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COUPLED-CLUSTERBASEDMETHODSFOR EXCITEDSTATEENERGIESANDGRADIENTS By STEVENRAYGWALTNEY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 1997 ACKNOWLEDGMENTS SoliDeoGloria,toGodalonebetheglory. IthankGodforcreatingauniversesofullofwonderandmysteryandforcreating meandallowingmetoexploreit. Ialsothankmywife.Charity,forherconstantsupport andencouragement,evenwhenitmeantthatshehadtodomorethanherfairshare. I loveyouandwillalwaysrememberhowmuchyousacrificedformeandforthisdegree. IwanttothankmyparentsforneverlettingmesettleforbeinglessthanIcouldbe, andIthankthemforpushingmewhenIwasgrowingupandthenlettingmegofar away when itwastime. Iwouldliketothankmyresearchadvisor,Prof.RodneyBartlett,fromwhomIhave learnedsomuch.IwouldalsoliketothankProf.ErnestDavidson.WhoknowsifIwould bedoingquantumchemistryifhehadnotallowedmetodoundergraduateresearchunder him,eventhoughIdidnotknowwhatIwasdoing.Thanksgoouttoallthemembersof theQuantumTheoryProject,tothemembersofDr.Bartlett'sgroup,andespeciallyto myofficematesthroughtheyears. Theenvironmentofcooperationhereiswonderful. ThankstoProf.MichaelZernerforhelpunderstandingthefreebaseporphinproblem. Finally,specialthanksgotoProf.MarcelNooijen. Ilearnedmuchinourdiscussions. TheworkpresentedherehasbeenfundedbytheNationalScienceFoundationthrough aGraduateResearchFellowshipandbytheUnitedStatesAirForceOfficeofScientific ResearchthroughgrantnumberF49620-95-1-0130andthroughAASERTgrantnumber F49620-95-I-0421. ComputertimefortheporphincalculationswasprovidedbytheCEWESDepart- mentofDefenseMajorSharedResourceCenter. TheCEWEScomputercenterisgrate- fullyacknowledgedfortheirhelpingettingthesecalculationsdone. TheEOM-CCSD porphincalculationswereperformedinconjunctionwithDr.HowardPrichardatCray Research/SGI. TABLE OFCONTENTS ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTERS LANOVERVffiWOFCURRENTMETHODSFOREXCITEDSTATES .... 1 Coupled-ClusterMethods 1 OtherCurrentExcitedStateMethods 3 GradientsforExcitedStateMethods 8 2.PARTITIONEDEQUATION-OF-MOTIONCOUPLED-CLUSTERMETHODS FOREXCITEDSTATEENERGIES 9 Introduction 9 Theory 13 PartitionedEOM-CCforExcitedStates 13 MBPT[2]GroundState 15 Calculations 17 BeAtom 17 ExampleMolecules 18 Conclusions 21 3.GRADIENTSFORTHEPARTITIONEDEQUATION-OF-MOTIONMBPT(2) METHODFOREXCITEDSTATES 28 Introduction 28 Theory 30 EOM-CCGradients 30 P-EOM-MBPT(2)Gradients 41 ApplicationtoDiatomicMolecules 46 VerticalandAdiabaticExcitationEnergies 48 BondLengthsandVibrationalFrequencies 50 ApplicationtoPolyatomicMolecules 51 TheSi StateofAmmonia 51 Trans-bentandVinylidenicIsomersontheSi SurfaceofC2H2 53 SimpleCarbonyls 54 Conclusions 55 4.TCHOUEPSLIEMDI-LCALRUISTYTETRRMAENSTFHOORDMEDEQUATION-OF-MOTION 69 Introduction 69 TheSTEOM-CCMethod 73 DiscussionAboutSTEOM-CC 78 5.THESPECTRUMOFFREEBASEPORPHIN 83 Introduction 83 ComputationalDetails 85 STEOM-CC 85 BasisSetandGeometry 86 Results 88 IonizedandElectronAttachedStates 88 ExcitedStates 89 Discussion 91 Conclusions 94 6.GRADIENTSFORTHESIMILARITYTRANSFORMED EQUATION-OF-MOTIONCOUPLED-CLUSTERMETHOD 110 Theory 110 Applications 120 7.AFINALWORD 127 BIBLIOGRAPHY 129 BIOGRAPHICALSKETCH 139 AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeofDoctorofPhilosophy COUPLED-CLUSTERBASEDMETHODSFOR EXCITEDSTATEENERGIESANDGRADIENTS By StevenRayGwaltney December1997 Chairman: RodneyJ.Bartlett MajorDepartment: Chemistry ' - ,. Coupled-clusterbasedmethodshavebecomewidelyusedforcalculatingenergiesand propertiesofmoleculargroundstates. Inthisdissertation,Ipresentanewmethod,the partitionedequation-of-motioncoupled-clustermethodbasedonasecondordermany- body perturbation theory ground state (P-E0M-MBPT(2)) forcalculating energies of molecular electronic excited states. I also describe how to calculate gradients for twoexcitedstatemethods, P-E0M-MBPT(2) andthesimilaritytransformedequation- of-motion coupled-cluster singles and doubles (STEOM-CCSD) method. These new methodsarethenappliedtoseveralexamplemolecules. VI CHAPTER 1 ANOVERVIEWOFCURRENTMETHODSFOREXCITEDSTATES Coupled-ClusterMethods Coupled-clustertheory'"''* provides a routinely applicable method forcalculating energiesandpropertiesofmoleculesintheloweststateofagivenspinandsymmetry. Byaddingtriple,'^-'^quadruple,'^''^andhigherexcitations,itispossibletoconvergeon theexactanswerwithinagivenbasisset. Theperturbativeinclusionoftripleexcitation effects'^"^' is now routine, meaning that energies and properties can approach near chemicalaccuracyformanymolecules. Unfortunately,thesemethodsdonot,ingeneral, workforelectronicexcitedstates. Theprimaryproblemwithdescribingexcitedstateswithnormalcoupled-cluster(CC) theory isits inherentsingle reference nature. The wavefunction forthestatecan be expressedasalinearcombinationofdeterminants,withthereferencedeterminanthaving acoefficientofone. Althoughinprinciplecoupled-clustertheorycanbemadeexact,the costofsuchacalculationwouldincreaseincostatthesamerateasfullconfiguration interaction(CI),i.e. asthesizeofthesystemfactorial.^^Therefore,thesetofexcitations usedinthecalculationaretruncated.InsuchatruncatedCC,aseconddeterminantwitha coefficientofmorethan0.2to0.3cancausetheaccuracyofthecomputedresultstodrop significantly. However,foranopen-shell singletexcitedstate,theseconddeterminant musthaveacoefficientofone. Coupled-cluster theory has been extended in order to be able to describe such multireferencestates. Multireferencecoupled-clustertheoriescanbedividedintotwo 1 2 main categories. They are the Hilbert space approaches^^"^' and the Foclc spaces approaches.^^"^^ AverysimplifiedviewofHilbertspacemultireferencecoupled-cluster theory isthateach determinant inthe active spacegets its own setofTamplitudes, andasmalleffectiveHamiltonianisdiagonalizedoverthespaceofthesewavefunctions. IntheFockspacecoupled-clustermethod,asingleexponentialoperatorisusedforall determinantsintheactivespace,butthisexponentialcontainsoperatorsthatchangethe numberofelectronsinthesystem. Hilbertspacecoupled-clustertheoryismoresuited tostudyingoneorafewstates,whileFockspacecoupled-clustertheoryismoresuited tostudyingenergydifferencesbetweenmanystatesandtocalculatingspectra. An alternative methodforcalculatingexcitationenergies within acoupled-cluster framework is to follow the response of the system to an external time dependent perturbation. Thiswas introducedby Monkhorst,"^^ andhas been developed intothe coupled-clusterlinearresponse(CCLR)method.'^^"^'^ Thesameequationscan alsobe derivedfrom an equations-of-motion^^ framework basedon acoupled-clusterground state. Thisgivestheequation-of-motion coupled-cluster(EOM-CC) method.^^^^ In EOM-CC (orCCLR) the ground state coupled-clusteroperator is used to perform a similaritytransformationoftheHamiltonian, whichisthendiagonalizedoveraspace ofexciteddeterminants. EOM-CCandCCLRareidenticalforexcitedstateenergies, but the approximations made by the two methods differ for oscillator strengths and polarizabilities.Typically,thegroundstateoperatorandthespaceofexciteddeterminants istruncatedtoincludeonlysingleanddoubleexcitations.ThisgivestheEOM-CCSD^^'^^ (orCCSDLR)^^'^"^method. Modelshavealsobeendevelopedwhichpartiallyinclude theeffectsoftripleexcitations.^^^^ 3 Recently,NooijenandBartletthavedevelopedanewmethodforcalculatingexcited stateenergiesandproperties. Itisthesimilaritytransformedequation-of-motioncoupled- cluster (STEOM-CC) method.^^6 For singly excited states, STEOM-CC is closely related to the Fock space coupled-cluster method,^^ but conceptually they are very different. In STEOM-CC, ionizationpotentialEOM-CCSD^'' andelectron attachment EOM-CCSD^^calculationsaredone. Informationaboutthestateswithonefewerand one more electron is then usedtoperform asecond similarity transformation on the Hamiltonian^^ suchthatthetermswhichdirectlycouplesingleanddoubleexcitations aresettozero. AdetaileddescriptionofSTEOM-CCwillbegiveninChapter4. y •. OtherCurrentExcitedStateMethods At. Anattempttocatalogall abinitiomethodscurrentlyusedforcalculatingexcited stateswouldbefutileandwillnotbeattempted. Instead,severalofthemethodswhich areusedmostoftenwillbereviewed. Butfirstitwillbeusefultotakeamoregeneral viewofexcitedstatesandbreakdownhowthevariousmethodsdescribeexcitedstates. Inthisview,theexcitedstateenergyisdividedintoseveralcomponents. Theyarethe zeroth orderdescription ofthe wavefunction, the orbital relaxation, andthe dynamic correlationintheexcitedstate. • Properlydescribingthezerothorderdescriptionsimplymeansbeingabletoinclude allofthedeterminantswhichplayadominantroleintheexcitedstate. Forexample, thismeansbeingabletogivethetwodeterminants inanopen-shell singletthesame weight. Theorbitalrelaxationreferstohowmuchtheorbitalsareallowedtorearrange themselvesinresponsetotheexcitation. 4 Thedynamiccorrelationfortheexcitedstateisfurtherbrokendownintothedynamic correlation forthe ground state and the differential dynamic correlation between the groundandexcitedstates. Thereasonforthisdivisionofthedynamiccorrelationisthat, especiallyforthesinglereferencetheories,firstagroundstateiscalculatedandthenthe excitedstatesarebuiltuponit. Thejustificationforthisdivisionisthatmostelectrons donottakepartintheexcitationandthereforestillinteractwitheachotherinthesame wayintheexcitedstateasinthegroundstate. Thesedefinitionsarenotunique,especiallywhenconsideringorbitalrelaxationversus differentialdynamiccorrelation. Thecoupled-clusterviewpointwillbetakenhere. The keytothecoupled-clusterviewpointisThouless' theorem,^^whichstatesthatanytwo non-orthogonaldeterminantscanbewrittenas |$,;)=e^'|$j), (1-1) whereT\ isasingleexcitationoperator. Therefore,anysetofsingleexcitationsorany productofsingleexcitationswillbeconsideredorbitalrelaxation,anddynamiccorrelation willbereservedfor"connected"doubleandhigherexcitations.'^ Table 1-1providesaconvenientrepresentationforhowtheselectedmethodsbreak down intothefourterms. A'V meansthatthemethodtreatsthisterm adequately. Twoplusesmeanthatthemethodtreatsthetermverywell. A"-"meansthattheterm is included, butpoorly. Finally, a"0" means thatthe term is completely neglected. Thequestionmarkmeansthathowwelltheterminquestionistreatedvariesbasedon specificsofthecalculation. Thefirstmethodtobeconsideredisprobablythesimplestofallabinitioexcitedstate methods. ItistheTamm-Dancoffapproximation,^'-"^^ alsoknownastheconfiguration