Understanding Fracture in Brittle Materials by Connecting Macroscopic Observations and Quantum Theory

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Title:
Understanding Fracture in Brittle Materials by Connecting Macroscopic Observations and Quantum Theory
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1 online resource (69 p.)
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english
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Strasberg, Matthew
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Materials Science and Engineering
Committee Chair:
MECHOLSKY,JOHN J,JR
Committee Co-Chair:
WHITNEY,ELLSWORTH D
Committee Members:
PHILLPOT,SIMON R
NINO,JUAN C
BARTLETT,RODNEY J

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Subjects / Keywords:
cot -- fracture -- oep -- zirconia
Materials Science and Engineering -- Dissertations, Academic -- UF
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Materials Science and Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Electronic Thesis or Dissertation

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Abstract:
Fracture is a fundamentally multi-scale process involving bond breaking which cascades to bulk phenomena. Connecting this quantum process to known engineering phenomena is limited by the quality of quantum theory and availability of appropriate experimental observables. New theory is develop to provide a correlated orbital framework which extends the concept of density functional theory to include exact kinetic energy and exact exchange, leaving electron correlation as the only unknown. Using this technique, bond breaking in zirconia is studied and related to the fractal nature of real fracture surfaces. While this is informative, other macroscopic issues dominate fracture in zirconia—stress induced phase transformations and R-curve behavior. These issues are experimentally analyzed in order to completely characterize the fracture process in zirconia.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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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 Matthew Strasberg.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: MECHOLSKY,JOHN J,JR.
Local:
Co-adviser: WHITNEY,ELLSWORTH D.

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lcc - LD1780 2013
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UFE0045045:00001


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DEVELOPMENTOFNEWQUANTUMMETHODSFORMATERIALSSCIENCE By MATTHEWSTRASBERG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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2013MatthewStrasberg 2

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Thisworkisdedicatedtothefuturerecipientsorvictimsof anyofmyengineering blunders.Unfortunately,thiswillnotprovidethesolacey oudesperatelydeserve,but mayitstandasatestamentthatatonepointIknewwhatIwasdo ing. 3

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ACKNOWLEDGMENTS Iwouldliketoacknowledgeeveryonethathasguidedmeandsu pportedmewhileI nishedthisdegree.SpecialthankstoDr.Mecholsky,Dr.Ba rtlett,Dr.Perera,Allyson Barrett,PaulRobinson,KarthikGopalakrishnan,RobertMo lt,andAlexBazante. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFFIGURES .....................................7 LISTOFABBREVIATIONS ................................8 ABSTRACT .........................................9 CHAPTER 1INTRODUCTION ...................................10 TheRoleofComputationalMaterialsScienceinUnderstandi ngFractureof BrittleMaterials ....................................10 ValidityofDensityFunctionalTheory ........................14 2OPTIMIZEDEFFECTIVEPOTENTIALINDENSITYFUNCTIONALTHEO RY .16 FoundationsofOptimizedEffectivePotential ...................16 LocalOEPThroughSecond-Order .........................17 OEP-MBPT(2) ....................................21 SelectingFockOperatoras H 0 ...........................23 abinitioDFT .....................................24 UsingOEPtoBuildaNon-LocalPotential .....................27 ComplementaryApproachestoImprovingOrbitalEigenvalue s .........31 3DEVELOPMENTOFAHIGHLY-CORRELATEDWAVE-FUNCTIONBASED THEORYFORMATERIALSSCIENCE .......................33 FoundationsofCorrelatedOrbitalTheory .....................33 GoalsandStructureofCOT .............................35 AnImplicitPotentialFromOrbitalOptimizations ..................37 BlochEquationtoDeneOOandVVRotations ..................40 DeningPerturbativeOrder .............................44 5

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Hermitization .....................................46 IntruderStates ....................................48 DeningOrbitalsfrom H eff ..............................49 PsuedocodeOutline .................................50 COTasanExtendableTool .............................51 4CANONICALMIXED-SECTORINTERMEDIATEHAMILTONIAN: HYBRIDIZATIONOFEQUATION-OF-MOTIONANDFOCK-SPACECOUPLED-CLUSTER ................................53 TheIntruderStateProblem .............................53 CanonicalBlochEquation:theInterconnectionBetweenFSC CandEOM ...53 CanonicalMixed-SectorIntermediateHamiltonian ................56 IdenticationofIntruderStates ...........................58 ComparisonwithMSIH ...............................59 5CONCLUSIONS ...................................61 REFERENCES .......................................63 BIOGRAPHICALSKETCH ................................69 6

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LISTOFFIGURES Figure page 3-1BlocheffectiveHamiltonian .............................43 3-2Diagrammaticrepresentationofrstorder n ....................45 3-3 PP blockof H eff B ...................................46 7

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LISTORABBREVIATIONS a o Measureofbasicfractureunit CCCoupledClusterCMSIHCanonicalMixed-sectorintermediateHamiltonianCOTCorrelatedOrbtialTheoryD Fractaldimensionalincrement DFTDensityFunctionalTheoryEOMEquation-of-motionHElectronicHamiltonianH 0 Zeroth-orderHamiltonian H e ff EffectiveHamiltonian KSKohn-ShamMBPTMany-bodyperturbationtheoryMSIHMixed-sectorintermediateHamiltonianNThenumberofelectronsinasystemOEPOptimizedEffectivePotentialVPerturbationoperator 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DEVELOPMENTOFNEWQUANTUMMETHODSFORMATERIALSSCIENCE By MatthewStrasberg December2013 Chair:JohnJ.Mecholsky,Jr.Major:MaterialsScienceAndEngineering Fractureisafundamentallymulti-scaleprocessinvolving bondbreakingwhich cascadestobulkphenomena.Connectingthisquantumproces stoknownengineering phenomenaislimitedbythequalityofquantumtheoryandava ilabilityofappropriate experimentalobservables.Newtheoryisdevelopedtoprovi deacorrelatedorbital frameworkwhichextendstheconceptofdensityfunctionalt heorytoincludeexact kineticenergyandexactexchange,leavingelectroncorrel ationastheonlyunknown. Improvedversionsofsecond-ordermany-bodyperturbation theoryarepresentedasan abinitioapproachtoquantummaterialscience. 9

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CHAPTER1 INTRODUCTION TheRoleofComputationalMaterialsScienceinUnderstandi ngFractureofBrittle Materials Thebasisofmodernfracturemechanicsconsiderstheconfou ndingeffectofastress appliedtoacrackinacontinuummaterial.Thisisdescribed bythestressintensity factor( K I ),afunctionofstressandcracksize.Fractureoccurswhen K I reachesacritical value K IC ,amaterialpropertyreferredtoasfracturetoughness.Fun ctionally,thisis convenientforengineeringdesignsinceoneproperty,frac turetoughness,encompasses allthemechanismsthatoccurduringfracture.Butthisform ulationisonlypartially supportedbytheory,namelyextensionsoftheGrifthsEner gyConsideration.Grifths approachmodelscrackgrowthasthecreationoftwonewfrees urfaces.Whenthe energyputintothesystembyanappliedloadequalstheenerg yneededtocreatetwo freesurfaces,thecrackwillpropagateandthematerialfra ctures[ 1 ].Thisapproach yieldsthenecessarystressforfractureasafunctionofcra cklength,butignoresthe atomicprocessesinvolved.Thisdeterministicviewoffrac ture,yieldingonlyconditions forfailure,issymptomaticoffracturetheoriesandleaves manyunansweredquestions. Howdoesacrackpropagate?Whyisafracturesurfacerough?I sitpossibletopredict fracturetoughnessfrommaterialstructure? Fractureexperimentscannotanswerthesequestionsdirect lyduetothespeed andscaleatwhichbondbreakingoccurs.Furthermore,simul ationsoffracturewith moleculardynamicsareseverelylimitedbysize,timeofsim ulation,anddescriptionof bondbreaking.Thishintsatafundamentalquestionofthevi abilityofcomputational modelingforfracture.Withoutmatchingphysicalexperime nts,modelingtodetermine materialpropertiesisonlysuggestivenotpredictive.Ino therwords,duetothe self-containednatureofatomicsimulations,simulations offracturelackthetransparency toextrapolatetothemacroscopicscale. 10

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Themissinglinkisapropertythatcanbedetectedquantitat ivelyinbothsimulations andphysicalexperiments.Notbycoincidence,oneoftheuna ddressedattributes offracture,theroughnessofthefracturesurface,issuspe ctedtotransversethese scalesinapredictablemanner.Roughnessisameasureofthe deviationsfroma two-dimensionalsurface,orsomepartialdimensiongreate rthan2-D.Thispartial dimension(orfractaldimension)isafractalconstructtha texhibitsself-afnity.Intheory thesefractalpatternsseenmacroscopicallyonthefractur esurfaceextendtotheatomic scale[ 2 ].Exploitingthispropertyhasthepotentialtorevisefrac turemechanicsand increasetherelevanceofquantummodelinginmaterialscie nce. Furthermore,thefractaldimensionhasbeendemonstratedt obequantitatively relatedtothefracturetoughness,providingameasurablec onnectionbetween crackpropagationandresistancetofracture[ 3 ].Thisimpliesthatthefracturepath isdependentonthearrangementofatomsandstrengthoftheb ondsconnectingthem. Basedonthispremise,asimulationthatproducesafracture surfacewiththesame fractaldimensionandsamefracturetoughnessasanexperim entalanalogcanbetaken asademonstrativefractureeventallowingforadetailedan alysisofcrackpropagation. Thiswouldexpandthepracticalapplicationsoffractureth eorytomaterialsthatare notadequatelyhandledbystressintensityfactorconsider ationorwhosesmallsize invalidatesanycontinuumbasedmodel.Ifthefractaldimen sioncanbequantitatively relatedtobondstrengthoreventhesequenceinwhichbondsb reak,thenthefracture toughnessofamaterialcanbepredictedpurelyfromasimula tion.Theincentiveis aframeworktodeterminethefracturetoughnessofnewmater ialswithoutactually fabricatingthem. Tothisend,thereareseveralexperimentalandcomputation alquestionsthatneed tobeansweredallrelatedtotheabilitytodeterminetheval idityofacomputational studyoffracturerelativetoexperiment.Fractureisafund amentallymulti-scaleproblem, aglobalstressinducesalocalstresscausingindividualat omstodissociatewhich 11

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transcendstomultipleatomsdissociatingsimultaneously leadingtoaverydynamic fractureprocessthatspanstheatomictothemacroscopic.I tmaybeeventuallypossible tocapturealltheseprocessesinasinglecomputationalmet hod,butthisisnotthe caseatthemoment.Apragmaticapproachwouldbetoseparate afractureeventinto constitutiveprocesseswhicharehandledbytailoredmetho ds.Thisapproachwould requireexperimentsdesignedforthelimitationsofcomput ationalsimulationsand correspondingquantitativeexperimentalmeasurements.F ractographyprovidesasetof suchmeasurements,whilenotanallinclusiveset,itisanec essaryset. Fractographyisthestudyandanalysisoffracturesurfaces withadistinctemphasis ontopographiccharacterization.Thisincludesamixofqua litativeandquantitative techniquescapableofidentifyingthecauseandoriginoffa ilure,failurestrength,fracture toughness,inuenceofenvironment,andqualityofprocess ingamongotherinformation usefulforfailureanalysis.[ 4 ]Independentoftypeofmaterial,brittlefracturesurface s showcharacteristicmirror,mist,andhacklepatternsthat emanatefromtheoriginof failure.Thesefeaturesareangerprintleftbythecrackfr ontasitpropagatesthrough thematerial.Thesefeatureshaveasizedependencethatinc reasesfartherfromthe originthatisaresultofthereleaseofstrainenergyduring fracture.Themirrorisa macroscopicallysmoothregionwhichleadsintothemistreg ionwhichisidentiedby visibleperturbationstothesurface.Asthecrackcontinue stopropagate,thehackle regionofsignicantlylargerfeaturesemerges.Thesefeat ureshavebeenexperimentally showntoexhibitself-similarityandscaleinvariance[ 5 – 7 ].Itwasdiscoveredthat thesefeaturesonthesurfaceexhibitameasurablefractals tatisticthatcorrelateswith thetoughnessofthematerial[ 8 ].Furtherexperimentationrevealedafundamental relationshipwiththefracturetoughness( K IC )ofamaterialandthemeasurablefractal dimensionalincrement( D )ofthefracturesurface[ 3 9 10 ]: K IC = E p a 0 D ,(1–1) 12

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whereEistheYoung'smodulusand a 0 isaparameterrelatedtothescaleofthe fractureprocess.Thefractaldimensioncanbemeasuredexp erimentallybyanalyzing cross-sectionsofcontoursofthefracturesurface[ 11 ]orareaanalysiswithatomicforce microscopy[ 2 12 ].Theparameter a 0 cannotbedirectlyexperimentallydetermined, butiscalculatedfromtheaboveequation.Studiesoffractu reofsinglecrystalsilicon determine a 0 tobeontheorderof5-10 Adependingonthedirectionofloading[ 10 13 ]. Theirresultsshowastrongcorrelationbetween a 0 andthelatticeparametersofthe diamondcubicunitcellforsilicon.Thisrelationshipwast estedwithsemi-empirical quantummechanictechniquesoncharacteristicunitsofdif ferentmaterials;forexample 3-,4-,5-,and6-memberringswereusedasananalogyforsili caglass[ 14 ].These structureswerepulledaparttosimulateafractureprocess ,fromthistheeffectivecrack extensionwasmeasured.Thismeasurementshowedgoodagree ment(84%correlation) withexperimentallydetermined a 0 values. Thefractaldimensionalincrement( D )and a 0 aretwoexperimentalparameters attwodifferentlengthscales,asurface( millionsofatoms)andalocalcluster respectively,thatdeterminethefracturetoughness,iden tifythemannerofcrack propagation,andprovideameanstoanalyzehowlocalbondin ginuencesmacroscopic fracture.Determiningeithercomputationallyisasignic antchallenge.Suchcalculations canbecarriedout,buttheirvalidityiscurrentlyhardtoqu alify.Andthisisbecausebond breaking,especiallyoflargeclusters,isstillahardands ometimesunsolvableproblem evenwiththemostrigorousquantumcalculationsletalonem oleculardynamics. Multi-referencequantummechanicalmethodsaremakinggre atstridestothisend [ 15 16 ],butthesemethodsarenotaccessibletomaterialscienced uetocomputational cost.Themoreaffordablesecond-orderperturbationtheor y(MBPT2),whileabinitio, attimeslacksthenecessarilyquality.Thenextfewchapter swillproposeimproved versionsofMBPT2inordertodevelopapracticalquantummet hodforstudyingbond 13

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breakinginmaterialsciencewiththeintentthiswillevent uallybeanapproachtostudy fracture.ValidityofDensityFunctionalTheory Densityfunctionaltheory(DFT)isalocalapproachtoquant umchemistryinwhich theenergyofasystemisdeterminedasafunctionoftheelect ronicdensityrather thanthewavefunction.Thevalidityofusingthedensityasa variableisprovenbythe Hohenberg-Kohntheorems,whichstatesthatthereisadirec trelationshipbetween theexternalpotentialofasystemandtheelectronicdensit y[ 17 ].Moreover,givena xednumberofelectronsinthesystem,theexternalpotenti aldenesauniqueground statedensityuptoanadditiveconstant.Thisallowsforava riationallikeprocedureto existforDFT.However,theelectronicenergyhasnoknownfo rmasafunctionalofthe electronicdensityandthedensityofsystemmapsbacktothe wavefunctioninaoneto manymanner[ 18 ].Inotherwords,thereisnoframeworkforasystematically improvable densityguaranteedtoconvergetothecorrectresult. Practicalapproachestohandlethisproblemhavebeendevel oped,namely Kohn-ShamDFT(KS-DFT)[ 19 ].KS-DFTproposesthattheN-electrondensitycan bedenedbyNspin-orbitalsallowingforacompactfunction alformforthedensity: ( r )= N X i ( r ) 2 .(1–2) Theelectronicenergyisgivenby E [ ]= T [ ]+ J [ ]+ E xc [ ]+ Z v ext ( r ) ( r ) dr (1–3) where T isthekineticenergy, J istheCoulombenergy, v ext istheexternalpotential, and E xc isanunknownenergyfunctionalusedtodenetheexchangean dcorrelation energies.ThemainadvantagesofKS-DFTarethatthekinetic energycanbedened asthekineticenergyofasingleSlaterdeterminantcompris edoftheKSorbitalsand 14

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thefunctionalderivativewithrespecttothedensityisrea dilydetermined[ 18 ].Thisjust leavesthe E xc termtobedetermined. However,theKS-densitycanneverachievetheexactresultd uetothelimited mathematicalrepresentation.Bychoosingthedensitytobe comprisedofNorthogonal orbitals,thedensityisequivalentofthetraceoftheone-m atrixofasingledeterminant. Whilethiscanbeagoodapproximation,theexactone-matrix cannotbedenedby asingleSlaterdeterminantmeaningtheexactdensitycanno tbedenedbyonlyN orbitals.Inotherwords,theexactdensityexistsinamathe maticalspaceofdimensionM greaterthenN,whileKS-DFTprojectsthedensityonlyontoN dimensions.Practically, thismeanstheenergyfunctionalshavetobefalsiedinorde rtogiveenergiesthat approachtheexactresult.Thisisakintoasemi-empiricala pproach.Ifthekinetic energyisdenedbyalocaloperatorinsteadofthenon-local operator,theapproach becomescompletelyclassical.Alternatively,iftheorbit alsarenon-orthogonal,the functionalformoftheKS-densitycanproducetheexactdens ity. 15

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CHAPTER2 OPTIMIZEDEFFECTIVEPOTENTIALINDENSITYFUNCTIONALTHEOR Y FoundationsofOptimizedEffectivePotential Theoptimizedeffectivepotential(OEP)frameworkprovide satooltooptimizethe many-bodyelectroncorrealtionproblemfromanorbitalper spectiveusingacentral potential[ 20 ].Thisapproachhasshownsuccessindeterminingexchangea nd correlationpotentialsfromwavefunctiontheory,acrucia lrequirementfordetermining thedensityfromanorbitalexpression.ConventionalKohnSham(KS)densityfunctional theory(DFT)suffersfromtheself-interactionerror,adir ectresultofattemptingto expresstheexchangeenergyintermsofthedensity.Thiserr orcascadesintothe determinationofcorrelationfunctionalswhichmustcompe nsatefortheinadequate handlingoftheexchange.ThiserrortranscendstheKSorbit aleigenvalueequations resultinginorbitaleigenvaluesthathavelimitedphysica lrelevanceandstruggleto satisfytheHOMOcondition[ 18 21 22 ].Withinthedensityframework,OEPapproaches acttoalleviatetheself-interactionerrorbyaleastsquar estypettingtothenon-local exchange;thishelpstorestoretheseparationbetweenexch angeandcorrelation representations[ 23 – 26 ].TheOEPframeworkallowstheexibilityforthecorrelati on energytobedenedfromwavefunctiontheory,densityfunct ionaltheory,densitymatrix functionaltheory,orotherorbitaldependentenergyfunct ionals. Takinganabinitioapproach,thesecond-ordermany-bodype rturbationtheory (MBPT)energyexpressionhasbeenusedtodeneOEP-MBPT2[ 27 28 ].This approachhassuccessfullyprovidedqualitativelymeaning fulorbitalpotentialsand quantitativeaccuratetotalenergies.Thetotalenergy,be ingasecond-orderexpression, isofsecond-orderquality.However,byselectingorbitals andorbitaleigenvaluesfrom alocalpotentialasthechoiceforzerothorderHamiltonian ( H 0 )causesdeviationinthe totalenergy.Thisdeviationisminorandneitheraconsiste ntimprovementorworsening tototalenergyrelativetothepurelyabinitiocoupledclus terresults[ 28 ].Thisoccursfor 16

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tworeasons:1)selectingthelocalOEPorbitaleigenvalues as H 0 propagatesanyerrors ofthelocalformulationintohigherordertermsand2)thede signationoftheoccupied andvirtualspacesaredenedbythelocalorbitaleigenvalu eequation.Toaddressthe rstissue,thecomplementary abinitio DFTapproachprovidesanequivalentframeto OEP-MBPT2,butwithamodied H 0 tousethesemi-canonicalFockoperator[ 29 – 31 ]. Tothesecondpoint,experimentingwithmixinginnon-local exchangeshowsimproved consistencyofthetotalenergy,butsuggeststhereareerro rsinthecorrelationpotential duetotheforcedlocalization[ 32 ]. ForcingtheOEPframeworktoproduceanon-localpotentialw illaddressthese inconsistencieswhilemaintainingtheefcacyoftheappro ach.Theuseofalocal potentialinOEPprovidesasystematicwaytoimproveDFTwhi leworkingwithin theknowntheoremsofDFT.Extendingtoanon-localpotentia l,doesnotviolate anyprinciplesofDFTbutprovidesextraexibility.Gilber tdemonstratedthatthe Hohenberg-Kohntheoremsarevalidfornon-localpotential sallowingforthetotalenergy tobeexpressedasafunctionaloftheone-particlereducedd ensitymatrix( r )[ 33 ].A parallelconstrainedsearchproofexistsforanidempotent r ,aone-matrixcomposedof thesamenumberoforbitalsaselectrons,asdoesexistforth eKSdensity[ 34 ]. AsimpliedderivationofthelocalOEPprocedurewillbedem onstratedalongwith connectiontoabinitioDFT.Thisderivationwillbeextende dtoprovideanon-local variant.Asecond-orderpotentialforboththelocalandnon -localcasewillbe constructed. LocalOEPThroughSecond-Order OEPapproachprovidesaframeworktooptimizeanenergyexpr essionthatisgiven asafunctionalofasetorbitals.Thiseliminatestheneedfo ranexplicitlydetermined wavefunction.Byplacinganemphasisonorbitalsratherthe density,densitymatrices, orthewavefunctionanyenergyexpressionconstructedfrom anycombinationofthese 17

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threequantitiescantreatedinanequivalentframework.Th isachievedbycouplingan orbitaleigenvalueequationandthemany-bodyenergyexpre ssionthroughafunctional derivativewithrespecttoaone-particlepotentialˆ u ( r ).Theorbitaleigenvalueequationis givenby: h q j ˆ g j p i = h q j ( ˆ h +ˆ u ) j p i = p qp (2–1) Whereˆ u ( r )islocaland ˆ h isthekineticenergyoperatorandexternalpotential.Inde xes i,j,k,...willindicateoccupiedorbitals;a,b,c,...will indicatevirtualorbitals;p,q,r,...are genericindexesrepresentinganyorbital.Inthecontextof DFT,thiswouldrefertothe KSequationwiththerstNorbitalsdeningthedensity ( r ). ( r )= N X i h i j ( r r 0 ) j i i = N X i i ( r ) i ( r )(2–2) TheKSequationisdeterminedbyimposingtheenergybeamini mumwithrespectto thedensity. E [ ( r )] ( r ) =0(2–3) Thisfunctionalderivativedenesˆ u ( r )andthustheorbitals,whichthendenethe densityandtotalenergy.OEPimposesamoregeneralconditi on,thattheenergyis stationarywithrespecttothepotentialˆ u ( r ) E ( f g f g ) ˆ u ( r ) =0,(2–4) whereEistheenergyofthesystem.Thedependenceonˆ u ( r )isnotexplicitinE,so thechainruleisappliedwithfunctionalderivativesovera llorbitalsandtheorbital eigenvalues: X p Z dr 0 E p ( r 0 ) p ( r 0 ) ˆ u ( r ) + c c + X p E p p ˆ u ( r ) =0.(2–5) 18

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FirstorderuncoupledperturbedHartree-Fock(UPHF)isuse dtothedenethe functionalderivatives p ( r 0 ) ˆ u ( r ) and p ˆ u ( r ) .Therst-orderUPHFequationsisgivenas: ( ˆ g p ) p +( ˆ u p ) p =0(2–6) Thevariationoforbital p inducedbyvariationinˆ u isdeterminedbyprojectingbyorbital q 6 = p : p ( r 0 )= X q 6 = p C pi q ( r 0 )= X q 6 = p h p j ˆ u ( r ) j i i i p p ( r 0 )(2–7) Thevariationintheeigenvalue p isgivenby: p = h p j ˆ u ( r ) j p i (2–8) Thefunctionalderivativesaredeterminedfromtheaboveex pressions: p ( r 0 ) u ( r ) = X q 6 = p q ( r ) p ( r ) p q q ( r 0 ) p ( r 0 ) u ( r ) = p ( r ) p ( r ) (2–9) InsertingtheaboveexpressionsintotheOEPconditioninEq uation 2–5 ,weobtain E ( f g f g ) u ( r ) = X p q 6 = p Z dr 0 q ( r ) p ( r ) q ( r 0 ) p q E p ( r 0 ) + p ( r ) q ( r ) q ( r 0 ) p q E p ( r 0 ) + X p p ( r ) p ( r ) E p ( r 0 ) (2–10) Thetotalenergyisasumofthekineticenergy( E T ),thenuclear-nuclearandthe nuclear-electron( E Ne ),andtheelectron-electroninteraction( V ee ).AswithKStheory, thekineticenergyofthesystemandthenuclear-electronin teractionisdenedbytheN occupiedorbitals. E T + E Ne = N X i h i j 1 2 r 2 + v ext j i i = N X i h i j ˆ h j i i (2–11) 19

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Thefunctionalderivativeofthispartoftheenergyonlyexi stsintheoccupied-virtual orbitalspaceduetothedependenceononlyoccupiedorbital sandthecancellationof occupied-occupiedtermsinthefunctionalderivative. ( E T + E Ne ) ˆ u ( r ) = X i a a ( r ) i ( r ) i a h i j ˆ h j a i + c c (2–12) ManipulatingtheOEPorbitalEquation 2–1 allowstheaboveequationtobeusedto imposeanexplicitdependenceonˆ u ( r ).Specically,insertingtherelation h i j ˆ h j a i = h i j ˆ u j a i intotheabovefunctionalderivative ( E T + E Ne ) u ( r ) = X i a a ( r ) i ( r ) i a h i j ˆ u j a i + c c = X i a Z dr 0 a ( r ) i ( r ) i ( r 0 ) a ( r 0 ) i a + i ( r ) a ( r ) a ( r 0 ) i ( r 0 ) i a ˆ u ( r 0 ) (2–13) Thetermprecedingˆ u ( r 0 )istheKSlinearresponsefunctionorthedensity-density responsefunction,whichwillbenotedas X ( r r 0 ). X ( r r 0 )= X i a a ( r ) i ( r ) i ( r 0 ) a ( r 0 ) i a + i ( r ) a ( r ) a ( r 0 ) i ( r 0 ) i a (2–14) Sincethestationaryconditioninsiststhefunctionalderi vativeofthetotalenergyiszero, thefunctionalderivativeofelectron-electroncontribut iontotheenergyisequaltothe negativeofthefunctionalderivativeof E T + E Ne .Thus,aframeworktodeneˆ u ( r )is established. ˆ u ( r 0 )= Z drX 1 ( r r 0 ) X p q 6 = p Z dr 0 q ( r ) p ( r ) q ( r 0 ) p q V ee p ( r 0 ) + c c + Z drX 1 ( r r 0 ) X p p ( r ) p ( r ) V ee p (2–15) X ( r r 0 )anditsinversecanbeexpressednominallyinanexpansiono forthonormal auxiliaryGaussianfunctions g p ( r ), X ( r r 0 )= X pq ( X ) qp g q ( r ) g p ( r 0 )(2–16) 20

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X 1 ( r r 0 )= X pq ( X 1 ) qp g q ( r ) g p ( r 0 )(2–17) where( X 1 ) qp isdeterminedbysingularvaluedecompositionof X ( r r 0 ).Similarly,ˆ u is determinedbyexpandingitinthesamebasis: ˆ u ( r 0 )= X p c p g p ( r 0 )(2–18) Whateverchoiceismadefor V ee denestheOEPpotentialˆ u ( r ).Thiscanincludeany DFTfunctional,aDMFTfunctional,orinthiscasetheelectr oncorrelationthrough second-orderperturbationtheory(MBPT2).OEP-MBPT(2) TherearetwochoicesforthezerothorderHamiltonian( H 0 ).Firstisaself-consistent formulationwhere H 0 isthesumoforbitaleigenvaluesfromtheOEPcondition.The otherchoiceistousethesumoftheFockorbitaleigenvalues .Thesearenotequivalent choices;selectingtheFockorbitalsandeigenvaluesconst itutesacorrectionfromFock orbitalsandresultsinadifferentfunctionalderivative; thisdetailwillbediscussedlater. Thederivationwillprogressassuming H 0 iscomprisedofOEPeigenvalues. V OEP MBPT 2 ee = 1 2 X ij h ij jj ij i + X ia h i j f j a ih a j f j i i ai + 1 4 X ijab h ij jj ab ih ab jj ij i abij (2–19) 21

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where ai = i a and abij = i + j a b .Thefunctionalderivativeof V OEP MBPT 2 ee is givenby: X p q 6 = p Z dr 0 p ( r ) q ( r ) q ( r 0 ) p q V ee p ( r 0 ) + c c = X i a i ( r ) a ( r ) i a 8<: X j h aj jj ij i + X b h a j f j b ih b j f j i i bi + X jb h ja jj bi ih b j f j j i bj + X jb h j j f j b ih ba jj ji i bj + 1 2 X jbc h aj jj bc ih bc jj ij i bcij + X j h a j f j j ih j j f j i i aj + 1 2 X jkb h jk jj ib ih ab jj jk i abjk 9=; + c c (2–20) Andthederivativewithrespecttotheorbitaleigenvalues: X p p ( r ) p ( r ) V ee p = X i i ( r ) i ( r ) 8<: X a h i j f j a ih a j f j i i ( i a ) 2 + 1 2 X jab h ij jj ab ih ab jj ij i ( i + j a b ) 2 9=; + X a a ( r ) a ( r ) 8<: X i h i j f j a ih a j f j i i ( i a ) 2 + 1 2 X ijb h ij jj ab ih ab jj ij i ( i + j a b ) 2 9=; (2–21) 22

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Collectingtermsoftheabovetwoequationsdenesˆ u ( r 0 )throughsecondorder: ˆ u ( r 0 )= Z drX 1 ( r r 0 ) X i a i ( r ) a ( r ) i a 8<: X j h aj jj ij i + X b h a j f j b ih b j f j i i bi + X jb h ja jj bi ih b j f j j i bj + X jb h j j f j b ih ba jj ji i bj + 1 2 X jbc h aj jj bc ih bc jj ij i bcij + X j h a j f j j ih j j f j i i aj + 1 2 X jkb h jk jj ib ih ab jj jk i abjk 9=; + c c Z drX 1 ( r r 0 ) X i i ( r ) i ( r ) 8<: X a h i j f j a ih a j f j i i ( i a ) 2 + 1 2 X jab h ij jj ab ih ab jj ij i ( i + j a b ) 2 9=; + Z drX 1 ( r r 0 ) X a a ( r ) a ( r ) 8<: X i h i j f j a ih a j f j i i ( i a ) 2 + 1 2 X ijb h ij jj ab ih ab jj ij i ( i + j a b ) 2 9=; (2–22) Onceˆ u ( r )isdetermined,thematrixelements h p j g j q i canbedeterminedallowingfor newdenitionoforbitalsandorbitaleigenvalues.This,in turn,denesanew H 0 allowing foraniterativedeterminationofˆ u ( r )andtheorbitals. SelectingFockOperatoras H 0 TheOEPprocedurecanbeframedasacorrectionfromtheFocko peratorand correspondingFockorbitals.Thisachievedbyselecting H 0 astheFockoperator;for simplicitysemi-canonicalFockorbitalswillbeselectedg iving H 0 asthediagonalFock elements: H 0 = X p h p j f j p i = X p p .(2–23) Thischoiceof H 0 causestheOEPproceduretobehaveonlyasanorbitaloptimiz ation witheachiterationdetermininganimprovedsetofsemi-can onicalFockorbitalsandan 23

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perturbingpotentialˆ u ( r ).TheOEPorbitalequationisnolongeraneigenvalueequati on: h i j g j j i = ij h a j g j b i = ab h i j g j a i = h a j g j i i =0 (2–24) Whiletheenergyisstilloptimizedwithrespecttoˆ u ( r ),thefunctionalderivativesinthe chainrulehavetobearemodied.TherelevantUPHFequation isgivenby: ( f p ) p +( u p ) p =0(2–25) where p isthediagonalelementfromtheFockmatrix. p ( r 0 ) u ( r ) = X q 6 = p q ( r ) p ( r ) p q q ( r 0 ) p ( r 0 ) u ( r ) = p ( r ) p ( r ) (2–26) Fromhere,thederivationofˆ u ( r )proceedsinthesamemannerasbefore,justwiththe modieddenominators p correspondingtothediagonalelementsoftheFockoperator Thisvariantmaintainsmoreofthenon-localqualityasalle lementsintheenergy expressionarecomingfromnon-localoperatorswiththeexc eptionoftheseparationof theoccupiedandvirtualorbitals.ThisschemeiscalledOEP -MBPT(2)SCinorderto differentiateitfromthedenitionofOEP-MBPT(2)present edintheprevioussection. abinitioDFT AnalternativeformulationoftheOEPframeworkisimposeco nditionsdirectlyon thecorrectionstothedensity.Specically,thatcorrecti onstotheKSdensity, delta arezero.Thisisalocalnaturalorbitalcondition.Thecorr ectionstothedensitycanbe determinedthroughdoubleperturbationtheorywiththesec ondperturbationoperator 24

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beingthedensityoperator, ˆ ( r r 0 ).Therstordercorrectionisgivenby: (1) = h 0 j R 0 V j 0 i + h 0 j VR 0 j 0 i = X ia i ( r ) a ( r ) ai h a j f j i i + c c = X ia i ( r ) a ( r ) ai h a j h j i i + X ia i ( r ) a ( r ) ai X j h aj jj ij i + c c = X ia i ( r ) a ( r ) a ( r 0 ) i ( r 0 ) ai u ( r 0 )+ X ia 8<: i ( r ) a ( r ) ai X j h aj jj ij i + c c 9=; (2–27) Fromhere,ˆ u ( r 0 )canbedeterminedinthesamemanneraswiththeOEPcaseby settingthecorrectiontothedensitytozero. ( r )= (1) ( r )+ (2) ( r )+ =0(2–28) Rearrangingtermsgives: u ( r 0 )= Z drX 1 ( r r 0 ) X ia 8<: i ( r ) a ( r ) ai X j h aj jj ij i + c c 9=; + Z drX 1 ( r r 0 )[ (2) ( r )+...] (2–29) Thesecondordercorrectiontothedensityisgivenby: (2) = h 0 j R 0 VR 0 V j 0 i + h 0 j VR 0 R 0 V j 0 i + h 0 j VR 0 VR 0 j 0 i (2–30) Therstandlasttermsin (2) aretheadjointofeachother,with h 0 j R 0 VR 0 V j 0 i given by: h 0 j R 0 VR 0 V j 0 i = X i a i ( r ) a ( r ) i a ( X b h a j f j b ih b j f j i i bi + X jb h ja jj bi ih b j f j j i bj + X jb h j j f j b ih ba jj ji i abij + 1 2 X jbc h aj jj bc ih bc jj ij i bcij + X j h a j f j j ih j j f j i i aj + 1 2 X jkb h jk jj ib ih ab jj jk i abjk 9=; (2–31) 25

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Thesecondtermisgivenby: h 0 j VR 0 R 0 V j 0 i = X i a i ( r ) a ( r ) X jb h j j f j b ih ba jj ji i abij bj + c c X i i ( r ) i ( r ) 8<: X a h i j f j a ih a j f j i i ( i a ) 2 + 1 2 X jab h ij jj ab ih ab jj ij i ( i + j a b ) 2 9=; + X a a ( r ) a ( r ) 8<: X i h i j f j a ih a j f j i i ( i a ) 2 + 1 2 X ijb h ij jj ab ih ab jj ij i ( i + j a b ) 2 9=; (2–32) Onesimplicationcanbemadebycombiningcertaintermsin h 0 j R 0 VR 0 V j 0 i and h 0 j VR 0 R 0 V j 0 i : X i a i ( r ) a ( r ) X jb ( h j j f j b ih ba jj ji i ai abij + h j j f j b ih ba jj ji i abij bj ) = X i a i ( r ) a ( r ) X jb ( h j j f j b ih ba jj ji i ( bj + ai ) ai bj abij ) = X i a i ( r ) a ( r ) i a X jb ( h j j f j b ih ba jj ji i bj ) (2–33) ComparingthetermsfromtheaboveexpressionstotheOEPEqu ations 2–22 shows thattheyareexactlythesame.Thisisacompletelyindepend entformulationofexactly thesameequations,withsameexibilityregardingthechoi ceof H 0 .Thisdemonstrates thattheOEPoptimizationistheequivalenttoastationaryd ensitycondition.Thisisnot coincidental.Thefunctionalderivativewithrespecttoˆ u ( r )isdenedusingthechain ruleandrstorderUPHF.UPHFisanorbitalperturbationont oponanorbitalequation, whichisbuiltintotheOEPframeworkimplicitly.Inotherwo rds,thisisanoptimizationof aperturbingoperatorinadoubleperturbationframework[ 35 ].Ifwedenethefollowing perturbations: V = H H 0 U =ˆ u ( r ) (2–34) 26

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Ithasbeendemonstratedthatanoptimalexpressionforthee xternalperturbationcanbe determinedbyminimizingthemixedperturbationexpressio n, h j U j i [ 36 37 ].Thisisa rstorderperturbationwithrespectto U andanychoiceofoforderin V ;ordersin U will benotedwithasubscriptandordersin V willnotedwithasuperscript.Thestationary conditionthroughsecond-orderin V isgivenby ( E 1 1 + E 2 1 ) ˆ u ( r ) = ˆ u ( r ) fh 0 j UR 0 V j 0 i + h 0 j VR 0 U j 0 i + h 0 j UR 0 VR 0 V j 0 i + h 0 j VR 0 UR 0 V j 0 i + h 0 j VR 0 VR 0 U j 0 ig (2–35) Theperturbation U willproducetermsoftheform h p j u j q i withacorresponding denominator.Thefunctionderivativeof h p j u j q i withrespecttotoˆ u ( r )is p ( r ) q ( r ).The resultisidenticaltothe abinitio DFTequations.Theconnectionbetween abinitio DFT andabinitioversionsofOEPhasbeendiscussedpreviously, butintermsofmodied functionalderivativediagrams[ 38 ]. Anadditionalnote,thederivativewithrespecttotheorbit aleigenvaluesisnotadding correlationbutisanoccupationnumbercorrection.Removi ngthesetermswillnot changetheconvergedresultfor(ˆ u ( r 0 ))asthesetermsmustsumtozeroinorderto preservethenumberofelectronsinthedensity. UsingOEPtoBuildaNon-LocalPotential ThelocalvariantofOEPhassignicantutilityasbothanorb italoptimizationscheme andttingschemetotranslatenon-localfunctionsintoloc alfunctionsforusewithDFT. Thettingfromnon-localtolocalallowsforamuchimproved treatmentofexchange thaninconventionalDFTfunctions.Thisaddressestheself -interactionerrorthat plaguesthedevelopmentoffunctionalsandconvolutesthed enitionoforbitalsand theirorbitaleigenvalues.However,asdiscussedbyNesbet ,forcingexchangetobelocal causesquantitativeerrorsproducingahigherenergythant heHartree-Fockcase.For example,withtheneonatomtheerrorisabout2milliHartree swiththeOEPenergy 27

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consistentlyhigherinenergythantheHFenergy(notconsid eringcorrelation)[ 39 ]. Thisisnotsurprising,HFisanenergyfromvariationallyop timizedwavefunctionwhile theOEPenergyisfromlocalttingofthatenergyexpression .Additionally,inorderto determinethelocalpotentialtheinverseofthedensityres ponse, X 1 ( r r 0 ),mustbe determinedthroughasingularvaluedecomposition[ 25 27 40 ].Thisprocedurecan causetheconvergenceofˆ u ( r )tobechallenging.Boththeseissuescanbealleviatedby insistingonanon-localpotentialˆ u ( r r 0 )instead.Thiscanbeextendedfurthertoforce ˆ u ( r r 0 )tobeacorrectiontotheFockoperator.Thenon-localOEPor bitalequationis givenby: h q j nl ˆ g j p i = h q j ( ˆ f +ˆ u ) j p i = p qp (2–36) Thenon-localOEPcondition,withthechainrule,isnow: E ( f g f g ) ˆ u ( r r 0 ) =0= X p Z dr 00 E p ( r 00 ) p ( r 00 ) ˆ u ( r r 0 ) + c c + X p E p p ˆ u ( r r 0 ) (2–37) Thisrequiresanewdenitionofthefunctionalderivatives withrespecttoˆ u ( r r 0 ).These arestilldeterminedfromrst-orderUPHF. p ( r 00 ) u ( r r 0 ) = X q 6 = p q ( r ) p ( r 0 ) p q q ( r 00 ) p ( r 0 ) u ( r ) = p ( r ) p ( r 0 ) (2–38) ThisisthesamedenitionimplicitlyusedintheworkbyPern altodetermineeffective potentialsinnaturalorbitalfunctionaltheory[ 41 ].Insertingtheaboveexpressioninto Equation 2–37 E ( f g f g ) u ( r r 0 ) = X p q 6 = p Z dr 00 q ( r ) p ( r 0 ) q ( r 00 ) p q E p ( r 00 ) + p ( r ) q ( r 0 ) q ( r 00 ) p q E p ( r 00 ) + X p p ( r ) p ( r 0 ) E p (2–39) 28

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Lookingattheoccupied-occupiedportionofthederivative : X i j 6 = i Z dr 00 ( j ( r ) i ( r 0 ) j ( r 00 ) i j E i ( r 00 ) + j ( r ) i ( r 0 ) i ( r 00 ) j i E j ( r 00 ) ) = X i j 6 = i j ( r ) i ( r 0 ) i j Z dr 00 ( E i ( r 00 ) j ( r 00 ) E j ( r 00 ) i ( r 00 ) ) (2–40) Theunfortunatelessonlearnedfromlocalderivationistha toccupied-occupiedand virtual-virtualportionsofthesefunctionalderivatives respectivelycanceloutasthe abinitiomany-bodyenergy(theHFenergyplusanylevelofco rrelation)isinvariantto occupied-occupied(OO)andvirtual-virtual(VV)orbitalr otations. Z dr 00 ( E i ( r 00 ) j ( r 00 ) E j ( r 00 ) i ( r 00 ) ) =0 Z dr 00 E a ( r 00 ) b ( r 00 ) E b ( r 00 ) a ( r 00 ) =0 (2–41) Inotherwords,onlytheoccupied-virtual(OV)andvirtualoccupied(VO)termspersistfor anyabinitiodenitionofthemany-bodyenergy, Furthermore,thederivativewithrespecttoorbitaleigenv aluesonlyindicatesa changeintheoccupationnumberandisirrelevanttothederi vationofpotential.The non-localOEPconditionisthengivenby: E ( f g f g ) u ( r r 0 ) = X i a i ( r ) a ( r 0 ) i a Z dr 00 E i ( r 00 ) a ( r 00 ) E a ( r 00 ) i ( r 00 ) + c c (2–42) Animportantpropertyofnon-localfunctionalderivative, incontrasttothelocalversion, isthateveryterm i ( r ) a ( r 0 )islinearlyindependent.Thiscanbeshownbyintegratingb y aspecic i ( r ) a ( r 0 ),whichmeanseachintegralinthesummustbezero. Z dr 00 E i ( r 00 ) a ( r 00 ) E a ( r 00 ) i ( r 00 ) =0 (2–43) 29

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Thisbypassestheneedforadensity-responseordensity-ma trixresponseoperatorand removesthedependenceon X 1 ( r r 0 ).Themany-bodyenergyisgivenby: E = X i h i j ˆ h j i i + 1 2 X ij h ij jj ij i + E c (2–44) where E c isthecorrelationenergy.Insertingthisdenitionintoth eintegralexpression above: Z dr 00 E i ( r 00 ) a ( r 00 ) E a ( r 00 ) i ( r 00 ) =0 = h a j h j i i + X j h aj jj ij i + Z dr 00 E c i ( r 00 ) a ( r 00 ) E c a ( r 00 ) i ( r 00 ) (2–45) UsingtheOEPorbitalEquation 2–36 ,itcanbeshownthat h a j h j i i = h a j u j i i + P j h aj jj ij i Usingthisrelationproducesadenitionfor h a j u j i i : h a j u j i i = Z dr 00 E c i ( r 00 ) a ( r 00 ) E c a ( r 00 ) i ( r 00 ) (2–46) Theexpressionfor h a j u j i i throughsecondisgivenby: h a j u j i i = X b h a j f j b ih b j f j i i bi + X jb h ja jj bi ih b j f j j i bj + X jb h j j f j b ih ba jj ji i bj + 1 2 X jbc h aj jj bc ih bc jj ij i bcij + X j h a j f j j ih j j f j i i aj + 1 2 X jkb h jk jj ib ih ab jj jk i abjk (2–47) Theaboveexpressioncanbederivedusingdoubleperturbati ontheoryinthesame mannerasintheabinitoDFTapproachbyreplacingthedensit yoperator( ˆ )withthe one-matrixoperator(ˆ r pq = j q ih p j ).Thisispurelyanalternativederivationthatallowsfor theuseofdiagrammaticperturbationtheoryinsteadoffunc tionalderivatives,butresults inthesamepotential.Unfortunatelyneitherapproachprov idesaprescriptiontodene acorrelatedpotentialintheoccupied-occupiedspaceorth evirtual-virtualspace.Atthis point,whathasbederivedisanaturalorbitalsecond-order whichproducesanorbital optimizedsecond-orderenergy(OO-MBPT2). 30

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ComplementaryApproachestoImprovingOrbitalEigenvalue s Therearelimitedadvantagestodeningapotentialˆ u ( r r 0 )intheOOorVVspaces. Rotationsinthesespaceshavenoeffectonthetotalenergy. Therearetwoequally validjusticationsforthisrotation:1)thesemi-canonic alFockrotationwhichforcesthe OOandVVblocksoftheFockmatrixtobediagonaland2)thedia gonalone-matrix condition.Theaimistoprovideacorrectiontotheeigenval uesbeyondHF,tothisend thesemi-canonicalrotationispreferred.TheworkofPerna ldetermines h p j u j p i bytaking thefunctionalderivativewithrespecttotheoccupationnu mber. h p j u j p i = E c n i (2–48) ThereisnoexplicitdependenceonoccupationnumberinMBPT approaches.Efforts havebeenmadetointroduceadependenceonoccupationnumbe rinthewavefunction framework,whichhasshownsomesuccessfordeterminingion izationpotentialsand electronattachments[ 42 ].Instead,anequivalentapproachtoconsideringoccupati on numberwillbeusedthatextendsthefunctionalderivativew ithrespecttotheorbitals using h p j u j p i asaLagrangianmultiplier: h p j u j p i = Z dr E c p ( r ) p ( r )= Z dr E c p ( r ) p ( r )(2–49) Insertingthesecond-orderenergyexpressiongivescorrec tionstotheorbital eigenvalues: h i j u j i i = X a h i j f j b ih b j f j i i bi + 1 2 X jbc h ij jj bc ih bc jj ij i bcij + X jb h j j f j b ih bi jj ji i bj + X jb h ji jj bi ih b j f j j i bj h a j u j a i = X j h a j f j j ih j j f j a i aj + 1 2 X jkb h jk jj ab ih ab jj jk i abjk (2–50) Thersttwotermsinbothexpressionsarecorrelationterms whilethelasttwotermsin h i j u j i i arecorrectionsto J and K orcoupledperturbedHFterms.Thedoubleexcitation 31

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termsintheaboveexpressionmatchdoubleexcitationcorre ctionsfromthefractional occupationframework[ 42 ].Thesearediagonalpiecesoftheone-electronportionof thesecond-ordercorrelatedHamiltonian, H (2) .Asresult,asmuchof H (2) aspossibleis beingputintheone-particleoperator ˆ g Again, H 0 canbeselectedastheFockoperator(OO-MBPT2)orcaninclud ethis correctiontotheorbitaleigenvaluescreatinganaugmente dsecond-order(A-MBPT2). 32

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CHAPTER3 DEVELOPMENTOFAHIGHLY-CORRELATEDWAVE-FUNCTIONBASEDTH EORY FORMATERIALSSCIENCE FoundationsofCorrelatedOrbitalTheory ThepurposeofCorrelatedOrbitalTheory(COT)istoprovide asingle-particle frameworkforroutineandaccuratequantumcalculations.T heutilityofsingle-particle equationshasalreadybeendemonstratedbythewidesuccess ofDFT,specically Kohn-ShamDFT(KS-DFT)[ 19 ].KS-DFTreliesonatwolevelformulation:1) amany-bodyenergygivenasafunctionaloftheelectronicde nsityand2)the non-interactingorbitalequationwhichisdeterminedbyaf unctionalderivativeofthe energywithrespecttoanassumedfunctionalformofthedens ity[ 18 ].Thisstructureof amany-bodyenergylinkedtoasingle-particleformulation persistsinanyorbitalbased quantummethod.WithDFT,however,themany-bodyenergyisu ndeterminedand cannotguaranteeconvergencetothecorrectanswer.Whilet hisisnotaoverwhelming deciency,itdoeshinderthepredictabilityofthemethod. Quantumsimulationsofmaterialsaredominatedbydensityf unctionaltheory (DFT)andsemi-empiricalmethods.Densitymatrixfunction altheoryhasemerged asalegitimatealternativetoDFT.Newapproachesincludec orrected-HartreeFock [ 43 ],GeminalFunctionalTheory[ 44 ],NaturalOrbitalFunctionalTheory[ 45 46 ],and DensityCumulantFunctionalTheory[ 47 ]tonamefew.Second-orderperturbation theory(MBPT2)isgaininganincreasedimportanceinmateri alsscienceduetherelative highaccuracyforthegivencomputationaleffort[ 48 – 50 ].MBPT2canbeviewedasa lineartypeansatztothemany-bodywavefunctionwhichcanb efunctionalizedthrough manipulationoforbitalsand H 0 .Despitethelonghistoryofthemethod(thepaper byMlletandPlessetwaspublishedin1934[ 51 ]),thereisstillsignicantexibility forcreatinganimprovedMBPT2suitableforthepredictives tudyofmaterialsand molecules. 33

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Orbitaloptimizationhasbeenexploredextensivelyinquan tumchemistrywith thepurposeofeitherimprovingthequalityofresultsorspe edofthecalculation. Variousorbitaloptimizationschemesexistfromdifferent variantsofHartree-Fock [ 52 ],optimizationofthevirtualspace[ 53 ],energyminimization[ 54 55 ],improved properties[ 56 ],reducedcomputationalcost[ 57 ],andimprovedconvergence[ 58 ]among othermotivations.Bruecknerorbitalscorrespondtomaxim umoverlapconditionwith theexactwavefunction[ 59 ].Theseorbitalscanbeobtainedthroughanoptimization of T 1 orindirectlythroughdirectmanipulationoftheone-parti cledensitymatrix,called theBrueckner-Hartree-Fockmethod[ 59 60 ].ABrueckner-Kohn-Shamschemehas beenproposedtocapitalizeonthephysicalrelevanceofBru ecknerorbitalsandthe correspondingqualityintreatmentofproperties[ 61 ].Orbitalsaredeterminedfroman effectivepotentialderivedthroughanenergyminimizatio nofaninnite-orderenergy expectation.TheresultisaKohn-Shamlikeeigenvalueequa tionresultinginBrueckner orbitals,whichthenareinsertedintoastandardDFTfuncti onal. OtherattemptshavebeenmadetoconstructanexactSCFtheor ywhichconnects anindependent-particlemodeltoamany-bodyformulation. Originallydeveloptostudy nuclearinteractions,aseriestotwoparticlepotentialsw eredevelopedfromscattering theory[ 62 63 ].However,thesetheoriesneverreachedapracticalimplem entation. Moresuccessfulhasbeenattemptstodetermineionizationp otentials(IP)andelectron afnities(EA)fromone-particleSCFlikeprocedures.Natu raltransitionorbitalsderive apotentialforcorrelatedorbitalscorrespondingtoIP'sa ndEA'sthroughcommutator relations[ 64 ].ThisisequivalenttotheExtended-Hartree-Fockprocedu re[ 65 66 ]. SimilarprocedureshavebeenusedtoobtainIP'sfromdensit ymatricesmanipulations [ 67 ]. Inconnectiontothetotalwavefunction,one-particleoper atorshavelimitedvalueas theexactwavefunctionisinvarianttoorbitalrotations.H owever,therelevancyoforbitals andtheireigenvaluesisincreasedatlowordersofperturba tiontheory.Koopmans' 34

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theoremindicatesthattheorbitaleigenvaluesfromtheHar tree-Fockprocedureprovide goodestimatestoIP'sandEA's.WhentheFockoperatorisuse dasthezerothorder Hamiltonian, H 0 ,inaperturbativeexpansion,differencesoftheseeigenva luesserveas weightsinthewavefunction.Ithasbeenshownthatmodifyin gthesevaluescanleadto improvedconvergenceorimplicithigherorderterms[ 68 69 ].Theroleof H 0 andwhat comprisesacriterionforgoodnesshasbeendiscussedwithn oclearconclusion[ 70 ]. Thisisthecrucialconnectionbetweenadevelopinganaccur ateone-particleoperator andthemany-bodyenergy.Tothisend,bymimickingKoopmans 'theorem,COTaimsto provideaphysicallysignicantchoicefor H 0 whilenotdeviatingfarfromthesuccessof standardHartree-Fockbasedperturbations. TheadditionalbenetdevelopingCOTbasedonMBPT2istheea seofproperty calculations.Coupled-clustertheoryisnon-hermitianwh ichcomplicatesthe determinationofdensitymatricesandproperties[ 71 ].Perturbationtheory,alternatively, whilehermitianislimitedthroughorder-by-orderexpansi onsandrequiressignicantly moreefforttoapproachthequalityofCCresults.Furthermo re,studiesin SCCS(T) showthatimposingsomehermitiancharacterisbenecialto exploringpotentialenergy surfaces,especiallyinstancesofbondbreaking[ 72 ].COT-MBPT2providesahermitian frameworkestablishedthroughsecond-orderperturbation theory,butalsofoldsinsome inniteorderqualitythroughthemodicationof H 0 GoalsandStructureofCOT COT,inthepresentformulation,iscompletelyabinitiopro ducingamany-body energythattraversesfromHFtheorytoCCtheory[ 73 – 75 ].Inthisstructurea single-particletheoryisestablishedsystematicallyina wavefunctionframework. Additionally,thecorrelatedorbitalsareconstructedtot ocorrespondtophysical observableswithorbitalenergiesequaltoionizationpote ntialsandelectronafnities,a constraintuniquetoCOT.ThisisingreatcontrasttoDFT,wh eretheorbitalscorrespond 35

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toanon-physical,non-interactingsysteminordertocaptu rethecorrectkineticenergy. Eventuallythiscanbeexpandedupontoutilizeknownphysic sfromotheraspectsof wavefunctiontheory,many-bodyGreen'sfunctions,oreven densityfunctionaltheory. TheconnectionbetweenCOTandelectronpropagatorapproac heshasbeenpreviously discussed[ 73 ]aswellasadetailedcomparisontothesecondorderDysonse lf-energy [ 75 ]. Hartree-Fock(HF)theorycanbecastasazerothorder,orunc orrelated,COT. CorrelationisaddedtotheFockoperatorasacorrelationpo tential.Thismaintainsthe simplicityoftheiterativedeterminationoforbitalsande igenvaluesasutilizedinHF theory.TheHFwavefunctionisdesignedassumingelectrons formashellstructure oforthonormalorbitalsarrangedinafermionicsystem.Tha tistosay,forasystemof Nelectrons,thewavefunctionisasingleSlaterdeterminan tcomprisedofNorbitals. TheinteractionofthewavefunctionwiththeelectronicHam iltonian(assumingthe Born-Oppenheimerapproximateform[ 76 ])canbeoptimizedwiththevariational principleprovidinganupperboundtotheenergyofthesyste m.Thisoptimization processconsistsofminimizingtheenergywithrespecttoor bitalshapeyieldinga variationallyoptimumwavefunction,amean-eldoperator ˆ f ,asetoforbitals f p g andorbitalenergies f p g thatcorrespondtoionizationpotentialsandelectronafn ities accordingtoKoopmans'Theorem[ 52 ]. ˆ f p = p p (3–1) HFtheoryassumesthateveryorbitalinteractswithameaneldfromtheotherN-1 electronsand,thus,lackscorrelation.Nevertheless,HFp rovidesaconvenient frameworktoextendtohigher-ordermethods,specicallyt heorbitalstructurethatis crucialtomany-bodyperturbationtheory,coupled-cluste rtheory,andconguration iterationmethods.Theutilityoftheorbitalstructureste msfrominherentcompression ofinformationfromthehighdimensionalityofthetotalwav efunctiontoasmallsetof 36

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one-electronorbitalsthatinteractwiththishighdimensi onalsystem.Tothisend,HF istoocrudeasthemean-eldapproximationeliminatesalld imensionalityassociated withcorrelation.COTusesanorbitaloptimizationprocedu rewithacorrelationpotential basedonmany-bodyperturbationtheoryandandFock-Spacet heoryresultinginanew systemoforbitaleigenvalueequations: ˆ g COT ( f g ) p =[ ˆ f +ˆ v c ( f g )] p = p p (3–2) AnImplicitPotentialFromOrbitalOptimizations COTinvolvesaniterativeupdateofbothorbitalsandorbita lenergies,whose convergenceindicatesthecompletionofthecalculation.T hisisachievedbyintegrating multiplewavefunctiontechniquestodeneorbitalrotatio nsinthreerespectiveblocks: occupied-occupied(OO),occupied-virtual(OV/VO),andth evirtual-virtual(VV). Occupiedorbitalswillbeindexedwithacharacter i j k ,...andvirtualorbitalswill besimilarlyindexedwith a b c ,....Theprocedurestartswithacurrentdenition oftheorbitalsandtheirenergies;attherstiterationthe seareFockorbitalsand energies.MinimizationtheHFenergyimposesthecondition thatthereisnointeraction betweenoccupiedandvirtualorbitalsacrosstheFockopera tor,alsocalledthe Brillouin-Bruecknercondition[ 77 ].Inotherwords,FockorbitalsaredenedbyanOV rotationwhichimposes: h a j ˆ f j i i =0. (3–3) ThisOVconditionwillbemimickedforthecorrelateddenit ionoforbitalsandtheCOT operator, ˆ g .Thatistosay: h a j ˆ g j i i =0. (3–4) Forthesecond-orderimplementationofCOT,whichwillbere ferredtoas COT-MBPT2,thetotalenergyisthesecondorderMBPT2energy withCOT-MBPT2 37

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orbitalsandeigenvalues. E COT MBPT 2 = h 0 j H 0 j 0 i + h 0 j V j 0 i + h 0 j V j 1 i = X i h i j h j i i + 1 2 X ij h ij jj ij i + X ia h i j f j a ih a j f j i i i a + 1 4 X ijab h ij jj ab ih ab jj ij i i + j a b (3–5) TheaboveexpressionisobtainedbyRaleigh-Schr ¨ odingerPerturbationTheorywhere theelectronicHamiltonianissplitintoazerothorderHami ltonianandperturbation: H = H 0 + V (3–6) H 0 isselectedtobetheCOToperator, g = f + v c ,whichattherstiterationistheFock operator.Theperturbation, V ,thenadjustswitheachsuccessiveupdateof g andhence H 0 Theenergycanbereformulatedintoanimmediatelyidentia bleHermitianform, namelytheHylleraassecond-orderfunctional.Byoptimizi ngwithrespecttoorbital choice,theOVorbitalrotationcandened.TheHylleraasfu nctionalisgivenby: E (2) = h 1 j V E (1) j 0 i + h 0 j V E (1) j 1 i + h 1 j H 0 E (0) j 1 i .(3–7) IntheCOTframework,emphasisisputontheoptimizationoft heorbitalsasameansto inuencethetotalwavefunction.Asaresult,functionalfo rmfortherst-ordercorrection tothewavefunctionisassumedandonlytheorbitalsarevari ed.Therst-order wavefunctionisgivenby: j 1 i =( ˆ T (1) 1 + ˆ T (1) 2 ) j 0 i (3–8) where ˆ T (1) 1 + ˆ T (1) 2 arerst-orderexcitationoperatorstakingthereferenced eterminanttoall singleexcitations( ˆ T (1) 1 )anddoubleexcitations( ˆ T (1) 2 )respectively.Thesearedetermined bytherst-ordergeneralizedmany-bodyperturbationtheo ry(GMBPT)expressions[ 78 ]: h ai j ( E 0 H 0 ) ˆ T (1) 1 j 0 i = h ai j V j 0 i (3–9) 38

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h abij j ( E 0 H 0 ) ˆ T (1) 2 j 0 i = h abij j V j 0 i (3–10) Inserting T 1 and T 2 intotheenergyfunctionalgivestheexpressions: E 2 = h 0 j ( ˆ T (1) 1 + ˆ T (1) 2 ) y V j 0 i + h 0 j V ( ˆ T (1) 1 + ˆ T (1) 2 ) j 0 i + h 0 j ( ˆ T (1) 1 + ˆ T (1) 2 ) y ( H 0 E (0) )( ˆ T (1) 1 + ˆ T (1) 2 ) j 0 i (3–11) Thisisnotainterestingresultatsecond-orderasthecorre latedHamiltonianwithout modicationisexpresslyHermitianwith T (1) y equaltothetransposeof T (1) .However, thisdiscussionisintroducedforthesakeofderivingtheOO andVVblocksoftheg operatorsincethisexpressionisnolongerHermitianforth eN-1andN+1spaces.For therestofthederivation,thestandardexpressionfor E 2 willbeused: E 2 = h 0 j V ( ˆ T (1) 1 + ˆ T (1) 2 ) j 0 i (3–12) TheOVandVOblocksofthegoperatoraredenedbyanminimiza tionoftheenergy withrespecttothezeroth-orderwavefunction.Sincetheen ergyexpressionisHermitian, optimizingtheleft-handprojectionofthezerothorderwav efunctionimpliesthatits complexconjugateisoptimizedaswell.Thevariationofthe energywithrespecttothe zeroth-orderwavefunctioncanbemanifestedbyanorbitalo ptimizationthroughaunitary optimization,namelyaBruecknerrotation: ( E 0 + E 1 + E 2 )= @ @ a i X ia h 0 j e ˆ a y i [ H + V ˆ T (1) ] j 0 i L (3–13) Thisisaniterativeprocedureinwhich isre-evaluatedaftereachorbitalrotation.Inthe limitofnearconvergence,theexponentialcanbeexpressed asalinearoperator: 0= @ @ a i h 0 j (1+ˆ a y i )[ H + V ˆ T (1) ] j 0 i L = h ai j [ V + V ˆ T (1) ] j 0 i L (3–14) Theaboveexpressiondenearelationbetween T 1 atrstandsecond-order, T (1) 1 + T (2) 1 = 0.Thisisthesecond-orderBruecknercondition.Thesecond -orderexpressionfor T 1 is 39

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givenby: h ai j V ˆ T (1) j 0 i = h ai j ( E 0 H 0 ) T (2) 1 j 0 i (3–15) TheVOrotationmatrixisthenegativetransposeoftheOVrot ationmatrixasdictated byorthogonalityconstraints.Sincethetotalenergyisinv arianttorotationswithin theOOandVVblocks,thesespacesarerigorouslydetermined bytheOVblock.At secondorder,thisisrotationisequivalenttonon-localOE Protation.TheOOandVV rotationswillbedenedbyIPandEAproblemsrespectively; thisishandledbythe Blochequation. BlochEquationtoDeneOOandVVRotations TheBlochequationisageneralizationoftheelectronicHam iltonianwhichis formulatedtodirectlyhandledegeneratereferencesaswel laschangingnumberof electronsfromaxedreference[ 79 80 ].OfparticularinteresttoCOTistheability tocalculatebothelectronicafnitiesandionizationpote ntialswithonesetoforbitals denedbytheneutralreference.Thisachievedthroughawav eoperatorframework inwhichtheexactelectronicwavefunctionforanysystemca nberepresentedbyan unknownoperator n actingonaknownreferencefunction[ 81 ]: H = H n = n E (3–16) where istheexactwavefunction, islinearcombinationofreferenceSlater determinants,and E istheexactenergyofthesystem.Thewavefunction is orthogonaltoallotherwavefunctions suchthatprojectionof n ontothe uniquely denes .Thisone-to-onecorrespondenceallowsfortheorthogonal ityoftheunknown many-bodywavefunctions f g tomanifestinknownfunctions f g .Each isalinear combinationofreferencedeterminants: = X C .(3–17) 40

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FortheN-1electroncasethesetreferencedeterminantsare alldeterminantsthat formedbyremovingoneorbitalfromthegroundstatereferen ce j 0 i ;theN+1referenceis thesetofalldeterminantswithonevirtualorbitaladded.T hesespaceswillbenotedby thefollowingsecondquantizedprojectionoperators: P N = j 0 ih 0 j P N 1 = X i i j 0 ih 0 j i y P N +1 = X a a y j 0 ih 0 j a (3–18) Sincetheoverlapofwavefunctionswithdifferentnumberso felectronsiszero,allthree projectorscanbecombinedintoageneralprojector P withoutalteringtherestofthe derivation.TheSchr ¨ odingerequationbecomes H n P = n P E (3–19) Therearetwounknownsinthewavefunction,thewaveoperato r Omega andthe coefcientsofthereferencestates C .Thewaveoperatorisdeterminedby perturbativetechniqueswhilethecoefcientsoftherefer encearedeterminedby formulatingtheBlochequationintoaneigenvalueproblem, asillustratedbyexpanding : H n P X C = E n P X C (3–20) ProjectingbyPontheleftproducesareducedspaceeffectiv eHamiltonianwhose eigenvaluesaretheenergies E .Since H eff B isnon-Hermitian,thematrix C denesthe right-handeigenvectorsinabi-orthogonalset.Thisisthe BlocheffectiveHamiltonian whichisasimilaritytransformationoftheexactelectroni cHamiltonianwiththesame eigenvalues[ 82 ]. H eff B = PH n P (3–21) 41

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Thereferencewavefunctions aretheeigenvectorsofthe H eff B .Thebenetof theBlocheffectiveHamiltonianistheabilitytodetermine theelectronicenergiesin areduceddimensionmatrixproblemratherthanthefullspac eofavailablestates. Thisisachievedthroughperturbativedeterminationof n andtheapplicationofmatrix consistencyconditions.Theuseofperturbationtheory(or coupled-clustertheory) pushestheeffortoftreatingcorrelationonto n andallowsforatractable P spacerather thanrequiringalarge,CI-likereferencespace.Thematrix consistencyconditions imposerestraintson n sothattheeffectiveHamiltonianonlyneedstobedenedfor the statesofinterest,the P space. Thecomplementaryspaceto P willbedenedas Q withprojector Q =1 P whichincludesallorbitalexcitationsfromdeterminantsi nthe P space.Implicitinthe abovederivationistheconditionthat n Q =0orthatthereisnoattempttosolveforthe wavefunctionsinthe Q space. Startingfromthewaveoperatorequationobscuresthemathe maticsenablingthe Blochequationwhichisjustasimilaritytransformofthetr ueelectronicHamiltonianwith rotationmatrix U : U =1+ Q n P U 1 =1 Q n P (3–22) ThetransformedHamiltonianisgivenby: H = H + HQ n P Q n PH Q n PHQ n P (3–23) andisshowninFigure 3-1 .Theeachblockof tildeH isgivenby: 42

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Figure3-1.SimilaritytransformationfortheBlocheffect iveHamiltonianallowsfor n and thecorrecteigenvaluestobedeterminedthoughblockdiago nalization. P HP = PHP + PHQ n P = PH n P P HQ = PHQ Q HP = QHP + QHQ n P Q n PHP Q n PHQ n P = QH n P Q n PH n P Q HQ = QHQ Q n PHQ (3–24) The P HP blockistheBlocheffectiveHamiltonian. n canbedeterminedbyinsisting Q HP =0whichblockdiagonalizes H .With n dened,theBlocheffectiveHamiltonianis completelydeterminedand,duetotheblockdiagonalizatio n,theeigenvaluesareexact. 43

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DeningPerturbativeOrder Thewaveoperator n isdeterminedbyperturbationtheory,specicallythe Brillouin-Wignerformulation[ 83 ].Thezeroth-orderapproximationto n is P ,the referencestateprojector.Higherordertermsmap P statesinto Q states. n = P + Q n (1) + Q n (2) +...,(3–25) TheinniteorderrealizationisFock-spacecoupled-clust er(FSCC)where n satisesan exponentialansatz[ 78 ]. n = e S = P + S + 1 2 S 2 +...(3–26) Theoperator S isanexcitationoperatorlike T butgeneralizedforanynumberof electronsandcanbeseparatedintooperatorsspecicforea chcase.Thepartof S that representstheN-electronwavefunction( S [0,0] )isthe T operatorfromCCtheory,whichis determinedbeforetherestof S .Assuch, T canbeextractedfromthetotalexponential: n = e T e S .(3–27) Therestoftheexponential, e S ,isdenedbytheBlochequation;theN-1portionis writtenas( S [0,1] )andtheN+1portionis S [1,0] .Tomakethenotationuniversalforboth theN-1andN+1systems,ageneral S willbeusedtostandinforeither S [0,1] or S [1,0] (theseoperatorsdonotinteractfortheequationsbelow,bu tthisisnotgenerallytrue fordifferentsectorsofFockSpace).Weareinterestedinse cond-orderenergiesand orbitalswhichrequires n tobedeterminedtorstorder.Bysplittingtheelectronic Hamiltonianintoazeroth-orderapproximation( H 0 )andarst-orderperturbation( V ),the BlochEquation 4–8 canbecastintoaperturbativeexpansion.Therstorderde nition of n isgivenbytheQPblock: 0= QH 0 Q n (1) P + QVP Q n (1) PH eff P (3–28) 44

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Beforedetermining n itisconvenienttonormalordertheBlochequationtosimpli fythe diagrammaticrepresentation. f H n g P = H n P h 0 j H n j 0 i = H n P h 0 j H (1+ T ) j 0 i = H n P ( E 0 + E 1 + E 2 ) (3–29) Theeigenvaluesfromthenormalorderedequationsbecomeen ergy differences–ionizationpotentialsfortheN-1systemande lectronafnitiesforthe N+1states.Normalorderingdoesnotchangethedenitionof n ,butdoessimplifythe derivation.Constructingarst-order n requirestruncationoftheexponentialformwhich createsalinearapproximation: n (0+1) =1+ T (1) + S (1) (3–30) Usingthisdenition,Equation 3–28 becomes 0= Q f H 0 gf S (1) g P + Q f V g P Q f S (1) g P f H eff g P (3–31) SincePisacompleteactivespace, Q islimitedtoallsingleexcitationsfromstatesin P Theelementsof S (1) aredeterminediterativelyuntilconvergence.Theexactfo rmofthe equationsarereadilydeterminedfromthediagrammaticrep resentation(Figure 3-2 ). 0= + (1) (1) Figure3-2.Diagrammaticrepresentationof n determiningequationsforthe(1,0)and (0,1)sectorsinskeletonform.The representsa H 0 interactionandthebox isthe H eff B interaction. Nowthat n throughrstorderisdetermined, H eff B canbedenedthroughsecond order.FromEquation 4–8 H eff B isthecontributionthroughsecondorderofthePP 45

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block. PH (2) eff B P = P f H 0 g P + P f V g P + P f V g n (1) P (3–32) Thezerothandrst-orderexpressionsjustrecoverthenorm alorderedHamiltonian whilethesecond-orderexpressionincorporatesthe Q space.Expanding n (1) usingthe linked-diagramtheorem[ 84 ]givesthesecondorder H eff B : PH (2) eff B P = P f H g P + P ( f V g T (1) 1 ) L P + P ( f V g T (1) 2 ) L P + P ( f V g S (1) 2 ) L P (3–33) The PP blockeffectiveHamiltonianthroughsecondorderisshownd iagrammaticallyin Figure 3-3 = f + (1) + f (1) + (1) + (1) f + (1) Figure3-3.Diagrammaticrepresentationof H eff B equationforthe(1,0)and(0,1)sectors inskeletonform.Theboxrepresents H eff B H eff B isdeterminedbyiterativelyupdating n andthen H eff B untilbothconverge.This interdependenceisarenormalizationprocessinherenttoF ock-Spaceapproaches. Onceconvergenceisreached, H eff B isdiagonalizedtoobtaineigenvaluesand eigenvectorsor,inthecontextofCOT,orbitalenergiesand orbitalrotationmatrices. Since H eff B ismanifestlynon-Hermitian,theconstructionoforbitals fromthe eigenvectorsof H eff B wouldrequireabi-orthogonalsetoforbitals.Alternative ly,the BlochequationcanbecastintoanHermitianformallowingfo ronesetoforbitals. Hermitization TheoriginalelectronicHamiltonianisHermitian,butenfo rcingtheexponentialansatz throughasimilaritytransformationcausestheresultinge ffectiveHamiltoniantobe non-Hermitian.Alternativesimilaritytransformationsr esultinthesameeigenvalues 46

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butdistinctlydifferenteffectiveHamiltonians,themost commonofwhichare:theBloch effectiveHamiltonian,the ˆ OkuboeffectiveHamiltonian(whichistheadjointoftheBlo ch effectiveHamiltonian),andthecanonicalHermitiandesCl oizeauxeffectiveHamiltonian [ 82 ].ThedesCloizeauxeffectiveHamiltonianisobtainedbyro tating H bythefollowing similaritytransformation: U = n ( n y n ) 1 2 P U 1 = P ( n y n ) 1 2 n y (3–34) ThisproducesthedesCloizeauxeffectiveHamiltonian: H eff dC = PU 1 HUP = P ( n y n ) 1 2 n y H n ( n y n ) 1 2 P (3–35) ThiscanbeiteratedliketheBlochequationwithwitheachit erationrequiring simultaneousdeterminationof n y and n .MeissnerandNooijendemonstratedthat thedesCloizeauxeffectiveHamiltonianexpressioncanbef ormulatedasasimilarity transformationactingontheBlocheffectiveHamiltonianr atherthanelectronic Hamiltonian[ 85 ]. H eff dC = P ( n y n ) 1 2 H eff B ( n y n ) 1 2 P (3–36) ThisallowsfortheHermitizationtobeimposedafterconver genceoftheBlocheffective HamiltonianbyasimilaritytransformationoftheBlochequ ation.Unfortunately,this rotationrequiresndingtheinverseof( n y n ),whichisnotguaranteedtoexist. AnalternativeHermitizationcanbeachievedbymimickingH ylleraasvariational approachfortheFock-Spaceformulation.Intraditionalpe rturbationtheory,theHylleraas equationsarederivedbyprojectingtheadjointoftherstorderwavefunctionontothe rst-orderequation,Equation 3–7 .Thisproducesanzerotermthatcanbeaddedto thesecondorderenergy.Thisstructurecanbereproducedwi ththeBlochequationby 47

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projectingwith P n (1) y ontotheequationdening n (1) (Equation 3–28 ). 0= PS (1) y Q f H 0 g Q f S (1) g P + PS (1) y Q f V g P PS (1) y Q f S (1) g P f H eff g P (3–37) IncorporatingboththisexpressionandtheHylleraasVaria tionalexpression, Equation 3–7 ,tothesecond-orderBlocheffectiveHamiltonianproduces aHermitian second-ordereffectiveHamiltonian: PH (2) eff P = PFP + P ( VT (1) ) L P + P ( T (1) y V ) L P + P ( T (1) y H 0 T (1) ) L P + P ( f V g S (1) ) L P + P ( S (1) y V ) L P + P ( S (1) y H 0 S (1) ) L P P ( S (1) y S (1) PH eff ) L P (3–38) Asinglesetoforthonormalorbitalscanbedeterminedfromt hiseffectiveHamiltonian, H eff ,whichisconstructedafterconvergenceoftheBlochequati on. IntruderStates SinceCOTisreliantonFock-Spacetools,intruderstatesar eafundamentalconcern. Intruderstatesrefertoexcitedstatesthatformneardegen eracieswiththestatesin thePspace.Thisresultsindivergentiterations,theoccas ionalzerodenominator,and spuriousresultsthataredifculttodetect[ 86 – 89 ].Theissuearisefromextracting eigenvaluesfromareduceddimensionalspaceproblem.Thed imensionalityis reducedbydetermining Omega perturbatively,whichusesenergydifferencesasa preconditioningtechnique.Asaresult,intruderstatesar egivenanarticiallyhigh signicancebecauseofthispreconditioning. Thereareseveralapproachestoaddresstheintruderstatep roblem.Approximate approachessuchasdenominatorshifts(IntruderStateAvoi dance[ 87 ])orneglect ofintruderstatesareefcient,butinaccurate.Intermedi ateHamiltoniansallowfor neutralizationofintruderstatesandexactresultsbysele ctinganactivespacewithinP [ 90 ]ordressingtheHamiltonian[ 85 91 ].Theonlydrawback,andaminoroneatthat,is thatchoosinganactivespacemeansthatCOTwouldnolongerb eanallorbitalmethod. TheMixed-SectorIntermediateHamiltonian,alternativel y,increasethePspaceinto 48

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theQspace[ 92 ].Unfortunately,thisapproachdependsonhavingasufcie ntlylarge auxiliaryspaceaddedtothePspace.AnewintermediateHami ltonianwasconstructed forthisproblem,theCanonical-MixedSectorIntermediate Hamiltonian(CMSIH).CMSIH hybridizesFock-SpaceandEOMtechniquestoallowthetreat mentofintruderstatesby addingthemtothePspace.Addingonlytheintruderstatesto thePspaceallowsfor anexactdeterminationof Omega andalleigenvalueswithoutthedependenceonan increasinglylargeQspace.Detailsofthismethodwillbedi scussedinthenextchapter. DeningOrbitalsfrom H eff Aspreviouslymentioned,theBlochequationcanbeusedtode ne occupied-occupiedrotationsthroughtheN-1electronequa tionandvirtual-virtual rotationsspacethroughtheN+1equation.Toelaborate,dia gonalizationofthe Hermitized H eff fortheN-1systemproducesasetorthogonaleigenvectorswh ich allowsforanewdenitionofoccupiedorbitals i .Thisispossiblebecausethereference functions, N 1 ,inthe P N 1 spacespanthesetofalldeterminantswithoneorbital removedfromtheNelectronreference.Theeigenvectorstha tdiagonalize H eff N 1 areorthogonallinearcombinationsofthesetof N 1 .Toillustrate,theeigenvector i is givenby i = X j C ji j j 0 i (3–39) Fromthissummation,aneworbital i canbedenedas: i = X j C ji j .(3–40) TheseorbitalsinherittheorthogonalityoftheN-1states. Thisissimilarlytrueforthe electronattachedstateswitharotationoverallvirtualor bitals.Thus,theeigenvector matrix C IP denesaunitarytransformationoftheoccupiedorbitals.T hisissimilarlytrue with C EA 49

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TheOVrotationiscarriedoutlikeatraditionalBruecknero rbitalrotationwhereeach orbitalisupdatedassumingalinearapproximation: i = i + X a t a [(1)+(2)] i a a = a X i t a [(1)+(2)] i i (3–41) Withtheexceptionatconvergence,thenewlyupdatedorbita lsarenotorthonormaland requireaGram-Schmidtprocedure. WhiletheOO,VV,andOVblocksarediscussedandformulateda sseparate problems,theyareinherentlyinterdependentmustbeupdat edsimultaneously.This isachievedthroughasinglematrixrotation(U)ofthemolec ularorbitals,whereUis givenby: U = 264 C IP T ( 1 )+( 2 ) 1 T ( 1 )+( 2 ) 1 C EA 375 (3–42) PsuedocodeOutline Convergenceofthecalculationisachievedwhentheorbital sandorbitalenergies becomestationarytoadesiredlevelofprecision.Theitera tiveprocessinvolvesa macro-iterativeprocedurewithnestedmicro-iterativest eps.Theprocedureisoutlined below.1.Dene H 0 = f + v c 2.Performsecond-orderenergycalculation. (a)CalculateHartree-FockEnergy(b)Buildrstorder T 1 and T 2 (c)Calculate E 2 3.Buildsecondorder T 1 50

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4.Calculatesecond-orderIonizationPotentials (a)IdentifyintruderstateswithCMSIH(b)Build S [0,1] 2 fornon-intruderstates (c)BuildanddiagonalizetheCMSIHintermediateHamiltoni an (d)Extract H [0,1] eff and S [0,1] 2 (e)Ifnotconvergedreturnto4.(a) 5.Calculatesecond-orderElectronAfnities (a)IdentifyintruderstateswithCMSIH(b)Build S [1,0] 2 fornon-intruderstates (c)BuildanddiagonalizetheCMSIHintermediateHamiltoni an (d)Extract H [1,0] eff and S [1,0] 2 (e)Ifnotconvergedreturnto4.(a) 6.Hermitize H [0,1] eff and H [1,0] eff (a)Build S [1,0] y 2 and S [0,1] y 2 (b)ConstructFS-HylleraasFunctional(c)Diagonalizetoobtain C IP and C IP 7.Updateorbitals8.Checkforconvergence,ifnotconvergedreturnto1 COTasanExtendableTool Therearetwologicalextensionsthatmaygreatlyincreaset hepowerofCOT.First, modicationofthemany-bodyenergyexpressiontofullyuti lizetheCOTorbitals.This canbeassimpleasextendingtheconceptofHF-DFT;usingCOT orbitalstodenea densitytobeusedinaDFTmany-bodyenergyexpression.Unli keKohn-Shamorbitals, COTarephysicalandaremotivatedfromamaximumoverlapcon ditionandasaresult willnotsufferfrommanyofthesameproblemsthatDFTorbita lsdo.Inamuchmore speculativesense,utilizingeithertoolsfromGreen'sfun ctionsordynamicalmeaneld 51

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theorymaygreatlyincreasetheusefulnessofCOT.Thesecon dtypeofextensionisthe allowanceforfunctionalizationattheIPandEAlevelwitho uttrivializingthewavefunction natureoftheequations.Therearethreetypesoffunctional sthatcanbeused: 1.Directfunctionalizationof H eff ,specicallyaddingtermsto PH n P suchasselect higherorderterms(LCCDorRPA)orexternalpotentialsfrom DFTorother sources. 2.Couplingfunctionsthatmanipulatetheinteractionwith n .Atthemomenttheseare onlytermsofthetype PHQ buttheycanincludeacorrelationoperator. 3.Wavefunctionmanipulationbyaddingselecthigherorder termsintheFS expansiontocreateabetterdescriptionof n TheprimaryintentofCOTisprovideaframeworktoviewquant umchemistry differentlythatmaintainsveriablecriteria.Inotherwo rds,provideaplatformfor experimentaltheorythathasamoremeaningfulmeasurethan totalenergy,namelyIP's andEA'sandtheircorrespondingorbitals.Thishasbeenach ieved. 52

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CHAPTER4 CANONICALMIXED-SECTORINTERMEDIATEHAMILTONIAN:HYBRID IZATIONOF EQUATION-OF-MOTIONANDFOCK-SPACECOUPLED-CLUSTER TheIntruderStateProblem Fock-spacecoupled-cluster(FSCC)isapowerfultoolwhich usesawave-operator todescribeexcitedstatesandstatesofdifferentparticle numberfromtheN-electron groundstatecorrelatedHamiltonian[ 93 94 ].Thisallowsforextensionofcoupled-cluster (CC)groundstatetools[ 95 – 97 ]tootheraspectsofchemistry.FSCCconstructsan effectiveHamiltonianinareduceddimensionmodelspaceth roughaperturbatively determinedwave-operator.Energiesandcorrespondingwav efunctionsofstatesinthe modelspacearedeterminedfromtheeffectiveHamiltonian. Equation-of-motionCC (EOM-CC),bycontrast,producesequivalentresultsbydiag onalizingthecorrelated Hamiltonianintheentirerelevantdeterminantspace[ 98 – 102 ].TheFSCCeffective Hamiltonianisacoupled-clusterformulationoftheBloche ffectiveHamiltonian,a similaritytransformationoftheelectronicHamiltonianw hichallowsforthereduced dimensiondescription[ 79 – 81 ]. EffectiveHamiltoniansingeneralandFSCCspecicallysuf ferfromconvergence issuesrelatedtointruderstatesorneardegenercieswiths tatesinthemodelspace.The issuesrelatedtointruderstatescanbehandledwithaninte rmediateHamiltonian,an extendedeffectiveHamiltonianwhichexactlydescribesst atesinthemodelspaceatthe costofapproximatelytreatingstatesinanintermediatere ferencespace[ 103 ].Thisis achievedthroughamodiedwave-operatorofwhichthereare multiplechoices[ 104 ]. Alternatively,intruderstatescanbeavoidedwithadresse dHamiltonianproducedby doublesimilaritytransformation[ 85 91 105 ]. CanonicalBlochEquation:theInterconnectionBetweenFSC CandEOM FSCCreferstotheapplicationofanexponentialansatztoth eBlochequation.The Blochequationutilizesawave-operator, n ,toprojectareferencestate( )toanexact 53

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wavefunction( ). = n = X n C (4–1) Eachreferencestateisalinearcombinationofknownrefere nceSlaterdeterminants ( ).Thebasisofallpossiblestatescanbesplitintotworelev antspacesand correspondingprojectionoperators: P containsallthereferencedeterminantsand Q =1 P isthecomplimentaryspace.Theenergyofeachreferencesta te,the coefcients( C ),andthewave-operator( n )arealldeterminedbytheBlochequation framework.TheBlochequationisderivedthroughasimilari tytransformationofthe electronicHamiltonianwhichyieldsaneffectiveHamilton ianthatconnectsthereference functionstomany-bodyenergies. PH eff P = PH n P = E (4–2) Manipulationofidentityconditionsof n allowstheBlochequationtobewrittenin compactform: H n = n H eff .(4–3) n isageneralizedwave-operatorwhichcanbecastintoanexpo nentialansatz: n = e S S = T + S [0,1] + S [0,1] +... (4–4) Thecoupled-clusterexponentialcanbeextractedfrom n : n = e T e S = e T n (4–5) ProjectingEquation 4–3 by e T yieldstheFSCCequations: H n = n H eff .(4–6) 54

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Theamplitudesof S aredeterminedbyprojectingeach Q stateontheaboveequation whichproducesasetofconsistencyequations. 0= Q H n Q n H eff .(4–7) Thisisequivalenttoblockdiagonalizationthesimilarity transformedH.Thesimilarity transformedHisgivenby: H = U 1 HU = PH n P + PHQ + Q ( H n n PH n ) P + Q ( H n H ) Q U = 264 P 0 Q n PQ 375 U 1 = 264 P 0 Q n PQ 375 (4–8) The PP blockis H eff andthe QP blockdenes n .Thiscanberecastintoanmatrix eigenvectorform: PU 1 HU P C P = PH eff PC P = C P E P HUC P = UC P E P (4–9) Theabovematrixequationissplitintocongurationalspac es P and Q [ 106 107 ]. 264 H PP H PQ H QP H QQ 375 264 C P C P 375 = 264 C P C P 375 E P (4–10) Expanding n toitsexponentialformandprojectingby e T theproducestheCanonical Blochframework[ 106 107 ]. 264 H PP H PQ H QP H QQ 375 264 C P C P 375 = 264 C P C P 375 E P (4–11) 55

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PartitioningtheseequationsrecoverstheFSCCequations. H PP C P + H PQ C P = C P E P H QP C P + H QQ C P = C P E P (4–12) Relabelingeigenvectorsas R P = C P and R Q = C P showsthattheaboveequationsare theEOM-CCequations[ 107 ]. 264 H PP H PQ H QP H QQ 375 264 R P R Q 375 = 264 R P R Q 375 E P (4–13) EOMapproachesutilizecongurationinteractiontoolswit hacorrelatedHamiltonian( H ) ratherthanthebareHamiltonian( H ).Inthiscontext,besidestheperturbationorderwith respecttodening T and S ,eachadditionalblockof H carriesanadditionalorder.The followingnotationwillbeusedwhere H ( N ) impliestheperturbationorderofeachblock. H ( N ) = P H ( N ) PP P + Q H ( N 1) QP P + P H ( N 1) PQ Q + Q H ( N 2) QQ Q (4–14) Theorderof n isambiguousasthedenitionisxedbythechoiceof H ( N ) .The discussionoforderbecomesirrelevantinaninniteorderC Cimplementation. CanonicalMixed-SectorIntermediateHamiltonian ThecomplementaryframeworksofFSCCandEOM-CCcanbemanip ulatedto avoidtheintruderstateprobleminFSCC.Intruderstatesar eonlyproblematicinthe determinationof n wheretheireffectisdemonstratedbydivergentamplitudes orover weightedamplitudeswhichproducespuriousresults.Howev er,thesestatesarereal anddohavecorrectamplitudesthatjustcan'tbedetermined throughtraditionalFSCC approaches.Alternatively,theEOM-CCstructurecanberec asttoselectivelydetermine theamplitudesfortheseproblematicintruderstates.This hybridizationofEOM-CCand FSCCwillbecalledtheCanonicalMixed-SectorIntermediat eHamiltonian(CMSIH). 56

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Thesetofalldeterminantswillbesplitintothreespaces: P thestandardsetofall referencestatesforthesectorofinterest, Z thesetofintruderstates,and Q 0 =1 P Z theremainingstates.Thedenitionof Q =1 P willpersist. n stillmapsstatesin P to exciteddeterminantsinboth Q 0 and Z n = P +( Q 0 + Z ) n P (4–15) n isseparatedintoaportiontobedeterminedperturbatively ( Q 0 n P )andportionto bedeterminedfromtheeigenvectorcoefcients( C Z = Z n PC P ).ThecanonicalBloch Equation( 4–11 )isrewrittentoreecttheredenedspaces. 2666664 H ( N ) PP H ( N 1 ) PZ H ( N 1 ) PQ 0 H ( N 1 ) Z 0 P H ( N 2 ) ZZ H ( N 2 ) ZQ 0 H ( N 1 ) Q 0 P H ( N 2 ) Q 0 Z H ( N 2 ) Q 0 Q 0 3777775 2666664 C P C Z Q 0 PC P 3777775 = 2666664 C P C Z Q 0 PC P 3777775 E P (4–16) Thematrixispartitionedintothreeequations: H ( N ) PP C P + H ( N 1 ) PZ C Z + H ( N 1 ) PQ 0 Q 0 PC P = C P E P (4–17) H ( N 1 ) ZP C P + H ( N 2 ) ZZ C Z + H ( N 2 ) ZQ 0 Q 0 PC P = C Z E P (4–18) H ( N 1 ) Q 0 P C P + H ( N 2 ) Q 0 Z C Z + H ( N 2 ) Q 0 Q 0 Q 0 PC P = Q 0 PC P E P (4–19) Equations 4–17 and 4–18 formareducedspaceeffectiveHamiltonian,ortheCMSIH: H ( N ) CMSIH = P H ( N ) ( P + Q 0 ) n P + Z H ( N ) ( P + Q 0 ) n P + P H ( N ) Z + Z H ( N ) Z = P ( H ( N ) PP + H ( N 1) PQ 0 Q 0 n ) P + Z ( H ( N 1) ZP + H ( N 2) ZQ 0 Q 0 n ) P + P H ( N 1) PZ Z + Z H ( N 2) ZZ Z (4–20) 57

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Equation 4–19 denes Q 0 n P : 0= Q 0 H ( N ) PC P + Q 0 H ( N ) Z PC P + Q 0 H ( N ) Q 0 PC P Q 0 PC P E P = Q 0 H ( N ) PC P Q 0 PC P C yP H ( N ) eff B C P =[ Q 0 H ( N ) P Q 0 PH ( N ) eff B P ] C P = Q 0 H ( N ) P Q 0 PH ( N ) eff B P = Q 0 H ( N 1 ) Q 0 P P + Q 0 H ( N 2 ) Q 0 Q 0 ( Z + Q 0 ) P Q 0 PH ( N ) eff B P (4–21) Onlyamplitudescorrespondingtodeterminantsin Q 0 aredenedbyEquation 4–21 However,theamplitudesaredenedbytheirinteractionwit h H eff B whichcontainsall amplitudesin n notjustthe Q 0 ones. TheCMSIHisaniterativeprocedurethatproducesexactFSCC eigenvalues,exact eigenvectors,andtheexact H eff B .Thecomputationalprocedureisgivenasfollows: 1.Identifyintruderstatesanddene Z 2.Load S amplitudesfrompreviousiteration,holdingamplitudesco rrespondingto statesin Z xed. 3.Build S usingEquation 4–21 4.Build H CMSIH usingEquation 4–20 andnewlydened S 5.Diagonalize H CMSIH 6.Extract H eff B = C P E P C 1 P 7.Extract Z P = C Z C 1 P 8.Checkforconvergence.Ifnotconvergedreturnto1. IdenticationofIntruderStates Themostrobustwaytoidentifyanintruderstatesisadirect analysisofenergy denominatorsthatoccurinthedeterminationof n .Specically,ifthemagnitudeofa denominatorislessthanoneatomicunit,thecorresponding statecanbeclassied 58

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asanintruderstate.Thiswilloverestimatethenumberofin trudersandclassify non-intruderstatesassuch,butthisguaranteessuccessan distherudimentary approachusedinthispaper.Thiscriteriacanbenedtunedo nacasebycasebasis, butthisisnotpreferred.Efforthasbeengiventotheidenti cationofintruderstatesin thecontextofmulti-referencesecond-orderperturbation theory(MRMP2)[ 86 87 108 ]. Inthiscontextanintruderstatecanbeclassiedasanydete rminantthatproduces asmallenergydenominatorrelativeachosenthresholdandh asanon-negligible contributiontotheenergy.Suchdeterminantscanbesubjec tedtoatwo-state perturbativeanalysistosurmisewhethereachdeterminant isproblematicthrough thecalculationofarepresentativeradiusofconvergence( R c ).Statesthatproducea small R c ,formallyanyradiuslessthanone,willcauseadivergentpe rturbation.This modeldoesnottranslatedirectlytothediscussionofintru derstatesinFSCCasMRMP2 isamulti-referenceRaleigh-Schr ¨ odingerperturbationwhereonlyonerootisdesired. Perhapsamodicationofthisapproachwillresultinanefc ientmethodtoidentifyall intruderstates. ComparisonwithMSIH Themixed-sectorintermediateHamiltonian(MSIH)isanano therintermediate Hamiltonianthatexpandsthereferencespace( P )toincludeanintermediatespace ( Z )whichincludesstatesfrom Q [ 92 109 ].Thisisachievedthroughanalternative wave-operator( R )andacorrespondingmodiedsimilaritytransformation( U R ). R = R + Q 0 RP U R =1+ Q 0 RP (4–22) TheamplitudesinRaredenedbyasetofconsistencyequatio nswiththeconvention wave-operatorfromtheBlochequation, n ;specically R n P = n P and Q 0 n HRP = Q 0 HRP .Fromtheserelations,anequationquantifyingthediffere ncebetween n and R canbedened.However,thechoiceof Z needstosufcientlylargesothat Q 0 RP = 59

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Q 0 n P causingtheworkingequationsotobedeterminedpurelyby n .Theamplitudesof n aregivenby[ 92 ]: Q 0 n H ( P + Q 0 ) n P = Q 0 H ( P + Q 0 ) n P (4–23) TheMSIHisgivenby: H I = PH ( P + Q 0 ) n P + ZH ( P + Q 0 ) n P + PHZ + ZHZ .(4–24) TherearethreeprimarydifferencebetweentheCMSIHandthe MSIH.First,the amplitudeequationfortheCMSIHisanexactformulationand doesnotrequirean increasinglylarge Z toobtainexactresults.Atthecorrectlimit,theMSIHampli tudes convergetotheCMSIHamplitudes.Theamplitudesfromthe Z statesareabsentfrom Equation 4–23 whichpreventscorrectcouplingwiththeseamplitudeslead ingtothe needforalargeenough Z spacetocreateabufferbetween P and Q 0 ,unlikeEquation 4–21 .Second,theMSIHdoesnotconsidertheorderoftheHamilton ian.Thisisaminor pointasthisisnotanissueforinniteorderFSCC.Lastly,t heMSIHdoesnotgiven correcteigenvectorswhiletheCMSIHdoesbydesign. 60

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CHAPTER5 CONCLUSIONS Severalnewapproachestoquantumchemistryarederivedtoe stablish functionalizedsecond-orderperturbationtheoryasanapp roachforconductingquantum mechaniccalculationsformaterialsscience. Anon-localextensiontoOEPispresented.Thisremovesthei nexacttreatmentof exchangethatoccursinlocalOEPand abinitioDFT .Thisbecomesaviabletoolfor orbitaloptimizationofanydensityfunctionalordensitym atrixfunctionalinadditiontoab initoapproaches.Asecond-orderimplementationisderive d(OO-MBPT2)alongwitha diagonalcorrectiontotheorbitaleigenvalues(A-MBPT2). Thiseigenvaluecorrectionis thediagonalelementsoftheone-electronpartofthecorrel atedeffectiveHamiltonian, H .Includingthispieceof H in H 0 resultsinaninniteordereffectatsecond-order.This frameworkprovidesaconsistentmeanstoprovideoptimized orbitalsanddensitiesfor creatinghybridfunctionals.Additionally,theequivalen ceoflocalOEPand abinitioDFT isdemonstratedthroughsecondorderperturbationtheory. Similarly,COTprovidesanorbitaloptimizationbasedonab initiotheory,correlated ionizationpotentialsandelectronafnities,andanimpro ved H 0 toincreasethe relevancyofthesecondordermany-bodyenergy.However,th erelevanceofCOTto quantumchemistryislimitedrelativetoitspotential.Att hemoment,themany-body energyislimitedtoperturbativeorcoupled-clusterexpre ssions.Inotherwords,despite beinganindependentformulationwithphysicallyimportan torbitals,thetotalenergy doesnotbreaknewgroundbeyondtheconceptofanimprovedse condorder. Thecanonicalmixed-sectorintermediateHamiltonianisan ewformofintermediate HamiltonianthatextendstheframeworkofFSCCtoutilizeEO M-CCtools.Thismethod allowsfordeterminationofallstateswith P (e.g.whenallorbitalsactiveinIPand EA)andtheircorrespondingeigenvectorsbyextendingthem odelspacetoinclude allintruderstates.Note,thisapproachhasallthefunctio nalityofFSCCandcan 61

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isolateindividualstatesorvalencestatesandonlyrequir es Z toincludeintruderstates correspondingtostatesin P .Properidenticationofintruderstatesisstillneededto renethedeterminationof Z 62

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BIOGRAPHICALSKETCH MatthewStrasbergwasborninPortland,Oregon.Aftergradu atingfromClarkstown HighSchoolNorthinNewCity,NewYork,MatthewattendedCor nellUniversity.At Cornell,heperusedadegreeinappliedphysicswithaintere stinthemechanical behaviorofmaterials.HisexperiencesatCornell,combine dwithresearchopportunities atBrookhavenNationalLaboratoryandStanfordLinearAcce leratorCenter,convinced topursueagraduatedegreeinmaterialscience.Heattended theUniversityofFlorida forhisgraduatestudiesandthisdocumentisatestamenttha thenished.Hereceived hisPh.D.fromtheUniversityofFloridainthefallof2013.M atthewcurrentlyalive.To thedeterminantoftheworld,thiswillchangeeventually.D onotdryyourtearswiththis document. 69