SINGLE-REFERENCECOUPLED-CLUSTERMETHODSFORSTRONGLY CORRELATEDSYSTEMS By IliasMagoulas ADISSERTATION Submittedto MichiganStateUniversity inpartialtoftherequirements forthedegreeof Chemistry{DoctorofPhilosophy 2021 ABSTRACT SINGLE-REFERENCECOUPLED-CLUSTERMETHODSFORSTRONGLY CORRELATEDSYSTEMS By IliasMagoulas Thedevelopmentofcomputationallytwavefunctionmethodsthatcanprovidean accuratedescriptionofstronglycorrelatedsystemsandmaterialsisattheheartofelectronic structuretheory.Ingeneral,strongmany-electroncorrelationctsarisefromtheentangle- mentofalargenumberofelectronsandarecharacterizedbytheunpairingofmanyelectron pairsandtheirsubsequentrecouplingtolow-spinstates,asinthecaseofMottmetal{ insulatortransitionswherethesystemtraversesfromaweaklycorrelatedmetallicphase toastronglycorrelatedinsulatingone.Althoughstrongcorrelationshaveanintrinsically multi-referencenature,multi-referenceapproachesarenotapplicableduetotheenormous dimensionalitiesoftheunderlyingmodelspaces.Therefore,inthisdissertation,wefocus onsingle-referencecoupled-cluster(CC)approaches,whicharewidelyrecognizedasthe de facto standardforhigh-accuracyelectronicstructurecalculationsandwhosesizeextensivity makesthemsuitableforthestudyofextendedsystemsandmaterials.However,itiswell establishedthatthetraditionalCCmethodologiesthatarebasedontruncatingthecluster operatoratagivenmany-bodyrank,givingrisetotheCCSD,CCSDT,CCSDTQ, etc. hier- archy,failtoprovidephysicallymeaningfulsolutionsinthepresenceofstrongcorrelations. Thus,inthisdissertation,weconsiderunconventionalsingle-referenceCCapproachescapa- bleofprovidinganaccuratedescriptionoftheentirespectrumofmany-electroncorrelation rangingfromtheweaklytothestronglycorrelatedregimes. Inthepartofthisdissertation,weexaminetheapproximatecoupled-pair(ACP) theories.TheexistingACPmethodsandtheirvariousmoretainalldoublyexcited clusteramplitudes,whileusingsubsetsofnon-lineardiagramsoftheCCD/CCSDequations. ThiseliminatesfailuresofconventionalCCapproaches,includingCCSDandevenCCSDTor CCSDTQ,instronglycorrelatedsituationscreatedbytheMottmetal{insulatortransitions, modeledbylinearchains,rings,orcubiclatticesoftheequallyspacedhydrogenatoms,and the ˇ -electronnetworksdescribedbytheHubbardandPariser{Parr{PopleHamiltonians thatmodelone-dimensionalmetallicsystemswithperiodicboundaryconditions.However, typicalACPmethodsneglectconnectedtriplyexcited( T 3 )clusters,whicharerequiredto producequantitativeresultsinmostchemicalapplications.Previousattemptstoincorporate theseclustersusingmany-bodyperturbationtheoryargumentswithintheACPframework haveonlybeenpartlysuccessful.Inthisdissertation,weaddressthisconcernbyemploying theactive-spaceideastoincorporatethedominant T 3 amplitudesintheACPmethodsina robust,yetcomputationallymanner.Furthermore,takingintoconsiderationthat thevariousdiagrammoACPapproacheswerederivedusingminimum- basis-setmodels,weintroduceanovelACPschemeutilizingbasis{set-dependentscaling factors,denotedasACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ),toextendtheACPmethodologies tolargerbasissets. Inthesecondpartofthisdissertation,wediscussanovelapproachtoextrapolating theexactenergeticsoutoftheearlystagesoffullinteractionquantumMonte Carlo(FCIQMC)propagations,eveninthepresenceofstrongcorrelations,bymerging theACPapproacheswiththerecentlyproposedcluster-analysis-drivenFCIQMC(CAD- FCIQMC)methodology.InthespiritofexternallycorrectedCCapproaches,intheCAD- FCIQMCmethodology,onesolvesCCSD-likeequationsfortheone-andtwo-bodyclusters inthepresenceoftheirthree-andfour-bodycounterpartsextractedfromtheFCIQMC stochasticwavefunctionsampling.Inthisdissertation,weextendCAD-FCIQMCtothe strongcorrelationregimebyrepartitioningtheCCequationssothatselectedcoupled-pair contributionsareextractedfromFCIQMCaswell. Foreachnewmethodologydescribedinthisthesis,wediscusstherelevantmathemat- icalandcomputerimplementationdetailsandprovidenumericalexamplesillustratingits performanceinchallengingstronglycorrelatedsituations. Copyrightby ILIASMAGOULAS 2021 Thisdissertationisdedicatedtomywife, ElènhLÔgda ,anddaughter, MarÐaMagoul‹ . PoluagaphmènecElenÐtsakaimikr€MarÐasaceuqarist¸p‹rapolÔpoukajhmerin¸comorfaÐnetethzw€mou! v ACKNOWLEDGMENTS Tobeginwith,Iwouldliketothankmydoctoraladvisor,ProfessorPiotrPiecuch.Asa Master'sstudentinGreece,IrememberthatthemeresightoftheFeynmandiagramsshown onthecoverofthe\ Many-BodyMethodsinChemistryandPhysics "bookbyR.J.Bartlett andI.Shavittmademethink\ Iwillneverbeabletounderstandcoupled-clustertheory ". UnderProfessorPiecuch'strainingIwasabletonotonlyunderstandthefoundationsof coupled-clusterandmany-bodytheories,includingdiagrammaticmethodstheyrelyon,but alsomakeanumberofnon-trivialmethodologicalcontributions,includingthosediscussed inthisdissertation.IamindebtedtoProfessorPiecuchforentrustingmewiththetask ofdevelopingandimplementingnewgenerationsofcoupled-clusterapproachesaimedatan exactornearlyexacttreatmentofstronglycorrelatedsystems,givingmeanopportunity togrowsubstantiallyasascientistunderhismentorship.Iamalsoverygratefulforhis continuoussupportandfordoinghisbesttoensurethatmyfamilyandIwouldhavea pleasantexperienceduringmyPh.D.studiesatMichiganStateUniversity. Iwouldliketothanktheremainingmembersofmyguidancecommittee,namely,Pro- fessorsKennethMerzJr.,JamesMcCusker,andRobertCukier,fortheirsupport,advice, andpatience. IwouldalsoliketothankProfessorsMarcosDantus,GaryBlanchard,JamesJackson, andBabakBorhanandtheirrespectiveresearchgroupsforthefantasticcollaborationre- gardingthephotoreactivityofthe FR0 -SBsuperphotobasethattheirexperimentalandour theoreticalgroupsstudiedtogether.Thisprojectnotonlygavemetheopportunitytolearn howtotlyperformsophisticatedelectronicstructurecalculationsforlargemolecules insolution,usingexplicitandimplicitsolvationmodelsandembeddingtechniques,butalso piquedmycuriosityregardingthevariousspectroscopicmeasurementsandtheirunderlying physicsandchemistry. IwouldalsoliketoexpressmygratitudetoformerandcurrentmembersofthePiecuch vi group,includingDr.JunShen,Dr.NicholasBauman,Dr.AdeayoAjala,Dr.J.Emiliano Deustua,Mr.StephenYuwono,Mr.ArnabChakraborty,andMr.KarthikGururangan.In particular,IwouldliketothankDr.NicholasBaumanforintroducingmetothecodes developedbythePiecuchgroupandforhisinvaluableassistanceduringmyinitialprojects asagraduatestudentatMichiganStateUniversity.IwouldalsoliketothankDr.Jun Shenforprovidingmewiththecoupled-clustercodesthatformedthebasisonwhichI implementedthenovelapproachesdiscussedinthisdissertation.IamindebtedtoDr.J. EmilianoDeustuaforhisassistanceandguidanceduringmystepsincodingusingthe FortranandPythonprogramminglanguagesaswellasforintroducingmetotheL A T E X typesettingsystem.IwouldalsoliketothankMr.StephenYuwonoforthecountlesshours ofdiscussionsregardingthevariousprojectsthatwetackledtogether. IwouldalsoliketothankmyundergraduatethesisadvisorProfessorAristidesMavridis andmyMaster'sthesisadvisorProfessorApostolosKalemos.Theyintroducedmetothe ofelectronicstructuretheoryandmademeappreciatetheimportanceofprecisionand ethicsinscienresearch.Theirguidanceduringmyinitialstepsinthishasbeen invaluableandIwillalwaysbethankfulforrecommendingpursuingaPh.D.underthe supervisionofProfessorPiecuch. Iwouldliketoacknowledgethesupportofvariousfriendsthathelpedtomakemyfam- ily'sandmystayinEastLansingaverypleasantexperience.Inparticular,Iwouldliketo thankDr.ChrysoulaVasileiou,ProfessorEliasStrangas,Dr.JaneTurner,ProfessorGeor- giosPerdikakis,ProfessorArtemisSpyrou,Dr.ChristosSidiropoulos,ProfessorXanthippi Chatzistavrou,Dr.ChristosGrigoriadis,Mr.IoannisZachos,Dr.GeorgiosPsaromiligkos, Mr.MichailPaparizos,Ms.Ana-MariaRaicu,Ms.AndrianaManousidaki,Mr.Dimitris Vardakis,Ms.ChristyNatalia,Ms.LauraCastroDiaz,andMr.ManosKokarakis.Iwould alsoliketothankDr.NikolaosEngelisforhissteadfastfriendshipsincewemetas freshmenbackinAthensin2007. Atthispoint,Iwouldliketothankmyfamilyfortheirsupportandlovealltheseyears. vii Inparticular,IwouldliketothankmyparentsThemistoklisMagoulasandMariaNikiforou (deceased),mybrotherThanosMagoulas,mywifeEleniLygda,andmydaughterMaria Magoulas.Maria'ssmilecancertainlybrighteneventhedarkestofmydays. viii TABLEOFCONTENTS LISTOFTABLES ................................... x LISTOFFIGURES ................................... xiii CHAPTER1INTRODUCTION ........................... 1 CHAPTER2PROJECTOBJECTIVES ....................... 22 CHAPTER3THEORY ................................ 23 3.1IntroductiontoCoupled-ClusterTheory....................23 3.2FailureofConventionalCoupled-ClusterMethodsinStronglyCorrelated Systems......................................30 3.3ApproximateCoupled-PairMethodswithanActive-SpaceTreatmentof Three-BodyClusters...............................40 3.4TowardtheFullInteractionLimitforStrongCorrelation...54 CHAPTER4NUMERICALRESULTS ........................ 67 4.1ApplicationoftheApproximateCoupled-PairMethodswithanActive- SpaceTreatmentofThree-BodyClusterstoModelMetal{InsulatorTransitions67 4.2ApproachingtheFullInteractionLimitforStrongCorrelation UsingSemi-StochasticIdeas...........................86 CHAPTER5CONCLUDINGREMARKSANDFUTUREOUTLOOK ....... 111 APPENDICES ...................................... 120 APPENDIXACOMPUTERIMPLEMENTATIONOFTHEAPPROXI- MATECOUPLED-PAIRMETHODSWITHANACTIVE- SPACETREATMENTOFTHREE-BODYCLUSTERS ... 121 APPENDIXBCOMPUTERIMPLEMENTATIONOFTHECLUSTER- ANALYSIS-DRIVENFULLCONFIGURATIONINTER- ACTIONQUANTUMMONTECARLOAPPROACHFOR STRONGLYCORRELATEDSYSTEMS ........... 138 BIBLIOGRAPHY .................................... 149 ix LISTOFTABLES Table4.1:AcomparisonoftheenergiesresultingfromthevariousCCapproaches withsinglesanddoublesandtheexactFCIdataforthesymmetricdis- sociationoftheH 6 /cc-pVTZringatselectedbonddistancesbetween neighboringHatoms R H{H (in A).......................76 Table4.2:Acomparisonoftheenergiesresultingfromthevariousactive-space triplesCCapproachesandtheexactFCIdataforthesymmetricdis- sociationoftheH 6 /cc-pVTZringatselectedbonddistancesbetween neighboringHatoms R H{H (in A).......................77 Table4.3:AcomparisonoftheenergiesresultingfromthevariousfulltriplesCC approachesandtheexactFCIdataforthesymmetricdissociationof theH 6 /cc-pVTZringatselectedbonddistancesbetweenneighboringH atoms R H{H (in A)...............................78 Table4.4:AcomparisonoftheenergiesresultingfromthevariousCCapproaches withsinglesanddoublesandtheexactFCIdataforthesymmetricdis- sociationoftheH 10 /DZringatselectedbonddistancesbetweenneigh- boringHatoms R H{H (in A)..........................79 Table4.5:Acomparisonoftheenergiesresultingfromthevariousactive-space triplesCCapproachesandtheexactFCIdataforthesymmetricdissoci- ationoftheH 10 /DZringatselectedbonddistancesbetweenneighboring Hatoms R H{H (in A)..............................80 Table4.6:AcomparisonoftheenergiesresultingfromthevariousfulltriplesCC approachesandtheexactFCIdataforthesymmetricdissociationofthe H 10 /DZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A)...................................81 Table4.7:AcomparisonoftheenergiesresultingfromthevariousCCapproaches includinguptotripleexcitationsandthenearlyexactLDMRG(500) dataforthesymmetricdissociationoftheH 50 /STO-6Glinearchainat selectedbonddistancesbetweenneighboringHatoms R H{H (inbohr)...82 x Table4.8:Convergenceoftheenergiesresultingfromtheall-electron i -FCIQMC, CAD-FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ =0 : 0001a.u.towardFCIfortheH 2 Omolecule,asdescribedbythe cc-pVDZbasisset,attheequilibriumgeometry R e andthegeometry obtainedbyasimultaneousstretchingofbothO{Hbondsbyafactorof 2.0.The i -FCIQMCcalculationswereinitiatedbyplacing1,000walkers ontheRHFdeterminantandthe n a parameteroftheinitiatoralgorithm wassetat3...................................104 Table4.9:Convergenceoftheenergiesresultingfromthe i -FCIQMC,CAD- FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ = 0 : 0001a.u.towardFCIforthesymmetricdissociationoftheH 6 ring, asdescribedbythecc-pVDZbasisset,attworepresentativevaluesof thedistancebetweenneighboringHatoms,including R H{H =1 : 0 A(the regionoftheminimumontheFCIPECshowninFigure4.5(a)character- izedbyweakercorrelations)and R H{H =2 : 0 A(theregioncharacterized bystrongcorrelationsinvolvingtheentanglementofallsixelectrons). The i -FCIQMCcalculationswereinitiatedbyplacing1,500walkerson theRHFdeterminantandthe n a parameteroftheinitiatoralgorithm wassetat3...................................105 Table4.10:SameasTable4.9forthesymmetricdissociationoftheH 10 ring,as describedbytheDZbasisset.InanalogytotheH 6 ring, R H{H =1 : 0 AcorrespondstotheregionoftheminimumontheFCIPECshown inFigure4.5(b)characterizedbyweakercorrelations,whereas R H{H = 2 : 0 Aistheregioncharacterizedbystrongcorrelationsinvolvingthe entanglementofalltenelectrons.......................106 Table4.11:ResultsoftheCAD-FCIQMCcalculationsbasedontheAS-FCIQMC wavefunctionsobtainedafterequilibrationrunsusing1billion(1B)and 2billion(2B)walkers..............................107 TableA.1:Thespin-orbitalHugenholtzandBrandowdiagramsandcorresponding algebraicexpressionsarisingfromthe D ab ij V N 1 2 T 2 2 C E term.Inthe lastcolumn,weprovidethecorrespondencebetweenthespin-orbitaldia- gramspresentedinthistableandthespin-adaptedonesshowninFigure 3.5........................................122 TableA.2:Thespin-integratedHugenholtzandBrandowdiagramsandcorrespond- ingalgebraicexpressionsarisingfromthe D AB IJ V N 1 2 T 2 2 C E term. Inthelastcolumn,weprovidethecorrespondencebetweenthespin- integrateddiagramspresentedinthistableandthespin-adaptedones showninFigure3.5...............................124 xi TableA.3:Thespin-integratedHugenholtzandBrandowdiagramsandcorrespond- ingalgebraicexpressionsarisingfromthe ˝ A e B I e J V N 1 2 T 2 2 C ˛ term. Inthelastcolumn,weprovidethecorrespondencebetweenthespin- integrateddiagramspresentedinthistableandthespin-adaptedones showninFigure3.5...............................127 TableA.4:Thespin-integratedHugenholtzandBrandowdiagramsandcorrespond- ingalgebraicexpressionsarisingfromthe ˝ e A e B e I e J V N 1 2 T 2 2 C ˛ term. Inthelastcolumn,weprovidethecorrespondencebetweenthespin- integrateddiagramspresentedinthistableandthespin-adaptedones showninFigure3.5...............................132 xii LISTOFFIGURES Figure3.1:Theground-statePECsofH 2 inthepresenceof0,2,4,and6Neatoms asobtainedwith(a)CCSD/cc-pVDZand(b)CISD/cc-pVDZ.TheNe atomswerelocatedsuchthattheydonotinteractamongthemselvesand withtheH 2 molecule.Inpanel(b),wealsoincludetheRHFPECfor comparison.AllPECshavebeenalignedsuchthatthecorresponding electronicenergiesattheinternuclearseparation R H{H =4 : 0 Aare identicalandsetat0kcal/mol........................26 Figure3.2:Acomparisonoftheenergiesresultingfromtheall-electronCCSD, CCSDT,andCCSDTQcomputationsfortheH 2 Omolecule,asde- scribedbythecc-pVDZbasisset[276],attheequilibriumandtwo displacedgeometriesobtainedbysimultaneousstretchingofbothO{H bondsbyfactorsof1.5and2.0[277].....................30 Figure3.3:SchematicrepresentationofthemodelsofcyclicpolyenesC N H N with (a) N =6,(b) N =10,and(c) N =14.Thepertinentmolecular orbitaldiagramsarepresentedaswell....................32 Figure3.4:Branchingdiagramillustratingthetotalspinasafunctionofthenumber of s = 1 2 coupledspins.Theintegersinsidethecirclesindicatethe numberofindependentspinstatesassociatedwithagiventotalspin andnumberof s = 1 2 spinscoupled.....................41 Figure3.5:Goldstone{Brandowdiagramsforthe 1 2 T 2 2 contributions (2) k ( AB;IJ ; S r ), k =1{5,totheCCequationsprojectedonthesingletpp-hhcou- pledorthogonallyspin-adapteddoublyexcited AB IJ E S r states.The intermediatespinquantumnumber S r in AB IJ E S r andthe correspondingdoublyexcitedclusteramplitudes [ S r ] t IJ AB is0or1.The occupiedorbitalindices M and N ,theunoccupiedorbitalindices E and F ,andtheintermediatespinquantumnumbers e S 1 r , e S 2 r ,and e S r are summedover..................................42 xiii Figure3.6:Ground-statePECscharacterizingthe D 6h -symmetricdissociationof theH 6 /STO-6Gring,asobtainedwiththevariousACCSDapproaches thatrelyonsubsetsofthenon-lineardiagramsshowninFigure3.5: (a)diagramsD1{D5areneglectedresultinginlinearizedCCSD,(b)all butoneofthediagramsD1{D5areneglected,(c)allbuttwoofthe diagramsD1{D5areneglected,(d)allbutthreeofthediagramsD1{D5 areneglected,(e)allbutfourofthediagramsD1{D5areneglected,and (f)alldiagramsareconsidered, i.e. ,fullCCSD.Thedashedhorizontal linedesignatestheexactdissociationlimitfortheemployedbasisset correspondingto6non-interactingHatoms................48 Figure3.7:Schematicrepresentationofthepartitioningoforbitalsintofourdis- jointgroups,namely,core(magenta),activeoccupied(olive),active unoccupied(blue),andvirtual(red),employedinactive-spaceSRCC approaches...................................53 Figure3.8:FlowchartoutliningthekeystepsoftheCAD-FCIQMCalgorithm....65 Figure4.1:Ground-statePECs[panels(a){(c)]anderrorsrelativetoFCI[panels (d){(f)]forthesymmetricdissociationoftheH 6 ringresultingfrom theCCSDandvariousACCSDcalculations[panels(a)and(d)],the active-spaceCCSDtandACCSDtcomputations[panels(b)and(e)], andthefullCCSDTandACCSDTmethods[panels(c)and(f)],using thecc-pVTZbasisset.Theactive-spacetriplesapproachesemployeda minimumactivespacebuiltfromthe1 s orbitalsofindividualhydrogen atoms.TheFCIPECisincludedinpanels(a){(c)tofacilitatethe comparisons..................................83 Figure4.2:Ground-statePECs[panels(a){(c)]anderrorsrelativetoFCI[panels (d){(f)]forthesymmetricdissociationoftheH 10 ringresultingfromthe CCSDandvariousACCSDcalculations[panels(a)and(d)],theactive- spaceCCSDtandACCSDtcomputations[panels(b)and(e)],andthe fullCCSDTandACCSDTmethods[panels(c)and(f)],usingtheDZ basisset.Theactive-spacetriplesapproachesemployedaminimum activespacebuiltfromthe1 s orbitalsofindividualhydrogenatoms. TheFCIPECisincludedinpanels(a){(c)tofacilitatethecomparisons.84 Figure4.3:Ground-statePECsforthesymmetricdissociationoftheH 50 linear chainresultingfromtheCCSDandACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ) calculations[panel(a)]andtheirfulltriplesextensions[panel(b)]using theSTO-6Gbasisset.ThenearlyexactLDMRG(500)PECofReference [175]isincludedinbothpanelstofacilitatethecomparisons.Theinsets showtheerrorsrelativetoLDMRG(500)inmillihartree( cf. Table4.7).85 xiv Figure4.4:Convergenceoftheenergiesresultingfromtheall-electron i -FCIQMC, CAD-FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ =0 : 0001a.u.towardFCIfortheH 2 Omolecule,asdescribedby thecc-pVDZbasisset,at(a)theequilibriumgeometryand(b)the geometryobtainedbyasimultaneousstretchingofbothO{Hbondsby afactorof2withoutchangingthe \ (H-O-H)angle(bothgeometries weretakenfrom[277]).The i -FCIQMCcalculationswereinitiatedby placing1,000walkersontheRHFdeterminantandthe n a parameter oftheinitiatoralgorithmwassetat3.Allenergiesareerrorsrelative toFCIinmillihartree,andtheinsetsshowthepercentagesoftriply (%T)andquadruply(%Q)exciteddeterminantscapturedduringthe i -FCIQMCpropagations...........................108 Figure4.5:Ground-statePECsforthesymmetricdissociationofthe(a)H 6 /cc- pVDZand(b)H 10 /DZsystemsresultingfromtheCCSD,CCSDT,and FCIcalculations................................108 Figure4.6:Convergenceoftheenergiesresultingfromthe i -FCIQMC,CAD- FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ = 0 : 0001a.u.towardFCIforthesymmetricdissociationoftheH 6 /cc- pVDZ[panels(a)and(b)]andH 10 /DZ[panels(c)and(d)]systemsat tworepresentativevaluesofthedistancebetweenneighboringHatoms, including R H{H =1 : 0 A[panels(a)and(c)]and R H{H =2 : 0 A[pan- els(b)and(d)].The i -FCIQMCcalculationswereinitiatedbyplacing 1,500walkersontheRHFdeterminantandthe n a parameterofthe initiatoralgorithmwassetat3.AllenergiesareerrorsrelativetoFCI inmillihartree,andtheinsetsshowthepercentagesoftriply(%T)and quadruply(%Q)exciteddeterminantscapturedduringthe i -FCIQMC propagations.TheCAD-FCIQMC[1{5]curveisabsentinpanel(d), sincethesolutionoftheCCequationsthedeterministicpart oftheCAD-FCIQMC[1{5]procedurefortheH 10 /DZsystemcouldnot becontinuedbeyond R H{H =1 : 75 A....................109 xv Figure4.7:Acomparisonoftheenergiesresultingfromthe i -FCIQMC,CAD- FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationsat50,000 [panels(a)and(d)],100,000[panels(b)and(e)],and150,000[pan- els(c)and(f)] i -FCIQMCiterationsusingtimestep ˝ =0 : 0001a.u., alongwiththecorrespondingfullydeterministicCCSD,CCSDT,and ACCSD(1 ; 3+4 2 )data,forthesymmetricdissociationoftheH 6 /cc-pVDZ [panels(a){(c)]andH 10 /DZ[panels(d){(f)]systemsatselecteddis- tancesbetweenneighboringHatoms, R H{H ,rangingfromtheweakly correlated(smaller R H{H )tothestronglycorrelated(larger R H{H )re- gions.The i -FCIQMCcalculationswereinitiatedbyplacing1,500 walkersontheRHFdeterminantandthe n a parameteroftheinitia- toralgorithmwassetat3.AllenergiesareerrorsrelativetoFCI inmillihartree.Theinsetsshowtheentirerangeoferrorsrelativeto FCI.......................................110 FigureA.1:Excerptofthecodethatcomputesthescaled 1 2 v EF MN t NM AF intermediate, whicheventuallymultiplies t I e J E e B andresultsintheD3contributionof the ˝ A e B I e J ( V NA T 2A T 2B ) C ˛ term( cf. TableA.3)............136 FigureA.2:ExcerptofthecodethatcomputesthescaledD1andD2diagrams arisingfromthe D AB IJ V NC 1 2 T 2 2B C E term( cf. TableA.2)......137 FigureB.1:PortionofalistcontainingSlaterdeterminantsandnumbersofsigned walkersinhabitingthemcorrespondingtothelasttimestep,namely, 160,000MCiterations,ofa i -FCIQMCsimulationforH 10 /DZwith R H{H =1 : 0 A.Thecolumncontainsserialnumbersforeachdeter- minant.Thesecondcolumncontainstheinformationaboutthesigned walkerpopulationswhiletheremainingcolumnslistthespin-orbitalin- dicesoccupiedinagivendeterminant.Odd(even)spin-orbitalindices correspondto ( )spinfunctions......................138 FigureB.2:Excerptofthecodethatcomputesthedeterministicdiag3 (2) 3 con- tributionofthe D AB IJ V NA 1 2 T 2 2A C E term( cf. TableA.2).......147 FigureB.3:Excerptofthecodethatcomputesthestochastic(1 diag3) (2),(MC) 3 contributionofthe D AB IJ V NA 1 2 T 2 2A C E term( cf. TableA.2).....148 xvi CHAPTER1 INTRODUCTION Everyattempttoreferchemicalquestionstomath- ematicaldoctrinesmustbeconsidered,nowandal- ways,profoundlyirrational,asbeingcontrarytothe natureofthephenomena. A.Comte, PositivePhilosophy ,translatedbyH. Martineau(CalvinBlanchard,NewYork,1858). TheapplicationsoftheScodingerequationtomolecularsystemsappearedjustone yearafterthepublicationofScodinger'sseminalpapers[1{6],whenBurrauexaminedthe groundstateoftheH + 2 species[7]andHeitlerandLondonstudiedthegroundstateofH 2 [8],thesimplestone-andtwo-electronmolecules,respectively,givingbirthtotheof quantumchemistry.Theprospectofelucidatingthenatureofthechemicalbondusingthe thennewlyformulatedquantummechanicsleadHeitlertoexclaim\Wecan,then,eatChem- istrywithaspoon"inalettertoLondonin1927[9].However,itwassoonrealizedthat theapplicationoftheScodingerequationtomany-electronsystemsleadstoformidable equationsthatcannotbesolvedanalytically[10].Thiscanbeunderstoodbyexamining theelectronicScodingerequation,thecentraltenetofquantumchemistry,which,from themathematicalpointofview,isaneigenvalueproblem: H ( X ; R )= E ( R ( X ; R ), where H istheelectronicHamiltonianforthesystemofinterest, isthemany-electron wavefunctioncharacterizingthe thelectronicstate,with =0denotingthegroundstate and > 0designatingexcitedstates,and E isthecorrespondingtotalelectronicenergy. ThankstotheuseoftheBorn{Oppenheimerapproximation[11],whichdecouplestheelec- tronicandnuclearmotions,theelectronicScodingerequationissomewhatsimplerthanthe underlyingmolecularScodingerequationfromwhichitisobtained,butitisstillahighly complexmulti-dimensionalmathematicalandnumericalproblemwhenrealisticsystemsare 1 examined.Indeed,themany-electronwavefunctiondependsexplicitlyonthespatialand spincoordinatesofallelectrons,collectivelydenotedas X ,andparametricallyorimplicitly onthenuclearcoordinates, R ,whilethetotalelectronicenergy,whencalculatedbysolving theelectronicScodingerequationatmultiplevaluesof R ,becomesafunctionofthenu- clearcoordinates.Formoleculesotherthansmallfew-electronsystems,solvingtheelectronic Scodingerequationbecomesamajorchallenge,evenwhenoneisinterestedinsingle-point calculations(calculationsatasinglenucleargeometry). Ingeneral,theelectronicHamiltoniancontainsone-andtwo-bodyterms.Inthisdis- sertation,wefocusonanon-relativisticdescription,wheretheelectronicHamiltonianfor asystemcomprisedof N electronsand M nucleitakestheform,inatomicunits, H = P N i =1 z i + P N i 0 9 meansexcitedstates;weusesymbol 1 todesignatetheunitoperator).Intheconventional treatmentofSRCCtheory,asequenceofapproximatemethodsisobtainedbytruncating T and R ataparticularexcitationrank m A 2,butonecannotproducequantitative resultsinthemajorityofchemistryapplicationswithout T 3 .Allpreviousattemptstoincor- porateconnectedtriplyexcitedclustersusingconventionalMBPT-likearguments,similar tothoseexploitedinCCSD[T][125],CCSD(T)[126],orCCSDT-1[119,120],withinthe ACPframeworkhaveonlybeenpartlysuccessful[190,192,193,219,231].Anothermajor problempertainstothefactthatthecombinationsofdiagramsthatresultintheimproved performanceofACPmethodsinthestronglycorrelatedregimeofminimum-basis-setmodel systemsarenotnecessarilyoptimumwhenlargerbasissetsareemployed. Inthisdissertation,wesuccessfullyaddressbothoftheseproblems.Wedealwiththeis- sueregardingthemissing T 3 physicsbyadoptingandfurtherdevelopingtheactive-spaceCC ideas[69,70,80{82,98{100,145{160]toincorporatethedominant T 3 amplitudesintheACP methodsinarobust,yetcomputationallymanner.Asshowninthisdissertation, theactive-spacetriplesACPapproachesareimmunetothevariousissuesplaguingconven- 16 tionalMBPT-basedconnectedtriplesenergycorrections,whichfromvanishinglysmall perturbativedenominatorsinthestrongcorrelationregime.Furthermore,theactive-space CCmethodologiesandtheirACPcounterpartsdevelopedinthisthesisresearchincorporate theleadinghigher{than{two-bodyclustersinaniterativemanner,allowingthelower-order clusteramplitudestorelaxinthepresenceofthedominanthigher-orderones.Inaddition, theactive-spaceCCandACPschemessystematicallyconvergetotheircorrespondingpar- entmethodsbysimplyincreasingthesizeoftheactivespace,thelimitbeingreachedwhen allorbitalsbecomeactive.Finally,weaddresstheconcernofextendingtheACPmethods tolargerbasissetsbyproposinganewACPvariantthatutilizesbasis{set-dependentscal- ingfactorsmultiplyingthepertinentCCSDdiagrams.Asshowninthisdissertation,this novelACPschemereducestoDCSDwhenaminimumbasissetisemployed, i.e. ,itremains exactforstronglycorrelatedmodelsystems,whilebecomingasymptoticallyequivalentto ACCSD(1,4), i.e. ,totheoriginalACP-D14schemeofPiecuchandPaldus[191]augmented withsingles,inaCBSlimit.Thisisadesiredbehaviorbecause,basedonournumerical observations,theACCSD(1,4)approachcorrectedforconnectedtriplesprovides,amongthe varioustestedtriples-correctedACPschemes,themostaccuratedescriptionfortheentire spectrumofcorrelation(fromweaktostrongcorrelations)forlargerbasissets. TheACPschemesexaminedinthisdissertation,especiallythoseincorporatingconnected three-bodyclusters,provideanaccuratedescriptionofelectroncorrelationincluding boththeweaklyandstronglycorrelatedregimes,andbecomeexactinthestrongcorrelation limitofmodelminimum-basis-setHamiltoniansorintheatomizationlimitoflatticesof hydrogenatoms,buttheymayhavelimitationsinsomesituations.Oneofthesuccessesof thisdissertationisthedevelopmentandimplementationofanovelsemi-stochasticapproach withCCSD-likecomputationalcostthatcanprovideFCI-qualityenergeticsfortheentire rangeofcorrelationincludingboththeweaklyandstronglycorrelatedregimes.Before weproceedwiththeinformationaboutthisnewsemi-stochasticmethodology,wegiveabrief introductiontoquantumMonteCarlo(QMC)approaches,onwhichthenovelsemi-stochastic 17 approachdevelopedinthisthesisresearchisbased. Ingeneral,MonteCarlo(MC)methodsprovidesolutionstoproblemsofeitherproba- bilisticordeterministicnaturebyvirtueofstochasticsampling.Theoriginalrealizationsof MCapproachesrequiredscientiststoperformaseriesofexperiments, i.e. ,samplings,and theoveralloutcomewasinferredbymeansofstatisticalanalysis.Onesuchexamplewas theexperimentaldeterminationofthemathematicalconstant ˇ in1873[245]using needle[246,247],butinstancesofsuchexperimentalmathematicscanbetracedbackto ancientBabylonandOldTestamenttimes[248].PivotalinthedevelopmentofmodernMC approacheswasthereplacementofcostlyandsometimesimpracticalorevenimpossibleex- perimentsbythenumericalprocessingofrandomorpseudorandomnumbers.Theinvention ofMCintheformthatisusedtodayisattributedtoFermi,whouseditinhisunpublished workonneutronmoderationintheearly1930s[249].However,theaccurateMCstudyof complexphysicalprocesses,whichrequirethecollectionofanenormousnumberofsampling points,wasnotpossibleuntiltheadventofelectronicdigitalcomputersinthelate1940s. Aroundthattime,thefoundationsofmodernMCmethodswerelaiddownatLosAlamosin thepioneeringworksofUlam,vonNeumann,Fermi,Metropolis,andothers[249{252].The applicationofMCmethodstoaddressproblemsofquantummechanicalnaturearetermed QMC(see,forexample,Reference[253]forareviewregardingtheuseofQMCmethodsto solvetheelectronicScodingerequation). WebeginourdiscussionofQMCapproachesstartingwithvariationalMC(VMC)[254, 255].Inanalogywithothervariationalapproaches,onestartswithatrialwavefunctionfor theproblemofinterestthatcontainsoneormoreparameters.Subsequently,oneinvokesthe variationalprinciple,solvingthevariousintegralsusingMCintegrationsamplingthewave- functionprobabilitydistribution,andevaluatingtheenergyexpectationvalueastheaverage ofthelocalenergyvalues,toobtaintheoptimumparametersthatresultinthelowest-energy wavefunctionofagivenfunctionalform.TheaccuracyoftheresultingVMCwavefunction andenergydependsonthequalityofthetrialwavefunction.AnalternativeQMCapproach 18 thatislesssensitivetothequalityoftrialwavefunctionisMC(DMC).Theba- sicideabehindtheDMCmethodologywasalreadystatedin1949,whenFermirecognized thatthetime-dependentScodingerequationinimaginarytimeresemblesaequa- tionthatcanbesolvedstochastically[250].Aslongastheinitialtrialwavefunctionhasa non-zerooverlapwiththeexactwavefunction,itisguaranteedthatbypropagatingthetime- dependentScodingerequationinimaginarytimetheexactwavefunctionwillbeprojected outintheimaginarytimelimit.OneoftheadvantagesofDMCisthattheunderly- ingcomputercodesareeasytoparallelizeacrossmultiplenodesandtheyrequirerelatively modestcomputationalresources.Inaddition,DMCschemesdirectlysampletherealspace of3 N electroniccoordinatesallowingthemtorecover,inprinciple,theexactsolutiontothe Scodingerequation,unlikeconventionalquantumchemistryapproachesoperatinginthe N -electronHilbertspacespannedbySlaterdeterminants,whicharelimitedbythesizeand qualityoftheone-and N -electronbasissets. Unfortunately,theDMCmethodsoutlinedaboveareplaguedbyamajorproblemthat makestheirapplicabilitytomolecularsystemsIfnoconstraintsareenforcedon thetrialwavefunction,propagatingthetime-dependentScodingerequationinimaginary timewilleventuallyprojectoutabosonicstate, i.e. ,thetruegroundstateofthespin- freeScodingerequation,albeitviolatingthePauliexclusionprinciple.Thisproblemcan beremediedbyusingwhatisknownasthedeapproximation[256{259].Inthis case,oneimposesthenodestructureofanapproximatewavefunction,obtainedusingcon- ventionalquantumchemistryapproaches,onthetrialwavefunction,essentiallyforcingthe DMCsolutiontobeantisymmetric.Ofcourse,byintroducingthenodesofanapproximate wavefunction,theDMCapproachcannolongeryieldtheexactsolutiontothemany-electron ScodingerequationandthequalityoftheDMCresultsmayheavilydependonthequality oftheapproximatewavefunctionanditsabilitytoproperlylocatethenodes. Recently,anovelQMCapproachwasformulatedthatalleviatestheissuesintroduced bythed-nodeapproximation.Thisisaccomplishedbypropagatingthetime-dependent 19 Scodingerequationinthe N -electronHilbertspacespannedbySlaterdeterminants,rather thanintherealspaceof3 N electroniccoordinates[260].ThefactthatSlaterdeterminants obeythePauliexclusionprinciplebyconstructionensuresthattheresultingwavefunction willbeantisymmetric.SimilartotheDMCcase,thestochasticsamplingofthe N -electron Hilbertspaceismadepossiblethroughawalkerpopulationdynamicsalgorithmthatplaces morewalkersonimportantdeterminantsandlesswalkersonthelessimportantones.The resultingapproachiscalledFCIQMC,since,inanalogytoFCI,thewalkersareallowedto exploretheentire N -electronHilbertspace,andthisprovidesthenumericallyexactsolution tothe N -electronScodingerequationinabasissetequivalenttoFCIinthelimitofe propagationtime[260].TheconvergenceoftheFCIQMCapproachcanbeslow,butthiscan beaddressedbyemployinganinitiatorapproach,givingrisetothe i -FCIQMCscheme,that dramaticallydecreasesthetotalwalkerpopulationrequiredforobtainingaccurateresults [261,262],andotherpowerfulalgorithms,suchastheadaptive-shiftapproachdevelopedin References[263,264]. AlthoughtheFCIQMCapproachisguaranteedtoprovidethenumericallyexactsolution inagivenbasisset,oneneedstensorhundredsofthousandsofMCtimesteps,called MCiterations,forthistobeaccomplished.Furthermore,inthepresenceofstrongmany- electroncorrelationtheconvergenceofFCIQMCisdeceleratedconsiderably.One ofthemajorsuccessesofthisdissertationistheintroductionofanovelsemi-stochastic electronicstructureapproachthatiscapableofextrapolatingFCI-qualityenergetics,even inthestrongcorrelationregime,outoftheearlystagesofFCIQMCpropagations.This methodology,calledcluster-analysis-drivenFCIQMC(CAD-FCIQMC)[265,266],isbased onthefactthat,forHamiltonianscontaininguptotwo-bodyterms,asisthecaseinquantum chemistry,thecorrelationenergydependsonlyontheone-andtwo-bodyclusters, T 1 and T 2 ,respectively,which,inturn,couplewiththeirthree-andfour-bodycounterparts,butno morethanthat,throughtheCCequationsprojectedonsinglesanddoubles.Thisimplies thatextractingtheexact T 3 and T 4 componentsoftheclusteroperator T andsolving 20 CCSD-likeequationsfor T 1 and T 2 intheirpresenceshouldyieldtheexact T 1 and T 2 and consequently,theexact,FCI,energy.Thisobservationisthedrivingforcebehindexternally correctedCCapproaches[219,220,230,243,267{274]inwhichoneextractsthethree-and four-bodyclustersfromanon-CCsourcethatprovidesagooddescriptionofthemany- electronsystemofinterest.Inourcase,wetakeadvantageofthe i -FCIQMCstochastic wavefunctionsamplingasagoodsourceofFCI-quality T 3 and T 4 .Althoughtheoriginal CAD-FCIQMCapproachprovidesnearlyexactenergeticsoutoftheearlystagesofthe i - FCIQMCpropagations,eveninthepresenceofelectronicquasi-degeneraciessuchasthose characterizingthedoublebonddissociationofH 2 O[265],itmayfailordisplayproblemsin thestrongcorrelationregime.Inthisdissertation,weextendtheCAD-FCIQMCapproach tostrongmany-electroncorrelationctsbyexploitingtheACPideaswithintheCAD- FCIQMCformalism,sothatonecanobtainFCI-qualityenergeticsfortheentirespectrum ofelectroncorrelation 21 CHAPTER2 PROJECTOBJECTIVES 1. ExtensionoftheACPmethodologies,especiallythosebasedontheACP-D13,ACP- D14,andACP-D1(3+4)/2ideas,toallowfortheexplicitincorporationof T 3 clusters viaactive-spaceCCconsiderations. 2. DevelopmentandtestingofanovelACPapproachusingbasis{set-dependentscaling factors,suitableforthestudyofsystemswithrealisticbasissets. 3. Developmentandimplementationofthesemi-stochasticCAD-FCIQMCmethodology anditsextensiontothestronglycorrelatedregime. 4. Applicationoftheabovenovelmethodologiestolow-dimensionalmodelsystemsrele- vanttometal{insulatortransitions,suchashydrogenclusters,aswellastoproblems ofchemicalinterest,suchasthedoublebonddissociationofH 2 Oandtheground-state energyofC 6 H 6 . 22 CHAPTER3 THEORY 3.1IntroductiontoCoupled-ClusterTheory ...believeitornot,itistakenfromoneofthe ChemistryJournals! RemarkmadebyR.McWeenyduringhis1967In- auguralLectureattheUniversityofrefer- ringtotheL-CCDdiagramsandCCDequations showninReference[45][quotetakenfromJ.Pal- dus,in TheoryandApplicationsofComputational Chemistry:TheFirstFortyYears ,editedbyC.E. Dykstra,G.Frenking,K.S.Kim,andG.E.Scuse- ria(Elsevier,Amsterdam,2005)pp.115{147]. AsmentionedintheIntroduction,intheSRCCformalismtheexactground-statewavefunc- tionofan N -electronsystemisexpressedas j 0 i = e T j i ;T = N X n =1 T n ; (3.1) where j i isanIPMreferencestatethatservesastheFermivacuum(usuallyaHFSlater determinant)and T istheclusteroperator.The n -bodycomponentof T isas T n = X i 1 < c regionbecoming complexwhen < c ,sothatthecorrelationenergiesaround c satisfytheLaurentseries E ( )= E ( c )+ E (1) ( c ) 1 2 + O ( c )[190].Itwasalsodemonstrated,byderiving theanalyticalformulafortheexactdoublyexcitedclusteramplitudesofthePPPandHub- bardHamitonianmodelsandbacksubstitution,thattheseexactamplitudesdonotsatisfy theCCDsystemat =0[191]. AsanailintheofconventionalSRCCapproaches,wewillproveanalytically thatinthestrongcorrelationregimetheonlymeaningfulCCschemeisFCC.Thiswas alreadyanticipatedsincetheearly1980s,whenPaldus etal. [188,278]showedthat,inthe ! 0 limitoftheHubbardandPPPmodelsofthe N =6and10cyclicpolyenes,the dominantquadruplesweretheconnectedones,byclusteranalysisofFCIwavefunctions. Furthermore,theyspeculatedthat T 2 n clusterswith n 3wouldplayanimportantrolefor largercyclicpolyenes,sincetheCISDTQapproachwasalreadystrugglingwithC 10 H 10 . Tosimplifythemathematicalmanipulations,weshallexaminethe12-siteat- tractivepairingHamiltonian[282{284],whichconstitutesasimplemodelforsuperconductiv- itybyphenomenologicallydescribingtheCooperproblemofboundelectronpairs[285{287]. TheattractivepairingHamiltonianhastheform[209] H = X P;˙ P a P˙ a P˙ G X PQ a P " a P # a Q " a Q # ;G 0 ; (3.15) and,aswasthecasewiththeHubbardandPPPHamiltonians,dependsontwoparameters, namely,theorbitalenergies P andtheinteractionstrength G .Byvaryingthevalueof G from0to 1 onecancontinuouslygofromtheweaklytothestronglycorrelatedregime.A simpleinspectionofEquation(3.15)revealsthattheattractivepairingHamiltonianpreserves thesenioritynumber,whichisthenumberofunpairedelectrons.Asaresult,allodd-number- excitedclustercomponents,whichnecessarilybreakthepairingofelectrons,arezero, i.e. , T 2 n +1 =0 ; 8 n 2 Z + [f 0 g ,whichconstitutesamajor 35 WebeginourproofbyexaminingthefullycorrelatedlimitoftheattractivepairingHamil- tonian.WeassumethatinthisregimeallSlaterdeterminantsbecomeexactlydegenerate withtheRHFone.ByexaminingthestructureofthecorrespondingFCIwavefunction, j 0 i = 1 + P n 2 Z + C 2 n j i ,wenoticethatthedegeneracyconditionforcesalldetermi- nantstohaveaweightof1,namely, c I " I # A " A # =1 ; 8 I 2 occ. ;A 2 unocc., c I " I # J " J # A " A # B " B # =1 ; 8 I< J 2 occ. ;AT 3 >T 4 > >T N .Thisconventionalparadigmisdestroyed,beingoftenreversed, inastronglycorrelatedregime,sothattheonlymeaningfulconventionalSRCCapproach capableofproperlydescribingstrongcorrelationsisFCC,whereallmany-bodycomponents of T mustbeincludedinthecalculations. 3.3ApproximateCoupled-PairMethodswithanActive-Space TreatmentofThree-BodyClusters ...theneglectofnonlinearnonfactorizablediagrams ...mustsimulatethehigherexcited-clustercontri- butions,suchas T 4 and T 6 . J.Paldus,M.Takahashi,andR.W.H.Cho,Int.J. QuantumChem.Symp. 18 ,237(1984). AcrucialstepinthediscoveryandsubsequentunderstandingofACPschemeswasthe developmentoftheorthogonallyspin-adaptedCCformalism[219,220,230,288{290].There- fore,beforeweproceedtothediscussionofACPapproaches,wepresentthesalientfeatures ofsingletspin-adaptedCCtheory,usingtheCCwithdoubles(CCD)[45,291,292]asan example. Thestepindevelopinganorthogonallyspin-adaptedCCformalismisto the A 1 :::A n I 1 :::I n ; f e S r g SM S ˛ statefunctionsthatspanthemany-electronHilbert space,whereindices I 1 , I 2 ;::: or I;J;::: ( A 1 ;A 2 ::: or A;B;::: )designateorbitalsthat areoccupied(unoccupied)intheclosed-shell( e.g. ,RHF)referencedeterminant, S and M S denotethetotalspinanditsprojectiononthe z -axis,respectively,and e S r representsthe relevantintermediatespinquantumnumbersresultinginthe j S;M s i spinstate(note thattheclosed-shellreferencedeterminantissingletspin-adaptedbyconstruction).Sincewe areinterestedinsingletspin-adaptedCCD,weneedtoconsiderdoublyexcited statefunctionsoftheform AB IJ ; f e S r g 00 E ,whichrequirethecouplingoffour s = 1 2 spins involvedin2-particle{2-holeexcitationviatheintermediatespins e S r .Althoughthereexista 40 1 2 3 4 0 1 2 1 3 2 2 1 1 1 2 1 2 3 1 Singlet Doublet Triplet Quartet Quintet Numberof s = 1 2 Spins TotalSpin Figure3.4:Branchingdiagramillustratingthetotalspinasafunctionofthenumberof s = 1 2 coupledspins.Theintegersinsidethecirclesindicatethenumberofindependentspinstates associatedwithagiventotalspinandnumberof s = 1 2 spinscoupled. fewwaysofcouplingthespinsofthefourspin-orbitals,ithasbeenshownthattheparticle- particle{hole-hole(pp-hh)schemeleadstothemostsymmetricandsimpleexpressions[293]. Inthepp-hhapproach,oneobtainstwointermediatespinsdenotedas S r pp and S r hh by separatelycouplingthespinsofthetwoparticleandthetwoholestates,respectively,that aresubsequentlycoupledtothe S;M S spinstate.Thefactthatthestateisa singletimposestherestriction S r pp = S r hh = S r .Therefore,asonemightanticipatefrom thebranchingdiagram[294]showninFigure3.4,therearetwoindependentclassesofdoubly excitedstatefunctions,namely,thosethatarisefromanintermediatesinglet, S r =0,andthoseoriginatingfromanintermediatetriplet, S r =1.Todistinguishbetween thetwo,theketsrepresentingthesingletpp-hhcoupledorthogonallyspin-adapteddoubly excitedstatefunctions AB IJ ; f S r S r g 00 E canbedesignatedas AB IJ E S r ,where S r =0or1. Havingthestatefunctions,wenowproceedtothediscussionofthe clusteroperator.InCCD,theclusteroperatorisapproximatedbyitstwo-bodycomponent, i.e. , T (CCD) = T 2 .Basedontheanalysispresentedinthepreviousparagraph,itcomesasno surprisethat T 2 ,asatwo-bodyoperator,hastwoindependentsinglet-coupledcomponents, i.e. , T 2 = T [0] 2 + T [1] 2 ,as T [ S r ] 2 j i = P I J;A B [ S r ] t IJ AB AB IJ E S r .Thus,theCC 41 A I J B N F M E e S 2 r e S 1 r S AB (D1) A I J B M F N E e S 2 r e S 1 r S AB (D2) B J E M F N A I e S r S r S AB (D3) B J N F M E I A e S r S r S IJ (D4) N B M F E I J A S r S r (D5) Figure3.5:Goldstone{Brandowdiagramsforthe 1 2 T 2 2 contributions (2) k ( AB;IJ ; S r ), k = 1{5,totheCCequationsprojectedonthesingletpp-hhcoupledorthogonallyspin-adapted doublyexcited AB IJ E S r states.Theintermediatespinquantumnumber S r in AB IJ E S r andthecorrespondingdoublyexcitedclusteramplitudes [ S r ] t IJ AB is0or1.The occupiedorbitalindices M and N ,theunoccupiedorbitalindices E and F ,andtheinter- mediatespinquantumnumbers e S 1 r , e S 2 r ,and e S r aresummedover. amplitudeequationsforsingletspin-adaptedCCDare 0= S r D AB IJ H N 1 + T 2 + 1 2 T 2 2 C E ) (3.20) 0= S r D AB IJ [ H N ( 1 + T 2 )] C E + 5 X k =1 (2) k ( AB;IJ ; S r ) ; (3.21) whereinthelaststepresultinginEquation(3.21)weisolatedthebilinear (2) k ( AB;IJ ; S r ), k =1{5 ; termsthatcorrespondto 1 2 T 2 2 contributionsandthatformthebasisofthevarious ACPapproaches.InFigure3.5,weprovidetheGoldstone{Brandowdiagramsassociated withtheseenon-linearterms. WenowproceedtothediscussionoftheACPapproaches,especiallyhowtheycameinto existence,thereasonsbehindtheirsuccessinthepresenceofstrongmany-electroncorrelation andtheirnovelextensionstoconnectedtriplesandrealisticbasissetsintroducedin thisdissertation.We,thus,beginwithabriefhistoricaloverviewofthedevelopmentofthe ACPfamilyofmethods. Inthelate1970sandearly1980s,Paldus,Jankowski,andAdamsperformedaseries ofinvestigationsaimedatdevelopingreliableapproximationstoorthogonallyspin-adapted 42 CCD[224{227].Themotivationbehindthesecanbetracedtothefollowingtwo facts.First,atthattime,theconstructionofintermediatesbydiagramfactorization(see, forexample,Reference[295])wasstillinitsinfancyandresearcherswerefacedwiththe computationallydemandingtaskofsolvingasystemofnon-linearCCequations.Thislead JankowskiandPaldustostatethat\ itseemsquiteunlikelythat "CCD\ canberoutinelyused inthenearfutureforlargerthan10{20electronsystemsorevensmallerproblemsrequiring veryextensiveorbitalbasissets "[225].Second,theperformanceoftheapproximations toCCDthatwereavailableatthattimewasoftenworsethanthatoffullCCDwhen electronicquasi-degeneracieswerepresent(see,forexample,References[224]and[225]and referencestherein).AmongtheschemesfacingproblemswerelinearizedCCD(L-CCD)[45], whichcompletelyneglectsthenon-linearcontributionsshowninFigure3.5,andcoupled electronpairapproximation(CEPA)methodologies[296{305]thatsimplifythequadratic 1 2 T 2 2 contributionsbyconsideringonlytheexclusionprincipleviolatingtermsthatcanbe expressedastheproductofapair-energycontributiontimesa T 2 matrixelement. Toaddresstheaboveproblems,whileattemptingtoavoidtheuseoffullCCD,Paldusand co-workersdevisedanapproximateCCDapproach,abbreviatedasACP-D45,thatreplaces the P 5 k =1 (2) k ( AB;IJ ; S r )termsinEquation(3.20)by (2) 4 ( AB;IJ ; S r )+ (2) 5 ( AB;IJ ; S r ) [224,225], i.e. ,onlydiagramsD4andD5showninFigure3.5areretained.Indoingso, theywerepartlyinspiredbytheCEPAfamilyofmethods,sinceonlydiagramsD4andD5 canproduceexclusionprincipleviolatingtermscontainingpairenergies.Inthisregard,the variousCEPAschemescanbeviewedasapproximationstoACP-D45.ThesameACP-D45 schemewasrediscovered(underatname)byChilesandDykstrain1981[228](see alsoReference[229])usingtargumentation.ByexaminingtheCCDequationsinthe limitofseparatedelectronpairs,ChilesandDykstraprovedthatdiagramsD1{D3cancel outandproposedtheACCDapproachneglectingdiagramsD1{D3,whichisidenticalto ACP-D45ofPaldus etal. TheperformanceoftheACP-D45schemeinthepresenceofelectronicquasi-degeneracies 43 wasoriginallytestedbyPaldusandco-workersonfour-electronspecies,includingtheBe atom[226,227],theminimum-basis-setH 4 models[225],and cis -butadieneasdescribedby thePPPHamiltonian[225].Thelasttwosystemsallowonetoexaminetheentirerangeof electroncorrelationeitherbychangingthenucleargeometry(H 4 )orbyvaryingthe parametersthePPPHamiltonian( cis -butadiene).Inallcases,theCCDmethod- ologyproducedenergeticsofgenerallygoodquality,slightlyovercorrelatingaftertheonset ofdegeneracies.TheperformanceoftheACP-D45approachignoringdiagramsD1{D3was remarkable,closelyreproducingtheCCDresultsintheweaklycorrelatedregimeandout- performingitinthepresenceofelectronicquasi-degeneracies.Infact,ACP-D45became practicallyexactinthestrongcorrelationlimitofthePPPmodelof cis -butadiene.Atthe sametime,theACP-D123scheme,whichisthecomplementofACP-D45keepingdiagrams D1{D3andneglectingtheD4andD5contributions,becamesingularinthestrongcorrela- tionregime,aswasthecasewithL-CCD.Furthermore,Paldusandco-workerssearchedfor othercombinationsoftheD1{D5non-lineardiagramsthatcouldpossiblyhandlethequasi- degeneracypresentinthecaseoftheBeatom[226].Amongthe32possibleACPvariants, ACP-D45performedthebest,faithfullyreproducingtheCCDcorrelationenergy.Anaddi- tionalinterestingobservationwasthattheresultsoftheACP-D123calculationswerevery closetothoseobtainedwithL-CCD.Thissuggestedthatamutualcancellationofdiagrams D1{D3takesplace,explainingwhyACP-D45wasanexcellentapproximationtoCCD. MotivatedbythesuccessoftheACP-D45method,Paldusandco-workersturnedtheir attentiontotheHubbardandPPPmodelsofcyclicpolyenes[187,188,220],which,as emphasizedintheprevioussection,exhibitsevereelectronicquasi-degeneraciesastheres- onanceintegral approaches0.Theydemonstratedthat,unlikeCCSD(equivalentinthis casetoCCD),whicheventuallyovercorrelatesorbecomessingular,theACP-D45approach waswell-behavedintheentirerangeofelectroncorrelationcharacterizingtheC 6 H 6 andC 10 H 10 cyclicpolyenes.Infact,ACP-D45providedtheexactcorrelationenergiesinthe strongcorrelationlimitoftheHubbardHamiltonian,whilebeingveryaccurateinthecaseof 44 thePPPmodel.Thefactthatthe T 2 n componentsoftheclusteroperator T with n 2be- comeimportantinthestrongcorrelationregimeofthesemodelHamiltonianssuggeststhat theACP-D45approachprovidesamechanismforsimulatingtheofthesehigher{than{ two-bodyclusters.Indeed,inthespiritofexternallycorrectedCCmethodologies,Paldus andco-workersprovedanalyticallythat,aslongasthePUHFapproach[215{218]isexact andsinglesdonotcontribute, i.e. , C 1 = T 1 =0,whichisthecaseforthestronglycorrelated limitoftheHubbardandPPPmodelsofcyclicpolyenes,the S r D AB IJ ( V N T 4 ) C E termwith T 4 extractedfromPUHFcancelsdiagramsD1{D3andmultipliesdiagramD5byafactorof 9whenprojectedonthedoublyexcitedstatefunctionswithanintermediate triplet, AB IJ E 1 ,[220].Inprinciple,thethree-bodyclustersshouldhavebeenextractedas well,butthePUHFwavefunctiondoesnotcontainsingletspin-adaptedconnectedtriples. Theresultingapproach,whichuses (2) 4 ( AB;IJ ; S r )+(2 S r +1) 2 (2) 5 ( AB;IJ ; S r )instead of P 5 k =1 (2) k ( AB;IJ ; S r )inEquation(3.20)andis,therefore,almostidenticaltoACP-D45, wastermedACP-D45 9 orACPQ,toemphasizethe defacto presenceof T 4 contribution inthelatteracronym.TheACPQapproach,beingformallyanapproximationtoCCD,is muchbetterthanCCD,becomingexactinthestrongcorrelationlimitofcyclicpolyenesas describedbytheHubbardandPPPHamiltonians.Furthermore,inthecaseoftheHubbard Hamiltonian,whereD5iszerowhenprojectionsonto AB IJ E 1 areconsidered,ACPQbecomes equivalenttotheoriginalACP-D45approach,explainingtheexactnessofthelatterinthe =0limitoftheHubbardHamiltonianmodel.Asalreadyalludedtoabove,intheiroriginal derivation[220],Paldusandco-workersassumedthatthePUHFwavefunction,expressedin termsoftheRHFSlaterdeterminantusingtheThoulesstheorem,hasnosingletmonoex- citationcomponent,which,althoughcorrectinthecaseoftheHubbardandPPPmodels ofcyclicpolyenessincetheRHForbitalsarecompletelydeterminedbysymmetryrendering themsimultaneouslyBruecknerorbitals,isnottrueingeneral.Piecuch etal. generalized theaforementionedderivationforcaseswheresingletmonoexcitationsdonotvanish[219]. Theydemonstratedthatnotonlythesinglet-coupledthree-bodyclusterscontinuedtobe 45 absentinthePUHFwavefunction,butalsothatthediagramcancellationtheACPQ approachaugmentedfor T 1 clustersremainsvalid.TheresultingACCSD 0 approach,which reducestoACCD 0 =ACPQwhen T 1 =0,providesexactelectronicenergeticswheneverthe PUHFstateanexactdescription. Asitturnsout,thecombinationofD4andD5diagramsisnotuniqueinprovidingexact resultsinthe =0limitofcyclicpolyenemodels.Paldusandco-workersrealizedthat, duetothephsymmetrycharacterizingtheHubbardandPPPmodelsofcyclicpolyenes, diagramsD3andD4areequivalent, i.e. , (2) 3 ( AB;IJ ; S r )= (2) 4 ( AB;IJ ; S r ),suggesting thatACP-D35 9 isalsoexactinthefullycorrelatedlimitofthesemodels.Infact,usingany combinationoftheform (2) 3 ( AB;IJ ; S r )+(1 (2) 4 ( AB;IJ ; S r )+(2 S r +1) 2 (2) 5 ( AB;IJ ; S r )(3.22) insteadof P 5 k =1 (2) k ( AB;IJ ; S r )inEquation(3.20)wouldbeexactinthestrongcor- relationlimitofcyclicpolyenesdescribedbythePPPandHubbardHamiltonians,for anyvalueoftherealparameter (theHubbardHamiltonianonlyifthe(2 S r +1) 2 fac- torat (2) 5 ( AB;IJ ; S r )isignored).Interestingly,PiecuchandPaldusdemonstrated,an- alyticallyandnumerically,thattheexactdoublyexcitedclusterscorrespondingtothe strongcorrelationlimitofthecyclicpolyenemodels,asextractedbyclusteranalysisof thePUHFwavefunction,satisfytheACPQequations,butarenotasolutionoftheCCD ones[191].Asaby-productoftheirtheyalsodemonstratedthatinthe =0limit (2) 1 ( AB;IJ ; S r )=(2 S r +1) 2 (2) 5 ( AB;IJ ; S r ).ThisobservationimpliesthattheACP-D14 approachproposedinReference[191],inwhichoneretainsonlythe 1 2 T 2 2 termsassociated withdiagramsD1andD4showninFigure3.5, i.e. ,replacing P 5 k =1 (2) k ( AB;IJ ; S r )in Equation(3.20)by (2) 1 ( AB;IJ ; S r )+ (2) 4 ( AB;IJ ; S r ),isexactinthefullycorrelatedlimit ofcyclicpolyenemodels,too.PiecuchandPaldusimplementedtheACP-D14methodology andshowedthatitperformsaswellastheACPQapproachfortheC N H N systemswith N =6 ; 10 ; 14 ; 18 ; and22fortheentirerangeof values.Bothareaccurateandcapableof removingbranchpointsingularitiesseeninCCDcalculationsfor N 14andbothbecome 46 exactwhen =0.Inaddition,andinlightofthephsymmetry,theidenticalbehavioris anticipatedforanyACPschemeusing (2) 1 ( AB;IJ ; S r )+ (2) 3 ( AB;IJ ; S r )+(1 (2) 4 ( AB;IJ ; S r ) ; (3.23) with 2 R ,insteadof P 5 k =1 (2) k ( AB;IJ ; S r )inEquation(3.20). AgraphicalillustrationoftheaboveobservationsisprovidedinFigure3.6,wherewe examinetheperformanceofall32combinationsofdiagramsD1{D5indescribingthe D 6h - symmetricdissociationoftheminimum-basis-set(STO-6G[306])H 6 ringusingcodesin- terfacedwiththeintegral,RHFandrestrictedopen-shellHF,andCCroutinesavailable intheGAMESSpackage[307{309],whichweredevelopedinthisthesisresearch.Inthis andsimilarhydrogenclusters,whichcanbeviewedasthesimplest abinitio analogsofthe PPPandHubbardHamiltonianmodelsofcyclicpolyenes,onecanexaminetheentirespec- trumofelectroncorrelationbyvaryingtheradius R ofthering,transitioningfroma weaklycorrelatedmetallicphaseatsmall R valuestoastronglycorrelatedinsulatingphase as R !1 ,mimickingtheMotttransitions.Todistinguishbetweentheoriginalorthogonally spin-adaptedACPapproachesandtheirvariousspin-integratedcounterpartsimplemented inthisthesisresearchthatincorporate T 1 clustersaswell,wecommonlydenotethelatter approachesasACCSD.IneachACCSDapproach,thesubsetofthee 1 2 T 2 2 diagramsthat areretainedintheCCequationscorrespondingtoprojectionsondoublyexciteddetermi- nants, ab ij E ,isprovidedinsideparentheses.Forexample,theACCSD(4,5)schemeincorpo- ratesonlydiagramsD4andD5, i.e. ,itisthespin-integratedanalogoftheoriginalsinglet spin-adaptedACP-D45approachextendedtosingles.Outofthe32possibleACCSDvari- ants,onlyACCSD(3),ACCSD(4),ACCSD(1,3),ACCSD(1,4),ACCSD(3,4),ACCSD(3,5), ACCSD(4,5),ACCSD(1,3,4),ACCSD(3,4,5),andACCSD(1,3,4,5)providequalitativelycor- rectPECs.AsanticipatedinlightoftheabovediscussionregardingtheHubbardandPPP modelsofcyclicpolyenes,onlyACCSD(1,3),ACCSD(1,4),ACCSD(3,5),andACCSD(4,5) becomeexactatthesymmetricH 6 ! 6Hdissociation, i.e. ,inthestronglycorrelated,limit. 47 Figure3.6:Ground-statePECscharacterizingthe D 6h -symmetricdissociationofthe H 6 /STO-6Gring,asobtainedwiththevariousACCSDapproachesthatrelyonsubsets ofthenon-lineardiagramsshowninFigure3.5:(a)diagramsD1{D5areneglectedresulting inlinearizedCCSD,(b)allbutoneofthediagramsD1{D5areneglected,(c)allbuttwoof thediagramsD1{D5areneglected,(d)allbutthreeofthediagramsD1{D5areneglected, (e)allbutfourofthediagramsD1{D5areneglected,and(f)alldiagramsareconsidered, i.e. ,fullCCSD.Thedashedhorizontallinedesignatestheexactdissociationlimitforthe employedbasissetcorrespondingto6non-interactingHatoms. ItisalsointerestingtonotethatanyschemethatincorporatesD2iseitherovercorrelating orplaguedbysingularities. Atthispoint,wewouldliketorecallthatthreeadditionalclassesofACPapproaches haverecentlyemerged,namely,2CC[232,233],pCCSD( , )[234],andDCSD[180].The 48 philosophybehindthesenewerfamiliesofACPschemesismoreinlinewiththeoriginal motivationofChilesandDykstra, i.e. ,itdoesnotfocusonthebehaviorinthestrongly correlatedregimeasisthecaseintheworkofPaldus, Czek,Piecuch,andco-workers summarizedabove.Theywerederivedbyseekingmointhediagramsarising fromthe 1 2 T 2 2 contributionstotheCCSDequationsthatwouldresultinapproachesthat areexactfortwo-electronsystemsandseparatedelectronpairs.Itis,thus,notsurprising thatallthreeclassesincludeACCSD(4,5)asaspecialcase.Infact,2CCisthesameas ACCSD(4,5), i.e. ,thereisnothingnewaboutit.ThepCCSD( , )schemeofReference [234],whereonereplacesthenon-lineartermsofEquation(3.20)by 3 X k =1 (2) k ( AB;IJ ; S r )+ 1+ 2 (2) 4 ( AB;IJ ; S r )+ (2) 5 ( AB;IJ ; S r ) ; (3.24) remainsexactfortwo-electronsystemsandseparatedelectronpairsforanyvaluesofthereal parameters and .Anotherparameterizationofthetermsquadraticin T 2 thatgivesrise toapproachesthatareexactfortwo-electronsystemsandseparatedelectronpairsis[180] (2) 1 ( AB;IJ ; S r )+(1+2 (2) 2 ( AB;IJ ; S r )+(1+ (2) 3 ( AB;IJ ; S r ) +(1+ (2) 4 ( AB;IJ ; S r )+(1+2 (2) 5 ( AB;IJ ; S r ) : (3.25) NotethatinthiscasediagramsD1andD2arenotnecessarilytreatedonanequalfooting. Setting = = 1 2 enforcesphsymmetry,evenifitisnotarealsymmetryofthesystemof interest,andyieldstheDCSD ACCSD(1 ; 3+4 2 )methodology,whichcorrespondstosetting inEquation(3.23)at 1 2 .However,oneneedstokeepinmindthatexactnessfortwo-electron systemsandseparatedelectronpairsdoesnotguaranteethatagivenmethodcanprovide qualitatively,letalonequantitatively,correctresultsinthepresenceofstrongnon-dynamical correlationInlightoftheabovediscussion,itis,thus,notsurprisingthatamong thesenewgenerationsofACPmethodsonlyDCSD ACCSD(1 ; 3+4 2 )iswell-behavedin situationscharacterizedbyelectronicquasi-degeneracies,sinceitEquation(3.23) with = 1 2 .Furthermore,theextensionsoftheseschemestoconnectedtriples( T 3 clusters), viathe3CC[232,233],pCCSDT[235],andDCSDT[235,242]approaches,cannothandle 49 strongcorrelations,becausetheyarebasedondiagramcancellations/moonthe triplesprojection, i.e. ,theyretainalldiagramsofCCSDTcorrespondingtoprojectionson doubles,whichisnotacorrectdesignforstrongcorrelations. TheabovediscussionimpliesthattheACP-D13,ACP-D14,ACP-D1(3+4)/2 DCD, ACP-D45,andACPQapproaches,obtainedbyconsideringsubsetsof 1 2 T 2 2 diagramswithin theCCDsystem,Equation(3.20),andtheirextensionsincorporating T 1 clustersaremore robustthanthetraditionalCCSD,CCSDT,CCSDTQ, etc. hierarchyinstronglycorrelated situations,butthemainrationalebehindtheirusefulnessisbasedonconsideringstrongly correlatedlimitsofhighlysymmetric,minimum-basis-set,modelHamiltonians.Itisfar fromobviousthatthesamecombinationsof 1 2 T 2 2 diagramsareoptimumwhenoneuses abinitio Hamiltoniansandlargerbasissets,requiredbyquantitativequantumchemistry. Inthisdissertation,weaddresstheissueofextendingtheACPframeworktolargerbasis setsandan abinitio descriptionbytakingadvantageofthefactthatEquations(3.22)and (3.23)continuousclassesofACPapproachesthatretaintheexactnessinthestrongly correlatedlimitofcyclicpolyenemodelsasdescribedbytheHubbardandPPPHamiltonians (theHubbardHamiltonianonlywhenthe(2 S r +1) 2 factorinEquation(3.22)isignored). Betweenthetwofamilies,wefocusontheonebasedondiagramsD1,D3,andD4, i.e. ,we useEquation(3.23),becausesuchmethodsarealsoexactinthe =0limitofthemore realisticPPPHamiltonian. AmongthevariousACPandACCSDapproachesoriginatingfromtheuseofEquation (3.23)insteadof P 5 k =1 (2) k ( AB;IJ ; S r )intheCCDorCCSDamplitudeequationsprojected ondoubles,onlyACP-D1(3+4)/2anditsACCSD(1 ; 3+4 2 )orDCSDextensionincorporating T 1 clusters,whereoneuses = 1 2 ,enforcephsymmetry.Naturally,thisisdesiredforthe HubbardandPPPmodelsofcyclicpolyenes,whichhaveanintrinsicphsymmetry,andisalso inthecaseofstronglycorrelatedsystemsdescribedby abinitio Hamiltonians,such asthehydrogenclustersexaminedinthisdissertation,aslongasoneusesaminimumbasis set,forwhichphsymmetryisapproximatelyHowever,thisdoesnotnecessarily 50 meanthat = 1 2 inEquation(3.23)isanappropriatechoiceforrealisticbasissets,especially when n o ˝ n u .Bynumericallyexaminingseveralstronglycorrelatedsystemstreatedwith variousbasissets,includingdissociatingringsandlinearchainscomposedofvaryingnumbers ofhydrogenatoms,wehavenoticedthattheACP-D14approximationaugmentedfor T 1 clusters, i.e. ,ACCSD(1,4),whichuses =0,worksbetterasthebasissetbecomeslarger thanACCSD(1 ; 3+4 2 ),especiallywhenoneincorporatesthethree-body T 3 clusterswithin theACPorACCSDframework.Thisnumericalobservationsuggeststhatonemightwant toscaleupdiagramD4bydecreasingthecot when n u getslarger.Tothatend,in thisdissertationproject,weintroducethemethodabbreviatedasACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ),whereweset = n o n o + n u inEquation(3.23)totakeintoaccounttheofthe dimensionalityofthebasissetinthecalculations.Thisvalueof hastwoadvantages.First, inthecaseof n o = n u ,whichistrue,forexample,intheHubbardandPPPmodelsofcyclic polyenesandthevarioushydrogenclustersasdescribedbyaminimumbasisset,thenovel methodisequivalenttoACCSD(1 ; 3+4 2 )orDCSD,whichiswell-behavedinthepresenceof strongmany-electroncorrelationwhentheminimumbasissetisemployed.Second, asoneapproachestheCBSlimit,theACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )schemebecomes ACCSD(1,4),whichisadesiredbehavioraswell,since,basedonournumericalobservations, ACCSD(1,4),especiallywhencorrectedfortheconnectedtriples,performsthebestoutof alltestedACPapproaches. Anotherissuethatissuccessfullyaddressedinthisdissertationistheincorporationof connectedtriplesintheACP(meaning,ACCSD)approaches.Asalreadymentionedearlier inthisthesis,thePUHFwavefunction,whichwasusedinderivingtheACPQandACCSD 0 equationsandexplainingtheoriginoftheexceptionalbehavioroftheseACPschemesin thepresenceofstrongcorrelations,doesnotcontainanyinformationregardingthethree- bodyclusters.Ofcourse,onecouldincorporateconnectedtriplesfully,yieldingCCSDT-like ACPschemes,buttheassociatedcomputationalcostswouldprohibittheapplicationofsuch triples-correctedmethodstorealisticstronglycorrelatedsystemsandmaterials.Histori- 51 cally,the T 3 physicswasincorporatedwithintheACPframeworkusingMBPTarguments, similartothoseexploitedintheCCSDT-1[119{122]andCCD+ST(CCD)[124]methodolo- gies.Naturally,theuseofMBPTalreadyimpliesthatsuchschemeswillhave inthepresenceofelectronicquasi-degeneraciesduetothevanishinglysmallperturbative denominators.Indeed,theresultingACPTQ[190,230]andACPQ+T(ACPQ)[192,231] methodologies,whicharetheACPQanalogsofCCDT-1andCCD+T(CCD),respectively, wereonlypartlysuccessful[190,192,193,219,231].Therefore,inthisstudy,weexplorethe variousdiagramselections/mocharacterizingtheACPfamilyofmethodswithin theactive-spaceCCframework[69,70,80{82,98{100,145{160],whichavoidsdangerous denominatorswhilebeingcomputationally Intheactive-spaceCCmethods,thespin-orbitalsusedinthecalculationsarepartitioned intofourdistinctgroups,namely,core,activeoccupied,activeunoccupied,andvirtualspin- orbitals,andthehigher-orderclustercomponents,suchas T 3 ,areapproximatedwiththe helpofactiveorbitals,whilethelower-orderones, i.e. , T 1 and T 2 ,aretreatedfully.Thishas severaladvantages.Byselecting,forexample,the T 3 clusteramplitudesusingactiveorbitals, wesavealotinthecomputerwithouttlossofaccuracywhencomparedtothe parentfullCCSDTapproach.Atthesametime,bykeepingthedominant T 3 amplitudesin thecalculations,weprovideamechanismforrelaxing T 1 and T 2 amplitudesinthepresence of T 3 ,whichisnotavailableinnon-iterativetriplesenergycorrectionsoftheCCSD[T]and CCSD(T)types. IntheCCapproachwithsingles,doubles,andanactive-spacetreatmentoftriples,ab- breviatedasCCSDt,whichismostrelevanttothisthesiswork,weapproximatethecluster operator T as T ˇ T (CCSDt) = T 1 + T 2 + t 3 ; (3.26) where T 1 and T 2 arethestandardone-andtwo-bodycomponentsof T ,treatedfully,and t 3 = X i > > > > > > > < > > > > > > > > : c 0 j 0 i ; for S = E 0 1 ; for S>E 0 ; 0 ; for S 0 ( p d < 0),thewalkerdies(iscloned)withprobability p d ( j p d j ).Itisalsoapparentthatthe growthofthewalkerpopulationdependsonthevalueoftheshiftenergy S .Forexample, inlightofEquation(3.31),ifarapidwalkerpopulationgrowthisdesired,theshiftenergy S issetatavaluethatislargerthan E 0 ;thisisdoneintheearlystagesoftheFCIQMC propagations.Ontheotherhand,onceatotalwalkerpopulationbecomestlylarge, onestartsusingtheenergyshift S tostabilizewalkerpopulationandreachconvergence. Asalreadyalludedtoabove,thestepofthewalkerpopulationdynamicsalgorithm involvestheannihilationofoppositelysignedwalkersinhabitingagivendeterminant.As withbirth/deathandspawning,thisisdoneateachtime ˝ duringthepropagation. Havingdiscussedthekeyelementsofthewalkerpopulationdynamicsalgorithm,we provideabriefoutlineofaFCIQMCsimulation.Westartwiththeoriginalalgorithm[260]. Webeginbyplacingacertainnumberofwalkersononeormorereferencedeterminants(in ourcase,one,RHF)andsettingtheenergyshift S abovetheexactground-stateenergy E 0 , 59 S>E 0 ,topromotethegrowthofthewalkerpopulation[ cf. Equation(3.31)].Atevery imaginary-timestep ˝ ,thethreemajorprocessesdrivingthewalkerpopulationdynamics areattempted,namely,spawning,birth/death,andannihilation.Initially,thesimulationis characterizedbyanexponentialgrowthofthewalkerpopulationuntilaplateauisreached, correspondingtoasystem-dependentcriticalpopulationofwalkers, N c .Theplateauis amanifestationofasteadystate,wheretherateofwalkercreationequalsthecombined rateofdeathandannihilation.Duringthisstageofthesimulation,theannihilationprocess theFCIQMCwavefunction,leadingtoaconvergedsignstructure.Oncetheproper signstructureisattained,thewalkerpopulationbeginstoriseagain.Atthispoint,the energyshift S isallowedtovaryinanattempttostabilizethewalkerpopulation.Basedon Equation(3.31),aconstantwalkerpopulationimpliesthat S ! E 0 and,thus,convergence isreached. TheperformanceoftheFCIQMCmethodologywasoriginallytestedontheNeatom andoligoatomicspecies,includingN 2 ,C 2 ,andH 2 O,asdescribedbytheaug-cc-pVDZand cc-pVDZbasissets,respectively.Itwasdemonstratedthat,evenwhenstartingwitha singlewalkerinhabitingtheRHFSlaterdeterminant,theFCIQMCsimulationwasableto convergetothedeterministicFCIenergywithsubmillihartreeaccuracy.Whatis,perhaps, moreinterestingisthefactthatthiskindofaccuracywasobtainedwithcriticalwalker populations,whichthecomputationalbottleneckofFCIQMC,lessthanthetotal numberofSlaterdeterminantsbytheone-electronbasis, N FCI .Thisobservation suggeststhatFCIQMCiscapableofprovidingFCI-qualityresultswithouthavingtosample theentiremany-electronHilbertspace.Althoughinmanycases f c N c N FCI ˝ 1,inthe originalFCIQMCwork[260]therewereafewchallengingsituationsinwhich f c wasas largeas0.9,implyingthatforsomesystemsapopulationofwalkerscomparabletothe dimensionalityoftheFCIproblemwasrequiredfortheconvergenceofthecorrectsign structureofthewavefunction.Inaddition,despitethefactthatthecriticalnumberof walkerswasalwayslessthanthedimensionofthemany-electronHilbertspace, N c ˝ .Itisguaranteedthatintheimaginary-timelimit E 0 ( ˝ )approachesthe exact,FCI,ground-statecorrelationenergy.Theaboveprocedureisgiveningraphicalform inFigure3.8. Ithasbeendemonstratedinourwork[265,266]thattheCAD-FCIQMCmethodologyis capableofproducingFCI-qualityenergeticsoutoftheearlystagesofFCIQMCpropagations, eveninchallengingsituationsinvolvingelectronicquasi-degeneracies,suchasthosecharac- terizingthe C 2v -symmetricdoublebonddissociationofH 2 O[265].However,CAD-FCIQMC, asdescribedabove,breaksdowninthepresenceofstrongmany-electroncorrelation AsimpleinspectionofEquations(3.29)and(3.30)revealstheoriginbehindthisfailure. At ˝ =0,theFCIQMCwavefunctionisequivalenttothereferenceSlaterdeterminant, usuallytheRHFdeterminant j i .Thus, T (MC) 3 ( ˝ =0)= T (MC) 4 ( ˝ =0)=0andthe CAD-FCIQMCapproachreducestoCCSD.Consideringthepoor,sometimesevensingu- lar,behaviorofCCSDinthestrongcorrelationregime,itisnaturalthatCAD-FCIQMC maystrugglewithstronglycorrelatedsystemsinvolvinglargenumbersofstronglycorrelated electrons. Inthisdissertation,weextendtheCAD-FCIQMCapproachtothestrongcorrelation regimebymergingitwiththeACPapproaches.Asdiscussedintheprevioussection,the keytotacklingstrongmany-electroncorrelationliesinthe 1 2 T 2 2 contributionstothe CCequationsprojectedondoubles.Tothatend,inordertoincorporatetheACPideas withintheCAD-FCIQMCframework,werepartitionEquation(3.30)suchthatselected 64 StartFCIQMC propagation Extract C (MC) n ( ˝ ), n =1{4,atagiventime ˝ usingEquation(3.37) UseEquation(3.39)to performclusteranalysis oftheFCIQMCwave- functionattime ˝ to extractthecorresponding T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ ) SolveCCSD-likesystemof equations,Equations(3.29) and(3.30),todetermine T 1 and T 2 inthepresence of T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ ) ComputeCAD-FCIQMC energyattime ˝ us- ingEquation(3.28) Convergence? E 0 ( ˝ ) ˇ E (FCI) 0 yes No Figure3.8:FlowchartoutliningthekeystepsoftheCAD-FCIQMCalgorithm. 65 coupled-paircontributionsareextractedfromFCIQMCaswell, D ab ij H N 1+ T 1 + T 2 + 1 2 T 2 1 + T (MC) 3 + T 1 T 2 + 1 3! T 3 1 + T (MC) 4 + T 1 T (MC) 3 + 1 2 T 2 1 T 2 + 1 4! T 4 1 C j i + 5 X i ˘ i (2) i +(1 ˘ i ) (2),(MC) i =0 : (3.40) Inotherwords,weassumethatinadditionto T 3 and T 4 ,whichareextractedfromthe FCIQMCwavefunctionassummarizedabove,weextractselected 1 2 T 2 2 diagramsresponsible forweakcorrelationsfromFCIQMCaswell(theyaremarkedinEquation(3.40)as (2),(MC) i terms).LikeitsCAD-FCIQMCpredecessorofReference[265],thisnewschemeisguaranteed toprovidenumericallyexact,FCI,energeticsintheimaginary-timelimit.Further- more,itrsanadditionalythatextendsitsapplicationtothestrongcorrelation regime.Tobeprecise,thesuccessbehindthismototheoriginalCAD-FCIQMC algorithmliesinthefactthatthe 1 2 T 2 2 partthatisresponsibleforgoodbehaviorinstrongly correlatedsystemsistreateddeterministically,whileitscomplement,whichis,moreorless, theweaklycorrelatedpartof 1 2 T 2 2 ,isextractedfromFCIQMC.Forexample,setting ˘ 1 =1 ; ˘ 2 =0 ;˘ 3 = ˘ 4 = 1 2 ,and ˘ 5 =0givesrisetotheCAD-FCIQMC[1,(3+4)/2]variant,where thenumbersinsquarebracketsdesignatethe 1 2 T 2 2 diagrams,showninFigure3.5inthecase ofthespin-adaptedCCformalism,thataretreateddeterministically.Consequently,the startingpointofCAD-FCIQMC[1,(3+4)/2]isACCSD(1 ; 3+4 2 ) DCSD,whichisalready well-behavedinthepresenceofstrongcorrelations.Theremainingmany-electroncorrela- tionareextractedfromFCIQMC.Withinthisframework,theoriginalCAD-FCIQMC approachisdesignatedusCAD-FCIQMC[1{5]. 66 CHAPTER4 NUMERICALRESULTS 4.1ApplicationoftheApproximateCoupled-PairMethodswith anActive-SpaceTreatmentofThree-BodyClusterstoModel Metal{InsulatorTransitions Webeginthediscussionofthenumericalresultsbyassessingtheperformanceofthede- terministicACPmethods,includingtheACCSD(1,3),ACCSD(1,4),ACCSD(1 ; 3+4 2 ),and ACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )approachesandtheiractive-spaceandfulltriplesex- tensions,developedandimplementedinthisdissertationproject,inchallengingsituations involvingstrongmany-electroncorrelationInparticular,weexaminetheirability toreproducetheexactornearlyexactPECscharacterizingthesymmetricdissociationsof theH 6 andH 10 ringsandtheH 50 linearchainofequidistanthydrogenatoms,inwhichthe degreeofentanglementoftheelectronsiscontinuouslyvariedasthesystemsdeparttheir weaklycorrelatedmetallicphasesaroundtherespectivepotentialminimaandapproachthe insulatingatomizationregionsgovernedbystrongcorrelationsofallelectronspresentinthe system.TofacilitatethediscussionandemphasizethepoweroftheACPmethods,without andwiththeconnectedtriples,inproperlyhandlingstrongcorrelations,wealsoprovidethe pertinentCCSD,CCSDt,andCCSDTPECs. AlloftheCCcalculationsreportedinthissubsectionwerebasedonRHFreference functions.TheCCSD,CCSDt,andfullCCSDTcomputationswereperformedusingthe active-spaceCCcodesdevelopedbyourgroup[144,161,162]thathavebeenrecentlyincor- poratedintheGAMESSpackage[307{309].ThevariousACCSD,ACCSDt,andfull ACCSDTcalculationswerecarriedoutusingalocalversionofGAMESS,wherewemade suitablemoinourCCSD,CCSDt,andCCSDTroutines.FortheH 6 andH 10 rings,weemployedthelargestbasissetsthatwouldallowustoperformtheexact,FCI 67 computations,namely,cc-pVTZ[276]inthecaseofH 6 andDZ[311,312]intheH 10 case. ThepertinentFCIcalculationswerecarriedoutusingthedeterminantalFCIcode[313{315] availableinGAMESS.WhenconsideringthesymmetricdissociationoftheH 50 linearchain, weusedtheminimumSTO-6G[306]basisset,sincethenearlyexactLDMRG(500)/STO- 6Gresultsareavailableforit[175](inthiscase,FCIcalculations,evenwithminimumbasis sets,arenotpossible).Theactive-spaceCCapproaches,includingCCSDtandACCSDt, employedactivespacesthatconsistedoftheorbitalsassociatedwiththe1 s shellsofthe hydrogenatoms, i.e. ,threeactiveoccupiedandthreeactiveunoccupiedorbitalsfortheH 6 ringand5activeoccupiedand5activeunoccupiedorbitalsinthecaseoftheH 10 ring.In thecaseoftheH 50 linearchain,whereweusedtheminimumSTO-6Gbasisset,theonly meaningfulactivespaceisthatincorporatingalloccupiedandunoccupiedorbitals,resulting incomputationswithafulltreatmentofconnectedtriples.Thus,inthiscase,weperformed theCCSDTandACCSDTcomputations,inadditiontoCCSDandACCSD.Duetothe factthattheH 50 /STO-6Gsystemischaracterizedby n o = n u ,meaninganapproximate phsymmetry,weonlyconsideredtheACCSD(1 ; 3+4 2 )andACCSDT(1 ; 3+4 2 )methodsusing = n o n o + n u inEquation(3.23).Allofourcalculationsforthesymmetricdissociationsof theH 6 andH 10 ringsemployedthefollowinggridofinternuclearseparationsbetweenneigh- boringhydrogenatoms:0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.0, 2.1,2.2,2.3,2.4,and2.5 A.ThePECscharacterizingthesymmetricdissociationoftheH 50 linearchainwerecalculatedatthegeometriesreportedinReference[175],namely,1.0,1.2, 1.4,1.6,1.8,2.0,2.4,2.8,3.2,3.6,and4.2bohr. Webeginbyexaminingthe D 6h -symmetricdissociationofthesix-memberedhydrogen ring,asdescribedbythecc-pVTZbasisset.Theground-statePECsoftheH 6 /cc-pVTZ systemresultingfromtheconventionalCCSDapproach,thevariousACPmethods,including ACCSD(1,3),ACCSD(1,4),ACCSD(1 ; 3+4 2 ),andACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ),their active-spaceandfulltriplescounterparts,andtheexactFCIdiagonalizationarepresented inTables4.1{4.3andFigure4.1. 68 AsshowninTables4.1{4.3andFigure4.1,theCCSDapproachandeventheactive{ space-basedCCSDtschemeanditsparentCCSDTcounterpartbreakdownratherquicklyas thedistancebetweenneighboringhydrogenatoms, R H{H ,increases.Infact,CCSDalready overcorrelatesaround R H{H =2.0 A, i.e. ,atabouttwicetheequilibrium R H{H distance,by almost25mE h ,whencomparedtotheexactFCIresults.Asonestretchesthe R H{H distance evenfurther,thedeviationfromFCIdramaticallyincreases,resultinginanunphysicalhump andthePECgoingdownhill,sothatwhenthelargestH{Hdistanceconsideredinthiswork, namely, R H{H =2.5 A,isconsidered,theunsignedbetweentheCCSDandFCI energiesreachesagargantuanvalueofmorethan237mE h .AddingtriplesontopofCCSD usingthefullCCSDTapproachresultsinatimprovementofthePECaround theequilibriumgeometry,witherrorsrelativetotheFCIenergeticsofabout0.1mE h . However,asoneapproachesthestrongcorrelationregime,CCSDTeventuallyfollowsthe samecatastrophicpathasCCSD.Onasidenote,itisworthmentioningthattheactive-space CCSDtapproach,usinganactivespacecomprisedofthesix1 s orbitalsoftheindividual Hatoms,faithfullyreproducestheparentCCSDTenergetics,withdeviationsfromCCSDT notexceeding ˘ 2mE h (seeTables4.2and4.3),butatasmallfractionofthecomputational Tobeprecise,theCPUtimeofaCCSDtiterationfortheH 6 /cc-pVTZsystemwas6 s,usingasinglecoreandwithouttakingadvantageofpointgroupsymmetry.Thisshouldbe contrastedwiththe98sperCCSDTiterationranthesamewayastheCCSDtcomputation. AquickinspectionofFigure4.1revealsthatallACPmethodsexaminedinthisstudy, withoutandwiththeconnectedtriples,providequalitativelycorrectPECsforthe D 6h - symmetricdissociationoftheH 6 /cc-pVTZring, i.e. ,theydonotovercorrelateasoneap- proachesthestrongcorrelationlimitandremainclosetoFCI.FocusingontheACPmethod- ologieswithuptotwo-bodyclusters,meaningtheACCSDapproaches,theACCSD(1,3) scheme,where =1inEquation(3.23),performsthebest,generatingaPECthatclosely reproducesitsFCIcounterpart,asillustratedbythesmallmeanunsignederror(MUE)and meansignederror(MSE)valuesrelativetoFCIof0.675mE h and 0 : 311mE h ,respec- 69 tively(seeTable4.1).Attheotherendofthespectrum,theACCSD(1,4)approach,where onesets =0inEquation(3.23),doesnotreproducetheFCIenergeticsasgoodasthe ACCSD(1,3)variant,havingMUEandMSEvaluesrelativetoFCIof9.349mE h and9.349 mE h ,respectively.Asonemighthaveanticipated,theACCSD(1 ; 3+4 2 ) DCSDmethod, whichtreatsdiagramsD3andD4showninFigure3.5onanequalfootingbysetting = 0.5inEquation(3.23),producesaPECthatismoreorlesstheaverageoftheACCSD(1,3) andACCSD(1,4)ones[seepanels(a)and(d)ofFigure4.1andTable4.1].Furthermore, theACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )PECcharacterizingthesymmetricdissociationof theH 6 /cc-pVTZsystemisalmostidentical,yetslightlybetterinreproducingtheexactFCI data,tothatresultingfromtheACCSD(1,4)calculations.Takingintoaccountthesizeof thecc-pVTZbasisset,where n o =3and n u =81=27 n o ,thisisnotasurprise,since,by construction,theACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )approachconvergestoACCSD(1,4) withincreasingbasissetsize( n u n o approaching 1 ). Fromtheabovediscussion,onemightcrowntheACCSD(1,3)schemethebestACPap- proachexaminedinthisdissertation.However,webelievethattheexcellentperformance oftheACCSD(1,3)methodinproducingtheFCI-qualityenergeticsforthemetal{insulator transitionintheH 6 /cc-pVTZsystemisratherfortuitous.Asalreadymentioned,theACP methodsareveryeinrecoveringnon-dynamicalcorrelationebytakingadvan- tageofvariousdiagramcancellationsintheCCD/CCSDequationsprojectedondoublyex- citedcstatefunctions.However,thesediagramcancellationsdonotdescribethe T 3 physics,neededtocapturedynamicalcorrelationsandobtainaquantitativedescription ofrealisticsystemsdescribedby abinitio Hamiltonians.Beforeproceedingtothedetailed discussionoftheperformanceofthevariousACPmethodswiththeconnectedtriplyexcited clustersconsideredinthiswork,wenotethattheACPapproacheswithanactive-space treatmentof T 3 components, i.e. ,theACCSDtschemes,closelyreproducetheirparentfull triplesACCSDTcounterparts.Atthesametime,theactive-spacetriplesframeworkreplaces theexpensive n 3 o n 5 u computationaltimestepsassociatedwithafulltreatmentofconnected 70 triplyexcitedclustersbytherelativelyinexpensive N o N u n 2 o n 4 u ones, i.e. ,thecomputational costsarereducedtothoseofCCSDtimesasmallprefactorontheorderofthenumberofsin- glesintheactivespace.Forexample,the98spersingleACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u ) iteration,ranonasinglecoreusing C 1 pointgroupsymmetry,arereducedtojust6swhen theactive-spacetriplesACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u )approachisemployed,consistent withthecomputationaltimesavingsobservedintheCCSDT/CCSDtcase.Thisisanim- portantobservation,especiallywhenonerealizesthattheultimategoalofourinthe longtermistoextendtheACPapproachestolargesystems,allowingustostudystrongly correlatedmaterials. TheinclusionofconnectedtriplesontopofthevariousACPschemesexaminedinthisdis- sertationresultsinadramaticimprovementoftheACCSD(1,4)andACCSD(1 ; 3 n o n o + n u + 4 n u n o + n u )energeticswhencomparedtotheexact,FCI,data.Tobeprecise,theMSEvalues characterizingtheH 6 /cc-pVTZPECsobtainedwiththeactive{space-basedACCSDt(1,4) andACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u )approachesare2.129mE h and1.818mE h ,respec- tively( cf. Table4.2).Asaresult,theACCSDt(1,4)and,especially,ACCSDt(1 ; 3 n o n o + n u + 4 n u n o + n u )PECscanbehardlydistinguishedfromtheFCIone.Tobetterappreciatethe ofthiswecomparetheACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u )andFCI CPUtimings,whichare6speriterationonasinglecore,withouttakingadvantageof pointgroupsymmetry,intheformercase,andmorethan7hperiteration,onthesame core,inthelattercase,inwhichweusedthe D 2h symmetryinthecalculations.Thefull treatmentofthree-bodyclustersfurtherreducesthealreadysmalldeviationsfromFCIeven further,asillustratedbytheMSEvaluesof1.189mE h ,forACCSDT(1,4),and0.875mE h , inthecaseofACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u )(seeTable4.3).Ontheotherhand, thePECscharacterizingthesymmetricdissociationoftheH 6 /cc-pVTZringgeneratedby theconnected-triplesextensionsoftheACCSD(1,3)andACCSD(1 ; 3+4 2 )approachesareofa poorerqualitywhencomparedtotheir =0[ACCSDt(1,4)/ACCSDT(1,4)]and = n o n o + n u [ACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u )/ACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u )]counterparts. 71 Tables4.1{4.3revealthefollowingrelationshipregardingtheabilityofthevariousACP schemesconsideredheretoreproducetheFCIenergetics:ACCSD(1,3) > ACCSDt(1,3) ˇ ACCSDT(1,3)andACCSD(1 ; 3+4 2 ) > ACCSDt(1 ; 3+4 2 ) ˇ ACCSDT(1 ; 3+4 2 ).Thisimplies thattheACCSD(1,3)andACCSD(1 ; 3+4 2 )variantsofACPapproachesarenotsystematically improvable,contrarytotheirACCSD(1,4)andACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )counter- parts,wherethefollowingaccuracypatternsareobserved:ACCSD(1,4) < ACCSDt(1,4) ˇ ACCSDT(1,4)andACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ) < ACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u ) ˇ ACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u ).Anadditionalimportantobservation,infavorofthe =0and = n o n o + n u variants,isthattheACCSD(1,4)andACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ) approachesrecover94{98%oftheexact,FCI,correlationenergycharacterizingthe D 6h - symmetricdissociationoftheH 6 /cc-pVTZringandtheirconnectedtriplesextensionsare practicallyexact,consistentlyretrievingabout99%oftheFCIcorrelationenergy. TofurtherexploretheusefulnessofACPmethods,especiallythosethatincorporate T 3 physics,infaithfullyreproducingFCI-levelenergeticsinthepresenceofstrongnon- dynamicalelectroncorrelationts,westudiedthe D 10h -symmetricdissociationofthe ten-memberedhydrogenringasdescribedbytheDZbasisset.Thischallengingsystem involvestheentanglementof10electrons,whichtranslatesintoanabout67%increasein thenumberofstronglycorrelatedelectronscomparedtothesmallerH 6 hydrogencluster. ThishasasevereontheperformanceoftheCCSD,CCSDt,andCCSDTapproaches, which,asillustratedinTables4.4{4.6andFigure4.2,failcatastrophically;theyovercorrelate ratherquicklyandnoconvergenceisobtainedfordistancesbetweenneighboringHatoms beyond1.7 A.Nevertheless,aswasthecasewiththesmallerH 6 ring,allACPmethods examinedinthiswork,withoutandwiththeconnectedtriples,producequalitativelycor- rectPECsforthemetal{insulatortransitioninthechallengingH 10 cluster(seeFigure4.2). EventhoughtheDZbasisset,where n o =5and n u =15,israthersmallcomparedtothat usedintheH 6 case,whichtranslatesintothevariousACPschemesbehavingmoreorless similarly( cf. Tables4.4{4.6andFigure4.2),westillobservethesamepatternsregardingthe 72 abilityoftheACPmethods,withoutandwiththeconnectedtriples,toreproducetheFCI energetics.Forexample,focusingontheACPapproacheswithuptotwo-bodyclusters,the ACCSD(1,3)energeticsaretheclosesttotheFCIones(MUE=MSE=12.572mE h ),fol- lowedbyACCSD(1 ; 3+4 2 )(MUE=MSE=14.315mE h ),ACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ) (MUE=MSE=14.980mE h ),andACCSD(1,4)(MUE=MSE=15.505mE h ),aswas thecasewiththesmallerH 6 ring.Theinclusionofasubsetofconnectedtripleexcitations, selectedviaanactivespaceof1 s orbitals,resultsinadramaticimprovementoftheresults, asillustratedbytheabout2{4timesreductionintheMUEandMSEvaluesandfavors theACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u )scheme(seeTables4.4and4.5).Consistentwith theH 6 case,orderingthevariousactive-spacetriplesACPapproachesbasedontheirabil- itytoreproducetheFCIdatayieldsACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u ) > ACCSDt(1,4) > ACCSDt(1 ; 3+4 2 ) > ACCSDt(1,3),withtheirMUEandMSEvaluesbeing3.770mE h and 3 : 770mE h forACCSDt(1 ; 3 n o n o + n u +4 n u n o + n u ),3.790mE h and 3 : 790mE h forACCSDt(1,4),4.110mE h and 4 : 110mE h forACCSDt(1 ; 3+4 2 ),and5.793mE h and 5 : 793mE h inthecaseofACCSDt(1,3).Thefullinclusionofthree-bodyclustersdoes notchangethescenery,since,duetothesmallsizeoftheDZbasisset,theactive-space approachesalreadyrecoverthelion'sshareof T 3 physics.Finally,theACCSD(1,4)and ACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )schemesconsistentlyrecoverabout96%oftheexact, FCI,correlationenergiescharacterizingthe D 10h -symmetricdissociationofH 10 /DZ,with theirconnected-triplescounterpartsbeingvirtuallyexact,recovering ˘ 99%oftheFCIcor- relationenergies,similartothesmallerH 6 ring.Therefore,theH 10 /DZresultsreinforceour conclusionthattheACCSD(1,4)andACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )methodsaresystem- aticallyimprovablebytheinclusionofconnectedtriplesandexhibitthebestperformance inreproducingthecorrespondingFCIPECs,onceconnectedtriplesareincluded. HavingestablishedtheabilityofthevariousACPapproaches,especiallythoseincorpo- ratingthree-bodyclusters,tocloselyreproducetheFCIPECscharacterizingthesymmetric dissociationofthesix-andten-memberedhydrogenrings,wenowproceedtoexaminethe 73 symmetricdissociationoftheH 50 linearchainintoindividualhydrogenatoms,whichis aproblemthatinvolvestheentanglementof50stronglycorrelatedelectrons.Asalready mentionedabove,evenwhenaminimumbasisset,suchasSTO-6G,isemployed,thesheer dimensionalityofthemany-electronHilbertspace,whichisspannedby1 : 21 10 27 singlet- spin-adaptedstatefunctions,rendersFCIcalculationsprohibitivelyexpensive. Asmentionedearlier,inthecaseoftheH 50 linearchain,wereliedonthenearlyexact LDMRG(500)/STO-6GdataofHachmann etal. [175]inassessingtheperformanceofthe ACPschemes,withoutandwiththeconnectedtriples.However,werecallthatthevarious ACPmethodologiesprovideessentiallyidenticalresultswhenappliedtohydrogenclusters describedbyminimumbasissets,asaconsequenceoftheapproximatephsymmetry.Fur- thermore,thesmallestmeaningfulactivespaceinthecaseofhydrogenclustersdissociating intoindividualatomsiscomprisedofthe1 s orbitalsofthehydrogenatoms,suggestingthat onlyfulltriplesapproachescanbeappliedwhenminimumbasissetsareemployed.We,thus, examinedonlytheACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )andACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u ) approachesinproperlydescribingthesymmetricdissociationoftheH 50 /STO-6Glinear chain,whichinthiscase,where n o = n u ,areequivalenttoACCSD(1 ; 3+4 2 ) DCSDand ACCSDT(1 ; 3+4 2 ). Figure4.3showsthePECscharacterizingthesymmetricdissociationoftheH 50 linear chaininto50hydrogenatomsusingCCSD,CCSDT,ACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ), andACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u ),alongwiththenearlyexactLDMRG(500)dataof Reference[175].AquickinspectionofFigure4.3immediatelyrevealsthecompletebreak- downoftheconventionalCCSDandCCSDTapproaches,whichfailtoconvergeevenfor geometriesincloseproximitytotheequilibriumone.Ontheotherhand,theACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )schemeanditsconnectedtriplesextensionprovidequalitativelycorrect PECs,closelyreproducingtheLDMRG(500)one.Thisisaremarkableobservation,espe- ciallywhenonerealizesthatthenumberofstronglycorrelatedelectronsisetimesgreater thaninthealreadychallengingH 10 ring.FocusingontheACCSD(1 ; 3 n o n o + n u +4 n u n o + n u ) 74 energeticsreportedinTable4.7,weseethattheMSEandMUEvalueswithrespectto thenearlyexactLDMRG(500)dataofReference[175]are51.33mE h .Eventhoughthis islargerthantheMSEvaluesrelativetoFCIcharacterizingtheACCSD(1 ; 3 n o n o + n u + 4 n u n o + n u )PECsintheH 6 andH 10 cases,thisisaverypromisingresult.Eveninthe H 50 torturetesttheACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )approachcapturesabout97% ofthenearlyexact,DMRG,correlationenergies.Thisisanimportantobservation,since theACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )schemeconsistentlyrecoveredapproximately96% oftheexactornearlyexactcorrelationenergiesforallexaminedsystems, i.e. ,indepen- dentofthenumberofstronglycorrelatedelectronsorthesizeofthebasisset.Fur- thermore,andmoreimportantly,oncethree-bodyclustersareincluded,thereisasignif- icantimprovementoftheACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )totalelectronicenergies,even whenthesymmetricdissociationoftheH 50 linearchainisconsidered.Tobeprecise,the ACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u )PECcanhardlybedistinguishedfromthenearlyexact LDMRG(500)one,reducingtheMUEandMSEvaluesof51.33mE h and51.33mE h in thecaseofACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )to19.14mE h and-6.08mE h ,respectively, whentheACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u )approachisused.Thisisyetanotherexam- plethatshowcasestheimportanceof T 3 physicswithintheACPframeworkinattaininga quantitativedescriptionofstronglycorrelatedsystems. 75 Table4.1:AcomparisonoftheenergiesresultingfromthevariousCCapproacheswithsinglesanddoublesandtheexactFCI dataforthesymmetricdissociationoftheH 6 /cc-pVTZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A). a R H{H CCSD ACCSD FCI (1,3)(1 ; 3+4 2 )(1 ; 3 n o n o + n u +4 n u n o + n u )(1,4) 0 : 63 : 422 0 : 2421 : 5573 : 1443 : 262 2 : 858958 0 : 73 : 593 0 : 3201 : 6183 : 3213 : 448 3 : 176147 0 : 83 : 823 0 : 4171 : 6953 : 5413 : 679 3 : 331124 0 : 94 : 103 0 : 5341 : 7883 : 8093 : 959 3 : 396176 1 : 04 : 424 0 : 6651 : 9064 : 1324 : 297 3 : 410069 1 : 14 : 774 0 : 8052 : 0594 : 5224 : 705 3 : 394673 1 : 25 : 139 0 : 9442 : 2635 : 0035 : 206 3 : 362943 1 : 35 : 489 1 : 0682 : 5435 : 6085 : 833 3 : 322850 1 : 45 : 758 1 : 1562 : 9346 : 3816 : 634 3 : 279466 1 : 55 : 802 1 : 1753 : 4797 : 3777 : 662 3 : 236119 1 : 65 : 309 1 : 0884 : 2198 : 6388 : 960 3 : 195040 1 : 73 : 647 0 : 8655 : 16510 : 16510 : 529 3 : 157716 1 : 8 0 : 380 0 : 5086 : 26611 : 87212 : 280 3 : 125051 1 : 9 8 : 826 0 : 0717 : 38813 : 56314 : 012 3 : 097433 2 : 0 24 : 7200 : 3508 : 33714 : 96315 : 445 3 : 074787 2 : 1 51 : 3310 : 6608 : 93115 : 81316 : 314 3 : 056686 2 : 2 89 : 8370 : 8039 : 07315 : 97516 : 478 3 : 042507 2 : 3 137 : 1970 : 7828 : 77615 : 46315 : 951 3 : 031571 2 : 4 187 : 8240 : 6358 : 13514 : 41714 : 876 3 : 023237 2 : 5 237 : 0740 : 4167 : 27713 : 02713 : 447 3 : 016948 MUE b 39 : 6240 : 6754 : 7709 : 0379 : 349| MSE c 34 : 095 0 : 3114 : 7709 : 0379 : 349| a TheFCIenergiesaretotalenergiesinhartree,whereasalloftheremainingenergiesareerrorsrelativetoFCIinmillihartree. b Meanunsignederror. c Meansignederror. 76 Table4.2:Acomparisonoftheenergiesresultingfromthevariousactive-spacetriplesCCapproachesandtheexactFCIdata forthesymmetricdissociationoftheH 6 /cc-pVTZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A). a R H{H CCSDt ACCSDt FCI (1,3)(1 ; 3+4 2 )(1 ; 3 n o n o + n u +4 n u n o + n u )(1,4) 0 : 62 : 548 1 : 2340 : 6152 : 2432 : 365 2 : 858958 0 : 72 : 419 1 : 6600 : 3452 : 1032 : 235 3 : 176147 0 : 82 : 235 2 : 247 0 : 0431 : 8802 : 023 3 : 331124 0 : 92 : 018 2 : 962 0 : 5151 : 6081 : 766 3 : 396176 1 : 01 : 804 3 : 757 1 : 0211 : 3391 : 514 3 : 410069 1 : 11 : 609 4 : 592 1 : 5191 : 1141 : 309 3 : 394673 1 : 21 : 414 5 : 454 1 : 9920 : 9531 : 170 3 : 362943 1 : 31 : 111 6 : 428 2 : 5040 : 8091 : 053 3 : 322850 1 : 40 : 669 7 : 437 2 : 9740 : 7661 : 040 3 : 279466 1 : 5 0 : 074 8 : 459 3 : 3750 : 8551 : 163 3 : 236119 1 : 6 1 : 438 9 : 451 3 : 6731 : 1041 : 451 3 : 195040 1 : 7 4 : 014 10 : 345 3 : 8371 : 5161 : 904 3 : 157716 1 : 8 8 : 888 11 : 062 3 : 8522 : 0512 : 478 3 : 125051 1 : 9 17 : 983 11 : 538 3 : 7442 : 6113 : 069 3 : 097433 2 : 0 34 : 481 11 : 755 3 : 5763 : 0593 : 535 3 : 074787 2 : 1 62 : 336 11 : 747 3 : 4323 : 2663 : 744 3 : 056686 2 : 2 102 : 675 11 : 583 3 : 3873 : 1603 : 624 3 : 042507 2 : 3 150 : 692 11 : 337 3 : 4832 : 7353 : 172 3 : 031571 2 : 4 199 : 899 11 : 068 3 : 7262 : 0412 : 442 3 : 023237 2 : 5 246 : 321 10 : 805 4 : 0941 : 1541 : 516 3 : 016948 MUE b 42 : 2317 : 7462 : 5851 : 8182 : 129| MSE c 40 : 649 7 : 746 2 : 4891 : 8182 : 129| a TheFCIenergiesaretotalenergiesinhartree,whereasalloftheremainingenergiesareerrorsrelativetoFCIinmillihartree.Allactive-spaceCC approachesemployedthreeactiveoccupiedandthreeactiveunoccupiedorbitals,correspondingtothe1 s shellsoftheindividualHatoms. b Meanunsignederror. c Meansignederror. 77 Table4.3:AcomparisonoftheenergiesresultingfromthevariousfulltriplesCCapproachesandtheexactFCIdataforthe symmetricdissociationoftheH 6 /cc-pVTZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A). a R H{H CCSDT ACCSDT FCI (1,3)(1 ; 3+4 2 )(1 ; 3 n o n o + n u +4 n u n o + n u )(1,4) 0 : 60 : 079 3 : 952 1 : 996 0 : 279 0 : 151 2 : 858958 0 : 70 : 095 4 : 233 2 : 123 0 : 277 0 : 139 3 : 176147 0 : 80 : 116 4 : 609 2 : 304 0 : 295 0 : 146 3 : 331124 0 : 90 : 140 5 : 074 2 : 530 0 : 325 0 : 162 3 : 396176 1 : 00 : 160 5 : 621 2 : 792 0 : 356 0 : 175 3 : 410069 1 : 10 : 166 6 : 241 3 : 079 0 : 374 0 : 174 3 : 394673 1 : 20 : 137 6 : 932 3 : 381 0 : 365 0 : 143 3 : 362943 1 : 30 : 031 7 : 692 3 : 686 0 : 307 0 : 059 3 : 322850 1 : 4 0 : 235 8 : 512 3 : 973 0 : 1720 : 106 3 : 279466 1 : 5 0 : 823 9 : 369 4 : 2130 : 0720 : 385 3 : 236119 1 : 6 2 : 046 10 : 216 4 : 3710 : 4570 : 808 3 : 195040 1 : 7 4 : 493 10 : 982 4 : 4110 : 9891 : 380 3 : 157716 1 : 8 9 : 245 11 : 585 4 : 3161 : 6312 : 061 3 : 125051 1 : 9 18 : 225 11 : 959 4 : 1092 : 2862 : 747 3 : 097433 2 : 0 34 : 617 12 : 084 3 : 8552 : 8173 : 296 3 : 074787 2 : 1 62 : 392 11 : 995 3 : 6353 : 0953 : 576 3 : 056686 2 : 2 102 : 685 11 : 760 3 : 5263 : 0483 : 514 3 : 042507 2 : 3 150 : 676 11 : 452 3 : 5672 : 6733 : 111 3 : 031571 2 : 4 199 : 857 11 : 129 3 : 7642 : 0192 : 421 3 : 023237 2 : 5 246 : 248 10 : 820 4 : 0921 : 1651 : 528 3 : 016948 MUE b 41 : 6238 : 8113 : 4861 : 1501 : 304| MSE c 41 : 531 8 : 811 3 : 4860 : 8751 : 189| a TheFCIenergiesaretotalenergiesinhartree,whereasalloftheremainingenergiesareerrorsrelativetoFCIinmillihartree. b Meanunsignederror. c Meansignederror. 78 Table4.4:AcomparisonoftheenergiesresultingfromthevariousCCapproacheswithsinglesanddoublesandtheexactFCI dataforthesymmetricdissociationoftheH 10 /DZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A). a R H{H CCSD ACCSD FCI (1,3)(1 ; 3+4 2 )(1 ; 3 n o n o + n u +4 n u n o + n u )(1,4) 0 : 62 : 4610 : 4120 : 9451 : 2041 : 458 4 : 581177 0 : 72 : 9420 : 4981 : 0711 : 3481 : 619 5 : 133564 0 : 83 : 5190 : 7081 : 3141 : 6061 : 890 5 : 400721 0 : 94 : 1571 : 0531 : 6891 : 9922 : 285 5 : 513021 1 : 04 : 8781 : 5912 : 2652 : 5822 : 886 5 : 538852 1 : 15 : 6912 : 4353 : 1643 : 5013 : 821 5 : 516586 1 : 26 : 5103 : 7444 : 5524 : 9175 : 257 5 : 468944 1 : 37 : 0135 : 7016 : 6207 : 0227 : 388 5 : 409818 1 : 46 : 2888 : 4519 : 5209 : 97210 : 370 5 : 347723 1 : 51 : 89611 : 98713 : 25813 : 77314 : 209 5 : 287756 1 : 6 13 : 10616 : 04117 : 57518 : 16818 : 647 5 : 232798 1 : 7 66 : 27620 : 06921 : 93022 : 61823 : 144 5 : 184271 1 : 8NC b 23 : 41025 : 65026 : 44627 : 026 5 : 142644 1 : 9NC b 25 : 53128 : 16329 : 07529 : 715 5 : 107796 2 : 0NC b 26 : 19129 : 17230 : 19730 : 903 5 : 079254 2 : 1NC b 25 : 46228 : 69129 : 80930 : 585 5 : 056349 2 : 2NC b 23 : 63626 : 96728 : 14628 : 985 5 : 038308 2 : 3NC b 21 : 09024 : 37225 : 56826 : 453 5 : 024332 2 : 4NC b 18 : 19321 : 29822 : 46823 : 370 5 : 013655 2 : 5NC b 15 : 24818 : 08719 : 19320 : 080 5 : 005591 MUE c NA d 12 : 57214 : 31514 : 98015 : 505| MSE e NA d 12 : 57214 : 31514 : 98015 : 505| a TheFCIenergiesaretotalenergiesinhartree,whereasalloftheremainingenergiesareerrorsrelativetoFCIinmillihartree. b Noconvergence. c Meanunsignederror. d CouldnotbedeterminedbecauseCCSDdoesnotconvergeatlargerdistances. e Meansignederror. 79 Table4.5:Acomparisonoftheenergiesresultingfromthevariousactive-spaceCCapproachesandtheexactFCIdataforthe symmetricdissociationoftheH 10 /DZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A). a R H{H CCSDt ACCSDt FCI (1,3)(1 ; 3+4 2 )(1 ; 3 n o n o + n u +4 n u n o + n u )(1,4) 0 : 60 : 455 1 : 751 1 : 181 0 : 904 0 : 633 4 : 581177 0 : 70 : 554 2 : 110 1 : 502 1 : 209 0 : 922 5 : 133564 0 : 80 : 416 2 : 726 2 : 081 1 : 773 1 : 473 5 : 400721 0 : 90 : 349 3 : 218 2 : 541 2 : 221 1 : 912 5 : 513021 1 : 00 : 181 3 : 735 3 : 019 2 : 686 2 : 369 5 : 538852 1 : 1 0 : 084 4 : 169 3 : 399 3 : 050 2 : 724 5 : 516586 1 : 2 0 : 650 4 : 477 3 : 634 3 : 266 2 : 933 5 : 468944 1 : 3 1 : 956 4 : 591 3 : 654 3 : 266 2 : 931 5 : 409818 1 : 4 4 : 961 4 : 457 3 : 399 2 : 994 2 : 672 5 : 347723 1 : 5 11 : 879 4 : 091 2 : 881 2 : 468 2 : 182 5 : 287756 1 : 6 28 : 679 3 : 630 2 : 233 1 : 824 1 : 609 5 : 232798 1 : 7 81 : 643 3 : 315 1 : 692 1 : 300 1 : 193 5 : 184271 1 : 8NC b 3 : 398 1 : 510 1 : 144 1 : 178 5 : 142644 1 : 9NC b 4 : 043 1 : 862 1 : 523 1 : 712 5 : 107796 2 : 0NC b 5 : 295 2 : 810 2 : 494 2 : 832 5 : 079254 2 : 1NC b 7 : 106 4 : 333 4 : 029 4 : 497 5 : 056349 2 : 2NC b 9 : 380 6 : 355 6 : 055 6 : 626 5 : 038308 2 : 3NC b 12 : 001 8 : 762 8 : 462 9 : 114 5 : 024332 2 : 4NC b 14 : 806 11 : 381 11 : 083 11 : 806 5 : 013655 2 : 5NC b 17 : 551 13 : 966 13 : 660 14 : 471 5 : 005591 MUE c NA d 5 : 7934 : 1103 : 7703 : 790| MSE e NA d 5 : 793 4 : 110 3 : 770 3 : 790| a TheFCIenergiesaretotalenergiesinhartree,whereasalloftheremainingenergiesareerrorsrelativetoFCIinmillihartree.Allactive-spaceCC approachesemployedeactiveoccupiedandeactiveunoccupiedorbitals,correspondingtothe1 s shellsoftheindividualHatoms. b Noconvergence. c Meanunsignederror. d CouldnotbedeterminedbecauseCCSDtdoesnotconvergeatlargerdistances. e Meansignederror. 80 Table4.6:AcomparisonoftheenergiesresultingfromthevariousfulltriplesCCapproachesandtheexactFCIdataforthe symmetricdissociationoftheH 10 /DZringatselectedbonddistancesbetweenneighboringHatoms R H{H (in A). a R H{H CCSDT ACCSDT FCI (1,3)(1 ; 3+4 2 )(1 ; 3 n o n o + n u +4 n u n o + n u )(1,4) 0 : 60 : 065 2 : 167 1 : 591 1 : 311 1 : 037 4 : 581177 0 : 70 : 094 2 : 602 1 : 986 1 : 688 1 : 398 5 : 133564 0 : 80 : 110 3 : 054 2 : 404 2 : 093 1 : 790 5 : 400721 0 : 90 : 101 3 : 485 2 : 805 2 : 482 2 : 172 5 : 513021 1 : 00 : 035 3 : 894 3 : 175 2 : 841 2 : 523 5 : 538852 1 : 1 0 : 170 4 : 262 3 : 491 3 : 141 2 : 815 5 : 516586 1 : 2 0 : 703 4 : 535 3 : 691 3 : 323 2 : 989 5 : 468944 1 : 3 1 : 991 4 : 631 3 : 693 3 : 305 2 : 969 5 : 409818 1 : 4 4 : 986 4 : 487 3 : 428 3 : 023 2 : 700 5 : 347723 1 : 5 11 : 897 4 : 115 2 : 904 2 : 490 2 : 205 5 : 287756 1 : 6 28 : 687 3 : 649 2 : 251 1 : 842 1 : 626 5 : 232798 1 : 7 81 : 631 3 : 330 1 : 706 1 : 313 1 : 206 5 : 184271 1 : 8NC b 3 : 408 1 : 520 1 : 153 1 : 187 5 : 142644 1 : 9NC b 4 : 049 1 : 867 1 : 528 1 : 717 5 : 107796 2 : 0NC b 5 : 296 2 : 811 2 : 494 2 : 833 5 : 079254 2 : 1NC b 7 : 101 4 : 329 4 : 025 4 : 493 5 : 056349 2 : 2NC b 9 : 371 6 : 346 6 : 045 6 : 618 5 : 038308 2 : 3NC b 11 : 988 8 : 750 8 : 449 9 : 100 5 : 024332 2 : 4NC b 14 : 790 11 : 365 11 : 068 11 : 791 5 : 013655 2 : 5NC b 17 : 533 13 : 927 13 : 644 14 : 459 5 : 005591 MUE c NA d 5 : 8874 : 2023 : 8633 : 881| MSE e NA d 5 : 887 4 : 202 3 : 863 3 : 881| a TheFCIenergiesaretotalenergiesinhartree,whereasalloftheremainingenergiesareerrorsrelativetoFCIinmillihartree. b Noconvergence. c Meanunsignederror. d CouldnotbedeterminedbecauseCCSDTdoesnotconvergeatlargerdistances. e Meansignederror. 81 Table4.7:AcomparisonoftheenergiesresultingfromthevariousCCapproachesincludinguptotripleexcitationsandthe nearlyexactLDMRG(500)dataforthesymmetricdissociationoftheH 50 /STO-6Glinearchainatselectedbonddistances betweenneighboringHatoms R H{H (ina.u.). a R H{H CCSDACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )CCSDTACCSDT(1 ; 3 n o n o + n u +4 n u n o + n u )LDMRG(500) 1 : 011 : 906 : 300 : 27 6 : 68 17 : 28407 1 : 216 : 289 : 120 : 29 9 : 08 22 : 94765 1 : 420 : 9912 : 78 0 : 10 11 : 37 25 : 59378 1 : 626 : 0117 : 82 1 : 71 13 : 26 26 : 71944 1 : 831 : 0024 : 87 6 : 96 14 : 62 27 : 03865 2 : 034 : 6034 : 30NC b 15 : 73 26 : 92609 2 : 4NC b 59 : 23NC b 22 : 04 c 26 : 16057 2 : 8NC b 86 : 24NC b 22 : 47 c 25 : 27480 3 : 2NC b 106 : 45NC b 18 : 78 c 24 : 56828 3 : 6NC b 113 : 43NC b 4 : 66 c 24 : 10277 4 : 2NC b 94 : 08NC b 71 : 86 c 23 : 74971 MUE d NA e 51 : 33NA f 19 : 14| MSE g NA e 51 : 33NA f 6 : 08| a TheLDMRG(500)energiesweretakenfromReference[175]andaretotalenergiesinhartree.Theremainingenergiesareerrorsrelativeto LDMRG(500)inmillihartree. b Noconvergence. c Wewereunabletoconvergetheseenergiesbeyond1 10 3 hartree.ThereportedvaluescorrespondtotherespectivelastCCiteration. d Meanunsignederror. e CouldnotbedeterminedbecauseCCSDdoesnotconvergeatlargerdistances. f CouldnotbedeterminedbecauseCCSDTdoesnotconvergeatlargerdistances. g Meansignederror. 82 Figure4.1:Ground-statePECs[panels(a){(c)]anderrorsrelativetoFCI[panels(d){(f)] forthesymmetricdissociationoftheH 6 ringresultingfromtheCCSDandvariousACCSD calculations[panels(a)and(d)],theactive-spaceCCSDtandACCSDtcomputations[panels (b)and(e)],andthefullCCSDTandACCSDTmethods[panels(c)and(f)],usingthecc- pVTZbasisset.Theactive-spacetriplesapproachesemployedaminimumactivespacebuilt fromthe1 s orbitalsofindividualhydrogenatoms.TheFCIPECisincludedinpanels(a){ (c)tofacilitatethecomparisons. 83 Figure4.2:Ground-statePECs[panels(a){(c)]anderrorsrelativetoFCI[panels(d){(f)] forthesymmetricdissociationoftheH 10 ringresultingfromtheCCSDandvariousACCSD calculations[panels(a)and(d)],theactive-spaceCCSDtandACCSDtcomputations[panels (b)and(e)],andthefullCCSDTandACCSDTmethods[panels(c)and(f)],usingtheDZ basisset.Theactive-spacetriplesapproachesemployedaminimumactivespacebuiltfrom the1 s orbitalsofindividualhydrogenatoms.TheFCIPECisincludedinpanels(a){(c)to facilitatethecomparisons. 84 Figure4.3:Ground-statePECsforthesymmetricdissociationoftheH 50 linearchainresult- ingfromtheCCSDandACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )calculations[panel(a)]andtheir fulltriplesextensions[panel(b)]usingtheSTO-6Gbasisset.ThenearlyexactLDMRG(500) PECofReference[175]isincludedinbothpanelstofacilitatethecomparisons.Theinsets showtheerrorsrelativetoLDMRG(500)inmillihartree( cf. Table4.7). 85 4.2ApproachingtheFullInteractionLimitfor StrongCorrelationUsingSemi-StochasticIdeas Inthissection,weexaminetheperformanceofthenovelCAD-FCIQMC[1,(3+4)/2] methodologyintroducedinthisdissertation.Inparticular,wecompareitsrateofcon- vergencetowardtheexact,FCI,energeticsagainsttheonecharacterizingtheunderlying FCIQMCsimulationsinchallengingstronglycorrelatedsituations.Beforeweproceed,how- ever,westdemonstratethatintheabsenceofstrongmany-electroncorrelationthe CAD-FCIQMC[1,(3+4)/2]approachyieldssimilarresultstotheoriginalCAD-FCIQMC[1{ 5]scheme.ThisisadesiredbehaviorsinceCAD-FCIQMC[1{5]ischaracterizedbyarapid convergence,muchfasterthanFCIQMCitself,towardtheexactenergyvaluesforproblems involvingweakormoderatelystrongcorrelations[265]. Tothatend,westudiedthe C 2v -symmetricdoublebonddissociationofH 2 Oasde- scribedbythecc-pVDZbasisset,whichservedasthetestinggroundoftheoriginalCAD- FCIQMC[1{5]algorithm[265].Inparticular,weconsideredtwogeometriesofH 2 O,namely, equilibrium, R O{H = R e ,andoneadditionalgeometryobtainedbythesimultaneousstretch ofbothO{Hbondsbyafactorof2.0, R O{H =2 : 0 R e ,whilekeepingthe \ H{O{Hangle toitsequilibriumvalue.Thegeometriesandcorrespondingall-electronFCIenergy valueswhereobtainedfromReference[277].The i -FCIQMCpropagations,whichprovided thesourceof T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ )clusters,wereperformedwiththeHANDEpackage [316].ForeachexaminedgeometryoftheH 2 Otriatomic,the i -FCIQMCstochasticwave- functionsamplingwasinitiatedbyplacing1,000walkersontheRHFSlaterdeterminant, whilethetimestepusedinpropagationswas ˝ =0 : 0001a.u.Toaccelerateconvergence, the n a parameteroftheinitiatoralgorithmwassetat3,meaningthat,atanygiventime step ˝ ,onlydeterminantsinhabitedbymorethan3walkerswereallowedtospawnprogeny onunpopulateddeterminants.Atevery1,000timesteps,weclusteranalyzedthe i -FCIQMC wavefunctionandsolvedtheCCSD-likesystemofequations,Equations(3.29)and(3.40), inwhich T (MC) 3 ( ˝ ), T (MC) 4 ( ˝ ),andselectedcoupled-paircontributionswereextractedfrom 86 i -FCIQMC,for T 1 and T 2 usingcodesdevelopedbythePiecuchgroup.Finally,boththe i -FCIQMCcalculationsandtheCCcomputationswereperformedusingthesamesetofone- andtwo-bodymolecularintegralsextractedfromGAMESS[307{309]. TheresultsoftheoriginalCAD-FCIQMC[1{5]scheme,theCAD-FCIQMC[1,(3+4)/2] methodologyintroducedinthisdissertation,andtheunderlying i -FCIQMCsimulationsfor thedoublebonddissociationofH 2 OareshowninFigure4.4and,forselectedtimesteps,in Table4.8.Asmighthavebeenanticipated,thedeviationbetweenthetwoCAD-FCIQMC variantsislargestattheinitialstepsofthesimulationandisgraduallybridgedasone approachestheimaginarytimelimit,owingtothefactthatbothschemesconvergeto theexact,FCI,energetics.At ˝ =0,theoriginalCAD-FCIQMC[1{5]methodologybecomes equivalenttoCCSD,whileitsCAD-FCIQMC[1,(3+4)/2]extensiontothestrongcorrelation regimereducestoACCSD(1 ; 3+4 2 ) DCSD.Attheweaklycorrelatedequilibriumgeometry, theCCSDandACCSD(1 ; 3+4 2 )dataarealreadyquiteclosetothepertinentFCIenergy value,thecorrespondingerrorsbeing3.744mE h and 0 : 731mE h ,respectively.Itis,thus, notsurprisingthattheCAD-FCIQMC[1{5]andCAD-FCIQMC[1,(3+4)/2]energeticscan hardlybedistinguishedfromeachotheratlaterpropagationtimes(see,forexample,Figure 4.4).Forthestretchedgeometry,thedeviationbetweentheCCSDandACCSD(1 ; 3+4 2 ) energeticsismorepronounced.AlthoughthedoublebonddissociationofH 2 Oinvolves theentanglementofonlyfourelectrons,thepresenceofquasi-degeneracieshasat impactontheCCSDenergyvalue,whichischaracterizedbyamorethan20mE h error relativetoFCI.Atthesametime,theACCSD(1 ; 3+4 2 )variantoftheACPschemes,which iswell-suitedforthestudyofelectronicquasi-degeneracies,adramaticallyimproved description,asisevidentbytheabout3mE h errorrelativetotheFCIdata.Nevertheless,the CAD-FCIQMC[1{5]methodologyquicklycatchesupwithCAD-FCIQMC[1,(3+4)/2]and afterabout20,000MCiterationstheyprovidetheessentiallyidenticalpicture( cf. Figure 4.4).Thus,intheabsenceofstrongmany-electroncorrelationbothoftheexamined vorsofCAD-FCIQMCprovidemoreorlessresultsofsimilarquality,eveninthepresence 87 ofquasi-degeneracies,suchasthosepresentinthedoublebonddissociationofH 2 O. WenowcomparetheperformanceoftheCAD-FCIQMCvariantsofinterestwiththat oftheunderlying i -FCIQMCsimulations.AsimpleinspectionofFigure4.4revealsthat boththeoriginalCAD-FCIQMC[1{5]schemeanditsCAD-FCIQMC[1,(3+4)/2]extension tothestrongcorrelationregimearecharacterizedbyarapidconvergencetotheexact, FCI,energetics,muchfasterthaninthecaseof i -FCIQMC,evenwhenbothO{Hbonds arestretchedtotwicetheirequilibriumvalue.AsshowninTable4.8,focusingonthe morechallenging R O{H =2 : 0 R e nuclearbothCAD-FCIQMC[1{5]andCAD- FCIQMC[1,(3+4)/2]arecapableofreproducingthedeterministicFCIenergytowithina millihartreealreadyatabout40,000MCiterations,withsubmillihartreeerrorsachievedat around100,000MCiterations.Atthesametime,theerrorrelativetoFCIcharacterizing theunderlying i -FCIQMCmethodologyat160,000MCiterations,thelastiterationbefore westoppedthe i -FCIQMCpropagation,is1.328mE h ,whichsuggeststhatoneneedstokeep samplingthemany-electronHilbertspaceforthe i -FCIQMCmethodologytoprovideresults comparabletoCAD-FCIQMC.Atthispoint,itisalsointerestingtonotethatthenoise inherenttothepurelystochastic i -FCIQMCsimulationspropagatestotheCAD-FCIQMC calculationsthroughtheextracted T (MC) 2 ( ˝ ), T (MC) 3 ( ˝ ),and T (MC) 4 ( ˝ )amplitudes.Al- thoughCAD-FCIQMC[1{5]andCAD-FCIQMC[1,(3+4)/2]cannoteliminatethestochastic noise,theydoreduceit,givingrisetoconsiderablysmootherenergetics( cf. Figure4.4). TheremarkableperformanceoftheCAD-FCIQMCfamilyofmethodscanbelargelyat- tributedtotheabilityofthe i -FCIQMCstochasticwavefunctionsamplingtoquicklyidentify theimportanttriplyandquadruplyexcitedSlaterdeterminantsthatentertheCCsystemof equationscorrectedforconnectedtriplesandquadruples,Equations(3.29)and(3.40).This ismosteasilyquanbyexaminingthepercentagesoftriplesandquadruplescaptured by i -FCIQMCasfunctionsofthepropagationtime ˝ .AscanbeseeninTable4.8,both variantsofCAD-FCIQMCarecapableofproducingFCI-qualityenergetics,stabletowithin amillihartree,once i -FCIQMCcapturesabout17%oftriplesandabout3%ofquadruples 88 (40,000MCiterations).Thisobservationimpliesthatnotonlydoes i -FCIQMCquickly capturethedominanttriplesandquadruples,butalsothattheirwalkerpopulations onthestructureoftheFCIwavefunctionalreadyfromtheearlystagesofthepropaga- tion.Atthesametime,theunderlying i -FCIQMCmethodologyhasonlyexploredatiny fractionofthemany-electronHilbertspace,havingcapturedjust0.02%ofthe451,681,246 S z =0totallysymmetricSlaterdeterminants,using C 2v pointgroupsymmetry,spanning themany-electronHilbertspace.TheseresultsindicatethattheCAD-FCIQMCmethodol- ogyiscapableofproducingaccurateestimatesoftheFCIenergiesoutoftheearlystagesof i -FCIQMCpropagationsusingcomputationallyCCSD-likecomputations,evenin thepresenceofquasi-degeneraciessuchasthosecharacterizingthedoublebonddissociation ofH 2 O. WearenowinapositiontoproceedtothediscussionoftheperformanceofCAD- FCIQMC[1,(3+4)/2]inthepresenceofstrongmany-electroncorrelationTothat end,weexaminethe D 6h -and D 10h -symmetricdissociationsofthesix-andten-membered hydrogenrings,respectively.InthecaseofH 6 weusedthecc-pVDZbasisset,whilefor thelargerH 10 hydrogenclusterweemployedtheDZbasis.Asalreadyemphasizedinthis dissertation,thedissociationofequallyspacedhydrogenclusterstoindividualHatomsis characterizedbyaMottmetal{insulatortransitionasthesystemtraversesfromaweakly correlatedmetallicphasetoastronglycorrelatedinsulatingphase,whichhasadevastating ontheperformanceofthehierarchyoftraditionalCCapproaches,including,forex- ample,CCSDandCCSDT.Forthesakeofcompleteness,thePECsoftheH 6 /cc-pVDZand H 10 /DZringsresultingfromCCSD,CCSDT,andFCIcalculationsarepresentedinFigure 4.5.AlthoughtheperformanceofCCSDandCCSDTindescribingtheH 6 andH 10 PECs wasdiscussedextensivelyintheprevioussection,herewemakeacoupleofremarksthatare usefulinthecontextoftheCAD-FCIQMCmethodology.InFigure4.5,weseethatCCSD closelyreproducestheFCIenergeticsaroundtheequilibriumgeometriesofbothH 6 and H 10 .ThissuggeststhattheoriginalCAD-FCIQMC[1{5]scheme,whichbecomesequivalent 89 toCCSDfor ˝ =0,willrapidlyconvergetowardtheexactdescriptionofbothhydrogen ringswhenconsidering,forexample,thestructureswherethedistancebetweenneighboring Hatomsis R H{H =1 : 0 A,correspondingtotheregionofminimumontheFCIPEC(cf. Figure4.5).Asonedepartsfromtheequilibriumregionandapproachestheatomization threshold,theperformanceofCCSDgraduallydeteriorates,resultinginanunphysicalde- scriptionofbothH 6 andH 1 0systemsasaconsequenceofovercorrelating.Asdiscussed earlierinthisdissertation,thefailureofCCSDismuchmoredramaticinthecaseofthe H 10 cluster,sincenoconvergencewasobtainedfordistancesbetweenneighboringhydrogen atomslargerthan1.75 A.Theseobservationsforeshadowthecatastrophethatwillbefall CAD-FCIQMC[1{5]inthestrongcorrelationregime.Inwhatfollows,wedemonstratethat thenovelCAD-FCIQMC[1,(3+4)/2]methodologyintroducedinthisdissertation,where, outofthee 1 2 T 2 2 Goldstone{BrandowdiagramsshowninFigure3.5,diagramsD1and anaverageofdiagramsD3andD4,whichareresponsibleforcapturingstrongcorrelations, aretreateddeterministicallywiththerestof 1 2 T 2 2 calculatedusing T (MC) 2 ( ˝ )extractedfrom FCIQMC,ispracticallyimmunetothepresenceofstrongmany-electroncorrelation andischaracterizedbythefastestconvergencetowardFCIamongtheexaminedapproaches. Asalreadymentioned,inthecomputationsregardingthesymmetricdissociationofthe H 6 andH 10 ringsweemployedthecc-pVDZandDZbasissets,respectively.Thegridof geometriesconsideredinourcalculationsconsistedofthefollowingdistancesbetweenneigh- boringhydrogenatoms:0.6,0.8,1.0,1.25,1.5,1.75,and2.0 A.Theunderlying i -FCIQMC simulationswereperformedwithHANDE.Forbothhydrogenclustersandforeachgeom- etryoftheaforementionedgrid,the i -FCIQMCpropagationwasinitiatedbyplacing1,500 walkersontheRHFSlaterdeterminantandsettingthe n a parameteroftheinitiatoralgo- rithmat3.Thetimestepusedinthesimulationswas ˝ =0 : 0001a.u.andweperformed CAD-FCIQMC[1{5]andCAD-FCIQMC[1,(3+4)/2]calculationsevery1,000timesteps.The clusteranalysisofthe i -FCIQMCwavefunctionandthesubsequentCCSD-likecomputations wereperformedwithcodesofthePiecuchgroup.AswasthecasewiththeH 2 Osystem, 90 thesamesetofone-andtwo-bodymolecularintegrals,extractedfromGAMESS,wasem- ployedforboththe i -FCIQMCsimulationsandsubsequentCCSD-likecomputations.The deterministicFCIdata,whichweneededtojudgetheperformanceofitspurelystochastic, i -FCIQMC,andsemi-stochastic,CAD-FCIQMC[1{5]andCAD-FCIQMC[1,(3+4)/2],ap- proximations,wereacquiredusingthedeterminantalFCIcodeavailableintheGAMESS package. TheresultsoftheCAD-FCIQMC[1{5]calculations,theCAD-FCIQMC[1,(3+4)/2]com- putations,andtheunderlying i -FCIQMCsimulationsfortheH 6 /cc-pVDZandH 10 /DZ systemsaresummarizedinFigures4.6and4.7andTables4.9and4.10.Webeginour discussionwiththesmallerH 6 ring,thedissociationofwhichinvolvestheentanglement ofsixelectrons.Panels(a)and(b)ofFigure4.6portraytheenergiesobtainedatthe i - FCIQMC,CAD-FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]levelsoftheoryasfunctions oftheimaginarytime ˝ atthestructurecharacterizedby R H{H =1 : 0 Aandthegeometryin whicheachH{Hbondisstretchedtotwicethisvalue,thelargestdistanceconsideredinour calculations,respectively.Focusingontheweaklycorrelated R H{H =1 : 0 Ageometry,we seethatboththeoriginalCAD-FCIQMC[1{5]andthenovelCAD-FCIQMC[1,(3+4)/2]ap- proachesdisplayaremarkablyfastconvergencetowardtheexact,FCI,energyvalue.Infact, Table4.9revealsthattheenergeticsofbothvorsofCAD-FCIQMCarecharacterizedby stablesubmillihartreeerrorsrelativetoFCIalreadyat20,000MCiterationsorwhenabout 26%oftriplesand4%ofquadruplesarecapturedbytheunderlyingstochasticwavefunction sampling.Despitethefactthatthe i -FCIQMCpropagationhasa110timeslargererror relativetoFCIat ˝ =0thanCAD-FCIQMC[1,(3+4)/2],itreproducestheFCIvalueto withinabout1.5millihartreeratherquickly(40,000MCiterations),althoughthestochastic noiseisquitet. Thepicturechangesdramaticallywhenoneexaminestheperformanceof i -FCIQMC, CAD-FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]atthestretchedgeometryinwhich R H{H =2 : 0 A.Asimpleinspectionofpanel(b)ofFigure4.6revealsthattheoriginalCAD- 91 FCIQMC[1{5]schemeischaracterizedbyaslowconvergence,comparabletothatofthe underlying i -FCIQMCpropagation,towardtheFCIenergyvalue.Suchbehaviorwasmore orlessanticipatedduetothefactthatat ˝ =0CAD-FCIQMC[1{5] CCSDovercorrelates bymorethan35mE h .Itis,however,interestingtonotethat,atleastforthedurationofthe particularsimulationshowninFigure4.6(b), i -FCIQMCapproachesFCIfromabovewhile CAD-FCIQMC[1{5]frombelow.Atthesametime,thenovelCAD-FCIQMC[1,(3+4)/2] methodologyintroducedinthisdissertationispracticallyimmunetothepresenceofstrong many-electroncorrelationAsdemonstratedinTable4.9,CAD-FCIQMC[1,(3+4)/2] faithfullyreproducestheFCIenergytowithinlessthan1mE h alreadyat60,000MCiter- ationsorwhen i -FCIQMCcapturesabout41%oftriplesand10%ofquadruples.Although thesepercentagesarelargerthanthoserequiredforsubmillihartreeaccuracyinthecaseof thecorrespondingequilibriumgeometry,theyarestillfarfromthetotalnumbersoftriples andquadruples.Infact,at60,000MCiterationsthe i -FCIQMCpropagationhasexplored only2%oftheentiremany-electronHilbertspace,whichisspannedby2,123,544 S z =0 totallysymmetricSlaterdeterminantsusing D 2h pointgroupsymmetry.Thisconstitutesa tremendousaccelerationtowardFCIeveninthepresenceofstrongnon-dynamicalcorrela- tionTheremarkableperformanceoftheCAD-FCIQMC[1,(3+4)/2]approachcanbe largelyattributedtothefactthatthe 1 2 T 2 2 Goldstone{BrandowdiagramD1andtheaverage ofdiagramsD3andD4(cf.Figure3.5),whichareresponsibleforcapturingstrongcorre- lations,aretreateddeterministicallywhiletherestof 1 2 T 2 2 isdeterminedusing T (MC) 2 ( ˝ ) extractedfrom i -FCIQMC.Consequently,evenat ˝ =0,CAD-FCIQMC[1,(3+4)/2]isal- readymuchclosertoFCIthanCAD-FCIQMC[1{5],havinganerroroflessthan9mE h ,and thegapisrapidlybridgedasthesimulationprogresses. Furtherinsightsintotheperformanceof i -FCIQMC,CAD-FCIQMC[1{5],andCAD- FCIQMC[1,(3+4)/2]canbegainedbyexaminingtheconvergenceoftheentirePECtoits FCIcounterpartasafunctionofMCiterations.Thiscanbefoundingraphicalforminthe caseoftheH 6 /cc-pVDZsysteminpanels(a),(b),and(c)ofFigure4.7for50,000,100,000, 92 and150,000MCiterations.Forthesakeofcomparison,wealsoincludetheinformation aboutthedeterministicCCSD,CCSDT,andACCSD(1 ; 3+4 2 )calculations.Tobeginwith, the i -FCIQMCapproachprovidesamoreorlessexactdescriptionoftheweaklyandmoder- atelycorrelatedportionsofthePECat100,000MCiterations.However,theenergiesofthe twopointsclosesttothedissociationlimit,correspondingtodistancesbetweenneighboring hydrogenatomsof R H{H =1 : 75and2.0 A,respectively,arefarfromtheirFCIcounter- parts.Infact,evenat100,000MCiterations i -FCIQMCisworsethanthedeterministic ACCSD(1 ; 3+4 2 )methodologyandoneneedstogotoatleast150,000MCiterationsfor i - FCIQMCtooutperformACCSD(1 ; 3+4 2 )intheentirerangeofgeometriesexaminedinthis dissertation. MovingontotheoriginalCAD-FCIQMC[1{5]scheme,asimpleinspectionofFigure 4.7(a)revealsthatitprovidesapracticallyexactdescriptionfortheweaklycorrelatedstruc- turescorrespondingto R H{H distancesof0.6,0.8,1.0,1.25,and1.5 Aalreadyat50,000MC iterations,asmighthavebeenanticipated.However,aftertheonsetofstrongcorrelations, CAD-FCIQMCsigtlyovercorrelatesevenat50,000MCiterations,asisevidentby thelargenegativeerrorswithrespecttoFCIforthe R H{H =1 : 75and2.0 Agridpoints(cf. insettoFigure4.7).Aswasthecasewiththeunderlying i -FCIQMCsimulations,oneneeds togoto150,000MCiterationsforCAD-FCIQMC[1{5]toprovideaPECofhigherquality thanACCSD(1 ; 3+4 2 ).Itis,however,worthmentioningthatCAD-FCIQMC[1{5]converges muchfastertotheFCI-levelenergeticsat R H{H =1 : 75 Athan i -FCIQMC. Aswasexpectedinlightoftheabovediscussion,theCAD-FCIQMC[1,(3+4)/2]method- ologyischaracterizedbythefastestconvergencetowardtheexact,FCI,energeticsforthe entirerangeofelectroncorrelationrangingfromtheweaklytothestronglycorre- latedregimes.Evenat50,000MCiterations,theCAD-FCIQMC[1,(3+4)/2]PECcharac- terizingthe D 6h -symmetricdissociationoftheH 6 /cc-pVDZringisfarsuperiortotheone ofACCSD(1 ; 3+4 2 ),closelyreproducingtheFCIdata.Indeed,at100,000MCiterationsone canhardlydistinguishtheCAD-FCIQMC[1,(3+4)/2]andFCIPECs. 93 Now,weturnourattentiontothelargerH 10 ring.Aswasmentionedearlierinthis dissertation,the D 10h -symmetricdissociationofH 10 involvestheentanglementof10elec- trons, i.e. ,thenumberofstronglycorrelatedelectronsisincreasedbymorethan66%when comparedtothesmallerH 6 cluster.Panels(c)and(d)ofFigure4.6showtheenergiesob- tainedusing i -FCIQMC,CAD-FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]asfunctions oftheMCiterationsatthestructurecharacterizedby R H{H =1 : 0 Aandthegeometry inwhicheachH{Hbondisstretchedtotwicethisvalue,thelargestdistanceconsidered inourcalculations,respectively.Whenconsideringtheweaklycorrelated R H{H =1 : 0 A geometry,itcomesasnosurprisethatbothCAD-FCIQMCvariantsexaminedinthisdisser- tationdisplayanequallyrapidconvergence,fasterthanthatoftheunderlying i -FCIQMC propagation,totheexact,FCI,energyvalue.Itisinterestingtonote,however,thatsub- millihartreeaccuraciesareattainedwhenthe i -FCIQMCstochasticwavefunctionsampling hascapturedabout42%oftriplesand9%ofquadruples(40,000MCiterations).Thisneeds tobecontrastedwiththe26%oftriplesand4%ofquadruplesthatwereneededtoobtain thesamelevelofaccuracyinthecaseofthe R H{H =1 : 0 AgridpointforH 6 /cc-pVDZ.This indicatestheseverityofelectroncorrelationinH 10 ,evenintheweaklycorrelated regime.Nevertheless,itisworthemphasizingthatat40,000MCiterationstheunderlying i -FCIQMCsimulationhasonlycaptured0.12%oftheentiremany-electronHilbertspace, whichisspannedby60,095,104 S z =0totallysymmetricSlaterdeterminantsusing D 2h pointgroupsymmetry.Thisisaveryinterestingobservation,becauseitimpliesthatevenif theentiremanifoldsoftriplyandquadruplyexcitedSlaterdeterminantsneedtobecaptured byFCIQMCsothatCAD-FCIQMCproducesFCI-qualityenergetics,theystillconstitutea tinyfractionofallSlaterdeterminantsspanningthemany-electronHilbertspace. Theonsetofstrongnon-dynamicalcorrelationshasadevastatingontheoriginal CAD-FCIQMC[1{5]scheme.Thisisshowcasedbythefactthatnoconvergencewasobtained for R H{H =2 : 0 A(cf.Table4.10).Infact,thesingularityplaguingtheCCSD-likeEquations (3.29)and(3.40)with ˘ i =1, i =1{5,wassoseverethatitpersistedfortheentire i -FCIQMC 94 simulation,independentofthepercentagesofcapturedtriplesandquadruples,andevenwhen weusedtheconvergedCAD-FCIQMC[1,(3+4)/2] T 1 and T 2 amplitudesasinitialguesses. DespitethesuccessoftheoriginalCAD-FCIQMC[1{5]approach,beingabletoprovidefast convergencetowardFCIinthepresenceofquasi-degeneraciessuchasthosecharacterizing thedoublebonddissociationofH 2 O,itcompletelybreaksdowninthepresenceofstrong correlations. Incontrast,thenovelCAD-FCIQMC[1,(3+4)/2]methodologyisnotonlywell-behaved inthestrongcorrelationregime,butalsocapableofprovidingFCI-qualityenergeticsoutof theearlystagesofthe i -FCIQMCsimulation[see,forexample,Figure4.6(d)].Asmentioned earlierinthisdissertation,thesuccessbehindtheCAD-FCIQMC[1,(3+4)/2]approachlies inthefactthat,byrepartitioningtheCCequationsprojectedondoubles,the 1 2 T 2 2 part responsibleforcapturingstrongcorrelationsistreateddeterministicallywhileitscompli- mentisextractedfromFCIQMC.At ˝ =0,theerrorwithrespecttoFCIcharacterizing CAD-FCIQMC[1,(3+4)/2] ACCSD(1 ; 3+4 2 )amountsto29.172mE h .Ataglance, suchadeviationseemsratherlarge,butweneedtokeepinmindthattheFCIcorrelation energyatthisgeometryis 454 : 768mE h .ThisimpliesthattheCAD-FCIQMC[1,(3+4)/2] schemeintroducedinthisdissertationcapturesabout94%oftheFCIcorrelationenergy alreadyat ˝ =0.Furthermore,aftercapturingabout70%oftriplesand36%ofquadru- ples,CAD-FCIQMC[1,(3+4)/2]steadilyrecovers99{101%oftheFCIcorrelationenergy. Althoughthepercentagesoftriplesandquadruplesarequitelarge,onehastokeepinmind thattheyconstituteatinyfractionoftheentiremany-electronHilbertspace.Infact,atthis pointofthesimulation,lessthan2%oftheSlaterdeterminantsspanningthemany-electron Hilbertspacehavebeencaptured.Itis,thus,astoundingthatwithsuchasmallsubspace ofthemany-electronHilbertspacerecovered,CAD-FCIQMC[1,(3+4)/2]closelyreproduces theFCIenergyvalue.Atthesametime,theunderlyingpurelystochastic i -FCIQMCap- proachisfarfromattainingsuchalevelofaccuracyevenat160,000MCiterations,thelast iterationbeforewestoppedthesimulation,wherealmostalltriplesandquadrupleshave 95 beencaptured. AcomparisonoftheconvergenceofthePECscharacterizingthe D 10h -symmetricdissoci- ationofH 10 obtainedusing i -FCIQMC,CAD-FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2] at50,000,100,000,and150,000MCiterationscanbefoundinpanels(d){(f)ofFigure4.7, respectively.AswasthecasewiththesmallerH 6 ring,oneneedstogoto150,000MC iterationsfortheerrorsrelativetotheFCIdatacharacterizingthe i -FCIQMCenergetics tobeconsistentlysmallerthanthoseofthedeterministicACCSD(1 ; 3+4 2 )approach.The sameappliestotheoriginalCAD-FCIQMCscheme,withtheexceptionofthe R H{H =2 : 0 Agridpointforwhichnoconvergencewasobtained.ThenovelCAD-FCIQMC[1,(3+4)/2] methodologyischaracterizedbyarapidconvergencetoFCIacrossallgridpointssam- plingthesymmetricdissociationofH 10 ,includingboththeweaklyandstronglycorrelated regimes.Indeed,CAD-FCIQMC[1,(3+4)/2]faithfullyreproducestheFCIPECalreadyat 100,000MCiterations,beingcharacterizedbysmallerthan1%relativeerrors. Sofar,wehaveseenthatbothvorsofCAD-FCIQMCconsideredinthisdissertation produceFCI-qualityenergeticsbasedoninformationextractedoutoftheearlystagesof FCIQMCpropagationsnotonlyinweaklycorrelatedsituations,butalsointhepresenceof quasi-degeneraciessuchasthosecharacterizingthe C 2v -symmetricdoublebonddissociation ofH 2 O.Furthermore,wedemonstratedthatinthestrongcorrelationregime,theoriginal CAD-FCIQMC[1{5]schemeischaracterizedbyaslowconvergencetowardtheexactenergy valuesorbreaksdowncompletely,dependingonthenumberofentangledelectrons.Onthe otherhand,thenovelCAD-FCIQMC[1,(3+4)/2]methodologyispracticallyimmunetothe presenceofstrongcorrelationsandarapidconvergencetowardFCI-qualityenergetics fortheentirespectrumofelectroncorrelationects.Asaillustrationofthepower ofCAD-FCIQMCapproachesingeneral,wesetouttodeterminethefrozen-coreFCItotal electronicenergyofC 6 H 6 ,thesimplestaromaticcompound,asdescribedbythecc-pVDZ basissetatitsequilibriumstructure.Inthiscase,thesizeofthemany-electronHilbert space,whichisspannedbyabout10 36 S z =0Slaterdeterminantswithouttakingadvantage 96 ofpointgroupsymmetry,rendersabrute-forceFCIcomputationimpossible.Infact,the dimensionalityofthepertinentHamiltonianmatrixexceedsthepresent-daycapabilitiesof state-of-the-artsupercomputersandmatrixdiagonalizationalgorithmsbymorethan25 ordersofmagnitude. Ataglance,itmayappearthatthenear-exactdescriptionoftheelectronicstructure oftheC 6 H 6 /cc-pVDZsystematitsequilibriumgeometryischild'splay,duetotheseemingly absentstrongcorrelations.Although,ingeneral,strongmany-electroncorrelationare almostinvariablyassociatedwithnon-dynamicalcorrelations,theminimum-energystructure ofC 6 H 6 ischaracterizedbystrongdynamicalcorrelationThiscanbeillustratedby examiningtheconvergenceofthehierarchyoftraditionalCCapproaches,includingCCSD, CCSDT,andCCSDTQ.Forexample,theincorporationofthree-bodyclusters,whichpre- dominantlyrecoverdynamicalcorrelations,ontopofthebasicCCSDapproachresultsin aloweringoftheCCSDcorrelationenergyby 36 : 45mE h [266]or 1 : 215mE h perelec- tron.Forthesakeofcomparison,the E (CCSDT) 0 E (CCSD) 0 ) =N correlationenergy perelectronfortheH 6 /cc-pVDZsystem,whichisthesimplest abinitio model ofthe ˇ -electronnetworkofC 6 H 6 /cc-pVDZ,isonly 0 : 5368mE h .Theofconnected quadruplesinthecaseofC 6 H 6 /cc-pVDZisfarfrombeingnegligible,asisevidentbythe factthattheCCSDTQcorrelationenergyof E (CCSDTQ) 0 = 862 : 37mE h islowerthan theCCSDToneby 2 : 47mE h [266,317]. Asalreadyalludedtoabove,thesituationisfurthercomplicatedbythesizeoftheproblem ofdistributingthe30valenceelectronsofC 6 H 6 tothe108correlatedorbitalsarisingfromthe cc-pVDZbasis.Toputitintoperspective,thedimensionofthemany-electronHilbertspace forC 6 H 6 /cc-pVDZisonlyafewordersofmagnitudesmallerthanthestatespaceofchess, i.e. ,thenumberofallpossiblelegalpositionsofchessmen,asestimatedbyShannon[318]. Consequently,thenumbersofcorrelatedelectronsandcorrelatedorbitalsrenderhigher-level SRCCapproaches,suchastheCCapproachwithsingles,doubles,triples,quadruples,and pentuples(CCSDTQP; T (CCSDTQP) = T 1 + T 2 + T 3 + T 4 + T 5 )[319],prohibitivelyexpensive 97 andthesameistrueformanyotherconventionalquantumchemistryapproaches. Theabovediscussionshowcasesthefactthatthenear-exactdescriptionofelectroncor- relationsintheC 6 H 6 /cc-pVDZsystemconstitutesagreatavenuefortestingthepowerof CAD-FCIQMC.Sofar,inperformingCAD-FCIQMCcomputations,wereliedoninformation extractedfrom i -FCIQMCpropagations.However,inthecaseofthemedium-sizedC 6 H 6 moleculethatisdominatedbystrongdynamicalcorrelation i -FCIQMCisanticipated tofacetduetothebiasingintroducedbytheinitiatorapproximation. Thisbecomesapparentoncewerealizethattheproperandaccuratedescriptionofdynam- icalcorrelationsrequiresthestochasticwavefunctionsamplingtocaptureavastnumberof excitedSlaterdeterminantsallofwhichareinhabitedbysmallwalkerpopulations.The undersamplingofthemany-electronHilbertspaceinducedbytheinitiatoralgorithmusually manifestsitselfinaveryslowconvergenceof i -FCIQMCwithrespecttothetotalwalker population,sometimessoslowthatitisessentiallyimpossibletoapproachtheexact,FCI, limit[310].Asamatteroffact,inthecaseofC 6 H 6 ,theconverged i -FCIQMC/cc-pVDZpro- jectedenergyliesbetweentheCCSDandCCSDTresults,beingmorethan20mE h higher thantheCCSDTQcorrelationenergy[310].Takingintoaccountthatintheabsenceof strongnon-dynamicalcorrelations,asinthecaseofC 6 H 6 initsequilibriumgeometry,CCS- DTQprovidesenergeticsthatareclosetotheircorrespondingFCIvalues,theconventional i -FCIQMCmethodologyfailstoconvergetotheexact,FCI,limitforthisseeminglydocile system.Thisclearlyindicatesthatthewavefunctioninformationprovidedby i -FCIQMCis unsuitableforinitializingCAD-FCIQMCcomputationsforC 6 H 6 . Onepossibilitytoaccountforthesystematicerrorassociatedwiththeinitiatorapproach istoemploya aposteriori energycorrectionbasedonsecond-orderperturbationtheory.In thecaseofC 6 H 6 /cc-pVDZ, i -FCIQMCaugmentedwiththeaforementionedperturbative correctiontlybridgesthegapbetween i -FCIQMCandCCSDTQ,asevidentby theconvergedcorrelationenergyof 860 : 7 0 : 7mE h [310].However,thisresultisstill higherthantheCCSDTQonebyatleast1mE h .Takingintoconsiderationthenumer- 98 icalobservationsthat,intheabsenceofstrongnon-dynamicalcorrelations,thehigh-level CCapproaches,althoughnotvariational,convergetoFCIfromabove,theperformanceof theperturbation{theory-corrected i -FCIQMCmethodologyisstillnotsatisfactory.Inad- dition,andmoreimportantlyforthecontextofthiswork,thecorrespondingwavefunction informationisthatobtainedintheunderlying i -FCIQMCsimulations,which,asalready emphasizedabove,constitutesapoorstartingpointforperformingCAD-FCIQMCcalcu- lations.TocircumventalloftheseinourCAD-FCIQMCcomputationsforthe C 6 H 6 /cc-pVDZsystem,wereliedonwavefunctionsresultingfromtherecentlyproposed AS-FCIQMCmethodologyofAlaviandco-workers[263].Asmentionedearlierinthisdis- sertation,theAS-FCIQMCmethodologyamelioratesthebiasintroducedbytheinitiator approximationina apriori manner,namely,bymakingsuitablemototheini- tiatoralgorithm.Asaresult,evenwitharelativelymodestwalkerpopulationof10 8 ,the AS-FCIQMCmethodologyproducedthecorrelationenergyforC 6 H 6 thatwaslowerthan thatofCCSDTQbyabout1mE h (cf.supportinginformationtoReference[266]),whichis alreadyveryencouraging. Asalreadymentioned,inthecomputationsfortheC 6 H 6 moleculeweemployedthecc- pVDZbasisset.Theequilibriumgeometryofbenzene,optimizedattheMP2/6-31G*level oftheory,wastakenfromReference[320].TheunderlyingAS-FCIQMCpropagationswere performedbyAlaviandGhanem[263]usingtheNECIcode[321,322].Thevariousdetails oftheAS-FCIQMCsimulationscanbefoundinthesupportinginformationtoReference [266].Here,ittosaythattheCAD-FCIQMCcalculationsreliedoninformationex- tractedfromtwoinstantaneousAS-FCIQMCwavefunctions:thewavefunctionobtainedat thelastMCiterationofanAS-FCIQMCsimulationwith1billion(1B)walkers,designated as (AS-FCIQMC) 1B ˛ ,andthewavefunctionobtainedattheendofaAS-FCIQMCpropagation with2billion(2B)walkers,abbreviatedas (AS-FCIQMC) 2B ˛ .Totestthenumericalstability ofourhighest-levelCAD-FCIQMCresultsusing2Bwalkers,weperformedanadditional CAD-FCIQMCcomputationinwhichwereplacedtheinstantaneous (AS-FCIQMC) 2B ˛ wave- 99 functionbythestateobtainedbyaveragingthelast100timestepsoftheaforementioned AS-FCIQMCsimulationwith2billionwalkers,denotedas (AS-FCIQMC) 2B (100-avg) ˛ .The clusteranalysisoftheAS-FCIQMCwavefunctionandthesubsequentCCSD-likecomputa- tionswereperformedwithcodesofthePiecuchgroup.Thesamesetofone-andtwo-body molecularintegrals,extractedfromtheMOLPROprogram[323,324],wasemployedforboth theAS-FCIQMCsimulationsandsubsequentCCSD-likecomputations. TheresultsofourCAD-FCIQMC[1{5]andCAD-FCIQMC[1,(3+4)/2]computationsfor theC 6 H 6 /cc-pVDZsystemaresummarizedinTable4.11.Thevariousacronymsshownin Table4.11areaugmentedbyeithera(1B)ora(2B)toindicatethetotalwalker populationusedinthepertinentAS-FCIQMCpropagation.Forthesakeofcomparison, wealsoincludethecorrelationenergiesobtainedfromtheunderlyingAS-FCIQMCsimu- lationsalongwiththeircorrespondingerrorbars.Thesewereobtainedbydiscardingthe datapointscorrespondingtothewalkergrowthandequilibrationperiodsandperforming blockinganalysis[325]ontherest(seethesupportinginformationtoReference[266]forthe details).Inaddition,inTable4.11,wealsoreporttheprojectiveenergycorrespondingto eachinstantaneousAS-FCIQMCwavefunction,designatedasCAD-FCIQMC-ext,whichis computedusingEquation(3.5)withtheinformationabouttheone-andtwo-bodyclusters extractedfromAS-FCIQMC, T (MC) 1 ( ˝ )and T (MC) 2 ( ˝ ),respectively.Thencebetween theCAD-FCIQMC-extandCAD-FCIQMCcorrelationenergiesprovidesavaluablediagnos- ticofthequalityoftheunderlyinginstantaneousAS-FCIQMCwavefunction,especiallyof its C (MC) n ( ˝ )with n =1{4components.Indeed,iftheinitialCAD-FCIQMC-extcorrelation energysigtlyfromtheCAD-FCIQMCresult, i.e. ,ifthe T (MC) 1 ( ˝ )and T (MC) 2 ( ˝ )amplitudesextractedfromAS-FCIQMCsubstantiallyrelaxduringtheprocessof solvingtheCCSD-likeEquations(3.29)and(3.40)inthepresenceoftheir T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ )counterparts,wecanconcludethattheAS-FCIQMCwavefunctionisnotcon- vergedyet.ThecorrelationenergiesreportedinthelasttworowsofTable4.11,designated asCAD-FCIQMC-ext(2B,100-avg)andCAD-FCIQMC[1{5](2B,100-avg),correspondtothe 100 (AS-FCIQMC) 2B (100-avg) ˛ wavefunctionthatwasobtainedbyaveragingthelast100time stepsoftheAS-FCIQMCsimulationwith2billionwalkers. WebeginourdiscussionbyfocusingontheresultsarisingfromtheAS-FCIQMCcal- culationwithatotalwalkerpopulationof1billion,startingwithAS-FCIQMC(1B)itself. TheAS-FCIQMC(1B)correlationenergyof 864 : 8 0 : 5mE h islowerthantheCCSDTQ onebyatleast2mE h .Atthesametime,theCAD-FCIQMC-extcorrelationenergyis 867 : 010mE h , i.e. ,about2mE h lowerthantheaforementionedAS-FCIQMC(1B)value. ThisobservationsuggeststhattheinstantaneousAS-FCIQMC(1B)wavefunctionisnotwell converged.Indeed,lettingthe T 1 and T 2 clustersrelaxinthepresenceoftheir T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ )counterpartsextractedfromAS-FCIQMC(1B)increasesthecorrelationenergyby about3mE h ,asevidentbytheCAD-FCIQMC[1{5](1B)andCAD-FCIQMC[1,(3+4)/2](1B) results.Furthermore,bothvorsofCAD-FCIQMCprovidecorrelationenergiesthatare abovetheupperlimitof 864 : 3mE h resultingfromtheunderlyingAS-FCIQMC(1B)com- putationsbyafractionofamillihartree.ThispotentiallysuggeststhatAS-FCIQMC(1B) slightlyoverestimatesthecorrelationenergyoftheC 6 H 6 /cc-pVDZsystem.Onasidenote, theCAD-FCIQMC[1{5](1B)andCAD-FCIQMC[1,(3+4)/2](1B)correlationenergiesarein excellentagreementwitheachother,asmighthavebeenanticipatedduetotheabsenceof strongnon-dynamicalcorrelation WenowproceedtotheresultsarisingfromthelargestAS-FCIQMCsimulationcon- sidered,namely,theoneusingatotalwalkerpopulationof2billion.Asimpleinspec- tionofTable4.11revealsthattheAS-FCIQMC(2B)correlationenergyof 863 : 7 0 : 3 mE h ishigherthanits1billioncounterpartbyabout1mE h .Infact,thereisanex- cellentagreementbetweentheAS-FCIQMC(2B)valueandtheCAD-FCIQMC[1{5](1B) andCAD-FCIQMC[1,(3+4)/2](1B)data.Thisshouldbecontrastedwiththefactthat theAS-FCIQMC(1B)valueisoutsidethe 0 : 3mE h errorbarsofthemoreaccurateAS- FCIQMC(2B)result.Theseobservationsfurthershowcasethespectacularperformanceof CAD-FCIQMCinacceleratingconvergencetowardtheexact,FCI,energetics.Movingon 101 toourhighest-levelCAD-FCIQMCresultsusing2billionwalkers,wenoticethattheCAD- FCIQMC-ext(2B)correlationenergyisinexcellentagreementwithitsAS-FCIQMC(2B) counterpart,suggestingthattheinstantaneous (AS-FCIQMC) 2B ˛ wavefunction,atleastits C (MC) 1 and C (MC) 2 components,arewellconverged.Thisisfurthercorroboratedbythe factthattheCAD-FCIQMC-ext(2B)resultcanhardlybedistinguishedfromboththe CAD-FCIQMC[1{5](2B)andCAD-FCIQMC[1,(3+4)/2](2B)energetics.Indeed,theCAD- FCIQMC-ext(2B),CAD-FCIQMC[1{5](2B),andCAD-FCIQMC[1,(3+4)/2](2B)correlation energiesagreewithoneanothertowithinabout0.02mE h .Thisclearlydemonstratesthatthe T (MC) 1 ( ˝ )and T (MC) 2 ( ˝ )clustersextractedfromtheinstantaneous (AS-FCIQMC) 2B ˛ wave- functionhardlyrelaxduringtheprocessofsolvingtheCCSD-likesystemofequations,Equa- tions(3.29)and(3.40),inthepresenceoftheir T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ )counterparts.In thespiritofexternallycorrectedCCapproaches,theslightrelaxationoftheone-andtwo- bodyclustersinthepresenceoftheirthree-andfour-bodycounterpartsextractedfromthe AS-FCIQMC(2B)wavefunctionimpliesthat T (MC) 3 ( ˝ )and T (MC) 4 ( ˝ )areofFCIquality. Asatestofthenumericalstabilityofourhighest-levelCAD-FCIQMCresultsus- ing2billionwalkers,weexaminetheCAD-FCIQMCcorrelationenergiesarisingfrompro- cessingthe (AS-FCIQMC) 2B (100-avg) ˛ wavefunction.Weseethatthereplacementofthe instantaneous (AS-FCIQMC) 2B ˛ wavefunctionbyits (AS-FCIQMC) 2B (100-avg) ˛ counterpart, obtainedbyaveragingthelast100timestepsoftheAS-FCIQMCsimulationwith2billion walkers,hasaminorontheCAD-FCIQMC-extandCAD-FCIQMC[1{5]correlation energies.Indeed,theCAD-FCIQMC-ext(2B,100-avg)andCAD-FCIQMC[1{5](2B,100-avg) datafromtheircounterpartsusingtheinstantaneous (AS-FCIQMC) 2B ˛ wavefunction byabout0.01mE h .TheremarkableconsistencybetweentheCAD-FCIQMCenergetics using2billionwalkersgivesusfurtherinourresults.Itisalsoworthmention- ingthattheCAD-FCIQMC[1{5](1B)andCAD-FCIQMC[1,(3+4)/2](1B)energetics fromtheircounterpartsusing2billionwalkersbyonlyabout0.5mE h .Thefactthatallof ourCAD-FCIQMCresultsreportedinthisdissertationarewithinthe 0 : 3mE h errorbars 102 characterizingthelargestAS-FCIQMCsimulationusing2billionwalkersisalsoveryreas- suring.Basedonourhighest-levelCAD-FCIQMC[1{5](2B),CAD-FCIQMC[1,(3+4)/2](2B), andCAD-FCIQMC[1{5](2B,100-avg)results,weestimatetheexact,FCI,correlationenergy oftheC 6 H 6 /cc-pVDZsystematitsminimum-energystructuretobe 863 : 4mE h . Atthispoint,itisworthmentioningthatthecalculationsfortheC 6 H 6 /cc-pVDZsystem reportedinthisdissertationformedpartofablindchallenge,reportedinReference[266], aimedatdeterminingitsfrozen-coreFCIenergy.InadditiontotheaforementionedAS- FCIQMCandCAD-FCIQMCapproaches,thevariousmethodologiesthatwereevaluatedin theblindchallengeincludedadaptivesamplingCI(ASCI)[326{329],semi-stochasticheat- bathCI(SHCI)[330{336],iterativeCIwithselection(iCI)[337{340],DMRG[198{201,341{ 348],many-bodyexpandedFCI(MBE-FCI)[317,349{351],andfullCCreduction(FCCR) [352].Oneofthemajorofthatinvestigationwasthat,withtheexceptionofthe ASCI,iCI,andSHCIresults,allcorrelationenergiesagreedwithoneanothertowithin0.9 mE h ,rangingfrom 863 : 7mE h to 862 : 8mE h .Itisalsointerestingtonotethatour CAD-FCIQMCresultliesmoreorlessinthemiddleoftheaforementionedinterval.Based ontheoftheblindchallenge,thefrozen-coreFCIcorrelationenergyoftheC 6 H 6 speciesatitsequilibriumgeometryasdescribedbythecc-pVDZbasissetisestimatedtobe intheneighborhoodof 863mE h . 103 Table4.8:Convergenceoftheenergiesresultingfromtheall-electron i -FCIQMC,CAD- FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ =0 : 0001a.u.toward FCIfortheH 2 Omolecule,asdescribedbythecc-pVDZbasisset,attheequilibriumgeom- etry R e andthegeometryobtainedbyasimultaneousstretchingofbothO{Hbondsbya factorof2.0. a The i -FCIQMCcalculationswereinitiatedbyplacing1,000walkersonthe RHFdeterminantandthe n a parameteroftheinitiatoralgorithmwassetat3. CAD-FCIQMC Iterations(I)%T b %Q c %FCI d f I e [1{5][1,(3+4)/2] i -FCIQMC R O{H = R e 0000 : 000 : 093 : 744 f 0 : 731 g 217 : 821 h 20,00011 : 81 : 380 : 013 : 95 0 : 073 0 : 0191 : 596 40,00016 : 42 : 280 : 016 : 52 0 : 211 0 : 126 2 : 217 60,00021 : 33 : 480 : 0210 : 3 0 : 046 0 : 2021 : 911 80,00026 : 55 : 120 : 0216 : 10 : 1890 : 210 0 : 686 100,00032 : 17 : 470 : 0425 : 7 0 : 036 0 : 0320 : 139 120,00038 : 010 : 30 : 0540 : 0 0 : 035 0 : 0610 : 597 140,00044 : 214 : 00 : 0863 : 50 : 0980 : 0870 : 080 160,00050 : 418 : 30 : 121000 : 0780 : 114 0 : 400 1 100100100| 76 : 241860 i R O{H =2 : 0 R e 0000 : 000 : 0122 : 032 f 3 : 041 g 363 : 954 h 20,00010 : 61 : 340 : 010 : 558 : 4855 : 65672 : 650 40,00016 : 62 : 720 : 021 : 290 : 138 0 : 23244 : 627 60,00024 : 55 : 030 : 032 : 85 0 : 2251 : 02619 : 660 80,00033 : 28 : 460 : 066 : 01 0 : 4250 : 81212 : 611 100,00042 : 513 : 20 : 1212 : 4 0 : 8161 : 1255 : 680 120,00051 : 719 : 20 : 2025 : 1 0 : 5550 : 5344 : 041 140,00060 : 926 : 60 : 3550 : 3 0 : 6660 : 4981 : 981 160,00069 : 534 : 90 : 57100 0 : 4340 : 3981 : 328 1 100100100| 75 : 951665 i a Theequilibriumgeometry, R O{H = R e ,andthegeometryobtainedbyasimultaneousstretchingofboth O{Hbondsbyafactorof2.0withoutchangingthe \ (H-O-H)angle, R O{H =2 : 0 R e ,weretakenfrom [277]andallelectronswerecorrelated.Unlessotherwisestated,allenergiesareerrorsrelativetoFCIin millihartree. b Percentagesoftriplyexciteddeterminantscapturedduringthe i -FCIQMCpropagations. c Percentagesofquadruplyexciteddeterminantscapturedduringthe i -FCIQMCpropagations. d Percentagesofalldeterminantsspanningtheentiremany-electronHilbertspacecapturedduringthe i - FCIQMCpropagations. e Walkerpopulationscharacterizingthe i -FCIQMCpropagationsreportedaspercentagesofthetotalwalker numbersat I = I max =160 ; 000,whichinthesp i -FCIQMCrunsreportedinthistableandFigure 4.4were1,169,396at R O{H = R e and10,146,724at R O{H =2 : 0 R e . f EquivalenttoCCSD. g EquivalenttoACCSD(1 ; 3+4 2 ). h EquivalenttoRHF. i TotalFCIenergyinhartreetakenfrom[277]. 104 Table4.9:Convergenceoftheenergiesresultingfromthe i -FCIQMC,CAD-FCIQMC[1{ 5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ =0 : 0001a.u.towardFCIforthe symmetricdissociationoftheH 6 ring,asdescribedbythecc-pVDZbasisset,attworep- resentativevaluesofthedistancebetweenneighboringHatoms,including R H{H =1 : 0 A (theregionoftheminimumontheFCIPECshowninFigure4.5(a)characterizedbyweaker correlations)and R H{H =2 : 0 A(theregioncharacterizedbystrongcorrelationsinvolving theentanglementofallsixelectrons). a The i -FCIQMCcalculationswereinitiatedbyplacing 1,500walkersontheRHFdeterminantandthe n a parameteroftheinitiatoralgorithmwas setat3. CAD-FCIQMC Iterations(I)%T b %Q c %FCI d f I e [1{5][1,(3+4)/2] i -FCIQMC R H{H =1 : 0 A 0000 : 000 : 583 : 328 f 1 : 256 g 138 : 122 h 20,00026 : 44 : 270 : 8413 : 30 : 2350 : 0476 : 029 40,00031 : 75 : 771 : 1118 : 50 : 2760 : 2430 : 598 60,00037 : 47 : 671 : 4425 : 0 0 : 124 0 : 096 0 : 225 80,00042 : 89 : 901 : 8633 : 5 0 : 0050 : 047 1 : 680 100,00047 : 712 : 42 : 2943 : 6 0 : 129 0 : 1460 : 511 120,00053 : 615 : 82 : 9358 : 4 0 : 022 0 : 050 1 : 338 140,00059 : 319 : 03 : 6175 : 8 0 : 107 0 : 1781 : 500 160,00064 : 023 : 34 : 501000 : 0850 : 129 0 : 347 1 100100100| 3 : 387731 i R H{H =2 : 0 A 0000 : 000 : 12 35 : 048 f 8 : 818 g 252 : 159 h 20,00023 : 93 : 810 : 832 : 82 34 : 2637 : 149119 : 438 40,00032 : 66 : 901 : 515 : 63 32 : 8842 : 90670 : 886 60,00040 : 710 : 42 : 339 : 60 27 : 8070 : 59643 : 687 80,00049 : 214 : 83 : 4115 : 9 17 : 0900 : 76226 : 814 100,00057 : 420 : 04 : 7725 : 1 11 : 8150 : 01916 : 263 120,00065 : 226 : 46 : 6140 : 1 9 : 077 0 : 40010 : 560 140,00072 : 333 : 48 : 8363 : 3 5 : 457 0 : 1146 : 543 160,00078 : 541 : 411 : 6100 2 : 4010 : 4194 : 015 1 100100100| 3 : 062318 i a Unlessotherwisestated,allenergiesareerrorsrelativetoFCIinmillihartree. b Percentagesoftriplyexciteddeterminantscapturedduringthe i -FCIQMCpropagations. c Percentagesofquadruplyexciteddeterminantscapturedduringthe i -FCIQMCpropagations. d Percentagesofalldeterminantsspanningtheentiremany-electronHilbertspacecapturedduringthe i - FCIQMCpropagations. e Walkerpopulationscharacterizingthe i -FCIQMCpropagationsreportedaspercentagesofthetotalwalker numbersat I = I max =160 ; 000,whichinthespc i -FCIQMCrunsreportedinthistableandFigures 4.6(a)and4.6(b)were257,301at R H{H =1 : 0 Aand1,271,883at R H{H =2 : 0 A. f EquivalenttoCCSD. g EquivalenttoACCSD(1 ; 3+4 2 ). h EquivalenttoRHF. i TotalFCIenergyinhartree. 105 Table4.10:SameasTable4.9forthesymmetricdissociationoftheH 10 ring,asdescribed bytheDZbasisset.InanalogytotheH 6 ring, R H{H =1 : 0 Acorrespondstotheregionof theminimumontheFCIPECshowninFigure4.5(b)characterizedbyweakercorrelations, whereas R H{H =2 : 0 Aistheregioncharacterizedbystrongcorrelationsinvolvingthe entanglementofalltenelectrons. CAD-FCIQMC Iterations(I)%T%Q%FCI f I a [1{5][1,(3+4)/2] i -FCIQMC R H{H =1 : 0 A 0000 : 000 : 144 : 8782 : 265162 : 374 20,00034 : 36 : 140 : 087 : 072 : 0831 : 52514 : 503 40,00042 : 09 : 120 : 1211 : 30 : 7740 : 7853 : 721 60,00049 : 212 : 60 : 1816 : 60 : 3670 : 457 0 : 260 80,00055 : 916 : 60 : 2523 : 60 : 0870 : 0151 : 991 100,00061 : 821 : 40 : 3433 : 30 : 3430 : 3821 : 096 120,00068 : 827 : 80 : 4849 : 1 0 : 0920 : 022 0 : 083 140,00074 : 734 : 70 : 6670 : 40 : 2310 : 285 0 : 388 160,00079 : 342 : 20 : 911000 : 0540 : 0511 : 394 1 100100100| 5 : 538852 R H{H =2 : 0 A 0000 : 000 : 00NC b 29 : 172454 : 768 20,00037 : 98 : 920 : 150 : 12NC b 23 : 004253 : 265 40,00055 : 121 : 70 : 680 : 56NC b 9 : 377163 : 979 60,00069 : 836 : 41 : 901 : 77NC b 3 : 77998 : 996 80,00080 : 751 : 74 : 114 : 46NC b 3 : 61263 : 082 100,00088 : 665 : 77 : 549 : 95NC b 3 : 56537 : 756 120,00093 : 577 : 412 : 521 : 4NC b 3 : 59221 : 909 140,00096 : 586 : 019 : 145 : 8NC b 1 : 70411 : 943 160,00098 : 191 : 827 : 3100NC b 1 : 8297 : 187 1 100100100| 5 : 079254 a Walkerpopulationscharacterizingthe i -FCIQMCpropagationsreportedaspercentagesofthetotalwalker numbersat I = I max =160 ; 000,whichinthespc i -FCIQMCrunsreportedinthistableandFigures 4.6(c)and4.6(d)were1,093,428at R H{H =1 : 0 Aand133,246,948at R H{H =2 : 0 A. b NCindicatesthatnoconvergencewasobtainedwhenthesolutionoftheCCequationsthedeter- ministicpartoftheCAD-FCIQMC[1{5]procedure[Equations(3.29)and(3.40),inwhich ˘ i , i =1{5,are allsetat1]wascarefullycontinuedfromtheweaklycorrelatedregion. 106 Table4.11:ResultsoftheCAD-FCIQMCcalculationsbasedontheAS-FCIQMCwavefunc- tionsobtainedafterequilibrationrunsusing1billion(1B)and2billion(2B)walkers. Calculation E /mE h 1BillionWalkers AS-FCIQMC(1B) 864 : 8 0 : 5 CAD-FCIQMC-ext(1B) 867 : 010 CAD-FCIQMC[1{5](1B) 864 : 089 CAD-FCIQMC[1,(3+4)/2](1B) 863 : 861 2BillionWalkers AS-FCIQMC(2B) 863 : 7 0 : 3 CAD-FCIQMC-ext(2B) 863 : 464 CAD-FCIQMC[1{5](2B) 863 : 453 CAD-FCIQMC[1,(3+4)/2](2B) 863 : 438 CAD-FCIQMC-ext(2B,100-avg) 863 : 460 CAD-FCIQMC[1{5](2B,100-avg) 863 : 439 107 Figure4.4:Convergenceoftheenergiesresultingfromtheall-electron i -FCIQMC,CAD- FCIQMC[1{5],andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ =0 : 0001a.u.toward FCIfortheH 2 Omolecule,asdescribedbythecc-pVDZbasisset,at(a)theequilibrium geometryand(b)thegeometryobtainedbyasimultaneousstretchingofbothO{Hbondsby afactorof2withoutchangingthe \ (H-O-H)angle(bothgeometriesweretakenfrom[277]). The i -FCIQMCcalculationswereinitiatedbyplacing1,000walkersontheRHFdeterminant andthe n a parameteroftheinitiatoralgorithmwassetat3.Allenergiesareerrorsrelative toFCIinmillihartree,andtheinsetsshowthepercentagesoftriply(%T)andquadruply (%Q)exciteddeterminantscapturedduringthe i -FCIQMCpropagations. Figure4.5:Ground-statePECsforthesymmetricdissociationofthe(a)H 6 /cc-pVDZand (b)H 10 /DZsystemsresultingfromtheCCSD,CCSDT,andFCIcalculations. 108 Figure4.6:Convergenceoftheenergiesresultingfromthe i -FCIQMC,CAD-FCIQMC[1{5], andCAD-FCIQMC[1,(3+4)/2]calculationswith ˝ =0 : 0001a.u.towardFCIforthesym- metricdissociationoftheH 6 /cc-pVDZ[panels(a)and(b)]andH 10 /DZ[panels(c)and(d)] systemsattworepresentativevaluesofthedistancebetweenneighboringHatoms,including R H{H =1 : 0 A[panels(a)and(c)]and R H{H =2 : 0 A[panels(b)and(d)].The i -FCIQMC calculationswereinitiatedbyplacing1,500walkersontheRHFdeterminantandthe n a parameteroftheinitiatoralgorithmwassetat3.AllenergiesareerrorsrelativetoFCIin millihartree,andtheinsetsshowthepercentagesoftriply(%T)andquadruply(%Q)excited determinantscapturedduringthe i -FCIQMCpropagations.TheCAD-FCIQMC[1{5]curve isabsentinpanel(d),sincethesolutionoftheCCequationsthedeterministicpart oftheCAD-FCIQMC[1{5]procedurefortheH 10 /DZsystemcouldnotbecontinuedbeyond R H{H =1 : 75 A. 109 Figure4.7:Acomparisonoftheenergiesresultingfromthe i -FCIQMC,CAD-FCIQMC[1{5], andCAD-FCIQMC[1,(3+4)/2]calculationsat50,000[panels(a)and(d)],100,000[panels (b)and(e)],and150,000[panels(c)and(f)]FCIQMCiterationsusingtimestep ˝ =0 : 0001 a.u.,alongwiththecorrespondingfullydeterministicCCSD,CCSDT,andACCSD(1 ; 3 n o n o + n u +4 n u n o + n u )data,forthesymmetricdissociationoftheH 6 /cc-pVDZ[panels(a){(c)] andH 10 /DZ[panels(d){(f)]systemsatselecteddistancesbetweenneighboringHatoms, R H{H ,rangingfromtheweaklycorrelated(smaller R H{H )tothestronglycorrelated(larger R H{H )regions.The i -FCIQMCcalculationswereinitiatedbyplacing1,500walkersonthe RHFdeterminantandthe n a parameteroftheinitiatoralgorithmwassetat3.Allenergies areerrorsrelativetoFCIinmillihartree.Theinsetsshowtheentirerangeoferrorsrelative toFCI. 110 CHAPTER5 CONCLUDINGREMARKSANDFUTUREOUTLOOK Inthisdissertation,wepresentedrecentadvancesinthedevelopmentofSRCCapproaches forstronglycorrelatedsystems,whichlieattheheartofcontemporaryquantumchemistry. Afterintroducingtheconceptofstrongcorrelation,wearguedthatthesheerdimensionalities oftheunderlyingmodelspacesrenderMRschemesinapplicabletothestrongcorrelation regime,emphasizingtheneedforrobust,yetcomputationallysingle-reference quantumchemistrymethodologies.Inthisdissertation,wefocusedonSRCCapproaches, which,overtheyears,havebeenestablishedasthe defacto standardforhigh-accuracy electronicstructurecalculations,eveninthepresenceofquasi-degeneraciessuchasthose characterizingsingleanddoublebonddissociations.Wepresentedvariousnumericalexam- plesfromtheliteraturethatdemonstratethatthehierarchyoftraditionalSRCCapproaches, includingCCSD,CCSDT,CCSDTQ, etc. ,completelybreaksdowninthestrongcorrelation regimeofmodelHamiltonians.InspiredbytheworkofScuseriaandco-workers[209],we provedanalyticallythatinthefullycorrelatedlimitofthe12-siteattractivepair- ingHamiltoniantheimportanceofthemany-bodycomponentsoftheclusteroperatoris reversed, i.e. , T 2 = T 4