MERGINGACTIVE-SPACEANDRENORMALIZEDCOUPLED-CLUSTERMETHODSVIATHECC(P;Q)FORMALISM,WITHAPPLICATIONSTOCHEMICALREACTIONPROFILESANDSINGLET{TRIPLETGAPSByNicholasP.BaumanADISSERTATIONSubmittedtoMichiganStateUniversityinpartialtoftherequirementsforthedegreeofChemistry-DoctorofPhilosophy2016ABSTRACTMERGINGACTIVE-SPACEANDRENORMALIZEDCOUPLED-CLUSTERMETHODSVIATHECC(P;Q)FORMALISM,WITHAPPLICATIONSTOCHEMICALREACTIONPROFILESANDSINGLET{TRIPLETGAPSByNicholasP.BaumanThedevelopmentofaccurateandcomputationallytwavefunctionmethodsthatcancaptureandbalancedynamicalandnon-dynamicalmany-electroncorrelationtodescribemulti-referenceproblems,suchaspotentialenergysurfacesinvolvingbondbreaking,biradicals,andexcitedstatescharacterizedbydominantmany-electronexcitations,isoneofthemaingoalsofquantumchemistry.Amongthepromisingapproachesinthisendeavorarethecompletelyrenormalizedandactive-spacecoupled-cluster(CC)andequation-of-motion(EOM)CCmethods.Whilethecompletelyrenormalizedandactive-spaceCCandEOMCCapproacheshavebeenverysuccessfulinmanyapplications,therearesomecaseswheretheydonotcapturethedynamicalornon-dynamicalmany-electroncorrelationinasatisfactorymanner.Inthisdissertation,weintroducetheCC(P;Q)formalism,whichalleviatesthisconcernbycombiningthecompletelyrenormalizedandactive-spacetogether.TheCC(P;Q)schemeprovidesasystematicapproachtocorrectingenergiesobtainedintheactive-spaceCCandEOMCCcalculationsthatrecovermuchofthenon-dynamicalandsomedynamicalmany-electroncorrelationfortheremaining,mostlydynamical,cor-relationmissingintheactive-spaceCCandEOMCCconsiderations.WediscussthedevelopmentoftheCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)methods,whichusetheCC(P,Q)formalismtocorrectenergiesobtainedwiththeCCandEOMCCapproacheswithsingles,doubles,andactive-spacetriples(CCSDt/EOMCCSDt)formissingtripleexcita-tions(CC(t;3)),ortocorrectenergiesobtainedwiththeCCandEOMCCapproacheswithsingles,doubles,andactive-spacetriplesandquadruples(CCSDtq/EOMCCSDtq)formiss-ingtriples(CC(t,q;3))ormissingtriplesandquadruples(CC(t,q;3,4)),oreventocorrectenergiesobtainedwiththeCCandEOMCCapproacheswithsingles,doubles,triples,andactive-spacequadruples(CCSDTq/EOMCCSDTq)forcorrelationduetothemiss-ingquadrupleexcitations(CC(q;4)).Byexaminingthedoubledissociationofwater,theBe+H2!HBeHinsertion,andthesinglet{tripletgapsinthestronglybiradical(HFH)systemandtheBNmolecule,wedemonstratethattheCC(t;3),CC(t,q;3),andCC(t,q;3,4)methodsreproducethetotalandrelativeenergiesobtainedwiththeparentfullCC/EOMCCapproacheswithsingles,doubles,andtriplesorsingles,doubles,triples,andquadruplestowithinfractionsofamillihartreeatthetinyfractionofthecomputercost,evenwhentheelectronicquasi-degeneraciesbecomesubstantial.TheCC(P,Q)formulationpromptedthedevelopmentoftCCSDt,CCSDtq,andCCSDTqprograms.Inthisdissertation,wedescribethetechniqueofspin-integrationforbothclosedandopenshells,andhowtheresultingequationsforCCSDTQwereautomaticallyderivedandimplementedinafactorizedform.Wealsodiscusshowtheofthecodewasimprovedbyremovingunnecessaryoperationsthrough,inparticular,thereorganizationoftherelevantloops.Finally,weexplainhowtheCCSDTQcodewastransformedtoobtaintheactive-spaceCCSDtqandCCSDTqapproaches,whicharethemostessentialpartsoftheCC(t,q;3),CC(t,q;3,4),andCC(q;4)calculations.CopyrightbyNICHOLASP.BAUMAN2016ThisisdedicatedtomylovelywifeJennifer.vACKNOWLEDGMENTSIwouldliketothankmyPhD.advisor,ProfessorPiotrPiecuch,forhispatienceandguidancethroughoutmygraduatecareer.Sincemydayinhisgrouphehasdrivenmetobeabetterscientistandshownmemypotential.ThroughmanylatenightsandlongdiscussionshehasprovidedmewithawealthofknowledgethatIneveranticipated.Iamforevergratefulforhiscontinuoussupportandeverythinghehasdoneforme.Iwouldliketothanktheothermembersofmycommittee,ProfessorBenjaminG.Levine,ProfessorRobertI.Cukier,andProfessorMarcosDantusforalltheirsupport,advice,andpatienceinoverseeingmygraduatestudy.IwouldalsoliketothankDr.JunShenfromourgroupwhoIoweagreatdealofgratitudetoo.Hehasnotonlymadethedevelopmentworkpresentedinthisdissertationpossible,buthasprovidedvaluableguidanceandhasalwaysbeenpatientandwillingtoansweranyquestionsImayhavehad.Last,butnotleast,IwouldliketoextendmythankstograduatestudentsfromthePiecuchresearchgroupIhavehadanopportunitytointeractwith,especially,Dr.JaredA.Hansen,Mr.AdeayoO.Ajala,Mr.JorgeEmilianoDeustua,andMr.IliasMagoulas,forthemanythoughtprovokingdiscussionsandallthehelptheyhaveprovidedmeovertheyears.viTABLEOFCONTENTSLISTOFTABLES....................................viiiLISTOFFIGURES...................................xChapter1Introduction................................1Chapter2ProjectObjectives............................18Chapter3MergingActive-SpaceandCompletelyRenormalizedCoupled-ClusterMethods.............................193.1Theory.......................................193.1.1CompletelyRenormalizedCoupled-ClusterApproaches........203.1.2Active-SpaceCoupled-ClusterMethods.................313.1.3TheCC(P;Q)Formalism.........................423.1.4CC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)Hierarchy........503.2NumericalResults.................................613.2.1TheC2v-SymmetricDoubleDissociationofWater..........613.2.2Be+H2!HBeHC2vInsertionPathway...............733.2.3Singlet{TripletGapinHFHAlongtheD1h-SymmetricDouble-DissociationPathway..................................813.2.4Singlet{TripletGapinBN........................100Chapter4AlgorithmicAdvances:tAutomatedImplementationofActive-SpaceCCSDtqandCCSDTqMethods,andTheirFullCCSDTQCounterpart.........................1114.1SpinIntegrationforClosedandOpenShells..................1124.2AutomatedApproachtoDerivationandComputerImplementationofCoupled-ClusterMethodsinFactorizedForm......................1254.3ImprovementsinviaLoopReorganization..............1374.4TransformationfromCCSDTQtotheCCSDtqandCCSDTqMethods...141Chapter5ConclusionsandFutureOutlook...................145BIBLIOGRAPHY...................................148viiLISTOFTABLESTable1.1:DependenceoftheCPUstepsonno,nu,No,andNuforthemostexpensivetermsforvariousCCapproximations.a...........5Table1.2:AcomparisonofvariousCCground-stateenergiesforthereactantandtransition-statespeciestheautomerizationofcyclobu-tadiene,basedonthegeometriesoptimizedintheMR-AQCCcalcu-lationsinRef.[254]aswellaspurelyelectronicbarrierheights(inkcal/mol).a;b...............................9Table3.1:ThecompletesetofprojectionsenteringtheCCSDTQamplitudeequationsorganizedaccordingtotheiractiveandinactivecharacter.Upper-caseboldlettersrepresentactivelabelsandlower-caseboldlettersrepresentinactivelabels(cf.thetextandFig.3.1fordetails).36Table3.2:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounterpartsandthecorre-spondingfullCIdatafortheequilibriumandtwodisplacedgeome-triesoftheH2Omolecule,asdescribedbytheDZbasisset[279]..69Table3.3:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounterpartsandthecor-respondingfullCIdatafortheequilibriumandfourdisplacedge-ometriesoftheH2Omolecule,asdescribedbythesphericalcc-pVDZbasisset[237]..............................71Table3.4:CoordinatesofpointsalongthesamplingpathC2v-symmetricpathdescribinginsertionofBeintoH2,introducedinRef.[283].StructureAisthelinearHBeHproduct,whilestructureIrepresentstheBe+H2reactant.StructureEisthetransitionstate............77Table3.5:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounterparts,andthecor-respondingfullCIdataforthegeometriesA{ItheC2v-symmetricinsertionpathwayofBeintoH2,asdescribedbythe[3s1p/2s]basisset,introducedinRef.[283].a.............78viiiTable3.6:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirpar-entCCSDTandCCSDTQcounterparts,andthecorrespondingfullCIdatafortheX1+gstateofthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atafewvaluesoftheH{FdistanceRH-F(inA).a...............91Table3.7:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirpar-entCCSDTandCCSDTQcounterparts,andthecorrespondingfullCIdatafortheA3+ustateofthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atafewvaluesoftheH{FdistanceRH-F(inA).a...............94Table3.8:AcomparisonoftheA3+uX1+ggapvaluesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounterparts,andthecorrespondingfullCIdataforthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atafewvaluesoftheH{FdistanceRH-F(inA).a..........97Table3.9:Equilibriumbondlengths(inA)forthelowesttripletandsingletstatesofBN,thecorrespondingadiabaticsinglet{tripletsplittingsTe(incm1),andenergiesrelativetoCCSDTandCCSDTQ(incm1),asobtainedwiththecc-pVDZbasisset.a...............109Table3.10:Equilibriumbondlengths(inA)forthelowesttripletandsingletstatesofBN,thecorrespondingadiabaticsinglet{tripletsplittingsTe(incm1),andenergiesrelativetoCCSDTandCCSDTQ(incm1),asobtainedwiththecc-pVTZbasisset.a...............110Table4.1:MatrixelementsforthestandardHamiltonianandamplitudesfortheclusteroperatorsthatappearinCCSDTQ,alongwiththeircor-respondingcasesforspin-integratedequations,andtherespectiveantisymmetrizersforthespincases...................121Table4.2:ExciteddeterminantsenteringintoEq.(3.11)forspin-orbitalimple-mentationsofCCSDTQandthecorrespondingspinvariantsforthespin-integratedimplementations.....................123Table4.3:Averageiterationtime,inseconds,forcalculationsoftheX1+gandA3+ustatesof(HFH),computedusingouroriginalimplementa-tionofCCSDTQ,theimprovedversionofCCSDTQ,andNWCHEMforcomparison.a............................140ixLISTOFFIGURESFigure1.1:Automerizationofcyclobutadiene.Theleftmostandrightmoststruc-turescorrespondtothedegeneratereactant/productminima,whereasthestructureinthemiddlerepresentsthetransitionstate.......10Figure3.1:Theorbitalusedintheactive-spaceSRCCmethods,suchasCCSDt,CCSDtq,andCCSDTq.Core,active,andvirtualorbitalsarerepresentedbysolid,dashed,anddottedlines,respec-tively.Fullandopencirclesrepresentcoreandactiveelectronsofthereferencedeterminantji(aclosed-shellreferencejiisassumed).33Figure3.2:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentfullCCSDTandCCSDTQcounterparts,fortheequilibriumandtwodisplacedgeometriesoftheH2Omolecule,asdescribedbytheDZbasisset[279].ThenumericalvaluesoftheerrorsarefoundinTable3.2.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.Forinterpretationofthereferencestocolorinthisandallotherthereaderisreferredtotheelectronicversionofthisthesis...........................70Figure3.3:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,fortheequilibriumandfourdisplacedgeometriesoftheH2Omolecule,asdescribedbythesphericalcc-pVDZbasisset[237].ThenumericalvaluesoftheerrorsarefoundinTable3.3.Toppanelshowsacom-parisonofCCSDTandCCSDTQwithfullCI.Middlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththepar-entCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.........72xFigure3.4:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculations,includinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,forthegeometriesA{ItheC2v-symmetricinsertionpathwayofBeintoH2,asdescribedbythe[3s1p/2s]basisset,introducedinRef.[283].ThenumericalvaluesoftheerrorsarefoundinTable3.5.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.InpresentingtheresultsforthetransitionstatestructureE,weusedthesamereferencej2iasthatusedintheF{Iregion....80Figure3.5:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,fortheX1+gstateofthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atseveralvaluesoftheH{FdistanceRH-F.ThenumericalvaluesoftheerrorsarefoundinTable3.6.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults...................................93Figure3.6:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,fortheA3+ustateofthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atseveralvaluesoftheH{FdistanceRH-F.ThenumericalvaluesoftheerrorsarefoundinTable3.7.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparedvariousapproximatequadruplesmethodswiththeparentCCSDTQresults...................................96xiFigure3.7:AcomparisonoftheA3+uX1+ggapvaluesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcita-tions,alongwiththeirparentCCSDTandCCSDTQcounterparts,forthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atseveralvaluesoftheH{FdistanceRH-F.ThenumericalvaluesoftheerrorsarefoundinTable3.8:ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults...................................99Figure4.1:Theone-bodyvertexrepresentingtheone-bodycomponentFN=fqpN[apaq]ofHN.............................115Figure4.2:Thetwo-bodyvertexrepresentingthetwo-bodycomponentVN=14vrspqN[apaqasar]ofHN.........................115Figure4.3:Theone-,two-,three-,andfour-bodyverticesrepresentingtheT1,T2,T3,andT4clusteroperators.....................116Figure4.4:Diagramrepresentingoneofthe12vnt22termsthatappearinthepro-jectionoftheCCequationsontodoublyexciteddeterminants....117Figure4.5:SampleoutputoftheLatexcorrespondingtotheHNT3Btermgeneratedbytheautomaticderivationcode.Seetextforadescriptionofthenotation..............................131Figure4.6:Thegeneralstructureofeachbinarytensorproductasproducedbytheautomaticderivationandimplementationprogram........133Figure4.7:Exampleofbinarytensorproductbetweenatwo-bodyHamiltonianoperatorandaT2AclusteroperatorthatproducesandintermediatelabeledS14,whichissummedwithotherintermediatesandHamil-tonianmatrixelementsinacollectiveintermediatelabeledX2....134Figure4.8:Anexampleofoneofthe\REORDER"routines.Thenumberinthetitleoftheroutinetheorderthattheindicesarepermutedto.135Figure4.9:Anexampleofoneofthe\SUM"routines.Thenumberinthetitleoftheroutinetheorderthattheindicesarepermutedto...136xiiFigure4.10:AnexampleoftheproductbetweenanintermediatelabeledU41andtheclusteroperatorT4B,whichisprojectedontotheexciteddeter-minanthabc~dijk~lj.ThearraycorrespondingtotheproductformedislabeledZ140andisaddedtotheresidualnamedV2B.........139Figure4.11:AnexampleoftheproductbetweenanintermediatelabeledX2andtheclusteroperatorT2A,whichisprojectedontotheexciteddeter-minanthabcijkj...............................142Figure4.12:AnexampleoftheproductbetweenanintermediatelabeledS5andtheclusteroperatorT4A,whichisprojectedontotheexciteddeter-minanthabcdijklj..............................143xiiiChapter1IntroductionOneofthemainchallengesinquantumchemistryistheaccuratedescriptionofquasi-degenerateelectronicstatesininstancesinvolvingbiradicals,bondbreaking,andexcitedstatescharacterizedbydominanttwo-electronexcitations.Thesesituationsareachallengeduetothestrongnon-dynamicalmany-electroncorrelationthathavetobeproperlybalancedwithdynamicalcorrelations.Theframeworkofcoupled-cluster(CC)theory[1{6]isidealforhandlingthesetypesofsituationsasitthebestbalancebetweenaccuracy[fastconvergencetotheexact,fullinteraction(CI),limit]andcomputationalcosts.WithintheCCframework,twotypicalroutestakentodescribesituationsinvolvingstrongnon-dynamicalcorrelationare:1)single-reference(SR)CCmethods,and2)multi-reference(MR)CCapproaches.Situationsinvolvingstrongnon-dynamicalcorrela-tions,suchasthosedescribedabove,areoftendescribedasMRproblems,soitwouldseemmostnaturaltoturntoMRmethodstosolvethem.Intheseapproaches,oneintroducesamulti-dimensionalmodelspaceconsistingofanumberofreferencedeterminants,whichareobtainedbydistributingactiveelectronsamongactiveorbitalsinvariouswaysandwhicharechosensuchastoprovideareasonablezeroth-orderdescriptionofthequasi-degenerateelectronicstate(s)ofinterest,asinthecomplete-active-spaceself-consisten(CASSCF)calculations.Onecanthen,forexample,usetheJeziorski-Monkhorstansatz[7]tocap-turetheremainingdynamicalcorrelationthroughexcitationsfromeachreference1determinant.Withinthisframework,onecanformulatethestate-univeral(SU)(see,e.g.,Refs.[7{47])orthestate-sp(SS)(seeRefs.[41,42,46{66]forrepresentativeexamplesandrecentadvances)MRCCmethods.However,thisformulationisjustonepossibledirectionintheMRframework,andunfortunately,unlikeSRCCmethods,thereisnounambiguouswayofwritingtheexponentialwavefunctionansatzforMRapproaches.Despitemorethanthreedecadesofactivedevelopment,MRCCmethodscontinuetofacevariousformalandpracticalchallenges.Forexample,thegenuineSUMRCCapproachesfromtheintruderstateorintrudersolutionproblem,singularities,andmultipleunphysicalstates[12{15,18,67].WhileSSMRCCapproximationsmaynotbebytheseissues,atleastinprinciple,theystillfromconvergenceproblems,particularlyinexcited-stateconsiderationsand,espe-cially,whenoneormorecotsatthereferencedeterminantsbecomesmall[41,66].Intheend,noneoftheexistingMRCCmethodsbasedontheJeziorski-MonkhorstansatzarecharacterizedbyaneaseofuseandapplicationofapproachesasthoseoftheSRCCansatz,andthesameistrueforaplethoraofotherMRCCmethodsthatuseotherformsofthewavefunction(see,e.g.,Refs[22]and[68]forreviews).Forthesereasonsandothers,thisworkfocusesontheapproachesoftheSRCCtype,withtheobjectiveofrecoveringstrongnon-dynamicalcorrelationdynamicallythroughexcitationsfromasinglereferencedeterminant.ThebasicSRCCapproximations,suchasCCwithsinglesanddoubles(CCSD)[69{72],canbeappliedtosystemscontainingdozensofnon-hydrogenatomsorhundredsofcor-relatedelectrons,andhundredsoreventhousandsofbasisfunctions,inpartduetotherelativelyinexpensiveCPUstepsthatscaleasn2on4u,whereno(nu)isthenumberofoccu-pied(unoccupied)orbitals,orasN6,withthesystemsizeN,whichcanbefurtherreducedtolinearscalingstepsvialocalcorrelationapproaches(see,e.g.,Refs.[73{80]andnumerous2referencestherein).AcomparisonofCPUtimescalingsofcanonicalCCSDwithotherCCmethodsmentionedinthisdissertationisshowninTable1.1.WhileCCSDisgenerallymoreaccuratethanitsCIcounterpart(i.e.,CISD),especiallyinlargersystemswherethelackofsize-extensivityofCISDbecomesamajorproblem,ithasseriouswithcapturingnon-dynamicalcorrelationcharacterizingchemicalreactioninvolv-ing,forinstance,bondbreakingandbiradicals,whilemissingimportantdynamicalcorrela-tions,especiallythoseduetoconnectedtriplesneededtoachieveaquantitativedescription.Theexcited-stateequation-of-motion(EOM)analogofCCSD,EOMCCSD[81{83],andthecorrespondingsymmetry-adapted-cluster(SAC)CI[84{87]andlinear-responseCC[88{93]counterpartsarecapableofdescribingexcitedstatesdominatedbyone-electrontransitions,butareoftennotaccurateenoughtoobtainaquantitativedescriptionofsuchstates,es-peciallywhenlargerpolyatomicspeciesareexamined(cf.,e.g.,Refs.[94{97]forselectedexamples;forathoroughevaluationofanumberofEOMCCmethods,includingEOM-CCSD,illustratingthisstatement,seeRefs.[98{104]).Furthermore,EOMCCSDanditsSAC-CIandlinear-responsecounterpartsfailatcharacterizingexcitedstateshavingcanttwo-orothermany-electroncontributions[104{119].Whilethehigher-orderdynamicalandstrongernon-dynamicalcorrelationcannotbecapturedusingtheabovelow-orderCC/EOMCCmethods,theycanberecoveredthroughtheexplicitandcompleteinclusionofhigher-rankcomponentsoftheclusteroperatorT,suchastheconnectedtriplyandquadru-plyexcitedclusters,T3andT4,respectively,intheground-stateSRCCwavefunctionansatzj0i=eTjiand,inthecaseofexcitedstates,throughtheinclusionoftheanalogoushigher-ordercomponentsofthelinearexcitationoperatorR,i.e.,R3andR4,intheEOMCCwavefunctionansatzji=ReTji,where=0designatesthegroundstate,>0labelsexcitedstates,andjiisthereferencedeterminant[inthisdocument,are-3strictedHartree-Fock(RHForROHF)Unfortunately,thefullincorporationofhigher-ordercomponentsofTandR,asintheCCapproachwithsingles,doubles,andtriples(CCSDT)[120,121],theCCmethodwithsingles,doubles,triples,andquadruples(CCSDTQ)[122{125],andtheirexcited-stateEOMCCcounterpartsabbreviatedasEOM-CCSDT[107,108,126{128]andEOMCCSDTQ[126,127,129,130],leadstolarge,oftenpro-hibitive,computationalcosts.Forexample,CCSDThasiterativeCPUoperationsthatscaleasn3on5u(N8)andCCSDTQ,withitsiterativen4on6u(N10)steps,isevenmoreexpensive,limitingtheuseofsuchschemestosystemsofonlyadozenorsocorrelatedelectrons(seeTable1.1).ThisrestrictionhasledtothedevelopmentofvariousmethodsthatapproximatetheofT3,T4,R3,andR4componentsinordertocombatthesesteepCPUtimescalings.Traditionalwaysofestimatingtheduetohigher-than-doublyexcitedcompo-nentsoftheclusteroperatorTandtheEOMCCexcitationoperatorRrelyonmany-bodyperturbationtheory(MBPT).IncludedinthisgrouparetheiterativeCCSDT-n[131{136]andCCSDTQ-n[137],ornon-iterativeCCSD[T][135,138,139],CCSD(T)[140],CCSD(TQf)[141],andsimilarapproaches[135,136,142{146],andtheirperturbativeexcited-stateEOMCC[147{149]andlinearresponseCC[150{153]extensions.WhilereducingthecomputationalcostsofthefullCCSDTandCCSDTQapproximationsandbeingusefulinpractice,theseapproachesstillhaveseriousincapturingnon-dynamicalcorrela-tioncharacterizingchemicalreactioninvolvingbondbreaking,biradicals,andexcitedstateshavingttwo-andothermany-electroncontributions,whichcomprisemanyoftheproblemsweareinterestedin,especiallyinareassuchasreactionmechanismsanddynamicsandphotochemistry.Therefore,morerobust,yetcomputationallyfeasible,approachesmustbeconsidered.4Table1.1:DependenceoftheCPUstepsonno,nu,No,andNuforthemostexpensivetermsforvariousCCapproximations.aCPUTimingScalingsMethodIterativeNon-iterativeCCSDn2on4uCCSDTn3on5uCCSDTQn4on6uCCSD(T)n2on4un3on4uCCSD(TQf)n2on4un3on4u+n2on5uCR-CC(2,3)An2on4un3on4uCR-CC(2,3)Dn2on4un3on4uCR-CC(2,4)AAn2on4un3on4u+n2on5uCR-CC(2,4)DAn2on4un3on4u+n2on5uCR-CC(2,4)DDn2on4un3on4u+n4on5uCR-CC(3,4)An3on5un2on5uCR-CC(3,4)Dn3on5un4on5uCCSDtNoNun2on4uCCSDtqN2oN2un2on4uCCSDTqNoNun3on5uCC(t;3)ANoNun2on4un3on4uCC(t;3)DNoNun2on4un3on4uCC(t,q;3)AN2oN2un2on4un3on4uCC(t,q;3)DN2oN2un2on4un3on4uCC(t,q;3,4)AAN2oN2un2on4un3on4u+n2on5uCC(t,q;3,4)DAN2oN2un2on4un3on4u+n2on5uCC(t,q;3,4)DDN2oN2un2on4un3on4u+n4on5uCC(q;4)ANoNun3on5un2on5uCC(q;4)DNoNun3on5un4on5uaNo(<noor˝no)designatesthenumberofactiveoccupiedorbitalsandNu(˝nu)designatesthenumberofactiveunoccupiedorbitals.Thecompletelyrenormalized(CR)CCandCR-EOMCCschemesandotherapproxi-mationsresultingfromthemoregeneraltheoreticalframeworktermedthemethodofmo-5mentsofCCequations(MMCC)[105,106,111{117,154{165]compriseoneoftheclassesofrobustandpromisingSRCCmethodsthatcanbeusedtosolveproblemswithlargernon-dynamicalcorrelations,suchasthosedescribedabove.Thesemethodsarebasedontheideaofaddinganaposteriori,non-iterative,andstatespcorrection,(A),duetohigher-orderexcitationsneglectedintheconventionalCC/EOMCCcalculationsbysometruncationlevelmA,tothecorrespondingCC/EOMCCenergiesE(A)(AreferstotheconventionalCC/EOMCCapproximationwewanttocorrect,sayCCSDorEOMCCSD).Forexample,CR-CCSD(T)[105,106,154{156]andCR-CC(2,3)[116,160{162]addacorrec-tiontotheCCSD(mA=2)energytoaccountfortheofconnectedtripleexcitations,whileCR-CCSD(TQ)[105,106,154{156]andCR-CC(2,4)[160,161]addacorrectionthatapproximatelyaccountsfortheoftripleandquadrupleexcitationsinanon-iterativemanner.ThesemethodsandtheirCR-EOMCCanalogs[105,106,113,114,116,117]eliminateorconsiderablyreducetheerrorsoftraditionalperturbativeCCschemes,suchasCCSD[T],CCSD(T),CCSD(TQf),etc.,intheregionsofpotentialenergysurfacesinvolvingbondbreaking,biradicals,andelectronicstatescharacterizedbytwo-electronexcitations.TheCR-CC,CR-EOMCC,andotherMMCCmethodsareveryeatcapturingdynamicalcorrelationswhileretainingtheeaseofuseof\black-box"approaches.WhiletheCR-CCandCR-EOMCCapproacheshavedemonstratedconsiderablesuccess(cf.,e.g.,Refs.[94{96,104{106,113,114,116{118,154{162,166{215]),eveninsystemsaslargeasmethylcobalamin[175]oroxygenmigrationonsiliconsurface[216],noneofthemareapplicabletoallMRsituations.AfundamentalproblemisthefactthattheCR-CC/CR-EOMCCapproaches,likeallnon-iterativeCC/EOMCCmethods,relyonasplower-orderaprioriCCorEOMCCcalculation(e.g.,CCSDorEOMCCSD),andthusthereisnonaturalmechanismallowingforthelower-orderclustercomponents,suchasT1andT2,or6theirexcited-stateR1andR2analogs,torelaxinthepresenceofhigher-orderTnandRcomponents,withn3,whichisimportantincasesofstrongernon-dynamicalcorrelations.Thisisdiscussedinmoredetaillaterinthisthesis,soatthistimeweonlyemphasizethatthisdecouplingcanhaveseriousconsequences,especiallywhenexaminingchemicalreactioninvolvingbiradicaltransitionstates,suchas,forexample,thetransitionstatecharacterizingthechallenging,andfrequentlystudied,automerizationofcyclobutadiene[31,65,164,217{232],showninFigure1.1andTable1.2,ortheequallychallengingtransitionstatethedisrotatorypathwayfortheisomerizationofbicyclo[1.1.0]butanetotrans-buta-1,3-diene,whichisoneofthetwolowest-energypathwaysthathavebeenstudiedinrecentyears[164,171,233{236].ExaminingTable1.2,wecanseethatfortheautomerizationofcyclobutadieneCR-CC(2,3)candescribethereactant/productminimawitherrorsoflessthanonemillihartreerelativetofullCCSDT,buthastroubledescribingthetransitionstate,witherrorsof14{15millihartreerelativetoCCSDTwhenthecc-pVDZandcc-pVTZbasissets[237]areused.Thisleadstoanoverestimationofthebarrierheightbyabout9kcal/mol,whichcanbeattributedtotheneglectofcouplingbetweenthelower-orderT1andT2componentsandT3.IfonewantstobeabletorelaxT1andT2orR1andR2clustersinthepresenceofhigher-orderclusteroperators,suchasT3andT4,thenalternativeapproaches,otherthanthenon-iterativecorrectionmethodsdiscussedsofar,mustbeconsidered.AnotherusefulclassofrobustandpromisingSRCC-likemethodsthatmayhandleMRsituationsisthatoftheactive-spaceCCandEOMCCapproaches[107{109,119,125,163{165,238{253],whichretaintherelativelyinexpensivecomputationalcostssimilartothebasic,yetinadequate,CCSDandEOMCCSDapproaches,whileallowingT1andT2orR1andR2torelaxinthepresenceofhigher-than-two-bodycomponentsofTandR,respectively,throughselectionofthedominantTnandRexcitationamplitudesusingactiveorbitals.7Thisresultsinahierarchyofmethods,suchasCCapproacheswithsingles,doubles,andactive-spacetriples(CCSDt/EOMCCSDt),oftentermed\littlet",andsingles,doubles,andactive-spacetriplesandquadruples(CCSDtq/EOMCCSDtq),usuallytermed\littletq".Theremayalsobeinstancesinvolvingstrongnon-dynamicalcorrelationthatrequirethefulltreatmentoftriplesandselectionofhigher-than-three-bodycomponentsofTandR,asinthecaseoftheCCsingles,doubles,triplesandactive-spacequadruples(CCS-DTq/EOMCCSDTq)approximation,sometimestermed\littleq".Thesemethodsnaturallyapproachtheirparentapproacheswhenallmolecularorbitalsinthebasissetbecomeactive.So,forexample,CCSDtandCCSDtqbecomefullCCSDTandCCSDTQ,respectively,whenallorbitalsareactive.Whileactive-spaceCCmethodsarecapableofcapturingthevastma-jorityofnon-dynamicalcorrelationtheyarenotastindescribingdynamicalcorrelationsasaretheCCSD(T),CR-CC(2,3),CCSD(TQf),CR-CC(2,4),orsimilarnon-iterativeapproaches.Asanexample,Table1.2fortheautomerizationofcyclobutadieneshowsthatCCSDtprovidesabalanceddescriptionofthereactantandtransitionstates,producingabarrierheightthatisonly0.3{1.1kcal/molfromthevaluegivenbyCCSDT,but,duetothemissing,mainlydynamical,correlationtotalelectronicenergiesre-sultingfromCCSDtcalculationsatthereactant/productandtransition-stategeometrieshaveerrorsof20{30millihartreerelativetoCCSDT.Asshownin[164],largererrorsofthismagnitudemaysometimesprovideafewkcal/molerrorsincharacterizingstationarypointenergeticsofchemicalreactionpathways.Itwouldbebesttoeliminatesuchlargeerrorstoavoidproblemsofthiskind.Theactive-spaceCC/EOMCCapproximationsarenolongerpure\black-box"methodssinceonehastoselectoccupiedandunoccupiedactiveorbitalspriortocalculation,butthisisnotthesituationofCASSCF-basedMRapproaches,sincetheCPUtimesoftheactive-spaceCC/EOMCCcalculationsscalelinearlyorquadraticallywith8thenumbersofactiveoccupiedandactiveunoccupiedorbitals,asopposedtoexponentialscalingsofCASSCF-basedmethods,i.e.,itistrivialtoincreasethenumberofactiveorbitalsbyatfactorwithoutrunningintoprohibitivecosts.Nevertheless,whiletheCR-CC/CR-EOMCCandactive-spaceCC/EOMCCapproacheshavetheirrespectivestrengths,itisclearthatbothtypesofmethodologieshaveweaknesseswhichneedtobeaddressed.Table1.2:AcomparisonofvariousCCground-stateenergiesforthereactantandtransition-statespeciestheautomerizationofcyclobutadiene,basedonthegeometriesopti-mizedintheMR-AQCCcalculationsinRef.[254]aswellaspurelyelectronicbarrierheights(inkcal/mol).a;bMethodReactantTransitionStateBarrierHeightcc-pVTZCCSD26.83747.97920.9CCSD(T)1.12314.19815.8CR-CC(2,3)D0.84814.63616.3CCSDt20.78620.2747.3CC(t;3)D-0.1370.0717.8CCSDT-154.244157-154.2320027.6cc-pVTZCCSD36.10655.20522.6CCSD(T)0.27812.29118.1CR-CC(2,3)D0.94113.79318.6CCSDt30.00728.2599.5CC(t;3)D-0.141-1.03810.0CCSDT-154.390763-154.37390210.6aAllresultsweretakenfromRef.[164].bTheCCSDTvaluesaretotalenergies,inhartree.Theremainingenergies,excludingthebarrierheights,representerrorsrelativetoCCSDT,inmillihartree.TheactivespaceusedintheCCSDt,CCSD(T)-h,andCC(t;3)calculationsconsistedoftheonehighest-energyoccupiedandonelowest-energyunoccupiedorbitalsOnewaytoimproveupontheoftheactive-spaceCC/EOMCCandCR-CC/CR-EOMCCmethods,whilestillretainingtheirrespectiveadvantages,isbycombining9Figure1.1:Automerizationofcyclobutadiene.Theleftmostandrightmoststructurescor-respondtothedegeneratereactant/productminima,whereasthestructureinthemiddlerepresentsthetransitionstate.thetwotypesofapproachesintoasingleformalism.Thisisaccomplishedbygeneraliz-ingtheexistingbiorthogonalMMCCtheory,suchthatonecancorrecttheCC/EOMCCenergiesobtainedwiththearbitrary,i.e.,conventional(e.g.,CCSD/EOMCCSDorCCS-DT/EOMCCSDT)aswellasunconventional(e.g.,CCSDt/EOMCCSDtorCCSDtq/EOM-CCSDtq)truncationsinTandRforessentiallyanysubsetofthemissingmany-electroncorrelationofinterest.Theresultingmomentexpansions,theCC(P;Q)formalism[163{165,251{253]developedinourgroup,enableonetocontemplateavarietyofnewschemesaddressingtheaboveconcernswiththeCR-CC/EOMCCandactive-spaceCC/EOMCCapproaches.Thisworkisfocusedontheensuinghierarchybasedoncorrect-ingenergiesobtainedinactive-spaceCCandEOMCCcalculationsformissing,primarilydynamical,higher-ordercorrelations.Forexample,thebasicCC(P;Q)method,abbreviatedasCC(t;3),correctstheenergyobtainedattheCCSDtlevelfortheremainingcorrelationduetoconnectedtripleexcitationsmissinginCCSDt,butpresentintheMMCC-basedCR-CC(2,3)non-iterativecorrection.Ithasbeenshown[163{165,251{253]thattheCC(t;3)methodimprovesuponboththeCCSDtandCR-CC(2,3)results,whileprovidingpotentialenergysurfacesalongbondbreakingcoordinatesincloseagreementwiththere-sultsobtainedusingthefulltreatmentoftriples,CCSDT,towithinasmallfractionofa10millihartreeandatafractionofthecomputationalcost.OthermethodsintheCC(P;Q)hi-erarchyincludeCC(t,q;3),CC(t,q;3,4),andCC(q;4),aswellastheirexcited-stateanalogs,inwhichenergiesobtainedusingactive-spaceCCSDtq/EOMCCSDtqarecorrectedformissinghigher-ordercorrelations,suchastriples(CC(t,q;3))ortriplesandquadruples(CC(t,q;3,4)),orinwhichenergiesobtainedusingactive-spaceCCSDTq/EOMCCSDTqarecorrectedformissinghigher-ordercorrelations,suchasquadruples(CC(q;4)),usingthenon-iterativeen-ergycorrectionssimilartothosethepreviouslydiscussedCR-CCapproaches.InthesamefashionasCC(t;3),theCC(t,q;3),CC(t,q;3,4),andCC(q;4)approacheshavebeenshown[251{253]toimproveuponbothoftheirstandardactive-spaceandCR-CCcompo-nents,reproducingresultsobtainedusingthefulltreatmentofquadruples,CCSDTQ,onceagaintowithinasmallfractionofamillihartree,andatafractionofthecomputationalcost.ThisthesisfocusesonthedevelopmentoftheCC(t,q;3),CC(t,q;3,4),andCC(q;4)approachesandthecorrespondingcomputercodesusedinRefs.[251{253].OurinterestindevelopingtheCC(t;3),CC(t,q;3),CC(t,q;3,4),CC(q;4),etc.hierarchyandtheunderlyingCC(P;Q)methodologyhasbeeninspired,tosomeextent,bytheearlierworkbyLiandPaldus[255,256],whodecidedtocorrectthereducedMRCCSD(RMR-CCSD)[257]energies,whichcontainsome,butnotalltriples,forthetriplescorrelationmissinginRMRCCSDusingthe(T)correctionsofCCSD(T),resultingintheRMR-CCSD(T)approach,andtheanalogousbyLietal.[258{261],whoproposedtodothesamewithintheactive-spaceCCSDtframework,onceagainusingthestandardCCSD(T)expressiontocorrecttheCCSDtenergiesforthesubsetoftriplesmissinginCCSDt.Inparticular,asexplainedinRefs.[163{165],theground-stateCC(t;3)approachcanbere-gardedasanimprovementovertheCCSD(T)-hmethodofLietal.[258{261],inwhichtheperturbative(T)correctionofCCSD(T)exploitedinCCSD(T)-htocorrecttheCCSDt11energiesforthesubsetoftriplesmissinginCCSDtisreplacedbythemuchmorerobustCR-CC(2,3)-typeexpressionresultingfromtheCC(P;Q)considerations.Similarly,themorerecentCCSD(T)q-happroachdevelopedinRef.[262],inwhichoneusesthe(T)correctionofCCSD(T)tocorrecttheCCSDtqenergiesforthesubsetoftriplesmissinginCCSDtq,isanapproximationtotheground-stateCC(t,q;3)method,whichweformallyproposedinRef.[163]andwhichweimplementandtest,alongwithitsCC(t,q;3,4)extension,inthepresentthesisresearch.WhilerecognizingthesimilaritiesbetweentheCC(t;3)andCC(t,q;3)approachesandtheirCCSD(T)-handCCSD(T)q-hcounterparts,assummarizedabove,weshouldalsopointoutthebetweentheformerandthelatterschemes.FirstandforemostisthefactthattheCC(t;3)andCC(t,q;3)approximationsarepartofalarger,system-aticallyimprovableCC(t;3),CC(t,q;3),CC(t,q;3,4),CC(q;4),etc.hierarchyintroducedinRefs.[163,164]andfurtherdevelopedhere,whichresultsfromtherigorousmomentenergyexpansionstheCC(P;Q)framework.TheCCSD(T)-handCCSD(T)q-hmeth-odsofRefs.[258{262]andtheirRMRCCSD(T)predecessorproposedinRefs.[255,256]arebasedonintuitive,adhocarguments,whichareaimedatcorrectingtherespectiveCCSDt,CCSDtq,andRMRCCSDenergiesforthemissingtripleexcitationwithoutaproofthatthe(T)correctionofCCSD(T)representsanappropriatemathematicalexpressiontodoit.Indeed,onemustkeepinmindthatthe(T)correctionofCCSD(T)wasoriginallyderived[140]andsubsequentlyrederived[160,161,263]asaquasi-perturbativecorrectiontoCCSD,notRMRCCSD,CCSDt,orCCSDtq.ItisalsounclearhowtoextendtheCCSD(T)-h,CCSD(T)q-h,andRMRCCSD(T)approachestohigher-order(i.e.,higher-than-tripleorhigher-than-quadruple)excitationsand,whatisespeciallyimportantinthiswork,whatformulastousetocorrecttheCCSD(T)q-handRMRCCSD(T)energiesforthesubsetsof12quadruplesstillmissingintheseschemes.NoneofthisisaproblemwhentheCC(P;Q)-basedCC(t;3),CC(t,q;3),CC(t,q;3,4),CC(q;4),etc.hierarchyisconsidered,sinceonecaneasilytherelevanthigher-orderschemesconvergingtotheexact,fullCIlimitandhavingwrelationshipswiththeparentschemesfromatraditionalCCSDT,CCS-DTQ,etc.CCsequenceviasuitablechoicesofthePandQsubspacesofthemany-electronHilbertspaceenteringtheCC(P;Q)expressionsreportedinRefs.[163,164].Furthermore,asshowninRefs.[163{165],itissafertousetheCC(P;Q)ideastodesignthenon-iterativecorrectionstotheactive-spaceCCmethods,suchasCCSDt,sincetheCC(P;Q)-inspiredcorrections,suchasthatCC(t;3),alwaysbringtheunderlyingCCSDtenergiestoacloseragreementwiththeparentfullCCSDTresults,whereastheCCSD(T)-inspiredcorrec-tiontoCCSDttheaforementionedCCSD(T)-happroachmaychangetheCCSDtenergiesinanoppositedirection,i.e.,awayfromCCSDT,whichisanundesiredbehavior.Finally,althoughthisstudyfocusesonextendingtheearlierCC(t;3)work[163{165]tothehigher-levelCC(t,q;3)andCC(t,q;3,4)schemes,asappliedtothegroundelectronicstates,itshouldalsobepointedoutthatunlikeRMRCCSD(T),CCSD(T)-h,andCCSD(T)q-h,whichapplytotheground-stateproblemonly,theCC(t;3),CC(t,q;3),CC(t,q;3,4),CC(q;4),etc.hierarchy,andtheunderlyingCC(P;Q)formalismhavenaturalextensionstoexcitedstateswithintheEOMCCframework[163].Inthisthesis,wefocusonthegroundelectronicstatesandCC(t,q;3),CC(t,q;3,4),andCC(q;4)approachesthatdescribethecombinedoftriplesandquadruples.ThekeyfeatureoftheCC(P;Q)formalism,whichwewouldliketoemphasizehere,isasfollows.Thenon-iterativecorrectionstotheCCSDt,CCSDtq,andotheractive-spaceSRCCenergiestheCC(t;3),CC(t,q;3),CC(t,q;3,4),andsimilarapproachesprovideuswithaconceptuallystraightforwardandrelativelyinexpensivemechanismforrelaxing13thesinglyanddoublyexcitedcomponentsoftheclusteroperatorT,T1andT2,respectively,inthepresenceofhigher-than-doublyexcitedclusters,suchasT3andT4,withouthavingtoturntotheprohibitivelyexpensivefullCCSDTorCCSDTQmethods.InanalogytotheconventionalCCSD(T)andCCSD(TQf)schemes,thepreviousgenerationsofthenon-iterativecorrectionsexploitingtheMMCCformalism,suchasthosetheCR-CC(2,3)andCR-CC(2,4)methods,relyonthevaluesofT1andT2determinedbysolvingthestandardCCSDequations.Asalreadypointedout,theT1andT2clustersobtainedwithCCSD,whereoneassumesthattheycanbedecoupledfromtheTncomponentsofTwithn3,andtheT1andT2clustersdeterminedinthepresenceofTn'swithn3canbequitet,particularlywhentheconnectedtriplesortheconnectedtriplesandquadruplesbecomelarger.Indeed,asdemonstratedinRefs.[163{165],andasillustratedinTable1.2,thereareclassesofchemicalreactionsinvolvingbiradicaltransitionstates,examplesofsinglet{tripletgapsinbiradicalsystems,andcasesofbondbreaking,wheretheneglectofthecouplingbetweenthelower-orderT1andT2componentsandT3leadstoratherlargeerrorsintheresultsoftheCR-CC(2,3)andothersimilarnon-iterative-basedcalculations,suchas[264,265]andCCSD(2)T[266],whicharecomparable,inabsolutevalue,tothoseobtainedwiththestandard(andfailing)CCSD(T)approach.AsdemonstratedinRefs.[163{165]andasillustratedinTable1.2,theCC(P;Q)-basedCC(t;3)method,inwhichtheT1andT2clustersthatareusedtodeterminethecorrectionduetotriplesaretakenfromtheCCSDtcalculations,whereonesolvesforT1andT2inthepresenceofthedominantT3contributionscapturedwiththehelpofactiveorbitals,addressesthisissue,substantialimprovementsintheCR-CC(2,3)andCCSD(2)T(alsoCCSD(T))resultsinthebondbreakingandbiradicalsituationswherethesemethodsperformpoorly,whileprovidingthetotalaswellasrelativeenergiesthatoftenarewithinsmallfractionsofamillihartree14relativetothecorrespondingfullCCSDTdatawithoutmakingthecalculationstlymoreexpensivethanintheCR-CC(2,3),CCSD(2)T,orCCSD(T)case.Oneofthemainobjectivesofthisworkistoshowthatoneobservessimilarimprovementsintheelectronicenergiesalongbondstretchingcoordinates,whentheCR-CC(2,4)correctionsduetotriplesandquadruples,whichrelyontheT1andT2clustersobtainedwithCCSD,arereplacedbytheCC(t,q;3)andCC(t,q;3,4)correctionstotheCCSDtqenergiesthatutilizetheT1andT2amplitudesoriginatingfromtheCCSDtqcalculations.Itistruethattheactive-spaceSRCCmethods,suchasCCSDtexaminedinRefs.[163{165]orCCSDtqexaminedinthiswork,whichcapturethecouplingofT1andT2withthehigher-orderTncomponentswithn3selectedwiththehelpofactiveorbitals,providehigh-qualityrelativeenergeticsinmanybiradicalandbondbreakingsituationstoo,but,asshowninRefs.[163{165],andasfurtherdemonstratedinthisthesis,theyoftenfailtoprovideaccuratetotalenergiesrelativetotheparentSRCCapproximationsduetotheneglectofthehigher-orderTncontributionsoftheprimarilydynamicalcharacterthatdonotinvolveactiveorbitalsandthismaynegativelytheresultingrelativeenergeticsofchemicalreactionpathways(see,e.g.,[164]).AsshowninRefs.[163{165]usingCC(t;3)andasdemonstratedinthisdissertationusingCC(t,q;3)andCC(t,q;3,4),theCC(P,Q)methodologyaddressesalltheseissues,sinceitprovidesuswithatransparentmechanismhowtocorrecttheCCSDtandCCSDtqenergiesforthesubsetsoftriple(CCSDt)ortripleandquadruple(CCSDtq)excitationsneglectedintheseactive-spaceapproaches.OtherwaysofaccountingfortherelaxationoftheT1andT2clustersinthepresenceofT3orT3andT4withintheSRCCframework,whichcanbeviewedasalternativestotheCC(P,Q)schemesconsideredinthisdissertationandtheearlier[163{165]work,anditsextensionsinRefs.[251{253],havebeenproposedinRefs.[267,268],wheretheauthorscombinedfullCCSDTcalculationsinanextendedactive15spacewiththetailoredCCframework[220,269,270]andinRefs.[271{273],wheretheauthorsexploitedtheLagrangianSRCCandEOMCCframeworksandtheappropriateperturbativeanalysisoftheresultingequations.OurfocushereisontheCC(P,Q)ideasoriginatingfromourgroup[163{165,251{253].Inthisdissertation,welaydownthetheoryoftheCC(P;Q)methodologyanddescribetheensuinghierarchyofmethods,namelyCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4),withafocusonthelatterthree,whichhavebeenimplementedinmyPh.D.research.Wedemon-strateandestablish,particularlyforapproachesinvolvingconnectedquadrupleexcitations,therobustness,utility,andaccuracyoftheCC(t,q;3),CC(t,q;3,4),andCC(q;4)approxima-tionsbyexaminingthedoubledissociationofwater,theBe+H2!HBeHinsertion,andthesinglet{tripletgapinthestronglybiradical(HFH)system[251],wherequadruplesareimportanttoachieveaquantitativedescription.Afterexaminingthesetestsets,weapplytheseapproachestoasmall,butsurprisinglyBNmolecule,whichservesasthe\torture"caseforexaminingtheCC(P;Q)methodsinthisdissertationandwhichrequiredthedevelopmentofhighlytcodesforthefullandactive-spacetreatmentofquadrupleexcitations.Asaresult,wedescribethetechniqueofspin-integrationforbothclosedandopenshells,andhowtheresultingequationsforfullCCSDTQanditsactive-spaceCCSDtqandCCSDTqcounterpartsareautomaticallyderivedandimplementedinahighlyt,fullyvectorizable,factorizedform.Wealsodiscusshowtheoftheresultingcodeswasfurtherimprovedthroughthereorganizationoftheunderlyingloopstructure.Finally,weexplainhowtheCCSDTQcode,whichcancompetewiththebestcodesofthiskindaround,wastransformedtoobtaintheactive-spaceCCSDtqandCCSDTqapproaches.Insummary,thefocusofthisdissertationistheextensionoftheCC(P;Q)hierarchytometh-odsinvolvingconnectedquadrupleexcitations,aswellasdemonstratingandestablishing16thattheseapproachesarecapableofreproducingresultsobtainedusingthefulltreatmentofquadruples,CCSDTQ,typicallytowithinafractionofamillihartree,atatinyfractionofthecomputationalofCCSDTQ,evenwhenelectronicquasi-degeneraciesbecomesubstantial,asisthecasewhenbondsarebrokenorchallengingcasesofsinglet{tripletgapsareexamined.17Chapter2ProjectObjectivesThemainobjectivesofthisthesisworkareA.IntroducetheCC(P;Q)theoryanddescribetheensuinghierarchyofmethods,namelytheCC(t;3),CC(t,q;3),CC(t,q;3,4)andCC(q;4)approaches.B.InvestigatetheperformanceoftheCC(P;Q)methodsbyexaminingtheinterestingbenchmarkproblemsofthedoubledissociationofwater,theBe+H2!HBeHin-sertion,andthesinglet{tripletgapinthestronglybiradical(HFH)systemandbyapplyingthesemethodstotheenormouslyBNmolecule.C.Describetheprocedureofspinintegrationforclosedandopenshellsystemsandhowthespin-integratedCCSDTQequationswerederived,factorized,andtranslatedintoFORTRANcodeusingaprogram,developedinthisworkaswell,thatcarriesouttheseproceduresautomatically.D.Discusshowtheperformanceofthespin-integratedCCSDTQcodewasfurtherim-provedbyreorganizingthecorrespondingloopstructure,andthendescribehowthecodewasmotoobtaintheCCSDtqandCCSDTqroutinesusedintheCC(t,q;3),CC(t,q;3,4)andCC(q;4)computations.18Chapter3MergingActive-SpaceandCompletelyRenormalizedCoupled-ClusterMethods3.1TheoryThisdissertationisconcernedwiththedevelopmentandapplicationofCCandEOMCCapproachesthathaveemergedfromtherecentlyproposedCC(P;Q)methodology,withafocusonground-statemethodsthattlyaccountfortheconnectedquadrupleexci-tations.WebeginbyreviewingtheconventionalCCandEOMCCtheories,theCR-CCandCR-EOMCCapproaches,andtheactive-spaceCCandEOMCCmethods.Then,wediscusshowtoimproveupontheoftheCR-CC/CR-EOMCCandactive-spaceCC/EOMCCapproximationsbycombiningthemtogetherviatheCC(P;Q)formalism.Weendthissectionbypresentingthedetailedequationsoftheresultinghierarchyofmethodsimplementedtodate,namely,theCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)approaches.193.1.1CompletelyRenormalizedCoupled-ClusterApproachesIntheSRCCtheory,theground-statewavefunctionj(A)0iofanN-electronsystemisexpressedusingtheexponentialansatzj0i=eTji;(3.1)whereTistheclusteroperator(theconnectedparticle-holeexcitationoperator)andjiisthereference(usuallyHartree{Fock)determinant.Typically,wetruncatethemany-bodyexpansionofTataconvenientlychosenexcitationrankmA6N,toobtainanapproximateT,i.e.,T'T(A).ThetruncatedclusteroperatorT(A)theapproximateCCmethodAisgivenbyT(A)=mAXn=1Tn;(3.2)withTn=Xi1<<ina1<<anti1:::ina1:::anEa1:::ani1:::in;(3.3)whereTnisthen-bodycomponentofT(A),ti1:::ina1:::anaretheclusteramplitudes,andEa1:::ani1:::in=nY=1aaai(3.4)istheusualn-bodyexcitationoperatorwithapandapdesignatingthecreationandanni-hilationoperatorsassociatedwiththespin-orbitalbasisfjpig.Hereandelsewhereinthisthesis,weusethestandardnotationinwhichtheindicesi1,i2,...ori,j,...(a1,a2,...ora,b,...)refertotheoccupied(unoccupied)spin-orbitalsinthereferencedeterminantji.IfmA=N,weobtainthefullCCwavefunction,whichisequivalenttothefullCIground20statecorrespondingtotheexactsolutionoftheelectronicScodingerequationinthebasisset.WhentheclusteroperatoristruncatedsuchthatmA<N,weobtainthewell-knownhierarchyofstandardCCapproximations(mA=2forCCSD,mA=3forCCSDT,mA=4forCCSDTQ,etc.).Inthecaseofexcitedstates,weobtaintheexcited-statewavefunctionjibyapplyingalinearparticle-holeexcitationoperatorRtoground-stateSRCCwavefunction,i.e.,ji=Rj0i;(3.5)wherej0iisbyEq.(3.1).ThisisreferredtoastheEOMCCformalism.Usually,theRlinearexcitationoperatoristruncatedatthesameexcitationlevelmAasthatusedintheunderlyingground-stateCCmethodA,sothatRisapproximatedbyitstruncatedcounterparttheEOMCCmethodA,R(A)=R(A)0+R(A)open=r01+mAXn=1R;(3.6)inwhich1istheunitoperatorandR=Xi1<<ina1<<anri1:::in1:::anEa1:::ani1:::in;(3.7)whereRisthen-bodycomponentofR(A)andri1:::in1:::anaretheexcitationamplitudesforthe-thexcitedstate.ByvaryingthetruncationlevelmAR(A)andthecorre-spondingT(A)operator,weobtaintheconventionalhierarchyofstandardEOMCCapprox-imations(mA=2forEOMCCSD,mA=3forEOMCCSDT,mA=4forEOMCCSDTQ,etc.).Forconsistencyofournotation,where=0correspondstothegroundstateand>0representsexcitedstates,weR(A)=0=1,sothatr=0;0=1andR=0forn>0.21Intheground-stateconsiderations,theclusteramplitudesti1:::ina1:::anT(A),Eq.(3.2),areobtainedbysolvingtheconventionalSRCCequations.WearriveattheseequationsbyinsertingtheCCwavefunctionj0i,Eq.(3.1),intotheelectronicScodingerequation,Hj0i=E0j0i;(3.8)andmultiplyingbothsidesofEq.(3.8)ontheleftbyeT(A)toobtaintheconnectedclusterformoftheScodingerequation,H(A)ji=E0ji;(3.9)whereH(A)=eT(A)HeT(A)=(HeT(A))C(3.10)isthesimilarity-transformedHamiltonianandthesubscriptCdenotestheconnectedpartofthecorrespondingoperatorexpression.Then,weprojectEq.(3.9)ontotheexcitedde-terminantsja1:::ani1:::iniEa1:::ani1:::injitoobtainthefollowingsetofequations:ha1:::ani1:::injH(A)ji=0;i1<:::<in;a1<:::<an;(3.11)wheren=1;:::;mA.TheexciteddeterminantsenteringEq.(3.11)correspondtothemany-bodycomponentsTnincludedinT(A).Forexample,intheCCSDcalculations(mA=2),weprojectEq.(3.9)onallsinglyanddoublyexciteddeterminants,jaiiandjabiji,respectively,whileforCCSDTQ(mA=4)weprojectEq.(3.9)onallsingly,doubly,triply,andquadruplyexciteddeterminants,jaii,jabiji,jabcijki,andjabcdijkli.Oncethesystemofequations,Eq.22(3.11),issolvedfortheclusteramplitudesti1:::ina1:::an,theground-stateenergyofthestandardCCmethodAiscalculatedusingE(A)0=hjH(A)jihjH(A)closedji;(3.12)whereH(A)closedisthe`closed'partofH(A),representedbytheconnecteddiagramsthathavenoexternalFermionlines.MovingontotheexcitedstateswithintheEOMCCframework,theexcitationampli-tudesri1:::in1:::anthen-bodycomponentsofR(A)withn=1;:::;mAareobtainedbysolvingthenon-hermitianeigenvalueprobleminvolvingthesimilarity-transformedHamil-tonianH(A),Eq.(3.10),inthespacespannedbytheexciteddeterminantsja1:::ani1:::inithatcorrespondtothetruncationlevelofR(A),ha1:::ani1:::inj(H(A)openR(A)open)Cji=!(A)ri1:::in1:::an;i1<:::<in;a1<:::<an;(3.13)whereH(A)open=H(A)H(A)closed=H(A)E(A)01(3.14)isthe`open'partofH(A),representedbydiagramshavingexternalFermionlinesand!(A)=E(A)E(A)0(3.15)istheverticalexcitationenergyobtainedwiththeEOMCCmethodA.Oncetheexcitationamplitudesri1:::in1:::anwithn=1;:::;mAR(A)open(>0)areknown,thecotr0thezero-bodycomponentR(A)0isdeterminedaposterioriusingthefollowing23equation:r0=hj(H(A)openR(A)open)Cji!(A):(3.16)LetusrecallthatthebasicCCSD/EOMCCSDmethodhasiterativeCPUstepsthatscaleasn2on4u,whicharepracticalenoughformanysituations.Unfortunately,thehigher-levelCCS-DT/EOMCCSDTandCCSDTQ/EOMCCSDTQapproacheshaveCPUstepsthatscaleasn3on5uandn4on6urespectively,whichareprohibitivelyexpensiveforsystemswithmorethanadozenorsocorrelatedelectrons.Thus,inordertoincorporatethephysicsassociatedwiththeT3,T4,R3,andR4operators,whichareneededtoobtainanaccuratedescriptionofdynamicalandnon-dynamicalcorrelationsinMRsituations,suchasbondbreaking,birad-icals,andtwoelectrontransitions,wemustresorttotheapproximatetreatmentsoftheseoperatorsthatreducetheiterativen3on5uandn4on6ustepstoamoremanageablelevel.AsexplainedintheIntroduction,oneofthebestapproachestothisproblemistheMMCCformalism,whichallowsonetocomeupwiththerelativelyinexpensivecorrectionstotheenergiesobtainedinthelow-orderCC/EOMCCcalculations,suchasCCSDorEOMCCSD,theCR-CCandCR-EOMCCapproaches,thataremorerobustinMRsituationsthanthetraditionalperturbativemethodsoftheCCSD(T)type.TheSRformulationoftheMMCCtheoryrelevanttothisthesisresearchisbasedontheideaofaddingtheaposteriori,non-iterative,andstate-spcorrections(A),duetohigher-ordermany-bodyexcitationsneglectedintheconventionalCC/EOMCCmethodA,tothecorrespondingCC/EOMCCenergiesE(A).TheMMCCcorrections(A)canbederivedusingoneoftheforms[105,106,111,112,116,116,154{156,158,160,161,163]oftheexpansiondescribingthebetweentheexactfullCIandCC/EOMCCmethodA24energies,i.e.,(A)EE(A):(3.17)Thename\MMCC"originatesfromthefactthatthe(A)correctionsareexpressedintermsofthegeneralizedmomentsoftheCC/EOMCCequations,designatedasMi1:::in1:::an(mA),characterizingthetruncatedCC/EOMCCmethodAwewanttocorrect.ThesemomentsareasprojectionsoftheCC/EOMCCequationswrittenforTapproximatedbyT(A)andRapproximatedbyR(A)ontheexciteddeterminantsja1:::ani1:::iniwithn>mAthatarenormallydisregardedintheCC/EOMCCcalculations,truncatedatmA-foldexcitations,i.e.,Mi1:::in0;a1:::an(mA)=ha1:::ani1:::inj(H(A))ji(3.18)forthegroundstate,andMi1:::in1:::an(mA)=ha1:::ani1:::inj(H(A)R(A))ji(3.19)forexcitedstates.InordertoderivetheMMCCcorrections(A),onetypicallybeginswiththeasymmetricenergyexpressionE=hjHR(A)eT(A)ji=hjR(A)eT(A)ji;(3.20)wherehjisthefullCIbrawavefunctionfortheground(=0)orexcited(>0)state,whichgivestheexactground-orexcited-stateenergy,E,independentofthetruncationlevelmAT(A)andR(A).WerecallthatR(A)istheunitoperatorintheground-state(=0)case.InthebiorthogonalMMCCtheory,whichinterestsusheremostand25whichleadstomethodsmentionedintheIntroduction,suchasCR-CC(2,3),CR-CC(2,4),CR-EOMCC(2,3),andCR-EOMCC(2,4),werepresenttheexactbrastatehjinEq.(3.20)inthefollowingmanner:hj=hjLeT(A);(3.21)wherethehole-particlelineardeexcitationoperatorLisgivenbyL=L(A)+L(A)mAXn=0L+NXn=mA+1L;(3.22)whereL0=01andL=Xi1<<ina1<<an`a1:::an1:::inEi1:::ina1:::an;(3.23)withEi1:::ina1:::anbyEi1:::ina1:::an=Ea1:::ani1:::iny=nY=1aiaa:(3.24)ByinsertingEq.(3.21)intoEq.(3.20)andimposingthenormalizationconditioninwhichhjL(A)R(A)ji=1,weimmediatelyobtainE=hjLH(A)R(A)ji:(3.25)InsertingtheresolutionofidentityintheN-electronHilbertspace,jihj+P+Q=1,whereP=mAPn=1Pn,Q=NPn=mA+1Pn,andPn=Pi1<<ina1<<anja1:::ani1:::iniha1:::ani1:::inj,betweenLandH(A),wearrive,aftersomemanipulations,attheformulafortheexactenergy26EthatthebiorthogonalMMCCtheory,E=E(A)+(A);(3.26)whereE(A)istheenergyofthe-thelectronicstateobtainedwiththeCC/EOMCCmethodAand(A)isthecorrectiontowardfullCIwhichtakesonthefollowinggenericform:(A)=NXn=mA+1Xi1<<ina1<<an`a1:::an1:::inMi1:::in1:::an(mA);(3.27)whereNisthehighestvalueofnforwhichmomentsMi1:::in1:::an(mA),Eq.(3.18)for=0orEq.(3.19)for>0,arenonzero.Inpracticalcalculationsbasedontheaboveequations,thesumoverninEq.(3.27)istruncatedatsomeexcitationlevelmB,wheremA<mBN.Forexample,intheCR-CC(2,3)andCR-EOMCC(2,3)approachesintroducedinRefs.[116,117,160{162]andfurtherdevelopedinRef.[96],theCCSDandEOMCCSDenergies,E(CCSD),arecorrectedfortheleadingcorrelationduetotripleexcitations,sothatmA=2andmB=3.Asaresult,theCR-CC(2,3)andCR-EOMCC(2,3)energyexpressionsareE(2;3)=E(CCSD)+(2;3);(3.28)where(2;3)=Xi<j<ka<b<c`abck(2)Mijk(2);(3.29)withMijk(2)representingthetriply-excitedmomentsoftheCCSD(=0)orEOMCCSD(>0)equations.InordertomakeEqs.(3.28)and(3.29)computationallymanageable,one27hastocomeupwiththeapproximateformofthe`abck(2)amplitudesthatoriginatefromthethree-bodycomponentoftheLoperator,Eq.(3.22)theexactbrastatehjinEq.(3.21),whichinthisparticularcase,wheremA=2,becomeshj=hjLeT(CCSD),withT(CCSD)=T1+T2representingtheclusteroperatorobtainedinCCSDcalculations.Thiscanbedoneinseveralways.InthespcaseofCR-CC(2,3)andCR-EOMCC(2,3)methodsofRefs.[116,117,160,161],the`abck(2)thatenterthecorrections(2;3),Eq.(3.29),arecalculatedusingtheexpression`abck(2)=hjL(CCSD)H(CCSD)jabcijki=Dijk(2)(3.30)whereDijk(2)=E(CCSD)habcijkjH(CCSD)jabcijki;(3.31)whichisobtainedbyperforminganapproximatequasi-perturbativeanalysisofthebraScodingerequationeigenvalueproblemhjH=Ehj,withhjgivenbyhjLeT(CCSD),constrainedtothesubspaceoftheHilbertspaceuptotripleexcitations(seeRefs.[116,117,160,161]forthedetails).TheL(CCSD)operatorintheaboveexpressionisthedeexcitationoperatorthebraCCSD/EOMCCSDstateh~(CCSD)j=hjL(CCSD)eT(CCSD),whichmatchestheCCSD/EOMCCSDketstatej(CCSD)i=R(CCSD)eT(CCSD)jiandwhichisasL(CCSD)=01+L1+L2,whereL1andL2aretherelevantone-andtwo-bodycomponents.H(CCSD)inEqs.(3.30)and(3.31)isthesimilarity-transformedHamiltonianofCCSD,H(CCSD)=eT(CCSD)HeT(CCSD)=(HeT(CCSD))c.Oneobtainsthesecomponents,orthedeexcitationamplitudeslaandlabthatthem,bysolvingtheleft-eigenstateCCSD/EOMCCSDequations.Thegeneralleft-eigenstateCC/EOMCC28systemcorrespondingtotruncationAatmA-foldexcitationshastheformofalinearsystem0hjH(A)openja1:::ani1:::ini+hjL(A)openH(A)openja1:::ani1:::ini=!(A)la1:::in1:::in(mA)i1<:::<in;a1<:::<ann=1;:::;mA(3.32)whereL(A)open=maPn=1ListhedeexcitationoperatorthebraCC/EOMCCstateh~(A)j=hj(01+L(A)open)eT(A)matchingtheketstatej(A)i=R(A)eT(A)ji.Theleft-eigenstateCCSD/EOMCCSDequationsareobtainedbysettingmAinEq.(3.32)at2.InperformingtheCR-CC(2,3)andCR-EOMCC(2,3)calculations,weoftendistinguishbetweenvariantsAandD.VariantDisthefullformofCR-CC(2,3)/CR-EOMCC(2,3),whereweusetheDijk(2)denominatorinitsmostcompleteEpstein{NesbetformgivenbyEq.(3.31).VariantAisobtainedbyreplacingEq.(3.31)bytheM˝ller{Plessetformula!(CCSD)(i+j+kabc),where!(CCSD)istheEOMCCSDexcitationenergy(0inthe=0ground-statecase)andp'saretheusualspin-orbitalenergies(diagonalelementsoftheFockmatrix).WewillusesimilarvariantsinourCC(P;Q)considerations.OnecanusesimilarideastodevelopotherCR-CC(mA,mB)/CR-EOMCC(mA,mB)meth-ods,suchasCR-CC(2,4)/CR-EOMCC(2,4)andCR-CC(3,4)/CR-EOMCC(3,4).IntheCR-CC(2,4)/CR-EOMCC(2,4)case,wheremA=2andmB=4,wecorrecttheCCSD/EOM-CCSDenergyforthecombinedoftriplesandquadruplesusingtheformulaE(2;4)=E(CCSD)+(2;4);(3.33)29where(2;4)=Xi<j<ka<b<c`abck(2)Mijk(2)+Xi<j<k<la<b<c<d`abcdkl(2)Mijkl(2):(3.34)IntheCR-CC(3,4)/CR-EOMCC(3,4)case,wheremA=3andmB=4,wecorrecttheCCSDT/EOMCCSDTenergyforcorrelationofquadruplesusingE(3;4)=E(CCSDT)+(3;4);(3.35)where(3;4)=Xi<j<k<la<b<c<d`abcdkl(3)Mijkl(3):(3.36)The`abck(2),`abcdkl(2),and`abcdkl(3)amplitudesenteringtheaboveexpressionsarede-terminedinasimilarwayasintheCR-CC(2,3)/CR-EOMCC(2,3)methodsusingquasi-perturbativeexpressionssimilartoEqs.(3.30)and(3.31).Again,ineachcase,wetypicallydistingushbetweenvariantsAandDinhandlingtherespectivedenominatorsanalogoustoDijk(2)enteringCR-CC(2,3)andCR-EOMCC(2,3),withvariantDimplyingthefullEpstein{Nesbet-liketreatmentandvariantAimplyingthemoreapproximateM˝ller{Plessettreatment.AlloftheaboveCR-CC/CR-EOMCCapproachestremendoussavingsinthecom-puterwhencomparedtotheiriterativeparents.Forexample,CR-CC(2,3)reducestheiterativen3on5uCPUstepsoffullCCSDTbytheiterativen2on4ustepsofCCSDandnon-iterativen3on4ustepsassociatedwiththedeterminationofthe(2;3)corrections.Similarly,CR-CC(2,4)replacestheiterativen4on6ustepsofCCSDTQbytheiterativen2on4ustepsofCCSDandnon-iterativen3on4uandn2on5u(variantA)orn4on5u(variantD)steps.CR-CC(3,4)replacestheiterativen4on6ustepsofCCSDTQbytheiterativen3on5ustepsofCCSDTand30non-iterativen2on5u(variantA)orn4on5u(variantD)steps.AllofthesecostreductionsbythevariousCR-CC(mA,mB)/CR-EOMCC(mA,mB)methodswithmA=2or3andmB=3or4aresummarizedinTable1.1.Thenon-iterativeCR-CC(ma,mb)approachesandtheirEOMCCextensionsareoftenasaccurateastheparentiterativemethodsatthefractionofthecost.Forexample,theCR-CC(2,3)approachrecoversthefullCCSDTresultsforpotentialenergysurfacesdescribingsinglebondbreakingtowithinamillihartreeorso,eliminatingfailuresofperturbativemethodssuchasCCSD(T).Unfortunately,asexplainedintheIntroduction,therearecaseswhereCR-CC(2,3)andotherCR-CC(mA,mB)methodshaveproblems.ThishappenswhentherelaxationoftheclusterandexcitationamplitudescorrespondingtotheunderlyingmethodAwearetryingtocorrectbecomestinthepresenceofhighermany-bodycomponentsofTandRoperators.Inallsuchcasesitisusefultoturntotheactive-spaceCCandEOMCCapproaches,whichprovidebetterclusterandexcitationamplitudesandwhichwediscussnext.3.1.2Active-SpaceCoupled-ClusterMethodsAsdiscussedintheIntroduction,ifwearetohandlesystemsinvolvingstrongernon-dynamicalcorrelationsinarobustmannerbyrecoveringsuchcorrelationsthroughtheexplicitin-clusionofhigher-than-doublyexcitedoperatorsTandRintheSRCCframe-work,thebiggestissuewefaceisthatoftheprohibitivecomputercostsoftheCCS-DT/EOMCCSDT,CCSDTQ/EOMCCSDTQ,andotherhigh-levelSRCCapproaches.TheCR-CC/CR-EOMCCcorrectionsdescribedinSection3.1.1areonewaytoaddressit,butonecannotsolveeveryMRprobleminthisway,soitisworthexploringalternatives,suchastheactive-spaceCCandEOMCCmethods.31Theactive-spaceCCandEOMCCapproximationsareaclassofrobustmethodsthatseektocapturestrongernon-dynamicalcorrelationthroughsuitableselectionofthedominantT3,T4,R3,R4,andotherhigher-than-doublyexcitedcomponentsofTandR,whileretainingrelativelyinexpensivecomputationalcosts.ThisisaccomplishedwiththehelpofactiveorbitalsthatareusedtoidentifydominantT3,T4,R3,R4,etc.amplitudes.Eachactive-spaceCC/EOMCCcalculationbeginsbypartitioningtheone-electronbasisofoccupiedandunoccupiedspin-orbitalsusedintheconventionalSRCCconsiderationsintofourdisjointgroupsofcoreorinactiveoccupiedspin-orbitals,labeledbylower-caseboldlettersi1,i2,:::,ori,j,:::,activespin-orbitalsoccupiedinthereferencedeterminantji,labeledbyupper-caseboldlettersI1,I2,:::,orI,J,:::,activespin-orbitalsunoccupiedinji,labeledbyupper-caseboldlettersA1,A2,:::,orA,B,:::,andvirtualorinactiveunoccupiedspin-orbitals,labeledbylower-caseboldlettersa1,a2,:::,ora,b,:::(seeFig.3.1).Inthefollowing,wecontinuetousesymbolssetinitalicsforoccupiedspin-orbitals(i1,i2,:::,ori,j,:::)andunoccupiedspin-orbitals(a1,a2,:::,ora,b,:::)iftheactive/inactivecharacterofagivenspin-orbitalisnotspTypically,intheactive-spaceCC/EOMCCmethods,thelow-orderclusterandexcitationoperators,suchasT1,T2,R1,andR2,aretreatedexactly,i.e.,allavailablespin-orbitalsareusedtothem,asinEqs.(3.2)and(3.6),butthehigher-orderoperators,suchasT3,T4,R3,andR4,aretreatedinanapproximatemanner,byselectingtheirdominantpartsthroughactivespin-orbitals,whichrepresentarelativelysmallsubsetofallspin-orbitals,thustlyreducingthecomputercosts.ThiswayofthinkingcanbeextendedtoanySRCCorEOMCC[107{109,119,125,163{165,238{253](even,MRCC[64,66])approach.Ingeneral,ifT˙andRarethehighest-ordermany-bodycomponentsofTandRconsidered,onecantreattheTnandRcomponentswithnˆ(ˆ<˙)exactly,asinEqs.(3.2)and32Figure3.1:Theorbitalusedintheactive-spaceSRCCmethods,suchasCCSDt,CCSDtq,andCCSDTq.Core,active,andvirtualorbitalsarerepresentedbysolid,dashed,anddottedlines,respectively.Fullandopencirclesrepresentcoreandactiveelectronsofthereferencedeterminantji(aclosed-shellreferencejiisassumed).(3.6),whileusingactivespin-orbitalstotheremaininghigher-orderTnandRmany-bodycomponentswithˆ<n˙.Thefreedomtochoosethehighest-orderoperatorsT˙andRandwhichoperatorstotreatexactlyandwhichapproximatelyusingactiveorbitalsleadstoahierarchyoftheactive-spaceCC/EOMCCmethods.Thesimplestofthesemethodsthatexplicitlyincorporateshigher-orderclusterandex-citationcomponentsbeyondthoseCCSDandEOMCCSDaretheCCapproacheswithsingles,doubles,andactive-spacetriples(CCSDt/EOMCCSDt),oftentermed\littlet".Inthiscase,wesolvetheSRCCamplitudeequations,Eq.(3.11),inwhichT(A)isreplacedbyT(CCSDt)=T1+T2+t3(3.37)andH(A)byH(CCSDt)=eT(CCSDt)HeT(CCSDt)=HeT(CCSDt)C;(3.38)33whereT1=Xi;atiaEai;(3.39)T2=Xi>j;a>btijabEabij;(3.40)andt3=XI>j>k;a>b>CtIjkabCEabCIjk:(3.41)Inotherwords,wesolvetheCCSDTequationsusingallsingles,alldoubles,andasubsetoftriplesbyEq.(3.41).Thetia,tijab,andtIjkabCamplitudesareobtainedbysolvingtheSRCCsystem,Eq.(3.11),inwhichT(A)=T(CCSDt),inasubspaceofthemany-electronHilbertspace,H,spannedbythesinglyexciteddeterminantsjaii=Eaiji,doublyexciteddeterminantsjabiji=Eabijji,andselectedtriplyexciteddeterminantsjabCIjki=EabCIjkjithatcorrespondtotheexcitationsinT(CCSDt),referredtoasH(t).Intermsofcore,active,andvirtualorbitalsdescribedaboveandshowninFig.3.1,thisisequivalenttoprojectingtheconnectedclusterformoftheScodingerequationwithT=T(CCSDt)onallsingly,alldoubly,andselectedtriplyexciteddeterminantsthatbelongtoclasses1{9onTable3.1.Inthecaseofexcitedstates,thecorrespondingEOMCCSDtanalogofCCSDtisbyreplacingtheR(A)operatorenteringtheexcited-statewavefunctionansatz,Eq.(3.5),byR(CCSDt)=r01+R1+R2+r3;(3.42)whereR1=Xi;ariEai;(3.43)R2=Xi>j;a>brijEabij;(3.44)34andr3=XI>j>k;a>b>CrIjkCEabCIjk:(3.45)Theri,rij,andrIjkCamplitudesandthecorrespondingverticalexcitationenergies!(CCSDt)aredeterminedbydiagonalizingthesimilarity-transformedHamiltonianofCCSDt,H(CCSDt),byEq.(3.38),inthespacespannedbyjaii,jabiji,andjabCIjkidetermi-nantscorrespondingtoclasses1{9inTable3.1.Themainbofrestrictingthehigher-orderT3andR3componentstotheiractive-spacet3andr3variantsisthetsavingincomputationalcostscomparedtotheparentfullCCSDTandEOMCCSDTapproaches.Ifnoandnuarethenumbersofoccupiedandunoccupiedorbitals,respectively,usedinthecorrelatedcalculation,andifNo(<noor˝no)andNu(˝nu)designatethecorrespondingnumbersoftheactiveoccupiedandactiveunoccupiedorbitals,theCCSDtandEOMCCSDtapproachesreplacetheiterativen3on5uCPUstepsoffullCCSDT/EOMCCSDTbythemuchlessexpensiveNoNun2on4usteps(seeTable1.1).Inotherwords,whenproperlyimplemented,theCCSDtandEOMCCSDtmethodsreplacethecomputationallyexpensiveN8iterativeCPUstepsofCCSDT/EOMCCSDT,bytherelativelyinexpensivestepsequivalenttothecostsofCCSDorEOMCCSDcalculationsmultipliedbyasmallprefactorequaltothenumberofsingleexcitationsintheactivespace.Thelow-orderpolynomialgrowthoftheCPUtimerepresentedbytheNoNuprefactoralsomeansthatitistrivialtoimprovetheresultsorcheckiftheyareconvergedbyincreasingtheactivespace.ThisshouldbecontrastedwiththegenuineMRmethods,whichareusu-allybasedonamodelspacebytheCASapproach,whichresultindimensionalitiesthatscalefactoriallywiththenumbersofactiveorbitalsandelectrons,inadditiontoothercostsofpost-CASSCFcalculations.Similarappliestomemoryrequirements,whichreduces35Table3.1:ThecompletesetofprojectionsenteringtheCCSDTQamplitudeequationsorganizedaccordingtotheiractiveandinactivecharacter.Upper-caseboldlettersrepresentactivelabelsandlower-caseboldlettersrepresentinactivelabels(cf.thetextandFig.3.1fordetails).ProjectionSingleDoubleTripleQuadruple1hAIjhABIJjhABCIJKjhABCDIJKLj2haIjhaBIJjhaBCIJKjhaBCDIJKLj3hAijhABIjjhABCIJkjhABCDIJKlj4haijhaBIjjhaBCIJkjhaBCDIJKlj5habIJjhabCIJKjhabCDIJKLj6habIjjhabCIJkjhabCDIJKlj7hABijjhABCIjkjhABCDIJklj8haBijjhaBCIjkjhaBCDIJklj9habijjhabCIjkjhabCDIJklj10habcIJKjhabcDIJKLj11habcIJkjhabcDIJKlj12habcIjkjhabcDIJklj13hABCijkjhABCDIjklj14haBCijkjhaBCDIjklj15habCijkjhabCDIjklj16habcijkjhabcDIjklj17habcdIJKLj18habcdIJKlj19habcdIJklj20habcdIjklj21hABCDijklj22haBCDijklj23habCDijklj24habcDijklj25habcdijkljfromthe˘n3on3urequirementsofCCSDT/EOMCCSDT,relatedtotheneedtostorealltriplyexcitedamplitudes,tothemuchmoremanageable˘NoNun2on2urequirementofCCS-Dt/EOMCCSDt.Whilethet3andr3operatorspresentedaboverequireaminimumofoneoccupiedandoneunoccupiedindextoberestrictedtotheactiveorbitals,onecanalwayscon-sideralternativewheremorespin-orbitalindicesarefromtheactiveset.Onecan,forexample,considertheso-calledCCSDt(II)[118,163,275]andCCSDt(III)[118,163,275]36orCCSDt0[242,243]approachesandtheirexcited-stateanalogs(asanote,theCCSDtap-proximationpresentedaboveisalternativelyreferredtoasCCSDt(I)[118,163,275]).Thesavingsthatsuchactive-spacemethodsprovideareevengreaterthanthosebytheaboveCCSDt/EOMCCSDtmodel.TheyinthechoiceoftheleveloftruncationinTandRandwhichoperatorstotreatexactlyandwhichtotreatapproximatelyallowsustoextendtheactive-spaceconsidera-tionsbeyondCCSDt/EOMCCSDttoincludeotherhigh-ordercorrelations,suchasquadru-ples,whichareofparticularinterestinthiswork.IntheCC/EOMCCapproacheswithsingles,doubles,andactive-spacetriplesandquadruples(CCSDtq/EOMCCSDtq),termed\littletq",thetruncatedclusteroperatortheground-statewavefunctionisasfollows:T(CCSDtq)=T1+T2+t3+t4;(3.46)andH(A)enteringtheSRCCamplitudeequations,Eq.(3.11),isreplacedbyH(CCSDtq)=eT(CCSDtq)HeT(CCSDtq)=HeT(CCSDtq)C;(3.47)whereT1,T2,andt3arebyEqs.(3.39){(3.41)andt4=XI>J>k>l;a>b>C>DtIJklabCDEabCDIJkl:(3.48)TheclusteramplitudesaredeterminedbysolvingEq.(3.11),inwhichT(A)isreplacedbyT(CCSDtq)inthespacespannedbythejaii,jabiji,jabCIjkiandjabCDIJklideterminants,usingtheprojections1{9inTable3.1.Inthecaseoftheexcited-stateEOMCCSDtqanalogofCCSDtq,wereplacetheR(A)operatorenteringtheEOMCCwavefunctionansatz,Eq.(3.5),37byR(CCSDtq)=r01+R1+R2+r3+r4;(3.49)whereR1,R2,andr3arebyEqs.(3.43){(3.45)andr4=XI>J>k>l;a>b>C>DrIJklCDEabCDIJkl:(3.50)TheexcitationamplitudesR(CCSDtq)andthecorrespondingverticalexcitationenergies!(CCSDtq)aredeterminedbydiagonalizingthesimilarity-transformedHamiltonianofCCSDtq,H(CCSDtq),givenbyEq.(3.47),inthespacespannedbythejaii,jabiji,jabCIjkiandjabCDIJklideterminants,usingprojections1{9inTable3.1.Wecanalsotreatthetri-excitedcomponentsofTandRexactly,whileapproximatingthequadruplescomponents,T4andR4,approximatelyusingactiveorbitals,whichishowtheCCSDTqandEOMCCSDTqapproachesareIntheCCSDTq/EOMCCSDTqmethodology,thetruncatedclusteroperatortheground-statewavefunctionisgivenbyT(CCSDTq)=T1+T2+T3+t4;(3.51)where,onceagain,T1andT2arebyEqs.(3.39)and(3.40),andT3istreatedfullyaswell,T3=Xi>j>k;a>b>c;tijkabcEabcijk;(3.52)butweuseactiveorbitalstothefour-bodyclustercomponents,t4=XI>j>k>l;a>b>c>DtIjklabcDEabcDIjkl:(3.53)38WethensolvetheSRCCamplitudeequations,Eq.(3.11),inwhichH(A)isreplacedbyH(CCSDTq)=eT(CCSDTq)HeT(CCSDTq)=HeT(CCSDTq)C;(3.54)inthespacespannedbythejaii,jabiji,jabcijkiandjabcDIjklideterminants,usingprojections1{16inTable3.1,andifweareinterestedincalculatingexcitedstatesusingEOMCCSDTq,wereplacetheR(A)operatorenteringtheEOMCCwavefunctionansatz,Eq.(3.5),byR(CCSDTq)=r01+R1+R2+R3+r4;(3.55)whereR1andR2arebyEqs.(3.43)and(3.44),R3=Xi>i>k;a>b>crijkEabcijk;(3.56)andr4=XI>j>k>l;a>b>c>DrIjklDEabcDIjkl:(3.57)TheexcitationamplitudesofEOMCCSDTqandthecorrespondingverticalexcitationener-gies!(CCSDTq)aredeterminedbydiagonalizingthesimilarity-transformedHamiltonianofCCSDTq,H(CCSDTq),byEq.(3.54),inthespacespannedbythejaii,jabiji,jabcijkiandjabcDIjklideterminants,usingprojections1{16inTable3.1.AsshowninTable1.1,thecomputationalsavingsbytheactive-spaceCCSDtq/EOMCCSDtqandCCSDTq/EOMCCSDTqapproachescanbehugecomparedtothepar-entCCSDTQandEOMCCSDTQmethods.ThemostexpensiveCPUstepsofthefullCCSDtqapproach,asdescribedabove,scaleasN2oN2un2on4u,i.e.,asasmallprefactorpro-39portionaltothenumberofdoubleexcitationswithintheactivespacetimesthen2on4ustepsofCCSD/EOMCCSD.SinceNo<no(or˝no)andNu˝nu,weobtainatsavingincomputercomparedtothen4on6ustepsofCCSDTQ/EOMCCSDTQ.Sim-ilarappliestothestoragerequirementsforthetriplyandquadruplyexcitedamplitudes,whichare˘n3on3uand˘n4on4u,respectively,intheCCSDTQ/EOMCCSDTQcase,butonly˘NoNun2on2uand˘N2oN2un2on2uinthecaseofCCSDtq/EOMCCSDtq.TheCCSDTqandEOMCCSDTQapproachespresentedabovealsoreducethecomputationalcomparedtoCCSDTQ/EOMCCSDTQ,withCPUstepsthatscaleasNoNun3on5u,i.e.,thenumberofsingleexcitationsinactivespacetimesthen3on5ustepsoftheCCSDTtype,andstoragerequirementsforquadruplesthatare˘NoNun3on3u.Indesigningthebestandmostcom-pleteCCSDtq/EOMCCSDtqandCCSDTq/EOCCSDTqapproaches,asdescribedabove,onefollowsthegeneralrecipeelaboratedonbelow,butonecanalwaysreducethecostsoftheCCSDtq/EOMCCSDtqandCCSDTq/EOMCCSDTqcalculationsfurtherbyrestrictingmoreoccupiedandunoccupiedindicesofthehigher-orderclusterandexcitationoperatorstotheactivespace,muchlikethecaseofthetvariantsofCCSDt.Infact,whilethequadrupleexcitationscorrespondingtoCCSDTqpresentedabovearebyamin-imumofoneactiveoccupiedandoneactiveunoccupiedindices,theinitialimplementationsofCCSDTq[243,276]employedat4operatorbyaminimumoftwoactiveoccupiedandtwoactiveunoccupiedindices.Onceagain,andunlikethegenuineMRmethods,thelow-orderpolynomialgrowthrepresentedbytheprefactorsofCCSDtqandCCSDTqmeansitisrathereasytoimproveourdescriptionofasystemofinterestbyincreasingtheactivespace.Thesuccessofactive-spaceSRCCmethodsinapplicationsinvolvingquasi-degeneratesituations,suchasbond-breaking,isrationalizedbyrecognizingthattheactive-spaceSRCC40frameworkmimicsagenuineMRCCdescription.Ingeneral,asshowninRef.[240],theactive-spaceSRCCtheorycorrespondstotheMRCCformalisminwhichweconsideruptoˆ-tupleexcitationsfromthemulti-dimensionalmodelspaceM0=spannji;A1I1E;A1A2I1I2E;:::;A1:::A˝I1:::I˝EoI1>I2>>I˝;A1>A2>>A˝(3.58)spannedbyallpossibleatmost˝-tupleexcitationsfromthereferencedeterminantji.InthatcasetheclusteroperatorToftheSRCCtheoryshouldhavetheformT=T1++Tˆ+tˆ+1++tˆ+˝;(3.59)whereT1;:::;Tˆaretreatedfully,tocapturealldynamicalcorrelationsoutsideM0,andtˆ+=XI1>>I>i1>>iˆa1>>aˆ>A1>>AtI1:::Ii1:::iˆa1:::aˆA1:::AEa1:::aˆA1:::AI1:::Ii1:::iˆ(3.60)for=1,:::,˝.InthefullCCSDtmethod,ˆ=2and˝=1,meaningthatCCSDtisanalogoustoMRCCmethodsinwhichweconsiderallsingleanddoubleexcitationsfromthemodelspacespannedbythereferencedeterminantjiandthesinglyexciteddeterminantsintheactivespace(plusalltheirproducts).InthefullCCSDtqapproach,ˆ=2and˝=2,correspondingtoallsingleanddoubleexcitationsfromthemodelspacespannedbythereferencedeterminantjiandthesinglyanddoublyexciteddeterminantsrelativetojiintheactivespace,whileinthefullCCSDTqapproachpresentedaboveˆ=3and˝=1,correspondingtoallsingle,double,andtripleexcitationsfromthemodelspacespannedbythereferencedeterminantjiandthesinglyexciteddeterminantsintheactivespace(plustheirvariousproducts).Whilethisisnotafocusofthiswork,itisimportanttomentionthattheabovegeneralizedtreatmentoftheclusteroperatortomimicMRCCalsoallowsusto41consideractive-spaceanalogsoftheSRCCmethodswithhigher-than-quadrupleexcitations.Wehavenowreviewedtheactive-spaceCC/EOMCCapproachesandtheMMCC-basedCR-CC/CR-EOMCCcorrections,andwealsoknowfromthediscussionintheIntroductionthatbothofthesemethodologiesfacechallengesinsomeMRsituations,suchaspotentialen-ergysurfacesinvolvingcertaintypesofbiradicaltransitionstates,soournexttaskistoshowhowtomergetheCR-CC/CR-EOMCCandactive-spaceCC/EOMCCmethodologiesintoasingleformalismwhich,asshowninthisthesis,enablesonetocalculatepotentialenergysurfacesalongbondbreakingcoordinatesandsinglet{tripletgapsinchallengingbiradicalspeciestowithinfractionsofamillihartreerelativetothehigh-levelCC(e.g.,CCSDTQ)orexact(fullCI)dataatthesmallfractionofthecomputercost.3.1.3TheCC(P;Q)FormalismTheoverallpurposeoftheCC(P;Q)theoryistocorrecttheresultsofCC/EOMCCcalcu-lationsinasubspaceoftheN-electronHilbertspace,referredtoastheP-space,fortheelectroncorrelationinthesubspacecalledtheQ-space.Forexample,iftheP-spaceisspannedbyallsinglyanddoublyexciteddeterminants,andtheQ-spacebythetriplyexciteddeterminants,thenweendupwiththepreviouslydiscussedCR-CC(2,3)approach.IftheP-spaceisspannedbyallsingles,doubles,andtriples,andtheQ-spacebyallquadru-ples,weobtainCR-CC(3,4),etc.However,bybeingmoreintheoftheP-andQ-spacesintheCC(P;Q)theory,wecancorrecttheresultsofCCorEOMCCcal-culationsusingnon-traditionaltruncationsintheclusterandexcitationoperatorsfortheelectroncorrelationmissinginsuchnon-traditionalcalculations.AnexampleofthetheCC(P;Q)methodrelevanttothisworkistheCC(t;3)approach,whichwediscussbelow,wheretheP-spaceisspannedbysingles,doubles,andasubsetoftriples,asbythe42active-spaceCCSDtapproach,andtheQ-spaceisspannedbytheremainingtriplesmissinginCCSDtbutpresentinCCSDT.Or,wecouldcorrecttheactive-spaceCCSDtqenergyforthemissingtriplesandquadruples,asintheCC(t,q;3,4)approach,alsodiscussedbelow.Inordertointroducethesemethods,wehavetotheunderlyingmathematicalconcepts.WebeginwiththeformaloftheP-spaceCC/EOMCCcalculations.LetusdesignateasubspaceoftheN-electronHilbertspaceofinterest,referredtoastheP-space,denotedH(P),whichisspannedbytheexciteddeterminantsjKi=EKji,whereEKistheelementaryexcitationoperatorwhichgeneratesjKifromthereferencedeterminantji.ThecorrespondingtruncatedoftheTandRoperatorsfortheP-spaceCC/EOMCCcalculations,designatedbyT(P)andR(P),respectively,canbewrittenasT(P)=XjKi2H(P)tKEK(3.61)andR(P)=R(P)0+R(P)open=r01+XjKi2H(P)rEK;(3.62)wheretKandraretheclusterandexcitationamplitudesobtainedintheCC/EOMCCcalculationsintheH(P)subspace.InanalogytotheconventionalSRCCformalism,thetKamplitudesT(P)aredeterminedbysolvingthesystemofnon-linear,energy-independentequations,hKjH(P)openji=0;jKi2H(P);(3.63)obtainedbyprojectingtheelectronicScodingerequationonthedeterminantsjKibe-longingtotheP-spaceH(P).Thecorrespondingground-stateenergyiscomputedusing43thefollowingexpression:E(P)0=hjH(P)ji;(3.64)whereH(P)=eT(P)HeT(P)=(HeT(P))C(3.65)istherelevantsimilarity-transformedHamiltonianandH(P)open=H(P)H(P)closed=H(P)E(P)01isthe`open'partofH(P)usingdiagramsofH(P)thathaveexternalFermionlines.Weobtaintheexcited-stateinformation,particularly,theamplitudesrR(P)andthecorrespondingexcited-stateenergiesE(P),bydiagonalizingthesimilarity-transformedHamiltonianH(P)inH(P)andsolvingthecorrespondingP-spaceEOMCCequations:hKj(H(P)openR(P)open)Cji=!(P)r;jKi2H(P);(3.66)where!(P)=E(P)E(P)0:(3.67)Theremainingr0zero-bodycotiscalculatedusingtheP-spaceanalogofEq.(3.16).TheabovesetofequationsencompassesvariouskindsofCCandEOMCCmethods,includingstandardandnonstandardtruncationschemesoftheTandRoperators.Ifwestoppedatthispoint,wewouldnotknowhowimportantthecorrelationoutsidetheP-spaceare.Thus,theonlywaytochecktheconvergencewouldbetoin-creasetheP-spacebyaddingmoredeterminantstoit,whichcanbeveryexpensive.TheCC(P;Q)theoryabetterandmoreeconomicalsolution.Thus,oncetheaboveP-spaceCC/EOMCCequationshavebeensolvedandtheinformationaboutthetruncatedformsofTandR,givenbyEqs.(3.61)and(3.62),obtained,wecorrecttheresultingen-44ergies,E(P),forthecorrelationinvolvingexciteddeterminantsjKifromanothersubspaceH(Q)(H(0)H(P))?,referredtoastheQ-space(H(0)istheone-dimensionalsubspacespannedbythereferencedeterminantji).ThisisaccomplishedbygeneralizingthepreviouslydiscussedbiorthogonalMMCCexpansion,givenbyEq.(3.27),toobtaintheQ-space-correctedenergiesE(P+Q),inthefollowingmanner[163,164]:E(P+Q)E(P)+(P;Q);(3.68)wherethenon-iterativecorrection(P;Q)duetotheQ-spacecontributionsmissingintheP-spaceCC/EOMCCcalculationsisas(P;Q)=XjKi2H(Q)rank(jKi)min(N(P);(Q))`(P)M(P);(3.69)withM(P)=hKj(H(P)R(P))ji(3.70)representingthegeneralizedmomentsoftheCC/EOMCCequationscorrespondingtothecalculationswithT=T(P)andR=R(P),associatedwiththeprojectionsoftheseequa-tionsonthejKideterminantsfromtheQ-spaceH(Q).Asinthecaseoftheconsiderationsdiscussedintheprevioussections,weformallyR(P)=0=1,sothatEq.(3.70)andtheequationsbelowcovertheground-state(=0)caseandexcited(>0)states.ThesymbolN(P)inEq.(3.69)isthehighestpossiblemany-bodyrankoftheexciteddeterminantjKirelativetothereferencejiforwhichthegeneralizedmomentsM(P)oftheP-spacecalculations,givenbyEq.(3.70),arestillstillnon-zero.Thesymbol(Q)inEq.(3.69)isthehighestmany-bodyrankoftheexciteddeterminantsjKiincludedinH(Q).For45example,ifwewanttouseEq.(3.69)tocorrecttheground-stateCCSDtenergiesforthetriplesmissinginCCSDt,asinCC(t;3),N(P)=10,but(Q)=3,sothat,asexpected,min(N(P);(Q))=3.IfwewanttouseittocorrectCCSDenergiesfortriplesandquadru-ples,asinCR-CC(2,4),N(P)=6,but(Q)=4,somin(N(P);(Q))=4,etc.AsshowninRef.[163],theQ-space-correctedenergyE(P+Q)byEq.(3.68)canbederivedbyapproximatingitsexactanalog,similartoEqs.(3.26)and(3.27),whichthe(P)betweenthefullCIenergiesEandthecorrespondingenergiesE(P)obtainedintheP-spaceCC/EOMCCcalculations,(P)EE(P)=XjKi2(H(0)H(P))?rank(jKi)N(P)`(P)M(P);(3.71)whichcanbederivedbyconsideringtheasymmetricenergyexpressionE=hjHR(P)eT(P)ji=hjR(P)eT(P)ji(3.72)andreplacingtheexactbrastatehjbytheP-spaceanalogofEq.(3.21):hj=hjLeT(P);(3.73)whereL=L(P)+L(P);(3.74)withL(P)=01+XjKi2H(P)`(EK)y(3.75)46andL(P)=XjKi2(H(0)H(P))?`(EK)y:(3.76)OneobtainsEq.(3.68)bylimitingthesummationovertheexciteddeterminantsjKifromtheorthogonalcomplementofH(0)H(P)inEq.(3.71)tothedeterminantsthatbelongtoH(Q)whichisasubspaceof(H(0)H(P))?,whilereplacingtheexactvaluesofthe`amplitudesenteringEq.(3.71),whichonecanobtainbysolvingthesimilarity-transformedformofthebraScodingerequation,hjLH(P)=EhjL;(3.77)intheentiremany-electronHilbertspaceH,bytheirapproximate`(P)valuesresultingfromconstrainingEq.(3.77)tothetotalsubspaceofinterest,i.e.,H(0)H(P)H(Q),basedontheinformationobtainedintheprecedingP-spaceCC/EOMCCcalculations.InanalogytotheCR-CC/CR-EOMCCmethodsdiscussedinSection3.1.1,inordertocomeupwithpracticalschemesEqs.(3.68){(3.70),wehavetoproposeaprocedurewhichwouldenableonetodeterminethe`(P)amplitudesthatmultiplymomentsM(P)intheofthecorrection(P;Q)inacomputationallymanageablefashion.Onesuchprocedure,proposedinRef.[163]andadoptedinthisthesis,isbasedonthereal-izationthatEqs.(3.73){(3.76)arereminiscentoftheexpressionsthattheCC/EOMCCbrastatesh~(P)j=hjL(P)eT(P);(3.78)resultingfromtheP-spaceCC/EOMCCcalculationsandformingabiorthogonalbasiswith47thecorrespondingCC/EOMCCketstatesj(P)i=R(P)j(P)0i=R(P)eT(P)ji:(3.79)Thus,wecanapproximatethedeexcitationoperatorLthatservesasasourceofamplitudes`(P)bysplittingitintotheknown,aprioridetermined,P-spacecomponentL(P)andtheunknowncomponentL(Q)thatprovidesinformationaboutthedesired`(P)amplitudescorrespondingtothedeterminantsjKi2H(Q)andenteringthecorrection(P;Q),Eq.(3.69).Theformercomponent,givenbyL(P)=L(P)0+L(P)open01+XjKi2H(P)l(EK)y;(3.80)isobtainedintheusualwaybysolvingthetruncated,P-space,leftCC/EOMCCeigenvalueproblem(cf.Eq.(3.77)),0hjH(P)openjKi+hjL(P)openH(P)openjKi=!(P)l;jKi2H(P):(3.81)TheunknownL(Q)contributiontotheLoperator,whichisnowapproximatedinthefollowingmanner:LˇL(P)+L(Q);(3.82)whereL(Q)=XjKi2H(Q)`(P)(EK)y;(3.83)canbeobtainedbyexaminingtheprojectedformoftheexactlefteigenvalueproblemgiven48byEq.(3.77)withintheH(Q)subspaceandexploitingtheinformationobtainedintheprecedingP-spaceCC/EOMCCcalculations.Indeed,byreplacingtheexactLoperatorEq.(3.77)byitsapproximateformgivenbyEq.(3.82)andbyrightprojectingtheresultingequationonthejKideterminantsfromH(Q),wecanshowthatthe`(P)amplitudesL(Q)satisfythesystemofcoupledlinearequationsthatlooksasfollows:hjL(P)H(P)jKi+XjK0i2H(Q)hK0jH(P)jKi`0(P)=E`(P);jKi2H(Q):(3.84)IfwefurtherapproximatetheexactenergyEinEq.(3.84)bytheenergyE(P)resultingfromtheprecedingP-spaceCC/EOMCCcalculationsandmovethediagonal,K0=K,contributionstothesummationontheleft-handsideofEq.(3.84)totheright-handside,asintheEpstein{Nesbetpartitioning,weobtainhjL(P)H(P)jKi+XjK0i2H(Q);K06=KhK0jH(P)jKi`0(P)=D(P)`(P);jKi2H(Q);(3.85)whereD(P)=E(P)hKjH(P)jKi(3.86)isthecorrespondingperturbativedenominator,analogoustothatusedintheCR-CC(2,3)andotherCC(ma,mb)approximations(seeinEq.(3.31)inSection3.1.1.).ThesystembyEq.(3.85)containscouplingsamongthejKideterminantsfromH(Q),bythematrixelementsofH(P)intheH(Q)subspace,andwecanattempt49tosolveititeratively.Thiswouldconstituteoneofthepossiblewaysofdeterminingthedesired`(P)amplitudesthatenterthecorrections(P;Q),Eq.(3.69).Alternatively,wecouldfollowthephilosophyoftheCR-CC(2,3)andotherCR-CC(ma,mb)schemesandignoretheelementsofthematrixrepresentingH(P)intheQ-spaceandreplacetheexpressiongivenbyEq.(3.85)bythesimplenon-iterativeformula`(P)=hjL(P)H(P)jKi=D(P);(3.87)anduseEq.(3.85),limitedtosmallblocksoftheH(P)matrixinvolvingthedegenerateQ-spacedeterminantsjKi,onlyifthereareorbitaldegeneraciesandonlyifweareinterestedinmaintainingthestrictinvarianceoftheresultingenergiesE(P+Q)withrespecttorotationsamongdegenerateorbitalsrelevanttonon-Abeliansymmetriesonly.Eqs.(3.68){(3.70),with`(P)byEq.3.87arethebasisexpressionsfortheCC(P;Q)methodsusedinthisthesisproject.TheCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)approacheswhicharebasedontheseexpressionandwhichareexploitedandfurtherdevelopedinthework,arediscussednext.3.1.4CC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)HierarchyWhilethegeneralformalismdescribedintheprevioussectionwaslaiddownforbothgroundandexcitedstates,thedetailedworkingequationsdescribingtheCC(P;Q)-basedCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)methodspresentedbelowfocusonground-stateconsid-erations,sincethisthesisprojectdealswiththedevelopmentofmethodsforquasidegenera-ciesinthegroundelectronicstatesorlowest-energystatesofagivenmultipliciity.Theground-stateCC(t;3)approach,initiallyimplementedandappliedbyDr.JunShen50fromourgroup[163{165]andreimplementedhere,startsbysolvingtheCCSDtamplitudeequations(seeEqs.(3.11)and(3.37){(3.41))anddeterminingtheCCSDtenergy,E(CCSDt)0=hjH(CCSDt)ji:(3.88)Then,followingthegeneralCC(P;Q)recipediscussedinSection3.1.2,weproceedtothecalculationofthedesiredCC(t;3)ground-stateenergyE(CC(t;3))0asE(CC(t;3))0=E(CCSDt)0+0(t;3);(3.89)where,thenon-iterativecorrection0(t;3)duetothetriplesmissingintheCCSDtconsid-erationisdeterminedusingtheformula(cf.Eqs.(3.68){(3.70))0(t;3)=Xjabcijki2H(T)H(t)`abc0;ijk(CCSDt)Mijk0;abc(CCSDt);(3.90)withtheMijk0;abc(CCSDt)momentsas:Mijk0;abc(CCSDt)=habcijkjH(CCSDt)ji:(3.91)The`abc0;ijk(CCSDt)amplitudesenteringEq.(3.90)arecalculatedas(cf.Eq.(3.87))`abc0;ijk(CCSDt)=hjL(CCSDt)0H(CCSDt)jabcijki=Dijk0;abc(CCSDt);(3.92)whereDijk0;abc(CCSDt)=E(CCSDt)0+habcijkjH(CCSDt)jabcijki:(3.93)51ThesummationovertriplyexciteddeterminantsjabcijkiinEq.(3.90)excludesthosethatarealreadyincludedintheCCSDtcalculations,whicharelistedonTable3.1asclasses10{16.ThedeexcitationoperatorL(CCSDt)0,enteringEq.(3.92)andgivenbyL(CCSDt)0=1+L0;1+L0;2+l0;3;(3.94)whereL0;1andL0;2aretheregularone-andtwo-bodycomponentsofL(CCSDt)0usingallcorrelatedspin-orbitalsandl0;3=XI>j>k;a>b>ClabC0;IjkEIjkabC(3.95)istheapproximatetriplydeexcitedactive-spacecomponent,whichtheCCSDtbrastateh~(CCSDt)0j=hjL(CCSDt)0eT(CCSDt);(3.96)isobtainedbysolvingtheleft-eigenstateCCSDtequations.InanalogytotheCR-CC(2,3)approachdiscussedinSection3.1.1,theEpstein{Nesbet-likeDijk0;abc(CCSDt)denominatorenteringEq.(3.92)andgivenbyEq.(3.93)themostcompletevariantDofCC(t;3)abbreviatedasCC(t;3)D.IfwereplacetheDijk0;abc(CCSDt)denominatorgivenbyEq.(3.93)bythecorrespondingM˝ller{Plesset-likeexpression,Dijk0;abc(CCSDt)=i+j+kabc;(3.97)wherepisthesingle-particleenergyassociatedwithspin-orbitalp(thediagonalelementsoftheFockmatrix)weendupwithvariantAofCC(t;3)labeledCC(t;3)A.Asshownin52Table1.1,theCC(t;3)AamdCC(t;3)DmethodshaveaCPUtimescalingsofNoNun2on4uintheiterativeactive-spaceCCSDtpartand˘n3on4uinthecalculationofthecorrespondingtriplescorrection0(t;3).InextendingtheCC(P;Q)hierarchytoincludeconnectedquadrupleexcitations,whichisthemainobjectiveofthisthesisproject,wecanstartbysolvingtheCCSDtqamplitudeequationsanddeterminingtheCCSDtqenergy,E(CCSDtq)0=hjH(CCSDtq)ji;(3.98)andthencorrecttheE(CCSDtq)0forthosetriplesandquadruplesthataremissinginCCSDtqtoobtaintheCC(t,q;3,4)energyE(CC(t;q;3;4))0asfollows:E(CC(t;q;3;4))0=E(CCSDtq)0+0(t;q;3;4);(3.99)where0(t;q;3;4)=Xjabcijki2H(T)H(t)`abc0;ijk(CCSDtq)Mijk0;abc(CCSDtq)+Xjabcdijkli2H(Q)H(q)`abcd0;ijkl(CCSDtq)Mijkl0;abcd(CCSDtq);(3.100)withtheMijk0;abc(CCSDtq)andMijkl0;abcd(CCSDtq)momentsgivenbyMijk0;abc(CCSDtq)=habcijkjH(CCSDtq)ji(3.101)53andMijkl0;abcd(CCSDtq)=habcdijkljH(CCSDtq)ji;(3.102)respectively.TheH(T)H(t)andH(Q)H(q)subspacesenteringEq.(3.100)arespannedbythetriplyandquadruplyexciteddeterminantsotherthanthosethatbelongtothejabcijkiandjabcdijklicategoriesincludedintheCCSDtqcalculations.Tobemoreprecise,thedeterminantsenteringtheH(T)H(t)andH(Q)H(q)subspacesarethosethatbelongtoclasses10{25inTable3.1.The`abc0;ijk(CCSDtq)and`abcd0;ijkl(CCSDtq)amplitudesenteringEq.(3.100)arecalculatedas`abc0;ijk(CCSDtq)=hjL(CCSDtq)0H(CCSDtq)jabcijki=Dijk0;abc(CCSDtq)(3.103)and`abcd0;ijkl(CCSDtq)=hjL(CCSDtq)0H(CCSDtq)jabcdijkli=Dijkl0;abcd(CCSDtq);(3.104)wheretheL(CCSDtq)0deexcitationoperatorresultingfromtheleft-eigenstateCCSDtqcalcu-lationsisasL(CCSDtq)0=1+L0;1+L0;2+l0;3+l0;4;(3.105)withl0;3givenbyEq.(3.95)andl0;4=XI>J>k>l;a>b>C>DlabCD0;IJklEIJklabCD:(3.106)IntheformallymostcompleteformulationoftheCC(t,q;3,4)approach,abbreviatedasCC(t,q;3,4)DD,thedenominatorsthatenterEqs.(3.103)and(3.104)aregivenbythe54Epstein{Nesbet-likeexpressions,Dijk0;abc(CCSDtq)=E(CCSDtq)0habcijkjH(CCSDtq)jabcijki(3.107)andDijkl0;abcd(CCSDtq)=E(CCSDtq)0habcdijkljH(CCSDtq)jabcdijkli:(3.108)Wecan,however,contemplateCC(t,q;3,4)schemes,whereoneorbothdenomi-natorsDijk0;abc(CCSDtq)andDijkl0;abcd(CCSDtq)isorarereplacedbytheM˝ller{Plesset-likeexpressions,Dijk0;abc(CCSDtq)=i+j+kabc(3.109)andDijkl0;abcd(CCSDtq)=i+j+k+labcd:(3.110)IfweusetheM˝ller{PlessetformsofDijk0;abc(CCSDtq)andDijkl0;abcd(CCSDtq),theresult-ingCC(t,q;3,4)methodisabbreviatedasCC(t,q;3,4)AA.IfweusethethemorecompleteEpstein{NesbetformfortheDijk0;abc(CCSDtq)denominatorenteringthetriplespartofthecorrection0(t;q;3;4)andtheM˝ller{PlessetformforDijkl0;abcd(CCSDtq),thecor-rectionduetoquadruples,weendupwithCC(t,q;3,4)DA.ItdoesnotmakemuchsensetoconsidertheCC(t,q;3,4)ADscheme,whereDijk0;abc(CCSDtq)isbythelesscompleteM˝ller{PlessetexpressionandDijkl0;abcd(CCSDtq)bythemoreaccurateEpstein{Nesbetform,but,asalreadymentioned,wecanusetheEpstein{Nesbetdenominatorinthetriplesandquadruplespartsof0(t;q;3;4),asintheaforementionedCC(t,q;3,4)DDapproximation.Thereis,however,oneissuethatmakestheCC(t,q;3,4)DDschemelessattractive,namely,theissueofcomputercosts.AsshowninTable1.1,theCC(t,q;3,4)AAandCC(t,q;3,4)DA55approachesarecharacterizedbytheiterativeCPUstepsofCCSDtqthatscaleasN2oN2un2on4uandthenon-iterativestepsof0(t;q;3;4)thatscaleasn3on4uinthetriplespartofitandn2on5uinthequadruplespartofit.Inotherwords,theAAandDAvariantsofCC(t,q;3,4)replacetheiterativeN10stepsoffullCCSDTQbytheiterativeN6-likeandnon-iterativeN7-typeoperations.ThesituationwithCC(t,q;3,4)DDis,however,t(seeTable1.1).TheCC(t,q;3,4)DDmethodhasidenticalN6-likeiterativestepsandidenticalN7-likestepsinthetriplespartof0(t;q;3;4)asinthecaseoftheAAandDAapproximations,butitisconsiderablymoreexpensiveinthequadruplespartof0(t;q;3;4),increasingthen2on5uscalingoftheAAandDAvariantsofCC(t,q;3,4)toan4on5u(N9-like)level.Forthisreason,ourfocusinthisworkisontheCC(t,q;3,4)DAapproach,withCC(t,q;3,4)AAbeingapossiblealternative.Asshowninthisthesis,theCC(t,q;3,4)approachcanbeveryaccurate,producingtheCCSDTQ-levelresultsatthefractionofthecost,butonecanimprovetheCC(t,q;3,4)descriptionfurtherbytreatingtriplesfullyviafullCCSDTqandcorrectingtheCCSDTqcalculationsforthequadruplesoutsidethe\littleq"setusingtheCC(P;Q)-stylecorrectiontheCC(q,4)theory.Inthiscase,wesolvetheCCSDTqamplitudeequationsanddeterminetheCCSDTqenergy,E(CCSDTq)0=hjH(CCSDTq)ji;(3.111)andthencorrectE(CCSDTq)0forthequadruplesoutsidethe\littleq"setusingtheformulaE(CC(q;4))0=E(CCSDTq)0+0(q;4);(3.112)56where0(q;4)=Xjabcdijkli2H(Q)H(q)`abcd0;ijkl(CCSDTq)Mijkl0;abcd(CCSDTq);(3.113)withMijkl0;abcd(CCSDTq)=habcdijkljH(CCSDTq)ji(3.114)representingthecorrespondingmomentsofCCSDTqrepresentingtheprojectionsoftheCCSDTqequationsonthemissingquadruplyexciteddeterminants.The`abcd0;ijkl(CCSDTq)amplitudesenteringEq.(3.114)arecalculatedas`abcd0;ijkl(CCSDTq)=hjL(CCSDTq)0H(CCSDTq)jabcdijkli=Dijkl0;abcd(CCSDTq);(3.115)wheretheL(CCSDTq)0deexcitationoperator,obtainedintheleft-eigenstateCCSDTqcalcu-lations,isasL(CCSDTq)0=1+L0;1+L0;2+L0;3+l0;4(3.116)andthedenominatorDijkl0;abcd(CCSDTq)inthemostcompleteformulationofCC(q;4)isgivenbyDijkl0;abcd(CCSDTq)=E(CCSDTq)0habcdijkljH(CCSDTq)jabcdijkli:(3.117)Onceagain,thedenominatorDijkl0;abcd(CCSDTq),Eq.(3.117),istheEpstein{Nesbet-likeexpressioncorrespondingtovariantDofCC(q;4),denotedCC(q;4)D,butwecanalsoreplace57Eq.(3.117)bythecorrespondingM˝ller-Plesset-likeequation,Dijkl0;abcd(CCSDTq)=i+j+k+labcd;(3.118)resultingintheAvariantofCC(q;4),denotedCC(q;4)A.TheCPUtimescalingassociatedwiththeCC(q;4)methodisNoNun3on5uintheactive-spaceCCSDTqcalculationsandn2on5u,whenCC(q;4)Aisemployed,orn4on5u,whenCC(q;4)Disused,inthequadruplescorrectionpart.InanalogytoCC(t,q;3,4)DAvsCC(t,q;3,4)DD,theCC(q;4)Amodelispreferredduetoitslessexpensiven2on5u(N7-like)stepscomparedton4on5u(N9-like)stepsofCC(q;4)D:WecanalsoutilizetheyoftheCC(P;Q)methodologyandconsidermethods\inbetween".Forexample,wecanassumethatalmostalldynamicalandnon-dynamicalcorrelationsarecapturedbytheactive-spaceapproachwithselectedtripleandquadrupleexcitations(CCSDtq)andwethencorrectthecorrespondingCCSDtqenergyE(CCSDtq)0fortheremainingtriples,ignoringthequadruplesoutsidethe\littleq"setmissinginCCSDtq.TheresultingCC(t,q;3)approach,inwhichtheenergyiscalculatedasE(CC(t;q;3))0=E(CCSDtq)0+0(t;q;3);(3.119)where0(t;q;3)=Xjabcijki2H(T)H(t);`abc0;ijk(CCSDtq)Mijk0;abc(CCSDtq);(3.120)isstudiedinthisworkaswell,beingareasonablecompromisebetweentheCC(t;3)andCC(t,q;3,4)methods,asdiscussedinthefollowingsection.TheCPUtimescalinginthe58CC(t,q;3)approachisN2oN2un2on4uintheactive-spaceCCSDtqcalculationsandn3on4uasso-ciatedwiththenon-iterativetriplescorrectionpart.Theground-stateCC(t;3)methodwasimplementedandappliedbyDr.JunSheninourgroup[163{165].ThedevelopmentandapplicationoftheCC(t,q;3),CC(t,q;3,4),andCC(q;4)methodsarethebasisforthisdissertation,althoughtheCC(t;3)codeswerepro-ducedinthisprojectaswell.Inourpresentimplementation,discussedfurtherinChapter4,allofthesemethodsareinterfacedwiththeRHF,ROHF,andintegraltransformationroutinesavailableintheGAMESSpackage[277,278].Sincewehavenotyetdevelopedthecodesthatcouldsolvetheleft-eigenstateCCSDt,CCSDtq,orCCSDTqequations,whichwouldnormallybeneededtodeterminethe`a1:::an0;i1:::in(A)(A=CCSDt,CCSDtq,orCCSDTq),amplitudesfortheground-statecorrectionsdescribedabove,wehaveintroducedafewsimpli-intheCC(P;Q)routines.Thus,weapproximatethesimilarity-transformedHamil-toniansoftheactive-spaceCCSDt,CCSDtq,andCCSDTqmethods,Eqs.(3.38),(3.47),and(3.54),respectively,whichentertheground-statemoments,Mijk0;abc(CCSDt),Eq.(3.91),Mijk0;abc(CCSDtq),Eq.(3.101),Mijkl0;abcd(CCSDtq),Eq.(3.102),andMijkl0;abcd(CCSDTq),Eq.(3.114),bytheCCSD-likeH(A)(2)=eT1T2HeT1+T2=(HeT1+T2)C;(3.121)inwhichthehigher-than-two-bodycomponentsoftherelevantT(CCSDt),T(CCSDtq),andT(CCSDTq)operatorsareneglected,although{andthisneedstobestronglyemphasized{theT1andT2amplitudesenteringEq.(3.121)originatefromthetrueCCSDt,CCSDtq,orCCSDTqcalculations,i.e.,theyareproperlyrelaxedinthepresenceoft3orT3andt4.Moreover,wereplacethefullofL(CCSDt)0,L(CCSDtq)0,andL(CCSDTq)0,Eqs.59(3.94),(3.105),and(3.116),respectively,whichenterthe`abc0;ijk(CCSDt),`abc0;ijk(CCSDtq),`abcd0;ijkl(CCSDtq),and`abcd0;ijkl(CCSDTq)amplitudesintheoftheCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)corrections,byL(A)0(2)=1+L0;1+L0;2;(3.122)whereA=CCSDt,CCSDtq,andCCSDTq,inwhichthehigher-than-two-bodycomponentsofL(A)0areneglectedandtheone-andtwo-bodycomponents,L0;1andL0;2,respectively,areobtainedbysolvingthelefteigenvalueprobleminvolvingH(A)(2),Eq.(3.121),inthespaceofsingleanddoubleexcitations.Inthisway,weaccountfortherelaxationoftheT1,T2,L0;1,andL0;2amplitudesinthepresenceofthehigher-than-two-bodycomponentsoftheCCSDt,CCSDtq,andCCSDTqclusteroperators,whichbecomestwhenthelattercomponentsbecomelarge,asisthecaseinthebiradicalandbond-breakingregionsofthepotentialenergysurface,butavoidcomplexcomputationalstepsrelatedtothefulluseofH(CCSDt),H(CCSDtq),andH(CCSDTq)Hamiltonians.TheinitialstudiesinvolvingCC(t;3)inthisapproximatemanner,reportedinRefs.[163{165],andcalculationsperformedinthisthesis,haverevealedthattheCC(t;3)resultsmatchthetotalandrelativeenergeticsoffullCCSDTalmostexactly,despiteitstruncatedform,byEqs.(3.121)and3.122.Inotherwords,replacingthetruesimilarity-transformedHamiltonianofCCSDtbyH(CCSDt)(2)andthetruedeexcitationoperatorL(CCSDt)0byL(CCSDt)0(2),asinEqs.(3.121)and(3.122),hasnonoticeableonthecalculatedenergies.WecanexpectthatthesameappliestotheCC(t,q;3),CC(t,q;3,4),andCC(q;4)calculations,andallofourteststodateindicatethatusingEqs.(3.121)and(3.122)insteadofthefullformsofL(A)0andH(A)(2)isbutwewillreturntothistopicinthefuture,onceroutinesenablingthe60completetreatmentofthesimilarity-transformedHamiltoniansanddeexcitationoperatorsofCCSDt,CCSDtq,andCCSDTq,enteringtheCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)correctionsaredeveloped.AdditionaltechnicaldetailsofourtimplementationoftheCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)methods,especiallythelatterthreeapproachesandtheunderlyingCCSDtqandCCSDTqandparentCCSDTQcodes,aregiveninChapter4.Inthenextsection,weillustratetheaccuraciestheCC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationscanprovide,especiallywhencomparedtotheparentCCSDTandCCSDTQdata,andfullCI.OurCC(q;4)codeshavebeenusedbyusaswell,butsincetheyaremoreexpensive,needingCCSDTqcomputations,andsincetheyhavebeenmostlyusedbyMr.IliasMagoulasfromourgroup,wefocushereontheCC(t;3),CC(t,q;3),andCC(t,q;3,4)computations.3.2NumericalResults3.2.1TheC2v-SymmetricDoubleDissociationofWaterTheperformanceoftheCC(t;3),CC(t,q;3),andCC(t,q;3,4)approaches,particularlythelattertwomethodsthatdescribequadrupleswasexaminedandevaluatedbyapplyingthemtoafewmolecularproblemsforwhichtheexact,fullCI,ornearlyexact,CCSDTQ,solutionsareknownornottotogenerate.Ourfocusisonsituationswhereconnectedquadruplesareimportant,whichisthecasewhendoublebondsarebrokenorwhencertaintypesofchemicalreactionsandbiradicalsareexamined.Oneparticularlycommonexampleusedtotesttheperformanceofnewquantumchemistrymethods,especiallyCCmethodswithquadruples,isthedoubledissociationofwater,inwhichweexaminetheground-statepotentialenergysurfaceasbothO{Hbondsaresimultaneouslystretchedbyuptotwoor61eventhreetimestheequilibriumbondlengths.ThevariousCCcalculationsofthedouble-dissociationofH2OwereperformedwiththeDZ[279]andcc-pVDZ[237]basissets.ForthesmallerDZbasisset,theresultsareshowninTable3.2andFigure3.2.Inthiscase,theequilibriumgeometry,abbreviatedasRO-H=Re,andcorrespondingfullCIenergyweretakenfrom[280].ThegeometriesthatrepresentasimultaneousstretchingofbothO{Hbondsbyafactorof1.5(RO-H=1.5Re)and2(RO-H=2Re)withoutchangingthe\(H{O{H)angleandthecorrespondingfullCIenergiesweretakenfromRef.[281].TheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsincludedthe3a1,1b2,4a1,and2b2orbitals,whicharethetwohighestoccupiedandthetwolowestunoccupiedorbitalsatRO-H=2Re.TheCCSD(T)andCCSD(TQf)energiescanbefound,forexample,inRef.[141],theCCSDTenergiesinRef.[120],theCCSDtqresultsinRef.[241],andtheCCSDTQresultsinRef.[124],althoughallofthesevalueswererecalculatedbyus.Forthelargercc-pVDZbasisset,theresultsareshowninTable3.3andFigure3.3.Inthiscase,thecorrespondinggeometriescoveringtheRO-H=Re3ReregionandthefullCIenergiesweretakenfromRef.[282].Again,theactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsincludedthevalence3a1,1b2,4a1,and2b2orbitals.Inallcalculations,allelectronswerecorrelatedandforthecc-pVDZbasissetsphericalcomponentsofthedorbitalswereemployed.Inadditiontotheenergiesor,tobemoreprecise,errorsrelativetofullCCSDT,fullCCSDTQ,andfullCIateachgeometry,Tables3.2and3.3alsoreporttheoverallmaximumunsignederror(MUE)andnon-parallelityerror(NPE)valuescharacterizingthevariousmethods.StartingwiththeDZbasisset(seeTable3.2andFigure3.2),wecanseethatthenon-iterativeCCSD(T)methodperformswellatequilibrium,reproducingtheparentCCSDTvaluetoasmallfractionofamillihartree.However,thissituationchangesdrasticallyasthe62O{Hbondsarestretched.BythetimewereachRO-H=2RetheerrorintheCCSD(T)calculationsisabout5.5millihartreecomparedtoCCSDT,divergingquicklyfromit.WhenCCSD(T)iscomparedtofullCI,theerrorisevengreater(7.699millihartree).AsforCR-CC(2,3),theAvariantperformssimilartoCCSD(T)attheequilibrium,andalsodivergesfromCCSDTasbothO{Hbondsarestretched,buttheerrorsarenotassevereasinthecaseofCCSD(T),withCR-CC(2,3)AdeviatingfromCCSDTbyabout2.7millihartreeatRO-H=2Re.TheDvariantofCR-CC(2,3)greatlyimprovesalloftheresults,producingsmallerrorsrelativetofullCCSDTthatdonotexceed0.3millihartree.Unfortunately,theCCSDTmethoditselfbeginstobreakintheRO-H=2Reregion,i.e.,wehavetoconsidermethodswithamorerobusttreatmentofquadruplestoobtainfurtherimprovement.Theactive-spaceCCSDtmethodcomparedtoCCSD(T),CR-CC(2,3)A,andCR-CC(2,3)D,providesasmallerNPEvalueof0.056millihartreerelativetoCCSDT,butthisisaresultoferrorcancellationatthevariousgeometries,sinceerrorsintheCCSDtresultsrelativetoCCSDT,of0.583{0.639millihartree,arenotsmall,indicatingsomemissingtriplescorrelations.TheagreementofCCSDtwithCCSDTcanbemethodicallyimprovedbyincreasingtheactivespaceintheformercase,butinthisworkwechoosethealternative,lessexpensiveapproach,basedoncorrectingtheCCSDtenergiesforthecorrelationduetothemissingtriplesusingtheCC(t;3)approach.Whenwecorrecttheactive-spaceCCSDtenergiesforthemissingtriplesusingtheCC(t;3)method,weseeadramaticimprovementoverboththeactive-spaceandcompletelyrenor-malizedapproaches.TheCC(t;3)AandCC(t;3)Dmethodshavemaximumerrors(MUEvalues)of0.295and0.212millihartree,respectively,whencomparedtoCCSDT.Thatisamajorimprovementoverthemaximumerrorsof5.489and2.717millihartreerelativetofullCCSDTobservedintheCCSD(T)andtheCR-CC(2,3)Acalculationsandasmallim-63provementoverCR-CC(2,3)DwhichgivesMUE=0.293millihartree.Atthesametime,theCC(t;3)correctionsarecapableofimprovingtheCCSDtresults,reducingtheMUEvaluecharacterizingCCSDt,of0.639millihartree,toa0.2-0.3millihartreelevel.Inadditiontothelowermaximumerrors,bothvariantsofCC(t;3)improvetheresultsofCR-CC(2,3)andCCSDtcalculationsforallthreegeometriesalmostperfectly,reproducingthefullCCSDTenergiesateachpointonthepotentialenergysurface.Unfortunately,asalreadypointedoutaboveandasshowninFigure3.2(toppanel),CCSDTitselfistandbeginstofailasweapproachtheRO-H=2Reregion.SothefactthatCC(t;3)agreeswithitalmostperfectlyisnotsolvingtheproblem.Weneedtoincorporateconnectedquadruplyexcitedclustersanddoitinarobustmannertoobtainanaccuratedescriptionofdoublebonddissociationinwater.ForthethreegeometriesexaminedintheDZcase,wecanseethatfullCCSDTQisessen-tiallyexact,withamaximumerrorofonly0.141millihartreerelativetofullCI(asopposedto2.210millihartreeintheCCSDTcaseatRO-H=2Re).TheCCSD(TQf)approachworkswellattheequilibrium,but,inthesamefashionasCCSD(T),itdivergesfromfullCCSDTQandfullCIastheO{Hbondsarestretched,producingerrorsofover6millihartreerelativetoCCSDTQatRO-H=2Re.TheAAvariantofCR-CC(2,4),whichcorrespondstousingM˝ller{Plessetdenominatorsforthetriplesandquadruplescorrections,performsbetterthanCCSD(TQf),withamaximumerrorofjustover1millihartreerelativetoCCSDTQ,andCR-CC(2,4)DAworksinasimilarway,butnoneoftheCR-CC(2,4)approachescancompetewiththeactive-spaceCCSDtqmethodwhenitcomestotheNPEvalues.InanalogytothecaseofCCSDtvsCCSDT,theCCSDtqcalculationsarecharacterizedbytheverysmallNPEvaluesrelativetotheparentCCSDTQapproach,of0.259millihartree,asopposedtoNPEsofover1millihartreeobservedintheCR-CC(2,4)calculations.Theactive-spaceCCSDtq64methodprovidesconsistenterrorsrelativetoCCSDTQthatrangefrom0.970millihartreeatRO-H=Reto1.229millihartreeatRO-H=2Re,butsometriplesandquadruplescor-relationsareclearlymissing.Insteadofsimplyincreasingtheactive-spacetoimprovetheseresults,wecorrecttheCCSDtqenergiesforthemissinghigher-ordercorrelationbyemployingtheCC(P;Q)framework.WhenwecorrecttheCCSDtqresultsforjustthemissingtriplesviatheCC(t,q;3)ap-proach,wereducethemaximumerrorrelativetofullCCSDTQof1.229millihartreeto0.888and0.805millihartreewhentheCC(t,q;3)AandCC(t,q;3)Dmethodsareemployed.TheCCSDtq-basedCC(t,q;3)AandCC(t,q;3)Dmethodsalsohavesmallermaximumandnon-parallelityerrorsthantheCCSD-basedCR-CC(2,4)AAandCR-CC(2,4)DAapproxima-tions,whichshowsthattheuseofbetterT1andT2amplitudesconstructingtheMMCC-stylecorrectionshelpstheoverallaccuracy.WhenwecorrecttheCCSDtqvaluesforthemissingtriplesandquadruples,wefurtherimproveouragreementwithCCSDTQ.Forthethreegeometries,theMUEvaluesrelativetofullCCSDTQcharacterizingtheCC(t,q;3,4)AAandCC(t,q;3,4)DAcalculationsareonly0.412and0.329millihartree,respectively,andthecor-respondingNPEsareequallysmall.Wecan,thus,concludethatfortheDZbasisset,theCC(t,q;3)andCC(t,q;3,4)approachesreproducetheparentandvirtuallyexactCCSDTQresultstoafractionofamillihartree,atatinyfractionofthecomputationalcostaswegofromtheiterativeN10CPUstepsofCCSDTQtoN6-likeiterativestepsofCCSDtqplustherelativelyinexpensiveN7-likestepsassociatedwiththenon-iterativeCC(t,q;3)andCC(t,q;3,4)corrections.Thelargercc-pVDZbasissetalsoallowsustoexaminehowwellthevariousCCmethodsperformaswefurtherstretchtheO-Hbondlengthsto2.5Reand3Re(seeTable3.3andFig-ureFigure3.3.Onceagain,aswebreakbothbonds,theCCSD(T)methodquicklydiverges65fromfullCCSDT,producinghuge50millihartreeerrorrelativetoCCSDTatRO-H=3Re.TheCR-CC(2,3)AandCR-CC(2,3)Dmethods,especiallyCR-CC(2,3)D,aremuchmorewellbehaved,withmaximumerrorsof8.922and1.652millihartree,respectively,relativetoCCSDT,butthisisnottoohelpful,sinceCCSDTitselfcompletelyfailswhenRO-H>2Re,givingerrorsrelativetofullCIthatexceed40millihartreeintheRO-H=3Reregion.AsinthecaseoftheDZbasisset,theactive-spaceCCSDtapproachreducestheNPEvaluesrelativetofullCCSDTcomparedtoCR-CC(2,3)calculations,butthisisaresultofcan-cellingrathersubstantialerrorsthatrangefrom0.860to2.216millihartreewhentheentireRO-H=Re3Reregionisexamined,indicatingmissingtriples.Whenwecorrecttheactive-spaceCCSDtresultsforthemissingtripleexcitationsthatareexcludedinCCSDtusingtheCC(t;3)corrections,theerrorsatallgeometriesrelativetoCCSDTreducetosmallfractionsofamillihartreeontheorderof0.1{0.5millihartrees,resultinginequallyimpressiveNPEvalues,butwemustkeepinmindthatCCSDTitselfcompletelyfailswhenbothO-Hbondsarestretchedbeyond2Re,i.e.,weneedtoincorporateT4clustersanddoitinarobustmannertoobtainamoreaccuratedescription.AsshowninTable3.3,upto2Re,CCSDTQisvirtuallyexact,whilebeginningtodeviatefromfullCIintheRO-H=2:5Re3Reregion,signalingtheneedforevenhigh-ordercorrelationsinordertoaccuratelydescribethesimultaneousO-Hbond-breakinginwater,butthisdeviationisnotnearlyasbigasintheCCSDTcaseandthepointofthisinvestigationistoevaluatehowwelltheCC(P;Q)methodsreproducetheirparentmethodsincomparisontopreviouslyestablishedapproximations,soCCSDTQremainsanimportantreferenceinjudgingothermethodsincludingquadruples.AscanbeseeninTable3.3andsimilarlytotheDZcase,theCCSD(TQf)approachdescribestheequilibriumgeometrywellcomparedtoCCSDTQ,butdivergesasthebondsarestretched,withamaximumerrorofapproximately14millihartree66atRO-H=3RerelativetoCCSDTQ.From2.5Reto3Re,CCSD(TQf)goesfromoverstabi-lizingthedissociatingwatermoleculebyover11millihartreetounderstabilizingitbynearly14millihartree,indicatingthemassivefailureofCCSD(TQf)inthebondbreakingregion.ThiserraticbehaviorofCCSD(TQf)canalsobeseeninFigure3.3.TheCR-CC(2,4)AAap-proachtendstohavebetteragreementwithCCSDTQ,butstillhasamaximumerrorofover6millihartreerelativetoCCSDTQ,whiletheCR-CC(2,4)DAmethodhasalargermaximumerrorof13.621millihartreerelativetoCCSDTQ,eventhoughitseemstobestabilizingatthelongerbondlengthswithrespecttofullCCSDTQ.Unliketheabovenon-iterativeap-proachesbasedontheCCSDamplitudes,theactive-spaceCCSDtqresultsdonotexhibitthedivergentbehavior,givingratherstableenergiesthatare1.497{2.656millihartreehigherinenergythantheirCCSDTQcounterparts,butitisquiteclearfromTable3.3thatCCSDtqisstillmissingsomedynamicalcorrelationsoftheT3andT4type,whichonewouldliketocapture.Wecanalwaystrytodoitbyincreasingtheactivespace,buthereweadvo-cateamoreappealingapproachofcapturingthemissingcorrelationsviatheCC(t,q;3)andCC(t,q;3,4)corrections.WhentheCCSDtqresultsarecorrectedforjustthemissingtriplesviatheCC(t,q;3)AandCC(t,q;3)Dapproaches,wereducethemaximumerrorrelativetoCCSDTQcharacterizingtheCCSDtqcalculationsof2.656millihartree,tomuchsmaller1.349and1.293millihartreeerrors.ThisclearlyshowsthattheCC(P;Q)correctionscanbeverye,whiledemon-stratingthatusingT1andT2amplitudesrelaxedinthepresenceoftriplesandquadruplesisalotbetterthanusingtheirunrelaxedCCSDcounterpartsexploitedinCR-CC(2,4)cal-culations.IftheCCSDtqenergiesarecorrectedformissingtriplesandquadrupleswiththeCC(t,q;3,4)approach,weimproveouragreementwithCCSDTQevenfurther.BoththeCC(t,q;3,4)AAandCC(t,q;3,4)DAmethodshavemaximumerrorsrelativetofullCCS-67DTQofjustover1millihartreeandthecorrespondingNPEvaluesareequallysmall.IfweconstrainourselvestotheRO-H=Re2Reregion,theagreementbetweenCC(t,q;3,4)andCCSDTQcalculationsisevenbetter,resultinginimpressivelysmallthatdonotexceed0.2{0.4millihartree.Ourcc-pVDZcalculationsthattheCC(t,q;3)andCC(t,q;3,4)approachescanreproducetheCCSDTQenergiesforthedouble-dissociationofwatertowithinamillihartreeorbetteratthesmallfractionofthecomputercosts.68Table3.2:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCS-DTQcounterpartsandthecorrespondingfullCIdatafortheequilibriumandtwodisplacedgeometriesoftheH2Omolecule,asdescribedbytheDZbasisset[279].MethodRO-H=ReaRO-H=1:5RebRO-H=2RebMUENPEFullCIc-76.157866-76.014521-75.905247CCmethodswithtriplesdCCSD(T)0.140(0.574)-0.008(1.465)-5.489(-7.699)5.489(7.699)5.629(9.164)CR-CC(2,3)A0.170(0.604)0.784(2.257)2.717(0.507)2.717(2.257)2.547(1.750)CR-CC(2,3)D-0.140(0.294)-0.293(1.180)-0.195(-2.405)0.293(2.405)0.153(3.585)CCSDte0.594(1.028)0.583(2.056)0.639(-1.571)0.639(2.056)0.056(3.627)CC(t;3)Ae0.073(0.507)0.105(1.578)0.295(-1.915)0.295(1.915)0.222(3.493)CC(t;3)De-0.066(0.368)-0.011(1.462)0.212(-1.998)0.212(1.998)0.278(3.460)CCSDT0.000(0.434)0.000(1.473)0.000(-2.210)0.000(2.210)0.000(3.683)CCmethodswithtriplesandquadruplesfCCSD(TQf)0.151(0.166)-0.047(0.094)-6.022(-5.914)6.022(5.914)6.173(6.080)CR-CC(2,4)AA0.041(0.056)0.238(0.379)-1.039(-0.931)1.039(0.931)1.277(1.310)CR-CC(2,4)DA-0.269(-0.254)-0.839(-0.698)-3.951(-3.843)3.951(3.843)3.682(3.589)CCSDtqe0.970(0.985)1.093(1.234)1.229(1.337)1.229(1.337)0.259(0.352)CC(t,q;3)Ae0.449(0.464)0.614(0.755)0.888(0.996)0.888(0.996)0.439(0.532)CC(t,q;3)De0.309(0.324)0.497(0.638)0.805(0.913)0.805(0.913)0.496(0.589)CC(t,q;3,4)AAe0.022(0.037)0.049(0.190)0.412(0.520)0.412(0.520)0.390(0.483)CC(t,q;3,4)DAe-0.118(-0.103)-0.068(0.073)0.329(0.437)0.329(0.437)0.447(0.540)CCSDTQ0.000(0.015)0.000(0.141)0.000(0.108)0.000(0.141)0.000(0.126)aTheequilibriumgeometryandthecorrespondingfullCIresultweretakenfromRef.[280].bThegeometriesthatrepresentasimultaneousstretchingofbothO{Hbondsbyfactorsof1.5and2.0withoutchangingthe\(H{O{H)angleandthecorrespondingfullCIresultsweretakenfromRef.[281].cThetotalfullCIenergiesinhartree.dFortheCCmethodswithuptotripleexcitations,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTand,inparentheses,relativetofullCI.eTheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedofthe3a1and1b2occupiedand4a1and2b2unoccupiedorbitals.fFortheCCmethodswithtriplesandquadruples,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTQand,inparentheses,relativetofullCI.69Figure3.2:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentfullCCSDTandCCSDTQcounterparts,fortheequilibriumandtwodisplacedgeometriesoftheH2Omolecule,asdescribedbytheDZbasisset[279].ThenumericalvaluesoftheerrorsarefoundinTable3.2.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.Forinterpretationofthereferencestocolorinthisandallotherthereaderisreferredtotheelectronicversionofthisthesis.7071Table3.3:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounterpartsandthecorrespondingfullCIdatafortheequilibriumandfourdisplacedgeometriesoftheH2Omolecule,asdescribedbythesphericalcc-pVDZbasisset[237].MethodRO-H=ReaRO-H=1:5ReaRO-H=2ReaRO-H=2:5ReaRO-H=3ReaMUENPEFullCIb-76.241860-76.072348-75.951665-75.917991-75.911946CCmethodswithtriplescCCSD(T)0.165(0.658)0.208(1.631)-2.415(-3.820)-17.812(-42.564)-50.386(-90.512)50.386(90.512)50.594(92.143)CR-CC(2,3)A0.413(0.906)1.402(2.825)5.210(3.805)8.922(-15.830)7.091(-33.035)8.922(33.035)8.509(36.840)CR-CC(2,3)D-0.149(0.344)-0.281(1.142)0.854(-0.551)1.652(-23.100)-0.430(-40.556)1.652(40.556)2.082(41.698)CCSDtd2.216(2.709)1.690(3.113)1.027(-0.378)0.860(-23.892)0.925(-39.201)2.216(39.201)1.356(42.314)CC(t;3)Ad0.261(0.754)0.130(1.553)0.168(-1.237)0.411(-24.341)0.528(-39.598)0.528(39.598)0.398(41.151)CC(t;3)Dd-0.135(0.358)-0.159(1.264)0.023(-1.382)0.337(-24.415)0.460(-39.666)0.460(39.666)0.619(40.930)CCSDT0.000(0.493)0.000(1.423)0.000(-1.405)0.000(-24.752)0.000(-40.126)0.000(40.126)0.000(41.549)CCmethodswithtriplesandquadrupleseCCSD(TQf)0.210(0.229)0.381(0.502)-3.933(-3.903)-11.315(-13.676)13.957(9.224)13.957(13.676)25.272(22.900)CR-CC(2,4)AA0.249(0.268)0.779(0.900)0.729(0.759)-5.402(-7.763)-6.101(-10.834)6.101(10.834)6.880(11.734)CR-CC(2,4)DA-0.312(-0.293)-0.904(-0.783)-3.627(-3.597)-12.672(-15.033)-13.621(-18.354)13.621(18.354)13.309(18.061)CCSDtqd2.656(2.675)2.123(2.244)1.497(1.527)1.627(-0.734)1.653(-3.080)2.656(3.080)1.159(5.755)CC(t,q;3)Ad0.701(0.720)0.561(0.682)0.636(0.666)1.241(-1.120)1.349(-3.384)1.349(3.384)0.788(4.104)CC(t,q;3)Dd0.305(0.324)0.271(0.392)0.491(0.521)1.177(-1.184)1.293(-3.440)1.293(3.440)1.022(3.961)CC(t,q;3,4)AAd0.142(0.161)-0.070(0.051)0.187(0.217)0.947(-1.414)1.104(-3.629)1.104(3.629)1.174(3.846)CC(t,q;3,4)DAd-0.254(-0.235)-0.360(-0.239)0.042(0.072)0.883(-1.478)1.048(-3.685)1.048(3.685)1.408(3.757)CCSDTQ0.000(0.019)0.000(0.121)0.000(0.030)0.000(-2.361)0.000(-4.733)0.000(4.733)0.000(4.854)aTheequilibriumgeometry,RO-H=Re,thegeometriesthatrepresentasimultaneousstretchingofbothO{Hbondsbyfactorsof1.5,2.0,2.5,and3.0withoutchangingthe\(H{O{H)angle,andthecorrespondingfullCIresultsweretakenfromRef.[282].bThetotalfullCIenergiesinhartree.cFortheCCmethodswithuptotripleexcitations,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTand,inparentheses,relativetofullCI.dTheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedofthe3a1and1b2occupiedand4a1and2b2unoccupiedorbitals.eFortheCCmethodswithtriplesandquadruples,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTQand,inparentheses,relativetofullCI.Figure3.3:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,fortheequilibriumandfourdisplacedgeometriesoftheH2Omolecule,asdescribedbythesphericalcc-pVDZbasisset[237].ThenumericalvaluesoftheerrorsarefoundinTable3.3.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.723.2.2Be+H2!HBeHC2vInsertionPathwayTheinsertionofBeintoH2isanotherexamplewhereonehastoincorporatetriplyandquadruplyexcitedclustersinarobustmannertoobtainanaccuratedescription,i.e.,thecasewellsuitedfortestingtheCC(P;Q)-basedCC(t,q;3)andCC(t,q;3,4)approachesdevelopedinthisthesis.WeexaminedninepointsalongtheC2v-symmetricBe+H2!HBeHinsertionpathway,labeledA{I,whichweretakenfromRef.[283].ThegeometriesareshowninTable3.4.StructureAcorrespondstothelinearHBeHproduct,whilestructureIcorrespondstotheBe+H2reactantsseparatedby6bohr.StructureEisthetransitionstate,whichisdescribedinmoredetailbelow.Calculationswereperformedusingthe[3s1p/2s]basissetdescribedinRef.[283]andallelectronswerecorrelated.Inordertohaveaccesstomoredecimalplacesforanaccurateerroranalysis,thecorrespondingfullCIenergiesweretakenfromRef.[48],sincethefullCIenergiesreportedininRef.[283]showonlyfourdecimalplaces,ascomparedtothesixdecimalplacesprovidedinRef.[48].TheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedofthehighestoccupiedandlowestunoccupied1b2and3a1orbitals.ForthestructuresA{E,thelowest-energyRHFreferencesolutionisprovidedbythej(1a1)2(2a1)2(1b2)2jdeterminant,whileforstructuresF{I,thelowest-energyRHFsolutionisgivenbythej(1a1)2(2a1)2(3a1)2jdeterminant,obtainedbyreplacingthehighestoccu-piedorbital,1b2,bythelowestunoccupiedorbital,3a1.ThisiswhyourchoiceofactivespacefortheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistsofthe1b2and3a1orbitals.InthefullCIwavefunctionexpansion,theA{Dregionisdom-inatedbythej(1a1)2(2a1)2(1b2)2jwhiletheF{Iregionisdominatedbythej(1a1)2(2a1)2(3a1)2jStructureErepresentsthetransitionstatewhereboth73becomequasi-degenerate,whichmakesthisstructurethehardestonetode-scribe.We,thus,usedthej(1a1)2(2a1)2(1b2)2jfurtherreferredtoasreferencej1iasareferencedeterminantforstructuresA{Dandthej(1a1)2(2a1)2(3a1)2jtion,furtherrefereedtoasrefrencej2i,forstructuresF{I.ToexaminethesensitivityofthevariousSRCCresultsonthechoiceofthereferencedeterminant,weperformedtwosetsofcalculationsforstructureE,oneusingj1iasareferenceandanotherusingj2iasareference.TheresultsofourcalculationsaregiveninTable3.5andFigure3.4.ThetransitionstatestructureEisthehardestonetodescribe,exhibitinglargererrorsthananyothergeometryforallmethodsexamined,somuchofourdiscussionwillfocusonit.Although,asshowninTable3.5andFigure3.4,oneneedstoincludequadruplesintheSRCCcalculationstoobtainaccurateresults,sinceevenCCSDTist,webeginourdiscussionwithacomparisonofthevariousapproximatetriplestreatmentswithCCSDT,toseehowfaithfultheycanbeinreproducingthefullCCSDTstate.TheCCSD(T)schemestrugglesatdescribingtheenergeticsofstructureE,producingalargeerrorof3.569millihartreerelativetoCCSDTwhendeterminantj1iisusedasreference,anda1.109millihartreeerrorrelativetofullCCSDTwhenthej2ireferenceisemployed.TheCR-CC(2,3)AapproachhasevenlargererrorsrelativetoCCSDT,butthemorecompleteCR-CC(2,3)treatment,representedbyvariantD,worksbetter,reducingthe3.569and1.109millihartreeerrorsrelativetoCCSDTintheCCSD(T)/j1iandCCSD(T)/j2icalculationsoftheEgeometryto3.305and0.824millihartree,respectively,makingtheresultingenergiessomewhatlessdependentonthechoiceofthereferencedeterminant,whencomparedtofullCI.Whenthej1ireferenceisemployed,theactive-spaceCCSDtapproachhasasmallererrorrelativetoCCSDTattheEgeometrythananyofthenon-iterativecorrectionstoCCSDandCCSDtcompeteswiththeCR-CC(2,3)Dwhenthej2ireferenceisemployed,retaining74theca.1millihartreeerrorrelativetoCCSDTattheEstructure,improvingCCSD(T).Asfortheremaininggeometries,theactive-spaceCCSDtcalculationstendtogiveerrorsrelativetoCCSDTthataretoolarge,orattheveryleastsimilartotheCCSD-basednon-iterativetriplesmethods,indicatingthatsomeimportanttriplesarestillmissing.Insteadofsimplyincreasingtheactivespace,whichisonewaytogoaboutit,wecorrecttheCCSDtenergiesforthemissingtripleexcitationsusingtheCC(P;Q)-basedCC(t;3)approach.WhentheCCSDtenergiesarecorrectedforthecorrelationduetothemissingtriplesusingtheCC(t;3)methodology,theagreementwithCCSDTisgenerallyverygood.WiththeexceptionofstructureE,thebetweentheCC(t;3)AorCC(t;3)DandCCSDTenergiesareontheorderoftensofmicrohartree.WhenthetransitionstatestructureEisexaminedandreferencej1iisemployed,theerrorsintheCC(t;3)AandCC(t;3)DresultsrelativetoCCSDTare1.469and1.199millihartree,respectively,whichisatimprovementoverthe2{5millihartreeerrorvaluesgivenbytheCCSD(T),CR-CC(2,3),andCCSDtmethods.Whenreferencej1iisadopted,theagreementofCC(t;3)AorCC(t;3)DwithCCSDTatstructureEisevenbetter,witherrorsresidinginthe0.3{0.5millihartreerange.TheCC(t;3)AandCC(t;3)DapproachesalsoprovidesubstantialimprovementsintheNPEvaluesrelativetoCCSDT,comparedtoCCSD(T),CR-CC(2,3),andCCSDtdata,sotheyaremoresystematicatdescribingthetriplesalongthereactionpathwaythanthestandardCCSD-basednon-iterativeandactive-spaceiterativeCCmethods.Thisis,however,nott,sinceonecannotobtainanaccuratedescriptionoftheBe+H2!HBeHreactionwithoutquadruples,especiallyintheregionofthetransitionstatestructureE.AsshowninTable3.5andFigure3.4,evenfullCCSDTfails,givingerrorsrelativetofullCIthatexceed2millihartree.Oneneedsconnectedquadrupleexcitationtoaddressthisproblem.BycomparingtheCCSDTQandfullCIenergiesatpartsA{I,wecanseethatfull75CCSDTQisanessentiallyexacttheoryinthiscase,reducingthe>2millihartreeerrorsintheCCSDTresultsatstructureEto16microhartreeorless(seeTable3.5andFigure3.4).ThequestionthenishowaccuratevariousapproximatetreatmentsoftriplesandquadruplesarecomparedtoCCSDTQ.ItisquiteclearfromTable3.5andFigure3.4thattraditionalperturbativecorrectionstoCCSD,representedherebytheCCSD(TQf)approximation,fail,givingerrorsrelativetoCCSDTQandfullCIatthetransitionstatestructureEthatexceed4.1millihartreewhenthej1iisusedasareferenceor1.7millihartreewhenthej2ireferenceisemployed.TheCR-CC(2,4)approaches,especiallyCR-CC(2,4)DA,helpbringingtheaboveerrortoa1{3millihartreelevel,butnoneofthenon-iterativecorrectionsduetotriplesandquadruplestoCCSDworkwell.Theiterativeactive-spaceCCSDtqmethodprovidesamorerobustdescription,reducingthe1{3millihartreeerrorsintheCR-CC(2,4)resultsrelativetofullCCSDTQattheEgeometryto1.430millihartree,whenthej1ireferenceisemployed,and0.908millihartree,whenj2iisusedasareference,reducingtheoverallNPEvaluesatthesametime,butthisisdoneattheexpenseoflosingaccuracyinthereactant(structuresF{I)andproduct(structuresA{D)regions,whereerrorsintheCCSDtqdescriptioncanbeashighas˘0:4millihartree,sinceCCSDtqneglectscertainclassesofdynamicaltriplesandquadruplesthatasmallactivespaceusedherecannotcapture.Itbecomes,therefore,importanttobringthemissinghigher-orderexcitationsviatheCC(P;Q)-basedCC(t,q;3)andCC(t,q;3,4)corrections.WhentheCCSDtqenergiesarecorrectedforthemissingtriplesviatheCC(t,q;3)AandCC(t,q;3)Dapproaches,the0.9{1.4millihartreeerrorsintheCCSDtqresultsforstructureErelativetoCCSDTQreduceto0.5{0.8millihartreewhenthej1ideterminantisusedasareferenceandto0.2{0.4millihartreewhenthej2ireferenceisemployed.TheCC(t,q;3)Dresults,whichgive0.2{0.5millihartreeerrors,areparticularlyimpressive.Fortheremaining76geometriesA{DandF{I,theCC(t,q;3)AapproachhasamaximumerrorrelativetoCCS-DTQof0.133millihartree,whiletheCC(t,q;3)Dapproximationworksevenbetter,givingamaximumerrorofonly37microhartree.VerylittlechangeswhentheCCSDtqenergiesarecorrectedforthemissingtripleaswellasquadrupleexcitationsviaCC(t,q;3,4)AAandCC(t,q;3,4)DAapproaches,i.e.,theCC(t,q;3,4)AAandCC(t,q;3,4)DAenergiesareasaccu-rateastheirCC(t,q;3)AandCC(t,q;3)Dcounterparts,whichindicatesthatitisttocorrectCCSDtqenergiesforthetriplesmissinginCCSDtqinthiscase,butwemustkeepinmindthattheCC(t,q;3)results,especiallythoseobtainedwithvariantD,arealreadyoutstanding.Wecan,thus,concludethattheCC(P;Q)-basedCC(t,q;3)andCC(t,q;3,4)correctionstoCCSDtq,especiallyCC(t,q;3)DandCC(t,q;3,4)DA,arecapableofreproduc-ingthevirtuallyexactfullCCSDTQandexactfullCIdatafortheBe+H2!HBeHreactiontowithinsmallfractionsofamillihartreeatthetinyfractionofthecomputerwhichisapromisingforthefutureapplicationsoftheCC(P;Q)formalism.Table3.4:CoordinatesofpointsalongthesamplingpathC2v-symmetricpathdescribinginsertionofBeintoH2,introducedinRef.[283].StructureAisthelinearHBeHproduct,whilestructureIrepresentstheBe+H2reactant.StructureEisthetransitionstate.PointCoordinatesforH2(X,Y,Z)aA(0.0,2.54,0.0)B(0.0,2.08,1.0)C(0.0,1.62,2.0)D(0.0,1.39,2.5)E(0.0,1.275,2.75)F(0.0,1.16,3.0)G(0.0,0.93,3.5)H(0.0,0.70,4.0)I(0.0,0.70,6.0)aBeislocatedat(0.0,0.0,0.0)andallvaluesareinbohr.77Table3.5:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCS-DTQcounterparts,andthecorrespondingfullCIdataforthegeometriesA{ItheC2v-symmetricinsertionpathwayofBeintoH2,asdescribedbythe[3s1p/2s]basisset,introducedinRef.[283].aji=j1i=j(1a1)2(2a1)2(1b2)2jbMethodABCDEFullCId-15.779172-15.737224-15.674818-15.622883-15.602919CCmethodswithtripleseCCSD(T)0.148(0.169)0.137(0.140)0.190(0.149)0.496(0.392)3.569(1.195)CR-CC(2,3)A0.150(0.171)0.140(0.143)0.188(0.147)0.547(0.443)4.828(2.454)CR-CC(2,3)D0.043(0.064)0.035(0.038)0.044(0.003)0.236(0.132)3.305(0.931)CCSDtf0.227(0.248)0.187(0.190)0.236(0.195)0.470(0.366)2.096(-0.278)CC(t;3)Af0.086(0.107)0.070(0.073)0.094(0.053)0.236(0.132)1.469(-0.905)CC(t;3)Df0.022(0.043)0.018(0.021)0.033(-0.008)0.138(0.034)1.199(-1.175)CCSDT0.000(0.021)0.000(0.003)0.000(-0.041)0.000(-0.104)0.000(-2.374)CCmethodswithtriplesandquadruplesgCCSD(TQf)0.161(0.161)0.142(0.141)0.182(0.181)0.465(0.464)-4.104(-4.106)CR-CC(2,4)AA0.165(0.165)0.141(0.140)0.154(0.153)0.473(0.472)3.149(3.147)CR-CC(2,4)DA0.058(0.058)0.036(0.035)0.010(0.009)0.162(0.161)1.626(1.624)CCSDtqf0.238(0.238)0.192(0.191)0.239(0.238)0.367(0.366)1.430(1.428)CC(t,q;3)Af0.096(0.096)0.074(0.073)0.098(0.097)0.133(0.132)0.808(0.806)CC(t,q;3)Df0.033(0.033)0.022(0.021)0.037(0.036)0.035(0.034)0.539(0.537)CC(t,q;3,4)AAf0.093(0.093)0.073(0.072)0.096(0.095)0.114(0.113)0.922(0.920)CC(t,q;3,4)DAf0.030(0.030)0.021(0.020)0.035(0.034)0.016(0.015)0.653(0.651)CCSDTQ0.000(0.000)0.000(-0.001)0.000(-0.001)0.000(-0.001)0.000(-0.002)78Table3.5(cont'd)ji=j2i=j(1a1)2(2a1)2(3a1)2jcEFGHIMUENPE-15.602919-15.624981-15.693194-15.736688-15.760878CCmethodswithtriplese-1.109(-3.314)0.166(0.224)0.058(0.100)0.034(0.049)0.005(0.009)3.569(3.314)4.678(4.509)2.191(-0.014)0.255(0.313)0.081(0.123)0.045(0.060)0.007(0.011)4.828(2.454)4.821(2.468)0.824(-1.381)0.037(0.095)-0.012(0.030)-0.007(0.008)-0.001(0.003)3.305(1.381)3.317(2.312)0.981(-1.224)0.308(0.366)0.175(0.217)0.109(0.124)0.021(0.025)2.096(1.224)2.075(1.590)0.494(-1.711)0.118(0.176)0.048(0.090)0.028(0.043)0.004(0.008)1.469(1.711)1.465(1.887)0.295(-1.910)0.026(0.084)-0.013(0.029)-0.004(0.011)-0.001(0.003)1.199(1.910)1.212(1.994)0.000(-2.205)0.000(0.058)0.000(0.042)0.000(0.015)0.000(0.004)0.000(2.374)0.000(2.432)CCmethodswithtriplesandquadruplesg-1.728(-1.728)0.182(0.198)0.075(0.074)0.040(0.039)0.006(0.006)4.104(4.106)4.569(4.570)-0.737(-0.737)0.255(0.271)0.093(0.092)0.051(0.050)0.008(0.008)3.149(3.147)3.886(3.884)-0.936(-0.936)0.037(0.053)-0.001(-0.002)-0.002(-0.003)0.000(0.000)1.626(1.624)2.562(2.560)0.908(0.908)0.315(0.331)0.185(0.184)0.117(0.116)0.023(0.023)1.430(1.428)1.407(1.405)0.412(0.412)0.125(0.141)0.058(0.057)0.036(0.035)0.006(0.006)0.808(0.806)0.802(0.800)0.209(0.209)0.033(0.049)-0.003(-0.004)0.004(0.003)0.002(0.002)0.539(0.537)0.542(0.541)0.465(0.465)0.127(0.143)0.054(0.053)0.033(0.032)0.005(0.005)0.922(0.920)0.917(0.915)0.263(0.263)0.035(0.051)-0.007(-0.008)0.001(0.000)0.001(0.001)0.653(0.651)0.660(0.659)0.000(0.000)0.000(0.016)0.000(-0.001)0.000(-0.001)0.000(0.000)0.000(0.016)0.000(0.018)aThe[3s1p/2s]basissetandthegeometriesAthroughItheC2v-symmetricBe+H2!HBeHinsertionpathway,wherestructureAcorrespondstotheHBeHproduct,structureItotheBe+H2reactantsseparatedby6bohr,andstructureEtothetransitionstate,weretakenfromRef.[283].Inordertohaveaccesstomoredecimalplacesforanaccurateerroranalysis,thecorrespondingfullCIenergiesweretakenfromRef.[48](thefullCIenergiesinRef.[283]showfourdecimalplaces,ascomparedtosixdecimalplacesprovidedinRef.[48]).bTheCCcalculationswereperformedusingthej(1a1)2(2a1)2(1b2)2jreferencedeterminant,whichisthelowest-energyRHFsolutionforthegeometriesA{E.Thej(1a1)2(2a1)2(1b2)2jdominatesthefullCIwavefunctionexpansionintheA{Dregion,whilebecomingquasi-degeneratewiththej(1a1)2(2a1)2(3a1)2jdeterminantatthetransition-statestructureEfromRef.[283]cTheCCcalculationswereperformedusingthej(1a1)2(2a1)2(3a1)2jreferencedeterminant,whichisthelowest-energyRHFsolutionforthegeometriesF{I.Thej(1a1)2(2a1)2(3a1)2jdominatesthefullCIwavefunctionexpansionintheF{Iregion,whilebecomingquasi-degeneratewiththej(1a1)2(2a1)2(1b2)2jdeterminantatthetransition-statestructureEfromRef.[283]dThetotalfullCIenergiesinhartree.eFortheCCmethodswithuptotripleexcitations,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTand,inparentheses,relativetofullCI.fTheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedofthe1b2and3a1orbitals,whichareoccupiedandunoccupied,respectively,inji=j(1a1)2(2a1)2(1b2)2jandunoccupiedandoccupied,respectively,inji=j(1a1)2(2a1)2(3a1)2j.gFortheCCmethodswithtriplesandquadruples,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTQand,inparentheses,relativetofullCI.79Figure3.4:Acomparisonoftheenergiesresultingfromvariousall-electronCCcalculations,includinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,forthegeometriesA{ItheC2v-symmetricinsertionpathwayofBeintoH2,asdescribedbythe[3s1p/2s]basisset,introducedinRef.[283].ThenumericalvaluesoftheerrorsarefoundinTable3.5.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.MiddlepanelshowsacomparisonofvariousapproximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.InpresentingtheresultsforthetransitionstatestructureE,weusedthesamereferencej2iasthatusedintheF{Iregion.803.2.3SingletTripletGapinHFHAlongtheD1h-SymmetricDouble-DissociationPathwayOurnextexampleisthelinear,D1h-symmetric,(HFH)anion,inwhichbothH{Fbondsaresimultaneouslystretched,andwhichhasbeenusedintheliteratureasaprototypemag-neticsystem,wheretwoparamagneticcenters,eachcarryinganunpairedspin,represented
bytheterminalhydrogenatoms,arelinkedviaapolarizablediamagneticbridgeconstituted
byF[284].ThespinsoftheparamagneticelectronsoftheHatomscanbeparallelorantiparallel,yieldingtwotspinstates,namely,asinglet,X1+g,whichisthegroundstate,andatriplet,A3+
u,whichistheexcitedstate.Thetotalelectronicenergiesofthesetwostatesandthegapbetweenthem,whichshouldapproachzeroasbothH{Fbondsarestretchedtoyandwhichprovidesinformationaboutthemagneticexchange
couplingconstantJasafunctionoftheH{FdistanceRHF,hasbeenstudiedusingavari-etyofabinitioanddensityfunctionaltheorymethodsinRefs.[162,165,260,284,285].ThisincludesourcalculationsreportedinRef.[164]and[165].whereweusedawidevarietyof
SRCCmethodswithuptotripleexcitations,includingCR-CC(2,3)[164]andCC(t;3)[165]
comparingtheresultswithCCSDTandfullCI.Someoftheseresults,whicharerelevantto
thiswork,arerestatedhere(seeTables3.6{3.8).AsshowninTables3.6{3.8andFigures3.5{3.7,themainchallengefortheCCtheory,ifwearetorelyontheSRCCformalismandutilizethespin-andsymmetry-adaptedRHF
(theX1+gstate)andROHF(theA3+
ustate)references,istheaccurateinclusionofhigher-than-doublyexcitedclusters,especiallyfortheX1+
gstate,whichhasamanifestlyMRcharacterinvolvingthedoublyexciteddeterminantcorrespondingtoexcitationsfromthehighestoccupiedmolecularorbital(HOMO)tothelowestunoccupiedmolecularorbital81(LUMO)(i.e.,(HOMO)2!(LUMO)2(˙2g!˙2u)),inadditiontotheRHFAsdemonstratedinRef.[162],theratioofthefullCIexpansioncotsatthe(HOMO)2!(LUMO)2andRHFstatefunctionscharacterizingtheX1+gstate,whichisequivalenttotheT2clusteramplitudecorrespondingtothe(HOMO)2!(LUMO)2doubleexcitation,sinceHOMOandLUMOhavetsymmetries,increases,inabsolutevalue,from0.38to1.17asRHFisvariedbetween1.5and4.0A.Thus,themoderatelybiradical(HFH)systematshorterH{FseparationsbecomesastrongbiradicalspeciesatlargerH{Fdistances.Becauseofthetbiradicalcharacterofthe(HFH)ionatalmostallH{FseparationsshowninTables3.6{3.8andFigures3.5{3.7,theA3+uX1+ggapisalreadyquitesmallandsensitivetotheelectroncorrelationtreatmentusedinthedeterminationoftheX1+gandA3+ustatesintheregionofshorterH{Fdistances,whilerapidlydecreasingasRHFbecomeslarger,causingtroublestothestandardCCSDandCCSD(T)approaches,whichareincapableofhandlingsuchachallengingsituation.Aswewilldiscussitbelow,asasseeninTables3.6{3.8andFigures3.5{3.7,thefullCCSDTmethodisalotmorerobust,butitisstillnotaccurateenoughtoprovideafullyquantitativedescription.ThefullCCSDTQapproachsolvestheproblem,butCCSDTQisveryexpensive,soitisimportanttoexaminethatvariousapproximatetreatmentsoftriplesandquadruplescandointhisregard.Followingtheearlierwork[162,165,260,284,285],weemployedthe6-31G(d,p)basisset[286,287]andsampledseveralvaluesoftheH{FdistanceRHFthelinearD1h-symmetric(HFH)systemrangingfromRHF=1:5AtoRHF=4:0A.ThefullCIenergiesweretakenfromRef.[162],buttheyarealsoavailableinRefs.[165,260,285].TheCCSD(T),CR-CC(2,3)A,CR-CC(2,3)D,andCCSDTvaluescanbefoundinRefs.[162,165,260,285],whiletheCCSDt,CR-CC(2,3)A,andCR-CC(2,3)DvaluescanbefoundinRef.[165].TheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)82calculationsconsistedoftwoactiveelectronsandtwoactiveorbitalscorrespondingtoHOMOandLUMO(the˙gand˙uvalenceorbitals).FollowingRefs.[162,165,285],thelowest-energycoreorbitalwasfrozeninthepost-SCFcalculations.TheresultsfortheX1+gstatecanbefoundinTable3.6andFigure3.5.TheresultsfortheA3+ustateinareshowninTable3.7andFigure3.6,andtheA3+uX1+ggapcalculationsaresummarizedinTable3.8andFigure3.7.AsinSections3.2.1and3.2.2,althoughourfocushereisontheCCmethodswithtriplesandquadruples,webeginoutdiscussionwithmethodstruncatedattriples.ThemostpopularSRCCapproachwithanapproximatetreatmentoftripleexcitations,CCSD(T),whichusesargumentsoriginatingfromMBPTtoestimatetheT3isincapableofhandlingtheA3+uX1+ggapin(HFH).ThedescriptionoftheX1+gstateiswhereCCSD(T)displaysacatastrophicfailure,asseenintheerrorsrelativetoCCSDT,whichgrowfrom0.435millihartreeatRHF=1:5Atoover40millihartreeatRHF=4:0A.AlreadyatRHF=1:875A,CCSD(T)hasalargererrorrelativetoCCSDTthananyoftheotherapproximatetriplesmethodsstudiedinthiswork.AsforthelargelySRA3+ustate,CCSD(T)doesnotexperiencethebreakdown,producingrathersmallerrorsrelativetoCCSDT,whichdonotexceed0.355millihartree,buttheCCSD(T)resultsfortheX1+gstateareverypoor.ThisunbalanceddescriptionofthesingletandtripletstatesleadstoA3+uX1+gseparationsthatarelargerthanthosegivenbyCCSDTanywherebetween148cm1atRHF=1:5Aand8957cm1atRHF=4:0A.Clearly,alternativeandmorerobustwaysofhandlingconnectedtripleexcitationsintheSRCCformalismareneededtoreproducethefullCCSDT-qualitydatafortheX1+gandA3+ustatesinthe(HFH)systemandthegapbetweenthem.AsshowninRefs.[162,285],theCR-CC(2,3)methodology,includingvariantsAandD83ofCR-CC(2,3)examinedinTables3.6{3.8andFigures3.5{3.7,greatimprovementsintheresultscomparedtoCCSD(T),particularlyforthequasi-degenerateX1+gstateandespeciallyatlargerH{Fseparations.ThelargeMUEandNPEvaluesrelativetofullCCSDTcharacterizingtheCCSD(T)energiesoftheX1+gstateintheRHF=1:54:0Aregion,of40.727and40.292millihartree,reducetothemuchbettervaluesof2.566and2.906millihartree,respectively,whentheCR-CC(2,3)Aapproachisemployed,and2.800and2.509millihartreewhentheCR-CC(2,3)Dapproximationisutilized.AlthoughCCSD(T)workswellfortheA3+ustate,producingsmallMUEandNPEvaluesrelativetofullCCSDTofonly0.335and0.250millihartree,respectively,bothvariantsofCR-CC(2,3)improvetheCCSD(T)resultsatallgeometriesexaminedfortheA3+ustateaswell.AsaresultoftheverygoodagreementoftheCR-CC(2,3)approacheswiththeCCSDTmethodforthetwostatesof(HFH)examinedhere,theoverallperformanceoftheCR-CC(2,3)AandCR-CC(2,3)DmethodsindescribingtheA3+uX1+ggapin(HFH)isalotbetterthantheperformanceofCCSD(T).WhencomparedtoCCSDT,theCR-CC(2,3)AapproachisslightlymoreaccuratethanitsCR-CC(2,3)DcounterpartintheRHF=2:1254:0Aregion,butCR-CC(2,3)DworksbetterwhenRHF<2:125A.IntheendbothCR-CC(2,3)approachesperforminasimilarmanner,givingtheMUEandNPEvaluesrelativetofullCCSDTcharacterizingtheA3+uX1+ggapinthe500600cm1range,whichisagreatimprovementoverCCSD(T),butonedoes,ofcourse,wonderiffurtherimprovementscanbemadebyturningtootherapproximatetreatmentsoftriples.Theactive-spacetreatmentoftriplesviatheCCSDtapproach,althoughprovidingmoreuniformerrorsatvariousH{FseparationsrelativetofullCCSDT,reducingtheMUEandNPEvaluescharacterizingtheA3+uX1+ggapto299and377cm1,respectively,isnotcompletelysatisfactoryeither.Indeed,thebetweentheCCSDtandCCSDT84A3+uX1+ggapsincreasefrom140cm1atRHF=1:5Ato299cm1atRHF=2:25A,todecreaseagainto78cm1atRHF=4:0A.WhiletheMUEandNPEvaluesrelativetoCCSDT,whichare299and377cm1,respectively,aresmallerthanthosegivenbyCCSD(T),CR-CC(2,3)A,andCR-CC(2,3)D,wearestillnotobtainingthedesirableagreementwithCCSDTconsideringthaterrorsresultingfromCCSDtcalculations,comparedtoCCSDT,intheRHF=2:254:0AregionarecomparabletothemagnitudeoftheA3+uX1+ggapitself.TheinabilityoftheCCSDtapproachtomoreaccuratelyreproducetheCCSDTsinglet{tripletgapvaluesinthe(HFH)systemstemsfromtherathertbetweenthetotalenergiesoftheX1+gandA3+ustatesresultingfromtheCCSDtandCCSDTcalculations,whichareaslargeas2.582and1.895millihartree,respectively.ThisdisagreementcouldberesolvedbyexpandingtheactivespaceintheCCSDtcalculations,butherewearemoreinterestinginexamininghowtheCC(P;Q)corrections,suchastheCC(t;3)correctiontoCCSDt,copewiththisissue.Asdesired,theCC(t;3)AandCC(t;3)DapproachesimprovethetotalenergiesoftheX1+gandA3+ustatesof(HFH),whencomparedtotheCR-CC(2,3)A,CR-CC(2,3)D,andCCSDtcalculations,bringingtheresultstoacloseragreementwiththefullCCSDTdata.TheMUEvaluesrelativetofullCCSDTcharacterizingtheCR-CC(2,3)A,CR-CC(2,3)D,andCCSDtenergiesoftheX1+gstatealongtheRHFcoordinate,of2.566,2.800,and2.582millihartree,respectively,arereducedtoamere0.357millihartree,whentheCC(t;3)Aap-proximationisemployed,and0.335millihartree,whentheCC(t;3)Dmethodisused.InadditiontothesmallMUEvalues,thecorrespondingNPEvaluesof0.420and0.253mil-lihartreefortheCC(t;3)AandCC(t;3)Dapproximationsaremajorimprovementsoverthe2.906,2.509,and1.814millihartreevaluesobtainedintheCR-CC(2,3)A,CR-CC(2,3)D,andCCSDtcalculations,nottomentionthecatastrophicallyfailingCCSD(T)approach,which85givesa40.292millihartreeerror.Thesituationforthe\easier",largelyA3+ustateissomewhattbecauseofthealreadylowMUEandNPEvaluesgivenbytheCR-CC(2,3)A,CR-CC(2,3)D,andCCSDtcalculations.Still,bothCC(t;3)AandCC(t;3)DprovideimproveddescriptionoftheA3+ustateovertheirCR-CC(2,3)andCCSDtcounterparts,resultingintheNPEandMUEvaluesrelativetoCCSDTof0.130and0.081millihartree,respectively,inthecaseofCC(t;3)Aand0.207and0.047millihartree,respectively,intheCC(t;3)Dcase.ThehighlyaccuratedescriptionofthelowestsingletandtripletstatesbytheCC(t;3)approachesresultsinthegreatlyimproveddescriptionoftheA3+uX1+ggap.BoththeCC(t;3)AandCC(t;3)DmethodsreproducetheCCSDTA3+uX1+ggapvaluestowithintensofwavenumbers,withthemaximumerrorsofonly62cm1forCC(t;3)Aand37cm1forCC(t;3)D.TheseextremelylowerrorsinthetotalenergiesoftheX1+gandA3+ustatesandthecorrespondingA3+uX1+ggapgivenbytheCC(t;3)schemessignifythattheCC(t;3)AandCC(t;3)Dmethodsarepromising,computationallytalternativestofullCCSDT,whichmayhelpthevariousapplicationswheretheCCSDTleveloftheoryist,but,asalreadypointedout,the(HFH)systemhastcorrelationbeyondCCSDT,especiallytheconnectedquadruples.Indeed,CCSDTgiveserrorsrelativetofullCI,whichareashighas2.276mil-lihartreefortheX1+gstate,0.389millihartreefortheA3+ustate,and420cm1forthecorrespondingA3+uX1+ggap.Thus,wemovenowtoSRCCmethodsthataccountfortriplesaswellasquadruplescorrelations.BycomparingtheCCSDTQandfullCIvalues,weimmediatelyseethatCCSDTQisessentiallyexact,givingerrorstofullCIthatdonotexceed0.148millihartreefortheX1+gstate,14microhartreefortheA3+ustate,and30cm1fortheA3+uX1+ggap.Un-fortunately,thefullCCSDTQcalculationsareusuallyprohibitivelyexpensive,soweneed86toexamineapproximatewaysofhandlingtriplesandquadruples,lookingformethodsthatcanreproducefullCCSDTQdataatthefractionofthecost.Thecompletelyrenormalizedmethodswithtriplesandquadruplestestedinthisstudy,namelyCR-CC(2,4)AAandCR-CC(2,4)DA,althoughnottoobad,arenotcapableofprovidingaccuraciesweareinterestedin.TheMUEandNPEvaluesrelativetofullCCSDTQof2.105and2.610millihartreechar-acterizingtheCR-CC(2,4)AAcalculationsfortheX1+gstateandthecorresponding3.267and2.889millihartreeMUEandNPEvaluesresultingfromtheCR-CC(2,4)DAcalculationsareacceptable,especiallygiventhechallengingnatureofthisstatewhenbothH{Fbondsaretlystretched,buttheyarenotasgoodaswedesire.ThesituationfortheSRA3+ustateismuchbetter,butthisisnott,sincetheMUEandNPEvaluesrelativetoCCSDTQcharacterizingtheA3+uX1+ggapasafunctionofRHFremainquitehigh,ontheorderof500{600cm1.Theactive-spaceCCSDtqapproachimprovestheoveralldescriptionoftheA3+uX1+ggap,reducingtheaboveMUEandNPEvaluestoa400{500cm1level,butitisquiteclearfromourtablesthatweneedtomoreifwearereachaccuraciesatthe0.1millihartreeand100cm1levels.WeneedtoawaytocorrecttheCCSDtqenergiesforthehigher-orderdynamicalcorrelationsthattheycannotdescribewhensmalleractivespacesareused.ThesolutiontothisisprovidedbytheCC(t,q;3)orCC(t,q;3,4)correctionsdevelopedinthiswork,whichwediscussnext.ForthemoredemandingX1+gstate,whentheCCSDtqenergiesarecorrectedforthemissingtriples,theresultingCC(t,q;3)AandCC(t,q;3)DvaluesareinverygoodagreementwithCCSDTQ.TheMUEvaluesrelativetoCCSDTQcharacterizingtheCC(t,q;3)AandCC(t,q;3)DcalculationsfortheX1+gstateare1.127and0.674millihartree,respectively,whichisatimprovementoverthe2.105,3.267,and3.349millihartreeMUEsob-tainedwithCR-CC(2,4)AA,CR-CC(2,4)DA,andCCSDtq.Similarremarksapplytothe87NPEvalues.TheCC(t,q;3)DapproachprovidesabetteroverallagreementwithCCSDTQthanitsCC(t,q;3)Acounterpart,althoughbothCC(t,q;3)methodsworkwell.Indeed,theMUEandNPEvaluesrelativetoCCSDTQresultingfromtheCC(t,q;3)Dcalculations,of0.574and0.590millihartree,respectively,aresomewhatbetterthanthe1.127and0.863millihartreevaluesobtainedwithCC(t,q;3)A.WhentheCCSDtqenergiesarecorrectedforthemissingtripleandquadrupleexcitationsviatheCR-CC(2,4)AAandCR-CC(2,4)DAap-proaches,theagreementwithCCSDTQisevenmoreimpressive.Bothapproacheshavemax-imumerrorsofonly0.366millihartreeandtheNPEvaluescharacterizingtheCC(t,q;3,4)AAandCR-CC(2,4)DAcalculationsareequallygood,especiallyinthelattercase,whereweobtain0.371millihartree.InthecaseoftheA3+ustate,whenweexaminetheCC(P;Q)methodscorrectingCCSDtqforthemissingtriplesandquadruples,wedonotwitnessthelargeimprovementinresultsaswiththeX1+gstate,sincetheA3+ustatehasaSRcharacterandmethodssuchasCR-CC(2,4)alreadyworkwellforit.TheCC(t,q;3)AandCC(t,q;3)DapproacheshavesmallMUEvaluesrelativetoCCSDTQof0.481and0.211millihartree,respectively,andallenergieslieaboveCCSDTQforbothmethods.Furthermore,bothapproachesbehaveverysystematicallyintheentireRHF=1:54:0Aregion,asbythelowNPEvaluesof81and71microhartree.Whenwecorrectforthemissingquadruplesaswell,viatheCC(t,q;3,4)AAandCC(t,q;3,4)DAapproaches,theenergiesareloweredslightlybelowtheCCSDTQones,buttheverysmallMUEandNPEvalues,ontheorderof0.1{0.4millihartree,remain.ThehighlyaccuratedescriptionoftheX1+gstateprovidedbytheCC(t,q;3)andCC(t,q;3,4)approximations,combinedwiththeequallysmallerrorsobtainedfortheA3+ustateleadstoimpressivelyaccurateA3+uX1+ggappredictions.TheMUEvaluesrela-88tivetoCCSDTQof465,642,and375cm1obtainedintheCR-CC(2,4)AA,CR-CC(2,4)DAandCCSDtqcalculationsarereducedtojust147and102cm1whentheCC(t,q;3)AandCC(t,q;3)Dmethodsareemployed,and108and76cm1whentheCC(t,q;3,4)AAandCC(t,q;3,4)DAapproachesareused.TheNPEvaluesimproveinasimilarmanner,soitishardtotellthebetweentheCC(t,q;3)andespecially,CC(t.q;3,4)gapvaluesandtheirvirtuallyexactCCSDTQcounterparts.Insummary,thechallenging(HFH)ion,wherethedegreeofbiradicalcharactercanbecontinuouslyvariedbysimultaneouslystretchingbothH{Fbonds,provedtobeanexcellentsystemtoexaminetheperformanceoftheCC(P;Q)methods.Whenitcomestotripleexcitations,theCR-CC(2,3)A,CR-CC(2,3)D,andCCSDtapproachesimprovetheerraticCCSD(T)data,butnoneofthemprovidessatisfactoryagreementwithCCSDTfortotalenergiesoftheA3+uandX1+gstatesandthecorrespondingenergygap.WhenwecorrecttheCCSDtenergiesforthecorrelationduetothemissingtriplesusingtheCC(t;3)methodology,boththeCC(t;3)AandCC(t;3)Dvariants,whichreplacetheexpensiveiterativeCPUstepsofCCSDTthatscalewiththesystemsizeasN8bytheiterativeN6-typeandnon-iterativeN7-typecalculations,reproducethetotalenergiesforbothstatestowithinasmallfractionofamillihartree.Asaresult,theCC(t;3)approachesaccuratelydescribethecorrespondingA3+uX1+ggap,witherrorsontheorderofonlytensofwavenumbersrelativetofullCCSDT.But,aswehavelearnedabove,onemustgobeyondtheCCSDTleveltoobtainanaccuratedescriptionofthesinglet{tripletgapin(HFH).Whenexaminingtheapproximatequadruplesmethods,wedemonstratedthatamongtheCR-CC(2,4)AA,CR-CC(2,4)DAandCCSDtqapproaches,nonearecapableofrepro-ducingtheCCSDTQdataasaccuratelyasdesired.However,whentheCCSDtqenergiesarecorrectedforthemissingcorrelationsduetotriplesortriplesandquadruples,weob-89tainthevirtuallyperfectagreementwithfullCCSDTQ,bothforthetotalenergiesoftheX1+gandA3+ustatesandthegapbetweenthem.TheCC(t,q;3,4)DAscheme,givinggapvaluestowithintensofcm1fromCCSDTQ,turnedouttobeparticularlye.TheexcellentagreementbetweentheCC(t,q;3)andCC(t,q;3,4)resultsandtheirCCSDTQcounterpartisveryencouraging,sinceCC(t,q;3)andCC(t,q;3,4)replacetheiterativeCPUstepsofCCSDTQthatscaleasN10withthetlylessexpensiveiterativeN6-typeandnon-iterativeN7-typecalculations.90Table3.6:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounter-parts,andthecorrespondingfullCIdatafortheX1+gstateofthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atafewvaluesoftheH{FdistanceRH-F(inA).aRH-FMethod1.5001.6251.7501.8752.000FullCIb-100.589392-100.584704-100.577669-100.570151-100.563055CCmethodswithtriplescCCSD(T)-0.435(0.827)-1.103(0.331)-2.207(-0.594)-3.857(-2.071)-6.122(-4.177)CR-CC(2,3)A2.209(3.471)2.448(3.882)2.566(4.179)2.505(4.291)2.239(4.184)CR-CC(2,3)D-0.343(0.919)-0.467(0.967)-0.686(0.927)-1.018(0.768)-1.455(0.490)CCSDtd2.532(3.794)2.541(3.975)2.557(4.170)2.575(4.361)2.582(4.527)CC(t;3)Ad0.261(1.523)0.293(1.727)0.323(1.936)0.347(2.133)0.357(2.302)CC(t;3)Dd-0.197(1.065)-0.164(1.270)-0.136(1.477)-0.113(1.673)-0.098(1.847)CCSDT0.000(1.262)0.000(1.434)0.000(1.613)0.000(1.786)0.000(1.945)CCmethodswithtriplesandquadrupleseCR-CC(2,4)AA1.820(1.908)2.027(2.131)2.105(2.225)2.002(2.136)1.701(1.844)CR-CC(2,4)DA-0.732(-0.644)-0.888(-0.784)-1.146(-1.026)-1.521(-1.387)-1.993(-1.850)CCSDtqd3.231(3.319)3.284(3.388)3.326(3.446)3.349(3.483)3.346(3.489)CC(t,q;3)Ad0.963(1.051)1.039(1.143)1.096(1.216)1.125(1.259)1.127(1.270)CC(t,q;3)Dd0.506(0.594)0.583(0.687)0.638(0.758)0.668(0.802)0.674(0.817)CC(t,q;3,4)AAd0.091(0.179)0.149(0.253)0.208(0.328)0.264(0.398)0.311(0.454)CC(t,q;3,4)DAd-0.366(-0.278)-0.307(-0.203)-0.249(-0.129)-0.194(-0.060)-0.142(0.001)CCSDTQ0.000(0.088)0.000(0.104)0.000(0.120)0.000(0.134)0.000(0.143)9192Table3.6(cont'd)RH-F2.1252.2502.5003.0004.000MUENPE-100.556686-100.551083-100.542059-100.531336-100.526513CCmethodswithtriplesc-8.994(-6.913)-12.411(-10.226)-20.360(-18.084)-34.862(-32.964)-40.727(-40.115)40.727(40.115)40.292(40.942)1.795(3.876)1.246(3.431)0.224(2.500)-0.340(1.558)-0.027(0.585)2.566(4.291)2.906(3.706)-1.937(0.144)-2.378(-0.193)-2.800(-0.524)-1.838(0.060)-0.291(0.321)2.800(0.967)2.509(1.491)2.566(4.647)2.513(4.698)2.283(4.559)1.618(3.516)0.768(1.380)2.582(4.698)1.814(3.318)0.353(2.434)0.332(2.517)0.244(2.520)-0.021(1.877)-0.063(0.549)0.357(2.520)0.420(1.971)-0.087(1.994)-0.082(2.103)-0.108(2.168)-0.335(1.563)-0.244(0.368)0.335(2.168)0.253(1.800)0.000(2.081)0.000(2.185)0.000(2.276)0.000(1.898)0.000(0.612)0.000(2.276)0.000(1.664)CCmethodswithtriplesandquadruplese1.237(1.385)0.692(0.838)-0.244(-0.115)-0.505(-0.434)-0.113(-0.101)2.105(2.225)2.610(2.659)-2.495(-2.347)-2.932(-2.786)-3.267(-3.138)-2.002(-1.931)-0.378(-0.366)3.267(3.138)2.889(2.772)3.307(3.455)3.225(3.371)2.937(3.066)2.139(2.210)1.093(1.105)3.349(3.489)2.256(2.384)1.102(1.250)1.054(1.200)0.910(1.039)0.515(0.586)0.264(0.276)1.127(1.270)0.863(0.994)0.664(0.812)0.642(0.788)0.561(0.690)0.206(0.277)0.084(0.096)0.674(0.817)0.590(0.721)0.346(0.494)0.366(0.512)0.353(0.482)0.097(0.168)-0.157(-0.145)0.366(0.512)0.523(0.657)-0.092(0.056)-0.046(0.100)0.005(0.134)-0.212(-0.141)-0.337(-0.325)0.366(0.325)0.371(0.459)0.000(0.148)0.000(0.146)0.000(0.129)0.000(0.071)0.000(0.012)0.000(0.148)0.000(0.136)aThefullCIenergiesweretakenfromRef.[162].AsinRef.[162],thelowest-energycoreorbitalwasfrozeninthepost-SCFcalculationsandthesphericalcomponentsofthedorbitalwereemployedthroughout.bThetotalfullCIenergiesinhartree.cFortheCCmethodswithuptotripleexcitations,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTand,inparentheses,relativetofullCI.dTheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedoftwoactiveelectronsandtwoactiveorbitalscorrespondingtotheHOMOandLUMO(the˙gand˙uvalenceorbitals).eFortheCCmethodswithtriplesandquadruples,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTQand,inparentheses,relativetofullCI.Figure3.5:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,fortheX1+gstateofthelinear,D1h-symmetric,(HFH)system,asde-scribedbythe6-31G(d,p)basisset[286,287],atseveralvaluesoftheH{FdistanceRH-F.ThenumericalvaluesoftheerrorsarefoundinTable3.6.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.Middlepanelshowsacomparisonofvariousap-proximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.93Table3.7:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounter-parts,andthecorrespondingfullCIdatafortheA3+ustateofthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atafewvaluesoftheH{FdistanceRH-F(inA).aRH-FMethod1.5001.6251.7501.8752.000FullCIb-100.545993-100.552773-100.555291-100.555097-100.553271CCmethodswithtriplescCCSD(T)0.240(0.600)0.253(0.625)0.269(0.652)0.289(0.678)0.308(0.697)CR-CC(2,3)A0.199(0.559)0.189(0.561)0.174(0.557)0.158(0.547)0.141(0.530)CR-CC(2,3)D-0.217(0.143)-0.197(0.175)-0.181(0.202)-0.173(0.216)-0.172(0.217)CCSDtd1.895(2.255)1.755(2.127)1.616(1.999)1.479(1.868)1.352(1.741)CC(t;3)Ad0.130(0.490)0.121(0.493)0.109(0.492)0.095(0.484)0.080(0.469)CC(t;3)Dd-0.207(0.153)-0.188(0.184)-0.174(0.209)-0.168(0.221)-0.167(0.222)CCSDT0.000(0.360)0.000(0.372)0.000(0.383)0.000(0.389)0.000(0.389)CCmethodswithtriplesandquadrupleseCR-CC(2,4)AA-0.001(0.012)-0.006(0.007)-0.016(-0.002)-0.027(-0.014)-0.042(-0.030)CR-CC(2,4)DA-0.418(-0.405)-0.392(-0.379)-0.371(-0.357)-0.358(-0.345)-0.355(-0.343)CCSDtqd2.242(2.255)2.114(2.127)1.985(1.999)1.855(1.868)1.729(1.741)CC(t,q;3)Ad0.477(0.490)0.480(0.493)0.478(0.492)0.471(0.484)0.457(0.469)CC(t,q;3)Dd0.140(0.153)0.171(0.184)0.195(0.209)0.208(0.221)0.210(0.222)CC(t,q;3,4)AAd-0.072(-0.059)-0.075(-0.062)-0.082(-0.068)-0.092(-0.079)-0.105(-0.093)CC(t,q;3,4)DAd-0.409(-0.396)-0.385(-0.372)-0.365(-0.351)-0.354(-0.341)-0.352(-0.340)CCSDTQ0.000(0.013)0.000(0.013)0.000(0.014)0.000(0.013)0.000(0.012)9495Table3.7(cont'd)RH-F2.1252.2502.5003.0004.000MUENPE-100.550520-100.547315-100.540796-100.531257-100.526513CCmethodswithtriplesc0.326(0.711)0.335(0.712)0.318(0.679)0.191(0.537)0.085(0.434)0.335(0.712)0.250(0.278)0.128(0.513)0.115(0.492)0.096(0.457)0.084(0.430)0.082(0.431)0.199(0.561)0.117(0.131)-0.166(0.219)-0.165(0.212)-0.167(0.194)-0.169(0.177)-0.180(0.169)0.217(0.219)0.052(0.076)1.241(1.626)1.149(1.526)1.034(1.395)1.031(1.377)1.121(1.470)1.895(2.255)0.864(0.878)0.069(0.454)0.059(0.436)0.049(0.410)0.065(0.411)0.082(0.431)0.130(0.493)0.081(0.083)-0.162(0.223)-0.160(0.217)-0.163(0.198)-0.167(0.179)-0.180(0.169)0.207(0.223)0.047(0.070)0.000(0.385)0.000(0.377)0.000(0.361)0.000(0.346)0.000(0.349)0.000(0.389)0.000(0.043)CCmethodswithtriplesandquadruplese-0.056(-0.044)-0.069(-0.058)-0.081(-0.070)-0.054(-0.044)-0.020(-0.010)0.081(0.070)0.080(0.082)-0.350(-0.338)-0.348(-0.337)-0.344(-0.333)-0.307(-0.297)-0.282(-0.272)0.418(0.405)0.136(0.133)1.614(1.626)1.515(1.526)1.384(1.395)1.367(1.377)1.460(1.470)2.242(2.255)0.875(0.878)0.442(0.454)0.425(0.436)0.399(0.410)0.401(0.411)0.421(0.431)0.480(0.493)0.081(0.083)0.211(0.223)0.206(0.217)0.187(0.198)0.169(0.179)0.159(0.169)0.211(0.223)0.071(0.070)-0.117(-0.105)-0.127(-0.116)-0.130(-0.119)-0.074(-0.064)-0.020(-0.010)0.130(0.119)0.110(0.109)-0.347(-0.335)-0.346(-0.335)-0.342(-0.331)-0.306(-0.296)-0.282(-0.272)0.409(0.396)0.127(0.124)0.000(0.012)0.000(0.011)0.000(0.011)0.000(0.010)0.000(0.010)0.000(0.014)0.000(0.004)aThefullCIenergiesweretakenfromRef.[162].AsinRef.[162],thelowest-energycoreorbitalwasfrozeninthepost-SCFcalculationsandthesphericalcomponentsofthedorbitalwereemployedthroughout.bThetotalfullCIenergiesinhartree.cFortheCCmethodswithuptotripleexcitations,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTand,inparentheses,relativetofullCI.dTheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedoftwoactiveelectronsandtwoactiveorbitalscorrespondingtotheHOMOandLUMO(the˙gand˙uvalenceorbitals).eFortheCCmethodswithtriplesandquadruples,thereportedenergyvalues,inmillihartree,areerrorsrelativetofullCCSDTQand,inparentheses,relativetofullCI.Figure3.6:AcomparisonoftheenergiesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,fortheA3+ustateofthelinear,D1h-symmetric,(HFH)system,asde-scribedbythe6-31G(d,p)basisset[286,287],atseveralvaluesoftheH{FdistanceRH-F.ThenumericalvaluesoftheerrorsarefoundinTable3.7.ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.Middlepanelshowsacomparisonofvariousapprox-imatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparedvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.96Table3.8:AcomparisonoftheA3+uX1+ggapvaluesresultingfromvariousCCcalcu-lationsincludinguptotripleanduptoquadrupleexcitationswiththeirparentCCSDTandCCSDTQcounterparts,andthecorrespondingfullCIdataforthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atafewvaluesoftheH{FdistanceRH-F(inA).aRH-FMethod1.5001.6251.7501.8752.000FullCIb95257008491133042147CCmethodswithtriplescCCSD(T)148(-50)298(65)544(274)910(603)1411(1070)CR-CC(2,3)A-441(-639)-496(-729)-525(-795)-515(-822)-461(-802)CR-CC(2,3)D28(-170)59(-174)111(-159)186(-121)281(-60)CCSDtd-140(-338)-173(-406)-206(-476)-240(-547)-270(-611)CC(t;3)Ad-29(-227)-38(-271)-47(-317)-55(-362)-61(-402)CC(t;3)Dd-2(-200)-5(-238)-8(-278)-12(-319)-15(-356)CCSDT0(-198)0(-233)0(-270)0(-307)0(-341)CCmethodswithtriplesandquadrupleseCR-CC(2,4)AA-400(-416)-446(-466)-465(-488)-445(-472)-383(-411)CR-CC(2,4)DA68(52)109(89)170(147)256(229)359(331)CCSDtqd-218(-234)-256(-276)-294(-317)-327(-354)-355(-383)CC(t,q;3)Ad-107(-123)-123(-143)-135(-158)-143(-170)-147(-175)CC(t,q;3)Dd-81(-97)-90(-110)-97(-120)-100(-127)-102(-130)CC(t,q;3,4)AAd-36(-52)-49(-69)-64(-87)-78(-105)-92(-120)CC(t,q;3,4)DAd-10(-26)-17(-37)-25(-48)-35(-62)-47(-75)CCSDTQ0(-16)0(-20)0(-23)0(-27)0(-28)97Table3.8(cont'd)RH-F2.1252.2502.5003.0004.000MUENPE1353827277170CCmethodswithtriplesc2046(1674)2798(2401)4538(4118)7693(7353)8957(8899)8957(8899)8809(8949)-366(-738)-248(-645)-28(-448)93(-247)24(-34)525(822)618(788)389(17)486(89)578(158)366(26)25(-33)578(174)553(332)-291(-663)-299(-696)-274(-694)-129(-469)78(20)299(696)377(716)-62(-434)-60(-457)-43(-464)19(-321)32(-26)62(463)94(437)-16(-388)-17(-414)-12(-432)37(-303)14(-44)37(432)54(388)0(-372)0(-397)0(-420)0(-340)0(-58)0(420)0(362)CCmethodswithtriplesandquadruplese-283(-313)-167(-197)36(10)99(86)20(20)465(488)564(574)471(441)567(537)642(616)372(359)21(21)642(616)621(595)-371(-401)-375(-405)-341(-367)-169(-182)80(80)375(405)455(485)-144(-174)-138(-168)-112(-138)-25(-38)34(34)147(176)181(209)-99(-129)-95(-125)-82(-108)-8(-21)16(16)102(130)118(146)-101(-131)-108(-138)-106(-132)-38(-51)30(30)108(138)138(168)-56(-86)-65(-95)-76(-102)-21(-34)12(12)76(102)88(114)0(-30)0(-30)0(-26)0(-13)0(0)0(30)0(30)aThefullCIenergiesweretakenfromRef.[162].AsinRef.[162],thelowest-energycoreorbitalwasfrozeninthepost-SCFcalculationsandthesphericalcomponentsofthedorbitalwereemployedthroughout.bThefullCIvaluesoftheA3+uX1+ggapincm1.cFortheCCmethodswithuptotripleexcitations,thereportedenergyvalues,incm1,areerrorsinthecalculatedA3+uX1+ggapsrelativetofullCCSDTand,inparentheses,relativetofullCI.dTheactivespaceusedintheCCSDt,CCSDtq,CC(t;3),CC(t,q;3),andCC(t,q;3,4)calculationsconsistedoftwoactiveelectronsandtwoactiveorbitalscorrespondingtotheHOMOandLUMO(the˙gand˙uvalenceorbitals).eFortheCCmethodswithtriplesandquadruples,thereportedenergyvalues,incm1,areerrorsinthecalculatedA3+uX1+ggapsrelativetofullCCSDTQand,inparentheses,relativetofullCI.98Figure3.7:AcomparisonoftheA3+uX1+ggapvaluesresultingfromvariousCCcalculationsincludinguptotripleanduptoquadrupleexcitations,alongwiththeirparentCCSDTandCCSDTQcounterparts,forthelinear,D1h-symmetric,(HFH)system,asdescribedbythe6-31G(d,p)basisset[286,287],atseveralvaluesoftheH{FdistanceRH-F.ThenumericalvaluesoftheerrorsarefoundinTable3.8:ToppanelshowsacomparisonofCCSDTandCCSDTQwithfullCI.Middlepanelshowsacomparisonofvariousap-proximatetriplesmethodswiththeparentCCSDTresults.BottompanelcomparesvariousapproximatequadruplesmethodswiththeparentCCSDTQresults.993.2.4Singlet{TripletGapinBNWhileitmayseemsimple,accuratedeterminationofthegroundstatesofthe12-electronisoelectronicseries,including,butnotlimitedtoBN,C2,BeO,CN+,andBO+,hashis-toricallybeenaverytaskforbothexperimentandtheory.Thisisbecausethetwolowest-energyelectronicstates,1+and3(1+gand3uinthecaseofC2),arenearlydegenerate.WiththeexceptionofBN,the1+(1+gforC2)statehasbeendeterminedtobeslightlylowerinenergy.InthecaseofBN,thegroundstateisof3symmetryandislowerthanthe1+statebylessthan200cm1.Asshowninthissection,arrivingataproperandbalanceddescriptionofthelowest-energy3(X3and1+(a1+)statesofBNandthegapbetweenthemisaHerculeantask.Fromthebeginning,BNgaveexperimentandtheorytrouble.TheinitialworkbyDouglasandHerzbergfocusedoninvestigatingthe3state,andwhiletheywereabletoobservethe1+stateintheirstudy,theycouldonlyspeculatethat3wasthelower-energystate[288].Inaddition,theirestimatedbondlengthofthe3statewasagrossunderestimationofthetruevalue.Itwasnotuntilseveralyearslaterthatadditionalexperiments[289{292]andthetheoreticalcalculations[293{297]supportthatthe3stateisthegroundstate.However,noneoftheexperimentsprovidedanestimatedenergyandtheorywasseverelyoverestimatingthea1+X3gap,evenashighas2.60eV[295].WhiletheseearlycalculationscouldnotproperlydescribetheenergeticsoftheX3anda1+states,fewofthemdidsuggestthatthebondlengthoftheX3statewastlylongerthanthatproposedbyDouglasandHerzberg[294{297].ThispromptedBredohlandcoworkerstoperformtheanalysiswhichrevealedalongerbondlengththatwasingoodagreementwiththeearlytheoreticalestimates[292].Soonafterthat,thefocuswasshiftedtowardaccurately100determiningthea1+X3gapbybothexperiment[298{300]andtheory[298,301{307],whichexploredseveralmethodsinanattempttodescribethelow-lyingstatesofBN.Allofthesecalculationsprovidedalotofintuitionontheimportantcorrelationthatmustbetakenintoaccount,suchastherequirementofatlylargebasisset(somesmallbasissetscanproduceawrongorderingofstates)andtheneedforextremelyaccuratemethods(low-levelmethodsdonotproducecorrectenergeticsandinsomecasesgetthewrongorderingofstates,evenwithatlylargebasisset).Asexperimentwassettlingonanadiabaticelectronic(Te)a1+X3gapoflessthan200cm1,givingresults,suchas15{182cm1[299]or15836cm1[300],computationalresourcesstartedincreasingandpreviouslyunattainabledesiredcalculationsbecameareality.Inthemostrecentyears,high-levelcalculationsemployingMRCIschemes[303{305,308{311],SRCCmethodswithuptohextupleexcitations[312{319],RMRCCapproximations[315,316],quantumMonteCarloapproach[320],andevenfullCI[314,317,318]havebeenperformedforthea1+andX3states.Manyofthesestudieshaveshownthatonecanobtainanaccuratedescriptionofbothstatesandacorrespondinga1+X3separationontheorderofabout200cm1,butonehastoworkveryhardtocomeclosetothebestavailableexperimentalestimates.Aftertheexhaustiveinvestigationsofitslow-lyingstates,assummarizedabove,BNnowservesasa\torture"moleculetotestnewquantumchemistrymethods.Inthisstudy,weexploretheperformanceofourCC(P;Q)methodology[163{165],withafocusonCC(t;3),CC(t,q;3),andCC(t,q;3,4)approachesbyinvestigatingthelow-lyingX3anda1+statesofBNandthecorrespondingadiabaticgapbetweenthem.Thesphericalcc-pVDZandcc-pVTZbasissets[237]wereemployedandtheresultsarecompiledinTables3.9and3.10,respectively.Wedidnotuselargerbasissets,sinceourgoalhasbeentocompareourCC(P;Q)calculationswiththefullCCSDTandCCSDTQdataanditisquitehardto101performfullCCSDTQcalculationswhenlargerbasissetsareemployed.Usingthecc-pVDZbasisset,theequilibriumbondlengthsforthetwostatesweredeterminedattheCCSD,CCSDt,CCSDT,CCSDtq,andCCSDTQlevelsoftheory,andtheresultsaresummarizedinTable3.9.FortheCCSDtandCCSDtqoptimizations,aswellastheCC(t;3),CC(t,q;3),andCC(t,q;3,4)single-pointcalculations,theactivespaceconsistedofthevalence1ˇ,5˙,2ˇ,and6˙orbitalsofBNcorrespondingtothe2psubshellsoftheBandNatoms.Intermsoftheseorbitals,theROHFandRHFwhichweusedinourvariousCCcalculations,arejfcoreg(1ˇ)3(5˙)1(2ˇ)0(6˙)0jfortheX3stateandjfcoreg(1ˇ)4(5˙)0(2ˇ)0(6˙)0jforthea1+state.TheCCSDtgeometrieswereusedforallapproximatetriplesmethods,i.e,allSRCCmethodswithtriplesotherthanCCSDT,whiletheCCSDtqgeometrieswereexploitedinthecalculationsincludingquadruplesotherthanCCSDTQ.Asforthecc-pVTZbasisset,equilibriumbondlengthsofthea1+andX3stateswereoptimizedattheCCSD,CCSDt,andCCSDTlevelsoftheoryand,onceagain,theCCSDtgeometrieswereusedforallapproximatetriplesmethods(seeTable3.10).TheCCSDT/cc-pVTZequilib-riumbondlengthscanbefoundinRef.[312],buttheywererecalculatedbyusinthisstudyaswell.SincetheCCSDt,CCSDT,CCSDtq,andCCSDTQgeometriesareallincloseagree-mentwithoneanother,whenusingthecc-pVDZbasisset,andsincetheSRCCgeometryoptimizationswithquadruplesusingcc-pVTZbasissetareratherexpensive,weusedtheCCSDt/cc-pVTZgeometriesinthecalculationswithapproximatetreatmentsofquadruplesandtheCCSDT/cc-pVTZgeometriesinthecalculationsusingCCSDTQ.Forallpost-SCFcalculations,thelowest-energymolecularorbitalsthatcorrelatewiththe1sorbitalsoftheBandNatomswerekeptfrozen.TheresultsinTables3.9and3.10showthatindependentofthebasisset,theCCSDapproachisincapableofprovidingaproperdescriptionoftheBNmolecule.Although102thisisnotanewitisworthcommentingonit.AsshowninTables3.9and3.10,CCSDprovidesnoticeablysmallerbondlengthsthanthehigher-orderCCSDTandCCSDTQmethods,withresultsfortheX3statebeingparticularlybad.TheCCSDapproachoverestimatestheadiabatica1+X3splittingbyabout3500cm1whencomparedtoCCSDTandnearly4000cm1whencomparedtoCCSDTQ.Clearly,theCCSDmethodcannotprovideanaccurateandbalanceddescriptionofmany-electroncorrelationforthetwostates.So,asdemonstratedinseveralearlierstudies,wemustturntothehigher-orderapproachesinordertoproperlydescribethea1+andX3statesofBNandthecorrespondingsplitting.AscanbeseeninTables3.9and3.10theCCSDTmethodprovidestimprove-mentsoverCCSD,reducingthea1+X3splittingfrom4196cm1obtainedwithCCSDto799cm1whenthecc-pVDZbasissetisemployed,andfrom4391cm1to834cm1whenoneusescc-pVTZ.WhencomparedwiththeCCSDTQdata,noneoftheseresultsisgoodyet,butbeforediscussingtheperformanceofvariousCCapproacheswithquadruples,letuscommentontheapproximatetreatmentsoftriplestoseehowwelltheydowhencomparedtotheirCCSDTparent.WestartwiththeCR-CC(2,3)approaches,sinceitiswellestablishedthatCCSDT(T)giveswrongstateordering,placingthesingletbelowthetriplet[315{317].AsweexaminetheCR-CC(2,3)calculations,wecanseethat,althoughtheyprovideamuchbetterdescriptionofthea1+andX3statesofBNthanCCSD,theresultsarestronglydependentonthevariantofCR-CC(2,3)used.TheCR-CC(2,3)Aapproachproducesenergiesthatareabout540{570andabout800cm1aboveCCSDTfortheX3anda1+states,respectively,whenthecc-pVDZandcc-pVTZbasissetsareemployed.Thisimbalanceindescribingcorrelationforthetwostatesresultsina1+X3gapsofabout1100cm1,almost300cm1abovethecorresponding103CCSDTvalues.GoingfromtheCR-CC(2,3)AapproachtotheCR-CC(2,3)Dmethod,thetotalenergiesoftheX3statelowerbyabout550cm1whenthecc-pVDZbasissetisusedandbyabout480cm1whenweusethecc-pVTZbasis,bringingtheresultingenergiestoamuchbetteragreementwithCCSDT.Unfortunately,theCR-CC(2,3)energiesofthea1+statelowerbyabout900{1160cm1whenwegofromvariantAtoD,somuchoftheimbalanceinelectroncorrelationbetweenthetwostatesremains,althoughtheCR-CC(2,3)Dvalueofthea1+X3gapobtainedusingthecc-pVTZbasisset,of670cm1,isinreasonableagreementwithCCSDT,whichgives834cm1.Onewould,however,liketoimprovethisresult.GiventhatT1andT2clustersthatentertheequationingthenon-iterativecorrectionsfortheCR-CC(2,3)methodsoriginatefromtheunderlyingpoorlyperformingCCSDcalculations,theCR-CC(2,3)approaches,eventhemorecompletevariantD,struggle.Letusthenturntotheactive-spaceCCSDtcalculations,whereT1andT2areiteratedinthepresenceofthedominanttriples,andCC(t;3)corrections.ItisclearfromTables3.9and3.10,thatCCSDtcalculationsarecapableofimprovingCCSDresults,bringingthemclosertoafullCCSDTlevelcomparedtoCCSD,ourexpectationsthatT1andT2amplitudesobtainedwithCCSDtarebetterthanthoseresultingfromCCSDcalculations,buttheCCSDtenergiesoftheX3anda1+statesandthegapbetweenthemarestillquiteinaccurate.Wecouldincreasetheactivespacetoimprovethissituation,butwebelievethatitismorettocorrecttheCCSDtresultsforthetriplesoutsideofthe\littlet"setusingtheCC(P;Q)-basedCC(t;3)methodology.Thisisbytheresultsinourtables.Indeed,theCC(t;3)AandCC(t;3)DschemesprovideanexcellentagreementwiththeparentCCSDTresultsfortheX3anda1+statesofBNandadi-abaticgapbetweenthem.TheCC(t;3)AschemeplacestheX3anda1+statesonly56{61and36{48cm1abovethecorrespondingCCSDTenergies,whenthecc-pVDZand104c-pVTZbasissetsareemployed.ThisperfectlybalanceddescriptionofthetwostatesgivenbytheCC(t;3)Aapproachleadstoa1+X3gapsthatareamere8{25cm1belowtheirvaluesgivenbyCCSDT.TheCC(t;3)Dschemeworksslightlyworse,buttheoverallagreementbetweentheCC(t;3)DandfullCCSDTdataisstillverygood.TheCC(t;3)Dvaluesofthea1+X3gap,of756{758cm1,areinexcellentagreementwiththeirCCSDTcounterparts,whichare799cm1whenthecc-pVDZbasissetisusedand834cm1whenweusecc-pVTZbasis.TheobservedexcellentagreementoftheCC(t;3)AandCC(t;3)DschemeswiththeparentCCSDTmethodinthecaseofthechallengingX3anda1+statesofBNisveryencouraging,sincetheCC(t;3)calculationsreplaceN8stepsofCCSDTbythemuchlessexpensiveN6-likeiterativestepsofCCSDtandnon-iterativeN7-likeoperationsassociatedwiththeCC(t;3)energycorrections.Wenowturntothevariousmethodswithquadrupleexcitations.Asalreadypointedoutabove,oneneedstoincorporateT4clustersinarobustmannertobringthea1+X3gapvaluesclosetotheavailableexperimentalestimates.ThisisbecauseT4inBNarehuge.Indeed,whenthecc-pVDZbasissetisemployed,thebetweentheCCSDTandCCSDTQenergiesare171cm1fortheX3state,501cm1forthea1+state,and330cm1fortheadiabatica1+X3gap.Whenweusethecc-pVTZbasissetthebetweentheCCSDTandCCSDTQenergiesfortheX3anda1+statesandthegapbetweenthemare195,663,and468cm1,respectively.Letusthenexaminehowvariousapproximatetreatmentsofthetriplesandquadruplesperform.TheCR-CC(2,4)correctionstoCCSDarenotrobustenoughtoprovidetrustworthydata.TheCR-CC(2,4)approach,whichweusedintheearlierstudy[316],createsanimpressionthatitworks,bringingthea1+X3gapvaluestoareasonableagreementwithCCSDTQandexperiment,givinggapsinthe250-300cm1range,butthismaybemisleading,sincethe105CR-CC(2,4)AAenergiesoftheindividualX3anda1+statesfromtheCCSDTQcounterpartsby399and184cm1,respectively,whenthecc-pVDZbasissetisusedand458and383cm1,respectively,whenweusecc-pVTZ.CR-CC(2,3)DAisevenworse,providingtheincorrectorderofbothstates.TheCCSDtqcalculationsimprovetheCCSDandCCSDtresults,buterrorsintheresultingenergiesoftheX3anda1+statesandtheadiabaticgapbetweenthemarestillratherlarge,indicatingthepresenceofthigh-orderdynamicalcorrelationbeyondtheCCSDtqlevel.Indeed,whenthecc-pVDZbasissetisemployed,theCCSDtqenergiesoftheX3anda1+statesandtheadiabaticgapbetweenthemdeviatefromthecorrespondingCCSDTQdataby555,874,and319cm1,respectively.Theseincreaseto1144,1867,and724cm1,respectively,whenthelargercc-pVTZbasissetisused,pointingtomassivedynamicalcorrelationsthatgrowwiththebasissetsize.Onceagain,wecouldtrytoimprovetheCCSDtqresultsusinglargeractivespaces,butherewearemoreinterestedintheenessoftheCC(P;Q)-basedCC(t,q;3)andCC(t,q;3,4)corrections,soweturnourattentiontotheCC(t,q;3)andCC(t,q;3,4)approaches.AswecanseeinTables3.9and3.10,theperformanceoftheCC(t,q;3)andCC(t,q;3,4)methodsisexcellent.Inthecaseofthecc-pVDZbasisset,theCC(t,q;3)AandCC(t,q;3)DcalculationscorrectingtheCCSDtqenergiesfortriplesoutsidethe\littlet"setreducethe555,874,and319cm1errorsrelativetoCCSDTQfortheenergiesoftheX3anda1+states,andthegapbetweenthemto138,193,and55cm1,respectively,intheCC(t,q;3)Acase,and47,67,and21cm1,inthecaseofCC(t,q;3)D.TheCC(t,q;3,4)AAandCC(t,q;3,4)DAapproaches,whichalsoincludethequadruplesoutsideofthe\littleq"set,workinasimilarway,withtheCC(t,q;3,4)AAapproachbeingsomewhatmoreaccuratethanCC(t,q;3,4)DA.Theuseofthelargercc-pVTZbasissetdoesnotchangetheseobservations.106TheCC(t,q;3)AandCC(t,q;3)Dmethodsreducethe1144,1867,and724cm1errorsintheenergiesoftheX3anda1+states,andthegapbetweenthemto181,337,and156cm1,respectively,whentheformerapproachisused,and37,139,and102cm1,whenthelattermethodisexploited.Onceagain,theCC(t,q;3,4)AAandCC(t,q;3,4)DAapproachesworkequallywell,withtheCC(t,q;3,4)AAmethodbeingmostaccurate,giving45,11,and56cm1errorsrelativetoCCSDTQindescribingtheX3anda1+energiesandthegapbetweenthem.Insummary,weinvestigatedthenearlydegenerateX3anda1+statesandthecorrespondingadiabatica1+X3gapfortheBNmolecule.WeevaluatedtheabilityoftheCR-CC(2,3),CCSDt,andCC(t;3)approachestocapturethecorrelationdueconnectedtripleexcitationsbycomparingthemagainstCCSDT.WewereabletoshowthatneitherCR-CC(2,3)norCCSDtprovideaccurateandreliableresults,butwhentheCCSDtenergiesarecorrectedforthecorrelationduetothemissingtriplesoutsideofthe\littlet"setviatheCC(t;3)methodology,weareabletoreproducethetotalelectronicenergiesoftheX3anda1+statesandthegapbetweenthemobtainedwithCCSDTtowithintensofwavenumbers.ThesizablecorrelationcontributionsduetoconnectedquadrupleexcitationsfortheX3anda1+statesprovideduswithanexcellentopportunitytoevaluatetheperformanceoftheCR-CC(2,4),CCSDtq,CC(t,q;3),andCC(t,q;3,4)approximations,whichwetestedagainstCCSDTQ.WeillustratedthattheCCSDtqapproachmissesatamountofthecorrelationduetotripleandquadrupleexcitations,especiallyforthelargercc-pVTZbasisset,whereastheCR-CC(2,4)methodologyhasbalancingthetwostates.WhentheCCSDtqenergiesarecorrectedforthemissingtripleorthemissingtripleandquadrupleexcitations,thecorrespondingCC(t,q;3)andCC(t,q;3;4)methodsprovide107thereliableandsystematicbehaviorwewerehopingtoBothschemesrecovertheduetoconnectedtripleandquadrupleexcitationsverywell,asevidencedbythetypicallysmallerrorswithrespecttoCCSDTQ,ontheorderoftensofwavenumbersfortheX3anda1+statesandtheadiabaticgapbetweenthem.GiventheobservedaccuraciesandthefactthattheCC(t,q;3)andCC(t,q;3,4)approachesreducetheiterativeN10stepsofCCSDTQtoiterativeN6andnon-iterativeN7levels,andconsideringthechallengingnatureofthelowestenergysingletandtripletstatesofBN,wecanconcludethatCC(t,q;3)andCC(t,q;3,4)methodsdevelopedinthisworkrepresentandimportantadvanceinelectronicstructurecalculations.108Table3.9:Equilibriumbondlengths(inA)forthelowesttripletandsingletstatesofBN,thecorrespondingadiabaticsinglet{tripletsplittingsTe(incm1),andenergiesrelativetoCCSDTandCCSDTQ(incm1),asobtainedwiththecc-pVDZbasisset.aBondLength(A)RelativetoCCSDTRelativetoCCSDTQMethodX3a1+TeX3a1+X3a1+CCSD1:33601:29344196:32368:95766:32540:06267:0CCSDtb1:3487c1:2970c1054:0471:8726:8642:91227:4CR-CC(2,3)A1083:7570:0791:8678:11292:4CR-CC(2,3)D410:218:6370:2189:7130:5CC(t;3)Ab791:155:647:7226:8548:4CC(t;3)Db757:935:876:9135:4423:7CCSDT1:3491d1:2962d799:0171:1500:7CCSDtqb1:3496e1:2983e788:3554:7873:5CR-CC(2,4)AA254:5399:3184:3CR-CC(2,4)DA453:257:8980:4CC(t,q;3)Ab523:9138:3192:7CC(t,q;3)Db490:046:767:3CC(t,q;3,4)AAb371:933:863:7CC(t,q;3,4)DAb338:157:8189:2CCSDTQ1:3503f1:2976f469:4aThelowest-energymolecularorbitalsthatcorrelatewiththe1sorbitalsoftheBandNatomswerefrozeninallcalculations.bTheactivespaceusedconsistedofthevalenceorbitalsthatcorrelatewiththe2psubshellsoftheBandNatoms.cTheCCSDtgeometrieswereusedfortheCCSDt,CR-CC(2,3),andCC(t;3)calculations.dTheCCSDTgeometrieswereoptimizedinthisworkwiththecc-pVDZbasisset.eTheCCSDtqgeometrieswereusedfortheCCSDtq,CR-CC(2,4),CC(t,q;3),andCC(t,q;3,4)calculations.fTheCCSDTQgeometrieswereoptimizedinthisworkwiththecc-pVDZbasisset.109Table3.10:Equilibriumbondlengths(inA)forthelowesttripletandsingletstatesofBN,thecorrespondingadiabaticsinglet{tripletsplittingsTe(incm1),andenergiesrelativetoCCSDTandCCSDTQ(incm1),asobtainedwiththecc-pVTZbasisset.aBondLength(A)RelativetoCCSDTRelativetoCCSDTQMethodX3a1+TeX3a1+X3a1+CCSD1:31771:27324391:33188:06744:93382:97407:8CCSDtb1:3365c1:2848c1372:01023:21560:81218:12223:6CR-CC(2,3)A1095:7544:2805:5739:11468:3CR-CC(2,3)D670:268:495:9263:2567:0CC(t;3)Ab809:560:735:8255:6698:6CC(t;3)Db756:383:3161:4111:5501:4CCSDT1:3367d1:2826d834:4194:9662:8CCSDtqb1:3365c1:2848c1089:91143:91867:3CR-CC(2,4)AA291:8457:8383:1CR-CC(2,4)DA133:718:1518:2CC(t,q;3)Ab522:4181:1337:0CC(t,q;3)Db468:236:9138:7CC(t,q;3,4)AAb310:744:711:0CC(t,q;3,4)DAb256:599:4209:4CCSDTQ1:3367e1:2826e366:4aThelowest-energymolecularorbitalsthatcorrelatewiththe1sorbitalsoftheBandNatomswerefrozeninallcalculations.bTheactivespaceusedconsistedofthevalenceorbitalsthatcorrelatewiththe2psubshellsoftheBandNatoms.cTheCCSDt/cc-pVTZoptimizedgeometrieswereusedfortheCCSDt,CR-CC(2,3),CC(t;3),CCSDtq,CR-CC(2,4),CC(t,q;3),andCC(t,q;3,4)calculations.dTheCCSDToptimizedgeometrieswerecomputedbyusandreplicatethepreviousCCSDT/cc-pVTZgeometriesinRef.[312].eTheCCSDToptimizedgeometrieswereusedfortheCCSDTQcalculations.110Chapter4AlgorithmicAdvances:tAutomatedImplementationofActive-SpaceCCSDtqandCCSDTqMethods,andTheirFullCCSDTQCounterpartTheCC(t;3)approachwasimplementedbyDr.JunShenfromourgroupwhoemployedaspin-integratedCCSDtprogramhecreated,coupledwithnon-iterativecorrectionsoftheCR-CCtypeobtainedfromalreadytmoin-houseprograms.Thespin-integratedCCSDtequationscamefromanautomaticderivationcodewrittenbyDr.JunShen,whichalsofactorizedtheseequationsandgeneratedthecorrespondingFORTRANcode.TheCC(t,q;3)andCC(t,q;3,4)schemesrequiredaCCSDtqprogram.However,thisautomaticderivationprocesswasinitiallydevelopedtoonlyhandleCCmethodsuptoCCSDT.Forthedoubledissociationofwater,theBe+H2!HBeHinsertion,andthesinglet{tripletgapinthestronglybiradical(HFH)system,wewereabletoperformtheCC(t,q;3)andCC(t,q;3,4)computationsbyusingapilotCCSDtqprograminconjunctionwiththeaforementioned111non-iterativecorrections.Unfortunately,thepilotcodewaswrittenimplementedusingspin-orbitalsandtheCCequationswereunfactorized,andduetothememoryandCPUtimesteprequirementsoftheprogram,wequicklyrealizedthatwithoutantCCSDtqcodewewouldbecomputationallyrestrictedtosystemswithsimilarnumbersofelectronsandorbitalsasthosedescribedabove.Irevisedtheautomaticderivationcode,originallywrittenbyDr.JunShen,toderivethesetofspin-integratedCCSDTQequationsandtheinitialworkingcode,whichwasmotoobtaintCCSDtqandCCSDTqcodes.Thissectionstartsbydiscussingthepowerfultechniqueofspinintegrationforclosedandopenshellsthroughanexampleandthendescribesbasicrulesforquicklyobtainingthespin-integratedtermsthroughdiagrammatictechniques.Thenwedescribehowthespin-integratedCCSDTQequationsareobtainedusinganautomaticderivationroutineandhowtheseequationsareautomaticallyturnedintoworkingFORTRANcodes.AfterthatwediscusshowthereorganizationofasetofloopsintheresultingcodereducedthenumberofunnecessaryCPUoperationsandleadtoatimprovementinthespeedofthecode.WealsodescribehowtheCCSDTQcodewasalteredtoobtainCCSDtqandCCSDTqcodes.4.1SpinIntegrationforClosedandOpenShellsWehaveseenthestructureoftheCCequationsinSection3.1.1,andonethingtonoticeisthattheHamiltonian,consistingofone-andtwo-bodyoperators,doesnotdependonthespinoftheelectrons.Whatspin-freeoperatorsallowustodoisintegrateoutthespinfunctionsfromourequations,andasaresultwenolongerhavetocarryarraysinourCCcodeswithallcombinationsofspin-orbitals,butratherarraysforspin-freeoperators,which112haveasmallernumberofelements,leadingtoatreductioninthecomputationalandmemorywhenproperlyimplemented.Thischaptermakesuseofdiagrams,whicharepowerfultoolstoderiveandorganizenumerousalgebraicexpressionsthatalmostanyaccuratemany-bodytheorygenerates,sowewanttogiveabriefoverviewofdiagramsandhowtointerpretthemalgebraically.His-torically,theuseofdiagramsoriginatedinquantumtheoryusingthetime-dependentformalism.However,asadvocatedbyCzekandPaldusinthelate1960sand1970s[3,4,321](cf.[322{327]foradditionalremarksandfurtherdetails),thetime-independentformulationistinthedevelopmentofquantumchemicalandothermany-bodymethodsthatrelyonthetime-independentScodingerequation.Diagramsareagraphicalrepresenta-tionofWick'stheorem,whichisabasictheoremforthealgebraicmanipulationsinvolvingoperatorsinthesecond-quantizedform.Thebriefdiscussionofdiagrammaticmethodsinthisdissertationfocusesonthetime-independentformulation.Itisimportanttonotethatthesequenceinwhichtheoperatorsact(i.e.,righttoleft)isimportant;thisisindicatedinthediagrambymeansofaso-calledformaltimeaxisasshownbelow:IfwewanttorepresenttheoperatorproductVNT1diagramatically,webeginwithadia-grammaticrepresentationofT1ontheright,followedbyadiagramrepresentingtheoperatorVNdrawntotheleftoftheT1diagram.ThesubscriptNthatwereareusingthenormal-orderformortheseoperators.Inderivingtheexplicitalgebraicexpressionsfortheground-stateCCequations(Eqs.(3.11)and(3.12))weobtainresultingdiagramsbycontractionsoffermionlinesrepresent-113ingtherelevantcreationandannihilationoperatorsthatenterthesecond-quantizedformsoftheoperators,andapplyingthediagrammaticrulestoconverttheresultingdiagramsbackintoalgebraicexpressions.WeusetheHugenholtzandthecorrespondingBrandowdiagrams[324{327]toderivetheexplicitmany-bodyexpressionsforalltermscorrespondingtotheground-stateCCequations(Eqs.(3.11)and(3.12)).Otherrepresentations,suchasthatofGoldstone,couldbeusedaswell,butGoldstonediagramsdonotmakeuseofanti-symmetrizedmatrixelements.Hugenholtz/Brandowdiagramsarepreferablewheneverwerelyonsecond-quantizedoperatorsusingantisymmetrizedmatrixelements,whichisassoci-atedwithfewerdistinctresultingdiagramsthantheGoldstonerepresentation.OneothercommontechniquewhenderivingtheCCequations,inwhichweproject(HNeT(A))Cjiintheexciteddeterminantsja1:::ani1:::ini,isthatwedonotdrawthediagramsrepresentingthebrastateha1:::ani1:::inj.Instead,wedrawallpermissbleresultingdiagramsfor(HNeT(A))Cjiwithnincomingandnoutgoingexternalfermionlineslabeledbyindicesi1;:::;inanda1;:::;an,correspondingtothedeterminantja1:::ani1:::inionwhichweproject[324{327].Thisgreatlyfacilitatestheprocessofdrawingtheresultingdiagramsandmakesthediagramsmuchlesscomplicated[324{327].Thediagramsmaybeinterpretedalgebraicallyusingthefollowingrules[321,324{327]:a.Eachexternallinepointingtotheleftislabeledwitha\particle"(unoccupied)spin-orbitallabela;b;c;d;:::andeachexternallinepointingtotherightwitha\hole"(occupied)spin-orbitallabeli;j;k;l;:::.IntheCCequationsanddiagrams,externallinesshouldalwaysbelabeledinacanonicalsequencetheparticle-holeexci-tationintheja1:::ani1:::inideterminantsonwhichweproject,i.e.,asa;i;b;j;c;k;etc.Theinternalholelinesarelabeledwithm;n;:::,whereastheinternalparticlelineswithe;f;:::.114b.Theone-bodyvertexrepresentingtheone-bodycomponentFN=fqpN[apaq]ofHN,carriesthenumericalvalueoftheFockmatrixelementhpjfjqi=fqp,wherepisanoutgoinglineandqisanincomingline.Forexample,thefollowingcarriesavalueofmatrixelementfba.Figure4.1:Theone-bodyvertexrepresentingtheone-bodycomponentFN=fqpN[apaq]ofHN.c.Thetwo-bodyvertexrepresentingthetwo-bodycomponentVN=14vrspqN[apaqasar]ofHNcarriesthenumericalvalueoftheantisymmetrizedinteractionmatrixelementvrspq=hpqjvjrsihpqjvjsri,wherepandqareoutgoinglinesandrandsareincominglines.Forexample,thefollowingisaBrandowdiagramthatcarriesavalueofvcdab=habjvjcdihabjvjdci.Ingeneral,vrspq=vsrpq=vrsqp=vsrqp.Figure4.2:Thetwo-bodyvertexrepresentingthetwo-bodycomponentVN=14vrspqN[apaqasar]ofHN.d.Theone-,two-,three-,andfour-bodyverticesrepresentingtheT1,T2,T3,andT4clusteroperators,i.e.,carrythenumericalvaluesofthetia,tijab,tijkabc,andtijklabcdampli-tudes,respectively.Thetwo-,three-andfour-bodyamplitudesareantisymmetricwithrespecttothepermutationofindicesi1;:::;inanda1;:::;an..115Figure4.3:Theone-,two-,three-,andfour-bodyverticesrepresentingtheT1,T2,T3,andT4clusteroperators.e.Allthespin-orbitallabelsaresummedoverinternallines,whichareobtainedbycon-tractingtheexternallinesofFN,VN,T1,T2,T3,andT4.f.Thesignofthediagramisdeterminedfrom(1)l+h,wherelisthenumberofloopsandhisthenumberofinternalholelinesinaBrandowrepresentation.g.Thecombinatorialweightfactoroftheconnecteddiagramisspby(12)z,wherezisthenumberofpairsof\equivalent"lines.Apairofequivalentlinesisasbeingtwolinesbeginningatthesamevertexandendingatanother,butalsosamevertex,andgoinginthesamedirection.ThisweightruleissptoaHugenholtz/Brandowrepresentation.Linesthatcarrylabels(suchastheexternallinestheha1:::ani1:::injbrastateonwhichweprojecttheCCequations)arealwaysregardedasnon-equivalent.h.Thealgebraicexpressionforeachdiagramshouldbeprecededbyasuitablecompleteorpartialantisymmetrizationoperator,permutingtheexternallinesinalldistinctwaystokeepthefullantisymmetryofaexpressionforaquantity,suchasthecluster116amplitudes,whichareantisymmetricwithrespecttopermutationsofindicesi1;:::;inanda1;:::;an.WhiletheHugenholtzrepresentationproducesfewerdistinctresultingdiagramsthantheGoldstoneone,theGoldstonerepresentationisusefulindevelopingaspin-adaptedformalismforspin-freeHamiltonians[321,324{326].Wewillusenon-antisymmetrizedmatrixelementsandclusteramplitudes,justaswiththeGoldstoneformulation,tocarryoutanexampleofspin-integrationandtobuildanunderstandingofthetechnique.Intheend,wewillrevisittheabovediagrammaticrulesforHugenholtz/Brandowdiagramsandextendthemforthedevelopmentofspin-integratedequations.Figure4.4:Diagramrepresentingoneofthe12vnt22termsthatappearintheprojectionoftheCCequationsontodoublyexciteddeterminants.LetusconsiderthetermcorrespondingtothediagraminFigure4.4,whichrepresentsoneofthe12vnt22termsthatappearintheprojectionoftheCCequationsontodoublyexciteddeterminants.IfwewereconsideringtheHugenholtz/Brandowformalism,thenthe117completeexpressionforthistermisAijvefmntimaetnjfb;(4.1)wherevefmn,timae,andtnjfbareantisymmetrizedmatrixelementsandAijisanantisymmetrizerasAij=1(ij);(4.2)where(ij)isthetranspositionofindicesiandj.However,forthisspinintegrationexample,weneedtoconsidernon-antisymmetrizedmatrixelementsandalsodroptheantisymmetrizerAij.Thenon-antisymmetrizedintegralformofthethreeelementsinEq.(4.1)arevefmn=hmnj^vjefi=Z m(x1) n(x2)^v e(x1) f(x2)dx1dx2;(4.3)timae=haej^t2jimi=Z a(x1) e(x2)^t2 i(x1) m(x2)dx1dx2;(4.4)tnjfb=hfbj^t2jnji=Z f(x1) b(x2)^t2 n(x1) j(x2)dx1dx2;(4.5)where p(xq)isaspin-orbitalandxq=f~rq;!qgisthecompositionofthethreespatialcoordinates(~rq)andanarbitraryspinvariable(!q)foragivenelectronq.Theeightindicesintheseexpressions,eachlabelingaspin-orbital(indicatedwithalower-caseletter)canberewrittenasaproductofaspatialorbital(indicatedwithanupper-caseletter)andaspinfunction, p(xq)=˚P(~rq)˙P(!q);˙P=or;(4.6)orinDiracnotation,jpi=jP˙Pi=jP˙Pi:(4.7)118WecanthereforerewriteEqs.(4.3{4.5)usingthisrelationshipasfollows:vefmn=hM˙MN˙Nj^vjE˙EF˙Fi;(4.8)timae=hA˙AE˙Ej^t2jI˙IM˙Mi;(4.9)tnjfb=hF˙FB˙Bj^t2jN˙NJ˙Ji:(4.10)Sincenoneoftheoperatorsdependonspin,thentheseintegralscanbeseparatedinintegralsthatcontainonlythespatialorbitalsandintegralsforthespinfunctionsasfollows:vefmn=hMNj^vjEFih˙M˙Nj˙E˙Fi;(4.11)timae=hAEj^t2jIMih˙A˙Ej˙I˙Mi;(4.12)tnjfb=hFBj^t2jNJih˙F˙Bj˙N˙Ji:(4.13)ThelastintegralsoverspinfunctionsintheseequationswillsimplyreducetoKroneckerdeltasgivingvefmn=hMNj^vjEFiMENF;(4.14)timae=hAEj^t2jIMiAIEM;(4.15)tnjfb=hFBj^t2jNJiFNBJ:(4.16)TwooftheKroneckerdeltasarerepeatedwhichleavesonlyfouruniqueKroneckerdeltasthathavetocorrespondtothetwospincases(and).WhenwestartedwithEqs.(4.3){(4.5)wehadeightindicesforeachspin-orbital.Thatmeansifoneweretonaivelycodethistermbasedonspin-orbitals,thenforagivencombinationofseparatespatialorbitals,each119containinganandspin-orbital,onewouldcarryout28or256productsforthisexpression.ButasoneseparatesoutthespinfunctionsandevaluatesthecorrespondingintegralsmanyofthesecombinationsaresimplyzeroandintheendweendupwithasetofKroneckerdeltasthatcorrespondtotheandspincases,whicharelessinnumberthanthenumberofindicesforaterm.Inthisexample,sincethereareonlyfouruniqueKroneckerdeltas,thenforagivencombinationofseparatespatialorbitalsonewouldonlycarryout24or16products,adrasticreductionfromthe256productsbasedonaspin-orbitalimplementation.Inaddition,wewentfrommatrixelementsforspin-orbitalstomatrixelementsthatdependonlyonthespatialorbitals,forwhichtherearehalfasmanyspatialorbitalsasspin-orbitals.Thismeansthereistheaddedbofsavingonmemoryinthecalculations.Fromourexample,wecanmakesomegeneralobservationssowecanestablishdiagram-maticrulesforquicklygeneratingexpressionswhichwillallowustotlyimplementthespin-integratedCCequations.Thekeyobservationfromourexampleisthateverypath,whetherclosedoropen,hasaKroneckerdelta,orasetofKroneckerdeltasthattogethersimplifytoasingleKroneckerdelta,associatedwiththetwospincases.Ifweweredealingwithaclosed-shellsystem,wecanfurthersimplifyourequationsbytakingadvantageofthefactthattheandspincasesforeachloopareequivalent.Inthiscasewecanmultiplytheexpressionbythefactor2l,wherelisthenumberofloopsinagivenGoldstonediagram.ThenonewouldsimplyevaluatethespincasesfortheopenpathscorrespondingtotheremainingKroneckerdeltas.Inthisthesiswork,wewantedtodevelopageneralapproachsowecandealwithclosed-andopen-shellmolecules,so,weneedtokeepallofthespincasesforeachpath.Soforeveryoperator,onewillneedseparatespincases.Table4.1liststhematrixelementsfortheone-andtwo-bodyHamiltonianoperatorsandtheampli-tudesfortheclusteroperatorsuptoT4correspondingtothevariousspincases.InTable120Table4.1:MatrixelementsforthestandardHamiltonianandamplitudesfortheclusteroperatorsthatappearinCCSDTQ,alongwiththeircorrespondingcasesforspin-integratedequations,andtherespectiveantisymmetrizersforthespincases.SpinOrbitSpinIntegratedAntisymmetrizersfqpfqpf~q~pvrspqvrspqApqArsvr~sp~qv~r~s~p~qApqArstiatiat~i~atijabtijabAijAabti~ja~bt~i~j~a~bAijAabtijkabctijkabcAijkAabctij~kab~cAijAabti~j~ka~b~cAjkAbct~i~j~k~a~b~cAijkAabctijklabcdtijklabcdAijklAabcdtijk~labc~dAijkAabctij~k~lab~c~dAijAklAabAcdti~j~k~la~b~c~dAjklAbcdt~i~j~k~l~a~b~c~dAijklAabcd4.1,theindicesforthespindonothaveaccents,whereastheindicesforthespinaretiatedbyatilde.Additionally,werecallthatwehaveappliedthespin-integration121techniquetoaGoldstonediagram,whichcorrelatestonon-antisymmetrizedmatrixelements.IfwegenerateallnecessaryGoldstonediagramsforagiventermandthenapplythespinintegrationtechnique,thenwecansumtheresultingtermstogivematrixelementsthatareantisymmetrizedforindicesthatsharethesameparticle-holecharacterandspinassignment.Inconventionalspin-orbitalimplementations,thisisavoidedbecauseonecangenerateex-pressionswithfullyantisymmetrizedmatrixelementsbyemployingHugenholtz/Brandowdiagrammatictechniques.However,inoursituation,wehavetspincasesforeachoperator,andtheindicesoftheseoperatorsareantisymmetrizedonlywithotherindicesthathavethesameparticle-holecharacterandthesamespinassignment.Table4.1liststhecorrespondingantisymmetrizationcharacterdescribinghoweachmatrixelementtransformsforthetspincasesofeachoperator.TheantisymmetrizersthatappearinTable4.1areasfollows:ApqApq=1(pq);(4.17)ApqrApqr=1(pq)(pr)(qr)+(pqr)+(prq);(4.18)ApqrsApqrs=1(pq)(pr)(ps)(qr)(qs)(rs)+(pq)(rs)+(pr)(qs)+(ps)(qr)+(pqr)+(prq)+(qrs)+(qsr)+(rsp)+(rsp)+(rps)+(spq)+(sqp)(pqrs)(pqsr)(prqs)(prsq)(psqr)(psrq);(4.19)where(pq)isthetranspositionofindicespandq,while(pqr)and(pqrs)designatethethree-andfour-indexcyclicpermutations.TheotherkeyobservationisthatclusteramplitudesaresolvedforbyprojectingtheCCequationsontoexciteddeterminantscorrespondingto122thespclusteroperatorsenteringagivenleveloftheory(seeEq.3.11).Inthecaseofspinintegratedequations,weprojectontoexciteddeterminantscorrespondingtothespincasesoftheclusteroperatorsinTable4.1.FortheCCSDTQmethod,thesetofexciteddeterminantsonwhichweprojectarelistedinTable4.2.Table4.2:ExciteddeterminantsenteringintoEq.(3.11)forspin-orbitalimplementationsofCCSDTQandthecorrespondingspinvariantsforthespin-integratedimplementations.Spin-OrbitSpin-Integratedhaijhaij,h~a~ijhabijjhabijj,ha~bi~jj,h~a~b~i~jjhabcijkjhabcijkj,hab~cij~kj,ha~b~ci~j~kj,h~a~b~c~i~j~kjhabcdijkljhabcdijklj,habc~dijk~lj,hab~c~dij~k~lj,ha~b~c~di~j~k~lj,h~a~b~c~d~i~j~k~ljAsmentionedabove,ifwesumthespin-integratedtermswhichusenon-antisymmetrizedmatrixelementswecanrewritetheequationsintermsofpartially,orinsomecasesfully,antisymmetrizedmatrixelements(seeTable4.1).InsteadofconsideringallGoldstonedi-agrams,whichforCCSDTQisadiscouragingtaskduetothesheernumberofdiagrams,wecanreformulatetheHugenholtz/Brandowdiagrammaticrulesmentionedpreviously.Di-agramscorrespondingtospin-integratedequationsmaybeobtainedandinterpretedusingthefollowingrules:a.Eachexternallinepointingtotheleftislabeledwitha\particle"(unoccupied)labela;b;c;d;:::forthespincasesor~a;~b;~c;~d;:::forthespincasesandeachexternallinepointingtotherightwitha\hole"(occupied)labeli;j;k;l;:::forthespincasesor~i;~j;~k;~l;:::forthespincases.IntheCCequationsanddiagrams,externallinesshouldalwaysbelabeledinacanonicalsequencetheparticle-holeexcitationintheha1:::am~am+1:::~ani1:::im~im+1:::~injdeterminantsonwhichweproject,i.e.,asa;i;b;j;c;k;123etc.(seeTable4.2).Theinternalholelinesarelabeledwithm;n;:::(or~m;~n;:::),whereastheinternalparticlelineswithe;f;:::(or~e;~f;:::).b.Theone-bodyvertexrepresentingtheone-bodycomponentFNofHNissplitintotwospincases:FN=fqpN[apaq]andFN=f~q~pN[a~pa~q].TheformercarriesthenumericalvalueoftheFockmatrixelementfqp=hpjfjqi,whilethelattercarriesthenumericalvalueoftheFockMatrixelementf~q~p=h~pjfj~qi,wherep(or~p)isanoutgoinglineandq(or~q)isanincomingline.c.Thetwo-bodyvertexrepresentingthetwo-bodycomponentVNofHNissplitintothreespincases:VN=14vrspqN[apaqasar],VN=vr~sp~qN[apa~qa~sar],andVN=14v~r~s~p~qN[a~pa~qa~sa~r].Thecarriesthenumericalvalueoftheantisymmetrizedinteractionmatrixele-mentvrspq=hpqjvjrsihpqjvjsri,thesecondvr~sp~q=hp~qjvjr~si,andthethirdv~r~s~p~q=h~p~qjvj~r~sih~p~qjvj~s~ri,wherep(or~p)andq(or~q)areoutgoinglinesandr(or~r)ands(or~s)areincominglines.d.Theone-,two-,three-,andfour-bodyverticesrepresentingtheT1,T2,T3,andT4clusteroperatorsaresplitintoseveralspincases.ThematrixelementscarriedbythetspincasesarelistedinTable4.1alongwiththeircorrespondingantisym-metrizersthatdescribehoweachelementtransformswithapermutationofindices.e.Allthespin-orbitallabelsaresummedoverinternallines,whichareobtainedbycon-tractingtheexternallinesofthetspincasesofFN,VN,T1,T2,T3,andT4.f.Thesignofthediagramisdeterminedfrom(1)l+h,wherelisthenumberofloopsandhisthenumberofinternalholelinesinaBrandowrepresentation.g.Thecombinatorialweightfactoroftheconnecteddiagramisspby(12)z,where124zisthenumberofpairsof\equivalent"lines.Apairofequivalentlinesisasbeingtwolinesbeginningatthesamevertexandendingatanother,butalsosamevertex,goinginthesamedirection,andhavingthesamespinassignment.Ifallthreerequirementsarenotmetthenthelinesareregardedasnon-equivalent.Linesthatcarrylabels(suchastheexternallinestheha1:::am~am+1:::~ani1:::im~im+1:::~injbrastateonwhichweprojecttheCCequations(seeTable4.2)arealwaysregardedasnon-equivalent.h.Thealgebraicexpressionforeachdiagramshouldbeprecededbyasuitablecompleteorpartialantisymmetrizationoperator,permutingtheexternallinesinalldistinctwaystokeepthefullorpartialantisymmetryofaexpressionforaquantitybyitscorrespondingspincase,suchasclusteramplitudes,whichareantisymmetricwithrespecttopermutationsofindicesi1;:::;imand~im+1;:::;~inaswellasa1;:::;amand~am+1;:::;~an(seeTable4.1).Theserulecangreatlyfacilitatetheprocessofdrawingtheresultingdiagramsforthespin-integratedequationsandallowsforeasytranslationbackintothealgebraiclanguage.4.2AutomatedApproachtoDerivationandComputerImplementationofCoupled-ClusterMethodsinFac-torizedFormFromthelastsectionwelearnedthatderivingthespin-integratedequationsforCCSDTQcanbeadauntingtask.Comparedtothefourtclusteroperatorsusedinstandardspin-orbital-basedimplementationsofCCSDTQ,thespin-integratedapproachrequiresfour-125teenclusteroperatorsforthevariousspincases.Asaconsequence,wemustprojecttheconnectedclusterformoftheScodingerequationontofourteenclassesofexciteddetermi-nantscomparedtoprojectingontofourclassesofexciteddeterminantscorrespondingtothestandardCCSDTQapproach.Also,insteadoftwooperatorsfortheHamiltonian,wenowconsidereseparateoperatorsafterspinintegration.Ifwehadtoderiveeverytermsfromthepossiblecombinationsfromthesesetsofoperatorsandprojectionsmanually,notonlywouldittakeanincrediblylargeamountoftime,butalsobeerror-proneandstillrequirestheintimidatingtaskoftranslatingtheequationsintoacomputerprogram.Asanote,theuseoftheword`projections'hereandthroughoutreferstotheexciteddeterminantontowhichweprojecttheconnectedclusterformoftheScodingerequationha1:::am~am+1:::~ani1:::im~im+1:::~injlistedinTable3.1forCCSDTQ.ThecodethatDr.JunShenwrote,andwhichwasmobymyselftoallowforthedevelopmentofCCSDTQ,providesameticulouslysystematicwayofderivingthespin-integratedCCSDTQequationsandsubsequentlytranslatingtheequationintoready-to-usecode.TheautomaticderivationprogramisdesignedtoonlytakethedesiredlevelofCCtheoryonewantstoderive,denotedbymA(recallthatmA=2forCCSD,mA=3forCCSDT,mA=4forCCSDTQ,etc.)asaninput.Fromthatsinglevariable,thecodeisabletogenerateallofthetermsforthatleveloftheorybyevaluatingallpossiblecombinationsoftheclusterandHamiltonianoperatorsforprojectionsontoeachofthetexciteddeterminantscorrespondingtospincasesoftheclusteroperators.Thisisdonebygeneratingafullycontractedexpressions(fullyconnecteddiagrams)whileobeyingthealgebraic/diagrammaticrulesofmany-bodymethods.Theinitialpartoftheprogramassignsvariousparameterstotheoperators,projections,andtheircorrespondingindicesinordertosystematicallytrackandeditthemthroughout126thecode.Tostart,eachspincaseoftheclusteroperatorsisassignedanindividualnumberfrom1tonT.Thatnumberisusedtodistinguishtheclusteroperatorsfromeachotherandisrelatedtotherank,rT,oftheclusteroperatorandtothevalue,sT,whichisthenumberofoccupiedandunoccupiedindicesthatcorrespondtospin.Forexample,thenumber1,intheseriesfrom1tonT,correspondstotheT1operatorwithamplitudestia,thenumber2correspondstotheT1operatorwithamplitudest~i~a,number3correspondstotheT2operatorwithamplitudestijab,andsoondownthelistofclusteroperatorsinTable4.1.Ingeneral,thetotalnumberoftclusteroperatorsnTisnT=mAXrT=1rTXsT=01:(4.20)ThiscanbevbyexaminingTable4.1(nT=5forCCSD,nT=9forCCSDT,nT=14forCCSDTQ).Sincethereareonlyuptotwo-bodycomponentsintheHamiltonian,labeledbyfourorbitalindices,thenwecanonlycontractuptofourclusteroperatorsofwhichtherearenToptions,plusonemoreforchoosingnottocontractwithaclusteroperatorbuttheprojectioninstead.Thatleaves(nT+1)4combinationsthatmustbeevaluatedinjunctionwiththeespincasesfortheHamiltonianoperatorsforeachprojectioncorrespondingtothenTclusteroperators.Whilecheckingeachcombinationtheprogramemploysaniftytrickinrecognizingthatanynumberfrom0to(nT+1)41canberepresentedas:IndT(1)(nT+1)3+IndT(2)(nT+1)2+IndT(3)(nT+1)+IndT(4);(4.21)whereIndT(i)istheindexfortheithclusteroperatorandhasvaluesfrom0tonT,where0correspondstonoclusteroperator,sowecaneasilydeterminewhichsetofspin-integrated127clusteroperatorscharacterizeanyofthe(nT+1)4combinations.SincethegeneratingcodeiswrittentobegeneralforanystandardlevelofCCtheory,thismathdeviseprovestobeausefultoolinmanycombinatoricscircumstances,suchascountingoperatorsandindices,whilesystematicallyderivingtheCCequations.Anotherobservationtomakeisthatwedonothavetoconsiderallcombinationsbecauseclusteroperatorscommutewitheachother,butratherthoseinwhichIndT(1)IndT(2)IndT(3)IndT(4).Aftersettingupallofthepossiblecombinationsofclusteroperators,whatisleftistoevaluatethemixofHamiltonianoperators,clusteroperators,andprojectionsinanattempttoobtainafullycontractedexpressionwiththesethreegroups.However,mostofthesecombinationscanbequicklyeliminatedbasedonthenumberoforbitalindicesthatlabeltheoperatorsandprojections.Forexample,ifthetotalnumberoforbitalindicesforanyoneofthethreegroupsmentionedabove(projection,Hamiltonian,orclusteroperators)ismorethanhalfofthetotalnumberofindicesforallthreegroupsinanyofthecombinations,thatmixcanbeskipped,becauseinordertoobtainafullycontractedexpression,onewouldhavetostartcontractingamongthatgroupofindices,whichisnotallowed.Soanyofthethreegroupscancontributeuptohalfofthetotalnumberofindices,buteventhenaslongasthereisaclusteroperatorinthetermbeingconsidered,thenumberofindicesfromtheprojectioncannotbehalfofthetotalnumberofindices,otherwisetheHamiltonianandclusteroperatorswouldhavetobeuncontracted,sothosecombinationscanbetossedaswell.Oftheremainingpossibilities,wehavetostartprobingthehole-particlecharacterandthespinassignmentsforindicesbeforewecaneliminateaterm.Inordertoanalyzetheindicesandstartconsideringcontractions,wehavetodesignawayofcarefullymonitoringallnecessaryinformationaboutagivenindex.Anarrayisallocatedthathasasmanyelementsasthetotalnumberofindicesfortheprojection,128Hamiltonian,andclusteroperatorsassociatedwithaparticularcombination.Eachelementofthearray,whichcorrespondstoaparticularindex,isassignedathreedigitnumber.Thenumbertellsiftheindexisorspin(=1,=2).Thesecondnumbertellsthesetofcharactersthatdescribewhethertheindexisaparticleorholeandwhetheritisfree,orundeclared(1:fa;b;c;dg,2:fi;j;k;lg,3:fe;f;g;hg,4:fm;n;o;pg,5:fu;vg,6:fv;wg).Thethirdnumbertellswhichspcharacterinthesetcorrespondstothatindex.Forexample,iftheindexwasgiventhelabelb,andcorrespondedtoaspincase,thenthethreedigitnumberassignedtotheparticularindexis212;thenumberis2becauseitcorrespondstospin,thesecondnumberis1becausewearedescribinganunoccupiedindex,andthethirdnumberis2becausebisthesecondletterinthatlistofcharacters.Alloftheindicesfortheprojectionare(characterlists1and2)andalwaysremainInitially,alloftheindicesfortheclusteroperatorsarefree(characterlists3and4),butcanchangetoindicesiftheycorrespondtoacontractionwithaindexfromtheprojection.Atthestarttheindicesfortheone-andtwo-bodytermsoftheHamiltonianoperatorareassignedcharactersfromlists5and6,becausedependingonwhetheritcontractswithanindexfromtheprojectorclusteroperatorsdeterminesiftheindexcorrespondstoaparticleorholeandifitisorfree.Nowthatthecodehasidenallpossiblecombinationsofclusteroperators,eliminatedthosethatobviouslydonotappearintheCCequations,andthenassignedtheindicestotheremainingcombinations,whatisleftistocarryoutWick'stheoremonthetcombinations.Theprogramsystematicallyandcarefullycariesoutthecontractionsinanattempttocreateafullycontractedexpression.Itiscarefultoconsiderthenecessarypermutationsofindicestoretaintheantisymmetryoftheclusteramplitudebeingcalculated.Theprogramtracksthesignandweightoftheterm,andifitcanformafullycontracted129expression,thenthattermissavedbywritingittoaLatexforthegivenprojectionforeasyviewing.Figure4.5showsanexampleoftheoutputbythecodeforaspHNT3combinationthatappearsintheprojectionoftheconnectedclusterformoftheScodingerequationontoha~bi~jjforCCSDTQ.Whenwritingthetermsout,theclusteroperatorsarelabeledwiththeirranks,butalsoaletter,whichisnecessaryforsuccessivealterationsoftheterms.TheletterAindicatesthatallindicescorrespondtothespin.EachsuccessivecharacterafterAcoincidewithreplacinganoccupiedandunoccupiedpairofindicesbythecompliment,justasillustratedintheorderofmatrixelementslistedinTable4.1.Asanexample,`tf3Bg(aebimj)',whichisfoundinFigure4.5,correspondstotheclusteramplitudetim~jae~b.Ifthereismorethanoneclusteroperatorforaterm,thentheyarewrittensuccessively,oneafterthenext.Thetspincasesfortheone-andtwo-bodymatrixelementsofHNarealsodistinguishedwhenwrittentotheLatexTheone-bodymatrixelementsfqpandf~q~parewrittenouttotheLatexintheform\Ffqpg",andforthecasewherebothindicescorrespondtothespinthistermisfollowedbyaletter`B'thatiscommentedoutinordertodistinguishit.Thetwo-bodymatrixelementvr~sp~qiswrittenoutintheform\<pqjrs>",whilethematrixelementsvrspqandv~r~s~p~qarewrittentotheLatexwiththeform\<pqjjrs>",whereonceagain,ifalltheindicesfollowedtothespinthenthistermisproceededbyaletter`B'thatiscommentedout.Withtheseformats,atermcanthenbepickedupanytimeandindicesandtheircorrespondingspincanbeassignedinstantlyfortheoperators.IfweweretonaivelyprogramthevarioustermsoftheCCequationsderivedbythecodeatthispoint,i.e.,onebyoneandwiththeexplicitloopswhatcorrespondtosummationsoverindicesthatlabelinternallines,wewouldhaveaCCSDTQroutinethathasCPUtimestepthatscaleasN12,ratherthanN10,whichisobtainedwithantimplementation1301\begin{eqnarray}2<\Psi_{i\tilde{j}}^{a\tilde{b}}|\hat{H}|\hat{T}_{3B}\Psi_0>3+\sum_{em}4F_{me}5t_{3B}(aebimj)6\nonumber\\&&7-\frac{1}{2}\sum_{emn}8<mn||ie>9t_{3B}(aebmnj)10\nonumber\\&&11+\frac{1}{2}\sum_{efm}12<am||ef>13t_{3B}(efbimj)14\nonumber\\&&15-\sum_{emn}16<mn|ej>17t_{3B}(aebimn)18\nonumber\\&&19+\sum_{efm}20<mb|ef>21t_{3B}(aefimj)22\nonumber\\&&23\end{eqnarray}Figure4.5:SampleoutputoftheLatexcorrespondingtotheHNT3Btermgeneratedbytheautomaticderivationcode.Seetextforadescriptionofthenotation.ofCCSDTQ.Inordertoobtainthelower-orderscalingtheequationshavetobefactorizedinwhichtheequationsarerewrittenintermsofbinarytensorproducts.Theprogramthatde-rivestheequationsalsoperformsthisfactorizationprocedure,breakingdownthetermsintobinaryproducts.WhentherearemultipleclusteroperatorscontractedwiththeHamiltonianoperators,theprogramwillformtheproductbetweentheHamiltonianandclusteroperatorwiththemostcomputationallyintensivesummationsoverindicesthatlabelinternallinesForexample,forthetermAijAabvefmntimaetjftnbtheproductbetweenvefmnandtimaewouldbecarriedoutsincethesummationoverindiceseandmismoredemandingtheneitherthesummationoverfcorrespondingtothesecondclusteroperatororthesummationover131ncorrespondingtothethirdclusteroperator.Theresultoftheproductofvefmnandtimaeistheintermediatexifan.Thisprocessisrepeated,formingaproductbetweentheintermediateandclusteroperatorcorrespondingtothenextmostcomputationallyintensivesummation,untilthelastclusteroperatorintheseriesisreached.Atthispoint,theprogramwilllookforothertermsthatsharethesameclusteroperatorandsameantisymmetrizersappliedtotheterm(inthisexampletheantisymmetrizersareAijandAab),andbeforecarryingouttheproduct,theprogramsumsthecorrespondingintermediatesandHamiltonianelementsforthetermsthatonlyhaveoneclusteroperatorintoacollectiveintermediateinordertoreducethenumberofoperationsthatwouldotherwisebecarriedout.Afterobtainingthefactorizedspin-integratedCCequations,thenthesetoftermsisconvertedintoFORTRANroutinesthatcanbereadintoacodepredesignedfortheiterativeprocedure.EveryproductoftwooperatorsdeterminedthroughthefactorizationiscarriedoutinthesamegeneralmannerwhichislaidoutinFigure4.6.Thestepsaretoreorganizetheindiceslabelingthetwooperatorsofthebinaryproductinordertotakeadvantageoftlinearalgebralibrariesthatcanbeintrinsicallyparallelizedmakinguseofshared-memorysystems.Wewanttotakeadvantageofthefastvector-matrixandmatrix-matrixmultiplicationroutines,butinordertodosoeveryarrayofaHamiltonian,intermediate,orclusteroperatorhastobearrangedaseitheravector(one-dimensionalarray)ormatrix(two-dimensionalarray).Ifalloftheindicesofanoperatorinagivenproductareinternalindicesthataresummedover,onecanrepresentitasavectorwhosesizeistheproductoftheindividualdimensionsoftheoriginalarray.So,forexample,thefemmatrixelementsintheproductfemtimaetermwouldbetransformedtoavectorinthisprocedurewiththedimensionnncorrespondingtothenumberofelementsforindiceseandm.Iftheindicesofanoperatorarebothinternalandexternal,onecanrepresentit132asasamatrix,whereonedimensionofthematrixisfortheinternalindicesbeingsummedoverandhasanextentequaltotheproductofeachdimensioncorrespondingtotheinternalindices.Theotherdimensionofthematrixisfortheexternalindicesandhasanextentequaltotheproductofeachdimensioncorrespondingtotheexternalindices.Forexample,thetimaematrixelementsintheproductfemtimaetermwouldbetransformedtoamatrix,whereonedimensioncorrespondstoindiceseandmandhasanextentequaltonn,andtheotherdimensionofthematrixcorrespondstoindicesaandiandhasanextentequaltonn.1ALLOCATEarrayA2CALL"REORDER"routineforarrayA3ALLOCATEarrayB4CALL"REORDER"routineforarrayB5ALLOCATEarrayC6DeclareI17DeclareI28DeclareI39CALLDGEMMorDGEMVroutine10DEALLOCATEarrayA11DEALLOCATEarrayB12CALL"SUM"routine(s)13DEALLOCATEarrayCFigure4.6:Thegeneralstructureofeachbinarytensorproductasproducedbytheautomaticderivationandimplementationprogram.Foramorespdescriptionofhoweachproductoftwoelementsiscarriedoutwecanexaminethegenerallayoutofthecodeforabinaryproduct(Figure4.6)andanactualexampleofthecode(Figure4.7)indetail.Forreasonsthatwillsoonberealized,theoperatorarraysmustbearrangedsuchthatthedimensionsareforinternalindicesofthatoperatorandtheremainingdimensionsarefortheexternalindicesoftheoperator.Theorderofinternalandexternalindicesamongthemselvesdoesnotmatter,withoneexception.1331ALLOCATE(D1(N0+1:N1,N0+1:N1,N1+1:N3,N0+1:N1))2CALLREORDER1243(N0,N3,N0,N3,N0,N3,N0,N3,3&N0,N1,N0,N1,N1,N3,N0,N1,IntR,D1)4ALOCATE(D2(N0+1:N1,N0+1:N1,N1+1:N3,N1+1:N3))5CALLREORDER4312(N1,N3,N1,N3,N0,N1,N0,N1,6&N0,N1,N0,N1,N1,N3,N1,N3,t2A,D2)7ALLOCATE(S14(N1+1:N3,N1+1:N3,N1+1:N3,N0+1:N1))8I1=K1*K39I2=K3*K310I3=K1*K111CALLDGEMMROUT(I1,I2,I3,D1,D2,S14)12DEALLOCATE(D1)13DEALLOCATE(D2)14CALLSUM2314(N1,N3,N1,N3,N1,N3,N0,N1,X2,S14,0.500)15DEALLOCATE(S14)Figure4.7:Exampleofbinarytensorproductbetweenatwo-bodyHamiltonianoperatorandaT2AclusteroperatorthatproducesandintermediatelabeledS14,whichissummedwithotherintermediatesandHamiltonianmatrixelementsinacollectiveintermediatelabeledX2.Sincetheinternalindicesthatarebeingsummedoverarecommonbetweenthetwooperatorsintheproductterm,theorderofinternalindicesinbotharrayshavetobethesame.Inordertorearrangetheelementstoachievethisreordering,wetakeatrivialapproachandallocateanewarraywiththesamerankandsizeastheoriginal,andthencallasubroutine\REORDER"thatcopiestheelementsinthedesiredorderinthenewarray.Anexampleofoneofthe\REORDER"routinescanbeseeninFigures4.8.Theelementsoftheandsecondoperatorarerespectivelycopiedtoarray\A"and\B"inthedesiredorder.Theresultofthebinarytensorproductisstoredinarray\C"whichisallocatedafterthereorderingofindicesforthetwooperatorsiscomplete.Atthispoint,arrays\A",\B"and\C"stillrangeinrankfromtwo-toeight-dimensionsandstillneedtotransformeddowntoone-ortwo-dimensionalarrays.Luckily,thiscanbeaccomplishedthroughtheprocessofcallingasubroutinethatisawrapperforcallingfastvector-matrixormatrix-matrix1341SUBROUTINEREORDER4312(M1,N1,M2,N2,M3,N3,M4,N4,2&K4,L4,K3,L3,K1,L1,K2,L2,A,B)3REAL*8A(M1+1:N1,M2+1:N2,M3+1:N3,M4+1:N4)4REAL*8B(K4+1:L4,K3+1:L3,K1+1:L1,K2+1:L2)5REAL*8C67DOI1=K1+1,L18DOI2=K2+1,L29DOI3=K3+1,L310DOI4=K4+1,L411B(I4,I3,I1,I2)=A(I1,I2,I3,I4)12ENDDO13ENDDO14ENDDO15ENDDO16ENDFigure4.8:Anexampleofoneofthe\REORDER"routines.Thenumberinthetitleoftheroutinetheorderthattheindicesarepermutedto.multiplicationroutines.Inthesubroutine,arrays\A",\B",and\C"aredeclaredasone-ortwo-dimensionalarrayssotheprogramwillautomaticallyassociatetheelementsof\A",\B",and\C",whicharemultidimensionaloutsidethesubroutine,withone-ortwo-dimensionalarraysinsidethesubroutinebysimplycallingthesubroutine.Inordertodothisthecodehastoknowwhatarethedimensionsoftheone-ortwo-dimensionalarraysforwhicharrays\A",\B",and\C"willtranslatedinto,sothreenumbersaredeclared(I1,I2,andI3).ThevariableI1isthedimensionfortheexternalindicesofarray\A",I2isthedimensionfortheexternalindicesofarray\B",andI3isthedimensionfortheinternalindicesofthetwoarrays.Ifeitherarray\A"or\B"hasnoexternalindices,thenthecorrespondingvariableisnotdeclared.Afterthevector-matrixormatrix-matrixmultiplicationiscarriedout,thenarrays\A"and\B"aredeallocatedsincetheyarenolongerneeded.Theresultsofthebinarytensorproductmaybeaddedwithotherintermediatesiffurtherproductswithadditionalclusteroperatorsareneededfortheterm,orisaddedtoarrayforthenewestiterationcluster1351SUBROUTINESUM2314(K1,L1,K2,L2,K3,L3,K4,L4,A,B,C)2REAL*8A(K1+1:L1,K2+1:L2,K3+1:L3,K4+1:L4)3REAL*8B(K2+1:L2,K3+1:L3,K1+1:L1,K4+1:L4)4REALC56DOI1=K1+1,L17DOI2=K2+1,L28DOI3=K3+1,L39DOI4=K4+1,L410A(I1,I2,I3,I4)=A(I1,I2,I3,I4)+C*B(I2,I3,I1,I4)11ENDDO12ENDDO13ENDDO14ENDDO15ENDFigure4.9:Anexampleofoneofthe\SUM"routines.Thenumberinthetitleoftheroutinetheorderthattheindicesarepermutedto.amplitudesiftheproductiswiththelastclusteroperatorintheseriesofclusteroperatorsforatermorsetofterms,andthisisdonebycallinga\SUM"subroutine.Anexampleofa\SUM"routinecanbeseeninFigure4.9.Iftheresultoftheproductisaddedtothearrayforthenewestiterationoftheclusteramplitude,thenaseriesof\SUM"routinesiscalledthenecessaryamountoftimestosatisfytheantisymmetrizerthatprecedestheterminordertoretaintheantisymmetryoftheclusteramplitudebeingupdated(seeTable4.1).Withthisgenerallayout,itiseasytopickupanyofthebinarytensorproductsandinstantlygeneratethecorrespondinglinesofFORTRANcode.1364.3ImprovementsinviaLoopReorganiza-tionThegeneralizedlayoutofeachproductoftwoarrays,illustratedinFig.4.6,isaneasywaytoquicklygeneratethecode,butrestrictingoneselftojustthisformatreduces,andthereisoneparticularsituationwhereisgreatlybythisform.Afterabinarytensorproductiscarriedout,the\SUM"routineaddstheresulttoanotherarrayandthisisdoneforallelementsofthearrayandacoupledofoptionsmayhappen.Iftheresultoftheproductisanintermediatethatcanbesummedwithotherintermediates,thenthisroutineisonlycalledonce.Anotherpossibilityisthattheintermediateformedbytheproductmaynotbesummedwithothers,butstillusedinlaterterms,andthereforethe\SUM"routineisnotnecessary.However,iftheproductisbetweenaHamiltonianoperator,intermediate,orsumofintermediatesandthelastclusteroperatorforatermorsetoftermsinthefactorizedCCequations,thenthe\SUM"routineiscalledasmanytimesasnecessarytosatisfytheantisymmetrizerassociatedwiththoseterms,astheresultisaddedtothearrayforthenewestiterationoftheclusteroperators.The\SUM"routineinthisinstanceiscalledanywherefromonetooverahundredtimesinCCSDTQ,dependingontheterm,andeachtimetheentirearraycorrespondingtotheresultoftheproductjustcarriedoutisaddedtothearrayforthenewestiterationoftheclusteroperatorbeingcalculated.But,werecognizedthatnotallindicesneedtobeupdatedforeveryterm.Rather,wecanconsiderupdatingonlytheuniquetermsbasedontheantisymmetryoftheclusteroperatorandthenemploythatantisymmetryattheendoftheroutinetoobtaintheremainingclusteramplitudes.Forexample,ifconsiderupdatingthetabcdijklamplitudes,thenratherupdateallvaluesofi;j;k;landa;b;c;dwecanchoosetoupdatejustthosethatsatisfytheinequalityi<j<k<l137anda<b<c<d,thenreplicatetheseamplitudesfortheremainingindicesattheendoftheroutine,whilebeingmindfuloftheirantisymmetry.Inthisexample,thenumberofoperatorsforeachsummationisreducedbyafactorof(4!)2or576.Thetspincasesfortheclusteroperatorshavetcorrespondingantisymmetrizersthatmustbetakenintoaccountwhendecidingtoupdatejusttheuniqueterms.Asanotherexample,ifweweretoupdatethetab~c~dij~k~lamplitudes,thentheprogramonlyneedstoupdateindicesi<j,~k<~l,a<b,and~c<~dthenumberofoperatorsforeachsummationisreducedbyafactorof(2!)4or16.AnexampleofthisrestructuringcanbeseeninFigure4.10wherethe\SUM"routinesarereplacedbyexplicitDOloopsovertheuniqueelements(inthiscasei<j<landa<b<ccorrespondingtothetabc~dijk~lamplitudes).Thistechniquewasputintoactiononlyforthebinaryproductsformedwiththelastclusteroperatorsforaterm,becausethoseinstancesaretheonesforwhichtheresultoftheproductisonlybyindicescorrespondingtotheexciteddeterminantbrastatewhichweprojecttheconnectedformoftheScodingerequationonto.Inaddition,thistechniquewasappliedtojusttheupdateroutinesfortheprojectionsontoquadruplyexciteddeterminants.Thesecasesarewherethecodebthemostfromthisrewritingofthe\SUM"routinessincewecantakethemostuseoftheantisymmetryoftheclusteroperatorsandalsobecausethe\SUM"routinesareusuallycalledseveraltimesinthesesectionsofthecode.Inthefollowingsection,whenwediscusstransformingtheCCSDTQroutinetoobtaintheCCSDtqandCCSDTqcodes,weincorporatethistechniqueaswerewritethebinaryproductinacompletelytwaythanthegenerallayoutpresentedinFigure4.6.ThebofthisrestructuringcanbeseeninTable4.3,wherewecomparetimingsofcalculationsfortheX1+gandA3+ustatesof(HFH)beforeandafterthechangesinthe\SUM"routines.TimingobtainedusingNWCHEMarealsoincludedforcomparison.1381ALLOCATE(F1(2&N0+1:N1,N2+1:N3,N0+1:N2,N2+1:N3,N0+1:N2,N0+1:N1))3CALLREORDER546123(N2,N3,N0,N2,N0,N1,N2,N3,N0,N1,N0,N2,4&N0,N1,N2,N3,N0,N2,N2,N3,N0,N2,N0,N1,U41,F1)5ALLOCATE(H2(N0+1:N1,N2+1:N3,N0+1:N2,N1+1:N3,6&N1+1:N3,N1+1:N3,N0+1:N1,N0+1:N1))7CALLREORDER61523478(N2,N3,N1,N3,N1,N3,N1,N3,8&N0,N2,N0,N1,N0,N1,N0,N1,N0,N1,N2,N3,N0,N2,N1,N3,9&N1,N3,N1,N3,N0,N1,N0,N1,t4B,H2)10ALLOCATE(Z140(N1+1:N3,N1+1:N3,N1+1:N3,N0+1:N1,11&N0+1:N1,N2+1:N3,N0+1:N2,N0+1:N1))12I1=K1*K2*K413I2=K1*K1*K3*K3*K314I3=K2*K4*K115CALLDGEMMROUT(I1,I2,I3,F1,H2,Z140)16DEALLOCATE(F1)17DEALLOCATE(H2)18DOi=N0+1,N1-2;DOj=i+1,N1-1;DOk=j+1,N1;DOl=N0+1,N219DOa=N1+1,N3-2;DOb=a+1,N3-1;DOc=b+1,N3;DOd=N2+1,N320V4B(d,c,b,a,l,k,j,i)=V4B(d,c,b,a,l,k,j,i)21&-Z140(c,b,a,k,j,d,l,i)22&+Z140(c,b,a,k,i,d,l,j)23&-Z140(c,b,a,j,i,d,l,k)24ENDDO;ENDDO;ENDDO;ENDDO25ENDDO;ENDDO;ENDDO;ENDDO26DEALLOCATE(Z140)Figure4.10:AnexampleoftheproductbetweenanintermediatelabeledU41andtheclusteroperatorT4B,whichisprojectedontotheexciteddeterminanthabc~dijk~lj.ThearraycorrespondingtotheproductformedislabeledZ140andisaddedtotheresidualnamedV2B.FortheoriginalCCSDTQprogram,beforethe\SUM"routineswerechanged,theiterationstookanaverageof2491and3601secondsfortheX1+gandA3+ustates,respectively.Asacomparison,forNWCHEMtheiterationstookanaverageof1888and2068seconds,sotheoriginalimplementationofCCSDTQwasbetween1.5and1.8timesslowerthanNWCHEMforthesesetofcalculations.Thischangestlybysimplyrewritingthe\SUM"routinesasdescribedabove.Afterthetechniquedescribedaboveisimplemented139Table4.3:Averageiterationtime,inseconds,forcalculationsoftheX1+gandA3+ustatesof(HFH),computedusingouroriginalimplementationofCCSDTQ,theimprovedversionofCCSDTQ,andNWCHEMforcomparison.aProgramX1+gA3+uCCSDTQ(Original)29413601CCSDTQ(Improved)553558NWCHEM18882068aCalculationswereperformedwiththe6-31G(d,p)basissetonasinglecomputercoreandthecoreelectronswerefrozeninallcalculations.theiterationreducedtoanaverageof553and558seconds,respective.Sothenewversionofthespin-integratedCCSDTQapproachisbetween5.3and6.5timesfasterthantheoriginalimplementationandis3.4{3.7timesfasterthanNWCHEMforthesesetofcalculations,andisgenerallytrueforothercalculationsaswell.Itsimportanttonotesincethecodeisgeneralforopen-andclosed-shellmolecules,thatthecalculationsfortheX1+gstillgotroughsamenumberofroutinesastheopen-shellcalculation,eventhoughmanyofthetermsareredundant.AsdiscussedinSection4.1,ifweweredealingwithaclosed-shellsystem,wecansimplifyourequationsbytakingadvantageofthefactthattheandspincasesforeachloopareequivalent.Despitegoingthroughextrasteps,theCCSDTQcodewiththeupdated\SUM"routinesisstillfasterthanNWCHEM.WhenwetranslatedtheCCSDTQcodetotheobtaintheCCSDtqandCCSDTqmethods,weabandoncallingthe\SUM"routines,butstillretainthelessonwelearnedfromthisrestructuring,whichisdiscussedinthefollowingsection.1404.4TransformationfromCCSDTQtotheCCSDtqandCCSDTqMethodsTheautomaticderivationprocessdescribedintheprevioussectionswasdoneforthefulltreatmentoftriply-andquadruply-excitedclusteroperators.Inotherwords,thebinarytensorproductfortwooperatorsiscarriedoutforallindicescorrespondingtotheoperators.Inordertoobtaintheactive-spaceCCSDtqandCCSDTqmethodswehadtoamecha-nismforreducingthenumberofoperatorssoasnottocarryouttheproductforallindices,butratherthosethatobeytheactive-spacelogicsetinplacebytheuser.Todothis,weabandonedthegenerallayoutseeninFigures4.6and4.7,andreplacethecodewithexplicitDOloops.Thisisonlydoneforthebinaryproductsinvolvingthelastclusteroperatorinthesetofclusteroperatorsforagiventerm,justaspreviousdonewiththe\SUM"routinesdescribedinthelastsection.ExamplesoftheexplicitloopsforprojectionsontotriplesandquadruplescanbeseeninFigures4.11and4.12respectively.Thereareseveraladvantagestorewritingtheproducts,whichagainareonlythosethatinvolvethelastclusteroperatorinthesetofclusteroperatorsforagiventerm,intheformatseeninFigures4.11and4.12.BystructuringtheproductswithDOloops,wecaneasilyintroducelogicstatements(lines2and4inFigures4.11and4.12)thatallowustobypasstheproductifagivensetofindicesdonotobeytheactive-spacelogicsetbytheuser.Beforerunningthecalculation,theusertheactivespacebydeclaringthenumberoftheoccupiedandunoccupiedandelectronsandorbitalsintheactivespacealongwithhowmanyindicesforthetripleandquadrupleexcitationstheywanttorestricttoonlybelongingtotheactivespace.Thenumberofrestrictedoccupiedandunoccupiedindicesforthetripleexcitationsisdeclaredseparatelyfromthenumberoccupiedandunoccupiedrestricted1411DOi=N0+1,N1-2;DOj=i+1,N1-1;DOk=j+1,N12IF(indocc(k,j,i).eq.1)CYCLE3DOa=N1+1,N3-2;DOb=a+1,N3-1;DOc=b+1,N34IF(indunocc(c,b,a).eq.1)CYCLE5SUM=0.0d06DOe=N1+1,N37SUM=SUM8&-X2(e,c,b,i)*t2A(e,a,k,j)9&+X2(e,c,a,i)*t2A(e,b,k,j)10&-X2(e,b,a,i)*t2A(e,c,k,j)11&+X2(e,c,b,j)*t2A(e,a,k,i)12&-X2(e,c,a,j)*t2A(e,b,k,i)13&+X2(e,b,a,j)*t2A(e,c,k,i)14&-X2(e,c,b,k)*t2A(e,a,j,i)15&+X2(e,c,a,k)*t2A(e,b,j,i)16&-X2(e,b,a,k)*t2A(e,c,j,i)17ENDDO18V3A(c,b,a,k,j,i)=V3A(c,b,a,k,j,i)+SUM19ENDDO;ENDDO;ENDDO20ENDDO;ENDDO;ENDDOFigure4.11:AnexampleoftheproductbetweenanintermediatelabeledX2andtheclusteroperatorT2A,whichisprojectedontotheexciteddeterminanthabcijkj.indicesforthequadrupleexcitations,givingtheuseryintheirchoiceoftheparticularactive-spaceapproach.Forexample,ifoneoccupiedandunoccupiedindexisrestrictedtotheactivespacefortripleexcitationsandtwooccupiedandunoccupiedindicesarerestrictedtotheactivespaceforquadruple,weobtaintheCCSDtqapproachpresentedinSection3.1.2.Iftheactivespaceincludesallorbitalsforthetripleexcitationsandoneoccupiedandunoccupiedindexisrestrictedtotheactivespaceforquadrupleexcitations,weobtaintheCCSDTqapproach.Thisyalsoallowsustoconsider\inbetween"approximations.Forexample,onemayconsideronlyrestrictingoneoccupiedandunoccupiedindextotheactivespaceforthetripleexcitations,whilerestrictingthreeoccupiedandunoccupiedindicestotheactivespaceforquadrupleexcitations,whichmaybeusefulinsituationswheretriple1421DOi=N0+1,N1-3;DOj=i+1,N1-2;DOk=j+1,N1-1;DOl=k+1,N12IF(indocc(l,k,j,i).eq.1)CYCLE3DOa=N1+1,N3-3;DOb=a+1,N3-2;DOc=b+1,N3-1;DOd=c+1,N34IF(indunocc(d,c,b,a).eq.1)CYCLE5SUM=0.0d06DOm=N0+1,N1;DOn=N0+1,N17SUM=SUM8&+(S5(j,m,i,n)*t4A(d,c,b,a,n,m,l,k)9&-S5(k,m,i,n)*t4A(d,c,b,a,n,m,l,j)10&+S5(l,m,i,n)*t4A(d,c,b,a,n,m,k,j)11&-S5(i,m,j,n)*t4A(d,c,b,a,n,m,l,k)12&+S5(i,m,k,n)*t4A(d,c,b,a,n,m,l,j)13&-S5(i,m,l,n)*t4A(d,c,b,a,n,m,k,j)14&+S5(k,m,j,n)*t4A(d,c,b,a,n,m,l,i)15&-S5(l,m,j,n)*t4A(d,c,b,a,n,m,k,i)16&-S5(j,m,k,n)*t4A(d,c,b,a,n,m,l,i)17&+S5(j,m,l,n)*t4A(d,c,b,a,n,m,k,i)18&+S5(l,m,k,n)*t4A(d,c,b,a,n,m,j,i)19&-S5(k,m,l,n)*t4A(d,c,b,a,n,m,j,i))/2.0d020ENDDO;ENDDO21V4A(d,c,b,a,l,k,j,i)=V4A(d,c,b,a,l,k,j,i)+SUM22ENDDO;ENDDO;ENDDO;ENDDO23ENDDO;ENDDO;ENDDO;ENDDOFigure4.12:AnexampleoftheproductbetweenanintermediatelabeledS5andtheclusteroperatorT4A,whichisprojectedontotheexciteddeterminanthabcdijklj.excitationsplayatroleandwewanttoincludealargesubsetoftheminouractive-spacecalculation,whileatthesametimeincorporatingaverysmallsubsetofquadruplesintheactive-spaceconsideration.Afterinputtingthisinformation,theprogramconstructstwoarraysfortheoccupiedandunoccupiedindiceswhichtellifaparticularcombinationofoccupiedandunoccupiedindicesisallowedbasedontheactive-spacebytheuser.Ifthecombinationofoccupiedindicesdoesnotmeettheactivespacerequirements,thentheloopscycletothenextcombination(line2inFigures4.11and4.12),andifthecombinationofunoccupiedindicesdoesnotmeettheactivespacerequirements,thentheloopscycletothenextcombination(line4inFigures4.11and4.12).Onlywhensetsofindicessatisfy143theactivespaceisthebinaryproductcarriedout.Thereareotherbtothisnewstructureaswell.Fromourearlierlessonwiththe\SUM"routinesdiscussedinSection4.3,weknowwedonothavetoupdateallvaluesoftheindices,butjusttheuniqueones,basedontheantisymmetryoftheoftheamplitudesweareupdating.Figure4.11isanexampleofaproductintheupdateroutineforthetabcijkamplitudes,whichcomputedusingjustthesetofindicesi<j<kanda<b<c,whileFigure4.12isanexampleaproductintheupdateroutineforthetabcdijklamplitudes,whichcomputedusingjustthesetofindicesi<j<k<landa<b<c<d.Inaddition,weincorporatethe\SUM"routinesintheloopbyexplicitlysummingtheproductofthetwotensorswiththeotherproductscorrespondingtoapplyingtheantisymmetrizerthatisneededtoretaintheantisymmetryofthenewestiterationoftheamplitudesbeingcalculated.Anotherbigbofthisnewformatisthatweremovetheneedtoreordertheindices,thereforewenolongerneedtoallocatearrays\A",\B",and\C".Asaresultofnothavingtoallocatethesearrays,thememoryneededtocarryoutthisproductisreducedbymorethanhalfoftheamount.Lastly,sincethisloopformatisprimitive,itallowsustoeasilyintroduceotheraspectssuchassymmetryadaptationandexplicitparallelizationinfuturedevelopmentwork.144Chapter5ConclusionsandFutureOutlookInthisdissertation,wepresentedtheCC(P;Q)formalism,whichprovidesasystematicap-proachtocorrectingenergiesobtainedintheactive-spaceCCandEOMCCcalculationsthatrecovermuchofthenon-dynamicalandsomedynamicalmany-electroncorrelationfortheremaining,mostlydynamical,correlationmissingintheactive-spaceCCandEOMCCconsiderations.WediscussedthedetailsoftheCC(t;3),CC(t,q;3),CC(t,q;3,4),andCC(q;4)methods,whichusetheCC(P,Q)formalismtocorrectenergiesobtainedwiththeCCandEOMCCapproacheswithsingles,doubles,andactive-spacetriples(CCSDt/EOM-CCSDt)formissingtripleexcitations(CC(t;3)),ortocorrectenergiesobtainedwiththeCCandEOMCCapproacheswithsingles,doubles,andactive-spacetriplesandquadruples(CCSDtq/EOMCCSDtq)formissingtriples(CC(t,q;3))ormissingtriplesandquadruples(CC(t,q;3,4)),andtocorrectenergiesobtainedwiththeCCandEOMCCapproacheswithsingles,doubles,triples,andactive-spacequadruples(CCSDTq/EOMCCSDTq)forcorre-lationduetothemissingquadrupleexcitations(CC(q;4)).Byexaminingthedoubledissociationofwater,theBe+H2!HBeHinsertion,andthesinglet{tripletgapsinthestronglybiradical(HFH)systemandtheBNmolecule,wefurtherestablishedthatthepreviouslydevelopedCC(t;3)schemeiscapablereproducingthetotalandrelativeenergiesobtainedwiththeparentCCSDTapproachtowithinfractionsofamillihartreeatatinyfractionofthecomputercost.Inaddition,wewerealsoabletoshow,forthetime,thattheCC(t,q;3)andCC(t,q;3,4)methodsaccuratelyandreliablyreproducethetotaland145relativeenergiesobtainedwiththeparentCCSDTQapproach,onceagaintowithinfrac-tionsofamillihartree,andatatinyfractionofthecomputercost,evenwhentheelectronicquasi-degeneraciesbecomesubstantial.Inourfutureendeavors,wewouldliketocontinueexaminingothermolecularsystemstofurthertesttheperformanceoftheCC(P;Q)methods,especiallythoseinvolvingquadrupleexcitations,andinparticulartheCC(q;4)schemewhichwasnotinvestigatedinthisstudy.CurrentareunderwaytostudytheperformanceofalloftheCC(P;Q)approachespresentedinthisdissertationbyexaminingthepotentialenergysurfacefortheberylliumdimer,Be2.Thesemethodsalsoprovidetheopportunitytostudyothermulti-referencemolecularproblemsinvolvingbiradicals,bondbreaking,andotherinstanceschar-acterizedbyquasi-degeneratestatesforwhichthefulltreatmentofhigher-ordercorrelationistooexpensive.ThefutureapplicationoftheCC(P;Q)arecontingentonhavingtalgorithms,especiallyfortheunderlyingactive-spacecalculationswhichcontinuouslybeingimproved.Inthisstudywedescribedhowthespin-integratedCCSDTQequationswerederived,factorized,andtranslatedintoFORTRANcodeusingaprogramthatcarriesouttheseproceduresautomatically.Welearnedthattheperformanceoftheresultingprogramcouldbetlyimprovedbyreorganizingsomeloops,removingunnecessaryoperations.Still,thereisalotofroomforfurtherimprovement.Inthefuturewewouldliketoexaminethecodeforadditionalunnecessaryoperationsanddetermineotherbottlenecksintheprogram.Forexample,inthecaseofclosed-shellsystems,wecantakeadvantagethattheandspincasesforeachloopareequivalentandgenerateaseparateandtcodeforjustclosed-shellcalculations.Thenewloopstructureintroducedfortheactive-spaceCCSDtqandCCSDTqalsoprovidestheopportunityforeasyfurtherimprovement.Forexample,we146wouldliketointroducespatialsymmetry-adaptationtofurtherspeedupthecode.Wewouldalsoliketointroduceexplicitparallelizationtotakeadvantageofsharedanddistributedmemorysystems,whichtheDOloopstructureoftheactive-spaceCCSDtqandCCSDTqmethodsallowsustoimplementmoreeasily.Finally,wewanttoaddressthefactthattheT3andT4clusteroperatorsareallocatedinmemoryforallvaluesoftheindicesthatcorrespondtotheoperators,notjustthosethatsatisfytheactive-spacelogicsetbytheuser.Inotherwords,thememoryrequirementsfortheCCSDtqandCCSDTqcalculationsarethesameasthefullCCSDTQmethod.Inthefuture,wewanttoreplacethearraysforT3andT4thatcontainallvaluesoftheindicesthatcorrespondtotheoperatorswiththosethatcontainonlytheelementsbytheactive-spaceopeningupmanymoreapplicationsoftheCC(P;Q)methods.147BIBLIOGRAPHY148BIBLIOGRAPHY[1]F.Coester,Nucl.Phys.7,421(1958).[2]F.CoesterandH.Kummel,Nucl.Phys.17,477(1960).[3]J.Czek,J.Chem.Phys.45,4256(1966).[4]J.Czek,Adv.Chem.Phys.14,35(1969).[5]J.CzekandJ.Paldus,Int.J.QuantumChem.5,359(1971).[6]J.Paldus,J.Czek,andI.Shavitt,Phys.Rev.A5,50(1972).[7]B.JeziorskiandH.J.Monkhorst,Phys.Rev.A24,1668(1981).[8]B.JeziorskiandJ.Paldus,J.Chem.Phys.88,5673(1988).[9]L.Meissner,K.Jankowski,andJ.Wasilewski,Int.J.QuantumChem.34,535(1988).[10]J.Paldus,L.Pylypow,andB.Jeziorski,inMany-BodyMethodsinQuantumChemistryLectureNotesinChemistry,Vol.52,editedbyU.Kaldor(Springer,Berlin,1989)p.151.[11]P.PiecuchandJ.Paldus,Theor.Chim.Acta83,69(1992).[12]J.Paldus,P.Piecuch,B.Jeziorski,andL.Pylypow,inRecentProgressinMany-BodyTheories,Vol.3,editedbyT.L.Ainsworthy,C.E.Campbell,B.E.Clements,andE.Krotschek(PlenumPress,NewYork,1992)p.287.[13]J.Paldus,P.Piecuch,L.Pylypow,andB.Jeziorski,Phys.Rev.A47,2738(1993).[14]P.Piecuch,R.Tobo 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