SYSTEMATICIMPROVEMENTSOFAB-INITIOIN-MEDIUMSIMILARITYRENORMALIZATIONGROUPCALCULATIONSByTitusDanMorrisADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofPhysics-DoctorofPhilosophy2016ABSTRACTSYSTEMATICIMPROVEMENTSOFAB-INITIOIN-MEDIUMSIMILARITYRENORMALIZATIONGROUPCALCULATIONSByTitusDanMorrisTheIn-MediumSimilarityRenormalizationGroup(IM-SRG)isanabinitiomany-bodymethodthathasenjoyedincreasingprominenceinnucleartheory,duetoitssoftpolynomialscalingwithsystemsize,andtheytotargetgroundandexcitedstatesofbothclosed-andopen-shellsystems.DespitemanysuccessfulapplicationsoftheIM-SRGtomicroscopiccalculationsofmedium-massnucleiinrecentyears,theconventionalformulationofthemethodanumberoflimitations.Keyamongstthesearei)largememorydemandsthatlimitcalculationsinheaviersystemsandrenderthecalculationofobservablesbesidesenergyspectraextremelyandii)thelackofacomputationallyfeasiblesequenceofimprovedapproximationsthatconvergetotheexactsolutionintheappropriatelimit,therebyverifyingthattheIM-SRGissystematicallyimprovable.Inthisthesis,IpresentanovelformulationoftheIM-SRGbasedontheMagnusexpansion.Iwillshowthatthisimprovedformulation,guidedbyintuitiongleanedfromadiagrammaticanalysisoftheperturbativecontentofttruncationsandparallelswithcoupled-clustertheory,allowsonetobypassthecomputationallimitationsoftraditionalimplementations,andprovidescomputationallyviableapproximationsthatgobeyondthetruncationsusedtodate.TheenessofthenewMagnusformulationisillustratedforseveralmany-nucleonandmany-electronsystems.ACKNOWLEDGMENTSThelistofpeoplewhoIhavebecomeindebtedtoalongthewayisvast,ifIhaveforgottenanyoneinthislist,Icanonlythankthemforpatience,andhopetheyunderstand.Tobeginwith,IowemyparentseverythingthatIam.Theyarebothspecialpeoplewhosupportedmealongthewayineverywayimaginable.TheyhavealwaysbeenthereformewithcallswhenIneedencouragement,andoccasionallyafewbuckswhenthingsweretight.Thereisnowaytounderstatetheirimportancetome.BeforemytimeatMichiganState,IlearnedmuchfromDr.WilliamOliverIII,whomentoredmewhileIearnedmyMastersdegreestudyingglassyliquidsattheUniversityofArkansas.HetaughtmepatiencebyhavingmelearnhowtoloadDiamondAnvilCells.Inaddition,hehelpednursealoveofphysicsinme,despitebeingconstantlymorebusythananysinglepersonshouldbe.Iwillalwaysthinkfondlyofmytimeinhislab,andTimRansom,whoworkedcloselywithmemylastyearinArkansas.IwouldberemissifIdidn'tmentionthemanyfriendsandlovedonesthathavebeenthereformefortoughtimesandgood.IhaveneverwantedforgoodfriendstogomournthecoldtemperaturesoverabeerinsideatCrunchy's,ortocelebratethereturnofthesuninthespringonapatio.IwouldliketomentionafewpeoplewhohavecomeandgoneduringmytimeherewhoIsincerelymiss,orwillmissuponleaving.SteveQuinn,ZachandJaydaMeisel,AnthonySchneider,NikkiLarsonareafewoftheclosefriendsImadeintown,andsomeofmyfavoritepeopletoseeonadanceor.OrenYair,KortneyKooper,andChrisProkophavecometobeclosefriendsthatIcanalwayscountonseeingatourweeklypoolnightattheRiv,andcountontolistentomymidweekproblems.InmylastyearhereIhavehadthepleasureofgettingtoknowDebraRichman,whointroducedmetothejoyofiiigettingthrashedatSeinfeldtrivia;shehasmademylastyearhereajoy.AllofthesepeopleIwillcertainlymissdearly,andwillhaveonlyfondmemoriesfortherestofmylife.Myresearchgrouphasbeenahugeblessing.Icouldnothaveaskedforabettergroupoffriendstoworkwith.DiscussionswithNathanParzuchowskihaveyieldedgreatinspirationandinsight,aswellasjustgenerallybeingfun.Hisrelentlessworkethichascertainlysetagoodexampleformetofollow,oralittlebitofshamewhenIwasnot.FeiYuan'sencyclope-dicknowledgeofallthingscomputershasbeeninvaluableanytimeacompiler/libraryerrorpoppedup.ThefrequentgroupluncheswithJustinLietz,AdamJones,andSamNovariohavealwaysbeenfun.Everyoneinthegrouphasbeenamazing.Mostofall,Ihavetothankmyadvisors,ScottBognerandMortenHjorth-Jensen,whobothdeservemydeepestthanks.Theyhavealwaysbeentheretoanswerquestionsaboutahostofissues.ScottandMortenhavealwayshadtimeforallthegraduatestudentsinthegroup,evenasthegrouphasgrownquitelarge.Evenduringtheirbusiesttimes,theyhavemadetimetokeepmeontrack.Ihavealwaysbeenencouragedtopursuemyownareasofinterest,andhelpedindoingso.Despitetheirclearleadingofthegroup,Ihaveneverfeltasanythingbutacolleagueandfriendtoboth.Idonotknowhowtheyaccomplishthis,butIfeelblessedbyit.HopefullybythetimeIhavestudentsofmyown,ifthateverhappens,I'llhaveouthowtocreatethissameatmosphere.Andinthelastyear,withthehiringofHeikoHergert,Ihavebeenblessedwithathirddeepwellofexpertisetodrawfrom.Thesethreehavemygreatestappreciation.Lastly,Iwouldliketothankmycommittee.Ihavehadeachasateacher,andcansaybeyondanydoubtthatevenbeforeservingonmycommittee,Iowedeachathanksformyownbetterunderstandingofphysicsthatwascrucialtomyresearch.Bothforthedirectionandsupporttheyhaveprovided,Iamexceedinglythankful.ivTABLEOFCONTENTSLISTOFTABLES....................................viiiLISTOFFIGURES...................................xChapter1Introduction...............................11.1BriefHistory...................................1Chapter2TheMany-BodyProblem.......................72.1Many-BodyScodingerEquation........................72.2NormalOrdering.................................92.3ManyBodyPerturbationTheory........................132.4CoupledClusterTheory.............................162.4.1ExponentialAnsatz............................172.5ApproximateTriples...............................202.5.1CCSD(T).................................202.5.2................................222.5.3CompletelyRenormalizedCCMethods.................23Chapter3In-MediumSimilarityRenormalizationGroup.........263.1IM-SRGFormalism................................273.1.1OverviewoftheSRG...........................273.1.2M-SchemeFlowEquationsforIM-SRG(2)...............293.1.3SymmetriesandtheFlowEquations..................313.1.4GeneralObservables...........................333.2ChoiceofGenerator................................343.2.1Decoupling................................343.2.2WhiteGenerators.............................363.2.3Imaginary-TimeGenerators.......................383.2.4WegnerGenerators............................393.2.5DecayScales................................413.3NumericalExplorations..............................443.3.1Implementation..............................443.3.2Convergence................................463.3.3ChoiceofGenerator...........................513.3.4Decoupling................................563.3.5Radii....................................603.4ChoiceofReferenceState............................633.4.1Overview.................................633.4.2HarmonicOscillatorvs.Hartree-FockSlaterDeterminants......633.5PerturbativeAnalysisoftheFlowEquations..................68v3.5.1Overview.................................683.5.2PowerCounting..............................713.5.3O(g)Flow.................................723.5.4O(g2)Flow................................733.5.5O(g3)Flow................................793.5.6EnergythroughO(g4)..........................823.5.7Discussion.................................863.5.8PerturbativeTruncations.........................89Chapter4MagnusFormulation..........................954.1Introduction....................................954.2Formalism.....................................964.3AnalyticalModel.................................984.4MAGNUS(2)Approximation...........................1014.5HamiltoniansandImplementation........................1024.6Results.......................................1054.7MAGNUS(2)Conclusion.............................115Chapter5ApproximatingtheIM-SRG(3)...................1165.1oftheIM-SRG(2)inChemistry..................1175.2TheIM-SRG(2*)Approximation........................1185.3TheMAGNUS(2*)Approximation.......................1205.4ApproximationstoMAGNUS(3).........................1245.5Applications....................................1285.5.1ElectronGasResults...........................1295.5.2NuclearResults..............................1325.5.3H2OResults................................1345.5.4NeonResults...............................1355.5.5C2Results.................................1365.5.6HFResults................................1375.6Summary.....................................138Chapter6OtherWork...............................1396.1ImprovingtheReference.............................1396.2BruecknerIM-SRGResults............................1426.2.1BMAGNUS(2*)ResultsforNeonandC2................1436.2.2BMAGNUS(2*)ResultsforHFandH2O................1446.2.3BruecknerSummary...........................1456.3ExtensionstoMR-IM-SRG............................1456.4ExtensionstoExcitedStateFormalism.....................151Chapter7SummaryandConclusions......................155viAPPENDICES......................................157AppendixAFundamentalCommutators.......................158AppendixBIM-SRG(3)FlowEquations.......................162AppendixCDiagramRules..............................165REFERENCES......................................170viiLISTOFTABLESTable3.1:IM-SRG(2)ground-stateenergiesofselectedclosed-shellnucleiforthethechiralN3LOinteractionbyEntemandMachleidt[1,2],with=1and=2:0fm1(cf.Fig.3.2).E14aretheenergiesobtainedforemax=14atoptimal~!,andEexareextrapolatedtobasissize(seetext),withextrapolationuncertaintiesindicatedinparentheses................................46Table5.1:ApproximationsmadeinthevariousMAGNUS(2*)[3]variants....129Table5.2:aFCIQMCresultsfromRef.[3].bCCDandCCDTresultsfromRef.[4].Groundstateof14electronscalculatedinabasissetofM=114planewaveswithvariousapproximations.TheFullCQuantumMonteCarlocorrelationenergyisreportedinHartree.Allotherenergiesarereportedasafractionofthecorrelationenergyrecoveredwithrespecttoquasi-exactFCIQMCresults........131Table5.3:aFromRef.[5].bCCSDandCCSD(T)resultsobtainedwithPSI4[6].cFromRef.[7]dFromRef.[8]AcomparisonofvariousCCground-stateenergiesobtainedforthecc-pVDZH2Omoleculeattheequilib-riumOHbondlengthRe=1.84345bohrandseveralnonequilibriumgeometriesobtainedbystretchingtheOHbonds,whilekeepingtheHOHangleat110.6.Thesphericalcomponentsofthedorbitalswereused.Inpost-RHFcalculations,allelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.Theremainingener-giesarereportedinmillihartreerelativetothecorrespondingfullCIenergies..................................135Table5.4:aFromRef.[9].AcomparisonofCCandMagnusIM-SRGground-stateenergiesobtainedforaNeonatominacc-pVDZbasisset.Inthesepost-HFcalculations,the1sorbitalwasfrozen.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedasafractionofthecorrelationenergyrecoveredrelativetoE=EHF-EFCI................................136Table5.5:aFromRef.[9].AcomparisonofCCandMagnusIM-SRGground-stateenergiesobtainedforC2attheequilibriumFCIbondlengthofre=1.27273Ainacc-pVDZbasisset,takenfromRef.[9].Inthesepost-HFcalculations,the1sorbitalswasfrozenontheCatoms.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedcorrelationenergyrecoveredrelativetoE=EHF-EFCI.137viiiTable5.6:aFromRef.[10].bCCSDandCCSD(T)resultsfrom[6].cFromRef.[8]AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedfortheequilibriumgeometryofRe=1.7328bohrandothernuclearseparationsofHFwithaDZbasisset.Inthesepost-HFcal-culationsallelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.TheremainingenergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergyvalues...........138Table6.1:aFromRef.[9].AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedforaNeonatom.Inthesepost-HFcalcula-tions,the1sorbitalwasfrozen.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedasafractionofthecorrelationenergyrecoveredrelativetoFCIinmH...........144Table6.2:aFromRef.[9].AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedforC2attheequilibriumFCIbondlengthofre=1.27273A.Inthesepost-HFcalculations,the1sorbitalswasfrozenontheCatoms.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedinmillihartreerelativetothefullCIenergy.................................144Table6.3:aFromRef.[5].bCCSDandCCSD(T)resultsobtainedwithPSI4[6].cFromRef.[7]dFromRef.[8]AcomparisonofvariousCCground-stateenergiesobtainedforthecc-pVDZH2Omoleculeattheequilib-riumOHbondlengthRe=1.84345bohrandseveralnonequilibriumgeometriesobtainedbystretchingtheOHbonds,whilekeepingtheHOHangleat110.6.Thesphericalcomponentsofthedorbitalswereused.Inpost-RHFcalculations,allelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.Theremainingen-ergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergies.InRef.[10],theauthorsnoticedthattherearetwoSCFsolutions,oneofwhichpoorlydescribestheweakH-Obonding.Inthechemistrysuite,PSI4[6],wecouldnotforcetheCCSDroutine,andthustheBCCDroutinetousethecorrectSCFstartingreference.Thisiswhytherearenoresultsreportedfor3:0Re..........146Table6.4:aFromRef.[10].bCCSDandCCSD(T)resultsobtainedwithPSI4[6].cFromRef.[8]AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedfortheequilibriumgeometryofRe=1.7328bohrandothernuclearseparationsofHFwithaDZbasisset.Inthesepost-HFcalculationsallelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.TheremainingenergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergyvalues...................................147ixLISTOFFIGURESFigure1.1:TakenfromRef.[11].Thechartofnuclidesandthereachofabinitiocalculationsin(a)2005and(b)2015.Nuclei(includingpotentiallyunboundisotopes)forwhichabinitiocalculationsbasedonhigh-precisionnuclearinteractionsexistarehighlighted.Essentiallyallofthe2015calculationsinclude3Nforces.Wenotethattheisforillustrativepurposesonly,andisbasedontheauthors'potentiallynon-exhaustivesurveyoftheliterature.................3Figure2.1:MatrixdimensionversusNmaxforstableandunstableOxygeniso-topes.Theverticalredlinesignalstheboundary,beyondwhichonemightexpectreasonableconvergencewithrespecttoNmax.Thehor-izontallinesshowthecomputationalpowerofafacilityexpectedtoconductthesediagonalizations.FiguretakenfromRef.[12]......9Figure2.2:DiagramdemonstratingthediagramaticformofEq.(2.26).ThedashedlineindicatestheresolventoperatorR0.............16Figure2.3:DiagramsdemonstratingtheapproximateT3constructedfromcon-vergedT2amplitudesintheCCSD[T]energycorrection.ThedashedlinerepresentstheresolventoperatorR0fromperturbationtheory..21Figure3.1:SchematicrepresentationoftheinitialandnalHamiltonians,H(0)andH(1),inthemany-bodyHilbertspacespannedbyparticle-holeexcitationsofthereferencestate.....................34Figure3.2:Convergenceof4He,16O,and40CaIM-SRG(2)ground-stateenergiesw.r.t.single-particlebasissizeemax,forachiralN3LONNinterac-tionwith=1(leftpanels)and=2:0fm1(rightpanels).Noticethetncesintheenergyscalesbetweentheleftandrightpanels.Graydashedlinesindicateenergiesfromextrap-olationtheemax10datasetstobasissize(seetextandRefs.[13,14])..............................47Figure3.3:Convergenceof78Ni,100Sn,and132SnIM-SRG(2)ground-stateen-ergiesw.r.t.single-particlebasissizeemax,forthechiralN3LONNinteractionwith=2:0fm1.Graydashedlinesindicateenergiesfromextrapolationtheemax10datasetstobasissize(seetextandRefs.[13,14]).........................49xFigure3.4:IM-SRG(2)ground-stateenergiesof40Caobtainedwithtchoicesofthegenerator,asafunctionof~!andthesingle-particlebasissizeemax.TheinteractionisthechiralN3LOpotentialwith=1(toppanels)and=2:0fm1(bottompanels),respectively.Thedashedlinesindicateextrapolatedenergies.FortheWegnergen-erator,theshadedareaindicatesthevariationfromusingtdatasetsfortheextrapolation(seetext)................52Figure3.5:IM-SRG(2)ground-stateenergiesof40Cafortheregular(left,asinFig.3.4)andrestrictedWegnergenerators(right,seetext),asafunctionof~!andthesingle-particlebasissizeemax.TheinteractionisthechiralN3LOpotentialwith=1(toppanels)and=2:0fm1(bottompanels),respectively.Thedashedlinesindicateextrapolatedenergies..........................54Figure3.6:DecouplingfortheWhitegenerator,Eq.(3.23),intheJˇ=0+neutron-neutroninteractionmatrixelementsof40Ca(emax=8;~!=20MeV,Entem-MachleidtN3LO(500)evolvedto=2:0fm1).Onlyhhhh;hhpp;pphh;andppppblocksofthematrixareshown...56Figure3.7:IM-SRG(2)ground-stateenergyof40Caasafunctionofthewpa-rameters,comparedtoMBPT(2),CCSD,andenergieswiththeIM-SRG-evolvedHamiltonianH(s).Weonlyshowpartofthedatapointstoavoidclutter.Calculationsweredoneforemax=10andoptimal~!=32MeV(top)and~!=24MeV(bottom),respec-tively,usingthechiralNNinteractionattresolutionscales.ThedashedlinesindicatetheIM-SRG(2)energies.......58Figure3.8:Convergenceof4He,16O,and40CaIM-SRG(2)chargeradiiw.r.t.single-particlebasissizeemax,forachiralN3LONNinteractionwith=1(leftpanels)and=2:0fm1(rightpanels).Thegraydashedlinesindicateexperimentalchargeradiifrom[15]..............61Figure3.9:Toppanel:IM-SRG(2)energyof40CawithaHF(solidlinesandsymbols)andaHOreferencestate(dashedlines,opensymbols),ob-tainedwiththeWegnergenerator.Bottompanel:OverlapoftheHFandHOreferencestates.........................64Figure3.10:IM-SRGdecouplingof1p1hexcitationsfortgeneratorchoices,startingfromaHOreferencestate.Theshowsthe40Caground-stateenergyasafunctionofthevalueofthewparameters.Theunitofsissuppressedbecauseitwiththechoiceofgenerator.ThegraylineindicatestheresultoftheHartree-Fockcalculationwiththesameinteractionandbasisparameters...............65xiFigure3.11:Schematicillustrationoftheenergywequation(3.61)fortheWhitegeneratorwithM˝ller-Plessetenergydenominators(Eq.(3.23))intermsofHugenholtzdiagrams(seetext).ThegreyverticesrepresentH(s),andthedoublelinesindicateenergydenominatorscalculatedwithf(s).Onthesecondline,thewequationisexpandedintermsofH(ss)(simpleblackvertices)andthecorrespondingenergydenominatorsfromf(ss)(singlelines).ThebracesindicatewhichtermofH(s)isexpanded,anddotsrepresenthigherorderdiagramsgeneratedbytheintegrationstepss!s..............67Figure3.12:AntisymmetrizedGoldstonediagramsfortheO(g2)eone-bodyHamiltonian(seetext).InterpretationrulesaresummarizedinAppendixC..............................75Figure3.13:AntisymmetrizedGoldstonediagramsfortheO(g2)etwo-bodyvertex(seetext).InterpretationrulesaresummarizedinAppendixC...............................76Figure3.14:AntisymmetrizedGoldstonediagramsfortheO(g2)ethree-bodyvertexW(seetext).InterpretationrulesaresummarizedinAppendixC...............................78Figure3.15:AntisymmetrizedGoldstonediagramsfortheO(g3)etwo-bodyvertex(seetext).Black(l)andgrayvertices(wg)correspondto[1](Eq.(3.77)),f[2](Eqs.(3.80){(3.82)),[2](Eqs.(3.85){(3.90)),andW[2](Eqs.(3.92){(3.94)),respectively.InterpretationrulesaresummarizedinAppendixC.......................79Figure3.16:ConnectedHugenholtzdiagramsforthefourth-orderenergycorrec-tionE(4)(Ref.[16])...........................84Figure3.17:ectoffourth-orderquadruples(4p4h)contributionE[4]C,Eq.(3.120)ontheground-stateenergiesof4He,16O,and40Ca(seetext):Com-parisonofIM-SRG(2)withandwithoutE[4]C,calculatedwiththeinitialHamiltonianH(0),toCCSDandCCSD(T).Allcalcula-tionsusedthechiralN3LOHamiltonianwith=1inanemax=14single-particlebasis.TheshownCCvaluesweretakenatoptimal~!.88xiiFigure3.18:Comparisonof40Caground-stateenergiesoftheregularIM-SRG(2)(solidlines)andperturbativeIM-SRG(2')truncations(dashedlines).ThedefaultWhitegeneratorIA,Eq.(3.23),wasusedinbothcases.TheinteractionisthechiralN3LOpotentialwith=1(leftandcenterpanels)and=2:0fm1(rightpanel),respectively.Thedashedlinesindicateextrapolatedenergies.FortheIM-SRG(2')trun-cation,theshadedareaindicatesthevariationfromusingtdatasetsfortheextrapolation(seetext)................91Figure3.19:ectofaddingthefourth-ordersingles(1p1h)contribution(cf.Eqs.(3.115),(3.118)and(3.128))totheIM-SRG(2')ground-stateenergyof40Ca(seetext).Thesinglescontributionsfort~!werecalculatedwiththeinitialHamiltonianH(0).AllshownresultswereobtainedforthechiralN3LOHamiltonianwith=1.............93Figure3.20:ctiveneutron(leftpanel)andproton(rightpanel)single-particleenergiesof40CafromIM-SRG(2)(solidlines)andIM-SRG(2')(dashedlines)calculationsusingthechiralN3LOinteractionwith=1inanemax=14single-particlebasis...................93Figure4.1:jH11(s)EgsjversussfortEulerstepsizescalculatedviadirectintegrationoftheSRGwequation,Eq.3.2,andusingtheMagnusexpansion,Eqs.4.6and4.8.AlsoplottedistheintegrationofEq.3.2withtheGordon-Shampineintegrator...........100Figure4.2:RelativeimportanceofthekthtermintheMagnusderivativeasde-bythelefthandsideofEq.4.16evaluatedintheNO2Bap-proximation.ThetoprowisforthehomogeneouselectrongasatWigner-Seitzradiiofa)rs=0:5andb)rs=5:0.Thebottomrowisfor16O,startingfromthechiralNNpotentialofEntemandMach-leidt[17],softenedbyafree-spaceSRGevolutionto(c)=2:0fm1and(d)=3:0fm1.TheelectrongascalculationsweredoneforN=14electronsinaperiodicboxwithM=114singleparticleor-bitals.The16Ocalculationsweredoneinanemax=8modelspace,with~!=24MeVfortheunderlyingharmonicoscillatorbasis....105Figure4.3:Magnitudeofthe0-bodycontributionsofthekthterminEq.4.8evaluatedintheNO2Bapproximation.ThetoprowisfortheelectrongasatWigner-Seitzradiiof(a)rs=0:5and(b)rs=5:0.Thebottomrowisfor16O,startingfromthechiralNNpotentialofEntemandMachleidt[17],softenedbyafree-spaceSRGevolutionto(c)=2:0fm1and(d)=3:0fm1.TheelectrongascalculationsweredoneforN=14electronsinaperiodicboxwithM=114singleparticleorbitals.The16Ocalculationsweredoneinanemax=8modelspace,with~!=24MeVfortheunderlyingharmonicoscillatorbasis....107xiiiFigure4.4:FlowingIM-SRG(2)andMAGNUS(2)HEGcorrelationenergy,E0(s)EHF,forWigner-Seitzradiiofa)rs=5:0andb)rs=0:5.ThesolidblacklinecorrespondstoIM-SRG(2)resultsusinganadaptivesolverbasedontheAdams-Bashforthmethod,whiletheotherlinescorre-spondtoMAGNUS(2)andIM-SRG(2)resultsusingtEulerstepsizes.Theredcirclesdenotethequasi-exactFCIQMCresultsofRef.[3]...................................108Figure4.5:FlowingIM-SRG(2)andMAGNUS(2)groundstateenergy,E0(s),for16OstartingfromtheN3LONNinteractionofEntemandMach-leidt[17]evolvedbythefree-spaceSRGtoa)=2:7fm1and=2:0fm1.ThesolidblacklinecorrespondstoIM-SRG(2)re-sultsusinganadaptivesolverbasedontheAdams-Bashforthmethod,whiletheotherlinescorrespondtoMAGNUS(2)andIM-SRG(2)re-sultsusingtEulerstepsizes.Thecalculationsweredoneinanemax=8modelspace,with~!=24MeVfortheunderlyingharmonicoscillatorbasis.........................109Figure4.6:ElectrongasmomentumdistributionscalculatedintheMAGNUS(2)approximation.ThecalculationsweredoneforN=14electronsinaperiodicboxwithM=778singleparticleorbitals.........111Figure4.7:TimingforMAGNUS(2)andIM-SRG(2)HEGcalculationsasthesingleparticlebasisisincreased.Thetwobottomcurvesareforrs=:5andthetopforrs=5.FortheMAGNUS(2)timing,thisincludesthecalculationsofthemomentumdistributions.......112Figure4.8:CenterofmassdiagnosticsforMAGNUS(2)calculationsof16Ostart-ingfromtheN3LONNinteractionofEntemandMachleidt[17]evolvedbythefree-spaceSRGto=2:0fm1.Seethetextfordetails.Thecalculationsweredoneinanemax=9modelspace...113Figure5.1:ThedarkcircleandsquarerepresentthebareHamiltonianandgen-eratorrespectively.Thelightcirclesinthecolumnrepresentthesecondorder1-bodyHamiltonianoriginatingfrom[;H]1B.Ifthesecondorder1-bodyvertexisexpandedintermsofbarequantities,the4asymmetricGoldstonediagramsontherightaretheresult..121Figure5.2:ThedarkcircleandsquarerepresentthebareHamiltonianandgen-eratorrespectively.Thelightcirclesinthecolumnrepresentthesecondorder3-bodyHamiltonianoriginatingfrom[;H]3B.Ifthesecondorderinduced3-bodyvertexisexpandedintermsofbarequantities,the4asymmetricGoldstonediagramsontherightaretheresult...................................122xivFigure5.3:Groundstatecalculationsfor14electronstoaboxatden-sitiesofrs=a0of.5,1,2,and5andperformedat3basissetsizesof114,186,and358withvariousmethods.AlthoughCCSDresultsarenotplotted,correctingtheMAGNUS(2)commutatorexpressionsasshowninEq.(5.4)makesMAGNUS(2*)indistinguishablefromCCSDonthesescales.Further,thetriplescorrectionduetotheMAGNUS(2*)[3]-DbindstheresultbackdowntoagreeverywellwithFCIQMCresultsfromRef.[3],whilethebaredenominatorsfoundinvariantAofMAGNUS(2*)[3]overbinddramatically....130Figure5.4:MAGNUS(2*)results,withMAGNUS(2*)[3]-Cforthelargestbasissetfor4HewiththechiralN3LOchiralinteractionbyEntemandMachleidt[1,2]softenedto=2:0fm1.Wenoticethatcor-rectingthecommutatorasshowninEq.(5.4)providesrepulsionthatbringsMAGNUS(2*)uptoCCSD.Further,thetriplescorrectionduetotheMAGNUS(2*)[3]-CbindstheresultbackdowntoagreewithCCresultsfromRef.[18].................133Figure5.5:MAGNUS(2*)results,withMAGNUS(2*)[3]-Cforthelargestbasissetfor16OwiththechiralN3LOchiralinteractionbyEntemandMachleidt[1,2]softenedto=2:0fm1.............133Figure6.1:Ground-stateenergiesoftheoxygenisotopesfromMR-IM-SRGandothermany-bodyapproaches,basedontheNN+3N-fullinteractionwith3N=400MeV,evolvedtotheresolutionscale=1:88fm1(=2:0fm1fortheGreen'sFunctionADC(3)results,cf.[19]).Blackbarsindicateexperimentaldata[20].SeeRef.[21]foradditionaldetails..................................149Figure6.2:MR-IM-SRGresultsforCatwo-neutronseparationenergies,forchi-ralNN+3Ninteractionswithdtinthe3Nsector,andarangeofresolutionscalesfrom=1:88fm1(opensymbols)to2:24fm1(solidsymbols).Blackbarsindicateexperimentaldata[20,22].SeeRef.[23]foradditionaldetails..............150Figure6.3:Excited-statespectraof22;23;24ObasedonchiralNN+3Ninteractionsandcomparedwithexperiment.FiguresadaptedfromRef.[24].TheMBPTresultsareperformedinanextendedsdf7=2p3=2space[25]basedonlow-momentumNN+3Ninteractions,whiletheIM-SRG[24]andCCectiveinteraction(CCEI)[26]resultsareinthesdshellfromtheSRG-evolvedNN+3N-fullHamiltonianwith~!=20MeV(CCEIanddottedIM-SRG)and~!=24MeV(solidIM-SRG).Thedashedlinesshowtheneutronseparationenergy.FiguretakenfromRef.[27]..................................153xvChapter1Introduction1.1BriefHistoryThequesttopredictandunderstandthepropertiesofnucleistartingfromtheunderlyingnuclearforcesgoesbacknearly60years,datingbacktothepioneeringworkofBrueckner,Bethe,andGoldstone[28{30].Incontrastpredictiveandaccurateabinitiomany-bodycalculationswerecommonplaceinquantumchemistrybythe1980s[31].Progresswasnotslowedtherebythechallengingaspectsofthenuclearproblemlikethelackofaconsistenttheoryforthestronginter-nucleoninteractions,andtheneedtoperformcomputationallyexpensive(anduncontrolled)resummationstohandlethenon-perturbativeaspectsoftheproblem.Consequently,formanyyearsnuclearabinitiotheorylanguishedasapredictiveforce,andcouldonlyexplaininsemi-quantitativetermshowsuccessfulphenomenologysuchastheshellmodelandSkyrmeenergy-densityfunctionalsarelinkedtotheunderlyingnuclearinteractions.Asexperimentalhaveshiftedtowardsexoticnuclei,therehasbeenanincreasedurgencytodevelopreliableabinitioapproachestocountertheinherentlimitationsofphe-nomenology.AsevidencedbyFig.1.1,tremendousprogresshasbeenmadeinrecentyears,wheretheinterplayoftthreads,namelyrapidlyincreasingcomputationalpower,etheory(EFT)descriptionsofinter-nucleoninteractions,andrenormalizationgroup(RG)transformations,haveenabledthedevelopmentofnewmany-bodymethodsand1therevivalofoldonestosuccessfullyattacktheseproblems[27,32{35].Remarkably,itisnowpossibletoperformquasi-exactcalculationsincludingthree-nucleoninteractionsofnucleiuptocarbonoroxygeninquantumMonteCarlo(QMC)andno-coreshellmodel(NCSM)calculations,andN=Znucleiupthrough28SiinlatticeetheorywithEuclideantimeprojection[12,36{38].Moreover,ahostofapproximate(butsystemat-icallyimprovable)methodssuchasCoupledCluster(CC),self-consistentGreen'sfunctions(SCGF),auxiliaryMonteCarlo(AFDMC),andtheIM-SRGhavepushedthefrontiersofabinitiotheorywellintothemedium-massregion,openingupnewdirectionstothechallengingterrainofopen-shellandexoticnuclei[21,23,24,26,39{46],withrecenthighlightsinthecalciumisotopes[22,47].RGmethodshaveplayedaprominentroleintheresurgenceofabinitiotheory.Akeytooptimizingcalculationsofnucleiisaproperchoiceofdegreesoffreedom.WhileQuantumChromodynamics(QCD)istheunderlyingtheoryofstronginteractions,themostlow-energydegreesoffreedomfornuclearstructurearethecolorlesshadronsoftraditionalnuclearphenomenology.Butthisrealizationisnotenough.Forlow-energycalculationstobecomputationallyt(orevenfeasibleinsomecases)weneedtoexcludeor,moregenerally,todecouplethehigh-energydegreesoffreedominamannerthatleaveslow-energyobservablesinvariant.Progressonthenuclearmany-bodyproblemwashinderedfordecadesbecausenucleon-nucleon(NN)potentialsthatreproduceelasticscatteringphaseshiftstypicallyhavestrongshort-rangerepulsionandstrongshort-rangetensorforces.Thisproducessubstantialcou-plingtohigh-momentummodes,whichismanifestedasstronglycorrelatedmany-bodywavefunctionsandhighlynonperturbativefew-andmany-bodysystems.Formanyyears,theonlyviableoptiontohandlethesefeaturesinacontrolledmannerwastousequasi-exactmethods2Figure1.1:TakenfromRef.[11].Thechartofnuclidesandthereachofabinitiocalculationsin(a)2005and(b)2015.Nuclei(includingpotentiallyunboundisotopes)forwhichabinitiocalculationsbasedonhigh-precisionnuclearinteractionsexistarehighlighted.Essentiallyallofthe2015calculationsinclude3Nforces.Wenotethattheisforillustrativepurposesonly,andisbasedontheauthors'potentiallynon-exhaustivesurveyoftheliterature.3suchasQMCorNCSM,whichlimitedthereachofabinitiocalculationstolightp-shellnu-clei.PowerfulmethodsthatscalefavorablytolargersystemslikeCCandmany-bodypertur-bationtheory(MBPT)werelargelyabandonedinnuclearphysics,butexportedtoquantumchemistry,wheretheyenjoyedimmediatesuccessandquicklybecamethegold-standardforabinitiocalculations[48{50].ThesuccessofCCandrelatedmethodsinquantumchemistrystemsfromthefactthatHartree-FockisarelativelygoodstartingpointduetotherelativelyweakcorrelationsinducedbytheCoulombinteraction,instarkcontrasttothenuclearcase.Additionally,thenuclearcaseisbesetwithwiththequalityofnuclearforces,andotherissuesthatplagueself-boundsystemslikecenterofmasscontamination.NewapproachestonuclearforcesgroundedinRGideasandtechniqueshavebeende-velopedinrecentyearsthatelymakethenuclearmany-bodyproblemlookmorelikequantumchemistry[33,37,51{55].TheRGallowscontinuouschangesin\resolution"thatdecouplethetroublesomehigh-momentummodesandcanbeusedtoevolveinteractionstonuclearstructureenergyandmomentumscaleswhilepreservinglow-energyobservables.Suchpotentials,knowngenericallyas\low-momentuminteractions,"aremoreperturbativeandgeneratemuchlesscorrelatedwavefunctions.Thishasplayedamajorroleinexpand-ingthereachofabinitiocalculationstomedium-massnuclei,sincemethodsthatexhibitpolynomialscalingcannowbeconvergedinmanageablemodelspaces.SeeRefs.[33,56,57]forrecentreviewsontheuseofRGmethodsinnuclearphysics.Aswillbeshowninthefollowing,theIM-SRGapproachextendstheRGnotionofdecouplingtothemany-bodyHilbertspacebyformulating\in-medium"wequations,thesolutionofwhichisequivalenttothepartialdiagonalizationorblock-diagonalizationofthemany-bodyHamiltonian[21,24,33,39,58].Becauseofitsfavorablepolynomialscalingwithsystemsize,andtheytotargetgroundandexcitedstatesofbothclosed-4andopen-shellsystems,theIM-SRGprovidesapowerfulabinitioframeworkforcalculatingmedium-massnucleifromprinciplesthatisgroundedinmodernRGprinciples.Despitetheinherentstrengthsofthismethod,itfromseveralshortcomings.Keyamongstthesearei)thelinearscalingwitheachadditionalobservableonewishestocalculate,ii)theneedtosolvealargesetofcoupledtialequationstohighnumericalaccuracy,iii)andtheinabilitytoapproximatetheofomittedtermsinthesimplesttruncationsoftheIM-SRGequations.TheaimofthisthesisistoshowhowanovelreformulationoftheIM-SRGusingMagnusexpansiontechniquesallowsonetocircumventallthreeoftheseweaknesses,atleastforclosed-shellsystems.IwillshowthattheIM-SRG,whencoupledwithatruematrixexponentialformalismviatheMagnusexpansion,providesacontrolled,non-perturbativeschemetothegroundstatesofnucleiandofquantumchemistrysystems.Moreover,therearepromisingindicationsthatthemethodslaidoutinthisthesiswillprovideimportanttoolsforderivingevalenceshellmodelHamiltoniansandoperatorsfromtheunderlyingnuclearforces,openingthedoortoanabinitiodescriptionofopen-shell,medium-massnuclei.Therestofthisthesisisorganizedasfollows.InChapter2,Istartwithabriefreviewofmany-bodyperturbationtheory(MBPT)andCCtheory,asmanyoftheimprovementsdescribedinthisthesisarebasedonanalyzingtheperturbativecontentoftheIM-SRGandunderstandingthesimilaritiesandfromCCtheory.InChapter3,thebasicele-mentsoftheIM-SRGmethodarereviewedinsomedetail,andasamplingofitssuccessesinnucleiarepresented.NotethatthediagrammaticanalysisoftheperturbativecontentoftheIM-SRGinSection3.5isespeciallyimportant.InadditiontoguidingmanyoftheimprovementsdetailedinChapter5,thiswasmymaincontributiontotherecentreviewarticle[11]thatmuchofthematerialinChapter3isbasedon.Chapter4describesthe5crucialreformulationoftheIM-SRGequationsusingtheMagnusexpansion,andshowshowthiseliminatesmanyofthecomputationallimitationsfacedbytheconventionalformulationduetolargememoryoverhead.Chapter5documentshowthesimplestIM-SRG(2)andMAGNUS(2)truncationsfailincertainquantumchemistrysystems,andusestheperturba-tiveanalysisofSection3.5tomotivateimprovedtruncations,whicharethenvalidatedforseveralelectronicandnuclearsystems.Chapter6highlightssomeinterestingopentopicsthatarepresently\indevelopment",suchasacomputationallyinexpensivemethodtoper-formcalculationsinBruecknerorbitalsandextensionstomorechallengingopen-shellandmulti-referenceproblems.Finally,conclusionsarepresentedinChapter7.6Chapter2TheMany-BodyProblemInlowenergynuclearsystems,quantumchemistry,andsolidstatephysics,onewishestobeabletounderstandemergentphenomenafromamicroscopicHamiltonian.Despitetheambiguityintheinter-nucleoninteractionsarisingfromschemeandscaledependence,thebasicmechanicsforsolvingallthreesystemsarethesameonceagiveninteractionissettledon.However,becausesolvingtheA-bodyScodingerequationinanexactandstraight-forwardmannerleadstoafactoriallyscalingprobleminA,approximatesolutionsaretheonlywaytomoveforward.Avarietyoftapproximatemethodshaveshownpromiseinrecentyears,butinordertomotivateboththestrengthsandoftheIM-SRGmethod,andtheto-be-presentedMagnusformulationofit,wewillbeginbyreviewingthebasicsofthemany-bodyproblemandtwoestablishedmethodsforapproximatelysolvingit.TheandmostwellknownoftheseisMBPT,whichwillbecoveredinSection2.3.FurtherbecauseofthestrongrelationshipbetweenIM-SRGandCCtheory,andhowCCtheorymotivatesimprovementsintheIM-SRGmethod,wewillalsopresenttheimportantelementsofCCtheoryinSection2.4.2.1Many-BodyScodingerEquationThemainprobleminnon-relativisticmany-bodyphysicsistothesolutionoftheScodingerequationforasystemofAinteractingelementaryparticles,generallyfermions.7Onecanwritedownthetime-independentgroundstatesolutionasHj0i=E0j0i:(2.1)Forthepurposesofthiswork,itisusefultostartwithaFock-spacesecondquantizedHamiltonianH=XpqTpqaypaq+14XpqrsV(2)pqrsaypayqasar+136XpqrstuV(3)pqrstuaypayqayrauatas;(2.2)whereV(2)pqrsandV(3)pqrstuareantisymmetrizedtwo-andthree-bodyinteractionmatrixele-ments,andacompletebasissetofSlaterdeterminantsfortheA-bodyHilbertspaceasjfp1:::pAgi=AYk=1aypkj0i;(2.3)giventhatthesingle-particlebasisstatetowhichthecreationoperatorsrefertoarecompleteintheone-bodyspace.ItisclearthenthatanyA-bodystatecanbewrittenintermsoftheseSlaterdeterminants,andinparticularthetruegroundstatehastheformj0i=Xp1<:::<pACp1:::pAjfp1:::pAgi:(2.4)Thisexpansiondemonstrateswhatwasmentionedearlier,thatastraightforwardvariationalcalculationoftheCp1:::pA'sbecomesintractablewithincreasingA.Ifthesingleparticlebasisistruncatedatnorbitalsandsymmetryisignored,thentherearenACp1:::pA'swhichmustbedetermined.Evenforverymoderatesystemsbeingoptimizedonaverylargecomputationalfacility,thisprocedureisintractable.Figure2.1demonstratesthefeasibility8Figure2.1:MatrixdimensionversusNmaxforstableandunstableOxygenisotopes.Theverticalredlinesignalstheboundary,beyondwhichonemightexpectreasonableconvergencewithrespecttoNmax.Thehorizontallinesshowthecomputationalpowerofafacilityexpectedtoconductthesediagonalizations.FiguretakenfromRef.[12].ofcalculatingtheoxygenisotopesinastraightforwardwaytakenfrom[12].TherearesomecommonconventionswhichwillbeworthwhiletointroduceinordertobeabletodiscussMBPT,CCtheory,andtheIM-SRGinthenextchapter.FirstIwillintroducetheconceptofanadequatelychosenreferencestate,whichwillleadimmediatelytotheconceptofnormalordering,whichisinvaluableforthepurposesofbookkeepingforthemethodsdiscussedinthiswork.2.2NormalOrderingReferencestatesareacommoningredienttomostmany-bodymethods.Usually,theirfunctionistocertaincharacteristicsofthesystemwewanttodescribe,e.g.,theproton9andneutronnumbersofanucleus,andtoprovideastartingpointfortheconstructionofamany-bodyHilbertspacethatissuperiortotheparticlevacuum.Describingmany-bodystatesasexcitationswithrespecttoasuitablychosenreferencestateallowsustoaccountforthecharacteristicenergyscalesofthetargetnucleus,andintroducesystematictruncationschemesbasedonthisinformation.Italsosuggeststheuseofnormal-orderingtechniquesinanaturalfashion(cf.2.2).UsingaSlaterdeterminantasthereferencestateisasuitablechoiceforsystemswithalargegapintheirexcitationspectrum,e.g.,closed-shellnuclei.AmongtheSlaterdeterminants,thosethatsatisfytheHartree-Fockconditionsforagivensystemarethemostnaturalchoices(cf.Sec.3.3),becausetheyminimizeboththeenergyandthebeyondmecorrelationenergyinavariationalsense.Onceasuitablesinglereferencejiischosen,onecaninvoketheconceptofnormalordering.Asmentioned,thisleadstoanaturalorganizationofthecompleteA-bodystatesintotheirlevelofexcitationawayfromji.Normal-orderedoperatorscanthenbebybeginningwiththemostsimpleofthese,aypaq:aypaq:+aypaq;(2.5)wherethecontractionisrelatedtothereferencestateji:aypaqhjaypaqjiˆqp=pqnp;(2.6)wherenpis1or0dependingonoccupationinji.ThisgeneralizeseasilytoA-body10operator:ayp1:::aypNaqN:::aq1:ayp1:::aypNaqN:::aq1:+ayp1aq1:ayp2:::aypNaqN:::aq2:ayp1aq2:ayp2:::aypNaqN:::aq3aq1:+singles+ayp1aq1ayp2aq2ayp1aq2ayp2aq1:ayp3:::aypNaqN:::aq3:+doubles+:::+fullcontractions:(2.7)Itisclearthathj:aypaq:jimustvanish,andasimilaristrueforgeneralnormal-orderedoperatorsinthereferencestatejihj:ayp1:::ap1:ji=0:(2.8)ItisthenpossibletoinvokeWick'stheorem(seee.g.[16]),whichisasimpleoftheofnormalorderinginEq.(2.7).Thisallowsfortheexpansionofproductsoftwoormorenormalorderedoperators::ayp1:::aypNaqN:::aq1::ayr1:::ayrMasM:::as1:=(1)MN:ayp1:::aypNayr1:::ayrMaqN:::aq1asM:::as1:+(1)MNayp1as1:ayp2:::ayrMaqN:::as2:+(1)(M1)(N1)aq1ayr1:ayp1:::ayrMaqN:::al1:+singles+doubles+::::(2.9)Thephasesappearastheayroperatorsareanti-commutedpastaq.SincejiisanA-body11stateandnotthetruevacuua,thereisanewtypeofcontraction,apayqhjapayqji=pqˆpq;(2.10)asexpectedfromthebasicfermionicanti-commutatoralgebra.Thisprovidesarobustframe-workforevaluatingtheproductofnormalorderedoperatorswiththeleastamountofterms.Alsobisthatitmotivatesadiagrammaticformalismthatmakesevaluationoftermsintuitive.AnimportantfromWick'stheoremthatwillberelevantintheformulationoftheIM-SRGisthataproductofnormal-orderedMandN-bodyoperatorshasthegeneralformAMBN=M+NXk=jMNjC(k):(2.11)Exploitingnormalordering,onecanexactlyrewritethehamiltonianEq.(2.2),H=E+Xpqfpq:aypaq:+14Xpqrspqrs:aypayqasar:+136XpqrstuWpqrstu:aypayqayrauatas::(2.12)Theindividualnormal-orderedcontributionsinEq.(2.12)arethengivenbyE=XihijTjii+12XijhijjV(2)jiji+16XijkhijkjV(3)jijki;(2.13)fpq=hpjTjqi+XihpijV(2)jqii+12XijhpijjV(3)jqiji;(2.14)pqrs=hpqjV(2)jrsi+XihpqijV(3)jrsii;(2.15)Wpqrstu=hpqrjV(3)jstui:(2.16)Iusetheconventionwherei;j;:::refertooccupiedorbitalsinji,a;b;:::referto12unoccupiedorbitalsinji,andp;q;:::refertoeither.InEqs.(2.13){(2.15),itisimportanttonotethatthezero-,one-,andtwo-bodypartsofthenormal-orderedhamiltonianallcontaincontributionsfromthehigher-bodyfree-spaceinteraction.Thissuggeststhatthedominantofcomputationallyexpensivethree-andhigher-bodyinteractionscanbeincludedintwo-andlower-bodyoperatorsvianormalordering.2.3ManyBodyPerturbationTheoryArmedwithaqualityreferencestateji,itisoftenpossibletoexpandthefullsolutionandpropertiesaroundthisreference.Todosoistofollowthepathofmanybodyperturbationtheory(MBPT).Ipresenthereabriefsummaryofthematerialprovidedin[16].TheHamiltonianispartitionedintoadiagonalandinteractionpartHj0i=(H0+HI)j0i=E0j0i(2.17)whereHI=HH0,andthezero-ordersolutionsbasedonH0areknownH0ji=E(0)0ji:(2.18)MultiplyingEq(2.17)ontheleftwiththereferenceh0jonearrivesatE(0)0+hjHIj0i=E0hj0i:(2.19)Invokingintermediatenormalizationwherehj0i=1,onearrivesattheworkingexpres-13sionforthecorrelationenergy,E=E0E(0)0=hjHIj0i:(2.20)Followingtheusualprescriptiontogeneratetherentversionsofperturbationtheory(e.g.Rayleigh-ScodingerandBrillouin-Wigner),onearrivesattheiteordersolutionj0i=1Xm=0fR0()(HIE0+)gmji;(2.21)withtheresolventoperatorR0()=QH0(2.22)whereQprojectsontotheorthogonalcomplementofji.Thisimmediatelyyieldsaper-turbativeexpansionfortheenergyofE=1Xm=0hjHIfR0()(HIE0+)gmji:(2.23)Fortheremainderofthisworkwemakethechoice=E(0)0whichcorrespondstothesizeextensiveRayleigh-Scodingerperturbationtheory.WeusethisdecisiontorewriteR0(E(0)0)asjustR0.E=1Xm=0hjHIfR0(HIE)gmji(2.24)Thus,aslongasH0anditsspectrumcanbefound,Eq.(2.24)canbesolvedorderbyorder.Thelackofsmallexpansionparameterinthisformalismindicatesthatconvergenceisnotguaranteed.EvaluationoftermsinEq.(2.24)isaidedbythehelpofdiagrammaticmethodsapplicablewhenworkingwithanormalorderedHamiltonian.Therulesforinterpreting14diagramsarefoundbothinAppendixCandRef[16].Formanysystems,itisnecessarytonon-perturbativemethodsthatresumcertainclassesofperturbativediagramstoordertoachieveadequateaccuracy.SizeextensivityisguaranteedinRayleigh-Scodingerperturbationtheory,asallunlinkeddiagramsappearingintheenergycancelorder-by-order[59].AswewillseeinChapter5,theintuitiongleanedfromdiagrammaticMBPTwillbecrucialindevelopingsystematicallyimprovableIM-SRGtruncations.Asanexample,thenon-trivialcontributiontotheenergythatarisesatsecondorderinMBPThastheformE[2]=hjfHIR0HIgCji;(2.25)wherethesubscriptedCrepresentsthatonlyconnectedtermscontribute.PlugginginthenormalorderedoperatorsinEqs.(2.14){(2.16)forHI,theexpandedversionofEq.(2.25)becomesE[2]=Xiafiafaiia+12!2Xijabijababijijab+13!2XijkabcWijkabcWabcijkijkabc(2.26)wherei1:::iNa1:::aN=E(0)0ha1:::aNi1:::iNjH0ja1:::aNi1:::iNi:(2.27)Similaranalysiscanbecarriedoutforeachorder,butthecostofcalculatingdiagramsevenatfourthorderscalessimilarlytomuchhigherqualitymany-bodymethods.15Figure2.2:DiagramdemonstratingthediagramaticformofEq.(2.26).ThedashedlineindicatestheresolventoperatorR0.2.4CoupledClusterTheoryAsmentionedabove,itisoftenimpossibletoarriveatasatisfactoryenergyorwavefunctionfromlow-orderperturbationtheory,necessitatingpartialresummationsofEq.(2.21)and(2.24).OnesuchresummationisCCtheory,whichhasadistinguishedhistoryinquantumchemisty,andiscommonlyregardedastheabinitiomethodwiththeoptimalcompromisebetweenaccuracyandcomputationalcost.Althoughcontemplatedbynu-clearphysicists,the\hardcore"inter-nucleoninteractionsusedatthattimemadeittoapplyCCtheorytonucleiwithoutperformingacomplicatedrearrangementoftheCCequationstoadequatelycapturethestrongshortrangecorrelations[48{50,60].Inchem-istryithasbecomeamaturemostnotablybecauseofitseaseofuse,anditsabilitytoapproximatemoreexactCCmethodsfromlessexpensiveoneswithoutrevertingtothemoreexactmethodscost.Thesefeatureswillbebriereviewedinthefollowingsections.162.4.1ExponentialAnsatzCCtheoryreliesandforemostontheabilitytoparametrizetheexactwavefunctionastheresultofanexponentialoperatoractingonareferenceSlaterdeterminantji,jCCi=eTji;T=T1+T2+:::;(2.28)withclusteroperatorsT1=Xaitai:ayaai:;(2.29)T2=14Xabijtabij:ayaaybajai::(2.30)...ItcanbeshownviathelinkedclustertheoremthatjCCiisanexactreformulationoftheMBPTwavefunction,andthustheCCansatzisanexactreformulationoftheScodingerequationifTisnottruncated[16,61].TheenergyandcotsoftheclusteroperatorsaredeterminedbysolvingthealgebraicsystemofequationshjeTHeTji=E0;(2.31)haijeTHeTji=0;(2.32)habijjeTHeTji=0;(2.33)...17wherejaii;jabiji;:::,areparticle-holeexcitedSlaterdeterminants.ThefullalgebraicformsofEqs.2.31-2.33canbefoundinRef.[16].ItisclearthatiftheseequationsarethesimilaritytransformedhamiltonianH=eTHeTnolongerconnectsjitoexcitedSlaterdeterminants.Itisalsoimportanttonotethatthesimilaritytransformationisnon-unitary,whichmakesthesimilaritytransformedHamiltonianHnon-Hermitian.Solvingforthehigher-bodyamplitudeslikethosefoundinEq.2.31-2.33becomesin-creasinglymoreexpensiveforeachhigher-bodyclusterthatissolvedfor.Thusitbecomesnecessarytotruncatetheclusteramplitudes,meaningthemethodbecomesanapproximatesolutiontotheScodingerequation.Itismostcommonthentosolvetheseequationsap-proximatelyforonlyTnwithn<mAwheremA<A.SolvingforuptoT1;T2;T3;:::hascommonlybecomeknownassingles(S),doubles(SD),triples(SDT),etc.Mostcommonly,TisapproximatedattheT1+T2level,whichisknownasCCsingles+doubles(CCSD),whichhasacostofn2on4uwherenoandnuarethenumberofoccupiedandunoccupiedsingleparticleorbitalsinthecalculation.OneofthemostimportantimplicationsofusingtheCCansatzisthatofrelativeinsen-sitivitytothechoiceofreference.AccordingtoatheorembyThouless[62],anytwoSlaterdeterminantsjAi;jBithatarenon-orthogonalandthereforehavenon-vanishingoverlaparerelated(uptoanormalizationconstantandphasefactor)byasimilaritytransformation:jBi˘expXaitai:ayaai:jAieT(1)jAi:(2.34)SincetheTiareonlyedintermsofparticle-holeexcitationoperators,itiseasytosee18thatclusteroperatorsoftparticlerankcommute,[Ti;Tj]=0;(2.35)becausecontractionsbetweenparticlecreationandholeannihilationoperatorsvanish.Con-sequently,theCCSDwavefunctioncanbewrittenasjCCSDiˇeT1+T2ji=eT2eT1ji;(2.36)andthusThouless'theorem(2.34)isdirectlybuiltintotheCCformalism.ThesingleSlaterdeterminantmappedtobyeT1jiwillingeneralbeofthebetterSlaterdeterminantsthatcanbechosen,regardlessoftheoriginalchoiceofji.IthasbeenshownthattheCCSDapproximationcontainscertainclassesofperturbativediagramsliketheso-calledparticle-particleladders,andhole-holeladderssummedtoorder[63].Further,theparticle-holeladders,andtheirinterferencewithparticle-particleandhole-holeladdersareincludedaswell.ThisseparatesitfrommoretraditionalresummationmethodsliketheBruecknerHartree-FockformalismortheRandomPhaseApproximation[63,64].TheperturbativecontentoftheCCSDmethodisimportanttonotenow,astheapproximationstotriplesthatareforthcomingwereoriginallymotivatedbyadesiretoincreasetheorderbyorderaccuracyofthemethod[16].AcloseinspectionoftheCCSDenergyshowsthatitiscompletethroughthirdorderinMBPT,andisincompletewithrespecttoMBPTbeginningwithconnectedtripleexcitationsatfourthorder.192.5ApproximateTriplesSincefullCCSDTiscomputationallyexpensiveevenwhenusingjustatwo-bodyHamilto-nian,scalingatn3on5p,variousapproximationshavebeendevelopedwhichtakeintoaccounttheleadingoftriplesonCCSD.Inthisthesis,Iwillfocusonthenon-iterativetriplescorrelationswhicharecalculatedusingfullyconvergedsinglesanddoublesamplitudesintheabsenceofT3.Theseapproximationssharestrongparallelsbothinformandphilosophytothethree-bodyIM-SRGapproximationsmadeinthisthesisinChapter5.2.5.1CCSD(T)ThemostcommonlyusedCCmethodinquantumchemistryandnuclearphysicsisCCsinglesdoublesplusperturbativetriples,anddenotedasCCSD(T)[65].HereIwillonlypresentanearlierversionofthisapproximation,referredtoasCCSD[T]orCCSD+T.IfstartingwithaHartree-Fockreferencestate,therequireddiagramstorestorefourthorderMBPTcontentarefoundinCCSD[T].Inthisapproximation,T3isapproximatedcorrectlyuptosecondorderinMBPT.Thisallowsforanenergycorrectioncorrectthroughfourthorder.T3hasthediagrammaticformfoundinFig.2.3,andalgebraicformofTabcijkˇhabcijkjfR0T2gCji:(2.37)ItcanbeshownthatallfourthordertripleexcitationdiagramsfoundinFig.3.16canbefoundinCCtheoryasECCSD[T]=13!2XabcijkjTabcijkj2ijkabc;(2.38)20Figure2.3:DiagramsdemonstratingtheapproximateT3constructedfromconvergedT2amplitudesintheCCSD[T]energycorrection.ThedashedlinerepresentstheresolventoperatorR0fromperturbationtheory.whereijkabc=i+j+kaab,andi=fii.ItisimportanttonotethatTabcijkmustbemadeantisymmetricbeforeusein(2.38).ByinspectingthediagramsinFig.2.3andcountingthenumberofparticleandholelines,onearrivesatthen3on4uscalingusuallyquotedbyaperturbativetriplescalculation.Thiswillbethesamescalingforallnon-iterativetripleswithinCCtheory.TheadditionoftheECCSD[T]termtoCCSDcalculationsimprovesagreementwithexactresultsatequilibriumgeometriesofmoleculesdramatically,butusuallybeginstofailatstretchedgeometriesevenforsinglebond-breakinginchemistryresults,andmoregenerallyanytimethesystemexhibitsastrongmulti-referencecharacter[66].Thesituationisbetter,butstillqualitativelysimilarforCCSD(T).Fornuclearsystems,itisexpectedthatthesecorrectionsbehavesimilarly,buttheissueofexplicitthree-bodyforcesmakestheissuesomewhatlesstransparent[67].212.5.2-CCSD[T]OneimportantdevelopmentinCCtheoryisitsrecastingasthesolutiontoabi-variationalminimization[16].ThisinvolvesidentifyingtherightandlefteigenvectorofthesimilaritytransformedHamiltonian.Thelefteigenstatecanbewrittenashj(1+;=1+2+:::;(2.39)wheretheentnaretrankde-excitationoperatorsas1=Xaiia:ayiaa:;(2.40)2=14Xijabijab:ayiayjabaa:::::ThisyieldstheformalCCenergyfunctionalasECC=hj(1+eTHeT)Cji:(2.41)IftheandTaretruncatedatthesamelevel,forexampleatthesinglesanddoubleslevel,thentheEq.2.31-2.33resultasthestationarityconditionsforthefunctionalasiaandijabarevaried.Additionally,onearrivesatasetofequationsifTiaandTijabarevaried.Thisnotonlyprovidesanavenueforgeneratingobservables,butalsoforfurtherapproximatingtheofhigherorderclusterandamplitudes.Again,isolatingtheperturbativecontenttorestorefourthordertermstoaenergycalculation,ECCSD[T]=hj2T3)Cji;(2.42)22whereT3isapproximatedinthesamewayasitwasforCCSD[T].ThiscorrectionallowsfortheinclusionoftdiagramsthanCCSD[T],andgenerallybehavesbetterwhendescribingbondbreaking,butagainbreaksdownforsystemswithastrongmulti-referencecharacter[16,68,69].2.5.3CompletelyRenormalizedCCMethodsArguablythemostcompletenon-iterativemethodforapproximatingfullCCSDTfromCCSDmethodsistherecentformulationofcompletelyrenormalizedCCcommonlylabelledasCR-CCLorjustCR-CCfromnowoninthiswork[8].CR-CCLimproveduponearliercompletelyrenormalizedmethodssuchasCR-CCSD(T)whichperformedmuchbetterthanthenon-iterativetriplespresentedabove,butwerenotrigorouslysizeextensive[70{72].WewillfollowthepresentationfoundinRef.[8].Herewemustsomenotationforexpedience.Asmentionedbefore,theclusteramplitudeistruncatedatsomeexcitationrank.IfwedenotethisbythenumbermA,thenT(ma)=PmAn=1Tn,andH(mA)=eT(mA)HeT(mA).CR-CCmethodsdispensewithpurelyperturbativeargumentsandinsteaduseamomentexpansioncombinedwithanovelparametrizationofthetruefullCI\bra"vectorh0jH=E0h0j=E0hjLeT(mA):(2.43)ItisimportantheretonoticethatalthoughtheLissimilarinformandfunctionto1+fromtheprevioussection,theyarenotidentical.Lisforhigher-bodycomponentsabovemA,whereas1+isonlyuptothesamelevelofapproximationasTintheCCtheory.Thiswouldmakeitunfeasibletoincorporatearesidualthree-bodyforceineitherofthepreviousapproximatetriplestreatments.Thisparametrizationleadstothe23asymmetricexpressionforthetruegroundstateenergyE0=hjLeT(mA)HeT(mA)jihjLeT(mA)eT(mA)ji=hjLH(mA)ji:(2.44)ThisallowsforinsertionofcompletenessintermsofexcitedSlaterdeterminantstoarriveatE0=AXn=0Xi1<:::<ina1<:::<anhjLja1:::ani1:::iniha1:::ani1:::injH(mA)ji=E(mA)+(mA;A):(2.45)ThelevelofexcitationsinsertedbetweenLandH(mA)canbetruncatedatanotherlevelmB.Ifwealsomaketheofthe\moments"ofthesimilaritytransformedmatrixelementasMa1:::ani1:::in(mA)=ha1:::ani1:::injH(mA)ji.ThisallowstheCR-CCenergycorrectiontobewrittenasafunctionofmAandmB,(mA;mB)=mBXn=mA+1Xi1<:::<ina1<:::<anla1:::ani1:::inMa1:::ani1:::in(mA):(2.46)HavinggiventheformalismunderpinningCR-CCmethods,InowturntotherelevantCR-CC(2,3)method.ThatmeansthatLˇ1+CCSD+L3.TheL3amplitudesareapproxi-matedbymultiplyingEq.(2.43)ontherightbyeT(CCSD)jabcijki,yieldinghj(1+CCSD)H(CCSD)jabcijki+hjL3H(CCSD)jabcijki=E0labcijk:(2.47)IfthefurtherapproximationismadethathdeflmnjH(CCSD)jabcijkiˇ0unlessijk=lmnandabc=def,andthatE0ˇECCSD,thenthelabcijkcanbeisolatedusingonlyknown24quantitiestoyieldlabcijk(CCSD)ˇhj(1+CCSD)H(CCSD)jabcijkiECCSDhabcijkjH(CCSD)jabcijki:(2.48)FullCCSDTtendstotrackfullCIresultsquitewellevenforsystemswithamoderatelylargemulti-referencecharacter.CR-CC(2,3)seemstogivecomparableaccuracytofullCCSDT,whiletheprevioustwofailtoqualitativelyreproducefullCCSDTinthesecases[8].ThisabilityofCR-CC(2,3)tomimicfullCCSDTdespitescalinginthesamemannerasCCSD(T)canbetrackeddowntotwomainfeatures.Theimprovemententersbyupdatingtheof\diagonal"thanisusedintheenergydenominatorsofbothCCSD(T)andCCSD(T)corrections.Thedenominatorofthetwomethodsarejusttheoller-PlessetdenominatorsofthebareHFenergies.Inthecompletelyrenormalizedtheorypresented,oneinsteadusestheenergydenominatorassociatedwiththefullsimilaritytransformedHamiltonian,anddiagonaltwo-bodytermsareincludedinthedenominators.InChapters5and6,resultswillbepresentedwiththelabelCCSD(2)T,whichisequivalenttoCR-CC(2,3),wherethedenominatorismadeupofonlyone-bodytermsfromthefullsimilaritytransformedHamiltonian.Secondly,thethree-body\moment"oftheCR-CC(2,3)correctioncontainsmanymoretopologiesnotfoundinthepreviousmethods,astheywereonlylinearintheT2amplitudes.25Chapter3In-MediumSimilarityRenormalizationGroupAsdiscussedinSec.1.1,therapidprogressinabinitionuclearstructureinrecentyearshasbeendriveninlargepartbythedevelopmentofrenormalizationgroupmethodstoproducesoft\low-momentum"interactionsfromunderlying\hard"interactionswithoutdistortinglow-energyobservables.Softpotentialsarehighlyadvantageousformany-bodymethodsthatrelyonexpandingwavefunctionsinabasisoflocalizedsingle-particleorbitals,astheconvergenceofcalculationswithrespecttobasissizeisdramaticallyimproved.Moreover,sincestrongshort-rangecorrelationsaresmoothedout,Hartree-Fockbecomesareasonablezerothorderstartingpointfornuclei,makingmethodslikeMBPTandcoupledclustertheorybasedonbuildingcorrelationsontopofa\simple"referencestateattractivemethodsfornuclearstructurecalculations.Aswillbeshowninthefollowing,theIM-SRGapproachextendstheRGnotionofdecouplinglow-andhigh-momentumdegreesoffreedomtothemany-bodyHilbertspacebyformulating\in-medium"wequations,thesolutionofwhichisequivalenttothepartialdiagonalizationorblock-diagonalizationofthemany-bodyHamiltonian[11,21,24,33,39,58].Becauseofitsfavorablepolynomialscalingwithsystemsize,andtheytotargetgroundandexcitedstatesofbothclosed-andopen-shellsystems,theIM-SRGprovidesa26powerfulabinitioframeworkforcalculatingmedium-massnucleifromprinciples.Inthepresentchapter,IpresentthebasicIM-SRGformalismandreviewsomeofthesuccessfulapplicationstonucleiinrecentyears.Afterthisintroductoryreviewandsurveyofpreviousresults,IpresentadetailedanalysisoftheperturbativecontentoftheIM-SRGinSec.3.5.Whilethissectionisrathertechnical,itprovidescrucialguidanceforunderstandingthesimilaritiesandwithcoupledclustertheory.Moreimportantly,theperturbativeanalysisinthischapterwillplayakeyroleinformulatingtheimprovedapproximationstobediscussedinChapter5.3.1IM-SRGFormalism3.1.1OverviewoftheSRGThemainideaoftheSimilarityRenormalizationGroup(SRG)istodrivetheHamiltonianH(s)towardsadiagonalorblock-diagonalformviaacontinuousunitarytransformation[73]H(s)=Uy(s)H(0)U(s):(3.1)TakingthederivativeofEq.(3.1)withrespecttothewparameters,weimmediatelyobtaintheoperatorwequationddsH(s)=[(s);H(s)];(3.2)27wherethegenerator(s)isrelatedtotheunitarytransformationU(s)by(s)=dUy(s)dsU(s)=y(s):(3.3)NotethattheformalsolutionforU(s)isgivenbythepath-orS-orderedexponentialU(s)=SexpZs0ds0(s0):(3.4)Fornow,thespformof(s)isunimportant;IwillrevisitpossiblechoicesinSection3.2,andhowtheycanbedesignedtodrivecertainpartsofthetransformedHamiltoniantozero.Intheoriginalapplicationstonuclearphysics,whichwerefertoasthefreespaceSRG,theaimwastosoftentheHamiltonianbychoosingtodecouplehigh-andlow-mo-mentummodes,drivingtheHamiltoniantowardsaband-diagonalformwithincreasingsinmomentumrepresentation,makingtheinteractionsmoretractableforabinitiocalculations[33,52,54,55].Ontheonehand,thefree-spaceSRGisconvenient,asitdoesnothavetobeperformedforeachtnucleusornuclearmatterdensity.Ontheotherhand,itisnecessarytoconsistentlyevolvethree-nucleon(andpossiblyhigher)interactionsthatareinducedduringtheevolutiontobeabletosoftentheinteractionstlyandmaintainapproximates-independenceofA>3observables.TheconsistentSRGevolutionofthree-nucleonoperatorsrepresentsattechnicalchallengethathasonlyrecentlybeensolvedinrecentyears.AninterestingalternativeistoperformtheSRGevolutionin-medium(IM-SRG)foreachA-bodysystemofinterestbynormal-orderingwithrespecttoanappropriateA-bodyreferencestate[11,21,24,33,39,58].Unlikethefree-spaceevolution,theIM-SRGhas28theappealingfeaturethatonecanapproximatelyevolve3;:::;A-bodyoperatorsusingonlytwo-bodymachinerythankstothereshoftermsbroughtaboutbynormal-ordering.Moreover,withasuitablenitionofthepartoftheHamiltoniantobedriventozero,theIM-SRGcanbeusedasanab-initiomethodinandofitself,ratherthansimplytosoftentheHamiltonianasinthefree-spaceSRG.3.1.2M-SchemeFlowEquationsforIM-SRG(2)IftheIM-SRGtransformationisperformedexactlyforanA-bodysystem,thetransformedHamiltonianwillinvolveuptoanA-bodyinteractionsregardlessoftheinitialparticlerankofthestartingHamiltonian.Withinthesecondquantizationformalismdiscussedabove,itissimpletoshowhowtheseinducedmany-bodyforceswouldoccurviathefollowingequation,[:ayaaybadac:;:ayiayjalak:]=ci:ayaaybayjalakad:+::::(3.5)InthefreespaceSRG,thecouplingbetweenthetparticlerankoperatorsis\one-way".Forinstance,thetwo-bodyinteractionsfeedintothewequationsforthethree-andhigher-bodyinteractions,buthigher-bodyoperatorsdon'trenormalizelower-rankoperators.Incontrast,fortheIM-SRGthesehigher-bodyforcescertainlyfeedbackintoevolutionofthelower-bodyinteractions.Tocontroltheproliferationofmany-bodyinteractions,practitionersoftheIM-SRGmethodhavetypicallyperformedasimpletruncationinwhichonlythenormal-orderedzero-,one-,andtwo-bodypartsof(s)andH(s)arekept,sothatH(s)ˇE(s)+f(s)+s);(3.6)(s)ˇ(1)(s)+(2)(s):(3.7)29ThisiscalledtheIM-SRG(2)truncation,andhasbeenverysuccessfulintreatingmediummassnuclei[39,58,74].Inthiswork,IwillalsodiscusstheIM-SRG(3),wherethethree-bodyforcesarekeptaswell.Thecommutatorexpressionwithfullthree-bodyforcesaregiveninAppendixB,butthefullIM-SRG(3)methodwithoutapproximationshasneverbeenimplementedinphysicalsystemsduetoitshighcomputationalcost.OneofthemainthrustsofmythesisisthedevelopmentofcomputationallytractableapproximateIM-SRG(3)calculations,seeChapter5.EvaluatingEq.3.2usingthegeneralcommutatorexpressionsfoundinAppendixA,oneobtainstheIM-SRG(2)equationsdEds=Xab(nanb)abfba+12Xabcdabcdcdabnanbncnd;(3.8)df12ds=Xa(1+P12)1afa2+Xab(nanb)(abb1a2fabb1a2)+12Xabc(nanbnc+nanbnc)(1+P12)c1ababc2;(3.9)d1234ds=Xaf(1P12)(1aa234f1aa234)(1P34)(a312a4fa312a4)g+12Xab(1nanb)(12abab3412abab34)Xab(nanb)(1P12)(1P34)b2a4a1b3;(3.10)whereni=1ni,andthes-dependencehasbeensuppressed.NotethatpermutationsymbolPij,whichrepresentsthefollowinginterchangePijg(:::;i;:::;j)g(:::;j;:::;i);(3.11)30hasbeenusedtosimplifytheexpressions.ThecoupledtialequationsinEqs.(3.8){(3.10)areintegratedfroms=0untilasuitabledecouplingconditionhasbeenachieved,withtheinitialvalueconditionH(0)=E(0)+f(0)+:(3.12)TheperturbativecontentofEqs.(3.8){(3.10)willbeanalyzedinSection3.5.Inthatpresentation,itwillbecomeclearthat,muchlikeCCSD,theIM-SRG(2)includes-orderladdersumsintheppandhhchannels(i.e.,BruecknerHartree-Focktypecorrelations),ringdiagramsumsinthephchannel(RPAtypecorrelations),pluscomplicated\interference"termsbetweenthevariouschannels.Furthermore,likecoupledclustertheory,theIM-SRGatanytruncationlevelisbasedonacommutatorexpression.Consequently,theHamiltoniancontainsonlyconnecteddiagrams[16,75]andoneobtainssize-extensiveresults.FromEqs.(3.8){(3.10),thetwo-bodywequationwhichcontainsdoublycontractedtwo-bodyoperatorsdominatesthecost.ThesetermsscalepolynomiallyasO(N6)withthesingle-particlebasissizeN.ThisissimilarincosttoCCSD,theSelf-ConsistentGreen'sFunctionApproach(SCGF)[19,76,77],orcanonicaltransformationtheory[78,79].3.1.3SymmetriesandtheFlowEquationsItisoftenusefultoimposeexplicitsymmetriesonahamiltonianinordertoreduceunneededtrivialThisisobviouslytrueforthewequationsaswell.Examplesincludespinsymmetryincoulombicsystems,translationalinvarianceinaplanewavebasis,andsphericalsymmetrywithnuclearsystems,allofwhichareutilizedinthiswork.IwillpresentonlythesphericallysymmetricJJ-coupledwequations,astheyaretheleasttrivialexample.31Ifthesingle-particleindicesaremadetorefertoradial,angularmomentum,andisospinquantumnumbersi=(kiliji˝i),thentheydonotdependontheangularmomentumpro-jectionmi.Thentheonlynon-diagonalpartoftheone-bodymatrixelementsbecometheradialquantumnumbers,e.g.,f12=fl1j1˝1k1k2l1l2j1j2˝1˝2:(3.13)Additionally,thetwo-bodymatrixcanbecoupledtoangularmomentumJtoyieldthe(computationally)IM-SRG(2)wequationsdEds=Xab^j2aabfba(nanb)+12XabcdJbJ2JabcdJcdabnanbncnd;(3.14)df12ds=Xa(1+P12)1afa2+1^j21XabJbJ2(nanb)abJb1a2fabJb1a2+12^j2112XabcJbJ2nanbnc+nanbnc(1+P12)Jc1abJabc2;(3.15)dJ1234ds=Xa1(1)Jj1j2P121aJa234f1aJa2341(1)Jj3j4P34a3J12a4fa3J12a4+12XabJ12abJab34J12abJab34(1nanb)+XabJ0(nanb)1(1)Jj1j2P12bJ028><>:j1j2Jj3j4J09>=>;J014abJ0ab32J014abJ0ab32;(3.16)where^j=p2j+1,indiceswithabarindicatetime-reversedstates,andtheandmatrixelementsinthelastlineofEq.(3.16)areobtainedbyageneralizedPandyatransform(see,32e.g.,[80]),OJ1234=XJ0^J028><>:j1j2Jj3j4J09>=>;OJ01432:(3.17)IthasbeenshownthatthisangularmomentumcouplingprocessdropstheNnumberofM-schemeorbitalsdowntoroughlytoroughlytoN2=3J-schemeorbitals[64]3.1.4GeneralObservablesWithintheIM-SRGframework,theconsistenttransformationofobservablesinadditiontotheHamiltonianisconceptuallyverysimple:TheoperatorO(s)isnormalorderedwithrespecttojiandtruncatedtothesametwo-bodylevel,andthensubjectedtoexactlythesametransformationviathesametialequationddsO(s)=[(s);O(s)];(3.18)ThismeansthatanadditionalsetofwequationsforeachO(s)needtobeintegratedconcurrentlywiththehamiltonian,roughlydoublingthesizeofthesystemofcoupledentialequations.Thismeansthatthemethodscaleslinearlywitheachadditionalobservableonewishestocalculate,renderingtheIM-SRG(aspresented)unsuitablefortreatingsystemswheremorethanafewadditionalobservablesareneeded.InChapter4,anovelformulationoftheIM-SRGusingMagnusexpansiontechniqueswillbepresentedthatbypassesthis(andotherrelated)computationallimitation.33hijH(0)jjihijH(1)jji0p0h1p1h2p2h3p3h0p0h1p1h2p2h3p3h0p0h1p1h2p2h3p3h0p0h1p1h2p2h3p3hFigure3.1:SchematicrepresentationoftheinitialandHamiltonians,H(0)andH(1),inthemany-bodyHilbertspacespannedbyparticle-holeexcitationsofthereferencestate.3.2ChoiceofGenerator3.2.1DecouplingAsmentionedpreviously,theIM-SRGtransformationcanbetailoredtodriveasuitablypartoftheHamiltoniantozero.Toseehowthisworks,letusconsiderthecaseofasinglereferencesystemsuchasthegroundstateofaclosed-shellnucleus.Fig.3.1showsaschematicrepresentationofthes=0normalorderedhamiltonianH(0)ontheleft,wherethereferencestatejiisthe0p0hstateatthetopleft.Toisolatethegroundstate,wedesireatransformationthatdrivestheHamiltoniantowardstheformshownintherightpanel,wherethe0p0hreferenceisnowaneigenstateofthetransformedhamiltonian.Notethataninitialthree-bodyinteractionwouldgeneratecouplingbetweenthenpnhand(n+3)p(n+3)hblocks,sothenotationofFig.3.1implicitlyassumesweonlyhavetwo-bodyinteractions.Clearly,thematrixelementsthatcouplethe0p0hreferencestatejitohigherexcitations34arehjH(0):aypah:ji=fph;(3.19)hjH(0):aypayp0ah0ah:ji=pp0hh0;(3.20)andtheirHermitianconjugates.Soforthisparticularproblem,asuitableforthepartsofthehamiltoniantobedriventozeroareHod(s)=Xphfph:aypah:+14Xpp0hh0pp0hh0:aypayp0ah0ah:+H.c.:(3.21)Duringthew,couplingisinducedbetweenthe0p0handhigherstates,e.g.hjH(s):ayp1:::aypAahA:::ah1:ji6=0:(3.22)Theseinducedforcesareforcedtovanishbyourtruncation,thustheeigenvalueobtainedinanytruncatedIM-SRG(n)typecalculationisnotvariational.Itisimportantthatthesizeandofthistruncationcanbeinvestigatedforsystematiccheckingandimprovement.Thematrixelements(3.19),(3.20)areapproximatelydriventozero,andthesinglezero-bodypartofthehamiltonianbecomestheeigenvalueoftheexactgroundstate.Alternatively,onecouldthinkofthetransformedhamiltonianasaunitarilyequivalenthamiltonianinwhichtheHartree-Focksolutionisexact.353.2.2WhiteGeneratorsAgeneratormustnowbechosen,withtherequirementthatitwilleliminateHodasthehamiltonianistransformedviatheIM-SRGwequations.TheclassofgeneratorswhichprovidesthebiggestnumericalbandmosttransparentconnectiontoMBPTisthatfoundintheworkofWhiteoncanonicaltransformationtheoryinquantumchemistry[58,78]:IA/B(s)Xphfph(s)A/Bph(s):aypah:+Xpp0hh0pp0hh0(s)A/Bpp0hh0(s):aypayp0ah0ah:H.c.:(3.23)Infuturesections,Iwillappealtotheobviousthree-bodygeneralizationsofthisgeneratorinordertomoveforwardwithapproximationstoIM-SRG(3).Anti-Hermiticityof(s)isguaranteedbythesignchangeintheenergydenominatorsundertranspositioninEq.(3.23).TherearetwotchoicesfortheenergydenominatorsemployedintheWhitegen-erators.Thesearetiatedbythesuperscriptsin(3.23).TheycorrespondtotheEpstein-NesbetandM˝ller-PlessetpartitioningsusedinMany-BodyPerturbationTheory(MBPT)(see,e.g.,[16]).White'sprescriptioninRef.[78]leadstotheEpstein-Nesbetcase:AphhphjHjphihjHji=fpfh+phph=Ahp;(3.24)App0hh0hpp0hh0jHjpp0hh0ihjHji=fp+fp0fhfh0App0hh0Ahh0pp0;(3.25)wherefp=fpp;fh=fhh,andApp0hh0pp0pp0+hh0hh0phphp0h0p0h0ph0ph0p0hp0h:(3.26)36TheM˝ller-Plessetcaseissimpler,yieldingBphfpfhBhp;(3.27)Bpp0hh0fp+fp0fhfh0Bhh0pp0:(3.28)Thesetwochoicesgiverisetoalmostnoceinpracticalcalculations,butthecon-nectiontoM˝ller-PlessetperturbationtheoryiseasiertoshowwiththeM˝ller-Plessettypedenominators.AlmostallresultspresentedinthisworkhavebeenproducedwithM˝ller-PlessettypeWhitegenerators.IfonewantstoworkwiththeJ-schemewequations(3.14){(3.16),itisnotunambigu-ouslyclearhowtotreatthetwo-bodymatrixelementsintheEpstein-Nesbetdenominators(3.24),(3.25)intheangularmomentumcouplingprocess.Usingthemonopolematrixele-mentsyieldastraightforwardsolutiontothisproblem,i.e.(0)abcdPJ(2J+JabcdPJ(2J+1)(3.29)inEqs.(3.24){(3.26).ThebigadvantageofWhite-typegeneratorsinpracticalcalculationsliesinthefactthatitsuppressesallmatrixatroughlythesamedecayscale(seeSection3.2.5).Thus,thesuppressionrateisnotafunctionofanyenergyormomentumscale,anditthereforedoesnotrepresentaproperRGw.Thisincontrasttotheimaginary-timeandWegnergenerators,whichwillbediscussedinthenextSection3.2.4.ThisRGdistinctionisunimportantintermsofresults,asanychoicechoiceofgeneratorthatdecouplesthereferencefromexcitationsisproducinganeigenstateregardlessoftheorganizationoforder37ofsuppression.Thebofthisuniformsuppressionofelementsistounderstate.BecausetheWhitetypegenerator'smatrixelementsaregivenbyratiosofenergies,fandappearlinearlyoftheright-handsideoftheIM-SRGwequations(3.8){(3.10).ThisclearlywouldyieldamuchlesssetofwequationsthantheWegnergenerator,wherethirdpowersoffandappear(seebelow),ortheimaginarytimeclass,wheresecondpowersappear.Thus,forsystemswheretheWhitetypegeneratorremainsawellobjectthroughoutthetransformation,thenumberofintegrationstepsrequiredtosolvetheIM-SRGwequationsaremanifestlylessthanothergenerators.However,ifoneencountersasystemswithvanishingenergydenominators[21,39],theWhitegeneratorswillbepoorlyInthiswork,allsystemspresentedcanbeapproximatedasclosed-shellsystems,andthustheWhitetypegeneratorsareingeneralawelloperator.3.2.3Imaginary-TimeGeneratorsOnecanalsomotivateasecondclassofgenerator,byappealingtosolutionsoftheimaginary-timeSchrdingerequation.UsingtheHamiltonian,Eq.(3.21),weIIA/B(s)XphsgnA/Bph(s)fph(s):aypah:+Xpp0hh0sgnA/Bpp0hh0(s)pp0hh0(s):aypayp0ah0ah:H.c.;(3.30)whereA/BareagaintheEpstein-NesbetandM˝ller-PlessetenergydenominatorsinEqs.(3.24){(3.28).Thesignfunctionsensurethatmatrixelementsaresup-pressedinsteadofenhancedduringthew.SolvingtheIM-SRG(2)equationsforthe38imaginarytimegeneratorshowsthatthedecayscaleforthesematrixelementsisapprox-imatelygivenbythediagonalenergybetweenthereferenceand1p1hor2p2hexcitations.ThisenergyisjustA/B,dependingonthechosenfordiagonal.ThusIIA/BgeneratesaproperRGw,organizedbyenergyinthechoicefor\diagonal".Asmentioned,thequadraticdependenceoftheIM-SRG(2)equationsonfandcreatesamoresetofequations,andthusrequiremoreintegrationstepstosolve.Thisgeneratorisrobustlyeveninthepresenceofvanishingenergydenominators,andthusprovidesagoodchoiceforsystemswheretheWhitegeneratorbecomesapoorlyobject.Thishasbeenfoundtobeveryusefulinthemulti-referenceIM-SRG(2)methodofHergertetal.[81].3.2.4WegnerGeneratorsThegeneratorwhichhasbeenmostformallyexploredistheWegnertypegenerators[73].Theseprovidearobustlygeneratorforanyofdiagonal",andnotjustgroundstatedecouplingpresentedabove.Intheoriginalwork,WegnerproposedthefollowinggeneratorIII(s)=[Hd(s);Hod(s)]:(3.31)UsingtheoftheHamiltonianforthiswork,Eq.(3.21),andthecom-mutatorsfromAppendixA,onecanarriveatthematrixelementsof(s).InIM-SRG(2)39calculations,onekeepsonlytwo-bodyandloweroperators,yielding12=Xa(1P12)fd1afoda2+Xab(nanb)(fdabodb1a2fodabdb1a2)+12Xabc(nanbnc+nanbnc)(1P12dc1abodabc2;(3.32)1234=Xan(1P12)(fd1aoda234fod1ada234)(1P34)(fda3od12a4foda3d12a4)o+12Xab(1nanbd12abodab34od12abdab34)Xab(nanb)(1P12)(1P34db2a4oda1b3:(3.33)Clearly,theWegnergeneratorinEqs.(3.32)and(3.33),andthewequations(3.9)and(3.10)arenearlyidenticalexceptforanti-hermiticityandhermiticityrespectively.ThusthesphericalJ-schemeexpressionsforIII(s)areeasilyobtainedfromEqs.(3.15)and(3.16).TheIM-SRGequationsreachapointwhen(s)vanishes,thetransformationceases.FortheWegnergenerator,apointats!1existsifHod(s)vanishesasrequired.Ithasbeenshownthat[73,82]ddstrHod(s)2=2try(s)(s)0(3.34)sincey(s)(s)ispositiveThisimpliesthatwiththischoiceofgenerator,Hod(s)isincreasinglysuppressedandH(s)isrenderedintoaformofHd(s).Similar,totheimaginarytimegenerator,theWegnergeneratorcreatesaproperRGwthatsupressesmatrixelementsbasedonthediagonalenergyencebetweenthereferenceand1p1hor2p2hexcitations.Whileformallyappealing,androbusttovanishing40energydenominators,thisgeneratorcreateswequationsthatdependonfandinacubicfashion.Thisleadstoextremelyequationsthatrequiremany,manystepstoarriveatadecoupledreference.Additionally,thecosttoconstructIIIisasmuchasevaluatingthewequations,makingeachtimestepapproximatelydoublethecostoftheprevioustwogenerators.3.2.5DecayScalesLetusexamineinmoredetailhowthegeneratorsgiverisetothetdecayscaleswehaveclaimedabove,andthuscreatevaryingdegreesofinsolvingtheIM-SRG.Aswitheverythingassumedabove,weidentifyadiagonalandandpartofthehamiltonian,H(s)=Hd(s)+Hod(s);(3.35)whereHod(s)istobesuppressedass!1.ItisthennaturaltoworkintheeigenbasisofHd(0).SincetheformofHd(s)doesn'tchange,itispossibletoassumethatitseigenbasisisinvariantunders,sothatateachstepoftheowHd(s)jni=En(s)jni:(3.36)Inthisbasisrepresentation,Eq.(3.2)becomesddshijHjji=Xk(hijjkihkjHjjihijHjkihkjjji)=EiEjhijjji+XkhijjkihkjHodjjihijHodjkihkjjji;(3.37)41andhijHodjii=0.ConsidernowaWhitetypegenerator,whichcanbewrittenashijIjji=hijHodjjiEiEj;(3.38)andspeEq.(3.2)tothefollowingddshijHjji=ijHodjji+XkEi+Ej2Ek(EiEk)(EjEk)hijHodjkihkjHodjji:(3.39)IfthetransformationgeneratedbytrulysuppressesHod,andifitisassumedthatHodeitherbeginssmallcomparedtoHd,orwillbecomesoduringthew,thenwecanneglectthesecondtermquadraticinHod.Thenitispossibletojustinspecttheterminthewequationsinordertoillustratehowdiagonalmatrixelementsarebeingsuppressed.Inthiscase,Eq.(3.39)impliesdEidss=0=2XkhijHodjkihkjHodjii(EiEk):ˇ0;(3.40)andtheenergiesstay(approximately)constant:Ei(s)ˇEi(0):(3.41)Consequently,Eq.(3.39)canbeintegrated,andonearrivesathijHod(s)jjiˇhijHod(0)jjies;s>s0;(3.42)42asalreadymentionedinSection3.2.2.Thisisnotsuggestingthatthequadratictermsareunimportant,justthatthetermiswhatsetsthedecayscale.Theimaginary-timegeneratorcanbewrittenashijIIjji=sgnEiEjhijHodjji;(3.43)andthewequationddshijHjji=EiEjhijHodjji+Xksgn(EiEk)+sgnEjEkhijHodjkihkjHodjji:(3.44)NotethatthesignfunctionintheofIIensuresthatonlytheabsolutevalueoftheenergybetweenthestatesjiiandjkiappearsintheterm.IntegrationofEq.(3.44)yieldshijHod(s)jjiˇhijHod(0)jjieEiEjjs;(3.45)andmatrixelementsaresuppressed,withadecayscalesetbyjEiEjj.Finally,weperformthesamekindofanalysisfortheWegnergeneratorhijIIIjji=hij[Hd;Hod]jji=(EiEj)hijHodjji:(3.46)ThewequationreadsddshijHjji=EiEj2hijHodjji+XkEi+Ej2EkhijHodjkihkjHodjji;(3.47)43andweobtainhijHod(s)jjiˇhijHod(s0)jjie(EiEj)2(ss0):(3.48)Thus,theimaginary-timeandWegnergeneratorsyieldproperRGtransformations,inthesensethatmatrixelementsbetweenstateswithlargeenergyEij=jEiEjjdecayatsmallerwparameterssthanstateswithsmallEij.TheWhitegenerator,ontheotherhand,actsonallmatrixelementssimultaneously.InSection3.3,itwillbeshownthatthesentchoicesdonotleadtolargeinthelargeslimit.3.3NumericalExplorationsInthissection,weillustratethegeneralpropertiesoftheIM-SRGwequationsinnumericalapplications,withspecialemphasisonacomparisonofthetgeneratorsthatwereintroducedintheprevioussections.Tosimplifymatters,weonlyuseatwo-bodyinteractionthroughoutthissection(see3.3.1fordetails).3.3.1ImplementationBaringafewcases,theIM-SRGhasonlyreallybeenimplementedfornuclearsystems,thereforeinthisreviewchapterofthemethodIonlypresentdetailsfortypicalnuclearcal-culations.Thedetailsthatwillberelevantforothersystemswillbeintroducedastheresultsappearinlaterchapters.Fornuclearsystems,theIM-SRGisimplementedinharmonicos-cillator(HO)spaces.Theprincipaladvantageofthisbasisinnuclearsystemsisthatonecanfactorizecenter-of-massandrelativedegreesoffreedomintheevaluationofmatrixelements(see,e.g.,for[83]).Thisisaninvaluablepropertyforself-boundsystemslikenuclei.FormethodsliketheNo-CoreShellModel,thispropertycanberetainedevenat44themany-bodylevelforasuitablechoiceofmodelspacetruncation[12].Formethodswhichusesingle-particlebasistruncations,liketheIM-SRG,coupledcluster,andself-consistentgreensfunctions,thisexactfactorizationofcenter-of-massandintrinsicwavefunctionsisanalyticallyspoiled,althoughitisstillobservedempirically.Werevisitthisinchapter4.Asmentionedbefore,theIM-SRGexplicitlyexploitssphericalsymmetryfornuclearapplicationsbyworkingwiththeJ-schemeIM-SRGwequationspresentedinSec.3.1.3.Inthesesphericallysymmetricbasissets,itispossibletoachieveconvergenceforsphericalnucleiofinterestinreasonablecalculations.Thisistrueevenfor\bare"interactionsfromchiralEFTliketheN3LOinteractionbyEntemandMachleidt,withaninitial=500MeV=c[1,2].Thisistheinteractionusedtoproducemostofthenuclearresultspresentedinthisthesis,bothatitsoriginalresolutionscale,indicatedby=1,andatalowerresolutionscale=2:0fm1,whichisgeneratedbyafree-spaceSRGevolutionorsoftening[33,53].ToobtainreferencestatesfortheIM-SRGcalculation,theHartree-FockequationsfortheintrinsicHamiltonian(2.2)areselfconsistentlysolved.TheintrinsicHamiltonianisthentransformedtotheHartree-FockbasisandnormalorderedwithrespecttotheHartree-Fockreferencestate,discardingtheresidual3Npartintheprocess(cf.Eq.(2.5)).Startingfromthezero-,one-,andtwo-bodymatrixelementsofthetruncatednormal-orderedHamiltonianasinitialvalues,theJ-schemewequations(3.14){(3.16)areintegratedwiththeCVODEsolverfromtheSUNDIALSpackage[84].ForWhiteandimaginary-timegenerators,wechoosetherecommendedAdams-Bashforth-Moultonpredictor-correctormethodfornon-systems,whilethebackwtiationmethodisusedforthewequationsintheWegnercase.Inordertodetermineatwhatvalueofstdecouplingisachieved,second-45Nucleus[fm1]E14[MeV]Eex[MeV]4He1-27.18-27.26(3)16O1-126.01-126.3(1)40Ca1-366.23-369(1)4He2.0-28.27-28.2716O2.0-165.68-165.6840Ca2.0-595.98-595.95(2)78Ni2.0-1319.41-1319.4(1)100Sn2.0-1953.96-1954.3(3)132Sn2.0-2752.03-2753(2)Table3.1:IM-SRG(2)ground-stateenergiesofselectedclosed-shellnucleiforthethechiralN3LOinteractionbyEntemandMachleidt[1,2],with=1and=2:0fm1(cf.Fig.3.2).E14aretheenergiesobtainedforemax=14atoptimal~!,andEexareextrapolatedtobasissize(seetext),withextrapolationuncertaintiesindicatedinparentheses.orderMBPTcorrectionforthewingHamiltonianH(s)isused.ThisisadirectmeasureoftheopartoftheHamiltonianasinEqs.(3.21).WhenthesecondorderMBPTcorrectiondropsbelow106MeV,thewisstoppedandtheresultingzero-bodyenergyisconsideredthefullgroundstateenergy.3.3.2ConvergenceInFig.3.2,theconvergenceoftheIM-SRG(2)ground-stateenergiesoftheclosed-shellnuclei4He;16O;and40Cawithrespecttothesingle-particlebasissizeemax(seeAppendix3.3.1)isshown.AllcalculationsshownusetheWhite-Epstein-NesbetgeneratorC,Eq.(3.23),anditshouldbeassumedunlessotherwisestatedthatthisisthegeneratorbeingused.ItshouldbenotedthatfortheunevolvedN3LOinteraction,theHartree-Focksolutionsforallthreenucleihavepositiveenergy.Nonetheless,theHFstatesstillleadtoreasonableandconvergedIM-SRG(2)energiesasshowninFig.3.2.WecancorrectfortheofusingaHObasisbyusingthemethodsdescribedinRefs.[13,14].AHObasiswithemaxhasultravioletandinfraredcutwhichare46-28-26-24-22-20-18.E[MeV]4He=1-28.3-28.2-28.1-28-27.9.E[MeV]=2:0fm1emaxl8n10u12s14-120-100-80.E[MeV]16O=1-166-164-162-160-158-156.E[MeV]=2:0fm116202428323640444852~![MeV]-350-300-250-200-150.E[MeV]40Ca=1(c)16202428323640444852~![MeV]-600-580-560-540.E[MeV]=2:0fm1Figure3.2:Convergenceof4He,16O,and40CaIM-SRG(2)ground-stateenergiesw.r.t.single-particlebasissizeemax,forachiralN3LONNinteractionwith=1(leftpan-els)and=2:0fm1(rightpanels).Noticethetintheenergyscalesbetweentheleftandrightpanels.Graydashedlinesindicateenergiesfromextrapolationtheemax10datasetstobasissize(seetextandRefs.[13,14]).47givenbyUVp2emax+7~=aHO;(3.49)LIRp2emax+7aHO;(3.50)whereaHO=p~=m!istheusualoscillatorlength,andmthenucleonmass.Withthesewecanperformasimultaneousofthedatafor(almost)allpairs(emax;~!)totheexpressionE(emax;~!)=E1+A0e2UV=A21+A2e2k1LIR;(3.51)wheretheenergyforbasissizeE1,thebindingmomentumk1,andtheAiaretreatedasparameters.FortheunevolvedN3LOinteraction,wefounditnecessarytoexcludetheemax=8datasettoobtainstablefor16Oand40Ca,mostlikelybecauseUVisclosetothecoftheinitialinteractionforemax=8andthelowervaluesof~!weareconsidering.TheresultingextrapolatedenergiesareindicatedbygraydashedlinesinFig.3.2,andtheyfallwithin1%orlessoftheenergiesforemax=14,thelargestbasissizewhichwasusedinactualcalculations.BothenergiesarereportedforeachnucleusinTable3.1.Forthelightnuclei4He,theIM-SRG(2)ground-stateenergyisabout2MeVbelowtheexactresultfromaNo-CoreShellModel(NCSM)calculationwiththesamechiralN3LOinteraction(see,e.g.,Ref.[54]).Further,IM-SRG(2)resultscanbecomparedtoCoupledClustercalculationswiththesameinteraction[64,85](alsoseeRef.[74]).TheIM-SRG(2)energiesaretlylowerthantheCCSDenergies,lowereventhanthe481624324048~![MeV]1:321:301:281:261:24.E[GeV]78Niemaxn10u12s141624324048~![MeV]1:951:901:85.E[GeV]100Sn1624324048~![MeV]2:752:702:65.E[GeV]132SnFigure3.3:Convergenceof78Ni,100Sn,and132SnIM-SRG(2)ground-stateenergiesw.r.t.single-particlebasissizeemax,forthechiralN3LONNinteractionwith=2:0fm1.Graydashedlinesindicateenergiesfromextrapolationtheemax10datasetstobasissize(seetextandRefs.[13,14]).results,aCCmethodwhichtakesperturbativetriplescorrectionsintoaccount.IwillsoonpresentaperturbativeanalysisoftheIM-SRGinSec.3.5,whichshowstheoriginofthethebetweenIM-SRG(2)andCCSD.Thisoverbindingcanbeexplainedbyasystematicundercountingofcertainrepulsivefourth-ordertermsintheIM-SRG(2)truncation,whichsimulatestheadditionalattractionthatisotherwisegainedfromincludingtriplescorrection.Forthe(comparably)hardinitialinteraction,theIM-SRG(2)overshootstheresults,whilethereducedimportanceofhigher-orderMBPTcorrectionsforsoftinteractionscausestheIM-SRG(2)resultstofallinbetweentheCCSDandresults(seeSecs.3.5andRefs.[58,74]).IntherightpanelsofFig.3.2,weshowthesamekindofconvergenceplotsforthechiralN3LOinteractionatthereducedresolutionscale=2:0fm1.Asexpected,thespeedoftheconvergenceisgreatlyenhancedbyusingasofterinteraction[33],whichisevident49fromthesignitlysmallerenergyscalesinthelowerpanels.InTab.3.1,wecanseethattheextrapolatedenergiesagreewiththeemax=14resultswithin0.01-0.1%.For4He,thereappeartobesomedeviationsfromtheotherwisevariationalconvergencepatternintheothercases.Ofcourse,theIM-SRGisnotstrictlyvariationalbecauseofthetruncationsinthewequations(3.8){(3.10).Inthepresentcase,however,thesedeviationsareontheorderofa10keVorless,andaremostlikelydominatedbynumericalartifactsfromintegratingthewequations.Forasoftinteraction,thelargesingle-particlebasissizeswehaveusedherearettoconvergenucleiwhicharemuchheavierthan40Ca.ThisisdemonstratedinFig.3.3,whereweshowtheconvergenceoftheIM-SRG(2)ground-stateenergiesoftheproton-orneutron-richexoticnuclei78Ni,100Sn,and132Sn.ThecorrespondingenergiesareincludedinTab.3.1.UsingonlyasoftenedchiralN3LOinteraction,thebindingenergyofthesenucleiisoverestimatedtly,continuingatrendwhichwasalreadynoticeablefor16OinFig.3.2.Thisoverbindingiscausedbytheshiftofrepulsivestrengthfromthetwo-bodyinteractiontoinducedthree-andhighermany-bodyforcesastheresolutionscaleislowered,ofcourse,andbyincludingatleasttheinducedthree-nucleonforces[21,74].Whiletheinclusionofthree-bodyoperatorscomeswithcomputationalchallenges,westressthattheseinducedtermshavelowresolutionscalesaswell,anddonottherateofconvergenceoftheIM-SRGground-stateenergiesadversely.Whilecomputationalissuespertainingtothestorageof3Nmatrixelementspresentachallenge,convergedcalculationswithNN+3NinteractionsfortheA˘100regionandbeyondhavenowbecomepossible[74,86,87].503.3.3ChoiceofGeneratorLetusnowstudytheofourchoiceofgeneratorontheIM-SRG(2)ground-stateener-gies.InFig.3.4,weshowtheIM-SRG(2)ground-stateenergiesforthevetgenera-torsdiscussedinSec.3.2.NotethatthepanelsfortheWhiteandimaginary-timegeneratorsshowcurvesforboththeEpstein-NesbetandM˝ller-Plessetchoicesfortheenergydenomi-natorsandsignfunctions,respectively.Theresultingground-stateenergiesfor40Caagreewithin15keV,whichamountstorelativedfrom106to104.Remarkably,thisagreementholdsforboththesoftenedandbareN3LOinteractions,andirrespectiveoftheusedbasisparametersemaxand~!.Theextrapolatedenergiesthereforealsoonlybyequallysmallamounts.ItisevidentfromFig.3.4thattheWhiteandimaginary-timegeneratorsgiveverysimilarresults.ForthebareN3LOinteraction,theextrapolated40Caground-stateenergiesare368:9MeVand367:7MeV,respectively,whichisaofabout0.3%.Forany~!inthestudiedrange,theenergybetweenthetwotypesofgeneratorsdropbelow1%fromemax=8onward.Asexpected,thebecomesmallerwhentheresolutionscaleoftheinteractionisloweredto=2:0fm1.Theextrapolatedenergiesare596:0MeVand595:6MeVfortheWhiteandimaginary-timegenerators,respectively,whichamountstoarelativeoforder104.Theextrapolatedvaluesarebyslightlylargererencesforsmallandlarge~!.Neartheenergyminimawithrespectto~!,wheretheresultsarebetterconverged,absolutearetypicallybelow10keV.Forthesoftinteraction,theresultsfortheWegnergeneratoragreeverywellwiththosefortheothergenerators:Theextrapolated40Caground-stateenergyis595:4MeV.Thesituationisquitetforthebareinteraction,though.Tounderstandwhatwesee,we5116202428323640444852~![MeV]600590580570560550.E[MeV]40Ca=2:0fm1emax810121416202428323640444852~![MeV]40Ca=2:0fm116202428323640444852~![MeV]40Ca=2:0fm1350300250200.E[MeV]40Ca=140Ca=140Ca=1White{IA/BIm.Time{IIA/BWegner{IIIFigure3.4:IM-SRG(2)ground-stateenergiesof40Caobtainedwithtchoicesofthegenerator,asafunctionof~!andthesingle-particlebasissizeemax.TheinteractionisthechiralN3LOpotentialwith=1(toppanels)and=2:0fm1(bottompanels),respectively.Thedashedlinesindicateextrapolatedenergies.FortheWegnergenerator,theshadedareaindicatesthevariationfromusingtdatasetsfortheextrapolation(seetext).considertheconvergencepatternthatispredictedfora(quasi)-variationaltheorybytheextrapolationformula(3.51)[13,14].Atemax,thederivativeofEq.(3.51)withrespecttotheoscillatorparameter~!indicatesthattheultraviolet(UV)andinfrared(IR)correctiontermsareminimizedatlargeandsmall~!,respectively.TheexponentsoftheUVandIRtermsbehavelike2UV˘emaxandLIR˘pemaxasemaxincreases,hencewe52expectIRcorrectionstodominateeventually.Consequently,wecaninferthattheminimumoftheenergywithrespecttotheoscillatorparametershouldmovetolarger~!untilUVconvergenceisachieved,andthentosmaller~!forIRconvergence.InFig.3.4,weonlyseetheenergyminimummovetowardsIRconvergenceatsmall~!,whichsuggeststhatthecalculationistlyconvergedintheUVregimealreadyforemax=8,thesmallestbasisshownintheFortheWegnergenerator,theminimumisstillmovingtolarger~!values,whichsuggeststhatthecalculationisnotyetconvergedintheUVregime,andaslowerconvergencewithbasisingeneral.Ifweusethedataforemax=8;10;12,whichbehavevariationally,anextrapolationtobasisusingEq.(3.51),yields370:7MeV,whichiscompatiblewiththeextrapolatedresultsfortheWhiteandimaginary-timegeneratorswithinuncertainties.Goingtoemax=14,wefaceacomplication:whiletheenergyminimummovestolarger~!,thecurveintersectsthoseforsmalleremax.Thisisnotruledoutapriori,becausetheIM-SRGisanon-variationalapproach,butmakestheassumptionsunderlyingtheextrapolationformula(3.51)questionable.Settingasidethefundamentalissueofapplicability,wehaveextrapolatedtheenergyusingtsubsetsofourcalculateddata,andtherebyobtaintheshadedbandinFig.3.4,whichrepresentsa10%variationoftheextrapolatedenergy.TobetterunderstandthebehavioroftheIM-SRGwfortheWegnergenerator,wehavetoconsiderhowitsstructurefromourotherchoicesfor.TheoftheHamiltonianHod(s),Eq.(3.21),isthesameinallthreecases,soweaimforthesame(oratleastsimilar)pointsofthew,where(1)=0.However,weknowthattheWhiteandimaginary-timegeneratorsaredirectlyproportionaltoHod,i.e.,theonlynon-vanishingmatrixelementsareofthetypesph=hpandpp0hh0=hh0pp0.TheWegnergenerator,ontheotherhand,hasmanyadditionalnon-zeromatrixelementscomingfromtheevaluation5316202428323640444852~![MeV]600590580570560550540.E[MeV]40Caemaxl8n10u12s14=2:0fm116202428323640444852~![MeV]40Ca=2:0fm1350300250200.E[MeV]40Ca=140Ca=1Wegner{IIIWegner(restricted){IVFigure3.5:IM-SRG(2)ground-stateenergiesof40Cafortheregular(left,asinFig.3.4)andrestrictedWegnergenerators(right,seetext),asafunctionof~!andthesingle-particlebasissizeemax.TheinteractionisthechiralN3LOpotentialwith=1(toppanels)and=2:0fm1(bottompanels),respectively.Thedashedlinesindicateextrapolatedenergies.ofthecommutator,analogoustotheIM-SRGwequationitself(cf.Eqs.(3.32),(3.33)).Itdoesnotcomeasasurprise,then,thatthegeneratorsinthewaytheybuildcorrelationfromthemany-bodyperturbationseriesintothewingHamiltonian|athatwillbeenhancedforinteractionsforwhichorder-by-orderconvergenceoftheMany-BodyPerturbationseriescannotbeguaranteed(cf.Secs.3.3.4and3.5).Forillustration,Fig.3.5comparesresultsfortheregularWegnergeneratorwiththosefora54restrictedversionbyIVij=IIIij;IVijkl=8>>><>>>:IIIijklforijkl=pp0hh0;hh0pp0;0else;(3.52)matchingthestructureoftheWhiteandimaginary-timegenerators.Wehaveexploredrestrictionsoftheone-bodypartaswell,buttheycausenonoticeablewhiletheimpactoftherestrictioninthetwo-bodypartist.TheconvergencepatternoftherestrictedIVisquasi-variationalforboththebareandsoftenedN3LOinteractions,andhastheenergyminimummovingtowardssmaller~!,sug-gestingthatthecalculationisconvergedintheUVregime,andnowconvergingintheIRregime.Theextrapolated40Cag.s.energiesare367:4MeVand595:3MeV,respec-tively,inverygoodagreementwiththeWhiteandimaginary-timegenerators,aswellastheunrestrictedWegnergeneratorIIIinthecaseofthesoftinteraction(alsocf.Fig.3.5).Thisstronglysuggeststhatourhypothesiswascorrect,anditisindeedtheadditionalnon-zeromatrixelementsinIIIwhichintroduceuncontrolledbehavior.Itremainstobeseenwhetherwecanreachadeeperunderstandingoftheunderlyingmechanism.Alikelyexpla-nationisthatthetruncationofthecommutator(3.31)toone-andtwo-bodycontributionsonly(Eqs.(3.32),(3.33))causesanimbalanceintheerresummationoftheMany-BodyPerturbationseries.Forthetimebeing,wehavetoadviseagainsttheuseoftheWegnergeneratorinIM-SRGcalculationswith(comparably)\hard"interactionsthatexhibitpoororder-by-orderconvergenceoftheperturbationseries.55V[MeVfm3]1050-5-10-15-20hhpp˙-hhpp?6s=0:0s=1:2s=2:0s=18:3Figure3.6:DecouplingfortheWhitegenerator,Eq.(3.23),intheJˇ=0+neutron-neutroninteractionmatrixelementsof40Ca(emax=8;~!=20MeV,Entem-MachleidtN3LO(500)evolvedto=2:0fm1).Onlyhhhh;hhpp;pphh;andppppblocksofthematrixareshown.3.3.4DecouplingAsdiscussedinSec.3.2.1,theIM-SRGisbuiltaroundtheconceptofdecouplingthereferencestatefromexcitations,andtherebymappingitontothefullyinteractinggroundstateofthemany-bodysystemwithintruncationerrors.Letusnowdemonstratethatthedecouplingoccursasintendedinasamplecalculationfor40CawithourstandardchiralN3LOinteractionat=2:0fm1.Fig.3.6showstherapidsuppressionofthematrixelementsintheJˇ=0+neutron-neutronmatrixelementsasweintegratetheIM-SRG(2)owequations.Ats=2:0,afteronly20{30integrationstepswiththeWhitegenerator,thepp0hh0(s)havebeenweakenedtly,andwhenwereachthestoppingcriterionforthewats=18:3,thesematrixelementshavevanishedtothedesiredaccuracy.Whilethedetailsdependonthespchoiceofgenerator,thedecouplingseeninFig.3.6isrepresentativeforothercases.Withthesuppressionofthematrixelements,themany-bodyHamiltonianisdriventotheformindicatedinFig.3.6.TheIM-SRGevolutiondoesnotonlydecouplethegroundstatefromexcitations,butreducesthecouplingbetweenexcitationsaswell.Thiscouplingisanindicatorofstrongcorrelationsinthemany-bodysystem,56whichusuallyrequirehigh-oreventreatmentsinapproachesbasedontheGoldstoneexpansion.AswehavediscussedinSec.3.1,theIM-SRGcanalsobeunderstoodassuchanon-perturbative,resummationoftheMany-BodyPerturbationseries,whichbuildstheofcorrelationsintothewingHamiltonian.Toillustratethis,weshowresultsfromusingtheIM-SRGHamiltonianH(1)inHartree-Fockandpost-HFmethodsinFig.3.7.Afterthesame20{30integrationstepsthatleadtoastrongsuppressionofthematrixelements(cf.Fig.3.11),theenergiesofallmethodscollapsetothesameresult,whichistheIM-SRG(2)ground-stateenergy.Byconstruction,thisistheresultthatwouldbeobtainedinaHartree-FockcalculationwiththeIM-SRGHamiltonian.Energycorrectionsduetocorrelationshavebeenre-summedintothezero-bodypartofH(1),andthereforeMBPT(2)oreitheroftheCCresummationsdonotcontributeadditionalcorrelationenergy.Thecollapseoftheground-stateenergiesoccursinthesamefashionforall(emax;~!),althoughtherateandmagnitudeofthechangeing.s.energywiththewparametersmaybequitetforeachmethod.LetustakeamoredetailedlookatFig.3.7.ForthebareN3LOinteraction,theemax=10resultsarenotyettlyconvergedwithrespecttoeitherthesingle-particlebasisandmany-bodyexpansions,hencetheground-stateenergychangesquitesitlywiths(cf.Fig.3.2).ForthesoftN3LOinteractionwith=2:0fm1,ontheotherhand,conver-gencew.r.t.basissizeisalreadyquitesatisfactoryatemax=10.Becausethisinteractionismoreperturbative,thesmallenergybetweenthetmany-bodymethods,inparticularthesecond-orderande-orderCCandIM-SRGresummations,indicatesgoodconvergenceofthemany-bodyexpansion1[33,51].Wewillreturntothissubjectin1AsdiscussedinSec.3.4,thereisacaveatattachedtothisstatement,namelythatorder-by-orderpertur-57340320300280260.E[MeV]40Ca=1104103102101110s600580560540520.E[MeV]40Ca=2:0fm1IM-SRG(2),E(s)MBPT(2)CCSDFigure3.7:IM-SRG(2)ground-stateenergyof40Caasafunctionofthewparameters,comparedtoMBPT(2),CCSD,andenergieswiththeIM-SRG-evolvedHamil-tonianH(s).Weonlyshowpartofthedatapointstoavoidclutter.Calculationsweredoneforemax=10andoptimal~!=32MeV(top)and~!=24MeV(bottom),respectively,usingthechiralNNinteractionattresolutionscales.ThedashedlinesindicatetheIM-SRG(2)energies.Sec.3.5.Toconcludethissection,wewanttodiscussthefourmainscenariosthatcanoccurwhenweuseIM-SRGHamiltoniansasinputforothermany-bodymethods.Weassumethatcalculationsareconvergedw.r.t.basissize,etc.1.FullIM-SRG,exactmany-bodymethod:ForexactmethodsliketheNo-CoreShellbativeconvergencestronglydependsaroundwhichreferencestatetheperturbationexpansionisconstructed.58ModelorNo-CoreFulln,theground-stateenergywouldbeasafunc-tionofs.ByperforminganuntruncatedIM-SRGcalculation,weessentiallysplitthediagonalizationofthemany-bodyHamiltonianintoapartthatisobtainedbysolvingtheIM-SRGwequation,andapartthatisobtainedwithtraditionaleigenvaluemethods,withsservingasanarbitraryseparationpoint.2.FullIM-SRG,approximatemany-bodymethod:Theground-stateenergyvarieswiths,butfors!1,theapproximatemany-bodymethodyieldstheexacteigenvalueduetotheuntruncatedIM-SRGtransformation.HereweseehowtheIM-SRGcanbeusedtoimprovetheinputHamiltonianforothermany-bodyapproaches.3.TruncatedIM-SRG,exactmany-bodymethod:Again,theground-stateenergyvarieswiths,andtheoverallvariationisameasureoftheextenttowhichtheIM-SRGtruncationviolatesexactunitarity.4.TruncatedIM-SRG,approximatemany-bodymethod:Thisisthemostcommon,andmostcomplicatedcase.BecauseoftheIM-SRGtruncation,theIM-SRGwillreproducetheexactground-stateenergyonlyapproximatelyinthelimits!1.Iftheapprox-imatemany-bodymethodcontainscontentbeyondthetruncatedIM-SRG,thentheresultmayactuallydegradetosomeextent,whereastheIM-SRGstillimprovesthere-sultintheoppositescenario,buttheuncertaintyofE(1)ishardtoquantifyunlessonealsousesexactmany-bodymethodsforcomparison.BothofthesescenariosarerealizedinFig.3.7:MBPT(2)islesscompletethantheIM-SRG(2),sotheMBPT(2)energyisimprovedtowardstheexactenergy.Notethatthisimprovementcancomeintheformofattractiveorrepulsivecorrections,becauseMBPT(2)typicallyunderestimatestheg.s.energyforthebareinteraction,butovershootswithsoftinteractions[33,58,88{5992].BothCCSDandfromtheIM-SRG(2)atfourthorderinMBPT(seeSec.3.5).CCSDtypicallyunderpredictsthenuclearbindingenergy,hencetheadditionalcorrelationenergyprovidedbytheIM-SRGimprovementshouldimproveagreementwithexactmethods.containsfourth-order3p3h(triples)cor-relations,whicharetypicallyattractive,andmissingintheIM-SRG(2)(cf.Sec.3.5).ThisexplainswhytheCCSD(T)ground-stateenergyactuallyincreases(i.e.,thebind-ingenergydecreases)withIM-SRG(2)inputHamiltoniansass!1forthesoftinteraction.Asmentionedabove,emax=10isnotyettlyconvergedinthecaseoftheground-stateenergiesshowninthetoppanel.Forlargerbases,theIM-SRG(2)againincreasestheground-stateenergyRef.[74].Partofthisincreaseisbenign,becauseCSD(T)isknowntooverestimateground-stateenergies[16,64,67,93{95].3.3.5RadiiInSec.3.1.4,wehavediscussedtheevaluationofobservablesotherthantheground-stateenergy,bysolvingadditionalsetsofwequationsalongwiththosefortheHamiltonian.Asanexample,weshowtheconvergenceofthechargeradiiof4He,16O,and40CainFig.3.8.Theresultsareobtainedbynormal-orderingandevolvingtheintrinsicprotonmean-squareradiusoperator,R2pXi121+˝(i)3(riR)2;(3.53)wheretheisospinoperatorprojectsonprotons,andRisthecenterofmass.Weobtainthechargeradiibyapplyingthecorrectionsduetothemean-squarechargeradiiofprotonand60=1=2:0fm11.61.71.81.9.R[fm]4He4Heemaxl8n10u12s142.22.42.6.R[fm]16O16O16202428323640~![MeV]2.62.833.23.43.6.R[fm]40Ca16202428323640~![MeV]40CaFigure3.8:Convergenceof4He,16O,and40CaIM-SRG(2)chargeradiiw.r.t.single-particlebasissizeemax,forachiralN3LONNinteractionwith=1(leftpanels)and=2:0fm1(rightpanels).Thegraydashedlinesindicateexperimentalchargeradiifrom[15].neutron(see,e.g.,[96]):RchrR2p+r2p+NZr2n=qR2p+(0:8775fm)20:1161fm2;(3.54)withvaluesofr2pandr2ntakenfrom[97].FocusingontheresultsforthebareN3LOinteractionwesatisfactoryconver-61genceofthechargeradiitothelevelof1%overawideregionofbasisparameters~!.Fortemax,thecurvesintersectinthevicinityofthe~!thatminimizestheground-stateenergies(cf..Fig.3.2).TheIM-SRG(2)resultforthechargeradiusof4Heisquiteclosetotheexperimentalvalue.Itissomewhatcounter-intuitive,however,thattheradiusisslightlyunderpredicted,whileabout1MeVbindingenergyismissing(seeTab.3.1).For16O,thebindingenergyissimilarlyclosetotheexperimentalone,butthechargeradiusisalreadytoosmallbyalmost10%,whileoverbindingandunderestimationoftheradiusareconsistentonasuplevelwith40Ca.UsingthesoftenedN3LOinteractionwith=2:0fm1asinput,convergenceoftheradiiimprovesdramaticallyoverthebareN3LOcase.OnthescalesshowninFig.3.8,resultsfromemax=10onwardsareallbutindistinguishable.Atthesametime,theunderestimationoftheradiibecomesworse,whichisconsistentwiththeincreasedbindingenergiesthatarereportedinSec.3.3.2.PartoftheproblemisthatthechangeoftheresolutionscaleoftheN3LOinteractioninduces3N;:::interactionswhichhavenotbeentakenintoaccount.Theseinducedinteractionsgiverepulsivecontributionstotheg.s.energy,andarethereforealsoexpectedtoincreasetheradiitosomeextentRefs.[19,21,42,74,86,87,98{100].Underachangeofresolutionscale,theradiusoperator(oranyotherobservable)shouldbetransformedconsistentlywiththeHamiltonian,causingittogaininducedmany-bodycontributions.SinceRGtransformationslikethefree-spaceSRG,andrelatedmethodslikeLee-Suzuki,aredesignedtodealwithhigh-momentum/short-distancephysics,theirontheradiusandotherlong-rangedoperators,andthereforethesizeofinducedcontributions,wasexpectedtobesmall[101{103].Arecentfree-spaceSRGstudysuggeststhatinducedcontributionsmaybesmallbutnotnegligibleinviewofthediscrepanciesbetweenexperi-mentalandcalculatedradiifromstate-of-the-artabinitiomany-bodycalculations[104].62Arelatedissueistheuseofsimpleone-bodyanstzelike(3.53)forthemean-squareprotonradiusandotherradiusortransitionoperators.Thesespformsneglecttwo-andhighermany-bodycontributionswhicharegeneratedbyexchangecurrents,forinstance,andshouldbeincludedinthe\bare"operatorintheplace.ChiralEFTprovidesaconsistentframeworktotreatthesectsonasimilarfootingastheinteractionitself[105{114],buttheexplorationofthesestructurallyricheroperatorsinnuclearmany-bodycalculationsisstillinitsinfancy[115].3.4ChoiceofReferenceState3.4.1OverviewAsexplainedinSec.3.2.1,theIM-SRGgeneratesamappingbetweenanarbitraryreferencestatejiandaneigenstatejioftheHamiltonian.Inasystem,i.e.,inabsenceofphasetransitions,andwithoutsymmetryconstraintsonthebasis,suchamappingalwaysexists,becausewecandiagonalizetheHamiltonianandconstructaunitarytransformationasthedyadicproductoftheexactgroundstateandthereferencestate,plussuitableadditionalstatestocompletethebasis.PerforminganevolutionwiththeuntruncatedIM-SRGwequationsisequivalenttosucha(partial)diagonalization2.3.4.2HarmonicOscillatorvs.Hartree-FockSlaterDeterminantsInprevioussections,wehaveexplainedthattheground-stateenergiesoftheuntruncatedIM-SRGwequationsdonotdependonthechoiceofreferencestate.Inpractice,theIM-SRG(2)truncationofthewequationsystem(Eqs.(3.8){(3.10))introducesanal2Problemscouldonlyoccurifweusedapathologicalgenerator.63580560540520500.E[MeV]40Ca=2:0fm116202428323640444852~![MeV]1051041031021011.hHFjHOiemaxl6n8s10Figure3.9:Toppanel:IM-SRG(2)energyof40CawithaHF(solidlinesandsymbols)andaHOreferencestate(dashedlines,opensymbols),obtainedwiththeWegnergenerator.Bottompanel:OverlapoftheHFandHOreferencestates.reference-statedependence.InFig.3.9,wecompareground-stateenergiesfor40CathatwereobtainedwithanaiveHOSlaterdeterminantandaHFSlaterdeterminant,respectively.Foroscillatorparameters16~!24MeV,thetwotypesofcalculationsessentiallyconvergetothesameground-stateenergies.Inthisrange,theHOandHFdeterminantshavetheirlargestoverlap,asshowninthelowerpanelofFig.3.9.Outsideofthiswindow,theoverlapdropssteeply,whichsuggeststhattheHFsingle-particlewavefunctionsappreciablyfromtheplainHOsingle-particlewavefunctions.Ofcourse,wehavetokeepinmindthatthese641061041021102s520518516514.E[MeV]40Ca=2:0fm1emax=6GeneratorlWhitenIm.TimeuWegnerFigure3.10:IM-SRGdecouplingof1p1hexcitationsfortgeneratorchoices,startingfromaHOreferencestate.Theshowsthe40Caground-stateenergyasafunctionofthevalueofthewparameters.Theunitofsissuppressedbecauseitwiththechoiceofgenerator.ThegraylineindicatestheresultoftheHartree-Fockcalculationwiththesameinteractionandbasisparameters.areexponentiallywhenthemany-bodyoverlapiscalculatedastheproductofsingle-particleoverlaps.Beyond~!=28MeV,theIM-SRG(2)energiesobtainedwithaHOrefererencestateactuallygrowwiththebasissizeemax,whichsuggeststhattheIM-SRGisnolongertargetingtheHamiltonian'sgroundstateinthosecases.ThisconclusionissupportedbyourinabilitytoobtainconvergedresultswithWhite-typegenerators(seeEq.(3.23))forthelarger~!values.TheIM-SRGwstallsbecauseofdivergencesinthegeneratormatrixelementsthatarecausedbysmallenergydenominators,whichcanbeviewedasindicatorsoflevelcrossingsinthespectrumoftheevolvingmany-bodyHamiltonian.SoweseethatunlikeCCtheory,truncatedIM-SRGcalculationsaresensitivetothequalityofthestartingreference.ThiscanbeeasilyexplainediftheunitarytransformationgeneratedbytheIM-SRG(2)isinspected.Inordertohavethisinsensitivitytoreferencechoice,wewouldneedtobeabletotakeadvantageoftheunitaryversionofThouless'thm65foundin(2.34),provenbyRowe,Ryman,andRosensteelinRef.[116],relatinganytwonon-orthogonalnormalizedSlaterdeterminantsjAi;jBiviajBi=expXphXph:aypah:Xph:ayhap:jAi:(3.55)Unfortunately,Eq.(3.55)doesnotapplytotheIM-SRGinastraightforwardfashion.AsmentionedinSec.3.1.1,theunitarytransformationgeneratedbytheIM-SRGisformallygivenbytheS-orderedexponentialU(s)=SexpZs0ds0(s0);(3.56)becausethegeneratordynamicallychangesduringthew.Itcanbeasaproductofunitarytransformations,U(s)=limN!1NYi=0e(si)si;Xisi=s;(3.57)ortheseriesexpansionU(s)=Xn1n!Zs0ds1Zs0ds2:::Zs0dsnSf(s1):::(sn)g:(3.58)Here,Sensuresthatthewparametersintheoperatorproductsappearingintheintegrandsarealwaysindescendingorder.UnliketheclusteroperatoroftheCCmethod,thegenerator(s)necessarilycontainsparticle-holede-excitationoperators,orelseitwouldnotbeanti-Hermitianasrequiredforaunitarytransformation.Thus,itispossibletohavenon-vanishingcontractionsbetweengeneratorcomponentsoftparticlerank,andcommutatorsof66ddsE=++=+++++:::++:::++:::|{z}js)j2|{z}jf(s)j2|{z}jW(s)j2Figure3.11:Schematicillustrationoftheenergywequation(3.61)fortheWhitegeneratorwithM˝ller-Plessetenergydenominators(Eq.(3.23))intermsofHugenholtzdiagrams(seetext).ThegreyverticesrepresentH(s),andthedoublelinesindicateenergydenominatorscalculatedwithf(s).Onthesecondline,thewequationisexpandedintermsofH(ss)(simpleblackvertices)andthecorrespondingenergydenominatorsfromf(ss)(singlelines).ThebracesindicatewhichtermofH(s)isexpanded,anddotsrepresenthigherorderdiagramsgeneratedbytheintegrationstepss!s.suchcomponentsdonotvanishingeneral:[(i)(s);(j)(s0)]6=0:(3.59)Asaresult,U(s)doesnotfactorizeautomatically,anditisthislackoffactorizationthatmakestheIM-SRG(2)methodsensitivetoreferencestates.WewillrevisitthissensitivitytoreferencestateinthecontextofchemistrysystemsinSec.6.1.673.5PerturbativeAnalysisoftheFlowEquations3.5.1OverviewTheexpressionsfortheWhite-typegeneratorsdiscussedinSec.3.2.2areamanifestlinkbetweentheIM-SRGandMany-BodyPerturbationTheory.Forthesakeofdiscussion,wefocusontheWhitegeneratorwithM˝ller-Plessetenergydenominators,keepingtheshort-handsph,pp0hh0,etc.,butdroppingthesuperscriptB.ThegeneratorwithEpstein-Nesbetenergydenominatorscanalwaysbeconnectedtothiscasebyseriesexpansion,e.g.,1fpfh+phph=1fpfhXkphphfpfhk:(3.60)Letusnowconsiderthewequationfortheground-stateenergy(3.8),butbroadenourperspectivebeyondtheIM-SRG(2)truncationtokeeptrackoftheinducedthree-bodycontribution(cf.Eq.(B.2)andthediscussioninSec.3.2.1).PluggingintheWhite-M˝ller-Plessetgeneratorwithexplicitthree-bodycontribution,weobtaindEds=2Xphjfphj2ph+12Xpp0hh0jpp0hh0j2pp0hh0+118Xpp0hh0jWpp0p00hh0h00j2pp0p00hh0h00:(3.61)Theright-handsideofEq.(3.61)hasthestructureofthesecond-orderMBPTcorrectiontotheground-stateenergy,butthematrixelementsandenergydenominatorsdependonthewparameters.Thus,Eq.(3.61)impliesthattheground-stateenergyE(s)isRG-improvedwithcontributionsfromhigherordersofMBPTduringthew.Inthefollowingdiscussion,wecharacterizealloperatorsintermsofthesamedimen-sionlessbook-keepingparameterg.WealsoassumethattheinitialHamiltonian68thehierarchyfd>>Wthroughoutthew.ThehierarchyofandW,inparticular,iscompatiblewiththenaturalhierarchyofchiraltwo-andthree-nucleonforces[32,33].Initially,E(0)=O(g0);fd(0)=O(g0);=O(g):(3.62)Ifwedonotincludeaninitialthree-bodyterm,andchooseaHFSlaterdeterminant(fod=ffph;fhpg)astheIM-SRGreferencestate,wealsohavefod(0)=0;W(0)=0:(3.63)Fromthewequations(3.8){(3.10)(or(B.2){(B.5)),wecanconcludethatcorrectionstos)areoforderO(g).Correctionstof(s)areO(g2)becausetheyaregeneratedbytermswhicharequadraticins),andthesamereasoningholdsfortheinducedandthree-bodymatrixelements,fod(s)=O(g2);W(s)=O(g2);fors>0(3.64)(alsocf.Sec.3.2.5).Thisestablishesthatthethreetermsintheowequation(3.61)areoforderO(g4);O(g2),andO(g4),respectively.InFig.3.11,theofintegratingEq.(3.61)byasinglestepss!sisillustratedschematicallyintermsofHugenholtzdiagrams(see,e.g.,[16,117]).ExpandingtheH(s)verticesintermsofH(ss)vertices,weseethatthes)termhascontributionsfromO(g2)throughO(g4).ExpandinginH(s2s)instead,wewouldgetadditionalhigherorderdiagrams,andsoforth.Thus,weperforma(partial)re-summationofthemany-body69perturbationseriesbyintegratingtheIM-SRGwequationsfroms=0to1.Fig.3.11showsthatalltopologiesforsecond-andthird-orderenergydiagramsaregener-ated,andwewilldemonstratebelowthatwebuilduptothecompleteenergythroughO(g3)whenweintegrateEq.(3.61).Thes)termalsogeneratesfourth-orderdiagramswithupto4p4h/quadruplesexcitations,butf(s)andW(s)termsclearlycontributeatfourthorderaswell.TheformerareincludedintheIM-SRG(2),whichisthereforethird-ordercorrect,similartoCoupledClusterwithsinglesanddoubles(CCSD).Toobtainaformallycorrectfourth-orderenergy,weneedtokeeptheinducedthree-bodyterms,e.g.,usetheIM-SRG(3)truncationorsomeappropriateapproximation,as,forinstance,inCCwithsingles,doubles,andperturbativetriples(CCSD(T)).Westress,however,thattheperturbativeanalysiswillnotprovideuswithameanstojudgetheIM-SRGtruncationerrorinnuclearphysicsapplications,asidefromaguaranteedlinearscalingoftheerrorwiththeparticlenumberAduetosizeextensivity[16,118].Intheremainderofthissection,wewillanalyzetheIM-SRGingreaterdetail.ThemaingoalofthisanalysisistoprovideanunderstandingofhowtheIM-SRGrelatestootherdia-grammaticmethodslikeMBPT,theSelf-ConsistentGreen'sFunctionapproach[19,76,119],ortheCoupledClustermethod,whichcanbeanalyzeddiagrammticallyalongthesamelinesastheIM-SRG(see,e.g.,[16]).Asmentionedabove,wechooseaHFSlaterdeterminantasthereferencestatejifortheIM-SRGandtheMBPTexpansion.Thenfph(s)vanishesfors=0(becauseoftheHFequations)ands!1(becauseoftheIM-SRGdecouplingcondition),andwewillonlyhavetodiscusscanonicalHFMBPTdiagramsinthelanguageof[16].Theinclusionofnon-HF(wherefph6=0)andnon-canonicalHFdiagrams(wherefpp0;fhh0arenon-diagonal)isstraightforwardbuttediousbecausetheirnumbergrowsmuchmorerapidlythanthenumber70ofcanonicalHFdiagrams[16].3.5.2PowerCountingInthefollowingdiscussion,wewillusesuperscriptstoindicatetheorderofindividualtermsintheIM-SRGwequations.LetusaddressthesubtletiesinthepowercountingthatwasinEqs.(3.62)and(3.64).ThenaturalorbitalsforaHFSlaterdeterminatjiaretheHForbitals,whichmeansthatf(0)isdiagonalintheparticleandholeblocksofthes.p.basis,andfph(0)=fhp(0)=0.Sincethesearethematrixelementstheone-bodypartofthegenerator(3.23),abvanishesaswell,andtheone-bodywequationats=0becomesdf12dss=0=Xabc(nanbnc+nanbnc)(1+P12)[1]c1ab[1]abc2+::::(3.65)Thus,correctionstofstartatO(g2)(cf.Sec.3.2.5),andwehavefpp0(s)=f[0]ppp0+f[2]pp0(s)+:::;(3.66)fhh0(s)=f[0]hhh0+f[2]hh0(s)+:::;(3.67)fph(s)=f[2]ph(s)+:::;(3.68)wherethenotationf[0]indicatesthatthetermdoesnotdependons.Itimmediatelyfollowsthatcorrectionsands-dependenceoftheM˝ller-Plessetenergydenominatorsalsoappearat71O(g2),ab(s)=[0]ab+[2]ab+:::;(3.69)abcd(s)=[0]abcd+[2]abcd+::::(3.70)Consequently,thegeneratormatrixelementsaregivenbyph=f[2]ph[0]ph+f[3]ph[0]ph+f[4]ph[0]ph+f[2]ph[2]ph[0]ph2+O(g5);(3.71)pp0hh0=[1]pp0hh0[0]pp0hh0+[2]pp0hh0[0]pp0hh0+[3]pp0hh0[0]pp0hh0+[1]pp0hh0[2]pp0hh0[0]pp0hh02+O(g4);(3.72)andtheirHermitianconjugates.Basedontheseconsiderations,wewillproceedtodiscusstheone-andtwo-bodywequationsatincreasingordersO(gn).Sincetheenergywequationdoesnotfeedbackintothewforfandwewilldiscussitseparatelyafterwards.3.5.3O(g)FlowAsshownintheprevioussection,correctionstotheone-bodyHamiltonianfonlybegintocontributeatO(g2),hence_f[1]12=0)f[1]12(s)=0;(3.73)wherethedotindicatesthederivativewithrespecttos.Thecontributiontothetwo-bodywcomesfromthelineofEq.(3.10):_[1]1234=Xan(1P12)(f[0]1a[1]a234)(1P34)(f[0]a3[1]12a4)o;(3.74)72wherewehaveusedEqs.(3.71)and(3.72),andf[1]=0.Sinceonlyhaspphhandhhppmatrixelementsandf[0]isdiagonal,wehave_[1]pp0hh0=f[0]p+f[0]p0f[0]hf[0]h0[1]pp0hh0=[0]pphh0[1]pp0hh0;(3.75)andananalogousequationfortheHermitianconjugate,while_[1]1234=0otherwise.Thus,thewequationscanbeintegratedeasily,andweobtain[1]abcd(s)=[1]abcd8>>><>>>:esforabcd=pp0hh0;hh0pp0;1otherwise;(3.76)with[1]abcdabcd(0):(3.77)3.5.4O(g2)FlowWebeginourdiscussionwiththesecond-ordercontributiontof.UsingEq.(3.76),theIM-SRGwequation(3.9)yields_f[2]pp0=12Xp00hh0[1]p00phh0[1]hh0p00p0+[1]p00p0hh0[1]hh0p00p=12Xp00hh0[1]p00phh0[1]hh0p00p00B@e2s[0]p00ph0hh0+e2s[0]p00p0h0hh01CA2f[2]pp0e2s:(3.78)73Thewequationsfortheothermatrixelementsoff[2](s)havethesamestructure,consistingofans-independentamplitudeandafunctioncontainingadecayingexponentialins.Withtheinitialvalueconditionf[2](0)=0,weobtainf[2]ab(s)=f[2]ab8>>><>>>:(1e2s)forab=pp0;hh0;sesforab=ph;hp:(3.79)Fors!1,theIM-SRGbuildsupandaddstheamplitudesf[2]pp0andf[2]hh0totheeone-bodyHamiltonian,whichpreciselycorrespondtothesecond-ordercontributionsfromMBPT.WecanexpressthemsuccinctlyintermsoftheantisymmetrizedGoldstonediagramsshowninFig.3.12:f[2]pp0=12(f1)pp0+p$p0;(3.80)f[2]hh0=12(f2)hh0+h$h0;(3.81)f[2]ph=(f3)ph+(f4)hp:(3.82)Therulesforinterpretingsuchdiagramsarederivedinmostmany-bodytexts,soweonlysummarizetheminAppendixCforconvenience.Fortheparticle-holematrixelements,wehavef[2]ph(0)=f[2]ph(1)=0;(3.83)becausewestartwithaHFSlaterdeterminantanddemandthatthereferencestateisagaindecoupledfrom1p1hexcitationsfors!1.Atintermediatestagesofthew,theamplitudesf[2]phandf[2]hpcontributetothebuild-upofhigher-orderMBPTdiagrams.74p0ph0hphhpf1f2f3f4Figure3.12:AntisymmetrizedGoldstonediagramsfortheO(g2)eone-bodyHamil-tonian(seetext).InterpretationrulesaresummarizedinAppendixC.Forthesecond-ordertwo-bodyvertex[2],thesamekindofanalysisyields[2]abcd(s)=[2]abcd8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:(1e2s)forabcd=p1p2p3p4;h1h2h3h4;p1h1p2h2;:::;(1es)forabcd=p1p2p3h;h1h2h3p;:::;sesforabcd=p1p2h1h2;:::;(3.84)wherethedotsindicateallallowedpermuationsandHermitianconjugatesoftheexplicitly75p3p4p1p21h1h2h3h42p3p1hp23p3p1p2h4p1h2p2h15p1h2p2h16h1ph2h37h1h2ph38p1h1h2p29h1p1p2h210p1h1p2h211p1h1p2h212Figure3.13:AntisymmetrizedGoldstonediagramsfortheO(g2)etwo-bodyvertex(seetext).InterpretationrulesaresummarizedinAppendixC.givenindices.Thecorrespondingamplitudesare[2]p1p2p3p4=121)p1p2p3p4+1)p3p4p1p2;(3.85)[2]h1h2h3h4=122)h1h2h3h4+2)h3h4h1h2;(3.86)[2]p1p2p3h=3)p1p2p3h+(1Pp1p24)p1p2p3h;(3.87)[2]p1h1p2h2=125)p1h1h2p2+6)h2p2p1h1;(3.88)[2]h1h2h3p=7)h1h2h3p+(1Ph1h28)h1h2h3p;(3.89)[2]p1p2h1h2=9)p1p2h1h2+10)h1h2p1p2+(1Pp1p211+12)p1p2h1h2;(3.90)wherewerefertothediagramsinFig.3.13.Expressionsfortheremainingcombinations76ofindicescanbeobtainedbyusingtheantisymmetryandHermiticityof[2]abcd.Equations(3.85){(3.90)aregiveninahybridform,i.e.,theycontainexplicitHermitianconjugatesandlinepermutationsofthediagrams.ThisallowsustoexpressouranalyticexpressionsfortheamplitudesintermsoftheminimalsetofdiagramsinFig.3.13.IfoneenvisionstheinverseproblemofconstructingtheIM-SRGwequationsfromdiagrams,onewouldofcourseincludeallpossiblediagramtopologies,andexpresstheamplitudespurelyassumsofdiagramsbeforederivinganalyticexpressions.AsintheschematicdiscussionoftheenergywequationinSec.3.5.1,wealsowanttokeeptrackofinducedthree-bodyterms.TheIM-SRG(3)wequationforthethree-bodyvertex,Eq.(B.5),revealsthatthereareO(g2)contributionsfromproductsof[1]abcd(s)and[1]abcd(s),hencewehavetoanalyzeW[2].However,wewilllimitthediscussiontothematrixelementsofW[2]whichcanactuallycontributetothefourth-ordercorrectionstotheground-stateenergy(seeFig.3.1andthediscussionSec.3.5.1).IntegratingtheO(g2)three-bodywequation,weobtainW[2]abcdef(s)=W[2]abcdef8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:(1e2s)forabcdef=p1p2h1h2p3p4;h1h2p1p2h3h4;:::;sesforabcdef=p1p2p3h1h2h3;:::;(3.91)77p1h2p2p3p4h2W1h1p2h2h3p1h4W2p1h1p2h2p3h3W3p1h1p2h2p3h3W4Figure3.14:AntisymmetrizedGoldstonediagramsfortheO(g2)ethree-bodyvertexW(seetext).InterpretationrulesaresummarizedinAppendixC.wherethedotsagainindicateallowedHermitianconjugatesandpermutationsofindices.IntermsofthediagramsshowninFig.3.14,theamplitudesareW[2]p1p2h1h2p3p4=12(W1)p1p2h1h2p3p4+(W1)h2p3p4p1p2h1;(3.92)W[2]h1h2p1p2h3h4=12(W2)h1h2p1p2h3h4+(W2)p2h3p4h1h2p1;(3.93)W[2]p1p2p3h1h2h3=P(p1p2=p3)P(h1h2=h3)(W3+W4)p1p2p3h1h2h3;(3.94)wherewehavethethree-bodypermutationsymbolsP(ij=k)1PikPjk;(3.95)P(i=jk)1PijPik:(3.96)78p1h1p2h2h1p1h2p2p1h1h2p2h1p1p2h2p1h1p2h2A1A2A3A4A5p1h1p2h2h1p1h2p2p1h1h2p2h1p1p2h2p1h1p2h2B1B2B3B4B5p1h1p2h2h1p1h2p2p1h1p2h2h1p1h2p2C1C2D1D2Figure3.15:AntisymmetrizedGoldstonediagramsfortheO(g3)etwo-bodyvertex(seetext).Black(l)andgrayvertices(wg)correspondto[1](Eq.(3.77)),f[2](Eqs.(3.80){(3.82)),[2](Eqs.(3.85){(3.90)),andW[2](Eqs.(3.92){(3.94)),respectively.InterpretationrulesaresummarizedinAppendixC.3.5.5O(g3)FlowTheanalysisofthethird-orderone-andtwo-bodywequationsisstraightforward,butthenumberofterms(ordiagrams)wehavetoconsiderincreasestly.Here,wecontentourselveswithanalyzing[3]pp0hh0(s),theonlymissingingredientforthediscussionoftheenergywequationthroughO(g4),asintheoverviewpresentedinSec.3.5.1.Usingour79resultsfromtheprevioussections,thetwo-bodywequationcanbewrittenas_[3]p1p2h1h2[3]p1p2h1h2+A+Dp1p2h1h2ses+B+Cp1p2h1h2e3ses;(3.97)whichissolvedby[3]p1p2h1h2(s)=A+Dp1p2h1h2s22esB+Cp1p2h1h2e3ses2+ses:(3.98)TheamplitudesAtoDaregivenbythediagramsshowninFig.3.15,whereblackandgreyindicesindicatetheandsecond-ordervertices,respectively:Ap1p2h1h2=1Pp1p2(A1)p1p2h1h2+1Ph1h2(A2)p1p2h1h2+(A3+A4)p1p2h1h2+1Pp1p21Ph1h2(A5)p1p2h1h2;(3.99)Bp1p2h1h2=[2]p1p2h1h2[2]p1p2h1h2+1Pp1p2(B1)p1p2h1h2+1Ph1h2(B2)p1p2h1h2+(B3+B4)p1p2h1h2+1Pp1p21Ph1h2(B5)p1p2h1h2;(3.100)Cp1p2h1h2=(1Ph1h2)(C1)p1p2h1h2+(1Pp1p2)(C2)p1p2h1h2;(3.101)Dp1p2h1h2=(1Ph1h2)(D1)p1p2h1h2+(1Pp1p2)(D2)p1p2h1h2:(3.102)AandBarecontainedinthestandardIM-SRG(2)truncation,whereasCandDareleading-orderinducedthree-bodyterms.Inparticular,theformerisaproductofW[2]andthe80two-bodygenerator,Cp1p2h1h2=12(1Ph1h2)Xp0p00h0W[2]p1h0p2p0h1p00[1]p0p00h0h2+12(1Pp1p2)Xh0h00p0W[2]h1p0h2h0p1h00[1]h0h00p0p2;(3.103)whilethelatterisaproductof[1]andthethree-bodygeneratorinstead:Dp1p2h1h2=12(1Ph1h2)Xh0h00p0[2]h0h2h00p1p2p0[1]h1p00h0h00+12(1Pp1p2)Xp0p00h0[2]p0p00p2h1h0h2[1]p1h0p0p00:(3.104)Thisdistinctionisoflittleconsequenceinthepresentanalysis,butmaybecomeimportantiftheHamiltonianandthegeneratorarenottruncatedtothesameparticlerank.Note,however,thatthediagramsforCandDhavettopologies:Theformercouplesthereferencestatetoanexcited2p2hstateviaintermediate2p2hexcitations,whereasthelatterhasintermediate3p3hstates.Byexpandingthegrey[2]verticesinFig.3.15intermsof[1],wecanalsoseehowtheIM-SRGwperformsanon-perturbativeresummationoftheMBPTseries,asindicatedinSec.3.5.1.ThediagramA3,forinstance,isexpandedas=+++:::;(3.105)81andcontainsladderdiagramsrow),aswellasdiagramswhereladderandpolarizationinterfere(secondrow).SuchinterferencediagramssettheIM-SRGapartfromthetraditionalG-matrixandRPAapproaches,whichonlyresumladdersandpolarizationdiagrams,respectively[120].3.5.6EnergythroughO(g4)Letusnowconsidertheenergywequation.AtO(g2),wehave_E[2]=12Xh1h2p1p2[1]h1h2p1p2[1]p1p2h1h2=12Xh1h2p1p2[1]h1h2p1p2[1]p1p2h1h2e2s:(3.106)IntegratingthisequationwithE[2](0)=0,weobtainE[2](s)=141e2sXh1h2p1p2[1]h1h2p1p2[1]p1p2h1h2[0]h1h2p1p2;(3.107)i.e.,E[2](1)isjustthestandardsecond-orderMBPTcorrectiontotheenergyofthereferencestate(cf.Fig.3.11).Likewise,thewequationfortheO(g3)energyreads_E[3]=12Xh1h2p1p2[1]h1h2p1p2[2]p1p2h1h2+[2]h1h2p1p2[1]p1p2h1h2=Xh1h2p1p2[1]h1h2p1p2[2]p1p2h1h2[0]h1h2p1p2e2s(3.108)82andintegrationyieldsE[3](s)=141(2s+1)e2sXh1h2p1p2[1]h1h2p1p2[2]p1p2h1h2[0]h1h2p1p2:(3.109)Fors!1,E[3](1)=14Xh1h2p1p2[1]h1h2p1p2[2]p1p2h1h2[0]h1h2p1p2;(3.110)andpluggingin[2]fromEq.(3.90),thisimmediatelybecomesE[3](1)=++;(3.111)thestandardthirdorderenergycorrection.AtO(g4),wehavetoconsiderproductsof[2]andthesecond-orderHamiltoniancontri-butionsf[2];[2];andW[2](cf.Fig.3.11),aswellasthecrosstermsE[4]31=12Xp1p2h1h2[3]h1h2p1p2[1]p1p2h1h2+[$=12Xp1p2h1h20@[1]h1h2p1p2[2]h1h2p1p2[0]h1h2p1p2+[3]h1h2p1p21A[1]p1p2h1h2[0]h1h2p1p2:(3.112)Thetermisduetotheexpansionoftheenergydenominatorin[3]tosecondorder(cf.Sec.3.5.2).However,itiseasytoseethatcontributionsfromthistermcancelinthesum,because[0=2]pp0hh0isantisymmetricundertranspositionwhile[1]pp0hh0issymmetric.Thus,83Singles(1p1h)S1S2S3S4Doubles(2p2h)D1D2D3D4D5D6D7D8D9D10D11D12Triples(3p3h)T1T2T3T4T5T6T7T8T9T10T11T12T13T14T15T16Quadruples(4p4h)Q1Q2Q3Q4Q5Q6Q7Figure3.16:ConnectedHugenholtzdiagramsforthefourth-orderenergycorrectionE(4)(Ref.[16]).84theenergywequationbecomes_E[4]=2s2e2sXph[2]hpf[2]ph+s22e2sXp1p2h1h2[2]h1h2p1p2[2]p1p2h1h2+s218e2sXp1p2p3h1h2h3[2]h1h2h3p1p2p3W[2]p1p2p3h1h2h3+s22e2sXp1p2h1h2A+Dh1h2p1p2[1]p1p2h1h2h1h2p1p2e4se2s2+sesXp1p2h1h2B+Ch1h2p1p2[1]p1p2h1h2h1h2p1p2:(3.113)Integratingandtakingthelimits!1,weobtainthefourth-orderenergycorrectionE[4](1)=12Xph[2]hpf[2]ph+18Xp1p2h1h2[2]h1h2p1p2[2]p1p2h1h2+172Xp1p2p3h1h2h3[2]h1h2h3p1p2p3W[2]p1p2p3h1h2h3+18Xp1p2h1h2hABh1h2p1p2+DCh1h2p1p2i[1]p1p2h1h2h1h2p1p2E[4]f+E[4]+E[4]W+E[4]A+E[4]B+E[4]C+E[4]D:(3.114)InFig.3.16,weshowallfourth-orderHugenholtzenergydiagramsforthecanonicalHFcase(seeSec.3.5.1andRef.[16])).ItisastraightforwardbutarduoustasktoidentifythediagrammticcontentoftheindividualcontributionstoE[4]bypluggingtheexpressionsfor85theamplitudesfromtheprevioussectionsinto(3.114).WeE[4]f=124Xi=1Si;(3.115)E[4]=1212Xi=1Di;(3.116)E[4]W=1216Xi=1Ti;(3.117)E[4]A=12 4Xi=1Si+12Xi=1Di!;(3.118)E[4]B=Q3+Q4+Q5+12(Q1+Q2+Q6+Q7);(3.119)E[4]C=12(Q1+Q2+Q6+Q7);(3.120)E[4]D=1216Xi=1Ti;(3.121)soE[4](1)containsallrequireddiagrams,andisindeedthecompletefourth-orderenergy.3.5.7DiscussionAsconcludedongeneralgroundsinSec.3.5.1,theIM-SRG(2)energyiscompletetothirdorderinMBPT,butmissescertaincontributionsinfourthorder.OurdetailedanalysisshowsthatE[4]IM-SRG(2)=E[4]f+E[4]+E[4]A+E[4]B=4Xi=1Si+12Xi=1Di+Q3+Q4+Q5+12(Q1+Q2+Q6+Q7);(3.122)86i.e.,IM-SRG(2)containsthecompletefourth-ordersinglesanddoublescontributions,aswellasthesymmetricandhalfoftheantisymmetricquadruplesdiagramsshowninFig.3.16.InthediscussionofFig.3.7inSec.3.3.4,wehaveobservedthattheIM-SRG(2)ground-stateenergyof40CaforthechiralNNHamiltonianwith=2:0fm1liesbetweenCoupledClusterresultsattheCCSDandCCSD(T)level[16,93,94].Overall,thethreemethodsagreewithinafewpercentofthetotalground-stateenergy.ThispatternhasconsistentlyemergedinallourIM-SRGcalculationsfornucleiwithsoftenedchiralinteractions(resolutionscales˘2fm1),bothwithandwithout3Nforces[21,23,58,74].Thediagrammaticcontentofthesemethodsthroughfourthorderexplainsthisbehavior,atleastqualitatively.Intermsofthequantities(3.115){(3.121)intheprevioussubsection,thefourth-orderenergycontributionstoCCSDandCCSD(T)areE[4]CCSD=E[4]f+E[4]+E[4]A+E[4]B+E[4]C=4Xi=1Si+12Xi=1Di+7Xi=1Qi;(3.123)andE[4]CCSD(T)=E[4]f+E[4]+E[4]W+E[4]A+E[4]B+E[4]C+E[4]D=4Xi=1Si+12Xi=1Di+16Xi=1Ti+7Xi=1Qi;(3.124)respectively.Inatypicalcalculation,CCSDground-stateenergiesaretoohighduetomissingcorrelationenergyfromattractivefourth-order3p3h(triples)thatareincludedinCCSD(T)throughE[4]W;D.Inallourcalculations,theasymmetricquadruplesdiagramsQ1;2;6;7(cf.Fig.3.16)arerepulsive.TheIM-SRG(2)misseshalfofthisrepulsion,namely8716202428323640444852~![MeV]2726252423.E[MeV]4He16202428323640444852~![MeV]125120115110105100.E[MeV]16OlIM-SRG(2)dsIM-SRG(2)+E[4]CCCSD16202428323640444852~![MeV]360340320300280260.E[MeV]40CaFigure3.17:offourth-orderquadruples(4p4h)contributionE[4]C,Eq.(3.120)ontheground-stateenergiesof4He,16O,and40Ca(seetext):ComparisonofIM-SRG(2)withandwithoutE[4]C,calculatedwiththeinitialHamiltonianH(0),toCCSDandCCSD(T).AllcalculationsusedthechiralN3LOHamiltonianwith=1inanemax=14single-particlebasis.TheshownCCvaluesweretakenatoptimal~!.theE[4]Cterm,andmocksupmissingattractionfromthetriplestermsE[4]W;Dinthisway.LetusnowconsidertheimplicationsofouranalysisforcalculationswiththeunevolvedchiralN3LOHamiltonian.ReferringbacktoFig.3.7again,thereisalargervariationbetweenthe40Caground-stateenergiesfromIM-SRG(2),CCSD,andCCSD(T).ThisisexpectedbecauseoftheHamiltonian'shigherresolutionscale,whichadverselythemany-bodyconvergence.WeanIM-SRG(2)ground-stateenergythatislowerthanthatofCCSD(T),whichcontainsthecompletefourth-orderenergyandisthereforeexpectedtobeabetterapproximationtotheexactresultfromtheMBPTpointofview.Asimilarobservationwasmadefor4HeinthepublishedIM-SRGstudy[58],wheretheIM-SRG(2)ground-stateenergyof27:6MeVwasfoundtobeabout2MeVlowerthantheCCSD(T)andexactNCSMresults.ThismotivatedthedevelopmentofaperturbativetruncationschemethatisdiscussedinSec.3.5.8,butnolongerusedinpractice.88InFig.3.17,weshowtheofaddingthefourth-orderquadruplestermE[4]CtotheIM-SRG(2)ground-stateenergiesof4He,16O,and40Ca.Inlightofourperturbativeanalysis,especiallyEqs.(3.122)and(3.123),itisnotsurprisingthattherepulsivecontributionsfromthistermshifttheground-stateenergiesincloseproximitytothetheCCSDresults,whichareshownforreference.Theagreementisnotexactduetoandhigher-orderintheperturbativecontentofIM-SRG(2)andCCSD.Finally,wewanttoremarkonthetoriginsoftheinducedthree-bodyverticeswhichcontributetoE[4]CandE[4]D,aspointedoutinthediscussionofEqs.(3.103)and(3.104)inSec.3.5.5.ThisisrelevantforasymmetrictruncationsofHandattparticlerank,andthedevelopmentofapproximationstothefullIM-SRG(3)schemebytheselectiveadditionoftermstotheIM-SRG(2)wequations.E[4]CisaproductofW[2]andthetwo-bodygenerator,whileE[4]Disaproductof[2]andthethree-bodygenerator.Thus,itisttoconsideronlytheinducedthree-bodyinteractionWtofullyincludethefourth-orderquadruples3.Afullinclusionoffourth-ordertriplesrequirestheinducedthree-bodyinteractionaswellastheuseofathree-bodygenerator.3.5.8PerturbativeTruncationsAsdiscussedrepeatedlythroughoutthiswork(see,e.g.,Secs.3.3.4,3.5.1),order-by-orderconvergenceofamany-bodyperturbationexpansionstronglydependsontheresolutionscaleoftheHamiltonian,andthechoiceofreferencestateonwhichtheperturbationseriesisconstructed.ThisisparticularlytrueforthecaseofnuclearHamiltonians[33,51,92,122,3InRef.[121],EvangelistaandGausshavedemonstratedthatE[4]CisnotincludedinamoCCSDschemeifintermediatetermsinthenestedcommutatorsareonlyexpandeduptotwo-bodyoperators.TheseintermediatescorrespondtothepiecesofWthatareinducedbythecommutatoroftwo-bodyoperators,hencethemechanismforgeneratingE[4]CisverysimilarinCCandIM-SRG.89123].Nevertheless,itisworthwhiletoattemptandorganizetheright-handsideoftheIM-SRGwequation|essentially,thefunctionoftheIM-SRGw(see,e.g.,[124,125])|intermsofaperturbativeexpansion,whichisacommonfeatureofRGapproachesthroughoutallofphysics.BasedonthepowercountingfromEqs.(3.62)and(3.64),anearlierwork[58]introducedaperturbativetruncationwhicheliminatestermsofO(g3)fromthewequations(3.8){(3.10):dEds=12Xabcdabcdcdabnanbncnd;(3.125)df12ds=Xa(1+P12)1afa2+Xabc(nanbnc+nanbnc)(1+P12)c1ababc2;(3.126)d1234ds=Xaf(1P12)f1aa234(1P34)fa312a4g+12Xab(1nanb)(12abab3412abab34)Xab(nanb)(1P12)(1P34)b2a4a1b3:(3.127)WewillrefertothistruncationschemeasIM-SRG(2')inthefollowing4.TheintegrationoftheIM-SRG(2')wequationyieldsathird-ordercompleteenergy,whilecertaincontributionsfromfourthorderonwardaremissing.Usingthesamede4(NotethatthelabelingwasreversedinRef.[58],whichprimarilyusedthisperturbativetruncationschemefornumericalcalculations.9016202428323640444852~![MeV]350300250.E[MeV]40Ca16202428323640444852~![MeV]40Ca16202428323640444852~![MeV]600590580570560550.E[MeV]40Caemaxl8n10u12s14IM-SRG(2),=1IM-SRG(2'),=1both,=2:0fm1Figure3.18:Comparisonof40Caground-stateenergiesoftheregularIM-SRG(2)(solidlines)andperturbativeIM-SRG(2')truncations(dashedlines).ThedefaultWhitegeneratorIA,Eq.(3.23),wasusedinbothcases.TheinteractionisthechiralN3LOpotentialwith=1(leftandcenterpanels)and=2:0fm1(rightpanel),respectively.Thedashedlinesindicateextrapolatedenergies.FortheIM-SRG(2')truncation,theshadedareaindicatesthevariationfromusingtdatasetsfortheextrapolation(seetext).asinEq.(3.114),wethatE[4]IM-SRG(2')=E[4]+(E[4]AE[4]f)+E[4]B=12Xi=1Di+Q3+Q4+Q5+12(Q1+Q2+Q6+Q7);(3.128)i.e.,theIM-SRG(2')doesnotcontainthefourth-ordersinglescontribution.Thisiscausedbytheabsenceofthesingle-particletermintheenergywequation(3.125),andthediagramsA1andA2fromtheamplitudeA(seeFig.3.15andEq.(3.99)).InFig.3.18,wecompare40Caground-stateenergiesobtainedwiththeregularandper-turbativetruncations.ForthesoftN3LOinteractionwith=2:0fm1,shownintherightpanel,thetwotruncationsgivealmostidenticalresults.Theagreementbetweenground-stateenergiesisonthelevelof104orbetter,withextrapolatedenergiesfor40Caering91byonly2keV.Forthebareinteraction,ontheotherhand,thetruncationschemesbehavequitedif-ferently.TheIM-SRG(2)ground-stateenergyhasaquasi-variationalconvergencepattern,whichallowsusastableextrapolationtoHObasissize.TheIM-SRG(2')trunca-tion'sground-stateenergyminimumisstillmovingtolarger~!fortheconsideredbases,indicatingalackofUVconvergence,andthevariationalpatternbreaksdownasweincreaseemaxfrom12to14.ExtrapolationfromtsubsetsofthecalculatedenergiesusingEq.(3.51)produceslargeuncertaintieswhichareindicatedbytheshadedbandinFig.3.18.Asdiscussedabove,theIM-SRG(2')ground-stateenergy,Eq.(3.128),doesnotcontainthefourth-ordersingles.InFig.3.19,wedemonstratethattheomissionofthiscontributionaccountsforthebulkoftheenergybetweenIM-SRG(2)andIM-SRG(2'),using40Caasanexample.Moreover,theadditionofthefourth-ordersinglesrestoresthevaria-tionalbehavioroftheground-stateenergyasafunctionofthesingle-particlebasissizeemax.ComparedtotheregularIM-SRG(2),theIM-SRG(2')wequationslackO(g3)contractionsoffandwiththetwo-andone-bodypartsof,respectively.Theofthisomissiononthetwo-bodymatrixelementishardtoanalyzeingreaterdetail,inpartduetotheirsheernumber.Totesttheimpactofthemissingtermsonthewingone-bodyHamiltonian,wecalculatetheBarangertivesingle-particleenergies(ESPEs)bydiagonalizingthef(1)inbothtruncations(see[126{128]).TheneutronandprotonsdandpfshellESPEsin40CaareshowninFig.3.20,andwethattheresultsobtainedwithIM-SRG(2)andIM-SRG(2')arepracticallyindistinguishable.WeconcludebyfollowingupontheperturbativeanalysisofthebetweenIM-SRG(2)andCCresultswiththeunevolvedchiralN3LOHamiltonianthatwasbeguninSec.3.5.7.InRef.[58],theoverestimationofthe4Heground-stateenergyinIM-SRG(2)cal-9216202428323640444852~![MeV]375350325300275250.E[MeV]16202428323640444852~![MeV]40CaIM-SRG(2')+singlesIM-SRG(2')IM-SRG(2)emaxn,10u,12s,M14Figure3.19:ofaddingthefourth-ordersingles(1p1h)contribution(cf.Eqs.(3.115),(3.118)and(3.128))totheIM-SRG(2')ground-stateenergyof40Ca(seetext).Thesinglescontributionsfort~!werecalculatedwiththeinitialHamiltonianH(0).AllshownresultswereobtainedforthechiralN3LOHamiltonianwith=1.16202428323640444852~![MeV]20020."[MeV]40Ca1s1=20d3=20d5=21p3=21p1=20f7=20f5=216202428323640444852~![MeV]20020."ˇ[MeV]40Ca1s1=20d3=20d5=21p3=21p1=20f7=20f5=2Figure3.20:eneutron(leftpanel)andproton(rightpanel)single-particleenergiesof40CafromIM-SRG(2)(solidlines)andIM-SRG(2')(dashedlines)calculationsusingthechiralN3LOinteractionwith=1inanemax=14single-particlebasis.culationswhencomparedtoCCSD(T)andexactNCSMresultswasthemainmotivationfortheinvestigationoftheIM-SRG(2')truncation.TheIM-SRG(2')resultcloselymatches93theCCSDresultfor4He,23:98MeV,butthepresentdiscussionrevealsthisagreementasaccidental,anartifactoftheomissionofattractivefourth-ordersinglesproducingasimilarchangeintheground-stateenergyastheadditionoftherepulsivequadruplestermE[4]C(seethediscussioninSec.3.5.7).Whilebothtruncationsworkequallywellfortlysoft,perturbativenuclearHamiltonians,theIM-SRG(2)truncationremainswell-behavedathigherresolutionscales,atthesamecomputationalcost,whichiswhywefavorthistruncationschemeinpracticalapplications.94Chapter4MagnusFormulation4.1IntroductionChapter3demonstratedtheenessoftheIM-SRG(2)methodforgroundstatecalcula-tionsofnucleiat(sub)shellclosures.However,wenowneedtoaddresstheprimarycomputa-tionallimitationsofthemethodasitwaspresented.TheIM-SRGcalculationspresentedinchapter3typicallyuseODEsolversbasedonhigh-orderRunge-Kuttaorpredictor-correctormethodstosolveEq.3.2.Theuseofthesehigh-ordermethodsisessentialastheaccumula-tionoftime-steperrorswilldestroytheunitaryequivalencebetweenH(s)andH(0),evenifnotruncationsaremadeinthewequations.State-of-the-artsolverscanrequirethestorageof15-20copiesofthesolutionvectorinmemory,whichbecomesproblematicforlargemodelspaces.Forexample,atypicalOxygencalculationinabasissetcorrespondingtoanemaxof12,IM-SRG(2)calculationsrequirearound30GBtorun.Thislargememoryfootprintisexacerbatedifonewantstocalculateadditionalobservables,roughlydoublingthememoryrequirementsassumingthesameNO2BtruncationasfortheHamiltonian.Moreover,theadditionalwequationsforeachobservablecanevolvewithratherttimescalesthantheHamiltonian,whichincreasethelikelihoodoftheODEsbecomingInthischapter,wewilldemonstratethatthesecanbecircumventedbyrecastingEq.(3.2)withtheMagnusexpansion[129].Thenewformulationisconvenientforestablishingimprovedtruncationstobediscussedinchapter5.Thepresentationfollowsarecentpublicationin95Ref.[130].4.2FormalismInthenotationofourpresentproblem,ourstartingpointisthetialequationobeyedbytheunitarytransformation,dU(s)ds=(s)U(s);(4.1)whereU(0)=1andUy(s)U(s)=U(s)Uy(s)=1.Thiscanbeformallyintegratedandwrittenasthetime-orderedexponentialU(s)=TseRs0(s0)ds0(4.2)1Zs0ds0(s0)+Zs0ds0Zs00ds00(s0)(s00)+:::(4.3)Eq.4.3isnotveryusefulinpracticalcalculationssincei)thereisnoguidanceonhowtheseriesshouldbetruncated,ii)onewouldneedtostoreformultiples-values,andiii)itisnotobvioushowtoconsistentlytransformtheHamiltonianandotherobservablesinafullylinked,size-extensivemannerwiththetruncatedseries.TheessenceoftheMagnusexpansionisthat,givenafewtechnicalrequirementson(s),asolutiontoEq.4.1oftheformU(s)=es)(4.4)exists,wherey(s)=s)and)=0[131].Inmostpreviousapplicationsofthe96Magnusexpansion,onetypicallyexpandss)inpowersof(s)as=1Xn=1n:(4.5)Combiningthiswiththeexactderivativedds=1Xk=0Bkk!adk()ad0()=adk()=;adk1()];(4.6)whereBkaretheBernoullinumbersandadk()therecursivelynestedcommutators,onecanobtainexplicitexpressionsforthen(s),1(s)=Zs0ds1(s1)2(s)=12Zs0ds1Zs10ds2[(s1);(s2)]...(4.7)Asexpected,rewritingthetime-orderedexponentialasatruematrixexponentialmovesthecomplicationsoftimeorderingintotheexpressionfors).TheutilityoftheMagnusexpansionliesinthefactthat,evenifistruncatedtolow-ordersin,theresultingtransformationinEq.4.4usingtheapproximateisunitary,incontrasttoanytruncatedversionofEq.4.2.Forlarge-scaleIM-SRGcalculations,theexpressionsinEq.4.7areoflimitedvaluesincetheyrequirethestorageof(s)overarangeofs-values.Therefore,inthepresentworkwe97insteadconstructs)bynumericallyintegratingEq.4.6,subjecttocertainapproximationsdiscussedbelow.ThetransformedHamiltonian,andanyotheroperatorofinterest,canthenbeconstructedbyapplyingtheBaker-Camb(BCH)formula,H(s)=eHe=P1k=01k!adk(H)(4.8)O(s)=eOe=P1k=01k!adk(O):(4.9)4.3AnalyticalModelBeforediscussinghowwetruncateEqs.4.6and4.8inpracticalcalculations,itisinstructivetostudyasimplematrixmodelthatcanbesolvedwithoutanytruncations.ConsidertheinitialHamiltonianH=T+V=0B@11111CA;(4.10)wherethediagonal\kineticenergy"termisT=0B@10011CA:(4.11)LetusnowtrytodiagonalizeHusingtheWegnergeneratorpresentedinchapter3,making(s)=[T;H(s)],solvingtheSRGequationsusingtheMagnusexpansionandbydirectintegrationofEq.3.2.NotethatforthischoiceofinitialH,both(s)ands)arereal,98antisymmetricmatricesthroughoutthew(s)=ig(s)˙2(4.12)s)=ig(s)˙2;(4.13)where˙2isthePaulimatrix.Consequently,Eq.4.6terminatesatthetermandEq.4.8canbesummeduptoallordersusingthewell-knownpropertiesofPaulimatrices.ThissimplicityalsoallowsforananalyticalsolutionforH(s)=gT(s)T+gV(s)V,wheregT(s)=p2tanh(p32s+arcsinh(1))(4.14)gV(s)=p2sech(p32s+arcsinh(1)):(4.15)Thelargememoryfootprintofhigh-orderadaptivesolversisthemaincomputationalchal-lengeinlarge-scaleSRGcalculations,soinadditiontousingaGordon-ShampineintegratortosolveEqs.3.2,wedemonstratewhathappenswhenanaiverderEulermethodisusedtointegrateEqs.3.2and4.6.TheresultsareshowninFig.4.1,whereweplotjH11(s)Egsj{whichshouldgotozeroatlarges{versussfortEulerstepsizess.Unsurprisingly,weseethatthedirectintegrationofEq.3.2accumulateslargetime-steperrors,withtheplateausatlargesdisplayingastrongdependenceontheEulerstepsize.EvenwhenEq.3.2isintegratedwithahigh-ordermethodwithveryconservativeabsoluteandrelativetolerancesof1e12,thesolutionfailstoproducetheexactanswer.TheMagnussolution,ontheotherhand,convergestoaansweratlargesthatisindependentofstepsizeandagreeswiththeexactresulttowithinmachineprecision.Evenmore,thenaiveEulerstepwiths=:001isindistinguishablefromtheanalyticalresult.Theinsensitivity99Figure4.1:jH11(s)EgsjversussfortEulerstepsizescalculatedviadirectintegra-tionoftheSRGwequation,Eq.3.2,andusingtheMagnusexpansion,Eqs.4.6and4.8.AlsoplottedistheintegrationofEq.3.2withtheGordon-Shampineintegrator.tothetimestepsizeisduetothefactthatwhileeachEulerstepinEq.4.6givesanerroroforderO(s2),theexponentiatedoperatorattheendoftheevolutionisstillunitary.ThisistheprimaryadvantageoftheMagnusexpansion;byreformulatingtheproblemtosolvewequationsfors)insteadofH(s),onecanuseasimplerderEulermethodanddramaticallyreducememoryusage.Onces)isinhand,thetransformationofH(s)andanyotherobservablesofinterestimmediatelyfollowsfromEq.4.8.NotethatincontrasttothedirectintegrationofEq.3.2,thedimensionalityofthewequationsdoesnotincreasewhenoneevolvesadditionalobservables.1004.4MAGNUS(2)ApproximationHavingillustratedtheadvantagesoftheMagnusexpansioninasimplemodel,wewouldnowliketoapplyitlarge-scalemany-bodycalculations.Beforegoingintonecessaryap-proximations,itisinstructivetohighlighthowtheMagnusformulationoftheIM-SRGmakestheconnectiontocoupledclustertheorymoretangible.Werecallfromchapter2thatclosedshellcoupledclustertheoryiscenteredinthephilosophyofdecoupingasinglereferencefromhigherexcitationsviaanon-hermitiansimilaritytransformedHamiltonianH=eTHeT.TheIM-SRGisalsobasedonthissamephilosophy,butwithintheMagnusapproach,themeansofdoingsobecomemoretransparent.WearenowsolvingthehermitianH(s)=es)Hes),whichbearsmorethanpassingsimilaritytoH.UnlikeincoupledclustertheorywheretheBCHformulaforthesimilaritytransformedHamiltonianterminatesatorder,bothEqs.4.6and4.8involveanseriesofnestedcommutatorsthatgenerateuptoA-bodyoperators.Thus,tomakeprogress,weintroducetheMAGNUS(2)truncationinwhichallcommutators(aswellass);(s)andH(s))aretruncatedtotheNO2Blevel.Evenwiththisapproximation,theexpressionsford=dsandH(s)involveannumberofterms.However,forbothEqs.4.6and4.8attheNO2Blevel,weempiricallythatthemagnitudeoftermsdecreasesmonotonicallyinkforallsystemsstudiedthusfar.Therefore,wenumericallytruncateEqs.4.6atthekthtermifBkkadk()kk!kk<deriv:(4.16)Forthetruncationof(4.8),wecoulduseasimilarcriteriaasforthederivativeexpression.However,sinceweareinterestedintheground-stateenergy,weuseasimplerconditionwhere101theseriesistruncatedwhenthezero-bodypieceofthekthtermfallsbelowsomethreshold,fadk(H)g0bk!<BCH:(4.17)Inthecalculationspresentedbelow,wewillndthattheresultsareinsensitivetolargevariationsinderivandBCH,whichwetakeasanaposterioriforourtruncations.4.5HamiltoniansandImplementationBeforepresentingtheresultsofIM-SRG(2)andMAGNUS(2)calculationsofthehomoge-neouselectrongas(HEG)and16O,wereviewsomedetailsofourimplementationsforbothsystems.Forthehomogeneouselectrongas,weperformourcalculationsfortheclosed-shellofN=14electronsinacubicboxwithperiodicboundaryconditions.Notethatifoneisinterestedinextrapolatingtothethermodynamiclimit,calculationsshouldbedoneforalargerclosed-shellofN=38;54;66;:::electrons,withsizecorrectionsforthekineticandpotentialenergytakenintoaccount.HereweneglectthesecorrectionssinceourprimarypurposeistodemonstratetheenessoftheMag-nusexpansion,andthequasi-exactFullInteractionQuantumMonteCarlo(FCIQMC)resultswecompareagainstalsoneglectthesecorrections[3].Therelevantsingleparticleorbitalsareplanewaveswithquantizedmomenta k˙(r)=1pL3eikr˜˙;(4.18)102whereL3istheboxvolume,˜˙isaspineigenfunction,andk=2ˇL(nx;ny;nz)wherenx;ny;andnzareintegers.WefollowcommonpracticeandusetheWigner-SeitzradiustocharacterizethedensityoftheHEG,rs=r0a0;(4.19)wherea0istheBohrradiusandr0isintermsofthedensityas43ˇr30=NL3:(4.20)WeuseabasissettruncationwhichkeepsMsingleparticlestateswithenergylessthansomeEc,althoughotherchoicesarepossible[132].Intheplanewavebasis,thekineticenergymatrixelementsarediagonalTi;j=12ki2ij;(4.21)andtheCoulombmatrixelementsaregivenbyVijkl=1L31q2˙i;˙k˙j;˙lq;kikkq;klkj:(4.22)Notethattheq=0termisomittedduetoitscancellationagainsttheinert,uniformpositivelychargedbackgroundthatisneededtomakethesystemchargeneutral[133].Sinceweareinterestedprimarilyinthecorrelationenergy,wehaveomittedtheMadelungterminallofourcalculations.103Forthecalculationsof16O,ourstartingpointistheintrinsicnuclearA-bodyHamiltonianH=11AT+T(2)+V(2);(4.23)whereT(2)isthetwo-bodypartoftheintrinsickineticenergy,andwerestrictourattentiontotwo-nucleoninteractionsonly.ResultsarepresentedforinputNNinteractionsderivedfromtheN3LO(500MeV)potentialofEntemandMachleidt[17]atseveraltfree-spaceSRGresolutionscales,=2:0;2:7,and3:0fm1.InbothMAGNUS(2)andIM-SRG(2)calculations,westartbynormalorderingtheHamiltonianwithrespecttotheHFgroundstate.InthecaseoftheHEG,translationalinvarianceimpliestheHForbitalsareplanewaves.Therefore,theHFreferencestateisjustaSlaterdeterminantcomprisedofthelowestenergydoublyoccupiedplanewavestatesforN=14electrons.For16O,wemustself-consistentlysolvetheHartree-Fockequationsbyex-pandingtheunknownHForbitalsinaharmonicoscillatorbasistruncatedtooscillatorstatesobeying2n+lemax,whereemaxistlylargesothattheresultsareapproximatelyindependentofthe~!valueoftheunderlyingoscillatorbasis.FortheNNinteractionsusedinthepresentcalculations,aofemax=8istlylargeformostpurposes.OnceaconvergedHFground-stateisobtained,theHamiltonianisnormal-orderedw.r.t.tothissolution,andtheresultingin-mediumzero-,one-,andtwo-bodyoperatorsserveastheinitialvaluesfortheMAGNUS(2)andIM-SRG(2)wequations.Thesearesubsequentlyinte-grateduntiltdecouplingisachieved,asdeterminedbythesizeofthesecond-ordermany-bodyperturbationtheoryMBPT(2)contributionofthewingHamiltonianH(s)tothegroundstateenergy.Weuseathresholdof106Hartree(MeV)fortheHEG(16O)calculations,respectively,whichcorrespondstorelativechangesinthewingground-state104Figure4.2:RelativeimportanceofthekthtermintheMagnusderivativeasbythelefthandsideofEq.4.16evaluatedintheNO2Bapproximation.ThetoprowisforthehomogeneouselectrongasatWigner-Seitzradiiofa)rs=0:5andb)rs=5:0.Thebottomrowisfor16O,startingfromthechiralNNpotentialofEntemandMachleidt[17],softenedbyafree-spaceSRGevolutionto(c)=2:0fm1and(d)=3:0fm1.TheelectrongascalculationsweredoneforN=14electronsinaperiodicboxwithM=114singleparticleorbitals.The16Ocalculationsweredoneinanemax=8modelspace,with~!=24MeVfortheunderlyingharmonicoscillatorbasis.energyof107orlessforbothsystems.4.6ResultsWebeginbyexaminingthenumericalevidencefortruncatingEqs.4.6and4.8byhand.InFigure4.2,weplotthelefthandsideofEq.4.16fortheHEG(toprow)and16O(bottomrow)asafunctionofthewparameter.Toassesstheroleofcorrelations,theHEGcalculationswereperformedattwoerentWigner-Seitzradii,rs=0:5andrs=5:0,andthe16O105calculationsweredoneusingNNinteractionsattwotresolutionscales,=2:0fm1and=3:0fm1.FortheHEG,thers=0:5contributionsarecompletelynegligiblebythek=2term,whichisnotsurprisingsincethekineticenergydominatesinthisweaklycorrelatedhigh-densityregime[133].Evenforthers=5:0case,wherecorrelationsandnon-perturbativestarttobecomesizable,onethatthesuccessivetermsinEq.4.6decreasemonotonically,thoughtheindividualtermsaresubstantiallylargerthanforthers=0:5case.Analogousresultsarefoundfor16O;theindividualtermsarelargerfortheharder=3:0fm1interactionsincethesystemismorestronglycorrelated,buttheysystematicallydecreasewithincreasingorderk.Figure4.3tellsasimilarstoryfortheBCHformula,wherethelefthandsideofEq.4.17isplottedasafunctionofthewparameter.Inallcases,weseetheimportanceofsuc-cessivetermsdecreasesmonotonically.Reassuringly,wethattheresultsinourcalculationsareessentiallyindependentoftheconvergencecriteriaprovidedderiv.104andBCH.104,wherethelatterisinunitsofHartree(MeV)fortheHEG(16O)calcula-tions,respectively.Forinstance,raisingbothconvergencecriteriafrom108to104changesthegroundstateenergyatthe1eV(107Hartree)levelinthe16O(HEG)calculations,respectively.AswasillustratedforthetoymodelinSec.4.2,thekeyadvantageoftheMagnusexpan-sionisthatonecanuseaEulermethodtoaccuratelysolvethewequations.WenowdemonstratethatthesameconclusionholdsforrealisticIM-SRGcalculations.Re-ferringtoFigs.4.4and4.5,weshowthe0-bodypartofthewingHamiltonianH(s)versusthewparameterfortheelectrongas1and16O.TheblacksolidlinesdenotetheresultsofastandardIM-SRG(2)calculationusingtheadaptiveODEsolverofShampineandGordon,1FortheHEG,weplotE0(s)EHF,whichapproachesthecorrelationenergyatlarges.106Figure4.3:Magnitudeofthe0-bodycontributionsofthekthterminEq.4.8evaluatedintheNO2Bapproximation.ThetoprowisfortheelectrongasatWigner-Seitzradiiof(a)rs=0:5and(b)rs=5:0.Thebottomrowisfor16O,startingfromthechiralNNpotentialofEntemandMachleidt[17],softenedbyafree-spaceSRGevolutionto(c)=2:0fm1and(d)=3:0fm1.TheelectrongascalculationsweredoneforN=14electronsinaperiodicboxwithM=114singleparticleorbitals.The16Ocalculationsweredoneinanemax=8modelspace,with~!=24MeVfortheunderlyingharmonicoscillatorbasis.whiletheothercurvesdenoteIM-SRG(2)andMAGNUS(2)calculationsusingaEulermethodwithtstepsizess.Fortheelectrongas,theexactFCIQMCresults[3]areshownforreference.Unsurprisingly,theIM-SRG(2)Eulercalculationsareverypoor,withthevariousstepsizesconvergingtotlarge-slimits.TheMAGNUS(2)calcu-lations,ontheotherhand,convergetothesamelarge-slimitinexcellentagreementwiththestandardIM-SRG(2)results.Theinsensitivitytostepsizeisduetothefactthatthetimesteperrorsaccumulateins)asopposedtoH(s).Attheendofthew,s)isstill107Figure4.4:FlowingIM-SRG(2)andMAGNUS(2)HEGcorrelationenergy,E0(s)EHF,forWigner-Seitzradiiofa)rs=5:0andb)rs=0:5.ThesolidblacklinecorrespondstoIM-SRG(2)resultsusinganadaptivesolverbasedontheAdams-Bashforthmethod,whiletheotherlinescorrespondtoMAGNUS(2)andIM-SRG(2)resultsusingerentEulerstepsizes.Theredcirclesdenotethequasi-exactFCIQMCresultsofRef.[3].ananti-hermitianoperator,andthetransformationinEq.4.8isunitary,uptotruncationerrorsintheNO2Bapproximation.GiventhattheMAGNUS(2)resultsareindependentofstepsizeovertherangeconsid-ered,onemighttrytokeepincreasingthestepsizetoreachthegroundstateinfewersteps.Thisunfortunatelyisnotpossible,asthewtendstodivergewithtoolargeofatime108Figure4.5:FlowingIM-SRG(2)andMAGNUS(2)groundstateenergy,E0(s),for16OstartingfromtheN3LONNinteractionofEntemandMachleidt[17]evolvedbythefree-spaceSRGtoa)=2:7fm1and=2:0fm1.ThesolidblacklinecorrespondstoIM-SRG(2)resultsusinganadaptivesolverbasedontheAdams-Bashforthmethod,whiletheotherlinescorrespondtoMAGNUS(2)andIM-SRG(2)resultsusingerentEulerstepsizes.Thecalculationsweredoneinanemax=8modelspace,with~!=24MeVfortheunderlyingharmonicoscillatorbasis.step.OneofthestrengthsoftheSRGapproachisthatthetransformationisadaptedastheHamiltonianistransformed.Withtoolargeofatimestep,werobthemethodofthisyandruntheriskofapplyinga\largerotation"oftheHamiltonianthatinduceslargethree-andhigher-bodycomponents.Thiswouldnotbeaproblemifweevaluatedthe109BCHandMagnusderivativeexpressionswithoutapproximation;themethodwoulditswaybacksincethelargerotationisstillunitaryifnotruncationsaremade.However,intheMAGNUS(2)approximationwemake,theneglectoftheinducedthree-andhigher-bodytermscanleadtoalackofconvergence.Empirically,wethatthisyisavoidedbyenforcingthatateachtimestepthematrixnormkHodkisdecreasing.Thiscanbeimplementedbyusingasimplemid-pointintegratoralgorithmanddecreasingthetimestepifkHodkhasincreasedbetweentheandsecondhalfofastep.AsademonstrationoftheutilityoftheMagnusexpansion,weturntotheevolutionofoperatorsotherthantheHamiltonian.IntheconventionalapproachbasedonthedirectintegrationofEq.3.2,thedimensionalityofthewequationsincreaseswitheachadditionaloperatortobeevolved.Incontrast,intheMagnusexpansionthedimensionalityofthewequationsdoesnotchange;theadditionalcomputationalexpenseshowsuponlyintheevaluationoftheBCHformulaforthetransformedoperator,Eq.4.9.ForagivenoperatorO,wehaveh0jOj0i=lims!1hjes)Oes)ji;(4.24)wherejiisthereferencestate.Therefore,the0-bodypieceofthetransformedoperatorapproachestheinteractinggroundstateexpectationvalueinthelarge-slimit.Asaproof-of-principle,weperformaMAGNUS(2)evolutiontoevaluatethegroundstateexpectationvalueofthemomentumdistributionoperator^nkaykakfortheHEG,andthegeneralizedcenterofmass(COM)Hamiltonianforthe16Onucleus,Hcm(~!)=Tcm+12mA~!2R2cm32~~!:(4.25)Figure4.6showstheMAGNUS(2)groundstatemomentumdistributionforasystemofN=110Figure4.6:ElectrongasmomentumdistributionscalculatedintheMAGNUS(2)approxi-mation.ThecalculationsweredoneforN=14electronsinaperiodicboxwithM=778singleparticleorbitals.14electronsinaperiodicboxforseveraltWigner-Seitzradii.Evenwiththeneglectofsizecorrectionsandtheextremelycoarsemomentumgridduetothesmallboxsizesconsidered,thequalitativebehavioragreeswithexpectationsfortheelectrongas;correlationsbecomemoreimportantatlargerrs,leadingtoastrongerdepletionofmodeswithk<kFandsmallerdiscontinuityattheFermisurface.WenotethattheMAGNUS(2)resultsareingoodagreementwiththeIM-SRG(2)calculationsbasedonEq.3.2aswellasresultsgeneratedbytheFeynman-Hellmantheorem,butatafractionofthecost.InadditiontoprovidingatmeansforevolvingoperatorsbeyondtheHamiltonian,Fig.4.7showsthattheMAGNUS(2)approximationgivesasmallbutrobustcomputationalspeedupforarangeofbasissets,evenincludingtheadditionalofgeneratingthemomentumdistributions,whichwerenotcomputedintheIM-SRG(2)timings.Foroursecondillustrationofoperatorevolution,weconsiderthegeneralizedCOMHamil-tonian,Eq.4.25.Incalculationsofnuclei,thegroundstateexpectationvalueofthisquan-111Figure4.7:TimingforMAGNUS(2)andIM-SRG(2)HEGcalculationsasthesingleparticlebasisisincreased.Thetwobottomcurvesareforrs=:5andthetopforrs=5.FortheMAGNUS(2)timing,thisincludesthecalculationsofthemomentumdistributions.tityisusefultodiagnosewhetherapproximatesolutionsoftheScodingerequationarecontaminatedbyspuriousCOMmotion.Sincenucleiareself-boundobjectsgovernedbyatranslationally-invariantHamiltonian,anexactsolutionoftheScodingerequationmustfactorizeintotheproductofawavefunctionforthephysicallyrelevantintrinsicmotiontimesawavefunctionfortheCOMcoordinate,ji=j iinj icm:(4.26)Asiswellknown,therearetwostrategiestorigorouslyguaranteethisfactorization;onecanworkinatranslationally-invariantbasisfromtheoutset,oronecanworkinaso-calledfullN~!modelspacecomprisedofallA-particleharmonicoscillatorSlaterdeterminantswithexcitationsuptoandincludingN~!.Neitherchoiceisoptimalsincetheformerislimitedtolightnucleiduetothefactorialscalingoftherequiredantisymmetrization,while112Figure4.8:CenterofmassdiagnosticsforMAGNUS(2)calculationsof16OstartingfromtheN3LONNinteractionofEntemandMachleidt[17]evolvedbythefree-spaceSRGto=2:0fm1.Seethetextfordetails.Thecalculationsweredoneinanemax=9modelspace.thelatterlimitsthechoiceofthesingleparticleorbitalstotheharmonicoscillatorbasisanddoesnotcarryovertomethodsthatarecapableofreachingheaviernuclei,suchascoupledclustertheoryandtheIM-SRGwhereitismorenaturaltothemodelspaceviaanenergycu(e.g.,2n+1emax)onthesingleparticlestates.InthecaseofcalculationswithanemaxthereisnoanalyticalguaranteethattheCOMandintrinsicwavefunctionsfactorize.InRef.[18],HagenandcollaboratorsgaveaningeniousprescriptiontodiagnosewhetherornotEq.4.26isinsuchcalculations.ThebasicideaistoassumethatthefactorizedCOMwavefunctionisinthegroundstateofHcm(~!)withlowesteigenvalue.Notethat~!6=!113ingeneral,where!isthefrequencyoftheunderlyingoscillatorbasis.Theprescriptionto~!involvessolvingaquadraticequation~~!=~!+23Ecm(!)r49(Ecm(!))2+43~!Ecm(!);(4.27)whereEcm(!)hjHcm(!)ji(4.28)=lims!1hjes)Hcm(!)es)ji(4.29)=lims!1es)Hcm(!)es)0b:(4.30)SincetherearetworootsofEq.4.27,wechoosetheonethatgivesasmallervalueforEcm(~!).Applyingthisprescriptiontoourcalculationsof16O,weobtaintheresultsshowninFig.4.8.Inthetoppanel,weseethattheexpectationvalueoftheCOMHamiltonianHcm(!)isapproximatelyzerofor!ˇ20MeV,butvariesparabolicallyandbecomesratherlargeawayfromthispoint.ThissuggeststhatifEq.4.26isthefrequencyofthefactorizedCOMGaussianshouldhave~!ˇ20MeV.Thisisbornoutinthebottompanel,wherethetworootsofEq.4.27areplottedasafunctionof~!.PickingtherootthatminimizesEcm(~!),wethatindeed~!ˇ20MeVoverawiderangeof!,andthatEcm(~!)ˇ0.SincetheexcitationenergyforthespuriousCOMmodeis~~!ˇ20MeV,whileEcm(~!)rangesbetween0:03-0:14MeVovertheentirerangeof~!,weconcludethatthefactorizationofCOM/intrinsicmotionistoanexcellentapproximation.1144.7MAGNUS(2)ConclusionInthesesimplesystems,wehaveshownthatalltheutilityofIM-SRG(2)calculationsinChapter3isavailablenowatcantlydecreasedcost,withtheoptionofinvestigatinganyobservablewithoutconductingthefullIM-SRG(2)calculationagain.Wehavethuscircumventedtwoofthelargestweaknessesofthemethod,thealreadymentionedneedtosolvealargesetoftialequationswithsmallerrors,andthelinearscalingindesiredobservables.Wewouldnowliketoaddresstheundercountingofacertainclassofdiagramsmentionedin3.5.WithintheMAGNUS(2)formulation,wehaveafairlystraightforwardwaytoproceedwiththis,whichwillbepresentedinChapter5.115Chapter5ApproximatingtheIM-SRG(3)Havingestablishedthattheexactsolutionofthevastmajorityofmany-bodyproblemsisbeyondthereachofeventhemostadvancedmethodsinchapter2,practitionersinsteadsettheiraimonsystematicapproximationsthat,insomelimit,approachthecompletesolution.Thisphilosophyisevidentincoupledclustertheory,whereonecantruncatetheclusteroperatorTatincreasingexcitationrank,givingahierarchyofimprovedapproxima-tions(CCD,CCSD,CCSDT,etc.).Moreover,sincetruncationsbeyondtheCCSDlevelcanbecomeextremelyexpensiveforlargersystems,muchrthasgoneintothedevelopmentofcomputationallycheapnon-iterativeapproximationstothesehighertruncations.ThecentralaimofthepresentchapteristodescribehowanalogousapproximationscanbeconstructedintheIM-SRG.InChapter4,wearguedthatthedirectintegrationoftheIM-SRG(2)wequationsinEq.(3.2)islimitedbythelargememorydemandsofthehigh-orderODEsolversthatareneededtocontroltheaccumulationofnumericalerrors.OneofthemajorthemesofmythesisistheuseoftheMagnusexpansiontoeliminatethesebyreformulatingtheproblemsothatthesetofwequationstobesolvedisgivenbyEq.(4.6).AsdiscussedinChapter4,thisreformulation,whichintheNO2BapproximationisdeemedMAGNUS(2),canbesolvedwithasimpleEulermethodandyieldsanearlyidenticaltransformationastheIM-SRG(2),butwithmuchsmallermemoryrequirements.Asaresult,theMAGNUS(2)makesitfeasibletoextendcalculationstoheaviersystemsand,moreimportantly,perform116calculationsofotherpropertiesbesidesspectra.Inadditiontoprovidingsubstantialcomputationalimprovements,theMagnusformula-tionprovidesaparticularlytransparentpathtowardsdevelopingapproximateMAGNUS(3)calculationsfromconvergedMAGNUS(2)calculations.Thedevelopmentofsuchapproxi-mateNO3Bcalculationsisessential,asthefullIM-SRG(3)/MAGNUS(3)methodsnaivelyscaleasn9,whichmakesthemintractableforlarge-scalecalculations.Inthepresentchapter,wewilldelveintoquantumchemistrytypesystemstodemonstratetheshortcomingsofIM-SRG(2)typecalculationsevenwithrespecttoCCSD,despitethefactthattheyshouldbenaivelyrelatedtoeachother.Fromthere,guidedbytheperturbativeanalysisofSection3.5,wewillintroducecomputationallyfeasibleapproximationstomitigatetheseshortcom-ings.Andthen,wewillintroduceafamilyofmethodsthatareanalagoustoCCSDplusnon-iterativeapproximatetriples.5.1oftheIM-SRG(2)inChemistryTheapplicationoftheIM-SRG(2)toab-initiomany-bodycalculations,thoughcouchedinsomewhattterminologyandnotation,wascarriedoutbyWhiteintreatinganH2OmoleculeinaDZPbasisset[78].Whilethemethodwasmoreorlessabandonedinchemistrysystems,thoughseetherelatedworkofEvangelistaandcollaborators[134]andChanetal.[79].Animportantinthatseminalworkwasthat,unlikeCCSDwhichgenerallyunderbindswithrespecttoFCI,IM-SRG(2)producedresultsthatweredramaticallyoverboundatequilibriumandnotconvergentatevenmodestbondlengths.Thiswaspeculiar,asthescalingandphilosophyofthemethodsaresimilar.PinpointingthefailureoftheIM-SRG(2)andMAGNUS(2)toproduceconvergedresults117inH2OandotherstretchedmoleculescanbeguidedmosteasilybytheperturbativeanalysisfoundinSec.3.5.Aswehavealreadypointedout,diagramsmakingupthose(3.120)areundercountedbyafactoroftwoinbothIM-SRG(2)andMAGNUS(2).TheseareproperlyaccountedforinCCSDcalculations,soinitialattemptstoresolvenon-convergencecenteredaroundrestoringthiscontenttothetransformedHamiltonian.Evangelistaetal.hadsimilarwhentheBCHoftraditionalcoupledclusterisappliedviaone-andtwo-bodyintermediates,asopposedtothefulldiagrammaticcontentoftheclusteramplitudesfoundin(2.31)-(2.33)[121].ThelackofthesediagramscauseddramaticoverbindingwithrespecttofullCCSDinH2Oandsimilarmolecularsystems.Asdiscussedinthefollowingandin3.5,thisisalsothereasonIM-SRG(2)/MAGNUS(2)resultstrackinbetweenCCSDandCCSD(T)calculationsforawiderangeofclosed-shellnuclei,2Dquantumdots,andthehomogenouselectrongas.5.2TheIM-SRG(2*)ApproximationInthetraditionalIM-SRGformalism,itisnotstraightforwardhowonemightrestoretheaboveundercountedtermswithoutstoringafullthree-bodyoperator,whichisasix-indextensor.Thisiscertainlysomethingwewishtoavoid,ascalculationsareoftenmemorylimitedevenwiththe15-20copiesofatwo-bodyHamiltonian.SincethesediagramsarefoundintheIM-SRG(2),butwiththewrongweight,maybeitispossibletoawaytore-weighttheofwhatisalreadyincludedwithoutchangingthecomputationalcosttoomuch.Indeed,thiscanbedone.Figure5.1andFigure5.2bothshowsimilarthird-ordercontributionstothetransformedHamiltonian,bothofwhichleadtoidenticalfourthorderenergycontributions.TheisfoundinIM-SRG(2)calculations,whilethesecond118entersviaanintermediatethree-bodyforcethatonewouldnaivelyneedtheIM-SRG(3)toinclude.AsimilarattemptatmockinguptwicethetotalofdiagramsfoundinFigure5.1wasmaderecentlybyEvangelistainRef.[134],simplybydoublingthepre-factorofthederivativetowingone-bodyHamiltonianinEq.(3.2).WhilethisbroughtresultsintogoodagreementwithCCSDatequilibrium,andstabilizedconvergenceforsomebond-lengths,itobviouslydistortsthesingle-particlespectra.Thisad-hocdistortionhassomeundesirableconsequences,particularlyiftheresultingtransformedHamiltonianistobeusedforsubsequentapplications(e.g.,thecomputationofexcitedstatesoravalenceshellmodelHamiltonian).Inthepresentwork,wedescribeanalternativeinwhich,ratherthandoubling\byhand"theprefactorofthesingle-particlespectrawequation,westorethechangetothesingleparticleHamiltonianseparatelyfromthetransformedHamiltonianitself.Thiscanbeeasilyaccomplishedbyintroducinganauxiliaryone-bodyoperator˜withthefollowingwequation,d˜qrds=12Xstu(nqnr+nqnr)(nsntnu+nsntnu)(1+Pqr)uqststur(5.1)subjecttotheboundaryconditions˜qr(0)=0:(5.2)ItisthenpossibletoamendtheowequationsinthefollowingwaydH(s)ds=[(s);H(s)+˜(s)]:(5.3)119Thisallowsforadirectsolutionof(3.2)thatcorrectlyaccountsforall(non-triples)fourthorderenergycontent.Wenotenow,thatalthoughtheinclusionof˜correctstheasymmetricfourthorderquadruplesdiagramscorrectly,itisnottechnicallyincorrectinthewayitcor-rectsthewingtwo-bodyvertex.Thiscanbeseenbycomparingtheresultingcontributionstothethird-ordervertexinFig.5.1andFig.5.2,sinceonehastriplecontractionsbetweenordergenerators,andonedoesnot.WewillrevisitwhetherthisisanadvantageornotinSec.5.3.Despitethisslightdiscrepancy,thisinclusionstabilizesthewawayfromequilibriumgeometriescomparedtothenaivere-weightingcorrectionin[134].Sincethisisnotatruethree-bodyIM-SRG,andstillscalesinthesamewayasIM-SRG(2),werefertoitasIM-SRG(2*).Unfortunately,wehavenotyetdiscernedawaytokeeptrackoftheinducedthree-bodyforcerequiredtoaccountforatriplestypecalculationinthedirectsolutionof(3.2).Fortunatelythough,themainthrustofmythesisworkhasbeenthedevelopmentoftheMagnusformulationoftheIM-SRG,whichamoretransparentpathtowardsapproximatingtheneglectedinducedthree-bodyinteractions.Forthisreason,wewillnotactuallypresentanyIM-SRG(2*)resultsinthiswork.5.3TheMAGNUS(2*)ApproximationTheMAGNUS(2)methodhasthesameundercountingofthefourth-orderquadruple-excitationdiagramsmentionedabove,andthereforeexhibitsthesameoverbindingandlackofstabilityforstretchedbonds.WenowpresenttheanalogousMAGNUS(2*)methodthatcorrectsfortheseundercountedterms.Aswewillsee,itismorenaturalintheMagnusformulationtoapproximatelycorrectfortermsthataremissingorundercountedintheMAGNUS(2)leveloftruncation.120Figure5.1:ThedarkcircleandsquarerepresentthebareHamiltonianandgeneratorrespec-tively.Thelightcirclesinthecolumnrepresentthesecondorder1-bodyHamiltonianoriginatingfrom[;H]1B.Ifthesecondorder1-bodyvertexisexpandedintermsofbarequantities,the4asymmetricGoldstonediagramsontherightaretheresult.WestartbyrecallingthattheMagnusformulationcentersaroundtheapplicationofanexponentviatheBaker-CambexpansioninEq.(4.8).IntheMAGNUS(2)approximation,everyinternalcommutatoristruncatedattheNO2Blevel.InthediagramsofFigs.5.1and5.2,wecanreplacethegeneratorwithwhichshowsthatthedia-gramsfrominternalone-andthree-bodytermsarisefromBCHterms;;H]1B]2Band;;H]3B]2Brespectively.WehaveattemptedtorestorethediagramsinFig.5.2astheyare,butitappearsthattheeone-bodyoperatorarisingfromthetriplecontractionofwithitself,whichappearsinthesecondcolumnofexpandeddiagrams,generallycausesproblemsatstretchedgeometriesofmolecules.Incontrast,fornuclearsystemsweincludethemastheyare,andseenoproblemswithconvergence,whichwasneveraproblemforthesesystemsevenintheIM-SRG(2)/MAGNUS(2)calculations.InkeepingwiththesamephilosophyastheIM-SRG(2*)method,wemodifytheex-121Figure5.2:ThedarkcircleandsquarerepresentthebareHamiltonianandgeneratorrespec-tively.Thelightcirclesinthecolumnrepresentthesecondorder3-bodyHamiltonianoriginatingfrom[;H]3B.Ifthesecondorderinduced3-bodyvertexisexpandedintermsofbarequantities,the4asymmetricGoldstonediagramsontherightaretheresult.pressionforeachadjointoperatorintheBCHformulawithaone-bodyoperator~˜kinthefollowingway,H(s)=eHe=1Xk=01k!fadk(H)(5.4)fadk(H)=adk(H)fork=0,1,(5.5)fadk(H)=;fadk1(H)+~˜k1]fork>1;(5.6)wheretheone-bodyoperator~˜kisas~˜kqr=12Pstu(nqnr+nqnr)(nsntnu+nsntnu)(1+PqruqstXk1stur(5.7)(5.8)122andXk=fadk(H)2B.AlthoughthiscorrectedBCHexpressionwasmotivatedbypertur-bativeconsiderations,itappearsthatthisgeneraltopologyofterms,i.e,thediagramsofFigs.5.1and5.2,butwiththefullwingvertices(whichthemselvesarenon-perturbativeresummations),areveryimportant.Toclarify,everypieceofthisdiagraminvolvesdiagonal"componentsoftheHamiltonian,andfeedsdirectlybackintothewofthesamevertices.Thisseemstopromoteitsimportancebeyondanaiveperturbativecountingofterms.Forinstance,thethirdsuchmonestedcommutator,whichcon-tributestoenergyatorder,changesanswersby.1-.5mHforthemoleculeswepresentresultsfor.Therefore,wekeepallsuchnestedcommutatorsinourcalculations.Thismightraiseconcernsthatotherneglectedtopologiesarisingfrom;;H]3B]1B;2Barenotneg-ligible.Tocheckthis,Ihaveimplementedalltermsarisingfromexpressionsofinternalthree-bodycommutatorsthatimmediatelyaretriplycontractedwiththetwo-bodyastheycanallbefactorizedintoanN6evaluations.However,thesetermstypicallytheanswer(atequilibrium)byanegligibleamountofaround.1mHorless.Wearenotclaimingthathigher-bodytermsinfurthernestedcommutatorsarealsonegligible,butgen-eratingthemwillcausethemethodtoscalethesameasthefullIM-SRG(3),andthereforetheyarenotconsideredinthiswork.Thisquestionwillbeexploredmoreinfuturework,atleastfortlysmallsystemssothatthefullcalculationscanbecarriedout,toverifythattheyarenotsizeable.ThismoedtruncationoftheBCHcommutatorsprovidesarobustconvergencepattern,whichagreeswithCCSDresultsatequilibriumgeometriesforeverymoleculeandbasissetwehaveinvestigatedinthiswork.Asbondsarestretched,theresultsfromMAGNUS(2*)begintounderbindwithrespecttoCCSDresults.However,thisunderbindingmayprovetobeagoodthing,giventhatthree-body(whichshouldbeattractive)havenotyet123beenfullyaccountedforintheabovemethod.Weshouldalsomentionthathere,thereisnothingthatismoreorlessvalidwhenapplyingthiscorrectedBCHexpressionforanyobservablewearedealingwith.Inotherwords,itshouldbegenerallyapplicableforanyoperatordominatedbyitsNO2Bandlowercomponents.Inspirit,thisapproximationcapturesmuchmorethanthethree-bodyMukherjeedecom-positionfoundincanonicaltransformationtheory[79].Theisthatthere,theyaredecouplingan\activespace",andthattheirtensordecompositionintroducesastatespeci-ytothetransformation.Itwouldbeinterestingtogobackanduseourapproximationpresentedinacompleteactivespacecalculationtoseehowitcompares.Fornow,thatisbeyondthescopeofthiswork.5.4ApproximationstoMAGNUS(3)InChapter2wereviewedthemainnon-iterativeCCSDplussomenon-iterativeapproximateinclusionofthree-bodyclusteramplitudes.Wepresentherethevariouswaysthatsimilarapproximationscanbeincludedourframework.Havingjustsummarizedhowtheofcertainthree-bodyoperatorsthatareinternaltotheBCHcanbeincluded,wenowturntoapproximatetheleadingofincludingallNO3Baswell,termedtheMAGNUS(3).LikeCCmethodspresentedearlier,weareonlyinterestedinnon-iterativemethodsthatarejustslightlymorecostlythantheMAGNUS(2*)calculationtheyarebasedupon.AllofourmethodsarereminiscentofCCtheory,butgiventhattheyareframedintermsofatransformedHamiltonianthatisHermitian,theyaremoreintuitive.Fortheremainderofthissection,anynon-subscriptedistheconvergedoperatorofaMAGNUS(2*)124decoupling.WedenoteH=H(1)=eH(0)einordertosimplifyequations.IfwewanttoincorporatephysicsbeyondtheNO2BHamiltonian,thenweneedtoallowforathree-bodyinteractionWinthefollowingway,HˇfE+f+gMAGNUS(2)+W:(5.9)ThecontentsofthebracketscontaintheHamiltonianwegenerateinourfullyconvergedMAGNUS(2*)transformation,andassuch,itiscompletely\diagonal."TheWpartoftheinteraction,ontheotherhand,canconnectourreferencetotriplyexcitedSlaterdeterminants.Fornow,wewillleavetheformofWunspGiventhatitispossibletogenerateit,themoststraightforwardwaytoimmediatelygiveanenergycorrectionduetoWisjustsecondorderperturbationtheory.WehavealreadygiventheformofthisinEq.(2.26),butforthereader'seasewereproduceithere.TheenergycorrectionE[3]canbewrittenasE[3]=13!2XijkabcWijkabcWabcijkijkabc=13!2Xijkabcijkabcabcijkabcijk:(5.10)AlthoughthisenergycorrectionwasmotivatedbyperturbationtheorywiththetransformedHamiltonian,thereisanalternatederivationinwhichitarisesinanotherway.WealsotakeamomenttonotethatthedenominatorijkabccanbemadefromthebareortransformedHamiltonian,andcanbechosenfreelytobeoller-PlessetorEpstein-Nesbetttype.Ifweintroduceasecondtransformatione,withelementsofthefollowingform,abcijk=Wabcijkabcijk;(5.11)125thenwecantransformtheMAGNUS(2*)HamiltonianasH=eHe:(5.12)Ifweassumethatthissecondtransformation'sBCHexpansionfollowsthesamepatternofmonotonicallydecreasingtermsasshowninFig.4.3,thenthebruntoftheenergycorrectionwillcomefromthefewterms.Isolatingthefewterms,wearriveatE[3]=[;H]0B+12[;[;H]]0B:(5.13)HerewewillhavetorestricttheHamiltonianHappearinginthesecondcommutatortoonlyhavethediagonalformusedtoelsethenumericalscalingtoestablishE[3]willrisefromitscheapestn3on4utoatleastn3on6uorhigherdependingonhowmuchofHisincludedinthiscommutator.Butkeepingtheprescriptionaspresented,thetwoformulasyieldidenticalresultsasdesigned.ThereasonfordemonstratingthesameenergycorrectionarisesfromBCHformalism,isthatitallowsforthestraightforwardinclusionofobservablesatthesamelevelofaccuracy,andinfutureworkwithotherapplicationsoftheMAGNUS(2)tovalencespaceandmultireferencemethods,wewillwillnotneedtoappealtoamorecomplicatedperturbationtheorytoproceed.WenowturntohowweapproximateW,anditisherethatwebegintoreallydrawanalogieswiththeCCmethodspresentedearlier.Ifwerecallhowthetriplesenergycor-rectiontoCCSD[T]wasestablished,T3wasapproximatedaslinearintheconvergedT2amplitudes,withthebareresolvent.Wecanformasimilarapproximation,W=;H]3B,andalsousingonlythebareHFenergiesinthedenominatortoestablishInordertodo126thisweneedonlyelementsoftheformWabcijk=;H]abcijk.;H]abcijk=(1PabPbc)(1PijPik)XlabillcjkXdabiddcjk$:(5.14)Notsurprisingly,thisisthetermthatdictatesthen3on4uscalingalreadymentioned,andisexactlythesameexpressionthatappearsincoupledclustertheoryinEq.(2.37)andFig.2.3.Onceestablished,E[3]correctstheEMAGNUS(2)energytofourthorderinMBPTwithregardstotheoriginalreference.Itisimportanttonoticethatthisenergyonlygoesbeyondfourthorderviatheuseofaniordertwo-bodyThiswillbethecrudestapproximationwecanmakeinthisformalism,andwedenoteitMAGNUS(2*)[3]-A,withenergycorrectionE[3]A.Fornextlevelofapproximation,insteadofusingtheoriginalbareHamiltonianforusageinourdenominators,wecanusethefullytransformedone-bodyHamiltonianresultingfromaMAGNUS(2*)calculation.ThustheresolventisupdatedbythefactthatoursingleSlaterdeterminantisamuchbetterapproximationaftertheMAGNUS(2*)transformation.ThistypeofcorrectionwillbedenotedbyMAGNUS(2*)[3]-Btomaketheconnectiontorenormalizedcoupledclustertheory.Wecangoonestepfurtherbynotlimitingourselvestoinducedthree-bodyinteractionsthatareonlylinearinThisisaccomplishedbyusingan\internalBCH"toestablishthetwo-bodypiecethatiscontractedwithtoformWC.ThisismotivatedbyreturningtothematrixadjointexpressionofHfromEq.5.4,whichonlybeginsatk=1foraninducedthree-bodyforceW=P1k=11k!~adk(H)3B=;H]3B+12;;H]]3B+::::(5.15)127Ifwewanttoapproximatethisasfullyaspossiblewithoutincreasingthesinglen3on4uitera-tion,itiseasytoformafullyrenormalizedinternalvertexbyrewritingW=;H+12;H]+:::]3B=;Xk=11(k+1)!~adk(H)2B]3B=;~H]3B:(5.16)ThismakesitpossibletouseEq.5.14,justbyreplacingHwith~H.Wesuggestthatthisre-placementmakesconnectsphilosophicallywiththecompletelyrenormalizedcoupledclusterCR-CC(2,3),aseverythingthatgoesintothetriplesmatrixelementsisconsistentlytrans-formed.WecallthemethodarisingfromthesechoicesMAGNUS(2*)[3]-C.Alltopologiesoftermsfoundinthethree-bodyMofCR-CC(2,3)canalsobefoundinWC,althoughad-mittedlysomeofthemwillbeundercounted.Thiscanbeseenasidenticaltermscanariseidenticallyfrom;;H]3B]3B,whichwillbeomittedinourtruncationscheme.Theseshouldbefurtherinvestigatedtoseehowlargetheyare.IthasbeenfoundthatusingEpstein-NesbetttypedenominatorsintheanalogousdenominatorsofCR-CC(2,3)leadtoresultsthattrackmuchmorecloselytofullCCSD[T][8].Wewillalsoexploreanotherscheme,wherethedenominatorfromMAGNUS(2*)[3]-Cismadetoincludediagonalele-mentsfromthefullytransformedtwo-bodyHamiltonianresultingfromaMAGNUS(2*).WewilldenotethisasMAGNUS(2*)[3]-D.5.5ApplicationsDespitethefactthatthesecorrectionshavebeenmotivatedbythefailureoftheIM-SRG(2)methodsinchemistrysystems,itisthatwestartelsewhere,asthestoryofthesuccessofthesemethodsinchemistrysystemsisstillnuanced.Wewillinvestigate128MethodWijkabcMAGNUS(2*)[3]-A;H]3BhabcijkjfjabcijkiMAGNUS(2*)[3]-B;H]3BhabcijkjfjabcijkiMAGNUS(2*)[3]-C;~H]3BhabcijkjfjabcijkiMAGNUS(2*)[3]-D;~H]3Bhabcijkjf+jabcijkiTable5.1:ApproximationsmadeinthevariousMAGNUS(2*)[3]variants.theabilityofthecorrectionspresentedabovetodramaticallyimprovecalculationsoftheelectrongas.Thenwewillpresentsomesimple,butrealisticnuclearcalculationsfor4Heand16O,thatshowgenerallyexpectedresults.Finallywewillshowresultsforahandfulofchemistryresults,withvaryinglevelofsuccess.WewillthenusesomeofthefailuresinchemistrysystemstohighlightanothertoolthattheMAGNUS(2*)classofmethodscanaddtotreatingsystemswhereHartree-Fockisnotanadequatestartingpoint.5.5.1ElectronGasResultsInChapter4,weexaminedtheabilityoftheMagnusformulationtoreproduceIM-SRG(2)calculationsforboththeelectrongasandnuclei.Inure5.3,webenchmarkresultsofMAGNUS(2)toquasi-exactcalculationsatavarietyofdensitiesandbasissetsizes.WhencomparedtoCCDcalculations,MAGNUS(2)(andIM-SRG(2))resultsarealwaysbetweentheexactFCIQMCandCCDcalculations.Asimilarpatternisfoundinthevastmajorityofnuclearcalculations,wherethenaiveIM-SRG(2)andMAGNUS(2)resultsfallbetweenCCSDandCCSD(T)calculations,seethediscussioninChapter3.WhenwenowapplytheMAGNUS(2*)totheelectrongas,weanswersthatarevirtuallyindistinguishablefrom129Figure5.3:Groundstatecalculationsfor14electronstoaboxatdensitiesofrs=a0of.5,1,2,and5andperformedat3basissetsizesof114,186,and358withvariousmethods.AlthoughCCSDresultsarenotplotted,correctingtheMAGNUS(2)commutatorexpressionsasshowninEq.(5.4)makesMAGNUS(2*)indistinguishablefromCCSDonthesescales.Further,thetriplescorrectionduetotheMAGNUS(2*)[3]-DbindstheresultbackdowntoagreeverywellwithFCIQMCresultsfromRef.[3],whilethebaredenominatorsfoundinvariantAofMAGNUS(2*)[3]overbinddramatically.CCD.ThisindicatesthattheapparenthigherqualityoftheIM-SRG(2)/MAGNUS(2)results(relativetotheanalogousCCDandCCSDresults)isgenerallyaresultofcancellationoftwoclassesoferrors,onerepulsiveomissionthatisbyincludingtheMAGNUS(2*)termsofEq.(5.4),andoneattractiveommisionthatiswiththeinclusionofapproximatetriples.WenowturntothetvariantsofMAGNUS(2*)[3],andasavisualguidetowhatapproximationisbeingmade,wereferthereadertoTable5.1.VariantsAandDoftheMAGNUS(2*)[3]correctionareplottedinFig.5.3,theothersarenotshownforclarity.ThesuccessoftheMAGNUS(2*)[3]approximationsisfairlyastonishing,evengiventhesim-plicityofthissystem.ThemostnaiveapproximationE[3]Acausesslightoverbindingatars=a0=:5,tofairlydramaticoverbindingatrs=a0=5.Thisisnotsurprisingasthesystemisexceedinglynon-perturbativeatrs=a0=5.AtthisdensityeventheplanewaveHFenergy130Methodrs=a0=:5rs=a0=1:0rs=a0=2:0rs=a0=5:0Hartree-Focka58.592713.60362.8786.2099FCIQMCa-.5169-0.4611-0.3842-0.2645CCDb0.99060.97140.93130.8445CCDTb1.00071.00321.0152MAGNUS(2)0.99720.99120.98020.9617MAGNUS(2*)0.99050.97140.93110.8434MAGNUS(2*)[3]-A1.00111.00561.02631.1234MAGNUS(2*)[3]-B1.00101.00431.01621.0401MAGNUS(2*)[3]-C1.00061.00211.00811.0151MAGNUS(2*)[3]-D1.00051.00161.00611.0107Table5.2:aFCIQMCresultsfromRef.[3].bCCDandCCDTresultsfromRef.[4].Groundstateof14electronscalculatedinabasissetofM=114planewaveswithvariousap-proximations.TheFullgurationQuantumMonteCarlocorrelationenergyisreportedinHartree.Allotherenergiesarereportedasafractionofthecorrelationenergyrecoveredwithrespecttoquasi-exactFCIQMCresults.isbound,soitissurprisingthatweachievesatisfactoryresultsatthatdensity.ThefourthorderMBPTtripleexcitationsforrs=a0=5produceresultsthatareoverboundbyalittleunderafullHartree.Thetriplesenergygapapproximatelydoubleswhenusingtransformedinsteadofuntransformedenergydenominators,thatisijkabcˇijkabc.ThisbofusingtransformeddenominatorscanbeseeninthedrasticimprovementfromAtoBinTable5.2.Theinclusionoftheinternallytransformed~HmakesanotherlargeimprovmentfromBtoC.AndnallyCtoD,wherethetransformedoller-PlessetdenominatorsarereplacedwithEpstein-Nesbettype,createsasmall,butmeasurablethatmakesE[3]DshowsverygoodagreementwithFCIQMCresults.Arecentpublicationproducedfulltriplesresultsforthesmallestofthesebasissetsizes,andthecomparisonofallmethodstotheFCIQMCareshowninTable5.2[4].MostinterestingisthatmethodsCandDoutperformevenfullCCDTresultsforthissystem.Thereisnoreasontoexpectthatthispatternwould131holdforallbasissetsizes.Unfortunately,thisclearandsystematicimprovementthatisseenintheperiodicelectrongasismuchmorecomplicatedtoseeinmorerealisticsystemslikenucleiandmolecules.5.5.2NuclearResultsFornuclearsystems,wheretheIM-SRG(2)methodhasfoundgreatsuccessintreatingmediummassnuclei,therehavebeenfewerproblemswithconvergence,buttheopenquestionofwhyIM-SRG(2)methods,whichscaleasCCSD,generallytrackstyperesults.Fortheresultsweareaboutpresent,weuseaNN-onlychiralN3LOchiralinteractionbyEntemandMachleidt[1,2]softenedto=2:0fm1.Figure5.4showsthesameconclusionthatwasdrawnfromtheelectrongas,thatis,itappearsthatwhencorrectedMAGNUS(2*)resultsagreeverycloselywithCCSD.For4He,MAGNUS(2*)comesfrombeingoverboundwithrespecttoresultstobeinginverycloseagreementwithCCSD.WhentheE[3]Ccorrectionisaddedtois,itisinverycloseagreementwiththeresults.Thiscanbeseenagainsimilarlyin16Ocalculations,showninFig.5.5.ThishasbeencodedupbyNathanParzuchowskiinhislargescalesphericalnuclearcode,andhehasperformedthesphericalcouplingoftheWoperatorssothatitcanbescaleduptotreatheaviernuclei.Thedetailsofthisprocess,whichisexactlythesameasrecentcalculationscanbefoundintheappendixof[64].Itwillbefruitfultoexplorethesenewapproximations,andinvestigatehowtheybehaveandcanbeextendedinthepresenceofafullresidualthree-bodyforce,inthesamespiritasrecentworkswhichanalyzemethodswithchiralthree-nucleonpotentials.132Figure5.4:MAGNUS(2*)results,withMAGNUS(2*)[3]-Cforthelargestbasissetfor4HewiththechiralN3LOchiralinteractionbyEntemandMachleidt[1,2]softenedto=2:0fm1.WenoticethatcorrectingthecommutatorasshowninEq.(5.4)providesrepulsionthatbringsMAGNUS(2*)uptoCCSD.Further,thetriplescorrectionduetotheMAGNUS(2*)[3]-CbindstheresultbackdowntoagreewithCCresultsfromRef.[18]Figure5.5:MAGNUS(2*)results,withMAGNUS(2*)[3]-Cforthelargestbasissetfor16OwiththechiralN3LOchiralinteractionbyEntemandMachleidt[1,2]softenedto=2:0fm1.1335.5.3H2OResultsWeturntochemistrysystems,wheretheoftheapproximationsismoretogauge.Wehavespentmuchofthisworkexplainingin3.5.7ofChapter3howIM-SRG(2)ismissingdiagramsthatCCSDkeeps,andthendiscussinghowthisledtothecatastrophicfailureoftheIM-SRG(2)inchemistrysystems.InTable5.3,weplottheresultsofseveralCCvsourcalculatedmagnusresultsforH2Oinacc-pVDZbasisset[135],atequilibriumbond(Re=1.84345bohr)andHOHangleat110.6forseveralsymmetricalstretchingsofthemolecule.Althoughthisisnotaneasysystemtotreatsinceitinvolvesdoublebondbreaking,wepresentittoaddresstheissueofconvergence.Theandmostimportantisthatinthistypicaldoublebondbreakingtestsystem,whereWhite'sseminalworkontheIM-SRG(2)failed,theMAGNUS(2*)calculationsarerobustlyconvergent[78]exceptatanO-Hbondlengthof3Re,whichwewillshowin6.1isanartifactofMAGNUS(2*)sensitivitytoreferencestates.WeseethatastheO-Hbondisstretched,MAGNUS(2*)isgenerallyunderboundwithrespecttoCCSD.Thismayactuallybebasitthelargetriplescorrectionthatwillbeaddedtoit,togivereasonableresults.WeseethatMAGNUS(2*)-A,B,CgiveresultsthatareunderboundwithrespecttoFCIbyafewmH,whiletheDvariantremainsquitecloseeveninthefaceoffailingCCSDTresults.Thisisaveryinterestingding,aswewillseeintreatingHydrogenFlouride,thattheMAGNUS(2*)[3]methodsfailwhileCCSDTremainswithinchemicalaccuracy.134MethodRe1:50Re2Re2:5Re3:0ReFullCIa-76.241860-76.072348-75.951665-75.917991-75.911946CCSDb3.74410.04322.03220.30710.849CCSDTa0.4931.423-1.405-24.752-40.126CCSD(T)b0.6581.631-3.820-42.564-90.512CCSD(2)Tc0.9062.8253.805-15.830-33.035CR-CC(2,3)d0.3441.142-0.551-23.100-40.556MAGNUS(2)-0.897N.C.N.C.N.C.N.C.MAGNUS(2*)3.79710.38425.16236.554N.C.MAGNUS(2*)-A0.6882.1954.131-1.646MAGNUS(2*)-B0.7942.7356.7254.865MAGNUS(2*)-C0.9823.3428.9528.825MAGNUS(2*)-D0.2990.9941.531-4.497Table5.3:aFromRef.[5].bCCSDandCCSD(T)resultsobtainedwithPSI4[6].cFromRef.[7]dFromRef.[8]AcomparisonofvariousCCground-stateenergiesobtainedforthecc-pVDZH2OmoleculeattheequilibriumOHbondlengthRe=1.84345bohrandseveralnonequilibriumgeometriesobtainedbystretchingtheOHbonds,whilekeepingtheHOHangleat110.6.Thesphericalcomponentsofthedorbitalswereused.Inpost-RHFcalculations,allelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.TheremainingenergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergies.5.5.4NeonResultsWenowturntoaatomicsystemoftheclosedshellsystemNeinacc-pVDZbasisset[135].ThisisjusttoshowthattheextremelyaccuratereproductionofFCIfoundinthehomogenouselectrongasofTable5.2wasnotaccidentalgiventhattherearenotpressingissuesofreferencestatedependenceinthissystem,orextremelytocapturestaticcorrelation.Weobservethatagain,MAGNUS(2*)reproducesCCSDquitewell,withtheapproximateMAGNUS(3)methodsperformingaswellorbetterthanfullCCSDTwithrespecttofullCI.Itshouldbeobserved,thatinthepublicationthattheCCresultsweretakenfrom,theunitaryvariantsofCChadsimilarperformanceinthecc-pVDZbasisset,135MethodFullCIaCCSDaCCSDTaCCSD(T)aEnergy-128.6790250.99350.99920.9990MethodMAG(2*)MAG(2*)[3]-AMAG(2*)[3]-BMAG(2*)[3]-CMAG(2*)[3]-DEnergy0.99360.99960.99940.99891.0004Table5.4:aFromRef.[9].AcomparisonofCCandMagnusIM-SRGground-stateenergiesobtainedforaNeonatominacc-pVDZbasisset.Inthesepost-HFcalculations,the1sorbitalwasfrozen.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedasafractionofthecorrelationenergyrecoveredrelativetoE=EHF-EFCI.andthenrecoveredslightlylessenergythanfullCCSDTinthelargercc-pVTZbasisset[9].5.5.5C2ResultsImusthighlightnowthatmyimplementationoftheIM-SRGandMAGNUSequationswasimplementedinsideaplug-inforthechemistrysuitePSI4[6],butcertainlynotataproductionlevel.Giventhattheunderstandingofhowpoint-groupsymmetrieswasnotunderstoodwhenwritingtheplug-in,thatsymmetryhasnotbeenexploited.Thatmakestypicalcalculationsforre,!e,andotherenergydependentquantitieslikethatinfeasibleforevenmodestbasissets.Itisagoalofthewriterstorewriteaproductionlevelplug-inforPSI4.ThisintroductionwasneededtoexplainwhyonlytheenergiesarebeingcomparedhereforC2,asystemwechoosetotreatbecauseithaslargemnalcontentevenatequilibrium[9,136].Inthissystem,weobservethatagain,MAGNUS(2*)reproducesCCSDquitewell,withtheapproximateMAGNUS(3)methodsperformingsimilarlytoCCmethods.136MethodFullCIaCCSDaCCSDTaCCSD(T)aEnergy75.72985329.9573.3712.042MethodMAG(2*)MAG(2*)[3]-AMAG(2*)[3]-BMAG(2*)[3]-CMAG(2*)[3]-DEnergy31.2343.6716.6349.420-2.448Table5.5:aFromRef.[9].AcomparisonofCCandMagnusIM-SRGground-stateenergiesobtainedforC2attheequilibriumFCIbondlengthofre=1.27273Ainacc-pVDZbasisset,takenfromRef.[9].Inthesepost-HFcalculations,the1sorbitalswasfrozenontheCatoms.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedcorrelationenergyrecoveredrelativetoE=EHF-EFCI.5.5.6HFResultsInTable5.6,wedemonstratetheoutcomeswhenHFinaDZPbasis[137]istreatedwithMAGNUS(2*)methods.Wegenerallymuchworseresultsthanwefoundinanyoftheprevioussystems.WithregardstofullCCmethods,CCSDTprovideschemicalac-curacyacrossthewholepotentialenergysurface,andCR-CC(2,3)approximatesthisverywell.CCSD(T)failsbadly,TheonlyplacewheresatisfactoryresultscomparedtoeitherCCSD(T),CR-CC(2,3)isachievedisatequilibriumH-Fbondlength.Everywhereelse,wedramaticallyoverboundresults,andnon-systematicresultsevenfortheMAGNUS(2*)basemethod,whereitbecomesveryclosetofullCIresultsat3Reandthenunboundat5Re.Wesuggestthatthis,likethefailuretoconvergeatlargebondlengthsforH2O,isaresultsofreferencestatedependenceandthefactthatHartree-Fockreferencesbecomemuchlessaccuratestartingpointsduringbondbreaking.137MethodRe2Re3Re5ReFullCIa-100.160300-100.021733-99.985281-99.983293CCSDb1.6346.04711.59612.291CCSDTa1.00071.00321.01520.431CCSD(T)b0.3250.038-24.480-53.183CCSD(2)Tc0.2291.452.1771.443CR-CC(2,3)c-0.1190.062-0.096-1.005MAGNUS(2)-0.897N.C.N.CN.C.MAGNUS(2*)1.5814.4950.98814.249MAGNUS(2*)[3]-A-0.115-3.170-29.319-24.377MAGNUS(2*)[3]-B-0.056-2.167-20.846-10.924MAGNUS(2*)[3]-C0.121-0.723-12.437-3.252MAGNUS(2*)[3]-D-0.324-4.080-28.661-31.390Table5.6:aFromRef.[10].bCCSDandCCSD(T)resultsfrom[6].cFromRef.[8]AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedfortheequilibriumgeometryofRe=1.7328bohrandothernuclearseparationsofHFwithaDZbasisset.Inthesepost-HFcalculationsallelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.TheremainingenergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergyvalues.5.6SummaryWethatfortheelectrongas,nuclearsystems,andtheverysimplechemistrysystems,thattheMAGNUS(2*)[3]methodsdoaverygoodjobofreproducingfullCIresultswhereavailable,andperformasexpectedwheretheyarenot.ThisrepresentsahugestepforwardfortheabilityoftheIM-SRGtodealwiththesesystemscheaplyathigheraccuracy,andtogetahandleontheexpectedcontributionsfromhigherordermethods.FormorecomplicatedchemistrysystemslikeH2OandHF,thereareotheropenquestions.WehaveshownthatthetruncatedIM-SRG(2)methodisingeneralsensitivetothequalityofstartingreferencestates.Itisbelievablethatthefailureofthemethodinthesesystemsisduetothis,inthenextsection,wewillexplorethispossibility.138Chapter6OtherWorkThemajorityofthisthesishasbeenfocusedonestablishingtheMagnusformulationoftheIM-SRGequationsasanewaytomoveforwardintreatingtheshortcomingsofthetraditionalsolutionbothinthegenerationofobservables,computationalandtheabil-itytoapproximatetheinclusionofthree-bodyforces.Wehaveshownthattheexponentialformalismthatunderpinsthenewformulationthepossibilityoffurtheradvances.Herewewillpresentafewareaswhereworkiscurrentandpromising,butfullconclusionsarenotyetreadytobedrawn.TheofthesewillbeinformedagainbycoupledclustermethodsbasedonthebestsinglereferencebyapproximatingBruecknerorbitals;thisissome-thingthatisevenmorenaturalinthecontextofMagnusIM-SRGaswewilldemonstrate.Wewillalsotaketheopportunityheretopresentthepossibilityofgeneralizingourapprox-imatethree-bodyinclusionmethodsinthecontextofexcitedstatemethodsbeingpursuedbyotherpractitionersoftheIM-SRGmethod.Finally,wewillpresenthowtheIM-SRGcanalsobeusedtomotivatemwavefunctionmethodsinthespiritofCIPSI.6.1ImprovingtheReferenceInChapter2and3,weshowedhowareferenceplaysacrucialroleasthestartingpointoftheIM-SRG,CC,andMBPTtypemethods.FortheIM-SRG,wehaveshownthatresultsaresensitivetothechoiceofreference.WehavealsoshownwhyCCmethodsaregenerally139notsensitivetostartingreference.Thequalityofapproximatetriples,eveninCCtheory,generallydependsonthequalityofthereference.Thuschemistryliteratureisfullofattemptstochoosethebestreference;oneofthemostinterestingisbyusingaSlaterdeterminantcomposedofnatural,orBruecknerOrbitals.ThisBruecknerreferencejBRicanbeasmeetingoneoftwoequivalentcriteria.TheisthattheoverlapoftheBruecknerreferencewiththetruegroundstateismaximized,thatis,hj0iislargestwhenji=jBRi[16].AsecondcommonlystatedconditionforthisstateisthatthefullCCgroundstatebuiltonaBruecknerreferencehasthefollowingproperty,jCCi=eTA+:::+T2+T1jBRi=eTA+:::+T2jBRi;(6.1)orthatT1vanishesinthefullyconvergedsolution[16].Torestateonemoretime,thismeanstheHamiltonianneedsnosingleparticlechangeofbasisandthegroundstatej0icontainsnosingleparticleexcitationswhenexpressedinBruecknerorbitals.Thisidealsetoforbitalsarefairlyeasytograspphilosophically,butareoftenasexpensivetopursueasafullycorrelatedsolutionitself.Althoughthereareseveralwaystoaccomplishthis,ithasbeenpursuedindepthwithinCCmethods[16,138,139].Theonewewillpresenthereistheonethatwillbringinsighttoourmethod,andtheonethatisimplementedinthePSI4[6]softwaresuiteunderthenameBCCDorBruecknerCCD.Init,fullCCSDcalculationsareconducted,andthentheresultingT1amplitudesareusedtogenerateanewsetoforthonormalorbitals.ThesearethenusedtotransformtheHamiltonianintothisnewbasis.ACCSDcalculationiscarriedoutagain,andthisprocedureisiterateduntiltheT1amplitudesarevanishinglysmall.Inthisway,theyapproximatetheBruecknerreferencewithinapproximatedCCtheory.Onecanseethatthisisaveryexpensiveprocedure,andgenerallythesesocalled140BCCDenergies,andCCSDenergiesareveryclosefornormalsystemspreciselybecauseofThoulesstheorem(2.34).However,whentheofT3areapproximatelyincludedinsteadofexactlyincluded,thestartingreferencebecomestial.Thereisalsosomeevidencethatinthecaseofsymmetrybreaking,BCCDcalculationscanprovidesomelargebaswell[139].ThesamerequirementismoretrickytoobserveintheMagnusformulationoftheIM-SRG,sinceaswehavementionedseveraltimes,e2B1B6=e2Be1B,oranytrankoperatorsforthatmatter.ItishoweverevidentthatevenintheIM-SRGformalism,jIMSRGi=eAB:::2B1BjBRi=eAB:::2BjBRi;(6.2)sinceanywillstillcreate1p1hexcitationfromjBRiwhichbyitiondonotbelong.TheMagnusformulationhasshownthatwedonotneedtosolveatialequationperfectlyinordertoarriveatourdesireddecoupledHamiltonian.ItturnsoutthattheMAGNUSformalismstillworksiffreedentirelyfromthetialequation.IfweforcethetransformationtoinsteadtaketheformofjBMAGNUS(2)i=e1Be2Bji;(6.3)andwecanstillaccomplishdecouplinginourformalism,thenji!jBRi,atleastagainwithinourMAGNUS(2*)approximation,whichshouldbesimilartotheBCCD.WealsodispensewiththeadditionalmatrixadjointtermsinEq.(4.6)besidesjustoperatoritself,sincewearenolongertryingtofollowthetialequation.Ifwenowlookathowthis141ispracticallycarriedout,H(s)=e2e1He1e2=1Xk=01k!adk2(e1He1)(6.4)e1He1=1Xk=01k!adk1(H):(6.5)Soweapplythetransformationintwosteps,applyingtheone-body1B,andthenthe2BtotheresultingHamiltonian.Becausethereisnoerrorinapplyingaone-body1Bwithinourmethod,thisrepresentsaperfectchangeofbasiswithoutapproximation.Thisisincontrasttoapplyingboththeone-andtwo-bodypiecestogether,as1Bthenappearsinthree-bodyintermediatesthataretruncated.Inthiswayweestablishanewmethod,whichwenameBruecknerIM-SRG,orforthisworkanditsestablishedtruncationscheme,BMAG-NUS(2*).Further,allthesameapproximateMAGNUS(2*)[3]methodscanbecarriedoverwithnonecessarygeneralization,thuswewillalsopresenttheBMAGNUS(2*)[3]methodswheretheirinterpretationsareobviouswiththeexceptionofwherethebareHartree-FockenergieswereusedinvariantA.ForBMAGNUS(2*)-A,thediagonalenergiesfromtheone-bodydiagonalHamiltonianofe1He1areusedtomakethedenominators.6.2BruecknerIM-SRGResultsForthissection,wewillpresentmostofthechemistrysystemstreatedfromChapter5,butnowwiththeanalagousBMAGNUS(2*)[3]methods.Iwillnotbepresentinganythingfortheelectrongas,asmomentumconservationmakestheplanewavestheonlybasissetonecanworkwithwithoutdramaticallyincreasingthecomputationalIwillalsonotpresentanyBruecknerresultsfornuclearsystems.Thisismostlyaresultofnothavinganynuclear142BCCDresultstodrawcomparisonswith,butafutureworkcenteredonthisBMAGNUS(2*)methodsinnuclearsystemsshouldbeforthcomingverysoon.Alongsidethechemistryre-sults,IwillpresenttheBCCD(T)resultsfromPSI4.Forthesemethods,itisalsointerestingtooutputtheBCCDandBMAGNUS(2*)referenceenergiesinordertocomparethecharac-teroftheorbitalsproducedfromthetwoapproximations,andthatBMAGNUS(2*)isasimilarapproximationasBCCD.WewillpresenttheseBMAGNUS(2)refinmostofthesystems.Givenmoretimeandunderstanding,itwouldalsobebforunderstandingtoactuallyproducetheoverlapofthetworeferences,butcoaxingPSI4tooutputitsBCCDorbitalsprovedbeyondourabilityinthetimeavailable.Webeginagainwiththeresultsfromtheverysimplesystemofacc-pVDZneonsystematequilibrium.Eventhoughthisaverysimplesystem,wecanbegintocharacterizesomeboftheBruecknerprocedure.6.2.1BMAGNUS(2*)ResultsforNeonandC2WebegintoseeapatternhereforwhathappenswithbothBMAGNUS(2*)typeresults.Almostuniversally,theBruecknerresultsareunderboundversustheMagnusresultsbasedonHartree-Fockreferences.WeseethisisparticularlyhelpfulinbringingtheBMAG-NUS(2*)[D]method,themostcompletemethod,backtoagreementwithexactvalues.ThisappearstobeageneralBecauseofthiswewillchoosetoonlypresenttheBMAGNUS(2*)-Dmethodfortheseresults.Unlessotherwisementioned,theotherresultsarefurtherunderboundwithrespecttopresentedresults.Similarly,weseeanoverboundresultforMAGNUS(2*)[3]-DriseuptoBMAGNUS(2*)[3]-Dandbecomevery,veryclosetoFCIresultsforC2inTable6.2.Fromtheseresults,itisreasonabletobelievethatintheabsenceofstrongstaticcorrelation,BMAGNUS(2*)[3]-Dcollectsalargerfractionof143MethodFullCIaCCSDaCCSDTaCCSD(T)aEnergy-128.6790250.99350.99920.9990MethodMAG(2*)MAG(2*)[3]-AMAG(2*)[3]-BMAG(2*)[3]-CMAG(2*)[3]-DEnergy0.99360.99960.99940.99891.0004MethodBCCDrefBCCD(T)BMAG(2*)refBMAG(2*)[3]-DEnergy-0.00130.9991-0.00100.9998Table6.1:aFromRef.[9].AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedforaNeonatom.Inthesepost-HFcalculations,the1sorbitalwasfrozen.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedasafractionofthecorrelationenergyrecoveredrelativetoFCIinmH.MethodFullCIaCCSDaCCSDTaCCSD(T)aEnergy75.72985329.9573.3712.042MethodMAG(2*)MAG(2*)[3]-AMAG(2*)[3]-BMAG(2*)[3]-CMAG(2*)[3]-DEnergy31.2343.6716.6349.420-2.448MethodSCFBCCDrefBCCD(T)BMAG(2*)refBMAG(2*)[3]-DEnergy343.396364.1571.665357.7040.516Table6.2:aFromRef.[9].AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedforC2attheequilibriumFCIbondlengthofre=1.27273A.Inthesepost-HFcalculations,the1sorbitalswasfrozenontheCatoms.ThefullCItotalenergyisgiveninHartree.TheremainingenergiesarereportedinmillihartreerelativetothefullCIenergy.correlationenergyevencomparedtofullCCSDT.ItisimperativethoughtotaketimetowriteaproductionlevelcodeinwhichsymmetryisexploitedinordertobenchmarkagainstfullIMSRG(3)resultsinreasonablebasissets.6.2.2BMAGNUS(2*)ResultsforHFandH2OInthetreatmentofHFintheDZbasis,andH2O,weseethatwhereBCCDresultsareavailable,thatthereferenceenergyofBMAGNUS(2*)refandBCCDrefareverycloseto144eachother.Further,inbondbreakingofHF,whereMAGNUS(2*)[3]-Dresultsbecomedramaticallyoverboundasat3Reandabove,theBruecknerBMAGNUS(2*)[3]-DisstableandreasonablyclosetoFCIanswers.TheseresultsstillarenotnearlyaswellbehavedasCR-CC(2,3)forthissystem.ButifweinsteadlookatH2O,notonlydowestillgetconvergenceat3Re,butwecontinuetogetreasonableresultsevenwhileCCSDTisextremelyoverboundfordoublebondstretching.6.2.3BruecknerSummaryInconclusion,weseethatBMAGNUS(2*)[3]resultsarereasonable,andalmostalwaysoutperformCCSD(T)results,butmaynotbecompetitiveCR-CC(2,3)resultsforeverymolecule.Rathergenerally,theBruecknerresultsappeartobeproducingareferencethatissimilarinqualitytoBCCDreferences,whichismostinterestingasproductionofthesereferenceshasacomputationalcostverysimilartoasingleMAGNUS(2*)calculation,whiletheseBCCDcalculationsrequiredabout10-50fullCCSDiterationsbeforetheofT1aresmallenoughtobeneglected.EvenifoldT1andT2areusedasthestartingpointofeachiteration,thisresultsinaBCCDcalculationthatisonetheorderof10timestheofasingleCCSDcalculation.Further,sinceweforce2Btokeepthestructureof=TyTforBMAGNUS(2*)results,itcanbesettoscaleexactlyasCCSD,n2on4uoverthen6ofMAGNUS(2*)withageneral6.3ExtensionstoMR-IM-SRGTheIM-SRGformalismandapplicationspresentedsofaruseasingleSlaterdeterminantasthereferencestate.Innuclearphysics,theseapproachesareonlyappropriateforthe145MethodRe1:50Re2Re2:5Re3:0ReFullCIa-76.241860-76.072348-75.951665-75.917991-75.911946CCSDb3.74410.04322.03220.30710.849CCSDTa0.4931.423-1.405-24.752-40.126CCSD(T)b0.6581.631-3.820-42.564-90.512CCSD(2)Tc0.9062.8253.805-15.830-33.035CR-CC(2,3)d0.3441.142-0.551-23.100-40.556SCF217.834269.982363.967476.756573.585MAGNUS(2*)3.79710.38425.16236.554N.C.MAGNUS(2*)-D0.2990.9941.531-4.497N.C.BCCDref218.758276.328383.342515.316BCCD3.88710.66822.12616.988BCCD(T)0.6821.803-3.753-46.317BMAGNUS(2*)ref218.621275.693383.154519.876634.684BMAGNUS(2*)3.87410.82926.83642.46947.968BMAGNUS(2*)-D0.3851.6004.5973.615-3.583Table6.3:aFromRef.[5].bCCSDandCCSD(T)resultsobtainedwithPSI4[6].cFromRef.[7]dFromRef.[8]AcomparisonofvariousCCground-stateenergiesobtainedforthecc-pVDZH2OmoleculeattheequilibriumOHbondlengthRe=1.84345bohrandseveralnonequilibriumgeometriesobtainedbystretchingtheOHbonds,whilekeepingtheHOHangleat110.6.Thesphericalcomponentsofthedorbitalswereused.Inpost-RHFcalculations,allelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.TheremainingenergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergies.InRef.[10],theauthorsnoticedthattherearetwoSCFsolutions,oneofwhichpoorlydescribestheweakH-Obonding.Inthechemistrysuite,PSI4[6],wecouldnotforcetheCCSDroutine,andthustheBCCDroutinetousethecorrectSCFstartingreference.Thisiswhytherearenoresultsreportedfor3:0Redescriptionofnucleiaround(sub-)shellclosures.Inopen-shellnuclei,correlationscausetheemergenceofphenomenalikenuclearsupidityorintrinsicdeformation.Withreference-stateconstructions,onecanattempttocapturetheseattheleveltosomeextent,bybreakingsymmetrieseithersponta-neouslyorexplicitly.PairingcorrelationscanbetreatedintheHartree-Fock-Bogoliubov146MethodRe2Re3Re5ReFullCIa-100.160300-100.021733-99.985281-99.983293CCSDb1.6346.04711.59612.291CCSDTa1.00071.00321.01520.431CCSD(T)b0.3250.038-24.480-53.183CCSD(2)Tc0.2291.452.1771.443CR-CC(2,3)c-0.1190.062-0.096-1.005MAGNUS(2*)1.5814.4950.98814.249MAGNUS(2*)[3]-D-0.324-4.080-28.661-31.390SCF138.329206.485299.388375.354BCCDref139.775221.576344.157454.695*BCCD2.0126.62210.696318.710*BCCD(T)0.2610.844-4.339317.288*BMAGNUS(2*)ref139.475222.492355.615445.704BMAGNUS(2*)1.9207.17016.97122.019BMAGNUS(2*)[3]-D0.0901.0714.6596.505Table6.4:aFromRef.[10].bCCSDandCCSD(T)resultsobtainedwithPSI4[6].cFromRef.[8]AcomparisonofCCandMagnusIMSRGground-stateenergiesobtainedfortheequilibriumgeometryofRe=1.7328bohrandothernuclearseparationsofHFwithaDZbasisset.Inthesepost-HFcalculationsallelectronswerecorrelated.ThefullCItotalenergiesaregiveninhartree.TheremainingenergiesarereportedinmillihartreerelativetothecorrespondingfullCIenergyvalues.(HFB)formalism,whichisformulatedintermsofSlaterdeterminantsoffermionicquasi-particlesthataresuperpositionsofparticlesandholes.Intrinsicdeformationwilldevelopifthesingle-particlebasisisnotsymmetryrestricted,e.g.,inanm-schemeformalism,androtationalsymmetrybreakingisenergeticallyfavored.Anm-schemeIM-SRGorCCcalculationmaybeabletoconvergetoasolutioniftheexcitationspectrumofthesymmetry-brokenreferencestatehasatlylargegap,i.e.,asingledominantIfsuchasolutionisfound,onemusteventuallyrestorethebrokensymmetriesthroughtheapplicationofprojectionmethods,whichhavealongtrackrecordinnuclearphysics(see,e.g.,[140{151]).Atthispoint,oneisnolongerdealingwith147asingle-referenceproblem,althoughtheprojectedstatesretainanimprintoftheoriginalsymmetry-broken(single-)referencestatesthatspracticalimplementations.Inthedomainofexoticneutron-richnuclei,thesingle-referenceparadigmmayalsobreakdown.Thecomplexinterplayofnuclearinteractions,many-bodycorrelations,and,inthedriplineregion,continuumcancausestrongcompetitionbetweenwithtintrinsicstructures.Thismanifestsinphenomenaliketheerosionandemergenceofshellclosures[22,23,44,152],ortheappearanceoftheso-calledislandsofinversion(see,e.g.,[153]).Theirdescriptionrequiresatruemulti-referencetreatment.TheMulti-ReferenceIM-SRG(MR-IM-SRG)iscapableofdealingwiththeaforemen-tioendscenarios[21,23,45].ItgeneralizestheIM-SRGformalismdiscussedinthisworktoarbitrarycorrelatedreferencestates,usingthemulti-referencenormalorderingandWick'stheoremdevelopedbyKutzelniggandMukherjee[154,155].Theideaofdecouplingthegroundstatefromexcitationsreadilycarriesover,exceptthatexcitedstatesaregivenby:ayiaj:ji;:ayiayjalak:ji;:::;andthesingle-particlestatesarenolongerofpureparticleorholecharacter.ThewequationformulationoftheMR-IM-SRGmakesitpossibletoavoidcomplicationsduetothenon-orthogonalityandpossiblelineardependencyofthesegeneralexcitations(see[45]formoredetails).Whileonlyone-bodydensitymatricesappearinthecontractionsofthestandardWick'stheorem,additionalcontractionsthatinvolvetwo-andhigher-bodydensitymatricesenterthatencodethecorrelationcontentofthereferencestate.IntheMR-IM-SRGframework,correlationsthatarehardtocaptureasfew-bodyexcitationsofthereferencestatecanbe148121416182022242628A1751501251007550.E[MeV]AOMR-IM-SRG(2)IT-NCSMCCSDADC(3)(=2:0fm1)Figure6.1:Ground-stateenergiesoftheoxygenisotopesfromMR-IM-SRGandothermany-bodyapproaches,basedontheNN+3N-fullinteractionwith3N=400MeV,evolvedtotheresolutionscale=1:88fm1(=2:0fm1fortheGreen'sFunctionADC(3)results,cf.[19]).Blackbarsindicateexperimentaldata[20].SeeRef.[21]foradditionaldetails.builtdirectlyintothereferencestate.InaapplicationsoftheMR-IM-SRGframework,spherical,particle-numberprojectedHFBvacuahavebeenusedtocomputetheground-stateenergiesoftheevenoxygenisotopes,startingfromchiralNN+3Nforces[21].ThisworkimprovedonpreviousShellModel[25,156]andCCstudies[99],thatincludedNN+3NinteractionsinMBPTorforthelatterwith3Nforcesinamorephenomenological,nuclear-matterbasednormalordering.BasedonaHamiltonianthatisentirelyedintheA=3;4systemandconsistentlyevolvedtolowerresolution,wefoundthatMR-IM-SRG,variousCCmethods,andtheimportance-truncatedNCSMconsistentlypredicttheneutrondriplinein24Oifchiral3Nforcesareincluded(seeFig.6.1),aspointedoutinthecontextoftheShellModelinRef.[156].Encouragedbythissuccess,wemovedontothecalciumandnickelisotopicchains[23],whereimportance-truncatedNCSMcalculationsarenolongerfeasible.ThesamefamilyofchiralNN+3NHamiltoniansthatsuccessfullyreproducetheoxygenground-stateenergies14938404244464850525456586062A10010203040.S2n[MeV]3N[MeV=c]/n350ds/l400ACaFigure6.2:MR-IM-SRGresultsforCatwo-neutronseparationenergies,forchiralNN+3Ninteractionswithtinthe3Nsector,andarangeofresolutionscalesfrom=1:88fm1(opensymbols)to2:24fm1(solidsymbols).Blackbarsindicateexperimentaldata[20,22].SeeRef.[23]foradditionaldetails.overestimatethebindingenergiesintheseisotopesbyseveralhundredkeVpernucleon,inMR-IM-SRGandCC(alsosee[86,87,98]),aswellasthesecond-orderGor'kovGreen'sFunctionapproach[44].Therevelationofthesehasledtoavarietyoftoimproveonthechiralinteractions[47,157{165].Contrarytotheground-stateenergies,chiralNN+3Nforcesreproducerelativequantitieslikethetwo-neutronseparationenergiesquitewell,asidefromtheexaggeratedN=20shellclosure(Fig.6.2).Inparticular,theyshowsignalsofsub-shellclosuresin52;54Ca,inagreementwithShellModelcalculationsbasedonNN+3NinteractionsinMBPT[22,152].TheseobservationsindicatewhichtermsinthechiralinputHamiltonianmaybet,andthisinformationcanbeusedinfutureoptimizations.OngoingworkwithinMR-IM-SRGrelevanttothisthesisfocusesonovercomingthesameshortcomingsthetraditionalIM-SRGfaced,andthattheMagnusformulationcircumventedinthesinglereferenceIM-SRGmethods.Initialinspectionindicatesthatmulti-referenceMAGNUS(2)calculationswillfaithfullyreproducetheirMR-IM-SRG(2)calculationsasit150didwithsinglereferenceIM-SRG(2).Further,itappearsthatonemightexpectthatthelargestmissedcorrectionstoMR-MAGNUS(2)wouldbecompletelyanalogoustotheMAG-NUS(2*)andMAGNUS(2*)[3]typecorrections.Thisisofcourseconjectureandwillneedfullinspectiontoverify.OnereasonwepresentedE[3]intermsofcommutatorsisthatitavoidstheappealtoperturbationtheory,whichbecomesveryexpensivewithamulti-referencetypestate.AsmentionedabovetheMR-IM-SRGavoidscomplicationsduetothenon-orthogonalityandpossiblelineardependencyofexcitationsthroughthisusageofgener-alizednormalordering.Thus,itwouldbeexpectedthatwecouldcomeupwithananalagousE[3]basedonleadingexpressionfromthemulti-referenceBCH.Thiswouldbeinthesamespiritofrecentmulti-referenceperturbationtheorybasedonthedrivensimilarityrenormal-izationgroupmotivatedperturbationtheoryfoundin[166].Further,usingthefactorizationschemeof[167],itwouldbepossibletofactorizetheseexpressionsinEq.5.13ton6scaling.Thisisonlyhelpfulforthemulti-referenceformalismwherethereisnodistinctionbetweenparticleandholestates,asthemostexpensivetermforaclosedshellsystemscalesasnon5uwhichisgenerallylargerthann3on4u.ThusitmightbepossibletocorrecttheMR-IMSRG(2)totheMR-MAGNUS(2*)[3]withoutscaling,yieldingaroundCCSD(T)accuracyevenforopenshellnuclei.6.4ExtensionstoExcitedStateFormalismForopen-shellsystems,ratherthansolvingthefullA-bodyproblem,itistofollowtheShellModelparadigmbyconstructinganddiagonalizinganeHamiltonianinwhichtheactivedegreesoffreedomareAvvalencenucleonstoafeworbitalsneartheFermilevel.BothphenomenologicalandmicroscopicimplementationsoftheShell151Modelhavebeenusedwithsuccesstounderstandandpredicttheevolutionofshellstructure,propertiesofgroundandexcitedstates,andelectroweaktransitions[168{170].RecentmicroscopicShell-Modelstudieshaverevealedtheimpactof3Nforcesinpredict-ingground-andexcited-statepropertiesinneutron-andproton-richnuclei[22,25,152,156,171{174].Despitethenovelinsightsgainedfromthesestudies,theymakeapproximationsthataretobenchmark.Themicroscopicderivationoftheevalence-spaceHamiltonianreliesonMBPT[175],whereorder-by-orderconvergenceisunclear.Evenwithtocalculateparticularclassesofdiagramsnonperturbatively[176],resultsaresensitivetotheHOfrequency~!(duetothecore),andthechoiceofvalencespace[25,171,172].Anonperturbativemethodtoaddresstheseissueswasdevelopedin[177{179],whichgeneratesvalence-spaceinteractionsandoperatorsbyprojectingtheirfullNCSMcounterpartsintoagivenvalencespace.Toovercometheselimitationsinheaviersystems,theIM-SRGcanbeextendedtoderiveevalence-spaceHamiltoniansandoperatorsnonperturbatively.Calculationswithoutinitial3Nforces[39]indicatedthatanabinitiodescriptionofgroundandexcitedstatesforopen-shellnucleimaybepossiblewiththisapproach.TheutilityoftheIM-SRGliesinthefreedomtotailorthedeofHodtoaspproblem.Forinstance,toconstructaShellModelHamiltonianforanucleuscomprisedofAvvalencenucleonsoutsideaclosedcore,weaHFreferencestatejiforthecorewithAcparticles,andsplitthesingle-particlebasisintohole(h),valence(v),andnon-valence(q)particlestates.TreatingallAnucleonsasactive,i.e.,withoutacoreapproximation,weeliminatematrixelementswhichcouplejitoexcitations,justasinIM-SRGground-statecalculations[21,58,74].Inaddition,wedecouplestateswithAvparticlesinthevalencespace,:ayv1:::ayvAv:ji,fromstatescontainingnon-valencestates.152Figure6.3:Excited-statespectraof22;23;24ObasedonchiralNN+3Ninteractionsandcom-paredwithexperiment.FiguresadaptedfromRef.[24].TheMBPTresultsareperformedinanextendedsdf7=2p3=2space[25]basedonlow-momentumNN+3Ninteractions,whiletheIM-SRG[24]andCCeinteraction(CCEI)[26]resultsareinthesdshellfromtheSRG-evolvedNN+3N-fullHamiltonianwith~!=20MeV(CCEIanddottedIM-SRG)and~!=24MeV(solidIM-SRG).Thedashedlinesshowtheneutronseparationenergy.FiguretakenfromRef.[27].AftertheIM-SRGderivationofthevalence-spaceHamiltonian,theA-dependentHamil-tonianisdiagonalizedinthevalencespacetoobtainthegroundandexcitedstates.Fortheoxygenisotopes,agooddescriptionoftheexperimentalspectraisfound(Fig.6.3).Re-cently,thesecalculationswereextendedtonearbyF,Ne,andMgisotopesshowingexcellentagreementwithnewmeasurementsin24F[180]andthatdeformationcanemergefromtheseabinitiocalculations[41].Futuredirectionsincludeextendingthevalencespace,whichwillgiveaccesstotheisland-of-inversionregionandpotentiallythefullsd-shell(andhigher)neutrondripline.TheresultsbeingproducedbyStrobergandcollaborators[41]arealreadyreliantontheMagnusformulationoftheIM-SRGdescribedhere;andquicklyitisbecomingclearthatevalencespaceobservableswillbereadilyavailablebecauseofit.Thiscouldhelp153toanswerlong-standingquestionsaboutawholehostofshell-modelphenomenologyfromprinciples.Itisimperativethatweinspecttheofthree-bodyforces,inducedandotherwise,intheseevalencespacesinteractions.Thegeneralizationofthiswork'sforgroundstatedecouplingswillnotgeneralizeeasilytothenewnon-trivialtionofusedtodecouplevalencespaces,butthepathforwardisstraightforwardandneedstobeinspected.154Chapter7SummaryandConclusionsThisworkinvestigatedtheIM-SRGmethod,whichhasseenincreasingrecentuseinnu-clearphysicsduetoitsyandrelativelygentlescalingwithsystemsize.Despiteitsamazingsuccessinnuclearphysics,itsinitialfailuretosuccessfullytreatevenfairlysimplechemicalsystemswasmorethanalittlepuzzling.Asasteptowardssolvingthispuzzle,IinvestigatedthetruncatedIM-SRG(2)'sperturbativecontent.Itwasfoundthatitunder-countedaclassoffourthorderquadrupoleexcitationdiagramsthatCCSDtheoryincludescorrectly.Beingthatthetwomethodsseemtohaveasimilarmachinery,computationalcostandphilosophy,itbecameoneofmygoalstoawaytorestorethiscontenttothemethodsothatitwouldpossibletobringthesuccessoftheIM-SRGtochemicalsystemsaswell.Incidentally,theoutstandingperformanceoftheIM-SRG(2)innuclearcalculationsisrelatedtothisundercountingof4th-orderterms,asitmimicsthepartialcancellationsthatoccurbetweentheserepulsivecontributionsandattractivetriplescorrelationsinCCSDandCCSDTcalculations.Inotherwords,theundercountingofthisclassofdiagramsmimicstheoftriplescorrelationsfornuclei,whichiswhytheIM-SRG(2)resultsfallinbetweenCCSDandCCSDTcalculationsforallnucleistudied.ItwasduringthisquesttoawaytorestorethefullcountingofthesetermsthatitwasfoundthattheIM-SRGwequationscouldberecastusingtheMagnusexpansion.ThisledtoformulationoftheIM-SRGequationswhichnotonlyalleviatedtheneedforsolvingthewequationswithexpensivehigh-orderODEsolvers,butalsoallowedforthegeneration155ofobservablesatnoadditionalcost.Thecalculationsshowingthesebhavebeenconductedfornucleiandtheelectrongas,withverypromisingresults.Asmentioned,thisformalismhasalreadyfounditswaytoseveralotherindependentpractitionersoftheIM-SRGformalism,particularlythosedevelopingvalencespaceinteractions,togreatsuccess.WiththeMagnusformulationinhand,itwasthenpossibletorevisitthesemissingtermsthatcausedthenaiveIM-SRG(2)andMAGNUS(2)truncationstofailforchemicalsystems.Notonlywasitpossibletorestoretheseterms,yieldingtheMAGNUS(2*)method,butitalsobecamepossibletoborrowinsightfromCCtheorytoengineerapproximatecorrectionsthatwouldbeduetoIM-SRG(3).WehaveshownthattheclassofapproximationstoIM-SRG(3)dubbedMAGNUS(2*)[3]treatstheelectrongas,nuclei,andsimplechemistrysystemsextraordinarilywell.Withmorecomplicatedchemistrysystems,thesemethodsdoaswellassomeapproximatenon-iterativetriplesmethodsinCCtheory,butnotall.Itseemsthatthemethodsabilitytomapthefullycorrelatedgroundstatetoameanpictureiscompromisedasthecorrelationsbecomemorecomplicated.ItislikelythatmorenuancedapproximationsofIM-SRG(3)orhighermaybeneeded.Regardless,forresultsofthemethod,thequalityofMAGNUS(2*)[3]resultsseemverypromisingindeed.ThusthisworkhasshownhowtheIM-SRGmethodhasbeenaugmentedwiththeMagnusformulationtobefaster,moreaccurate,morerobust,andmoreversatileinthetreatmentofobservables.Theoutlookforfuturedevelopments,bothinnuclearandchemicalsystemsispromising,betheyintheBruecknerformulation,themulti-referenceformulation,orpursuingnewapproximationstotheIM-SRG(3)whenderivingvalenceinteractionswithintheIM-SRG.Weexpectthatnotonlyaretheseendeavorsnowpossible,buttheywilllikelybeaccomplishedsoon.156APPENDICES157AppendixAFundamentalCommutatorsForconvenience,wecollecttheexpressionsforthefundamentalcommutatorswhicharerequiredforthederivationoftheIM-SRGwequationsandWegner-typegenerators.Wewriteone-,two-,andthree-bodyoperatorsasA(1)=XijAij:ayiaj:;(A.1)A(2)=1(2!)2XijklAijkl:ayiayjalak:;(A.2)A(3)=1(3!)2XijklmnAijklmn:ayiayjaykanamal:;(A.3)wherethetwo-andthree-bodymatrixelementsareassumedtobefullyanti-symmetrized.Single-particleindicesrefertonaturalorbitals,sothatoccupationnumbersareni=0;1,andweusethenotationna=1na.WealsorecallthatthecommutatoroftwooperatorsofrankMandNcanonlyhavecontributionsofrankjMNj;:::;M+N1,[A(M);B(M)]=M+N1Xk=jMNjC(k):(A.4)158[A(1);][A(1);B(1)](1)=XijXa:ayiaj:AiaBajBiaAaj(A.5)[A(1);B(1)](0)=XijAijBji(ninj)(A.6)[A(1);B(2)](2)=14XijklXa:ayiayjalak:(1Pij)AiaBajkl(1Pkl)AakBijal(A.7)[A(1);B(2)](1)=XijXab:ayiaj:(nanb)AabBbiaj(A.8)[A(1);B(3)](3)=136XijklmnXa:ayiayjaykanamal:(1PijPik)AiaBajklmn(1PlmPln)AalBijkamn(A.9)[A(1);B(3)](2)=XijklXab:ayiayjalak:(nanb)AabBbijakl(A.10)159[A(2);][A(2);B(2)](3)=136XijklmnXa:ayiayjaykanamal:P(ij=k)P(l=mn)AijlaBakmnBijlaAakmn(A.11)[A(2);B(2)](2)=14XijklXab:ayiayjalak:ˆ12(AijabBabklBijabAabkl)(1nanb)+(nanb)(1PijPkl+PijPkl)AaibkBbjal˙(A.12)[A(2);B(2)](1)=12XijXabc:ayiayj:AciabBabcjBciabAabcj(nanbnc+nanbnc)(A.13)[A(2);B(2)](0)=14XijklninjnknlAijklBklijBijklAklij(A.14)[A(2);B(3)](3)=172XijklmnXab:ayiayjaykanamal:(1nanb)P(ij=k)AijabBabklmnP(l=mn)AabmnBijklab(A.15)[A(2);B(3)](2)=18XijklXabc:ayiayjalak:(nanbnc+nanbnc)1PijPikPjlPkl+PikPjlAbcakBaijbcl(A.16)[A(2);B(3)](1)=14XijXabcd:ayiaj:(nanbncndnanbncnd)AcdabBabijcd(A.17)160[A(3);][A(3);B(3)](3)=136XijklmnXabc:ayiayjaykanamal:ˆ16(nanbnc+nanbnc)(AijkabcBabclmnBijkabcAabclmn)+12(nanbnc+nanbnc)P(ij=k)P(l=mn)(AabkcmnBcijablAcjkabnBiablmc)˙(A.18)[A(3);B(3)](2)=14XijklXabcd:ayiayjalak:ˆ16(nanbncndnanbncnd)(AaijbcdBbcdaklAbcdaklBaijbcd)+14(nanbncndnanbncnd)(1Pij)(1Pkl)AabicdlBcdjabk˙(A.19)[A(3);B(3)](1)=112XijXacde:ayiaj:(nanbncndne+nanbncndne)(AabicdeBcdeabjBabicdeAcdeabj)(A.20)[A(3);B(3)](0)=136Xijklmn(ninjnknlnmnnninjnknlnmnn)AijklmnBlmnijk(A.21)161AppendixBIM-SRG(3)FlowEquationsTheIM-SRG(3)wequationscanbederivedusingthefundamentalcommutatorsfromAppendixA.ThepermuationsymbolsPij;P(ij=k);andP(i=jk)havebeennedinEqs.(3.11),(3.95),and(3.96).Thenormal-orderedHamiltonianisgivenbyH(s)ˇE(s)+f(s)+s)+W(s):(B.1)TheparticleranksoftheindividualcontributionsofHandthegeneratorareobviousfromtheindicesoftheassociatedmatrixelements.ddsE=Xab(nanb)abfba+12Xabcdabcdcdabnanbncnd+118XabcdefabcdefWdefabcnanbncndnenf(B.2)162ddsfij=Xa(1+Pij)iafaj+Xab(nanb)(abbiajfabbiaj)+12Xabc(nanbnc+nanbnc)(1+Pij)ciababcj+14Xabcd(nanbncnd)(abicdjcdabWabicdjcdab)+112Xabcde(nanbncndne+nanbncndne)(abicdeWcdeabjWabicdecdeabj)(B.3)ddsijkl=Xan(1Pij)(iaajklfiaajkl)(1Pkl)(akijalfakijal)o+12Xab(1nanb)(ijababklijababkl)Xab(nanb)(1Pij)(1Pkl)bjalaibk+Xab(nanb)aijbklfbaWaijbklba+12Xabc(nanbnc+nanbnc)(1PikPjlPijPkl+PikPjl)(aijbclbcakWaijbclbcak)+16Xabcd(nanbncndnanbncnd)(aijbcdWbcdaklbcdaklWaijbcd)+14Xabcd(nanbncndnanbncnd)(1Pij)(1Pkl)abicdlWcdjabk(B.4)163ddsWijklmn=XanP(i=jk)iaWajklmnP(l=mn)alWijkamnoXanP(i=jk)fiaajklmnP(l=mn)falijkamno+XaP(ij=k)P(l=mn)(ijlaakmnijlaakmn)+12Xab(1nanb)(P(i=jk))(ijabWabklmnijababklmn)12Xab(1nanb)(P(lm=n))(ablmWijkabnablmijkabn)Xab(nanb)P(i=jk)p(l=mn)(bialWajkbmnbialajkbmn)+16Xabc(nanbnc+nanbnc)(ijkabcWabclmnWijkabcabclmn)+12Xabc(nanbnc+nanbnc)P(ij=k)P(l=mn)(abkcmnWcijablcjkabnWiablmc)(B.5)164AppendixCDiagramRulesForconvenience,webriefysummarizetherulesforinterpretingtheantisymmetrizedGold-stoneandHugenholtzdiagramsthatappearintheperturbativediscussionoftheIM-SRGinSec.3.5.Detailedderivationscanbefoundinstandardtextsonmany-bodytheory,e.g.,inRef.[16,117,120],aswellasinRefs.[175,181,182],whichareparticularlyusefulfordiagrammtictreatmentsofenuclearHamiltonians.1.Solidlinesrepresentsingle-particlestates(indices),withup-anddownwardpointingarrowsindicatingparticle(">"F)andholestates(""F),respectively.2.InteractionverticesarerepresentedasdotsinHugenholtzdiagrams,hijfjji=ji;hijjjkli=klij;hijkjWjlmni=lmnijk;(C.1)wherethetwo-andthree-bodymatrixelementsarefullyantisymmetrized.Throughoutthiswork,wewillalsousetheshort-handnotationfij=hijfjji;ijkl=hijjjkli;etc.Forthediscussionoftheeone-andtwo-bodyHamiltonians,weswitchfromHugenholtzdiagramstoantisymmetrizedGoldstonediagramsforclarity(see,e.g.,Ref.[16]).Tothisend,theHugenholtzpointverticesarestretchedintodashedinter-165actionlines,hijfjji=ji;hijjjkli=klij;hijkjWjlmni=lmnijk:(C.2)Notethatthematrixelementsarestillantisymmetrized:Eachofthediagramsshownhererepresentsallallowedexchangesofsingle-particlelines/indicesinthebraandket.Thisisintherulesforprefactorsthatweadoptinthefollowing[16].3.Assignafactor1=2ndforndequivalentpairs,i.e.,pairsofparticleorholelinesthatstartatthesameinteractionvertexandendatthesameinteractionvertex.Likewise,assign1=6ntforntequivalenttriplesconnectingthesameinteractionvertices.4.Assignaphasefactor(1)nl+nh+nc+nexhtoeachdiagram,wherenlisthenumberofclosedfermionloops,nhthetotalnumberofholelines,ncisthenumberofcrossingsofdistinctexternallines,andnexhthenumberofholelinescontinuouslypassingthroughthewholediagram(i.e.,nexh=0forenergydiagrams).5.Foreachintervalbetweeninteractionswithparticlelinesp1;:::;pMandholelinesh1;:::;hNmultiplytheexpressionwiththeenergydenominator1+PNi=1"hiPMi=1"pi;(C.3)whereistheunperturbedenergyofthestateenteringthediagramrelativetothereferencestate,readingfrombottomtotop(e.g.,=0forenergydiagrams).Through-outthiswork,theenergiesaregivenbythediagonalmatrixelementsoftheone-body166partoftheHamiltonian"i=fii;forHartree-Fockreferencestates,fisdiagonal,ofcourse.ThesumoverintermediateparticleandholelinesinthedenominatoristheunperturbedenergyoftheexcitedMpNhstateinaM˝ller-Plessettypeperturbationtheorywithrespecttothereferencestate.IntheEpstein-Nesbetcase,itisreplacedwiththediagonalmatrixelementoftheHamiltonianinthesamestate,i.e.,hj:ayhN:::ayh1apM:::ap1:H:ayp1:::aypMah1:::ahN:jiE0=MXi=1"piNXi=1"hi+additionalterms;(C.4)whereE0=hjHji.6.Sumfreelyoverallinternalsingle-particleindices.Letusdemonstratetheuseofthediagramrulesforafewexamples.Forthethird-orderparticle-ladderdiagram,p1p2p3p4h1h2=18Xp1p2p3p4h1h2h1h2p3p4p3p4p1p2p1p2h1h2("h1+"h2"p1"p2)("h1+"h2"p3"p4):(C.5)Herenc=nexh=0,nh=2,andthenumberofclosedfermionloopsisnl=2,namelyp1!p3!h1!p1andp2!p4!h2!p2.Fortheparticle-holediagram,wehaveh1p1h3p3p1h2=Xp1p2p3h1h2h3h3h2p1p3h1p3h3p2p1p2h1h2("h1+"h2"p1"p2)("h2+"h3"p1"p3);(C.6)167withnc=nexh=0,nh=3,andtwoclosedloops(nl=2),p1!h3!h1!p1andp2!p3!h2,givinganegativesign.Sincetheinteractionverticesareconnectedbyoneparticleandoneholelineeach,nd=0,andthepre-factoris1.Forthesecond-ordereHamiltonian,diagramf4inFig.3.12translatesintohpp0p00h0=12Xp0p00h0ph0p0p00p0p00hh0"h+"h0"p0"p00:(C.7)Readingfrombottomtotop,wehave=0justlikeinanenergydiagram.Todeterminethephase,wenotethatthereisonefermionloop(p00!h0!p00),therearetwoholelines,oneofwhichisexternalandpassingthroughthediagramviah!p0!p.Thusnl=1;nh=2;nexh=1,andnc=0,sothephasefactoris+1.Thereisonepairofequivalentparticlelines,nd=1,givingrisetothepre-factor12.Asanexampleforasecond-ordertwo-bodyinteraction,weconsiderdiagram3inFig.3.13:p3p1hh0h00p2=12Xh0h00p1p2h0h00h0h00p3h"h0+"h00"p1"p2;(C.8)wherenl=nc=0;nh=3,andthereisoneexternalholeline(nexh=1)passingthroughthediagram,h!h00!p,givingaphasefactor+1.Thereisonepairofequivalentholelines(nd=1),andthestartingenergyis=p3,whichexplainsthesymmetrypre-factorandenergydenominator,respectively.168Ourexampleisaninducedthree-bodyinteraction,diagramW3inFig.3.14.Theexpressionisp1h1p2h2p3h3h0=Xh0p1p2h1h0h0p3h3h3"h1+"h0"p1"p2;(C.9)where=0,thephasefactoris1becausenl=nc=0;nh=4;nexh=3.Duetothelackofequivalentlines,theoverallpre-factorofthediagramis1.169REFERENCES170REFERENCES[1]D.R.EntemandR.Machleidt,Phys.Rev.C68,041001(2003).[2]R.MachleidtandD.Entem,Phys.Rept.503,1(2011).[3]J.J.Shepherd,G.H.Booth,andA.Alavi,Chem.Phys.136,244101(2012).[4]A.S.Hansen,Coupledclusterstudiesofsystems,Master'sthesis,UniversityofOslo,Oslo,Norway(2015).[5]J.Olsen,P.Jorgensen,H.Koch,A.Balkova,andR.J.Bartlett,Chem.Phys.104(1996).[6]J.M.Turney,A.C.Simmonett,R.M.Parrish,E.G.Hohenstein,F.A.Evangelista,J.T.Fermann,B.J.Mintz,L.A.Burns,J.J.Wilke,M.L.Abrams,N.J.Russ,M.L.Leininger,C.L.Janssen,E.T.Seidl,W.D.Allen,H.F.Schaefer,R.A.King,E.F.Valeev,C.D.Sherrill,andT.D.Crawford,WileyInterdiscip.Rev.Comput.Mol.Sci.2,556(2012).[7]S.Hirata,P.-D.Fan,A.A.Auer,M.Nooijen,andP.Piecuch,Chem.Phys.121(2004).[8]P.PiecuchandM.Wloch,Chem.Phys.123,224105(2005),10.1063/1.2137318.[9]F.A.Evangelista,Chem.Phys.134,224102(2011),10.1063/1.3598471.[10]K.B.Ghose,P.Piecuch,andL.Adamowicz,Chem.Phys.103(1995).[11]H.Hergert,S.K.Bogner,T.D.Morris,A.Schwenk,andK.Tsukiyama,Phys.Rept.621,165(2016).[12]B.R.Barrett,P.Naatil,andJ.P.Vary,Prog.Part.Nucl.Phys.69,131(2013).[13]R.J.Furnstahl,G.Hagen,andT.Papenbrock,Phys.Rev.C86,031301(2012).[14]S.N.More,A.om,R.J.Furnstahl,G.Hagen,andT.Papenbrock,Phys.Rev.C87,044326(2013).[15]I.Angeli,At.DataNucl.DataTables87,185(2004).[16]I.ShavittandR.J.Bartlett,Many-BodyMethodsinChemistryandPhysics:MBPTandCoupled-ClusterTheory(CambridgeUniversityPress,2009).[17]D.EntemandR.Machleidt,Phys.Rev.C68,041001(2003).[18]G.Hagen,T.Papenbrock,M.Hjorth-Jensen,andD.J.Dean,Rep.Prog.Phys.77,096302(2014).171[19]A.Cipollone,C.Barbieri,andP.Naatil,Phys.Rev.Lett.111,062501(2013).[20]M.Wang,G.Audi,A.Wapstra,F.Kondev,M.MacCormick,X.Xu,andB.Chin.Phys.C36,1603(2012).[21]H.Hergert,S.Binder,A.Calci,J.Langhammer,andR.Roth,Phys.Rev.Lett.110,242501(2013).[22]F.Wienholtz,D.Beck,K.Blaum,C.Borgmann,M.Breitenfeldt,R.B.Cakirli,S.George,F.Herfurth,J.D.Holt,M.Kowalska,S.Kreim,D.Lunney,V.Manea,J.Menendez,D.Neidherr,M.Rosenbusch,L.Schweikhard,A.Schwenk,J.Simonis,J.Stanja,R.N.Wolf,andK.Zuber,Nature498,346(2013).[23]H.Hergert,S.K.Bogner,T.D.Morris,S.Binder,A.Calci,J.Langhammer,andR.Roth,Phys.Rev.C90,041302(2014).[24]S.K.Bogner,H.Hergert,J.D.Holt,A.Schwenk,S.Binder,A.Calci,J.Langhammer,andR.Roth,Phys.Rev.Lett.113,142501(2014).[25]J.Holt,J.Menendez,andA.Schwenk,Eur.Phys.J.A49,1(2013).[26]G.R.Jansen,J.Engel,G.Hagen,P.Navratil,andA.Signoracci,Phys.Rev.Lett.113,142502(2014).[27]K.Hebeler,J.D.Holt,J.Menendez,andA.Schwenk,AnnualReviewofNuclearandParticleScience,Ann.Rev.Nucl.Part.Sci.(2015).[28]K.A.BruecknerandC.A.Levinson,Phys.Rev.97,1344(1955).[29]H.A.Bethe,Phys.Rev.103,1353(1956).[30]J.Goldstone,Proc.Roy.Soc.Lond.AMath.Phys.Eng.Sci.239,267(1957).[31]H.F.S.III,QuantumChemistry:TheDevelopmentofAb-InitioMethodsinMolecularElectronicStructureTheory(DoverPublicationsInc.,2004).[32]E.Epelbaum,H.-W.Hammer,andU.-G.Meiˇner,Rev.Mod.Phys.81,1773(2009).[33]S.K.Bogner,R.J.Furnstahl,andA.Schwenk,Prog.Part.Nucl.Phys.65,94(2010).[34]H.-W.Hammer,A.Nogga,andA.Schwenk,Rev.Mod.Phys.85,197(2013).[35]A.B.Balantekin,J.Carlson,D.J.Dean,G.M.Fuller,R.J.Furnstahl,M.Hjorth-Jensen,R.V.F.Janssens,B.-A.Li,W.Nazarewicz,F.M.Nunes,W.E.Ormand,S.Reddy,andB.M.Sherrill,Mod.Phys.Lett.A29,1430010(2014).[36]J.Carlson,S.F.Pederiva,S.C.Pieper,R.Schiavilla,K.E.Schmidt,andR.B.Wiringa,Rev.Mod.Phys.87,1067(2015).[37]R.Roth,J.Langhammer,A.Calci,S.Binder,andP.Naatil,Phys.Rev.Lett.107,072501(2011).172[38]T.A.ahde,E.Epelbaum,H.Krebs,D.Lee,U.-G.Meiˇner,andG.Rupak,Phys.Lett.B732,110(2014).[39]K.Tsukiyama,S.K.Bogner,andA.Schwenk,Phys.Rev.C85,061304(2012).[40]G.R.Jansen,A.Signoracci,G.Hagen,andP.Naatil,(2015),arXiv:1511.00757[nucl-th].[41]J.D.H.S.K.B.S.R.Stroberg,H.HergertandA.Schwenk,arXiv:1511.02802[nucl-th].[42]V.a,C.Barbieri,andT.Duguet,Phys.Rev.C87,011303(2013).[43]V.a,C.Barbieri,andT.Duguet,Phys.Rev.C89,024323(2014).[44]V.a,A.Cipollone,C.Barbieri,P.Naatil,andT.Duguet,Phys.Rev.C89,061301(2014).[45]H.Hergert,inpreparation(2015).[46]S.A.Lovato,J.Carlson,andK.E.Schmidt,Phys.Rev.C90,061306(2014),arXiv:1406.3388[nucl-th].[47]G.Hagen,A.Ekstrom,C.Forssen,G.R.Jansen,W.Nazarewicz,T.Papenbrock,K.A.Wendt,S.Bacca,N.Barnea,B.Carlsson,C.Drischler,K.Hebeler,M.Hjorth-Jensen,M.Miorelli,G.Orlandini,A.Schwenk,andJ.Simonis,NaturePhys.advanceonlinepublication,(2015).[48]F.Coester,Nucl.Phys.7,421(1958).[49]F.CoesterandH.Kummel,Nucl.Phys.17,477(1960).[50]J.CizekandJ.Paldus,Int.J.Quant.Chem.5,359(1971).[51]S.K.Bogner,R.J.Furnstahl,S.Ramanan,andA.Schwenk,Nucl.Phys.A773,203(2006).[52]S.K.Bogner,R.J.Furnstahl,S.Ramanan,andA.Schwenk,Nucl.Phys.A784,79(2007),arXiv:nucl-th/0609003.[53]S.K.Bogner,R.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