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Extending the Capabilities of Accurate Ab Initio Methods

Permanent Link: http://ufdc.ufl.edu/UFE0044477/00001

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Title: Extending the Capabilities of Accurate Ab Initio Methods Novel Algorithms and Massively Parallelizable Implementations for Feasible Calculations
Physical Description: 1 online resource (179 p.)
Language: english
Creator: Watson, Thomas J Jr
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: algorithms -- coupled-cluster -- parallel-implementation -- perturbation-theory -- quantum-chemistry
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The most general methods in electronic structure theory, capable of attaining chemical accuracy, are  coupled cluster (CC) methods.  Unfortunately, the ``gold-standard'', CCSD(T), formally scales as N^7, where N is the size of the chemical system.  Consequently, practical brute force computations  with these methods on systems of O(10) atoms is out of the question with the serial programs  that have been developed.  This wall needs to be overcome to allow feasible   calculations for moderately sized chemically relevant systems with high accuracy. In this vein, I present different approaches to tackle this scaling problem.   First, I use an exact, in the limit, method that reduces the computational effort of calculations.   Specifically, a formally exact procedure for  decoupling the four-component relativistic Dirac equation is implemented, in a general way in the ACESII and ACESIII program packages, and used in conjunction with  IP-EOM-CCSD methods to improve core ionization energies.   Another way to overcome the scaling of CC methods is to simply develop new \textit{ab initio} theoretical models that are more computationally tractable.  I propose a new method with mean-field cost, $N^4$, that obtains core ionization energies to a high degree of accuracy as opposed to increasing the complexity of IP-EOM-CC methods, which would  include the same amount of relaxation as the proposed method in the limit only with exponential scaling, $N^N$.  Furthermore, the proposed method is unambiguous in describing orbital relaxation effects, the dominant energetic effect in core ionizations of organic molecules.  Therefore, a perturbation theory analysis allows quick corrections to the computationally tractable IP-EOM-CCSD method. The final way to push CC methods further is to use modern computer architecture and parallelize over processors.  Equation of motion CC (EOM-CC) methods are  implemented in this way to allow  benchmark studies on biologically relevant molecules.  EOM-CC properties, including dipole, second, and transition moments are implemented in this work, as well as the EOM-CCSD(T) method (and other triples extensions), an analogue of the very  successful CCSD(T) method, but for excited states.  Gradients for EOM-CCSD theory are implemented as well, paving the way for routine studies of potential energy surfaces and dynamics of excited states within the CC framework.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Thomas J Jr Watson.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Bartlett, Rodney J.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044477:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044477/00001

Material Information

Title: Extending the Capabilities of Accurate Ab Initio Methods Novel Algorithms and Massively Parallelizable Implementations for Feasible Calculations
Physical Description: 1 online resource (179 p.)
Language: english
Creator: Watson, Thomas J Jr
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: algorithms -- coupled-cluster -- parallel-implementation -- perturbation-theory -- quantum-chemistry
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The most general methods in electronic structure theory, capable of attaining chemical accuracy, are  coupled cluster (CC) methods.  Unfortunately, the ``gold-standard'', CCSD(T), formally scales as N^7, where N is the size of the chemical system.  Consequently, practical brute force computations  with these methods on systems of O(10) atoms is out of the question with the serial programs  that have been developed.  This wall needs to be overcome to allow feasible   calculations for moderately sized chemically relevant systems with high accuracy. In this vein, I present different approaches to tackle this scaling problem.   First, I use an exact, in the limit, method that reduces the computational effort of calculations.   Specifically, a formally exact procedure for  decoupling the four-component relativistic Dirac equation is implemented, in a general way in the ACESII and ACESIII program packages, and used in conjunction with  IP-EOM-CCSD methods to improve core ionization energies.   Another way to overcome the scaling of CC methods is to simply develop new \textit{ab initio} theoretical models that are more computationally tractable.  I propose a new method with mean-field cost, $N^4$, that obtains core ionization energies to a high degree of accuracy as opposed to increasing the complexity of IP-EOM-CC methods, which would  include the same amount of relaxation as the proposed method in the limit only with exponential scaling, $N^N$.  Furthermore, the proposed method is unambiguous in describing orbital relaxation effects, the dominant energetic effect in core ionizations of organic molecules.  Therefore, a perturbation theory analysis allows quick corrections to the computationally tractable IP-EOM-CCSD method. The final way to push CC methods further is to use modern computer architecture and parallelize over processors.  Equation of motion CC (EOM-CC) methods are  implemented in this way to allow  benchmark studies on biologically relevant molecules.  EOM-CC properties, including dipole, second, and transition moments are implemented in this work, as well as the EOM-CCSD(T) method (and other triples extensions), an analogue of the very  successful CCSD(T) method, but for excited states.  Gradients for EOM-CCSD theory are implemented as well, paving the way for routine studies of potential energy surfaces and dynamics of excited states within the CC framework.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Thomas J Jr Watson.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Bartlett, Rodney J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044477:00001


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EXTENDINGTHECAPABILITIESOFACCURATEABINITIOMETHODS:NOVELALGORITHMSANDMASSIVELYPARALLELIZABLEIMPLEMENTATIONSFORFEASIBLECALCULATIONSByTHOMASJ.WATSONJR.ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012ThomasJ.WatsonJr. 2

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Tomywife,Sondra 3

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ACKNOWLEDGMENTS MyworkhereatUFhasbeenbothchallengingandrewarding,andwouldnothavebeensowithoutthedirectionofmyadvisor,ProfessorBartlett.IhavetothankDr.Bartlettforallowingmetofreelypursuecountlessdirections,somefruitful,othersnecessarytohelpmeredirectmyambitions.Dr.Bartlettneverletmeslowdown,pushedmewhenIthoughtIcouldnotbepushed,andwasalwaysfreetotalkatlengthaboutthefuturedirectionsofmywork.Criticaltomyunderstandingofallthingsgraduateschool,quantumchemistry,andlifehereinGainesville,wereDrs.AndrewTaube,JoshMcClellan,TomHughes,andDanSindhikara.Ihavetothankthesegentlemenfortransitioningmetowardsrobustandrigorousscienticendeavors;andhelpingremovetheintimidationIfeltasarstyeargraduatestudent.IowemuchofmysanityinmylateryearsatUFtoJasonSwails,BillyMiller,MatthewStrasberg,AlexBazante,RobertMolt,JonvanderHest,andAnnMelnichuk.Theymadethedaytodayoperationsmorethanenjoyableandofferedsomuchinsightfromtheiruniqueperspectivesgiventheirresearchprojectsandbackgrounds.IcannotevenbegintoexpressmygratitudetowardsDrs.AjithPerera,VictorLotrich,andDmtriLyakh.Myprogrammingskillswouldbesubpar,myunderstandingofhowquantumchemistryisimplementedintorobustcomputercodeswouldberudimentary,andingeneral,mypragmaticquantumchemistryskillswouldbelackingifnotforthesegentlemen.QTPofferssuchtremendousopportunitiesviatheopennessandavailabilityofthefacultymemberswithinthelabrynthoftheNPB.IwouldliketoespeciallythankDrs.ErikDeumens,AdrianRoitberg,andYngveOhrnfortheirtrulygeneroushelpandadvice.Also,Dr.KennieMerzforgraciouslysupportingmeforasummer,Dr.PeterSzalayforhisunderstanding,advice,andcollaboration,andMrs.JudyParker,whowasalwaysreadytolendanearandofferdiplomaticsolutions. 4

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Finally,Iwouldliketothankmyfamily,especiallymywife,Sondra,mybrothers,Dave,Chris,andDr.ChrisFureyaswellasmysister,Jen,foralltheirsupportduringmytimehere. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 10 LISTOFSYMBOLS .................................... 12 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1AbInitioQuantumChemistry ......................... 15 1.2TheHartree-FockMean-FieldApproximation ................ 23 1.2.1FormalScalingandRate-DeterminingSteps ............. 25 1.2.2Scalar-RelativisticHamiltonians: ................... 26 1.3IncludingElectronCorrelation ......................... 27 1.3.1CongurationInteraction ........................ 27 1.3.2Coupled-ClusterTheory ........................ 29 1.4ExploringMoreSectorsofFockSpace .................... 37 1.5ImplementationinRobustComputerCodes ................. 38 1.6OutlineofthisStudy .............................. 44 2DECOUPLINGTHEDIRACEQUATIONFORMEANFIELDCOSTAPPLIEDTOCOREIONIZATIONS .............................. 46 2.1Preliminaries .................................. 46 2.2Theory ...................................... 49 2.3Implementation ................................. 55 2.4ResultsandDiscussion ............................ 56 2.4.1OrganicMolecules ........................... 56 2.4.2NobleGases .............................. 57 2.4.3HighlyStrippedIons .......................... 60 2.5Conclusions ................................... 62 3ANUNTRADITIONALAPPROACHTOCOREIONIZATIONENERGIES .... 73 3.1Preliminaries .................................. 73 3.2Theory ...................................... 76 3.2.1DirectESCFRealizedThroughVariationalCCS ........... 76 3.2.2Third-OrderRelaxationCorrection .................. 81 3.2.3RelaxationwithECCMethods .................... 82 3.3Implementation ................................. 83 6

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3.4ResultsandDiscussion ............................ 83 3.4.1IP-VCCS ................................. 83 3.4.2Third-OrderCorrectiontoIP-EOM-CCSD .............. 86 3.5ConclusionsandOutlook ........................... 88 4MASSIVELYPARALLELIMPLEMENTATIONOFEOM-CCMETHODSFOREXCITEDSTATES .................................. 96 4.1Preliminaries .................................. 96 4.2GeneralTheory ................................. 102 4.3ExpectationValueFormforOne-ElectronProperties ............ 103 4.3.1Implementation ............................. 107 4.3.2ResultsandDiscussion ........................ 109 4.4PerturbativeTriplesforAccurateEnergies. .................. 110 4.4.1ParallelizationStrategiesforEOM-CCSD(T) ............. 119 4.4.2ApproximatingtheLeftHandEigenvector .............. 121 4.4.3ComparisonofTriplesMethods .................... 122 4.4.3.1Computationaldetails .................... 122 4.4.3.2Results ............................ 123 4.4.4ApplicationtoNucleobasesandTheirComplexes .......... 126 4.5GradientTheoryforAccurateCriticalPointGeometries .......... 128 4.5.1Implementation ............................. 133 4.5.2DemonstrablePerformance ...................... 139 4.5.3ExcitedStateGeometriesforNucleobases .............. 139 4.6ConclusionsandOutlook ........................... 140 5CONCLUDINGREMARKS ............................. 160 APPENDIX:SUPPLEMENTARYMATERIAL ....................... 163 A.1DataforCorrelationPlotsofVariousTriplesMethods ............ 163 REFERENCES ....................................... 170 BIOGRAPHICALSKETCH ................................ 179 7

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LISTOFTABLES Table page 2-1ConvergenceofDKpotentialforArgon ....................... 64 2-2IonizationpotentialsofArgonwithIP-EOM-CCSD/DK .............. 65 2-3ConvergenceoftheDKpotentialforKrypton ................... 66 2-4IonizationpotentialsofKryptonwithIP-EOM-CCSD/DK ............. 67 2-5ConvergenceoftheDKpotentialforXenon .................... 68 2-6IonizationpotentialsofXenonwithIP-EOM-CCSD/DK .............. 69 2-7Ionizationpotentialsofhelium-likeionswithIP-EOM-CCSD/DKinPartridgebasissets ....................................... 70 2-8Ionizationpotentialsofberyllium-likeionswithIP-EOM-CCSD/DKinPartridgebasissets ....................................... 71 2-9Ionizationpotentialsofhelium-likeionswithIP-EOM-CCSD/DK/WTBS ..... 71 2-10Ionizationpotentialsofberyllium-likeionswithIP-EOM-CCSD/DK/WTBS ... 72 3-1DemonstrationoftheexactIP-VCCSmethod ................... 91 3-2DemonstrationoftheapproximateIP-VCCSmethod ............... 92 3-3DoublecoreionizationenergiesoforthoaminophenolwithIP-VCCS ...... 92 3-4IP-VCCSappliedtobrokensymmetryreferences ................. 93 3-5IP-VCCSappliedtopropenetodemonstratelackofspincontamination .... 93 3-6DeviationofIPsfromexperimentwithdouble-basisset ............. 93 3-7DeviationofIPsfromexperimentintriple-basissets .............. 94 3-8Effectoftriplesoncoreionizationenergies .................... 94 3-9ThirdorderrelaxationcorrectiontoIP-EOM-CCSD ................ 95 4-1Demonstrationof^=^Tyand^Lk=^Rykapproximationincomputingoscillatorstrengths. ....................................... 153 4-2Assessmentof^Lk=^RykinEOM-CCSD(T)calculations. .............. 154 4-3ParametersusedinEOM-CCSD(T)performancecurves ............. 154 4-4TimingsforEOM-CCSD(T)appliedtothecytosine-guanineWatson-Crickbasepair .......................................... 154 8

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4-5Effectofmono-hydrationonexcitationenergiesofcytosine ........... 155 4-6Excitationenergiesandoscillatorstrengthsofhydratedcytosine ........ 156 4-7Excitationenergiesofcytosineandcytidine .................... 157 4-8Excitationenergiesofguanineandguanosine ................... 158 4-9Energiesandoscillatorstrengthsforcytosineandguaninebasepair ...... 159 A-1Comparisonofperturbativetriplesmethods .................... 163 9

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LISTOFFIGURES Figure page 2-1AveragedifferencebetweenDK5andDK0fordifferentmethods. ........ 63 2-2DifferencebetweenDK5andDK0forasubsetofmolecules. .......... 63 2-3AverageerrorofdifferentDKpotentialsandmethods. .............. 63 3-1Groundstategeometryoforthoaminophenolmolecule. ............. 90 3-2ContributionstocoreIPusingEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSD(correlationandrelaxation)andIP-VCCS(relaxation)withcc-pCVDZbasisset. .................. 90 3-3DemonstrationofconvergenceofIP-VCCS. .................... 90 4-1Errorinoscillatorstrengthusing^Lk=^Ryk. ..................... 142 4-2CorrelationplotforcalculatedsingleexcitedstateswithCCSDvs.CCSDT-3. 142 4-3CorrelationplotforcalculatedsingleexcitedstateswithEOM-CCmethodsvs.CCSDT-3. ....................................... 142 4-4CorrelationplotforcalculatedsingleexcitedstateswiththelinearresponseCCmethodsvs.CCSDT-3. ............................. 143 4-5CorrelationplotforsingleexcitedstateswithCCSD(T)vs.CCSDR(3). ..... 143 4-6MaximumerrorplotusingEOM-CCSD(T)andCCSDR(3) ............ 143 4-7EfciencyforcompleteEOM-CCSD(T)calculationoverarangeofCPUs. ... 144 4-8EfciencyforthetriplespieceoftheEOM-CCSD(T)calculationoverarangeofCPUs. ....................................... 144 4-9BlockwaittimesintheEOM-CCSD(T)calculations. ............... 144 4-10ScalingcurvesforcompleteEOM-CCSD(T)calculation. ............. 145 4-11ScalingcurvesforthetriplespieceoftheEOM-CCSD(T)calculation. ...... 145 4-12StructureofthemicrohydratedcytosineclustersoptimizedattheMP2/aug-cc-pVDZbasis. ......................................... 146 4-13MP2/aug-cc-pVDZoptimizedstructureofcytidineusedinthisstudy. ...... 147 4-14MP2/aug-cc-pVDZoptimizedstructureofguanosineusedinthisstudy ..... 147 4-15MP2/aug-cc-pVDZoptimizedstructureofguanine-cytosineWatson-Crickpairusedinthisstudy ................................... 147 4-16EOM-CCSDgradientscalingcurveforAdenineupto512processors. ..... 148 10

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4-17EOM-CCSDgradientscalingcurveforCytosineupto1024processors. .... 148 4-18EOM-CCSDgradientscalingcurveforThymineupto512processors. ..... 148 4-19EOM-CCSDgradientscalingcurveforGuanineupto512processors. ..... 149 4-20EOM-CCSDgradientscalingcurveforCytosineGuaninestackupto2048processors. ...................................... 149 4-21Densitydifferencesforthelowestexcitationonguanine. ........... 150 4-22Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforguanine .................................... 150 4-23Densitydifferencesforthelowestexcitationoncytosine. ........... 150 4-24Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforcytosine .................................... 151 4-25Densitydifferencesforthelowestexcitationonthymine. ........... 151 4-26Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforthymine .................................... 151 4-27Densitydifferencesforthelowestexcitationonstackedcytosine-guanine. 152 4-28Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforstackedcytosine-guanine. .......................... 152 11

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LISTOFSYMBOLS HFHartree-FockCICongurationinteractionCCCoupledclusterEmethodDifferenceoftwoenergiescomputedforagivenmethodCCSCCtheorylimitedtosinglesubstitutionsCCSDCCtheorylimitedtosingleanddoublesubstitutionsCCSD(T)CCSDwithaperturbativetreatmentoftriplesubstitutionsHSimilaritytransformedHamiltonianofCCtheoryEOM-CCEquationofmotionCCtheoryEOM-CCSEOM-CCtheorylimitedtosinglesubstitutionsEOM-CCSDEOM-CCtheorylimitedtosingleanddoublesubstitutionsEOM-CCSD(T)EOM-CCSDwithaMller-PlessetperturbativetreatmentoftriplesEOM-CCSD(~T)EOM-CCSDwithaperturbativetreatmentoftriplesEE-EOM-CCExcitationenergyEOM-CCtheoryIP-EOM-CCIonizationpotentialEOM-CCtheoryEA-EOM-CCElectronattachmentEOM-CCtheoryDKnn-thorderDouglass-Kroll-HessVCCVariationalCCtheoryVCCSVCCtheorylimitedtosinglesubstitutionsIP-VCCS-EExactformulationofionizationpotentialVCCStheoryIP-VCCSApproximateformulationofionizationpotentialVCCStheory 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEXTENDINGTHECAPABILITIESOFACCURATEABINITIOMETHODS:NOVELALGORITHMSANDMASSIVELYPARALLELIZABLEIMPLEMENTATIONSFORFEASIBLECALCULATIONSByThomasJ.WatsonJr.August2012Chair:RodneyJ.BartlettMajor:ChemistryThemostgeneralmethodsinelectronicstructuretheory,capableofattainingchemicalaccuracy,arecoupledcluster(CC)methods.Unfortunately,thegold-standard,CCSD(T),formallyscalesasN7,whereNisthesizeofthechemicalsystem.Consequently,practicalbruteforcecomputationswiththesemethodsonsystemsofO(10)atomsisoutofthequestionwiththeserialprogramsthathavebeendeveloped.Thiswallneedstobeovercometoallowfeasiblecalculationsformoderatelysizedchemicallyrelevantsystemswithhighaccuracy.Inthisvein,Ipresentdifferentapproachestotacklethisscalingproblem.First,Iuseanexact,inthelimit,methodthatreducesthecomputationaleffortofcalculations.Specically,aformallyexactprocedurefordecouplingthefour-componentrelativisticDiracequationisimplemented,inageneralwayintheACESIIandACESIIIprogrampackages,andusedinconjunctionwithIP-EOM-CCSDmethodstoimprovecoreionizationenergies.AnotherwaytoovercomethescalingofCCmethodsistosimplydevelopnewabinitiotheoreticalmodelsthataremorecomputationallytractable.Iproposeanewmethodwithmean-eldcost,N4,thatobtainscoreionizationenergiestoahighdegreeofaccuracyasopposedtoincreasingthecomplexityofIP-EOM-CCmethods,whichwouldincludethesameamountofrelaxationastheproposedmethodinthelimitonly 13

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withexponentialscaling,NN.Furthermore,theproposedmethodisunambiguousindescribingorbitalrelaxationeffects,thedominantenergeticeffectincoreionizationsoforganicmolecules.Therefore,aperturbationtheoryanalysisallowsquickcorrectionstothecomputationallytractableIP-EOM-CCSDmethod.ThenalwaytopushCCmethodsfurtheristousemoderncomputerarchitectureandparallelizeoverprocessors.EquationofmotionCC(EOM-CC)methodsareimplementedinthiswaytoallowbenchmarkstudiesonbiologicallyrelevantmolecules.EOM-CCproperties,includingdipole,second,andtransitionmomentsareimplementedinthiswork,aswellastheEOM-CCSD(T)method(andothertriplesextensions),ananalogueoftheverysuccessfulCCSD(T)method,butforexcitedstates.GradientsforEOM-CCSDtheoryareimplementedaswell,pavingthewayforroutinestudiesofpotentialenergysurfacesanddynamicsofexcitedstateswithintheCCframework. 14

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CHAPTER1INTRODUCTION Theunderlyingphysicallawsnecessaryforthemathematicaltheoryofalargepartofphysicsandthewholeofchemistryarethuscompletelyknown,andthedifcultyisonlythattheexactapplicationoftheselawsleadstoequationsmuchtoocomplicatedtobesoluble.P.A.M.Dirac,TheQuantumTheoryoftheElectron 1.1AbInitioQuantumChemistryDirac'sfamousquote[ 1 ]hasappearedinnumeroustextbooks,articles,dissertations,etc...,emphasizingthetroublingrealityofmodernquantumchemistryandphysics.Anabintio,orfromthebeginningexactquantummechanicaldescriptionofelectrons,thenecessaryparticlesfordescribingchemicalbonding,reactivity,properties,excitations,andphenomenaingeneralwithoutexperimentalinput,isanintractable,ratherunobtainable,goal.However,allisnotlost.Diraccontinues, Itthereforebecomesdesirablethatapproximatepracticalmethodsofapplyingquantummechanicsshouldbedeveloped,whichcanleadtoanexplanationofthemainfeaturesofcomplexatomicsystemswithouttoomuchcomputation.Thisassertioncarriesalittlemorehopethantheprevious.Infact,itpreciselydescribestheremarkableeffortsmadeinthenameofmodernquantumchemistry.Judiciousandcarefullychosenapproximationsaremadewiththegoalofpredictingexperimentalresults.Approximateabinitiomethodscanprobetransientspecieswhereexperimentcannot,providechemicalinsightforthoseexperimentsthataredifcult,dangerous,orevenimpossible,andcanhelpelucidatecomplicatedexperimentaldatatoprovidedenitiveconclusions.Tobegin,abinitoquantumchemistryisconcernedwithcalculatingthedistributionofelectronsinmolecules.Thenon-relativistic,time-dependentdescriptionofquantum 15

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particles,electronsinthiscase,isgovernedbytheSchrodingerequation )]TJ /F5 11.955 Tf 11.95 0 Td[(i@ @t(x,t)=^H(t)(x,t).(1)Foratime-independentHamiltonian,^H,onewithnoexternaleldsorpotentials,thewavefunctiondescribingtheparticleswithcoordinatesx,isseparableas (x,t)=(x)(t).(1)InsertingthisintoEquation( 1 )andperformingaseparationofvariables,yieldsthetime-independentSchrodingerequationforboundenergeticstates ^H(x)=E(x)(1)withaphasefactordescribingthetimeevolutionoftheparticlegivenas (t)=e)]TJ /F6 7.97 Tf 6.59 0 Td[(iEt.(1)AnexperimentallyobservablevalueorpropertyhAiisdenedbytheexpectationvalueofthecorrespondingHermitianoperator^A, hAi=RdxeiEt(x)^A(x)e)]TJ /F6 7.97 Tf 6.58 0 Td[(iEt RdxeiEt(x)(x)e)]TJ /F6 7.97 Tf 6.59 0 Td[(iEt=Rdx(x)^A(x) Rdx(x)(x)=hj^Aji hji(1)Here,forconvenience,IuseDirac'sbra-ketnotation[ 2 ],wherethebra-kethj^Ajiistheinnerproductoftheketjivector,denedintherelevantxedparticleinnitedimensionalHilbertspace,andthebrahj,thecorrespondinglinearfunctionalinthedualspace.[ 2 ]Ofprimaryimportanceisthegroundstateenergyofasystem.Assuminganexact,normalizedwavefunction(hji=1),thenon-relativisticenergyisgivenby Eexact=hj^Hji(1) 16

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Since^Hisahermitian,andthereforepositivesemi-denite,operator[ 3 ],thenforanynormalizedtrialwavefunction,j~i,thefollowingpropertyholds h~j^H)]TJ /F5 11.955 Tf 11.95 0 Td[(Eexactj~i0(1)ThisistheRayleigh-Ritzvariationalprinciple,[ 4 ]andprovidesarigorousupper-boundtotheenergyforapproximatewavefunctions.Forhydrogenandhydrogenicions,systemswithoneelectronandonenucleus,A(withchargeZAandmassMA),theHamiltonianis ^H=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2r21)]TJ /F4 11.955 Tf 21.55 8.09 Td[(1 2MAr2A)]TJ /F5 11.955 Tf 28.84 8.09 Td[(ZA jr1)]TJ /F12 11.955 Tf 11.96 0 Td[(RAj,(1)andcontainsthekineticenergyoftheelectronandnucleusintherstandsecondterms,respectively,andtheelectron-nucleusattractivepotentialenergy.Inpractice,sincethetimescaleoftheelectronicredistribution,ormovement,isextremelyfastcomparedtothetimescaleofthemovementofthenucleus,likeiesaroundacake,1thereisnokineticenergytermforthenuclei.ThisisthebasisfortheBorn-Oppenheimerapproximation.Infact,forsystemswithmorethanonenuclei,thenucleus-nucleuspotentialenergybetweennucleiAandB,^VNN(R)=PNnAPNnB>AZAZB jRA)]TJ /F9 7.97 Tf 6.59 0 Td[(RBj,simplybecomesanadditivescalar,andistrivialtocomputeusingthepredenedcharges,Z,andcoordinatevectorsR.Italsoisessentialfortheconstructionofpotentialenergysurfaces(PES)toobtaincriticalpointssuchasgroundstategeometries,transitionstates,andintermediatestructures.WithintheBorn-OppenheimerapproximationthehydrogenicHamiltonianthenbecomes ^H=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2r21)]TJ /F5 11.955 Tf 28.83 8.09 Td[(ZA jr1)]TJ /F12 11.955 Tf 11.95 0 Td[(RAj,(1) 1R.J.Bartlett,privatecommunication,2007 17

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WiththisHamiltonian,Equation( 1 )isexactlysolublewithexponentiallydecaying,one-electrondistributionfunctions,oratomicorbitals.Ingeneral,theseatomicorbitalfunctionsareproductsofapolynomialradialfunctionandasphericalharmonicfunctionfortheangularpartcentered,orpositioned,atR nlm(r)=Ylm(,)Rn(r)e)]TJ /F16 7.97 Tf 6.59 0 Td[(njr)]TJ /F9 7.97 Tf 6.59 0 Td[(Rj.(1)TheseexponentiallydecayingfunctionsinsphericalcoordinatesarecalledSlaterorbitals.Incartesiancoordinates,Slaterorbitalstaketheform lmn(r)=xlymzne)]TJ /F16 7.97 Tf 6.58 0 Td[(jr)]TJ /F9 7.97 Tf 6.59 0 Td[(Rj,(1)wherefl,m,ngarenolongertheusualquantumnumbers,althoughtheprincipalquantumnumberisthesumoftheseexponents.However,formolecularsystems,itwillbeconvenienttoexpresstheseSlaterorbitalsasasumoverGaussians, xlymznejr)]TJ /F9 7.97 Tf 6.59 0 Td[(RjXdxlymznejr)]TJ /F9 7.97 Tf 6.59 0 Td[(Rj2=(r).(1)Thesetoffg,orfjig,arecalledcontractedGaussianfunctions.AlthoughthesecontractedGaussianfunctionsaresmoothatthenucleus,ratherthanhavingacusp,anddecaytoorapidlyfartherawayfromthenucleus,thettingtechniquesusedtodeterminethecoefcientsdandexponentsdosotoveryreasonableaccuracyinintermediateranges.ContractedGaussianbasisfunctionssimplifyintegralevaluationinmolecularsystems,aswillbeshownlater.ForamolecularsystemscomprisedofNnnucleiandNeelectrons,withnucleusAspeciedbyitsatomicnumber,ZA,andatomicmass,MA,thetime-independentHamiltonian,withintheBorn-Oppenheimerapproximation,is ^H=)]TJ /F6 7.97 Tf 15 14.94 Td[(NeXi1 2r2i)]TJ /F6 7.97 Tf 15.66 14.94 Td[(NeXiNnXAZA jri)]TJ /F12 11.955 Tf 11.95 0 Td[(RAj+NeXiNeXj>i1 jri)]TJ /F12 11.955 Tf 11.95 0 Td[(rjj+^VNN(R).(1) 18

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Thersttermcorrespondtothekineticenergyoftheelectrons,followedbytheattractivenucleus-electronpotentialenergyandtherepulsiveelectron-electronandnucleus-nucleuspotentialenergyterms.Bysubtractingoutthenucleus-nucleuspotentialenergyterm,theelectronicenergy,Eeeisdeterminedvia ")]TJ /F6 7.97 Tf 15 14.94 Td[(NeXi1 2r2i)]TJ /F6 7.97 Tf 15.66 14.94 Td[(NeXiNnXAZA jri)]TJ /F12 11.955 Tf 11.95 0 Td[(RAj+NeXiNeXj>i1 jri)]TJ /F12 11.955 Tf 11.96 0 Td[(rjj#j(x)i=Eeej(x)i.(1)Unfortunately,thelasttermofEquation( 1 ),theelectron-electronrepulsionterm,makesanexactanalyticalsolutionimpossible.Thisisknownastheelectronicstructureproblem.However,sincej(x)iisanabstractvectorinHilbertspace,itcanbeexpandedinacompletesetoforthogonalbasisvectorsinthatspace. j(x)i=XIcIjI(x)i.(1)Theindividualbasisvectors,jI(x)i,arethemselvesrepresentedbyaniteantisymmetricproductofone-electronbasisfunctionscalledmolecularorbitals,f pi(xi)g,as jI(x)i=1 p Ne!AjNeYp pi(xi)i.(1)Theantisymmetryofthisproductfunctionisessentialduetotheindistinguishabilityoftheelectronsasdictatedbyquantummechanics.[ 2 ]Whentwoparticlesareswitched,thewavefunctionmustchangesign.AsanexampleofEquation( 1 ),forasystemwithtwoelectrons,thewavefunctionmusthavetheform j(x1,x2)i=1 p 2j 1(x1) 2(x2)i)-223(j 1(x2) 2(x1)i.(1) 19

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ThiscanbeaccomplishedbyexpressingeachjIiasthedeterminantofamatrixofmolecularorbitals,witheachrowcorrespondingtoanelectronnumber jI(x)i=1 p Ne! i(x1) j(x1) k(x1) i(x2) j(x2) k(x2) i(x3) j(x3) k(x3)............(1)Sinceswappingrowsofamatrixchangesthesignofthedeterminant,antisymmetryisbuiltintothebasisvectorsfromthebeginning.Anothercriticalresultofquantummechanicsthatneedstobeincorporatedinthewavefunctionisthemsquantumnumber,orspin=f,g,oftheelectron.Inthenon-relativisticlimit,thespinandspatialcomponentsofthemolecularorbitalscanbedecoupledresultinginmolecularorbitalsas (xi)=(ri)(i),(1)withribeingthefx,y,zgcartesiancoordinatesoftheithelectron.Thisisaratherniceresultsinceitallowsmolecularorbitals,ordistributionfunction,tobegraphedandvisualizedafterspinisintegratedout.Although,itisimportanttonote,theseorbitalsdonotcorrespondtoanyobservablequantityinthelaboratory.Itiscomputationallyinfeasibletouseacompletebasisforthesetofmolecularorbitalsfig;instead,anitebasismustbeused.Themostcommonlyusedbasisformolecularsystemsisanexpansionoftheaforementionedatomicorbitals(AOs),orcontractedGaussianfunctionsfjig,distributedoverthesystem,typicallycenteredonthenucleiofthemolecule.Thereare,ofcourse,otherchoicesforabasis.Finite,real-spacegrids[ 5 ]aregainingwide-spreaduseformolecules,andforperiodicsystems,planewavesarenecessarybecauseoftheperiodicboundaryconditions.[ 6 ] 20

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ThemolecularorbitalsarethenexpandedasalinearcombinationoftheAOsasi(r)=Xci(r)jii=Xciji (1)Theelectronicenergythenofanarbitrarynormalizeddeterminantcomprisedoftheseorbitals,undertheconstrainttheMOsthemselvesarenormalized,hijji=ij,isgivenasE=hj^Hji (1)=h1ijnj^Hj1ijni (1)=NeXihii++1 2NeXijhijjjiji (1)wherehij=hij^hjji=hi(r1)j)]TJ /F4 11.955 Tf 19.13 8.09 Td[(1 2r21)]TJ /F13 11.955 Tf 11.95 11.36 Td[(XAZA jr1)]TJ /F12 11.955 Tf 11.96 0 Td[(RAjjj(r1)i (1)hijjjkli=hijjkli)-222(hijjlki (1)hijjkli=hi(r1)j(r2)j1 jr1)]TJ /F12 11.955 Tf 11.95 0 Td[(r2jjk(r1)l(r2)i (1) (1)TheseformulafollownaturallywiththeorthogonalityoftheMOsandtheproperinclusionofantisymmetry.Since^hisaone-electronoperator,itactsononeketMO,leavingtherestuntouchedandduetoorthogonalityoftheMOs,canonlyformtheinnerproductwiththecorrespondingbraMOhijnj^h(r1)jijni=(hijhjjhnj)^h(r1)(jiijjijni) (1)=hij^h(r1)jiihjjjihnjni (1)=hij^h(r1)jii. (1) 21

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Thesamefollowsfortheelectron-electronrepulsionoperator,exceptthattheantisymmetricnatureofthedeterminantyieldstwocontributions,thecoulombandexchangepieces.Theexchangepieceispurelyaquantummechanicaleffectarisingfromtheindistinguishabilityoftheelectrons.SincetheMOsarelinearcombinationsofAOs,theintegralsareactuallyevaluatedintheAObasisandtransformedtotheMObasis.Forexample,thetwo-electronintegralintheMObasisis hijjkli=Xcicjhjickcl.(1)Explicity,thisintegral,with,,,andcenteredonnucleiA,B,C,andD,respectively,is hji=ZdxdydzA(r1)B(r2)C(r1)D(r2) jr1)]TJ /F12 11.955 Tf 11.96 0 Td[(r2j(1)However,sincetheAOsarecontractedGaussianfunctions,wecanusethewell-knownGaussianproducttheoremtoconstructaGaussiancenteredatPfromtwoGaussianscenteredatAandB,respectively. e)]TJ /F16 7.97 Tf 6.58 0 Td[(jri)]TJ /F9 7.97 Tf 6.58 0 Td[(RAj2e)]TJ /F16 7.97 Tf 6.58 0 Td[(jri)]TJ /F9 7.97 Tf 6.58 0 Td[(RBj2=EABe)]TJ /F6 7.97 Tf 6.59 0 Td[(pjri)]TJ /F9 7.97 Tf 6.58 0 Td[(RPj2(1)wherep=+ (1)P=RA+RB + (1)EAB=e)]TJ /F17 5.978 Tf 10.25 3.69 Td[( +(RA)]TJ /F9 7.97 Tf 6.59 0 Td[(RB)2 (1)Therefore,the4-centerintegralreducesto ZdxdydzA(r1)B(r2)C(r1)D(r2) jr1)]TJ /F12 11.955 Tf 11.95 0 Td[(r2j=ZdxdydzI(x1,y1,z1)I(x2,y2,z2)P(r1)Q(r2) jr1)]TJ /F12 11.955 Tf 11.96 0 Td[(r2j(1) 22

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WiththecartesianpolynomialpieceofthecontractedGaussianbeingabsorbedinI(xi,yi,zi)as I(xi,yi,zi)=(xi)]TJ /F5 11.955 Tf 11.96 0 Td[(Ax)l(yi)]TJ /F5 11.955 Tf 11.96 0 Td[(Ay)m(zi)]TJ /F5 11.955 Tf 11.96 0 Td[(Az)n(1)TheSlatertypeorbitalsofEquation( 1 )areexpandedinGaussiansbecauseofthisproperty.Itsimpliestheintegralevaluationtremendously.Thereisanassociatedlossofaccuracyasthisexpansionisincompleteandcontainsunphysicalcharacteristicsatthenucleus.OneusuallychoosesGaussianbasissetsoptimizedfortheparticularsystemunderinvestigationtoreducethiserror.TherearelibrariesoftheseGaussianbasissetsforeachatomintheperiodictable[ 7 8 ].Sometimesitisconvenienttogenerateone'sownbasisset,optimizingthecontractioncoefcientsandexponents.ItisalsopossibletoconvergetothecompletebasissetlimitbyincreasingtheGaussianbasissetincareful,predenedways[ 9 10 ]byaddingpolarizationand/ordiffusefunctions.Forexample,excitedstatesofmoleculesaresometimesverydelocalized.Diffusefunctions,functionsthatextendfaroutfromthenucleus,arenecessarytoaccuratelydescribetheseRydbergtype,delocalizedstates.However,thesemaynotbenecessaryforionizedorcationicsystems,sincethechargedistributionwillmostlikelynotbedistributedtoofarfromthenucleus.Itisanextremelyimportantpoint,asonewouldliketocapturethemostamountofnecessaryphysics,withaslittlecomputationaleffortaspossible. 1.2TheHartree-FockMean-FieldApproximationThesimplest,non-empirical,abinitiomethodforcomputingtheelectronicenergyofmolecularsystemsistheindependentparticlemodel(IPM).Specically,minimizingtheelectronicenergywithrespecttothemolecularorbitalsdenestheHartree-FockIPMapproximationandconceptuallycorrespondstoeachelectronrelaxingintheeldgeneratedbytheremainingelectrons.Weseekasingledeterminantwavefunction,minimizedwithrespecttotheMOstobethelowestenergysolutionofEquation( 1 )subjecttotheconstraintthattheMOs 23

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remainorthogonal.TheLagrangian L="NeXihij^hjii+1 2NeXijhijjjiji)]TJ /F7 11.955 Tf 19.26 0 Td[(ji(hijji)]TJ /F7 11.955 Tf 19.26 0 Td[(ij)#,(1)follows,withthejibeingtheundeterminedmultipliersenforcingtheMOorthogonalityconstraint.ThisminimizationyieldsthecanonicalHartree-Fockequations )]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2r2)]TJ /F13 11.955 Tf 11.96 11.36 Td[(XAZA jri)]TJ /F12 11.955 Tf 11.95 0 Td[(RAj+Ji(r1))]TJ /F5 11.955 Tf 11.95 0 Td[(Ki(r1)!jii=ijii(1)Thecoulombandexchangeoperators,J(r)andK(r),respectively,areorbitaldependentanddenedasJi(r)jii=NeXjZdr0j(r0)j(r0) jr)]TJ /F12 11.955 Tf 11.96 0 Td[(r0ji(r)=Zdr0(r0) jr)]TJ /F12 11.955 Tf 11.96 0 Td[(r0ji(r) (1)Ki(r)jii=NeXjZdr0j(r0)j(r) jr)]TJ /F12 11.955 Tf 11.96 0 Td[(r0ji(r0)=Zdr0(r,r0) jr)]TJ /F12 11.955 Tf 11.95 0 Td[(r0ji(r0). (1)ThecoulomboperatorjustiestheIPM,ormean-eldterminology,asitconceptuallysmearsoutthechargedistributionofallotherelectrons.Theexchangeoperatoriswritteninamoresuggestiveformtodemonstrateitsnon-localnature,notetheinterchangeofr0withr,andisresponsibleforthecorrelationofelectronsofparallelspin.ThelefthandsideofEquation( 1 )iscalledtheFockoperator,written^fifororbitali,andisdependentontheorbitals.LeftprojectingbymolecularorbitalhjjyieldstheFockmatrixhjj^fijii=ihjjii (1)=iij (1)Theorbitaldependenceresultsinanon-linearequationthatbeginswithaninitialguessforthesetfjpigandissolveduntilself-consistencyisreached.Atrst,thesecoupledintegro-differentialequationsweresolvednumerically.Numericalapproachesarestillasubjectofinteresttoday[ 11 ],sincetheyofferaroutetoarbitraryprecision, 24

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howevertheydonotprovidehigherenergyorthogonalfunctionsforthegenerationofexciteddeterminants,whereanelectroninanoccupiedorbitaljii,isreplacedbyahigherenergyvirtualorbitaljai.Amorepragmaticapproach,thatgeneratestheorthogonalhigherenergymolecularorbitals,wasdevelopedbyRoothaanandHall[ 12 ].Assuggestedabove,themolecularorbitalsinEquation( 1 )areexpandedinanatomicorbitalbasis NAOX^fijic=NAOXjic.(1)LeftprojectionwithhjyieldstheRoothaan-HallorHartree-Fock-RoothaanequationsNAOXhj^fijic=NAOXhjic (1)FC=SC (1)withS=hjibeingtheatomicorbitaloverlapmatrix.BeginningwithaninitialCvectors,fromHuckel'smethod,previouslydenedsmallbasisvectors,localizedatomicHFcalculations,etc...,theFockmatrixisconstructedanddiagonalized,yieldinganewsetofvectors,C0,whicharetransformedtoeliminatetheoverlapmatrix.Thisprocedureisrepeateduntilaconvergencethresholdisreached.ThelowestNemolecularorbitalscomprisetheHartree-Fockdeterminant,j0i,andexciteddeterminants,jIiareformedfromtheswappingoccupiedorbitalswiththeremainingM=NAO)]TJ /F5 11.955 Tf 12.55 0 Td[(Nevirtualorbitals.Theseexciteddeterminantsformthebasisforrelaxingtheenergyfurther,whichwillbepresentedafteradetourtoanapproximaterelativisticextensionoftheone-electronoperatorsin^fi. 1.2.1FormalScalingandRate-DeterminingStepsTheconstructionoftheFockmatrixintheAObasisformallyscalesasN4AOascanbeseenintheconstructionofthe^J)]TJ /F4 11.955 Tf 13.74 2.66 Td[(^Kmatrices hj^J)]TJ /F4 11.955 Tf 13.74 2.65 Td[(^Kji=XhjjiD(1) 25

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withDbeingthechargedensity[ 4 ]generatedfrombacktransformingthemolecularorbitalscontainedintheoperators.Havingtoloopoverthefourindicesf,,,gcausestheN4AOscaling.However,thispieceoftheFockmatrixcanbecomputedinblocks,thatcanbedistributedoverprocessors.Aswillbeshownlater,forlargebasissets,withaproperlyparallelizableprogramandalgorithm,thisbruteforceconstructionoftheFockmatrixcanceasetobetherate-determiningstep.Whenthishappens,thenewrate-determiningstepbecomestheserialdiagonalizationoftheFockmatrixtoobtainanewsetofmolecularorbitalcoefcients.ThediagonalizationscalesasN3AOandcannotbedividedintoindependentblocksofdata,henceremainsserial.Therefore,thedevelopmentoftechniquesthatavoidthisserialdiagonalizationarecriticaltopushtheeldfurthertowardstheabinitiotreatmentofverylargemoleculeswithO(10,000)toO(100,000)atoms. 1.2.2Scalar-RelativisticHamiltonians:Asthenuclearchargeincreases,anextensiontoarelativisticframeworkisnecessary.TheelectronbeginstomoveclosetothespeedoflightandDirac'sequation ^HD=E(1)becomestheequationdescribingtheboundenergeticstatesoftheelectron.TheDiracHamiltonianisa4x4matrixandisafour-componentspinor.Four-componentspinorbasissetsmustbeconstructedtorepresent,withthelargecomponentkineticallybalancedwiththesmallcomponent.[ 13 ]Thisisnecessarytoavoidvariationalcollapseofthepositiveenergysolutionstoboundpositronicstates,ornegativeenergysolutions,sincetheenergyspectrumcontainsbothpositiveandnegativeenergysolutions.Also,themean-eldapproximationobtainedbyminimizingtheDiracenergywithrespecttothefour-componentbasisfunctionsrepresentingismuchmorecostlythanthenon-relativisticcalculation. 26

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Toavoidthesedifculties,manytechniqueshavebeendevelopedthatattempttodecouplethelargeandsmallcomponentsforone-electronoperatorsoftheDiracHamiltonian.Consequently,veryniceexpressionsareobtainedfortransformedone-electronoperatorsinthenon-relativisticSchrodingerequation-thekineticenergyandnucleus-electronpotentialenergyoperators.Thesetransformationtechniquesareseparatedtospinfree(onecomponent)andspindependent(twocomponent)additivecorrections.Thespinfreetransformations,ingeneral,tintotheexistingnon-relativisticstructure,computercodesincluded,sincethesameone-electronoperatorsin^Harealsoin^fi. 1.3IncludingElectronCorrelation 1.3.1CongurationInteractionHartree-Focktheoryrecoversroughly99%ofthetotalelectronicenergyofthesystem.Unfortunately,theremaining1%iscriticaltodescribesystemsaccurately.Asanexample,Hartree-Fockcannotcapturedispersion,orLondon,forces.Itwillshownominimumenergystructureexistsforthemethanedimer,whereweknowinpractice,thatthereisaminimum,albeitaveryshallowone.Thismissingenergeticcontributionisknownaselectroncorrelation,andisthesudden,instantaneousinteractionofelectronswitheachother.ElectroncorrelationcanbeincludedbyconsideringallpossiblereplacementsoftheoccupiedorbitalsintheHFreferencedeterminantwithhigherenergyvirtualorbitals.Rigorously,theexactwavefunctioninagivenbasissetfpgcanbedescribedasalinearcombinationoftheseexciteddeterminantsji=c0j0i+Xiacaijaii+Xi
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Theleadingterm,j0i,istheHFreferencedeterminant,andindermediatenormal-ization,hj0i=1,wasusedtoarriveatEquation( 1 ).Theremainingtermscorrespondtosingle,double,etc...exciteddeterminants,whereoccupiedorbitaliisreplacedbyvirtualorbitalainthesingles,andsoonandsoforthwithaprobability(coefcient)ofthatexcitationtakingplace.ThisexpansionisknownasCongurationInteraction(CI),andinthelimitofallpossiblesubstitutionsistheFullCI.Itisthebestanswerinagivenbasisset,invarianttoorbitalchoice,anupperboundtotheexactenergy,andrigorouslydenesthecorrelationenergyas Ecorr=E)]TJ /F5 11.955 Tf 11.96 0 Td[(EHF(1)withEbeingtheexactnon-relativisticelectronicenergy.However,thecostofperformingaFullCIcalculationgrowsexponentiallywiththesizeofthebasisset,makingitonlyfeasibleforverysmallmolecularsystemswithsmallbasissets.AnytruncatedCIcalculationwillinherentlycontainerrorsthatwillgrowwiththesystemsizeasitisnotasizeextensivemethod.Asizeextensivemethodisonewheretheenergyscalesproperlywiththenumberofparticles;truncatedCIdoesnotscaleproperlywithsystemsize. Formalscaling.TheFullCImethodbringsinallthepossibledeterminantsthatcanbegeneratedfromthenumberofelectronsoccupyingeachofthemolecularorbitals,thusexhaustingthespacespannedbythenitebasisset.Inthislimit,thechoiceoforbitalsisarbitrary,asallpossiblecombinationsareincorporated,andthereforeincludesallmean-eld,ororbitalrelaxation,andelectroncorrelationeffects.ThescalingofthismethodiscompletelyandutterlyinfeasiblesojudiciousapproximationsaremadeviatruncatingtheFullCIexpansiontoaspecicexcitationlevel,[ 4 ]state-speciciterativematrixdiagonalizationroutinesthatavoidcompletediagonalizationoftheFullCImatrix,[ 4 ],montecarloalgorithmstoseekonlythemostimportantdeterminants,[ 14 ]andpossiblymanyothers. 28

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Whentruncated,thenumberofdeterminantsintheCImatrixisapproximatelygiven,withbinomialcoefcients,as Ndet0B@Nem1CA0B@nvm1CA1 (m!)2Nmenmv(1)withagivenexcitationlevel,m,numberofelectrons,Ne,andnumberofvirtualorbitalsunoccupiedinthereferencedeterminant,nv.EvenmoredisastrousfortruncatedCImethodsisthelackofsizeextensivity.ExploitingthefactthattheHamiltoniancontainsatwo-electronoperatoryieldingasparseCImatrixcanhelptoreducethescaling,andapplyingcorrectionstoreducesizeinexstensivitycanhelp,butthemethodstillremainscomputationallyunsustainabletoday. 1.3.2Coupled-ClusterTheoryTheFullCIwavefunctioncanbewritteninanexponentialformas ji=exp^T1+^T2++^TNj0i(1)wherethe^TnexcitationoperatorsarehavethesameformastheCIexcitationoperators,namely^T1j0i=Xiataijaii^T2j0i=Xi
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ThisoffersrapidconvergencetotheFullCIenergyforanytruncationofthe^Tnoperators.Truncatingatdoubleexcitations,expt(^T)=expt(^T1+^T2)correspondstotheCCSDmodel.Todemonstratethefastconvergence,theCIcoefcientscanbedecomposedintotheseclusteramplitudesas^C1=^T1 (1)^C2=^T2+1 2^T21 (1)^C3=^T2^T1+1 3!^T31 (1)^C4=1 2^T22+1 2^T2^T21+^T21^T2+1 4!^T41 (1)TruncatingtodoublesintheCCmodelincorporatesindependenthigherexcitationsthatCIclearlydoesnotcontaininthetruncatedspace.TheinclusionoftheseindependenthigherexcitationsmakesCCtheorysizeextensiveforanytruncation.CCtheoryalsohasthepropertyoforbitalinvarianceintheoccupied-occupiedandvirtual-virtualspacesandinclusionofhigherexcitationsconvergestotheFullCIenergy.TheSchrodingerequationwiththeCCansatzis^Hji=Eji (1)^HeTj0i=EeTj0i. (1)ProjectionontothereferencespacedenestheCCenergyexpressionh0j^Hji=Eh0ji (1)h0j^Hji=E (1)h0j^HeTj0i=E (1)whereintermediatenormalizationwasused.Complicationsarisewhenattemptingtodeterminethe^Tamplitudes.ProjectingEquation( 1 )ontotheexciteddeterminants 30

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yieldshabcijkj^Hji=Ehabcijkji (1)habcijkj^HeTj0i=EhabcijkeTj0i (1)habcijkj^HeTj0i=Etabcijk. (1)TheorthogonalityoftheexcitedbasisvectorswasusedtogettoEquation( 1 )fromEquation( 1 ).Theenergydependenttermscanallbeeliminatedbyacarefulanalysisoftheequations,andrestrictingthecontractionsbetween^Hand^Ttoonlythosethatareconnected. Traditionalcoupled-clustertheory.AmoretransparentwaytoseetheremovaloftheenergyintheamplitudeequationsistoleftprojectEquation( 1 )byexp()]TJ /F4 11.955 Tf 11.29 2.66 Td[(^T),e)]TJ /F9 7.97 Tf 7.97 1.78 Td[(^T^He^Tj0i=ECCe)]TJ /F9 7.97 Tf 7.97 1.78 Td[(^Te^Tj0i (1)=ECCj0i (1)Thissimilaritytransformationdoesnotchangetheenergyeigenvaluespectrumof^Hbutitdoesbreakthehermiticityoftheequation,andECCisnolongeranupperboundtotheexactenergy.ItisstillguaranteedtoconvergetotheFullCIenergyifnotruncationismade.Sincethereferencedeterminantisorthogonaltoallexciteddeterminants,theamplitudeequationsbecomehabcijkje)]TJ /F9 7.97 Tf 7.97 1.77 Td[(^T^He^Tji=ECChabcijkj0i (1)=0. (1)ThisisthetraditionalwaytheCCequationsaresolved[ 15 ].Truncationofthe^Toperatorsdenestheprojectionspace,theamplitudeequationsaresolved,andtheenergyiscalculatednally.ThenewsimilaritytransformedHamiltoniancanbeexpandedintoaseriesofnestedcommutatorsbetween^Hand^TusingtheBaker-Campbell-Hausdorffformula[ 15 ]leavingonlytermsthatcontaincontractions 31

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between^Hand^T,orconnectedterms e)]TJ /F9 7.97 Tf 7.97 1.77 Td[(^T^He^T=^He^TCH(1)withthesubscriptCreferringtoconnectedtermsonly.Consequently,wecansubtractoutthereferenceenergyfromthebeginningandobtainadirectexpressionforthecorrelationenergyEcorr=h0j^HeTC)]TJ /F5 11.955 Tf 11.95 0 Td[(E0j0i (1)=h0jHNj0i (1)Duetothepresenceofexcitationoperators,^Tm,withaparticlerank(orexcitationnumberm)greaterthan1,thesimilaritytransformedHamiltoniancontainsone-andtwo-bodyoperators,butalsothree-,four-,andhigherbodyoperators H=fpq+Wpqrs+IIIpqrstu+IVpqrstuvw+,(1)whichareformedfromtheconnectedtensorcontractionsbetweenthebareHamiltonianandtheclusteramplitudes.Inmatrixform,inthebasiscontainingtheground,single,double,triple,etc...(0,S,D,T,...), 266666666664H00H0SH0DH0T0HSSHSDHST0HDSHDDHDT0HTSHTDHTT...............377777777775(1)TheHmatrixisclearlynothermitianandthereforeWpqrs6=Wrspq.Thisdoublesthestoragerequirements.Also,thethree-andhigherbodymatrixelementsshouldneverbestored,andaretypicallynotstored,butrathercomputedontheywhenneeded. 32

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Anotherconsequenceofthenon-hermiticityofHisthatthefunctionalformhasleftandrighthandbiorthogonaleigenvectorswhichyieldthesameeigenvalue.ThereforetheCCenergyfunctionalhasthegeneralform ECC=h0jLHRj0i(1)Forthegroundstate,L=1+^andR=1,leavingthegroundstateenergyfunctionalasECC=h0j1+^Hj0i (1)=h0jHj0i+h0j^Hj0i (1)=h0jHj0i+0 (1)The^operatorisadeexcitationoperator,similartothe^Toperator;consequentlyh0j^Hj0i=h0j^(P+Q)Hj0i (1)=h0j^PHj0i+h0j^QHj0i (1)=h0j^j0ih0jHj0i+XIh0j^jIihIjHj0i (1)=0h0jHj0i+XIh0j^jIi0 (1)=0. (1)Intheaboveequation,aninsertionofaresolutionoftheidentity, 1=P+Q=j0ih0j+XIjIihIj,(1)wasperformed,whereindexIcorrespondstoexciteddeterminants.Thesecondtermcorrespondstotheamplitudeequations,whichhavebeenshowntobe0.Ingeneral,thePandQ-spacescanbedened,orpartitioned,tosuittheproblem.Thisgeneralizedenergyfunctionalwillbeusedlaterforthecomputationofexcited,ionized,andelectron 33

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attachedstates.TocomputeperturbativetripleseffectsthePandQ-spaceswillbepartitioneddifferently. Variationalcoupled-clustertheory.TheSchrodingerequationhasthepropertythatforanytrialwavefunctionj~i,theenergyobtainedfromtheexpectationvalueoftheHamiltonian h~j^Hj~i h~j~i=E0,(1)isnecessarilyanupperboundthetheexactenergyobtainedfromtheexactwavefunction. E0E(1)Thisvariationalprinciple[ 4 ]followsfromthefactthattheHamiltonianoperatorispositivesemi-denite[ 3 ].ThispropertyoftheHamiltonianislostwiththesimilaritytransformationinEquation( 1 ),breakingthisvariationalcondition.Thisisnotusuallyaterribleproblem,sinceforproblemssufcientlydescribedwithasingledeterminant,thenon-variational,orperturbativetraditionalcoupledclustermethodconvergesrapidlytothecorrectanswer.KeepingthisvariationalconditioninCCtheoryrequiresEVCC=h0je^Ty^He^Tj0i h0je^Tye^Tj0i (1)=h0je^Ty^He^TCj0i, (1)wherethenumeratorwasfactorizedtoleaveonlyconnectedterms.[ 16 ]Inthelimitofnotruncation,thisisequivalenttothetraditionalCCmethod.TheconnectedformofEquation( 1 )doesnotterminateandmustbetruncatedatjudiciouslychosen,predenedorders.[ 17 ]VariationalCClimitedtosinglesubstitutions(VCCS)isaspecialcaseasitcorrespondstoHartree-FocktheoryviaThouless'Theorem.[ 18 19 ]Thouless'theoremstatesthattheHartree-Fockgroundstatedeterminantcanbewrittenasarotationofany 34

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arbitrarysinglereferencedeterminant j0i=e^T1j~0i.(1)TheHartree-Fockenergycanbewrittenas EHF=h~0je^Ty1^He^T1Cj~0i(1)Recently,ageneralalgorithmfordeterminingtheVCCSamplitudeswaspresentedbySimunekandNoga[ 20 ].Thecentralidea,asmoredetailedequationswillbepresentedinChapter 2 ,istoconstructtheVCCSdensityasthesumofareferencedensityandrelaxationcorrectionstoit D(t,ty)pq=Xipiiq+h~0je^Ty1fpyqge^T1Cj~0i.(1)SincethedensityisafunctionofthetaiamplitudestheFockenergyequationcanbeminimizedwithrespecttotheamplitudesas E tai=XpqFpqD(t,ty)qp tai=0.(1)ThisgivesrisetoatransformationmatrixthatenforcesthegeneralizedBrillouinconditionandyieldsanupdatedtvectorthatisusedtoconstructthenextiterationdensitymatrix.ThisisratherniceasitavoidstheserialdiagonalizationofHartree-Focktheorydiscussedintheprecedingsectioninfavorofveryfastmatrixmultiplications.Itdoesneedasufcientlyclosestartingguessthough,whichinmostcasestodatehavebeenlowerlevelHartree-Fockcalculations[ 20 ]orafewSCFiterationsinthesamebasisset[ 21 ].Theformermaynotconvergeinallcases,andthelatteronlyreducestheproblemofserialdiagonalization,anddoesnoteliminateit. Formalscaling.Typically,theformalscalingofCCmethodstruncatedtoexcitationsofm,isgivenasnmonm+2v,withnmonmvstoragefortheamplitudes.Very 35

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often,inmolecularcalculations,thenumberofvirtualorbitalsismuchgreaterthanthenumberofoccupiedorbitals.Consequently,therate-determiningstepisthevirtualMOcontraction Xcdhabjjcditabcdijkl7!tabcdijkl(1)ThefastestalgorithmsforthiscontractionaredoneusingtheAObasis,wheretheintegralsarecomputedon-the-y.Thefc,dgindicesofthe^TamplitudesarebacktransformedandcontractedwiththeAOtwo-electronintegrals.Thiscanbeperformedwithindependentblocksofthearraysdistributedoverprocessors.Inpracticalcalculations,CCSDperformsquitesatisfactorily.Aperturbativetriplesextension,CCSD(T),(non-iterative)isconsideredthegold-standardofcomputationalmethods,oftenobtainingveryreliablethermochemicalquantitiesofinterestinexperimentalchemistry.Thismethodconstructsanapproximate^T3vectorfrom^T2andthetwo-electronintegrals tabc[2]ijk=P(a=bc)P(k=ij)Xdtadijhbcjjdki)]TJ /F5 11.955 Tf 19.26 0 Td[(P(i=jk)P(c=ab)Xltabilhcljjjki(1)withsuperscript[2]indicatingitisaperturbativeapproximation,andtheP(p=qr)operatordenedbyitsactiononanarray P(p=qr)t(pqr)=(1)]TJ /F5 11.955 Tf 10.84 0 Td[(P(pq))]TJ /F5 11.955 Tf 10.84 0 Td[(P(pr))t(pqr)=t(pqr))-129(t(qpr))-129(t(rqp)(1)Asaresult,itisoftennotnecessarytogobeyondtheseapproximatetripleexcitationsformoleculesinthegroundstateneartheequilibriumgeometry.Thismethodscalesasn3on4v+n4on3vasevidentfromtheamplitudeequation.Itcanbeimplementedinaparallelizableway,andrepresentsaneconomicalroutetowardsbenchmarkaccuracy.Anyinclusionofhigherexcitationsbecomesunfeasibleveryfast. 36

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1.4ExploringMoreSectorsofFockSpaceThepreviousmethodswereformulatedintheN-electronHilbertspace.AgeneralizationoftheHilbertspacetoincludeanynumberofelectronsiscalledFockspace.[ 22 ]TobeginexploringmoresectorsofFockspace(electronionized,attachedorexcitedsectors)ageneralizationofCCtheoryisnecessary.[ 15 23 ]Ratherelegantly,othersectorsofFockspacearegeneratedwithanoperatoractingonthegroundstatewavefunctiontocreatethestateofinterest.Thestateofinterest,jki,iscreatedas jki=^Rkj0i=[jkih0j]j0i(1)Todeterminetheexactstructureof^R,webeginwiththefollowingenergyequations^Hj0i=^He^Tj0i=E0e^Tj0i (1)^Hjki=^H^Rkj0i=^H^Rke^Tj0i=Ek^Rke^Tj0i (1)Leftprojectionwithexp()]TJ /F4 11.955 Tf 11.29 2.66 Td[(^T)followedbycommuting^Rwith^TyieldsHj0i=E0j0i (1)Hjki=H^Rkj0i=Ek^Rkj0i (1)Finally,multiplyingEquation( 1 )ontheleftwith^RkandsubtractingthetwoyieldshH,^Rkij0i=Ek^Rkj0i (1)H^RkCj0i=!k^Rkj0i (1)whichisaneigenvalueequationfor!kwiththeeigenvectors,^Rkj0i.Thisformulationiscompletelygeneral.Infact,theonlydifferencebetweenEquation( 1 )forallstatesisthesectorsofHusedinthematrixdiagonalization.Forinstance,buildingHintheparticle-hole(ph),particle-particle-hole-hole(pphh),etc...sectors(ie.singly,doubly,etc...)yieldstheexcitationenergyequationofmotioncoupledclustermethod(EE-EOM-CC).[ 24 25 ]Furthermore,thep,pph,...andh,phh,...,correspondtothe 37

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ionizationpotential(IP-EOM-CC)[ 24 26 27 ]andelectronattachment(EA-EOM-CC)[ 28 29 ]variants.Infact,allsectorsareavailable,allowingthestudyofmultiplyionizedandattachedstates(MI/MA-EOM-CC),[ 30 ]howevertheformalscalingincreasesbyfactorsofn1o,n1v,andn1on1vforeachadditionalelectronionization,attachment,andexcitation,respectively.Forlargemolecularsystems,theEOM-CCmethodsarelimitedtosingleanddoublesubstitutions(EOM-CCSD)forcomputationaltractability.Insomecases,tripleexcitationsarenecessarytoaccuratelycharacterizeexcitedstateenergies.[ 31 33 ]Thesehigherexcitationscanbeaddedperturbatively,similartothegroundstatederivationofCCSD(T).[ 15 ]ThiswillbeexploredindetailinChapter 4 1.5ImplementationinRobustComputerCodesAbinitioquantumchemistrywouldbeoflittleuseinpracticalreal-worldchemicalapplicationswithouttheuseoftodayspowerfulcomputersystemsandarchitectures.Thecomputationalrequirementsoftheaforementionedmethodsaredeterminedbythescalingofthemethodasafunctionofthesizeoftheatomicorbitalbasissetdescribingthemolecularorbitals,thenumberofelectronsinthesystem,andthecorrespondingchargeandspinmultiplicity.Evenforthesimplestmethod,theindependent-particlemodel,therequirementscanincreaserapidlyandmustbemappedintosufcientlyfastandrobustcomputercode.Withthedevelopmentoffasttraditionalcomputerprocessorunits(CPU),graphicalprocessingunits(GPU),andparalleldistributionovertheseresources,itseemsappropriatetomodifyDirac'squoteas Itthereforebecomesdesirablethatapproximatepracticalmethodsofapplyingquantummechanicsshouldbedeveloped withsufcientparallelizationstrategies ,whichcanleadtoanexplanationofthemainfeaturesofcomplexatomicsystems whileproperlydistributingdataoverCPUsand/orGPUsto avoidtoomuch`direct'computation.Inthisvein,averysuccessfulsolutiontothemassiveparallelizationnecessarytodescriberelevantchemicalphenomenawiththesemethodsistheACESIIIquantum 38

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chemicalsoftwarepackage.[ 34 ]ACESIIIhasbeenbuilttoallowhigh-levelCCcalculationsonO(100)electronswithO(1,000)ofbasisfunctionsthroughefcientparallelizationstrategiestorunonO(10,000)toO(100,000)computecores,orprocessors.[ 35 ]Currently,serial,orsingleprocessorjobs,arecapableofonlyO(10)ofelectronswithO(100)basisfunctions,limitingtheapplicabilityoftheprogramtosmallmoleculesin,atbest,moderatelysizedbasissets.Thedesignphilosophyisi)toseparatecomputerscienceorientedtasksfromcommonquantumchemistryspecictasksandii)singleoatingpointnumbersaretoosmalladataelementtoperformworkwithoronandblocksofnumbersshouldformthebasicelementthatispassedbetweenprocessorsandworkedwithoron. Separationoftasks.Tothequantumchemist,certaintasksareperformedondataarraysratherroutinely.Forexample,manyofthearraysintheaforementionedmany-bodymethodsaredividedbydiagonalelementsoftheFockoperator.Thesearecalledenergydenominators.Conveniently,thesuperinstructionarchitecture(SIA)ofACESIIIcanhandlethesecommontasksefcientlyandeasilythroughtheuseofasuperinstructionprocessor(SIP).TheSIPreadscodewritteninACESIIIhomebrewhigh-levelsymbolicsuperinstructionassemblylanguage(SIAL,pronouncedsail),andexecutessuperinstructions.Thesearecalledsuperinstructionsbecause,usingtheenergydenominatorsasanexample,thequantumchemistprogramsarathersimpleexpression PARDOa,i GETT1(a,i) EXECUTEENERGY DENOMINATORT1 ENDPARDOa,i readbytheSIP.TheSIPloadstheT1arrayinthesuperinstructionENERGY DENOMINATOR,determinestheblocksizesandrangesofthearray,andperformstheenergydenominator 39

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as T1(a,i)=T1(a,i) i)]TJ /F7 11.955 Tf 11.96 0 Td[(a.(1)Alldatafetching,MPImessagepassing,etc...areperformedunderthehoodallowingthequantumchemisttowriteveryefcientcomputercode,withoutbeingdraggeddownbysomeofthemoretechnicalaspectsofthecomputerscience.Inpracticethough,ahybridapproachgeneratesthemostefcientcode,andaknowledgeofhowtowrite,use,andmanipulatethesuperinstructionsisverybenecial. Distributionofdata.Ratherthandistributingthefollowing4x4arrayofoatingpointnumbers 266666664x11x12x13x14x21x22x23x24x31x32x33x34x41x42x43x44377777775(1)onecandistributethefollowing2x2arrayofoatingpointblocks 264X11X12X21X22375(1) 40

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whereblockindices1and2containsimpleindicesf1,2gandf3,4g,respectively,yieldingtheXarraysasX11=264x11x12x21x22375 (1)X12=264x13x14x23x24375 (1)X21=264x31x32x41x42375 (1)X22=264x33x34x44x34375 (1)Fromthebeginning,onealreadyhasablockedspacewhenworkingwithmolecularorbitals-occupied,O,andvirtualV.ThisautomaticallyforcesOO,OV,VO,andVVblocksofone-electronoperatormatrixelements;withasimilarstructurefortwo-electronoperatormatrices.However,whenrunningcomputationswithO(102)toO(103)basisfunctions,itisnecessarytofurthersubdividetheseblockstomaximizeparallelizationandreducethecomputationaltime.Forexample,theratedeterminingstepforaHartree-FockcomputationisbuildingtheFockmatrix,specically,the^J)]TJ /F4 11.955 Tf 13.74 2.65 Td[(^KpieceoftheFockoperator, XVP7!f(1)Forsimplicity,assume,using10basisfunctions,thispieceoftheFockbuildtakes1second.With100basisfunctions,theidealtimeisthen10,000seconds,andwith1,000basisfunctionstheidealtimeis100,000,000seconds.Now,assumingperfectparallelization,byblocking100basisfunctionsinto10blocksof10basisfunctionseach,distributedperfectlysuchthateachprocessorisoperatingon1block(100 41

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processors),thenideally,thiswouldtake1second.Thesamefollowsforthe1,000basisfunctionexample.Inpractice,however,sincequantumchemistryisextremelydataintensive(largeformalscalingofdevelopedmethods),thecomputationsareperformedonlessthantheidealnumberofprocessorsandeachprocessorhasanassortmentofindependentdatablocks.Asameasureofperformancethough,ifonedoublesthenumberofCPUsinacomputation,onehopestoseeafactorof2reductionincomputingtime,andsoonandsoforth.Theideallimitofanx-foldincreaseinCPUstoanx-folddecreaseintimeislinearscaling.Factorsincludingtheproximityofprocessorstooneanother,thepotentialimbalanceofCPUspeedcomparedtooneanother,datainput(reading)andoutput(writing)todisk(I/O),andaccesstomemorycanplayimportantrolesinattaining,orfallingshortof,linearscaling.Thequestionthenarisesastowhynotdistributetheoatingpointnumbersacrossasmanyprocessorsasareavailable.Theaforementionedcommunicationanddata/messagingpassingfactorstakeconsiderabletimetoperform.Withpoorlywrittencode,ortheuseofverysmalldataelements,onecouldbewaitinglongerfordatapassingandaccessratherthanactualcomputation.ACESIIIsoftwareperformsoperationswithblocksofnumbers,andduringthoseoperationsitissimultaneouslypassinginformationtoandfromCPUs,effectivelyhidingcommunicationunderwork.Ideally,blocksizesarechosensuchthatallcommunicationaddszerotimetothecomputation.Theonlytimeisthatoftheactualblockcomputingoperationsthemselves.Inpractice,blocksizesarechosentoallowformaximumbenetofthiscapability.Asademonstration,thefollowingispsuedocodeforaserialcomputationusingEquation( 1 ) DO=1,Numberofbasisfunctions DO=1,Numberofbasisfunctions FOCK(,)=0.0 42

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DO=1,Numberofbasisfunctions DO=1,Numberofbasisfunctions FOCK(,)+=V(,,,)P(,) ENDDO ENDDO ENDDO ENDDO TheO(N4)scalingisseenexplicitly.InACESIIIsialcode,thisisimplementedas PARDO,,, REQUESTV(,,,) GETP(,) Txx(,)=V(,,,)P(,) FOCK(,)+=Txx(,) ENDPARDO,,, TheREQUESTandGETcommandscorrespondtoretrievingdatastoredondiskandinmemory,respectively.Eachprocessorisassignedblockindicesandperformsthecontractionoverthisdataretrievalas DO=a,b DO=a,b DO=a,b DO=a,b Txx(,)=V(,,,)P(,) FOCK(,)+=Txx(,) ENDDO ENDDO ENDDO 43

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ENDDO Theindicesa,b,...,arethesimpleindicescontainedintheblockofsizejb)]TJ /F7 11.955 Tf -428.66 -23.91 Td[(aj,andeachprocessorhasdifferentvaluesfortheseallowingforthemassiveparallelizabilityofACESIIIsoftware.TheprogramsdevelopedinthisworkhavebeenimplementedwiththisstructureinACESIII.Idemonstratetheperformanceoftheimplementedmethodswithscalingcurves,timings,andattimes,Iletthesizeofthesystemsunderinvestigationspeakforthemselvesconcerningperformance. 1.6OutlineofthisStudyInthenextthreechaptersIwilldiscussaccurateabinitiotechniquesusedtogaininsightintochemicalprocesses.Themainfocusofthisworkistheexpedientcalculationoftotalenergies,properties,andgeometrieswithveryhighaccuracyeitherthroughround-aboutmethodsordirectmassivelyparallelizedalgorithms.InChapter 2 ,Idescribeawell-knownandusedformallyexactprocedurefordecouplingthefour-componentDiracequationinanorder-by-orderschemeandtheimplementationofthisDouglas-Kroll-HesstransformationintheACESIIandACESIIIprogrampackages.ThistransformationallowsasignicanttimesavingscomparedtothefullDiracequation,butuntilnowhasnotbeensystematicallytestedwithpostHartree-Fockmethods,specicallyforcoreionizations.InChapter 3 ,Idevelopamethodthatcapturestheindependentparticleeffectoforbitalrelaxationtoinniteorderinadirectfashionandapplyitspecicallytocoreionizations,althoughitcanbeappliedtoelectronexcitedandattachedstates.Incontrast,currently,fullorbitalrelaxationcanonlybeincorporatedintheFullCI(orFullCC)limit(alongwithcorrelation),orbyindirectmeansthatareunwieldytouseinpractice.TheproposedmethodscalesasN4,andisclearlyatremendousbenetovertheFullCI,orperturbativeapproximationswithhigherexcitationsinCCtheory,althoughitisstatespecic. 44

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HavingdevelopedandimplementedschemestoefcientlyobtaintheinterestingquantitiesincoreionizationsIdepartandfocusonmassivelyparallelimplementationsofEOM-CCtheory.Withtheageofserialcomputingcomingtoanend,itiscriticaltotheeldtopushtheseaccurateCCmethodsfurtherthroughtheuseofmoderncomputerarchitectures.Irederive,inadifferentlight,aperturbativetreatmentoftripleexcitationswithinEE-EOM-CCSDtheoryandpresentresultsonsystemscurrentprogramscannotbegintodoinChapter 4 .AfterndinginterestingresultsfornucleobasesystemsinDNA,IdiscussgradienttheoryforEOM-CCSD,anddiscusstheparallelimplementation.ScalingcurvesanddemonstrablesystemsarepresentedtodenethenewbenchmarkofsystemsizeswithCCtheory. 45

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CHAPTER2DECOUPLINGTHEDIRACEQUATIONFORMEANFIELDCOSTAPPLIEDTOCOREIONIZATIONS 2.1PreliminariesThemutualrelationshipthatexistsbetweentheoryandexperimenthasneverbeenmoreevidentthanwhenelectronbindingenergies,orionizationpotentials(IPs),areinvolved.Theorbitalpicture,orshellmodel,isthefoundationforourconceptualunderstandingoftheelectronicstructureofatomsandmolecules.ByapproximatingtheN-electronwavefunctionasaproductofdoubleandsinglyoccupiedorbitals,forspinrestrictedandunrestrictedsystems,oneimmediatelyobtainsabasisforaquantitativeunderstandingoftheionizationprocess.Experimentally,principalionizationenergiescanbemeasuredusingX-RayPhotoelectronSpectroscopy(XPS),originallydevelopedbySiegbahnet.al[ 36 37 ]primarilyforcoreorbitals,UltravioletPhotoelectronSpectroscopy(UPS)techniquesforthevalencestates[ 38 ],andevenX-raytwo-photonphotoelectronspectroscopyfordoublecoreionizations[ 39 ].Theseexperimentsofferquantitativestructuralinformationwhenpairedwithasufcientlyaccuratetheoreticalmethod,andcanthereforecaptureamolecule'sngerprint,determinethestructureofchemicalspecies,andverifyproposedmechanismsofchemicalreactions[ 40 45 ].Howaccurateissufcientlyaccurate[ 46 ]?Ifweconsidercoreelectrons,weseechemicalshiftsofupto5eV[ 26 ]forelectronsindifferentenvironmentsduetotheprimarily,butnottotally,atomiccharacterofthecoreorbital.Thisiswithintheerrorrangeofthemostaccurateandpracticallyusefulmethodstoday.Separatingtherelativistic,relaxation,andcorrelationcontributionswillallowasystematicstudyaimedatrevealingthedecienciesofthesemethods.InthisChapter,IdiscusstheimplementationoftheDouglas-Kroll-Hesstransformationtostudythescalar-relativisticeffectsincoreionizationprocessesfororganicmolecules,wherespin-orbitisassumedtobesmall,withCCtheory.Wealsostudythebreakdownofincludingonlyscalar-relativistic 46

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effectsforcoreionizationsinheavyatomsandhighlystrippedionswiththegoalofndinganexpedientroutetowardsaccurateanswersforsystemswhererelativisticeffectsarenon-negligible.Webeginwiththeequationofmotioncoupledclustertheoryforionizationpotentialslimitedtosingleanddoublesubstitutions(IP-EOM-CCSD)[ 24 27 ].IP-EOM-CCSDisanefcientandaccuratemethodtocomputesingleionizationenergiesofmolecules.Itisadirectmethod,needingonlytheN-electronreferencefunctionandcorrespondingcoupledclustersolutions,andapplicabletotheentirespectrumasasinglecalculationviadiagonalizationof H(H-bar),[ 15 ]fromtraditionalCCtheory,intheholeandparticle-particle-holesectorsdenedinChapter 1 .Manyvalenceionizationsformoleculesandionshavebeenstudiedandbenchmarkedwiththismethod,butsystematicstudiesformid-valenceandinner-shellionizationsarefarfewer.[ 29 47 48 ].Themid-valenceregioncanbenotoriouslydifculttoobtainquantitativeaccuracyduetothebalanceneededbetweencorrelation,relaxation,andrelativisticeffects.Theinner-shellelectronsarealsodifculttogetcorrectduetotheexceedinglylargerelaxationandpotentiallylargerelativisticeffectspresentasIwillshow.IP-EOM-CCSDcanincorporateasignicantamountofrelaxationbecauseoftheinclusionofthelinear,singleexcitationoperator,^Rk.[ 29 ]Ofcourse,thegroundstatesolutioniscorrelatedwithinthisframeworkaswell.Itcannot,however,includerelativisticeffects.Thesemustbeincludedattheindependentparticlelevelofapproximationforincorporationintothecalculation.Averysuccessfulmethodthathasbeenusedrecentlytoaddscalar-relativisticeffectsistheDouglas-Kroll-Hess(DK)transformation.[ 49 50 ]ThistransformationseekstodecouplethepositiveandnegativeenergysolutionsoftheDiracequationbyblockdiagonalizingtheone-electronoperatorsinanorderbyorderunitarytransformationinmomentumspace.Thisallowsonetofocusontherelevant,positiveenergysolutions. 47

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Italsoconvergesrapidlyduetolargedenominatorspresentinmomentumspace.[ 49 ]Interestingly,therst-ordertransformationdrasticallyoverestimatesthescalar-relativisticeffects,thesecond-ordertransformationthenover-shootstheexactscalar-relativisticeffects,thethird-ordertransformationbringstheenergeticsontrack,andthefourthandfth-ordertransformationsessentiallyironsoutthewrinkles,convergingthepotential.Thisresultisratherwell-known,[ 50 52 ]andisthereasonIimplementedtheDKtransformationtofth-order,eventhoughthereareinniteordertechniquesforexactdecoupling.[ 53 ]Fortotalenergiesofhighlychargedone-electronions,thefth-orderscalar-relativisticDKtransformationprovidesquantitativeaccuracycomparedtousingthefullDiracHamiltonianforsystemsuptoZ=100.[ 49 ]Theseareexcellenttestsystemsbecausetherearenospin-orbiteffectsandinprinciple,willconvergetotheexactDiraclimit,astheyhavenotwo-electronterm,whichisignoredintheDKtransformation.Toreallytestthelimitsofthefth-orderDKtransformationwithrespecttototalenergies,WolfandReiher[ 49 ]haveappliedthethefth-ordermethodtomanyelectronatomsincludingneutralandsinglychargedsilver,Ag0andAg+,andneutralandsinglychargedgold,Au0andAu+.ThesesystemsdemonstratetheeffectivenessoftheDKmethodsincetheycontainnon-negligiblespin-orbiteffects,duetotheirsizeandnuclearcharge,andcontainmorethanoneelectron.Thedifferentpropertiesofgoldascomparedtosilver,includingitscharacteristicshinycolor,isactuallyduetorelativisticeffects,andthereforeisagreatfundamentaltestforapproximaterelativisticmethods.Forthesilversystems,thefth-orderDKtransformationveryaccuratelyreproducedthetotalenergycomparedtofull4-componentDirac-Fock-Coulombcalculationsfortheneutralandcation.Thenon-relativisticenergyforAg0isroughly117a.u.inerror!TheHOMOenergyoftheneutralsilveratomwascomparedtothecomputedionizationenergy!=E(Ag+))]TJ /F5 11.955 Tf 12.41 0 Td[(E(Ag0)andshowntobeveryconsistent,beinginerrorbyonly0.1eV.Thenon-relativistictotalenergyofAu0wasinerrorofthefullyrelativisticresult 48

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by1,174a.u.!Thescalar-relativisticeffectsreducedthaterrorto27a.u.ThedifferencebetweentheKoopmans'valueandtheEvaluefortheionizationenergyisonly0.3eV.Clearly,theDKtransformationprovidesanexcellentstartingpointforcorrelatedcalculations,inparticularIP-EOM-CCSD.Recently,thesetwomethodshavebeenmerged,[ 54 55 ]tocomputethevalenceionizationenergiesofsystemscontainingheavyatoms.Benchmarkresults,agreementwithexperimentunder1kcalmol)]TJ /F9 7.97 Tf 6.59 0 Td[(1,wereobtainedforthevalenceionizationenergiesunderinvestigationafterincludingspin-orbiteffects.ThisisnotsurprisingwhenonelooksatthevalenceionizationenergiesobtainedbyWolfandReiherforsilverandgold.[ 49 ].However,applicationtocoreionizationenergiesisnon-existent.Therearestudiesconcerningthecoreionizationenergiesofmolecules,[ 39 56 57 ]however,theseemploydifferentHamiltoniansincludingscalar-relativisticeffectsandusedifferentmethods,mostnotablypropagatormethods,tocomputethecoreIP.ByexaminingtheconvergenceoftheDKpotential,wedenitivelybenchmarkthescalar-relativisticeffectsinthecoreIPsoforganicmoleculesanddemonstratetheyarenecessarytoobtainaccurateenergies.WealsoapplytheIP-EOM-CCSD/DKmethodtoincreasinglyheavynoblegasestodetermineitsbreakingpoint.Ournaltestisonhighlystrippedions,whicharerelevantinplasmaresearch.[ 58 ] 2.2TheoryTherelativisticextensionoftheSchrodingerequationwasrealizedbyDirac[ 1 ]inattemptingtotreatbothspaceandtimesymmetricallyusingEinstein'srelation[ 2 ] H=)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(c2p2+m2c21=2(2) 49

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Bywritingtheexpressioninthesquarerootasaperfectsquare,wecanobtainaHamiltonianthatislinearinspaceandtime^HD=cp+()]TJ /F4 11.955 Tf 11.96 0 Td[(1)mc2 (2)=0B@mc2I2cpcp)]TJ /F5 11.955 Tf 9.3 0 Td[(mc2I21CA (2)whereI2isa2x2identitymatrixandarethePaulispinmatricesfx,y,zg =0B@01101CA^x+0B@0)]TJ /F5 11.955 Tf 9.3 0 Td[(ii01CA^y+0B@100-11CA^z(2)TheDiracHamiltonian,^HD,isa4x4matrixand,consequently,actsonafour-component(4-component)Lorentzspinor,,withupper(L)andlower(S),forlarge(positive)andsmall(negative)components,respectively.Byaddingtheelectron-nucleuscoulombpotential,^V,theone-electronDiracequationis 0B@(^V+mc2)I2cpcp(^V)]TJ /F5 11.955 Tf 11.96 0 Td[(mc2)I21CA264LS375=E264LS375(2)Forlargemolecularsystems,theDiracequationbecomescomputationalinfeasible.Thegoalthen,istodecouplethelargeandsmallcomponentstoobtainrelativisticeffectswithstandardmean-eldscaling ^Hbd=U^HDUy=0B@h+00h)]TJ /F13 11.955 Tf 13.07 43.16 Td[(1CA.(2)BydeningamatrixXthatrelatesthelargecomponenttothesmallcomponent, S=XL,(2) 50

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thentheunitarymatrixcanbewritten U=U(X)=0B@(1+XyX))]TJ /F9 7.97 Tf 6.59 0 Td[(1=2(1+XyX))]TJ /F9 7.97 Tf 6.59 0 Td[(1=2Xy(1+XXy))]TJ /F9 7.97 Tf 6.58 0 Td[(1=2X(1+XXy))]TJ /F9 7.97 Tf 6.58 0 Td[(1=21CA(2)andtheupperdiagonalblockoftheDiracHamiltonianisgivenas h+=1 p 1+XyX)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(V+cpX+Xycp+Xy(V)]TJ /F5 11.955 Tf 11.96 0 Td[(mc2)X1 p 1+XyX(2)However,ananalyticalenergy-independentXmatrixisnotknown.Consequently,afewtechniqueshavebeendevelopedthatattempttosolveforthis.NumericalprocedureshavebeenimplementedbyBarysz,Sadlej,andSniders(BSStransformation)[ 59 60 ]withsomesuccess.Though,ananalyticalexpressionismoredesirable.Inthatspirit,zerothandrstorderregularapproximations(ZORAandFORA)havebeenderivedandimplemented[ 61 ]andextendedtoinniteorder(IORA)generalizationsbyDyallandLenthe[ 62 ].ThesemethodsexpandXinEquation( 2 )inpowersof1=c,creatinganeffectiveHamiltonianmatrix.[ 63 ]Inthelimit,thesemethodsarenotexactandhaveassociatederror.Inpractice,however,theZORAandFORAmethodsperformquitesatisfactoryafterscalingthetotalenergyexpressions.[ 61 64 66 ]Anotherwaytoblock-diagonalizetheDiracHamiltonianistochooseasuitableunitarytransformationorder-by-order^Hbd=U^HDUy (2)=Uy4Uy3Uy2Uy1Uy0^HDU0U1U2U3U4 (2)=0B@h+00h)]TJ /F13 11.955 Tf 13.06 43.16 Td[(1CA. (2) 51

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ThemostgeneralparameterizationforUicanbewrittenasapowerseriesofananti-hermitianoperatorW, Ui=Ui(Wi)=ai,01+1Xk=1ai,kWki.(2)Then,then-thordertransformedHamiltonianisgivenas^Hn=Uyn)]TJ /F9 7.97 Tf 6.59 0 Td[(1^Hn)]TJ /F9 7.97 Tf 6.58 0 Td[(1Un)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (2)="a(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1),01+1Xk=1a(n)]TJ /F9 7.97 Tf 6.58 0 Td[(1),kWkn)]TJ /F9 7.97 Tf 6.59 0 Td[(1#^Hn)]TJ /F9 7.97 Tf 6.59 0 Td[(1"a(n)]TJ /F9 7.97 Tf 6.58 0 Td[(1),01+1Xk=1a(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1),kWkn)]TJ /F9 7.97 Tf 6.59 0 Td[(1# (2)Thezeroth-ordertransformationisthefree-particleFoldy-Wouthuysen(fpFW)transformation[ 67 ],whichiswell-behavedandexactforfreeelectrons.ThezerothorderfpFWtransformationmatrixis[ 49 ] U0=Ap(1+Rp)(2)withAp=s Ep+mc2 2Ep,Ep=p p2c2+m2c4 (2)Rp=cp Ep+mc2=Rpp. (2)Thisyieldstherst-orderblock-diagonalizedHamiltonianas ^H1=)]TJ /F7 11.955 Tf 5.47 -9.68 Td[(Ep)]TJ /F5 11.955 Tf 11.95 0 Td[(mc2+(Ap(V+RpVRp)Ap)+(Ap[Rp,V]Ap),(2)withthelasttwotermscorrespondingtoadiagonal(D1)andoff-diagonal(O1)form,respectivelywithsubscriptidenotingtheorderinV,theelectron-nucleuspotential.Notethatthesecondtermisactually Ap(V+RpVRp)Ap=Ap(V+RppVpRp)Ap(2) 52

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andthereforerequiresAOintegralsoftheformhpVpi=hjpVpji (2)=hpjVjpi (2)=hrjVjri, (2)orsecondderivativetypeintegralsofthenucleus-electronpotentialenergy.Notethattheseexpressionsarewell-denedinmomentum(p-)space.TheunitarytransformationmatrixisobtainedbydiagonalizingthekineticenergyAOintegrals,yieldingadiagonalrepresentationforEp.Consequently,thetransformationisperformedinp-space,andbacktransformedtorealspaceafterwards.TheDKtransformationisthestepwiseeliminationofthislowestorderinVoff-diagonalpieceviaaseriesofunitarytransformations O1=Ap[Rp,V]Ap.(2)Asabriefaside,wecanrewriteEquation( 2 )inamoretransparentform,separatingtermsinordersof1=c, ^H1=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(1)mc2+V+p2 2m+)]TJ /F7 11.955 Tf 9.29 0 Td[(p4 8m3c4)]TJ /F4 11.955 Tf 13.15 8.09 Td[([p,[p,V]] 8m2c2.(2)ThersttwotermsofthisFWexpansioncorrespondtotherestenergyoftheelectronandthenon-relativisticresult,respectively.TheDarwin,mass-velocity,andspin-orbitcouplingtermsarecontainedinthelastterm,whichcanberecasttoshowthisas 1 8m2c2(V))]TJ /F7 11.955 Tf 11.95 0 Td[(p4 8m3c2+1 4m2c2[(rV)p],(2)withbeingthe4-componentextensionoftheaforementionedPaulimatrices. 53

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Toremovethisoff-diagonalpotential,Uisparameterizedinthefollowingway^Hbd=U^HDUy=Uy4Uy3Uy2Uy1^H1U1U2U3U4 (2)=1XiDi (2)=1Xi0B@Di+00Di)]TJ /F13 11.955 Tf 13.06 43.15 Td[(1CA=1Xi0B@Dsfi++Dsdi+00Dsfi)]TJ /F4 11.955 Tf 9.75 3.11 Td[(+Dsdi)]TJ /F13 11.955 Tf 13.06 44.47 Td[(1CA (2)Thespinfree(onecomponent)andspindependentterms(twocomponent)havebeenseparatedinthelasttermofEquation( 2 ).Wewillfocusonlyonthespinfreeterms,thescalar-relativisticpiece,asthisiswhathasbeenimplemented.FollowingthenotationWolfet.al[ 49 ],thespin-freeDKn(upton=5)HamiltoniansaregivenasDsf2+=1 2[W1,O1] (2)Dsf3+=1 2[W1,[W1,D1]] (2)Dsf4+=1 8[W1,[W1,[W1,O1]]]+1 2[W2,[W1,D1]] (2)Dsf5+=1 2[W2,[W2,D1]]+1 2[W2,[W1,[W1,O1]]]+1 2[W2,W1O1W1])]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 8[W21,[W21,D1]]+0.14644661[[W2,W31],D0] (2)with,transparentforprogramming,D0=Ep)]TJ /F5 11.955 Tf 11.96 0 Td[(mc2 (2)D1(i,j)=AiVijAj+AiRiVijRjAj (2)W1(i,j)=O1(i,j) Ei+Ej (2)W2(i,j)=W1(i,k)D1(k,j))-222(D1(i,k)W1(k,j) Ei+Ek (2)Theprocedurethen,istodiagonalizethekineticenergyAOintegrals,transformthehpVpiintegrals,computeEp,Ap,andRp,andconstructtheDKHamiltonians(in 54

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p-space)uptothedesiredorder.Followingitsconstruction,theDKHamiltonianisbacktransformedwiththekineticenergyeigenvectors,resultinginthecoordinatespacerepresentationofthescalar-relativisticHamiltonian.Withnotransformationoftheelectron-electronpotentialterm,thenalelectronicDKHamiltonian,uptoordern,is ^HDKn=^TDKn+^VNeDKn+^Vee,(2)withthecorrespondingFockoperator ^F=^TDKn+^VNeDKn+J(r1))]TJ /F5 11.955 Tf 11.96 0 Td[(K(r1).(2)Sincethetransformedquantitiesareindependentoftheorbitals,theyonlyneedtobecomputedonceandstoredatthebeginningoftheSCFprocedure.Whenperformingpost-SCFcomputations,nochangesneedtobemadetothecode,sincetheMOcoefcientsandorbitalenergiesaredeterminedwiththeDKHamiltonian(ie.theformoftheHamiltonianisunchanged,onlytheindividualelementsofthematrixaredifferent). 2.3ImplementationTheDKtransformationwasimplementeduptofthorderasalibraryandthereforeisincorporatedintheACESIIandACESIIIprogrampackages.TheDKmaindriverneedsonlytheatomicorbitalkineticenergy,nucleus-electronpotentialenergy,andtheDKspecicpVpintegrals,thecorrespondingarraysizes(numberofbasisfunctions),andsufcientscratchspaceoninput.ThepVpintegralswereimplementedinthevmolintegralpackage.InACESII,theDKtransformationhasbeenimplementedtakingfulladvantageoftheAbelianpointgroupsymmetryavailableintheprogram.Essentially,eachirreduciblerepresentationoftheone-electronhamiltonianistransformedtothedesiredorder.ACESIII,however,doesnotincludeAbelianpointgroupsymmetryinitsstructure,sotheentireintegralarraysaretransformedinonecall.Thisisnotreallyaproblemsincethe 55

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integralarraysarealreadystoredstaticallyoneveryprocessorandonlyhavedimensionofNAOxNAO.Sincethetransformationisaseriesofmatrixmultiplications,thescalingisalwayssmallerthanthatoftheSCForcorrelatedmethodandeffectivelyaddsnotimetothecomputation.Thecontractioncoefcientsandexponentsforcurrentgeneralavailablebasissets[ 7 8 ]areoptimizedusingnon-relativisticmethods.Consequently,thesecontractedbasissetsare,atbest,ill-suitedtobeappliedwiththeDKtransformationandeitherneedtobereoptimizedoruncontracted.Therefore,thepre-processingprogram,xjoda,wasmodiedwithanewkeywordtoreadingeneralcontractedGaussianbasisfunctions,anduncontractthemforthecalculation.Thisisactuallybenecialforthecalculationformorethantheaforementionedreasons.ContractedGaussianstypicallycontainlargeexponentprimitivestorepresentSlaterorbitalsnearthenucleus,andarefreedwhenuncontractedtomoreaccuratelydescribetheregionnearthenucleus[ 68 ](ie.morepronouncedlocalizationofcoreelectronsnearthenucleus).Also,itgeneratesmorefunctionstodescribethemolecularsystem,whichareoptimizedduringtheSCFprocedure. 2.4ResultsandDiscussionForconsistencythefamilyofbasissetschosenfortheinvestigatedsystemsaretheWell-TemperedBasissets(WTBS)ofHuzinagaet.al[ 69 70 ]Thesebasissetscontainmanyprimitivesandarethereforesufcientlylargeenoughtoprovidedenitiveresults.Theyalsocontainverylargeexponentprimitivefunctionstomoreaccuratelydescribetheregionwhererelativisticeffectsarelargest. 2.4.1OrganicMoleculesTheaveragescalar-relativisticstabilizationgainedwiththeDKpotentialforthedifferentmethodsareshowninFigure 2-1 .Thesearetotalenergydifferencestohighlighttheeffectofthetransformedpotentialonvariousmethodstocomputethecorrelationenergy.Thestabilizationisrelativelyuniformacrossthetestedcorrelated 56

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methods,indicatingtheseparabilityoftherelativisticandcorrelationeffects.Inotherwords,thecorrelationenergyisthesameforthenon-relativisticandrelativisticpotentials(forCCSDandbeyond),eventhoughtheorbitalsareverydifferent.Calculationswerealsoperformedwiththeuncontractedtriple-basisset,whichdoesnotincludeverytightprimitivefunctions.Thisdemonstratestheneedforverytightbasisfunctionsnearthenucleus,whicharenotincludedthecc-pCVTZbasisset.Onaverage,thisbasissetovershootsthescalar-relativisticcorrectionsasshowninFigure 2-1 .TheDKtransformationdoesnotcorrectallcoreionizationpotentialsunidirectionally.AfewselectmoleculesareshowninFigure 2-2 .Thereappearstobenoaprioriwaytosystematicallydeterminewhenthiswouldoccur.Ofthefewthataredestabilized,allbutammoniaareorganiccompoundswithasingleoxgyen-carbonmonoxide,formaldehyde,andmethanol.However,dimethylether(CH3-O-CH3)containsoneoxygenandisstabilized,breakingthistrend,andofcourseammoniadoesnotteither.ThedirectIP-EOM-CCSDionizationenergyis,onaverage,alwaysimprovedwithaconvergedDKpotentialcomparedtoexperimentalvalues,demonstratedinFigure 2-3 .Thisisveryencouragingasdirectmethodsaremorepracticalthantheirindirectanalogues,andtheDKmethoddoesnotchangethecomputationaleffortforthesecalculations.Theindirectanalogue,ECCSD,thoughmuchbetterthanIP-EOM-CCSDforreasonsdiscussedinChapter 3 ,isonaveragemarginallyworsewiththeDK5potential. 2.4.2NobleGases Argon.Argonisagood,simplestartingpointtoassesstheaccuracyofIP-EOM-CCSDwithaDKpotentialforneutralsystems.Therearethreetotallysymmetricms=0shellsandoneexperimentallydetectablespinsplitms=1toms=1=2,3=2shell.Directcomparisontoexperimentatthevalence(2p)levelispossiblesincethespin-orbitsplittingbetween2p1=2and2p3=2isnegligible.[ 71 ]Also,relativisticeffectsshouldbenoticeableforthetotalenergyandcoreorbitalenergies. 57

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TheconvergenceoftheDKpotentialtofthorderfortheSCFandcorrelationenergies,Koopmansenergy,andcorrelationcontributionobtainedwithIP-EOM-CCSDisshowninTable 2-1 .Thepotentialisconvergedbythirdorder,andthereforeunambiguoslydemonstratestheutilityofIP-EOM-CCSD.Clearly,theDKpotentialisnecessaryattheSCFlevel,stabilizingthenon-relativisticresultby58.6eV.TheusualtrendfortheDKtransformationisseen,rstorderovershootstheSCF,secondorderbringsitback,andthirdorderisessentiallyconverged.Thecoreorbitalenergies,wherethetranformedpotentialhasthegreatestimpact,behavesthesameway.Notethatthesignischanged,sincetheionizationpotentialsarereported,ratherthantheorbitalenergies.ThetransformedpotentialhaslittleeffectonthecorrelatedportionofthecalculationwithCCSD.TheKoopmans'ionizationpotentials,inparticularthecore,varysignicantlyfromDK0toDK1,andnon-negligiblyfromDK1toDK2,butthecorrelationenergyandcorrelationcontributiontotheIPareessentiallyconvergedatrstorder.ComparisonoftheionizationpotentialswithexperimentarepresentedinTable 2-2 .Koopmans'IPsarepresentedtoshowtheorbitalshiftsateachorderinthetransformedpotential,whichcanbedrasticasdemonstratedinTable 2-1 .AgreementwithexperimentisquitegoodwithIP-EOM-CCSDfortheentirespectrumatanyorder.Thisindicatesthatargonmaynotbethebestexampletostudythelimitsofthiscomputationalcombination.Thevalencelevel,though,worsenswhenincludingscalar-relativisticeffects.Weseemtoberemovingasourceoffortuitouserrorcancellationbetweenrelaxation,correlation,andrelativityforthevalenceshell.Koopmanstheoremisapproximately10eVtoohighovertheexperimental2p1=2shellinthemid-valenceregion,butisremediedwithIP-EOM-CCSD.ItbringstheIPwithintherangeofthespin-orbitsplittingfromexperiment.Forthe2selectrons,thenon-relativisticresultis1eVclosertoexperimentthanDK5;thoughthisseemstobealackoferrorcancellationbetweenrelativisticandrelaxationeffects,thelatternotbeingdescribedasfully.Relaxationandcorrelationinthe1slevelaloneisnotsufcientto 58

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quantitativelydescribetheionization,thoughtheDK5resultisnotmuchbetter.TheDK0andDK5resultdifferfromexperimentby7.0eVand5.7eV,respectively,showingasmallimprovementwithaDKpotential. Krypton.TheconvergenceoftheDKpotentialuptofthorderforkryptonisshowninTable 2-3 .Thesametrendsareseenforkryptonasforargonexceptconvergenceisreachedbyfourthorder.Onlythevalence4pshellisconvergedwithnorelativisticeffects.TheremainingKoopmans'IPschangedrasticallyatlowerordersandtheSCFenergyisstabilizedby986eVatconvergence.AgainweseethatthecorrelationenergyandcontributiontotheIPsarelesssensitivetotheorderofthepotential,andareconvergedafterrstorderforthemid-valencetovalenceshells,andsecondorderforthecoreselectrons.ComparisonwithexperimentisshowninTable 2-4 .TheKoopmans'valuesareshownbecausetheorbitalshiftcausedbytheDKpotentialisthedriverforqualitativeaccuracy.Thenon-relativisticKoopmans'andIP-EOM-CCSDresultforthecoreorbital,whererelativityshouldbemostimportant,differfromexperimentby175.6eVand216.4eV,respectively.TheIP-EOM-CCSDresultmovesinthewrongdirection.Atfthorder,thetwodifferfromexperimentby)]TJ /F4 11.955 Tf 9.29 0 Td[(78.7eVand)]TJ /F4 11.955 Tf 9.3 0 Td[(29.6eVrespectively.ItseemstheIP-EOM-CCSDrelaxestheorbitalofinterest,andthereforerequiresanappropriateinitialstate,oneincorporatingtherelativisticstabilizationoftheinner-shellelectrons.Effectively,allionizationenergiesobtainedfromIP-EOM-CCSDimprovewithaDKreference.ThevalencelevelisunchangedandtheIPformid-valencesplitlevelseithermovesintoorremainswithintherangeofthesplitting.Theimprovementisnotnearquantitativeaccuracythough,andsuggestsmoreaccurateresultscanbeobtainedbyincorporatingrelativityintheionizedstate.Although,itisevidentthattheDKpotentialisnecessaryfortheIPtobeginapproachingexperimentalvalues. Xenon.TheconvergenceoftheDKtotalenergiesandorbitalenergiesforxenonareshowinTable 2-5 .Relativisticeffectsareseentobeverylargeforthissystem,and 59

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theDKconvergencebehavesthesamewayasinargonandkrytpon.WecannotsaywithcertaintythattheSCFenergyandcoreorbitalenergiesareconvergedatfthorder.Thetotalcorrelationenergyisconvergedatrstorderbutthecorrelationcontributionstotheionizationenergiesconvergemuchmoreslowlyduetotheamountofrelaxationforeachcoreionization.Thenon-relativisticandIP-EOM-CCSDionizationenergyforthe1scoreelectrondifferfromexperimentby1245eVand1307.32eV,respectively,showninTable 2-6 .UsingthefthorderDKpotential,theydifferby)]TJ /F4 11.955 Tf 9.3 0 Td[(164.1eVand)]TJ /F4 11.955 Tf 9.3 0 Td[(98.0eV,respectively.AgainweseethatIP-EOM-CCSDsimplyincorporatesmissingrelaxationfortheionizationofinterestandtheDKpotentialisneededtoshifttheIP,causingittochangesigncomparedtoexperimentsoIP-EOM-CCSDcanbringitclosertoexperiment,ratherthanfartheraway.Clearly,forxenon,theionizationenergiesforthecoreelectronsarefarfromthetruth,indicatingabreakdownoftheIP-EOM-CCSD/DKcombination.Theyareordersofmagnitudeclosertoexperiment,though,andbegintomovetowardsprovidingqualitativeresults.TheIP-EOM-CCSD/DKresultsprovidequalitativeaccuracyforthemid-valenceregion.Again,theresultsmoveintothecorrectrange,betweenthesplitting,whenthenon-relativisticresultsfailtodoso. 2.4.3HighlyStrippedIonsSincethenuclearchargeforhighlystrippedionsislarge,thePartridgefamilyofbasissets[ 72 ]wasusedtorstassessthequalityoftheWTBSused.ThePartridgefamilyofbasissetsarecompletelyuncontracted,haveverytightprimitivefunctions,andincreaseinsize,whichhelpsdeterminewhetherthebasissetisconverged.TheresultsforheliumlikeionsandberylliumlikeionsareshowninTable 2-7 andTable 2-8 ,respectively.Forallcases,eventhelargestnuclearcharge,thereisverylittledifference 60

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betweentheIPsasafunctionofthebasissetsize.SincetheWTBScontainsmorefunctions,thesystemshouldbeveryadequatelydescribedwithit.ResultswiththelargerWTBSforhighlystrippedHe-likeionsareshowninTable 2-9 .Thesesystemswerestudiedbecausetheyaretwo-electronsystems,whereIP-EOM-CCSDisequivalenttofullcongurationinteraction(FCI),andtherearenospin-orbiteffectsintheoccupiedlevels.Nearquantitativeagreementwiththebesttheoreticalestimate(within1eV),includingQEDeffects,[ 73 ]isobtaineduptoanionizationenergyofaround5,000eV,thoughthehighernuclearchargeresultsaredrasticallyimprovedandveryqualitativeoverthenon-relativisticresult.Thenon-relativisticandfthorderresultsfornuclearchargeZ=20differfromthebesttheoreticalestimateby24.1eVand)]TJ /F4 11.955 Tf 9.3 0 Td[(1.2eV,respectively.FornuclearchargeZ=30,thetwodifferby132eVand)]TJ /F4 11.955 Tf 9.3 0 Td[(4.6eV,respectively,demonstratingtheneedfortheDKpotential..Foracomputationallytractablemethod,theIP-EOM-CCSDwithaDK5potentialisexceedinglygoodforthesesystems.ResultswiththelargerWTBSforBe-likeions,whereIP-EOM-CCSDisnotequivalenttoFullCI,butspin-orbitisalsonotanissuefortheoccupiedelectrons,areshowninTable 2-10 .Againwesee,wherethecoreelectronsareupto5,000eV,excellentagreementwithexperimentandthebesttheoreticalestimate.[ 74 ]WhenthecoreelectronIPismuchlarger,theDKpotentialstillprovidesverygoodqualitativeresults.ForthecoreionizationofNi24+,thenon-relativisticresultdiffersfromthebesttheoreticalestimateby149.2eV,whereastheDK5resultdiffersby5.39eV.SimilarresultsareobtainedforZn26+,wherethenon-relativisticandDK5resultsdifferfromthebestestimateby187eVand6.9eV,respectively.Interestingly,forthesetwoBe-likeions,the1s22sDK5ionizationenergy,wellbelow5,000eV,differfromexperimentby)]TJ /F4 11.955 Tf 9.3 0 Td[(2.5eVand)]TJ /F4 11.955 Tf 9.3 0 Td[(8.1eV,respectively.Thereappearstobeconsistenterrorinthespectrumforsystemswithveryhighenergycoreelectrons,thoughusingtheDKpotentialprovidesresultsordersofmagnitudebetterthannotusingone.Perhaps,thesehighenergy 61

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systemsindicate,notnecessarilyabreakdown,buttheinsufcientrelativistictreatmentoftheionizedstate.Though,thisprovidesaveryefcientroutetowardsthenearquantitativestudyofplasmas. 2.5ConclusionsIP-EOM-CCSDcalculationswithDKreferencepotentialshavebeensystematicallystudiedandproposedasameanstoobtainnearquantitativeaccuracyforhighlystrippedionsandacorrectqualitativedescriptionofionizationenergiesforsystemswithelectronsmovingnearthespeedoflight.Calculationsonhighlystrippedionsdemonstratethatforsystemswithorbitalenergiesupto5,000eVquantitativeresultsareobtainable.Evenwhentheorbitalenergiesbecomeextremelylargeinmagnitude,verygoodqualitativeresultsareobtained.Forneutralsystemswithheavyatoms,despitetheIP-EOM-CCSD/DKcombinationnotbeingquantitativelyaccurate,itisnecessaryforevenaqualitativedescriptionoftheionizationprocess.TheDKpotentialshiftsalltheorbitalenergies,exceptfortheoutermostvalenceshell,inthedirectionneededforIP-EOM-CCSDtoappropriatelyrelaxandcorrelatetheionization. 62

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Figure2-1. AveragedifferenceintotalionizationenergiesbetweenDK5andDK0fordifferentmethodsineV.Thenegativeaverageunsigneddifferenceisreportedfordirectcomparisontotheaveragedifference. Figure2-2. DifferencebetweenDK5andDK0totalionizationenergieswithdifferentmethodsforasubsetofmoleculesineV. Figure2-3. AverageerrorofdifferentDKpotentialsandmethodscomparedtoexperimentineV. 63

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Table2-1. ConvergenceofSCFandcorrelatedmethodswithrespecttoDouglas-Kroll-HessordersforArgon.EnergyunitsareineV.TheWTBSisemployed. DK1-DK0DK2-DK1DK3-DK2DK4-DK3DK5-DK4 SCFenergy-58.5959.330-0.1270.0040.000Totalcorr.-0.0740.0000.0000.0000.0003pSCF-0.0240.0050.0000.0000.0003pcorr.-0.0010.0000.0000.0000.0003sSCF0.278-0.0330.0000.0000.0003scorr.-0.0170.0020.0000.0000.0002pSCF0.0330.047-0.0010.0000.0002pcorr.-0.033-0.0010.0000.0000.0002sSCF2.675-0.3170.0040.0000.0002scorr.-0.9340.155-0.0020.0000.0001sSCF17.046-4.0560.055-0.0020.0001scorr.-0.3450.0190.0000.0000.000 64

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Table2-2. Ionizationpotentials(ineV)ofArgonusingdifferentDouglas-Kroll-HessreferenceswiththeWTBS. IP-EOM-CCSDKoopmansOrbitalTheor.Est.aDK0DK1DK2DK3DK4DK5DK0DK1DK2DK3DK4DK5 3p1=215.8115.1215.0915.1015.1015.1015.1016.0816.0616.0616.0616.0616.063s1=229.3133.2433.5033.4733.4733.4733.4734.7635.0435.0035.0035.0035.002p3=2248.47249.87249.87249.91249.91249.91249.91260.45260.49260.53260.53260.53260.532p1=2250.562s1=2324.36325.36327.10326.94326.94326.94326.94335.30337.98337.66337.67337.67337.671s1=23206.073199.073215.773211.733211.783211.783211.783227.553244.603240.543240.603240.603240.60 a BesttheoreticalestimatefromRef[ 71 ] 65

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Table2-3. ConvergenceofSCFandcorrelatedmethodswithrespecttoDouglas-Kroll-HessordersforKrypton.EnergyunitsareineV.TheWTBSisemployed. DK1-DK0DK2-DK1DK3-DK2DK4-DK3DK5-DK4 SCFenergy-1209.283229.906-6.7040.449-0.084Totalcorr.-0.531-0.0010.0000.0000.0004pSCF-0.0480.021-0.0010.0000.0004pcorr.0.009-0.0030.0000.0000.0004sSCF1.098-0.1650.0050.0000.0004scorr.-0.2150.048-0.0010.0000.0003dSCF-2.2250.118-0.0040.0000.0003dcorr.0.206-0.0780.0020.0000.0003pSCF2.0390.146-0.0040.0000.0003pcorr.-1.435-0.0200.0010.0000.0003sSCF11.596-1.5480.046-0.0030.0013scorr.-3.2450.606-0.0230.0020.0002pSCF13.0670.707-0.0210.0010.0002pcorr.-1.412-0.0090.0000.0000.0002sSCF67.704-9.5710.286-0.0190.0042scorr.1.4460.955-0.0300.0020.0001sSCF347.525-100.0032.905-0.1940.0361scorr.-4.9920.787-0.0450.003-0.001 66

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Table2-4. Ionizationpotentials(ineV)ofKryptonusingdifferentDouglas-Kroll-HessreferenceswiththeWTBS. IP-EOM-CCSDKoopmansOrbitalTheor.Est.aDK0DK1DK2DK3DK4DK5DK0DK1DK2DK3DK4DK5 4p3=214.0113.9213.8813.9013.9013.9013.9014.2614.2214.2414.2414.2414.244p1=214.644s1=227.4026.9427.8327.7127.7127.7127.7131.3732.4732.3132.3132.3132.313d5=293.7295.9493.9293.9693.9693.9693.96104.09101.86101.98101.98101.98101.983d3=294.913p3=2214.40219.04219.65219.77219.77219.77219.77226.71228.75228.90228.89228.89228.893p1=2222.213s1=2292.79290.62298.97298.03298.05298.05298.05295.23306.82305.28305.32305.32305.322p3=21678.401687.381699.031699.731699.711699.711699.711714.581727.651728.361728.341728.341728.342p1=21730.922s1=21924.661870.301939.451930.831931.091931.071931.071902.161969.861960.291960.581960.561960.561s1=214325.9814109.6014452.1414352.9214355.7814355.5914355.6214154.4314501.9514401.9514404.8514404.6614404.70 a BesttheoreticalestimatefromRef[ 71 ] 67

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Table2-5. ConvergenceofSCFandcorrelatedmethodswithrespecttoDouglas-Kroll-HessordersforXenon.EnergyunitsareineV.TheWTBSisemployed. DK1-DK0DK2-DK1DK3-DK2DK4-DK3DK5-DK4 SCFenergy-7186.9411490.375-65.4166.578-1.965Totalcorr.-1.4540.0070.0000.0000.0005pSCF-0.0670.045-0.0020.0000.0005pcorr.0.019-0.0080.0000.0000.0005sSCF2.129-0.3660.017-0.0020.0015scorr.-0.3340.093-0.0040.0000.0004dSCF-3.2450.204-0.0090.0010.0004dcorr.0.200-0.0240.0010.0000.0004pSCF3.1240.253-0.0120.0010.0004pcorr.-1.387-0.0200.0010.0000.0004sSCF18.146-2.7150.123-0.0130.0044scorr.3.476-8.776-0.0720.0070.0823dSCF-11.3330.837-0.0380.004-0.0013dcorr.-0.070-0.0080.0000.0000.0003pSCF21.9840.892-0.0420.004-0.0013pcorr.6.0691.7620.046-0.0050.0013sSCF89.194-13.0670.591-0.0610.0183scorr.-1.2050.406-0.2910.032-0.0092pSCF96.3023.402-0.1530.016-0.0052pcorr.-2.4500.223-1.3041.390-1.3992sSCF417.760-66.4182.989-0.3090.0912scorr.1.794-2.074-0.0910.009-0.0031sSCF2027.390-644.19328.161-2.8160.8451scorr.-4.0250.258-0.4070.075-0.021 68

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Table2-6. Ionizationpotentials(ineV)ofXenonusingdifferentDouglas-Kroll-HessreferencesintheWTBS. IP-EOM-CCSDKoopmansOrbitalTheor.Est.aDK0DK1DK2DK3DK4DK5DK0DK1DK2DK3DK4DK5 5p3=212.1412.1712.1312.1612.1612.1612.1612.4412.3812.4212.4212.4212.425p1=213.395s1=223.2922.2424.0423.7723.7823.7823.7825.7027.8327.4627.4827.4827.484d5=267.5170.4067.3567.5367.5267.5367.5375.5972.3472.5572.5472.5472.544d3=269.504p3=2145.50158.96160.70160.93160.92160.92160.92163.49166.62166.87166.86166.86166.864p1=24s1=2213.20208.86230.48218.99219.04219.04219.13213.78231.93229.21229.33229.32229.333d5=2676.48692.82681.42682.25682.21682.21682.21710.73699.40700.23700.20700.20700.203d3=2688.993p3=2940.70935.97964.02966.67966.68966.68966.68958.43980.41981.31981.26981.27981.273p1=21002.203s1=21148.871074.781162.771150.111150.411150.381150.391093.241182.431169.361169.951169.891169.912p3=24787.574799.604893.454897.074895.624897.024895.624837.714934.014937.414937.264937.274937.272p1=25107.582s1=25453.445115.095534.645466.155469.055468.755468.845152.215569.975503.555506.545506.235506.321s1=234562.8333255.5135278.8834634.9434662.7034659.9634660.7833317.5735344.9634700.7734728.9334726.1134726.96 a BesttheoreticalestimatefromRef[ 71 ] 69

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Table2-7. Ionizationpotentialsofhelium-likeionswithIP-EOM-CCSD/DKinPartridgebasissets.TherelativisticcolumncorrespondstotheDK5potential.AllvaluesareineV. ZBasissetRelativisticNon-relativisticTheor.est.a Partridge-11762.1931759.2581762.09212Partridge-21762.1941759.259Partridge-31762.1951759.259Partridge-12438.3932432.7402438.13414Partridge-22438.3952432.741Partridge-32438.3962432.742Partridge-13224.9993215.0683224.52216Partridge-23225.0013215.070Partridge-33225.0023215.070Partridge-14122.5214106.2424121.75918Partridge-24122.5234106.243Partridge-34122.5254106.244Partridge-15131.5485106.2615130.42520Partridge-25131.5525106.263Partridge-35131.5525106.264Partridge-111875.49411739.09611871.01830Partridge-211875.48711739.078Partridge-311875.49011739.078 a BesttheoreticalestimatefromRef.[ 75 ] 70

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Table2-8. Ionizationpotentialsofberyllium-likeionswithIP-EOM-CCSD/DKinPartridgebasissets.TherelativisticcolumncorrespondstotheDK5potential.AllvaluesareineV. IonCong.BasissetRelativisticNon-rel.Theor.est.aExp.b P11+1s2s2Partridge-12667.4832660.6782667.9Partridge-22667.4352660.627Partridge-32667.3072660.5091s22sPartridge-1560.797559.081560.79Partridge-2560.798559.082Partridge-3560.798559.082Ca16+1s2s2Partridge-14925.7104902.3314926.75Partridge-24925.8744902.503Partridge-34925.7964902.4101s22sPartridge-11087.4941081.3131087.36Partridge-21087.4951081.314Partridge-31087.4961081.314Ni24+1s2s2Partridge-110001.2159904.28410006.89Partridge-210001.6449907.075Partridge-310001.3569904.4131s22sPartridge-12297.4582270.7122295Partridge-22297.4662270.719Partridge-32297.4642270.717Zn26+1s2s2Partridge-111556.43411426.89911563.5Partridge-211556.78511420.075Partridge-311556.51111426.9621s22sPartridge-12672.0992636.0802664Partridge-22672.1002636.080Partridge-32672.1002636.080 a BesttheoreticalestimatefromRef.[ 76 ] b Ref.[ 76 ] Table2-9. Ionizationpotentials(ineV)ofhelium-likeionsusingIP-EOM-CCSDwithdifferentDouglas-Kroll-Hessreferences.TheWTBSisemployedandallvaluesareineV. ZDK0DK1DK2DK3DK4DK5Theor.est.a 121759.31762.81762.21762.21762.21762.21762.092142432.72439.72438.42438.42438.42438.42438.134163215.13227.43225.03225.03225.03225.03224.522184106.24126.74122.54122.54122.54122.54121.759205106.35138.35131.55131.65131.65131.65130.4253011739.111919.011874.511875.611875.611875.611871.018 a BesttheoreticalestimatefromRef.[ 75 ] 71

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Table2-10. Ionizationpotentials(ineV)ofberyllium-likeionsusingIP-EOM-CCSDwithdifferentDouglas-Kroll-HessreferencesintheWTBS. IonCong.DK0DK1DK2DK3DK4DK5Theor.est.aExp.b P11+1s2s22638.82669.22667.52667.52667.52667.52667.91s22s518.4560.9560.8560.8560.8560.8560.79Ca16+1s2s24901.24933.54927.14927.14927.14927.14926.751s22s1081.31088.11087.51087.51087.51087.51087.36Ni24+1s2s29857.710032.610000.810001.510001.510001.510006.891s22s2189.42300.62297.42297.52297.52297.52295Zn26+1s2s211376.411599.311555.611556.611556.611556.611563.51s22s2548.42676.52672.02672.12672.12672.12664 a BesttheoreticalestimatefromRef[ 76 ] b Ref.[ 76 ] 72

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CHAPTER3ANUNTRADITIONALAPPROACHTOCOREIONIZATIONENERGIES 3.1PreliminariesInChapter 2 thescalar-relativisticenergeticeffectsincoreionizationsweresystematicallystudiedandshowntobenon-negligiblebuteasytoobtaininanefcientmannerwiththeDKHtransformation.Inthischapteramethodwillbedevelopedthatincludesrelaxationeffectstoinniteorderatamean-eldcomputationalcost.Furthermore,thestructureofthismethodlendsitselfnicelytoanorder-by-orderanalysisofrelaxation,andallowsonetocomputetherelaxationenergytoanyorder.Thisiscritical,sinceIwillshowthatthelargesourceoferrorinIP-EOM-CCSDisduetohigherorderrelaxationeffects.Becauserelaxationissolargeforcoreionizations,ascomparedtocorrelation,thebestapproachistoseparatethesecontributionsandsolveforthemindependently.[ 77 ]Numeroustheoreticalmodelshavebeendevelopedinthisvein,butpracticalmethodsareonlyconsistenttosecondorderinrelaxation.Forexample,ifwedenetherelaxationenergytoinniteorder,!Rk,as !Rk=k)]TJ /F13 11.955 Tf 11.96 9.68 Td[()]TJ /F5 11.955 Tf 5.48 -9.68 Td[(ESCFk(N)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F5 11.955 Tf 11.96 0 Td[(ESCF(N)(3)wherekisthekthoccupied-orbitalenergyfortheNelectronsystem,andEkSCF(N)]TJ /F4 11.955 Tf 12.3 0 Td[(1)andESCF(N)arethe(N)]TJ /F4 11.955 Tf 13.04 0 Td[(1)andN-electronSCFenergies,respectively,thenanapproachsuchastheTransitionOperatorMethod(TOM)originallydevelopedbySlater[ 78 ]andfurtherdevelopedbyPickup,Goscinski,andPurvis[ 79 ]iscorrectthroughsecondorderonlyandstillcontainsconsiderableerrorcomparedtoexperiment.However,Ortizet.alhaveobtainedveryreasonableresultsforcoreionizationsusingtheTOMwavefunctionasthestartingpointforElectronPropagator(EP)computations.[ 80 ]However,theTOMwavefunctioncorrespondstoactitioussystem 73

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wheretheionizedorbitalisoccupiedbyhalfanelectronandalsoyieldsadeterminantthatiscomprisedoforbitalsthatdifferfromtheN-electronreferencedeterminant.MoredirectmethodsthatattempttoobtainrelaxationwithcorrelationorderbyorderareGreensfunctionsmethods.[ 81 87 ]andtheirrelativisticextensions[ 56 88 ].ThesemethodsareaviableapproachtocomputetheIPsdirectlyandaccountforrelaxationandcorrelation;however,therearesomedifcultiesassociatedwiththem.Themethodsareonlyinexpensiveatlowordersofperturbationtheory,areenergydependent,anditisdifculttoknowiftheperturbationseriesconverges.Forcomparison,onecanexaminetheEPtechniquesagainstvariousinniteorderequationofmotioncoupled-cluster(EOM-CC)methods[ 89 91 ],sometimescalledCoupledClusterGreen'sFunctions.[ 92 ]SuchEOM-CCmethodsdecoupletheionizationandelectronattachmenttermsintheone-particleGreen'sFunctionsthusofferingamoreattractivetheorywhereinnite-ordereffectsenableittoexceedtheaccuracyofmostEPtechniques.Unfortunately,EOM-CCSDis,onaverage,1to3eVinerrorforcoreIPscomparedtoexperiment.Iwillpresentevidencethatthisisdueprimarilytothelineartreatmentofrelaxation(singleexcitations)intheionizedstate,asopposedtothefullexponentialtreatmentofsinglesinthegroundstateCCsolution.Inclusionofhigherexcitations,whichinthelimityieldtheFull-CIresult,willobviouslyreducetheerror,butcomesatahigherandimpracticalcomputationalprice.Ofcourse,onecanmoveawayfromthesedirectmethodsandcomputetheIPindirectlyas!k=Ek(N)]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F5 11.955 Tf 11.95 0 Td[(E(N) (3)=ESCFk(N)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+Ek(N)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F13 11.955 Tf 11.96 9.68 Td[(ESCF(N)+E(N) (3)=ESCFk(N)]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F5 11.955 Tf 11.95 0 Td[(ESCF(N)+[Ek(N)]TJ /F4 11.955 Tf 11.95 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[(E(N)] (3)=ESCFk+Ecorrk (3)=!relaxk+!corrk (3) 74

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Thisrigorouslydenesrelaxationenergytoinniteorderinanindependent-particlewaydeningtheESCFmethod.Bycomputingtheenergyofthe(N)]TJ /F4 11.955 Tf 13.03 0 Td[(1)-electronsystem,onecanincorporaterelaxationtoinniteorderattheSCFlevel.Also,anyamountofcorrelation,accordingtothemethod,canbecomputedforbothstates.ThisideallyseparatestherelaxationfromthecorrelationandcanprovideveryaccuratecoreIPs.[ 46 77 93 ]However,convergingtheSCFforthecoreholestatecanbetediousanddifcult,sincevariationalcollapseofthewavefunctionishardtoavoid.Thecorrelatedcomputationcanalsobefraughtwithconvergenceissues.InthespeciccaseofCCtheory,thehighlynonlinearequationsofCCtheoryarenotevenguaranteedtoconvergetothecorrectsolution,thoughthisseldomhappensincarefulapplication.Essentially,theproblemathandfortheseindirectmethodsistheadequateandefcientsolutionforthe(N)]TJ /F4 11.955 Tf 12.14 0 Td[(1)-electronsystem.Therefore,ourobjectiveistoseparaterelaxationfromcorrelationandcomputeitdirectly,efciently,andaccurately.Sincethedominantcontributionforacoreionizationofanorganicmoleculeistherelaxation,wepresentaDirectESCFprocedure(namedIP-VCCS)anddeterminethisquantitytoinniteorderatthecostofanSCFcomputation.ThismethodisbuiltintheN-electronmolecularorbitalbasis,thusremovingtherequirementtooptimizetheorbitalsforthe(N)]TJ /F4 11.955 Tf 12.05 0 Td[(1)-electronstate,andconvergesextremelywell.Infact,thepresentmethodallowsonetoobtainthe(N)]TJ /F4 11.955 Tf 12.31 0 Td[(1)-determinantasasimplerotationoftheN-electrononeifoneseekstheionizeddeterminant.Anexactequivalencerequiresonetochangetheorbitaloccupation,however,althoughthiscanbeavoidedtoaverygoodapproximation-andthisispresentedaswell.Thepresentmethodcanlocalizethecoreelectronsonsymmetryequivalentatoms,necessaryformoleculeswithsymmetriccenters[ 94 99 ].Itdoessoaprioriandthusavoidstheneedforastabilityanalysis,whichwouldhavetondthebrokensymmetrysolutionsoflowerenergy.Italsoappearsnottosufferfromspincontamination,whichispresentandproblematicinthetraditionalSCFmethod. 75

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Furthermore,IdemonstratethatIP-EOM-CCSDisconsistenttosecondorderonlyinrelaxation.Consequently,Ievaluateathird-orderrelaxationcorrectionthatcanbeaddedtotheIP-EOM-CCSDionizationenergy.Unfortunately,tackonperturbativecorrectionssuchasthesedonotyieldawavefunction,makingthecomputationofpropertiesandoscillatorstrengthsill-dened. 3.2Theory 3.2.1DirectESCFRealizedThroughVariationalCCSThebasicideaoftheDirectESCFmethodistheconstruction,andsolution,ofaconnectedwaveoperator,whichcreatesthe(N)]TJ /F4 11.955 Tf 9.5 0 Td[(1)-determinantfromtheN-electrononeusingThouless'theorem[ 18 ].Thouless'theoremstatesthatanyarbitrarydeterminantcanberotatedtotheHartree-Fockdeterminantforthestateofinterestviae^T1asjki=kj0i (3)=e^T1ke^T01j~0i (3)with ^T1=Xaitaifayig(3)andorbitalindiceswithi,j,k,...,a,b,c,...,andp,q,r,...correspondingtooccupied,virtual,andarbitraryorbitals,respectively,referencedtotheN-electronvacuum.Conceptually,e^T01j~0irepresentstheNelectrongroundstateHFdeterminant,j0iande^T1krepresentsk,thewaveoperatorresponsibleforrotatingtheN-electrondeterminanttothe(N)]TJ /F4 11.955 Tf 12.41 0 Td[(1)-electronHFdeterminantforthekthionizedstate.ByusingtheN-electronHFdeterminant,j0i,asj~0i,then^T01=0,andweareleftwith jki=kj0i=e^T1kj0i(3)withthecorrespondingenergyequation Ek=h0jkye^Ty1He^T1kj0i h0jkye^Ty1e^T1kj0i(3) 76

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andoneparticledensitymatrix[D(k)]pq=h0jkye^Ty1pyqe^T1kj0i h0jkye^Ty1e^T1kj0i (3)=Xih0jkypyqkj0iipqi+h0j(kye^Ty1fpyqge^T1k)Cj0i (3)Equations( 3 )and( 3 )arevariantsofavariationalcoupledclusterwithsingles(VCCS)model.TheVCCSequationswereanalyticallysolvedbyNogaet.al.[ 20 ]Thesecondtermofequation( 3 ),duetotheconnectedpropertyoftheexponential,canberecastash0j(kye^Ty1fpyqge^T1k)Cj0i=h0jkye^Ty1e^T1Lfpyqgkj0iC+h0jkye^Ty1e^T1Lfpyqge^Ty1e^T1Lkj0iC (3)=XrsXrsh0jkyfrysgfpyqgkj0i+XrstuXrsXtuh0jkyfrysgfpyqgftyugkj0i (3)withtheXmatrixbeingageometricsuminvolving^T1andcanbecomputedtoarbitraryprecision.[ 20 ]Ourgoalnowistoincludethekdependencesoasnottochangethereferencedeterminant.Asimpleevaluationofthematrixelementsofequation( 3 )withequation( 3 )usingsecondquantizedalgebratechniquesyieldsthefollowing[D(k)]pq=Xipiiq)]TJ /F7 11.955 Tf 11.96 0 Td[(pkkq+Xpq(1+kp2occ)]TJ /F7 11.955 Tf 11.96 0 Td[(q2vrtk))]TJ /F5 11.955 Tf 11.95 0 Td[(Xp2occkXkq2occ)]TJ /F5 11.955 Tf 11.95 0 Td[(Xp2vrtkXkq2vrt (3)=Xipiiq+h~X(k)ipq (3)WherethereferencedensityiseasilyextractedasPipiiq+Xpqandtherest,beingthecorrectionduetoionization,aretermstreatingkasavirtualorbital.Thisismosteasily 77

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seeninmatrixform ~X(k)=266664XooXokXovXkoXkkXkvXvoXvkXvv377775(3)whereo,k,andvcorrespondtotheoccupied,ionized,andvirtualspacesrespectively.ThuswearriveatapracticalworkingpointfortheDirectESCFmethod.Wewillomittheexplicitdependenceofthedensityonktosimplifytheexpressions.Tosolvefor^T1,werstformthedensitymatrix.Byusingthegeometricseriesnatureofthedensitymatrixbuiltfrom^T1,wecanwritetheoccupied-occupiedblockofthedensitymatrixas Di(m+1)j=ij+XlDi(m)l(Xatlataj)(3)Theremainingblocksofthedensitymatrixarecomputedwiththeoccupied-occupiedblockasDai=XjtajDji (3)Dia=XjDijtja (3)Dab=XjDajtjb (3)BymakingtheexpectationvalueofthedensitymatrixwiththeFockoperatorstationarywithrespecttothetaiamplitudes, XpqFpqDqp @tai=0(3)oneobtainsatransformationmatrix,Q,thattransformstheFockmatrixateachiterationclosertotheFockmatrixfortheN)]TJ /F4 11.955 Tf 12.38 0 Td[(1electronsystemthroughthedensitydependent^J)]TJ /F4 11.955 Tf 13.73 2.66 Td[(^KoperatoroftheFockmatrix. ~F=QTFQ(3) 78

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whereQij=ij+h~X(k)iij (3)Qia=)]TJ /F13 11.955 Tf 11.29 13.27 Td[(h~X(k)iia (3)Qai=h~X(k)iai (3)Qab=ab)]TJ /F13 11.955 Tf 11.96 13.28 Td[(h~X(k)iab (3)Computationally,theFockmatrixisconstructedfromtheobtaineddensity.ThisFockmatrixisthentransformedwithQtogenerateanewFockmatrix,onewhichapproachessatisfyingtheBrillouincondition.ThisavoidsdiagonalizationoftheFockmatrix,andalsoallowsonetoremainintheN-particlebasis.OncethetransformedFockmatrixisobtained,onecanupdatethe^T1amplitudesusing ta(m+1)i=ta(m)i+~Fa(m)i i)]TJ /F7 11.955 Tf 11.95 0 Td[(a(3)ItiseasilyseenthatthetotalN)]TJ /F4 11.955 Tf 12.11 0 Td[(1-Hartree-Fockenergycanbecomputedateverystep,usingtheFockmatrix,F,constructedfromthecurrentdensityas EN)]TJ /F9 7.97 Tf 6.59 0 Td[(1SCF=Xpq)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(hpq+Fpq[D(k)]qp(3)However,ifweexpandtheFockoperatoranddensitymatrixEN)]TJ /F9 7.97 Tf 6.59 0 Td[(1SCF=Xpqhpq+Fpq+[~G(k)]pq Xipiiq+h~X(k)iqp! (3)=ENSCF+k+Xpq hpq+Xj6=khpjjjqji+Xrshprjjqsi[~X(k)]sr![~X(k)]qp (3)oneimmediatlyseestheimplicitHartree-FockenergyfortheN-electronstate,Koopmans'theorem,andtherelaxationcorrectiontoKoopmans'theorem.Subtracting 79

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ENSCFfrombothsidesyieldstheDirectESCFionizationenergy, !k=k+Xpq hpq+Xj6=khpjjjqji+Xrshprjjqsi[~X(k)]sr![~X(k)]qp(3)ComparingthistoEquation( 3 ),weseethedirectandcompleteexpressionforrelaxationisrecoveredwiththisenergyexpression.ThismethodwillbenamedIP-VCCS-E,wheretheEcorrespondstoanexactformulationcomparedtoESCF.ItmustbementionedatthispointthatexactequivalencetotheESCFmethodisobtainedwhenkistreatedasavirtualorbital.Thisiseasilyseenviathe^T1operator,whereitexcitesoccupiedelectronsinthedeterminanttovirtualorbitals,thusfacilitatingrelaxation.However,ifkisnotavirtualorbitalindex,then^T1cannotexciteintoit.Thisaffectstheoccupationnumberofthedeterminant,butitdoesnotaffecttheorbitalbasis.Still,itismoredesirabletoworkinthesameorbitalbasisandoccupation,i.e.inthesameHilbertspace.Tothatend,anexcellentapproximationcanbemade,whichremovestheneedtochangeoccupation.Ifoneallowsnocouplingbetweentheionizedorbitalsandtherest-asuddenionization-thenonecanworkinthesameHilbertspacethroughoutthecomputation.Byeliminatingthecouplingterms,andconsequentlyprohibitingapossiblevariationalcollapse,thecorrectionmatrix,~X(k),becomes h~X(k)ipq=)]TJ /F7 11.955 Tf 9.3 0 Td[(pkkq+Xpq,(3)orinmatrixform ~X(k)=264Xoo)]TJ /F7 11.955 Tf 11.95 0 Td[(okkoXovXvoXvv375(3)ThisapproximateformulationwillbeabbreviatedasIP-VCCS.Theequationsareidenticalforelectronattachedstates,exceptthatthekdependentoccupiedcorrection,becomesacdependentvirtualaddition,h~X(c)ipq+pccq+Xpq.Thismethodcanpresumablyworkwellforcoreexcitedstatesaswell,wherethecorrectionish~X(k,c)ipq+pccq)]TJ /F7 11.955 Tf 12.55 0 Td[(pkkq+Xpq,althoughcorrelationwillbenecessary 80

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fortheelectronattachmentprocessinthecoreexcitation,sincetheattachmentsusuallyoccurin,ornear,theLUMOenergyregion. 3.2.2Third-OrderRelaxationCorrectionIP-EOM-CCSDincludesinniteordereffectsduetorelaxationandcorrelationandperformsverywellforvalenceionizations;however,itshowssomedecienciesinthecoreregion[ 29 ]owingtothelargerelaxationeffectsaccompanyingcoreionization.[ 39 100 ]Theionizedstateiscreatedwithalinearoperator ^Rk=Xkrkf^kg+Xiakraikf^ay^i^kg(3)withthesecondtermcreatingtheholestate,followedbysinglesubstitutions,(relaxation).Onewondersifaddingquadratic,cubic,...,wouldincorporatethenecessaryrelaxationtoobtaintrulyquantitativeresults.Inthatvein,werecastthepreviouslyderivedrelaxationoperator,relax.,asafunctionof^S1insteadof^T1(soasnottoconfuseitwiththegroundstateCCamplitudes) relax.=e^S1f^kg.(3)Truncationofrelax.tolineartermsisanapproximationofEq.( 3 ).Withthis,wecanwritethefullyrelaxedenergyforthekthionizedstateas[ 101 ]Ek=h0jyrHNrj0i=h0jyrrj0i (3)=h0jf^kyge^Sy1HNe^S1f^kgCj0i (3)ThefullyrelaxedenergyexpressionforIP-VCCS,oranyVCCSmethod,is Ek=1 2Xpq)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(hpq+fpqDqp(3) 81

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wherefpqandDqparethecompletefockoperatoranddensitymatrix,respectively,asfunctionsoff^Sy1,^S1g.Thisallowsustoexpandinordersofperturbationtheoryas E(0)k+E(1)k++E(n)k=1 2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(h(0)pq+f(0)pq+f(1)pq++f(n)pq)]TJ /F5 11.955 Tf 12.95 -9.69 Td[(D(0)qp+D(1)qp++D(n)qp(3)WithD(0)qp=Piqiip)]TJ /F7 11.955 Tf 9.3 0 Td[(qkkp,thezerothorderenergyissimplyKoopmans'approximation, E(0)k=E(N))]TJ /F7 11.955 Tf 11.95 0 Td[(k(3)Bycomparison,IP-EOM-CCS,thedirectanalogueofthiszerothorderanalysis,yieldsKoopmans'approximationalso.Consequently,sinceIP-EOM-CCSDcontainsthelinearionizationoperator,(ie.consistenttosecondorderinrelaxationinthedensityinouranalysisofIP-VCCS),wecanevaluateathird-orderrelaxationcorrectionas E(3)k=1 2Xpq)]TJ /F5 11.955 Tf 10.46 -9.68 Td[(h(0)pq+f(0)pqD(3)qp+f(1)pqD(2)qp+f(2)pqD(1)qp+f(3)pqD(0)qp(3)thatcanbeaddedtoanyIP-EOM-CCSDcomputation,withD(1)qpD(1)ai+D(1)ia=sai+sia (3)D(2)qpD(2)ij+D(2)ab+c.c.=Xdsidsdj+Xlsalslb+c.c. (3)D(3)qpD(3)ai+D(3)ia=Xldsalsldsdi+Xdlsidsdlsla (3)andthecorrespondingf(n)pqfpq[D(n)]. 3.2.3RelaxationwithECCMethodsToanalyzetheerrorsinIP-EOM-CCSDwecompareagainstionizationpotentialsobtainedindirectlybytotalenergydifferences !k=Ek)]TJ /F5 11.955 Tf 11.95 0 Td[(E0(3)Wetesttwomethods,onewherethereferenceionizedstate,jki,isvariationallyoptimized,yieldingfullyrelaxedorbitals,beforeperformingaCCSDcorrelated 82

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calculation.ThiswillbelabeledtheErelaxedCCSDmethod.Asreferencedin 2 ,amoredirectcomparisonwithIP-EOM-CCSDcanbemadewiththeEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDmethod.ThereferenceionizedstateisleftasaKoopmans'determinant,thusgeneratingaquasi-RHF(QRHF)reference[ 46 102 ]usedinthecorrelatedCCSDcalculation.ThishastheadvantagethatthesameorbitalsusedinthismethodareusedinIP-EOM-CCSD,andamoredirectanalysisofrelaxationandcorrelationisthereforepossible.Although,thesamelineofreasoningtoobtaincorrectionstoIP-EOM-CCSDcannotbeusedwiththisEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDapproach.Inthelimitoffullsubstitutions,allrelaxationwillbeincluded,butthereexistsnopracticalwaytocomparetruncatedvariationalversustruncatedtraditionalcoupledclusterapproaches.[ 103 104 ] 3.3ImplementationTheACESIIprogrampackage[ 105 ]wasusedtocomputethecoreionizationenergieswiththeErelaxedCCSD,Enon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSD,ECCSDT,andIP-EOM-CCSDmethods.TheIP-VCCSalgorithm[ 101 ]wasimplementedinthemassivelyparallelizableprogrampackage,ACESIII[ 34 ],alongwiththethird-orderenergycorrection.Thecorrelationconsistentpolarizedcore-valencedoubleandtriple-basissetswereusedinthecalculations.FortheIP-VCCSmethodformoleculeswithsymmetryequivalentcenters,thegroundstateN-electrondeterminantwascomputedwithatraditionalHartree-Fockalgorithm,yieldingsymmetric,delocalizedorbitals.Alinearcombinationwasthenmadewiththeganduorbitalsthatlocalizedthemonequivalentcenters.TheIP-VCCSalgorithmwasthenappliedtothesereferencedeterminantstodeterminethecoreIP,whichmatchedbrokensymmetrySCFcomputationsonthe(N)]TJ /F4 11.955 Tf 11.95 0 Td[(1)-electronstates. 3.4ResultsandDiscussion 3.4.1IP-VCCSExamplemoleculesaretakenfromtheG2testset;thegeometriesareunmodied.Fororthoaminophenol,thegeometryisoptimizedusingsecond-ordermany-body 83

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perturbationtheory[ 15 ]withtheaugmentedcorrelation-consistentdouble-basisset[ 106 ].TheoptimizedstructureisshowninFigure 3-1 .Thismoleculeischosenbecauseoftherecentinterestinitscoreionizationspectrum[ 57 ].TheESCFcalculations,whereboththeNand(N)]TJ /F4 11.955 Tf 12.87 0 Td[(1)-statesarevariationallyoptimized,areperformedwiththeACESIIprogrampackage[ 105 ]forreasonsthatwillbementionedlater.Theexact,comparedtoESCF,calculationwiththe(N)]TJ /F4 11.955 Tf 12.75 0 Td[(1)occupation(IP-VCCS-E)andtheapproximate(IP-VCCS)calculationsareperformedwiththeACESIIIprogrampackage[ 34 ].Thecorrelation-consistentcore-valencefamilyofbasissets[ 106 ]areemployed,uptothequadruple-level.ResultsobtainedwithIP-VCCS-EforthesinglecoreIPsofsmallmoleculesareshowninTable 3-1 forasubsetofmolecules.ExactagreementwithESCFresultsareobtained,asrequired,thoughtheresultsarereportedtothreedecimalplacesforreadability.Themethodconvergessmoothlyandrapidly,exhibitingnosignsofpossiblevariationalcollapseforthesesystems.However,sincethechangeinoccupationnumberisanundesirablefeatureofthistheory,asonewouldliketodescribetheground,excited,ionized,andattachedstatesequallyandconsistently.Resultsofthesuddenapproximation,IP-VCCSareshowninTable 3-2 .WeseeexcellentagreementforsingleIPscomparedtotheexactanaloguewithinthisapproximation.Theaverageabsoluteerrorsforthedoubleandtriple-basissetsare0.007eVand0.008eV,respectively.ThecomparisonofconvergencebetweenESCFandIP-VCCSispresentedinFigure 3-3 .Althoughthisisaratherunfaircomparison,astheESCFcomputationsareperformedwithcomputationalsymmetryinACESII[ 105 ]andtheDirectESCFcomputationsareperformedinC1symmetryinACESIII[ 34 ].Also,fortheESCF,upto25historiesarestoredfortheDIISalgorithm,whereasonly5historiesarestoredintheDirectESCFroutine.And,sincetheESCFkcomputationisextremelysusceptibletovariationalcollapse,ateveryiteration,theoccupationnumbershadtoberereadand 84

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themaximumoverlapmethodofRef.[ 107 ]isusedtohelpavoidanycollapse.Evenwiththeseprecautions,onlytwo-thirdsofthetestsetconvergedusingtheconventionalESCFprocedure,comparedto100%convergencewithIP-VCCS.Takingallthisintoaccount,itisratherremarkablethatthedirectmethodonlydiffers,onaverage(includingozone)byafactorof1.4.Infact,sincesuchpoorconvergenceisexperiencedfortheESCFmethodusingthedoubleandtriple-basissets,weperformedthedirectcalculationusingthequadruple-basissetandmaintainedexcellentconvergence.ResultsfortheextensionofIP-VCCStodoublecoreionizationpotentialsfortheorthoaminophenolmoleculeareshowninTable 3-3 ,Forthesinglecoreionizationenergiesonoxygenandnitrogenverygoodagreement,again,isobtainedcomparedtoESCF;providingmoreevidencefortheutilityoftheapproach.Thedoublecoreionizationpotentials,however,areroughly0.3eVofffromtheESCFmethodforionizationsonthesamecenter,butonly0.03eVofffortheionizationsinnitrogenandoxygen.OfcoursetheIPobtainedwiththeapproximatemethodisalwayslessthantheexactvalue(assumingthereisnospincontamination),duetotheconstraintontheionizedorbital,butevenfortheworstcaseof0.3eV,thisisunder1%oftherelaxationenergy.Obviously,anyperturbativecorrectionstocorrecttheionizationenergiesshouldquicklyrecoverthislackofrelaxation,andthereforethemethodisstillquiteusefulandpowerfulfordoublecoreionizations. Spincontamination.Interestingly,forpropenethismethoddiffersfromtheESCFresultby0.1eV,aspresentedinTable 3-5 .ThisapparentpeculiarityisduetomoderatespincontaminationforthetraditionalESCF.Forpropene,thespinmultiplicityfortheEHFkstateis2.24,comparedtotheexpecteddoublet.GreatcareistakenintryingtoconvergetothedoubletcoreholestatefortheEHFk,demonstratingthedifcultyinobtainingaproperreferencefunctionforthesecalculations.Consequently,thisverynicelyillustratesthepowerandutilityoftheIP-VCCSmethod,whichmaintains 85

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properspinsymmetrythroughoutthecomputation,duetotheexponentialansatz,andreproducesresultsveryclosetoexperiment. Brokenspatialsymmetry.Itisawellknowresultthatformoleculeswithsymmetryequivalentcenters,thecoreorbitalshavetobelocalizedoneachcenter,thusbreakingthepointgroupsymmetryofthewavefunction.[ 94 99 ]ForESCFcalculations,astabilityanalysisisperformed,whichdeterminesiftherearelowerenergysolutions,androtatestheorbitalsaccordinglytoconvergetothissolution.ThisrequiresthesecondderivativesoftheSCFwavefunction.Thecoreorbitals,beingessentiallyatomic,arerathereasytolocalizefromthebeginningthoughas 1sA=B=1g1u,(3)withsymmetriccentersAandB.ComparisonbetweenthesymmetryconstrainedandsymmetrybrokensolutionsareshowninTable 3-4 .ThebrokensymmetrysolutionsarealsoequaltotheonesobtainedwiththeESCFmethod.Eventhoughthesymmetrybreakingruinsthequalityofthewavefunction,theenergyismuchclosertotheexactenergyduetothevariationalprinciple. 3.4.2Third-OrderCorrectiontoIP-EOM-CCSDThecomparisonsofagivenmethodusingdouble-andtriple-qualitybasissetstoexperimentalvaluesareshowninTable 3-6 andTable 3-7 ,respectively.Koopmans'valuesarelistedforcomparisonpurposes,sincethesevaluesarecompletelyunrelaxedIPs.AlsounreliablearetheErelaxedCCSDresults,evidentusingbothbasissets.SincethismethodisveryimpracticaltouseduetopoorconvergenceintheSCFandCCSDequations,wewillfocusonthemorepromisingEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSD,IP-VCCS,andIP-EOM-CCSDmethodstoobtainrelaxedandcorrelatedcoreIPs.SomewhatdeceivingistheverygoodagreementwithexperimentobtainedwithIP-VCCSandthedouble-basisset.Although,completelyrelaxed,thereappearstobeafortuitouscancellationoferrorbetweentheincompletebasissetdescription 86

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andcorrelationeffects,whichbecomesapparentwhenexaminingthetriple-resultsforIP-VCCS.Infact,thecorrelatedanalogofIP-VCCS,theEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSDmethod,performspoorlyusingthedouble-basisset,beingapproximately2.1eVinerror,butthenperformsquitesatisfactorilyforthetriple-basisset,with0.3eVaverageerror.Theincreaseinthesizeofthebasisset,combinedwiththeaccuratedescriptionofcorrelationbeginstoprovidequantitativeresults.Also,itdoesnotseemnecessarytobreakpointgroupsymmetryformoleculeswithsymmetryequivalentcenterswhenperformingaEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDcomputation.Itisnecessary,though,forIP-VCCScomputations.TheeffectsoftriplesubstitutionsonthecoreIPsofselectmoleculesareshowninTable 3.5 .Thistestsetisrathersmall,butsupportsthehypothesisthathighersubstitutionswillconvergeveryslowlytotheFullCIresultsincethecoreisnotastronglycorrelatedregion.Therefore,relaxationisthedominantremainingeffect.ComparingKoopmans'valuewithIP-VCCSyieldstheinniteorderrelaxationenergyandcomparingafullycorrelatedEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSDwithonewheretheionizedorbitalisfrozenyieldsthecorrelationandrelaxationenergy.ThesecontributionsareshownforasubsetofmoleculesinFigure 3-2 .Formoleculeswithnosymmetryequivalentcenters,thecontributionsfromEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDarealwaysbelowthoseofthetotalrelaxationenergy.ThisstronglysuggeststhattheEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDismissingaverysmallamountofrelaxationenergy.TherelaxationenergyisalmostexactlyhalfofthecontributionsfromEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDformoleculeswithsymmetryequivalentcenters.Thissimplydemonstratestheneedtobreaksymmetryforthesetypesofmolecules,andlocalizetheionizedorbitaltoonecenter,ratherthanhalfonbothorbitalsinanSCFprocedure.ThedirectanalogueofEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSD,IP-EOM-CCSD,issystematicallybelowtheexperimentalvalues,onaverageof3.41eVand1.78eVforthedouble-andtriple-basissets,respectively.Theagreementisbetterwiththelargerbasisset,butappearstoconvergeslowlytotheexperimentalvalue.Iftriplesubstitutionsarenotimportantforthe 87

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ECCmethods,thenitfollowstheyarenotimportantwithintheIP-EOM-CCframework.Therefore,themajordifferencebetweenIP-EOM-CCSDandEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSDistheinclusionof^T1inalinearansatz,versustheexponentialansatz,andconsequently,theproductsofsinglesubstitutionswhichincorporaterelaxationtoinniteorder.HavingestablishedthatthemajorerrorinEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDandIP-EOM-CCSDmethodsisthelackofhigherorderrelaxationeffects,wedemonstratetheincreaseinaccuracybyaddingadirect,consistent,third-orderenergycorrectiontoIP-EOM-CCSDinTable 3-9 .Formoleculeswithsymmetryequivalentcenters,boththesymmetryconstrainedandsymmetrybrokenresultsarereported.EventhoughthesymmetrybrokensolutionsuseareferencefunctiondifferentthantheoneusedintheIP-EOM-CCSDcomputation,theenergycorrectionyieldsmoreaccurateresultscomparedtothesymmetryconstrainedresults.Onaverage,theerrorcomparedtoexperiment,withatriple-basisset,decreasesfrom1.83eVto0.04eV,andtheaverageunsignederrordecreasesfrom1.83eVto1.01eV.Forsuchasimplecorrection,theresultsaremuchmorequantitative. 3.5ConclusionsandOutlookAnewdirectmethodforelectronionized,attached,andexcitedstateshasbeentestedforcoreionizations.TheIP-VCCS-EfaithfullyreproducestheESCFresultandtheapproximateIP-VCCSreproducestheexactresulttobetterthan0.01eVonaverageforcc-pCVDZandcc-pCVTZbasissets.Therefore,thesingleparticleperturbation,orbitalrelaxation,isdescribedtoinniteorderdirectly.Interestingly,thoughnotsurprisingly,thisoffersaveryattractiveroutetowardsdeterminingcoreexcitationenergies.Forexample,ifthefullyrelaxed,coreionizedHamiltoniancanbedescribedas ~Hk=kyeTy1HeT1k(3)then,thecorrelatedcoreionizedHamiltoniancanbewrittenas ~Hk=~HkeT2.(3) 88

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ThisisobviouslyananalogueofH(H-bar)fromCCtheory[ 15 ],limitedtodoubles(CCD),butspecictoachosencoreionizedstate,andstandardEOM-CCmethodscanbeappliedtocomputetheelectronattachedportion,completewithcorrelation. ~HkRcCj0i=!ckRcj0i;8c(3)Futureworkwilladdresscomputingtransitionmoments.Thiswouldprovideaseamlessroutetowardstheextremelydifcultproblemofcoreexcitationspectra.Arelaxationenergycorrectionbasedonathird-orderanalysisoftheionizedstatedensitymatrixbetweenIP-VCCSandIP-EOM-CCSDhasbeendevelopedandappliedtocoreionizationenergies.TheerrorsinIP-EOM-CCSDforcoreionizationshavealsobeendemonstratedbycomparingitwithEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSDandIP-VCCS.Formoleculeswithsymmetryequivalentcenters,verygoodagreementwithexperimentisobtainedbycomputingthecorrectionfromthesymmetrybrokenreferencefunction.TheaverageerrorinIP-EOM-CCSD,dueessentiallytomissinghigherorderrelaxationeffectsdecreasesfrom1.83eVto0.04eV,andtheaverageunsignederrordecreasesfrom1.83eVto1.01eV.Evenhigherordercorrectionscanbederivedandimplementedandfutureworkisaimedinthisdirection. 89

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Figure3-1. Groundstategeometryoforthoaminophenolmolecule. Figure3-2. ContributionstocoreIPusingEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSD(correlationandrelaxation)andIP-VCCS(relaxation)withcc-pCVDZbasisset. Figure3-3. DemonstrationofconvergenceofIP-VCCS. 90

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Table3-1. ComparisonbetweentheexactVCCSmethod(IP-VCCS-E)andtheESCFmethodforcoreIPsofdifferentmoleculeswithdifferentbasissets.AllvaluesareineV. cc-pCVDZcc-pCVTZMoleculeESCFIP-VCCS-EESCFIP-VCCS-E Ethane297.989297.989297.476297.476Methane291.577291.577290.612290.612Cyanogen418.355418.355417.698417.698Fluorine709.573709.573708.781708.781Formicacid540.987540.987539.890539.890Hydrogencyanide407.337407.337406.098406.098Ketene540.146540.146538.901538.901Nitrogen420.186420.186419.555419.555Ozone547.511547.511546.624546.624Vinyluoride692.957692.957691.624691.624 91

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Table3-2. ComparisonbetweentheESCFandtheapproximatedirectESCFmethodforthecoreIPofdifferentmoleculesusingdifferentbasissets.AllvaluesareineV. cc-pCVDZcc-pCVTZcc-pCVQZMoleculeESCFDirectESCFESCFDirectESCFDirectESCF Acetaldehyde538.337538.324537.120537.020Acetylene299.561299.558298.917298.914298.882Ethane297.989297.987297.476297.473297.442Diuoromethane704.988704.986704.368704.365704.301Methane291.577291.567290.612290.599290.540Carbonmonoxide542.460542.448541.219541.204541.080Cyanogen418.355418.355417.698417.697417.642Dimethylacetylene299.463299.464298.981298.982298.957Fluorine709.573709.570708.781708.778708.656Formaldehyde539.120539.108537.940537.925537.827Formicacid540.987540.975539.890539.875539.777Hydrogenperoxide551.745551.742551.228551.225551.172Hydrogencyanide407.337407.330406.098406.088397.434Hydrazine414.853414.851414.409414.406414.396Ketene540.146540.133538.901538.885538.776Methanol539.071539.059537.971537.956537.871Nitrogen420.186420.184419.555419.552419.510Nitromethane551.044551.043550.535550.534550.494Ozone547.511547.499546.624546.611555.368Oxirane538.529538.517537.422537.406537.306Vinyluoride692.957692.945691.624691.608691.489Averagedifference-0.007-0.008Averageunsigneddifference0.0070.008 Table3-3. ComparisonbetweentheIP-VCCS-EandtheIP-VCCSmethodsfordesignatedcoreIPsoforthoaminophenolusingthecc-pVQZbasisset.AllvaluesareineV.Thecc-pCVQZbasissetisemployed. ESCFIP-VCCS-EIP-VCCS DoubletsingleIPsO538.232538.232538.217N404.594404.594404.580SingletdoubleIPsOO1165.7631165.7631165.425NN887.152887.152886.830NO947.592947.592947.563 92

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Table3-4. DifferenceofdifferentsymmetryreferenceresultsfromexperimentusingIP-VCCS.AllvaluesareineV.Thecc-pCVTZbasissetisusedinallcalculations. IP-VCCSMoleculeSymbolKoopmansSymmetryconstrainedBrokensymmetry AcetyleneHCCH-14.91-7.820.36CarbondioxideO=C=O-20.80-11.260.66FluorineF-F-22.46-12.092.18NitrogenNN-17.18-9.730.47OxygenO=O-18.15-8.082.94 Table3-5. ComparisonbetweentheESCFandIP-VCCSmethodforthecoreIPforpropene,whichexhibitsstrongspincontaminationinthetraditionalEHFk(inparenthesis)computation.AllvaluesineV MoleculeExp.acc-pCVDZcc-pCVTZ Propene290.68291.178(291.055)290.208(290.117) a Ref.[ 108 ] Table3-6. Differencesofthegivenmethodwiththecc-pCVDZbasissetfromexperimentalvalues.AllvaluesareineV. MoleculeSymbolKoopmansIP-VCCSa,bErelaxedCCSDbEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSDIP-EOM-CCSD AcetonitrileCH3CN-18.37-0.55-3.27AmmoniaNH3-17.22-0.56-2.61-1.42-2.64MethaneCH4-14.39-0.74-2.53-1.27-2.36CarbonmonoxideCO-19.940.04-3.10-1.66-3.42FormaldehydeH2C=O-20.440.36-14.41-5.29-3.55FuranC4H4O-21.78-0.49-4.43HydrogencyanideHCN-18.43-0.98-3.18-2.08-3.48HydrogenuorideHF-20.870.07-2.61-1.58-3.03MethanolCH3OH-20.090.04-1.47-3.16NitrousoxideN2O-20.87-0.27-1.140.07-3.99NitrogentriuorideNF3-23.98-0.77-5.07PyrroleC4H5N-18.59-0.40-3.34WaterH2O-19.45-0.24-2.92AcetyleneHCCH-15.12-8.46(-0.80)8.47(-3.23)-2.42-2.92CarbondioxideO=C=O-20.87-11.86(-0.54)12.70-3.58-3.89FluorineF-F-22.57-12.88(0.73)14.48(-2.42)-3.09-3.63NitrogenNN-17.27-10.36(-0.78)10.14(-3.70)-2.66-2.88OxygenO=O-20.62-8.70(0.41)10.22(-3.11)-1.02-3.46Averagesignederror-19.49-0.34(-0.30)2.20(-3.82)-2.11-3.41Averageunsignederror19.490.42(0.49)4.75(3.82)2.123.41 a Parentheticalnumbersrefertobrokensymmetrysolutions b Valuesforsymmetryequivalentcentersarenotincludedintheaverages 93

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Table3-7. Differenceofagivenmethodwiththecc-pCVTZbasissetfromexperimentalvalues.AllvaluesareineV. MoleculeSymbolKoopmansIP-VCCSa,bErelaxedCCSDEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDIP-EOM-CCSD AmmoniaNH3-17.160.47-1.460.21-1.30MethaneCH4-14.120.230.17-1.17CarbonmonoxideCO-19.841.28-1.450.35-1.63FormaldehydeH2C=O-20.391.54-1.82HydrogencyanideHCN-18.280.26-2.00HydrogenuorideHF-21.091.35-1.170.47-1.20MethanolCH3OH-20.151.14NitrousoxideN2O-20.840.91-2.29NitrogentriuorideNF3-23.860.64-3.37WaterH2O-19.570.86-1.420.27-1.35AcetyleneHCCH-14.91-7.82(0.36)-7.71-1.06-1.60CarbondioxideO=C=O-20.80-11.26(0.66)-11.72-1.83-2.18FluorineF-F-22.46-12.09(2.18)-14.04-1.14-1.65NitrogenNN-17.18-9.73(0.47)-9.39-1.20-1.44OxygenO=O-18.15-8.08(2.94)-9.490.60-1.88Averagesignederror-19.25-0.87(1.02)-6.43-0.32-1.78Averageunsignederror19.250.87(1.02)6.430.731.78 a Parentheticalnumbersrefertobrokensymmetrysolutions b Valuesforsymmetryequivalentcentersarenot Table3-8. ComparisonofCCSDTwithCCSDandIP-EOM-CCSDusingthecc-pCVDZbasisset.AllvaluesareineV. MoleculeEnon)]TJ /F6 7.97 Tf 6.58 0 Td[(relaxedCCSDTEnon)]TJ /F6 7.97 Tf 6.59 0 Td[(relaxedCCSDIP-EOM-CCSDExp.a Ammonia407.02406.99408.21405.57Fluorine699.16699.78700.32696.69HydrogenFluoride695.68695.76697.21694.18Water541.30541.34542.72539.79 a Ref.[ 108 ] 94

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Table3-9. ComparisonofIP-EOM-CCSDandthirdordercorrectedIP-EOM-CCSDwithexperimenta.AllvaluesareineV. MoleculeIP-EOM-CCSDIP-EOM-CCSD+VCCS(3)b Acetaldehyde-2.221.05Acetylene-1.56-1.37Ammonia-1.340.45Carbondioxide-2.09-1.87Carbonmonoxide-2.030.31Carbonyluoride-2.11-1.87Cyanogen-2.06-0.64Diuoromonoxide-3.14-1.37Formaldehyde-1.871.40Formicacid-2.000.18Hydrogencyanide-2.233.70Hydrogenuoride-1.070.48Hydrogenperoxide-0.66-0.47Ketene-2.360.94Methane-1.210.33Methylamine-1.380.48Ozone-2.871.65Oxirane-2.000.56Tetrauoroethene-1.97-1.96Triuoromethane-1.77-1.58Water-1.450.33Ethane-1.21-0.97(-0.21)Fluorine-1.67-1.57(0.46)Hydrazine-1.35-1.10(-0.17)Nitrogen-1.37-1.23(1.13)Oxygen-2.59-2.46(-0.50)Avg.error-1.83-0.25(0.04)Avg.unsignederror1.831.17(1.01) a Ref.[ 108 ] b Parentheticalnumbersrefertobrokensymmetrysolutionforthethird-ordercorrection. 95

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CHAPTER4MASSIVELYPARALLELIMPLEMENTATIONOFEOM-CCMETHODSFOREXCITEDSTATES 4.1PreliminariesTheresultspresentedinChapters 2 and 3 demonstratesuccessfulapproachesforquantitiesofchemicalinterestwithonlymean-eldscaling.Incontrast,addinghigherexcitationsinCIorCCtheorywillguaranteetherightanswerinthelimit,butatdrasticincreasesincomputationalcost.However,formoleculeswelldescribedbyasinglereference,CCSDperformssatisfactorily,andCCSDwithperturbativetriples(CCSD(T))isconsideredthegold-standard;eachwithn2on4vandn3on4vformalscalingrespectively.Therefore,pushingthesemethodsforward,toperformcalculationsonmoderatelysizedmolecules,stillremainsanareaofintenseresearch.[ 34 35 ]Makinguseofmodernsupercomputers,equippedwithO(1,000)toO(100,000)ofcomputecoresandsharedmemoryoncomputenodesisthelogical(andnecessary)routetowardsthisgoal.TheACESIIIprogrampackage[ 34 ]hasbeendesigned,veryefciently,withthiscapabilityandthusoffersaveryattractiveenvironmentforthemassiveparallelizationofcoupledclusteralgorithms.Infact,manyrecentstudieshavebeenperformedonsystemsofdemonstrablesizewithcoupledclustermethods.[ 109 112 ]However,anewprogrammingparadigmrequiresnewcodesforwell-establishedmethods,butwithadifferentmindsetastohowthedatashouldactuallybedistributed.OnesuchexamplewithACESIII,isthattheintegralpackageatthebeginning,althoughextremelyefcient,onlycomputedthenecessaryintegralsfortotalenergies(kineticenergy,nucleus-electronattraction,electron-electronrepulsion,andAOoverlapintegrals),[ 113 ]andwasratherlimitedinapplicabilityduetothelackofpropertyintegralsforobservables.Consequently,followingtheblockeddesignphilosophyofFlocke[ 113 ],arbitraryordermomentintegrals(xpyqzr)integralsandarbitraryderivativesoftheseintegralswereimplementedintheintegralpackage.Thisopenedthedoorfor 96

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routinepropertyanalysisforthemassivelyparallelMBPT(2)andCCSDprogramsthatalreadyexisted,giventhatthegroundstatedensitiesarecomputedforeachmethod.Similarly,onlytherighthandeigenvectorsolutionsofHwereprogrammedinACESIII,[ 110 ]allowingonlythecalculationofexcitationenergies,withnoinformationaboutthestrengthofthetransitioninthespectrum,theavenueforscanningtheexcitedstatepotentialenergysurface,orevenextremelyhighaccuracywithperturbativetriples,allofwhichneedthelefthandeigenvector.Infact,inthedemonstrableperformancepaperbyKus[ 110 ],theexcitationenergiesofthecytosine-hydroxyladductaregivenbytheEE-EOM-CCSDmethod,andthelineintensitiesaregivenattheCISleveloftheory.ThegoalthenofthischapteristoextendthemassivelyparallelgroundstatecapabilitiesofACESIIItoexcitedstates,usingtheequationofmotioncoupledclustermethodology.Specically,onewouldlikeageneralprogramforoscillatorstrengths,orlineintensitiesforUVspectra,benchmarkaccuracyforexcitationenergies,givenbytheEE-EOM-CCSD(T)method,andtheabilitytoscanandcharacterizecriticalpointsonapotentialenergysurface. Simulatingultra-violetspectra.Theoreticallydeterminedexcitationenergiesarenecessarytoaideexperimentalspectra,anddosowiththeaccurateassignmentsofthestatesbycomputingthecorrespondinglineintensity,oroscillatorstrength.Infact,thecontroversyconcerningthestructureofwaterclusters[ 114 116 ]pointsdirectlytotheneedofaccurateexcitationenergiesandoscillatorstrengths.Saykallyet.al[ 116 ]arguethatthetheoreticalmethodsusedtosimulatethespectraofstructuresobtainedbyZwierandShin[ 114 115 ]couldgenerateanynumberforanoscillatorstrength,andthereforearenotpreciseenoughtoprovidethequantitativedetailnecessary.SinceserialEOM-CCSDprogramsarenotabletobeusedonsystemsofmorethan10heavyatomsinmodestsizedbasissets,studiesoflargerwaterclustersarenotpossiblewithoutamassivelyparallelimplementation.Furthermore,obtainingoscillatorstrengthsrequiresthesolutionofthelefthandeigenvectorofH,whichraises 97

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thecomputationalcostbyafactorof2.Inthelimit,though,^Lk=^Ryk.Therefore,foranytruncationoftheEOM-CCequations(where^Lk6=^Ryk),onecanwrite^Lk=^Ryk+^Lk)]TJ /F4 11.955 Tf 13.31 2.66 Td[(^Ryk=^Ryk+^Lk, (4)andassumethatthesecondtermisnegligible.Then,^Rykrepresentsagoodapproximationto^Lk.Thiswouldsavethisfactorof2inthecomputationofoscillatorstrengths.Thisapproximationistestedwiththegoalofaccurateandpreciseresultswithhalfthecomputationalcost. Benchmarkexcitationenergies.OfprimaryimportancetobiologicalsystemsarethenucleobaseandsugarextendednetworksofDNA.[ 117 120 ].Thenucleobasesincludechromophores,andconsequentlyeasilyaccessibleexcitedstates,whichisasubjectofintensecomputationalresearch.[ 121 128 ]Duetothesizeofthesesystems,mostlylowerlevelapproachessuchasCASPT2,TDDFT,MCSCF,andCC2havebeenusedtoexaminetheverticalexcitationspectrum.Thesemethodstypicallyhavelarge,lesssystematicerrorcomparedtoEE-EOM-CCSDandEE-EOM-CCSD(T).[ 117 ]However,understandingthespectralpropertiesofDNAandRNArequirestheaccuratecalculationsonnucleobasedimers,hydratednucleobases,andnucleosidesystems,ratherthanthenucleobasemonomers.Forgroundstatesdominatedbyasinglereferencewavefunction,CClimitedtosingleanddoublesubstitutionswithanoniterativeperturbativetreatmentoftriples(CCSD(T))[ 129 130 ]isknownasthegoldstandardoftheoreticalmethodsfornitemolecularsystems.TheCCSD(T)methodhasbeenextendedtoincludefactorizedquadrupleseffects(CCSD(TQf))[ 131 132 ]andhasbeenformulatedusingthecompletelyrenormalizedvariantsofPiecuchet.al.[ 133 134 ]Also,awholeperturbativehierarchyontheCCgroundstate,convergenttotheFullCIlimit,hasbeendeveloped,beginningwiththeCCSD(2)methodofGwaltneyet.al[ 135 137 ]andextendedtohigherordercorrectionsintheclusteroperatorsandperturbationtruncations,mand 98

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n,respectively(CC(m)PT(n))byHirataet.al.[ 138 ]AlsorelatedasapproximationstotraditionalCCtheoryarethelinearresponseCCnfamilyofmethodsofChristiansen,Koch,andJrgensen[ 139 141 ]Similarly,successfulnoniterativeperturbativeapproachescanbeextendedtoexcitedstatesviaEOM-CCmethodology.Akintothegroundstate,completelyrenormalizedvariantsofPiecuchet.al[ 142 148 ]havebeendevelopedforexcitedstates(includingEOM-CCSD(T),EOM-CCSD(T)L,r-EOM-CCSD(T)).However,thesemethodsareeithernotsizeconsistentornotinvarianttoorbitalrotationsbyvirtueofusingofEpstein-Nesbetenergydenominators.[ 149 150 ]AnoniterativeperturbativeapproximationtoCC3,[ 139 140 ]namedCCSDR(3),[ 151 152 ]performsquitesatisfactorilycomparedtoCC3.[ 153 ]TheCC(m)PT(n)hierarchyofHirataet.al[ 138 ]havebeenextendedforexcitedstatesinasimilarfashionanddenotedEOM-CC(m)PT(n).[ 33 154 155 ]ThishierarchyconvergestotheFullCIlimit,butincurssevereformalscalingbottlenecksasmandnareincreased.ThespecicEOM-CC(2)PT(2)Tvariantisformallyequivalent,fortheexcitedstate,astheEOM-CCSD(~T)methodofWattsandBartlett.[ 31 156 ]thoughthereissomeambiguityinthetreatmentofthegroundstateforthelattermethod.[ 33 ]Takingtheleadingordertermsinthetwo-electronperturbation,^W,ofEOM-CCSD(~T)givestheEOM-CCSD(T)method[ 157 ]forexcitedstates,ananalogueoftheCCSD(T)forgroundstates.Wenotethatintheoriginalderivation[ 156 ]givenbyWattsfortheEOM-CCSD(~T)method,furtherapproximations,ontopofdeningtheordersofthecomplete^Rkand^Lkeigenvectors,aremadebasedonorderargumentsconcerningtheexcitationenergyitself(ie.!CCSDTbeingreplacedby!CCSD)anditcoupledwiththeperturbativeoverlapterm(!CCSDTh^L3kj^R3ki).However,thesereplacementsareneverneeded,sinceallthesetermscancelwithacarefulexaminationoftheamplitudeequationsandaproperlypartitionedHamiltonian.Furthermore,theorder-by-orderanalysisandrederivationgivenbyGwaltneyet.al[ 31 ]containsan!CCSDdependenceinthedenominator,andthisis 99

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removedbyasuitablypartitionedHamiltonian.WepartitionthesimilaritytransformedHamiltonianofCCtheory,HinasimilarspirittoStantonandGauss'partitioningscheme[ 158 160 ]toarriveattheEOM-CCSD(~T)energyexpressionandamplitudeequationswithnoapproximationotherthanthelimitationtotriplesubstitutionsintheQ-space.AsimpleLowdintypepartitioning[ 161 ]isalsoapplied,analogoustoStanton'sanalysisofCCSD(T)[ 162 ]andextensionbyStantonandCrawford[ 163 ],whichbolstersthepresentedderivation.Furthermore,onecantaketheleadtermsin^WtoarriveatEOM-CCSD(T)oronecanperformaMller-Plesset[ 164 ]typepartitioningofH,[ 158 160 162 163 ]transparentwithLowdinpartitioning,toarriveattheEOM-CCSD(T)fourthordercorrectiontotheexcitationenergy.Thesenoniterativeperturbativetriplesmethodshavemostlybeentestedonsmallprototypicalsystems.Recently,arobusttestsetconsistingofover24moleculesandover120excitedstateshasbeenproposedandusedinassessingtheaccuracyofTDDFTandMRCImethods[ 165 ]aswellasCASPT2,CC2,EOM-CCSD,andCC3methods[ 166 ].ThistestsetwasalsousedforbenchmarkingtheperturbativeCCSDR(3)methodcomparedtoitsiterativeanalogue,CC3.[ 153 ]Onewondershowwellanyoftheseotherperturbativemethodswouldperformoverthiswidevarietyofchromophores.Inthatvein,theEOM-CCSD(~T)andEOM-CCSD(T)arerederivedinadifferentlight,implementedinamassivelyparallelway,andcomparedagainstheformallyrobustEOM-CCSDT-3method[ 31 156 ]withtheaforementionedtestset.Forcompleteness,theCCSDR(3)andCC3methods,whicharelinearresponseanaloguesoftheformermethods,arecomparedagainstEOM-CCSDT-3.Inaratherinterestingturnofevents,theEE-EOM-CCSD(T),havingbeenshowntohavelesssatisfactoryperformanceforexcitedstatescomparedtotheremarkableperformanceofCCSD(T)forgroundstates[ 33 ]forsmallmoleculesessentiallyreproducestheiterativeandrobustEE-EOM-CCSDT-3methodforthelargernucleobase 100

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systemsforalltypesoftransitions.[ 117 ]Also,theEE-EOM-CCSD(~T)method,whichperformedremarkablywellforaclassofsystems[ 32 ]performsfarlesssatisfactorilyforgeneraltransitionsofavarietyoforganicmolecules,[ 167 ]typicallyovershootingthecorrecttriplesenergy.Asaresult,sincetheEE-EOM-CCSD(T)methodisafactorof2fasterthantheEE-EOM-CCSD(~T)variantandmoreaccurateoverawide-rangingtestset,thisisthemethodofchoiceforbenchmarkingtheexcitedstatesofthenucleobasesystems.Sincethismethodalsofollowsfromtheenergyfunctional,the^Lk=^Rykistestedtoseeksignicanttimesavings.Alongwithanewderivationandmassivelyparallelimplementation,benchmarkresultsarepresentedformicrohydratedcytosine,cytidine,guanosine,andtheWatson-Crickcytosine-guaninebasepair. Characterizingthepotentialenergysurface.Theaccurateandfeasiblecharacterizationofgroundandexcitedstatepotentialenergysurfacesofchemicallyrelevantsystemsisoneofthemostimportantresearchtopicsinmodernquantumchemicalmethods.TheEE-EOM-CCSDmethodisconsideredablack-boxmethod,capableofhighaccuracy,andretainstheopportunityforsystematiccorrections.However,therearecurrentlyonlyserialimplementationsoftheEE-EOM-CCSDgradients,greatlylimitingtheapplicabilityofthemethodtosmallsystems.Amassivelyparallelimplementationisnecessarytotacklelargersystems.Specically,questionsremainabouttherelaxationmechanismsalongthepotentialenergysurfaceofDNAandRNAafterabsorptionofultra-violetlight.[ 117 168 ].Lowerlevelmethods,includingsemi-empirical[ 169 173 ]andCASSCF,[ 168 174 176 ]MR-CIS,[ 168 177 ],andTDDFT[ 169 178 179 ]moleculardynamicshavebeenused,andwithoutanaccurateaccountoftherelativeenergiesoftheinvolvedelectronicstates,thepictureobtainedcanbequiteunreliable.[ 112 ].Therefore,themassivelyparallelimplementationoftheEE-EOM-CCSDgradientsaregiven,withdemonstrablescalingcurves.Thesizesofthesystemsalsospeakforthe 101

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applicabilityofthecode.Idemonstratetheperformanceonthenucleobasescytosine,guanine,thymine,andadenine,aswellasthecytosine-guaninestackeddimer.ThisallowstheroutineuseofEE-EOM-CCSDtheoryforaccuratescanningofpotentialenergysurfacesofchemicallyrelevantmolecules.Analyzingthegroundandexcitedstatedensitiesandthedifferencebetweenthetwoisessentialforadeeperunderstandingoftheinvolvedtransitionsandtheelectronicredistribution.Tothisend,thedensitiesanddensitydifferencesarecomputedandstoredondisk,inamolden[ 180 ]readableformat. 4.2GeneralTheoryInEOM-CCmethods,theHamiltonianthatisdiagonalizedintheappropriatesectorofFockspace,H,isnothermitian,havingbothleftandrighteigenvectors.Thesearechosentobebiorthogonal kl=h~kjli=h0j^Lk^Rlj0i(4)yieldingtheCCenergyfunctional Ek=h0j^LkH^Rkj0i.(4)WhentheCCequationsarelimitedtosingleanddoublesubstitutionsonly,the^Lkand^Rkeigenvectorshavetheform^Rkj0i=(r0+^R1k+^R2k)j0i=r0j0i+Xiaraijaii+1 4Xijabrabijjabiji (4)h0j^Lk=h0j(l0+^L1k+^L2k)=h0jl0+Xaihaijlia+1 4Xabijhabijjlijab. (4)NotethelinearvectorsresembleaCIansatz,wherether0termisnecessarytodescribeexcitedstatesofthesamesymmetryasthegroundstate;l0iszerobyconstructionforallexcitedstates(l0=1forthegroundstate).[ 15 ] 102

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TheCCfunctionalofEquation( 4 )satisesthegeneralizedHellman-Feynmanproperty,[ 15 ]sincelinearvariationsinthe^Lkand^Rkeigenvectorsiszero,andsetsupasuitableperturbationparameterizationattheEOM-CCSDlevelwiththeP-spacedenedbytheground,singly,anddoublyexciteddeterminantsP=fj0i,jaii,jabijgandtheQ-spacecontaininghigherexciteddeterminants.Consequently,properties,includingtransitionmomentsandoscillatorstrengths,canbecomputedwithlittlecomputationaleffort,andperturbativecorrectionsduetohigherexcitationscanbecomputedaswellforbenchmarkqualitysinglyexcitedstates.Also,thefunctionalcanbedifferentiatedwithrespecttogeometricalperturbationsallowingthecomputationofgradientsandcriticalpointsonapotentialenergysurface. 4.3ExpectationValueFormforOne-ElectronPropertiesTheexpectationvalueofanylinearoperatorinEOM-CCtheoryis=h0j^Lk^Rkj0i (4)=Tr[] (4)withbeingthematrixrepresentationoftheoperatorinthemolecularorbitalbasis,andthereducedn-particledensitymatrix,,givenby pq...rs=h0j^Lkfpyqy...srge^TC^Rkj0i(4) 103

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Thereduced1-particledensitymatrixisparticularlysimple,givenineinsteinnotation[ 25 ]asij=)]TJ /F5 11.955 Tf 9.29 0 Td[(r0ljctci+1 2ljkcdtcdik)]TJ /F5 11.955 Tf 11.95 0 Td[(ljcrci)]TJ /F13 11.955 Tf 11.95 16.86 Td[(tcirdk+1 2rcdikljkcd=)]TJ /F5 11.955 Tf 9.29 0 Td[(r0~Dji)]TJ /F5 11.955 Tf 11.96 0 Td[(ljcrci)]TJ /F4 11.955 Tf 11.58 0 Td[(~rcdikljkcd (4)ab=)]TJ /F5 11.955 Tf 9.29 0 Td[(r0lkbtak+1 2lklbctackl+lkbrak+takrcl+1 2rackllklbc=)]TJ /F5 11.955 Tf 9.29 0 Td[(r0~Dab+lkbrak+~rackllklbc (4)ia=r0lia+likacrck (4)ai=r0lkc(tcaki)]TJ /F5 11.955 Tf 11.96 0 Td[(tcitac)+1 2(tcdiktal+tadkltci)lklcd+tai+lkc(racik)]TJ /F5 11.955 Tf 11.95 0 Td[(takrci)]TJ /F5 11.955 Tf 11.95 0 Td[(raktci)+lklcd(tcaki)]TJ /F5 11.955 Tf 11.96 0 Td[(tcitac)rdl)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2lklcd)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(tadklrci+radkltci)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2lklcd)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(tcdilrak+rcdiltak (4)andistheonlynecessaryquantityforagivenone-electronproperty.Inpractice,suitableintermediatesareformed,orfactorizationisperformed,thatcanbechosentoreducediskspacerequirementsorincreaseparallelizability.Asanexample,itwouldbeunwisetoprogramthetadkltcipieceofEquation( 4 )asanintermediate.Thestoragerequirementswouldben3on3vandtheresultingtensorcontractionwithlklcdwouldscalethesame,orroughlyN6.Rather,theintermediatelklcdtadklisformedinann2on3vcontractionstepwithonlyn2vstorage(ifstored)andthencontractedwithtciinasimplenon2vcontractionstep.However,onehasthechoicetocomputetheintermediatesaheadoftimeandstoretheminmemoryorondisk,orcomputetheintermediatesonthey.Inaserialprogram,theintermediatesaregenerallycomputedontheyandusedimmediatelyafter,thusavoidingthestorage.Thiscanhurtaparallelprogram.Inpseudocode,theontheycontractionis PARDOa,c 104

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Yac=0.0 DOk DOl DOd REQUESTlklcd REQUESTtadkl Yac=Yac+lklcdtadkl ENDDOd ENDDOl ENDDOk DOi GETtci Xai=Yactci ai+=Xai ENDDOi ENDPARDOa,c Thisimplementationonlyparallelizesoverblocksaandcandworse,requestsdataintheinnermostloop.Thebetterwaytoperformthiscontraction,todemonstratewithpseudocode, PARDOd,k,l,c REQUESTlklcd DOa REQUESTtadkl PUTYac+=Yac+lklcdtadkl ENDDOa ENDPARDOd,k,l,c 105

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EXECUTESIP BARRIER PARDOi,a,c GETYac GETtci Xai=Yactci ai+=Xai ENDPARDOi,a,c Intheabove,theintermediateYisformed,parallelizedover4indices,andstoredinmemory.OnceallblocksofYhavebeencomputedandstored(attheSIP BARRIER),asimple,completelyparallelizedcontractionisperformedtocomputethecontributiontothedensity.Theintermediatearraycansubsequentlybedestroyed,sotheblockofmemoryisfreedandabletobeusedagain.Ofparticularimportanceinexcitedstatecalculationsistransitionmomentsandthecorrespondingoscillatorstrengths.SinceHhasleftandrighteigenvectors,leftandrighttransitionmoments(0kandk0,respectively)arecomputedforthedipolestrength,D,asD=h0j^L0^Rkj0ih0j^Lk^R0j0i (4)=Tr[0k]Tr[k0] (4)=0kk0, (4)whichisusedtocomputetheoscillatorstrength,orlineintensityf,as f=2 3!kD.(4)TheEE-EOM-CCSDmethodhasbeenverysuccessfulindeterminingexcitationenergiesandcomputingthecorrespondingoscillatorstrengths.[ 25 ]However,this 106

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expectationvalueformularequiresthesolutionof^Rkand^Lk,whichraisesthecomputationaleffortbyafactoroftwo.Itwouldbeniceifthiseffortcouldbereduced.Inthatvein,weexperimentedwithreplacing^Lkby^Ryktocomputethedensityandproperties.ThiscanbejustiedbynotingthatinthelimitofFullCIorCC,^Lk=^Rk.Wetestthehypothesisthatthelefteigenvectorshouldbewellrepresentedbytherighteigenvector,truncatingatthesinglesanddoubleslevel. 4.3.1ImplementationTheeigenvalueequationtodeterminethelefthandeigenvectorwasimplementedinACESIII.The^Reigenvectorsareusedasinitialguesses,reducingthenumberofiterations,onaverage,neededtoconvergethediagonalizationalgorithm.Thelargestcontribution(n2on4v) 1 2Xcdhabjjcdilijcd7!lijab(4)iscomputeddirectlybybacktransformingthevirtualindicesofthe^Lkamplitudearrayandcontractingwithtwo-electronAOintegralsgeneratedon-the-y.Thisincreasesscalability,asthehabjjcdiintegralarraysaresolarge,andwouldneedtobereadoffdiskeveryiteration.Itreducesdiskstorageandperformssignicantlybetterthantheindirectalgorithm[ 110 ]atthecostofmoreoatingpointoperations.Intermediateswereformedatthecostofstorageduetothelargegaininparallelization.Notonlybecausethecomputationofeachintermediateismoreefcientlyparallelized,buttheintermediatescanbecomputedindependentlyfromeachother,givingfurthergainsinperformance.Afteralltheindependentintermediatesarecomputed,theyareusedinthenalcontractionstoformthedensity,andimmediatelydeletedtorestorespace.TheEOM-CCSDdensitycodewasimplementedinageneralwaytoallowtheuseof^Lkor^Rykamplitudes.Infact,sincethecomputationaltimeofcomputingthedensityissmall,whenusingappropriateintermediates,incomparisontotherightorleftEOMcomputation,aftercomputingtherightandlefthandside,thepropertiesarecomputed 107

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withthe^Rykapproximationand^Lkeigenvector,respectively.Meaning,ifonlytherighthandcodeisrun,thentheoscillatorstrengthscorrespondtothe^Rykapproximation. Blockedmomentintegrals.Thexp,yq,andzrintegralsnecessaryforcalculatingthedipoleandtransitionmoments,foranyleveloftheory(providedadensitymatrixiscomputed),wereimplementedintheACESIIIprogrampackagemakinguseofitsblockstructure.Thiswasdoneinthemostgeneralwaytoallowthecomputationofarbitrarymomentintegrals.AquickdescriptioncanbegivenstartingwithageneralmomentintegralcenteredatCbetweentwoAOscenteredatAandB hjM()ji=Zdr1A,la,ma,na(r1)MCs,t,u()B,lb,mb,nb(r1)(4)withA,la,ma,na(r1)=(x)]TJ /F5 11.955 Tf 11.95 0 Td[(Ax)la(y)]TJ /F5 11.955 Tf 11.96 0 Td[(Ay)ma(z)]TJ /F5 11.955 Tf 11.96 0 Td[(Az)nae)]TJ /F16 7.97 Tf 6.59 0 Td[(jr1)]TJ /F9 7.97 Tf 6.59 0 Td[(RAj2 (4)B,lb,mb,nb(r1)=(x)]TJ /F5 11.955 Tf 11.95 0 Td[(Bx)lb(y)]TJ /F5 11.955 Tf 11.96 0 Td[(By)mb(z)]TJ /F5 11.955 Tf 11.95 0 Td[(Bz)nbe)]TJ /F16 7.97 Tf 6.58 0 Td[(jr1)]TJ /F9 7.97 Tf 6.58 0 Td[(RBj2 (4)MCs,t,u()=(x)]TJ /F5 11.955 Tf 11.95 0 Td[(Cx)s(y)]TJ /F5 11.955 Tf 11.95 0 Td[(Cy)t(z)]TJ /F5 11.955 Tf 11.95 0 Td[(Cz)u (4)Focusingononecartesiancoordinate,sincetheyareseparable,wecanrewritethemomentintegralashjM()ji=Zdr1A,a(r1)MCs()B,b(r1) (4)=Zdr1MCs()Pp,a,b(r1) (4)=Zdr1Pp,a,b,s(r1) (4)withthemodiedoverlapGaussiangivenas Pp,a,b,s(r1)=(x)]TJ /F5 11.955 Tf 11.95 0 Td[(Ax)a(x)]TJ /F5 11.955 Tf 11.95 0 Td[(Bx)b(x)]TJ /F5 11.955 Tf 11.95 0 Td[(Cx)se)]TJ /F6 7.97 Tf 6.59 0 Td[(pjr1)]TJ /F9 7.97 Tf 6.59 0 Td[(RPxj2(4) 108

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Withthisform,itisclearthattheintegralsreducetomodiedoverlapintegrals.SincetheACESIIIprogrampackagecontainsaveryefcientintegralpackagewithoverlapintegrals,[ 113 ]andarbitraryderivativeoverlapintegrals,thenecessarymodicationsweremadetoincorporatemomentintegralstoanypowerandarbitraryderivativemomentintegralstoanypower.ThisimplementationgreatlyincreasedtheapplicabilityofACESIII,andgiventheseintegralsyieldchemicalproperties,resultedinagreaterchemicalunderstanding,ofDNAnucleobasesandcomplexes[ 110 112 117 ]andlargechemicalexplosivecompounds.[ 111 ] 4.3.2ResultsandDiscussionVariousbiologicallyrelevantmoleculesinanassortmentofbasissetsformedthetestsettoexamineusingtherighthandEOMeigenvector,^Ryk,inplaceofthethetruelefthandeigenvector,^Lkforcomputingoscillatorstrengths.Includedinthisarethefournucleobases(adenine,cytosine,thymine,andguanine)withaugmenteddouble-andtriple-basissets,thenucleosidescytidineandguanosineinanaugmenteddouble-basisset,hydratedcytosinemoleculeswithaugmenteddouble-basissetsandtwoWatson-Crickbasepairs,adenine-thymineandcytosine-guanineinAhlrichs'triple-basisset[ 181 ].ThelowestelevenortwelverootsarecomputedattheEOMlevelresultinginatestsetof187excitedstates.ThedeviationofthetotaloscillatorstrengthscomparedtotheformallycorrectexpectationvalueformulaisshowninFigure 4-1 .Thebrightstatesareseparatedfromthedarkstatessinceonaverage,theyappeartodiffermoresignicantlyfromthedarkstates.Mostdarkstateshaveanintensityunder10)]TJ /F9 7.97 Tf 6.59 0 Td[(3,andthisapproximationfaithfullycapturesthis.Also,sincethemagnitudeofthebrightstatesinrelationtoeachotheristheimportantquantity,theminordeviationsseenforthemarerelativelyinsignicant.Onaverage,thebrightanddarkstatesdifferfromtheexactby0.003and0.0003units,respectively.Furtheranalysiswithsmallmoleculesused^=^Tytoexploremoretimesavings.ThisalsoperformedexceptionallywellasdemonstratedinTable 4.6 .Askewedstructure 109

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ofammoniawasusedsinceithasC1symmetry,andtheapproximateoscillatorstrengthsareinalmostperfectagreementwiththeexactones.Thelambdaequationsarerelativelyfasttosolve,comparedtotheEOMequations,possiblymakingthisapproximationunnecessary.Nevertheless,itdoessaveafactorof2inthegroundstatecalculation.ThisdegreeofaccuracydemonstratesqualityoftheCCwavefunction.InthelimitoftheexactFullCIresult,theleftandrighthandeigenvectorsareequivalent.[ 25 ]Therefore,thisaccuratedescriptionoftheoscillatorstrengthsreectsthefastconvergencepropertyofCC,orinotherwords,itillustratesthat^Ryk^LkattheCCSDlevel.Morenotably,withcondence,onecancompletelyneglectthelefthandEOMequationsifseekingtheoscillatorstrengthsandrst-orderpropertiesalone.Thissavesafactorof2intheexcitedstatecalculations,withnoapproximationsmadetotheexcitationenergyitself. 4.4PerturbativeTriplesforAccurateEnergies.TheP-spaceforEE-EOM-CCSDisspannedbythereferencedeterminantwithaprojector,j0ih0jandtheQ-spaceisspannedbythesingly,jSi,anddoubly,jDi,exciteddeterminantswithaprojectorQ=jSihSj+jDihDj (4)=Xiajaiihaij+1 4Xijabjabijihabijj (4)ThispartitionstheHamiltonian,H,explicitlyas H=264HPP HPQ HQP HQQ375=2666640 H0SH0D 0 HSSHSD0 HDSHDD377775,(4)clearlyindicatingthatonlytheQ-spaceofHneedstobediagonalized.ThisiscriticalwhenexaminingperturbativetriplescorrectionstoEE-EOM-CCSD. 110

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ItwillbeusefultodenethefollowingquantitiesusedintheperturbativetriplescorrectiontoEE-EOM-CCSD.WewillredenetheP-spacetobespannedbytheground,singly,anddoublyexciteddeterminantswithaprojectorP=j0ih0j+jSihSj+jDihDj (4)=j0ih0j+Xiajaiihaij+1 4Xijabjabijihabijj (4)andtheQ-spacewillbelimitedtotriples,QQ3=1 36PijkPabcjabcijkihabcijkj.ThecompletesimilaritytransformedHamiltonian,H,ispartitionedintheP-andQ-spacesasH=(P+Q)H(P+Q) (4)=HPP+HPQ+HQP+HQQ, (4)andpartitionedinordersas H=H(0)+W(1)(4)withthezerothorderHamiltonian H(0)=PHP+Q(!CCSD+foo+fvv)Q(4)andW(1)containingtherestoftheHmatrixintheappropriatespaces.Notethatthesubscriptsareusedtoreduceclutter.ThefreedomtoincludetheEE-EOM-CCSDexcitationenergy,!CCSD,andthe(semi)canonicaloccupied-occupiedandvirtual-virtualFockmatrices,fooandfvv,respectivelyintheQ-spacewasused,asitwilldene(andsimplify)theresultingenergydenominatorswhilemaintainingorbitalinvariance.[ 33 182 ]ThispartitioningwilldenetheEE-EOM-CCSD(~T)methodoriginallydevelopedbyWattsandBartlett[ 31 ]andfurtherexaminedbyGwaltneyet.al[ 156 ]withHgiven 111

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explicitlyas H=264HPP HPQ HQP HQQ375=2666666640H0SH0D H0T0HSSHSD HST0HDSHDD HDT HT0HTSHTD HTT377777775(4)WenoteinpassingthattheQ3-projectionswouldbezeroifthefulltriplesweresolvedforinthegroundstate.ThishasbeenalludedtobyShiozakiet.al[ 33 ]wheretheyaddresstheambiguityoftriplesinthegroundstateforthismethod.Theexactleftandrighthandeigenvectorsuptotriplesforstatek,^Lkand^Rk,areh0j^Lk=h0j^Lk(P+Q)=h0j^LP+h0j^LQ (4)=h0j(^L1k+^L2k)+h0j^L3k (4)^Rkj0i=(P+Q)^Rkj0i=^RPj0i+^RQj0i (4)=(r0k+^R1k+^R2k)j0i+^R3kj0i (4)Withthesedenitions,theexcitationenergycanbewrittenas !k=!=!P+!Q=!CCSD+!Q.(4)ThestartingpointforperturbativecorrectionstotheexcitationenergyistheEE-EOM-CCSDenergyfunctionalpartitionedwiththedenedPandQ-spaces,!h0j^Lk^Rkj0i=h0j^LkH,^Rkj0i (4)(!P+!Q)^LP^RP+^LQ^RQ=h0j^Lk)]TJ /F4 11.955 Tf 7.04 -7.03 Td[(H^RkCj0i (4)(!P+!Q)^LP^RP+^LQ^RQ=^LP^LQ264HPPHPQHQPHQQ375264^RP^RQ375 (4)=^LPHPP^RP+^LPHPQ^RQ+^LQHQP^RP+^LQHQQ^RQ (4) 112

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Thersttermsontheleftandrighthandsidecancel.Wearefreetosubtract^LQ!CCSD^RQfrombothsidestoremovethesecondtermandthereforearriveatthefollowingexpressionfortheenergycorrection,usingthenormalizationconditionh0j^LP^RPj0i=1,!Q1+!Qh0j^LQ^RQj0i=h0j^LPHPQ^RQ+^LQHQP^RP+^LQ(HQQ)]TJ /F7 11.955 Tf 11.96 0 Td[(!CCSD)^RQj0i. (4)Tofurthersimplify,weexaminetheprojectedequationsfortherighthandsideQ!^Rkj0i=Q)]TJ /F4 11.955 Tf 7.04 -7.03 Td[(H^RkC0i (4)(!CCSD+!Q)^RQj0i=HQP^RPCj0i+HQQ^RQCj0i (4)!Q^RQj0i=HQP^RPCj0i+(HQQ)]TJ /F7 11.955 Tf 11.95 0 Td[(!CCSD)^RQCj0i (4)InsertionofthisnalexpressioninEquation( 4 )simpliestheenergycorrection(droppingtheconnectedsubscript)!Q1+!Qh0j^LQ^RQj0i=h0j^LPHPQ^RQ+!Q^LQ^RQj0i (4)!Q=h0j^LPHPQ^RQj0i. (4)Allthatremainsistodene^RQ.Tothisend,weusetheHamiltonianseparatedbythedenedordersandcollectthelowestordercorrection,inW(1),to!Q.Applyingthistotheamplitudeequation( 4 ),showingexplicitlytheordersandtriplesQ-space!Q3^RQ3j0i=HQ3P^RPj0i+(HQ3Q3)]TJ /F7 11.955 Tf 11.95 0 Td[(!CCSD)^RQ3j0i (4)(!(2)Q3)]TJ /F4 11.955 Tf 13.52 2.66 Td[(H(0)Q3Q3+!(0)CCSD)^R(1)Q3j0i=H(1)Q3P^RPj0i (4)(foo+fvv)^R(1)Q3j0i=W(1)Q3P^RPj0i (4)^R(1)Q3j0i=D3W(1)Q3P^RPj0i. (4) 113

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Theenergydenominator,D3,usingthedenitionsgiveninEquation( 4 )isD3=Q3H(0)Q3Q3)]TJ /F7 11.955 Tf 11.96 0 Td[(!CCSDQ3 (4)=Q3(!CCSD+foo+fvv)]TJ /F7 11.955 Tf 11.96 0 Td[(!CCSD)Q3 (4)=1 36XijkXabcjabcijkihabcijkj i+j+k)]TJ /F7 11.955 Tf 11.96 0 Td[(a)]TJ /F7 11.955 Tf 11.95 0 Td[(b)]TJ /F7 11.955 Tf 11.95 0 Td[(c (4)The^R(1)Q3isdenedexplicitlyasQ4^R(1)Q3=Q3D3r0W(1)j0i+(W(1)^R1k)Cj0i+(W(1)^R2k)Cj0i (4)=Q3D3(W(1)^R1k)Cj0i+(W(1)^R2k)Cj0i (4)withthesecondordercorrectiontotheEE-EOM-CCSDexcitationenergygivenas !(2)=h0j^LPW(1)PQ3(W(1)^R1k)C+(W(1)^R2k)Cj0i(4)SincewehavelimitedourselvestoQ3,thecorrectionscalesn3on4v.Notethatther0termvanishes,asitisascalarandcannotbeconnectedtoW.Wenowhighlightsomeofthenewfeaturesofthisderivation.TheoriginalderivationgivenbyWattset.al[ 31 ]makesapparentapproximationssuchasreplacing!Q(!CCSDT)]TJ /F9 7.97 Tf 6.58 0 Td[(3intheirderivation)by!CCSDandneglectingtheoverlaph0j^L3k^R3kj0ibasedonthenegligiblemagnitudeoftheseterms,however,theyareseentobeunnecessarywiththispartitioning.Thesecondderivation,andorder-by-orderanalysis,givenbyGwaltneyet.al,[ 156 ]isverysimilarinspirittothepresentedderivation,butstillcontainsthe!CCSDdependenceinthedenominator,shownheretovanish.Infact,amuchsimplerderivationthathighlightstheseresultscanbedonebyusingLowdinpartitioning.[ 161 163 ]First,weneedtoexpressthefunctionalforminaform 114

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thatismoreeasilymanipulated,!k=h0j^LkH,^Rkj0i (4)=h0j^LkH^Rkj0i)-223(h0j^Lk^RkHj0i (4)=h0j^LkH^Rkj0i)-223(h0j^Lk^Rk(P+Q3)Hj0i (4)=h0j^LkH^Rkj0i)-223(h0j^Lk^Rkj0iE0)-222(h0j^Lk^RkjQ3ihQ3jHj0i (4)=h0j^Lk)]TJ /F4 11.955 Tf 7.04 -7.03 Td[(H)]TJ /F5 11.955 Tf 11.96 0 Td[(E0^Rkj0i)]TJ /F5 11.955 Tf 19.26 0 Td[(r0h0j^Lk3Wj0i. (4)NoticethatinEquation( 4 )wewillbeincludingtheHT0blockdescribedatthebeginningofthissection,andwewillcarryalongthescalar,)]TJ /F5 11.955 Tf 9.3 0 Td[(r0h0j^Lk3Wj0i.Withthisform,wecanperformaLowdintypepartitioningonthedisconnectedfunctionalintheP-spaceas !exact=h0j^LPH^RPj0i+h0j^LPHPQ)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(!exact)]TJ /F4 11.955 Tf 13.52 2.65 Td[(HQQ)]TJ /F9 7.97 Tf 6.58 0 Td[(1HQP^RPj0i)]TJ /F5 11.955 Tf 15.54 0 Td[(r0h0j^Lk3Wj0i(4)ThelowestordercorrectiontothisexpressionintheQ3-space,usingthepredeneddenitionsofHgiveninEquation( 4 )andleftandrighteigenvectors,is !(2)=h0j^LPH(1)PQ3(!CCSD)]TJ /F4 11.955 Tf 13.52 2.66 Td[(H(0)Q3Q3))]TJ /F9 7.97 Tf 6.59 0 Td[(1H(1)Q3P^RPj0i)]TJ /F5 11.955 Tf 19.26 0 Td[(r0h0j^Lk3Hj0i(4)SubstitutionwiththeaforementionedHamiltonianand^RQ3(disconnected)expressionyield!(2)=h0j^LPH(1)PQ3(!CCSD)]TJ /F4 11.955 Tf 13.52 2.65 Td[(H(0)Q3Q3))]TJ /F9 7.97 Tf 6.59 0 Td[(1H(1)Q3P^RPj0i)]TJ /F5 11.955 Tf 19.26 0 Td[(r0h0j^Lk3Hj0i (4)=h0j^LPW(1)PQ3D3W(1)Q3P^RPj0i)]TJ /F5 11.955 Tf 19.26 0 Td[(r0h0j^Lk3Hj0i (4)=h0j^LPW(1)PQ3^RQ3j0i)]TJ /F5 11.955 Tf 19.27 .01 Td[(r0h0j^Lk3Hj0i (4)=h0j^LPW(1)PQ3(W(1)^R1k)C+(W(1)^R2k)Cj0i (4)whichisidenticaltoEquation( 4 ). 115

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Thismethodclearlycouplestripleswithsinglesviathe(W(1)^R1k)Ccontribution,whichisassumedtobeessentialfordominantsinglyexcitedstates.[ 31 32 156 ]However,thisassumesthehigherexcitedQ-spaces,specicallytheQ4contributiontobenegligible.ThisseemstobeareasonableassumptionforsmallmoleculesasitapproachestheFullCIresultforthosecases.[ 31 33 156 ].However,aswillbeshownlater,thisapproximationalwaysovercorrectstheexcitationenergyinthelargetestsetused.TheQ4W(1)^R1kCj0icontributionishypothesizedtobenon-negligibleforlargersystemsduetothesimilarcontributionintheQ3-space(Q3W(1)^R1kCj0i).IncludingthisQ4contributionwouldscaleasn4on5v,makingitinfeasibleforanysystemofmoderatesize.Aconvenientapproximationistocountordersinthetwo-electronperturbation,^W,inEquation( 4 ) ^W=Xiafiafayig+1 4Xpqrshpqjjrsifpyqysrg=^fov+^V,(4)DoingthisremovestheQ3W(1)^R1kCj0iasitcannotbeconnected,andyieldstheperturbed^RQ3vectoras ^R(1)Q3j0i=D3^W(1)^R2kCj0i(4)Thisapproximation,namedEE-EOM-CCSD(T),performsverysatisfactorily,possiblybecausetheunbalancedterms(Q3butnotQ4)inEE-EOM-CCSD(~T)arenotincluded.ItcanalsobeadaptedtocomputercodeswithagroundstateCCSD(T)implementation,sincetheonlydifferenceisthe^Ramplitudesreplacethe^Tamplitudesandthe^Lamplitudesreplacethe^amplitudesintheprogrammableequations.ThereisalsonoambiguitywiththetriplescontributioninthegroundstatewiththeEE-EOM-CCSD(T)method.ThisisevidentbyemployingLowdin'spartitioningagainoftheenergyfunctional[ 161 163 ] !exact=h0j^LPH^RPj0i+h0j^LPHPQ)]TJ /F7 11.955 Tf 5.48 -9.68 Td[(!exact)]TJ /F4 11.955 Tf 13.52 2.65 Td[(HQQ)]TJ /F9 7.97 Tf 6.58 0 Td[(1HQP^RPj0i(4) 116

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withtheeigenvectorsasdenedbefore,excepttheHamiltonianispartitionedasH[0]=PHP+Q!CCSD+^foo+^fvvQ (4)H[1]=^W+[(^foo+^fvv)^T1]C+[(^foo+^fvv)^T2]C (4)H[2]=(^W^T1)C+(^W^T2)C (4)Inthisway,wecancountgeneralizedorders(indicatedwithbrackets)inaMoller-Plessetfashion,[ 158 160 164 ]using^Wasperturbationparameter.Meaning,wecanusethewellknownorder-by-orderexpansioningroundstateCCtheoryappliedtotheexcitedstate.Forthegroundstate,thelowestordercorrectionwiththispartitioning,rstshownbyStanton[ 162 163 ]andlaterbyTaube,[ 182 ]totheenergyisgivenbyE[3]=h0j^LPH[1]jQ3ihQ3jH[0]jQ3i)]TJ /F9 7.97 Tf 6.58 0 Td[(1hQ3jH[2]^RPj0i+h0j^LPH[2]jQ3ihQ3jH[0]jQ3i)]TJ /F9 7.97 Tf 6.58 0 Td[(1hQ3jH[1]^RPj0i (4)However,forexcitedstatesthelowestordercorrectionis E[2]=h0j^LPH[1]jQ3ihQ3jH[0]jQ3i)]TJ /F9 7.97 Tf 6.58 0 Td[(1hQ3jH[1]^RPj0i(4)Forthegroundstate,theleftandrightvectorsaredenedas^LP=^1+^ (4)^RP=^1 (4) 117

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andconsequently,thesecondterminEquation( 4 )disappearssincethereisnoQ3projectionfromH[1]j0i.Thisleaves,showingexplicitgeneralizedordersin^W,E=h0j^([1]1+^[1]2^W(1)jQ3iD3hQ3j(^W(1)^T[1]2)Cj0i (4)=h0j^[1]1^V(1)+^[1]2f(1)ov+(^[1]2^V(1))CjQ3iD3hQ3j(^V(1)^T[1]2)Cj0i (4)=1 36XijkXabcijk[2]abc1 abcijktabc[2]ijk (4)WhengoingfromEquation( 4 )toEquation( 4 ),werecognizethatthe^and^T2amplitudesrstariseinrst-order,andthereforetheenergycontributionisfourth-orderin^W.Forexcitedstates,weusethesecond-ordercorrectioninEquation( 4 )with^LP=^L1k+^L2k (4)^RP=r0+^R1k+^R2k, (4)toarriveatE[2]=h0j(^L1k+^L2k)H[1]jQ3iD3hQ3jH[1](r0+^R1k+^R2k)j0i (4)=h0j(^L1k+^L2k)H[1]jQ3iD3hQ3jH[1]^R2kj0i (4)=h0j^L1k^V+^L2kfov+(^L2k^V)CjQ3iD3hQ3j^W^R2kj0i (4)=1 36XijkXabclijk[1]abc1 abcijkrabc[1]ijk. (4)Withthe^L3kand^R3kamplitudesgivenexactlyasingroundstateCCSD(T),justwith^Rreplacing^Tand^Lreplacing^.WiththeperturbedHamiltonianinsertedasdenedabove,thisistheexpressionfortheEOM-CCSD(T)correction.The!CCSDkdependenceinthedenominatoriseliminatedbytheappropriatechoiceofH[0]andthetriplesinthegroundstate,^T3,donotenterwhenusingthisMller-PlessetperturbationexpansionofH,sincetheywouldrstappearinH[2].Furthermore,theexpandedspaceisonlyQ3,sincetherecanbenoconnectionsmadetoQ4atthisorder.Recognizingthatthe^LPand 118

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^RPvectorsarethemselvesrst-orderin^W,andthe^R3kand^L3kvectorsareoneorderhigher,theenergyexpressionisactuallyfourth-order.Muchakintotheexpectationvaluecalculationofproperties,itisjustiabletouse^Rykinplaceof^Lkintheaboveexpressions.Notethatthisisnottrueforalinearresponsetreatment,oraderivativeofthefunctionalwithrespecttoaperturbation.Thiswillbedetailedlater,inthegradientsection.ThisreplacementwouldsaveafactoroftwointheEE-EOM-CCSDenergyequationandinthetriplescalculation,sinceonlyQ3RwouldneedtobecomputedandnotLQ3 4.4.1ParallelizationStrategiesforEOM-CCSD(T)Thedetailsoftheimplementationofthetriplescorrectedexcitationenergyareimportant,asthemethodformallyscalesasn3on4v.ThetriplesmethodisideallysuitedforACESIII,sincethedatapassingisforarraysof,atmost,sizenon3v,andtheentiretimeofthecomputationisspentinthecontractionstep.TheexpressionforthecorrectiontotheenergyisidenticalforEOM-CCSD(~T)andEOM-CCSD(T) ![4]=Xi
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ThepermutationoperatorsP(p=qr)accountforantisymmetrizationandaredenedbytheiractiononanarbitrarytensor,X,asP(p=qr)=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(P(pq))]TJ /F5 11.955 Tf 11.96 0 Td[(P(pr))Xpqrstu (4)=Xpqrstu)]TJ /F5 11.955 Tf 11.96 0 Td[(Xqprstu)]TJ /F5 11.955 Tf 11.95 0 Td[(Xrpqstu (4)Forlargesystems,storingblocks(notetheprimedquantities)ofdimensionn0o3n0v3isnotagoodstrategyforimplementation.Instead,onestoresblocksofn0on0v3n2o,wheretheoccupiedsquaredstorageresultsfromsimpleindicesjandk.Essentially,thevectorsarecomputedandstoredasrABCIjk(withcapitalindicesreferringtoblocksofsimpleindices).Also,alltermsresultingfromtheactionofthepermutationoperatorareindependentandcanbecomputedseparately.Consequently,wecancomputeall9permutationsineachtriplesamplitudeatthesametime.ThisishowtheEOM-CCSD(T)methodwasimplemented.TheEE-EOM-CCSD(~T)amplitudeequationsareabitmoretedious.Explicity,theyaregiveninspin-orbitalform,usingtheaforementionedexpansionofH,asrabc[2]ijk=(P(i=jk)XdIIIabcdjkrdi)]TJ /F5 11.955 Tf 11.96 0 Td[(P(a=bc)XlIIIlbcijkral+XldIVlabcdijkrdlP(a=bc)P(i=jk)XldIIIlbcdjkradil+1 2XdeIIIabcdjerdeik+1 2P(b=ac)XlmIIIlbmijkraclm)+(P(a=bc)P(k=ij)XeWbcekraeij)]TJ /F5 11.955 Tf 11.96 0 Td[(P(i=jk)P(c=ab)XlWmcjkrabim)=A+Blijk[2]abc=P(a=bc)P(k=ij)XelijaeWekbc)]TJ /F5 11.955 Tf 11.95 0 Td[(P(i=jk)P(c=ab)XllimabWjkmc (4)+P(k=ij)P(c=ab)hijjjabilkc+P(k=ij)P(c=ab)lijabfkc 120

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Theconstructionoftherighthandeigenvector,althoughintimidating,isonlytwotimesmoreexpensivetocomputethanintheEE-EOM-CCSD(T)variant.Eachofthecomputationallyexpedientterms,AandB,canbeprogrammedas ^R3k=P(a=bc)P(k=ij)XeYbcekXaeij)]TJ /F5 11.955 Tf 11.95 0 Td[(P(i=jk)P(c=ab)XlYmcjkXabim(4)tomakeuseofthepre-existingcode.TermBisexactlythesameasintheEE-EOM-CCSD(T)variant,withX^R2kandthetwo-electronbareintegralsarereplacedbyHintegrals,YW2.TheAtermcanbewrittenwiththeintermediates,inspin-orbitalform,Yabci=XdWabcdrdk+P(ab)XkdWakcdrbdik)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ab)XkWkbcirak)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2XktabkiXldhcdjjklirdl)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Xkl(rabkl+P(kl)P(ab)raktbl)Wklci (4)Ykaij=)]TJ /F13 11.955 Tf 11.29 11.36 Td[(XlWklijral+P(ij)XldWklidradjl)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ab)XdWkadjrdi)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2XdtdaijXlehdejjklirel+1 2Xcd(rcdij+P(ij)P(cd)rcitdj)Wkacd (4)Xabij=tabij (4)andusethesamecodeforBandEE-EOM-CCSD(T).ThisishowtheEE-EOM-CCSD(~T)isimplementedinACESIII. 4.4.2ApproximatingtheLeftHandEigenvectorTotestthelimitsofthislefthandapproximation,thetriplescorrectionwascomputedwith^Lk=^Ryk.Resultsforthewatermoleculeinanaugmenteddouble-basisset,areshowninTable 4-2 forveroots.Thisapproximationisquitebad,andinerrorbyaboutafactorof2.Todeterminethesourceoftheerror,thetriplescontributiontotheenergywasseparatedintoonlythosecomingfrom^L1k=^Ry1k(singles)and^L2k=^Ry2k(doubles).TheseresultsarealsoshowninTable 4-2 .Thesinglescontributionapproximationisinexcellentagreementwiththeexactsinglesresult,furthersupportingthe^Lk=^Rykapproximationforone-electronproperties.Thedoublescontribution,however,differs 121

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signicantly(relativetothetotaltriplescorrection),andisalwaysoppositeinsigntothesinglescontribution.Consequently,thedifferenceofthesetwolargernumberstodeterminethetotaltriplescorrectionspropagatestheerrorfurther,resultinginthepooragreementwiththisapproximation. 4.4.3ComparisonofTriplesMethods 4.4.3.1ComputationaldetailsTheEE-EOM-CCSD(T)andEE-EOM-CCSD(~T)methodswereimplementedinthemassivelyparallelprogrampackage,ACESIII.[ 34 ]TheCFOUR[ 183 ]programpackagewasusedfortheEE-EOM-CCSDT-3calculations,andhastheadvantagetoruninthecomputationalpointgroupsymmetry.Thetestsetcomprisedofasubsetofthe121singletvalencestatesfor24moleculesusedintheCCSDR(3)performancestudiesofSaueret.al.[ 153 ],duetotheexpenseofEOM-CCSDT-3.AllgeometrieswereobtainedfromMP2/6-31G*optimizationswiththecoreelectronsfrozen.ThesubsequentEOMcalculationswereperformedwiththeTZVPbasisset[ 181 ]withcoreelectronsfrozentoallowdirectcomparisontoCCSDR(3)andCC3results.[ 153 ]AllEOM-CCSD(T)andEOM-CCSD(~T)calculationsonthetestsetwereperformedon256computecoresusingtheARSCmachine,Chugach.[ 184 ]Asaquickdemonstrationoftimings,thelargestcalculation,Naphthalene,with238basisfunctionsand24valenceelectronstookonly2.5hoursfortheEE-EOM-CCSD(T)andthesmallestcalculation,ethene,with50basisfunctionsand6valenceelectronstook6.7minutes.Morerepresentativetimingsaredemonstratedwithcytosine,guanine,andtheircorrespondingWatson-Crickbasepair;thelatterchosentodemonstratetheapplicabilityoftheprogram.Thenucleobases-cytosine,adenine,thymine,andguanine-inaug-cc-pVDZbasissets[ 106 ]weretestedoverarangeofcomputecoresstartingwith32andendingwith512todemonstratetheefciencyandscalingproperties.TheparametersforeachmoleculeareshowninTable 4-3 ,withtheblocksizes[ 34 ]beingkeptconstantacrosstheCPUrange. 122

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4.4.3.2ResultsAcorrelationplotwasconstructedforvisualdemonstrationofEOM-CCSDcomparedtoEOM-CCSDT-3andshowninFigure 4-2 .Notsurprisingly,tripleseffectscannotbeneglected.[ 153 ]Thelargestdeviationwiththiscomparisonisfrom21A1g!stateofall-E-hexatrienewithadeviationof0.72eV.Therefore,weexaminetheperturbativetriplescorrectedmethods.ThecorrelationplotsfortheEOM-CCperturbativetriplesmethodscomparedtoEOM-CCSDT-3areshowninFigure 4-3 .TheEOM-CCSD(~T),whichhasbeenshowntolowerEOM-CCSDexcitationenergiesformanystates,[ 31 32 156 ]continuestodosoforthistestset,overshootingtheEOM-CCSDT-3inessentiallyallcases,withanaveragedeviationof0.18eV,andamaximumdeviationof0.32eVinthe11A001n!excitedstateofs-triazine.WeexpectthatthecontributionoftheQ4space,whichhasbeenneglected,wouldbringtheexcitationenergyclosertothatofEOM-CCSDT-3.Although,Shiozakiet.alincorporatethequadruplescontributionforsmallmoleculesincludingCH+,C2,andH2CO,andshowthatthequadruplescontributionactuallylowerstheexcitationenergyevenmore.[ 33 ]However,thosemoleculesaresufcientlysmallforbroadconclusions,asiswitnessedinthisstudywhenEOM-CCSD(~T)isperformedwithlargermoleculesandthegroundstateistreateddifferentlybetweenthetwomethods.Thehypothesisremainsthatthefollowingsubstantialterm,presentinQ3forthismethod rabcijk=XmramIIImbcijk+XereiIIIabcejk(4)cancelswithasimilarterminQ4 rabcdijkl=XmramIVmbcdijkl+XereiIVabcdejkl(4)Infact,excludingthiscontributionintheEOM-CCSD(~T)computationyieldsresultsessentiallyidenticaltotheEOM-CCSD(T)method.Totestthisthough,meansimplementinganapproximate^R4array.Consequently,themethodwouldscale 123

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n4occupiedn5virtual,whereastheproperinclusionoffulltripleswouldonlyscaleasn3occupiedn5virtual.TheEOM-CCSD(T)methodperformsquitewellforthislargetestset;betterthanexpected,asmentionedbyShiozakiet.al[ 33 ]alludingtoearlierstudies,[ 31 32 156 ]andmostlikelyisduetothewiderangeofmoleculesinthisset.Theaverageabsolutedeviationis0.06eV,withintherangeofdesiredaccuracyforexcitedstates.Themaximumdeviation,howeveris0.15eVandisforthe31A1!stateoffuran.ComparisonwithCCSDR(3)isinsightful,asthelargestdeviationstypicallyoccurinaside-by-sidemanner.Forinstance,thelargestdeviationofCCSDR(3)isforthe21A1g!stateofall-E-hexatriene,thestatewithlargestdeviationinEOM-CCSD.ThedeviationsforCCSDR(3)andEOM-CCSD(T)are0.16eVand0.13eV,respectively.ThistrendisdemonstratedinFigure 4-6 .ThemaximumdeviationforeachsystemisplottedforbothmethodsaswellastheEOM-CCSD(T)deviationfortherootthatcorrespondstorootfromCCSDR(3)withmaximumdeviation,todemonstratethattheexcitedstaterootswithmaximumdeviationaretypicallythesameforeachmethod.TheCCSDR(3)methodperformsslightlybetter,onaverage,thoughbotharewithintherangeofaccuracynecessaryforanaccuratedescriptionofexcitedstates,butthemaximumerrorsineachtypicallycoincide.Themaximumdeviationsofonlyvemolecules,outofthe24,donotcoincide.ThetwocorrelationplotsinFigure 4-4 comparethelinearresponsetriplesmethodswithEOM-CCSDT-3.Thereislittledifferencebetweenthemethods,supportingpreviousworkofSaueret.al[ 153 ].TheiterativeN7CC3methodistooprohibitivetobeusedforlargersystems,sowechoosetofocusontheperturbativeCCSDR(3)correction.TheaverageabsolutedeviationfromEOM-CCSDT-3is0.03eVwithamaximumdeviation,asmentionedearlier,of0.16eVforthe21A1g!stateofall-E-hexatriene.Forfurtherdetail,acorrelationplotforEOM-CCSD(T)comparedtoCCSDR(3)isshowninFigure 4-5 .TheabsoluteaveragedeviationofEOM-CCSD(T) 124

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fromCCSDR(3)is0.08eVwithamaximumdeviationof0.66eVforthe21B1u!stateofall-E-octatetraene.However,theCCSDR(3)differsbyCC3forthisstateby0.85eV.TheEOM-CCSD(T)differsfromCC3byonly0.19eV.Saueret.alrecognizethatall-E-octatetraeneprovidedafewoutliers,butEOM-CCSD(T)doesnotseemtofollowthistrend.WithnostatisticallydistinguishabledifferencebetweenCCSDR(3)andEOM-CCSD(T)overthistestset,wediscussthemassivelyparallelperformanceoftheEOM-CCSD(T)method.AsthismethodiscomputationallyequivalenttothegroundstateCCSD(T)equations,[ 162 163 182 ]exceptthatthe^and^Tamplitudesarereplacedby^Lkand^Rkamplitudes,respectively,theexistinggroundstateCCSD(T)codecanaccommodatethismethodwithminorchanges.Tothisend,moreappropriatetestsystemsthatdemonstratethecapabilitiesofACESIIIwerechosenforrepresentativetimings.Theseincludethecytosine,adenine,andguaninenucleobases,aswellasthecytosine-guanineWatson-Crickbasepair.Themoredemonstrablesystem,theWatson-Crickbasepair,timingsareshowninTable 4-4 ,alongsidethenucleobasebuildingblocks.ThenalcolumnreportsthetotaltimeinhoursfortwelveEOM-CCSDsolutions,fortherightandlefteigenvectors,aswellastwelveEOM-CCSD(T)calculations.Thisisslightlymisleadingasthecalculationsarepartitionedintotheandsummations,overvirtualindices,andcomputedseparately.ThesetwocontributionstotheEOM-CCSD(T)energycanbecomputedindependently,thoughthetotaltimeforbotharereported.Also,theEOM-CCSD(T)energyforalltwelverootswerecomputedseparately,witheachonetakingonlyafractionofthattotaltime.Inessence,thenumberofrootsandthedifferentsummationscanbeparallelizedoverforsignicanttimesavings,andthe368hoursisactuallyaquiteinatedtime.Thescalingandefciencyofthecodewasperformedonthefourmentionednucleobases.Theefciencyofthetotalcalculationsforeachsystemareshownin 125

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Figure 4-7 ,andtheefciencyforjustthetriplespieceisshowninFigure 4-8 .Therstgraphineachgurecorrespondtothecompletecalculation.Cytosineandguaninebehaveasexpected-thelargerthesystem,thebettertheperformanceacrosscores-butthymineandadenineexhibitwhatappearstobesuper-scalingperformance.However,thisisanartifactofthechosenblocksizesbeinginadequateforthesmallerprocessorrange.Todemonstratethis,theaveragetimetheprocessorswaitedforblocksofdata(calledblockwaittime),tothenbeginoatingpointoperations,acrosstherange,forthetotalcalculationandthetriplespieceareshowninFigure 4-9 .Ideally,theblockwaittimeforthetotalcalculationshouldresemblethatofthetriplespiece,thelargerthemolecule,thelongerthewaitonsmallnumbersofprocessors.However,theEOM-CCSDequationstooklongertoconvergeforadenine,thusinatingtheblockwaittime.On64processors,though,anoptimumamountofthedataappearstoberesidinginmemory,yieldinglowerblockwaittimesandtheapparentsuper-scaling.Theefcienciesforthetotalcalculationandtriplespiece,subtractingouttheblockwaittime,areshowninthesecondgraphs,andexhibittheexpectedbehavior.Notethatthetriplesefciencydoesnotchange,illustratingthenatureofiterativeportionsofthecalculationcomparedtononiterativeonesandspeakstothegoodperformanceofthetriplescode.Thescalingcurvesforthetotalcalculation,withandwithoutblockwaittime,areshowninFigure 4-10 andthetriplescurvesareshowninFigure 4-11 .Asthesystemgetslarger,thescalinggetsbetter,asindicatedbythegraphthatdoesnotincludetheblockwaittime.Thisistrueforthetotalcalculationandthetriplespiece,however,againadenineisseentoperformverywellonlybecausetheblocksizesarenotoptimumfor32processorsandthenumberofEOM-CCSDiterationsislargerthanforanyothernucleobase. 4.4.4ApplicationtoNucleobasesandTheirComplexesToactuallyapplytheaforementionedapproximationsforbenchmarkdatasetsonbiologicallyrelevantmolecules,weexaminedtheexcitedstateenergiesandorderings 126

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ofthefreenucleobases,hydratedcytosine,cytidine,guanosine,andcytosine-guaninebasepair.ThegroundstateMBPT(2)/aug-cc-pvDZoptimizedgeometriesforthelargermoleculesinthisbenchmarkstudyareshowninFigures 4-12 4-13 4-14 ,and 4-15 .ThecharacterizedexcitedstatesofthehydratedcytosinestructureBwithEOM-CCSD(T)energiesareshowninTable 4-5 .Theexcitedstatesdonotchangeorderwhenmicro-hydratedwithonewatermolecule.Moreimportantly,thetriplescorrectiononthefreebaseisessentiallyunchangedwhenaddingthewatermolecule.Thisisratherniceasitimpliesthetriplescorrectioncanbecarriedovertothecomplexfromthefreenucleobase,offeringsignicanttimesavingsforlargercomplexes.TheoscillatorstrengthsoftheexcitationsasafunctionofthenumberofwatermoleculesaroundcytosineareshowninTable 4-6 usingthe^Rykapproximation.Themostintensetransition,thethirdtransition,inthefreenucleobasedecreaseswitheachaddedwatermoleculefrom0.412to0.267.Aftermicrosolvatedwithvewaters,thefourthstatebecomesthemostintensetransition,growingfrom0.180to0.557.Thethirdmostintensetransition,thesecondtransition,isrelativelyunchangedbytheadditionofwatermolecules.Anylowerlevelparameterizedmethodshouldreectthissolventeffectwhenbeingdeveloped.ThetriplescorrectionforcytidineremainsessentiallythesameasthetriplescorrectiononcytosineasshowninTable 4-7 ,againsuggestingthatatriplescorrectionappliedtothefreebaseissufcientforchemicalaccuracy.Therearemoredrasticchangestothetransitionintensitieswhenaddingthesugar.Thebrighteststate,thethirdtransition,isreducedfrom0.412to0.191,similartotheeffectofhydration.Fourdarkstates,however,becomemuchmoreintenseaftertheadditionofthesugar.Therstincreasesfrom0.049to0.129,thesecondRincreasesfrom0.003to0.115withanncomponentresponsiblefortheintensity.ThethirdandfourthR,withacomponentresponsibleforthetransition,increasesbothfrom0.0to0.107and0.109,respectively. 127

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Guanineremainslargelyunaffectedbytheadditionofthesugar.ResultsareshowninTable 4-8 .Thetriplescorrectiononthefreebaseandthesugar,again,remainsunchanged.Therearealsoonlyminorchangestotheoscillatorstrengthsinguanosinecomparedtoguanine.Thecytosine-guanineWatson-CrickbasepairresultsareshowninTable 4-9 .Theexcitedstatesarestabilizedinthiscomplexasthetwolowestexcitedstatesofthefreebases,cytosinerstwithanexcitationenergyof4.74eVandguaninesecondat4.93eV,changeordering.Thelowestexcitationonthebasepairisaonguanine,withanenergyof4.67eV,followedbyaexcitationlocalizedoncytosine,withanenergyof4.86eV.Theoscillatorstrengthsremainunchanged,however.Thischangeinorderingreectstheneedforbenchmarkdatasetsdescribingthesesystems.Theaforementionedpreviousstudiesdonotreproducethisorderingintheirstudy.Acharge-transferstateexists.Itisthefthstate.Itwouldbeveryinterestingtocharacterizethepotentialenergysurfaceofthisexcitedstateasitwouldhelpelucidatethecomplicatedrelaxationmechanismsthesecomplexesundergo. 4.5GradientTheoryforAccurateCriticalPointGeometriesRecallthattheEOM-CCenergyfunctionalisgivenasEk=h0jLkHRkj0i (4)=h0jLke)]TJ /F9 7.97 Tf 7.96 1.77 Td[(^T^He^TCRkj0i (4) 128

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Thechangeinelectronicenergywithrespecttoaperturbation,,suchasthechangeinnuclearcoordinates,isthefollowingenergyderivative@Ek @=h0j@Lk @HRkj0i+h0jLkH@R @j0i)-221(h0jLk@^T @HRkj0i+h0jLkH@^T @Rkj0i+h0jLk e)]TJ /F9 7.97 Tf 7.97 1.77 Td[(^T@^H @e^T!CRkj0i=A+B+C+D+E. (4)Thersttwoterms(AandB)vanishasaconsequenceofthegeneralizedHellman-Feynmantheorem.Thethirdandfourthtermsremainsincethe^Tamplitudesarenotstationarywithrespecttoaperturbationontheexcitedstateandwouldrequirethesolutionfor3Ndegreesoffreedomforgeometricalperturbations.However,insertingaresolutionoftheidentityyieldsC+D=h0jLk@^T @(P+Q)HRkj0i+h0jLkH(P+Q)@^T @Rkj0i (4)=!k h0jLk@^T @PRkj0i+h0jLkP@^T @Rkj0i!)-221(h0jLk@^T @QHRkj0i+h0jLkHQ@^T @Rkj0i (4)=h0jLk@^T @QHRkj0i+h0jLkHQ@^T @Rkj0i (4)=h0jLkHQ@^T @Rkj0i. (4)TheP-spaceisdenedasbefore P=j0ih0j+Xiajaiihaij+1 4Xijabjabijihabijj(4) 129

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andtheQ-spacecontainsallhigherexciteddeterminants.Thefollowingdenitionswereused!k=h0jLkHRkj0i=h0jLkPHPRkj0i (4)Rk=PRk (4)Lk=LkP (4)0=h0jLk@^T @Q=h0jLkQ0=h0jLkQ (4)wherethelastequalityholdsbecause@^T=@isstrictlyanexcitationoperator,sowhenactingontheQ-space,onlyQ-spacedeterminantscanbegenerated,which^LPcannotconnectto.Unfortunately,inthepresentform,Equation( 4 )stillrequiresthesolutionof@^T=@forall3Ndegreesoffreedom.However,wecanusethedenitionofRkandthefactthatitcommuteswith^Tandderivativesof^T,toisolatethederivativeofthe^Tamplitudesh0jLkHQ@^T @Rkj0i=h0jLkHQRk@^T @j0i (4)=h0jLkHQRkP@^T @j0i (4)=h0jP@^T @j0i (4)wheretheshorthandnotationwithh0jP=h0jLkHQRkPwasused,followingthederivationofStantonet.al.[ 185 ]Beforegivingtheexplicitspin-orbitalprogrammableequationsforEquation( 4 ),weexaminethederivativeofthegroundstateCCequationstodetermineaperturbationindependentformforthederivative^Tamplitudes.ItwillbeconvenienttorecasttheP-spaceas P=j0ih0j+jgihgj,(4) 130

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withjgihgjcontainingthespacespannedbythesinglyanddoubleexciteddeterminants.ThederivativeofthegroundstateCCSDamplitudeequationsarethen0=@ @)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(hgjHj0i (4)=hgjH@^T @j0i)-222(hgj@^T @Hj0i+hgj e)]TJ /F9 7.97 Tf 7.97 1.77 Td[(^T@^H @e^T!Cj0i (4)=hgjH(P+Q)@^T @j0i)-223(hgj@^T @(P+Q)Hj0i+hgjHj0i (4)=hgjHjgihgj@^T @j0i)]TJ /F4 11.955 Tf 19.26 0 Td[(ECChgj@^T @j0i+hgjHj0i (4)=hgjH)]TJ /F4 11.955 Tf 11.96 0 Td[(ECCjgihgj@^T @j0i+hgjHj0i (4)=hgjHNjgihgj@^T @j0i+hgjHj0i (4)Thederivative^Tamplitudesaretheneasilyseentobegivenby hgj@^T @j0i=hgjHNjgi)]TJ /F9 7.97 Tf 6.59 0 Td[(1hgjHj0i(4)Itshouldbenotedthattoarriveatthisexpression,thelinear^Toperators,denedtogeneratesingleanddoubleexcitationsonly,cannotconnectthereferencedeterminanttothetriplyorhigherexciteddeterminantsintheQ-space.Consequently,allQ@^T=@j0itermsarezero.Also,withtheP-spacedenedinEquation( 4 ),thefollowingrelationsholdhgjHP=hgjHj0i+hgjHjgi=0+hgjHjgi (4)PHj0i=h0jHj0i+hgjHj0i=ECC+0. (4)Theshorthandnotation,He)]TJ /F9 7.97 Tf 7.97 1.77 Td[(^T@^H @e^TC,wasusedforthenaltermtoreduceclutter. 131

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InsertionofEquation( 4 )intoEquation( 4 )yieldsh0jLkHQRk@^T @j0i=h0jjgihgj@^T @j0i (4)=h0jjgihgjHNjgi)]TJ /F9 7.97 Tf 6.58 0 Td[(1hgjHj0i (4)RatherthaninverttheverylargehgjHNjgimatrix,wedenetheperturbationindependentoperatorZas h0jZjgi=h0jjgihgjHNjgi)]TJ /F9 7.97 Tf 6.59 0 Td[(1(4)andusestandardlinearalgebratechniquestodetermineZ.Thegradientexpressionisthen,inclosedanalyticalform,@Ek @=h0jZHj0i+h0jLkHRkj0i (4)=Tr[(Z)@f @]+Tr[(Lk,Rk)@f @]+Tr[)]TJ /F4 11.955 Tf 6.94 0 Td[((Z)@V @]+Tr[)]TJ /F4 11.955 Tf 6.94 0 Td[((Lk,Rk)@V @] (4)=XpqDpq@fqp @+1 4XpqrsDrspq@hpqjjrsi @. (4)Thisformisconvenient,asonecanexploitpre-existinggroundstateCCgradientprogramsgiventheone-electronandtwo-electrondensitymatrices.Insteadofthegroundstatetwo-particledensitymatrix )]TJ /F6 7.97 Tf 6.78 4.94 Td[(pqrs=h0j(1+^)fpyqyrsge^TCj0i,(4)the^arrayisreplacedbyZ+r0^Lkforexcitedstategradients,allowingtheuseofthegroundstatecode.ThegroundstateCCgradientcodeinACESIIIhasagreattrackrecordalready.Itscalesverywellandcanhandlelargesystemsinthemassivelyparallelenvironmentenablingthedeterminationofcriticalpointsonpotentialenergysurfacesoflargermolecules.However,EE-EOM-CCSDgradientshavenotbeenimplementedinarobustforminanyprogramcapableofscalingtoO(10,000)toO(100,000)processors.Therefore,giventhesuccessofthegroundstategradientcode, 132

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Iwillfocusonlyonthemassivelyparallelimplementationofthepresentedequations;theformationofandtheone-andtwo-particledensitymatrices.Theuseof^Ryinplaceof^LisnotanoptioninEE-EOM-CCSDgradients.Thersttwoterms,AandB,inthederivativeoftheenergyfunctional(Equation( 4 ))arezerobecauseofthefollowingI+II=h0j@Lk @HRkj0i+h0jLkH@Rk @j0i (4)=h0j@Lk @(P+Q)HRkj0i+h0jLkH(P+Q)@Rk @j0i (4)=!kh0j@Lk @Rkj0i+h0jLk@R @j0i (4)=!k@ @[h0jLkRkj0i] (4)=!k@ @[1] (4)=0 (4)Thissimplicationisnotpossibleifusing^Rybecauseh0j^Ry!k6=h0j^RyHduetothenon-hermiticityofH. 4.5.1ImplementationDetailsfortheone-particledensitymatrixaregiveninsection 4.3 .Theconstructionofandthetwo-particledensitymatrixusemanyofthesameintermediates.Withtheprimarygoalofparallelperformance,theseintermediateswereformedinanindependentstepandstoredondiskormemory,whereappropriate,ratherthancomputedon-the-yandusedimmediately.Remember,thelattercanhurtparallelperformance.Thechoiceofintermediatesisnotcompletelyunambiguous.GiventhedesignofACESIII,thefollowingintermediatesspecictotheexcitedstatewereconstructed,in 133

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order,as(L2R2)aibj=Xkcraciklkjcb (4)(L2T2)aibj=Xkctaciklkjcb (4)(WR2)bijk=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2XcdWbicdrcdjk (4)(WR1)ba=XkcWbkacrck (4)(L2R2)ijkl=Xcdlijcdrcdkl (4)(L2T2)ijkl=Xcdlijcdtcdkl (4)(WR1)ai=XkcWikacrck (4)(WR1)ij=XkcWikjcrck (4)(L2R1)ia=Xkclikacrck (4)(L1R2)ai=Xkcraciklkc (4)(WR2)ai=1 2XkcdWakcdrcdik)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2XklcWklicrackl (4)(L2R2)ba=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(1 2Xklclklacrbckl (4)(WR2)ba=)]TJ /F4 11.955 Tf 10.49 8.08 Td[(1 2XklcWklacrbckl (4)(L2R2)ij=1 2Xkcdlikcdrcdjk (4)(WR2)ij=1 2XkcdWikcdrcdjk (4)Notethatnointermediatesofsizenon3vareformed.Thelargestintermediatesareofsizen2on2v.Thesearesmallerandcanconsequentlybepassedbetweenprocessorsfaster.Also,alltwo-indexintermediateswerestoredinlocalarraysinmemoryforfasteraccess 134

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tothesequantities.Giventheseintermediates,useddirectlyintheformationofandthetwo-particledensitymatrix,moreintermediatesforthelatterwereconstructedasXij=Xc(L2R1)ictcj+(L2R2)ij+Xclicrcj (4)Xba=Xk(L2R1)katbk+(L2R2)ba+Xklkarbk (4)Xai=(L1R2)ai)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2XlcdrcdilXklklcdtak)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2XlcdtcdilXklklcdrak)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xkcraklkctci)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xkctaklkcrci+1 2XkldradklXclklcdtci+1 2XkldtadklXclklcdrci (4)Yij=1 2Xkcdlikcdtcdjk+Xclictcj (4)Yba=Xklkatbk)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Xklclklactbckl (4)Yai=Xkctaciklkc)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2XlcdtcdilXklklcdtak)]TJ /F13 11.955 Tf 11.95 11.35 Td[(Xkctaklkctci+1 2XkldtadklXclklcdtci (4)~rabij=rabij+raitbj+tairbj (4)abij=tabij+P(ij)P(ab)taitbj (4)(L2)klij=Xcdlklcdcdij (4)(L2~r)klij=Xcdlklcd~rcdij (4)Allofthetwo-indexintermediatesarestoredlocallyoneachprocessor,andweavoidthestorage,again,ofnon3vorlargerfour-indexquantities.Then4oarraysarestoreddistributedinmemoryovertheprocessors. 135

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Havingdenedtheintermediates,theamplitudesaregivenasia=Xkcd(L2R2)idckWckad)]TJ /F13 11.955 Tf 11.95 11.36 Td[(XcdWcida(L2R2)dc)]TJ /F13 11.955 Tf 11.96 11.36 Td[(XklcWiklc(L2R2)lcak+1 2Xklclklac(R2W)ickl+Xkclikac(R2W2)ck+XkcWacik(R2L1)ik+Xkcfck(L2R2)ikac+1 4XklmWklma(R2L2)mikl)]TJ /F13 11.955 Tf 11.96 11.36 Td[(XklWlika(R2L2)kl+Xclic(R2W)ca+Xcfic(R2L2)ca)]TJ /F13 11.955 Tf 11.96 11.36 Td[(Xklka(R2W)ik)]TJ /F13 11.955 Tf 11.96 11.35 Td[(Xkfka(R2L2)ik+lia+fia (4)ijab=P(ij)P(ab)XbcWkjcb(L2R2)ikac+P(ij)XcWcjab(L2R1)icP(ij)P(ab)Xkclikac XdrdkWcjfb+XlrclWljkb!)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ij)XkWkjab(R2L2)ik)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ij)Xklkjab(R2W)ik)]TJ /F5 11.955 Tf 11.95 0 Td[(P(ab)XcWijcb(R2L2)ca)]TJ /F5 11.955 Tf 11.95 0 Td[(P(ab)Xclijcb(R2W)ca+P(ab)Xclijac(R1W)cb)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ij)Xklikab(R1W)jk+Wijab+P(ij)P(ab)lia(R1W)jb+P(ij)P(ab)fia(R1L2)jb+lijab Xkcrckfkc+1 4XklcdrcdklWklcd!)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ab)XkWijkb(R1L2)ka+1 4Xkllklab(R2W)ijkl+1 4XklWklab(R2L2)ijkl)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ab)XkclkarckWijcb)]TJ /F5 11.955 Tf 11.95 0 Td[(P(ab)Xkclijcbrckfka)]TJ /F5 11.955 Tf 11.95 0 Td[(P(ij)XkclicrckWkjab)]TJ /F5 11.955 Tf 11.96 0 Td[(P(ij)Xkclkjabrckfic)]TJ /F13 11.955 Tf 11.96 11.36 Td[(XkcdlijcdrckWkdab)]TJ /F13 11.955 Tf 11.95 11.36 Td[(XklclklabrckWijcl (4)Intheequationsfor,theorderofthetermsisimportant,andshedslightontheparallelimplementationandperformance.Forexample,therstterminijabcanbecomputed,parallelizedoverfa,b,c,kg,andthenaltermisparallelizedoverfi,k,lg.Thedesignphilosophywastoimplementthefatloopsrstanddescendtotheskinnyloopsinthecode.Inthisway,themostworkisdistributedrst,andsoon,tooptimizethedataow. 136

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Thesamephilosophyisusedintheconstructionofthetwo-particledensitymatrix.Theindependentblocksofthedensitymatrixisgivenas)]TJ /F6 7.97 Tf 6.77 4.94 Td[(klij=1 2Xcd~rcdijlklcd (4))]TJ /F6 7.97 Tf 6.77 4.94 Td[(abcd=1 2Xkl~rabkllklcd (4))]TJ /F6 7.97 Tf 6.77 5.45 Td[(ijka=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2Xclijcarck (4))]TJ /F6 7.97 Tf 6.78 4.93 Td[(kaij=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 2Xc~rcaijlkc)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Xc(L2R1)kccaij+1 2Xl(L2~r)klijtal+1 2Xl(L2)klijral1 2P(ij))]TJ /F4 11.955 Tf 5.48 -9.68 Td[((R2L2)kitaj+(L2T2)kiraj)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2P(ij)Xd)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(tdi(L2R2)kadj+rdi(L2T2)kadj (4))]TJ /F6 7.97 Tf 6.77 4.93 Td[(ciab=1 2Xklkiabrck (4))]TJ /F6 7.97 Tf 6.77 4.93 Td[(abci=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(1 2Xk~rabkilkc)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Xkldlklcdtdi~rabkl)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2Xkldlklcdrdiabkl)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2P(ab))]TJ /F4 11.955 Tf 5.48 -9.68 Td[((L2R2)actbi+(L2T2)acrbi+1 2P(ab)Xktak(L2R2)kbci+1 2P(ab)Xkrak(L2T2)kbci+1 2Xk(L2R2)kcabki (4))]TJ /F6 7.97 Tf 6.78 5.45 Td[(jabi=ljbrai+(L2R2)jbrai+(L2R2)jabi+Xkcljkbc(rcitak+tcirak) (4) 137

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)]TJ /F6 7.97 Tf 6.77 4.94 Td[(abij=abij+P(ij)P(ab))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Xaitbj+Yairbj+Xkc(tacik)]TJ /F5 11.955 Tf 11.96 0 Td[(taktci)(L2R1)kctbj)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2P(ij)Xk)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(tabkjXki+rabkjYki)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2P(ab)Xc)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(tcbijXac+rcbijYac+1 4Xkl)]TJ /F4 11.955 Tf 5.48 -9.68 Td[((L2~r)klijabkl+(L2)klij~rabkl+P(ij)P(ab)Xkc)]TJ /F4 11.955 Tf 5.48 -9.68 Td[((L2R2)acik(tcbkj)]TJ /F4 11.955 Tf 11.96 0 Td[(2tcjtbk)+(L2T2)acik(rcbkj)]TJ /F4 11.955 Tf 11.96 0 Td[(2(rcjtbk+tcjrbk)) (4)Thestructureofthecodethenbecomes,separatingtheindependentstepsinanefcientway, COMPUTEINTERMEDIATES EXECUTEBARRIER COMPUTE EXECUTEBARRIER SOLVEFORZ(=ZH) EXECUTEBARRIER COMPUTE)]TJ /F1 11.955 Tf 14.08 0 Td[(INTERMEDIATES EXECUTEBARRIER COMPUTE)]TJ ET BT /F1 11.955 Tf 42.26 -526.37 Td[(EXECUTEBARRIER USEEXISTINGGROUNDSTATEGRADIENTCODE DONE 138

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4.5.2DemonstrablePerformanceTotesttheperformanceofthecode,scalingcurvesweregeneratedforgradientcalculationsonadenine,thymineandguanineinanaugmenteddouble-basisseton256and512processors.TheseresultsareshowninFigures 4-16 4-18 ,and 4-19 .Eachgradientcalculationisafactorof1.5fasterwithinthisrangeprocessorcount.Therangewasextendedforcytosineinanaugmenteddouble-basissetto512and1024processors.Again,thereisafactorof1.5speedincreasewithinthisrange.Mostnotably,thegradientcodewasrunon1024and2048processorsforthestackedcytosine-guaninecomplex.Thissystemhas29atoms,136electrons(noneweredroppedfromthecorrelatedcalculation),and527basisfunctions.Thissystemalsoexperiencedaspeedupof1.5timeswhendoublingtheprocessorcount.Theconsistencyamongsystemsandprocessorcountisveryencouraging,sincethereisnodegradationofperformanceasthesystemandnumberofprocessorsgetslarger. 4.5.3ExcitedStateGeometriesforNucleobasesMethodstovisualizetheorbitalsinvolvedintheexcitationprocessaswellasthechangeindensitybetweenthegroundandexcitedstatesarecriticalforadeeperunderstandingofthechemistry.Twomethodsweemploytodeciphertheorbitalsinvolvedinthetransitionare 1. Largeraiweightsforspecicaandiorbitalsareausefulmetric. 2. Naturalorbitalsofthedensitydifference.Therstcanbecomeambiguousifthereareafewdominantraielements.Thesecondisunambiguousinassigningthecharacterofthestatessincethenaturalorbitalsobtainedfromthedensitydifferencecontainalargepositiveoccupationinthegroundstateorbitalandalargenegativeoccupationintheexcitedstateorbital.Therefore,thenaturalorbitalsweregeneratedforthegroundandexcitedstates,aswellasthedensitydifferences,andwrittentodiskinaMolden[ 180 ]readableformat.However,ACESIII(andACESII)didnothavethiscapability,anditconsequentlywasadded. 139

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Todemonstratethepracticalityofthisdensitydifferenceapproach,thedifferenceinelectrondensitiesforcytosine,thymine,guanine,andthecytosine-guaninestackareshowninFigures 4-23 4-25 4-21 ,and 4-27 ,respectively.Therightsideoftheguresindicatetheregionofspacetheelectronoccupiedinthegroundstate,theleftsideindicatestheregionofspacetheelectronoccupiesintheexcitedstate.Thegeometriesareallgroundstatestructuresandallexcitedstatesarethelowesttransitions.Theseguresareshownsimplytodemonstratethetypesofsystemsthatcanberoutinelystudiedinablack-boxmannerbythecommunity. 4.6ConclusionsandOutlookAccuratestudiesofexcitedstatesrequirestheequationofmotioncoupledclustermethod.Increasingtheapplicabilityofthismethodrequiresapproximationsthatresultinnegligiblelossofaccuracyandtheefcientmassiveparallelizationofdataacrosscomputecores.Inthisvein,foraccurateoscillatorstrengthsusedindecipheringUVspectra,itissufcienttocomputeonlytherighthandeigenvectorandusethattocomputerst-orderproperties,includingtransitionmomentsandoscillatorstrengths.Thenecessaryintegralshavebeenimplementedinablockformthatallowsdistributionofbasisfunctionsoverprocessorstoallowmassivelyparallelcalculationswiththisapproximation.Thereisessentiallynolossofaccuracywiththismethod,thussavingafactorof2incomputertimeandresources.Furthermore,triplesubstitutionsarenecessaryforaccurateexcitationenergies.Thenon-iterativeEOM-CCSD(T)withnoomegashiftwastestedagainsttheiterativebenchmarkshortofthefulltriples,EOM-CCDST-3,withexcellentperformance.ThemoreformallyjustiedEOM-CCSD(~T)methodperformsworse,however,thisisattributedtoalackofconsistencywithouttheinclusionoftheQ4spaceintheperturbation.Regardless,theEOM-CCSD(T)methodwasimplementedinamassivelyparallelwayintheACESIIIprogrampackagewithveryencouragingscalingresults.The 140

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sizesofthesystemsstudiedspeakforthemselvesconcerningtheperformanceoftheprogram.Extremelyaccurateoscillatorstrengthsandexcitationenergieshavebeenpresentedfornucleobasesandnucleobasecomplexes.ThesedatasetscanservetoparameterizelowerlevelmethodstoenabletheaccuratecalculationofDNAandRNAstrands.Onepossiblescheme,demonstratedhere,isforexcitationslocalizedonnucleobases,thetriplescorrectionsonthefreenucleobaseissufcienttoaccuratelydescribetheenergeticsofthecomplex;providedatleasttheEOM-CCSDlevelisperformedonthecomplex.GradientsfortheEOM-CCSDmethodhavebeenimplementedinthemassivelyparallelizableACESIIIprogrampackageenablingaveryaccuratecharacterizationofpotentialenergysurfacesoflargemolecules.Specically,excitedstategeometrieshavebeenreportedfornucleobasesandthecytosine-guaninestackedstructure.Theprogramsscaleswell,allowingtheuseofO(10,000)processorstostudyevenlargerchemicallyrelevantmolecules.Adiabaticexcitationsarealittlemorecomplicatedtoobtain,asitrequiresthecalculationofthehessianmatrix.Forthelargestsystemstudiedhere,thehessiancanbecomputednumericallybyrunninggradientcalculationson3(29))]TJ /F4 11.955 Tf 12.42 0 Td[(6=81differentpoints.Unfortunately,thisisatremendouscomputationaltask.Eachgradientcalculationon1024processorstakesroughly20hours.Eachgradientpointcanbecomputedindependentlythough,buttheavailablecomputationalresourceswillstilllimitthesehessiancalculations.Although,thefoundationislaidforextensionstogradientsformulti-ionized/attachedEOM-CCSDmethods[ 30 ]enablinganefcientroutetowardsroutinemulti-referencestudies.Greatcarewastakentoproducearobusttransparentcodewiththegoaloftransferabilitytotheseextendedmethods. 141

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Figure4-1. Errorinoscillatorstrengthusing^Lk=^Rykforcomputingthedensityinatomicunits. Figure4-2. CorrelationplotforcalculatedsingleexcitedstateswithCCSDvs.CCSDT-3. A B Figure4-3. CorrelationplotforcalculatedsingleexcitedstateswithEOM-CCmethodsvs.CCSDT-3.A)correspondstoEOM-CCSD(T),andB)correspondstoEOM-CCSD(~T). 142

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A B Figure4-4. CorrelationplotforcalculatedsingleexcitedstateswiththelinearresponseCCmethodsvs.CCSDT-3.A)correspondstoCCSDR(3),B)correspondstoCC3. Figure4-5. CorrelationplotforcalculatedsingleexcitedstateswithCCSD(T)vs.CCSDR(3). Figure4-6. Maximumdeviations(errors)fromEOM-CCSDT-3foreachsystem,orderedbyincreasingbasissetsize,usingtheEOM-CCSD(T)andCCSDR(3)methods.Fordetailedcomparisonbetweenthetwomethods,theerrorintherootobtainedwithEOM-CCSD(T)correspondingtotherootwithmaximumerrorusingCCSDR(3)isalsoplotted. 143

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A B Figure4-7. EfciencyplotsforthecompleteEOM-CCSD(T)calculationforvariousnucleobasesinanaug-cc-pVDZbasissetover32-512rangeofCPUs.A)isthetotaltime,B)doesnotincludetheblockwaittime,thetimewaitingfordatatoarriveatacomputecore.Seetextfordetailsontheover-efciencyofadenineandthymine. A B Figure4-8. EfciencyplotsforthetriplespieceoftheEOM-CCSD(T)calculationforvariousnucleobasesinanaug-cc-pVDZbasissetover32-512rangeofCPUs.A)isthetotaltime,B)doesnotincludetheblockwaittime,thetimewaitingfordatatoarriveatacomputecore. A B Figure4-9. BlockwaittimesforeachprocessorcountintheEOM-CCSD(T)calculations.A)isforthetotalcalculation,B)isforthetriplespieceonly.Sinceadenineblockwaiton32processorsissolarge,theinsetshowstherangefrom64to512processors. 144

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A B Figure4-10. Alog-logplotdepictingthescalingbehaviorofthetotalEOM-CCSD(T)calculationforvariousnucleobasesinanaug-cc-pVDZbasisset.A)isthetotaltime,B)doesnotincludetheblockwaittime,thetimewaitingfordatatoarriveatacomputecore.Notethatthex-axisdisplaysthenumberofprocessors,butisactuallythelogofthatnumbertohaveequaldistancesbetweenthedifferentprocessorcounts.They-axisforeachnucleobasewasshiftedforclarity. A B Figure4-11. Log-logplotsdepictingthescalingbehaviorofthetriplespieceoftheEOM-CCSD(T)calculationforvariousnucleobasesinanaug-cc-pVDZbasisset.A)isthetotaltime,B)doesnotincludetheblockwaittime,thetimewaitingfordatatoarriveatacomputecore.Notethatthex-axisdisplaysthenumberofprocessors,butisactuallythelogofthatnumbertohaveequaldistancesbetweenthedifferentprocessorcounts.They-axisforeachnucleobasewasshiftedforclarity. 145

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A B AA BB AB ABB AABB AABBC Figure4-12. StructureofthemicrohydratedcytosineclustersoptimizedattheMP2/aug-cc-pVDZbasis.ThestructuresareseparatedbythenumberofwaterspositionedatsitesA,B,orC. 146

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Figure4-13. MP2/aug-cc-pVDZoptimizedstructureofcytidineusedinthisstudy. Figure4-14. MP2/aug-cc-pVDZoptimizedstructureofguanosineusedinthisstudy Figure4-15. MP2/aug-cc-pVDZoptimizedstructureofguanine-cytosineWatson-Crickpairusedinthisstudy 147

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Figure4-16. EOM-CCSDgradientscalingcurveforAdenineupto512processors. Figure4-17. EOM-CCSDgradientscalingcurveforCytosineupto1024processors. Figure4-18. EOM-CCSDgradientscalingcurveforThymineupto512processors. 148

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Figure4-19. EOM-CCSDgradientscalingcurveforGuanineupto512processors. Figure4-20. EOM-CCSDgradientscalingcurveforCytosineGuaninestackupto2048processors. 149

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Figure4-21. DensitydifferencesforthelowestexcitationusingEOM-CCSDinanaug-cc-pVDZbasissetonguanine. Figure4-22. Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforguanine Figure4-23. DensitydifferencesforthelowestexcitationusingEOM-CCSDinanaug-cc-pVDZbasissetoncytosine. 150

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Figure4-24. Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforcytosine Figure4-25. DensitydifferencesforthelowestexcitationusingEOM-CCSDinanaug-cc-pVDZbasissetonthymine. Figure4-26. Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforthymine 151

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Figure4-27. DensitydifferencesforthelowestexcitationusingEOM-CCSDinanaug-cc-pVDZbasissetonstackedcytosine-guanine. Figure4-28. Comparisonofexcitedvs.groundstategeometriesinanaug-cc-pVDZbasissetforstackedcytosine-guanine. 152

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Table4-1. ComputedoscillatorstrengthsaattheEE-EOM-CCSD/aug-cc-pVTZleveloftheorywithvariousapproximations. MoleculeExc.energy(eV)EOMosc.str.bApprox.osc.str.cAbsolutedifference Carbonmonoxide8.420.07450.07260.001913.450.04110.04680.005611.990.05910.06280.003711.250.01170.01670.005011.790.21640.22240.0060Hydrazine5.370.00440.00450.00026.610.00070.00080.00016.080.03340.03430.00097.040.03660.03850.00208.100.02710.02800.00108.500.00840.00840.00008.660.00210.00190.00028.830.00250.00250.00008.890.04490.04580.0009Ozone8.480.00160.00140.00025.060.13510.10710.02808.840.03720.03190.005310.300.00950.01030.00088.980.00010.00000.00018.510.00150.00300.00149.120.02520.02560.0004Skewammonia6.210.07250.07400.00157.800.01690.01680.00018.010.00650.00650.00009.190.00390.00420.00039.240.00930.00920.00019.630.01010.01010.00009.970.01160.01190.000210.500.00210.00250.000410.650.00480.00490.0001Water7.540.05230.05440.002110.780.00270.00290.000112.870.02650.02730.00089.920.10000.10230.002311.340.00100.00110.000013.240.00010.00010.000011.660.01880.01900.0002Averageerror0.0019 a Onlyoscillatorstrengthsgreaterthan10)]TJ /F18 5.978 Tf 5.76 0 Td[(4wereusedincomparison. b OscillatorstrengthcomputedusingLkandforthelefttransitiondensity. c OscillatorstrengthcomputedusingRykandTyforthelefttransitiondensity. 153

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Table4-2. ComparisonofusingRyforLintriplescorrectionsforthewatermoleculeinanaug-cc-pVDZbasisset.AllvaluesareineV.Seetextfordetails. Correction!LkLkLkRTkLkL1kLkRT1kLkL2kLkRT2k 6.640.070.120.440.44-0.36-0.328.420.070.120.430.44-0.36-0.329.080.070.110.400.40-0.33-0.2910.600.090.140.450.45-0.36-0.3110.960.070.120.430.44-0.36-0.32 Table4-3. ParametersforeachnucleobaseusedtodemonstrateperformanceofEOM-CCSD(T)calculations.DetailsonblocksizescanbefoundinRef.[ 34 ] BlocksizesMoleculeElectronsaBasisfunctionsAtomicorbitalOccupiedVirtual Cytosine21229332130Thymine24261331733Adenine25275351830Guanine28298332033 a Thenumberofvalenceelectronsarereported,sincethecoreelectronswerefrozen. Table4-4. ParametersandtimingsoftheACESIIIcalculationsoncytosineandguanineandthecorrespondingWatson-Crickbasepair.Allcalculationsemploytheaug-cc-pVDZbasisset. NumberofElapsedtime(sec)TotaltimecSystematomsval.els.bf.CPUsCCSDEOMa(T)binhours Cytosine13422293251536945843643232179308012821012921722256155900135845121417601161Guanine1656298322157211782590064122610272138171286684654769325641727434771512276173032315.2Cytosine-Guanine299852751243101236098000368 a TimingsforconstructionofHandboththeleftandrightEOMsolutionsforoneexcitedstate b Timingsforoneexcitedstatesolution. c TimingsforallstepsincludingEOM-CCSD(T)considering12excitedstatesolutions 154

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Table4-5. Changeoftheexcitationenergies(eV)ofcytosinebyinteractionwithwater(structureB)calculatedattheEOMEE-CCSDandEOMEE-CCSD(T)levelsa transitioncytosinecytosine-water(B)btypeassignmentCCSDCCSD(T)CCSDCCSD(T)c 1()!4.944.74-0.06-0.070.011(n)nN!5.465.250.200.180.021(R)!R5.565.490.080.070.012())]TJ /F9 7.97 Tf 6.59 0 Td[(1!5.865.62-0.17-0.170.002(R))]TJ /F9 7.97 Tf 6.59 0 Td[(1!R6.04d5.910.00-0.030.032(n)nO!26.06d5.960.120.15-0.033(R)!R6.196.080.180.180.003(n)nO!6.345.90-0.05-0.060.013()!26.506.35-0.03-0.030.004(R)!R6.516.430.100.100.001(nR)nO,nN!R6.706.570.100.11-0.014())]TJ /F9 7.97 Tf 6.59 0 Td[(1!26.886.69-0.11-0.13-0.02 aMP2/aug-cc-pVDZgeometry,aug-cc-pVDZbasis.bChangeofexcitationenergywithrespecttoisolatedcytosine.cDifferenceofthetripleshiftbetweenmonomerandhydratedmolecule.dThesetwostatesarestronglymixed. 155

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Table4-6. Excitationenergies(eV),oscillatorstrengths(a.u.)andtransitiontypesaofcytosineanditsvariouswatercomplexes.EOM-CCSD/aug-cc-pVDZresultsb typecytosinecytosine+waterABAABBABABBAABBAABBC 1()4.940.0494.850.0654.880.0664.880.0794.930.0684.930.0774.980.0865.000.0925.090.1041(n)5.460.0025.420.0045.660.0015.450.0035.750.0065.770.0025.860.0055.900.0035.930.0141(R)5.560.0045.790.0175.640.0055.810.0145.660.0045.850.0065.870.0055.890.0056.010.0032()5.860.1425.830.1305.690.1715.850.1255.640.1825.720.1695.680.2005.710.1815.720.1842(R)6.04c0.0036.180.0116.180.0026.210.0016.180.0026.260.0106.270.0116.290.0166.430.0132(n)6.06c0.0066.150.0006.040.0006.180.0146.120.0006.280.0026.360.0056.410.0006.640.0443(R)6.190.0066.440.0906.370.0156.490.0656.420.0146.590.0036.650.0116.660.0316.830.0043(n)6.340.0006.400.0036.290.0006.470.0016.390.0006.530.0006.630.0076.710.0027.070.0073()6.500.4126.340.4146.470.4166.310.4596.470.3776.390.3586.380.3146.350.2866.320.2674(R)6.510.0056.620.0046.610.0026.590.0036.630.0086.670.0496.700.0976.690.0086.870.0031(nR)6.700.0266.800.0226.850.0265(R)6.820.0007.090.0067.080.0067.100.0034()6.880.1806.810.1926.770.2246.800.2296.690.2496.720.2976.640.2996.630.3676.610.5574(n)6.870.0476.830.0176.940.008 aFororbitalbasedassignmentcomparewith 4-5 .bOscillatorstrengthfromrighthandvectoronly;cThesetwostatesarestronglymixed. 156

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Table4-7. Excitationenergies(eV)ofthelowest12transitionsofcytosineandcytidine transitionCytosineCytidinetypeassignmentCCSDCCSD(T)CCSDCCSD(T)CCSDaCCSD(T)ab 1()!4.940.0494.744.840.1294.63-0.10-0.110.011(n)nN!5.460.0025.255.490.0065.290.030.04-0.011(R)!R5.560.0045.495.770.0095.670.210.180.032())]TJ /F9 7.97 Tf 6.58 0 Td[(1!5.860.1425.625.810.1425.58-0.05-0.04-0.012(R))]TJ /F9 7.97 Tf 6.58 0 Td[(1!R6.04c0.0035.916.14d0.1155.990.100.080.022(n)nO!26.06c0.0065.966.16d0.0516.020.100.060.043()!26.500.4126.356.24d0.1916.120.050.040.013(R)!R6.190.0066.086.460.0116.340.270.28-0.013(n)nO!6.340.0005.906.470.0086.050.130.15-0.024(R)!R6.510.0056.436.55e0.1076.420.04-0.010.055(R)!R6.820.0006.736.67e0.1016.54-0.15-0.190.041(nR)nO,nN!R6.700.0266.574())]TJ /F9 7.97 Tf 6.58 0 Td[(1!26.880.1806.686.820.174-0.06 aRelativeexcitationenergywithrespecttocytosine.b)DifferenceofthetripleshiftbetweenmonomerandhydratedmoleculecThesetwostatesarestronglymixed,essentiallyamixtureofthetwodesignations.dThesethreestatesaremixedcombinationofthecorrespondingcytosinestatescausingintensityborrowingfromthethirdstate.eThereisalsoacomponent!2whichintroducesoscillatorstrengthinbothofthesesates. 157

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Table4-8. Excitationenergiesofthelowest12transitionsofguanineandguanosine transitionGuanineGuanosinetypeassignmentCCSDCCSD(T)CCSDCCSD(T)CCSDaCCSD(T)ab 1(R)!R4.920.0034.815.160.0345.030.240.220.021()!5.110.1144.935.050.1254.87-0.06-0.060.002(R)!R5.320.0055.235.900.0065.800.580.570.012()!25.610.2975.435.540.2465.36-0.07-0.070.001(n)nO!5.650.0005.515.560.0925.40-0.09-0.110.023(R)!R5.850.0015.765.990.0015.880.140.120.024(R)!R6.010.0015.946.170.0156.040.160.100.065(R)!R6.290.0016.226.340.0046.200.05-0.020.076(R)!R()6.320.0106.246.230.0016.10-0.09-0.140.053()!36.490.0256.316.39c0.0336.22-0.10-0.09-0.012(n)nN!26.620.0036.466.410.0166.24-0.21-0.220.017(R)!R6.680.0056.606.680.0066.500.00-0.100.10 aRelativeexcitationenergywithrespecttoguanine.b)Differenceofthetriplesshiftbetweenmonomerandhydratedmolecule.cIncludesstrongRydbergcomponent. 158

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Table4-9. Excitationenergies(eV)andoscillatorstrengthsofthelowesttransitionsofcytosine,guanineandtheirWatson-Crickpair.EOM-CCSD/aug-pVDZcalculations transitionCytosineGuanineGCpairmonomersGCpairtypeassignmentCCSDCCSD(T)b 1()G!5.110.1144.890.0774.934.670.042()C!4.940.0495.070.0974.744.860.013()G!25.610.2975.450.4475.435.270.004()C)]TJ /F9 7.97 Tf 6.59 0 Td[(1!5.860.1425.550.1745.625.300.011(n)CnN!5.460.0025.790.0025.255.570.012(n)GnO!5.650.0005.910.0005.515.730.041(R)G!R4.920.0034.920.0004.814.84-0.032(R)G!R5.320.0055.370.0065.235.270.013(R)G!R5.850.0015.660.0035.765.550.024(R)G!R6.010.0015.76a0.0005.945(R)C!R5.560.0045.86a0.0025.495.760.035()CTG!C5.680.0045.40 a)Thenaturalorbitalscorrespondingtotheholetohavesubstantialcontributiononbothcytosineandguanine.b)DifferenceoftheshiftduetriplesbetweenmonomerandGCpair. 159

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CHAPTER5CONCLUDINGREMARKSThemostaccurateelectronicstructuremethodstodate,capableofchemicalaccuracy,quicklyhitthewallofapplicabilityduetotheirlargeformalscaling.Extendingthecalculationstoincludefulleffectsofrelativitymakethesituationmoredire.Successfulapproaches,withthegoalofperformingaccuratecalculationsonincreasinglylargemolecules,shouldincludeavarietyofoptimizationtechniquesincludingthedevelopmentofnew,efcienttheoreticalmethods,andexploitingmoderncomputerarchitecturestospeedupcalculations.Theworkpresentedinthisstudyappliesbothofthesestrategiestofurtherpushtheapplicabilityofcoupledclustertheory.TheDouglas-Kroll-HesstransformationisaformallyexactprocedurethatcanincorporatealltherelativisticinformationfromtheDiracequationinaverycomputationallyefcientmanner.IthasbeentestedinconjunctionwithIP-EOM-CCSDinanattempttoimprovetheoreticallydeterminedcoreionizationenergiestoaideindecipheringexperimentaldata.Theimplementationonlyincludesthescalar-relativisticeffects,asIhaveignoredthespin-dependenttermsduetothehard-codedsymmetryoftheexistingquantumchemistrycoupledclusterprograms.However,itallowsnearquantitativeaccuracyformoleculeswithlownuclearchargeandqualitativeaccuracyformoleculeswithhighernuclearcharge.Theinclusionofthespindependenttermswouldmostlikelyimprovetheresultstowardsquantitativeaccuracy,butisbeyondthescopeofthepresentstudy.Infact,thefth-oderDKHtransformationperformswithquantitativeaccuracyforhighlystrippedionswherespin-orbiteffectsarenotpresentuptoaveryhighenergythresholdof5,000eV.Thisenablestheefcientstudyofhighenergyplasmas,andhasbeenimplementedefcientlyintheACESIIandACESIIIprogrampackages.Fororganicmolecules,whererelaxationeffectsarethedominantenergeticeffectsincoreionizationenergies,amethodwasdevelopedtocomputethisrelaxationquantitytoinnite-order,directlyusingvariationalcoupledclustertheory.Adirectformulation 160

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hasbeenneededsincethepioneeringworkofBaguset.al[ 186 ].ThisIP-VCCSmethodconvergesrapidly,avoidssymmetrydilemmasapriori,andreducesspincontaminationerrorsobtainedinthetraditional,indirectway.Itisacompletelysizeextensivemethodandhasonlymean-eldcomputationalcost.Consequently,thewholeionizationspectrumcanbeobtainedwithnoccN4AOformalscaling.Thismethodparallelizesverywell,asitavoidsanyserialdiagonalization,whichispresentinatraditionalSCFalgorithm,andhasthusbeenimplementedintheACESIIIprogrampackage.Inadeparturefromcoreionizationenergies,IembarkonamissiontooffertheveryaccurateEOM-CCmethodologyinamassivelyparallelway.HavingimplementedthelefthandeigenvectorsolutionsinACESIII,theroutinestudyofpropertiesandtransitionintensitiescanberoutinelystudiedoneverlargermolecules.Withthegoalofcomputationalefciency,eveninamassivelyparallelenvironment,inmind,usingtherighthandeigenvectorsinplaceofthelefthandeigenvectorsprovidesaquantitativeapproximationtooscillatorstrengths,andreducesthecomputationaleffortbyafactorof2.Furthermore,onecansimplyavoidthecomputationofthe^equationsofCCtheorytoobtainnearquantitativetheoreticallineintensities,withnoapproximationtotheexcitationenergyitself.Inordertoobtainmorequantitativeexcitationenergies,thenoniterativeperturbativetriplesmethods,EOM-CCSD(T)andEOM-CCSD(~T),havebeenrederivedinadifferentlightviaLowdinpartitioning,andimplementedinamassivelyparallelwayinACESIII.IshowthattheEOM-CCSD(~T)method,whichshouldincorporatethequadruplesspaceinthecomputation,performslesssatisfactorilythanitsrstorderapproximation(inthetwo-electronperturbation),EOM-CCSD(T).ThenewderivationhighlightsthelackofapproximationsintheEOM-CCSD(T)method,andhelpsexplainitsexcellentperformancecomparedtoEOM-CCSDT-3.Theperformanceofthecodeisequallygood,andisshowntoperformbetter,thelargerthesystemgets. 161

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Toexploretheexcitedstatepotentialenergysurfacesoflargemolecules,theEOM-CCSDgradientswereimplementedintheACESIIIenvironment.Thecodeshowsverygoodperformanceacrossawiderangeofprocessors,enablingtheroutinestudyofmoleculesthatarefarbeyondthereachofserialimplementations.Ihavetestedtheimplementationonbiologicallyrelevantmolecules,amajorfocusinthiswork.Thisopensthedoortoadiabaticexcitationenergies,excitedstategeometries,transitionstatesalongtheexcitedstatesurface,andevenexcitedstateabinitiomoleculardynamics.Moreover,thegroundworkislaidfortheextensionofmulti-ionized/multi-attachedEOM-CCmethodsfortheroutineexplorationofstatesdominatedbymultiple,degenerate,referencedeterminants. 162

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APPENDIX:SUPPLEMENTARYMATERIAL A.1DataforCorrelationPlotsofVariousTriplesMethods TableA-1.ExcitationenergiesofmoleculeswithvariousCCmethods.AllvaluesareineV.TheTZVPbasissetwasusedforallmethods. MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 Acetamide5.725.675.515.695.695.707.447.267.147.227.857.757.597.697.677.739.058.898.778.85Acetone4.444.394.224.394.404.419.139.028.858.989.269.199.019.179.179.189.869.789.599.669.659.72Benzene5.195.014.915.125.075.096.746.696.456.706.687.657.617.287.457.458.468.318.17Benzoquinone3.072.822.642.902.753.192.932.753.012.854.934.704.394.694.595.895.725.365.655.626.766.536.34 163

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TableA-1.Continued MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 6.556.156.026.095.826.786.326.197.296.776.627.557.277.067.627.357.167.367.277.746.906.777.867.207.07Butadiene6.726.706.416.566.586.607.416.946.876.956.776.888.768.728.488.61Cyclopentadiene5.865.845.545.725.735.757.056.746.646.766.616.718.958.928.628.728.698.75Cyclopropene7.247.237.007.107.107.126.976.946.786.896.906.91Ethene8.518.548.298.368.378.398.768.738.648.738.788.748.648.719.709.659.559.6211.1911.1511.1011.1411.4111.3411.2511.30 164

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TableA-1.Continued MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 Formaldehyde3.973.953.803.943.953.969.269.219.059.199.189.209.779.719.499.58Formamide5.665.635.475.655.655.677.527.367.227.328.528.408.248.308.278.358.738.608.508.5711.3311.1510.9711.0610.93Furan6.806.746.436.646.606.636.896.666.536.716.626.698.838.758.468.578.538.60Hexatriene5.725.665.345.565.585.616.616.015.916.045.725.887.817.777.477.66Imidazole6.796.656.426.646.586.636.716.616.386.597.016.916.766.876.827.277.136.927.157.107.148.158.017.857.987.938.308.097.988.07 165

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TableA-1.Continued MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 8.698.608.348.498.458.52Propamide5.745.695.525.715.725.737.457.287.157.247.807.707.537.647.627.688.698.548.428.52Pyrazine4.424.264.044.314.244.305.144.984.795.075.025.045.305.104.915.115.055.146.035.815.645.865.745.827.187.126.847.107.077.097.146.866.686.866.758.298.207.908.088.058.348.277.898.098.068.148.538.418.278.398.988.908.748.90Pyridazine4.123.953.733.993.923.994.764.544.364.574.494.595.355.165.025.285.225.256.005.795.585.845.745.826.706.496.306.496.416.517.096.996.706.996.936.967.797.727.367.587.557.63 166

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TableA-1.Continued MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 8.118.037.667.867.827.918.848.658.528.65Pyridine5.265.084.905.125.055.125.275.104.975.205.155.175.735.555.365.555.505.586.946.886.626.886.856.867.817.747.417.617.597.667.947.897.547.727.707.788.588.428.328.40Pyrimidine4.704.524.314.564.504.575.134.954.754.974.935.005.495.315.185.425.365.397.177.096.827.107.067.097.977.887.557.777.748.248.177.838.028.018.08Pyrrole6.616.426.286.476.406.456.876.816.536.746.716.748.438.368.078.208.178.23Triazine4.964.804.574.814.784.894.994.774.584.834.764.835.024.834.644.874.814.89 167

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TableA-1.Continued MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 5.845.635.495.765.715.747.517.407.147.447.417.438.217.947.797.957.808.288.197.848.078.048.12Naphthalene5.215.074.795.085.034.414.184.074.344.276.776.656.366.606.576.235.985.866.095.986.536.236.046.266.076.556.456.106.356.336.976.846.586.816.797.367.237.107.577.377.207.747.527.357.747.607.477.767.257.137.296.90Norbornadiene5.805.735.465.655.645.676.686.586.366.516.496.547.877.787.587.657.647.707.877.827.547.737.717.75Octatetraene5.074.984.644.934.945.985.415.325.404.97 168

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TableA-1.Continued MoleculeCCSDCCSD(T)CCSD(~T)CCSDR(3)CC3CCSDT-3 6.886.256.156.916.066.756.636.496.976.786.566.726.507.056.816.607.016.817.627.517.407.697.517.367.747.617.497.787.657.537.977.857.728.117.957.817.957.91 169

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BIOGRAPHICALSKETCH TomWatsonJr.wasborninProvidence,RhodeIslandin1985.Hissixthgradephysicalscienceteacher,Mr.Moore,atBurrillvilleMiddleSchoolwastheliveliestteacherthere,andasaresulthebegangravitatingtowardsscience.DuringhistimeattheWoonsocketHighSchool,pursuingapathspecictomedicalschool,herecalledhisfonddaysinmiddleschool.ThankstotheguidanceofMr.Brown,Mrs.Pierannunnzi,andMrs.Pichette,heforgotaboutthatpathandfocusedonhisrstacademiclove,chemistry,ofanykind,andattendedRhodeIslandCollege(RIC).HisoriginalintentionsforteachingandencouraginghighschoolstudentstowardsacareerinchemistrywereveryquicklyremoveduponjoiningDr.GennisondeOliveira's(Dr.D)computationalchemistryresearchgroupatRIC,whereheplayedaroundwithbasissetsanddensityfunctionals(whatelsecananundergraduatedo).Workingwith,talkingwith,andrunninganOlympicTriathlonwithDr.Dbroughtouthispassionforquantumchemicalresearch(anddisdainforrunning),withaveryspecicgoalofincorporatingitintheundergraduatecurriculum,sinceitissofundamentaltochemistry.Yearningforamorerigorousandrobustwaytostudychemicalsystems,heattendedtheUniversityofFloridatoworkunderthedirectionofDr.RodneyJ.Bartlett.Hefurtherrenedhisintereststowardsthedevelopmentandimplementationofnewtheorywithnewtechnologies.Healsomarriedhiscollegesweetheart,SondraTrafford,hisrstyearthere,andhadhisrstdaughter,AylaGraceWatson,inhisthird.Hehopestobeactivelyinvolvedinusingnewcomputationaltechnologiestoencourageundergraduatestudentstowardsthenoblepursuitofadvancingscience.Hehopestobeaasencouragingashismentorstothefuturegenerationsnotyetexcitedaboutchemistry. 179