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DensityMatrixRenormalizationGroup (DMRG)-basedApproachesin ComputationalChemistry Thispageintentionallyleftblank
DensityMatrix Renormalization Group(DMRG)-based Approachesin Computational Chemistry HaiboMa
SchoolofChemistryandChemicalEngineering, NanjingUniversity,Nanjing, P.R.China
UlrichSchollwo¨ck
DepartmentofPhysics,Ludwig-MaximilianUniversityofMunich, Munich,Germany
ZhigangShuai
DepartmentofChemistry,TsinghuaUniversity,Beijing,P.R.China SchoolofScienceandEngineering,TheChineseUniversityof HongKong,Shenzhen,P.R.China
Elsevier
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1.Densitymatrixrenormalizationgroup
1.2Infinite-systemdensitymatrixrenormalizationgroup 7
1.3Finite-systemdensitymatrixrenormalizationgroup 13 References 18
2.Tensornetworkstates:matrixproductstatesand relatives 19
2.1Tensordecompositions 20
2.1.1Singularvaluedecomposition20
2.1.2Frequentlyencounteredtensordecompositions21
2.2Schmidtdecompositionandquantumentanglement
2.3Matrixproductstate
2.3.1Buildingmatrixproductstate28
2.3.2Overlaps,expectationvaluesandmatrixelements33
2.3.3Addingtwomatrixproductstates34
2.3.4Bringingamatrixproductstateintocanonicalform35
2.3.5ApproximatecompressionofanMPS36
2.3.6Goodquantumnumbers37
2.4Matrixproductoperator 38
2.4.1ApplyinganMPOtoanMPS40
2.4.2AddingandmultiplyingMPOs40
2.5GroundstatecalculationswithMPS 41
2.5.1Thebasicalgorithm41
2.5.2Excitations46
2.5.3“Singlesite”vs“twosite”47
2.5.4MPOrepresentationofHamiltonians48
2.5.5ComparingDMRGtovariationalMPSgroundstate searches50 References
3.Densitymatrixrenormalizationgroupfor
3.2SemiempiricalmodelHamiltonian
3.3Symmetrizeddensitymatrixrenormalization groupalgorithm
3.3.1Particlenumber Ntot and Sz symmetry64
3.3.2Spin-flipsymmetry64
3.3.3Spatialsymmetry65
3.3.4Electron-holesymmetry66
3.4Applications 68
3.4.1Theelectronicstructureofthegroundstateof cyclicpolyene68
3.4.2Theexcitedstatesordering,excitonbinding,and opticalpropertiesofpolyene70
3.4.3Solitonstructureofexcitedstatesofpolyene77
3.4.4Intramolecularsingletfissionindonor acceptortype conjugatedcopolymer80
3.4.5Pariser Parr Popledensitymatrixrenormalization groupforsystemsbeyondone-dimension83
4.Densitymatrixrenormalizationgroupforabinitio
4.8Componentanalysisofdensitymatrixrenormalization groupwavefunction
5.Densitymatrixrenormalizationgroupwithorbital optimization 149
5.1Orbitalrotation 149
5.2Themulticonfigurationalself-consistentfieldmethods 152
5.2.1Energy,gradient,andHessian152
5.2.2Super-configurationinteractionmethod:afirst-order multiconfigurationalself-consistentfieldimplementation156
5.2.3Second-ordermulticonfigurationalself-consistentfield method157
5.2.4Simultaneousoptimizationofconfigurationinteraction coefficientsandorbitalrotations162
5.3Densitymatrixrenormalizationgroupself-consistentfield methods 164
5.4Excitedstatecalculation 165
5.5Analyticgradientandgeometryoptimization 168
5.6Molecularspectra 172
5.7BeyondBorn Oppenheimerapproximation 178 5.8Applications 181
5.8.1ElectroniclandscapeoftheP-clusterofnitrogenase181
5.8.2Mechanismforphotochromicring-openingreactionof spiropyran183 References 184
6.Post-densitymatrixrenormalizationgroup 189
6.1Fundamentalsformultireferencequantumchemical calculations 189
6.1.1Staticanddynamicelectroncorrelation190
6.1.2Contractionapproximations191
6.2Densitymatrixrenormalizationgroup-multireference configurationinteraction 198
6.2.1Densitymatrixrenormalizationgroup-fullyinternally contracted-multireferenceconfigurationinteraction198
6.2.2Densitymatrixrenormalizationgroup-externally contracted-multireferenceconfigurationinteraction199
6.2.3Uncontractedmatrixproductstate-multireference configurationinteraction200
6.3Densitymatrixrenormalizationgroup-multireference perturbationtheory 201
6.3.1Recapitulationofmultireferenceperturbationtheory201
6.3.2Densitymatrixrenormalizationgroup-completeactive spacewithsecond-orderperturbationtheory206
6.3.3Densitymatrixrenormalizationgroup-second-order N-electronvalencestateperturbationtheory208
6.3.4Densitymatrixrenormalizationgroup-ENPT2210
6.3.5Matrixproductstates-perturbationtheory211
6.3.6Othervariants214
6.4Densitymatrixrenormalizationgroup-coupledcluster theory 215
6.4.1Recapitulationofcoupledclustertheory215
6.4.2Densitymatrixrenormalizationgroup-alternativemultireferencecoupledcluster217
6.4.3Densitymatrixrenormalizationgroup-canonical transformation220
6.5Hybridizationofdensitymatrixrenormalizationgroup withdensityfunctionaltheory 223
6.5.1Recapitulationofdensityfunctionaltheory224
6.5.2Densitymatrixrenormalizationgroup-short-range densityfunctionaltheory225
6.5.3Densitymatrixrenormalizationgroup-pairdensity functionaltheory228
6.6Densitymatrixrenormalizationgroup-adiabatic connection 230
6.7Embeddingdensitymatrixrenormalizationgroupin environments 231
6.7.1Densitymatrixrenormalizationgroup-in-density functionaltheory232
6.7.2Polarizableembeddingdensitymatrixrenormalization group233
6.7.3Combiningdensitymatrixrenormalizationgroupwith referenceinteractionsitemodel234
6.8Summaryandoutlook 236
7.DMRGinfrequencyspace
7.1Introduction
7.2Spectralfunctioninlinearresponseregime 249
7.3Algorithmsatzerotemperature 251
7.3.1Lanczosdensitymatrixrenormalizationgroup251
7.3.2Correctionvectordensitymatrixrenormalizationgroup254
7.3.3Dynamicaldensitymatrixrenormalizationgroup256
7.3.4Chebyshevmatrixproductstates258
7.3.5Analyticlinearresponsedensitymatrixrenormalization group263
7.4Finitetemperaturealgorithms 264
7.4.1Lanczosdensitymatrixrenormalizationgroup264
7.4.2Dynamicaldensitymatrixrenormalizationgroup265
7.4.3Chebyshevmatrixproductstate266
7.5Applications 267
7.5.1Electronsystem267
7.5.2Electron phononsystem271
7.6Summaryandoutlook 275
8.Time-dependentdensitymatrixrenormalization
8.1Overview
8.1.1Time-dependentdensitymatrixrenormalization groupandnonadiabaticdynamics279
8.1.2Relationbetweentime-dependentdensitymatrix renormalizationgroupandmultilayer multiconfigurationtime-dependentHartree280
8.1.3Reviews,software,andotherresources283
8.2Timeevolutionalgorithms 284
8.2.1Propagationandcompression285
8.2.2Time-dependentvariationalprinciple288
8.2.3Timesteptargeting296
8.3Finitetemperaturealgorithms
8.3.1PurificationinanenlargedHilbertspace297
8.3.2Minimallyentangledtypicalthermalstates301 8.4Applications 302
8.4.1Excitonandchargetransferdynamics303
8.4.2Excitedstatedynamicsandspectra305
8.4.3Chargetransport307
8.4.4Electrondynamics307 8.5Summaryandoutlook
Thispageintentionallyleftblank
Preface Quantumchemistrystrivestosolvethemolecule’snonrelativistic SchrodingerequationorrelativisticDiracequationforthepurposeofaccuratelyunderstandingandpredictingitsdiversechemicalandphysicalproperties,includingthestructure,spectroscopy,andreactivityaswellasthe optoelectronicandmagneticresponses.Becausenumerousandcomplicated interactionsbetweentheelectronsandnucleiareinvolvedinamolecular system,accuratelysolvingthequantummany-bodyproblemisstillthemajor difficultyforquantumchemistry.Inthepastyears,thedevelopmentsofa varietyofpost-Hartree-Fock(HF)methods[e.g.,truncatedconfiguration interaction(CI),many-bodyperturbationtheory(PT),andcoupledcluster (CC)approaches]builtonHFmean-fieldsingle-determinantreferencewavefunctionhaveenabledelectronicstructurecalculationswithchemicalaccuracy(B1kcal/mol)forchemicalsystemswithhundredsofweaklycorrelated electrons.Challengesarisehowever,whenthetraditionalweakcorrelation assumptionbreaksdown.Inmanychemicalproblemssuchasbondbreaking/ formationinchemicalreactionsandtransitionmetalcatalysisinbiological photosynthesis,therearemanyenergeticallynear-degeneratefrontiermolecularorbitals,makingitimpossibletoapproximatetheelectronicwavefunctionbyusingonlyoneleadingcomponent.Insuchcases,inorderto describethestrongcorrelationstherein,allpossibleimportantdeterminants orconfigurationstatefunctions(CSFs)havetobefirstidentified.Forexample,thewidelyusedcompleteactivespace(CAS)methodsexpandthewavefunctionusingallpossibledeterminantsorCSFswithinanactivespace constructedfromapreselectedsetofactiveorbitals.Unfortunately,itis almostimpossibletoobtaintheexactsolutionforlargeactivespaces,asthe dimensionoftheconfigurationspacegrowsexponentiallywiththeincrease ofthesystemsize.Nowadays,thelargestexactlysolvableactivespaceis20 electronsin20orbitals(20e,20o).Thisgreatlyhinderedthesimulationof manyrealisticchemicalsystemsoflargeconjugatedmoleculesorpolynucleartransitionmetalcomplexes.Forexample,calculatingtheelectronic structureoftheMn4CaO5 clusterinphotosystemIIofphotosyntheticreactionorMn12 single-moleculemagnetrequiresatleast35or60activeorbitals,evenifonlyMn 3d valenceorbitalsandbridgingoxygen 2p orbitalsare includedintheactivespace.
Totacklethisso-calledcurseofdimensionality,sinceitsinventionin 1992byWhite,thedensitymatrixrenormalizationgroup(DMRG)hasbeen widelyknownbythequantumphysicscommunityasthecurrentlymost powerfulnumericalmethodinthestudyofone-dimensional(1D)strongly correlatedquantumlattices.DMRG’sextraordinarilyhighperformancecan beascribedtoitsefficientcompressionandlocalizedrepresentationofquantumstatesinitswavefunction’sentangledmatrixproductstate(MPS)formulationortheequivalenttensortrainstructureinmathematicslanguage.In 1997DMRGwasforthefirsttimeintroducedintotheoreticalchemistry communitybyShuaietal.,beingutilizedtosolvethesemiempiricalquantumchemicalHamiltonianforstudyingtheexcitedstatesinconjugatedpolymers.Laterin1999DMRGwasfurtherappliedtohelpthesolutionofab initioquantumchemicalHamiltoniansbyWhiteandMartin.Itshouldbe notedthattheDMRGimplementationinquantumchemistryisquitedifferentfromthatincondensedmatterphysics.Theorbitalsinchemicalmolecularsystemsdon’thavespatialtranslationsymmetrywhichisusuallypresent inthequantumlatticesiteswithinthecondensedmatterphysicsmodels. AnotheraspectisthattheabinitioquantumchemicalHamiltonianhasfourcenterandlong-rangeinteractionterms,whereascondensedmatterphysics models(likeHeisenbergandHubbardmodels)oftenhaveonlytwo-center andnearest-neighborones.Inaddition,inmanycasesaDMRGquantum chemicalcalculationhastobedoneinmomentumorenergyspaceinsteadof realspace,becauseusuallythereisnoevident1Dspatialtopologyforthe molecule’sactiveorbitals.AlltheseissuesmadetheapplicationofDMRG intoquantumchemistrybecomehighlynontrivial.
Fortunately,contributedbythecontinuouseffortsofmanyquantum chemistrygroups(includingXiang,Shuai,White,Chan,Reiher,Legeza, Zgid,Yanai,Kurashige,Wouters,Ma,etal.),DMRGhasbecomeoneofthe biggestbreakthroughsinquantumchemistryinthelastquartercenturyto tacklethechallengeofsimulatingstronglycorrelatedsystems.DMRGis nowwidelyusedasabenchmarkreferencewhentestingnewquantumchemicalmethodsforstrongelectroncorrelationproblems.Itevolvesfroma purelyapproximatefullCIsolvertobeingfullyadaptedtoavarietyofCAS andmultireference(MR)methods.Nowadays,theadvancedimplementation ofDMRGquantumchemistrycodehasgreatlyextendedthesolvableactive spacesize,from20orbitalsbyconventionalCASmethodtoaround100 orbitalsbyDMRG.Thedevelopmentofpost-DMRGmethodsbycombining DMRGwithMR-CI,MR-PT,andMR-CCaswellasdensityfunctionaltheorycanfurtherincludethedynamicelectroncorrelationenergyoutsidethe activespace,makingthecalculationmorequantitativelyaccurateforrealistic molecules.Theincorporationofgradientandresponsetheoryalsogreatly expandstheapplicationtoolboxforchemicalproblems,capableofdescribing variousgeometricalandspectroscopyproperties.Encouragingly,inthepast fewyears,time-dependentDMRG(TD-DMRG)wasalsosuccessfullyused
Preface xiii
andfurtherdevelopedbyquantumchemiststosimulatethereal-timenonadiabaticquantumdynamicsinmanychemicalproblems,rangingfrommodelingexcitondynamicsinphotovoltaicandphotosyntheticsystemsto simulatingvibrationallyresolved1Dandtwo-dimensionalelectronicspectroscopyinmolecularaggregatesandtohandlingcomplexproblemsuchas carrierandspintransportinmolecularmaterials.Ofcourse,DMRGquantum chemistrystillhasplentyofroomfordevelopmentbeforebeingarobust, user-friendly,andmultifunctionalmethodforpopularizingitsapplications. Theseinclude,butarenotlimitedto,designingnewpost-DMRGapproaches toaccountdynamiccorrelationswithoutusinghigh-order n-electronreduced densitymatrices,embeddingDMRG,orTD-DMRGinlargerchemicalenvironmentsandimplementingmassivelyparallelDMRGcalculations.
Aimingtopresentacomprehensiv ereviewandsummaryoftheoutstandingprogressintherapidlyde velopingDMRGquantumchemistry fieldinthelastquartercenturyan dinspirenewideasfordescribing stronglycorrelatedsystems,thisbookexploresthefundamentaltheories andalgorithmsofDMRG-basedquant umchemistryapproaches,detailing recentideasandkeydevelopmentsandprovidinganup-to-dateviewofthe currentunderstanding.WenoticethatDMRGmethodwasoriginatedin solid-statephysicsandonlylaterhasb eentransferredto quantumchemistry.Forresearchscientistsbyconventionalquantumchemistrytraining, languageandconceptsofDMRGareusu allynotfamiliar,sothereiscertainlearningbarrier.Therefore,wetrytoorganizethebookinapedagogicalmannertofacilitatethestudybygraduatestudents,tograspthe importantconceptsliketherelationshipbetweenDMRGalgorithmand MPSformulation,etc.WeexpectthisbookwillbeusefulforgraduatestudentsandresearcherswhoareinterestedindevelopingDMRG-basedmethodsforquantummany-bodyproblemsinchemistryorthosewhoare interestedinusingthestate-of-the -artDMRGmethodtodealwithchallengingchemicalproblemsofelectronicstructure anddynamics.
Inthisbook,Chapters1and2introducethefundamentalsandconcepts ofDMRG,MPS,matrixproductoperator(MPO),andtensornetworkstate aswellastheirrelationshipwithquantuminformationtheory.Mostofthe techniqueshavebeendevelopedbyquantumphysicistswiththespecialproblemsofquantummany-bodytheoryinmind,whichdifferfromthosein quantumchemistry.Wepointoutandhighlightthedifferencesforquantum chemists,pavingthewayforthemorechemistry-orientedexpositionsinlater chapters.
InChapters3and4theimplementationschemesofDMRGforsemiempiricalandabinitioquantumchemicalHamiltonians,therelatedtechnical detailsfortreatingMPOconstructions,implementationofsymmetry,aswell asselectingandorderingactiveorbitalsaredescribed.Asapreparationfor laterchapters,wealsodiscusshowtoperformwavefunctioncomponent analysisandcomputeone-andmany-bodyRDMs.
InChapter5weintroducethemethodsofDMRGself-consistentfield,in whichtheDMRGbasis(i.e.,molecularorbitals)isfurtheriterativelyand variationallyoptimizedinamolecularenvironment.Thealgorithmsforcalculatingthegradientsandthegeometryandspectroscopypropertiesaswell astheexcitedstatesarealsounraveled.Chapter6coversthedescriptionsfor variousabinitiopost-DMRG(DMRG-MR-CI,DMRG-MR-PT,DMRG-MRCC,etc.)approachestofurtheraccountfordynamicelectroncorrelations. Thetechniquestoincorporateenvironmentaleffectsarealsobriefly discussed.
Chapters7and8discusstheDMRGmethodsfordynamicalandrealtimepropertiesinthefrequencydomainandtimedomain.InChapter7we willintroducethefrequency-domainDMRGmethodsforthedynamical responseproperties,includingtheLanczos-DMRG,correctionvector DMRG,dynamicalDMRG,andChebyshevmatrixproductstate.In Chapter8thetime-domainTD-DMRGmethodsfornonadiabaticquantum dynamicsareintroduced.Thecommonlyusedtimeevolutionschemesare described.Thealgorithmstoincorporatetemperatureeffect,includingthermofielddynamicsandminimallyentangledtypicalthermalstate,arealso presented.Inbothchapters,severalapplicationsrangingfrompureelectron dynamicstovibronicdynamicsarecovered.
Weoweagreatdebtofgratitudetonumerouscollaborators,colleagues, andstudentswhohavehelpedtoshapeourthinkingandwhohaveprovided advicesinthepreparationofthisbook.Wecannotlistalltheirnameshere becausetheyarenumerousandwearesuretomisssome.Butwearetruly gratefultothemassomeoftheirsightshavepercolatedtheirwayintothis book.AspecialthankshouldbegiventoDr.JiajunRen,Dr.ZhenLuo,Dr. LuisCarlosVasquezCardenas,TongJiang,WeitangLi,andYifanCheng forhavingreadvariouspartsofthebookandprovidinginputs.Wewouldbe gratefultoreceiveerrataandwillmaintainanup-to-datelistoferrataonour websites.Pleasefeelfreetocontactanyoftheauthors: haibo@nju.edu.cn (HM), schollwoeck@lmu.de (US),or zgshuai@tsinghua.edu.cn (ZS).
UlrichSchollwo ¨ ck ZhigangShuai
HaiboMa
Chapter1 Densitymatrixrenormalization group 1.1Introduction
Thedensitymatrixrenormalizationgroup(DMRG)hasitsorigininthefield ofstronglycorrelatedquantumsystemsastheyappearincondensed-matter physics.Wewillsetoutbydiscussingitsgeneralframeworkfromthisperspective;thiswillallowustounderstandmoreclearly,whatmakes“original”DMRGsomewhatdifferentfromDMRGasadaptedtobeusefulfor problemsinquantumchemistry.Thisshouldmakethelanguageofmost seminalDMRGpapersinphysicsmoreaccessibletothereaderwithachemistrybackground;numerousparalleldevelopmentsseemtohavehappenedin condensed-matterphysicsandquantumchemistrythatcouldhavebeenmore mutuallyfruitfulifacommonlanguageexisted.
InthecaseofDMRG,therehasbeenachangeinthewayitisrepresented andthoughtaboutwithinphysicsitself;roughlyspeaking,apointofview anchoredinstatisticalphysicsandrenormalizationgrouptheory(asindicated bythename)hasgivenwaytothinkingofDMRGaspredominantly(butnot exclusively)avariationalmethod.Incondensed-matterphysics,theassociated changeofnotationsandcodesiscomparativelyeasyandhasbeenlargely achieved.Ithasopenedthewaytoimportantnewalgorithmicdevelopments, withDMRGbeing(only)oneofagroupofalgorithms.Inquantumchemistry, thistransitionturnsouttobemorecomplicated.Thereformulationisunderway,butalotoftherelevantliteratureisintheoldlanguage.Thefoundations ofthisnewapproachwillbecoveredinChapter2.
Wesetoutfromthe N -particletime-independentSchrodingerequation ^ H ψ 5 E ψ.Infirstquantizationandreal-spacerepresentation,thewavefunctiondependson3N coordinatesinspace, ψðr1 ; ...; rN Þ,andtheHamiltonian reads
Thefirstterm,wheretheoperator ri actsoncoordinate ri ,containskinetic energy,whichwetaketobenonrelativistic.Weassumethattheparticles
2 DMRG-basedApproachesinComputationalChemistry
interactthroughatwo-bodyinteraction V thatonlydependsonpositions.The N particlesaresupposedtobeelectrons,identicalfermions,suchthatthe wavefunctionmustmeetthefermionicantisymmetrizationrequirement
Notethatweignorespinforthemomentwherewearejustinterestedin thegeneralstructureoftheproblem.InordertosolvetheSchrodingerequationforHamiltonian(Eq.1.1),westartfromthesingle-particleHilbert-space ℋ1 .The N -particleHilbert-space ℋN isthenthetensorproductof N singleparticleHilbert-spaces ℋ1 ,
Inordertoproceed,wehavetogivebasestothesespaces.Intherealspacerepresentation,wechooseasingle-particlebasis fφk ðrÞg of ℋ1 ,a countablyinfinitesetofsquare-integrablefunctions.Nonumericalapproach canhandlethisinfinity,andwehavetoinvokeabasissettruncationto B basisstates(B $ N for N fermionicparticles).Thetruncatedbasisthen inducesabasisof ℋN astheproducts fφk1 ðr1 Þ
k2 ðr2 Þ...φkN ðrN Þg.Themost general N -particlewavefunctiontakestheform
which,ingeneral,willnotbeantisymmetric.Antisymmetrizationisimposedin thefirstquantizationbytheintroductionofSlaterdeterminants:wechoose N outof B basisfunctions,indexed ðk1 ; ...; kN Þ where k1 , k2 , ... , kN .Then
andthe N -particlewavefunctiontakestheform
wherethesumnowonlyrunsovertheordered N -tuples ðk1 ; ...; kN Þ.This approach,alsoextendedtoincludespins,iscoveredextensivelyinallquantumchemistryliterature,forinstance, SzaboandOstlund(1996).Itisuseful aslongas N issmallandwecantruncatethebasistosomesmall B. Ultimately,thelimitationrestsinthenumberofSlaterdeterminantsthat havenonnegligiblecoefficientsinEq.(1.6).
Incondensed-matterphysics,firstquantizationisoftenreplacedbythelanguageofsecondquantization,andmostof,ifnotallof,theliteratureon DMRGusesit.Thisissimplybecauseinthebulkmatter N B1023 ,andsecond quantizationisoftenaveryconvenientwaytoworkaroundthisproblem.
Westillhavesingle-particleand N -particleHilbert-spaces,butnowwealso introducedtheFockspace,thedirectsumofall N -particleHilbert-spaces,
(Wewilldiscuss ℋ0 inasecond.)Again,westartfromasingle-particle basis φk σ ðrÞ,wherenowwehaveintroducedaspin-degreeoffreedom σ .In thephysicsofstronglycorrelatedsystems,whereDMRGoriginated,the usualchoiceisWannierfunctions φinσ ðrÞ 5 φnσ ðr ri Þ:assumewehavea (say,cubic)latticewithlatticesites i,thenonecanconstructorthonormal functions φnσ ,whicharerepeatedidenticallyoneachlatticesite.Theconstruction,whosedetailswillnotbeusedhere,ensuresthat
Abasistruncationisagainrequired,and n islimitedtosomesmallnumber.InmanyDMRGapplications,likeforthesingle-bandHubbardmodel, wehaveasingle φðrÞ,thesameforspin-upandspin-down,towhichwe attachspinandrepeatitonlatticesites,leadingto φim and φik on-site i, whichformallycorrespondstoanorbitalinchemistry,whichcanaccommodateonespin-upandonespin-downelectron.
WenowconstructthebasisoftheFockspace ℱ fromthissingle-particle basis.Assumewehave L latticesites,soifwetakeonespin-upandone spin-downWannierfunction,wehave2L orthonormalbasisfunctionsand can,therefore,accommodateupto2L particles.Wenowintroduce(goingto theabstractket-notation)occupationnumberbasisstates jn1m n1k ...nLm nLk
, indexingsite(orbital)numberandspinorientation;the n givethenumberof electronsintheseorbitals,soall niσ A0; 1.Thecorrespondingfirst-quantized representationwouldbeaSlaterdeterminant.Themostgeneralquantum statethenreads
withoutfixingparticlenumber N atthemoment.Antisymmetrizationisnow introducedatthelevelofoperators,namelythecreationandannihilation operators ^ cy i;σ and ^ ci;σ :theycreateorannihilateanelectronwithspinorientation σ atsite i.Antisymmetrizationisensuredbyorderingtheorbitals(sites) arbitrarilyanddefiningtheactionoftheoperatorsas
4 DMRG-basedApproachesinComputationalChemistry
where n , isthesumoverall n “before” niσ intheorderedoccupationnumber state.Theresultistakentobe0ifthenewoccupationnumber niσ = 2f0; 1g.(We willnotprovetheclaimsmadehere,butrefertoanynumberofexcellenttextbookssuchas FetterandWalecka(2003) or BruusandFlensberg(2004)).
Ifwedefineavacuumstate j[i 5 j0; 0; 0; i,whichisinfacttheonly elementof ℋ0 ,anystate
i iscreatedfromitas
(Infact,onecouldalsostartfromthefilledFermiseaasreferencestate. Itonlymatterstohavethesignscorrectandrelativetoeachother,asglobal phasesdonotmatter.)Thereasonwhysecondquantizationisveryattractive isthattheHamiltonians,expressedincreationandannihilationoperators, becomequitesimpleandfermionicantisymmetrizationistakencareofautomatically:thekineticenergytermbecomesasumoftermswhereanelectron hopsfromsite(orbital) j tosite(orbital) i withsomeamplitude Tij ;inthe languageofoperators,itisannihilatedonsite j andcreatedonsite i. Similarly,two-bodyinteractions,whichwemaythinkofastwoelectrons beingscatteredoutoftheiroriginalstatesintotwonewones,becomesums overtwocreationandtwoannihilationoperators,
where Vijkl isalsoaparamaterfordescribinginteractionstrengths(assuming thatthesingle-particlewavefunctionsdonotdependonspin,butthegeneralizationissimple).
Asinteraction,weassumedtheCoulombinteraction,butthisisofcourse moregeneral.Inaddition,notethereversedorderofindicesinthetwoannihilationoperatorsin(13).
DMRGdealswithstatesandoperatorsinthisform.ManyofitsapplicationsconcernvariantsoftheHeisenbergmodel(onlylocalizedspindegrees offreedomwheretheproblemofantisymmetrizationdisappears)andthe Hubbardmodel,butmanyotherHamiltonianshavebeenstudied.Letus, therefore,introduceamoregenericnotation:Weconsiderasystemof L sites (orbitals)(Fig.1.1),whichhave d localdegreesoffreedom;thestatesare
FIGURE1.1 Ourtoymodel:achainoflength L withopenends,whereaspin-1 2 sitsoneach siteandinteractswithitsnearestneighbors.
denoted jσ i (ingeneral, σ will,therefore,labelmorethanjustspin).Fora chainof L spins 1 2, d 5 2,and jσ iAfjmi; jkig.TheprototypicalHeisenberg Hamiltonianreads
inonedimension.Inasingle-bandHubbardmodelwithnearest-neighbor hoppingandon-siteCoulombrepulsion,weconsider(insteadofonespin-up andonespin-downorbital)asinglespatialorbitalthatcanaccommodateup to2electrons.Its d 5 4statesaredenotedby j0i, jmi, jki, jmki,empty, withonespin-uporspin-downelectron,orwithoneeachforbothspinorientations.TheHamiltonianreads
Ingeneral,withtheexceptionofbosonic(inphysics,mainlyphononic; inchemistry,mainlyvibrational)modes, d isasmallnumber;thebosonic caserequiresspecialattention;ultimately,thebosonicoccupationnumbers havetobelimitedtosomemaximumandconvergenceofresultsunder changesofthismaximumchecked.Ourcomputationalbasisisthenformed bythetensorproductofthe L localbasesofdimension d each,
Itisanorthonormalbasis,
wherethelocalbasisstatesareorthogonalbetweensites,
0for i ¼ j.Themostgeneralquantumstatenowreads
Asantisymmetrizationistakencareofatthelevelofoperators,except forglobalnormalizationthereisnoconstraintonthecoefficients
Theobviousproblem(bothinfirstandsecondquantization,whichisjust smartbook-keeping)istheexponentiallylargenumberofexpansioncoefficients c
;here d L .Oneideaoftacklingthisissueistodecimatethe basisinsuchawaythatonlythosedegreesoffreedomthatarerelevantto theproblemathandarekept.Thisisthefundamentalideaoftherenormalizationgroupmethod,whichachievesthischoiceiteratively.Duringthese iterations,asequenceofeffectiveHamiltoniansisgeneratedbyprojecting ontothereducedstatespace.Thebasisstatesofthereducedstatespaceswill ingeneralbecomplicatedsuperpositionsoftheoriginalcomputationalbasis
6 DMRG-basedApproachesinComputationalChemistry
statessuchthatoperators(localoperators, n-pointcorrelators,andsoforth) havetobetransformedandprojectedaswell.Rescalingandrenormalizing arefurtherimportantstepswhichare,however,notimportantinourcontext. Buthowdowechoosethedegreesoffreedomtoberetained?
Akeyrenormalizationgroupmethodisthenumericalrenormalization group(NRG)of Wilson(1975),whichisveryperformantinthestudyofthe Kondoeffectformagneticimpuritiesinconductorsandmoregenerallyofthe physicsveryclosetotheFermisurfaceofsolids.(Theapplicationtoelectronic problemsisintroducedin Krishna-murthyetal.(1980) andanexcellentmore recentreviewis Bullaetal.(2008)).Withoutgoingintothedetails,NRGdiscretizestheelectronicconductionband,whichcontainstheFermisurfacelogarithmicallyinenergyspacebygroupingtogetherallofthe(dense)energy levelsofthebandwithinsomeenergyintervalwithanexponentiallyfineresolutionclosetotheFermienergy:thewidthoftheintervalsbecomesexponentiallysmall.Eachofthebandintervalsisthenassociatedwithexactlyone Hubbard-likeorbital,whichcancontainuptotwoelectronsandrepresentsall theenergylevelsofthebandintervalwithinappropriateapproximations:After asequenceofastutesimplifications,onefinallyarrivesataninfinitelylong chainofHubbard-likeorbitalswithexponentiallydecreasingnearest-neighbor single-particlehoppingasintheHubbardmodel,wherethehoppingis,however,constant.ThoseclosetotheleftendareclosetotheFermienergy,and furtherawayifwemoveright.Theelectronsintheorbitalsonsites i . 1do notinteract:theyrepresenttheeffectivelyfreeelectronsofaconductor.There is,however,aHubbard-type U -interactionon-site i 5 1,whichrepresentsthe magneticimpuritybecauseelectronsatthisimpurityarestronglylocalized and,therefore,“see”eachother.
NRGnowconsiderstheHilbert-spaceformedbytheleft-mostsites, whichhaveajointcomputationalbasis jm‘ i ofdimension D,whichwetake tobesomesmall,manageablepowerof d 5 4(D 5 4096 5 d 6 orsoisnot untypical)suchthatwebeginwith ‘ 5 logd D sites,6inourexample.We callthesesixsitestogetherwithablock.Nowweaddone(seventh)site, ‘ 1 1,withbasisstates jσ ‘ 11 i.Nowthe(old)blockplustheaddedsiteis takentobethe(new)block,withacomputationalbasis fjm‘ i jσ ‘ 11 ig.In thisvein,wecanaddsiteaftersite,butthenthebasisexplodesexponentially.NRGintroducesatruncationprescriptiontoobtainareduced D-dimensionalbasis fjm‘11 ig alsoforthenewblock,thatis,onein d states isretained.ThechoiceisgivenbydiagonalizingtheHamiltonianonthe blockof ‘ 1 1sitesandretainingthe D lowest-energyeigenstates(moreprecisely:thoseclosesttotheFermienergy;insuitablemappings,theseare thoselowestinenergy).Inthenewbasis,creationandannihilationoperators takenewforms,sotheHamiltonianhastobetransformed(andprojected)as well;aswehavediscardedbasisstate,theprojectionontothenewbasis involvesalossofinformation.(NRGhasimportantadditionalscalingsteps, whichneednotconcernushere.)
Densitymatrixrenormalizationgroup Chapter|1 7
NRGsolvedtheKondoproblem,whichhadbeenoneofthebigmysteriesofcondensed-matterphysics(withhindsight,notsurprisinglyso:itis inaccessibletoperturbativeapproaches)andwasoneofthereasonsfor Wilson’sNobelprizein1982.Notsurprisingly,itwasattemptedtoapply thesameprocedureofiterativegrowthofblocksofsitesanddecimationby alow-energyprescriptiontofindthegroundstatesofone-dimensionalquantumsystemssuchastheHeisenbergorHubbardchains.Thisdidnotwork despitethesuperficialsimilarity.Thelogarithmicdiscretizationprocedureof NRG,whichresultsinexponentiallydecayingscalesinthehoppingelements,leadstoanexponentialseparationofenergyscalesasoneincreases theblock,whichmakestheiterativeapproachpossible.BoththeHubbard andHeisenbergchainaretranslationallyinvariant,andcontributionsfrom “furtherdownthechain”willcontinuetobeofthesamesize.DMRGgrew outofthisdilemmaandprovidedaverypowerfulsolution.
Thisprehistoryexplainssomeaspectsoftheoriginalformulationof DMRG(White,1992),whichconsistsoftwosubsequentsteps,theinfinitesystemDMRG,whichisformallyquitesimilartoNRG(thoughconceptually verydifferent),andthefinite-systemDMRG,whichwasconsideredarequired add-ontoimprovenumericalprecision,butsomehownottheessenceofthe method(formoredetailssee,e.g., White(1993) or Schollwo ¨ ck(2005)).Inthe laterformulationsofChapter2,theemphasisiscompletelyreversed:infinitesystemDMRGisoneofseveralconceivablewarm-upprocedures,whilethe finite-systemDMRG,withsomeminormodifications,isrecognizedtobea variationalstateoptimizationwithinaconstrainedstatespace.DMRGturns outtobehighlysuccessfulforone-dimensionalquantumsystems,soletus focusinthefollowingonaone-dimensionalHubbardorHeisenbergmodel, thatis,electronsorspinsonachainoflatticesites.
1.2Infinite-systemdensitymatrixrenormalizationgroup Infinite-systemDMRGstartsexactlylikeNRGbyconsideringan(initially) smalloldblockof ‘ sitestowhichonesiteisadded,formingthenewblock, truncatingthebasis.Whatischangedisthedecimationprocedure.InNRG, onecouldignore“whatcomeslater”becauseoftheseparationofenergy scales.Here,inatypicallytranslationallyinvariantHubbardorHeisenberg chain,wehavetoimaginethattheoldblockplusonesiteispartofa thermodynamicallylargechain.Noterightawaythattypicalproblemsof quantumchemistrysaytheelectronicstructureofamoleculedonothave thisimportantsimplifyingaspect.Both Tij and Vijkl areingeneralverycomplicated.Inthermodynamiclanguage,theentirechainistheuniverse,the blockplussitethesystem S,andtherestofthechaintheenvironment E .A completedescriptionofthesystemasembeddedintheuniverseisprovided bythereduceddensityoperator ^ ρS 5 trE jψihψj,where jψi isthestateofthe universe.Diagonalizationofthereduceddensityoperatorprovidesabasis
forthesystem.Theimportanceofthebasisstatesisgivenbytheassociated eigenvaluesofthereduceddensityoperator,whichsuggestsatruncation scheme:keepthosestatesasnewbasisstates,whichhavethelargestweight.
Theproblemis,ofcourse,thatwedonotknow jψi andthattheenvironmentisthermodynamicallylarge.Wesimulateitnowbythebestapproximationwehave,namelyusingtheblockandsitealsoastheenvironment. CallingtheleftandrightblocksAandB,weobtaina(small)universeor superblockA B,wherethebulletsstandfortheindividualsites.Itisa chainoflength2‘ 1 2forblocksoflength ‘ .Ofcourse,wecanimaginethis procedurealsoforchainswheretheleftandrightblocksaredifferent,for instance,becausetheHamiltonianisnottranslationallyinvariant.Wegrow bothatthesametime,oneactingastheenvironmentfortheotherone.
Ingeneral,statesofthesuperblockA Bread
where fjσ A ig arethestatesoftheleftsingle-siteand fjσ B ig thoseoftheright one.
Letusassumethatwearelookingforthegroundstateofouruniverse. Weapproximateitbythegroundstateofthesuperblock,thenearestweget totheuniverse.Groundstate jψi minimizestheenergy
withrespecttotheHamiltonianofthesuperblock.ByimplementingthenormalizationviaaLagrangianmultiplier,thisminimizationproblemisturned intoaneigenvalueproblem
wheretheLagrangianmultiplier E0 isthenthegroundstateenergyofthe superblock.
Thisisalargeeigenvalueproblem:inmanycases, D 1000, d 5 4,and thevectordimensionis D2 d 2 5 16 3 106 inthiscase.Thereisnowayto solveitbyoneoftheusual“direct”algorithms,whichcandealwith,say,up todimension105 .Insuchacase,theonlywayoffindingthegroundstate anditsenergyisviaaniterativesparsematrixeigensolver,thatis,the LanczosorJacobi-Davidsonalgorithms.ThisrequiresthattheHamiltonian ^ H A B canindeedbebroughtintosparseform,thatis,ifexpressedasa matrix,onlyaverysmallfractionofthematrixelementsarenonzeroandwe knowtheirposition.Thisthenallowstocarryoutthebasicandmostcostly operationofiterativesparseeigensolvers,thematrix-vectormultiplication ^ H A B jφi effectively.Wewillshowthatthisisthecaseinamoment,and takeitforgrantednow.
Thereduced D-dimensionalbasis fjmA ig ofthenewblockA isnow determinedbyminimizingthedistancebetween jψi and j ~ ψi,thestate jψi projectedontothenewbasis,inthe2-norm.Onefindsaresultthatcanalso beunderstoodintuitively:weformthereduceddensityoperatorforA ,
whichintheuntruncatedproductbasisofA hasthematrixelements
ðρA Þii0 5 Pj ψij ψ⁎ i0 j .Thereduceddensitymatrixishermitian,andcanbediagonalizedwithrealnonnegativeeigenvalues,whichsumto1fornormalized jψi;theeigenvectorsformanorthonormalbasis.The D retainedbasisvectorsforA aresimplythoseeigenvectorsthathavethelargesteigenvalues; inotherwords,wekeepthestateswiththelargeststatisticalweight.Froma statisticalphysicsperspective,thisisverynatural.Bisgrownatthesame timebyusingthesameprocedure.Operatorshavetobetransformed (approximately)intothenewbasis(bases),apointwewillreturnto.This growthprocedureisrepeateduntilthesuperblockhasreachedthefinalsize L,givingusanapproximationtothegroundstate.Ofcourse, L cannotreach thethermodynamiclimit,butusuallytherelevantinformationcanalreadybe obtainedfromquitesmall L,saya(few)hundredsites.
Take L 5 100Hubbardsites.Thenthecompletebasishasdimension 4100 1:6 3 1060 .If D 5 1000,ourfinalbasishasdimension1:6 3 107 , about1053 smaller.WhyisDMRGsosuccessful?Oneobservesthat,atleast forone-dimensionaltranslationallyinvariantchains,evenformoderate D (say,afewhundred)thetruncationerror ε,thesumofthestatisticalweights ofthediscardedstatesisonly10 10 orevenlessateachgrowthstep,such thatthefinalwavefunctionisindeedanexcellentapproximationtothetrue one.Thisisaconsequenceofthetypicallylowentanglementofground statesofone-dimensionalquantumsystems,whichwewillbrieflydiscussin Chapter2.Infact,itisnumericallyadvantageous,perhapsalittlebitmore complicatedtoimplement,nottouse D inordertocontroltheapproximation,butrathertofixasmallmaximallyallowed ε andchoose D dynamicallytomeetthisrequirement.Thisensuresamoreevenlydistributed qualityoflocalquantitiesandsimplifiesextrapolationsin ε-0,theexact limit,ifwerunseveralDMRGrunsatdifferent ε.Mostquantitiescanbe extrapolatedeasilyin ε-0butnotin D (where D-N wouldbeexact).
Letusnowlookattheimplementationofoperators,inparticular,alsothe Hamiltonian.Forthemoment,weignorethatthefermioniccreationandannihilationoperatorscarryanontrivialsignthatdependsontheoccupationofother sitesandconsideranoperator ^ O actingpurelylocallyonsite ‘ ,withmatrix elements
isthesitejustbeingaddedintoblockA.When blockAgrowsfrom ‘ 1-‘ itsmatrixelementsreadinthenewblockbasis
Here, fjm‘ ig and fjm‘ 1 ig areorthonormalbasesoftheblocksoflength ‘ and ‘ 1.OperatorsthatwerealreadypartofblockA,alsoneedtobe transformed:
Itisimportanttorealizethatthesumin Eq.(1.24) mustbesplitas
reducingthecalculationalloadfrom OðD4 d Þ to2OðD3 d Þ.Thisisalsotypical ofmanyothercalculationsinDMRG.
Whiletheoperatorsalreadyinsomeblockcanbehighlydelocalized(in fact,theblockbasisisdelocalizedandthenotionoflocalitybecomesmeaningless),theinitialsteppresupposedalocaloperatoractingonjustonesite (thesitebeingadded).
InHamiltonians,operatorproducts ^ O ^ P occur,forinstance, ^ cy mi ^ ci11m for thenearest-neighborhoppingofanup-spinelectron.Wecanalsoimaginea longer-rangeterm.Letusassumethat i and i 1 1willultimatelybothbein blockA.Letusignorethefermionicsignforthemoment.Thenthecorrect wayistoobtainablockexpressionfor ^ cy mi asforthe ^ O discussedpreviously. WhenblockAhasincorporatedsite i,wehaveinthecurrentblockbasisthe matrixelements
Wenowreachsite i 1 1atthenextstep(oraftersomestepsinthecase ofalonger-rangedinteraction).Intheblock-sitebasis
isaproduct ofmatrixelements,whichbecome,inthe(new)blockbasis
Inallfurthersteps,thiscompoundoperatorlookslikeasingle-siteoperator. Allthislooksquitecumbersome,andthenotationintheMPSlanguageof Chapter2willbemuchsimplerwhenbeingusedtoit.
Fortypicalnearest-neighborhopping/interactionHamiltoniansof condensed-matterphysics,theHamiltonianwillthenalwaysbeoftheform
Weseeafirstsimplication,highlyimportantfortheuseofthesparse matrixalgorithms.Naively,withavectordimension D2 d 2 ,amatrix-vector
multiplicationwouldscaleasthedimensionsquared,thatis, D4 d 4 .Ifwe apply ^ H intheformjustgiven,itisinfactlargelydiagonal.Themostexpensivetermsaretheterms ^ H A and ^ H B ,which,however,arediagonalinthe otherblockandsite,respectively,suchthatthemultiplicationcostisonly D3 d 3 .Giventhat Dd isoftenafewthousand,thisspeed-upisalreadydrastic. Evenfornext-nearest-neighborhoppings/interactions,thecostremainsthe same.Butwecandoevenbetter,bysuitablebracketingofthemultiplicationsasbefore,toturnthisintotwooperationsofcost D3 d 2 (whichusually dominates)and D2 d 3
Inquantumchemistry(andofcourseinsomeapplicationsincondensedmatterphysics),wehavemuchlonger-rangedhoppings,forinstance,dueto two-operatorterms Tij ^ c y is ^ c js ,where s givesthemagnetization(spin).Inthis case,thematrixelementcanstillbedoneas
whichisasequenceoftwo OðD3 d 2 Þ multiplications(insteadofonenaive OðD4 d 2 Þ calculation).
Morecomplicatedly,wealsohavetoconsidertermslike Vijlk ^ cy
whichincondensed-matterphysicsusuallyshowupwhengoingtomomentum space(withasuitableinterpretationofthelabels).Inquantumchemistry,these termsarefrequentandnumerous(oftheorder L4 ).Thereare,therefore,two questions:Whatisthecostofthe“worst”contribution?Canwereducethe numberofterms?Again,suitablebracketingofthesumsalongthelinesgiven abovereducesthecosttooperations,whichatworstscaleas D3 d 2 (assuming Dcd ).Theissueofreducingtheirnumberissomewhattrickier.Thekeyidea wasdevelopedby Xiang(1996) inthecontextofmomentum-spaceDMRG, butfromapurelyformalpointofview(andignoringthesimplificationsina momentum-spaceHamiltonianthatcomefrommomentumconservation)the challengeisthesameasinquantumchemistry.Wewillpostponeitsdiscussiontothefinite-systemDMRGalgorithmtoputitintothemostgeneralperspective,butitwillturnoutthatafactor L2 canbegained.
Inanycase,theultimateevaluationofexpectationvaluesisgivenatthe endofthegrowthprocedureas
Bracketingturnsthisintoanoperationoforder OðD3 d 2 Þ.Localoperators thathappentobeononeofthesites canbeevaluatedevenmoreefficiently(cost D2 d 3 ,with Dcd ).
