COUPLED-CLUSTERANDEQUATION-OF-MOTIONCOUPLED-CLUSTER THEORIES:APPLICATIONSTOPHOTOCHEMISTRYANDCATALYSISAN D ALGORITHMICADVANCES By JaredA.Hansen ADISSERTATION Submittedto MichiganStateUniversity inpartialfulllmentoftherequirements forthedegreeof Chemistry-DoctorofPhilosophy 2015 ABSTRACT COUPLED-CLUSTERANDEQUATION-OF-MOTION COUPLED-CLUSTERTHEORIES:APPLICATIONSTO PHOTOCHEMISTRYANDCATALYSISANDALGORITHMICADVANCES By JaredA.Hansen Understandingelectronicexcitation,photoelectron,and multiphotonionizationspectra, particularlythoseinvolvingdarkandstronglycorrelated states,ortransitionmetals,as wellascatalytic,structural,andelectronicpropertieso fgoldnanoparticlesposesignicant challengesfortheoryandexperiment.Theexistingexperim entaltechniquesmaynotbesuf- cientlypowerfultoprovidedenitiveinformationonthei rown,whereasanaccuratetreat- mentoftherelevantmany-electroncorrelationeectsrequ iredintheoreticalanalysesmaybe farfromobvious.Inthisdissertation,wedescribeseveral high-level abinitio computational studiesemployingthecompletelyrenormalized(CR)andact ive-spacecoupled-cluster(CC) andequation-of-motionCC(EOMCC)approachesandtheexten sionsoftheEOMCCtheory toopen-shellsystemsaroundclosedshellsdeningtheelec tron-attached(EA)andionized (IP)EOMCCframeworks,whichdemonstratethetransformati verolethesenovelelectronic structuremethodsdevelopedinourgrouphaveplayedinunde rstandingthepreviouslyunex- plainedexperimentsandphenomena.Theyinclude(i)challe ngingelectronicspectraofthe CNC,C 2 N,N 3 ,andNCOmoleculesandthephotoelectronspectrumofAu 3 nanoparticle examinedwiththeEA-andIP-EOMCCapproaches,especiallyt hoseinventedinourgroup, (ii)thediscoveryofthedoublyexcitedstateofazulenebel owtheionizationthreshold,which mediatesthe1+2 0 multiphotonionizationexperimentsresultinginclearRyd bergnger- printspectra,wheretheCR-EOMCCformalismdevelopedinou rgroupplayedacrucialrole, (iii)thedetailedinvestigationofthemechanismandenerg eticsoftheaerobicoxidationof methanolonAu 8 particle,whichbenetedfromtheapplicationoftheground -stateCR-CC methodology,developedbyourgroupaswell,and(iv)denit iveCR-CCandactive-space CCstudiesshowingthatthegroundstateof1,2,3,4-cyclobu tanetetraone,whichischaracter- izedbydenselyspacedlow-lyingstates,isatriplet,inagr eementwiththerecentlyrecorded photodetachmentspectrum.Thesecutting-edgecomputatio nalstudiesareaccompaniedby advancesinCC/EOMCCalgorithmsandmethodologies,includ ingthedevelopmentofpar- allelnumericalenergygradientsandsecondderivativesfo rfastgeometryoptimizationsand harmonicvibrationalfrequencycalculationsatanyCC/EOM CClevel,allowingustoestab- lishthegeometriesandrelativeenergiesofthelow-energy isomersofthecontroversialAu 8 particle,andtheimplementationoftheunrestrictedHartr ee-Fock-based(UHF-based)CR- CC(2,3)approach,allowingustoshowthatunlikethepopula rCCSD(T)approach,which isverysensitivetothetypeofthereferencedeterminantem ployedinthecalculations,fail- inginbond-breakingsituationswhentherestrictedHartre e-Fock(RHF)referenceisused anddisplayingpoorbehavioratintermediatenuclearsepar ationswithUHFreferences,its CR-CC(2,3)counterpartprovidesarobustdescriptionrega rdlessofthereferencetype(RHF orUHF),withthespin-adaptedRHF-basedCR-CC(2,3)result sbeingmostaccurateinthe examinedbonddissociationcases. ThisisdedicatedtomylovelywifePamanddaughterKaitlyn. iv ACKNOWLEDGMENTS IwouldliketothankmyPhD.advisor,ProfessorPiotrPiecuc h,forhispatienceovermany longhoursofintensedebateanddiscussionsashetaughtand guidedme.Ihavelearned whatitmeansandwilltaketobeabetterscientistthankstoh isguidinghandandexample. Iamforevergratefulforhiscontinuoussupportandeveryth inghehasdoneforme. Iwouldliketothanktheothermembersofmycommittee,Profe ssorJamesF.Harrison, ProfessorJamesE.Jackson,andProfessorBenjaminG.Levin efortheiradvice,suggestions, support,andpatienceinoverseeingmygraduatestudies.Pa rticularly,Iwouldliketothank ProfessorBenjaminG.Levineforhelpingmetocontributeto theCIOptcodeandformany usefuldiscussionswehadthatallowedmetobetterundersta ndtechnicalissuesrelatedto parallelnumericalderivatives.Ialsooweadebtofgratitu detoProfessorMasahiroEhara, who,throughProfessorPiotrPiecuch,invitedmetospendaf ewmonthsattheInstitutefor MolecularScienceinOkazaki,Japan,aswellasforseverals timulatingdiscussionsthathave resultedinimprovementsofsomeoftheresultspresentedin thisthesis. TothecurrentandpastmembersofthePiecuchresearchgroup ,Dr.JesseJ.Lutz,Dr. WeiLi,Dr.JunShen,Mr.NicholasP.Bauman,Mr.AdeayoO.Aja la,andMr.Jorge EmilianoDeustua,forallthehelptheyhaveprovidedmeover theyearsandthemany discussionswehavehadaboutscienticmatters. v TABLEOFCONTENTS LISTOFTABLES ................................... viii LISTOFFIGURES .................................. ii Chapter1Introduction ............................... 1 Chapter2Projectobjectives ............................ 10 Chapter3Applicationsoffullandactive-spaceelectronat tachedandion- izedequation-of-motioncoupled-clustermethods ........ 11 3.1Theory.......................................12 3.2Geometriesandadiabaticexcitationenergiesofthelow -lyingvalencestates ofCNC,C 2 N,N 3 ,andNCO...........................19 3.2.1Computationaldetails.......................... 19 3.2.2Results...................................24 3.2.2.1EA-EOMCCresultsforCNCandC 2 N............25 3.2.2.2IP-EOMCCresultsforNCOandN 3 .............42 3.3Coupled-clusterinterpretationofthephotoelectrons pectrumofAu 3 ....55 3.3.1Introductoryremarks........................... 55 3.3.2IP-EOMCCextrapolationscheme................... .57 3.3.3Results...................................64 Chapter4Applicationsofcompletelyrenormalizedcoupled -clusterand equation-of-motioncoupled-clusterapproaches .......... 75 4.1Theory.......................................75 4.2Discoveryofthedoublyexcitedstatethatmediatesthep hotoionizationof azulene.......................................83 4.2.1Backgroundinformationandperformedcalculations. .........83 4.2.2Results...................................91 4.3AerobicoxidationofmethanoltoformicacidonAu 8 :Benchmarkanaly- sisbasedoncompletelyrenormalizedcoupled-clusterandd ensityfunctional theorycalculations................................1 00 4.3.1Backgroundinformationandscopeofthework........ .....100 4.3.2Molecularmodel.............................105 4.3.3Electronicstructuremethods.................... ..108 4.3.4Resultsanddiscussion.......................... 114 4.3.4.1Coupled-clustercalculations................. .124 vi 4.3.4.2BenchmarkingDFTagainstthereferenceCR-CC(2,3) ,Ddata129 4.4Coupled-ClusterandMultireferenceCongurationInte ractionStudiesofthe Low-LyingElectronicStatesof1,2,3,4-Cyclobutanetetra one..........136 4.4.1Backgroundinformationandscopeofthework........ .....136 4.4.2Computationaldetails.......................... 145 4.4.3Resultsanddiscussion.......................... 153 Chapter5Algorithmicadvances .......................... 165 5.1Parallelnumericalderivatives.................... ......165 5.1.1Gradientsforgeometryoptimizations............. .....168 5.1.2Hessiansforharmonicvibrationalanalyses........ .......176 5.1.3Applicationtothelow-energyisomersofAu 8 .............180 5.2UnrestrictedimplementationofCR-CC(2,3).......... ........191 5.2.1Algorithm.................................193 5.2.2Numericalexamples...........................19 5 5.2.2.1TheHFmolecule........................195 5.2.2.2TheF 2 molecule........................197 5.2.2.3TheH 2 O 2 molecule......................205 5.2.2.4TheC 2 H 6 molecule.......................208 Chapter6Conclusionsandfutureoutlook ................... 215 BIBLIOGRAPHY ................................... 219 vii LISTOFTABLES Table3.1:Totalandadiabaticexcitationenergiesforthegroundandlow-lying excitedstatesofCNC,asobtainedwiththedierentEA-EOMCC approachesusingtheDZP[4s2p1d]andcc-pV xZ( x=D,T,Q)basis setsandextrapolatingtotheCBSlimit................26 Table3.2:Totalandadiabaticexcitationenergiesforthegroundandlow-lying excitedstatesofC 2NobtainedwiththevariousEA-EOMCCap- proachesusingtheDZP[4s2p1d]andcc-pV xZ( x=D ;T;Q)basis setsandextrapolatingtotheCBSlimit................33 Table3.3:Comparisonoftheoptimizedequilibriumgeometriesforthelow-lying statesofCNCandC 2N,asobtainedwiththeEA-EOMCCandSAC- CI-SDT- R/PSapproachesusingtheDZP[4 s2p1d]andcc-pV xZ( x=D,T,Q)basissets............................39 Table3.4:Totalandadiabaticexcitationenergiesforthegroundandlow-lying excitedstatesofNCO,asobtainedwiththedierentIP-EOMCC approachesusingtheDZP[4s2p1d]andcc-pVxZ(x=D,T,Q)basis setsandextrapolatingtotheCBSlimit................44 Table3.5:Totalandadiabaticexcitationenergiesforthegroundandlow-lying excitedstatesofN 3,asobtainedwiththedierentIP-EOMCCap- proachesusingtheDZP[4s2p1d]andcc-pVxZ(x=D,T,Q)basissets andextrapolatingtotheCBSlimit..................48 Table3.6:Comparisonoftheoptimizedequilibriumgeometriesforthelow- lyingstatesofN 3andNCO,asobtainedwiththeIP-EOMCCand SAC-CI-SDT- R/PSapproachesusingtheDZP[4s2p1d]andcc-pVxZ (x=D,T,Q)basissets..........................53 viii Table3.7:Scalarrelativisticionizationenergies(ineV)ofAufromAu com- putedusingtheIP-EOMCCapproaches;comparisonwithexperiment andexperimentallyderivedestimates.................59 Table3.8: Vertical a(V)andadiabatic b(Ad)IEs(ineV)ofAu 3withrespecttoitsground statecomputedusingthescalarrelativisticIP-EOMCCSD(2 h-1 p)/aug-cc-pV xZ- PP(+CV)( x=D,T)andIP-EOMCCSD(3 h-2 p)/aug-cc-pVDZ-PPapproaches. 66 Table4.1:Bondlengths(in A)deningthecalculatedandexperimentalgeome- triesofazuleneinitsgroundelectronicstate S0(X1A1)(forthe meaningofvariousdistances,seeFig.4.1)...............95 Table4.2:Verticalexcitationenergies(ineV)andRELvaluesforexcitedstates ofazulenefoundinthiswork......................96 Table4.3:Theabsolutevaluesofthereference( r0)andleadingsinglyexcited (ria)anddoublyexcited( rij ab )amplitudes,alongwiththecorrespond- ingorbitalinformation,deningtheEOMCCSDwavefunctionsof variousexcitedstatesofazuleneobtainedusingthe6-31G(d)and cc-pVDZbasissets............................97 Table4.4:Oscillatorstrengthsforthevarioustransitionsinazuleneobtained fromtheEOMCCSDcalculationsusingthe6-31G(d)andcc-pVDZ basissetsandtheavailableCASSCF(10,10)/6-31G(d)[262]andex- perimental[273]results.........................98 Table4.5:Energydierences(inkcal/mol)forthevariousspeciesalongpathway IcomputedusingselectedCCandDFTapproachesandtheBS1basis set. a...................................118 Table4.6:Energydierences(inkcal/mol)forthevariousspeciesalongpathway IcomputedusingselectedCCandDFTapproachesandtheBS2basis set. a...................................119 Table4.7:Energydierences(inkcal/mol)forthevariousspeciesalongpathway IcomputedattheCCSDandCR-CC(2,3)levelsusingtheBS1basis setandtwodierentschemesforobtainingthecanonicalROHForbitals.121 Table4.8:Comparisonsofenergydierences(inkcal/mol)forthevariousspecies alongpathwayIobtainedusingtwodierentwaysofextrapolating theCC/BS2energetics, arepresentedbyEqs.(4.28)and(4.29)....123 ix Table4.9: Energiesofthefourlowest-energyelectronicstatesofC 4O4atthe UB3LYP/6-31G(d),CASSCF(2,2)/6-31G(d),andCASSCF(16,16)/6- 31G(d)geometriesobtainedinRef.[327]. a..............148 Table4.10:OrbitalstructureofCASSCF(16,16)andMRCI(Q)(14,13)activespaces. 151 Table4.11:BondlengthsofoptimizedC 4O4D4hgeometries(in A). a......152 Table4.12:Energiesofthelow-lyingelectronicstatesofC 4O4atthegeometries obtainedwithCCmethods. a.....................156 Table5.1:Walltime(hours)comparisonofparallelandserialnumericalgeom- etryoptimizationsofseveralmolecules.................170 Table5.2:ComparisonoftheLM-SR1andBFGSquasi-Newtonalgorithmsfor CR-CC(2,3),D/TZVPgeometryoptimizations. a...........174 Table5.3:Detailsofthewalltimes(hours)characterizingtheCCSD(T)/SBKJC(1f) parallelnumericaloptimizationoftheAu 8S1isomeremployingthe D2hsymmetry(4degreesoffreedom),performedusingnine8-core nodes a..................................184 Table5.4:Bondlengths(in A)oftheS 1,S 3,S 4,andS 6isomersofAu 8ob- tainedfromgeometryoptimizationsusingMP2,CCSD(T),andsome representativeDFTapproaches(seeFig.5.1forthemeaningofthe geometricalparameters).........................186 Table5.5:Relativeenergies(inkcal/mol)oftheS 1,S 3,S 4,andS 6isomers ofAu 8,withrespecttotheS 1isomer,obtainedintheMP2and CCSD(T)calculations..........................190 Table5.6:Comparisonoftheenergies(inmillihartree)ofRHF-andUHF-based CCSDandvarioustriples-correctedCCapproximationswiththecor- respondingfullCIdata afortheequilibriumandfourdisplacedge- ometries( Re=1 :7328bohr)oftheHFmolecule,asdescribedbythe sphericalDunningDZbasis.......................198 xTable5.7: Comparisonoftheenergies(inmillihartree)ofRHF-andUHF-based CCSDandvarioustriples-correctedCCapproximationswiththeir parentCCSDTapproachdatafortheequilibriumandfourdisplaced geometries( Re=1 :7328bohr)oftheHFmolecule,asdescribedby thesphericalDunningDZbasis.....................199 Table5.8:Comparisonoftheenergies(inmillihartree)ofRHF-andUHF-based CCSDandvarioustriples-correctedCCapproximationswiththecor- respondingfullCIdatafortheequilibriumanddisplacedgeometries oftheF 2molecule,asdescribedbythesphericalcc-pVDZbasisset.202 Table5.9:Comparisonoftheenergies(inmillihartree)ofRHF-andUHF-based CCSDandvarioustriples-correctedCCapproximationswiththeir parentCCSDTapproachfortheequilibriumanddisplacedgeometries oftheF 2molecule,asdescribedbythesphericalcc-pVDZbasisset.203 Table5.10:Comparisonoftheenergies(inmillihartree)ofRHF-andUHF-based CCSDandvarioustriples-correctedCCapproximationswiththeir parentCCSDTapproachfortheequilibriumanddisplacedgeometries oftheH 2O2molecule, aasdescribedbythesphericalcc-pVDZbasis.207 Table5.11:Comparisonoftheenergies(inmillihartree)ofRHF-andUHF-based CCSDandvarioustriples-correctedCCapproximationswiththeir parentCCSDTapproachfortheequilibriumanddisplacedgeometries oftheCH 3CH 3molecule, aasdescribedbythesphericalcc-pVDZ basis...................................210 LISTOFFIGURES Figure3.1:OrbitallevelsoftheCH +andOH ionsandaschematicrepresenta- tionoftheelectronattachmentandionizationprocessesthatleadto theformationoftheCHandOHradicalsfromtheCH +andOH ref- erenceclosed-shellsystems.ValenceshellsofCH +andOH thatplay adominantroleintherelevantelectronattachmentandionization processesandthatareusedintheactive-spaceEA-andIP-EOMCC calculationsareemphasizedwiththehelpofdottedframes.....18 Figure3.2:VerticalIEsofAu 3(redbars)superimposedonthephotoelectron spectrumfromFig.1(c)inRef.[222]:(a)IP-EOMCCSD(2 h-1 p)/aug- cc-pVDZ-PPcalculations;(b)IP-EOMCCSD(2 h-1 p)/aug-cc-pVTZ- PP+CVcalculations;(c)IP-EOMCCSD(3 h-2 p)/aug-cc-pVDZ-PPcal- culations;(d)extrapolatedIEsdeterminedusingEq.(3.21).Forthe actualsymmetriesandenergiesofthecalculatedstatesofAu 3,see Table3.8.................................68 Figure4.1:The C2v-symmetricstructureofazuleneanditsbondlengths.The largeblackspheresrepresentcarbonatoms,whereasthesmallgray spheresrepresenthydrogens.......................84 Figure4.2:Thephotoelectronspectrumofazuleneusingaprobepulsecentered at400nm,recordedforfourpumpexcitationenergies(thecorre- spondingvibrationalenergiesin S2giveninparentheses),namely, 201nm(2.61eV;Ref.[142]),268nm(1.07eV;Ref.[243]),283nm (0.82eV;Ref.[243]),and335nm(0.14eV;Ref.[243]),andapump- probedelaytimeof500fs(forotherpump-probedelaytimes,which yieldsimilarspectra,seeRef.[243]).Theunstructureddirection- ization(65%ofphotoionizationevents)hasbeensubtracted.Each spectralprolerepresentsangerprintoftheRydbergstatesfrom whichazuleneisphotoionizedafterabsorbingthesecondprobepho- ton.TheelectronicenergiesoftheseRydbergstatesaremarkedby RA,RB,etc.Twohorizontalaxesnearthetopshowthevibrational energiesinRydbergstatesforthepumpwavelengthsof201and335 nm....................................88 iiFigure4.3:Theproposedschematicsofthe1+2 0ph otoionizationexperiment [243].TheRydbergstates,fromwhichazuleneisphotoionized,are populatedbytheelectronicrelaxationfromthepostulateddoubly excitedstatelocatedbelowtheionizationthreshold D0,markedby .Excitationofthedoublyexcitedstatefrom S2,relaxationinto theRydbergstates,andphotoionizationtakeplacewithintheprobe pulseduration(100fs).........................89 Figure4.4:SchematicrepresentationofthereactionpathwayIforthemethanol oxidationtoformicacidontheAu 8clusterproposedinRef.[300].107 Figure4.5:EnergydiagramforreactionpathwayIcharacterizingtheoxidation ofmethanoltoformicacidontheAu 8particleresultingfromCR- CC(2,3),D(greenboldfont),M06(reditalicfont),B3LYP(black romanfont),and !-B97X-D(blueromanfont)calculationsusingthe BS2and,inparentheses,BS1basissets.TheCR-CC(2,3),D/BS1en- ergiesandalloftheDFTenergiesrepresenttrulycomputeddata.The CR-CC(2,3),D/BS2energieswereobtainedbyextrapolationbased onEq.(4.28),inwhichthedierencebetweentheCCSDener- giesobtainedwiththeBS2andBS1basissetsisaddedtotheCR- CC(2,3),D/BS1energy.TheenergiesshownfortheA1,B1,C1,and D1intermediatesarerelativetotheCH 3O+O 2+Au 8reactants. TheenergiesshownfortheproductsarerelativetotheD1interme- diate.TheenergiesshownforthetransitionstatesTS A1,TS B1, an dTS C1arethecorresponding(TS A1 A1),(TS B1 B1),and (TS C1 C1)barrierheights.Allenergiesareinkcal/mol.Also shownarethemolecularstructuresdeningthestationarypoints alongpathwayIresultingfromtheM06geometryoptimizationscar- riedoutinRef.[300]...........................117 Figure4.6:ElectrondistributionsamongthenearlydegenerateHOMOandLUMO makingupthefourlowest-energyelectronicstatesofC 4O4.....139 Figure4.7:Orbitalclassicationusedintheactive-spaceCCapproaches,such asCCSDt.Fullcirclesrepresentcoreandactiveelectronsofthe referencedeterminant ji,whichistheformalreferencedeningthe Fermivacuumfortheactive-spaceCCcalculations..........144 Figure5.1:TheS 1,S 3,S 4,andS 6isomersofAu 8examinedinthisstudy,along withtheselectedgeometricalparametersincludedinTable5.4...181 iiiFigure5.2:EnergydierencesofthevariousRHF-andUHF-basedcoupled- clusterapproacheswithsomeformoftripleswithrespecttothefull CIresultsfor(a)theHFmoleculeand(b)theF 2diatomicspeciesat variousbondlengths...........................204 Figure5.3:EnergydierencesofthevariousRHF-andUHF-basedapproximate triplescoupled-clustermethodswithrespecttotheirparentCCSDT resultsfor(a)theHFmolecule,(b)theF 2diatomicspecies,(c)the H2O2species,and(d)theC 2H6polyatomicmoleculeatvariousbond lengths..................................212 ivChapter1 Introduction Coupled-cluster(CC)theory[1{6],whichisknowntooerth ebestcompromisebetween computationalcostandaccuracyandwhichresultsinapprox imationsthatleadtoarapid convergencetotheexact,fullcongurationinteraction(C I)limit,hasbecomethe defacto standardforhigh-accuracymolecularelectronicstructur ecalculationsforgroundandexcited statesandpropertiesotherthanenergy(seeRefs.[7{11]fo rreviews).However,inorderfor theground-stateCCmethodsandtheirvariousexcited-stat eextensionstobesuccessful, byprovidingchemicalofnear-spectroscopicaccuracies,o neneedstondcomputationally ecientandrobusttreatmentsofhigher-than-doubleexcit ations,suchastriplyortriply andquadruplyexcitedclusters.Indeed,thebasicCCmethod withsinglesanddoubles (CCSD)[12{15],whichhasrelativelyinexpensiveCPUsteps thatscaleas n 2 o n 4 u or N 6 ( n o and n u are,respectively,thenumbersofoccupiedandunoccupiedo rbitalsusedinthecorrelated calculationsand N isameasureofthesystemsize),althoughgenerallymoreacc uratethanits CI(i.e.,CISD)counterpart,especiallyinlargersystems, wherethelackofsize-extensivityof CISDbecomesamajorproblem,isinsucientinobtainingacc urateground-stateproperties, especiallyenergetics(reactionenergies,activationbar riers,chemicalreactionproles,etc.). Atthesametime,theextensionofCCSDtoexcitedstatesviat heequation-of-motionCCSD (EOMCCSD)approach[16{18]anditssymmetry-adapted-clus ter(SAC)CI[19{22]and 1 linear-responseCC[23{28]counterparts,althoughuseful inaqualitativecharacterization ofexcitedstatesdominatedbyone-electrontransitions,i sgenerallynotaccurateenoughto obtainaquantitativedescriptionofsuchstates,especial lywhenlargerpolyatomicspeciesare examined(cf.,e.g.,Refs.[29{32];forathoroughevaluati onofanumberofEOMCCmethods, includingEOMCCSD,illustratingthesame,seeRefs.[33{39 ]).EOMCCSDanditsSAC-CI andlinear-responsecounterpartsalsofailwhentheelectr onicallyexcitedstatesofinterestare characterizedbysignicanttwo-electronorhighermany-e lectrontransitions[39{54].When theconnectedtripleexcitationsareexplicitlyincludedi ntheCCandEOMCCconsiderations viatheCCSDTapproach[55,56]anditsexcited-stateEOMCCS DTextension[42,43,57], thedescriptionoftheground-stateenergeticsandpropert iesandexcitedstatessignicantly improves,yieldingoftenthevirtuallyexactresults(see, e.g.,Refs.[40{43,54{61]).However, itisalsoaccompaniedbyasteepincreaseintheiterativeCP Utimescalingandexcitation amplitudestoragerequirementscharacterizingtheCCSDT/ EOMCCSDTapproximation, from n 2 o n 4 u ( N 6 )and n 2 o n 2 u ( N 4 )intheCCSD/EOMCCSDcaseto n 3 o n 5 u ( N 8 )and n 3 o n 3 u ( N 6 ),respectively,inthecaseofCCSDT/EOMCCSDT,limitingit sapplicabilitytosystems withuptoadozenorsocorrelatedelectronsandsmallerbasi ssets.Thus,ifoneistomakeuse oftheCC/EOMCCmethodologiesinaccuratemolecularelectr onicstructurecalculations, theCC/EOMCCschemesthatcanaccountfortheeectsoftripl esinanapproximate,cost eective,andyetreliablemannerneedtobeemployed. Onewaytheeectsoftripleandotherhigher-than-doubleex citationscanbeincluded intheCC/EOMCCapproacheswithouthavingtodealwiththepr ohibitivecomputational costsofthefullCCSDT/EOMCCSDTmethodsisthroughtheuseo factiveorbitals,asdone 2 intheground-stateCCSDt[62{67]andexcitedstateEOMCCSD t[42{44]approaches(cf. Refs.[54,68]forreviews).WhilethisallowsforfullCCSDT /EOMCCSDT-qualityresults atthecostofCCSD/EOMCCSDtimesaprefactorproportionalt othenumbersofactive occupiedandactiveunoccupiedorbitalsusedtoselectthet riples,theapproachisnolonger, strictlyspeaking,apurecomputationalblackboxasonehas toselecttheactiveorbitals involvedinidentifyingthedominanttriples.Thesameappl iestohigher-ordermethodsin thiscategory,suchasCCSDtq[54,63{66,69]. Onecanalsocontemplateapproachesforidentifyingthemos timportanttriples(and higher)contributionsthroughthemany-bodyperturbation theory(MBPT)analysis,asis commonlydonewhendevelopingapproximatequasi-perturba tiveCC/EOMCCapproxima- tions,bestrepresentedbytheoldestmethodsinthiscatego ry,suchasCCSD[T][70,71], CCSD(T)[72],andCCSDT-1[71,73].Methodsofthistypearec omputationalblackboxes andreducecomputercostsoftheirparentfullapproximatio nsbyordersofmagnitude,while providinganaccuratedescriptionofmoleculesneartheequ ilibriumgeometries.Thisisbest symbolizedbythesuccessoftheCCSD(T)approach,whichrep lacestheiterative n 3 o n 5 u CPU stepsofCCSDTbytheiterative n 2 o n 4 u CPUstepsofCCSDandthenoniterative n 3 o n 4 u steps associatedwiththedeterminationofthetriplescorrectio ntotheCCSDenergy,whileoering theCCSDTlevelofaccuracyintheground-statecalculation saslongasonedoesnotsigni- cantlystretchorbreakchemicalbonds.Problemsemerge,ho wever,whenonewantstoapply methodsoftheCCSD(T)typetobondbreakingorbiradicals,o rconsiderexcitedstates,espe- ciallythosehavingthemoresubstantialcontributionsfro mtwo-electrontransitions(cf.,e.g., Refs.[39{44,54,66{68]).Inresponsetothesechallenges, severalalternativesorextensions 3 ofCCSD(T)thatimprovetheCCSD(T)resultsinbondbreaking andothermultireference situationsandthatimproveEOMCCSDresultsforexcitedsta teswithoutsignicantlyin- creasingcostsandeaseofuseofCCSD(T)calculationshaveb eenformulatedovertheyears. SomeexamplesofthesealternativestoCCSD(T)ofrobustext ensionsofCCSD(T)-likeap- proachestoexcitedstatesincludethenoniterativetriple sortriplesandquadruplesmethods fortheground-statecomputationsdenedtheCCSD(2)orCC( 2)PT(2)andCCSD(2) T approaches[74{77](seeRefs.[78{80]forrelatedideas),a ndtheirexcited-stateanalogs intheformoftheEOMCC(2)PT(2)method[76]anditssize-int ensiveEOMCCSD(2) T modication[45],thelinear-responseCCSDR(3)scheme[81 ,82],anditsiterative,still ratherinexpensiveCC3parent[81{84],theEOMCCSD(T)[85] ,EOMCCSD( ~ T)[86],and EOMCCSD(T 0 )[86]hierarchyobtainedfromtheperturbativeanalysisof theEOMCCSDT equations,andawidevarietyofnoniterativecorrectionst oCCSDorEOMCCSDenergies resultingfromthemethodofmomentsofCC(MMCC)equations[ 40,41,46{52,87{95],such asCR-CC(2,3)[51,93{95],CR-EOMCC(2,3)[51,52],andther ecentlyimplemented -CR- EOMCC(2,3)scheme[31].TheMMCCapproaches,suchasCR-CC( 2,3),CR-EOMCC(2,3), and -CR-EOMCC(2,3)areparticularlypromising,sincetheyret aintheblack-boxnature ofthepopularground-stateCCSD(T)approximationandredu cethelargecostsofthefull CCSDT/EOMCCSDTcalculationstothemoremanageableiterat ive n 2 o n 4 u ( N 6 )andnon- iterative n 3 o n 4 u ( N 7 )computationalsteps,whileprovidingahighlyaccuratede scriptionof chemicalreactionpathwaysinvolvingsinglebondbreaking ,biradicals,andexcitedstates dominatedbyone-andtwo-electrontransitions[39,44,46{ 51,51{53,93,94,96{99].TheCR- CC(2,3)andrelatedapproachesarecapableofproducinghig hlyaccurateresultsforavast 4 arrayofmolecularsystemswithuptoabout100correlatedel ectrons,whenthecanonicalfor- mulationisemployed,orhundredsoreventhousandsofelect rons,whenthelocalcorrelation ideasareimplemented[99{103]. WhiletheCR-CCandCR-EOMCCapproacheshavedemonstratedc onsiderablesuccess (cf.,e.g.,Refs.[29{31,40,41,48,49,51,51{53,88,89,93 ,94,96{99,104{149]),theyarenotfree fromalltheproblems.Thereare,forexample,casesofbirad icalreactionmechanisms,where theydonotworkaswellasdesired,resultingintheunderest imationofelectronicenergiesof singletbiradicals,duetosignicantcouplingofsingles, doubles,andconnectedtriplesinsuch situations,whichtheCR-CC(2,3)andothernoniterativeco rrectionstoCCSDneglect.This issuecanbeaddressedbycombiningtheaforementionedacti ve-spaceCC/EOMCC(e.g., CCSDt)andCR-CC/CR-EOMCC(e.g.,CR-CC(2,3))ideasinthef ormoftheso-called CC( P ; Q )methodology[68,150,151].TheresultingCC(t,3)approac hcombiningCCSDt iterationswithnoniterativecorrectionsduetocertainca tegoriesoftriplesmissinginCCSDt providesspectacularresultsindescribingbondbreaking, biradicalreactionmechanisms,and singlet-tripletgapsinbiradicals[68,150,151].Thereis ,however,anotherproblem,which noneoftheabovemethodsaddressesinasatisfactorymanner ,namelytheissueofproper spinadaptation,whichisdiculttoachieveinconventiona lCCtheorywhenappliedtoopen- shellstates.Forexample,theusualopen-shellimplementa tionsoftheCR-CC(2,3)andCR- EOMCC(2,3)andotherCC/EOMCCschemes,whichutilizeaspin -integratedspin-orbital formalisminderivationsandcomputerimplementationsand whichmakeuseofeitherthe unrestrictedorrestrictedopen-shellHartree-Fock(UHFo rROHF,respectively)reference determinants,donotproperlyaccountforthespinsymmetry whenelectronicstatesof 5 interestarenotsinglets,and,asaresult,maysuerfromth espin-contaminationproblem (cf.,e.g.,Ref.[152])orotherissues,suchasthenon-anal yticbehaviorandsubstantialloss ofaccuracyintheregionofHartree-Fockinstabilities[15 3]. Issuesrelatedtospin-adaptionofCC/EOMCCcalculationsf oropen-shellsystems,such asradicalsandbiradicals,canbeaddressedbyturningtoth ealternativehierarchyof EOMCCapproaches,termedtheelectron-attached(EA)[154{ 162]andionized(IP)[157{169] EOMCCtheoriesaswellastheiranalogousdoublyattached(D EA)anddoublyionized (DIP)approaches[170{178].Inparticular,theEA-andIP-E OMCCmethodsenableone toperformorthogonallyspin-adaptedcalculationsforthe groundandexcitedstatesofthe ( N 1)-electronopen-shellsystemsaround N -electronclosedshellsbyapplyingthelinear electron-attachingorionizingoperator, R ( N 1) ,tothecorrelatedgroundstateoftheref- erence N -electronclosed-shellcoreobtainedwiththesingle-refe renceCCapproach.Be- causethesemethodsuseaclosed-shellreferencedetermina ntandbecauseoneobtainstar- get( N 1)-electronstatesbydiagonalizingthesimilaritytransf ormedHamiltonianobtained intheclosed-shellCCcalculationsfortheunderlying N -electronsystem,whichcommutes with S 2 and S z ,theyprovideaconvenientformalismforperformingorthog onallyspin- andsymmetry-adaptedcalculationsforradicalsandionsof closed-shellspecies,eliminating theissuesassociatedwithspin-contaminationandmakingt heinterpretationofthecal- culatedelectronicstatesmuchmoretransparentthaninthe traditionalROHForUHF basedCC/EOMCCcalculations.Inanalogytoourearlierdisc ussedCCSD/EOMCCSDor CCSDT/EOMCCSDT,thebasiclow-orderEA-andIP-EOMCCappro ximationsincluding upto2-particle-1-hole(2 p -1 h )and2-hole-1-particle(2 h -1 p )excitations,herereferredtoas 6 EA-EOMCCSD(2 p -1 h )andIP-EOMCCSD(2 h -1 p ),althoughusefulincalculationsofthe electronanitiesandionizationpotentialsofclosed-she llmolecules,respectively,areinsuf- cienttoprovideanaccuratedescriptionforthegroundand ,especially,excitedstatesof radicalsandsimilaropen-shellsystems[154,157{160,179 ].Theinclusionofhigher-order componentsof R ( N 1) ,suchas3-particle-2-hole(3 p -2 h )and3-hole-2-particle(3 h -2 p ),ex- citations,greatlyimprovestheresults,butalsosignica ntlyincreasesthecomputational costs,limitingtheuseoftheresultingEA-EOMCCSD(3 p -2 h )andIP-EOMCCSD(3 h -2 p ) approachesandtheirEA-andIP-EOMCCSDTanalogs[156,167, 168]tosmallmolecu- larsystems.Toremedythisproblem,theaforementionedact ive-spaceCCandEOMCC methodologies[42{44,54,62{69]methodologieshavebeene xtendedtotheEA-EOMCCand IP-EOMCCapproaches[157{161].Theactive-spaceEA-andIP -EOMCCschemesuseonly asmallsubsetofallhigherthan2 p -1 h andhigherthan2 h -1 p excitations,respectively,which arechosenthroughasuitablydenedsetofactiveorbitals. Inanalogytoparticle-conserving active-spaceCC/EOMCCmethods,suchasCCSDtorEOMCCSDt,t hissignicantlyre- ducesthecomputationalcostsassociatedwithinclusionof allhigher-orderexcitationsin theEA-andIP-EOMCCapproaches,butsincethemostimportan thigher-ordercontribu- tionsarestillaccountedfor,theactive-spaceEA-andIP-E OMCCmethodsarecapable ofproducinghighlyaccurateresultsofthesamequalityast heirparentapproximations. TheEA-EOMCCandIP-EOMCCapproacheswithanactive-spacet reatmentof3 p -2 h and 3 h -2 p excitations[157{159],designatedhereandelsewhereinth isdocumentastheEA- EOMCCSDt f N u g andIP-EOMCCSDt f N o g methods,where N o and N u arethenumberof activeoccupiedandactiveunoccupiedorbitals,respectiv ely,havedemonstratedtheirreliabil- 7 ityinreproducingtheresultsforthegroundandexcitedsta tesofradicalsobtainedwiththeir moreexpensiveparentapproximations,EA-EOMCCSD(3 p -2 h )andIP-EOMCCSD(3 h -2 p ), respectively,andotherhigh-level abinitio approachesatasmallfractionofthecomputer cost[157{160,180].Similarremarksapplytotherecentlyd evelopedDEA-andDIP-EOMCC methodswithanactive-spacetreatmentof4 p -2 h and4 h -2 p excitations,whichareparticu- larlyusefulinexaminingtheelectronicspectraofbiradic alspecies[177,178]. Inthisdissertation,wehaveusedtheEA-andIP-EOMCCappro acheswithfulland active-spacetreatmentsof3 p -2 h and3 h -2 p excitationstoinvestigateseveralchallenging situationsinvolvingsmallandyetcomplicatedopen-shell molecularspecies,suchasthe adiabaticexcitationspectraoftheCNC,C 2 N,N 3 ,andNCOmolecules[180]andthepho- toelectronspectraofAu n particles,demonstratingtherobustnessandutilityofthe seap- proachesinsuchchallengingcases.Inadditiontotheelect ronicstructureofthesesystems, wehaveusedparallelnumericalderivativesfortheEA-andI P-EOMCCmethods,developed inthisPhDresearch,toexaminetheabilityoftheEA/IP-EOM CCmethodswithupto 3 p -2 h /3 h -2 p excitationstoprovideaccuratenuclearcongurationinfo rmation,includingex- citedstates.Forseveralotherclosed-andopen-shellspec ies,wehaveusedtheCR-CC(2,3) and -CR-EOMCC(2,3)approachesandparallelnumericalderivat ivesbasedontheCR- CC(2,3)andCR-EOMCC(2,3)levels,developedinthisthesis worktoo,tosolveimportant chemicalandspectroscopicproblems,providingdenitive conrmationoftheexistenceof thehighly-correlateddoublyexcitedstateofazulenebelo wtheionizationthreshold,which drivesthe1+2 0 multiphotonionizationexperimentsthatleadtoaclearRyd bergnger- print[142],examiningthedetailsofthemechanismandener geticsoftheaerobicoxidation 8 ofmethanolonAu 8 nanoparticle[181],andobtainingahighlyaccuratedescri ptionofthe low-lyingstatesofthedeceptivelysimple1,2,3,4-cyclob utanetetraone[182]conrmingpre- cisephotodetachmentexperimentsperformedrecently.Asa lreadyalludedtoabove,aspart oftheseinvestigationswehavedevelopedandimplementedp arallelnumericalderivative routineswhichcanbeusedwithanymolecularelectronicstr ucturemethod(includingand CC/EOMCCapproach)inconjunctionwithoptimizationalgor ithmroutinestohelpspeed uptheevaluationofgeometriesandharmonicvibrationalfr equencies.Thishasallowed ustoprovidedenitiveinformationaboutthegeometriesan drelativeenergiesofthelow- energyisomersofthecontroversialAu 8 particleatthehighCCSD(T)level[183].Finally, returningtotheissueofUHF vs RHF(restrictedHartree-Fock)referencedeterminantin CCcalculations,wehaveexaminedtheeectsofusingspin-s ymmetrybrokenUHFrather thanspin-adaptedRHFreferencewavefunctionontheCR-CC( 2,3)resultsindescribing bond-breakingonsingletpotentialenergysurfaces,showi ngthatthespin-adaptedclosed- shellCR-CC(2,3)codesbasedonRHFreferencesprovidetheb estoverallandmostaccurate results.Insummary,thefocusofthisdissertationhasbeen theapplicationofnovelhigh-level CC/EOMCCapproachesdevelopedinourgrouptochallenginga ndexperimentallyrelevant chemicalsituations,consideredtobehighlycomplexforco nventionalquantumchemistry, andthedevelopmentofnewcodesandalgorithmsforhigh-lev elgeometryoptimizationsand frequencycalculationsanduseofunrestrictedreferences inCR-CCcomputations. 9 Chapter2 Projectobjectives Themainobjectivesofthisworkare: A.Applicationofthefullandactive-spaceEA-andIP-EOMCC methodstotheenergies andgeometriesofthegroundandexcitedstatesofCNC,C 2 N,NCO,andN 3 .Inter- pretationofthephotoelectronspectrumofAu 3 usingtheIP-EOMCCmethodologies. B.ApplicationoftheCR-CC(2,3)and -CR-EOMCC(2,3)approachestohighly-correlated systems,includingthedoublyexcitedstateofazulenebelo wtheionizationthreshold mediatingthe1+2 0 multiphotonionization,theaerobicoxidationofmethanol toformic acidonAu 8 ,andthelow-lyingelectronicstatesof1,2,3,4-cyclobuta netetraone. C.Developmentandtestingofparallelnumericalgradients andsecondderivativesforge- ometryoptimizationsandvibrationalharmonicfrequencie susingvariousCC/EOMCC approximations,includingCCSD(T),CR-CC(2,3),andCR-EO MCC(2,3),andthe EA/IP-EOMCCmethodswithupto3 p -2 h and3 h -2 p excitationstreatedfullyorvia activeorbitals.Examiningalternativealgorithmsforacc eleratingtheconvergenceof thegeometryoptimizationmethodsusingnumericallydeter minedenergyderivatives. D.ImplementationandbenchmarkingofunrestrictedCR-CC( 2,3). 10 Chapter3 Applicationsoffullandactive-space electronattachedandionized equation-of-motioncoupled-cluster methods Inthischapter,wediscusstheapplicationoftheEA-andIP- EOMCCapproacheswithup to3 p -2 h and3 h -2 p excitations,treatedfullyorwithactiveorbitals,tochal lengingopen- shellproblems.Section3.1providesthetheoreticalbackg roundoftherelevantEA-and IP-EOMCCmethodologies.Section3.2discussestheirappli cationtoaseriesofchalleng- ingtriatomicradicals,CNC,C 2 N,N 3 ,andNCO,includinggeometryoptimizationsand adiabaticexcitationenergies,basedontheresultsreport edinRef.[180].Finally,inSec- tion3.3,wedemonstratetheutilityoftheIP-EOMCCapproac hesinprovidinganaccurate interpretationofnotonlypeakpositions,butalsopeakwid thsandintensities,forthepre- viouslyunexplainedphotoelectronspectrumofAu 3 ,relyingonourcalculationsreportedin Ref.[184]. 11 3.1Theory IntheEA/IP-EOMCCtheories,oneobtainstheground( =0)andexcited( > 0)states j ( N 1) i ofan( N 1)-electronsystembyapplyingalinearelectron-attachin gorionizing operator R ( N 1) totheCCgroundstate j ( N ) 0 i = e T j i (3.1) oftherelatedclosed-shell N -electronsystem,sothat j ( N 1) i = R ( N 1) j ( N ) 0 i R ( N 1) e T j i : (3.2) Here, T = N X n =1 T n ; (3.3) where T n = X i 1 < k; A j>k;b 0labelsexcitedstates,whichintheCCSD/EOMCCSDcaseareg eneratedby thedeexcitationoperators L ˇ L (CCSD) = L ; 0 + L ; 1 + L ; 2 ,where L ; 0 = ; 0 1 ; (4.14) L ; 1 = X i;a l a i a i a a ; (4.15) and L ; 2 = X i 0)energies E (CCSD) arecalculatedas (2 ; 3)= X i 0.AsshowninRefs.[31,52],theenforcementofstrictsizei ntensivityofthe (2 ; 3)triplescorrectionstotheEOMCCSDexcitationenergies( ! (CCSD) issizeinten- sive[28,242])requiresthatwereplacethecompletemoment M ijk ;abc ,Eq.(4.20),inEq. (4.19)for (2 ; 3)byitstruncatedanalogignoringtheground-state r ; 0 h abc ijk j H (CCSD) j i contribution,i.e., M ijk ;abc = h abc ijk j H (CCSD) ( R ; 1 + R ; 2 ) j i : (4.25) WecontinueusingEq.(4.21)forthedeexcitationamplitude s ` abc ;ijk ,whichinthecaseof excitedstatestargetedby -CR-EOMCC(2,3)arecalculatedas(cf.Eq.(4.14)) ` abc ;ijk = h j ( L ; 1 + L ; 2 ) H (CCSD) j abc ijk i =D abc ;ijk : (4.26) Again,ifthe Dabc ;ijk denominatorenteringEq.(4.26)isgivenbytheEpstein-Nes bet-type expression,Eq.(4.22),weobtainthemorecompletevariant Dof -CR-EOMCC(2,3),ab- breviatedas -CR-EOMCC(2,3),D.IfwereplaceEq.(4.22)for Dabc ;ijk inEq.(4.26)by theM˝ller-Plesset-typeexpressiongivenbyEq.(4.23),we obtainthesimpliedAvariant, abbreviatedas -CR-EOMCC(2,3),A,whichis,asexplainedinRef.[31](cf., also,Ref.[52]), equivalenttotheEOMCCSD(2) T approachofRef.[45]and,ifwelimitourselvestovertical excitationenergies,totheEOMCCSD( ~ T)methodofRef.[86].Typically,weusethe -CR- EOMCC(2,3)approachtocalculateexcitationenergies.Ifw eare,however,interestedin total -CR-EOMCC(2,3)energies,wesimplyadd ! (2 ; 3),Eq.(4.24),totheground-state CR-CC(2,3)energy.Asaresult,the -CR-EOMCC(2,3)methodisextensiveintheground state(usingtheCR-CC(2,3)expressionsfortheground-sta teenergy)andsizeintensivein 82 describingexcitationenergies,satisfyingthekeydeside rataoftheCC/EOMCCtheories. 4.2Discoveryofthedoublyexcitedstatethatmediates thephotoionizationofazulene 4.2.1Backgroundinformationandperformedcalculations Typically,whentalkingaboutmolecularelectronicspectr oscopy,thefocusisondipole- allowedtransitionstostateswithpredominantone-electr onexcitationcharacter,whichmake upthemajorityofphotoabsorptionspectra.Itiswellknown ,though,thatmolecularsystems canpossessstronglycorrelatedexcitedstatesdominatedb ytwo-electrontransitions.Such states,whilebeingspectroscopicallydarkwhenpopulated fromthegroundstates,canbe usefultoolsforprobingwiderrangesofvibrationalenergi esandnoveltypesofphotoinduced chemicaldynamics.Asapartofthisthesiswork,weexamined theelectronicspectrum ofazulenecombininghigh-level abinitio quantumchemistrywithexperimenttoprovethe existenceofadoublyexcitedstatebelowtheionizationthr esholdwhichcandrivethein- triguingmultiphotonionizationdynamicsexaminedinRef. [243],resultinginclearRydberg ngerprintspectra. Theelectronicstructureofazulene,whosemolecularcong urationisshowninFig.4.1, isnoteworthyduetoitsatypicaluorescence,whichoccurs fromthesecond-excited S 2 state, insteadofthelower-energy S 1 state.Assuch,azuleneisatextbookexceptiontoKasha's rule[244],which,althoughoriginallyformulatedforcond ensedphasematter,appliesto manygasphasemoleculesaswell,saysthat\theemittinglev elofagivenmultiplicityis 83 !"#$%& '(&"&#&$Figure4.1:The C 2 v -symmetricstructureofazuleneanditsbondlengths.Thela rgeblack spheresrepresentcarbonatoms,whereasthesmallgraysphe resrepresenthydrogens. 84 thelowestexcitedlevelofthatmultiplicity."Duetothisu nusualbehavior,therehavebeen manystudies,bothexperimentalandtheoretical,onthepho tochemistryofazulene,including severalexperimentalstudiesdealingwiththespectroscop yanddynamicsinvolvingthe S 1 and S 2 states[245{248].Theearlytwo-photonionizationexperim entsviathe S n , n =2 4, statesofazulene[249]shouldbementionedhere,too.Muchl essisknown,however,about highervalencestatesofazulene[250{253],otherthanthei rutilityinpreparinghighlyexcited moleculesforobservationofphotoinducedunimoleculardy namics[254{256]. Recently,Blanchetandcoworkerscarriedoutanextensivee xperimentalstudyofthe photoionizationofazulene[243].Thephotoelectronspect raofazulenerecordedintheir two-color,three-photon,1+2 0 time-resolvedphotoelectronimagingexperimentsareshow n inFig.4.2,whichcombinesthephotoelectronspectrafromR ef.[243],wherethepump wavelengthswerevariedfrom268to335nmandprobepulseswe rexedat400nm,with theanalogousspectrumcorrespondingtothe201nmpumpobta inedinRef.[142](see Ref.[257]forarelatedstudy).Theynoticedthatthe1+2 0 time-resolvedphotoelectron spectraareinvariant(apartfromtheintensity)withrespe cttothepump-probedelaytime andwavelengthofthepumppulse(cf.Fig.4.2),whichledthe mtosuggestthatthe1+2 0 photoionizationisdrivenbyanunstabledoublyexcitedele ctronicstatelocatedbelowthe ionizationthreshold[243].InthestudycarriedoutinRef[ 243]Blanchetandcoworkersstated \...wehopeourobservationsmightinspiretheoriststotak eupthechallengetocalculatethe geometryandelectroniccongurationofthedoublyexcited statesonpolyaromaticsystems, suchasazulene",astheyhadnodirectwaytoproveexperimen tallythatsuchastateexisted. Animportantnoteaboutthepostulateddoublyexcitedstate inthemiddleoftheRydberg 85 partofazulene'selectronicspectrum,isthatitshouldbed istinguishedfrom\superexcited" resonancestatesabovetheionizationthresholdthattheno rmal,basis-set-based,realspace abinitio quantumchemistryapproachesforelectronicallyboundsta tes,includingthose employedinthisdissertation,cannotdescribe.Asdepicte dinFig.4.3,thepostulated doublyexcitedstate,markedby ,isproposedtobepopulatedbyaprobetransitionfrom the S 2 state,whichitselfispopulatedbytheinitialpumpphotono rbyfastrelaxationfrom the S n stateswith n> 2reachedbythepumpenergies.Oncepopulated,thedoublyex cited stateisproposedtorapidlyrelaxintothevibrationallyhi ghlyexcitedRydbergstates,from whichazulenecanbephotoionizedafterabsorbingthesecon dprobephotontoproducea Rydbergngerprint. 86 Theavailableexperimentalinformationsuggeststhatthep ostulateddoublyexcitedstate islocatednotonlybelowtheionizationthresholdof7.41eV ,butitmust,infact,appear below6.81eV(acombinationofthelowest-energypumpphoto nat335nmanda400nm probephoton).ItshouldalsobelocatedabovethesecondRyd bergstateat5.19eV,marked inFig.4.2as R B (cf.Table4.2),sincethengerprintofthisstateisclearl yseeninthelower- energypartsofallthephotoelectronspectrashowninFig.4 .2,independentofthepump wavelength.Furthermore,torationalizetheopticallyind ucedanisotropyassociatedwith thepumpexcitation[257],thepostulateddoublyexcitedst ateshouldrepresentatotally symmetricsingletexcitationaccessiblefrom S 2 .Deleuzecarriedanexhaustiveinvestiga- tionusingalgebraic-diagrammaticconstructioncalculat ionsattheADC(3)leveloftheory forazuleneandotherpolycyclicaromatichydrocarbonsand theirionizationspectra[258]. Heshowedthattherstshake-upionstateoftheazulenecati on,whichliesatverylow energy,consistsofadominantorbitalcongurationof ::: (3b 1 ) 2 (2a 2 ) 0 (4b 1 ) 0 (3a 2 ) 1 withre- specttotheneutralground-stateelectronconguration, ::: (3b 1 ) 2 (2a 2 ) 2 (4b 1 ) 0 (3a 2 ) 0 .Since thepostulateddoublyexcitedstateliesnearstheionizati onlevelandtheRydbergstates areassumedtoconvergetothiscationstate,thepostulated doublyexcitedstateshould haveasimilarelectroncongurationdominatedbythe(HOMO ) 2 ! (LUMO+1) 2 and (HOMO 1) 2 ! (LUMO) 2 transitions[243].Itshouldbementionedthatthemultipho ton ionizationofazulenefrom S 2 viaunstablesuperexcitedvalencestatesrelaxingtotheRy d- bergstatespriortothenalphotoionizationeventhasalso beendiscussedinRef.[259],but withoutdeterminingaprecisemakeupofthehypotheticalsu perexcitedstates.Moreover, superexcitedstatesarelocatedabovetheionizationthres hold,whereasthedoublyexcited 87 Ekin /eV0.00.51.11.52.02.5 photoelectrons /arb. units 201 nm 268 nm 283 nm 335 nm RARBRCRDREElectronic energy of Rydberg states /eV 4.55.05.56.06.57.0 Evib (Rydberg states at 201 + 400 nm) /eV 4.54.0 3.53.02.5 2.0 1.5 1.0 0.5 0.0 Evib (Rydberg states at 335 + 400 nm) /eV Figure4.2:Thephotoelectronspectrumofazuleneusingapr obepulsecenteredat400nm, recordedforfourpumpexcitationenergies(thecorrespond ingvibrationalenergiesin S 2 giveninparentheses),namely,201nm(2.61eV;Ref.[142]), 268nm(1.07eV;Ref.[243]), 283nm(0.82eV;Ref.[243]),and335nm(0.14eV;Ref.[243]), andapump-probedelay timeof500fs(forotherpump-probedelaytimes,whichyield similarspectra,seeRef.[243]). Theunstructureddirectionization(65%ofphotoionizatio nevents)hasbeensubtracted. EachspectralprolerepresentsangerprintoftheRydberg statesfromwhichazuleneis photoionizedafterabsorbingthesecondprobephoton.Thee lectronicenergiesofthese Rydbergstatesaremarkedby R A , R B ,etc.Twohorizontalaxesnearthetopshowthe vibrationalenergiesinRydbergstatesforthepumpwavelen gthsof201and335nm. 88 S0S2S3S4~ 50 fs ~ 50 - 100 fs RERD** D0e-e-~ 50 fs PUMP PROBE PROBE Figure4.3:Theproposedschematicsofthe1+2 0 photoionizationexperiment[243].The Rydbergstates,fromwhichazuleneisphotoionized,arepop ulatedbytheelectronicrelax- ationfromthepostulateddoublyexcitedstatelocatedbelo wtheionizationthreshold D0 , markedby .Excitationofthedoublyexcitedstatefrom S 2 ,relaxationintotheRydberg states,andphotoionizationtakeplacewithintheprobepul seduration(100fs). 89 statepostulatedinRef.[243]andsoughtinthisthesisrese archisexpectedtoliebelowthe ionizationthreshold.Wenotethatdoublyexcitedstatesat energiesaslowas4.95eVhave beendetectedbythemagneticcirculardichroismexperimen tsonazulenederivatives[260]. Todetermineiftheproposeddoublyexcitedstateindeedexi sts,wecarriedoutalarge numberof -CR-EOMCC(2,3)calculations,includinglow-lyingvalenc estatesandseveral statesinthehigher-energyregionneartheionizationthre shold[142].AsexplainedinSection 4.1,the -CR-EOMCC(2,3)approachistherigorouslysize-intensive modicationoftheCR- EOMCC(2,3)methodwhichis,inturn,theextensionoftheCR- CC(2,3)approachexcited electronicstates.JustlikeCR-EOMCC(2,3)anditspredece ssors[48,49],ormethodsmen- tionedinSection4.1,suchasEOMCCSD( ~ T)andEOMCCSD(2) T ,the -CR-EOMCC(2,3) approachcorrectstheverticalexcitationenergiesobtain edwiththeEOMCCSDscheme fortheeectsoftripleexcitationsthatarenecessarytoac curatelydescribeexcitedstates dominatedbytwo-electrontransitionswithintheEOMCCfra mework.Asmentionedin theIntroduction,EOMCCSDdescribestheenergeticsofexci tedstatesdominatedbytwo- electrontransitionspoorly,pushingthemtomuchhigheren ergies(cf.,e.g.,Refs.[18,40{43, 46{52,54,85,86]).Methods,suchasEOMCCSD( ~ T),EOMCCSD(2) T ,andespecially -CR- EOMCC(2,3)providethenecessaryenergylowering.The -CR-EOMCC(2,3)approachhas afewvariants,discussed,inparticular,inSection4.1,in cludingthemorecompletevariant DandthesimpliedvariantAequivalenttoEOMCCSD(2) T andEOMCCSD( ~ T),butwe onlyshowvariantDcalculations,sincetheresultsobtaine dwithvariantAarerathersimilar anddonotalterourmainconclusions. Our -CR-EOMCC(2,3)calculationsforazulenewereperformedus ingthe6-31G(d)and 90 cc-pVDZbasissets,whichwerethelargestwecouldreasonab lyaccommodatewhencalcu- latingsomanyexcitedstatesofazuleneatsuchahighEOMCCl evel.Theinitial -CR- EOMCC(2,3)/6-31G(d)calculationswereperformedatthegr ound-stategeometryoptimized attheMP2/6-31G(d)levelwithGAMESS,wherethe -CR-EOMCC(2,3)andotherEOMCC routinesdevelopedbythePiecuchgroup[31,48,49,51,52,1 45]areincorporated.Thenal -CR-EOMCC(2,3)/cc-pVDZcalculationswereperformedatth eimprovedground-statege- ometryobtainedusingthenumericalCR-CC(2,3)/cc-pVDZgr adientsinGAMESS,consis- tentwiththe -CR-EOMCC(2,3)/cc-pVDZdescriptionofexcitedstates,wh ichweveried withourparallelnumericalderivativesdiscussedinSecti on5.1AllcalculationsusedtheRHF referenceandthe10lowest-energycoreorbitalscorrespon dingtothe1sshellsofthecarbon atomswerefrozeninthepost-SCFconsiderations.Although basissetsusedherecannotde- scribetheRydbergstates,theyareacceptableforrepresen tingone-andtwo-electronvalence transitions,includingthedoublyexcitedstatewehaveatt emptedtond,aslongasoneuses ahigher-levelEOMCCmethodologywitharobusttreatmentof tripleexcitations,suchas -CR-EOMCC(2,3)(intheFranck-Condonregionofthepumpand probetransitions,thein- teractionwiththeRydbergstatesisnotexpectedtostrongl yperturbtheverticalexcitation energyofthedoublyexcitedvalencestate). 4.2.2Results TheresultsofourMP2/6-31G(d)andCR-CC(2,3)/cc-pVDZgeo metryoptimizationsper- formedpriortothenal -CR-EOMCC(2,3)work,areshowninTable4.1alongwiththe availableexperimentalandpreviouslyobtainedtheoretic aldata.Ourequilibriumgeome- 91 triesareingoodagreementwiththeothertheoretical[261{ 264]andexperimentallydeter- mined[265{268]results.Table4.2showsthetenlowestexci tedstateswefoundalongwith theavailablemultireferencecompleteactive-spaceselfc onsistenteld(CASSCF)[269,270], CASSCFbasedsecond-orderperturbationtheory(CASPT2)[2 71,272],andexperimental verticalexcitationenergies[243].Thenumberofelectron icstateswehadtoconvergehad tobemuchlargerthaninthecaseoftheearlierCASSCFandCAS PT2calculationsofthe low-lyingexcitationsinazulenedominatedbyone-electro ntransitions[262],whichshows theadvantageofusingoursingle-referenceEOMCCideasove rthemultireferenceCASSCF- basedthinking,sincechoosinganadequateactive-spaceto encompassallofthesinglet excitedstatesofazuleneandconsideredinthisworkwouldb eproblematicandprohibitively expensive.Additionalinformationaboutthedominantexci tationamplitudesdeningthe EOMCCSDwavefunctionsforthecalculatedexcitedstatesis giveninTable4.3.The EOMCCSDvaluesofthedipoleoscillatorstrengthscharacte rizingthecalculatedelectronic transitions,alongwiththeavailableexperimental[273]a ndCASSCF[262]data,canbe foundinTable4.4. AsshowninTable4.2,our -CR-EOMCC(2,3)resultsforthelow-lyingvalenceexcited statesinazuleneareinverygoodagreementwiththeavailab leCASPT2(10,10)[262]and experimental[250,251]data.Inmostcases,our -CR-EOMCC(2,3)excitationenergiescor- respondingtothelow-lyingvalencestatesshowaslightlyb etteragreementwithexperiment thanthoseobtainedwithCASPT2.Theverticalexcitationen ergiesinTable4.2areaccom- paniedbythereducedexcitationlevel(REL)diagnosticint roducedinRef.[49],resulting 92 fromEOMCCSDcalculations,whichisdenedas(cf.Eq.(26)i nRef.[49]) REL= 2 P n =0 n h ˚ j ( R ;n ) y R ;n j ˚ i 2 P n =0 h ˚ j ( R ;n ) y R ;n j ˚ i = P i;a ( r i a ) 2 +2 P i?<@;<=A?>BCDDEFG HCDDIFJKLMNON PQRSTU VWXYZ[\]^_`abcdefghcdifjklmnopqlmrostuvwxyz{|{}~•†‡†…—–ƒ⁄ ‹›−‰„“ ”‘’‚™fiflŁŒŠŸŽıłœšž€łœž¡¢£¤¥¦§££¥¨©ª«¬®¯°±²³´µ¶·¸¶¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÄÉÆÊËÌÍÎÏÐÑÒÎÑ ÓÔÕÖ×ØÙÙ×ÚÛÜÝÞßàáâãäåæçèéê ëçìéíîïðñòóôôõö÷øùúù ûøüúýþÿÿÿ ˘ ˇˆ˙˝˛˚˜ !˛"˜˛#$%&' ()*+,-./0.123456789:;789<=>?@A B>>@CDFigure4.5:EnergydiagramforreactionpathwayIcharacter izingtheoxidationofmethanol toformicacidontheAu 8 particleresultingfromCR-CC(2,3),D(greenboldfont),M0 6 (reditalicfont),B3LYP(blackromanfont),and ! -B97X-D(blueromanfont)calculations usingtheBS2and,inparentheses,BS1basissets.TheCR-CC( 2,3),D/BS1energiesand alloftheDFTenergiesrepresenttrulycomputeddata.TheCR -CC(2,3),D/BS2energies wereobtainedbyextrapolationbasedonEq.(4.28),inwhich thedierencebetweenthe CCSDenergiesobtainedwiththeBS2andBS1basissetsisadde dtotheCR-CC(2,3),D/BS1 energy.TheenergiesshownfortheA1,B1,C1,andD1intermed iatesarerelativetothe CH 3 O +O 2 +Au 8 reactants.Theenergiesshownfortheproductsarerelative totheD1 intermediate.Theenergiesshownforthetransitionstates TS A1,TS B1,andTS C1arethe corresponding(TS A1 A1),(TS B1 B1),and(TS C1 C1)barrierheights.Allenergies areinkcal/mol.Alsoshownarethemolecularstructuresde ningthestationarypointsalong pathwayIresultingfromtheM06geometryoptimizationscar riedoutinRef.[300]. 117 118 Table4.5:Energydierences(inkcal/mol)forthevariouss peciesalongpathwayIcomputedusingselectedCCandDFT approachesandtheBS1basisset. a CCSDCR(2,3),A b CR(2,3),D b M06 c M06-LB3LYPBP86B97-DTPSS ! -B97X-D A1 reactants-23.3-24.7-22.0-15.8-18.9-12.9-19.6-19.2-2 0.7-24.6 TS A1 reactants21.314.116.116.07.722.22.812.14.720.0 TS A1 A144.638.838.231.826.635.122.431.225.444.6 B1 reactants-36.1-38.7-36.2-34.0-29.9-24.8-29.7-29.3-2 8.0-30.7 TS B1 reactants-2.3-9.9-7.7-5.0-9.65.6-9.9-3.6-9.13.6 TS B1 B133.928.828.529.020.330.519.825.718.934.3 C1 reactants-54.3-57.1-54.7-47.6-46.5-35.5-42.3-37.5-4 4.5-44.9 TS C1 reactants-34.9-40.4-38.4-36.1-38.0-24.9-38.1-33.5-3 6.1-32.2 TS C1 C119.416.816.311.58.410.74.24.08.512.7 D1 reactants-117.1-119.8-117.0-117.9-109.8-102.5-105.5 -108.3-100.9-113.8 products reactants-98.1-99.8-98.9-99.9-97.8-89.0-92.1-85.9-8 6.3-93.6 products D119.020.018.118.018.013.513.422.414.620.2 MUE d 2.1(5.0)2.2(0.5)0.0(0.0)2.8(3.9)5.3(9.2)12.1(3.6)6. 9(12.2)7.7(7.4)7.9(10.1)6.0(5.3) NPE e 6.7(3.3)1.9(0.3)0.0(0.0)8.1(6.9)16.6(3.7)13.1(7.6)2 5.7(7.1)21.2(9.5)27.5(5.0)13.9(10.0) a ForthedenitionsofthevariousspeciesalongpathwayI,se eFigs.4.4and4.5.TheCCresultscorrespondtotheGuest-Sa unders canonicalizationschemeforROHForbitals[314]. b CR(2,3),A=CR-CC(2,3),AisequivalenttotheCCSD(2) T approachofRef.[77].CR(2,3),D=CR-CC(2,3),D. c DatatakenfromRef.[300]. d MeanunsignederrorrelativetoCR-CC(2,3),Dcalculatedus ingtheenergiesofA1,TS A1,B1,TS B1,C1,TS C1,D1,andproductsrelative tothereactantsand,inparentheses,usingthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)energydierences. e NonparallelityerrorrelativetoCR-CC(2,3),Dcalculated usingtheenergiesofA1,TS A1,B1,TS B1,C1,TS C1,D1,andproductsrelative tothereactantsand,inparentheses,usingthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)energydierences. 119 Table4.6:Energydierences(inkcal/mol)forthevariouss peciesalongpathwayIcomputedusingselectedCCandDFT approachesandtheBS2basisset. a CCSDCR(2,3),A b CR(2,3),D b M06 c M06-LB3LYPBP86B97-DTPSS ! -B97X-D A1 reactants-16.6-18.0-15.3-18.7-21.9-14.9-21.3-20.4-2 2.0-26.6 TS A1 reactants28.521.323.313.65.420.91.911.33.518.6 TS A1 A145.039.338.632.327.335.823.231.725.545.2 B1 reactants-32.2-34.7-32.2-35.9-31.8-26.3-31.0-30.1-2 9.4-32.1 TS B1 reactants6.2-1.40.8-8.2-11.34.9-11.0-4.0-10.32.9 TS B1 B138.433.333.027.220.531.219.926.119.135.0 C1 reactants-47.4-50.3-47.8-52.0-51.0-39.5-47.1-41.6-4 9.0-48.6 TS C1 reactants-29.1-34.6-32.6-39.7-40.8-27.3-41.0-35.7-3 9.0-34.5 TS C1 C118.315.715.212.310.212.26.15.810.114.1 D1 reactants-114.8-117.4-114.6-121.5-112.2-105.1-108.2 -110.4-103.6-116.2 products reactants-97.9-99.6-98.6-103.4-94.2-92.5-95.8-89.2- 90.0-96.7 products D116.917.916.018.218.012.612.421.213.619.5 MUE d 2.1(5.0)2.2(0.5)0.0(0.0)6.1(4.9)6.9(9.6)5.2(2.6)7.4 (12.5)5.9(7.7)8.5(10.7)3.1(3.2) NPE e 6.7(3.3)1.9(0.3)0.0(0.0)6.3(3.4)22.3(7.4)11.9(1.2)2 7.8(6.3)21.4(2.4)30.8(8.8)13.5(7.7) a ForthedenitionsofthevariousspeciesalongpathwayI,se eFigs.4.4and4.5.TheCCresultscorrespondtotheGuest-Sa unders canonicalizationschemeforROHForbitals[314]. b CR(2,3),A=CR-CC(2,3),AisequivalenttotheCCSD(2) T approachofRef.[77].CR(2,3),D=CR-CC(2,3),D.Extrapola tedusingEq. (4.28). c DatatakenfromRef.[300]. d MeanunsignederrorrelativetoCR-CC(2,3),Dcalculatedus ingtheenergiesofA1,TS A1,B1,TS B1,C1,TS C1,D1,andproductsrelative tothereactantsand,inparentheses,usingthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)energydierences. e NonparallelityerrorrelativetoCR-CC(2,3),Dcalculated usingtheenergiesofA1,TS A1,B1,TS B1,C1,TS C1,D1,andproductsrelative tothereactantsand,inparentheses,usingthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)energydierences. Table4.7comparestheCCSD/BS1,CR-CC(2,3),A/BS1,andCR- CC(2,3),D/BS1results obtainedwiththetwotypesofcanonicalROHForbitalsconsi deredinthisstudy.Examining Table4.7itisimmediatelyobviousthatforeachCCmethodan alyzedinthiswork,theim- pactofthedierentROHFcanonicalizationschemesonthere sultingenergeticsisnegligible, asthedierencesbetweentheCCSD,CR-CC(2,3),A,andCR-CC (2,3),Denergiesobtained withtheRoothaanandGuest-Saundersapproachesdonotexce ed0.1,0.2,and0.3kcal/mol, respectively.Thus,intherestofthediscussioninthissec tion,theCCresultsbasedonthe frequentlyexploitedGuest-Saunderscanonicalizationap proach,summarizedinTables4.5 and4.6,willbeused. Table4.8comparestheenergydierencesforthevariousspe ciesalongpathwayIresulting fromthetwodierentwaysofextrapolatingthelarger-basi s-setCC/BS2energeticsdiscussed inSection4.3.3andrepresentedbyEqs.(4.28)and(4.29).A lthoughmuchofthediscus- sionbelowfocusesonthemoreaccurateprocedureforextrap olatingtheCR-CC(2,3)/BS2 energeticsbasedonEq.(4.28),inwhichthebulkoftheelect roncorrelationenergycaptured byCCSDisdeterminedusingthetargetBS2basis,reducingth euseofBS1tothetriples correctionsonly,itisinterestingtoobservethatonecano btainsimilarresultsbyreplacing CCSDinthebasissetextrapolationschemebythemuchlessex pensiveUMP2orUMP3 approach,eliminatingtheneedtocarryoutthelarger-basi s-setCCcalculationsaltogether. AsshowninTable4.8,theenergydierencesforthevariouss peciesalongpathwayIre- sultingfromthetrueCCSD/BS2calculationsandtheirextra polatedcounterpartsusingEq. (4.29)agreetowithin ˘ 3kcal/molintheUMP2caseand ˘ 2kcal/molwhentheUMP3 energiesareused,thoughthereareseveralcaseswherethet rueCCSD/BS2resultsandthe 120 Table4.7:Energydierences(inkcal/mol)forthevariouss peciesalongpathwayIcomputed attheCCSDandCR-CC(2,3)levelsusingtheBS1basissetandt wodierentschemesfor obtainingthecanonicalROHForbitals. CCSDCR-CC(2,3),A a CR-CC(2,3),D R b GS c R b GS c R b GS c A1 reactants-23.3-23.3-24.4-24.7-21.9-22.0 TS A1 reactants21.321.314.314.116.416.1 TS A1 A144.644.638.738.838.338.2 B1 reactants-36.1-36.1-38.6-38.7-36.1-36.2 TS B1 reactants-2.2-2.3-9.8-9.9-7.7-7.7 TS B1 B133.933.928.828.828.428.5 C1 reactants-54.3-54.3-57.0-57.1-54.5-54.7 TS C1 reactants-34.9-34.9-40.2-40.4-38.3-38.4 TS C1 C119.419.416.716.816.216.3 D1 reactants-117.1-117.1-119.7-119.8-117.1-117.0 products reactants-98.1-98.1-99.7-99.8-98.7-98.9 products D119.019.020.020.018.318.1 a EquivalenttotheCCSD(2) T approachofRef.[77]. b ResultsobtainedusingtheRoothaancanonicalizationsche meforROHForbitals[313]. c ResultsobtainedusingtheGuest-Saunderscanonicalizati onschemeforROHForbitals[314]. 121 UMP3-extrapolatedenergeticsagreetowithinlessthan0.5 kcal/mol.Inmostcases,includ- ing,forexample,therate-determining(TS A1 A1)barrier,whichmeasurestheamount ofenergyneededtogofromtheO 2 andmethoxymoleculescoadsorbedontheAu 8 particle totheinitialtransitionstateTS A1,andtheoverallreactionenergy,theagreementiseven better,withthedierencesbetweenthetrueandUMP n -basedextrapolatedCCSD/BS2 dataoscillatingaround1{2kcal/molorless.Sincewecurre ntlyhavenoaccesstothetrue CR-CC(2,3)/BS2data,wecannotmakeanalogouscomparisons ,but,inviewofthefact thattheconnectedtriplyexcitedclustersaccountforasma llfractionofthecorrelationef- fectsrelativetotheCCSDcontributions,itisrathersafet oassumethatsimilarlevelsof accuracyapplytotheCR-CC(2,3)/BS2energiesextrapolate dusingEq.(4.29),whichare presentedinTable4.8aswell.Therelativelysmalldieren cesbetweenthetrueCCSD/BS2 energiesandtheirUMP n -basedextrapolatedcounterpartsobtainedusingEq.(4.29 )auto- maticallyguaranteethatthesamesmalldierencesmustbeo bservedwhencomparingthe CR-CC(2,3),A/BS2andCR-CC(2,3),D/BS2energeticsextrap olatedusingEq.(4.29)with theircounterpartsdeterminedusingtheextrapolationsch emedenedbyEq.(4.28). 122 123 Table4.8:Comparisonsofenergydierences(inkcal/mol)f orthevariousspeciesalongpathwayIobtainedusingtwo dierentwaysofextrapolatingtheCC/BS2energetics, a representedbyEqs.(4.28)and(4.29). CCSDCR-CC(2,3),A b CR-CC(2,3),D CCSD c UMP2 d UMP3 d CCSD e UMP2 d UMP3 d CCSD e UMP2 d UMP3 d A1 reactants-16.6-13.6-16.9-18.0-15.0-18.3-15.3-12.3-1 5.7 TS A1 reactants28.531.730.521.324.523.323.326.525.3 TS A1 A145.045.347.439.339.541.638.638.841.0 B1 reactants-32.2-32.6-30.8-34.7-35.1-33.3-32.2-32.6-3 0.8 TS B1 reactants6.24.17.6-1.4-3.50.00.8-1.32.2 TS B1 B138.436.738.433.331.733.333.031.333.0 C1 reactants-47.4-51.6-45.4-50.3-54.5-48.2-47.8-52.1-4 5.8 TS C1 reactants-29.1-31.0-27.1-34.6-36.5-32.5-32.6-34.5-3 0.6 TS C1 C118.320.618.315.718.015.715.217.515.2 D1 reactants-114.8-117.1-112.8-117.4-119.7-115.5-114.6 -116.9-112.7 products reactants-97.9-98.9-97.0-99.6-100.6-98.7-98.6-99.7- 97.8 products D116.918.115.817.919.116.716.017.314.9` a AlloftheCCenergiesusedinthebasissetextrapolationsco rrespondtotheGuest-Saunderscanonicalizationschemefo rROHForbitals[314]. b EquivalenttotheCCSD(2) T approachofRef.[77]. c ResultsofthetrueCCSD/BS2calculations. d ExtrapolatedbyaddingthedierencebetweentheUMP n ( n =2 ; 3)energiesobtainedwiththeBS2andBS1basissetstotheapp ropriate CC/BS1energy,asinEq.(4.29). e ExtrapolatedbyaddingthedierencebetweentheCCSDenerg iesobtainedwiththeBS2andBS1basissetstotheappropriat eCR- CC(2,3)/BS1energy,asinEq.(4.28). Clearly,ifthelarger-basis-setCCSDcomputationsareao rdable,asisthecaseinthis study,wheretheaerobicoxidationofmethanolcatalyzedby theAu 8 nanoclusterisex- amined,theextrapolationoftheCR-CC(2,3)/BS2resultsth atcombinestheCCSD/BS1, CCSD/BS2,andCR-CC(2,3)/BS1data(Eq.(4.28))istherecom mendedapproach.How- ever,onemayencounterdicultieswithperforminglarger- basis-setCCSDcalculationsfor theanalogousreactionscatalyzedbylargerAu n particles,suchasAu 20 ,whichwasin- vestigatedusingDFTinRef.[299].UsingtheMP n -based( n =2 ; 3)extrapolation,asin Eq.(4.29),toobtainthelarger-basis-setCR-CC(2,3)info rmation,wheretheeectsofthe one-electronbasisontheCR-CC(2,3)energiesareextracte dfromtherelativelyinexpensive UMP2orUMP3calculations,mayoeragoodalternativetothe procedurebasedonEq. (4.28).SincewehaveaccesstothemoreaccurateCR-CC(2,3) /BS2-leveldataextrapolated usingthemoreaccurateEq.(4.28),werelyonthisequationi ntheremainingdiscussion, though,oneshouldkeepinmindthattheCR-CC(2,3)/BS2-lev elinformationobtainedusing thelessdemandingEq.(4.29)isnotmuchdierent. 4.3.4.1Coupled-clustercalculations AccordingtotheCCSD,CR-CC(2,3),A,andCR-CC(2,3),Dmeth odsandinagreementwith theexperimental[202]andpreviouslyobtainedtheoretica l[299,300]data,thereactionex- aminedhereisexothermicandtheconversionfromthemethox yspeciestoformaldehydevia transitionstateTS A1istheratedeterminingstep,independentofthebasisset employed inthecalculations.AsshowninTable4.6,ourbestCR-CC(2, 3),D/BS2estimates,obtained usingEq.(4.28),placetheinitialtransitionstateTS A1,correspondingtothehydrogen transferfromthemethoxyspeciestothemolecularoxygento formintermediateB1,at 124 23.3kcal/molabovetheisolatedreactants(38.6kcal/mola bovestructureA1representing O 2 andCH 3 O coadsorbedonAu 8 )andconsiderablyabovetheremainingtwotransition statescharacterizingpathwayI,namely,22.5kcal/molabo veTS B1and55.9kcal/molabove TS C1.AsshowninTable4.5,theuseofthesmallerBS1basislowe rstheinitialbarrier relativetotheisolatedreactantsto16.1kcal/mol,mostli kelybecauseofthelargerBSSE, butthe(TS A1 TS B1),and(TS A1 TS C1)dierencesof23.8and54.5kcal/mol, respectively,resultingfromtheCR-CC(2,3),D/BS1calcul ations,whicharenotaectedby BSSE,arealmostthesameasintheBS2-basedcomputations.W ecan,thus,conclude,with greatdealofcondence,thatafterpassingtheinitialbarr ierdenedbyTS A1,theoxidation ofmethanoltoformicacidcatalyzedbytheAu 8 particleisenergeticallyanoverall\down- hill"processthatproceedsviatwoadditionalbarriersloc atedconsiderablybelowTS A1, resultingintheproductsbeinglocatedalmost100kcal/mol belowthereactants.Allthree CCtheoriesusedforthisstudyprovidethesamepictureinth isregard. WhiletheenergiesofthevariousspeciesdeningpathwayIr elativetothereactants obtainedwiththeCCapproachesaresomewhatsensitivetoth ebasisset,withdierences betweentheestimatedBS2andcalculatedBS1datarangingfr omlessthan1kcal/molfor theproductsto ˘ 8kcal/molfortheTS B1transitionstate(mostlyduetotheBSSEeects, whichaecttheenergiesoftheisolatedCH 3 O ,O 2 ,andAu 8 reactants,particularlyinthe calculationsemployingthesmallerBS1basis),thekeyacti vationbarriers,denedasthe (TS A1 A1),(TS B1 B1),and(TS C1 C1)energydierences,especially(TS A1 A1),andtheoverallreactionenergydependrelativelylitt leonthebasisset.Forexample, theCR-CC(2,3),D/BS2estimatesofthe(TS A1 A1),(TS B1 B1),and(TS C1 125 C1)barrierheightsandtheoverallreactionenergybasedon Eq.(4.28)are38.6,33.0, 15.2,and-98.6kcal/mol,respectively.Theseshouldbecom paredtotheresultsoftheCR- CC(2,3),D/BS1calculations,whichgive38.2,28.5,16.3,a nd-98.9kcal/mol,respectively, forthesamequantities,inoverallgoodagreementwiththeC R-CC(2,3),D/BS2estimates. EventhemoreprimitiveestimatesoftheCR-CC(2,3),D/BS2e nergeticsrelyingonthebasis setdependenceextractedfromtheUMP2/BS1andUMP2/BS2ort heUMP3/BS1and UMP3/BS2calculations,Eq.(4.29),givesimilarvaluesoft he(TS A1 A1),(TS B1 B1),and(TS C1 C1)energydierencesandthereactionenergy(38.8,31.3,1 7.5,and- 99.7kcal/mol,respectively,intheUMP2-basedcaseand41. 0,33.0,15.2,and-97.8kcal/mol, respectively,whenusinginformationfromtheUMP3calcula tions;seeTable4.8). AllofthisimpliesthattheCR-CC(2,3),Dvaluesofthe(TS A1 A1),(TS B1 B1), and(TS C1 C1)barriers,whicharenotaectedbyBSSEandwhichdonotch ange muchwhenweswitchfromthesmallerBS1basissettothelarge rBS2basis,especiallyin the(TS A1 A1)and(TS C1 C1)cases,andthecorrespondingreactionenergy,which displaysaweakbasissetdependenceaswell,areparticular lyusefulforjudgingthevarious DFTfunctionalsconsideredinthisstudy.Someotherenergy dierencesalongpathwayI listedinTables4.5and4.6maybesomewhatlessrobustbecau seoftheirstrongerbasisset dependence,butthesophisticated abinitio leveloftheoryrepresentedinthisworkbythe CR-CC(2,3),D/BS2approach,whereweusearatherlargebasi ssetintheunderlyingCCSD calculationsandtheadvancedCR-CC(2,3)treatmentoftheh igh-ordercorrelationeects beyondCCSD,iscertainlyhighenoughinthepresentcasetob enchmarkDFT,wherethe variationintheresultsobtainedwithvariousfunctionals ismuchgreaterthanthatobtained 126 intheCR-CC(2,3)considerations.Aspointedoutaboveanda sshowninnumerousearlier applicationsandbenchmarkstudies,suchasthosereported inRefs.[51,93{95,98,99,105,290{ 292,323,324],theCR-CC(2,3)methodology,particularlyi tsvariantD,iscapableofproviding resultsinthechemicalaccuracy( ˘ 1{2kcal/mol)rangeforactivation,reaction,andbinding energieswhenbasissetsoftriple- qualitywithpolarizationanddiusefunctions,suchas BS2,areemployed.ObviouslytheCR-CC(2,3)resultsemploy ingthesmallerBS1basisare notasaccurateandmaycarrylargererrors,buttheycanstil lbeusefulinassessingtrends inDFT.Indeed,inagreementwiththeearliertestapplicati onsinvolvingCR-CC(2,3),such asthosepresentedinRefs.[98,99,323,324],theaverageun signeddierencebetweenthe triple- -levelCR-CC(2,3)/BS2anddouble- -levelCR-CC(2,3)/BS1data(whichinourcase isthesameastheaverageunsigneddierencebetweentheCCS D/BS2andCCSD/BS1 resultsbecauseoftheuseofEq.(4.28))obtainedusingthee nergiesofA1,TS A1,B1, TS B1,C1,TS C1,D1,andproductsrelativetothereactantsisabout5kcal /mol.The analogousdierencebetweentheCR-CC(2,3)/BS2andCR-CC( 2,3)/BS1resultsbasedon themostimportant(TS A1 A1),(TS B1 B1),and(TS C1 C1)barriersisonly2 kcal/mol.ThismeansthateventheCR-CC(2,3)/BS1calculat ions,whicharecertainlynot convergedwithrespecttotheone-electronbasisset,butst illoerawell-balancedtreatment ofdynamicalandnondynamicalcorrelationeectsthatthel ower-ordermethodsbasedon DFTdonotalwaysdescribe,maybequitevaluable.Tables4.5 and4.6indeedshowthat thisisthecase,withmostoftheDFTfunctionalsproducinge rrorsintherelativeenergies andvariationsintheresultsforalmosteverystructurealo ngpathwayIthatexceedbya widemarginthe2{5kcal/molaveragebasisseterrorscharac terizingtheCR-CC(2,3)/BS1 127 data.However,inexaminingtheDFTdata,wewillrelyonourb estCR-CC(2,3),D/BS2 energetics,whichusesalargerbasissetandcarrierssmall ererrors,toensureourconclusions aremoredenitive. AscanbeseeninTable4.5,theMUEvaluescharacterizingthe CCSDandCR-CC(2,3),A calculationsrelativetoCR-CC(2,3),D,whichareontheord erof2kcal/mol,aresimilar, supportingourearlierassertionthattheCR-CC(2,3),Dres ultscanberegardedasreasonably wellconvergedwiththemany-electroncorrelationeectsi nagivenbasisset.However, asisgenerallythecase,theCCSDapproachaloneisnotsuci entlyaccuratetoprovide areliabledescriptionoftheactivationbarriers.Indeed, thedierencesbetweentheCR- CC(2,3),D/BS1andCCSD/BS1activationenergiescharacter izingpathwayIrangefrom about3to6kcal/mol,withthelargestdierencebeingobtai nedforTS A1andTS B1( ˘ 5 6kcal/mol).Furthermore,asobservedinotherapplication s,theCCSDapproach,which neglectsconnectedtripleexcitations,overestimatesthe activationenergies.Itissucientto useCCSDtoobtainanaccuratedescriptionoftheA1,B1,andC 1intermediatesandthe overallreactionenergy(thedierencesbetweenCR-CC(2,3 ),DandCCSDareinthiscase ontheorderof1kcal/molorsmaller),butoneneedstoincorp orateconnectedtriplesinthe CCcalculationstoobtainaconsistentlyaccuratedescript ionoftheentirereactionpathway. ThisisseenbycomparingtheCR-CC(2,3),AandCR-CC(2,3),D datacalculatedwiththe BS1basisset,summarizedinTable4.5.AlthoughtheMUEvalu escharacterizingtheCCSD andCR-CC(2,3),AcomputationsrelativetoCR-CC(2,3),Dar esimilar,thereisagreatdeal ofconsistencybetweentheindividualCR-CC(2,3),AandCR- CC(2,3),Denergies,diering byabout1{2kcal/molatallstationarypoints.Whenwerepla ceCR-CC(2,3),AbyCCSD, 128 comparingwithCR-CC(2,3),D,thisisnolongerthecaseand, asalreadymentioned,the agreementisnolongerasgood. 4.3.4.2BenchmarkingDFTagainstthereferenceCR-CC(2,3) ,Ddata WenowturntothecomparisonofourbestCR-CC(2,3),Dresult swiththoseobtainedfrom thevariousDFTfunctionals,focusingontheresultsobtain edwiththelargerBS2basisset, assummarizedinTable4.6andFig.4.5.Inourpublishedwork [181],weconcludedthat amongtheseveralrepresentativeDFTmethodsconsideredin Ref.[181],theM06approach usedintheearliercalculationsforthesamereactionpathw ay[300]andthehighlypopular hybridB3LYPfunctionalaregenerallymostaccurate.Indee d,theoverallMUEandNPE valuesrelativetoCR-CC(2,3),D/BS2usingallstationaryp ointsalongpathwayIare6.1 and6.3kcal/mol,respectively,intheM06caseand5.2and11 .9kcal/mol,respectively,for B3LYPwhichisabetterperformancecomparedtoallotherDFT calculationsincludedin Ref.[181](i.e.,alllistedinTables4.5and4.6otherthan ! -B97X-D).Withtheadditionaland previouslyunpublished ! -B97X-Dresultsincludedinthisdissertation,thesituati onseems tobechanging,sincetheMUEandNPErelativetoCR-CC(2,3), D/BS2characterizing ! - B97X-Dof3.1and13.5kcal/mol,respectively,lookcompeti tive,whencomparedtoM06 andB3LYP,butthisisonlytheinitialimpression.Whilethe MUEforthe ! -B97X-Dresults appearstobebetterthanthosefortheM06andB3LYPapproach es,withanNPElargerthan NPEscharacterizingM06andB3LYPandcloseto14kcal/mol,i tishardtomaketheclaim that ! -B97X-Dperformsbetter.Obviously,noneoftheDFTmethods lookgoodcompared with,forexample,theCCSDMUEvalueof2.1kcal/mol,butthe M06and,especially, B3LYPfunctionalsperformsubstantiallybetterthanthose characterizingtheremaining 129 DFTcalculationspresentedinTable4.6.Indeed,bothhybri dfunctionals,M06andB3LYP, particularlythelatter,performwellindescribingthekey (TS A1 A1),(TS B1 B1),and (TS C1 C1)barrierheights,whichare35.8,31.2,and12.2kcal/mol ,respectively,when theB3LYP/BS2approachisused,and32.3,27.7,and12.3kcal /mol,respectively,when theM06/BS2methodisemployed,inreasonablyaniceagreeme ntwiththecorresponding CR-CC(2,3),D/BS2benchmarkdata,whichare38.6,33.0,and 15.2kcal/mol,respectively, forthesamethreebarrierheights.Thisisreectedintheco nsiderableimprovementsin theMUEandNPEvaluesrelativetoCR-CC(2,3),D/BS2resulti ngfromthecorresponding B3LYPandM06calculations,whicharereducedto2.6and1.2k cal/mol,respectively,in theformercase,and4.9and3.4kcal/mol,respectively,int helattercasewhenwelimit ourselvestothe(TS A1 A1),(TS B1 B1),and(TS C1 C1)activationenergies.The B3LYPdescriptionofthesethreebarriers,whichisbettert hanthatprovidedbyCCSD,is especiallyencouraging,placingtheinitial(TS A1 A1)barrierat35.8kcal/mol,thatis only2.8kcal/molapartfromourbestCR-CC(2,3),D/BS2esti mate;theCCSDapproach overestimatesthesamebarrierbyabout6kcal/molandM06un derestimatesitbymore orlessthesameamount.Thepreviouslyunpublished ! -B97X-D/BS2resultsforthethree barrierheights,(TS A1 A1),(TS B1 B1),and(TS C1 C1),are45.2,35.0,and 14.1kcal/mol,respectively,i.e.,theyarenotnearlyasgo odasthoseobtainedwithB3LYP. Assuch,wedonotincludethisfunctionalintherestofthisd iscussionandfocusonthe functionalsincludedinourpublishedstudy[181]. ThereisalsogoodagreementbetweentheB3LYP/BS2andCR-CC (2,3),D/BS2energies oftransitionstatesTS A1,TS B1,andTS C1relativetotheisolatedreactants,whichare 130 20.9,4.9,and 27 : 3kcal/mol,respectively,intheformercaseand23.3,0.8,a nd 32 : 6 kcal/mol,respectively,inthecaseofthelatterapproach, indicatingthattheB3LYPfunc- tionaliscapableofprovidingareliabledescriptionofthe bondrearrangementsinvolvedin catalyticreactionsonmetallicnanoparticlesofthetyper epresentedbytheoxidationpro- cesscatalyzedbygoldclustersexaminedinthisstudy.Itlo oksasthoughtheenergyofthe TS B1transitionstaterelativetothereactantsresultingfro mtheB3LYP/BS2calculations isinlargererror,butwemustkeepinmindthatB3LYPistheon lyfunctionalinTable4.6, otherthan ! -B97X-Dnotincludedintheourpublishedstudy[181],andwh ichincapableof properlycharacterizingthekey(TS A1 A1)barrierthatproducesasmallpositivebarrier inthiscase,withtheCR-CC(2,3),D/BS2approachdoingthes ame.M06doesnotseemto workwellinthiscase,butweshouldnotreadtoomuchintoit, sincetheM06/BS2result forthephysicallymoresignicant(TS B1 B1)energydierence(27.7kcal/mol)isstillin reasonableagreementwiththeCR-CC(2,3),D/BS2result(33 .0kcal/mol).Ofallfunction- alstestedhere,theonlyonethatcanimprovetheoverallpat hwayIdescriptionbyM06is B3LYP. Interestingly,ourndingthattheB3LYPandM06functional sprovidethebestdescrip- tionofthekeyactivationbarriersrepresentedbythe(TS A1 A1),(TS B1 B1),and (TS C1 C1)energydierencesremainstruewhenwecomparetheDFTan dCR-CC(2,3),D resultsobtainedwiththesmallerBS1basisset,showninTab le4.5.Thisisaconsequence of(1)thewell-establishedfactthatDFTconvergesfastwit hrespecttothebasissetand (2)thepreviouslydiscussedobservationthattheCCvalues ofthesethreeenergydierences areessentiallyfreefromBSSEandlargelyinsensitivetoth esizeofthebasisemployedin 131 theCCcalculations.AspointedoutaboveandshowninTable4 .8,onecansuccessfully correcttheresultsofthesmall-basis-setCR-CC(2,3)calc ulationsbycarryingoutadditional UMP2orUMP3calculationswithalargerbasissetandbyextra polatingthelarger-basis-set CR-CC(2,3)-leveldatausingEq.(4.29)withouttheneedtop erformanylarger-basis-setCC computations.Theseremarksmaybeusefulinfutureworkonc atalyticsystemslargerthan theoneexaminedhere,wherelarger-basis-setCCcalculati onsmaynolongerbefeasible, butacombinationofsmaller-basis-setCR-CC(2,3)andlarg er-basis-setUMP n ( n =2 ; 3) computations,run,forexample,onmultiplecores,maystil lbemanageable,allowingoneto testvariousDFTapproachesinaverymeaningfulmanner. ConsideringthegenerallygoodagreementbetweentheCR-CC (2,3),D,B3LYP,andM06 energeticscharacterizingpathwayI,wecancertainlyconc ludethatthepreviouslyreported M06calculationsonthemechanismofmethanoloxidationtof ormicacidontheAu 8 and Au 20 particles[299,300]canberegardedasquitereliable,alth oughitwouldbedesirable torepeatthecalculationsforthelargercatalyticsystemi nvolvingAu 20 usingtheB3LYP approach,whichhasherebeenshowntoimprovetheresultsfo rtheactivationbarriers characterizingtheanalogousAu 8 -containingsystem.Unfortunately,noneoftheotherDFT functionalsexaminedinthepresentstudyoersatisfactor yperformance.Forexample, althoughthepureGGA-typeanalogofM06,denotedasM06-L,i scharacterizedbyan overallMUEvaluerelativetoCR-CC(2,3),Dof6.9kcal/mol, whichisonlyslightlyworse thanthatobtainedwithM06whentheBS2basisisemployed,it lowerstheactivationenergies relativetoM06by6kcal/molwhenthe(TS A1 A1)energydierenceisconsideredand7.2 kcal/molinthe(TS B1 B1)case.Asaresult,theMUEvaluerelativetoCR-CC(2,3),D 132 characterizingtheM06-Lcalculationsforthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)barriersincreasebyfactorsof2relativetoM06whenthe BS2basissetisemployed andmorethan2inthecaseoftheBS1basis.TheoverallNPEval uerelativetoCR- CC(2,3),Dbasedonallofthestationarypointsinvolvedinp athwayIcharacterizingthe M06-LcalculationswiththeBS2basisset,of22.3kcal/mol, representsanincreasebya factorof3.5relativetoM06and2relativetoB3LYP.Similar remarks,asdiscussedabove, applytothelong-rangecorrectedhybrid ! -B97X-Dresults,whoseMUEissmallerthanthat forbothB3LYPandM06whentheBS2basissetisused,butitsNP Evaluesaresubstantially largerthanthosecharacterizingM06andB3LYP,especially whenwefocusonthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)activationbarriers. Similarbehaviorisobservedwhenotherpurefunctionalsar econsidered.Theoverall MUEvaluesrelativetoCR-CC(2,3),D/BS2characterizingth eBP86/BS2andTPSS/BS2 calculations,of7.4and8.5kcal/mol,respectively,donot appeartobemuchhigherthanthe M06result(6.1kcal/mol),butthisismisleading,sincethe correspondingNPEvaluesbased onallstationarypointsalongpathwayIare27.8and30.8kca l/mol,respectively,indicating averyinaccuraterepresentationofpathwayIbytheBP86and TPSSfunctionals.This againparticularlytruewhenwelookatthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)barriersresultingfromtheBP86andTPSScalculations, whichareunderestimated by ˘ 13{15kcal/molinthecaseofthe(TS A1 A1)and(TS B1 B1)dierencesand about5kcal/mol(TPSS)or9kcal/mol(BP86)whenthe(TS C1 C1)energydierenceis examined.Asaresult,theMUEvaluesrelativetoCR-CC(2,3) ,DcharacterizingtheBP86 andTPSScalculationsforthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)activation 133 barriersincreaseto12.5and10.7kcal/mol,respectively, whentheBS2basisisemployed (12.2and10.1kcal/mol,respectively,whenoneusesBS1).S imilartrendsareobservedwhen welookattheenergiesofthetransitionstatesTS A1,TS B1,andTS C1relativetothe isolatedreactants,whicharemuchtoolowaswell.Forexamp le,theactivationbarriers relativetothereactantscorrespondingtothetransitions tateTS A1,whichdenestherate- determiningstepinvolvingtheconversionofthemethoxysp eciestoformaldehyde,are1.9 and3.5kcal/molfortheBP86andTPSScalculations,respect ively,whentheBS2basisset isemployed.Theseshouldbecomparedwiththevaluesof23.3 kcal/molobtainedinthe CR-CC(2,3),D/BS2calculationsand20.9kcal/molobtained withB3LYP/BS2. OnecanimprovetheaboveresultsobtainedwithpureGGAsthr oughtheuseofthe popularGrimme'sempiricaldispersioncorrections,asint heB97-Dcase,wherethe(TS A1 reactants)and(TS A1 A1)energydierencesincreasefrom1.9and23.2kcal/molwh en theBP86/BS2methodisusedto11.3and31.7kcal/mol,respec tively,whentheB97-D/BS2 approachisexploited,bringingtheresultsclosetotheM06 level,buttheoveralldescription ofpathwayIbyB97-Disnotasgoodasthatprovidedbythehybr idM06andB3LYP functionals.Forexample,whileimprovingthe(TS A1 A1)and(TS B1 B1)barriers resultingfromtheBP86calculations,theB97-Dfunctional isincapableofchangingthe poorBP86resultsforthe(TS C1 C1)energydierence.B97-Dalsoworsenstheoverall reactionenergyobtainedwithBP86,whichis 95 : 8kcal/molaccordingtotheBP86/BS2 calculationsand 89 : 2kcal/molaccordingtothecorrespondingB97-Dcomputatio ns,where theCR-CC(2,3),Dapproachgives 98 : 6kcal/molwhentheBS2basissetisemployed.The overallNPEvaluerelativetoCR-CC(2,3),D/BS2characteri zingtheB97-D/BS2calculations, 134 of21.4kcal/mol,isalmostasbadasthatobtainedwiththeM0 6-Lscheme,althoughone observesanimprovementcomparedtothepureBP86functiona l,whosecorrespondingNPE is27.8kcal/mol.ThesameholdstruewhenwecomparetheMUEv aluescharacterizingthe B97-Ddescriptionofthe(TS A1 A1),(TS B1 B1),and(TS C1 C1)barrierswith theBS1andBS2basissets,whicharebetterthanthoseprovid edbyBP86,butnotasgood astheresultsoftheM06andB3LYPcalculations. Insummary,wehaveshownthatourhighest-levelCR-CC(2,3) calculationsconrmedthe earlierproposals[299,300]thattheoxidationofmethanol toformicacidonAu 8 proceeds exothermicallyandthattheratedeterminingstepforthere actionistheinitialconversion ofthemethoxyspeciestoformaldehyde.Theinitialrate-de terminingtransitionstate,which correspondstohydrogentransferfromthemethoxyspeciest othemolecularoxygen,is placedabout20kcal/molabovethereactants,lessthan40kc al/molabovetheO 2 and CH 3 O speciescoadsorbedonAu 8 ,andconsiderablyabovetheremainingtwotransition statesalongthereactionpathway.Thepreviouslyexploite d[299,300]M06hybridfunctional showsreasonableagreementwithCR-CC(2,3),butB3LYPimpr ovesthedescriptionofthe activationbarrierscomparedwiththeM06approach,bringi ngtheMUEandNPEvalues relativetoourbestCR-CC(2,3),D/BS2estimatescharacter izingthekeyactivationbarriers from4.9and3.4kcal/mol,respectively,whentheM06method isemployedto2.6and1.2 kcal/mol,respectively,whenoneusesB3LYP.Clearly,exam iningmethanoloxidationto formicacidonAu 20 usingCR-CC(2,3)bytheapproachinwhichthelarger-basis- setCR- CC(2,3)dataareextrapolatedbycombiningthesmaller-bas is-setCR-CC(2,3)resultswith thelarger-basis-setCCSD,UMP2,orUMP3information,asha sbeendonehere,orbytaking 135 advantageoftherecentlydevelopedmultilevellocalcorre lationCR-CC(2,3)methodology [101],whichisapplicabletolargerreactivesystemsconta iningtransition-metalatoms[99], wouldbehelpfultoo.Wehopetheinformationprovidedherei nwillbeusefulforfuture theoreticalstudiesofreactiveprocessescatalyzedbytra nsition-metalnanoparticles. 4.4Coupled-ClusterandMultireferenceConguration InteractionStudiesoftheLow-LyingElectronic Statesof1,2,3,4-Cyclobutanetetraone 4.4.1Backgroundinformationandscopeofthework 1,2,3,4-cyclobutanetetraone(C 4 O 4 )isasmall,butsurprisinglycomplexmoleculethatposes severalchallengesfortheoryandexperiment.Theearlythe oreticalsuggestionthatthis D4 h -symmetricspecieshasatripletgroundstateoftheB 2 u symmetry,withaclosed-shell singletoftheA 1 g symmetrylocatedonlyafewkcal/molhigher[325{328],hasr ecentlybeen strengthenedthroughphotodetachmentexperiments[329,3 30].Therehave,however,been seriousproblemsinobtainingthisresultcomputationally ,sinceeventhemostsophisticated electronicstructuretreatmentsencountersignicantdi cultieswithdeterminingwhether thegroundstateofC 4 O 4 isatripletwithnine ˇ electronsorasingletwitheightorten ˇ electrons[325{327,331](cf.Fig.4.6fortheconguration sdeningthefourlowest-energy electronicstatesofC 4 O 4 ). Moleculeswithsmallgapsbetweenthelowestsingletandtri pletstateshavehistorically 136 beenachallengeforbothexperimentalandtheoreticalmeth ods.Forexample,themethylene biradical,whichisoneofthemostcelebratedcasesinthisa rea,causedmuchcontroversy, especiallyinthe1960sand1970s,duenotonlytothesmallne ssofitssinglet-tripletgap, butalsobecauseofthedirectchallengebyvarioustheoreti calapproaches[332{334]tothe initialHerzberg'sexperimentalndings[335]suggesting thatthetripletgroundstatehada lineargeometry(cf.Refs.[177,178,192,271,272,336{353 ]forselectedexamplesof abinitio calculationsofthegroundandexcitedstatesofmethylene, whichiscertainlynotcomplete norexhaustive;see,also,Refs.[354,355]forpersonalacc ountsbySchaeferandHarrison, whomadepioneeringcontributionstocomputationalstudie sofmethylene).Thus,even fortinymolecules,suchasCH 2 ,determiningtheorderingofclose-andlow-lyingelectron ic statescanbeachallengeforsingle-andmultireferencequa ntumchemicalapproaches.C 4 O 4 , whichhasmanymoreelectronsandeightnon-hydrogenatomsa ndwhichisrelatedtoother similarlychallenginglargerorganicmolecules,suchaste trakis-annelatedcyclooctatetraene thathasbeenpredictedtohaveasmallgapbetweenthelowest singletandtripletelectronic states[356],hasatinysinglet-tripletgapsmallerthanth atinmethylene,whilehavingnear- degenerateHOMOandLUMO,resultingintwoadditionalsingl etstateswhicharealsovery closeinenergy.Thus,therearefourlow-lyingnearlydegen erateelectronicstatesin1,2,3,4- cyclobutanetetraonethatonehastodealwith,whichisamaj orchallengeforquantum chemistry.Followingthepreviousstudies[325{328],thes estatesaredenotedas8 ˇ ( 1 A 1 g ),a closed-shellsingletarisingfromthe sigma -type b 1 g HOMObeingdoublyoccupied,having eight ˇ electronsinitselectronconguration,10 ˇ ( 1 A 1 g ),anotherclosed-shellsingletwhere the ˇ -type a 2 u LUMOofthe8 ˇ ( 1 A 1 g )referenceisdoublyoccupiedandthe sigma -type b 1 g 137 HOMOisempty,havingten ˇ electronsinitsdominantelectronconguration,9 ˇ ( 3 B 2 u ),the tripletstateoriginatingfromthesinglyoccupiedHOMOand LUMO,havingnine ˇ electrons initsleadingelectronconguration,and9 ˇ ( 1 B 2 u ),anopen-shellsingletcounterpartofthe 9 ˇ ( 3 B 2 u )state,havingnine ˇ electronsinitsdominantelectronconguration(seeFig.4 .6). Determiningthesinglet-tripletgapinC 4 O 4 alongwiththeorderingofthefourlowest- energyelectronicstateshasproventobeverydicultforth evarioustheoreticalandex- perimentalstudies[325{331,357,358].TheinitialDFT,CA SSCF,CASPT2,andCCSD(T)- typecalculationsgavestronglyvariedresultsdependingo nthegeometryemployedand themethodandbasissetthatwereused[327,328].Giventhec hallengerepresentedby thelow-lyingelectronicstatesofC 4 O 4 ,whichrequirearobustandwell-balanceddescrip- tionofdynamicalandnondynamicalcorrelationeects,thi sisnotsurprising.TheDFT methodology,asimplementedinthepopularquantumchemist rycodes,isknowntohave problemswithelectronicneardegeneracies,CASSCFdoesno tcapturedynamicalcorrela- tions,CASPT2over-stabilizessingletstateswithastrong biradicalcharacterrelativetothe correspondingsingle-referenceclosed-shellsingletand high-spintripletstates,andCCSD(T), whichprovidesanaccuratedescriptionofdynamicalelectr oncorrelationeectsfornonde- generategroundstatesofmoleculesneartheequilibriumge ometries,failswhenappliedto biradicalsandothercasesofelectronicneardegeneracies ,wherenondynamicalcorrelation eectsbecomemoresignicant.Thesubsequentnegativeion photoelectron(NIPE)spec- troscopy[329]experimentdeterminedthegroundstatetobe the9 ˇ ( 3 B 2 u )state,beingabout 6 : 27 0 : 5kJ/mollowerinenergythanthelowestsingletstate,inagr eementwithsomeof thetheoreticalresults,butnoneoftheexperimentalenerg ydierencesandevenbesttheo- 138 Figure4.6:Electrondistributionsamongthenearlydegene rateHOMOandLUMOmaking upthefourlowest-energyelectronicstatesofC 4 O 4 . 139 reticalpredictionswereinquantitativeagreement.Furth ermore,theoriginalexperimental interpretation[329]assignedaseriesofpeaksseenintheN IPEspectrumofC 4 O 4 tothe open-shellsinglet9 ˇ ( 1 B 2 u )state[329],whileinafollow-uppaper,whichincludedade tailed Franck-Condonanalysis[330],thepeakswerereassignedas avibrationalprogressioninthe lowest-energy 1 A 1 g state,namely8 ˇ ( 1 A 1 g ),showingthateventheexperimentalinterpreta- tionmaybeuncertaininsomeaspects. Morerecently,totrytogiveamoredenitiveresultforthes inglet-tripletgapinC 4 O 4 ,the high-levelDEA-andDIP-EOMCCapproaches,inwhichtheEOMo peratorsattachingtwo electronstoorremovingtwoelectronsfromthecorrespondi ngclosed-shellcoresaretruncated at3 p -1 h and3 h -2 p components,respectively,wereappliedtothe8 ˇ ( 1 A 1 g )and9 ˇ ( 3 B 2 u ) statesinC 4 O 4 [331].TheauthorsofRef.[331]foundthateventheDEA-EOMC C(3 p -1 h ) andDIP-EOMCC(3 h -1 p )approximationswerenotsucienttoprovidereliableresu lts,with the8 ˇ ( 1 A 1 g )beingpredictedtobethegroundstateinsomecases.Thisis notentirely surprising,though.FortheDEA-andDIP-EOMCCmethodstopr ovidestableresults, especiallyinsituationsasoomplexasthatcreatedbyC 4 O 4 ,oneneedstoincludesome formof4 p -2 h and4 h -2 p correlations,respectively,ashasbeenshownforothercha llenging systemswithlow-lyingnearlydegenerateelectronicstate s[177,178].Thiswouldexplain dicultiesinstabilizingtheresultsoftheDEA-andDIP-EO MCCcalculationsreported inRef.[331],whichdidnotinclude4 p -2 h and4 h -2 p excitations,butwewouldhaveto provethisviadirectcomputationsusing,forexample,thea ctive-spaceDEA-EOMCC(4 p -2 h ) andDIP-EOMCC(4 h -2 p )approachesdevelopedinourgroup[177,178].OurpresentD EA- EOMCC(4 p -2 h )andDIP-EOMCC(4 h -2 p )codeshaveapilotcharacterand,assuch,are 140 hardtoapplytothelarger-basissetcalculationsforC 4 O 4 ,butthismaybeaninteresting directiontopursueinfuturework. AllofthisshowsthatC 4 O 4 isaremarkablychallengingmoleculeforboththeoryand experiment.Asfarastheoryisconcerned,onefamilyofCCme thods,whichhasbeen showntoprovideabalanceddescriptionofdynamicalandnon dynamicalcorrelationeects, particularlyinthecaseofbiradicalsandnear-degenerate electronicstates,isthepreviously discussedCR-CChierarchy,includingmethodssuchasCR-CC (2,3),whichweusedinthe earliersectionsofthisdissertation.Itisrelevanttomen tioninthiscontextthattheCR- CC(2,3)approachhasalreadybeenusedtodeterminethesing let-tripletgap[95]and,more recently,usingitsCR-EOMCC(2,3)extension,fourlowest- energystatesofmethylene[192], achievinggreatsuccess.Thus,aspartofthisdissertation ,weexaminedthefourlowest- energystatesofC 4 O 4 ,namely,8 ˇ ( 1 A 1 g ),9 ˇ ( 3 B 2 u ),9 ˇ ( 1 B 2 u ),and10 ˇ ( 1 A 1 g ),investigating theroleofbothbasissetandnucleargeometrytodeterminet heirordering,byemployingthe CR-CC(2,3),AandCR-CC(2,3),Dapproachesthatseektomini mizethedierencebetween theexact,fullCI,andCCSDenergiesbyaddingrobustnonite rativecorrectionsduetotriply excitedclusterstotheCCSDenergyexploitingmomentsofCC SDequationsdiscussedin Section4.1.WethencomparetheCR-CC(2,3)resultswithmul tireferenceCIcalculations usingthepopularMRCI(Q)method[359,360],theactive-spa ceCCSDtapproach[62{67,361], whichisknowntopreciselyreproducerelativeenergiesofi tsparentfullCCSDTcalculations, andtheavailableexperimentaldata[329]. WehavealreadydiscussedtheCR-CC(2,3)methodologyinSec tion4.1,butwehavenot saidmuchaboutCCSDt.Wedothisnow.TheCCSDtmethodisbase donthemoregeneral 141 active-spaceCCideasdevelopedinRefs.[62{64,66,69]for thegroundstateand[42{44] forexcitedstates(forreviewsoftheactive-spaceCC/EOMC CtheoriesandtheirEA/IP, DEA/DIP,etc.extensions,andadditionalreferences,seeR efs.[54,157{159,177,178]). CCSDtallowsonetoobtainmolecularelectronicstatesthat overallareofthefullCCSDT qualityatasmallfractionofthecomputationalcostinvolv edintheCCSDTcalculations. Intheactive-spaceCC/EOMCCmethods,thespin-orbitalsar edividedintofourdisjoint groups,namely,(i)coreorinactiveoccupiedspin-orbital s,labeledbyboldlower-caseletters i , j , ::: ,(ii)activeoccupiedspin-orbitals,labeledbyupper-cas eboldletters I , J , ::: , (iii)activeunoccupiedspin-orbitals,labeledbyupper-c aseboldletters A , B , ::: ,and(iv) virtualorinactiveunoccupiedspin-orbitals,labeledbyb oldlower-caseletters a , b , ::: (see Fig.4.7).Wecontinuetodesignatetheoccupiedandunoccup iedorbitalsinthereference determinant j i bytheitaliclower-casecharacters i , j , ::: and a , b , ::: ,respectively,ifthe active/inactivecharacterofthespin-orbitalsisnotspec ied.Wethenusetheabovede- compositionofspin-orbitalsintoactiveandinactivesubs etstocapturethemostimportant higher-than-doubleexcitations,suchastriples(CCSDt)o rtriplesandquadruples(CCS- Dtq),reducingtheoverallcostofhigh-levelCCcomputatio nswiththoseexcitations,such asCCSDTorCCSDTQ.InthespeciccaseofCCSDt,whichisanap proximationtofull CCSDT,theone-andtwo-bodyclusteroperators, T 1 and T 2 ,aredenedthroughexactly thesamesetofamplitudesasusedinthestandardCCSDcalcul ations,discussedinSection 4.1,withtheirlabelsrunningoverallspin-orbitals.Inot herwords, T 1 and T 2 aretreated fully.Adierentapproach,however,isadoptedwhenitcome sto T 3 clusters(triples),which inCCSDtarereplacedbytheir t 3 (\littlet")counterpartsdenedthroughasmallsubset 142 ofalltriplyexcitedclusteramplitudesthatcarryatleast oneactiveoccupiedandatleast oneactiveunoccupiedspin-orbitalindices,toobtainthef ollowingdenitionofthecluster operator T : T (CCSDt) = T 1 + T 2 + t 3 = X i;a t i a a a a i + X i>j;a>b t ij ab a a a b a j a i + X I >j>k;a>b> C t I jk ab C a a a b a C a k a j a I ; (4.30) where a p ( a p )arethesamecreation(annihilation)operatorsasdenedp reviouslyassociated withthespin-orbitalbasisset fj p ig .Theground-statewavefunctionisthenwrittenas j (CCSDt) 0 i = e T (CCSDt) j i andwesolveasetofnonlinearequationssimilartoCCSD, namely, h a i j H (CCSDt) j i =0, h ab ij j H (CCSDt) j i =0,and h ab C I jk j H (CCSDt) j i =0,where H (CCSDt) isthesimilaritytransformedHamiltonianofCCtheorywrit tenforthecluster operator T (CCSDt) and j a i i , j ab ij i ,and j ab C I jk i arethesingly,doubly,andtriplyexcited determinants,respectively,correspondingtothecluster amplitudesdening T (CCSDt) .As onecansee,theCCSDtequationsformasubsetofthefullCCSD Tequationsandbecome equivalenttothelatterequationswhenallspinorbitalsar echosenasactive. ThekeyideaoftheCCSDtcalculationsisthatonedoesnothav etousetheentireset oftriplyexcitedamplitudes t ijk abc anddeterminants j abc ijk i toobtaintheresultsofthefull CCSDTquality,whichthedenitionof t 3 operatorshowninEq.(4.30)guarantees(as showninnumerousapplications).Becauseoftheconsiderab lereductioninthenumberof triplesusedbyCCSDt,themostexpensivestepsofCCSDtscal eas N o N u n 2 o n 4 u ,where n o ( N o )and n u ( N u )arethenumbersofall(active)occupiedandunoccupiedspi n-orbitals usedinthepost-SCFcalculations.Sinceonetypicallyhast hat N o 0, andanapproximateinverseHessian H 0 computethethegradient 5 f k ofthefunction beingminimized f ,where k isthenumberofthecurrentoptimizationcycle. 2.If jj5 f k jj > computethesearchdirection d k = H k 5 f k andthenewsetof parameters x k +1 = x k + k d k where k isasuitablychosenparameterwitheithera xedorvariablevalue,dependingontheimplementationand canbedeterminedfrom, forexample,aline-searchortrust-regioncalculationast heminimizerof f ( x k + k d k ). 3.UpdatetheapproximateinverseHessian, H k usingthefollowing,oroneofitsother 171 equivalentvariations H k +1 =( 1 ˆ k s k y T k ) H k ( 1 ˆ k y k s T k )+ ˆs k s T k (5.4) where s k = x k +1 x k , y k = 5 f k +1 5 f k and ˆ =( y T k s k ) 1 . 4.Iterateuntil jj5 f k jj < . Oneofthereasonsforitspopularityisthatitpreservesthe symmetricand(semi)positive denitepropertiesof H k (e.g., x T H k x> 0 ; 8 x 2< n ,since H k issymmetric),suchthatits eigenvaluesremainpositiveandrealthroughouttheoptimi zationprocess.Thesymmetry andpositivedenitenessarepreservedthroughtheuseofar ank-twoupdate,Eq.(5.4). Asimplerrank-onematrixupdateto H k ,whichmaintainsitssymmetrythoughnot necessarilyitspositivedeniteness,wasproposedinRef. [389{391]calledthesymmetricrank- one(SR1)approach.Ithasbeenshownthatinmanycasesthisv ersionoftheSR1algorithm outperformstheBFGSmethod[389{392]andthat,unlikewith BFGS,thesequenceof matricesgeneratedbytheSR1process H k convergetothetrueHessian H undercertain conditions[391].Inspiteofthemanyexamplesshowingthat theSR1algorithmrarely violatesthepositivedenitenessoftheHessianwithitsup date,ithasfailedtogarner asmuchuseorattentionastheBFGSapproach.Ingeometryopt imizationsofchemical systemstherehasappearedonestudy,inwhichtheSR1approa chwasemployed[393]using theHartree-Fockleveloftheory.Particularlyforthemole culeswithalargernumberof degreesoffreedom,itwasshownthattheSR1methodcandrast icallyreducethenumber ofoptimizationcyclesrequiredtolocateastationarypoin tcorrespondingtoaminimumon 172 thepotentialenergysurfaceofthemolecule.Therehasbeen nofollowupbytheauthorof Ref.[393]andnooneelsehaspursuedthistopicintermsofge ometryoptimizationsfor chemicalspeciesandsystems. WhiletheoriginallyproposedSR1algorithm[389{391]that interestsusinthisthesis workdoesnotpreservethepositivedenitenatureof H k ithasbeenshownthatdeviations frompositivedenitenessintheapproximateupdateto H k arerelatedtothecondition numberofthematrixandpositivedenitenesscantherefore berestoredusinganoptimal scalingfactor[394,395].Usingthisidea,theauthorsofRe f.[396]proposedwhattheycalled ascaled\memoryless"SR1method,thoughitisactuallyalim ited-memory(LM)algorithm asshownbelow,inwhichthepositivedenitenatureof H k ispreservedthroughoutthe optimization.InordertomaketheirSR1approachaLMmethod theyusedthesimpletrick ofreplacingtheinverseHessian H k bytheidentitymatrix 1 multipliedbyanappropriate scalingfactor k ,whichresultsintheoriginalupdate H k +1 = H k + ( s k H k y k )( s k H k y k ) T y T k ( s k H k y k ) (5.5) simplifyingto H k +1 = k 1 + ( s k k y k )( s k k y k ) T y T k ( s k k y k ) (5.6) where k = s T k s k s T k y k 2 4 s T k s k s T k y k ! 2 s T k s k y T k y k 3 5 1 = 2 (5.7) andinsteadofhavingtostoretheinverseHessianmatrixofs ize n ( n +1) 2 oneonlyneedstore 173 Table5.2:ComparisonoftheLM-SR1andBFGSquasi-Newtonal gorithmsforCR- CC(2,3),D/TZVPgeometryoptimizations. a OptimizationCycles Molecule(symmetry)DoF b LM-SR1BFGS Benzene( D 6 h )297 Cyclopentadiene( C 2 v )71012 Acetone( C 2 v )9714 Formamide( C s )12315 E-Butadiene( C 2 h )101224 all-E-Hexatriene( C 2 h )142050 < Pyridine( C 2 v )11841 Thymine( C s )29628 a Asmanycoresassingle-pointenergiesneededtocomputethe numericalgradientwereused(2DoF+1). b Numberofdegreesoffreedom(DoF)optimized theseveralvectorseachofsize n .Thestepdirection d k = H k 5 f k = H k g k thenbecomes d k = k 1 g k + k 1 s T k 1 g k k 1 y T k 1 g k y T k 1 s k 1 k 1 y T k 1 y k 1 ! y k 1 s T k 1 g k k 1 y T k 1 g k y T k 1 s k 1 k 1 y T k 1 y k 1 ! s k 1 : (5.8) UsingthesamegeneraloutlineasfortheBFGSalgorithmabov ewiththe H k and d k updates beingreplacedbythoseinEqs.(5.6)and(5.8)theauthorsof Ref.[396]wereabletoshow thattheirLM-SR1algorithmwasabletoperformatleastaswe ll,ifnotbetter,thanthe standardBFGSandLM-BFGS[397]methods. Encouragedbythepreviousstudy[393]andtheseimprovemen ts[396]totheSR1al- gorithm,theLM-SR1andaversionoftheLM-BFGSapproachesw ereimplementedinthe CIOptprogram.TheLM-SR1andBFGSalgorithmfromRef.[379] wereusedinconjunc- 174 tionwiththeparallelnumericalgradientroutinealsoavai lableinCIOptandinterfaced withGAMESStooptimizethegeometryof8moleculesofvaryin gsize,namely,benzene, cyclopentadiene,acetone,formamide,E-butadiene,all-E -hexatriene,pyridine,andthenucle- obasethymineattheCR-CC(2,3),DleveloftheoryusingtheT ZVPbasis[398].Theresults oftheseequilibriumgeometryoptimizationsareshowninTa ble5.2.Thegradientswere convergedto10 4 andthedierenceinenergyfortheSR1andBFGSoptimizedstr uctures werelessthan10 5 inallcases.Forbenzenewhereweemployedthe D6 h spatialsymme- tryandonlytwodegreesoffreedom(DoF)wereoptimizedtheB FGSapproachreachesthe minimuminfeweroptimizationcyclesthantheLM-SR1method .WhenthenumberofDoF becomeslarger,asinthecaseofthymine,weseeamorepronou nceddierencebetween thetwoalgorithms.Probablythemoreimpressivecaseistha tofall-E-hexatrienewherethe LM-SR1approachtakes20optimizationcyclestoreachamini mum,whiletheBFGSmethod tookmorethan50forthesamesystem.Thenucleobasethymine ,whichhasalargenumber ofDoF(29),particularlywhenoptimizingthegeometrywith suchhigh-levelapproaches asCR-CC(2,3),whereourLM-SR1algorithmperformsquitewe ll,reachingtheminimum energynuclearcongurationinjustsixcycleswhereasthee quivalentBFGSoptimization took28cycles.Clearlymorestudiesneedtobecarriedout,b uttheseinitialresultsarequite encouraging.TheLM-SR1algorithmhasalsobeenusedonacou pleoftestcases,namely ammoniaandethylene,usingthe -CR-EOMCC(2,3),DandCASSCFapproaches,respec- tively,toevaluateitsperformanceinlocatingminimumene rgycrossingsbetweendierent electronicstatesofagivenmoleculewithencouragingresu lts,whichwillbefollowedupin futurestudies. 175 5.1.2Hessiansforharmonicvibrationalanalyses Asmentionedabove,higher-ordernumericalderivativesbe yondgradientscanbecomevery expensive,especiallywhenruninserial.Ontheotherhand, asseenbythelackofan- alyticHessiansandhigher-orderanalyticderivativesfor high-level abinitio CC/EOMCC approaches,ecientparallelnumericalsecondandhigherd erivativesforusewithanylevel oftheoryhasthepotentialtoopennewavenuesforpredictin gandinterpretingvarious experimentalresults.Aspartoftheworkdoneforthisdisse rtation,anecientparallel numericalsecondderivativeroutineforcomputingharmoni cvibrationalfrequencieswasde- velopedandimplementedintheCIOptpackage.Inimplementi ngthenumericalsecond derivativealgorithmwemadeuseofthefactthattheHessian isasymmetricmatrixwhose entriesare H ij = @E @x i @x j = @E @x j @x i ; (5.9) where E istheenergyand x i and x j representthedegreesoffreedom.Byusingthis symmetryweneedonlycompute M ( M +1) 2 entriesratherthanthefull M 2 Hessian H (note: weareusing H torepresenttheHessianwhereasintheprevioussectionweu sedittodenote theinverseHessian;inSection5.1thisisdoneinkeepingwi ththestandardnotationused innumericalanalysiswheretheHessianitselfisrepresent edby B .Inthissectionwedo notuse B fortheHessian,butrather H soastoavoidconfusionwithWilson's B -matrix discussedbelow).Toparallelizethisportionofthealgori thmitwasdecidedthatattening thetwo-dimensionalHessiantoonedimensionwouldbethemo stecientasitwouldallow foruseoftheMPIroutinesalreadyavailableinCIOptwithmi nimalmodication.Toatten theentire H matrix,whichisan M M matrix,oneusesthetrivialformulawherethetwo- 176 indexelement H ij ismappedtooneindexwith k = j 1+( i 1) M ,where i denotesthe rowand j thecolumnof H ,where H ij resides.However,whereweareonlyinterestedin thoseelementsalongandabove/belowthediagonalthemappi ngformulahastobeslightly modiedfor i> 1suchthat k = j +( i 1) M i ( i +1) 2 ; (5.10) wherethelasttermontheright-handsideaccountsforthefa ctthatwearesubtracting thelower-diagonalindicesofan i i sub-matrix.Usingthissameidea,thehigher-order derivatives,suchasthethirdandfourthderivatives,canb emoreeasilyparallelizedaswell. Forexample,thethreeindicesofthethirdderivative i;j; and k canbeattenedintoaunique indexvia ` = k +( j 1) M j ( j 1) 2 + 1 2 i X =2 [ M ( M 2 +3)+( 1)( 2)] ; (5.11) andforfourthderivativeswehave m = ` +( k 1) M +( j 1) M 2 +( i 1) M 3 M X = M i +2 i X =1 ( 1) ( +1) 2 : (5.12) Thesewillbeusefulforfutureworkasthethirdandfourthde rivativesareneededtocompute variousspectroscopicpropertiesbeyondtheHarmonicappr oximation. Foreachofthediagonalelementsof H weneedtocomputetwouniquesingle-point 177 energies @ 2 E ( x 1 ;:::;x i ;:::;x n ) @x 2 i = E ( x 1 ;:::;x i + h x i ;:::;x n ) h 2 x i 2 E ( x 1 ;:::;x i ;:::;x n ) h 2 x i + E ( x 1 ;:::;x i h x i ;:::;x n ) h 2 x i ; ( i = j ) (5.13) aswellastheenergyattheunperturbed/equilibriumgeomet ry.Foreachoftheo-diagonal elementsweneedfouruniquesingle-pointenergies @ 2 E ( x 1 ;:::;x i ;:::;x j ;:::;x n ) @x j @x i = E ( x 1 ;:::;x i + 1 2 h x i ;:::;x j + 1 2 h x j ;:::;x n ) h x j h x i E ( x 1 ;:::;x i + 1 2 h x i ;:::;x j 1 2 h x j ;:::;x n ) h x j h x i E ( x 1 ;:::;x i 1 2 h x i ;:::;x j + 1 2 h x j ;:::;x n ) h x j h x i + E ( x 1 ;:::;x i 1 2 h x i ;:::;x j 1 2 h x j ;:::;x n ) h x j h x i ; ( i 6= j ) : (5.14) Thusthereareatotalof2 M +4 M ( M 1) 2 +1=2 M 2 +1single-pointsneededtoevaluate theHessian H numerically.Whendiscussingthenumericalderivatives,p articularlysecond andhigher,itmustberememberedthatthesearemeanttoberu ninparallelandnot sequentially,since,forexample,ifweconsideratriatomi csystem,whichhas9degreesof freedominthethreedimensionalCartesianspace,thenatot alof162single-pointenergies needtobecomputed.Itiswellknownthatfornonlinear(line ar)moleculesthereareonly 178 M 6( M 5)degreesoffreedomthatneedtobeconsideredforharmonic vibrational frequencyanalysis,wecanseethatthistranslatesinto,fo rthecaseofatriatomicmolecule, only18(32)single-pointenergycomputations.Therearese veralapproachesforreducing theDoFofamolecularsystemtotheminimalnumber[399{406] ,allofwhichmakeuse ofavariantofWilson's B -matrix[407].Implementingaroutinethatgeneratesaform of Wilson's B -matrixforecientparallelnumericalsecondderivatives hasbeenstartedand willbenishedaspartoffuturework. Toevaluatetheharmonicvibrationalfrequenciesoncethen umericalHessianhasbeen computed,followingRef.[407],theHessianismassweighte dbymultiplyingeachelement H ij by1 = ( m i m j ) 1 = 2 where m i isthemassofatom i .Themass-weightedHessianisthen diagonalizedproducingthesetofnormalmodes Q andmass-weightedforce-constants ( =1 ;:::;M ),wherethelatterareusedtocomputetheharmonicvibratio nalfre- quenciesusing = 1 2 ˇ p : (5.15) AnalgorithmtodeterminetheAbeliansymmetryofthevibrat ionalfrequencieswasim- plemented,wheretheirreduciblerepresentationtowhicht henormalmodebelongedwas determinedusing[408] P q Q = ` q h X R ˜ q ( R ) P R Q ; (5.16) where P q aretheprojectionoperatorsofthepointgrouprepresentat ion, ` q isthedimension oftheirreduciblerepresentationofthegroup, h isthedimensionofthepointgroup, ˜ q ( R ) arethecharactersforagivenirreduciblerepresentation q ofthegroup, R aretheopera- tionsofthepointgroup,and P R aretheoperators/matrixrepresentationscorresponding 179 to R .Iftheabovesumisnon-zerothenthe Q normalmodebelongstothatirreducible representation.ThisformulaaswrittenonlyappliestoAbe lianpointgroups.Fornon- Abeliangroupsthesituationbecomesmuchmorecomplicated whereasonecancomputethe harmonicvibrationalfrequenciesusinganAbeliansymmetr ygroupandthenusingcorrela- tiontablesdeterminethesymmetryofthenormalmodefreque nciesinthehigher-symmetry non-Abeliangroup. TheabovealgorithmshavebeenimplementedintheCIOptprog rampackageandhave beentestedonasmallnumberoftestcasesshowingpromising results,thoughmorestudies needtobecarriedoutwithmoreexamplesdemonstratingthei rutility. 5.1.3Applicationtothelow-energyisomersofAu 8 Thediscoverythatsmall(2nm diameter 4nm)goldparticlesAu n canselectivelycat- alyzechemicalreactions,asalreadydiscussedinSections 3.3and4.3,suchastheepoxi- dationofpropene[409],hasinspiredalotofactivityamong experimentalistsandtheorists towardunderstandingtheoriginsofthiscatalyticbehavio r.Severalfactors,includingsurface roughening,mayplayanimportantroleinthecatalyticacti vityofAu n clusters,sincenon- planarityoftheclusterslocalizestheelectrondensityan dpromotesreactivity[410].Because oftheimportanceofsurfacerougheninginthecatalyticact ivityofgoldclusters,itisessential todeterminethenumberofgoldatomsintheAu n particleforwhichtheplanar{to{non- planarturnoveroccursandthenon-planarisomersbegintod ominateasthelowest-energy species.IntheearlierstudybyOlsen etal .[214],calculationsusingtheCCSD(T)approach suggestedthatthemoststablestructuresofAu n wereplanarfor n =6andnon-planarfor 180 Figure5.1:TheS 1 ,S 3 ,S 4 ,andS 6 isomersofAu 8 examinedinthisstudy,alongwiththe selectedgeometricalparametersincludedinTable5.4. n =8,incontrastwithDFTcalculations,whichtypicallypred ictAu 8 tofavortheplanar conguration(cf.,e.g.,Refs.[204{207],andreferencest herein;see,also,theintroductionto Ref.[214]foradditionalremarks). WhentheinitialCCSD(T)calculationsforAu 8 werereported[214],issues,suchas theuseoflargerbasissets,geometryrelaxation,andthenu mbersofcorrelatedelectrons usedinCCcalculationscouldnotbeaddressedduetoprohibi tivecomputationalcosts. SincethenseveralotherCC,MP2,andDFTcalculationshavea ppearedintheliterature [215{217,411{414],usuallyindisagreementwiththendin gsinRef.[214],includingthe CCSD(T)calculationswithlargerbasissetsandsomecoreco rrelationsthatleantowardthe conclusionthatthelowest-energyAu 8 isomershouldindeedbeplanar[215{217]. However,thetopicstillremainsopen,sinceallCCcalculat ionsforAu 8 todaterely onlow-ordermethods,suchasMP2orDFT,whichdonotnecessa rilyprovidethecorrect energetics,todeterminenucleargeometries.Therealsoar eindicationsthatclusterswith7or moregoldatomsmaybenon-planar[415].Inviewofallthisan daspartofthisthesisworkwe 181 performednewCCSD(T)calculationsforAu 8 employinglargerbasissetsthanthoseusedin theinitialcalculations[214]andexaminingtheroleofcor e-valence(CV)correlations,ashas beendoneinRefs.[215{217],andaddressingtheissueofgeo metryrelaxationatthesame timebyreoptimizingthelow-energystructuresoftheAu 8 particleusingCCSD(T)[183]. Clearly,theissueofgeometryrelaxationisachallengingo neattheCCtheoryleveldueto highcomputationalcosts,but,asdemonstratedinthisdiss ertation,wecantakeadvantage oftheecientparallelnumericalgradientsforgeometryop timizations,whichcanworkwith anyCCmethod,includingtheparallelimplementationofCCS D(T)utilizedinthisstudy. Aspartoftheworkforthecompletionofthisdissertation,w eundertookasystematic investigationoftheroleofgeometryoptimization,basiss et,andsemi-core(i.e.,CV)electron correlationsintheCCSD(T)calculationsoftherelativeen ergeticsofafewlowest-energy Au 8 isomersinthehopeofprovidingamoredenitiveanswerasto whetherornotthemost stablestructureofAu 8 mightbenon-planar.Wefocusonthefourlowest-energyisom ers ofAu 8 ,showninFig.5.1,aspredictedinRef.[214]bythesmallbas issetMP2geometry optimizationsandCCSD(T)single-pointcalculations,ree xaminedlaterinRef.[217].For eachofthefourstructures,twosetsofgeometryoptimizati ons,correlatingineachcasethe 5 d 10 6 s 1 valenceelectrons,werecarriedout.Intherstset,furthe relaboratedonbelow,we usedtheparallelanalyticgradientsofMP2,availableinth eGAMESSpackage[190,191]. Inthesecondsetofgeometryoptimizations,furtherdiscus sedbelowaswell,weusedthe dual-levelparallelismappliedtothenumericalCCSD(T)de rivatives,inwhichthecoarse- grainnite-dierencemodelavailableintheCIOptprogram suite[235,378],utilizingas manynodesasthenumberofenergiesneededtodeterminetheg radient,wascombinedwith 182 thene-grainsingle-pointCCSD(T)calculationsoneachno de.Inrunningtherequired single-pointCCSD(T)computations,weusedthehighlyscal ableparallelCCSD(T)codes, availableinGAMESSanddevelopedinRefs.[416,417],which arebasedontheCCSD(T) algorithmfromRef.[238].Bycombiningtheembarrassingly parallelnite-dierencemodel withthehighlyscalableCCSD(T)approach,asdescribeabov e,wewereabletodeterminea singleenergygradientwhenemployingtheCCSD(T)/SBKJC(f )approachinabout9hours whenrunningonninenodeseachwith8cores(cf.Table5.3),w hileattheCCSD(T)/cc- pVDZ-PPlevelin ˘ 41hrs,when4corespernodewereused,or ˘ 18hrs,when9cores wereutilizedineachoftherequiredsingle-pointcalculat ions.Thedierenceintimings betweenthetwobasesreectsonthedierenceintheirsizea ndqualitywiththelarger cc-pVDZ-PPbeingthebetterofthetwo,asshownintheresult sbelow.Thenumbersof thesingle-pointcalculationsneededtodetermineasingle energygradientwere5forthe D4 h -symmetric S 1 structure(9whenrunasa D2 h conguration),9forthe T d -symmetric S 3 and D2 d -symmetricS 6 isomers,and15forthe C s -symmetric S 4 conguration.Byusing thepreviouslyconvergedMP2geometries,wetypicallyneed ed5{10CCSD(T)optimization cyclestoconvergethegradientto10 4 . WestartedourexaminationoftheMP2andCCSD(T)energetics usingtheMP2-optimized geometries.ThesmallbasissetMP2geometryoptimizations usingthestandardSBKJCba- sisandtheassociatedscalarrelativisticECP,whichwasop timizedforgold[418],augmented byasinglesetofffunctions(exponent=0.89),referredtoa sSBKJC(1f),werecarriedout inRef.[214].TheresultingCCSD(T)/SBKJC(1f)//MP2/SBKJ C(1f)energieshaveledto theaforementionedconictingndingsabouttheorderingo fthevariousstructuresofAu 8 . 183 Table5.3:Detailsofthewalltimes(hours)characterizing theCCSD(T)/SBKJC(1f)parallel numericaloptimizationoftheAu 8 S 1 isomeremployingtheD 2 h symmetry(4degreesof freedom),performedusingnine8-corenodes a . NodeNumber OptimizationCycle012345678Average 19.679.609.509.529.809.829.829.529.489.64 29.439.539.429.309.429.589.859.439.909.54 39.809.639.539.609.809.6210.029.809.579.71 49.879.779.679.729.709.6510.109.679.589.75 59.429.489.489.489.429.589.839.489.439.51 69.379.639.379.359.379.389.689.429.409.44 79.359.409.379.379.379.409.709.409.359.41 89.409.729.459.439.439.5010.009.629.359.54 Average9.549.609.479.479.549.579.889.549.519.57 a Thiscalculationwasrunonnodesconsistingoftwofour-cor eIntelXeonE5620s(Westmerefamily)at2.4 GHzwith24GBofRAMatMichiganStateUniversity'sHighPerf ormanceComputingCenter. Thus,inordertoexaminetheroleofthebasissetusedtodete rminetheMP2geometries, wereoptimizedallfourAu 8 structuresshowninFig.5.1usingtheMP2methodandthe cc-pVDZ-PPbasis[229]combinedwiththescalarrelativist icECPcreatedforusewiththe correlation-consistentbasissets,whichwasoriginallyd evelopedinRef.[230]andlatermod- iedtoincludetheeectsofthe h functionsinthepseudopotentialinRef.[231].Using theseimprovedMP2/cc-pVDZ-PPgeometries,single-pointC CSD(T)calculationswerecar- riedoutexploitingthecc-pV x Z-PP( x =D,T)basissets,combinedonceagainwiththe ECPofRefs.[230,231],correlatingthesamenumberofelect ronscorrespondingtothe5 d and6 s shellsofthegoldatomsasinthegeometryoptimizations.To investigatetherole oftheCVcorrelationeects,themoreextensiveCCSD(T)/cc -pV x Z-PP+CV( x =D,T) calculationscorrelatingthe5 s 2 5 p 6 semi-coreand5 d 10 6 s 1 valenceelectronswereperformed aswell.Throughoutthissectionandasdoneinotherportion softhisdissertation,weuse thenotationinwhichthecc-pV x Z-PPacronymreferstothecalculationscombiningthe 184 cc-pV x Z-PPbasisset[229]andtheECPofRefs.[230,231].Ifthe5 s 2 5 p 6 electronsare correlatedinadditiontothevalence5 d 10 6 s 1 electrons,thecorrespondingcomputationis abbreviated(followingRef.[217])ascc-pV x Z-PP+CV. HavingexaminedtheroleofthebasissetontheMP2geometrie s,wewenttothenext stepinwhichweperformedtheCCSD(T)geometryoptimizatio nsemployingtheSBKJC(1f) andcc-pVDZ-PPbases,andtheaccompanyingECPs.Inthisway ,wecouldexaminethe signicanceofhigher-ordercorrelationeectsonthecalc ulatedgeometriesandtheeectof thebasissetonthegeometryoptimizationsattheCCSD(T)le veloftheory.Single-point CCSD(T)/cc-pV x Z-PPandCCSD(T)/cc-pV x Z-PP+CV( x =D,T)calculations,using,once again,therelativisticECPsfromRefs.[230,231],werethe ncarriedouttodeterminethenal relativeenergeticsofthefourAu 8 structuresshowninFig.5.1.Wehavenotconsidered theeectsofrelativityonthecalculatedgeometriesanden ergiesbeyondthosecapturedby ECPs,suchasspin-orbitcoupling,sinceitiswellestablis hedthatthespin-orbitcoupling doesnotaltertherelativestabilityofAu n clusterswith n 20,whilehavingnegligibleeect ontheirgeometries[419](see,also,Refs.[215,411,412]) . 185 186 Table5.4:Bondlengths(in A)oftheS 1 ,S 3 ,S 4 ,andS 6 isomersofAu 8 obtainedfromgeometryoptimizationsusing MP2,CCSD(T),andsomerepresentativeDFTapproaches(seeF ig.5.1forthemeaningofthegeometricalparameters). S 1 S 3 S 4 S 6 MethodBasis r 1 r 2 r 1 r 2 r 1 r 2 r 3 r 4 r 5 r 1 r 2 r 3 r 4 MP2 a SBKJC(1f)2.6972.5922.8042.6722.7042.6542.6862.7582. 6122.6552.7032.7622.779 CCSD(T) a SBKJC(1f)2.7712.6432.8652.7432.7662.7112.7552.8322. 6742.7142.7722.8192.908 MP2 a cc-pVDZ-PP2.7072.5882.8852.6772.7042.6722.6932.7662 .6122.6642.7132.7782.810 CCSD(T) a cc-pVDZ-PP2.7822.6432.8962.7512.7682.7232.7662.8502 .6762.7282.7832.8292.964 LC- ! PBE b cc-pVDZ-PP2.7592.626 ! B97X b cc-pVDZ-PP2.8172.663 TPSS b cc-pVDZ-PP2.7582.633 CAM-B3LYP b cc-pVDZ-PP2.8062.648 B3LYP b cc-pVDZ-PP2.8302.672 a Thiswork.AlthoughtheMP2/SBKJC(1f)geometrieswereorig inallyobtainedinRef.[214],theywererecalculatedinRef .[183]aspartof theworkdoneforthisdissertation. b TakenfromRef.[219]. TheMP2andCCSD(T)geometriesoptimizedinthisstudy,alon gwithselectedDFT resultsfortheS 1 isomertakenfromRef.[219],arepresentedinTable5.4.For theMP2 results,thesizeofthebasishasusuallyaslighteectonth ecalculatedgeometries,with themajorityofthedierencesbetweentheMP2/SBKJC(1f)an dMP2/cc-pVDZ-PPopti- mizationsfallingintothe ˘ 0.01 Arange,andwiththelargestdierencesbeing0.031and 0.081 A.SimilarremarksapplytotheCCSD(T)geometriesobtained usingthetwodierent basissetsexploitedhere,withtheaverageandmaximumdie rencesbeing0.014and0.056 A,respectively.ComparingtheMP2andCCSD(T)geometriesw ithinagivenbasis,wesee thatfortheSBKJC(1f)results,thedierencesrangefrom0. 051to0.129 A,for0.069 Aon average,whileforthecc-pVDZ-PPbasistheyareaslargeas0 .154 A,again0.069 Aonaver- age.Generally,theCCSD(T)calculationsmaketheAu{Aubon dsinAu 8 longer.Thismight beaconsequenceofMP2overbindingtheAu 8 cluster,whichhassomecharacteristicsofthe weaklyboundsystems,includingthepotentiallyimportant roleofnon-additivedispersion forces[205,420{422].Indeed,thebindingenergypergolda tomobtainedintheMP2/cc- pVTZ-PP+CV//MP2/cc-pVDZ-PPcalculations,averagedover thefourstructuresofAu 8 examinedinthiswork,is57.43kcal/mol,asopposedto45.26 kcal/molobtainedinthecor- respondingCCSD(T)/cc-pVTZ-PP+CV//MP2/cc-pVDZ-PPcomp utations(45.16kcal/mol whentheCCSD(T)/cc-pVDZ-PPgeometriesareused).However ,asshownbelow,thesedif- ferencesbetweentheMP2andCCSD(T)geometries,althoughs ignicantinsomecases,are notsucientlylargetoaectthenalconclusionsregardin gtheplanarity vs non-planarity oftheAu 8 particle.Whatseemstobemoreimportantistheelectroncor relationtreatment appliedtotherelativeenergiesofthevariousstructures. 187 Therelativeenergetics,showninTable5.5,demonstrateth atevenwhenthegeometries areoptimizedwithalargercc-pVDZ-PPbasissetandtheCVco rrelationsareincluded inthecalculations,theMP2resultsarecompletelyunrelia ble.Theuseofthecc-pVTZ- PPbasisset,withorwithoutCVcorrelations,inMP2calcula tionsdoesnothelpeither. TheMP2calculationsarrangetheS 6 andS 1 isomersasthelowestandhighestinenergy, respectively,whilestronglyfavoringthenon-planarAu 8 structuresovertheplanarS 1 isomer. Thisimmediatelyimpliesthatoneneedstoaccountforthehi gher-ordercorrelationeects toobtainreliableenergeticsofthevariousAu 8 structures. TheCCSD(T)/cc-pV x Z-PPandCCSD(T)/cc-pV x Z-PP+CV( x =D,T)single-pointcal- culations,usingtheMP2/cc-pVDZ-PPgeometries,lowerthe energydierencesoftheS 3 , S 4 ,andS 6 structures,relativetoS 1 ,byapproximately1{2kcal/molcomparedtotheircoun- terpartsusingtheMP2/SBKJC(1f)geometriesreportedinRe f.[217],buttheydonotalter themainconclusionsofRef.[217](orRefs.[215]and[216]) thatS 1 isaglobalminimum. AccordingtoourbestCCSD(T)/cc-pVTZ-PP+CVcalculations employingthegeometries optimizedattheCCSD(T)/cc-pVDZ-PPlevel,theenergyorde ringofthelowestfourisomers ofAu 8 isS 1 < S 3 ˇ S 4 < S 6 ,inagreementwiththendingsofRef.[217].However,itis importanttoemphasizethattheuseofthesmallercc-pVDZ-P PbasissetintheCCSD(T) calculations,althoughreasonableingeometryoptimizati ons,isnotsucienttoprovidea reliabledescriptionoftherelativeenergiesofthevariou sAu 8 structures. Indeed,asshowninTable5.5,theuseofthecc-pVDZ-PPbasis setintheCCSD(T) calculationsinsteadofcc-pVTZ-PPlowerstheenergiesoft heS 3 ,S 4 ,andS 6 structuresrela- tivetoS 1 by4{5kcal/mol,evenwhentheCCSD(T)/cc-pVDZ-PPgeometri esareemployed, 188 bringingS 3 ,S 4 ,andS 6 closetoS 1 whentheCVeectsareignored,andplacingS 3 ,S 4 , andS 6 belowS 1 whentheCVeectsareaccountedfor.Onehastousethebasiss etof thecc-pVTZqualityintheCCSD(T)calculationstostabiliz eS 1 asaglobalminimum.The CVeectsplaysomerole,loweringtheenergiesofS 3 ,S 4 ,andS 6 relativetoS 1 byabout 1{2kcal/molattheCCSD(T)/cc-pVTZ-PPlevel,buttheydono taltertherelativeenergies whenthecc-pVTZ-PPbasissetisusedintheCCSD(T)calculat ions(notethattheydo alterthemwhenthesmallercc-pVDZ-PPbasissetisemployed ).Thisisverydierentfrom theourstudyoftheAu 3 photoelectronspectrum[184]wheretheCVcorrelationshad a signicanteectonthenalresults,whichdemonstratesth eneedtocarefullyconsiderthe systemonewishestostudyastohowitneedstobetreatedinor dertoobtainanaccurate descriptionofsuch.Similarappliestotheroleofgeometry optimization.Theuseofthe betterCCSD(T)/cc-pVDZ-PPgeometriesinsteadoftheMP2/S BKJC(1f)geometriesem- ployedinRef.[217]lowerstheenergiesoftheS 3 ,S 4 ,andS 6 structuresrelativetoS 1 by 1{2kcal/mol,butthisisnotsucienttochangetheoveralle nergyordering,aslongasthe cc-pVTZ-PPbasissetisusedinthenalsingle-pointCCSD(T )calculations. Insummary,weperformedgeometryoptimizationsattheMP2/ cc-pVDZ-PP,CCSD(T)/ SBKJC(1f),andCCSD(T)/cc-pVDZ-PPlevelsoftheoryemploy ingrelativisticECPsforthe fourAu 8 lowest-energyisomersconsideredintheearlierCCstudies [214{217].Wethenused theresultinggeometriestocarryoutthesingle-pointMP2/ cc-pVDZ-PP,MP2/cc-pVTZ- PP,CCSD(T)/cc-pVDZ-PP,andCCSD(T)/cc-pVTZ-PPenergyca lculations,adoptingthe sameECPsasusedinthegeometryoptimizationsandcorrelat ingthe5 d 10 6 s 1 valenceand 5 s 2 5 p 6 5 d 10 6 s 1 semi-coreandvalenceelectrons.AccordingtoourbestCCSD (T)calcula- 189 Table5.5:Relativeenergies(inkcal/mol)oftheS 1 ,S 3 ,S 4 ,andS 6 isomersofAu 8 ,with respecttotheS 1 isomer,obtainedintheMP2andCCSD(T)calculations. Method/BasisIsomerEnergy GeometryOptimizationSingle-pointCalculationS 1 S 3 S 4 S 6 MP2/SBKJC(1f)MP2/SBKJC(1f) a ; b 0.0-25.1-23.4-30.8 MP2/cc-pVDZ-PP b 0.0-16.9-15.8-22.3 MP2/cc-pVDZ-PP+CV b 0.0-19.9-19.2-25.5 MP2/cc-pVTZ-PP b 0.0-15.4-16.3-21.8 MP2/cc-pVTZ-PP+CV b 0.0-16.5-17.8-23.3 CCSD(T)/SBKJC(1f) a ; b 0.0-4.7-2.5-3.2 CCSD(T)/cc-pVDZ-PP b 0.03.65.25.6 CCSD(T)/cc-pVDZ-PP+CV b 0.00.11.31.8 CCSD(T)/cc-pVTZ-PP b 0.07.57.59.1 CCSD(T)/cc-pVTZ-PP+CV b 0.06.05.77.2 CCSD(T)/SBKJC(1f) c CCSD(T)/SBKJC(1f)0.0-6.29-4.04-5.08 CCSD(T)/cc-pVDZ-PP0.00.371.991.98 CCSD(T)/cc-pVDZ-PP+CV0.0-3.05-1.72-1.74 CCSD(T)/cc-pVTZ-PP0.05.195.416.49 CCSD(T)/cc-pVTZ-PP+CV0.03.933.854.86 MP2/cc-pVDZ-PP c MP2/cc-pVDZ-PP0.0-17.32-16.37-22.52 MP2/cc-pVDZ-PP+CV0.0-19.81-19.31-25.24 MP2/cc-pVTZ-PP0.0-14.39-15.25-20.95 MP2/cc-pVTZ-PP+CV0.0-15.04-16.33-21.92 CCSD(T)/cc-pVDZ-PP0.01.473.043.41 CCSD(T)/cc-pVDZ-PP+CV0.0-1.81-0.56-0.11 CCSD(T)/cc-pVTZ-PP0.06.526.547.83 CCSD(T)/cc-pVTZ-PP+CV0.05.335.006.28 CCSD(T)/cc-pVDZ-PP c CCSD(T)/cc-pVDZ-PP0.00.161.721.79 CCSD(T)/cc-pVDZ-PP+CV0.0-3.08-1.74-1.72 CCSD(T)/cc-pVTZ-PP0.05.635.726.91 CCSD(T)/cc-pVTZ-PP+CV0.04.604.395.57 a FromRef.[214]. c FromRef.[217]. c FromRef.[183]doneaspartoftheworkforthisdissertation . 190 tionswiththecc-pVTZ-PPbasisset,accountingfortheCVco rrelations,andusingthe CCSD(T)/cc-pVDZ-PPoptimizedgeometries,thelowest-ene rgystructureofAu 8 isthepla- narS 1 isomershowninFig.5.1.Thisisinagreementwithanexperim entalstudy[423] publishedafterourinitialstudywaspublishedinwhichthe yshowed,usingFar-IRspec- troscopy,thatthelowestenergyisomerofAu 8 isindeedtheplanarS 1 structure.Thenext twoisomers,S 3 andS 4 ,arenearlydegenerateandlieabout4{5kcal/molaboveS 1 ,andthe fourthstructurestudiedinthiswork,designatedasS 6 ,isabout6kcal/molaboveS 1 .Geom- etryrelaxationplaysarathersmallroleonthecalculatede nergies,loweringtheS 3 ,S 4 ,and S 6 structuresrelativetoS 1 by1{2kcal/mol,butdoesnotalterthemainconclusionsabou t theplanarityofthelowest-energyisomerofAu 8 ,aslongasoneusestheCCSD(T)approach andthelargerbasissetofthetriple-zetaquality,suchast hecc-pVTZ-PPbasisemployedin thisdissertation.Atthesametime,useoftheMP2methodpro videscompletelyincorrect energeticresults,evenwhenthelargerbasissetsandtheCV correlationsareincluded,in agreementwiththendingsofRef.[217].Thus,oneneedstou sehigh-levelmethods,such asCCSD(T),capableofaccountingforhigher-ordercorrela tioneects,toobtainareliable descriptionatthe abinitio wavefunctiontheorylevel.Thismayberelatedtothefactth at goldparticlesarecharacterizedbysubstantialnon-addit ivedispersioneects,whichcannot becapturedbyMP2. 5.2UnrestrictedimplementationofCR-CC(2,3) ThesuccessofCCtheoryistypicallyassociatedwiththepop ularsinglereference(SR) CCSD(T)approach,whichprovidesanaccuratedescriptiono fdynamicalelectroncorrela- 191 tioneectsfornondegenerategroundstatesofmoleculesne artheequilibriumgeometries(cf. Section5.1.3).ItiswellknownthattheRHF-basedCCSD(T), RCCSD(T),approximation failswhenappliedtobiradicalsandbondbreakingsituatio nswherenondynamicalcorrelation eectsbecomemoresignicant.Itisalsowellknownthatone canremedytheunphysical characterizationmanifestinthepotentialenergysurface (PES)alongbondbreakingcoor- dinatesofsinglebondbreakingsituationsintheRCCSD(T)c omputationsbyswitchingto theUHF-basedCCSD(T),UCCSD(T),approach.Whiletheovera llenergeticsofthePES alongbondbreakingcoordinatesareimproveditisdoneatth esacriceofintroducingother unphysicalcharacteristics,suchasspincontaminationan dnon-analyticbehaviorofthePES. TheCR-CCmethods,suchastheoriginalCR-CCSD(T)[40,41,8 7{89]andnewerrig- orouslysize-extensivelefteigenvalueapproaches,CR-CC (2,3),aswellastheirhigher-order extensions[40,41,87{89,93,94,424],havebeenshowntoim provethepoorRCCSD(T)re- sults,particularlyinregionsofthePESinvolvingsingleb ondbreakingevenwhenaRHF determinantisusedasthereferencewavefunction(seeRef. [68]andreferencesthereinfor furtherdetails).TheCR-CCSD(T)andCR-CC(2,3)approache sareofparticularinterestas theyretainthesameblack-boxeaseofuseanditerative N 6 andnoniterative N 7 CPUscaling costsasCCSD(T).WhiletheperformanceoftheCR-CCSD(T)an dCR-CC(2,3)approxi- mationsemployinganRHFwavefunction,RCR-CCSD(T)andRCR -CC(2,3),respectively, hasbeenextensivelyexamined,itisnotwellknownifusinga nUHFreferencedeterminant willoersomeimprovementofthealreadyquitegoodCR-CCPE Ssalongbondbreakingco- ordinatesforclosed-shellsingletmolecularsystems.Rec entlytheUHF-basedCR-CCSD(T), UCR-CCSD(T),approachwasusedtoexaminesinglebondbreak ingonasingletPES[425], 192 concludingthat,whileUCR-CCSD(T)computationsgivequal itativecorrectresults,there- sultsatstretchedintermediatebondlengthsoverestimate therelativeenergies,thesameas seenwithUCCSD(T)calculations.Thisbehaviorisaknownfe atureofspincontaminated UHF-basedCCresults[426]. Aspartofthisthesis,weexaminedwhetheremployinganUHFr eferencewavefunctionin theCR-CCcalculationsofsinglebondbreakingintoopen-sh ellfragmentsonsingletPESswill oeranyimprovementsoverthealreadyaccurateRHF-basedC R-CCresultsbyexamining bondbreakinginthetwomodeldiatomicsystemsusedforbenc hmarking,namely,HFand F 2 aswellastwopolyatomicsystems,whichserveasmodelsforl argermoleculeswhere importantO{OandC{Csinglebondbreakingeventscanplaycr ucialroles,theH 2 O 2 ,and C 2 H 6 molecularsystems,comparingthemwiththeexact,fullcon gurationinteraction(CI), andfullCCSDTresults.WealsocompareourUHF-basedCR-CCr esultswiththeirRHF- basedcounterpartsshowingthatthespin-adaptedRHF-base dCR-CC(2,3)resultsprovide themostaccuratedescriptionatallpointsalongthePESs. 5.2.1Algorithm InordertocomputetheUHF-basedCR-CC(2,3)resultsweneed edtotransformtheintegrals intheatomicorbital(AO)basistothemolecularorbital(MO )basis.Thisisdoneusingtwo- andfour-indexintegraltransformations,withthelatterb eingthemoreexpensivestep.The four-indexintegraltransformationofthetwo-bodyintegr alscanberepresentedalgebraically as ( pq j rs )= X ;;ˆ;˙ C p C q C rˆ C s˙ ( j ˆ˙ ) ; (5.17) 193 where( pq j rs )and( j ˆ˙ )arethetwo-electronintegralsintheMOandAObases,respe c- tively,andthe C coecientsarethestandardexpansioncoecientsobtained fromdiagonal- izingtheFockmatrix.Weonlyconsideredthe - , - ,and - blocksofthetwo-electron integralsasothercombinationsarezero.Ifimplementedna ively,loopingoverallp,q,r, s, ;;ˆ ,and ˙ usingnestedloopsthiswouldbean O ( N 8 )algorithmandwouldnotbe veryuseful.Indevelopingourownroutinewemadeuseofthef actthatmodernFortran allowsfortheuseofupto8-dimensionalarrays,storingthe two-bodymatrixelementsina 4-dimensionalarray.Wewouldthentakeasliceofthefourdi mensionalarrayandperform theappropriatematrix-matrixmultiplicationusingcalls totheBLASDGEMMroutine.We alsomadeuseofthethreadingcapabilitiesoftheIntelMKLl ibrary,allowingforalevel ofparallelisminourroutine.Thealgorithmistriviallypa rallelizableinthewaythatit iswrittenandwillbepursuedinfuturestudies.Onecritici smofthistypeofapproach isthat,dependingontheorderofcalls,itmightresultinal argenumberofcachemisses ascomparedtoaroutinewherethe4-dimensionalarrayisat tenedtotwooronedimen- sions,whichforlargeproblemswouldresultinadropinperf ormance.Evenforaproperly optimizedprogrammakinguseofthe4-dimensionalarraythe remightstillbealossinef- ciencycomparedtoonethatattensthearrayduetothecont iguousnatureofcurrent CPUmemory,particularlywhenruninserial.However,asmen tionedabove,itiswritten suchthatitisan\embarrassinglyparallel"algorithmmaki ngiteasytoparallelize.Once theone-andtwo-bodyintegralsweretransformedtotheMOba sistheywerereadintoour group'sin-houseCCprogrampackage,calledCC PACKAGE,whichweusedtocompute theUHF-basedCR-CC(2,3)resultsdiscussedinthenextsect ion. 194 5.2.2Numericalexamples WeusedtheNWChemsuitetocarryouttheUHF-basedCR-CCSD(T )calculationsand theGAMESSpackagefortheRHF-basedCCSD(T),CR-CCSD(T),a ndCR-CC(2,3)com- putations,whichspin-adaptedroutineswereimplementedb yourgroup.TheRHF-and UHF-basedCCSDTcalculationsreportedhereweredoneusing in-housespin-integratedCC codes.ToperformtheUHF-basedCR-CC(2,3)calculationswe usedamodiedversionof GAMESStoobtaintheproperone-andtwo-bodyintegralsandc oecients,whichwerethen transformedwithourownfour-indexintegraltransformati onroutine,asdiscussedabove,and nallycarriedouttheUHF-basedCR-CC(2,3)computationsu singourin-houseCCroutines, whichworkforanySRwavefunction.Thespecicdetailsofba sissetsandnumberoffrozen coreorbitalsarediscussedbelowwiththeresultsofeachex ample. 5.2.2.1TheHFmolecule FollowingpreviousCCstudies[87,88,90,93]wecomputedth ePESoftheHFmolecule asdescribedbytheDZbasisset[185]andcorrelatingallele ctrons,forwhichthefullCI energies[361]areavailableforcomparison.Theresultsof ourRHF-andUHF-basedCCand CR-CCcalculationsforHFarepresentedinTables5.6and5.7 andgraphicallyinFigs.5.2(a) and5.3(a).AsmentionedaboveandasiswellknowntheUHF-ba sedCCSD(T)resultsarea dramaticimprovementovertheirRHF-basedcounterparts,r educingthemeansignederror (MSE)fromaround-20.0millihartreeto0.5{0.7millihartr eewhencomparedtoRCCSDT andUCCSDT,respectively,withthesamedegreeofimproveme ntcomparedtofullCI.The meanunsignederror(MUE)fortheRHFandUHF-basedCCSD(T)c alculationsreduces 195 from ˘ 7to0.5{0.7millihartreecomparedtoRCCSDTandUCCSDT,res pectively.The nonparallelityerror(NPE)showsamassivedecreeswhenthe UHFreferencedeterminantis employedintheCCSD(T)computations,droppingfromabout5 0millihartreetoaround1 millihartreecomparedtotheirrespectiveparentCCSDTapp roachesandfullCI. ExaminingtheCR-CCresults,thealreadyreasonableMSE,MU EandNPEvaluesfor theRHF-basedapproachesareimproveduponslightlywhenth eUHFreferencedeterminant isemployed.OurleastaccurateCR-CCapproach,whichstill providesverygoodresults, CR-CCSD(T)providesreasonableMSEandMUEvaluesontheord erof1(1.5)millihartree comparedtoCCSDT(fullCI)whentheRHFwavefunctionisused asareferenceandan NPEofabout1.4(1.6)millihartreecomparedtofullCCSDT(f ullCI).SwitchingtotheUHF wavefunctionasareferencetheMSEandMUEvaluesimproveby about0.01(0.4)milli- hartreecomparedtoUCCSDT(fullCI).Inthiscase,though,w eseeaslightincreaseinthe NPEgoingfrom1.43(1.60)to1.85(2.58)millihartreecompa redtoRCCSDTandUCCSDT (fullCI),respectively.Thisisamanifestationofthewell knownspin-contaminationproblem intheCCcalculations[426]and,aswewillshowinthislette r,ispresentinalloftheapprox- imatetriplesCCapproachesexaminedinthisworkwhentheUH Freferencewavefunctionis employed.TheCR-CC(2,3),A[CCSD(2) T ]approachshowsslightlybetterperformancewith theRHF-basedCR-CC(2,3),AMSEandMUEvaluesbeing ˘ 0 : 9(1.3)millihartreecom- paredtoRCCSDT(fullCI)andNPEontheorderof1.5(2.0)mill ihartree.TheUHF-based CR-CC(2,3),AshowasimilarbehavioraswithitsUHF-basedc ounterpartCR-CCSD(T), loweringtheMSEandMUEby0.01millihartreeandtheNPEincr easingfromaround1.5 (2.0)to1.7(2.4)comparedtoRCCSDTandUCCSDT(fullCI),re spectively.Ourbest,CR- 196 CC(2,3),D,computationsprovidethebestoverallagreemen twhencomparedtotheirparent CCSDTresultsandtheexact,fullCI,values.TheRHF-basedC R-CC(2,3),DMSEandMUE -0.89(-0.29)and0.37(0.27)millihartreeareinverygooda greementwiththeRHF-based CCSDT(fullCI)withtheUHF-basedCR-CC(2,3),Doersomeim provement,particularly fortheMSE,changingitsvalueto-0.04(0.18)millihartree withrespecttoUCCSDT(full CI).TheincreaseintheNPEwhenusingtheRHF vs UHFreferencedeterminantisnot assevereasintheCR-CCSD(T)andCR-CC(2,3),Acasesdueinp arttothemorerobust treatmentusingthefullexpressionofthedenominatorfort heCR-CC(2,3),Dcalculations. 5.2.2.2TheF 2 molecule TheverychallengingF 2 molecule,longknowntohavealargedegreeofnondynamicalc or- relationeectsandtobeunboundwhendescribedattheUHFle veloftheory,isidealfor testingapproximatetriplesCCapproachesasitrequiresap roperbalanceofthedynamical andnondynamicalcorrelationeectsforanaccuratedescri ptionofitsPES.Itisalsowell knownthatCCSDTcanaccuratelydescribesinglebondbreaki ngsituations,astheinclusion oftheeectsduetotriplyexcitedclustersarerequiredtop roperlyaccountfortheimportant correlationsinsinglebondbreaking.ThusF 2 haslongprovidedawaytotestthereliability androbustnessofapproximatetriplesCCmethodologiesand theirabilitytobalancedy- namicalandnondynamicalcorrelations.Inthecalculation sreportedhereonF 2 weusedthe sphericalcomponentsof d basisfunctionsofthecc-pVDZbasisset[186],freezingthe core 1 s orbitalsofeachFatom.TheresultsatafewF{Fdistancesand thestatisticsforallthe F 2 resultsshowninTables5.8and5.9,andFigs.5.2(b)(compar edtofullCI)and5.3(b) (comparedtoCCSDT). 197 Table5.6:Comparisonoftheenergies(inmillihartree)ofR HF-andUHF-basedCCSD andvarioustriples-correctedCCapproximationswiththec orrespondingfullCIdata a for theequilibriumandfourdisplacedgeometries( R e =1 : 7328bohr)oftheHFmolecule,as describedbythesphericalDunningDZbasis. Method R e a 2 R e a 3 R e a 5 R e a MSE b MUE c NPE d CCSD RHF1.636.0511.6012.297.897.8910.66 UHF1.635.892.471.232.802.804.66 CC(T) e RHF0.330.04-24.48-53.18-19.327.2753.51 UHF0.331.261.220.040.710.711.22 CR(T) e RHF0.502.032.101.651.571.571.60 UHF0.502.741.340.151.181.182.58 CR(2,3),A e ; f RHF0.231.452.181.441.321.321.95 UHF0.232.401.18-0.010.950.952.41 CR(2,3),D e RHF-0.120.06-0.10-1.00-0.290.271.07 UHF-0.120.360.83-0.340.180.411.17 T e RHF0.170.860.960.430.600.600.78 UHF0.170.630.18-0.100.220.270.73 FullCI g -100.160300-100.021733-99.985281-99.983293 a FullCIresultsweretakenfromRef.[361]. b MeansignederrorrelativetofullCI. c MeanunsignederrorrelativetofullCI. d NonparallelityerrorrelativetofullCI. e FortheCCmethodswithuptotripleexcitations,thereporte denergyvalues,inkcal/mol,areerrors relativetofullCI.CC(T) CCSD(T);CR(T) CR-CCSD(T);CR(2,3),A CR-CC(2,3),A;CR(2,3),D CR-CC(2,3),D;T CCSDT. f EquivalenttoCCSD(2) T approachofRef.[77]. g ThetotalfullCIenergiesinhartree. 198 Table5.7:Comparisonoftheenergies(inmillihartree)ofR HF-andUHF-basedCCSDand varioustriples-correctedCCapproximationswiththeirpa rentCCSDTapproachdatafor theequilibriumandfourdisplacedgeometries( R e =1 : 7328bohr)oftheHFmolecule,as describedbythesphericalDunningDZbasis. Method R e a 2 R e a 3 R e a 5 R e a MSE a MUE b NPE c CCSD RHF1.465.1910.6411.867.297.2910.40 UHF1.465.262.291.332.592.593.93 CC(T) d RHF0.15-0.82-25.44-53.61-19.937.2953.77 UHF0.150.631.040.140.490.490.90 CR(T) d RHF0.331.181.141.220.970.970.89 UHF0.332.111.170.260.960.961.85 CR(2,3),A d ; e RHF0.060.591.221.010.720.721.16 UHF0.061.771.000.090.730.731.71 CR(2,3),D d RHF-0.29-0.79-1.05-1.44-0.890.370.76 UHF-0.29-0.270.65-0.24-0.040.360.94 T d ; f RHF-100.160127-100.020878-99.984324-99.982862 UHF-100.160127-100.021105-99.985103-99.983395 a MeansignederrorrelativetoCCSDT. b MeanunsignederrorrelativetoCCSDT. c NonparallelityerrorrelativetoCCSDT. d FortheCCmethodswithuptotripleexcitations,thereporte denergyvalues,inkcal/mol,areerrors relativetotherespectiveRHF-andUHF-basedfullCCSDT.CC (T) CCSD(T);CR(T) CR-CCSD(T); CR(2,3),A CR-CC(2,3),A;CR(2,3),D CR-CC(2,3),D;T CCSDT. e EquivalenttoCCSD(2) T approachofRef.[77]. f ThetotalCCSDTenergiesinhartree. 199 ExaminingtheRHF-andUHF-basedCCSD(T)resultsweseethew ellknownbehavior, withtheRHF-basedCCSD(T)calculationsprovidingunphysi calresultsatlargerinternu- cleardistances,resultinginanMSEofabout-7(-5)milliha rtree,MUEof ˘ 7( ˘ 7)milli- hartree,andaverylargeNPEvalueof40(41)millihartreewi threspecttotheRHF-based CCSDT(fullCI)approach.TheCCSD(T)resultsusingtheUHFd eterminantasarefer- encewavefunctionimprovetheRHF-basedCCSD(T)computati ons,changingtheMSEand MUEbothtoabout3(4)millihartree,withthemostdramaticc hangeintheNPEtothe valueofapproximately10(11)millihartreecomparedtothe parentfullUCCSDT(fullCI) method,demonstratingonceagainthestrongdependenceoft heCCSD(T)resultsonthe referencedeterminantemployed. Again,wecanseethequalityoftheCR-CCresultsarenotplag uedbythisrelianceonthe typeofdeterminantusedasreference,withtheCR-CCSD(T)a ndCR-CC(2,3),Aapproaches providingverysimilarresults,withtheCR-CC(2,3),Avari antimprovingtheCR-CCSD(T) calculationsbyabout1millihartreeasmeasuredbytheMSE, MUE,andNPEvalueswhen theRHFwavefunctionisusedandbyabout0.6millihartreefo rtheUHF-basedresults, comparedtobothCCSDTandthefullCIenergies.Ifwecompare theirRHF-andUHF- basedcomputationsweagainseeaslightimprovementinthei rMSEandMUEvaluesof lessthan1millihartree.ExaminingtheNPEvaluesfortheRH F-basedandCR-CCSD(T) andCR-CC(2,3),Aapproaches,whichareontheorderofabout 7millihartree,weseethat switchingtotheirUHF-basedcounterparts,theNPEsincrea seby4{5millihartreedueto spincontaminationoftheresultsattheintermediateinter nuclearbonddistances.Examining ourbestCR-CC(2,3),Dresults,weseeexcellentagreemento fourRHF-basedresultswith 200 CCSDT(fullCI),providingMSEofabout0.6(2.3),MUEofappr oximately0.9(2.3),and NPEof ˘ 2 : 1(4.0)millihartree.WhenweemploytheUHFdeterminantthe overallstatistics ofourCR-CC(2,3),D,whileimprovingontheUCCSD(T)result s,areslightlyworsethan theRHF-basedCR-CC(2,3),Dcalculations,withMSEandMUEo fabout2(3)millihartree withrespecttoCCSDT(fullCI)andagainamuchlargerNPEof ˘ 9( ˘ 10)millihartree, whichisstillanimprovementoftheUCCSD(T)calculations. AsshowninFig.5.3(b)alloftheUHF-basedapproximatetrip lesapproacheshavelarge errorsfromtheirparentCCSDTmethodwiththeCR-CC(2,3),D variantgivingthesmallest errorof8.5millihartree(5.4kcal/mol)inthisintermedia testretchedbondlengthregion.In thissameregiontheRHF-basedCR-CC(2,3),Dapproachprovi deserrorsofabout1milli- hartree( < 1kcal/mol).WhiletheUHF-basedCR-CCandCCSD(T)computat ionsapproach thefullCCSDTresultsintheasymptoticregionthelargeerr orssimilarinsizetosomebar- riersforchemicalreactionsmaketheiruseproblematic.Th eRHF-basedCR-CC(2,3),D,on theotherhand,providesareasonabledescriptionofthePES atallinternucleardistances withtheerrorsintheasymptoticregionbeing ˘ 1kcal/mol,typicallycalledchemicalac- curacy.AnotherimportantfeatureoftherobustRHF-basedC R-CC(2,3),Dapproach,as demonstratedinFig.5.2(b),whichshowstheerrorsrelativ etofullCI,isthatatallpoints alongthePESitscomputedenergiesfollowverycloselythef ullCCSDTvalues,beingshifted byanapproximatelyequalamount,demonstratingthesystem aticallyimprovablenatureand robustnessofthisapproach. 201 202 Table5.8:Comparisonoftheenergies(inmillihartree)ofR HF-andUHF-basedCCSDandvarioustriples-correctedCC approximationswiththecorrespondingfullCIdataforthee quilibriumanddisplacedgeometriesoftheF 2 molecule,as describedbythesphericalcc-pVDZbasisset. Method1.141.201.301.36 R e a 1.501.601.802.002.202.402.808.00MSE b MUE c NPE d CCSD RHF5.816.618.379.5810.7413.1416.2723.5630.7637.1041 .9746.9250.6223.1923.1944.81 UHF5.816.618.379.9811.9615.7219.5420.8714.157.984.7 32.422.0710.0210.0218.80 CC(T) e RHF0.881.031.401.591.722.042.271.89-0.71-5.18-10.66 -21.21-38.71-4.906.8740.98 UHF0.881.031.402.002.794.126.1511.7810.235.672.810. 630.303.833.8311.48 CR(T) e RHF1.581.842.482.883.244.075.127.419.1010.2910.9110 .427.185.895.899.33 UHF1.581.842.483.484.917.3910.0613.6810.795.963.040 .840.505.125.1213.17 CR(2,3),A e ; f RHF1.321.562.162.532.853.624.586.607.958.769.068.18 4.784.924.927.74 UHF1.321.562.162.793.695.688.3813.0510.595.802.890. 700.374.544.5412.68 CR(2,3),D e RHF0.420.530.871.061.201.672.223.363.964.304.373.61 2.622.322.323.95 UHF0.430.530.861.271.833.155.1910.529.695.452.670.4 6-0.073.233.2410.59 T e RHF0.750.861.191.341.451.792.082.592.522.422.381.74 1.021.701.701.84 UHF0.750.861.191.401.571.892.061.971.371.040.990.36 0.151.201.201.91 FullCI g 7.1848.1185.1095.1799.2099.8195.1080.9068.8261.6558 .2355.7755.45 a ExperimentalequilibriumF{FinternuclearbondlengthR e =1.41193 AtakenfromRef.[427]. b MeansignederrorrelativetofullCI. c MeanunsignederrorrelativetofullCI. d NonparallelityerrorrelativetofullCI. e FortheCCmethodswithuptotripleexcitations,thereporte denergyvalues,inkcal/mol,areerrorsrelativetofullCI. CC(T) CCSD(T); CR(T) CR-CCSD(T);CR(2,3),A CR-CC(2,3),A;CR(2,3),D CR-CC(2,3),D;T CCSDT. f EquivalenttoCCSD(2) T approachofRef.[77]. g ThetotalCEEISfullCIenergies E ,reportedas-(199+ mE ),areinhartree,takenfromRef.[428]. 203 Table5.9:Comparisonoftheenergies(inmillihartree)ofR HF-andUHF-basedCCSDandvarioustriples-correctedCC approximationswiththeirparentCCSDTapproachfortheequ ilibriumanddisplacedgeometriesoftheF 2 molecule,as describedbythesphericalcc-pVDZbasisset. Method1.141.201.301.36 R e a 1.501.601.802.002.202.402.808.00MSE b MUE c NPE d CCSD RHF5.065.747.188.249.2911.3714.1920.9728.2434.6839. 5945.1849.6121.4921.4944.54 UHF5.065.747.188.5910.3913.8317.4818.9112.786.943.7 42.071.928.828.8216.84 CC(T) e RHF0.130.160.220.250.270.280.18-0.70-3.23-7.61-13.0 5-22.95-39.72-6.606.6840.00 UHF0.130.160.220.601.222.234.109.828.874.631.820.28 0.162.632.639.68 CR(T) e RHF0.840.981.301.541.792.313.044.826.597.878.538.68 6.174.194.197.85 UHF0.840.981.302.093.345.508.0111.719.424.912.050.4 80.363.923.9211.36 CR(2,3),A e ; f RHF0.580.700.971.191.411.862.494.015.436.336.676.44 3.763.223.226.10 UHF0.580.700.971.402.123.796.3311.089.224.761.900.3 40.223.343.3410.86 CR(2,3),D e RHF-0.32-0.33-0.32-0.28-0.25-0.100.130.771.451.871. 991.871.610.620.872.08 UHF-0.31-0.33-0.32-0.130.261.263.138.558.324.411.69 0.10-0.222.032.238.88 T g RHF6.4447.2583.9193.8397.7598.0593.0278.3166.3059.2 355.8554.0354.43 UHF6.4447.2583.9193.7797.6397.9293.0578.9367.4560.6 157.2455.4155.30 a ExperimentalequilibriumF{FinternuclearbondlengthR e =1.41193 AtakenfromRef.[427]. b MeansignederrorrelativetoCCSDT. c MeanunsignederrorrelativetoCCSDT. d NonparallelityerrorrelativetoCCSDT. e FortheCCmethodswithuptotripleexcitations,thereporte denergyvalues,inkcal/mol,areerrorsrelativetotheresp ectiveRHF-and UHF-basedfullCCSDT.CC(T) CCSD(T);CR(T) CR-CCSD(T);CR(2,3),A CR-CC(2,3),A;CR(2,3),D CR-CC(2,3),D;T CCSDT. f EquivalenttoCCSD(2) T approachofRef.[77]. g ThetotalCCSDTenergies E ,reportedas-(199+ mE ),areinhartree. 204 Figure5.2:EnergydierencesofthevariousRHF-andUHF-ba sedcoupled-clusterapproacheswithsomeformoftriples withrespecttothefullCIresultsfor(a)theHFmoleculeand (b)theF 2 diatomicspeciesatvariousbondlengths. 5.2.2.3TheH 2 O 2 molecule Asmentionedabove,theH 2 O 2 moleculeservesasarepresentativemodelforO{Osingle bondbreaking,asitissmallenoughthathigh-levelCCSDTca lculationscanbeperformed forassessmentoftheperformanceofthevariousapproximat etriplesCCapproaches.As theCR-CC(2,3)andCCSD(T)methodsarestrictlysizeextens ivetheseresultscanbeused toextrapolatetheirperformanceinlargermoleculesinvol vingO{Obondcleavage,where CCSDTcalculationswouldbecomputationallyprohibitive. Weshouldalsomentionthat, whiletheCR-CCSD(T)approachisnotstrictlysizeextensiv e,itsviolationofthisproperty issmallandthus,theconclusionsdrawnhere,andasalready showninRef.[425],willalso beapplicablewhenexamininglargermolecules.Weusethecc -pVDZbasisset,freezingthe coreorbitals,andemployingspherical d functions,keepingtherestofthegeometryxedas westretchtheO{Obondlength,followingthesameprocedure asinRef.[425].Theresults atafewO{Odistancesandthestatisticsforallofourcalcul ationsarepresentedinTable 5.10andatalargernumberofO{ObondlengthsintheSuppleme ntaryMaterialandinFig. 5.3(c). WhiletheRHF-basedCCSD(T)resultsexhibitthesameunphys icalbehavioritiswell knownforinbondbreakingsituations,itisabitsurprising toseethattheUHF-based CCSD(T)resultsdolittletoimprovetheiroverallquality. ComparedtotheirparentCCSDT calculations,theRHF-basedandUHF-basedCCSD(T)computa tionsprovideMSEsof-4.4 and3.5millihartree,respectively,andMUEvaluesof4.6an d3.5millihartree,respectively. Inthepreviouscases,HFandF 2 ,thelargestimprovementwasespeciallyevidentintheNPE resultsfortheRHF-andUHF-basedCCSD(T)calculations.Fo rH 2 O 2 ,however,weseethere 205 isalmostnooverallimprovementwhentheUHFreferencedete rminantisused,resultingin NPEsof19.6and17.4millihartree,respectively.Thisisag oodexampleofwhereeven resortingtothespincontaminatedUHFreferencewavefunct ionwilldolittletoimprovethe poorRHF-basedCCSD(T)resultsforPESsalongsinglebondbr eakingcoordinates. TheCR-CCSD(T)andCR-CC(2,3),Aapproachesprovideverysi milarresults,withthe CR-CC(2,3),AvariantimprovingtheCR-CCSD(T)calculatio nsbyabout2{3millihartreeas measuredbytheMSE,MUE,andNPEvalueswhentheRHFwavefunc tionisusedandby about1millihartreefortheUHF-basedresults,comparedto theirparentCCSDTenergies. ComparingtheirRHF-andUHF-basedresults,weseetheMSEan dMUEloweringby ˘ 1 andincreasingby0.03millihartreefortheCR-CCSD(T)andC R-CC(2,3),Aapproaches, respectively.Inbothcasesthereisasignicantincreasei nNPEwhentheUHFdeterminant isemployed,by10.6millihartreeintheCR-CCSD(T)caseand 12.0millihartreeforCR- CC(2,3),A,resultinginNPEvaluesonthesameorderasforth eCCSD(T)calculations ( ˘ 20millihartree).TheRHF-basedCR-CC(2,3),Dresults,ont heotherhand,provide resultsontheorderof1millihartreefortheMSEandMUEwith itsNPEvalueremaining around2.5millihartreecomparedtoitsparentfullCCSDTap proach.Again,theeects ofspincontaminationfortherecouplingoftwodoubletOHra dicalsintoasingletwhen usingtheUHFreferencewavefunctionismanifestintheslig htlylarger,thoughimproved comparedtoUCCSD(T),MSEandMUEvaluesofabout3millihart ree.TheNPEforthe UHF-basedCR-CC(2,3),Dcomputationsisslightlylessthan thatforUCCSD(T),butstill quitelarge,beingapproximately17millihartree. 206 207 Table5.10:Comparisonoftheenergies(inmillihartree)of RHF-andUHF-basedCCSDandvarioustriples-correctedCC approximationswiththeirparentCCSDTapproachfortheequ ilibriumanddisplacedgeometriesoftheH 2 O 2 molecule, a asdescribedbythesphericalcc-pVDZbasis. Method1.101.201.301.40 R e a 1.501.601.701.801.902.002.102.202.302.402.502.602.7 02.802.903.00MSE b MUE c NPE d CCSD RHF6.547.017.768.789.0110.1011.7613.7916.2218.9922. 0225.1828.3331.3434.1336.6438.8440.7442.3643.7244.8 723.7223.7238.33 UHF6.547.017.768.789.0110.1012.9116.1718.8629.5918. 8016.2913.3110.558.306.615.454.714.294.073.9610.621 0.6225.64 CC(T) e RHF0.210.210.230.260.270.300.330.340.280.08-0.34-1. 07-2.18-3.68-5.54-7.69-10.03-12.45-14.84-17.13-19.2 7-4.374.6119.60 UHF0.210.210.230.260.270.301.653.195.3217.589.459.2 87.845.974.202.781.761.110.740.550.473.493.4917.37 CR(T) e RHF1.231.331.501.761.822.102.563.163.894.755.686.63 7.538.328.999.529.9210.2010.3910.5110.575.835.839.2 4 UHF1.231.331.501.761.822.104.486.969.2320.6711.4710 .568.736.654.783.302.251.591.201.010.924.934.9319.7 5 CR(2,3),A e ; f RHF0.780.860.991.181.221.441.792.262.843.524.264.99 5.666.236.676.987.187.297.337.327.284.194.196.50 UHF0.780.860.991.181.221.442.704.657.1119.1610.5610 .018.346.334.493.042.001.340.960.770.684.224.2218.4 7 CR(2,3),D e RHF-0.22-0.24-0.28-0.28-0.27-0.18-0.12-0.030.190.48 0.821.181.481.741.952.092.182.232.272.282.280.931.0 82.52 UHF-0.22-0.24-0.28-0.28-0.27-0.180.712.264.4816.718 .688.687.375.583.852.471.490.860.500.320.222.993.13 16.96 a Theequilibriumgeometry,with R e = R O-O =1 : 419952 A optimizedattheM06-2X/MG3SlevelinRef.[425].Theotherg eometriesrepresent astretchingoftheO{Obondwithoutchangingtheothergeome tricparameters,andaretakenfromtheSupportingInformat iontoRef.[425]. b MeansignederrorrelativetoCCSDT. c MeanunsignederrorrelativetoCCSDT. d NonparallelityerrorrelativetoCCSDT. e FortheCCmethodswithuptotripleexcitations,thereporte denergyvalues,inkcal/mol,areerrorsrelativetotheresp ectiveRHF-and UHF-basedfullCCSDT.CC(T) CCSD(T);CR(T) CR-CCSD(T);CR(2,3),A CR-CC(2,3),A;CR(2,3),D CR-CC(2,3),D;T CCSDT. f EquivalenttoCCSD(2) T approachofRef.[77]. ThedierenceswithrespecttotheirparentCCSDTapproacha reshowninFig.5.3(c), withallUHF-basedapproximatetriplesapproachesshowing evenlargerdierenceswiththe UCCSDTcalculations,rangingfrom16{20millihartree(10{ 13kcal/mol),intheintermediate O{Obondlengthregion.Thus,itseemsthatanyimprovements gainedbyturningtothe UHF-basedCCapproacheswithanapproximatetreatmentoftr iples,particularlyinthe asymptoticregionofthePESalongsinglebondbreakingcoor dinates,arefaroutweighed bytheinaccuraciesintroducedduetospincontaminationat theintermediateinternuclear distances.WhileonecannotusetheRHF-basedCCSD(T)appro achfortheseproblems either,theRHF-basedCR-CC(2,3),Dmethoddoesprovidever ygoodagreementwiththe fullCCSDTresultsnearequilibrium,attheintermediatebo ndlengths,andintheasymptotic region,wherethelargesterrorsdonotexceed2.3millihart ree(1.5kcal/mol). 5.2.2.4TheC 2 H 6 molecule InanalogytoH 2 O 2 ,theC 2 H 6 moleculeisareasonablemodelforC{Csinglebondbreaking, whichissmallenoughtobeabletoaordthefullCCSDTcalcul ationsforgaugingthe reliabilityofthevariousnoniterativeapproximatetripl esCCmethods,whichcanbeused toextrapolatetheirperformanceinlargermolecularsyste mswhereCCSDTcomputations arenotroutinelyaordable.Weagainfollowtheprocedureo fRef.[425],freezingthecore orbitalsinallCCcalculations,employingthecc-pVDZbasi ssetandspherical d functions, andnotallowingthegeometrytorelaxaswestretchtheinter nuclearC{Cbond.Theresults atafewC{Cinternucleardistancesandthestatisticsforal lofourcomputationsforC 2 H 6 arepresentedinTable5.11andFig.5.3(d). TheCCSD(T)resultsagainshowsomeimprovementwhentheUHF determinantisused 208 overtheRHFreferencewavefunction,withtheMSEchangingf rom-1.0to2.3millihartree intheRHF-andUHF-basedCCSD(T)computations,respective ly.Muchlikeinthecase ofH 2 O 2 though,theMUEshowsnochangebeingabout2.3millihartree inbothcases. WherethereisalargeimprovementisintheirNPEvalues,whi chchangefromaround22 to7millihartreeemployingtheRHFandUHFdeterminants,re spectively,intheCCSD(T) calculations. TheCR-CCmethodsareinterestinginthiscaseinthatonlyth eCR-CCSD(T)com- putationsshowanoverallimprovementintheMSEandMUEvalu eswhentheUHFde- terminantisusedasareferencewavefunctionandthisimpro vementisonlyminor( ˘ 0 : 3 millihartree).IntheCR-CC(2,3),AresultstheMSEandMUEi ncreasebyanevensmaller amount(about0.08millihartree)andtheCR-CC(2,3),DMSEa ndMUEincreasingby0.7 millihartreewhentheUHFwavefunctionisemployed.Inspit eofthisincreasewhenswitch- ingtotheUHFreferencefunctionourCR-CC(2,3),Dresultss tillmanagetoimprovethe UCCSD(T)calculations,withourCR-CC(2,3),AandCR-CCSD( T)approachesgivingcom- parableresultstoUCCSD(T).TheNPEsforallUHF-basedCCap proachesexaminedinthe thiswork,whichincludesomeformofapproximatetripleexc itations,rangefrom6.6to8.5 millihartree,withourbestUCR-CC(2,3),Dapproachprovid ingthesmallestNPEvalue,im- provingtheUCCSD(T)approximationsdescriptionoftheC{C singlebondbreakingPESin C 2 H 6 .WhiletheUCR-CC(2,3),DmethodimprovestheUCCSD(T)resu lts,theRHF-based CR-CC(2,3),Dvariantprovidesthebestoveralldescriptio n,withallerrorsnotexceeding2.2 millihartreeandinsomecasesbeingsignicantlylower. 209 210 Table5.11:Comparisonoftheenergies(inmillihartree)of RHF-andUHF-basedCCSDandvarioustriples-corrected CCapproximationswiththeirparentCCSDTapproachforthee quilibriumanddisplacedgeometriesoftheCH 3 CH 3 molecule, a asdescribedbythesphericalcc-pVDZbasis. Method1.101.201.301.401.50 R e a 1.601.701.801.902.002.102.202.302.402.502.602.803.0 03.203.504.005.00MSE b MUE c NPE d CCSD RHF8.548.488.528.638.818.869.069.389.7810.2610.8511 .5512.3813.3514.4715.7417.1620.3523.8227.2231.6836. 6740.5815.9215.9232.10 UHF8.548.488.528.638.818.869.069.389.7810.2610.8511 .5613.1014.7516.3217.5518.1617.0713.9110.717.716.41 6.2811.0711.0711.88 CC(T) e RHF0.540.560.570.590.620.630.650.680.730.770.830.89 0.961.021.081.121.120.900.19-1.16-4.35-11.07-20.94- 1.002.2622.06 UHF0.540.560.570.590.620.630.650.680.730.770.830.90 1.832.853.995.286.587.986.844.602.080.940.862.262.2 67.43 CR(T) e RHF1.991.992.022.072.132.152.222.332.462.622.823.07 3.363.714.124.595.126.297.518.599.7510.4610.054.414 .418.47 UHF1.991.992.022.072.132.152.222.332.462.622.823.08 4.686.207.598.809.709.998.165.643.001.831.734.144.1 48.26 CR(2,3),A e ; f RHF1.321.331.341.371.411.421.461.531.611.721.852.02 2.232.482.793.143.534.415.316.076.807.046.382.982.9 85.71 UHF1.321.331.341.371.411.421.461.531.611.721.852.03 2.823.825.036.377.638.797.435.082.501.351.253.063.0 67.46 CR(2,3),D e RHF0.080.080.090.100.100.100.120.140.170.210.260.30 0.350.430.530.660.841.181.591.952.272.382.300.710.7 12.20 UHF0.080.080.090.100.100.100.120.140.170.210.260.31 0.811.512.453.604.866.726.064.051.620.430.031.471.4 76.64 a Theequilibriumgeometrywith R e = R C-C =1 : 5227 A istakenfromRef.[429].Theothergeometriesrepresentast retchingoftheC{C bondwithoutchangingtheothergeometricparametersweret akenfromRef.[425]. b MeansignederrorrelativetoCCSDT. c MeanunsignederrorrelativetoCCSDT. d NonparallelityerrorrelativetoCCSDT. e FortheCCmethodswithuptotripleexcitations,thereporte denergyvalues,inkcal/mol,areerrorsrelativetotheresp ectiveRHF-and UHF-basedfullCCSDT.CC(T) CCSD(T);CR(T) CR-CCSD(T);CR(2,3),A CR-CC(2,3),A;CR(2,3),D CR-CC(2,3),D;T CCSDT. f EquivalenttoCCSD(2) T approachofRef.[77]. ExaminingtheenergydierencesofthevariousRHF-andUHF- basedCCapproaches comparedtotheirparentCCSDTmethod,asshowninFig.5.3(d ),wecanagainseethat, whileallUHF-basedCCmethodswithanapproximateformoftr iplesprovideresultswhich approachthefullCCSDTenergiesasymptotically,attheint ermediatestretchedC{Cbond distancestheyagainproduceerrorsontheorderof7to10mil lihartree(4{6kcal/mol). Again,itisonlytheRHF-basedCR-CC(2,3),Dapproachwhich isabletoprovideanaccurate descriptionofthePESalongtheC{Csinglebondbreakingcoo rdinateatallinternuclear separations,withthelargesterrorsnotexceeding2.4mill ihartree(1.5kcal/mol),producing resultsofthechemicalaccuracy( ˘ 1kcal/mol)oftentoutedasthestandardbywhichto measuretheperformanceof abinitio electronicstructuremethods. 211 212 Figure5.3:EnergydierencesofthevariousRHF-andUHF-ba sedapproximatetriplescoupled-clustermethodswith respecttotheirparentCCSDTresultsfor(a)theHFmolecule ,(b)theF 2 diatomicspecies,(c)theH 2 O 2 species,and (d)theC 2 H 6 polyatomicmoleculeatvariousbondlengths. Aspartoftheworkdoneforthisdissertationwehaveexamine dtheperformanceofthe UHF-basedCR-CCapproacheswithapproximatetriplescorre ctions,namely,CR-CCSD(T), CR-CC(2,3),A,andCR-CC(2,3),D,fordescribingpotential energysurfacesalongsinglebond breakingcoordinatesforclosed-shellsingletmoleculesd issociatingintoopen-shelldoublet species,fortheHF,F 2 ,H 2 O 2 ,andC 2 H 6 molecularspecies,comparingtheCR-CCcom- putationswiththefullCIandtheparentCCSDTdataaswellas withtheirRHF-based counterparts. UnlikethepopularCCSD(T)approach,whichfailstoaccurat edescribePESsalongbond breakingcoordinateswhentheRHFdeterminantisusedasare ferencewavefunctionand whichgenerallyisconsideredtoberescuedbyturningtothe UHFreferencedeterminant, introducingotherunphysicaleects,suchasspincontamin ationandnonanalyticbehaviorof thePES,ourCR-CCmethodsarenotverysensitivetotherefer encedeterminantemployed (RHF vs UHF).OurbestRHF-basedCR-CC(2,3),Dresultsimprovedram aticallytheRHF- basedCCSD(T)calculationsandtheUHF-basedCR-CC(2,3),D approach,whilenotas markedasintheRHFcasethereisstillanimprovementoverth eUHF-basedCCSD(T) resultsforthecomputedPESsparticularlyintheregionofi ntermediatestretchedbond lengthswherethespincontaminationbeginstorenderthere sultsuseless,providingerrors ontheorderofabout10kcal/mol,ormoreinsomechallenging cases,forallUHF-based approximatetriplesapproaches. FromourresultsinthisworkwecannotrecommendtheUHF-bas edCCapproacheswith approximatetriplesforanaccuratedescriptionofPESsalo ngbondbreakingcoordinatesdue totheunphysicaleects,whichinchallengingcasescanbeq uitelarge,ofspincontamination 213 providingerrorsontheorderofsomechemicalreactionbarr iers.ThiseectfromusingUHF- basedapproximatetriplesCCmethodsdestroysoneoftheimp ortanthallmarkattributes thathasledtotheirbeingusedforchallengingchemicalsit uations,specicallythatofbeing abletoprovidequantitativelycorrectresultscomparedto theirmuchmoreexpensiveparent CCSDTapproaches.WhileonecannotusetheRHF-basedCCSD(T )approximationfor anaccuratedescriptionofPESsalongbondbreakingcoordin atesourrobustRHF-based CR-CC(2,3),Dvariantcanprovideanexcellentquantitativ edescription,comparedtoboth itsparentCCSDTandtheexactfullCIresults,ofsinglebond breakingonasingletPES, whileavoidingtheproblematicbehavioroftheRHF-andUHF- basedCCSD(T)method.In thefuturewewouldliketoexaminetheperformanceoftheUHF -basedCR-CCapproaches comparedtotheirROHF-basedanalogsforbond-breakingino pen-shellspecies. 214 Chapter6 Conclusionsandfutureoutlook Inthisdissertation,wehavedescribedseveralhigh-level abinitio computationalstudiesem- ployingtheCR-CC/CR-EOMCCandactive-spaceCCapproaches andtheextensionsofthe EOMCCtheorytoopen-shellsystemsaroundclosedshellsde ningtheEA-EOMCCand IP-EOMCCframeworks,demonstratingthetransformativero lethesenovelelectronicstruc- turemethods,developedinourgroup,haveplayedinunderst andingpreviouslyunexplained experimentsandphenomena.UsingtheEA-andIP-EOMCCappro aches,especiallythe higher-order3 p -2 h and3 h -2 p approachesinventedinthePiecuchresearchgroup,wehave computedthechallengingelectronicspectraoftheCNC,C 2 N,N 3 ,andNCOmoleculesand thephotoelectronspectrumofAu 3 ,providingforthersttimeanaccurateinterpretation ofthespectrumofthelatter.TheCR-EOMCCformalismdevelo pedinourgroupplayed acrucialroleinthediscoveryofthedoublyexcitedstateof azulenebelowtheionization threshold,whichmediatesthe1+2 0 multiphotonionizationexperimentsresultinginclear Rydbergngerprintspectra.EmployingtheCR-CC(2,3)meth odology,developedinour groupaswell,wecarriedoutadetailedinvestigationofthe mechanismandenergeticsof theaerobicoxidationofmethanolonAu 8 particleconrmingtheearlierDFT-basedpro- posalsthatthereactionproceedsexothermicallyandthatt herate-determiningstepforthe reactionistheinitialconversionofthemethoxyspeciesto formaldehyde.Wealsocarried 215 outdenitiveCR-CCandactive-spaceCCstudiesshowingtha tthegroundstateof1,2,3,4- cyclobutanetetraone,whichischaracterizedbydenselysp acedlow-lyingstates,isatriplet, inagreementwiththerecentlyrecordedphotodetachmentsp ectrum.Inadditiontothese applicationstochallengingandchemicallyrelevantprobl emswediscussedthedevelopment ofparallelnumericalenergygradientsandsecondderivati vesforfastgeometryoptimizations andharmonicvibrationalfrequencycalculationsatanyCC/ EOMCClevel,allowingusto establishthegeometriesandrelativeenergiesofthelow-e nergyisomersofthecontroversial Au 8 particle.Wealsodiscussedtheimplementationoftheunres trictedHartree-Fock-based (UHF-based)CR-CC(2,3)approach.Weshowthatunlikethepo pularCCSD(T)approach, whichisverysensitivetothetypeofthereferencedetermin antemployedinthecalculations, failinginbond-breakingsituationswhentherestrictedHa rtree-Fock(RHF)referenceisused anddisplayingpoorbehavioratintermediatenuclearsepar ationswithUHFreferences,its CR-CC(2,3)counterpartprovidesarobustdescriptionrega rdlessofthereferencetype(RHF orUHF).Wealsoshowedthatthespin-adaptedRHF-basedCR-C C(2,3)resultsprovidethe mostaccuratedescriptionofthesinglepotentialenergysu rfaceforsinglebondbreaking coordinatesthanallUHF-basedapproximatetriplesCCappr oachesintheexaminedcases. Infuturestudies,wewouldliketostudycatalyticreaction semployinglargergoldnanopar- ticles,likethenon-planarAu 20 cluster,bimetalliccatalysts,suchasPd:Ausystems,aswe ll assilvercontainingcatalyticparticles.Instudyingthes etypesofsystemswewilllikely needtomakeuseoflocalcorrelationapproaches[99{103]an d/orextrapolationtechniques, suchasthoseexploredaspartofthisdissertation.Further explorationofphotochemical problemsinvolvingmultiphotonprocessesanddetermining reactionpathwaysontheground 216 andexcitedstates,whichinvolvelocatingandcharacteriz ingminimumenergycrossingsand seamsofconicalintersections,usingtheCIOptpackage,wh ichisabletoperformthesefor anyleveloftheory,shouldbeinvestigated.Thiswillprese ntitsownsetofchallengesand problemsastheEOMCCwavefunctionansatzresultsinanon-H ermitianeigenvalueproblem andwhichisknowntohaveissuesdescribingthetopologyand locationoftheseminimum energycrossingpoints[430{433].Thedevelopmentofstrin g-basedapproachestosearchfor transitionstatesonthegroundandexcitedstatepotential energysurfacesaswellastheir usetolocateconicalintersectionsandtoexploretheirsea mswillalsobepursuedinfuture workbuildingonwhathasalreadybeendoneasapartofthisdi ssertation. ThedevelopmentofMMCCcorrectionsforthenon-particleco nservingEA-,IP-,DEA-, DIP-EOMCCapproachesshouldalsobepursuedasitwasorigin allyproposedaspartofthis dissertationfortheEA-andIP-EOMCCmethodologies,thoug hmorethanworkingoutthe relevantequationshasnotbeendone.Thisisalsoastepinth edirectionofextendingthe CC(P;Q)typeapproachestotheactive-spaceEA-,IP-,DEA-, andDIP-EOMCCmethods. Thiswillhopefullyallowfortheextremelyhigh-levelstud yofchallengingopen-shellmolec- ularsystems,whereaccuratemethodologiesgreatlyhelpin thepredictionandinterpretation oftheirvariousproperties. Thepursuitofhigher-orderparallelnumericalderivative algorithmswillalsobeunder- takeninfuturestudiesalongwiththecompletionofa B -matrixroutineforreducingthe numberofdegreesoffreedomtotheminimumrequiredforaccu ratelydescribingmolecular species.Justasthisdissertationhasbeencarriedoutandf acilitatedthroughthehelpand supportofmorethanjusttheauthorofthisdissertationsow illthefuturestudiesandde- 217 velopmentsbecarriedoutasacontinuedcollaborativeeor twiththemanynescientistsI havemetandwillyetmeetandworkwith. 218 BIBLIOGRAPHY 219 BIBLIOGRAPHY [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.A 5 ,50(1972). [7]J.Gauss,In EncyclopediaofComputationalChemistry ,editedbyP.v.R.Schleyer, N.L.Allinger,T.Clark,J.Gasteiger,P.A.Kollman,H.F.Sc haefer,III,andP.R. 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