半經驗方法Semiempiricaltheory密度泛函理論DensityFunctionalTheory(DFT)快速計算方法第一性原理計算ExactSolutionHFMP2CCSD(T)DZTZQZBasisSetExpansionFullCIWaveFunctionExpansionHF

LimitTypicalCalculations

BasisSetAllpossibleconfigurationsNeedtospecifymethodandbasissetwhendescribingacalculationInteractionbetweenbasissetandcorrelationmethodrequirepropertreatmentofbothforaccuratecalculations.dilemmaaccuracycostWhynotusebestavailablecorrelationmethodwiththelargestavailablebasisset?chemicalaccuracyComputationalCostC2H6C4H101天27=128天

等待并行ComputationalCostAMP2calculationwouldbe100xmoreexpensivethanHFcalculationwithsamebasisset.ACCSD(T)calculationwouldbe104xmoreexpensivethanHFcalculationwithsamebasisset.TriplingbasissetsizewouldincreaseMP2calculation243x(35).Increasingthemoleculesize2x(sayethanebutane)wouldincreaseaCCSD(T)calculation128x(27).為什么從頭算方法慢?N粒子基Slaterdeterminant6重積分!NxNmatrix為什么從頭算方法慢?單粒子基GTOSTOGTO1,2-electronintegrals6重積分!為什么從頭算方法慢?Howmanyintegrals?電子數HFCI(H2O)2dimer億億億為什么從頭算方法慢?太多電子！太多積分！I/Obottleneck硬盤積分數據IO解決方案:不直接計算積分,用參數代替半經驗方法Semiempirical不直接解方程密度泛函理論DensityFunctionalSemi-empiricalMOMethods基本思路ThehighcostofabinitioMOcalculationsislargelyduetothemanyintegralsthatneedtobecalculated(esp.twoelectronintegrals).Semi-empiricalMOmethodsstartwiththegeneralformofabinitioHartree-Fockcalculations,butmakenumerousapproximationsforthevariousintegrals.Manyoftheintegralsareapproximatedbyfunctions

withempiricalparametersthatareadjustedtoimprovetheagreementwithexperiment.Semi-empiricalMOMethods基本思路Coreorbitalsarenottreatedbysemi-empiricalmethods,sincetheydonotchangemuchduringchemicalreactionsOnlyaminimalsetofvalenceorbitalsareconsideredoneachatom(e.g.2s,2px,2py,2pzoncarbon)HFlevel沒有電子相關valence只有價電子STO-V3G最小基組functions擬合實驗Semi-empiricalMOMethods(1918-1997)non-benzenoid

aromaticity

"AMolecularOrbitalTheoryofOrganicChemistry",I,II,III,IV,V,VI,JACS,1952,3345-3350-3353-3354-3356-3363MOPACprogramMolecularOrbitalPACkageHuckelmethodErichHuckelextendedHuckelmethodRoaldHoffmannCNDO/2,INDO,NDDOJohnPopleMINDO,MNDO,AM1,PM3,RM1andSAM1Lysergicaciddiethylamide(LSD)49atomsgeometryoptimization:1974@CDC6600week2006@pc1minnowseconds~10000atomsExtendedHückelMethodH----HamiltonianmatrixCi----Columnvectorofthemolecularorbitalcoefficientsi----OrbitalenergyS----OverlapmatrixH

Ci=i

S

CiH

----Chooseasaconstant(valenceshellIP)H=KS(H+H)/2近似:Wolfsberg-Helmholtzconstant,1.75R.

Hoffmann,J.Chem.Phys.39,1397(1963).ExtendedHückelMethodconformation

cited1151ZeroDifferentialOverlap(ZDO,零級微分重迭)

TwoelectronrepulsionintegralsareoneofthemostexpensivepartsofabinitioMOcalculationsNeglectintegralsiforbitalsarenotthesameApproximateintegralsbyusings

orbitalsonly

CompleteNeglectofDifferentialOverlapCNDO[CNDO/1,CNDO/2]J.A.Pople,D.P.SantryandG.A.Segal,J.Chem.Phys.,1965,43,S129.totalnumberofsuchintegrals[N(N+1)/2][N(N+1)/2+1]/2N4/8N(N+1)/2N2/2全略微分重迭(H2O)220000200cited:478

IntermediateNeglectofDifferentialOverlapINDO間略微分重迭J.A.Pople,D.L.Beveridge,andP.A.Dobosh,J.Chem.Phys.47,2026(1967)keepintegralswhenA=B=C=DnowrarelyusedMINDO,ZINDO,SINDOINDOcited:415

ModifiedIntermediateNeglectofDifferentialOverlapBingham,R.C.,Dewar,M.J.S.andLo,D.H.J.Amer.Chem.Soc.,1975,97,1285.MINDO,MINDO/1,MINDO/2,MINDO/3MINDO/3參數化MINDO/3參數化生成焓偶極矩ZINDO/1,ZINDO/s

Zerner's

IntermediateNeglectofDifferentialOverlapMichaelZerner(1940-2000)groundstategeom.excitedstatesUVspectra

Symmetricorthogonalised

INDOSINDO,SINDO/1D.N.NandaandK.Jug,,TheoreticaChimicaActa,57,95,(1980)dorbitalsfor2ndrowelementNeglectofDiatomicDifferentialOverlapNDDO忽略雙原子微分重迭J.A.Pople,D.L.Beveridge,andP.A.Dobosh,J.Chem.Phys.47,2026(1967)keepintegralswhenA=B&C=DThebasisofmostsuccessfulsemiempiricalmethodsMNDOAM1SAM1RM1PM3PM6

ModifiedNeglectofDifferentialOverlapMNDODewar,M.J.S.andThiel,W.,J.Amer.Chem.Soc.,1977,99,4899.MNDO/dThiel,W.andVoityuk,A.A.,J.Phys.Chem.,1996,100.616.+dbasisfunctionsMNDOCThiel,W.,J.Amer.Chem.Soc.,1981,103,1413.+correlationsDewar,M.J.S.andThiel,W.,J.Amer.Chem.Soc.,1977,99,4899.databaseparameterizationDewar,M.J.S.andThiel,W.,J.Amer.Chem.Soc.,1977,99,4899.cited:372Thiel,W.andVoityuk,A.A.,J.Phys.Chem.,1996,100.616.cited:87Thiel,W.,J.Amer.Chem.Soc.,1981,103,1413.cited:68notwelltested.

AustinModel1AM1

SemiempiricalabinitioModel1SAM1Dewar,M.J.S.,Zoebisch,E.G.,Healy,E.F.andStewart,J.J.P.,J.Amer.Chem.Soc.,1985,107,3902.Tetrahedron,1993,23,5003.MNDO+AM1/dPt-oligoolefinsbindingenergy

ParameterizedModelnumber3PM3thesameformalismandequationsastheAM1method,butcorerepulsionfunction:PM3usestwoGaussianfunctionsAM1usesbetweenoneandfourGaussians/elementStewart,J.J.P.J.Comput.Chem.1989,10,209.Stewart,J.J.P.J.Comput.Chem.1989,10,221.Stewart,J.J.P.J.Comput.Chem.1991,12,320.cited:4982

ParameterizedModelnumber3Stewart,J.J.P.J.Mol.Model.2004,10,155.Stewart,J.J.P.J.Mol.Model.2007,13,1173.PM6cited:261geometricaloptimization!BScatStrathclydeUniversity,Glasgow,Scotland,in1969PhDatStrathclydeUniversity,Glasgow,Scotland,in1972DScatStrathclydeUniversity,Glasgow,Scotland,in1995AuthoredthefirstMOPACwhileworkinginProfessorMichaelDewar'sgroup,1983.BeenworkingonMOPACnowfor27years.Authoredover140papers.In1999,wasreportedtobethe15thmost-citedchemistintheworld.WorkedattheFrankJ.SeilerResearchLaboratoryattheAirForceAcademyinColoradoSpringsfrom1984-1991.Becameaconsultant(asoleproprietor)in1991,andworkedasaconsultanttoFujitsuuntil2004.Hasbeenanindependentdevelopersincethen.HasseveralPCs,andworksoutofaroominthebasementofhishouseinColoradoSprings.Hasnostudentsorco-workers,butcommunicatesviatheInternet.Hehastwocats,awife,andasnow-blower,noneofwhichwork.ScienceorTechnique?Semi-empiricalmethods:heavilyparameterizedmethodsFit-an-elephantFreemanDysonEnricoFermi(1901-1954)(1923-)meson–protonscatteringcalculatednumbersagreedprettywellwithFermi'smeasurednumbers"Therearetwowaysofdoingcalculationsintheoreticalphysics.Oneway,andthisisthewayIprefer,istohaveaclearphysicalpictureoftheprocessthatyouarecalculating.Theotherwayistohaveapreciseandself-consistentmathematicalformalism.Youhaveneither."IndesperationIaskedFermiwhetherhewasnotimpressedbytheagreementbetweenourcalculatednumbersandhismeasurednumbers.Hereplied,“Howmanyarbitraryparametersdidyouuseforyourcalculations?”Ithoughtforamomentaboutourcut-offproceduresandsaid,“Four.”Hesaid,“IremembermyfriendJohnnyvonNeumannusedtosay,withfourparametersIcanfitanelephant,andwithfiveIcanmakehimwigglehistrunk.”ScienceorTechnique?Semi-empiricalmethods:heavilyparameterizedmethodsFit-an-elephantFreemanDysonEnricoFermiFit-an-elephantFreemanDysonEnricoFermiScienceorTechnique?heavilyparameterizedSemi-empiricalmethodsindependentofexperimentsexperiment-dependenttruth&onlytruthuseful&usable密度泛函理論DensityFunctionalTheoryDFTThewavefunctionitselfisessentiallyuninterpertable.Reduceproblemsize:WavefunctionsforN-electronsystemscontain4Ncoordinates.Wavefunctionbasedmethodsquicklybecomeintractableforlargesystems,evenwithcontinuedimprovementincomputingpower,duetothecoupledmotionoftheelectrons.Adesiretoworkwithsomephysicalobservableratherthanprobabilityamplitude.MotivationElectronicEnergyComponentsTotalelectronicenergycanbepartitioned:E=ET+ENE+EJ+EX+ECET,ENE,&EJarelargestcontributorstoEEX>EC

ET=KineticenergyoftheelectronsENE=CoulombattractionenergybetweenelectronsandnucleiEJ=CoulombrepulsionenergybetweenelectronsEX=Exchangeenergy,acorrectionfortheself-repulsionsofelectronsEC=CorrelationenergybetweenthemotionsofelectronswithdifferentspinsThomas-Fermi-Dirac(TFD)ModelEnergyisafunctionoftheoneelectrondensity,Nuclear-electronattraction&electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergyXModelTFDdoesnotpredictbondingandthetotalenergiesareinerrorby15-50%.IfthevalueinSlater’sExistreatedasparameter,thenbetterresultsareachieved.TheXmodel(aka.Hartree-Fock-Slater)uses=3/4.AlthoughXhasbeensupercededbymodernfunctionals,itisstillusefulforinorganicsystemsandpreliminarycalculations.TheNobelPrizeinChemistry1998“forhisdevelopmentofthedensity-functionaltheory"WalterKohn(1923-)1925-2004TheoreticalBasiscanbewrittenasasingleSlaterdeterminantoforbitals,butorbitalsarenotthesameasHartree-FockEXCtakescareofelectroncorrelationaswellasexchangeEnergyisafunctionalofthedensityE[]Thefunctionalisuniversal,independentofthesystemTheexactdensityminimizesE[]Appliesonlytothegroundstate

HohenbergandKohn(1964)KohnandSham(1965)

VariationalequationsforalocalfunctionalTheHohenberg-KohnTheorem

propertiesareuniquelydeterminedbytheground-stateelectron

In1964,HohenbergandKohnprovedthat:molecularenergy,wavefunction

andallothermolecularelectronic

probabilitydensity

namely,Phys.Rev.136,13864(1964)

.”“Formoleculeswitha

nondegenerate

groundstate,theground-state

Densityfunctionaltheory(DFT)attemptstoandotherground-statemolecularproperties

fromtheground-stateelectrondensity

calculate

probabilitydensityandotherproperties”emphasizesthedependenceoftheexternalpotential

differs

fordifferentmolecules.“Forsystemswithanondegenerategroundstate,theground-stateelectrondeterminestheground-statewavefunctionandenergy,,whichHowever,thefunctionalsareunknown.isalsowrittenasThefunctionalindependentoftheexternalonispotential.TheHohenberg-kohnvariationaltheorem“Foreverytrialdensityfunctionthatsatisfiesandforall,thefollowinginequalityholds:,isthetrueground–stateenergy.”whereTheKohn-Shammethod

Ifweknowtheground-stateelectrondensity

molecularpropertiesfromfunction.,theHohenberg-Kohntheoremtellsusthatitispossibleinprincipletocalculatealltheground-state,withouthavingtofindthemolecularwave

1965,KohnandShamdevisedapracticalmethodforfinding

andforfinding

from.[Phys.Rev.,140,A1133(1965)].Theirmethod

iscapable,inprinciple,ofyieldingexactresults,butbecausetheequationsof

theKohn-Sham(KS)methodcontainanunknownfunctionalthatmustbeapproximated,theKSformationofDFTyield

approximateresults.electronsthateachexperiencethesameexternalpotential

theground-stateelectronprobabilitydensity

equaltotheexactofthemoleculeweareinterestedin:.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingthatmakesofthereferencesystemSincetheelectronsdonot

interactwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

自由電子氣模型Thus,theground-statewavefunctionofthereferencesystemis:

isaspinfunctionorbitalenergies.areKohn-ShamForconvenience,thezerosubscriptonisomittedhereafter.Defineasfollows:ground-state

electronickineticenergysystemofnoninteractingelectrons.(either)isthedifferenceintheaveragebetweenthemoleculeand

thereference

Thequantityrepulsionenergy.units)

for

theelectrostaticinterelectronicistheclassicalexpression(inatomicRememberthatWiththeabovedefinitions,

canbewrittenasDefinetheexchange-correlationenergyfunctionalbyNowwehaveside

are

easytoevaluatefromgetagoodapproximationto

totheground-stateenergy.

Thefourthquantity

accurately.

ThekeytoaccurateKSDFT

calculationofmolecular

propertiesisto

Thefirstthreetermsontherightisarelativelysmallterm,butisnoteasytoevaluate

andtheymakethe

maincontributionsThusbecomes.Nowweneedexplicitequationstofindtheground-stateelectrondensity.sameelectrondensityasthatinthegroundstateofthemolecule:isreadilyprovedthatSincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,itground-stateenergybyvaryingtominimizethefunctional

canvarytheKSorbitals

minimizetheaboveenergyexpressionsubjecttotheorthonormalityconstraint:TheHohenberg-Kohnvariationaltheoremtellusthatwecanfindthe

soas.Equivalently,insteadofvaryingweThus,theKohn-Shamorbitalsarethosethatwiththeexchange-correlationpotential

definedby(Ifisknown,itsfunctionalderivative

isalsoknown.)CommentsontheDFTmethods:(1)TheKSequationsaresolvedinaself-consistentfashion,liketheHFequations.(2)ThecomputationtimerequiredforaDFTcalculationformallyscalesthe

third

power

ofthenumberofbasisfunctions.(3)ThereisnoDFmolecularwavefunction.(4)TheKSorbitalscanbeusedinqualitativeMOdiscussions,liketheHF

orbitals.TheKSoperatorexchangeoperatorsintheHFoperatorarereplacedbytheeffectsofbothexchangeandelectroncorrelation.isthesameastheHFoperator

exceptthatthe,whichhandles(5)Variousapproximatefunctionals

DFcalculations.Thefunctionalandacorrelation-energyfunctionalAmongvariousCommonlyusedandPW91(PerdewandWang’s1991functional)Lee-Yang-Parr(LYP)functionalareusedinmolecularapproximations,gradient-corrected

exchangeandcorrelationenergyfunctionalsarethemostaccurate.PW86(PerdewandWang’s1986functional)B88(Becke’s1988functional)P86(the

Perdew1986correlationfunctional)

(6)NowadaysKSDFTmethodsaregenerallybelievedtobebetterthantheHFmethod,andinmostcasestheyareevenbetterthanMP2

iswrittenasthesumofanexchange-energyfunctional

ConstructingDensityFunctionalsExactformisunknown.Hohenberg-Kohnisonlyanexistenceproof.Densityfunctionalshavetheform:ForLSDA:a=b=c=0Forpurefunctionals:a=0Systematicimprovementoffunctionalsispossible,butcomplicatedbythefactthatexactconstraintsandpropertiesofsaidfunctionalsarestillbeingelucidated.IncreasingChemicalAccuracyDecreasingComputationalCostsAccuracyvs.ComputationalCostLSDAGGAMeta-GGAX1951Dirac1930G96B86B88PW91PBE1996RPBE1999revPBE1998xPBE2004PW86mPWTPSS2003BR89PKZB1999Exchange,ExCS1975LSDAGGAMeta-GGAW38xPBE2004PW86PBE1996PW91LYP1988B95TPSS2003PKZB1999B88VWN1980PZ81PW92CAData1980Correlation,EcCalculatingExcTermsExchange-correlationfunctionalsmustbenumericallyintegratednotasrobustasanalyticmethods.Energiesandgradientsare1-3timesthecostofHartree-Fock.Frequenciesare2-4timesthecostofHartree-Fock.Someofthiscomputationalcostcanberecuperatedforpuredensityfunctionalsbyemployingthede