Optimization of nonlinear wave function parameters

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OptimizationofNonlinearWaveFunctionParameters

RONSHEPARD,MICHAELMINKOFF

ChemistryDivision,ArgonneNationalLaboratory,Argonne,Illinois60439

Received4April2006;accepted17May2006

Publishedonline4August2006inWileyInterScience(http://wendang.chazidian.com).DOI10.1002/qua.21140

ABSTRACT:Anenergy-basedoptimizationmethodispresentedforourrecently

developednonlinearwavefunctionexpansionformforelectronicwavefunctions.Thisexpansionformisbasedonspineigenfunctions,usingthegraphicalunitarygroup

approach(GUGA).Thewavefunctionisexpandedinabasisofproductfunctions,allowingapplicationtoclosed-shellandopen-shellsystemsandtogroundandexcitedelectronicstates.Eachproductbasisfunctionisitselfamulticon?gurationalfunctionthatdependsonarelativelysmallnumberofnonlinearparameterscalledarcfactors.Theenergy-basedoptimizationisformulatedintermsofanalyticarcfactorgradientsandorbital-levelHamiltonianmatricesthatcorrespondtoaspeci?ckindofuncontractionofeachoftheproductbasisfunctions.Theseorbital-levelHamiltonianmatricesgiveanintuitive

representationoftheenergyintermsofdisjointsubsetsofthearcfactors,theyprovideforanef?cientcomputationofgradientsoftheenergywithrespecttothearcfactors,andtheyallowoptimalarcfactorstobedeterminedinclosedformforsubspacesofthefullvariationproblem.Timingsforenergyandarcfactorgradientcomputationsinvolvingexpansionspacesof?1024con?gurationstatefunctionsarereported.Preliminaryconvergencestudiesandmoleculardissociationcurvesarepresentedforsomesmallmolecules.©2006Wiley

Periodicals,Inc.IntJQuantumChem106:3190–3207,2006

Keywords:nonlinear;wavefunction;optimization;GUGA

methodisbasedonthegraphicalunitarygroupapproach(GUGA)ofShavitt[3–9].Thisnewap-proachisintendedtobeusedinMCSCF[10,11]andcon?gurationinteraction(CI)[6,12]wavefunc-tions,anditisbeingdevelopedwithintheCOLUM-BUSProgramSystem[12–14],whosemainempha-sisistheaccuratecomputationofglobalpotentialenergysurfacesofgroundandexcitedstates.Thewavefunctionisexpandedinabasisofproductfunctions,andeachproductfunctiondependsonarelativelysmallnumberofnonlinearparameters.Inthepreviouswork,wedevelopedrecursiveproce-duresforef?cientcomputationoftheoverlapbe-tweentwobasisfunctionSMN??M?N?,Hamilto-

1.Introduction

I

npreviouswork[1,2],wehaveintroducedacomputationalmethodbasedonanewtypeofexpansionbasisforelectronicwavefunctions.This

Correspondenceto:R.Shepard;e-mail:shepard@tcg.anl.govContractgrantsponsor:U.S.DepartmentofEnergy(Of?ceofBasicEnergySciences,DivisionofChemicalSciences,Geo-sciencesandBiosciences).

Contractgrantnumber:W-31-109-ENG-38.

©2006WileyPeriodicals,Inc.*ThisarticleisaUSGovern-mentworkand,assuch,isinthepublicdomainintheUnitedStatesof

America.

InternationalJournalofQuantumChemistry,Vol106,3190–3207(2006)©2006WileyPeriodicals,Inc.

OPTIMIZATIONOFNONLINEARWAVEFUNCTIONPARAMETERS

walks,soitisconvenienttoorganizethegraphbasedonstorageofthenodes;thestorageoftheconnectingarcs,andotherinformationdiscussedbelow,associatedwitheachnodeiscalledadistinctrowtable(DRT).EachCSFmaybeassignedacon-tiguousintegerindexthatmaybecomputedasasummationoftheintegerarcweightsassociatedwiththearcs:

n?1

FIGURE1.Graphicalrepresentationofthefourpossi-blearctypesinaShavittgraph.Stepd?0corre-spondstoanemptyorbital,d?1toasinglyoccu-piedorbitalthatincreasesthespinby1?2,d?2toasinglyoccupiedorbitalthatdecreasesthespinby1?2,andd?3toadoublyoccupiedorbital.a,b,cvaluesofthetopnodeofthearcaregivenrelativetotheval-uesofthebottomnodeofthearc.

m?1?

p?0

?y

n?1

dpjp

?1?

p?0

?y

??p?

.(1)

??N?,transitionnianmatrixelementsHMN??M?H

one-particlereduceddensitymatricesDMNpq,andtransitiontwo-particlereduceddensitymatricesMNdpqrs.Fromthesequantities,groundandexcitedelectronicstateenergiesmaybecomputedalongwiththeexpectationvaluesofotherarbitraryone-andtwo-electronoperators.Inthepresentwork,weexaminetheoptimizationofthenonlinearwavefunctionparameterstominimizetheenergy.

2.Method

Wesummarizebrie?ythemethodandnotationthathavebeenintroducedpreviously[1,2].EachnodeofaShavittgraph,indexedbyj,isassociatedwithanintegertriple(aj,bj,cj)andcorrespondsto?2spineigenfunctionwitheigenvalueSj(Sj?1)anS

withbj?2Sj,toaspeci?cnumberofelectronsNj?2aj?bj,andtoasubspaceoftheorbitalsofdimen-sionnj?aj?bj?cj.TheintegertriplecorrespondstoarowofaPaldusABCtableau[15–20].TheShavittgraphisadirectedgraphwithasingletail(source)nodelocatedata?ctitiouslevel0corre-spondingtothephysicalvacuum,andasinglehead(sink)atthehighestlevelcorrespondingtotheNandSofinterest.ThenodesoftheShavittgraphareconnectedbyarcs.ThefourarctypesareshowninFigure1.Eachcon?gurationstatefunction(CSF)expansiontermcorrespondstoawalkfromthegraphtailtothegraphhead.Thiswalktouchesonenodeateachlevel,andittouchesonlythesinglearcateachlevelthatconnectsthenodebelowittothenodeaboveitinthatwalk.ACSFcantherebyberepresentedbydenotingeitherthesetofnodesinthecorrespondingwalkorthesequenceofsteps(thestepvector)inthatwalk.InatypicalShavittgraph,anindividualnodemaybetouchedbymany

Weadopttheconventionthatjpisthenodeindexofthebottomofthearcinthewalkofinterestatlevelp,anddpisthestepnumberassociatedwiththearc.Inthisway,thepairofindices(d,j)specifyanarc.Inthefollowing,itissometimesconvenienttode-notea(d,j)pairbyasinglearcindex?,and?(p)inEq.(1)isthearcatlevelpinthewalk.FromtheinformationstoredintheDRT,itisstraightforwardtoconstructthestepvectorfromagivenCSFindexm,ortodothereverseanddeterminetheintegerCSFindexmfromagivenstepvector.

Inaproductbasisfunction,anumericalarcfac-torisassignedtoeachofthearcsinagivenShavittgraph.Thesearcfactorsaredenotedindividuallyas?djwhere,analogoustotheydjnotationofthearcweightsgivenabove,jistheindexofthenodeatthebottomofthearcanddisthestepnumberofthearc.TheCSFcoef?cientxmassociatedwithapartic-ularwalkmistakenastheproductofthearcfactorsinthatwalk.Thatis,inanalogytoEq.(1):

n?1

xm?

p?0

??

n?1

dp,mjp,m

?

p?0

??

??p,m?

.(2)

Becauseonearcfactorisassociatedwitheachor-bitallevelinthisproduct,therearealwaysexactlynarcfactorsthatcontributetoeachoftheCSFcoef?cients.Aproductbasisfunction,denoted?M?,isthende?nedintermsoftheseCSFcoef?cientsas

Ncsf

?M??

m?1

??.?x?m

Mm

(3)

Therecanbeseveralsetsofarcfactors,eachasso-ciatedwithacorrespondingproductfunctionthroughEqs.(2)and(3).ThemappingofasetofarcfactorstothevectorofCSFcoef?cientswillbedenotedasxM?L(?M).Thisisa

many-to-one

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SHEPARDANDMINKOFF

mappingbecausemorethanonesetofarcfactors?maptothesameCSFcoef?cientvectorx.Thislackofuniquenessmaybeeliminated[1]byintroducinganarcphaseandnormalizationconventionbasedonthelowerwalkpartialproductfunctionsassoci-atedwitheachnodeoftheShavittgraph.Asetofarcfactors?Mthatsatisfythisnormalizationcon-ventionisinstandardform,andsuchasetofarcfactorsmaybecharacterizedbyasmallernumberofessentialvariationalparameters?M.Thenumberoftheseessentialvariables[1]isgivenbyN??Narc?Nrow?1,whereNarcandNrowarethenumberofarcsandnodes,respectively,intheShavittgraph.Thesestandardformarcfactorsalsohaveasimpleandintuitivemathematicalandphysicalinterpre-tation[1].

Forthespecialcaseoffull-CIShavittgraphs,therelevantgraphstatistics(seeRefs.[1,3,15]andAppendixA)aregivenbyb?1n?1n?1Ncsf?acwithn?Nrangingfromn?2ton?46.Notein

particularthatthelastrowcorrespondstoanex-pansionspaceof?5.5?1024CSFs;thisis?9.2NA,or?9moles,ofCSFs,manyordersofmagnitudelargerthancanbetreatedwithtraditionalCImeth-ods,yetonlyamodesteffortofafewsecondsisrequiredtocomputeHMNforthisexpansion.Thegoaloftheproductfunctionapproachistoelimi-natealleffortthatdependsonNcsfthatisassociatedwithwavefunctionmanipulation,interpretation,andoptimization.ThisgoalisapracticalnecessityforwavefunctionexpansionssuchasthoseinthebottomhalfofTableIorlarger.

Figure2presentsanexampleofaShavittgraphfora3-electron,3-orbital,doublet,full-CIexpan-sion,whichconsistsof8CSFs.Thecorrespondingproductfunctionisgivenintermsoftheseexpan-sionCSFs(usingstep-vectornotation)as

???x?m

8

m

????

?6????1??2??3?a?c??1?

1

1

???3

(4)

?M??

m?1

Nrow??a?1??b?1??c?1?

(5)

???3,1?1,2?0,5??310????3,1?0,2?1,6??301????1,1?3,3?0,5??130????1,1?2,3?1,6??121????1,1?1,3?2,7??112????1,1?0,3?3,8??103?

(6)

???0,1?3,4?1,6??031????0,1?1,4?3,8??013?.

(12)

Althoughasingleproductbasisfunctionisa

complicatedmulticon?gurationalfunctionthatiscapableofdescribingbonddissociationprocesses,electroncorrelation,andhasotherinterestingfea-tures[1],itisnotpossibleingeneraltowriteanarbitrarylinearcombinationofCSFsasasingleproductfunction.Amore?exiblewavefunctioniswrittenasthegenerallinearcombination

Narc??2a?1?b?2c?1???1??4??5?

??a?c????1??2??1?

N??b?3ac?a?c???ac?a?c?

?2???1????2??3???a?c??3??2??,

1

(7)

with??min(a,c).Forthespecialcaseofsingletfull-CIexpansionswithn?N,thesegraphstatisticsaregivenby

n?11

1Ncsf?

2n

??

2

(8)(9)(10)(11)

????

Nrow??n?2??n?3??n?4?/24Narc?n?n?2??2n?5?/12N??n?n2?3n?2?/8.

?c?M?,

N?

MM

(13)

Forlargen,theseexactrelationsgiveapproxi-matelyNcsf?(8/?)4n/n2,Nrow?n3/24,Narc?n3/6?4Nrow,N??n3/8?3Nrow,n?log4(Ncsf),and??Nrow/n??(N2).TableIshowsthesevalues

inwhichtheN?productfunctions?M?formanexpansionbasis.Theoptimizationofthelinearex-pansioncoef?cientsctominimizetheenergyex-pectationvaluetakestheformofageneralizedsymmetriceigenvalueequation

Hc?ScE,

(14)

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OPTIMIZATIONOFNONLINEARWAVEFUNCTIONPARAMETERS

TABLEI______________________________________________________________________________________________

Statisticsforsingletfull-CIwavefunctionexpansions.n?N246810121416182022242628303234363840424446

a

Ncsf

3201751,76419,404226,5122,760,61534,763,300449,141,8365,924,217,93679,483,257,3081,081,724,803,60014,901,311,070,000207,426,250,094,4002,913,690,606,794,77541,255,439,318,353,700588,272,005,095,043,5008,441,132,926,294,530,000121,805,548,126,430,067,9001,766,594,752,418,700,032,40025,739,723,541,439,406,257,200376,607,675,256,599,252,232,0005,531,425,230,331,301,517,157,500

Nrow51430559114020428538550665081910151240149617852109247028703311379543244900

N?21339861602674136048461145150719382444303137054472533863097391859099121136312949

t(HMN)a0.000.000.000.000.000.000.010.040.070.130.210.340.540.821.211.752.493.464.666.278.2511.1914.43

t(E?)b0.000.000.000.010.050.160.441.243.489.2925.6765.49140.61250.45423.87676.761.07E31.62E32.38E33.48E34.93E36.88E39.47E3

t(E?;FD)c0.000.000.020.100.643.2012.3936.24118.44297.70632.941.32E32.64E34.97E38.97E31.57E42.66E44.37E46.89E41.08E51.64E52.54E53.74E5

??N?matrixelement.Timesareinsecondsona2.5-GHzPowerMacG5toconstructasingle?M?H

TimesinsecondstoconstructtheanalyticgradientvectorE?(?0)??E(?)/??(mM)??0forN??1,usingtheG[u]andS[u]arrays.c

Timesinsecondstoconstructthegradientwitha?nite-differenceapproximation,t(E?;FD)?2N??t(HMN).

b

??N?andSMN??M?N?.Theef?-withHMN??M?H

cientcomputationofthemetricmatrixSandoftheHamiltonianmatrixHhasbeendescribedprevi-ously[1,2].Giventwosetsofessentialvariables?Mand?Nandthecorrespondingarcfactors?Mand?N,thecomputationofanoverlapSMNelementscalesas?(?n),andthecomputationofaHamilto-nianmatrixelementHMNscalesas?(?n4).Thesca-larfactor??Nrow/n?Narc/nscalesbetween?(N0)and?(N2),dependingonthecomplexityoftheunderlyingShavittgraph,anditcorrespondstotheaverageincrementaleffortperorbitallevelintherecursivecomputationofHMNandSMN.ThereisnocomponentofthiseffortthatscalesasNcsf.ThroughtheRitzvariationalprinciple,thelowesteigenval-uescomputedfromtheproductfunctionbasisinEq.(14)areupperboundstothecorrespondingeigenvaluesoftheunderlyinglinearCSFexpansionspace,whichinturnareupperboundstotheexactfull-CIeigenvalues.Consequently,thisgeneralap-proachisapplicabletobothgroundandexcitedelectronicstates.Thewavefunction???isaspin

eigenfunctionbecausetheproductbasisfunctions

?2;thereforethismethod?M?areeigenfunctionsofS

doesnotsufferfromspincontaminationorspininstabilities.

If?M?and?N?aretwoarbitraryproductfunctionsde?nedbythearcfactors?Mand?N,aHamilto-nianmatrixelementintheproductfunctionbasismaybewrittenusingstandardGUGAnotationas??N?HMN??M?H

?

?h??h

p,qp,q

pq

?pq?N??1?M?E2

MNDpq?1

2

pq

p,q,r,s

?g

p,q,r,s

?g

d

pqrs

?M?e?pqrs?N?

MN

pqrspqrs

?Tr?hDMN??1Tr?gdMN?,2

(15)

wherehpqandgpqrsaretheone-andtwo-electron

Hamiltonianintegralsindexedbythemolecularor-?pqandebitalindicesp,q,r,ands.OperatorsE?pqrs?

?pqE?rs??rqE?psarethegeneratorsandthenormal-E

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SHEPARDANDMINKOFF

?cientsfortwostates??I?and??J?fromEqs.(13)and(14),thestatetransitiondensitymatricesmaybecomputedas

IJ?pq?E?qp??J??Dpq?1??I?E2

M,N

?ccD

IJ

MN

MNpq

IJdpqrs?1??I?e?pqrs?e?pqsr?e?qprs?e?qpsr??J?4

?

M,N

?ccd

IJMN

MNpqrs.(17)

FIGURE2.Shavittgraphforathree-electron,three-orbital,doubletfull-CIexpansion.Nodeindexisde-notedbythecircledvalues,andarcweightydjbythesquareboxes;arcfactor?djiswrittennexttoitscorre-spondingarc.

ordergeneratorproducts.Thecommutationrela-?pq,E?rs]??rqE?ps??psE?rqresultsintheoper-tion[E

atoridentitye?pqrs?e?rspq.Becauseoftheindexsymmetryoftheone-andtwo-electronintegrals(hpq?hqp,gpqrs?gpqsrandgpqrs?grspqwithallquantitiesassumedtobereal),wechoosetouseinEq.(15)thesymmetrizedone-andtwo-particletransitiondensitymatrices,de?nedas

MN?pq?E?qp?N??1?M?EDpq

2

MN

??M?e?pqrs?e?pqsr?e?qprs?e?qpsr?N?.dpqrs

Fromthesequantities,arbitraryexpectationvalues

(I?J)andtransitionproperties(I??J)maybecomputed.

AsdiscussedindetailinRef.[2],thetransitiondensitymatricesarecomputedintermsofsegmentvaluesofShavittloops.Inthepresentwork,itisconvenienttoignorethespecialtreatmentoftheupperandlowerwalksforeachShavittloop,orforalltheupperandlowerwalksforgroupsofShavittloops.Inthiscase,thesegmentfactorscorrespond-ingtothesegmentswithinanupperorlowerwalkareassociatedwiththesegmenttypeD.Theseg-menttypeD,whichisactuallyanaliasforthesegmenttypeRL0,hasthesegmentvalues?[Tu(dbra,dket,bbra,bket;D)]??dbra,dket?bbra,bket???bra,?ket;thatis,thesegmentvalueisunitywhenthebraandketarcsarethesameandithasavalueofzerootherwise.Withthisconvention,eachShavittloopvalueconsistsofaproductofexactlynseg-mentfactorsfornmolecularorbitals.

ThetransitiondensitymatricesinEq.(16)arecomputedasasequenceofsparsematrixproductsofsegmentfactors,denotedFMN???(Qu).Thisfactor-izationismosteasilyrealizedfromtheauxiliarypairgraphrepresentationoftheShavittloops.Fig-ure3showsthisrelationforatypicalone-particleShavittloop.Aone-particledensityelementiswrit-tenas

MNDpq?FMN?Q0?FMN?Q1?...FMN?Qp?1?FMN?Qp?...

1

4

(16)

?...FMN?Qq?1?FMN?Qq?...FMN?Qn?2?FMN?Qn?1?,

(18)

withQu?Dforu?0...(p?2)andforu?q...(n?1).Asimilarrelationholdsforthetwo-particledensityconstruction.Inthisexpression,pandqaretheorbitallevelsassociatedwiththedensityelement.FMN(Qu)isasparserectangularmatrixindexedbythenodepairs?atleveluandnodepairs??atlevel(u?1).Thesegmentfactor

is

Theseexpressionsde?nethenormalizationandin-dexingconventionsofthedensitymatricesinthiswork(seealsoRefs.[2,11]).Theadjointoperator

??pq?E?qpidentityEresultsintheidentitiesDMN?

DNManddMN?dNM.Giventheone-andtwo-particletransitiondensitymatrixelements,itisstraightforwardtocombinethesequantitieswiththeappropriateHamiltonianintegralstocomputethematrixelementHMN.Giventheexpansioncoef-

3194INTERNATIONALJOURNALOFQUANTUMCHEMISTRYDOI10.1002/quaVOL.106,NO.15