDFT,andtransitionstatesarenot radicallyrearranged.Asisnotedin[126],B3LYPtendstounderestimateenergybarriers; ourcoupled-clusterresultssupportthisconclusion,asthemajorityofthetransition statesweredeterminedtobehigherinenergythanpredictedbyDFT.Theshiftsarenot uniform,however,leadingtoareorderingofseveralofthehigh-lyingtransitionstates. 103

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Thelower-lyingtransitionstateswereleftunchangedinorder,leadingtothesame conclusionsaboutthekineticallyfavoredchannel.Thetransitionstatefortheelimination ofHONOviaaconcertedreactionhasthelowestbarrierby10kcal/molinB3LYP,and by8kcal/molforbothCCSDTandCCSDT.Theconcertednatureofthistransition statemightraiseconcernabouttheapplicabilityoftheperturbativeCCSDTmethod, whichfailsforRHF-basedbondbreaking,butrecentwork[123]shows,surprisingly, thatCCSDTandCCSDTwhichamelioratestheRHFfailuretendtoreproduce transitionstateswithequalaccuracy. FromtheB3LYPcalculations,theeliminationofwateristhemost thermodynamicallystableproductbymorethan16kcal/mol.Ontheotherhand,the coupled-clustercalculationspredictanenergygapbetweentheeliminationofwaterand theeliminationofHNOofonly6 : 8kcal/mol[CCSDT]or6 : 3kcal/mol[CCSDT]. TheeliminationofwaterisexoenergeticinB3LYPbymorethan7 : 5kcal/mol,while itisendoenergeticby1 : 5kcal/molbybothCCmethods.Whencomparingtothe energiesoftheintermediates,theglobalminimumontheCCpotentialenergysurface isnow1,1-nitrosoethanolINT7andethylnitriteINT3isslightlylowerinenergythan nitroethane.Thecoupled-clustercalculationsalsosuggestthattheeliminationofHNOis lessfavorablekinetically,asthebarriersalongthereactionpathwayarehigherrelativeto thosefromB3LYP. 2.5Conclusion Theapplicationofmethodsthatreducebasissetsizewillalwaysbelimitedunless analyticalgradientsareavailable.FormethodssuchasFNO-CC,wherethebasisset reductionisbasedonanauxiliarycalculationforthemoleculeataparticulargeometry, theinclusionoforbitalrelaxationtermsissubstantiallymorecomplicatedthanitis formoretraditionalmethodsthatsimplymodifytheorbitaleigenvalueequations.In ourcase,becauseofthedependenceonaMBPTdensitymatrix,thereisanorbital relaxationcontributiontothetwo-particledensitymatrixthatisnew.Becauseofthe 104

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Table2-14.Relativeenergiesinkcal/molofimportantstationarypointsforthe decompositionofnitroethane.Allspeciesareattheirappropriatelyoptimized structuresandenergiesarerelativetothatofnitroethaneincludingzero-point energycorrections.TheB3LYPDFTresultsarefrom[126]andusethe 6-311+Gdf,2pbasis.ResultsforCCSDTandCCSDTarefromthis workusingacc-pVTZbasissetwith60%ofthevirtualorbitalskeptbythe FNOprocedure.Onlyvalenceelectronswerecorrelated.Specieslabels correspondtothoseinFigures2-10to2-12. SpeciesB3LYPCCSDTCCSDT TransitionStates: TS542 : 1148 : 2948 : 32 TS659 : 4064 : 8364 : 94 TS835 : 3538 : 0738 : 46 TS952 : 5057 : 6257 : 72 TS1063 : 0860 : 1560 : 81 TS1164 : 7467 : 2767 : 42 TS1255 : 9367 : 1068 : 32 TS1361 : 4160 : 4863 : 30 TS1431 : 8331 : 0731 : 31 TS1570 : 4166 : 2068 : 26 Intermediates: INT31 : 60 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 12 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 27 INT59 : 6414 : 6614 : 61 INT7 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 54 )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 : 32 )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 : 54 Products: CH 3 CH 2 +NO 2 52 : 3257 : 1256 : 95 CH 2 CH 2 +HONO15 : 6218 : 3518 : 26 CH 3 CHO+HNO8 : 625 : 264 : 98 CH 3 CNO+H 2 O )]TJ/F15 11.9552 Tf 9.298 0 Td [(7 : 721 : 571 : 34 CH 3 CH 2 O+NO36 : 2234 : 5334 : 00 P454 : 4053 : 8854 : 61 P824 : 4721 : 9921 : 90 105

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one-andtwo-particlenatureofalltheinteractionsintheHamiltonian,themostgeneral suchtruncationprocedureshouldonlycontributeorbitalrelaxationeectstobothdensity matrices. Despitethecomplexityoftheorbitalrelaxationterms,weareabletoshowthatjust asinthecaseforHartree-Fockorbitals,onecanseparatetheperturbation-dependent integralderivativesfromtheperturbation-independentorbitalrelaxation.Therefore,one needstosolvetheCPFNOequationsorequivalently,thez-vectorequationsonceinstead offoreachperturbation.ThentheCCresultsfollowwithsubstantialsavingsintime thatcanapproachanorderofmagnitude,dependinguponthelevelofCCcorrelation. Unfortunately,thepricepaidforthiscomputationalsavingistheneedtostoreseveral quantitiesofthedimensionoftwo-electronintegrals.Propercombinationofthetermsin anintegral-directformalismmaybeabletocircumventthatcomplication. TheFNOprocedureinitiatestheoptimizedvirtualspaceOVOSmethod[90,91, 94,95],whichimposestheadditionalconstraintoftryingtoobtainthelowestMBPT energy[90]ormaximizestheoverlapbetweenthetruncatedanduntruncatedMBPT wavefunctions[95,154].ThisconstraintcaneasilybeaddedtotheanalyticalFNO gradientprocedurepresentedheretoenableOVOSstructuresandhessianstobeobtained analytically.Infact,theOVOSmethodisanexampleofthegeneralissueofimposing additionalconditionsonavirtualspacetofullladesiredobjective. ApplicationoftheFNOtruncationmethodstothetestsetofmoleculesshowed thatwhileacc-pVDZorcc-pCVDZbasisisinadequatetobepredictiveforgeometries, atruncatedcc-pVTZbasisofthesamenumberofactiveorbitalsissubstantiallybetter. Itisalwayspreferabletousethelargestpossiblebasissetandthenreduceitseective virtualorbitalspacedimensionviatheFNOmethodthantocompromiseonthesizeof theunderlyingbasisset.Resultsforvibrationalfrequenciesaremoremixed,withoutthe clearpreferenceforFNOtruncationsoveruntruncatedsmallerbasissets.Thisconclusion maypartlybeduetothelimitedsetofmoleculesstudied,butitalsosuggeststhatsome 106

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furtherdevelopmentofthepropertreatmentofvibrationalfrequencieswithinFNO-CC maybeneeded. OurresultssupportthegeneralconclusionsreachedbyDenis,etal.,[126]about thedecompositionofnitroethane.Theone-stepeliminationofHONOappearstobe favoredkinetically,withthebarrierforthatreaction8kcal/mollowerthanthatforthe directbondssion.However,thereareimportantdierencesintheenergetics,withthe energiesalongthepathwaysinitiatedbytheisomerizationtoethylnitritebeingmost aected.ThegapbetweenthethermodynamicallyfavoredproductsCH 3 CNO+H 2 O andCH 3 CHO+HNOisreducedtoroughly6kcal/mol,versus16kcal/molfromB3LYP calculations. 107

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CHAPTER3 GENERALIZEDCOUPLED-CLUSTERSINGLES,DOUBLESANDPERTURBATIVE TRIPLESMETHODS PortionsofthischapterareexcerptedandadaptedwithpermissionfromA.G.Taube andR.J.Bartlett,J.Chem.Phys., 128 ,04411008[129] c 2008,American InstituteofPhysicsandA.G.TaubeandR.J.Bartlett,J.Chem.Phys., 128 ,044111 [123] c 2008,AmericanInstituteofPhysics. 3.1Bond-Breaking 3.1.1Introduction Unfortunately,spin-restrictedHartree-FockRHFbasedCCSDTfailssignicantly awayfromequilibriumstructuresbecauseofitsperturbativeinclusionoftriples:the energiesproducedarefaruptohundredsofkcal/molbelowthefullconguration interactionFCIenergy.Onepossiblesolutionistousespin-andspatial-symmetry unrestrictedHartree-FockUHFbasedCCSDT.Thischoiceusuallyxestheasymptotic breakdownofCCSDT,however,italsoleadstolargererrorsintheintermediatespin re-coupling"regime.Thesefailuresarenotidleproblems;ifonewouldliketoperform moleculardynamicsorMonteCarlocalculationsofasystem,one'senergymodelmustbe accurateforarbitrarygeometries,notsolelygeometriesnearequilibria. Giventhepotentialrangeofapplications,itisnotsurprisingthattherehavebeen severalattemptstoimproveuponCCSDTforthesesituations.Asofyet,nonehas reachedthewide-spreadacceptanceofCCSDTitself.Thisfactispartlyduetothe stringentrequirementthatanynewmethodmustnotdisruptthesuccessesofCCSDT. TherehavebeenmayattemptstoimproveuponCCSDT,startingwithCCSDT introducedbyKucharskiandBartlett[127],lateralsostudiedbyCrawfordandStanton [128].Other,lateronesincludethecompletelyrenormalizedmethodsofPiecuch,et. al.,[155]andmethodsbasedonLowdinpartitioningofthecoupled-clustereective Hamiltonian,suchasthoseintroducedbyHirata,et.al.,[156]andbyGwaltneyand Head-Gordon[157]. 108

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ThecompletelyrenormalizedCCSDT[CR-CCSDT]methodintroducesanoverlap inthedenominatortothetriplescorrectionthat,whilecorrectingtheasymptoticbehavior ofCCSDT,introducessignicanterrornearequilibrium.Also,becauseoftheglobal natureofthedenominator,CR-CCSDTisnotextensive,violatingtherationaleforall CCmethods.Recentwork[158{161]hasreformulatedthesemethodsbyutilizingthe coupled-clusterleft-handeigenoperator, L .Thoughthisnewerversionismuchcloserto theearlier L CCSDTmentionedabove,unfortunatelyitalsomakesthetriples correctionorbitalrotationdependent. TheCCSDmethodofGwaltney[157]andtheCCSD T methodofHirata[156] bothderivefrompartitioningthecoupled-clustereectiveHamiltonianandkeepingthe lowestordercorrectionstotheenergyinperturbationtheory.Inbothcases,thecorrection isextensiveandorbitallyinvariant,andbothmethodssignicantlyimproveCCSDTat longbonddistances.Inspiteofthesesuccesses,theresultsnearequilibriumareworse thanconventionalCCSDT,andbothmethodsrequireadditionalapproximationsto keepthecomputationalscaling O n 3 N 4 .Bothmethodsalsorequiretheentireeective Hamiltonianmatrix,whichisnotasreadilyavailableasthebareHamiltonian,itself. Theactualnumericalimprovementsgainedbytheabovecanbelargelytracedto theolderCCSDT,whichrstintroducedalefthandeigenvector,,bybasingthe non-iterativeTcorrectionontheCCfunctional[127].ThederivationbyCrawfordand Stanton[128]wasfromaLowdinpartitioningviewpointsimilartothejusticationof CCSDTin[162];thoughthepartitioningisimplicitinthefunctional[42].Aswillbe shownbelow,itcanalsobederivedbyperturbationtheoryfromCCSD T .Infact, CCSDTcanbeviewedastheminimalcorrectionbeyondCCSDT,andsinceit onlyrequires,andhastobeobtainedinanyCCanalyticalgradientapplication, itisparticularlyattractive.Unliketherenormalizedapproachesabove,itisrigorously size-extensive,showsoccupied-occupiedandvirtual-virtualorbitalinvariance,anddenes densitymatricesthataretiedtoenergyderivatives.Withoutthelatterconnection 109

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anyarbitraryoperatorcouldbeusedtoleftmultiplytheCCsolution.Italsoimproves CCSDTatlongbonddistanceswithoutdegradingtheequilibriumperformance. Oneoftheareasofchemistrywherecomputationcanplaythegreatestroleisin explainingthereactionsofenergeticmaterials.Thesemoleculesarehigh-energyandtheir decompositiondynamicsaresorapidthatitisdicultforexperimenttocharacterizethe pathwaysthatcontributetodecomposition.Therefore,identifyingandclassifyingpotential stationarypointsonthepotentialenergysurfacesofenergeticmaterialsbecomesatask thatiswell-suitedforquantumchemistry.Oneoftheprototypicalenergeticmaterials is1,3,5-trinitrohexahydro-1,3,5-triazineRDX,andthereisdisputeintheliterature aboutthepossibleimportanceofaconcertedtransitionstateasadecompositionpathway. ResultsfromDFTB3LYP6-31dindicatethattheactivationbarrierisnotthelowest ofthepotentialpathways[118],butitiscloseenoughinenergytootherbarriersthat higherlevelsoftheoryarenecessarytoverifythisconclusion. Becausethistransitionstateisconcerted,onemightexpectthatthereissignicant quasi-degeneratecharacter.Therefore,thequalityofresultthatcouldbeexpected fromCCSDTisindoubt.However,thisisexactlythetypeofproblemthat CCSDTshouldbecapableofhandling,andweshouldbeabletodeterminewhether quasi-degenerateeectsarerelevantforthisparticulartransitionstate,aswellaswhether thetransitionstateisasignicantcontributortothedecompositionofRDX. 3.1.2Theory Derivationof CCSDT :CCSDTisformulatedasanon-iterativecorrection totheconvergedCCSDsolutionthatshouldapproximateCCSDT.ForCC,the wavefunctionisapproximatedbytheCCansatz j i = e T j 0 i {1 where j 0 i isasingle-determinantreferencefunction.Thisfunctionismostoftena Hartree-Fockwavefunction,butthegeneralizedCCSDTadmitsanyreferencefunction 110

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[52].The T operatorisanexcitationoperator,which,forCCSDT,iscomposedof T = T 1 + T 2 + T 3 {2 where,usingthenotationthat i j k areoccupiedorbitals, a b c areunoccupiedvirtual orbitals,and p q r arearbitraryorbitals, T n = 1 n 2 X ijk abc t abc ijk a y ib y jc y k ; {3 where T n isan n -particleexcitationoperator.Thischoiceforwavefunctiondenesthe CCSDTenergyfunctional E CC = h 0 j + H N j 0 i + h 0 j H j 0 i ; {4 withthenon-hermitianeectiveHamiltonian H N denedbythesimilaritytransformation H N = e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T H N e T = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(H N e T C : {5 ofthenormal-orderedbareHamiltonian H N = X pq f pq f p y q g + 1 4 X pqrs h pq jj rs if p y q y sr g = H N + H N ; {6 where f pq areFockmatrixelementsand h pq jj rs i areantisymmetrictwo-electronintegrals, and H N = X ij f ij f i y j g + X ab f ab f a y b g = F oo + F vv {7a H N = X ia f ia )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(f a y i g + f i y a g + 1 4 X pqrs h pq jj rs if p y q y sr g = F ov + W: {7b Thede-excitationoperatoris,forCCSDT, = 1 + 2 + 3 n = 1 n 2 X abc ijk ijk abc i y aj y bk y c : {8 ExpandingtheCCfunctional,Equation3{4,tofourth-ordercontributionsandisolating 111

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[2] 3 and T [2] 3 gives E [4] CCSDT = E [4] CCSD + h 0 j [1] 2 H N T [2] 3 C j 0 i + h 0 j [1] 1 WT [2] 3 C j 0 i + h 0 j [2] 3 H N T [2] 3 C j 0 i + h 0 j [2] 3 WT [1] 2 C j 0 i : {9 Onecaneliminate [2] 3 byusingtheequationfor T [2] 3 H N T [2] 3 C + WT [1] 2 C =0 ; {10 whichleavestherstthreetermsofEquation3{9asthedenitionofCCSDT. Furthermore,thisissimplythegeneralizednon-HFCCSDT[52],exceptfor 1 and 2 replacing T y 1 and T y 2 WecanalsoconsideraderivationofCCSDTviapartitioningtorelateittoother methodsthathavebeensuggested.Itisstraightforwardtoshowthatageneralfunctional canbeformulatedbasedontheCCSDeectiveHamiltonian H SD thatisequivalentto FCI, E exact = h 0 jL H SD Rj 0 i ; {11 where L and R aretheleft-andright-handeigenvectorsofthenon-hermitianCCSD eectiveHamiltoniandiagonalizedinthespaceofalldeterminants,analogoustothe EOM-CCtreatmentofexcitationenergies[163,164].Onecanthendeneaprojectorto thereferenceandthesinglyanddoublyexciteddeterminants, ^ P ^ P = j 0 ih 0 j + X ia j a i ih a i j + 1 4 X ijab j ab ij ih ab ij j {12 anddeneitsorthogonalprojector ^ Q = ^ I )]TJ/F15 11.9552 Tf 15.435 3.022 Td [(^ P .Theseprojectors,inturn,denetwo spaces:thespaceofdeterminantsin ^ P P -space,andthespaceofdeterminantsthatspan ^ Q Q -space.FollowingLowdin[165],theexactenergycanthenbewritten E exact = h 0 jL P H SD R P j 0 i + h 0 jL P H SD ^ Q )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(E exact )]TJ/F15 11.9552 Tf 14.99 3.022 Td [( H SD )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ^ Q H SD R P j 0 i ; {13 112

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where L P = ^ P L ^ P R P = ^ P R ^ P {14 aretheprojectionsoftheexactleft-andright-eigenstatesintothe P -space.Wethen perturbativelyapproximatethesecondtermofEquation3{13.Thepartitioningdenedin Equations3{7inducesaperturbativeorderingoftheeectiveHamiltonian H N H N = H [0] N + H [1] N + H [2] N + {15 ByadditionallydeningthesolutionoftheCCSDequationsinthe P -spacetobeof zerothorder,then H [0] N = ^ P H SD ^ P + ^ Q E CCSD + F oo + F vv ^ Q {16a H [1] N = F ov + W +[ F oo + F vv T 1 ] C +[ F oo + F vv T 2 ] C {16b H [2] N = WT 1 C +[ F ov + W T 2 ] C {16c L [0] P =1+{16d R [0] P =1 : {16e Notethat T 1 isbeingtreatedasarst-orderterm,asitwouldbeinthegeneralnon-HF case.Usingthesedenitions, E [0] = h 0 j + H SD j 0 i = E CCSD {17 andthelowestordernon-vanishingcorrectionto E CCSD is E [3] = E CCSDT = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( 1 3! 2 X ijk abc h 0 j H [1] N j abc ijk ih abc ijk j H N j def lmn i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 h def lmn j H [2] N j 0 i : {18 TheenergyexpressioninEquation3{18isinvariantunderanyrotationwithineitherthe occupiedorthevirtualspace,astheinnite-orderCCSDmethodis.However,tomake theCCSDTnon-iterative,onemustchoosetheFockoperatortobediagonalwithin 113

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theoccupiedandvirtualspaces,thatissemi-canonicalorbitals.Thentheinversein Equation3{18istrivial,andthisexpressioncanbesimpliedto E CCSDT = 1 3! 2 X ijk abc h 0 j F ov + W j abc ijk i 1 abc ijk h abc ijk j WT 2 C j 0 i {19 where abc ijk = f ii + f jj + f kk )]TJ/F21 11.9552 Tf 11.955 0 Td [(f aa )]TJ/F21 11.9552 Tf 11.955 0 Td [(f bb )]TJ/F21 11.9552 Tf 11.955 0 Td [(f cc ; {20 whichisequivalenttoEquation3{9.Ifoneneedstokeeptheorbitalsnon-canonical,e.g. inthecaseoflocalizedorbitals,thenonlyEquation3{18applies,whichmustbesolved iterativelytomaintainorbitalinvariance. Comparisontoothermethods :Asmentionedintheintroduction,there havebeenseveraldierentattemptstoimproveCCSDTsothatitbetterhandles bond-breakingandotherquasi-degeneratesituations.Thesemethodsincludecompletely renormalizedCCSDT[CR-CCSDT],itsrecentextensivemodicationCR-CCSDT L andCCSD T .Allofthesemethodsare O n 3 N 4 non-iterativeandimproveupon CCSDTbond-breakingvideinfra,butnotallsatisfythedesideratathatonewouldlike forageneralpurposemethod. InCR-CCSDT[155],aglobaloverlapdenominatorisintroducedintotheCCSDT energyexpressiontotempertheover-correctionofCCSDTatlongbonddistances. Unfortunately,thepresenceofthisglobaldenominatorimmediatelybreaksextensivityfor thecorrection,makingthemethodmorelikeCIthanCC. ArecentmodicationtoCR-CCSDTistherigorouslyextensiveCR-CCSDT L [158,160,161].Inthismethod,asinCCSDTandCCSD T ,theleft-handCCSD eigenvectorisusedtomoderatetheovercorrectionofCCSDT.TheCR-CCSDT L energycorrectioncanbewritten E CR-CCSDT L = 1 3! 2 X ijk abc h 0 j H SD j abc ijk i 1 abc ijk h abc ijk j H SD j 0 i {21 114

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where abc ijk = E CCSD )-222(h abc ijk j H j abc ijk i : {22 Unlikethebaredenominator abc ijk ,thisnewdenominator abc ijk isnotoccupied-occupied andvirtual-virtualorbitallyinvariant.ItissimilartothechoiceofEpstein-Nesbet partitioningforsecond-orderperturbationtheory.Therefore,duetoarbitraryrotations amongthevirtualorbitals,forexample,theenergycorrectioncouldchange,particularly asafunctionofgeometry,makingitunsuitableforPES.Innite-ordercoupled-cluster methodsareindependentoforbitalrotations,andonewouldpreferthattobethecasefor aperturbativeapproximationaswell,ashasbeenthecaseforthenon-HFgeneralization ofCCSDT[52]forsometime. BoththeCCSD T [156]andCCSD[157]methodsarebasedonLowdin partitioningoftheeectiveHamiltonian.Inbothcases,theeectiveHamiltonianis splitintotwo: H N = H [0] N + H [1] N : {23 ForCCSD T H [0] ischosentobethesameasforCCSDT,whileforCCSD, H [0] is chosentobe ^ P H SD ^ P + ^ Q )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(E CCSD + H 1 ;oo + H 1 ;vv ^ Q {24 where H 1 istheone-particlepartoftheeectiveHamiltonian.Becausetheeective Hamiltonianissplitintoonlytwopiecesratherthanaperturbationseriesasisdonein CCSDT,thecorrectiontoCCSDinbothcasesincludescontributionsfromquadruple excitations.Calculatingquadruplesaddssignicantcomputationalexpensetothese methodscausinga O n 4 N 5 scaling.Toavoidthisexpensetheseapproachesuseadditional approximations.InthecaseofCCSD,theapproximatefactorizationofthequadruples term,introducedbyKucharskiandBartlett[166],isused,whichdropsthecomputational complexityfrom O n 4 N 5 to O n 3 N 4 ,howeverthereisalsoanadditional O N 6 term conventionalCCSDT{andCCSDT{scaleas O n 3 N 4 .Becauseofthisscaling,it 115

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ismorereasonabletocompareCCSDwithmethodssuchasCCSDTQ f thatalsohave a O N 6 termandexplicitlyincludequadrupleexcitations. Ontheotherhand,themostcloselyrelatedmethodtoCCSDTisCCSD T ThismethodwasrstintroducedunderthenameCCPTin[167].Thefullenergy correctionfromtheCCSD T partitioningisactuallyCCSD TQ andincludesatriples andquadruplesterm E CCSD T = 1 3! 2 X ijk abc h 0 j H SD j abc ijk i 1 abc ijk h abc ijk j H SD j 0 i {25a E CCSD Q = 1 4! 2 X ijkl abcd h 0 j H SD j abcd ijkl i 1 abcd ijkl h abcd ijkl j H SD j 0 i : {25b Thequadruplestermisdiscardedtokeepthecomputationalcomplexity O N 7 and, therefore,CCSD T doesnotcorrespondtoawell-denedperturbationorder.However, ifoneperturbativelyexpandsthisenergyusingthepartitioningdenedabove,thenthe lowestnon-vanishingtermistheCCSDTcorrection.Therefore,itisreasonabletocall CCSDTtheminimalextensionofCCSDT. 3.1.3Implementation CCSDThasbeenimplementedwithinACESII[54,55]byaddingtothe capabilitiesofthepre-existingCCSDTcode.Utilizingthedirectproductformulation ofsymmetry,ourimplementationusesrealabelianpointgroupsymmetrytoaccelerate calculations.Wecalculatethecorrectionbyformingforeachsetof f i
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and ijk abc [2] = 1 abc ijk P a=bc j k=ij X d h bc jj dk i ij ad )]TJ/F21 11.9552 Tf 11.955 0 Td [(P c=ab j i=jk X l h lc jj jk i il ab + P i=jk j a=bc i a h jk jj bc i + P i=jk j a=bc f ia jk bc i ; {28 where P p=qr =1 )-222(P pq )-222(P pr P p=qr j s=tu = P p=qr P s=tu {29 with P pq interchangingthelabels p and q .Thetotalenergycorrectioncanthenbe constructedas E CCSDT = X i
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forthesucceedinggeometries.Theenergiesaredisplayedasfunctionsofbondlengthin unitsoftheequilibriumbondlength.ForUHFcalculations,thesymmetryisnotenforced andwefollowallHFinstabilitiestoremaininalocalminimum. Singlebond-breaking :Singlebond-breakingshould,inprinciple,betheeasiest testofbond-breaking.However,asisshowninFigure3-1,RHFCCSDTiscompletely unsuitableforreachingthedissociationlimitofeventheserelativelysimplecases.In thisgure,errorsfromFCI[168]forbond-breakingofHFareshown.Therestricted Hartree-FocksolutionforHFbond-breakingapproachestheionizedH + +F )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(staterather thanH +F .Therefore,correlationisattemptingtobringtheasymptoticbehaviorfrom theionstotheneutrals.NeitherCCSDTorCCSDTissucientinthiscasetobe predictivealongtheentirepotentialenergysurface,however,ratherthanerrorsoftensof kcal/mol,CCSDTstayswithin7kcal/moloftheFCIresultacrosstheentirerangeof bondlengths. Figure3-2isacomparisonoffourofthealternativetriplecorrectionmethodsfor bond-breakinginF 2 inacc-pVDZbasis[113].Thereferenceforthesecalculationsis CCSDT,whichforasinglebondshouldbeanexcellentapproximationtotheFCI. Forallfourmethods,theimprovementoverCCSDTissubstantial,however,there aresignicantdierences.Atequilibrium,CCSDTisthebestmethodandmost closelymatchesCCSDT.ForCR-CCSDT,CCSD T andCCSDT,theerrors fromCCSDT[169]aremorenegativethefurtheralongthepotentialenergysurfaceone goes.Ofthosethree,CCSDTstaysclosesttoCCSDTovertherangeconsidered. Qualitatively,itappearsthatCR-CCSDT L isthebestmethodforthismolecule atallbonddistancesbeyond2 R e ;afterthatpointtheerrorsareeectivelydistance independent.Interestingly,however,theenergyatequilibriumisbelowthatofCCSDT. Doublebond-breaking :Thequalityofdoublebond-breakingresultsaremore instructivethanthoseforsinglebond-breakingbecauseofthecorrespondingincreased dicultyofdescribingthesemolecules.Dicarbideisanotoriouslydicultmolecule. 118

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Figure3-1.Hydrogenuoridebond-breakingerrorsinkcal/molfromFCIforRHF CCSDTandRHFCCSDTina6-31G**basis.Thehorizontalaxisisin unitsoftheequilibriumbondlengthofHF, R e =0 : 9 A,andonlyvalence electronsarecorrelated.FCIresultsarefrom[168]. RecentFCIresults[170]allowfordirectcomparisonwiththecorrectanswerinthe reasonablylarge6-31G*basisset,asshowninFigure3-3.CCSDT,CCSDTand CR-CCSDT[170]alldisplayerrormaximaatapproximately1 : 6 R e .ThoughCCSDT isthebestchoicenearequilibrium,CCSDTisonlyacoupleofkcal/molmoreinerror, andunlikeCCSDT,canstillyieldqualitativelycorrectresultsupto2 R e andbeyond. CR-CCSDTbehavesbestatlongbondlengths,butis7kcal/molinerroratequilibrium. Oneoftheparadigmaticexamplesofdoublebond-breaking"isthesymmetric stretchofH 2 O.InFigure3-4A,errorsfromFCI[171]inacc-pVDZbasisareshownfor RHFCCSDT,CCSDT,CR-CCSDT,CCSD T andCR-CCSDT L [156,160]. 119

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Figure3-2.FluorineF 2 bond-breakingerrorsinkcal/molfromRHFCCSDTforRHF CCSDT,RHFCCSDT,RHFCCSD T andRHFCR-CCSDT L in cc-pVDZbasis.Thehorizontalaxisisinunitsoftheequilibriumbondlength ofF 2 R e =2 : 66816 a 0 ,andonlyvalenceelectronsarecorrelated.CCSDTand CCSD T resultsarefrom[169]andCR-CCSDT L resultsarefrom[160]. CCSDTandCR-CCSDT L performequivalentlyforthisexample{errorsdierby lessthan1 : 5kcal/molacrossthePES.CCSD T isslightlybetterasymptotically,with errorsreducedbyafewkcal/molrelativetoCCSDT,butworseat2 R e Foramorechemicallyinterestingexample,welookatthebond-breakingofethylene. Theplanarethylenemoleculeisdoubly-bondedand,subjecttoanRHFreference, shouldbemuchmorediculttodescribethefartheroneisalongthepotentialenergy surface.ThetotalenergycurveforthisbondstretchingisshowninFigure3-5A.By twistingthetwomethylenegroups90 outofplanefromeachother,ethylenebecomes asinglybondedbiradical.Inthiscase,becauseitisonlyaneectivesinglebond,one 120

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Figure3-3.DicarbideC 2 bond-breakingerrorsinkcal/molfromFCIforRHF CCSDT,RHFCCSDT,andRHFCR-CCSDTina6-31G*basis.The horizontalaxisisinunitsoftheequilibriumbondlengthofC 2 R e =1 : 243 A, andonlyvalenceelectronsarecorrelated.FCIandCR-CCSDTresultsare from[170]. wouldexpectthatbond-breakingwouldbemoreeasilydescribedatlongrange,butthe biradicalnaturewouldmaketheequilibriummorediculttodescribe.Resultsforthis caseareshowninFigure3-5B.Inbothcases,itisclearthatforbond-breakingCCSDT performssignicantlybetterthandoesCCSDT.Forthecaseofaclosed-shellequilibrium structurenon-twistedethyleneCCSDTandCCSDTarethesameatequilibrium, andonlybegintodisagreeatroughly2 R e .Ontheotherhand,forthebiradicaltwisted ethylenecase,CCSDTis4 : 4kcal/molhigherinenergy. Theinterconversionbetweenthesetwoconformationsofthemoleculeasplottedin Figure3-6.ThoughaFCIreferenceisnotavailable,itisknownthattheinterconversion 121

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A B Figure3-4.Waterbond-breakingerrorsinkcal/molfromFCIinacc-pVDZbasisby variousapproximatetriplesmethods.Thehorizontalaxisisinunitsofthe equilibriumbondlengthofH 2 O, R e =1 : 84345 a 0 ,thebondangleisxedat 110 : 6 andonlyvalenceelectronsarecorrelated.ASpin-restricted Hartree-FockresultsforCCSDT,CCSDT[128],CR-CCSDT[156], CCSD T [156]andCR-CCSDT L [160]arecompared.BSpin-unrestricted Hartree-FockresultsforCCSDT,CCSDT,andCCSD T [156]are shown.FCIresultsarefrom[171]. 122

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A B Figure3-5.EthyleneC 2 H 4 bond-breakingtotalenergiesforRHFCCSDTandRHF CCSDTinaDZPbasisset[14].ForbothAandB,thehorizontalaxisisin unitsoftheequilibriumCCbondlength, R CC =1 : 334 A,andtheCHbond length R CH =1 : 081 AandHCCbondangle : 32 arekeptxed.Only valenceelectronsarecorrelated.APlanarethyleneisstretched,keepingthe twomethyleneunitsinthesameplane.BThetwomethyleneunitsarekept 90 out-of-plane. 123

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Figure3-6.EthyleneC 2 H 4 out-of-planetorsiontotalenergiesforRHFCCSDTand RHFCCSDTinaDZPbasisset.Thehorizontalaxisisindegrees,witha maximumatthe90 twistedgeometry,whiletheCCandCHbondlengthsare keptxedat R CC =1 : 334 Aand R CH =1 : 081 A,respectively,andtheHCC bondangleisxedat121 : 31 .Onlyvalenceelectronsarecorrelated. shouldnotdisplayacuspasitpassesthrough90 ,butratherbesmooth.Therefore,it isslightlysurprising,giventheassumedquasi-degeneratenatureofthe90 state,that CCSDTapparentlyperformsslightlyworsethanCCSDTforthistorsion. Triplebond-breaking :ComparingtheresultsforCOinacc-pVTZbasisset[113] showninFigure3-7betweenRHFCCSD,CCSDTandCCSDT,whatisstriking ishowpoorlyCCSDTbehaves.Itdoesnotsimplyvariationallycollapse,butrather approachesacompletelywrongasymptoteandintroducesanarticialmaximumonthe potentialenergycurve.CCSD,ontheotherhand,isatleastnotacompletequalitative failure,andCCSDTmaintainsthatproperty. 124

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Figure3-7.CarbonmonoxideCObond-breakingcomparisonoftotalenergiesinE h usingRHFCCSD,RHFCCSDTandRHFCCSDTinacc-pVTZbasis. ThehorizontalaxisisinunitsoftheequilibriumbondlengthofCO, R e =1 : 128 A,andonlyvalenceelectronsarecorrelated. TojudgetheasymptoticqualityofRHFandUHFCCSDTandCCSDTresults forCO,wecalculatethedissociationenergiesfromthesemethodsintwoways.To calculatethedirect"dissociationenergy,theUHF-CCenergyoftheCandOatoms arecalculatedandtheequilibriumenergyoftheappropriatemethodsubtracted.The extrapolateddissociationenergyisthenobtainedbytakingtheenergyforeachmethod at4 R e asthatofthedissociatedatoms.Longerbonddistanceswerenotusedbecauseof dicultiesconvergingtheRHFCCSDequations.Becausetheextrapolatedresultsare takenatonly4 R e thereisstillresidualinteractionbetweenthetwoatoms.Agaugeof thatresidualinteractionisgivenbytheUHFCCSDTandUHFCCSDTresults, 125

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Table3-1.CarbonmonoxideCOdissociationenergyinkcal/molcalculatedusingtwo methodsseetextinacc-pVTZbasisandthedierencebetweenthesetwo resultsforCCSDTandCCSDT. R e CO=1 : 128 A,andonlyvalence electronsarecorrelated Method D e direct D e extrap D e RHF CCSDT279 : 326 : 9252 : 4 CCSDT250 : 4263 : 212 : 8 UHF CCSDT250 : 4254 : 94 : 4 CCSDT250 : 4254 : 84 : 4 Experiment[150]256 : 1 whichshowthatthereisstill4 : 4kcal/molofinteractionenergybetweentheatoms.Using theseresultsasareference,anyenergydierencegreaterthan4 : 4kcal/molcanbetraced toimpropertreatmentoftheasymptoticregionofthepotentialenergycurve. TheresultsinTable3-1showthattheRHFCCSDTarecompletelyunusablefor thecalculationofdissociationenergies,aswouldbeexpectedfromlookingatthepotential energycurveinFigure3-7.However,theRHFCCSDTresultsarestillinerrorby morethan12kcal/mol[150]. Inanyexaminationofbond-breaking,onemustconsiderthecaseofN 2 .InFigure3-8, theFCIdata[172]forN 2 inacc-pVDZbasisareplottedalongwithseveralofthedierent triplescorrectionmethods.Fromthesedatapoints,noneofthemethodslookparticularly bad.EvenCCSDTremainsreasonablywell-behaved. However,inTable3-2,thelongerrangebehaviorofN 2 isplottedforCCSDT, CCSDTandCCSDT.Inthiscase,itisclearthatCCSDTfailsutterlyat2 R e CCSDTimprovesthisresultsignicantly,andevenout-performsCCSDT.However, errorsarestillontheorderof60kcal/mol{clearlynotchemicalaccuracy! InFigure3-9,acomparisonofthenon-parallelityerrorsismadebetweenCCSD, CCSDT,andCCSDTforHF,C 2 ,N 2 andH 2 O.Tocompensateforthefactthatthese potentialenergycurvesareofdierentlengthsforthedierentmolecules,theaverage non-parallelityhasbeencalculatedusingaweightingfactordependingonthelengthofthe 126

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Figure3-8.TotalenergiesfornitrogenN 2 bond-breakinginacc-pVDZbasisbyFCIand variousapproximatetriplesmethods.Thehorizontalaxisisinunitsofthe equilibriumbondlengthofN 2 R e =2 : 118 a 0 ,andonlyvalenceelectronsare correlated.FCIresultsarefrom[172],CCSDT,CR-CCSDTand CCSD T arefrom[156]. Table3-2.ErrorsfromFCI[172]inkcal/molforN 2 inacc-pVDZbasisvalence electronsarecorrelatedatlargebonddistances.Non-parallelityerrorsfrom R e to2 R e R e =2 : 118 a 0 Method1 : 7 R e 2 R e NPE CCSDT )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 850 )]TJ/F15 11.9552 Tf 9.299 0 Td [(103 : 546108 : 378 CCSDT0 : 972 )]TJ/F15 11.9552 Tf 9.299 0 Td [(60 : 40567 : 276 CCSDT )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 : 103 )]TJ/F15 11.9552 Tf 9.299 0 Td [(70 : 99077 : 442 127

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Figure3-9.Weightednon-parallelityerrorsforRHFCCSD,RHFCCSDTandRHF CCSDT,denedinEquation3{31.FCIresultsarefrom[168]forHF,[170] forC 2 ,[171]forH 2 Oand[172]forN 2 curvecomputed h E NPE i = 1 M M X A =1 E NPE A R max A R e A : {31 Theseaveragenon-parallelityerrorsandtheircorrespondingstandarddeviationsaretto gaussianfunctionstogiveapictorialrepresentationofthespreadofresultsovermultiple molecules.Ofcourse,allofthesemethodsperformmorepoorlythemorecomplexthe bondingsituation,butCCSDThasbothasmalleraverageerroraswellasatighter distributionoferrorsthanCCSDorCCSDT.ThepracticaleectoftheCCSDT failureatlongbonddistancesisillustratedbythefactthatCCSDisabetterchoicefor minimizingnon-parallelityerrorsthanisCCSDT. 128

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UHFbond-breaking :MostoftheresultspresentedaboveareforRHFreferences. Duetothenatureoftheunderlyingsymmetryoftheproblemandcomputational advantages,onewouldprefertouseRHFifpossible.However,astheexamplefor COdissociationenergiesshowedTable3-1,UHFcanpotentiallyimprovethings considerably.WhatisinterestingiscomparingtheUHFCCSDTresultsandUHFresults forCCSDTandCCSD T .InTable3-1,theresultsforUHFCCSDTandUHF CCSDTarewithin1kcal/molofeachother.EvenmorestrikingisFigure3-4B,that comparestheUHFbond-breakingforH 2 Osymmetricstretch{theerrorsforCCSDT, CCSDTandCCSD T allvirtuallyontopofoneanother.Thisresultisnotisolated. FurtherexaminationsshowthatUHFCCSDTandUHFCCSDTarealmostidentical inallcases.ThisresultstandsinsharpcontrasttoRHFcomparisons,raisingthequestion ofwhythismightbethecase. Tosimplifythediscussion,wewillfocusonthecomponentsthatarisepurelyfrom doubles:thiscorrespondstoCCSD[T]orthesimilarvariantCCSD[T],wheretermsthat involve T 1 or 1 aredropped.IntheHartree-Fockcase,wedene ~ T 3 = T 2 W C ~ 3 = 2 W C : {32 Then,theenergyexpressionsforthebracketsvariantscanbewritten E CCSD[T] = X ijk;abc ~ t abc ijk ~ t abc ijk = abc ijk {33a E CCSD[T] = X ijk;abc ~ ijk abc ~ t abc ijk = abc ijk : {33b Takingacuefromtheanalysesofbond-breakinginMBPT,where HOMO LUMO atlargeseparation,itwouldbereasonabletoexpectthatthecausesoffailureofRHF CCSDTaresmalldenominators abc ijk .Ifoneassumesthatgenerallythemagnitudeof ~ t abc ijk isindependentofthemagnitudeof abc ijk thentheproblematictermsshouldbethosefor which ~ t abc ijk happenstobelargeand abc ijk happenstobesmall.Therefore,forCCSDTto 129

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Table3-3.ThemostsignicantasjudgedbytheircontributiontotheCCSDTenergy T [2] 3 amplitudesforF 2 inacc-pVDZbasisvalenceelectronsarecorrelated. Energycontributionsareinkcal/mol. R FF =5 : 336 a 0 =2 R e RHFUHF E ~ t abc ijk ~ ijk abc abc ijk E ~ t abc ijk ~ ijk abc abc ijk )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 99 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 166 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 056 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 96 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 06 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 022 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 022 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 15 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 99 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 165 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 056 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 96 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 06 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 022 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 022 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 15 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 34 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 141 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 050 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 31 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 04 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 019 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 018 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 20 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 39 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 074 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 028 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 36 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 020 : 0140 : 014 )]TJ/F15 11.9552 Tf 9.298 0 Td [(6 : 53 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 39 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 074 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 028 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 36 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 020 : 0150 : 015 )]TJ/F15 11.9552 Tf 9.298 0 Td [(6 : 43 improvethings,itmustbetruethatwhen ~ t abc ijk islargeand abc ijk issmall,that ~ ijk abc mustbe smallenoughtomoderatethatterm.IntheUHFcase,however,onewouldexpectthere tobenosmalldenominators,and,therefore,thedieringmagnitudesof ~ T and ~ wouldbe inconsequential. Thissimpleexplanationturnsouttobefalse.Ananalysisofthe ~ T and ~ amplitudes showsthatthebiggestdierencebetweenRHFCCSD[T]andRHFCCSD[T]isnotdue tomoderationofsmalldenominators.Rather,itistheactualmagnitudeof ~ T 3 thatisthe sourceofthefailure.ForthecaseofF 2 inacc-pVDZbasisat R FF =5 : 336 a 0 ,theseresults areshowninTable3-3,orderedbytheirrelativeimportancetotheCCSD[T]correction. IntheRHFcase,thebiggestcontributionsto E CCSD[T] arefromlarge ~ T 3 amplitudes. ComparingtotheUHFcase,whilethedenominatorshavenotchangedsignicantly focusingonthe spincase,forexamplethemagnitudeof ~ T 3 hasbecomemuch smaller.InboththeRHFandUHFcases,the ~ 3 amplitudesarethesamemagnitude. Therefore,thereasonthatRHFCCSDTissignicantlybetterthanCCSDTatlong bonddistancesisthatRHF ~ ismuchclosertoUHF ~ thanis ~ T Innoneofthecasesanalyzedisasmall T 3 denominatorarelevantconcerntothe turnoverofCCSDT.IncaseswhereRHFCCSDTcontinuestoturnover,e.g.N 2 boththe T 3 and 3 valuesdivergesignicantlyfromtheirUHFcounterparts,andthe denominatorstillremainsapproximatelythesameforbothRHFandUHF. 130

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Table3-4.ErrorsfromFCI[173]forNH 2 ROHFandUHFbond-breakinginaDZPbasis inkcal/mol.Geometriesarefrom[173]. Method 2 B 1 2 A 1 R e 1 : 5 R e 2 R e R e 1 : 5 R e 2 R e ROHF CCSDT[52]0 : 3571 : 088 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 2900 : 3461 : 3847 : 157 CCSDT0 : 3911 : 330 : 730 : 3681 : 5798 : 607 UHF CCSDT[52]0 : 3432 : 3684 : 7160 : 3341 : 3766 : 295 CCSDT0 : 3762 : 5634 : 8470 : 3561 : 5647 : 410 2 S +12 : 0002 : 1272 : 8952 : 0002 : 0012 : 216 FCIE h )]TJ/F15 11.9552 Tf 9.299 0 Td [(55 : 743 )]TJ/F15 11.9552 Tf 9.299 0 Td [(55 : 605 )]TJ/F15 11.9552 Tf 9.298 0 Td [(55 : 505 )]TJ/F15 11.9552 Tf 9.298 0 Td [(55 : 689 )]TJ/F15 11.9552 Tf 9.299 0 Td [(55 : 518 )]TJ/F15 11.9552 Tf 9.299 0 Td [(55 : 415 Fundamentally,theseresultscanbesummarizedsuccinctly:althoughtheRHFCCSD energyisqualitativelyadequate,theRHFCCSD wavefunction isqualitativelywrongat bond-breakingseparations,andthiscanbetracedtosmall ab ij denominators.Thisfailure oftheRHFCCSDwavefunctionexplainswhymethodssuchasCCSDTandCCSD T improveuponCCSDT:theinclusionoftheleft-handeigenvector,whichduetoitsCI natureislesswrong,"correctsforsomeofthefailuresoftheCCSDwavefunction.Onthe other,intheUHFcase,theUHFCCSDwavefunctionanditscounterpartaregood descriptionsatlongbonddistances.Theyarebothequallypoordescriptionsinthespin re-couplingregion,whichiswhythesemethodsfailthere.Ofcourse,thefailureofthe CCSDwavefunctionhasminimalimpactunlesstriplesareincludedperturbatively. Non-Hartree-Fockbond-breaking :Byexplicitlyincludingthecontraction ofwith F ov into 3 ,non-Hartree-Fockreferencefunctionscanbehandledjustas wellasHartree-Fockreferencefunctions[52].Thisexibilityallowsustoexamine bond-breakinginhigh-spinopen-shellsystemsusingrestrictedopen-shellHartree-Fock ROHFreferences.InTable3-4thedependenceonbondlengthforNH 2 inaDZPbasis fortwoelectronicstatesofdierentsymmetriesissummarized. ThedierencebetweenCCSDTandCCSDTaresmallforbothROHFand UHFreferencesatallgeometries;errorsonlydierbyatmost1 : 5kcal/molforROHF andby1kcal/molforUHF.InthesecondtolastrowofTable3-4,themultiplicity 131

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Table3-5.ErrorsfromFCI[175]formethylradicalCH 3 2 A 00 2 bond-breakinginaDZP basisinkcal/mol. R e =2 : 06 a 0 ,HCH=120 ,C 3v symmetry. Method R e 1 : 5 R e 2 R e ROHF CCSDT0 : 3251 : 3266 : 893 CCSDT0 : 3591 : 53710 : 939 UHF CCSDT0 : 3131 : 6759 : 864 CCSDT0 : 3461 : 86410 : 208 2 S +12 : 0012 : 0263 : 345 FCIE h )]TJ/F15 11.9552 Tf 9.298 0 Td [(39 : 721 )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 483 )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 303 oftheUHF-CCSDstateisdeterminedviatheprojectedformofthespinexpectation value.AshaspointedoutpreviouslyforROHFcases,theprojectedspinvalueistrivially satised[142],butnotthe S 2 expectationvalue[174].Thereforeitisimportanttonot over-interpretthedierenceinspincontaminationbetweentheUHFandROHFresults. However,withthatcaveat,atthelevelofthespinprojection,itappearsthattheROHF basedmethodsbetterrepresentthestateat2 R e AsimilarexampleisillustratedinTable3-5,whereresultsforbond-breakingof thedoubletCH 3 radicalinaDZPbasisillustratetheaccuracyofCCSDTfor bond-breakingofanopen-shellsystem.Theconclusionsaresimilar{ROHFandUHF resultsforCCSDTandCCSDTareveryclosetoeachother.Also,atlongbond distances,thespinmultiplicityoftheUHF-CCSDwavefunctionisfarfromthatofthe ROHF-CCSDwavefunctionasevaluatedusingtheprojectedexpectationvalue. OneconsiderationthatmaybeconfusingisthatifCCSDTfailsforRHF bond-breakingandCCSDTimprovesthatbehavior,thenwhyisitthatROHF CCSDTbond-breakingismuchmoreaccurateandROHFCCSDTdoesnotimprove uponit?InthenormalROHF-coupled-clusterimplementation[52],whiletheROHF referencefunctionisspin-restricted,theexcitationoperatorsarespin-unrestricted,i.e. theycanbedierentfor and spins.Therefore,ROHF-CCismuchmoresimilar toUHF-CCthanitistoRHF-CC,andthediscussionaboveofRHFCCSDTversus UHFCCSDTapplies.However,ifoneweretoapplytheCCSDTderivation 132

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Table3-6.Errorskcal/molfromFCI[171]forlowestH 2 O + statesofeachsymmetry. Geometriesarefrom[171]. Method 2 A 1 2 B 1 2 B 2 Average ROHF CCSDT[171]0 : 290 : 330 : 280 : 30 CCSDT0 : 290 : 350 : 300 : 31 UHF CCSDT[171]0 : 290 : 340 : 280 : 30 CCSDT0 : 300 : 350 : 300 : 31 CCSD T [156]0 : 360 : 430 : 360 : 38 CR-CCSDT[156]0 : 410 : 480 : 410 : 43 FCIE h )]TJ/F15 11.9552 Tf 9.298 0 Td [(75 : 73291 )]TJ/F15 11.9552 Tf 9.298 0 Td [(75 : 55823 )]TJ/F15 11.9552 Tf 9.299 0 Td [(75 : 80689N/A andmethodologytotherecentspin-restrictedcoupled-clustermethodsofGaussand SzalaySR-CC[176,177],thenweconjecturethataCCSDTvariantwouldperform signicantlybetterthanaCCSDTcorrectiontoSR-CCSD.Recentworkhasshown improvementsforF + 2 bondbreakingusingROHFCR-CCSDT L aswell[161]. TocompareCCSDTforsomeopen-shellsystemstoothermethodsthathave beenpublishedintheliterature,weexamineerrorsfromFCIfortheloweststatesofeach symmetryforH 2 O + ,asisshowninTable3-6.Thesestatesaremildlymultireference, withvaryingpercentagesofimportanceoftheunderlyingHFstate,inthecaseofthe UHFresults.Allofthe O N 7 methodsarewithin1kcal/moloftheFCIresultsforthese totalenergies.However,itisalsoclearthatCCSDTisbetterthanbothoftheother alternativesatrepresentingallthreeofthesestates.Theabilitytoapplythismethodto aROHFreferencefunctionalsoshowsclearlythatspin-contaminationoftheunderlying functionisnotanimportantfactoratequilibriumforthismolecule. Activationenergies :Thoughsurprising,thefailuresofCCSDTawayfrom equilibriaappearnottoextendtoenergiesoftransitionstates,atleastformost well-characterizedtransitionstates.TocomparetheimpactofCCSDTonthese transitionstates,weappliedbothCCSDTandCCSDTtoTruhlar'ssetofreactions withreliableexperimentalbarrierheights[178,179].Allcalculationswereperformed intheMG3basis[180{182],andopen-shellmoleculeswerecalculatedusinganROHF 133

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Figure3-10.TheRDXconcerteddissociationtriplewhammy"transitionstate. reference.TheresultsarecollectedinTable3-7.ThedierencebetweenCCSDTand CCSDTisremarkablysmall.Overall,theaverageabsoluteerrordierenceislessthan 0 : 05kcal/mol.Recentreducedmultireferencecoupled-clusterRMR-CCSDTresultsof LiandPaldus[183]similarlyshowagreementwithCCSDTtoroughly1kcal/molfor barrierheight,indicatingthequalityoftheCCSDTnumbers. RDXconcertedtransitionstate :Theproposedconcertedtriplewhammy" transitionstate,illustratedinFigure3-10,breaksRDXintothreemethylenenitramine fragments[118].Becauseitiscomputationallyinfeasibleatthistimetodoafulltransition statesearchusingCCSDTforRDX,weusethegeometriesoptimizedattheB3LYP 6-31dlevel[118].CalculationsareperformedatRHFCCSDTandRHFCCSDTin afrozencoreDZPbasissettruncatedusingfrozennaturalorbitalsto60%ofthevirtual space[76].FortheRDXequilibriumandtheconcertedtransitionstate,thesechoiceslead to42activeoccupiedorbitalsand120activevirtualorbitals,formethylenenitramineitis 14activeoccupiedorbitalsand40activevirtualorbitals. TheresultsofthesecalculationsaresummarizedinTable3-8andshowngraphically inFigure3-11.AllresultsareosetsothatRDXequilibriumenergiesaresettobe0. Therearetwormconclusionsthatcanbemadefromtheseresults:rst,despitethe concertednatureofthistransitionstate,CCSDTandCCSDTagreetowithin0.5 kcal/mol.Therefore,onecantrusttheCCSDTenergiesforthistransitionstate.Second, thecoupled-clusterresultsindicatethattheactivationbarrierofthispathwayishigher 134

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Table3-7.Barrierheightsforwell-characterizedreactionsfrom[178,179].Energiesand errorsfrombestexperimentalestimatesareinkcal/mol. Reaction BarrierHeightError CCSDTCCSDTCCSDTCCSDT H+N 2 O OH+N 2 V z f 18 : 8019 : 040 : 660 : 90 V z r 83 : 7384 : 240 : 511 : 02 H+FH HF+H V z f 43 : 5743 : 631 : 391 : 45 V z r 43 : 5743 : 631 : 391 : 45 H+ClH HCl+H V z f 19 : 6319 : 731 : 631 : 73 V z r 19 : 6319 : 731 : 631 : 73 H+FCH 3 HF+CH 3 V z f 31 : 9231 : 981 : 541 : 60 V z r 57 : 6957 : 740 : 670 : 72 H+F 2 HF+F V z f 2 : 372 : 630 : 100 : 36 V z r 105 : 96106 : 63 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 220 : 45 CH 3 +FCl CH 3 F+Cl V z f 6 : 466 : 82 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 97 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 61 V z r 61 : 8162 : 351 : 642 : 18 F )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(+CH 3 F FCH 3 +F )]TJ/F15 11.9552 Tf 57.502 2.885 Td [(V z f )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 29 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 28 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 95 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 94 V z r )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 29 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 28 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 95 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 94 F )]TJ/F19 11.9552 Tf 9.077 -4.338 Td [( CH 3 F FCH 3 F )]TJ/F15 11.9552 Tf 39.846 2.885 Td [(V z f 13 : 3513 : 34 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 03 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 04 V z r 13 : 3513 : 34 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 03 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 04 Cl )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(+CH 3 Cl ClCH 3 +Cl )]TJ/F15 11.9552 Tf 40.91 2.884 Td [(V z f 2 : 372 : 52 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 73 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 58 V z r 2 : 372 : 52 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 73 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 58 Cl )]TJ/F19 11.9552 Tf 9.077 -4.338 Td [( CH 3 Cl ClCH 3 Cl )]TJ/F15 11.9552 Tf 23.254 2.885 Td [(V z f 13 : 0313 : 15 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 58 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 46 V z r 13 : 0313 : 15 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 58 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 46 F )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(+CH 3 Cl FCH 3 +Cl )]TJ/F15 11.9552 Tf 49.369 2.885 Td [(V z f )]TJ/F15 11.9552 Tf 9.298 0 Td [(14 : 50 )]TJ/F15 11.9552 Tf 9.298 0 Td [(14 : 41 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 96 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 87 V z r 22 : 2822 : 292 : 172 : 18 F )]TJ/F19 11.9552 Tf 9.077 -4.339 Td [( CH 3 Cl FCH 3 Cl )]TJ/F15 11.9552 Tf 31.713 2.884 Td [(V z f 2 : 302 : 36 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 59 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 53 V z r 31 : 7031 : 692 : 082 : 07 OH )]TJ/F15 11.9552 Tf 7.084 -4.338 Td [(+CH 3 F HOCH 3 +F )]TJ/F15 11.9552 Tf 36.696 2.885 Td [(V z f )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 : 89 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 84 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 11 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 06 V z r 16 : 2316 : 31 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 10 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 02 OH )]TJ/F19 11.9552 Tf 9.077 -4.338 Td [( CH 3 F HOCH 3 F )]TJ/F15 11.9552 Tf 19.04 2.885 Td [(V z f 10 : 6710 : 71 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 29 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 25 V z r 47 : 8647 : 850 : 660 : 65 H+CO HCO V z f 4 : 074 : 130 : 900 : 96 V z r 21 : 9922 : 07 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 69 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 61 H+C 2 H 4 CH 3 CH 2 V z f 2 : 682 : 750 : 961 : 03 V z r 41 : 7541 : 960 : 000 : 21 CH 3 +C 2 H 4 CH 3 CH 2 CH 2 V z f 6 : 927 : 130 : 070 : 28 V z r 32 : 5032 : 78 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 47 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 19 HCN HNC V z f 48 : 4748 : 530 : 310 : 37 V z r 33 : 5633 : 690 : 450 : 58 AverageAbsoluteError0 : 850 : 89 135

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Table3-8.ConcertedreactionpathwaycriticalpointenergiesforRDXinkcal/molfor B3LYPina6-31dbasis[118]andCCSDTandCCSDTcalculationsina 60%truncatedDZPbasisset. Method E act E rxn B3LYP59.445.9 CCSDT65.843.6 CCSDT66.243.8 inenergythantheactivationbarriercalculatedfromDFT.Tentativelythismeasureof theenergywouldmaketheconcertedtransitionstateatbestaminorcontributiontothe decompositionofRDX.Unfortunately,becauseofthesmallsizeofthebasisset,andthe lackofafullyself-consistenttransitionstatesearch,wecannotdenitelysaythatthe concertedpathwayisnotrelevant.Furthermore,entropiceectsmaybeimportantinthis pathway.GiventhatCCSDTseemstobeperformingwellforthistransitionstate,the cheaperCCSDTresultswouldbethebestchoiceforfurthercalculations. 3.1.5Conclusion ThegreatsuccessoftheCCSDTmethodhasbeeninitshigh-accuracy,desirable theoreticalproperties,andblack-boxnaturethatmakesitstraight-forwardtoapply. CCSDTmaintainsalloftheseproperties,whilealsoimprovingthedescriptionof potentialenergysurfacesasawhole.WhencomparingCCSDTtootherextensive methodsthathavebeenproposedtoimproveuponCCSDT,itisunclearwhether oneissubstantiallysuperiornumerically.Partofthisuncertaintyisthatthereare limitedreferencedataformoleculesfarfromequilibrium;FCInumbersarerelatively scarceandrestrictedtoverysmallmolecules.Giventhefactthatnomethodstands abovetheothers,theoreticalreasoningcanbeusedtosuggestapreferredmethodology. BecauseCR-CCSDTisnotextensiveandisrelativelyinferiornearequilibrium,it cannotberecommended.CR-CCSDT L isextensive,butthenatureofthedenominator inthetriplescorrectionmeansthatdierentchoicesoforbitalswillleadtodierent energies.Forsomesituationsthisorbitaldependencemaynotbeaseriousdrawback, 136

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Figure3-11.ReactionprolecomparisonforRDXconcertedtransitionstate suchaswhenusedwithpredeterminedlocalizedorbitals,butitisnotassatisfyingas methodsthatallowarbitraryrotationsamongtheoccupiedorunoccupiedorbitals; virtuallyanecessityfortheimplementationofanalyticalgradients.BothCCSD T and CCSDTappeartosatisfyallthefundamentalrequirementsthatonecoulddesireinan approximatemethod.However,wesuggestthatthenaturalconnectionthatCCSDT hastobothdiagrammaticperturbationtheory,theCCfunctional,CCdensitymatrices, andLowdinpartitioningistoitscredit.Also,theneglectofthe Q partofCCSDby purelycomputationalscalingargumentsisunfortunate.ThatisnottosaythatCCSD T doesnothaveitemstorecommendit.Recentworkhasshownthatthereisanatural generalizationofCCSD T toexcitedstatesthatallowsatotalenergytobeidentied withtheEOM-CCSD ~ T correction[164],whichisdesirable. 137

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InthecourseoftheinvestigationoftheimprovementofRHFCCSDToverRHF CCSDT,andtherelativelysurprisinglackofimprovementforUHFCCSDTover UHFCCSDT,wehaveshownthatthefailureofRHFCCSDTatlongbonddistances ismorecomplicatedthanonemightinitiallyexpect.Itisnotsimplythepresenceofsmall denominatorsduetoquasi-degeneracyintheTcorrectionthatleadstocollapseofthe CCSDTenergy.Rather,itisthe T 2 contractionthatisover-weightedatthesebond lengths.Theunderlyingcauseofthelarge T 2 values does derivefromasmalldenominator, butratherthanasmall abc ijk ,itistheimplicit ab ij denominatorwithin T 2 thatbecomestoo small.Therefore,tofundamentallyremedythefailureoftriplesmethodsatlongbond distancesonecannotfocussolelyonthetriplescorrection,butmustalsotakeintoaccount themisbehavioroftheRHFCCSD wavefunction ,thoughtheRHFCCSD energy appears adequate. ItshouldbenotedthattheCCSDTmethodisgeneralizabletohigherordersin variousways[127,184,185].Inlightoftheresultspresentedhere,itislikelythatmethods suchasCCSDTQ,etc.,couldbeexcellentchoicesforhigh-accuracydescriptionsof potentialenergysurfaceswhileretainingtheadvantagesofsingle-referencecoupled-cluster theory. FurtherworkisnecessarytodenitelydeterminewhethertheRDXconcerted transitionstateisapossiblyimportantdecompositionpathway.Basedonthecalculation thusfaritseemsunlikely. 3.2Forces 3.2.1Introduction InthepriorsectionaboutCCSDT[127,128],wejustiedthatusinginformation fromtheleft-handcoupled-clusterCCgroundstateeigenvector,oneisableto improvethewidelyusedcoupled-clustersingles,double,andperturbativetriplesmethod [CCSDT][51],anditsnon-HFgeneralization[52],forbond-breaking.Foraquantum chemicalmethodtobegenerallyuseful,itmustbeabletoproducemorethansingle-point 138

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energies.Determiningcriticalpoints,suchasequilibriumstructuresandtransitionstates, isoneofthemostimportanttasksforcomputationalchemistry. MorerelevanttothecaseofCCSDT,characterizingreactionpathways,driving moleculardynamicssimulations,andperformingdetailedanalysesofelectronicspectra atlargeseparationsrequirenotonlyenergies,butforcesandoccasionallyhessian information.Inthesecases,itisrequisitetohaveaccurateforcesawayfromequilibrium. Todothisinacomputationallyecientandstablemannerrequiresanalyticalderivatives foramethodthatisapplicabletotheentirepotentialenergysurface;CCSDTissucha method. AnalyticalderivativesforCCSDT[127]wereoriginallydiscussed,butnot implemented,byCrawfordandStanton[128].Wepresentanalternativederivationof theCCSDTanalyticalderivativesbaseduponastationaryformulationofCCSDT. ThestationaryformoftheequationsallowsustousethegeneralizedHellman-Feynman theoremGHFtoeasilydeterminethecorrectexpressionforderivativesandfordensity matricesforCCSDT.Theseequationsaresubstantiallymorecomplicatedthanthose forCCSDT,andtheinterpretationoftheindividualtermsisobscure.However,by comparingtosimilarstationaryformulationsofEOM-CCSDandnormalCCSDT, anidenticationofthemeaningofeachofthetermsintheCCSDTfunctionalcan bemade.TheanalyticalderivativesanddensitymatricesforCCSDThavebeen implementedandtestedonaseriesofmoleculesbothatandawayfromequilibrium. Oneareaofpotentialfutureresearchistheuseofcoupled-clustermethodstodrive moleculardynamics.Forexample,providingdetailedunderstandingofpossiblereaction pathwaysorwithinaquantummechanical-molecularmechanicalQM/MMhybrid wheretheelectronicstructureofsomeactiveregionaremodeledusingcoupled-cluster theory.Inthesecases,itisnecessarytogenerateforcesfromthequantummechanical modelatpointsthatareawayfromanystationarypoint.Giventheknownfailuresof CCSDTawayfromequilibrium,weexaminewhetherCCSDTcouldpotentiallybe 139

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abletodrivemoleculardynamicsproperlybyfocusingonthequalityofsimplediatomic forcecurves,whichallowfordirectinterpretationoftheresults. Asimplemodelsystemforthedissociationbehaviorofnitro-group-containing energeticmaterialsisnitromethaneNMT[1].Thekineticsofdecompositionare importantforunderstandingmorecomplicatedmaterialsusedasexplosivesand propellants,suchasRDXand1,3,5,7-tetranitrooctahydro-1,3,5,7-tetrazocineHMX. BecauseNMTisanenergeticmaterial,smallchangesinthepositionandstructureof transitionstatescouldleadtosignicantchangesinthekineticbehavior[120,121].Much oftherecenttheoreticalworkonmodelingthissystemwassummarizedin[121].There areseveralpossiblepathwaysforthedecompositionofNMT;oneofwhichinvolvesthe initialmigrationofthemethylgroupfromthenitrogentooneoftheoxygensofthenitro group,formingmethylnitriteMNTfollowedbydissociationfromMNT[1,120{122]. Thisrearrangementapparentlyproceedsviaaconcertedmechanism,withtheC-Nbond breakingsimultaneouslywithaC-ObondformingandaN-O bondbreaking.The multiplebondingnatureofthetransitionstatesuggeststhatCCSDTresultscouldbe subjecttotheweaknessinherentinthatmethod,thereforewere-investigatethenatureof theNMT-MNTtransitionstateandthereactionpathwaythatleadstoit. 3.2.2Theory Thestrongesttoolthatcanbeusedfordetermininggradientexpressionsfora quantummechanicalequationisthegeneralizedHellman-Feynmantheorem.Ifallofthe variousparametersdenoted z arestationaryinanenergyfunctional E ,thentheGHF states E = E stat [ H; z ]{34a @E @ = E stat [ @H @ ; z ]{34b where isanexternalperturbation,forexampleachangeinnuclearposition.The gradientisthusindependentoftheactualperturbationexceptthroughthechangeinthe 140

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Hamiltonian.Asmentionedabove,theGHFonlyapplieswhentheenergyfunctionalis stationarywithrespecttovariationsofallofitsparameters.Initerativecoupled-cluster theory,theintroductionofmakes T stationary,andallowstheGHFtobeapplied[130]. Itwasshownin[52]thatbyasuitablemodicationoftheequations,thesamecould besaidforCCSDT,andtheirargumentsholdingeneralityforperturbativemethods thatarebasedsolelyonutilizingthe T amplitudes.However,inCCSDT,the amplitudesarerequiredtobethoseforCCSD.Therefore,thereisnofreedomtochange theequationstotakeintoaccounttheperturbativecorrection,andtheconjugatenatureof T andisbroken.Withoutmodication,theCCSDTenergyexpressionismanifestly non-stationaryandtheGHFcannotbeapplied. Toremedythisproblem,itisnecessarytocompletelydecouplethevariationsof fromthoseof T .FromthenormalCCenergyfunctionalitisknownthat T canbedened bythepartialderivativeoftheCCenergyfunctionalwithrespectto.Conversely,can bedenedbythepartialderivativeoftheenergyfunctionalwithrespectto T .Therefore, todecouplethetwo,followingtheanalysisofCrawfordandStanton[128],weintroduce twonewconjugatevariables:conjugateto T andconjugateto.Thesevariableswill allow T andtosatisfytheirstandardequations,butwithoutrequiringtheirvariations tobecoupledtoeachother.Iftherewerenoperturbativecorrection,thisaugmentedCC energyfunctionalcouldthenbewritten E decoupled = h 0 j ++ H N j 0 i + h 0 j + )]TJ/F15 11.9552 Tf 8.515 -6.662 Td [( H N C j 0 i + h 0 j H j 0 i : {35 Byconstructionthestationarypointofthisfunctionalisidenticalwiththatforthe standardCCenergyfunctional,soatthispointnothingisgained.However,this equationcanbefurtheraugmentedtoincludetheCCSDTperturbativecorrection inacompletelystationarymanner. Itisclearertopostulatethisnewenergyfunctionalrst,andthenjustifythatitdoes, infact,reproducetheCCSDTenergyatstationarity.Theaugmentedenergyfunctional 141

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is E CCSDT = h 0 j ++ H N j 0 i + h 0 j + )]TJ/F15 11.9552 Tf 8.514 -6.662 Td [( H N C j 0 i + h 0 j 1 WT [2] 3 j 0 i + h 0 j 2 F ov + W T [2] 3 j 0 i + h 0 j [2] 3 F oo + F vv T [2] 3 j 0 i + h 0 j [2] 3 WT 2 C j 0 i + h 0 j H j 0 i {36 where F indicateselementsoftheoccupied-occupiedoo,virtual-virtualvvor occupied-virtualovpartsoftheFockoperator,and W representstwo-electronintegrals. Imposingstationaryconditionsforvariationsoftheconjugateparameteryields @ E @ a i = h a i j H N j 0 i =0 @ E @ ab ij = h ab ij j H N j 0 i =0 {37 whicharesimplythestandardCCSDamplitudeequations,whoseexplicitformhavebeen reportedbefore[186].Similarly,stationaritywithrespecttoleadsto @ E @ a i = h 0 j + )]TJ/F15 11.9552 Tf 8.514 -6.661 Td [( H N a a i C j 0 i =0 @ E @ ab ij = h 0 j + )]TJ/F15 11.9552 Tf 8.514 -6.661 Td [( H N a ab ij C j 0 i =0 {38 whicharetheequationsforCCSD[130].Finally,dierentiatingEquation3{36with respectto [2] 3 @ E @ ijk abc [2] = h abc ijk j F oo + F vv T [2] 3 j 0 i + h abc ijk j WT 2 C j 0 i =0;{39 alinearequationthatcanbesolvedexplicitlyas t abc ijk [2] = 1 abc ijk P a=bc j k=ij X d h bc jj dk i t ad ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(P c=ab j i=jk X l h lc jj jk i t ab il # ; {40 wherethedenominatoristhefollowingcombinationofdiagonalelementsofthefock matrix abc ijk = f ii + f jj + f kk )]TJ/F21 11.9552 Tf 11.955 0 Td [(f aa )]TJ/F21 11.9552 Tf 11.955 0 Td [(f bb )]TJ/F21 11.9552 Tf 11.955 0 Td [(f cc ; {41 142

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whichagreeswiththedenitionofthesecond-order T 3 amplitudeinCCSDT[127]. Atstationarity, E stat CCSDT = h 0 j + H N j 0 i + h 0 j H j 0 i + 1 3! 2 X ijk abc h 0 j F ov + W j abc ijk i 1 abc ijk h abc ijk j WT 2 C j 0 i = E CCSDT : {42 Theconjugatevariables,,and [2] 3 areunnecessarytocalculatetheenergy,butare neededtocalculatethegradient;theformsandexplicitspin-orbitalequationsareshown inAppendixB.Onecanmakethisexpressionsomewhatmorecompactbysolvingan additionallinearequation, @ E @t abc ijk [2] = h 0 j 1 W j abc ijk i + h 0 j 2 F ov + W j abc ijk i + h 0 j [2] 3 F oo + F vv j abc ijk i =0 ; {43 whichhasthesolution ijk abc [2] = 1 abc ijk P a=bc j k=ij X d h bc jj dk i ij ad )]TJ/F21 11.9552 Tf 11.955 0 Td [(P c=ab j i=jk X l h lc jj jk i il ab + P i=jk P a=bc i a h jk jj bc i + P i=jk P a=bc f ia jk bc i : {44 Thenthestationarypointofthefunctionalcanbesummarizedtobe E CCSDT = h 0 j + H N j 0 i + h 0 j [2] 3 F oo + F vv T [2] 3 j 0 i + h 0 j H j 0 i : {45 Becausethisfunctional,Equation3{36,isstationarywithrespecttovariationsofall ofitsparameters,theGHFnowapplies.OnecanreplacetheHamiltonianbyitsperturbed counterpart,anddothesamefortheFockoperatorandtwo-electronintegrals, @E CCSDT @ = E stat CCSDT [ H ]= h 0 j ++ H N j 0 i + h 0 j + )]TJ/F15 11.9552 Tf 8.515 -6.662 Td [( H N C j 0 i + h 0 j 1 W T [2] 3 j 0 i + h 0 j 2 F ov + W T [2] 3 j 0 i + h 0 j [2] 3 F oo + F vv T [2] 3 j 0 i + h 0 j [2] 3 W T 2 C j 0 i + h 0 j H j 0 i : {46 143

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Thesuperscript indicatesthatperturbedatomicorbitalquantitieshavebeen transformedtotheunperturbedmolecularorbitalbasis.FollowingthestandardCC gradientformulation,thiscanbereformulatedintermsofone-andtwo-particledensity matrices: @E @ = X pq pq @f pq @ + X pqrs )]TJ/F22 7.9701 Tf 7.314 -1.794 Td [(pq;rs @ h pq jj rs i @ {47 where pq = h 0 j ++ )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p y qe T C j 0 i + h 0 j + )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(p y qe T C C j 0 i + h 0 j 2 p y qT [2] 3 j 0 i + h 0 j [2] 3 p y qT [2] 3 j 0 i {48 and )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(pq;rs = h 0 j ++ )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(p y q y sre T C j 0 i + h 0 j + )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(p y q y sre T C C j 0 i + h 0 j 1 p y q y srT [2] 3 j 0 i + h 0 j 2 p y q y srT [2] 3 j 0 i + h 0 j [2] 3 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(p y q y srT 2 C j 0 i : {49 Fullspin-orbitalexpressionsfortheelementsofthedensitymatricesareintheappendix. Orbitalrelaxationtermsarehandledidenticallyasinothercoupled-clustermethods [130],and,usingthenotationfromthatreference,thenalgradientexpressionis @E CCSDT @ = X pq D pq f pq + X pqrs )]TJ/F22 7.9701 Tf 7.315 -1.793 Td [(pq;rs h pq jj rs i + X pq I pq S pq {50 where D pq = pq + D or pq {51 isthedensitymatrixincludingorbitalrelaxation D or pq terms, I pq istheintermediate matrixcomposedofcombinationsofthedensitymatricesandunperturbedintegrals, and S pq istheperturbedatomicorbitaloverlapintegralstransformedtotheunperturbed molecularorbitalbasis. TounderstandthestructureoftheCCSDTfunctionalandgradient,itishelpfulto comparethemtoequationsformorefamiliarmethods.OnecanreformulateCCSDTina 144

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functionalform,aswasdoneaboveforCCSDT,arrivingattheequation E CCSDT = h 0 j + H N j 0 i + h 0 j T y 1 WT [2] 3 j 0 i + h 0 j T y 2 F ov + W T [2] 3 j 0 i + h 0 j [2] 3 F oo + F vv T [2] 3 j 0 i + h 0 j [2] 3 WT 2 C j 0 i + h 0 j H j 0 i : {52 Thetermwrittenhereas T [2] 3 isdenedas T [2] 3 c andtheterm [2] 3 is T [2] 3 c + T [2] 3 d where c and d indicateconnected"anddisconnected,"respectively,in[52].Inthisform, however,itisclearthatifonemakesthesubstitutionoffor T y thenthelastfourterms ofEquation3{52andthoseofEquation3{36areidentical. Similarly,equation-of-motioncoupled-clustertheoryEOM-CCforexcitedstatescan bewritteninfunctionalformas[187,188] E EOM-CC = h 0 j Z H N j 0 i + h 0 jL H N Rj 0 i : {53 ComparingthisequationwithEquation3{36,onerecognizesthatthersttwotermsof Equation3{46areidenticaltothoseforEOM-CCwiththedenitions, Z =1++{54a L =1+{54b R = : {54c Therefore,theCCSDTfunctionalcanbeviewedasacombinationofthatfor conventionalCCSDTwiththefunctionalforEOM-CC.Thisviewpointalsosuggests thattheCCSDTfunctionalispotentiallyagoodstartingpointforperturbative correctionstoEOM-CCSDexcitationenergies. 3.2.3Implementation TakingadvantageofthefactthatACESIIhasbothnormalCCSDTandEOM-CC analyticalgradients,CCSDTcouldbeimplementedwithintheACESIIprogram system[54,55]withrealabelianpointgroupsymmetry.Allequationswerespin-summed foreciency.Gradientsrequiresemi-canonicalorbitalsfornon-Hartree-Fockreferences, 145

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butotherwisearegeneralforallreferencesimplementedwithinACESII.Toeliminate theneedtosolveforthetriplesamplitudeequationsiteratively,theorbitalsandtheir perturbationsarerequiredtoremainsemi-canonical. 3.2.4ResultsandDiscussion Evenmorethanfortheexampleofenergies,therearefewreferencecalculations thathavedatafromthepotentialenergysurfaceawayfromequilibrium.Becauseofthe high-qualityofCCSDT,itisdiculttondnaturalexampleswhereCCSDTis signicantlybetterthanisCCSDT.Weattempttoavoidthisproblembyfocusingon exampleswheresomeanalyticalknowledgecanbeusedtoevaluatetheresults,orwhere thesystemsarewell-characterized. Forcecurves :DrivingmoleculardynamicsrequirestheforceateachMDstep.In general,thegeneratedgeometrieswillnotbeclosetoequilibriumortotransitionstates. ToevaluatethecapabilityofusingCCSDTtobethegeneratorforMDforces,we examinethebehaviorofCCSDTforcesatmultiplepointsalongapotentialenergy surface.Wefocusedonlookingatthebond-stretchingofsimplediatomics.Inthesecases, itisknownthattheforceshouldremainpositiveforallbonddistancesgreaterthanthe equilibriumbonddistancei.e.themoleculeshouldpullbacktowardtheequilibriumand havenozerosotherthanatequilibrium.Asisclearinthegures,thisisnotthecasefor CCSDT,orevenforCCSDT. Anynegativevalueoftheforceatbonddistancesgreaterthanequilibrium correspondstoanartifactualrepulsiveforce.Wecomparethemaximummagnitudes oftheseartifactualforcesasameasureofhowbadlytheforceisdescribedawayfrom equilibrium. Giventhatthesecurvesarethederivativesofbond-breakingcurvesand,asshown inSection3.1,bondbreakingiswell-describedbyCCSDTbutnotbyCCSDT,the forcecurvesarenotsurprising.InFigure3-12theforcecurveforhydrogenuorideina cc-pVTZbasissetisplottedforRHFCCSDTandCCSDT. 146

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Figure3-12.HFForcecurveforRHFCCSDTandRHFCCSDTinacc-pVTZbasis. ThehorizontalaxisisinunitsoftheequilibriumbondlengthofHF, R e =1 : 7328 A,andonlyvalenceelectronsarecorrelated. Astheplotshows,thereisasignicantarticialmaximumintroducedbyCCSDT at2 : 6 R e .Thistransitionstate"isreducedagreatdealintheCCSDTcalculations, but,unfortunately,noteliminated.Thereisaweakarticialtransitionstateat3 R e Comparingthemaximumrepulsiveforceneartheseartifactualstationarypoints,for CCSDTthereisarepulsiveforceof18 : 1mE h =a 0 whileforCCSDTtheartifactual forcemaximumisanorderofmagnitudesmaller,2 : 4mE h =a 0 Figure3-13showsasimilarforcecurveforcarbonmonoxide,alsoinacc-pVTZbasis set.ThiscaseshowsevenmoreclearlytheadvantageofCCSDToverconventional CCSDT.TheCCSDTdoesnothaveanyartifactualrepulsiveforcecomponentpast 3 R e .Ontheotherhand,CCSDTshowsasignicantartifactualrepulsiveforcewith 147

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Figure3-13.COForcecurveforRHFCCSDTandRHFCCSDTinacc-pVTZbasis. ThehorizontalaxisisinunitsoftheequilibriumbondlengthofCO, R e =1 : 128 A,andonlyvalenceelectronsarecorrelated. maximummagnitude294mE h =a 0 .Thisrepulsiveforceisstrongerthanthemaximum attractiveforceforCCSDTthepeakat1 : 3 R e of248mE h =a 0 .ForthecaseofCO, CCSDTalsoshowsanadditionalspuriouslong-rangeminimumataround2 : 5 R e Equilibriumgeometries :CCSDThasbecomethestandardreferencemethodfor properties{includinggeometries{atequilibrium.Therefore,anecessaryrequirementis forCCSDTtonotadverselyaecttheCCSDTequilibriumgeometries.Weapplied CCSDTtothebenchmarksmallmoleculedatasetofBaketal.,[50]togaugethis accuracy. AsissummarizedinTable3-9,theperformanceofCCSDTisvirtuallyidentical tothatofCCSDTforpredictingequilibriumbondlengths.Deviationsbetweenthe 148

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Figure3-14.Errorinbondlengthsinpmforthesetofmoleculesconsideredin[50]ina cc-pCVQZbasisforCCSDTandCCSDT.Thegaussiancurvesaretto themeanandstandarddeviationoftheset. statisticsofthetwomethodsarelessthan0 : 5pmforallbasissetsandallstatistical measures.PlottedinFigure3-14arethettedgaussiandistributionsfortheCCSDT andCCSDTdataforthecc-pCVQZbasisset.Thisgraphicaldisplaymakesclearthat thedistributionofaccuraciesforbothmethodsareapproximatelyequivalent. Similarly,forbondangles,Table3-10summarizesthedataforCCSDTand CCSDT.Formostofthemolecules,thedierencebetweenCCSDTandCCSDT issmall.However,CCSDTseemstodescribetheH 2 O 2 andNH 3 bondanglesalready knowntobediculttocalculateaccuratelylesswellthanCCSDT. TheseresultsjustifythecontentionthatCCSDTdoesnothurttheCCSDT descriptionofequilibriumgeometries.Thebehaviorwithrespecttoconvergencewithbasis 149

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Table3-9.Comparisonofoptimizedequilibriumbondlengthsformultiple correlation-consistentbasissetsforCCSDTandCCSDT.Averageswere calculatedoverthesetofmoleculesfrom[50].Forcalculationsusingacc-pVXZ basis,onlyvalenceelectronswerecorrelated;forcalculationsusingacc-pCVXZ basisallelectronswerecorrelated. isthesignedmeanerror, abs isthe meanabsoluteerror, max isthemaximumabsoluteerror,and std isthe standarddeviation.Allnumbersareinunitsofpm. BasisSet CCSDTCCSDT abs max std abs max std cc-pVDZ1 : 681 : 684 : 510 : 801 : 681 : 684 : 080 : 75 cc-pVTZ0 : 020 : 200 : 450 : 230 : 010 : 220 : 730 : 27 cc-pVQZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 100 : 130 : 610 : 17 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 120 : 150 : 720 : 19 cc-pCVDZ1 : 611 : 614 : 420 : 781 : 621 : 623 : 970 : 75 cc-pCVTZ0 : 170 : 220 : 490 : 180 : 140 : 240 : 850 : 27 cc-pCVQZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 040 : 090 : 590 : 16 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 060 : 110 : 610 : 18 Table3-10.Comparisonofoptimizedequilibriumbondanglesformultiple correlation-consistentbasissetsforCCSDTandCCSDT.Averageswere calculatedoverthesetofmoleculesfrom[50].Allelectronswerecorrelated. isthesignedmeanerror, abs isthemeanabsoluteerror, max isthe maximumabsoluteerror,and std isthestandarddeviation.Allnumbersare inunitsofdegrees. BasisSet CCSDTCCSDT abs max std abs max std cc-pCVDZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 811 : 813 : 661 : 22 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 752 : 447 : 613 : 42 cc-pCVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 830 : 832 : 680 : 80 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 221 : 495 : 692 : 54 cc-pCVQZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 520 : 522 : 330 : 72 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 031 : 185 : 152 : 29 withoutH 2 O 2 andNH 3 cc-pCVDZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 311 : 312 : 580 : 81 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 351 : 352 : 580 : 79 cc-pCVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 480 : 480 : 830 : 26 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 500 : 500 : 840 : 28 cc-pCVQZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 210 : 210 : 410 : 13 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 230 : 230 : 510 : 16 setisalsothesameforCCSDTandCCSDT,withbondlengthsoverestimatedin doublebasissetsandslightlyunderestimatedinquadruplebasissets. Transitionstategeometries :Afterequilibriumgeometries,themostcritical regionsofapotentialenergysurfacearearoundtransitionstates.Identifyingthe locationofthesestationarypointsismoredicultthanislocatingthelocalminima thatcorrespondtoequilibria,bothintermsoftheoptimizationalgorithmtondextrema 150

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Table3-11.ComparisonofrootmeansquaredistancesRMSDin a 0 amongoptimized transitionstatestructuresforQCISD[179],CCSDTandCCSDTinan MG3basis,forasetofwell-characterizedreactions[178,179]. Methods RMSD a 0 max std QCISD-CCSDT0 : 0200 : 1140 : 033 QCISD-CCSDT0 : 0370 : 0990 : 054 CCSDT-CCSDT0 : 0080 : 0150 : 007 andintheelectronicstructuredescriptionatthesepoints.Totestboththequalityof CCSDTandCCSDTtransitionstatestructures,weoptimizedalloftherelevant molecularspeciesfromtheMinnesotasetofreactionswithreliableexperimentalbarrier heights[178,179].TheinitialguessstructuresweretheQCISDoptimizedstructuresin anMG3[180{182],andallopen-shellmoleculeswereoptimizedusinganROHFreference function.Thereareatotaloffourteentransitionstatesforthereactionsconsidered. Wecomparedtheoptimizedstructuresproducedbythesethreemethodsusing root-mean-squaredistancesRMSD.TheRMSDbetweentwostructuresofasingle molecule,onecalculatedbymethod X andtheotherbymethod Y is RMSD X;Y = 1 N atoms N atoms X A =1 j R A X )]TJ/F36 11.9552 Tf 11.955 0 Td [(R A Y j 2 1 = 2 : {55 Thisformulaisnotinvarianttorotationsandtranslationsoftheindividualmolecules, thereforetheKabschprocedure[189]asimplementedinVMD[190]wasusedtooptimally alignthemolecules.TheresultingRMSDvaluesarethereforewell-dened.Itisimportant tonotethatunliketheenergies,theQCISDstructurescannotbeconsideredunambiguous referencestructures;rather,theyareapointofcomparisonfortheothersetsofstructures. TheresultssummarizedinTable3-11showthegreatsimilaritybetweenallthreeof thesemethodsfordescribingtransitionstates.Itseemsthatsomehowthesespeciesavoid theknownfailureofCCSDTforbondbreaking,despitethefactthatourpictureofbond breakinginmoleculesisbuiltaroundtransitionstates!Onfurtherreection,however, theseresultsmaybelessstrangethantheyrstappear.AswasestablishedinSection3.1, 151

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Figure3-15.Schematicoftransitionstatefortherearrangementofnitromethaneto methylnitrite thedierencebetweenCCSDTandCCSDTwhenusingaUHForROHFreference functionissmall.Nineoutofthefourteentransitionstatesareopen-shell,soforthose transitionstatesCCSDTandCCSDTshouldagree.However,evenforRHFreference casesCCSDTandCCSDTaremuchthesame.TheaverageRMSDfortheRHF transitionstateswas0 : 002 a 0 andforROHFtransitionstateswas0 : 010 a 0 .Onepossible explanationisthatthetransitionstatesfromthisdatasetarenotcomplicatedenough toshowthefailuresofCCSDT,soweconsiderasomewhatmoredicultsituation,as discussedinthenextsection. Nitromethanetomethylnitriterearrangement :Arelativelysimpleexample ofaconcertedreactionistherearrangementofnitromethaneNMTtomethylnitrite MNT[121].Alongthereactioncoordinate,theCNbondisbreakingsimultaneously withtheformationoftheCObond.Werstoptimizedthetransitionstateforthis rearrangementusingCCSDTandCCSDTinacc-pVDZbasiswiththevalence orbitalscorrelated,aswellasthereactantandproduct.Thisbasissetwaschosento matchtheCCSDTcalculationsperformedin[121].Allminimaandtransitionstates wereveriedbycalculatingthehessiansfortheoptimizedgeometries.Themoleculeat itstransitionstateisshownschematicallyinFigure3-15.Table3-12showstheresultsof thetransitionstateoptimizations.ThegeometricaldierencesbetweenB3LYPandthe coupled-clustermethodsaresmallbutsignicant:bondlengthchangesofupto0 : 07 A.On theotherhand,thedierencesbetweenCCSDTandCCSDTaremuchsmaller,with amaximumdeviationofonly0 : 004 A. 152

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Table3-12.NitromethanetomethylnitriteoptimizedtransitionstateresultsforB3LYP, CCSDTandCCSDT.Energiesarezero-pointcorrected. ParameterB3LYP[121]CCSDTCCSDT BarrierHeightkcal/mol E z f 6467 : 2467 : 47 E z r 6669 : 5369 : 88 BondLengths A CN1 : 9641 : 8951 : 891 CO1 : 9991 : 9701 : 969 ON1 : 2981 : 3231 : 321 ON1 : 1981 : 2061 : 206 BondAngles ONC73 : 073 : 173 : 2 ONO a 118 : 8118 : 9 a Angleunavailable. Atleastforanexamplelikethissimpleoneitisclearthatdespitemultireference characterinthetransitionstate,CCSDTworkswellforthetransitionstateitself.One mightconsiderthepossibilitythatalthoughCCSDTworkswellatreactantandthe transitionstate,atpointsalongthereactioncoordinatebetweenthosetwolimits,itmight fail,asisthecaseforbondbreakinginadiatomicmolecule.Therefore,wecalculatedthe reactionpathbetweenNMTandthetransitionstatealongtheC-N-Oangulardistortion. Angularstepswere2 : 5 from120 neartheNMTequilibriumto70 nearthetransition stateoftherearrangement.Ateachxedchoiceofangle,alloftheotherdegreesof freedomwereallowedtorelaxtotheirminimumenergyvalue.Allcalculationswere performedinacc-pVDZbasiswithvalenceelectronscorrelated.InFigure3-16theresults ofthesecalculationsareplotted,withallenergiesshiftedrelativetothereactantNMT equilibrium.ThereactionpathwaysforCCSDTandCCSDTfalldirectlyontop ofeachother.WeransimilarcalculationsusingtheC-Nbonddistanceasareaction coordinate,withthesameresult. 153

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Figure3-16.NitromethanetoMethylnitritereactionpathcomputedbyCCSDTand CCSDT Examiningthedataobtainedforthisreaction,aswellastheresultsfromthe transitionstatesdiscussedabove,itisclearthatfortheseexamplesatleast,CCSDT doesnotshowthesamelevelofaberrantbehaviorfortransitionstatesthatitdoesfor pure"bondbreaking. 3.2.5Conclusion ThegeneraldevelopmentofenergiesandgradientsforCCSDThasbeen presented.ItisthesimplestextensionbeyondCCSDTthatretainssize-extensivity, occupied-occupiedandvirtual-virtualorbitalinvariance,andoersdemonstrable improvementforbondbreaking.Tofacilitatetheunderstandingandderivation ofgradients,CCSDThasbeenreformulatedasthestationarypointofanew functional.Asasidebenet,CCSDThasalsobeenwritteninstationaryfunctional 154

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form.ComparingthefunctionalexpressionforCCSDT,CCSDTandEOM-CCSD, showsthattheCCSDTfunctionalcanbeseensymbolicallyasacombinationoftwo othermethods.Thekeytheoreticaltoolthatwasusedwastosupplementthestandard coupled-clusterequationswithadditionalconjugateparametersthatallowbothand T tobestationary,andthegeneralizedHellmann-Feynmantheoremtobeapplied.This approachismoregeneralthanjustforCCSDT,andcanbeappliedinanysituation wheretheoperatorscontributetotheenergy,suchasCCSD T Formoleculesnearequilibrium,thedierenceintheoptimizedgeometriesbetween CCSDTandCCSDTisnegligible.Thisresultisencouraginggiventhesuccessthat CCSDThasdescribingequilibriumgeometries.Whatismoresurprisingisthesimilarity inbehaviorofCCSDTandCCSDTderivativesawayfromequilibrium.Aswasshown inSection3.1,theenergydierenceatthetransitionstatebetweenthesetwomethods isfoundtobenegligiblefortheexampleswehaveinvestigated.Besidesthosereported here,theseincludeseveraltransitionstatesforthedecompositionofnitroethane,theCope rearrangement,andseveralothers.Wendthistobearatherremarkablefeatureofa perturbativemethodlikeCCSDT,foritdeeslogicthateventhoughCCSDTfails dramaticallyforbondbreaking,forawealthoftransitionstates,thereisnofailure. 3.3Brueckner CCSDT 3.3.1Introduction Theresultsforpotentialenergysurfaces,asshowninSection3.1,fromCCSDT canbesummarizedintwomainconclusions.WhenusinganRHFreferenceCCSDT improvesthebondbreakingbehaviorconsiderably.However,asymptoticallythe CCSDTresultsdonothavethesameaccuracyasnearequilibrium.Thestructure oftheRHFreferencefunctionisanimproperdescriptionoftheelectronicstructure atthedissociationlimit,makingitunsurprisingthatCCSDTdoesnotsucceed there.ItwouldbefarbetterforaUHFreferencetobeusedintheasymptoticregion. However,aswasdiscussedinSection3.1.4,CCSDTisnotimprovedoverCCSDT 155

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forspin-unrestrictedreferences.Inpractice,theCCSDTresultsexhibitapronounced deviationfromFCIinthespin-recouplingregionofthepotentialenergysurface,as CCSDTdoes.ThedissociationbehaviorisexcellentforbothUHFCCSDTandUHF CCSDT. Ifthespin-symmetrywasmaintainedfurtheralongthepotentialenergysurfacethan itiswithUHF,thenonewouldexpectCCSDTtoimprovetheresultsoverCCSDT. Ifthespin-symmetrybreakingpointoccurredafterthespinre-couplingregion,thenthe unrestrictedCCSDTisacceptable.Doesareferencefunctionexistthatdelaysthe symmetrybreakingpoint?IthasbeensuggestedthattheBruecknerorbitals[34,191{195] arelesssusceptibletosymmetrybreakingthanareHartree-Fockorbitals[196{199]. Anapproximateexpectationvaluefor S 2 wasusedtojudgewhenthespin-restricted /spin-unrestrictedtransitionoccurredwasusedin[198].Forthesinglebondsthey considered,theBruecknerCCDwavefunctionremainedspin-restrictedsignicantlyfurther alongthepotentialenergysurfacethandidCCSDbasedonthesameinitialUHForbitals. Whenattemptingtodeterminewhetherthespin-restrictedorspin-unrestricted solutionispreferable,inHartree-FocktheoryoneusestheHartree-Fockinstability conditions[20,22].Ifthespin-restrictedHartree-Focksolutionisfoundtobeasaddle pointinthespaceofspin-symmetrybreakingorbitalrotations,thenthereisalower energyUHFsolutionnearbyinorbitalspace.Onewouldliketohavethesameanalysis capabilityfortheBruecknerorbitalssowehaveimplementedtheBruecknerinstability conditionsofPaldus,etal.[200]tomorerigorouslyidentifythelocationoftheBrueckner CCDsymmetrybreakingpoint.Theseconditionsapparentlyhaveonlybeusedonce beforecomputationally[201],wherethefocuswasonmodelproblemsandexcitedstates, ratherthanonpotentialenergysurfaces. Afterverifyingthattheinstabilitypointwasfurtheralongthepotentialenergy surfaceforBruecknerorbitalsthanforHartree-Fockorbitals,wewantedtobuild CCSDTontopoftheBruecknerorbitals.Therearetwowaysofdoingthis 156

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construction:therstistoviewtheBruecknerorbitalsasanarbitrarysetoforbitals pluggedintotheCCequationsandusetheconventionalnon-Hartree-FockCCSDT equationstogenerateanenergy.Becausethisapproachinvolvesnosingleexcitations, eitherfrom T 1 or 1 ,onecantermitBCCDT.Thesecond,and,inourview,more theoreticallyjustiedapproach,istorecognizethattheBCCansatzis not thesameas theCCSDansatz,andtheequationsandtheCCSDTshouldbemodied.Despite thefactthat T 1 0,whenproperlytreated, 1 6 =0.Therefore,wecallthatvariant BCCSDT,todenotethefundamentalroleoftheBruecknerorbitalsinthisapproach. TheinclusionoftheS"inthenameindicatesthatthesinglesequationsforhavetobe solved. 3.3.2Theory Stability :TheBruecknerorbitalsaredenedtobethebest",inthesenseof closenessinthetwo-normoftheHilbertspace,single-determinantapproximationto theexactwavefunction[193,200].ApproximateBruecknerorbitalscanbedetermined givenanyreferencefunctionandthatcriterion.Inthecaseofcoupled-clustertheory,the Bruecknercriterionisequivalenttorequiringthat T 1 vanish[195,202].Following[200],we willre-derivethiscondition,andtheaccompanyingBruecknerstabilityconditions. Thegoalistomaximizetheoverlapbetweenanarbitrarynormalized single-determinantandthecoupled-clusterwavefunction.Thecoupled-cluster wavefunctionisdenedintheconventionalway, j i = e T j 0 i {56 where,usingthenotationthat i j k areoccupiedorbitals, a b c areunoccupiedvirtual orbitals,and p q r arearbitraryorbitals, T = N e X n =1 T n = 1 n 2 X ijk abc a y ib y jc y k : {57 Arbitrarysingle-determinantsareparameterizedbyaunitarysingle-particlerotation[19]. 157

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Deningtheantihermitianoperator as,assumingrealorbitals = X ia a i \010 a y i )-222(f i y a g {58 thentheexponentialorbitalrotationoperatoris e = U {59 where U isunitary.Normal-orderingforboth T and iswithrespecttoanarbitrary normalizedreferencestate.TheBruecknerorbitalscanbecharacterizedasthesetof orbitalsgeneratedbytheBruecknerorbitalrotationoperator, B B =argmax h 0 j e T y e j 0 i h 0 j e T y e T j 0 i 1 = 2 j B i = e B j 0 i : {60 Onecandenetheoverlapfunctionalas S B = h 0 j e T y e j 0 i h 0 j e T y e T j 0 i 1 = 2 : {61 AroundtheBruecknerdeterminant,where canbechosentobezerobychoosingthe referencestate j 0 i = j B i ,theoverlapfunctionalshouldbestationarywithrespectto variationsof @ S B @ a i =0 = h B j e T y e a y i j B i h B j e T y e T j B i 1 = 2 =0 =0 : {62 Eliminatingthenon-vanishingdenominator,thisequationisequivalentto t a i =0 : {63 Therefore,the T 1 =0conditionisequivalenttothemaximumoverlapcondition. Fororbitalsthatsatisfy T 1 =0tobetrulyBruecknerorbitals,theoverlapmustbea maximum,notasaddlepointorminimum.Whethertheorbitalsaremaximaorotherwise canbedeterminedbywhethertheorbitalrotationhessianisnegative-deniteornot. AroundtheBruecknerdeterminant,theorbitalrotationhessianis @ S B @ a i b j =0 = h B j e T y e )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(f b y j g)-222(f j y b g a y i j B i h B j e T y e T j B i 1 = 2 =0 : {64 158

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Becausethedenominatorispositivedenite,itisirrelevantinthestabilityanalysis,and canbeignored.UsingWick'stheorem,thisequationsimpliesto H B ia ; jb = t ab ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( ij jb : {65 Thepresenceofapositiveeigenvalueforthismatrixindicatesthattheorbitalsfoundare notmaximumoverlap,butonlyastationarypoint.Thatmeansthatthereisanother solutionoftheBruecknerequations,connectedbyacontinuoustransformationofthe orbitals,that is amaximum.Followingthepositiveeigenvectorofthehessianshould leadtothatsolution.Conceptually,thefactthattheBruecknerorbitalsdependon T 2 is satisfying,becauseitdirectlyshowstherolethatcorrelationplaysintheconstructionof theBruecknerorbitals. Bruecknerfunctional :ToproperlyformulatetheCCSDTequationsfor Bruecknerorbitals,onemustderivetheequationsforBruecknerCC.Intheoriginal presentationofBCCderivatives[196]theBruecknerorbitalswereasanorbitalrotation term.However,aspresentedin[203],theresponseoftheBruecknerorbitalscanbe groupedwiththeresponseofthecoupled-clusteramplitudethemselves.Followingthis approach,developedinSection1.3.3,leadstotheBCCfunctional E BCC = h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T 0 e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T 0 j 0 i + h 0 j H j 0 i {66 where T 0 doesnotinclude T 1 and isdenedasinEquation3{58. does include 1 OnecandenetheBruecknerCCeectiveHamiltonianby ~ H N = e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e {67a ~ H N = e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T 2 ~ H N e T 2 {67b where ~ H istheHamiltonianinarotatedsetoforbitals.Imposingstationaritywithrespect toleadstoequationsfor T 0 and specifyingtoBCCD, h a i j ~ H N j 0 i =0 h ab ij j ~ H N j 0 i =0 : {68 159

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Byrotatingtheorbitalsaftersolvingtheseequationssuchthat n =0,theBrueckner orbitalsaredenedattheconvergenceofthisprocess.Inpractice,whatwedoistosolve theconventionalCCequations h a i j H N j 0 i =0 h ab ij j H N j 0 i =0{69 generatinganon-zero T 1 .Then,thisvalueisusedtorotatethereferencefunction j 0 i andtheintegralsin H ,andtheprocessisrepeateduntil T 1 islessthanathreshold. ThesamesimilaritytotheconventionalCCequationscannotbeexploitedwhen attemptingtosolvefor.TheequationsaredeterminedbymakingEquation3{66 stationarywithrespecttovariationsin and T 2 h 0 j + 1 + 2 e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T 2 [ i y a; ~ H N ]+[ ~ H N ;a y i ] e T 2 j 0 i =0{70a h 0 j + 1 + 2 [ ~ H N ;a y ib y j ] j 0 i =0 : {70b Thespin-orbitalformsoftheseequationsareinAppendixC. ThederivationofCCSDTinSection3.1.2appliesequallywelltotheBCC functional,leadingtothe E [3] = E BCCSDT = 1 3! 2 X ijk abc h 0 j ~ H [1] N j abc ijk ih abc ijk j ~ F oo + ~ F vv j def lmn i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 h def lmn j ~ H [2] N j 0 i : {71 IftheBruecknerorbitalsarechosentobesemi-canonical,thentheinversein Equation3{71istrivial, E BCCSDT = 1 3! 2 X ijk abc h 0 j ~ F ov + ~ W j abc ijk i 1 ~ abc ijk h abc ijk j ~ WT 2 C j 0 i : {72 Equation3{72iswhatisusedintheprogram. InSection3.2.2weshowedthatonceafunctionalisestablisheditisrelatively straightforwardtoderivethederivativesbyaugmentingthefunctionalsothat,at stationarity,theperturbativesolutionisdetermined.Usingthedenitionsthat ~ F indicates 160

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elementsoftheoccupied-occupiedoo,virtual-virtualvvoroccupied-virtualovparts oftheFockoperatorintheBruecknerbasis,and ~ W representstwo-electronintegralsin theBruecknerbasis,andifavariablehasnosubscriptitisassumetoincludesingles anddoubleexcitationsorde-excitations,thentheaugmentedenergyfunctionalforthe BruecknerorbitalCCSDTis, E BCCSDT = h 0 j ++ ~ H N j 0 i + h 0 j + ~ H N C j 0 i + h 0 j 1 ~ WT [2] 3 j 0 i + h 0 j + h y 1 ~ H C e T 2 i C j 0 i + h 0 j [2] 3 ~ WT 2 C j 0 i + h 0 j 2 ~ F ov + ~ W T [2] 3 j 0 i + h 0 j [2] 3 ~ F oo + ~ F vv T [2] 3 j 0 i + h 0 j H j 0 i : {73 Imposingstationaryconditionsforvariationsoftheconjugateparametersand reproducetheBCCenergyandequations.DierentiatingEquation3{73withrespectto [2] 3 yieldsthe T [2] 3 whichagreeswiththeBCCSDTenergyinEquation3{71. 3.3.3Implementation TheACESIIprogramsystem[54,55]alreadyhadapre-existingBruecknerCC code.However,itwasnotnumericallystableenoughtobeabletotreatpotentialenergy surfaces.Therefore,severaladditionalchangesweremadetothecode.Convergenceaway fromequilibriumrequiredtheuseofconvergeacceleratorsfortheBruecknerrotations. Forthistoworkinanintuitivefashion,itwasimperativethattheorbitalsbedetermined byasequenceofunitaryrotations.ThisschemedivergedfromthewaythatACES hadpreviouslyimplementedBruecknerrotationsviaalinearapproximationtothe unitaryoperatorandreorthonormalization.Ahigh-orderPadeapproximanttothe matrixexponentialbasedonAlgorithm2.3of[204]wasusedtoparameterizetheunitary operator.Extrapolationforconvergenceaccelerationrequiredastableandecient algorithmforthematrixlogarithmaswell,usingAlgorithm7.1of[205].Inbothofthese algorithms,theorderofthePadeapproximantsoftherelevantoperatorswasdetermined byanestimateofthetwo-normofthematrixitself. 161

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FortheBCCSDTcorrectiontobesolvednon-iteratively,onemust semi-canonicalizetheorbitals.Ifthissemi-canonicalizationweretobedoneiteration byiterationoftheBruecknerorbitals,thentheextrapolationwouldfail.Therefore, insteadtheBruecknercalculationwasconvergedinthestandardnon-canonicalbasis, thenafterconvergenceasemi-canonicaltransformationwasperformed. Evenwiththeseconvergenceaccelerationtechniquesaddedtothecode,itbecomes diculttoremainonasinglepotentialenergysurface.Thisstate-jumpingoccursdespite usinginitialguessesforthe T amplitudesfromthepriorpointonthepotentialenergy surface.IfthenonlinearCCequationsarenearsingular,thananysmallperturbation willbedrasticallyamplied,pushingthe T amplitudesawayfromtheirinitialgood approximation.AnexactsingularityintheequationswouldmaketheBCCequations ill-posed;anearsingularitywouldmakethemill-conditioned[4].Ourexperiencesuggested thattheBCCequationswereill-conditioned,ratherthanill-posed.Moreover,usinga traditionalregularizationtechniqueSection1.4wouldleadtohigherenergiesthanthe unregularizedsolutionwould.GiventhatthegoaloftheBCCSDTapproachisto attempttoapproachchemicalaccuracy,thatpenaltyisunacceptable. Instead,wereformulatetheCCequationsinawaythatisinspiredbyTikhonov regularization.ThetraditionalmethodforsolvingtheCCequationsisbasedonthe iterativeupdatescheme,usingCCDforsimplicity, t ab ij [ n +1] = 1 ab ij F [ t [ n ] ] ; {74 where F isafunctionalofallofthe T 2 amplitudes.If ab ij approacheszero,thenthisupdate schemewillbeunstable,andcouldleadtostate-jumping.Instead,weusetheiterative update t ab ij [ n +1] = ab ij )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ab ij 2 + 2 F [ t [ n ] ]+ 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ab ij 2 + 2 t ab ij [ n ] ; {75 where 2 isapositiverealnumber.Thechoiceof 2 hasnotbeenoptimized,but empiricallywefoundthat 2 =0 : 1wassucienttoaidconvergenceinallthecases 162

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weexamined.IfthelastterminEquation3{75wasmissingthenthisschemewouldbea regularizationoftheCCDequations.Toshowthatthisnewschemedoesconvergetothe rightanswer,wenotethatatconvergence t ab ij [ n ] = t ab ij [ n +1] = t ab ij {76 thentheupdateschemeis t ab ij = ab ij )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ab ij 2 + 2 F [ t ]+ 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ab ij 2 + 2 t ab ij : {77 or )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ab ij 2 + 2 ab ij t ab ij = F [ t ]+ 2 ab ij t ab ij : {78 Subtracting 2 ab ij t ab ij fromtheleft-handandright-handsidesofEquation3{78and simplifyingyields t ab ij = 1 ab ij F [ t ] ; {79 whichisthecorrectCCresult. Asshownin[206],itispossibletoiteratetheBruecknerorbitaliterations simultaneouslywiththeCCamplitudeiterations.Usingthisapproach,thereisno computationalpenaltyforusingBCC.Unfortunately,todosorequiresanintegral-direct implementation,thatisnotinACESII.Therefore,thereisalinearincreaseinthe computationaltimeduetohavingtoiteratetheCCequationsandtheBrueckner iterationsinastepwisefashion.AfutureimplementationwithinACESIII[207]would allowforanintegral-directformoftheBCCequations.BruecknerCCSDTwillstillbe slightlyslowerthanHFCCSDTduetotheaddedtermsintheequations.Several termsareautomaticallyzerointheBCCDequationsthatarenotzerointheCCSD equations.Ourcodedoesnotfullytakethisadvantageintoaccount,butafterfull optimization,aBCCSDTwilltakethesameamountoftimeasaHFCCSDT calculation. 163

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Table3-13.Comparisonofthepointofspin-unrestrictedinstabilityforHartree-Fock referencesandBruecknerCCDreferencesforseveralmolecules.Alldistances areinmultiplesoftheequilibriumbonddistance.Thegeometriesandbasis setswereasfollows:forHF, R e =0 : 9 Aandthebasisis6-31G**,forF 2 R e =2 : 66816 a 0 andthebasisiscc-pVDZ,forH 2 O, R e =1 : 84345 a 0 ,thebond angleisxedat110 : 6 inacc-pVDZbasis,forC 2 R e =1 : 243 Aandthebasis is6-31G*,forN 2 R e =2 : 118 a 0 andthebasisiscc-pVDZ. MoleculeHFB HF1 : 5 > 5 : 0 F 2 1 : 0 > 5 : 0 H 2 O1 : 52 : 5 C 2 0 : 52 : 1 N 2 1 : 11 : 9 3.3.4ResultsandDiscussion InTable3-13thecomparisonbetweenthelocationwheretheHartree-Fock spin-restrictedtospin-unrestrictedinstabilityoccursiscomparedtothatfromaBrueckner CCDcalculation.TheBCCDinstabilitiesuniformlyoccuratlongerbondlengthsthan dotheHFinstabilities.Thisresultagreeswiththeconjectureandtheconclusionreached in[198].TheBCCDinstabilitiesfollowanadditionalpattern;asthebondorderofa moleculeincreases,theBCCinstabilityoccursearlieralongthebondbreakingcoordinate. Thattrendwouldagreewithchemicalintuitionthatwouldsaythatitisharderto describethebreakingofamultiply-bondedsystemthanasingly-bondedmoleculefor example.However,forHFthetrendisnotasclear.RestrictedHFdescribesF 2 sopoorly thattheUHFinstabilityoccursearlierforF 2 thanitdoesforN 2 NowthattheBruecknerorbitalshavebeenveriedtoleadtoadditionalspinstability, wecantestthesecondpartoftheconjecture:thatusingaBruecknerreferencewillleadto improvedCCSDTresults.First,welookedattwosingly-bondedmolecules.Hydrogen uorideisanexampleofafairlytypicalsinglebond.InFigure3-17,weplottheRB CCSDTresults,aswellastheRBCCDTresultsandtheRHFandUHFresultsfor bothCCSDTandCCSDT.ForbothRHFandRB,CCSDTfailsstrongly.The failureofBruecknerorbitalsforCCDTisnotsurprising;theBruecknerorbitalswillnot 164

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Figure3-17.HydrogenuorideHFbond-breakingerrorsinkcal/molfromFCIfor RHFCCSDT,RHFCCSDT,RBCCDT,RBCCSDT,UHF CCSDTandUHFCCSDTina6-31G**basis.Thehorizontalaxisisin unitsoftheequilibriumbondlengthofHF, R e =0 : 9 A,andonlyvalence electronsarecorrelated.FCIresultsarefrom[168]. xthequasi-degeneracythatleadstoanomalous T 2 amplitudes,whichisthecauseofthe failureofRHFCCSDT.Ontheotherhand,thecombinationofBruecknerorbitalswith CCSDTisaconsiderablyimprovement.Thenon-parallelityerroracross4 : 5 R e isless than1 : 5kcal/mol.BecausetheBruecknerreferenceisstillspin-restricted,theasymptotics ofRBCCSDTarenotasgoodastheUHFCCSDTorCCSDTresults.However, thedeviationat4 : 5 R e islessthan0 : 5kcal/mol. TheresultsforF 2 arequitedierent,however.TheBruecknerorbitalsarenot unstableatanypointconsideredforF 2 inthisbasis,aswasthecaseforhydrogen uoride.Unlikethecaseofhydrogenuoride,Figure3-18showsthattheBrueckner 165

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orbitalsleadtominimaldierencefromtheRHForbitals,forbothCCSDTand CCSDT.Infact,forCCSDTtheRHForbitalsyieldaslightlybetterdissociation curvethanRBorbitalsascomparedtoRHFCCSDT.Becausethecomparisoniswith CCSDT,andnotFCI,itispossiblethatRBCCSDTisclosertoFCIthanRHF CCSDT.Ifso,thedierenceissmall.Thedissociationofuorineisanoutlierin anotherway,aswell,becausetheasymptoticbehavioroftheUHFbasedCCSDTand CCSDTisactuallypoorerascomparedtoCCSDTthanaretheRBCCSDT results.Wedonotyethaveafullunderstandingofthesedata. Formultiply-bondedsystemswemightexpecttoseemoresignicantdierences betweenBruecknerandHartree-Fockbasedresults;ascanbeseeninFigures3-19 and3-20,thisexpectationisborneout.InbothH 2 OFigure3-19andN 2 Figure3-20 theBruecknerorbitalsbreakspinsymmetry.Inthewaterexample,thisspinsymmetry breakingallowstheasymptoticbehavioroftheUBCCSDTtomatchthatofUHF CCSDTandCCSDT.Atthesametime,becauseinthespin-recouplingregionthe Bruecknerorbitalsarenotyetunstable,theUBCCSDTfaroutperformstheUHF CCSDTresultswheresymmetryhasalreadybroken.Thenon-parallelityerrorfor thismoleculeislessthan5kcal/molover3 R e usingUBCCSDT,ascomparedto9 : 5 kcal/molforUHFCCSDT. ForN 2 ,theBruecknerresultsalsosignicantlyoutperformtheUHFresults.The asymptoticbehaviorforbothBruecknerandHartree-Fockorbitalsisthesame,butin theintermediateregiontheBruecknerresultsaremorethan5kcal/molbetterthan theUHFresults.ThebondbreakingcurvesfortheBruecknerorbitals,however,are morejaggedthanthosefromUHF.Thissharpchangeisduetothelocationofthespin instability.BecausetheUBsolutionbecomespreferredatalargeNNdistance,theenergy dierencefromswitchingtothespin-unrestrictedsolutionissignicant.Thisbehavior isnotideal;onewouldprefertohaveasmootherbondbreakingcurve.However,we believethatincludingtripleexcitationsintotheBruecknerorbitalprocedureBCCDT 166

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Figure3-18.Fluorinebond-breakingerrorsinkcal/molfromRHFCCSDTinacc-pVDZ basisbyRHF,UHF,andRBreferencesusingCCSDTandCCSDT.The horizontalaxisisinunitsoftheequilibriumbondlengthofF 2 R e =2 : 66816 a 0 andonlyvalenceelectronsarecorrelated.CCSDTresultsare from[169]. wouldimprovetheBruecknerorbitalsenoughtosmooththisdierence,whencombined withaperturbativetreatmentofquadruples.Yet,theUBCCSDTresultisthebest single-referenceperturbativetriplesCCcalculationintheliterature. 3.3.5Conclusions WehaveshownthatthecombinationofBruecknerorbitalsandCCSDT appeartocomplementeachotherwell.TheBruecknerorbitalsdelaysymmetry breakingconsiderably;forsinglybondedexamplestheyappeartocompletelyeliminate spin-symmetrybreaking.Atthesametime,becausethespin-symmetrydoesnot break,theorbitalsremainquasi-degeneratefurtheralongthepotentialenergysurface, 167

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Figure3-19.Waterbond-breakingerrorsinkcal/molfromFCIinacc-pVDZbasisby UHFCCSDT,UHFCCSDT,UBCCDTandUBCCSDT.The horizontalaxisisinunitsoftheequilibriumbondlengthofH 2 O, R e =1 : 84345 a 0 ,thebondangleisxedat110 : 6 andonlyvalenceelectrons arecorrelated.FCIresultsarefrom[171]. contaminatingthe T 2 amplitudesoftheBCCDequations.Thiscontaminationmakes theBCCDTcorrectionfartoolarge,andresultsexhibitapronouncedturnover.On theotherhand,CCSDTstronglymoderatestheinuenceoftheseill-behaved T 2 amplitudes,leadingtoaboundedtriplescorrection.Atlongrange,thesymmetrybreaking allowstheUBCCSDTresultstomatchthosefromUHFCCSDT,whichare excellent. AnoutlierfromthisgeneraltrendisF 2 .DespitethefacttheBruecknerorbitalsdo nothaveaninstabilityatanypointonthepotentialenergysurface,theF 2 curveturns over.Thereasonforthisbehavioriscurrentlyobscure.Itiswell-knownthatthesmall 168

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Figure3-20.NitrogenN 2 bond-breakingerrorsinkcal/molfromFCIinacc-pVDZ basisbyUHFCCSDT,UHFCCSDT,UBCCDTandUBCCSDT. ThehorizontalaxisisinunitsoftheequilibriumbondlengthofN 2 R e =2 : 118 a 0 andonlyvalenceelectronsarecorrelated.FCIresultsarefrom [172]. physicalsizeofF 2 makesitadicultmoleculeforanymethodtoaddress.Onepossibility isthatitistheactualelectronstructureoneachF atomthatisleadingtothefailuresof BCCSDT.Single-determinantreferencefunctionshaveadiculttimeaddressingF 2 molecule,andUHFCCSDTandCCSDTarenotexceptions. Weintroducedseveralmodiedandnewnumericaltechniquestobeableto convergetheBruecknerequationsatlargeinteratomicseparations.Thecombination ofahigh-orderPadeapproximantforthematrixexponentialandlogarithm,aswell asconvergenceacceleration,appeartoleadtoquitegoodconvergenceoftheBCCD equationsnearequilibrium.However,atevenlargerbonddistances,thereareseveral 169

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closelyingBruecknersolutions.Byusingamodiedregularizationprocedure,we arebetterabletofollowasinglestateandavoidstate-jumpingtonearbystates.The stabilizedreformulationoftheCCequationsinEquation3{75improvesconventionalCC calculationsaswellasBCCcalculations,however,itisessentialintheBCCcase. Anintegral-directformoftheBCCDcodewillbenecessarytomakeBCCSDT computationallycompetitivewithconventionalHFbasedCCSDT,but,inprinciple, theBCCSDTcodewouldhaveminimaladditionalcomputationalcost. 170

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CHAPTER4 HERMITIANCOUPLED-CLUSTERTHEORIES 4.1MotivationandIntroduction Theaccomplishmentsofcoupled-clustertheoryaremanifold,andtheresults presentedinChapters2and3arehopefullyasteptowardfurthersuccesses.Overthe lasttwodecades,increasingcomputerpowerallowedCCmethodstoadvanceacrosstwo fronts:increasingaccuracyinsmallmoleculesbyusinglargerbasissetsandhigher excitationsandaddressinglargermolecules.Theageofserialcomputationiscoming toanend,andtheneedtousemassivelyparallelandmultiplecorecomputersishere. ThevastmajorityofCCcalculationshavebeenserialand,unfortunately,CCisadicult methodtoparallelizeeciently.Thenonlinearitiesintheequationsleadtotightcoupling betweendierentamplitudes,whichnecessitatessignicantdatapassingbetweenvarious nodesofacluster[207,208]. Theseissueswerediscussedatarecentconferenceoncoupled-clustertheory.One oftheconclusionswasthattobeabletoecientlyparallelizeCC,onlyobjectsofthe dimensionsofdoubleexcitations O n 2 N 2 canbestored.ForcasessuchasCCSDT, wheretheamountofstoragerequiredis O n 3 N 3 ,theabilitytotransferthedatafrom processortoprocessorbecomesprohibitive.Unfortunately,thereisalimittowhatlevelof accuracyonecanachievefromconventionalCCwiththatlimitation.Eventheiterative triplesmethodsthatdonotrequirestorageofthe T 3 amplitudes,suchasCCSDT-3,are limitedandthereisnopathwaytoimprovethem.Toreachultra-highaccuracywithCC usingaparallelcomputer,itappearsthatitmightbebettertouseamethodthatismore expensiveintermsofoperatorsifthestoragerequirementsaresmaller.Alternativeforms oftheCCapproach,whichhavebeendisregardedduetotheirhigheroperationscaling, maybebetterpossibilitiesforparallelizationandhighaccuracy. Whileparallelizationwillallowfortheachievementofhigheraccuracy,andwith thejudicioususeoflocalizationandalternativeCCmethods,beabletoaddresslarger 171

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systems,thereisanotheraxistothechemicalproblem:physicaltime.Becauseofthe costofcoupled-clustertheory,ithasbeenimpossibletorunadynamicalsimulation. Idealparallelizationonlyleadstoalinearspeedupwiththenumberofprocessorsthough [207]showsthatsuper-linearscalingispossibleinsomecases,whichmeansthatthere isalimittothereturnduetoparallelization.Torunamoleculardynamicssimulation, thousandsofpointswillneedtobecalculatedrapidly,andinsuccession,sothatthe naiveparallelizationofdierentpointsondierentprocessorsisnotgoingtohold. Itisimperativethatthesimulationscaleverywelltohundreds,ifnotthousandsof processors,andbefasterthanaconventionalCCcalculationtobeabletoachieveany sortofdynamicalresult.Theseconstraintsagainsuggestrethinkingthecoupled-cluster framework,andseeingifotherchoicescanbemade. WhiletherehavebeenseveralapproachestoalternativeCCansatze,onegroupthat IbelieveholdsparticularpromiseisthehermitianCCmethods.Amongotherreasons, byconstructionthesehermitianapproachessatisfythegeneralizedHellman-Feynman theoremwithoutrequiringthesolutionofasetofequations.Whenconsideringrunning amoleculardynamicssimulation,forceswillbeneededateverystep.Avoidingthe equationscouldthenspeeduptheoverallcalculationtimebyafactorof2.Ithasbeen suggestedthroughoutthehistoryofCCthathermitianCCmethodsintroduceindirect eectsduetohigherexcitations,withoutincreasingthestoragecost.Ofcourse,this statementisnotassimpleasitsounds,andwillconsideredindepth.InthisChapter, Iwillrstdealwiththetheoreticalstructureofhermitiancoupled-clustertheoriesin general.Then,inSection4.3,IwillfocusonthehermitianlinearizedCCmethodLinCC, whichIbelievecouldbeusedtorunmoleculardynamicsusingCC. 4.2HermitianCoupled-ClusterTheory Conventionalcoupled-clustertheoryhasthreeimportantpropertiesthathave contributedsignicantlytoitssuccesses.Theseare[209]size-extensivity,inclusion ofhigherexcitationsthroughproductsof T operators,andanitesetofequations. 172

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Ofcourse,thesepropertiesarenotindependentofeachother.Furthermore,after solvingtheequations,theCCequationsarestationary,meaningthatthegeneralized Hellman-FeynmantheoremcanbeappliedSection1.3.3.Ifpossibleonewouldliketwo additionalproperties:avariationalupperboundlikeincongurationinteractionand satisfyingtheHellman-Feynmantheoremwithoutneedingtosolvetheequations.Most importantlyforimprovingresults,onewouldliketoincludecontributionsfromhigher excitationsindirectlyintolowerexcitationequations,asCCdoesrelativetoCI[210]. Unfortunately,itappearsthatthereisnoansatzthatsatisesalloftheseconditions. However, hermitian coupled-clustertheoriessatisfymostofthem.Ifamethodisboth hermitianandstationary,thenitwillsatisfytheHellman-Feynmantheoremwithout theneedtosolveanadditionalsetofequations.Thehermitiancoupled-clustertheories discussedbelowincludecontributionsfromhigherexcitationsaswell[211{213].Aswill beshown,thesemethodsarealsovariationalandextensive.Unfortunately,theequations nolongerarenite.TruncationsofthesemethodsSection4.2.4invariablylosesomeof theirproperties,butthemethodsarestillafertilegroundforexploration.Therearethree approachestohermitiancoupled-clustertheory:expectation-valuecoupled-clustertheory XCC,unitarycoupled-clustertheoryUCCandstrongly-connectedXCCSC-XCC;I willdiscusseachoftheseinturn. 4.2.1Expectation-ValueCoupled-ClusterTheory Expectation-valuecoupled-clustertheoryXCC[212,214{216]isbuiltbydirectly manipulatingtheexpectationvalueoftheHamiltonian, E VCC = h 0 j e T y H N e T j 0 i h 0 j e T y e T j 0 i + h 0 j H j 0 i : {1 Ifonedirectlymakesthisequationstationarywithrespecttovariationsin T y ,then clearlyitsatisestheRayleigh-Ritzprinciple,andisavariationalupperboundonthe groundstateenergy.Therefore,usingthisformoftheequationisknownasvariational coupled-clusterVCC[217].NotethatunliketheconventionalCCexpectationvalue,this 173

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expressionisnotmanifestlylinked,i.e.theoperatorsinvolvedarenotconnectedtoeach other.InXCC,thisequationisreformulatedtobecompletelyconnected.Thenumerator ofEquation4{1canbefactoredinthefollowingwayAppendixD h 0 j e T y H N e T j 0 i = h 0 j e T y H N e T C j 0 ih 0 j e T y e T j 0 i : {2 CancelingthedenominatorinEquation4{1yieldstheXCCfunctional E XCC = h 0 j e T y H N e T C j 0 i + h 0 j H j 0 i : {3 Thisexpressionismanifestlylinked.BecausethefactorizationinEquation4{2doesnot dependonanythingaboutthewavefunctionotherthanitbeinganexponential,theXCC functionalisidenticaltotheVCCfunctionalforallvaluesof T .Thus,theamplitude equationswillbethesameforXCCandVCCaswell.Then,thegroundstateenergycan bedenedby E 0 =min T y h 0 j e T y H N e T C j 0 i + h 0 j H j 0 i {4 whichimpliestheamplitudeequations hKj e T y H N e T L j 0 i =0 : {5 InCI,satisfactionoftheequivalentexpression hKj H N + C j 0 i = E hKj + C j 0 i {6 impliesthattheenergyisnotonlystationarybutalsoaminimum,sincetheCIexcitation hessianis H CI KM = hKj H N )]TJ/F15 11.9552 Tf 11.956 0 Td [( E jMi 0 ; {7 apositivesemi-deniteoperator,withtheonlyzerooccurringatthegroundstateenergy itself.Ontheotherhand,theXCChessianis H XCC KM = hKj e T y H N e T L jMi + hKj a M e T y H N e T L j 0 i : {8 174

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Thersttermcanbeshowntobepositivesemi-denite,butthesecondtermisnot necessarilypositive.Ifallexcitationsareincludedin T FCC,thenthesecondterm vanishesandthehessianwillbepositivesemi-denite,guaranteeingaminimumas thestationarypoint.However,fortruncatedexcitationoperators,thesecondtermis non-vanishinganditispossiblefortheequationstoconvergetoastatethatisnota minimum. TheXCCapproachhasafundamentalproblem.Boththeenergyandamplitude equationsareinniteseries.Toseethisnotethattheonlyrestrictionontheenergyis thatthetermsbeconnected.Thatmeansthatthefollowingtermisallowedintheenergy expectationvalue 1 5!6! h 0 j nh )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(T y 6 T 5 i C H N o C j 0 i {9 andtermswithevenhigherpowersof T and T y .Therefore,inpractice,onemusttruncate theXCCfunctionalviasomecriterionforuseinaprogram.IwilldiscussvariousXCC truncationschemesinSection4.2.4. 4.2.2UnitaryCoupled-ClusterTheory AnalternativetoXCCisunitarycoupled-clustertheoryUCC.InUCCone recognizesthatifthecorrelationoperatorwereunitarythen,because U y U = I for anyunitaryoperator,thedenominatorinEquation4{1woulddisappear.Anyunitary operatorcanbewrittenasanexponentialofananti-hermitianoperator,sothefollowing ansatzcanbeusedforUCC E UCC = h 0 j e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e j 0 i h 0 j e )]TJ/F22 7.9701 Tf 6.586 0 Td [( e j 0 i + h 0 j H j 0 i = h 0 j e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e j 0 i + h 0 j H j 0 i = h 0 j H N e C j 0 i + h 0 j H j 0 i {10 wherethecorrelationoperator is = N e X n =1 n n = 1 n 2 X ijk abc abc ijk \010 a y ib y jc y k )]TJ/F26 11.9552 Tf 11.956 9.684 Td [( i y aj y bk y c ; {11 175

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soeachindividual n isbothanexcitation and de-excitationoperator. y = )]TJ/F21 11.9552 Tf 9.298 0 Td [( ,whichis thedenitionofananti-hermitianoperator.Thisdenitionofthecorrelationoperatoris equivalenttotheparameterization = T )]TJ/F21 11.9552 Tf 11.955 0 Td [(T y : {12 Because[ T;T y ] 6 =0thisexpressionisnotterminating.TheBaker-Campbell-Hausdor expansionoftheUCCansatzis H N e C = H N +[ H N ;T ]+ 1 2 [[ H N ;T ] ;T ]+ H N ; [ T y ;T ] + h:c: : {13 Becauseanynumberof T operatorscancontractwith T y operators,thisseriesisobviously notnite,meaningthat,likeXCC,onemusttruncateUCCtouseitinapractical program.DependingonhowonetruncatesUCC,theremaybenodierencefromXCC [215]. 4.2.3Strongly-ConnectedXCC TheXCCenergyfunctionalisbothclosedandlinked,andthereforeconnected, E XCC = h 0 j e T y H N e T C j 0 i + h 0 j H j 0 i : {14 Therefore,theamplitudeequationsareclearlylinked,butbecausetheyareopentheyare notnecessarilyconnected, hKj e T y H N e T L j 0 i =0 : {15 However,itcanbeshownthatonlyconnectedtermscontributetotheamplitude equations.Ifonlyconnectedtermscontributetotheamplitudeequationsthenthe energywillbemanifestlysize-extensive. Adisconnected,butlinked,operatorcanbeclosedinoneoftwoways:eitherit canbeclosedtoyieldaconnected,linkedclosedoperatororitcanbeclosedtoyield anunlinkedproduct.Ifanopendiagramisconnected,thenclosingtheoperatorcan onlyyieldlinkedterms.Ifanoperatorisdisconnecteditcanbewrittenasaproduct ofconnectedoperators.Lookingspecicallyatadoubleexcitation,adisconnectedbut 176

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linkedpiecewouldhavethealgebraicstructure ^ D rs pq = ^ D r p ^ D s q : {16 Nowthiscanbeclosedbyaproductoftwooperators X pqrs L p r L q s ^ D r p ^ D s q = X pr L p r ^ D r p X qs L q s ^ D s q {17 whichistheproductoftwoclosedoperators,andisthereforeunlinked.Ifanoperatoris connected,thenonecannotwriteitinproductform, X pqrs L p r L q s ^ C rs pq ; {18 therefore,thecorrespondingcloseddiagramisnecessarilylinked.Theamplitudeequations forXCCcanthenbefactoredintoconnectedanddisconnectedpieces.Dene X a i = h a i j e T y H N e T L j 0 i X ab ij = h ab ij j e T y H N e T L j 0 i : {19 andsoforthforhigherexcitations.Now,itisclearthat X a i cannotbewrittenasa product,soitisnecessarilyconnected,i.e. X a i = X a i c .Fordoubleexcitations,onecan factortheexpressionintoconnectedanddisconnectedpieces, X ab ij = h ab ij j e T y H N e T C j 0 i + 1 2 P ij j ab X a i c X b j c = X ab ij c + 1 2 P ij j ab X a i c X b j c ; {20 where P pq j rs = P pq P rs and P pq =1 )-272(P pq ,with P pq interchangingthelabels p and q .Ofcourse,thesolutionofthesinglesequationsimplythat X a i = X a i c =0,so X ab ij = X ab ij c .Movingtotriples, X abc ijk = h abc ijk j e T y H N e T C j 0 i + P i=jk j a=bc X a i c X bc jk c + 1 3! P ijk j abc X a i c X b j c X c k c ; {21 where P p=qr =1 )-222(P pq )-222(P pr and 177

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P pqr =1 )-222(P pq )-222(P pr )-222(P qr + P ab P ac + P ac P ab P pqr j stu = P pqr P stu : {22 Giventhelowerexcitationequations, X abc ijk = X abc ijk c .Atanarbitraryexcitationlevel, thedisconnectedtermswillbeproductsoftheconnectedtermsfromloweramplitude equations,andwecanchoosetheloweramplitudeequationstobeconnectedandsatised. Byinduction,everyexcitationlevelcanbewritteninconnectedform,i.e. hKj e T y H N e T L j 0 i = hKj e T y H N e T C j 0 i : {23 Thisrelationonlyholdsatthestationarypointoftheenergyfunctional,however, onecandeneanewenergyfunctionalusingonlyconnectedamplitudeexpressions,which willhavethesamestationarypoint.However,becausetheamplitudeequationshave beendevelopedbyonlydierentiatingwithrespectto T y ,thenewfunctionalwouldbe non-hermitian.Theseconnectedtermsintheamplitudeequationsthatbreakhermitian symmetrymustalsobezero,sincethefunctionalitselfishermitian.Eliminatingthose extraterms,andgivenEquation4{23,onecanfactortheenergyfunctionalintotwo pieces: h 0 j e T y H N e T C j 0 i = h 0 j e T y H N e T SC j 0 i + h 0 j e T y H N e T WC j 0 i {24 where SC indicatestermswheretheconnectivityissuchthatremovinganyone T or T y willnotmakethetermdisconnectedstronglyconnected"and WC indicatesthose termsthatdobecomedisconnectedproductsuponremovingasingle T or T y weakly connected".Recognizingthatatthestationarypointthe WC termsarezerobothinthe energyandamplitudeequations,onecanremovethemfromthefunctional,deninganew functional E SC-XCC = h 0 j e T y H N e T SC j 0 i + h 0 j H j 0 i {25 whoseamplitudeequationsare 178

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h 0 j a y K e T y H N e T SC j 0 i =0 : {26 Thisreformulation,whilecontainingfewertermsthanXCC,isstillaninniteseries,so thesameproblemsariseaswithconventionalXCCorUCC.However,bythestructure ofthefunctional,foranytruncationoftheexponential,boththeenergyandamplitude equationswillbeconnected.AsimilarderivationofSC-XCCwasperformedin[218]. 4.2.4Truncations Theinniteseriesnatureofallthreeoftheseapproachesmaketheminfeasible asacomputationaltechnique.Thegoalistondconvenientapproximationstothese methodsthatmaintaintheirproperties.Mostimportantly,anyapproximationmustbe size-extensiveand,sothatitsatisesthegeneralizedHellman-Feynmantheoremwithout needingtosolvefor,hermitian.Onemustignorethevariationalconditiontosatisfy thesetwoproperties. Inpractice,approximationstoXCCandUCCmethodshavefallenintotwoclasses: therstisperturbativetechniques,wheretermsareincludedthroughcertainordersin theelectron-electroninteraction.ThesemethodsarebestexempliedbytheXCC approach[209{211,216].Becausethoseapproachesaretiedtothemagnitudeofthe perturbationwhichisafunctionofhowwelltheHartree-Fockreferencedescribesthe systeminquestiontheywillinevitablybreakdownwhereHartree-Focktheoryispoor. Unfortunately,itisinexactlytheseregionswhereamethodbeyondconventionalCCare necessary.Therefore,Ibelievethatother,non-perturbativetruncationtechniquesare preferable. Thesimplestwaytotruncatetheseseriesnon-perturbativelyistoreplaceexp T anditsconjugatebyanapproximationtotheexponentialfunction.Onecouldchooseto truncatetheexponentialatlinearorder,yieldingahermitianlinearizedCCLinCC, E LinCC = h 0 j )]TJ/F15 11.9552 Tf 10.461 -9.683 Td [(1+ T y H N + T SC j 0 i + h 0 j H j 0 i : {27 179

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Thepolynomialapproximationthrougha p thorderapproximationtotheexponentialis E [ p ] SC-XCC = h 0 j H j 0 i + p X m =0 p X n =0 1 m n h 0 j )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(T y m H N T n SC j 0 i {28 andthestationaryequations p )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 X m =0 p X n =0 1 m n h 0 j h a y K )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m H N T n i SC j 0 i =0 : {29 Thestrongly-connectednatureofthesetruncationsleadstocompletelyconnected amplitudeandenergyexpressions,guaranteeingextensivity. Therearealsonon-perturbativebasedtruncationsofthesefunctionals,assuggested byKutzelnigg[212].Intheseapproaches,onekeepsanumberof T and T y terms,giving T animplicitorderparameter,andthenkeepallofthetermsthroughagivenorderinthat parameter.Thisisequivalenttomakingapolynomialapproximationtoexp inUCC, leadingtotheexpression E p SC-XCC = h 0 j H j 0 i + p X m =0 m X n =0 1 m )]TJ/F21 11.9552 Tf 11.955 0 Td [(n n h 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(n H N T n i SC j 0 i {30 andstationaryequations, p )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 X m =0 m X n =0 1 m )]TJ/F21 11.9552 Tf 11.955 0 Td [(n n h 0 j h a y K )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(n H N T n i L j 0 i =0 : {31 Notethatunlessstrong-connectivityisenforced,theseexpressionscanhavedisconnected contributionstotheamplitudeequations.Thesedisconnectedcontributionstothe amplitudeswillleadtounlinkedcontributionstotheenergy. Apriori ,thereislittleobviousreasontopreferoneofthesemethodsof non-perturbativeexpansiontoanother.However,therstchoicepolynomial approximationcanbeusedtogenerateaLinCCmethod,whichcanbeviewedasusinga CIparameterizationofthewavefunction,butusingaconnectedexpectationvalue.Onthe otherhand,theorderparameterapproximationsfocusontheenergyfunctional,notthe wavefunction.Therefore,thereisnonaturalanalogywithotherwavefunctionmethods. 180

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WhiletheXCC,UCC,andSC-XCCapproachsareallequivalentinfullexponential form,onlypolynomialtruncationsthatmaintainstrong-connectivitywillbesize-extensive. Therefore,itisnecessarytotruncateusingtheSC-XCCformoftheequations. 4.2.5BeyondtheGroundState Equation-of-MotionExcitedStates :Theoreticaldevelopmentofhermitian approachestoCChavefocuseduponthegroundstate.However,CCalsodealswith excitedstatesandproperties.IntheliteraturebothXCC[219]andUCC[220]havebeen showntohaveconsistentexcitedstategeneralizations.Becausethenon-commutingnature ofUCCcomplicatesthedevelopment,IwillfocusonXCC-basedmethods. Unlikethecaseforgroundstates,thereisasignicantdierencebetweenCImethods forexcitedstatesandCCmethodsforexcitedstates.Becauseoftheeigenvaluestructure oftheCIequations,approximationstoexcitedstateswithinCIcorrespondtoorthogonal solutionsoftheeigenvalueequation.Forcoupled-clustermethods,ontheotherhand, theequationsareeithernonlinearforCC,XCC,etc.orlinearforLinCCvariants,see Section4.3.InthecaseofLinCC,whentheequationsarewell-posed,thereisonlyone solution.Inthenonlinearcase,alternativesolutionscanbefound,butthesestatesmay notcorrespondtophysicalexcitedstates[221{226]. ForCI, j 0 i = ^ C j 0 ij m i = ^ C m j 0 i ; {32 sothegroundstateandexcitedstateshavethesamelinearstructureandobeythesame eigenvalueequation.ForCC,theparameterizationsaredierent, j 0 i = e T j 0 ij m i = ^ R m e T j 0 i : {33 TheintrinsicasymmetryintheCCapproachrelativetotheCIapproachmeansthatone cannoteasilymapbetweenthetwoforexcitedstates.DespitethefactthatforFVCC, theexcitedsolutionsmustexactlymatchthoseofFCI,foranyotherapproximationeven 181

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truncatedVCC,oneneedstoorthogonalizetheexcitedstateswithrespecttotheground state.Therefore,theproperexcitedstateansatzfromtheVCCviewpointcanbewritten m =min R m h 0 j e T y R m y H N R m e T j 0 i h 0 j e T y R m y R m e T j 0 i )]TJ 13.15 8.088 Td [(h 0 j e T y H N e T j 0 i h 0 j e T y e T j 0 i : {34 Itwasshownin[219]thatonecanapplytheXCCfactorizationAppendixDtothis functionalaswell.Afterfactorization,oneisleftwiththefullyconnectedexcitation energyexpression m = h 0 j h e T y R m y H N R m e T i C j 0 i h 0 j e T y R m y R m e T C j 0 i : {35 Makingthisequationstationarywithrespecttovariationsin R m y hKj h e T y H N e T R m i L j 0 i = m hKj h e T y e T R m i L j 0 i ; {36 whichisanonlinearequationfor m .UnlikeEOM-CCorCI,thisequationisnotan eigenvalueequation;comparethisequationtoEquation1{124. ResponseTheory :Linearandhigherorderresponsetheorycanbedevelopedby examiningthetime-dependentgeneralizationsofcoupled-clustertheory.Denetheaction functionalas[71,72] A XCC = Z t 1 t 0 d E XCC {37 wherethetime-dependentenergyfunctionalis, E XCC = h 0 j e T y H N )]TJ/F21 11.9552 Tf 11.955 0 Td [(i @ @ e T C j 0 i + h 0 j H j 0 i = h 0 j h e T y H N )]TJ/F21 11.9552 Tf 11.955 0 Td [(i T e T i C j 0 i + h 0 j H j 0 i ; {38 where T isthetime-derivativeof T .BecauseIamintendingthisdevelopmentfor understandingresponseproperties,Iamfreetochoose j 0 i tobetime-independent. TheproperstationaryconditionsfortheactionaretheEuler-Lagrangeequations, @ E XCC @t K )]TJ/F21 11.9552 Tf 16.36 8.088 Td [(d d @ E XCC @ t K =0 @ E XCC @t K )]TJ/F21 11.9552 Tf 16.36 8.088 Td [(d d @ E XCC @ t K =0 : {39 182

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Inthetime-independentcasethesecondtermvanishes,whichyieldsthestandard stationaryequationsusedbefore.Thesetwoequationscanbecombinedintoastationary equationfortherealandimaginarypartsofthewavefunction.Iwillassumefor conveniencethat T isreal.Becauseofhermicity,assuming T realmeansthatonlya singleequationneedstobesolved.Indynamicalsimulationsthisassumptionwillnothold, butitissucienttorevealthestructureoftheequations. Expandingthederivatives,thestationaryequationbecomes hKj h e T y H N )]TJ/F21 11.9552 Tf 11.955 0 Td [(i T e T i L j 0 i =0 : {40 Assumethat H iscomposedofatime-independentHamiltonian H anda time-dependentperturbation V .Tosimplifythenotation,Iwilldroptheexplicit dependenceon .Then,throughzerothorderintheperturbation,theenergyfunctionalis E XCC = h 0 j h e T y H N )]TJ/F21 11.9552 Tf 11.955 0 Td [(i T e T i C j 0 i + h 0 j H j 0 i : {41 Makingthisequationstationarywithrespecttothezeroth-orderparameters, hKj h e T y H N )]TJ/F21 11.9552 Tf 11.955 0 Td [(i T e T i L j 0 i =0 ; {42 whichcanbesolvedbychoosing T tosatisfythetime-independentstationarycondition and T =0. Therst-orderenergyfunctionalis E XCC = h 0 j h e T y H N T e T i C j 0 i + h 0 j h e T y T y H N e T i C j 0 i + h 0 j h e T y V N e T i C j 0 i )]TJ/F21 11.9552 Tf 11.955 0 Td [(i h 0 j h e T y T T e T i C j 0 i)]TJ/F21 11.9552 Tf 19.261 0 Td [(i h 0 j h e T y T e T i C j 0 i + h 0 j V j 0 i : {43 Thersttwotermsofthisequationvanishduetothezeroth-orderstationaryconditions. Thethirdtermcanbechosentobezerobyaconstrainton T :Fouriertransformingthe rst-orderenergy, 183

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E XCC = h 0 j h e T y V N e T i C j 0 i + h 0 j h e T y T e T i C j 0 i + h 0 j V j 0 i : {44 Byrequiringtherst-orderperturbedwavefunction, T e T ,tobeorthogonaltothe zeroth-orderwavefunction,thenthetermdependentontherst-orderwavefunctioncanbe removedfromthefunctional,leaving E XCC = h 0 j h e T y V N e T i C j 0 i + h 0 j V j 0 i ; {45 whichistheexpectedrstorderresult.Thesecond-orderenergyfunctionalis E XCC = h 0 j h e T y H N T 2 + T e T i C j 0 i + h 0 j h e T y T y 2 + T y H N e T i C j 0 i +2 h 0 j h e T y T y H N T e T i C j 0 i +2 h 0 j h e T y )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y V N + V N T e T i C j 0 i )]TJ/F21 11.9552 Tf 11.955 0 Td [(i h 0 j h e T y T y 2 + T y T + T + T T 2 + T e T i C j 0 i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 i h 0 j h e T y T y T T + T T + T y T e T i C j 0 i : {46 Usingthelower-orderstationaryconditionsandtheorthogonalityoftheperturbed wavefunctions, E XCC = h 0 j h e T y H N T 2 e T i C j 0 i + h 0 j h e T y T y 2 H N e T i C j 0 i +2 h 0 j h e T y T y H N T e T i C j 0 i +2 h 0 j h e T y )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y V N + V N T e T i C j 0 i )]TJ/F21 11.9552 Tf 11.955 0 Td [(i h 0 j h e T y T T +2 T y T e T i C j 0 i : {47 Imposingstationaritywithrespectto t K givestheamplitudeequation, 2 hKj h e T y T y H N e T i L j 0 i +2 hKj h e T y H N T e T i L j 0 i +2 hKj h e T y V N e T i L j 0 i)]TJ/F15 11.9552 Tf 19.261 0 Td [(2 i hKj h e T y T e T i L j 0 i =0 : {48 Fouriertransformingthisequation,andlookingforpoleswhere V =0, 184

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hKj h e T y T y H N e T i L j 0 i + hKj h e T y H N T e T i L j 0 i = hKj h e T y T e T i L j 0 i : {49 Unlikethecasefornon-hermitianCC,thersttermofEquation4{49cannotbe directlymappedintotheEOMequation,Equation4{36.Therefore,unlikethecaseof conventionalcoupled-clustertheory,therearedierencesbetweenthelinearresponse andequation-of-motionexcitationenergies.Thisdierenceisanalogoustothatbetween responsesecond-orderpropertiesandEOM-basedsecond-orderproperties[73]. Thisdierencebetweenthetwoapproachesissomewhattroubling.Itisclearthatthe LR-XCCapproachincludeshigher-ordercontributionsthandoestheEOM-XCCapproach. Unfortunately,thestructureoftheequationsalsomakesLR-XCCmorecomplicated. Numericalapproximationswillhavetobetestedtodeterminewhetherthebenetsof LR-XCCinsometruncatedformareworththeaddedexpense. MultireferenceTheory :Asapointoftheoreticalconsistency,itwouldbesatisfying ifmultireferenceCCcouldbegeneralizedtoahermitian,XCC-likeframework.In thissection,IdevelopanalternativeSU-MRXCCframework,similartothestandard Jeziorski-Monkhorstapproach[227].Palandco-workers[228]developedanXCC-like methodologyforFock-spacevalence-universalmultireferenceCC,therefore,hermitian state-universalMRXCCwouldcompletetheparallelbetweennon-hermitianCCand hermitianXCC. Multireferencecoupled-clustertheorycanbewrittenintermsofawaveoperator anditsdeningequation,theBlochequationareformulationoftheSchrodingerequation inthemodelspace H = H : {50 AgeneralhermitianwaveoperatorforacompletemodelspaceCMSis[229{233] = X e T j 0 i S )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 h 0 j e T y {51 185

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where S = h 0 j e T y e T j 0 i : {52 TheCMSdenitionwouldgenerallyrequiretheconditionthat h 0 j e T j 0 i = ; {53 i.e.therearenocomponentsofthe wavefunctionthatlieinthereference .While thisconditionwouldseemtoberelevantinthehermitiancaseaswell,thede-excitations complicatethings.WhileEquation4{53willhold, S 6 =0ingeneral.Yet,requiring S =0wouldseemtobeamorephysicalrequirement.Factoringtheoverlap, S = h 0 j e T y e T cl j 0 ih 0 j e T y e T op j 0 i = S cl S op : {54 Theterm S op correspondstoopentermsintheexpansion,whicharecontributionsthat directlylinkreference j 0 i and j 0 i .Inthecasewhere = thentheoverlapisan expectationvalue,andonlycloseddiagramscansurvive.Therefore, S = S cl : {55 Theopentermswilldestroyextensivity;theyaresimilartoindirectinternalexcitations thatbreakextensivityinthecaseofnon-hermitianincompletemodelspaces.Forgeneral modelspaces,LiandPaldus[234{239]showedthatusingtheconnectivityconditions"or C-conditionsoneisabletoreintroduceextensivityinanincompletemodelspace.These conditionsintroduceexplicitinternalexcitationsandchoosesthevaluesofthosevariables soEquation4{53holds.Iintroduceananalogyofthoseconditionsbyrequiring S op =0 8 6 = : {56 Theinternalexcitationsareinaone-to-onemappingwiththeseopenoverlapelements,so theseequationscanallbesatised.Thisconditionimpliesthat S =0forall 6 = 186

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UsingthenewC-conditions,thehermitianwaveoperatorhastheform = X e T j 0 ih 0 j e T y S cl : {57 InsertingthiswaveoperatorintotheBlochequationandprojectingon e T j 0 i He T j 0 i = X e T j 0 i )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(S cl )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 h 0 j e T y He T j 0 i : {58 AsshowninAppendixD,onecanfactorthetransitionmatrixelementintotwopieces yielding h 0 j e T y He T j 0 i = h 0 j e T y He T L; op j 0 i S cl + h 0 j e T y He T cl j 0 i S )]TJ/F21 11.9552 Tf 11.956 0 Td [( = h 0 j e T y He T L; op j 0 i S cl : {59 OnecanthenidentifytheeectiveHamiltonianasthematrixwhoseelementsare H e = h 0 j e T y He T L; op j 0 i ; {60 leadingtoaclearlyhermitianmatrix.Oncetheamplitudesthatdenetheeective Hamiltonianhavebeendetermined,thenitcanbediagonalized,yieldingtheproper multireferencewavefunction, H e c = cE {61 where c arethereferencecoecientsand E isthediagonalmatrixofenergies.Thenal wavefunctionswillbeinformforstate p j p i = X c p e T j 0 i : {62 TheBlochequationcanbesimpliedto He T j 0 i = X e T j 0 i S cl S cl H e : {63 Projectingontheleftby hK j e T y ,where hK j isanexciteddeterminantrelativeto h 0 j andisnotoneoftheotherreferences,onehas 187

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hK j e T y He T j 0 i = X hK j e T y e T j 0 i S cl S cl H e : {64 Theleft-handsideofthisequationcanbefactoredintoEquationD{17 hK j e T y He T j 0 i = hK j e T y He T L j 0 i S + H e hK j e T y e T j 0 i : {65 Thissecondtermappearsonbothsidesoftheequation,andcanbeeliminated hK j e T y He T L j 0 i S = X 6 = hK j e T y e T j 0 i S cl S cl H e : {66 Inasimilarfactorization,andfollowingfromthedevelopmentinAppendixD, hK j e T y e T j 0 i = hK j e T y e T L; op j 0 i S cl : {67 Ofcourse, S cl = S cl ,so hK j e T y He T L j 0 i = X 6 = hK j e T y e T L; op j 0 i S cl 2 S cl S cl H e : {68 ThisequationisquitesimilartothestandardJeziorski-Monkhorst[227]state-universal multireferencecoupled-clusterformulation.Inthatapproach,theeectiveHamiltonianhas theform H e = h 0 j )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(He T C j 0 i ; {69 andtheBlochequationis hK j )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(He T C j 0 i = X 6 = hK j e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T e T j 0 i H e : {70 Thereisonekeydierence:theratioofoverlapsontheright-handsideofEquation4{68. Eachoftheseoverlaptermsisindividuallyunlinked;however,iftheratioislinked, thentheamplitudeequationislinked,andthisisafullyconsistentformulationof state-universalmultireferenceXCCSU-MRXCC.Thusfar,Ihavebeenunableto rigorouslyshowthattheoperatoris,infact,linked.If j 0 i = j 0 A ij 0 B i and 188

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T = T A + T B forall ,where A and B aretwofragmentsofthemolecule,then S cl = h 0 j e T y e T cl j 0 i = h 0 A j e T y A e T A cl j 0 A ih 0 B j e T y B e T B cl j 0 B i = S cl A S cl B {71 Similarly, S cl = S cl A S cl B {72 Therefore,theratiosplitscleanlyintoaproductoverunitson A and B S cl 2 S cl S cl = S cl A 2 S cl A S cl A S cl B 2 S cl B S cl B {73 Thisisnotaproofofproperscalingwithparticlenumber,howeveritissuggestiveofthe properlinkedbehavioroftheamplitudeequations. 4.2.6Conclusion Ihavepresentedpurelytheoreticalinvestigationsofthepossibilityofusinghermitian coupled-clustertheories.Whilethehermitiancoupled-clustermethodsallareextensive, variational,andsatisfythegeneralizedHellman-Feynmantheorem,theyarealsoall inniteseries.Therefore,onemusttruncatetohaveamethodthatisusableinpractice. Thoughtherearethreehermitiancoupled-clustervariantsdiscussedhere,theycanall bemadeequivalentwhenlookingatniteordertruncationsoftheexpressions.For thesetruncationstoremainrigorouslysize-extensive,thebestchoiceistotruncate strongly-connectedexpectation-valuecoupled-clustertheory.Thelineartruncationofthis methodwillbediscussedindetailinSection4.3.IbelievethataquadraticSC-XCCcould beagoodchoicetoinvestigatemoreclosely.ItwouldencompassCCSD,withadditional hermitianterms,withoutsignicantlyincreasingcomputationalscaling. Theexcitedstateprobleminthesehermitianmethodsismorecomplicatedthan itisfornon-hermitianCC.Theequation-of-motionvariantofXCCdoesnotgivethe sameexcitationenergiesasdoeslinear-responseXCC.Whetherthisdierencewillbe 189

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numericallyimportantisunknownatthispoint.Comparingthelinearizedversionofthese twoequationscouldgiveinsightintothisproblem.Forstate-universalmultireference coupled-cluster,Idevelopedanewframeworkthatshouldyieldahermitianeective Hamiltonian.ThisframeworkisbasedonageneralizationofPaldus'sC-conditionsand theapplicationofseveralfactorizations.Theendresultappearstoscaleproperlywith particlenumber,butproofofsize-extensivityisstillwanting. 4.3LinearizedCoupled-ClusterTheory 4.3.1Introduction Despitethemanysuccessesofcoupled-clustertheoryCC,therearestilllimitations thatkeepCCfrombeingmoregenerallyapplicable.Ifonewantedtoperformafully ab initio coupled-clustermoleculardynamicsMDsimulationthecalculationofbothenergies andforcesmustbefast,sinceateachpointoftheMDtheforcewillbeneeded,and,to makethemethodpractical,themethodmustscaletolargenumberofprocessors.Bothof theserequirementsarediculttoaddresswithconventionalcoupled-clustertheory.The highlynonlinearnatureoftheequationsmakesthemdiculttoparallelizeandrequires largeamountsofdatapassing.BecauseconventionalCCisnon-hermitian,onemust solveanadditionalsetoflinearequationstheequationsifforcesareneeded.The nonlinearcoupled-clusterstructurealsoleadstodicultiesinanalyzingtheerrorsmadein varioustruncationsofelementsintheequations.Theuseoflocalizationapproximations, forexample,aremadewithoutbeingabletoaddrigorouserrorboundstotheequations [140,240]. Linearizedcoupled-clusterLinCCtheoriesoerseveraladvantagesoverconventional CC.WhilethescalingofLinCCsinglesanddoublesisthesameasCCsinglesand doubles,thelinearizedvariantisfaster.Thesimplicityofthelinearapproachmeansthat onecaneasilyparallelizeLinCCanddosoinamoreecientwaythaninconventional CC.HermitianLinCCdoesnotrequirethesolutionofequationsforforces.Also,the linearequationsmakeitpossibletoguaranteethatthewavefunction"isspin-symmetry 190

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adaptedwithoutintroducingthecomplexityofthenonlinear,non-commutingoperators [241].ThisadditionalsymmetrywouldfurtherreducethecomputationalcostofLinCC relativetoconventionalCC.Becauseofthesimplicityofthestructureoftheequations, theinuenceofvariousapproximationschemes,suchasusinglocalizedorbitals,canbe understoodinamuchmorerigorouswaythaninthenonlinearequations.Potentially,one mightbeabletoprovideboundsontheerrorintroducedbylimitingexcitationstocertain distanceranges. Unfortunately,thereisabenettothenonlinearformulationofCCthatisnot presentinLinCC.Whenaneardegeneracyispresentonapotentialenergysurface,the nonlinearequationsofconventionalCCareabletoamelioratethepotentialsingularity, allowingmeaningfulpotentialenergycurvestobecalculatedwithCC.Thelinear equationsofLinCCareunabletocontrolthesingularity[39],makingLinCCuseless forpotentialenergysurfaces.Innumericalanalysis,singularitiesinequationsare oftenindicativeofill-posedorill-conditionedequations.Whenconfrontedwithan ill-conditionedmatrix,onecanuseregularizationtechniques[4,75]tomaketherelevant equationmorenumericallystable.IattempttoremovethesingularityproblemofLinCC byusingamodiedTikhonovregularization[75]. TheregularizedLinCCSDisappliedtosmalltestbond-breakingexamplestoexamine whethersmoothandaccuratepotentialenergysurfacesarepossible.LinCCSDisalso testedonthecalculationofequilibriumgeometriesofseveralsmallmoleculesandonthe calculationofactivationbarriersandheatsofreactionforseveralwell-characterizedsmall moleculereactions. Finally,anoteonnomenclature.Linearizedcoupled-clustertheoryhasbeenknown byseveralnamesintheliterature.PrimarilyithasbeenknownasLCC[39],butLCC methodshavealsobeenknownbyLCPMET[46],CPA 0 [242],DMBPT 1 [243,244], andCEPA-0[245,246].BecauseLCChasbecomethepreferrednotationforlocalized coupled-clustertheory[240],IprefertouseLinCCtoavoidconfusion. 191

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4.3.2Theory Theprocessoflinearizationofthecoupled-clusterfunctionalcanseemapurely computationaltrick.However,linearizedCCexistsinanaturalframeworkof approximations.Thecoupled-clusteransatzencodestheinformationthataparticularly ecientwaytoreproducetheexactwavefunctionisbyusinganexponentialoperatingon asingledeterminantreferencefunction.Insertingthisansatzintothestandardquantum mechanicalexpectationvalueexpressionandapplyingtheRayleigh-Ritzvariational principleyieldsvariationalcoupled-clusterVCC. E VCC = h 0 j e T y H N e T j 0 i h 0 j e T y e T j 0 i + h 0 j H j 0 i : {74 Unfortunately,whileVCCishermitianandanupperboundtotheexactground-state energy,itisneithercomputationallytractablenormanifestlyextensive.Linearization ofthisfunctionalreplacingeachexponentialbyitslinearpartisalsoanexact parameterizationofthewavefunction;infact,itiscongurationinteractionCI.Of course,CIisnotextensive,butthatshowsthenaturalrelationshipbetweenlinear methodsandexponentialmethods.Similarly,theextendedcoupled-clusterapproach pioneeredbyArponenandBishop[59,247,248]includeshigherexcitationsthan conventionalcoupled-clusterbyincludinganexponentialde-excitationoperatoron theleft: E ECC = h 0 j e )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(H N e T C j 0 i + h 0 j H j 0 i : {75 Linearizationofexpyields E CC = h 0 j + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(H N e T C j 0 i + h 0 j H j 0 i ; {76 thenormalcoupled-clusterfunctional. Turningtothedevelopmentofafullylinearmethod,onecanbeginwith strongly-connectedexpectation-valuecoupled-clustermethodSC-XCC[218].The SC-XCCenergyfunctionalis 192

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E SC-XCC = h 0 j e T y H N e T SC j 0 i + h 0 j H j 0 i : {77 Truncatingtheexponentialsthroughlowestorderin T ,leadstowhatIcallLinCC, E LinCC = h 0 j )]TJ/F15 11.9552 Tf 10.461 -9.684 Td [(1+ T y H N + T SC j 0 i + h 0 j H j 0 i : {78 Thismethodisclearlyhermitian,extensive,andatstationaritytheenergywillbe E LinCC,stat = h 0 j H N T SC j 0 i + h 0 j H j 0 i ; {79 whichislinearin T Anon-hermitianLinCCisdenedbylinearizingtheECCfunctionalEquation4{75 inbothand T E LinECC = h 0 jf +[ H N + T ] C g C j 0 i + h 0 j H j 0 i : {80 IwillrefertothisvariantasLinECC.Keepingonlysingleanddoubleexcitations,LinCC andLinECCareidenticalforHFreferences.Fortripleexcitationsandhigher,thesetwo approacheswilldierforallreferences. ForbothLinCCandLinECC,theenergyexpressionatstationarityisexplicitly, E stat = X ia f i a t a i + 1 4 X ijab h ij jj ab i t ab ij {81 evenifhigherexcitationsthandoubleexcitationsareincluded.Onecanwritethe amplitudeequationsinarelativelysimpleform.ForLinCCSD,thesinglesamplitude equationis a i t a i = f a i + X a 6 = b f ab t b i )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X i 6 = j f ji t a j + X jb t b j h ja jj bi i + X jbc t bc ij h aj jj bc i)]TJ/F26 11.9552 Tf 19.261 11.357 Td [(X jkb t ab jk h jk jj ib i : {82 InthecaseofLinECCSD,thereisoneadditionalterminthesinglesequation, a i t a i =RHSLinCCSD+ X jb t ab ij f j b : {83 193

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ThedoublesequationforLinCCSDis ab ij t ab ij = h ab jj ij i)]TJ/F21 11.9552 Tf 19.261 0 Td [(P ij X k 6 = j t ab ik f kj + P ab X c 6 = b t ac ij f cb + 1 2 X cd t cd ij h ab jj cd i + 1 2 X kl t ab kl h kl jj ij i + P ij j ab X kc t ac ik h kb jj cj i)]TJ/F21 11.9552 Tf 19.261 0 Td [(P ab X k t a k h kb jj ij i + P ij X c t c i h ab jj cj i : {84 ThedoublesequationforLinECCSDisidentical.InLinCC,thereisnoneedtosolveaset ofequationsforthelagrangianmultipliersduetothehermicityoftheequations.Forthe non-hermitianLinECC,thereis.Inthiscase,theseequationsareexactlythesameasthe T equationsforLinCC,witheach T replacedbya. BothLinCCSDandLinECCSDwillscale O n 2 N 4 where n isthenumberofelectrons and N isthenumberofvirtualorbitals.ThisscalingisthesameasCCSD.However,fewer T amplitudesareneededforeach T equation,minimizingtheamountofdatapassing necessaryinanyparallelcode. PotentialEnergySurfaces :ThepotentialthatLinCCSDtotakeaconcrete examplehasforecientcomputationalcalculationshasbeenknownforquitesometime. However,onereasonthatitwassupersededbythefullynon-linearCCSDisthefailure ofLinCCSDawayfromequilibrium.Thisfailureisduetoasingularitythatarisesinthe LinCCSDequations.WritingtheLinCCSDequationsinmatrixform, At = g {85 where g arethebareintegrals, A arethecouplingsbetween T amplitudes,and t arethe T -amplitudesthemselves.If A becomesill-conditionedorsingular,thenthisequationwill bediculttosolve.Numericalevidenceshowsthatthisisthecaseasonemovesaway fromequilibrium.Thelinearnatureoftheequationsmakesthemespeciallysensitiveto degeneraciesinthemany-bodyspace. Ithaslongbeenknownthatasabondisbroken,theLinCCSDequationsbecome singular.Innumericallinearalgebra,whenamatrixequationbecomessingular,a 194

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standardtechniquetoremovethesingularityisknownas regularization .Thetwomost commonregularizationtechniquesaresingularvaluedecompositionSVDregularization andTikhonovregularization,aswerediscussedinSection1.4.Aspresented,bothmethods ofregularizationrequiretheSVDofthe A matrix,whichiscomputationallyimpossible inLinCCSD.Thedimensionofthe A scalesas n 2 N 2 n 2 N 2 .Therefore,theSVDwould requireassumingsparsitywasnotused n 6 N 6 operations,farbeyondthecomputational costofLinCCSDwithouttheSVD.Onemustreformulatetheproblemintermsthatcan beecientlysolvedviathestandardsolutionmethod. Becauseofthedimensionofthe A matrix,LinCCSDisalwayssolvedinaniterative scheme,despitethefactthatitslinearitypermitsadirectsolution.Themosttypical solutionschemeisasfollows.Thespin-orbitalequationsforLinCCDforsimplicityina canonicalbasisare h ab jj ij i)]TJ/F21 11.9552 Tf 18.914 0 Td [( ab ij t ab ij + 1 2 X kl h kl jj ij i t ab kl + 1 2 X cd h ab jj cd i t cd ij )]TJ/F21 11.9552 Tf 11.782 0 Td [(P ab j ij X kc h kb jj ci i t ac ik =0 : {86 Tosolvethisiteratively,theequationisrearrangedas h ab jj ij i + 1 2 X kl h kl jj ij i t ab kl + 1 2 X cd h ab jj cd i t cd ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(P ab j ij X kc h kb jj ci i t ac ik = ab ij t ab ij : {87 Thensettingeachsideoftheequationtoadierentiterationnumber, h ab jj ij i + 1 2 X kl h kl jj ij i t ab kl n + 1 2 X cd h ab jj cd i t cd ij n )]TJ/F21 11.9552 Tf 11.955 0 Td [(P ab j ij X kc h kb jj ci i t ac ik n = ab ij t ab ij n +1 : {88 Solvingthisequationfor t ab ij n +1 t ab ij n +1 = 1 ab ij h ab jj ij i + 1 2 X kl h kl jj ij i t ab kl n + 1 2 X cd h ab jj cd i t cd ij n )]TJ/F21 11.9552 Tf 9.299 0 Td [(P ab j ij X kc h kb jj ci i t ac ik n # : {89 195

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Assumingthatthesingularityisdominatedbyanear-degeneracyintheeigenvalues, inparticularthatforcertainvaluesof i;j;a;b ab ij 0,thenitwouldmakesensethat regularizationofthematrixcomposedpurelyofthoseeigenvaluesshouldbeabletosolve theproblemsofLinCCSD.Ifwedenethematrix E suchthat E ijab ; ijab = ab ij thenone canregularizethisequationusingeithertheSVDmethodortheTikhonovmethod.Then, theresultingiterativeequationswouldbe,fortheSVDcase t ab ij n +1 = j ab ij j)]TJ/F21 11.9552 Tf 17.932 0 Td [( cut 1 ab ij h ab jj ij i + 1 2 X kl h kl jj ij i t ab kl n + 1 2 X cd h ab jj cd i t cd ij n )]TJ/F21 11.9552 Tf 9.299 0 Td [(P ab j ij X kc h kb jj ci i t ac ik n # ; {90 whereistheHeavisidefunction,andfortheTikhonovcase, t ab ij n +1 = ab ij )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ab ij 2 + 2 h ab jj ij i + 1 2 X kl h kl jj ij i t ab kl n + 1 2 X cd h ab jj cd i t cd ij n )]TJ/F21 11.9552 Tf 9.299 0 Td [(P ab j ij X kc h kb jj ci i t ac ik n # : {91 InvertingEquation4{91,thelinearizedfunctionalbecomes E LinCC,reg = h 0 j )]TJ/F15 11.9552 Tf 10.461 -9.683 Td [(1+ T y H N + T SC j 0 i + 2 h 0 j T y 1 E 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(H N )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 T 1 j 0 i + 2 h 0 j T y 2 E 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(H N )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 T 2 j 0 i + h 0 j H j 0 i : {92 Notethattheadditionaltermvanishesatstationarityfromtheenergy,asonewould hope.Toperformtheregularizationfornon-HForbitals,theinverseof E 0 )]TJ/F21 11.9552 Tf 13.315 0 Td [(H N requiressemi-canonicalorbitals.Perturbedcanonicalorbitalsarerequiredforregularized derivativesaswell. Derivatives :Aswasdiscussedabove,thereisnoneedtosolveasetofequations forLinCC,andforLinECC,theequationsarethe T equationsforLinCC.Becausefor bothchoicestheenergyisstationary,thegeneralizedHellman-Feynmantheoremapplies. 196

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Onecandenetheone-andtwo-particledensitymatricesviatheequations,forLinCC pq = h 0 j )]TJ/F15 11.9552 Tf 10.461 -9.684 Td [(1+ T y p y q + T SC j 0 i ; )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(pq;rs = h 0 j )]TJ/F15 11.9552 Tf 10.461 -9.684 Td [(1+ T y p y q y sr + T SC j 0 i ; {93 whichisexplicitlyforLinCCSDincludingonlythecorrelationcontribution ij = )]TJ/F26 11.9552 Tf 11.291 11.357 Td [(X a t a i t j a )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 X kab t ab ki t kj ab ai = t a i {94a ab = X i t a i t i b + 1 2 X ijc t ac ij t ij bc {94b )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ij;kl = 1 2 X ab t ab ij t kl ab )]TJ/F22 7.9701 Tf 7.315 -1.794 Td [(ij;ka = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 X b t k b t ba ij + t b i t kj ab {94c )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ia;jb = t a i t j b )]TJ/F22 7.9701 Tf 7.315 -1.793 Td [(ij;ab = t ab ij {94d )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ab;ci = 1 2 X j )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(t j a t bc ji + t b j t ji ac )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ab;cd = 1 2 X ij t cd ij t ij ab : {94e ForLinECC, pq = h 0 j + p y q + T C j 0 i ; )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(pq;rs = h 0 j + p y q y sr + T C j 0 i {95 andthespin-orbitalequationsforLinECCSDaredeterminedbyreplacingthe T y termsin theLinCCSDdensitymatrices,using ia = t a i + X b t ab ij j b ; {96 andsymmetrizingthedensitymatrices. Fortheregularizedcase,thereareadditionaltermsinderivatives,butbecausethe equationsarestillstationarythereareofarelativesimpleform.Iftheorbitalsarekept semicanonical,thentheadditionalcontributiontothedensitymatricesare ii += 2 X a t a i t i a a i 2 + 2 2 X kab t ab ki t ki ab )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [( ab ik 2 {97a ab += )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 X i t a i t i b a i 2 )]TJ/F21 11.9552 Tf 13.151 8.087 Td [( 2 2 X ijc t ac ij t ij ac )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( ac ij 2 {97b whichcanbeaddedtotheexpressionsinEquations4{94. 197

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4.3.3Implementation BoththeTikhonov-regularizedandunregularizedhermitianLinCCDandLinCCSD andnon-hermitianLinECCDandLinECCSDhavebeenimplementedwithintheACES IIprogramsystem[54,55]forbothenergiesandgradients.Forallofthevariantsof linearizedcoupled-clustertheory,energiesandgradientsareavailableforarbitrary single-determinantreferences.Utilizingthedirectproductformulationofsymmetry,our implementationusesrealabelianpointgroupsymmetrytoacceleratecalculations. 4.3.4Results PotentialEnergySurfaces :Itisnecessarytoverifythattheregularizationworks properly,andtoidentifyaproperchoiceof 2 .InFigure4-1,thepotentialenergysurface forregularizedandunregularizedbond-breakingofhydrogenuoridearecomparedto thatofaUHFCCSDcalculation.InFigure4-1Aitisclearthattheunregularizedand regularizationparameter 2 =0 : 01E 2 h showproblems.Thesingularityoftheunregularized caseissomewhatmodiedbytheslightlyregularizedresult,butthereisstillaprofound dipinthepotentialenergysurface. Ontheotherhand,focusingonFigure4-1B,theexcellentqualityoftheregularized LinCCSDcurvewitharegularizationparameter 2 =0 : 1E 2 h isclear;itmatchesthe nonlinearCCSDresultclosely.Themajordeviationthatoccursisinthespin-recoupling region,wherethelackoforbitalrelaxationinLinCCSDislikelythesourceoftheerror. Thequalityoftheregularizedresultsaresuchthattheyholdformultiply-bonded systemsaswell.Onemightexpectthattheaddedcorrelationindescribinga multiply-bondedsystemwouldmakeitdicultforLinCCSDtodescribetheentire potentialenergycurve.InFigure4-2,Ishowtwoexamplesofmultiplybondedsystems. IntheFigure4-2A,thesymmetricbondstretchingcurveofH 2 OiscomparedforUHF CCSD,UHFLinCCSDwithoutregularizationandUHFLinCCSDwitharegularization 2 =0 : 1E 2 h ;theregularizedresulttrackscloselytheCCSDresults.Evenfortriply-bonded carbonmonoxide,Figure4-2B,theregularizedresultisaclosematchtoUHFCCSD. 198

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A B Figure4-1.Hydrogenuoridebond-breakingusinghermitianunrestrictedHFlinearized coupled-clustersinglesanddoublescomparedtoUHFCCSD.Allresultsarein a6-31G**basis,andthehorizontalaxisisinunitsoftheequilibriumbond lengthofHF, R e =0 : 9 A.ATheunregularizedLinCCSD,LinCCSDwitha regularization 2 =0 : 01E 2 h ,andwitharegularization 2 =0 : 1E 2 h comparedto UHFCCSD.BFocusingonUHFLinCCSDwitharegularizationof0 : 1E 2 h andUHFCCSD. 199

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Theseresultsareencouraging;withanon-optimizedchoiceofregularization parameter : 1E 2 h thebehaviorofLinCCSDisquiteacceptable.WhileUHFCCSDis notFCI,beingabletousealinearized,hermitianapproachandagreewiththenonlinear, non-hermitianresultswouldallowforlargersystemstobetreatedateectivelytheCCSD leveloftheory.Inalloftheresultspresentedbelow,theregularizationparameterhasbeen xedtothevalue 2 =0 : 1E 2 h Beforemovingontoproperties,itisimportanttoshowthelimitationsofthe regularizationapproach.Usingaspin-restrictedHFreferencebecomesproblematicfor regularizedLinCCSD.InFigure4-3Icomparethebond-breakingcurvesofhydrogen uoridewithRHFCCSDtothatfromRHFLinCCSDwithoutregularization,andwith regularizations 2 =0 : 01E 2 h and 2 =0 : 1E 2 h .WhenexaminingFigure4-3A,itappears thatthesameregularizationworkswellforRHFLinCCSD,asitdidforUHFLinCCSD. Unfortunately,inFigure4-3Bitisclearthattheregularizationhasadownside.At equilibrium,theregularizedresultisne,however,pastthepointwheretheRHF/UHF instabilityis,theregularizedRHFLinCCSDsolutionisqualitativelywrong.Thisresult makessense;usinganRHFreferencefunctionmeansthatintheasymptoticregionthe referenceisqualitativelywrong,andneardegeneracieswilloccur.Theregularization procedureremovesthoseneardegeneracies,whicharenecessarytogettheasymptotic RHFresultcorrect.ThedeviationfromtheCCSDresultearlieronthecurveisreects thefactthatorbitalrelaxationismoreimportantforRHForbitalsthanUHForbitals. Geometries :Intheprevioussection,IshowedthatregularizedLinCCSDwasa gooddescriptionoftheentirepotentialenergysurfacewhenusingaUHFreference.For themethodtobeuseful,itmustalsobeagooddescriptionofequilibriumproperties.I investigatedtheabilityofLinCCSDtoreproduceequilibriumgeometries,usingthesetof moleculesfrom[50].Becauseinthisrangeofgeometries,RHFdoesnotexhibitanynear degeneracies,geometryoptimizationswereperformedusingregularizedRHFLinCCSD. TheresultsarecollectedinTables4-1and4-2.Theaverageresultsforbothbondlengths 200

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A B Figure4-2.Multiplebond-breakingusinghermitianunrestrictedHFlinearized coupled-clustersinglesanddoublescomparedtoUHFCCSD.AThe unregularizedLinCCSDandwitharegularization 2 =0 : 1E 2 h comparedto UHFCCSDforH 2 Oincc-pVDZbasis,withthebondanglexedat110 : 6 andthebondlengthsinmultiplesoftheequilibrium R e =1 : 84345 a 0 .BThe unregularizedLinCCSDandwitharegularization 2 =0 : 1E 2 h comparedto UHFCCSDforCOincc-pVTZbasis,withthebondlengthsinmultiplesof theequilibrium R e =1 : 128 A. 201

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A B Figure4-3.Hydrogenuoridebond-breakingusinghermitianspin-restrictedHFlinearized coupled-clustersinglesanddoublescomparedtoRHFCCSD.Allresultsarein a6-31G**basis,andthehorizontalaxisisinunitsoftheequilibriumbond lengthofHF, R e =0 : 9 A.ATheunregularizedLinCCSD,LinCCSDwitha regularization 2 =0 : 01E 2 h ,andwitharegularization 2 =0 : 1E 2 h comparedto RHFCCSD.BFocusingonRHFLinCCSDwitharegularization 2 =0 : 1E 2 h andRHFCCSD. 202

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Table4-1.Comparisonofoptimizedequilibriumbondlengthsformultiple correlation-consistentbasissetsforLinCCSD.Averageswerecalculatedover thesetofmoleculesfrom[50].Forcalculationsusingacc-pVXZbasis,only valenceelectronswerecorrelated;forcalculationsusingacc-pCVXZbasisall electronswerecorrelated. isthesignedmeanerror, abs isthemean absoluteerror, max isthemaximumabsoluteerror,and std isthestandard deviation.Allnumbersareinunitsofpm. BasisSet FullwithoutF 2 abs max std abs max std cc-pVDZ1 : 721 : 728 : 351 : 461 : 481 : 483 : 400 : 68 cc-pVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 020 : 322 : 310 : 57 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 100 : 251 : 130 : 35 cc-pCVDZ1 : 661 : 668 : 221 : 451 : 411 : 413 : 330 : 68 cc-pCVTZ0 : 120 : 262 : 310 : 550 : 040 : 181 : 320 : 35 Table4-2.Comparisonofoptimizedequilibriumbondanglesformultiple correlation-consistentbasissetsforLinCCSD.Averageswerecalculatedover thesetofmoleculesfrom[50].Allelectronswerecorrelated. isthesigned meanerror, abs isthemeanabsoluteerror, max isthemaximumabsolute error,and std isthestandarddeviation.Allnumbersareinunitsofdegrees. BasisSet FullwithoutH 2 O 2 andNH 3 abs max std abs max std cc-pVDZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 971 : 974 : 941 : 49 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 331 : 332 : 440 : 80 cc-pVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 890 : 894 : 241 : 29 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 380 : 380 : 630 : 22 cc-pCVDZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 971 : 974 : 951 : 50 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 331 : 332 : 440 : 81 cc-pCVTZ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 001 : 004 : 251 : 28 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 490 : 491 : 040 : 29 andbondangles,theaverageerrorscomparefavorablywiththosefromCCSDTand CCSDT,seeTables3-9and3-10. Thestandarddeviationsandmaximumerrorsforbothbondlengthsandbondangles arenotasgoodasfromtheCCSDTandCCSDTmethods.RemovingF 2 fromthe setofmoleculesconsideredintheaveragesbringsthestandarddeviationsintomuchcloser agreementwiththeTmethods.Ashasbeendiscussedbefore,F 2 isaverydicult moleculetotreatproperly,andLinCCSDappearstobeunabletooptimizethestructure tothelevelofaccuracythatonewouldexpect. Onemightexpectthattheregularizationwouldbeunimportantforthegeometries, however,thatwasfoundnottobethecase.InseveralexamplesF 2 included, 203

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unregularizedgeometryoptimizationswerequalitativelyincorrect.ForF 2 ,the unregularizedLinCCSDequilibriumgeometrywasmorethan100%incorrect.This factsuggeststhattheinuenceofthesingularityisstrongeronthederivativethanitison theenergy. ActivationBarriers :Oneofthemaingoalsofquantumchemistryisthestudyof reactions.InTable4-3,IshowtheresultsfromapplyingregularizedLinCCSDtotheset ofTruhlar'swell-characterizedreactions[178,179].Thesecalculationswereperformed atthexedgeometriescalculatedusingQCISDusingtheregularizedLinCCSD.For closed-shellmolecules,anRHFreferencewasused,andforopen-shellmoleculesanROHF referencewasused.Unlikethecaseofequilibriumgeometries,theactivationbarriers deviatesignicantlyfromtheCCSDTandCCSDTresults.Mostimportantly,there areerrorsintheactivationenergyofupto4 : 5kcal/mol.Becauseoftheexponential dependenceofrateconstantsontheactivationenergy,thislargeanerrorisunacceptable. Fullgeometryoptimizationsandtransitionstatesearchesarenecessarytoseeifitis themismatchbetweenQCISDgeometriesandLinCCSDthatisthecauseoftheproblem, orifLinCCSDisunabletoaccuratelyreproduceactivationbarriers.Ifitisafailureof LinCCSD,thenitmightbepossibletoaddacorrectionduetotriplesatthestationary pointstoimprovetheenergetics.Furthertreatmentoforbitalrelaxationmaybenetthe calculationoftheopen-shellsystems,inparticular. 4.3.5Conclusion Linearizedcoupled-clustertheoryhasalotofadvantagesduetoitssimplicityand speed.Unfortunately,duetothepresenceofsingularitiesintheequations,LinCCSD hasnotbeenawidelyusedmethod.Ihavepresentedresultsthatshowthatusinga relativelysimplediagonalTikhonovregularization,oneisabletoremovethesingularity fromUHFLinCCSD,yieldingsmoothpotentialenergycurves.Furthermore,thesecurves closelymatchUHFCCSDresults,evenformultiply-bondedmolecules.Optimizationof theTikhonovparameterhasnotbeendone,butthevalueidentiedinthetextappears 204

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Table4-3.Barrierheightsforwell-characterizedreactionsfrom[178,179].Energiesand errorsfrombestexperimentalestimatesareinkcal/mol. Reaction BarrierHeight E rxn EnergyErrorEnergyError H+N 2 O OH+N 2 V z f 19 : 901 : 76 -67.81-2.73 V z r 87 : 704 : 48 H+FH HF+H V z f 44 : 131 : 95 00 V z r 44 : 131 : 95 H+ClH HCl+H V z f 20 : 552 : 55 00 V z r 20 : 551 : 55 H+FCH 3 HF+CH 3 V z f 34 : 514 : 13 -26.100.54 V z r 60 : 613 : 59 H+F 2 HF+F V z f 0 : 81 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 46 -102.431.48 V z r 103 : 24 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 94 CH 3 +FCl CH 3 F+Cl V z f 7 : 620 : 19 -55.21-2.47 V z r 62 : 832 : 66 F )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(+CH 3 F FCH 3 +F )]TJ/F15 11.9552 Tf 57.502 2.885 Td [(V z f )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 340 : 00 00 V z r )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 340 : 00 F )]TJ/F19 11.9552 Tf 9.077 -4.338 Td [( CH 3 F FCH 3 F )]TJ/F15 11.9552 Tf 39.846 2.885 Td [(V z f 14 : 140 : 76 00 V z r 14 : 140 : 76 Cl )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(+CH 3 Cl ClCH 3 +Cl )]TJ/F15 11.9552 Tf 40.91 2.884 Td [(V z f 4 : 121 : 02 00 V z r 4 : 121 : 02 Cl )]TJ/F19 11.9552 Tf 9.077 -4.338 Td [( CH 3 Cl ClCH 3 Cl )]TJ/F15 11.9552 Tf 23.254 2.885 Td [(V z f 14 : 480 : 87 00 V z r 14 : 480 : 87 F )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(+CH 3 Cl FCH 3 +Cl )]TJ/F15 11.9552 Tf 49.369 2.885 Td [(V z f )]TJ/F15 11.9552 Tf 9.298 0 Td [(13 : 48 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 94 -37.27-4.62 V z r 23 : 793 : 68 F )]TJ/F19 11.9552 Tf 9.077 -4.339 Td [( CH 3 Cl FCH 3 Cl )]TJ/F15 11.9552 Tf 31.713 2.884 Td [(V z f 2 : 950 : 06 -30.11-3.38 V z r 33 : 053 : 43 OH )]TJ/F15 11.9552 Tf 7.084 -4.338 Td [(+CH 3 F HOCH 3 +F )]TJ/F15 11.9552 Tf 36.696 2.885 Td [(V z f )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 730 : 05 -20.44-0.33 V z r 17 : 700 : 37 OH )]TJ/F19 11.9552 Tf 9.077 -4.338 Td [( CH 3 F HOCH 3 F )]TJ/F15 11.9552 Tf 19.04 2.885 Td [(V z f 11 : 580 : 62 -37.36-0.17 V z r 48 : 931 : 73 H+CO HCO V z f 4 : 671 : 50 -17.951.56 V z r 22 : 62 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 06 H+C 2 H 4 CH 3 CH 2 V z f 3 : 521 : 80 40.21-0.18 V z r 43 : 721 : 97 CH 3 +C 2 H 4 CH 3 CH 2 CH 2 V z f 8 : 801 : 95 -25.690.43 V z r 34 : 491 : 52 HCN HNC V z f 48 : 450 : 29 14.77-0.28 V z r 33 : 680 : 57 AverageAbsoluteError1 : 561.04 205

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toworkuniversally.Equilibriumgeometriesarealsoinexcellentagreementbothwith experimentandwithCCSDTandCCSDTresults.TheexceptionisF 2 .Thismay indicatethatmoreorbitalrelaxationthrough,forexample,aBruecknerorbitalprocedure Section3.3maybenecessaryforsuchahighlycorrelatedmolecule.Activationbarriers areunfortunatelynotaswell-describedaseithertheoverallshapeofthepotentialenergy surfaceorequilibriumgeometries.Fulloptimizationsandtransitionstatesearchesstill needtobedonetoprovethatthisisafailureofLinCCSDandnotamismatchbetween LinCCSDenergeticsandQCISDgeometries. LinCCSDhasthepotential,duetotheregularization,ofbeingauseful,fast, approximateCCmethod.Inparticular,thehermitiannatureofLinCCSDmeans thatthecalculationofforces,forexampleforMD,ismuchfasterthaninconventional coupled-clustertheory.Becausethe T amplitudeequationsarelesstightlycoupledthan inconventionalCC,ecientparallelizationiseasierwithintheLinCCframework.Even addinganadditionalcorrectionduetotriplesmaybepossiblewithoutsignicantly degradingtheeciencyofthemethod. 206

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CHAPTER5 CONCLUSIONS Coupled-clustertheoryisthemostpowerfultechniqueforpredictivecalculationsof small-andmedium-sizedmoleculesofchemicalinterest.Despitethesesuccesses,there arestillanumberofproblemsthatkeepCCfrombeingevenmorewidelyused.Thetwo mostprominentoftheseareitscomputationalcostandfailuresalongpotentialenergy surfaces.Theseproblemsarenotunrelated;thefailuresforpotentialenergysurfacescan alwaysbeaddressedbyincludinghigherexcitations,butatthepenaltyofmuchhigher computationalcost.Theworkpresentedinthisstudyaddressesbothoftheseissuesby generalizingthestructureofcoupled-clustertheory. Frozennaturalorbital-CCdirectlyfocusesonspeedingupcalculationsviabasisset truncation.Forenergies,thisisrelativelystraightforward,butforgradients,theseparation betweendirectresponseofthewavefunctiontotheperturbationandtherelaxationof theorbitalsthatisinducedbytheperturbationisnon-trivial.Itwasnecessarytousethe interchangetheoremthatiscommonlyusedincoupled-perturbedHartree-Focktheoryin anewway,sothatonewouldnotneedtosolvefortheperturbedFNOcoecientsina perturbation-by-perturbationmanner.TheseresultsestablishthatFNO-CCisauseful methodtospeedcalculationsofenergiesandgeometryoptimizations.Furtherworkis neededtobeabletoseeifvibrationalfrequenciescanbemoreaccuratelytreated,aswell asthepossibilitythatdiscontinuitiesmayoccurasonemovesacrossapotentialenergy surface. Themostpopularhigh-accuracyversionofcoupled-clustertheoryisCCSDT,which includesallsingleanddoubleexcitations,aswellasanapproximationtotripleexcitations. Thecomputationalcostofthismethod,whilehigh,isfeasibleformanymoleculesof chemicalinterest.Unfortunately,awayfromequilibriumCCSDThaspathologiesthat leadtolargeerrors.IhaveshownthattherearetwoindependentpathologiestoCCSDT: adegeneracy-basedfailureofperturbationtheorybasedonarestrictedHartree-Fock 207

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referenceandaconvergenceproblemforspin-unrestrictedperturbationtheorywhenusing anunrestrictedHartree-Fockreference. ThedegeneracyproblemcanbeamelioratedbyusingtheCCSDTmethod, thatwasdevelopedinChapter3.ToderivederivativesforCCSDT,Iintroduceda generalizationofthetypicalCCstationaryfunctionalexpression,allowingthegeneralized Hellman-Feynmantheoremtobeapplied.TheCCSDTmethodwasestablishedto reproduceCCSDTqualityresultsatequilibriumanddrasticallyreducetheerrorsas onemovesacrossapotentialenergysurface.TheimprovementoverCCSDTholdsonly whenusinganRHFreference,however.Thisresultcanberationalizedbyrecognizing thatCCSDTonlyattemptstoxthedegeneracyprobleminCCSDT,notthe spin-symmetryproblem.Therefore,IimplementedBruecknerCCSDT,whichby virtueofthereducedsymmetrybreakingoftheBruecknerdeterminant,convertsmany casesofspin-symmetrybreakingproblemsintodegeneracyproblems.Intheprocessof analyzingthiscase,Iusedageneralstationaryformulationofcoupled-clustertheorythat canproperlyincludetheimpactofanarbitraryreferencechoiceintothedenitionofboth thecoupled-clusterequationsandtheirderivatives. Inamoreradicaldeparturefromconventionalcoupled-clustertheory,Ihave reconsideredthepossibilitiesofhermitiancoupled-clustertheories.Aswasshown,if itwerecomputationallyfeasiblestrongly-connectedexpectation-valueCCSC-XCC wouldbeanidealcomputationalmethod:hermitian,variational,extensive,satisesthe Hellman-Feynmantheorem,potentiallyexact.Thismethodologywouldlikelyremoveany problemacrossthepotentialenergysurface.Additionally,Ishowedthattherearestill unsolvedproblemsintheconsistentgeneralizationofXCCmethodstotheexcitedstate andtothestate-universalmultireferencecase. Unfortunately,addressingthefullSC-XCCisimpossible.LinearizedSC-XCC LinCChasbeeninvestigatedbefore,butthepresenceofasingularityasonestretches abond,andevenwhenperformingageometryoptimization,hasinhibiteditsuse.Ihave 208

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shownthatthissingularitycanberemovedfromtheequationsbyusingaregularization technique;inparticular,thatTikhonovregularizationoftheLinCCSDequationsprovide ageneral-purpose,fast,hermitianCCmethodwithoutasingularity.Bothgeometries andpotentialenergysurfaceswereshowntobesurprisinglyaccurateforawide-rangeof molecules.Thetuningoftheregularizationparameterisstillnotoptimized,andfuture workwillbenecessarytochoosethebestsuchvalue. 209

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APPENDIXA ORBITALFUNCTIONALSFORDIFFERENTREFERENCEFUNCTIONS Oftheinnitenumberofreferencefunctionsthatonecanchoosetousewith coupled-clustertheory,thereareafewthatstandoutintheirutility.InthisAppendix,I introducethevarioustermsthatneedtobeaddedtotheCCenergyfunctionaltoallowa consistentdenitionofpropertiesandderivativesincludingorbitalrelaxationusingthese references.ThetermsinEquation1{111arethosenecessarytoforcetheHartree-Fock orbitalstobecanonical.Thesetermsareactuallymoregeneralthanthat,andcanbe usedtokeeparbitraryorbitals semi-canonical ;theFockmatrixismadediagonalinthe occupied-occupiedandthevirtual-virtualsubspaces.Therefore,inallthefollowingone canchoosetoaddthosetermstoanyofthefunctionals. Inallofthesecases,theconditionsbelowareforastationarysolution not for aminimumormaximumdependingonthecondition.Toguaranteethatoneisat thepropertypeofstationarypoint,itisnecessarytoevaluatethestabilityconditions [19,21,22,200]ofthereferencefunctionanddeterminethecharacterofthestationary point. RestrictedOpen-ShellHartree-Fock :Restrictedopen-shellHartree-Fock referencesROHFaredesignedforhigh-spinopen-shellsystems.Aslongasthespatial partofthe and orbitalsarexedtobeidentical,theaufbauprinciplewillleadtoa symmetry-adaptedreferencefunction.ForstandardorbitalswhichdiagonalizetheROHF Fockmatrix",therelevantgeneralizedBrillouinconditions[249,250]are f i a + f i a =0 ; A{1 f i s = f s a =0 ; A{2 where s isasinglyoccupiedorbital.Theseconditionsaresimpletoincorporateina functionalframework,aswasdoneforRHFandUHF.However,ifonewantstoallowfor semi-canonicalROHForbitalsasisnecessaryfornon-iterativeROHF-CCSDT[52] 210

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thentheseconditionswillnothold.Instead,onemustrecognizethattheseconditionsare aspeciccaseofthemorefundamentalconditionthat h CSF a i j H j ROHF i =0 ; A{3 i.e.thatallsinglyexcitedcongurationstatefunctionsCSFsdonotmixthrough theHamiltoniantotheROHFreferencefunction.BecausetheHamiltonianis spin-independent,CSFsthatareofdierentspinthantheROHFfunctionautomatically satisfythisrequirement;therelevantsingly-excitedCSFsarethoseofthesamespinas theROHFfunction.ThisconditiondoesnotdependonanyspecicformoftheROHF orbitals,butinsteadonthespacespannedbythe and orbitals.Theorbitalfunctional forROHFcanbewritten E ROHF = h 0 j + ^ P S 2 e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e j 0 i + h 0 j H j 0 i A{4 where ^ P S 2 istheprojectortotheproperspin-adaptedcongurationstatefunction[18]. Theoperator takestheformwhere i;a arespatialorbitallabels = X ia a i )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(f a y i g + f a y i g)-222(f i y a g)-222(f i y a g + X sp p s )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(f s y p g + f s y p g)-222(f p y s g)-222(f p y s g A{5 where p arenotsingly-occupiedorbitals. Becausethisconditiondoesnotdependonanyparticularformoforbitals,oneisfree toaugmentitwithasemi-canonicalfunctionalterm.Notethatforsemi-canonicalROHF orbitals,thisformofthefunctionalisnotparticularlycomputationallyecient.The implementationoftheprojectionoperatorinageneralformisquitedicult[17].Instead itsimplyactsasatoolfortheoreticaldevelopment. Quasi-RestrictedHartree-Fock :Insomesituations,thecorrelationsarestrong enoughinasystemthatanRHF,UHForROHFcalculationgivepoordescriptionsofthe 211

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electronicstructureofamolecule.Onewaytocircumventthisproblemistouseorbitals fromanother,related,electronicstateasthereferencefunctionwithdierentoccupation. Forexample,ifonewereinterestedinstudyingN )]TJ/F20 7.9701 Tf 0 -7.879 Td [(2 ,itmaybesensibletostartwith orbitalsfromN 2 andaddan electron.Thisapproachisknownasquasi-restricted Hartree-FockQRHF[142].Incaseswherethenumberofelectronsisdierentforthe orbitalsthanfortheproblemofinterest,thereare two relevantHamiltonians.Specializing tothecasewhereweareaddingasingleelectrontoaclosedshell E QRHF/CC = h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T e )]TJ/F20 7.9701 Tf 6.91 0 Td [(~ H N e ~ e T j 0 i + h 0 j s y ~ e )]TJ/F20 7.9701 Tf 6.91 0 Td [(~ ~ H N e ~ s j 0 i + h 0 j e )]TJ/F20 7.9701 Tf 6.909 0 Td [(~ H N e ~ j 0 i + h 0 j H j 0 i A{6 where ~ and~ havethenumberofoccupiedorbitalscorrespondingtotheauxiliary Hamiltonian ~ H .Theoperator s iseitheraparticleorholeannihilationoperator, dependingonwhethertheQRHFreferencehasonemoreoronelesselectronthanthe stateofinterest.Itisbychoosingaparticularorbital s thatonetakestheHartree-Fock solutionforstate1andusesitasareferenceforstate2{whichhasadierentnumberof electrons. Coupled-ClusterNaturalOrbitals :Forsomereferencesitisnotstraightforward toderiveafunctionalthatproperlysatisesthegeneralizedHellman-Feynmantheorem. Naturalorbitalsofagivendensitymatrixaretheorbitaleigenfunctionsoftheone-particle densitymatrix[251];generalizednaturalorbitalsarenon-canonicalnaturalorbitals,which satisfytheconditionforaCCwavefunction 0= N )]TJ/F22 7.9701 Tf 16.804 14.944 Td [(N X i =1 h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( i y ie e T j 0 i : A{7 Thestationarypointofthisconstraintischaracterizedbytheconditions, 0= h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T e )]TJ/F22 7.9701 Tf 6.587 0 Td [( )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(a y i e e T j 0 i A{8 0= h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T e )]TJ/F22 7.9701 Tf 6.587 0 Td [( )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(i y a e e T j 0 i : A{9 212

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BecausetheCCexpectationvalueexpressionisnothermitian,ingeneralthedensity matrixissymmetrized,yieldingthesymmetrizednaturalorbitalcondition h 0 j + e T e )]TJ/F22 7.9701 Tf 6.587 0 Td [( )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(a y i + i y a e e T j 0 i =0 : A{10 Thefullfunctionalcouldthenbewritteninstandardorbitalsas E NO=CC = h 0 j + e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T j 0 i + X ai i a h 0 j + e T e )]TJ/F22 7.9701 Tf 6.587 0 Td [( )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(a y i + i y a e e T j 0 i + h 0 j e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e j 0 i + h 0 j H j 0 i : A{11 Takingthederivativewithrespectto K hKj e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T j 0 i + X ai i a hKj e T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(a y i + i y a e e T j 0 i =0 : A{12 FortheGHFtoapplythisequationmusthold,butclearlyitcannot,giventhatonly thersttermrepresentsthecoupled-clusterequations.Therefore,therewouldbe anon-Hellman-Feynmancorrection.ToeliminatethistermviaaHellman-Feynman framework,onemustdecoupletheequationsfromthosethatdenethenaturalorbitals. Onecandothisbyaddingtwoauxiliaryvariables[123],anexcitationoperatoranda conjugatede-excitationoperator.Theaugmentedfunctionalcanthenbewritten, E 0 NO=CC = h 0 j ++ e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T j 0 i + h 0 j + )]TJ/F15 11.9552 Tf 8.515 -6.662 Td [( H N C j 0 i + X ai i a h 0 j + e T e )]TJ/F22 7.9701 Tf 6.586 0 Td [( )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(a y i + i y a e e T j 0 i + h 0 j e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e j 0 i + h 0 j H j 0 i : A{13 DierentiatingwithrespecttoyieldstheCCequations,dierentiatingwithrespectto yieldstheequations,anddierentiatingwithrespecttoyieldsthenaturalorbital equations.Thestationarypointofthisfunctionalisthen E stat = h 0 j e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e e T j 0 i + h 0 j e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e j 0 i + h 0 j H j 0 i ; A{14 whichisthecorrectCCenergyinthenaturalorbitals.Theequationsnecessaryfor 213

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calculatingthederivativesoftheCCnaturalorbitalscanbedeterminedbyrequiring stationaritywithrespecttovariationsin T ,and Bruecknerorbitals :Oneparticularchoiceofreferencefunctionthatplaysan importantroleinChapter3aretheBruecknerorbitals.Bruecknerorbitalsaredened abstractlyasthoseorbitalsthatmaximizetheoverlapbetweenthereferencefunctionand theexactwavefunction.Equivalently, B =argmax kh j e j 0 ik A{15 j B i = e j 0 i : A{16 Ifonesubstitutesthecoupled-clusterexpansionintothisexpression,itcanbeshownthat T 1 =0denestheBruecknerorbitals.Therefore,onecandropthe T 1 and 1 terminthe coupled-clusterequations,andmodifythereferencepieceofthefunctional E BCC = h 0 j + 0 e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T 0 e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T 0 j 0 i + h 0 j e )]TJ/F22 7.9701 Tf 6.587 0 Td [(T 0 e )]TJ/F22 7.9701 Tf 6.587 0 Td [( H N e e T 0 j 0 i + h 0 j H j 0 i A{17 where T 0 doesnotinclude T 1 .Dening 1 =,thesetwopiecescanbejoinedinto E BCC = h 0 j + e )]TJ/F22 7.9701 Tf 6.586 0 Td [(T 0 e )]TJ/F22 7.9701 Tf 6.586 0 Td [( H N e e T 0 j 0 i + h 0 j H j 0 i : A{18 Notethatthelackofareferenceterminthefunctionaleliminatesthedistinctionbetween relaxed"andresponse"densitymatrices.Becausethefunctionalstilldependson anunderlyingatomicorbitalbasis,theoverlapmatrixwillappear,butsimplyasan additionalsingleparticlecontributiontotheoverallformofthegradient. 214

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APPENDIXB EQUATIONSFORANALYTICALDERIVATIVESOFCCSDT TodeterminetheanalyticalderivativeofCCSDT,itisnecessarytondthe stationaryvaluesof,and [2] 3 .Thestationaryequationfor [2] 3 wassolvedabove Equation3{43.Theexcitationoperatoristheconjugatevariableto,thereforethe equationsdeningare, @ E @ i a = h a i j )]TJ/F15 11.9552 Tf 8.514 -6.662 Td [( H N C j 0 i + h a i j WT [2] 3 j 0 i =0 ; @ E @ ij ab = h ab ij j )]TJ/F15 11.9552 Tf 8.515 -6.661 Td [( H N C j 0 i + h ab ij j F ov + W T [2] 3 j 0 i =0 : B{1 Similarly,istheconjugatede-excitationoperatorto T ,thereforeitisdenedby, @ E @t a i = h 0 j )]TJ/F15 11.9552 Tf 8.515 -6.662 Td [( H N a a i C j 0 i + h 0 j + )]TJ/F15 11.9552 Tf 13.496 -6.662 Td [( H N a a i C C j 0 i =0 ; @ E @t ab ij = h 0 j )]TJ/F15 11.9552 Tf 8.515 -6.661 Td [( H N a ab ij C j 0 i + h 0 j + h )]TJ/F15 11.9552 Tf 8.514 -6.661 Td [( H N a ab ij C i C j 0 i + h 0 j [2] 3 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Wa ab ij C j 0 i =0 : B{2 where a a i and a ab ij areexcitationsoperators. Inthefollowingitisnecessarytousepermutationoperators: P qr =1 P qr P qr j st = P qr P st B{3 where P qr permutesthelabels q and r .Repeatedindicesarealwayssummedover. ExpandingEquationsB{1intospinorbitalform,thenon-linearequationsforare h ad d i )]TJ/F15 11.9552 Tf 12.073 3.155 Td [( h li a l + h ladi d l + 1 2 h alde de il )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 h lmid ad lm + 1 4 h lm jj de i t ade ilm [2] =0B{4 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij h abdj d i )]TJ/F21 11.9552 Tf 11.956 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab h lbij a l + h labdij d l + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab h bd ad ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij h lj ab il + 1 2 h abde de ij + 1 2 h lmij ab lm + P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab j ij h lbdj ad il + 1 2 P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij h albdej de il )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab h lmbidj ae lm + f dl t abd ijl [2] + 1 2 P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab h al jj de i t deb ilj [2] )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij h lm jj id i t abd lmj [2] =0 : B{5 215

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Intheseequations, h indicatesmatrixelementsofthefulleectiveHamiltonian H Theequationsforaresomewhatmorecomplicated,butinfullspin-orbitalformare h ad i d )]TJ/F15 11.9552 Tf 12.073 3.155 Td [( h li l a + h dila l d + 1 2 h deal il de )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 h idlm lm ad + 1 4 i a bc jk h jkbc + j b bc jk h ikac )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 j a bc jk h ikbc )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 i b bc jk h jkac + 1 4 jk bc bc jk h ia + ij ab bc jk h kc )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 jk ac bc jk h ib )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 ik bc bc jk h ja + 1 2 jk bd bc jk h dica )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 jl bc bc jk h kila )]TJ/F21 11.9552 Tf 11.955 0 Td [( jl ba bc jk h iklc + ij db bc jk h dkac + ij ad bc ij h dkbc )]TJ/F21 11.9552 Tf 11.955 0 Td [( il ab bc jk h jklc + 1 2 il cb bc jk h jkla )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 jk da bc jk h dibc =0B{6 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab h cb ij ac )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij h jk ik ab + P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij j ab h ai j b + 1 2 h abde ij de + 1 2 h lmij lm ab + k c c k h ijab + P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij j ab i a c k h kjcb )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab k a c k h ijcb )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij i c c k h kjab + P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij ki cd c k h djab )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( ab kl ca c k h ijlb + kl ba c k h ijlb )]TJ/F21 11.9552 Tf 11.955 0 Td [( ij dc c k h dkab + P )]TJ/F15 11.9552 Tf 7.084 1.794 Td [( ab ij ad c k h dkbc )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( ij il ab c k h jklc + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij j ab ik ad c k h djcb )]TJ/F21 11.9552 Tf 11.956 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij j ab il ac c k h kjlb )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab ij ac c k h kb )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij ik ab c k h jc + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij j ab ik ac c k h jb + 1 4 kl cd cd kl h ijab + 1 4 ij ab cd kl h klcd + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij j ab lj db cd kl h ikac + 1 4 ij cd cd kl h klab + 1 4 kl ab cd kl h ijcd )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab ij ac cd kl h jkbd )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij ik ab cd kl h jlcd )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab kl bd cd kl h ijac )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ij jl cd cd kl h ikab + ij ab c k h kc + 1 2 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab h cd jj bk i ijk acd [2] )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij h jc jj kl i ikl abc [2] =0 : B{7 Oncetheseequationshavebeensolved,theresultantoperatorscanbecontracted toformdensitymatrices.Tosimplifytheseequations,itishelpfultodeneafew intermediatequantities, ` i a = i a + i a ` ij ab = ij ab + ij ab B{8a ab ij = t ab ij + t a i t b j )]TJ/F21 11.9552 Tf 11.956 0 Td [(t b i t a j s ab ij = ab ij + t a i b j )]TJ/F21 11.9552 Tf 11.955 0 Td [(t b i a j B{8b V ijkl = 1 2 X ab ab ij ` kl ab + s ab ij kl ab U ijkl = 1 2 X ab ab ij kl ab B{8c V abcd = 1 2 X ij cd ij ` ij ab + s cd ij ij ab U abcd = 1 2 X ij cd ij ij ab B{8d 216

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V iajb = 1 2 X kc t bc ik ` jk ac + bc ik jk ac U iajb = 1 2 X kc t bc ik jk ac B{8e G ab = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 X ijc t bc ij ` ij ac + bc ij ij ac Q ab = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 X ijc t bc ij ij ac B{8f G ij = 1 2 X kab h t ab ik ` jk ab + ab ik jk ab i Q ij = 1 2 X kab t ab ik jk ab : B{8g Elementsofthehermitizedone-particledensitymatrix: ij = )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 P + ij X a t a i ` j a + a i j a + 1 4 P + ij X kab h t ab ki ` jk ab + ab ki jk ab i )]TJ/F15 11.9552 Tf 16.077 8.088 Td [(1 12 X klabc jlk abc [2] t abc lki [2] B{9a ia = 1 2 t a i + a i + ` i a + X jb ` j b )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t ab ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(t a j t b i + j b )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( ab ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( a j t b i )]TJ/F21 11.9552 Tf 11.955 0 Td [( b i t a j )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 X jkbc h ` jk bc )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t bc ik t a j + t ac jk t b i + jk bc )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( bc ik t a j + ac jk t b i + t bc ik a j + t ac jk b i i + 1 4 X jkbc jk bc t abc ijk [2] B{9b ab = 1 2 P + ab X i t a i ` i b + a i i b + 1 4 P + ab X ijc t ac ij ` ij bc + ac ij ij bc + 1 12 X ijkcd ijk adc [2] t dcb ijk [2] : B{9c Elementsofthehermitizedtwo-particledensitymatrix: )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ij;kl = 1 8 P + ij;kl V ijkl B{10a )]TJ/F22 7.9701 Tf 7.315 -1.794 Td [(ij;ka = )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 8 X b ` k b ba ij + k b s ba ij + t b k ` ij ba + b k ij ba + 1 8 X l [ V ijkl t a l + U ijkl a l ] + 1 4 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij X b V kbia t b j + U kbia b j )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 8 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij G ik t a j + Q ik a j + 1 4 X lbc h kl bc t abc ijl [2] + t bc kl ijl abc [2] i B{10b )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ia;jb = 1 4 P + ib;ja V iajb + 1 8 P + ib;ja X kc t c i t b k ` jk ac + c i t b k jk ac + t c i b k jk ac )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 8 P + ib;ja t b i ` j a + b i j a B{10c 217

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)]TJ/F22 7.9701 Tf 7.315 -1.794 Td [(ij;ab = 1 8 ab ij + s ab ij + ` ab ij + 1 16 X kl ab kl V ijkl + s ab kl U ijkl + 1 4 X kc bc jk t abc ijk [2] )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 8 P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( ij X k ab kj G ik + X c )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(` k c t c i + k c c i + s ab kj Q ik + X c k c t c i !# + 1 8 P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( ab X c cb ij G ca )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X k )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(` k c t a k + k c a k + s cb ij Q ca )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X k k c t a k !# )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 8 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij j ab X kc t ac ki +2 t a k t c i X ld V jdlb + ` k c t c j + k c c j # )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 8 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ij j ab X kc ac ki +2 a k t c i +2 t a k c i X ld U jdlb + k c t c j # + 3 2 P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( ij j ab X kc t b k t c j t a i ` k c + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( b k t c j t a i + t b k c j t a i + t b k t c j a i k c B{10d )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ab;ci = 1 8 X j ` j c ab ji + j c s ab ji + t c j ` ji ab + c j ji ab )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 8 X d V cdab t d i + U cdab d i + 1 4 P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ab X j V j t a j + U g a j )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 8 P )]TJ/F15 11.9552 Tf 7.084 1.793 Td [( ab G ca t b i + Q ca b i + 1 4 X jkd dc jk t abd ijl + t dc jk abd ijl B{10e )]TJ/F22 7.9701 Tf 7.314 -1.793 Td [(ab;cd = 1 8 P + ab;cd V abcd : B{10f 218

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APPENDIXC EQUATIONSFORFORBCCD Followingthenotationof[252],the 1 equationforBCCDis D a i i a = f ia + F 0 ia + X e i e F ea )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X m m a F im + 1 2 X mef im ef f W efam + X me m e f W eima + f W 0 eima )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 X mne mn ae f W mnie )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X ef G ef + G fe h ei jj fa i )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X mn G mn + G nm h mi jj na i + X me X f G fe t f m )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X n G mn t e m h im jj ae i )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 X mef V iemf h ea jj mf i +2 X mne V mena h em jj ni i C{1 where F ea = )]TJ/F21 11.9552 Tf 11.956 0 Td [( ea f ea )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 X mnf t ef in h mn jj ef i C{2 F im = )]TJ/F21 11.9552 Tf 11.956 0 Td [( mi + 1 2 X nef t ef in h mn jj ef i C{3 G mn = 1 2 X oef t ef mo no ef G ef = )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 X mng t fg mn mn eg C{4 F 0 ia = 1 2 X mef t ef im h am jj ef i)]TJ/F15 11.9552 Tf 20.456 8.088 Td [(1 2 X mne t ae mn h mn jj ie iV menf = 1 2 X go t fg mo no eg C{5 and f W efam = h ef jj am i + X n F na t ef mn + 1 2 X no h am jj no i t ef no + P )]TJ/F15 11.9552 Tf 7.085 1.793 Td [( ef X ng h en jj ag i t fg mn C{6 f W eima = h ei jj ma i + X nf t ef mn h ni jj fa i C{7 f W mnie = h mn jj ie i)]TJ/F26 11.9552 Tf 19.261 11.358 Td [(X f F if t ef mn + 1 2 X fg h fg jj ie i t fg mn + P )]TJ/F15 11.9552 Tf 7.085 1.794 Td [( mn X of h mf jj io i t ef no C{8 f W 0 eima = h ea jj mi i + X f t ef mi f af )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X n t ea mn f in + 1 2 X fg t fg mi h ea jj fg i)]TJ/F26 11.9552 Tf 19.261 11.357 Td [(X nf t ef ni h na jj mf i )]TJ/F26 11.9552 Tf 11.956 11.357 Td [(X nf t fa mn h en jj fi i + 1 2 X no t ea no h no jj mi i)]TJ/F26 11.9552 Tf 19.261 11.357 Td [(X nf t ef mn h na jj if i : C{9 219

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APPENDIXD FACTORIZATIONSOFCOUPLED-CLUSTERFUNCTIONALS FactorizationoftheVariationalCoupled-ClusterEnergyFunctional :The variationalcoupled-clusterVCCenergyfunctionalis E VCC = h 0 j e T y H N e T j 0 i h 0 j e T y e T j 0 i + h 0 j H j 0 i ; D{1 whichisnotmanifestlyconnected.Onecanshowthatthisfunctionalisequivalenttothat forexpectation-valueCCXCC, E XCC = h 0 j e T y H N e T C j 0 i + h 0 j H j 0 i ; D{2 whichismanifestlyconnected.Following[215],thenumeratorofEquationD{1is h 0 j e T y H N e T j 0 i = 1 X m;n =0 1 m n h 0 j )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(T y m H N T n j 0 i : D{3 Bythedenitionofnormal-ordering,theonlytermsthatcansurviveinanexpectation valueofthereferencefunctionarethosethatareclosed.Therefore, 1 X m;n =0 1 m n h 0 j )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m H N T n j 0 i = 1 X m;n =0 1 m n h 0 j )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(T y m H N T n cl j 0 i ; D{4 wherethesubscriptclindicatesonlycloseddiagrams.Foragiven m and n ,onecan contract k =0 ;:::;mT y 'swiththeHamiltonianand l =0 ;:::;nT 's.Theremaining m )]TJ/F21 11.9552 Tf 11.955 0 Td [(k T y 'sand n )]TJ/F21 11.9552 Tf 11.955 0 Td [(l T 'smustbecontractedwitheachother, 1 X m;n =0 1 m n h 0 j )]TJ/F21 11.9552 Tf 10.46 -9.684 Td [(T y m H N T n cl j 0 i = 1 X m;n =0 1 m n m X k =0 n X l =0 m k n l h 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(T y k H N T l i C; cl j 0 ih 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(T y m )]TJ/F22 7.9701 Tf 6.586 0 Td [(k T n )]TJ/F22 7.9701 Tf 6.586 0 Td [(l i cl j 0 i D{5 wherethecoecientsarisefromthenumberofwaysthatyoucanconnectthe l nT operatorswiththeHamiltonian.Since,bydenition, m k =0 k>m D{6 220

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thesummationcanbeextendedto 1 X m;n =0 1 m n m X k =0 n X l =0 m k n l h 0 j h )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T y k H N T l i C; cl j 0 ih 0 j h )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(k T n )]TJ/F22 7.9701 Tf 6.587 0 Td [(l i cl j 0 i = 1 X m;n =0 1 m n 1 X k;l =0 m k n l h 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y k H N T l i C; cl j 0 ih 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.586 0 Td [(k T n )]TJ/F22 7.9701 Tf 6.586 0 Td [(l i cl j 0 i D{7 withoutarestrictiononthe k and l summations.Substituting p = m )]TJ/F21 11.9552 Tf 12.328 0 Td [(k and q = n )]TJ/F21 11.9552 Tf 12.327 0 Td [(l intothisexpression 1 X m;n =0 1 m n 1 X k;l =0 m k n l h 0 j h )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T y k H N T l i C; cl j 0 ih 0 j h )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(k T n )]TJ/F22 7.9701 Tf 6.587 0 Td [(l i cl j 0 i = 1 X k;l =0 1 X p = )]TJ/F22 7.9701 Tf 6.586 0 Td [(k 1 X q = )]TJ/F22 7.9701 Tf 6.586 0 Td [(l 1 p + k q + l p + k k q + l l h 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y k H N T l i C; cl j 0 ih 0 j )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(T y p T q cl j 0 i : D{8 For p< 0then p + k
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h 0 j e T y H N e T j 0 i = h 0 j e T y H N e T C j 0 ih 0 j e T y e T j 0 i : D{12 ThesecondtermintheproductisthesameasthedenominatorinEquationD{1,proving that E VCC = E XCC : D{13 FactorizationoftheAmplitudeequations :Thefollowingidentityisuseful,both toproveequivalencesofthestationaryequationsforXCCandVCCandtoformulatea consistentmultireferenceXCC hKj e T y H N e T j 0 i = hKj e T y H N e T L j 0 ih 0 j e T y e T j 0 i + hKj e T y e T op j 0 ih 0 j e T y H N e T C j 0 ih 0 j e T y e T j 0 i : D{14 Theright-handsideofEquationD{14canbeviewedasthederivativeofanexpectation value, hKj e T y H N e T j 0 i = @ @t K h 0 j e T y H N e T j 0 i : D{15 Then,usingtherelationinEquationD{12,theexpectationvalueis @ @t K h 0 j e T y H N e T j 0 i = @ @t K h h 0 j e T y H N e T C j 0 ih 0 j e T y e T j 0 i i : D{16 Expandingthederivative, @ @t K h h 0 j e T y He T C j 0 ih 0 j e T y e T j 0 i i = hKj e T y H N e T L j 0 ih 0 j e T y e T j 0 i + h 0 j e T y H N e T C j 0 ihKj e T y e T j 0 i : D{17 Thersttermmatchesthersttermontheleft-handsideofEquationD{14;thesecond termmustbefurtherfactored.Focusingonthatsecondterm, hKj e T y e T j 0 i = @ @t K h 0 j e T y e T j 0 i = @ @t K h 0 j e T y e T cl j 0 i ; D{18 222

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whereinthelaststepIhaveusedthedenitionofnormal-orderingtoeliminateany openterms.Theoverlapbetweentwoexponentialwavefunctionscanbewritteninthe exponentialform[9,253], h 0 j e T y e T cl j 0 i = e h 0 j e T y e T C; cl j 0 i : D{19 Applyingthederivativetotheexponential, @ @t K e h 0 j e T y e T C; cl j 0 i = e h 0 j e T y e T C; cl j 0 i hKj e T y e T L; op j 0 i D{20 = h 0 j e T y e T cl j 0 ihKj e T y e T L; op j 0 i : D{21 ThisrelationnishestheproofofEquationD{14. GeneralizationofFactorizationtoMultireferenceStates :Thefollowing identityisusedinthederivationofahermitianmultireferencecoupled-clustertheory h 0 j e T y He T j 0 i = h 0 j e T y He T L; op j 0 ih 0 j e T y e T cl j 0 i + h 0 j e T y He T C; cl j 0 ih 0 j e T y e T j 0 i )]TJ/F21 11.9552 Tf 11.955 0 Td [( : D{22 Thecaseof = followsidenticallyfromthesingle-referenceproof.Therefore,Iaddress here 6 = ,so =0.Thereference j 0 i canbeviewedasanexcitationofthereference j 0 i ,i.e. j 0 i = jK i .Becausethisexcitationmaynotbeinthespanoftheexcitations availableto T ,Iintroduceanewoperator C thatgeneratesthatrelationship j 0 i = jK i = @ @c K C j 0 i : D{23 Then, h 0 j e T y He T j 0 i = @ @c K h 0 j C y e T y He T j 0 i : D{24 Focusingontheundierentiatedexpression,onlyclosedpartsoftheexpectationvaluewill survive h 0 j C y e T y He T j 0 i = h 0 j C y e T y He T cl j 0 i : D{25 223

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Theoperator T isnotapureexcitationoperatorwithrespectto j 0 i ,insteadithas excitationandde-excitationparts. T cannotcontracttoitselfbecauseallofthetermsin T arenormal-orderedwithrespecttoacommonreference.Expandingtheexponentials, h 0 j C y e T y He T cl j 0 i = 1 X m;n =0 1 m n h 0 j C y )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m HT n cl j 0 i ; D{26 Theformlooksstrikinglysimilartothatofthesingle-referencecase,exceptthat C y can contractwithanyof T y becausetheyareorderedwithrespecttodierentreferences, H or T .Thatmeansthatamongclosedtermsthe C y willeitherbeinatermthatis connectedtotheHamiltonianoronethatisconnectedpurelyto T operators.Therefore, therearetwosetsoftermsinthematrixelement, 1 X m;n =0 1 m n h 0 j C y )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(T y m HT n cl j 0 i = 1 X m;n =0 1 m n m X k =0 n X l =0 m k n l h 0 j h C y )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(T y k HT l i C; cl j 0 ih 0 j h )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.587 0 Td [(k T n )]TJ/F22 7.9701 Tf 6.586 0 Td [(l i cl j 0 i + h 0 j h )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T y k HT l i C; cl j 0 ih 0 j h C y )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T y m )]TJ/F22 7.9701 Tf 6.586 0 Td [(k T n )]TJ/F22 7.9701 Tf 6.586 0 Td [(l i cl j 0 i : D{27 Therestoftheanalysiscanproceedasinthesinglereferencecaseleadingto h 0 j C y e T y He T j 0 i = h 0 j C y e T y He T C; cl j 0 ih 0 j e T y e T cl j 0 i + h 0 j e T y He T C; cl j 0 ih 0 j C y e T y e T cl j 0 i : D{28 Thendierentiatingwithrespectto c K h 0 j e T y He T j 0 i = h 0 j e T y He T L; op j 0 ih 0 j e T y e T cl j 0 i + h 0 j e T y He T C; cl j 0 ih 0 j e T y e T op j 0 i : D{29 Substitutinginthedenitionsoftheoverlaps,thiscanbesimpliedto h 0 j e T y He T j 0 i = h 0 j e T y He T L; op j 0 i S cl + h 0 j e T y He T C; cl j 0 i S : D{30 224

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BIOGRAPHICALSKETCH AndrewG.Taubewasbornin1981,inNewYorkCity.Withinafewyearshis familysettledinNorthernVirginia,whereheattendedpublicschoolsandgraduated fromThomasJeersonHighSchoolofScienceandTechnologyTJHSSTinAlexandria, Virginiain1999.Duringthesummerbeforehissenioryearofhighschool,heworkedon anIntelScienceTalentSearchprojectunderthesupervisionofDr.JohnLiebermannat TJHSSTthatsolidiedhisdesiretodoresearchinchemistry.Itwasduringthisproject thatherstencounteredcomputationalmethodsforchemistry,whichwouldbecomethe focusofhislaterresearch. StayinginACCcountryforcollege,AndrewattendedDukeUniversity,graduating magnacumlaudewithaB.S.inChemistry,withdistinction,andaB.S.inMathematics. Duringhisundergraduateyears,heworkedwithProfesorRichardA.PalmerintheDuke Chemistrydepartmentonapplyingtime-resolvedstep-scanFourier-transformInfrared Spectroscopytoinvestigatechargetransferintransitionmetalcomplexes.Afterhis sophomoreyearofcollege,hereturnedtotheD.C.areaandworkedintheNational CancerInstituteoftheNationalInstitutesofHealthinBethesda,MD.Hisworkthere helpedtodevelopthedatacollectionandanalysissoftwareforanewtypeofmedical imaging,electronparamagneticresonanceimaging,foruseincancerdiagnosis. Aftergreatlyenjoyinghisphysicalchemistrywork,andfondlyrememberingthe computationalworkhehaddoneinhighschool,Andrewthoughtthatworkinginquantum chemistrymaybewhathewantedtodo,sobeforehissenioryearofcollegeheentered aResearchExperienceforUndergraduatesprogramintheQuantumTheoryProjectat theUniversityofFlorida,workingwithProfessorRodneyJ.Bartlett.HistimeatUF convincedhimthatelectronicstructuretheorywastherighteldforhim,andthatProf. Bartlettwastherightpersontoworkwith. 236