1ParrRGYangWDensityFunctionalTheoryofAtomsandOxfordUniversityPressNewYork19892DreizlerRMGrossEKUDensityFunctionalTheorySpringerVerlagNewYork19903ZieglerTChemRe6514 ID: 166521
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WhatDotheKohnShamOrbitalsandEigenvaluesMean?RalfStowasserandRoaldHoffmann*ContributionfromtheDepartmentofChemistry,BakerLaboratory,CornellUniIthaca,NewYork14853-1301edJuly29,1998.ReisedManuscriptReceiedFebruary16,1999Shamorbitalsandeigenvaluesarecalculatedwithgradient-correctedfunctionalsforasetofsmallmolecules(HO,N,CrH,andPdCl),varyingbasissetsandfunctionals.ThecalculatedKohnSham(KS)orbitalshapes,symmetries,andtheorderandabsoluteenergyoftheassociatedeigenvaluesareinvestigatedandcomparedwiththoseofHartreeFock(HF)andone-electronextendedHuÈckel(eH)calculations,aswellasexperimentalionizationpotentials.TheshapeandsymmetrypropertiesoftheKSorbitalsareverysimilartothosecalculatedbyHFandeHmethods.Theenergyorderoftheoccupiedorbitalsisinmostcasesinagreementamongthevariousmethods.Theorderofemptyorbitalsofaminimalbasissetissometimesinterchanged,withinthatgrouporwithsomeorbitalsresultingfromalargerbasiscalculation.OveralltheKSorbitalsareagoodbasisasBaerendssuggestedforqualitativeinterpretationofmolecularorbitals.FortheShameigenvalueswefindanapproximatelylineardependencyofIP)fortheoccupiedaswellasfortheunoccupiedorbitaleigenvalues.WesuggestanscalingforquantitativeinterpretationofKSeigenvalues,atleastifthesearecalculatedutilizingcommonlyusedfunctionals.1.IntroductionChemistshavefoundorbitalsusefulatomicorbitals,mo-lecularorbitals,theorbitalsofmolecularfragments.Orbitalsprovideanaturallanguageforanofthecomplexrealityofthemoleculesoftheinorganicandorganicworld.AsHartreeFockself-consistent-fieldtheoryanditsimprove-mentthroughconfigurationinteraction(HF-SCFandCI)evolved,theªrealityºoforbitalswasbothstrengthenedandweakened.ThestrengtheningwasderivedfromKoopman'stheorem,whichprovidedasimpleandnaturalconnectionbetweenorbitalenergyandionizationpotential(enhancedbythedevelopmentofphotoelectronspectroscopy,whichef-fectivelyallowedustomeasureeasilyionizationpotentials(IP's)otherthanthefirst).However,thelimitationofthesingle-configurationviewpointweakened,sotospeak,therealityoforbitals.Ifmanystatesofamoleculearenotwelldescribedbyone(ortwo)configurations,theorbitalidealosesitsutility.Weareinatimeofascendencyofdensityfunctionaltheoryforthecomputationoftheelectronicstructureofmolecules.Themethodologycontainsorbitalstobesure,theKohn(KS)orbitalswhichwewilldenoteas,withassociated.Butthefundamentalvariable,whichdeterminesallobservables,isthetotalelectrondensityFromthebeginningoftheutilizationofthedensityfunctionalmethod,thesignificanceoftheKohnShamorbitalshasbeendeemphasized,perhapsbecauseititsverydifficulttoextractquantitativeinformationfromtheseorbitals.Theyhavebeenoftenviewedasjustanauxiliaryconstruct,anecessarybutnotnecessarilymeaningfulwaytobuilduptheall-importanttotaldensity.Herethenisthecruxoftheproblemchemistsknowthatorbitalsareuseful,butthephysicistsandchemistswhousedensityfunctionaltheorysofruitfullyhavebyandlargeshiedawayfromattributingtoKohnShamorbitalstherealitythat(wethink)theydeserve.Aswithanygeneralstatement,thisisanexaggeration.Therearetheoreticiansprovidedawelcomeplacefororbitalsinthedensityfunctionalschemeareprominentexamples.Baerends'impressiveworkpinpointsthephysicalsignificanceofKSorbitalsbyªsplittingtheexchange-correlationpartoftheKSpotentialintoapartthatisdirectlyrelatedtothetotalenergyandaso-calledresponsepartthatisrelatedtoresponseoftheexchange-correlationholetodensitychangeº.Baerendsandco-workersinfactarguethattheKSorbitalsaresuitableforqualitative,chemicalapplications.Meanwhile,withoutwaitingforthejustificationthatBaerends'argumentsprovide,thechemicalcommunityhasmainlyinthepastfewyearstoapplyKSorbitalsinrationalizingchemicalphenomena,asweareusedtodoingwithextendedHuÈckel(eH)orbitals. (1)Parr,R.G.;Yang,W.Density-FunctionalTheoryofAtomsand;OxfordUniversityPress:NewYork,1989.(2)Dreizler,R.M.;Gross,E.K.U.DensityFunctionalTheory;Springer-Verlag:NewYork,1990.(3)Ziegler,T.Chem.Re,651.(4)Kohn,W.;Sham,L.J.Phys.Re,1133. (5)Perdew,J.P.;Norman,M.R.Phys.Re,5445.(6)Perdew,J.P.;Levy,M.Phys.Re.Lett.,1884.(7)Kohn,W.;Vashishta,M.R.InTheoryoftheInhomogeneousElectron;Lundquist,S.,March,N.H.,Eds.;Plenum:NewYork;p79.(8)Levy,M.;Perdew,J.P.;Sahni,V.Phys.Re,2745.(9)Tozer,D.J.;Handy,N.C.;Green,W.H.Chem.Phys.Lett.,183.(10)Kohn,W.;Becke,A.D.;Parr,R.G.J.Phys.Chem.(11)Zhao,Q.;Parr,R.G.Phys.Re,2337.(12)Zhao,Q.;Parr,R.G.J.Chem.Phys.,543.(13)Baerends,E.J.;Gritsenko,O.V.;vanLeeuwen,R.InApplicationofDensity-FunctionalTheory;Laird,B.B.,R.B.,Eds.;(14)Baerends,E.J.;Gritsenko,O.V.J.Phys.Chem.,5383.(15)Bickelhaupt,F.M.;Baerends,E.J.;Ravenek,W.Inorg.Chem.,350.(16)Rosa,A.;Baerends,E.J.NewJ.Chem.,815.(17)DeKoch,R.L.;Baerends,E.J.;Hengelmolen,R.,289.(18)Sargent,A.L.;Titus,E.P.,65.(19)Garland,M.T.;Saillard,J.-Y.;Ogliaro,F.;Otero,M.;Roman,E.Inorg.Chim.Acta,253.(20)Ehlers,A.W.;Baerends,E.J.;Bickelhaupt,F.M.;Radius,U.Eur.J.,inpress.(21)Bickelhaupt,F.M.;Radius,U.;Ehlers,A.W.;Hoffmann,R.;Baerends,E.J.NewJ.Chem.,1.(22)Hoffmann,R.J.Mol.Struct.(THEOCHEM),1.J.Am.Chem.Soc.10.1021/ja9826892CCC:$18.001999AmericanChemicalSocietyPublishedonWeb03/25/1999 Inthispaperweconfronttheproblemdiscussedaboveempiricallyanddirectly,wehope.Weasktwoquestions:(1)AretheKSorbitalsdifferentinnumber,symmetryproperties,andshapefromtheorbitalsofaHF-SCFcalculationorfromthoseofaone-electronscheme?(2)WhatshallwemakeoftheShameigenvaluesgivenbycurrentlypopularpotentials?Howaretheyrelatedtoionizationpotentials?DotheyprovideanenergyorderingoftheorbitalsthatresemblesordiffersfromthatgivenbyHF-SCF,one-electronschemes,orexperimental2.MethodologyKSandHF-SCFcalculationswereperformedbymeansoftheGaussian94programpackage.ForallHFandDF(functionalBP86)calculationsonfirst-andsecond-rowmol-eculesthe6-31G*basissetwasused.Fortransition-metalcomplexesweworkedwiththecomparablesizeLANL1DZbasissetwithaLosAlamosECP.ForthevariationofbasissetsandfunctionalstheSTO-3G,3-21G,6-31G,6-31G*,6-311G,6-311G*,6-311G**basissetsandBHandH,X-BP86,BLYP,PW91,HFS,andSVWN(LDA)exchangecorrelationfunctionalswereinvestigated.eHorbitalenergyvalueswerecalculatedwiththeprogramYAeHMOPandtheparametersetgiveninTable1.Wewillstudythefollowingmoleculesindetail:HO,N,andPdCl.IncaseoftheDFandHFcalculationsweperformafullgeometryoptimization;forthehypotheticalwelookedatsingle-pointcalculationonaMP2/double-TheeHresultsarefortheexperimentalgeometries(H)andfortheoptimizedgeometryofCrH2.1.OrbitalShape:TheCaseofWater.Ofcourse,thenumberandsymmetrypropertiesoftheKSorbitals,thecanonicalHForbitals,andeHorbitalsarethesamefortheoccupiedle(theeHorbitalsdonotcontainthecore1slevels).Coulditbedifferent?CouldyouimaginethattheKSorbitalsof,forexample,aNeatomarenot1s,2s,2p-like?OnewouldjustifiablythinktheeHorbitalsareapoorapproximation(toeithertheHF-SCForDFTorbitals),butevenaspoorasthatapproximationmightbe,thequalitativedescriptionoftheorbitals,reallydeterminedasitisbytheirnodalstructure,shouldbethesame.Weneedtobealittlemoreprecise,however,indefiningseveraltypesoforbitals:groupI,corelevels(e.g.1sonoxygen);groupII,occupiedvalenceorbitals;groupIII,unoc-cupiedvalenceorbitals;groupIV,higherunoccupiedorbitals,mainlyaresultofthelargerbasis.Figure1showstheone-electronenergiesoftheseorbitalsforeH,DFT,andHF-SCFcalculationsforawatermolecule.TheeHcalculationsbydefinitionuseavalenceorbitalbasisset;therefore,theylackgroupIIVorbitals.TheHF-SCFandDFTcalculationshaveorbitalsofallkinds,thenumberofgroupIVorbitalsdepending,ofcourse,onthisbasissetsize.TheborderbetweengroupIIIandIVlevelscannotbesharp.Inparticular,thereareexcitedstateswhicharereasonablywell-decribedbyvalence-stateMO'sandothers(e.g.Rydbergstates)whichneedalargebasisforanaccuratedescription.Thereisnodoubtthateachmethodgives(asidefromthecorestates)forwaterfouroccupiedvalenceenergylevelsof (23)ManyofthesequestionshavebeenaddressedbyR.G.ParrandwhoseexcellentworkprovidesaquantitativeaccountoftherelationshipbetweenKSandHForbitalsandsuggeststhattheEHmethodbeviewedasaparticularempiricizationoftheKSmethod.(24)Trucks,G.W.;Head-Gordon,M.;Gill,P.M.W.;Wong,M.W.;Foresman,J.B.;Johnson,B.G.;Schlegel,H.B.;Robb,M.A.;Replogle,E.S.;Gomperts,R.;Andres,J.L.;Raghavachari,K.;Binkley,J.S.;Gonzalez,C.;Martin,R.L.;Fox,D.J.;Defrees,D.J.;Baker,J.;Stewart,J.J.P.;Pople,J.A.Gaussian94;Gaussian,Inc.,Pittsburgh,PA,1994.(25)Landrum,G.A.:YetAnotherextendedHuÈckelMolecularOrbitalPackage.YAeHMOPisfreelyavailableontheWWWattheURLhttp://(26)Thesimplestconceivable18-electrontransition-metalcomplex)isnotstableinDFandHFgeometryoptimizations,presumablyduetothehighnegativecharge.Tocalculatethisprototype,weusedthereforetheoptimizedCrHgeometryanddidasingle-pointSCFcalculationforthe6(27)Kang,S.K.;Tang,H.;Albright,T.A.J.Am.Chem.Soc.,1971. (28)Herzberg,G.L.MolecularSpectraandMolecularStructure.III.ElectronicSpectraandElectronicStructureofPolyatomicMolecules;VanNostrandReinhold:NewYork,1966.(29)Huber,K.P.;Herzberg,G.L.MolecularSpectraandMolecularStructure.IV.ConstantsofDiatomicMolecules;VanNostrandReinhold:NewYork,1979.TablesofInteratomicDistancesandConfigurationsinMoleculesandIons;SpecialPublicationNo.11;TheChemicalSociety:London,1958. Table1.ParametersUsedintheEHCalculations atomorbitalH1s13.61.3N2s26.01.95013.41.950O2s32.32.27514.82.275Cl3s26.32.18314.21.733Cr4s8.661.75.241.711.224.950.50601.800.6750Pd5s7.322.193.752.15212.025.9830.55352.6130.6701 Figure1.CalculatedoccupiedvalenceandvirtualorbitalsforwaterbyBP86/6-31G*,RHF/6-31G*,andeHmethods,aswellastheexperimentalvalues.TheFermilevelisindicatedbyadottedline;thecompressionoftheKSlevelsrelativetoHFlevelsishighlightedbydashedlines.ShamOrbitalsandEigenaluesJ.Am.Chem.Soc.,Vol.121,No.14,1999 symmetry.Wewilldiscusstheenergiesoftheselevelsandthesignificanceoftheeigenvaluesinthenextsection;herewewanttofocusontheirshape.Figure2showsacontourdiagramofthefouroccupiedvalenceMO'sofwatercomputedbythethreemethods(twobasissetsforHF-SCFandDFT).Aretheseorbitalssimilarordifferent?TheeHorbitalsarbitrarilyexcludethecoreandusenodelessSlaterfunctions;thus,theeHorbitalsarerecognizablydifferent,especiallynearthenuclei.Theotherfoursetsoforbitalsare,onthescaleofthefigure,nearlyindistinguishable(wecouldexaggeratethedifferencewithadensitydifferencemap),andawayfromthenucleitheyarenotthatdifferentfromtheeHorbitals.2.2.OrbitalEnergiesandTheirOrdering,andtheRelationshiptoIonizationPotentials:Water.ThegeneralliteratureofdensityfunctionaltheorystatesthattheenergyofthehighestoccupiedKSorbital(HOMO)hasphysicalsignificance,inthesensethatthevalueoftheHOMOisintheoryequaltothefirstionizationpotentialInpractice,withthecommonlyusedfunctionalsimplementedinstandardquantumchemistryprogrampackages,evenHOMOenergiesdiffersignificantlyfromexperimentaldata(e.g.IP14.52eV6.50eVforthenitrogenatom).Thisdeviationarisesfromtheinsufficientcancellationoftheself-interactionerrorintheHartreetermtermF(ri)][W(ri,rj)]-[F(rj)]bytermsoftheoppositesignintheexchange-correlationfunctionals.Newfunctionalsandap-proximationsfortheKSpotentialhavebeendeveloped(e.g.LSDSIC,OEP,KLI)whichcorrecttheself-interactionerrorandleadtopreciseagreementoftheenergywiththefirstIP;hopefullytheseapproximationswillbeimplementedinquantumchemistrypackagessoon.WebeginourstudywiththeinfluenceofthebasissetchosenontheKSenergylevels.InFigure3theKSorbitalenergiesofthewatermolecule,calculatedwiththeBP86functionalanddifferentbasissets,areshown.TheabsoluteKSorbitalenergyvaluesare(withtheexceptionoftheinadequateSTO-3G (31)Theproofgivenin1982thatIPhasbeenrecentlyquestionedbyKleinman.(32)Levy,M.Phys.Re,1133,12044.(33)Perdew,J.P.;Parr,R.G.;Levy,M.;Balduz,J.L.,Jr.Phys.Re,1691.(34)Kleinman,L.Phys.Re,12042.(35)Moore,C.E.AtomicEnergyLe;NBSCircular;NationalInstituteofStandardsandTechnology:Washington,DC,1998;p467.(36)Trickey,S.B.Phys.Re.Lett.,881.(37)Grabo,T.;Gross,E.K.U.Chem.Phys.Lett.,141(38)Chen,J.;Krieger,J.B.;Li,Y.;Lafrate,G.J.Phys.Re,3939. Figure2.Calculatedcontourplots(plane)ofthea,andborbitalsofwater(forb0intheplane;thisorbitalishenceplottedwithanoffsetof0.5…)withBP86/3-21G,BP86/6-31G*,RHF/3-21G,RHF/6-31G*,andeHmethods. Figure3.CalculatedoccupiedvalenceandvirtualKSorbitalsforwaterusingaBP86functionalbutvaryingthebasisset.TheoccupiedcoreorbitalsaredenotedastypeI,theoccupiedvalenceorbitalsastypeII,thevirtualorbitalsarisingfromaminimalbasisastypeIII,andallothervirtualorbitalsastypeIV;theFermilevelisindicatedbyadottedJ.Am.Chem.Soc.,Vol.121,No.14,1999StowasserandHoffmann minimalbasisset)roughlyindependentofthebasissetfortypeIIandIIIorbitals.ThesymmetriesandorderofallKSorbitalsarenotinfluencedbyexpandingthebasisset.Next(cf.Figure4)westudytheinfluenceoftheexchange-correlationfunctionalusedontheKSorbitalenergiesofthewatermolecule.WeincludedtheBHandH,Xalpha,BP86,BLYP,PW91,HFS,andSVWN(LDA)exchangecorrelationfunctionals.Thebasisisfixedasacommon6-31G*setofTheKSenergylevelsdrawninFigure4showsubstantialabsoluteenergyshifts,dependingontheappliedfunctional.However,therelativespacingofthelevels(e.g.theHOMOLUMOgap)isapproximatelyconstantwithintheentireset(exceptforthehybridfunctionalsB3LYPandBHandH).Thevariationintheorbitalenergiesmaybetracedtodifferentself-interactionerrorsoftheappliedfunctional.Iftheabsoluteshiftsarecorrectedbyasuitablescaling,e.g.byaconstantenergyrelativetotheexperimentallyaccessibleHOMOenergy(asmeasuredbytheionizationpotential),thentheorbitalenergiesofallexaminedfunctionals(exceptthehybridfunctionalB3LYPandtheBHandHfunctional)matchverywell.However,ifnoscalingisapplied,onehastobeawarethatdifferentcontem-poraryfunctionalsmayleadtoorbitalenergiesdifferentbyupto10eVfromeachotherforagivenvalenceorbital.Areviewer(whomwethank)aptlyremarks:ªThefinitebasissetcanbearbitrarilylargeuntilitsfinitenesscauses`negligible'errorsgivenenoughcomputingpower.ImprovingapproximateusableXC-potentialsisanongoingmajortheoreticalchallengeº.Thereismoretobelearnedfromtheorbitalenergiescomputedbythevariousmethods.InFigure1theKSorbitalenergiesofwater(calculatedwiththeBP86/6-31G*functional/basissetcombination)wereshown,alongwiththeorbitalenergylevelsofHartreeFock(RHF/6-31G*)andextendedHuÈckel)calculationsandtheexperimentallydeterminedverticalIP's.Asfarastheoccupiedorbitalenergiesgo(cf.Figure1;thegapbetweenfilledandunfilledlevelsisindicatedbyadottedline),theHFresultsmatchwelltheexperimentalIP's.TheeHresultsareinacceptableagreementwithexperi-mentaldata(cf.Figure1)aswell.However,theKSenergylevelsareshiftedbyaconstant(e.g.fortheHOMOeV)tohigherenergy,relativetotheHFresults.Duringpreparationofthispaperwebecameawareofworksimilarinmotivationtoours,byPolitzerandAbu-Awwad.TheseauthorsexaminetherelationshipbetweenvariousIP'sandtheKSandHF's.TheirresultsfortheHOmoleculearesimilartoours,andingeneraltheyfindthatthecalculatedKS'sdiffersignificantlyfromtheexperimentalIP's,aswedo.ItmustbementionedrightawayherethatpeoplewhouseDFTcalculationsandareinterestedinIP'sorultravioletphotoelectronspectroscopy(UPS)donotgenerallyfocuson'sbutcalculateIP'sas(cationradical),choosingtheappropriatecationradicalstate.Theresultsareingeneralquitesatisfactory.Still,onewouldliketoseeifaªKoopman's-theoremlikeºassociationofIP'swithKScalculatedwithcommonfunctionalsmightwork.2.3.AScalingRelationship.Wefindasystematic(ifasyetmysterious)relationbetweenKSenergies,calculatedwithgradientcorrectedfunctionals,andIP's(whichareclosetotheHFvalues).InFigure5weplottheenergydifferencebetweentheHFandKSorbitalenergiesvstheHForbitalenergies(whichareclosetotheIP'sfortheoccupiedlevels).Onecanseealinearenergyshiftfortheoccupiedorbitalenergies(leftpartofthegraph;rhomboidsymbols).Incalculationsonextended (39)Siegbahn,K.;Nordling,C.;Johansson,G.;Hedman,J.;Heden,P.F.;Hamrin,K.;Gelius,U.;Bergmark,T.;Werme,L.O.;Manne,R.;Baer,ESCAAppliedtoFreeMolecules;North-Holland:Amsterdam,1969.(40)Politzer,P.;Abu-Awwad,F.Theor.Chem.Acc.,83.(41)Politzer,P.;Abu-Awwad,F.;Murray,J.S.Int.J.QuantumChem.inpress.(42)Thislineardependencyisnotnecessarilyvalidforthe(yetunknown)trueKSeigenvalues. Figure4.Calculatedoccupiedvalenceandvirtualorbitalsforwater,assuminga6-31G*basisandvaryingthefunctional.TheFermilevelisindicatedbyadottedline. Figure5.DifferencesofRHF/6-31G*andDF(BP86,B3LYP)/6-31G*calculatedorbitalenergiesplottedvsRHF/6-31G*orbitalenergiesforShamOrbitalsandEigenaluesJ.Am.Chem.Soc.,Vol.121,No.14,1999 systemssimilarlineardependenciesbetweenKSandso-calledquasi-particlebandenergiesappear.MostintriguinginFigure5(rightpartofthegraph;graph;symbols)isthatthislinearenergyshiftapparentlyalsoappliestothevirtualorbitals!ThisisconsistentwiththefactthatcalculatedvirtualKSenergylevelsareusuallyfoundtobeoflowerenergythanthoseofHFToobtainacorrespondencebetweenKS(calculatedwithgradient-correctedfunctionals,e.g.BP86)andHForbitalenergies,anempiricalscalingoftheformappearstobeadjustsfortheconstantshift(self-interactionerror),accountsphenomenologicallyforthelinearscalingwhichwefind.Wedefinedinthatwaythattheslopesoftheinterpolationlines(cf.Figure5)fortheoccupiedandvirtualorbitalenergydifferences,respectively,istheinterceptofthecrossingofthetwointerpolationlines.Specificcalculationofgivesthefollowingfor4.68eV,andTheotherdatasetinFigure5(symbolizedbytriangles)showswhattranspiresifonesusestheB3LYPfunctional.Thedeviationasoneexpectsfromthehybridcharacterofthefunctional(intermediatebetweenHFandDFT)smallerthanthatofthepuredensityfunctionals.ThisisinagreementwiththefactthatbandgapsofhybridfunctionalsliebetweenthecalculatedDFandHFbandgapenergiesandwithcalculationsintheusinghybridfunctionalsandthe6-31G**basisThecalculatedcoreorbitalenergy510.87eVofthewatermoleculeisverydifferentfromtheHFresult559.17eV.AgaintheHFenergyrepresentsareasonablygoodreference,becausethesumoftheexperimentaldeterminedcore539.89eVandanestimationoftherelaxation20eVgivesanenergyvalueof560eV,whichisclosetotheHFresult.ThecalculateddifferenceliesroughlyontheextrapolatedlineinFigure5Qualitatively,theorderinenergyofthecalculatedoccupiedandvirtualHOorbitalsisconsistentforallmethodsapplied(cf.Figure5).OrbitalsoftypeIVappearabovetheorbitalstypeIII.2.4.TheCaseofNThenextmoleculewewanttofocusonisN,becausetwoofthevalenceorbitalsofN),areclosetogetherinenergy.Figure6showstrendssimilartothosefoundforthewatermolecule.ThecalculatedHForbitalenergiesmatchbestwiththeexperimentalvalues;eHone-electronenergiesareacceptablyclose,andtheKSorbitalsshowagaintheenergyshift.Clearlyonecanalsoseethisshiftinthecompression(indicatedbydashedlines,cf.Figure6)oftheKSorbitalenergyspacingrelativetotheHFenergiesplotted.Thecalculatedscalingparameters5.4eV,andarerelativelyclosetothevaluesfortheHOmolecule.Interestinglytheexperimentallydetermined(andverysensi-tive)orderofhighestoccupiedlevelsªiswell (43)AnupwardshiftofKSeigenvalues,calculatedwithLDAandGGA,isnotedin:vanLeuween,R.;Baerends,E.J.Phys.Re,41,(44)Hybertson,M.S.;Louie,S.G.Phys.Re,5390.(45)Baerends,E.J.;Ellis,D.E.;Ros,P.Chem.Phys.,41.(46)Baerends,E.J.;Ros,P.Chem.Phys.,412.(47)Baerends,E.J.;Ros,P.Int.J.QuantumChem.(48)Salzner,U.;Lagowski,J.B.;Pickup,P.G.;Poirier,R.A.J.Comput.,1943. (49)Thomas,T.D.;Shaw,R.W.,Jr.J.ElectronSpectrosc.Relat.,1081.(50)Clark,D.T.;Cromarty,B.J.;Sgamellotti,A.J.ElectronSpectrosc.Relat.Phenom.,1. Figure6.CalculatedoccupiedvalenceandvirtualorbitalsforthenitrogenmoleculebyBP86/6-31G*,RHF/6-31G*,andeHmethodsandexperimentalvalues.ThecompressionoftheKSlevelsrelativetoHFlevelsishighlightedbydashedlines. Figure7.CalculatedoccupiedvalenceandvirtualorbitalsforCrHbyBP86/LANDZL1,RHF/LANDZL1,andeHmethods.TheFermilevelisindicatedbyadottedline.J.Am.Chem.Soc.,Vol.121,No.14,1999StowasserandHoffmann reproducedbyKSaswellaseHcalculations.IntheHFresultsordering(cf.Figure6)isinterchanged.ThesymmetriesandshapesoftheorbitalscalculatedwithDF,HF,andeHmethodsareastheyshouldbeandareconsistentwitheachother.Thesametrendsareobservedincalculations(notreportedhere)onmoleculessuchasO,ethane,ethylene,singletCHandtripletCH2.5.ATransition-MetalCase,CrHTostudyabitmorecomplicatedorbitalordering,whereenergylevelsofdifferenttypesarestillclosertogether,weinvestigatedtheorbitalenergiesofthesimplestconceivable18-electronMLcomplex,thehypotheticald(cf.Figure7).Forsuchaprototypeoctahedralorganometalliccomplex,onewouldexpectfilledalevels,localizedmainlyontheligands,andabelowesplittingofthemetal3orbitals.Thecrystalfieldsplittingshouldbesubstantial,with(expectedHOMO)belowtheunfilledeAllthemethodsgivethisgeneralorbitalpattern.AstrikingdifferencehereisthatalltheKSandHFlevelsareup(indeedatpositiveenergy),whiletheEHlevelsaredowninenergy.Thisisaconsequenceofthelargenegativechargeonthemolecule.TheextendedHuÈckelmethodignoreselectronrepul-sion;therefore,itsenergylevelsarelowinenergy.Anisolatedisunrealistic;oncetheion(evenifitdidexist)issurroundedbycountercations,itslevelswillmovedowninIfwecomparethevirtualorbitallevels(cf.Figure7),weseethattheorderingofthecalculatedvirtualKSenergylevelsisidenticalwiththatofHForbitalsbutissignificantlydifferentfromthatofeHenergies.ThisisillustrateddramaticallybytheintheDF/HFcalculationsthisorbitalisnottheexpectedesymmetry,butatorbitalinstead.WefindtheªLUMOº(mainly)atmuchhigherenergy,abovethetorbitalsoftypeIII,andanotherthigherordertypeIVorbital(cf.Figure6).ThisisaniceasmentionedatthebeginningofthepaperofthefactthattheborderlinebetweentypeIIIandtypeIVorbitalsisnotsharp.Thereisnodoubtfromthesecalculationsthatallchemicallyexpectedorbitalscanbefoundinthevariouscomputationalschemes(eH,KS,andHF).WecanmapalleH-likeorbitals(whichwewanttoassumeaschemicallymeaningful;however,onecanlookatthemalsoasanarbitrarilychosenreferencesystem)toorbitalsintheKS(andHF)picture.TheKSandHForbitalenergypositionsrelativetoeHorbitalscanbeinter-changedamongthemselves(e.g.theemptytanda(cf.Figure7);orbyvirtualtypeIVorbitals).2.6.ASecond-RowTransition-MetalComplex,PdClSofarwehaveshownthatKSorbitalsareabletodescribeHForbitalsqualitativelyand,aftersuitablescaling,alsoquantita-tively(theHFandKStermscanbeexchangedinthissentence).Wefinishthispaperbydiscussingthecalculatedenergylevelsofarepresentativesecond-rowtransition-metalcomplex,thesquare-planarPdCl.TheKSeigenvaluesofPdClinFigure8)showagain,impressively,thecompressionandtheshiftoftheKSorbitalsrelativetoHForbitals.ThissystematiclineardependencycanbenicelyseeninFigure9.Wecalculatedthescalingparametersas1.54eV,and0.26.Thevalueofdifferssignificantlyfromthatfoundfor,butthescalingparameterfortheoccupiedorbitals, (51)Cade,P.E.;Sales,K.D.;Wahl,A.C.J.Chem.Phys. (52)AccordingtothesimplecrystalfieldconsiderationsandeHcalculations.Wewanttobealittlecarefulhereandrefrainfromsayingwhichorderisªcorrectº. Figure8.CalculatedoccupiedvalenceandvirtualorbitalsforPdClbyBP86/LANDZL1,RHF/LANDZL1,andeHmethods.TheFermilevelisindicatedbyadottedline.ThecompressionoftheKSlevelsrelativetoHFlevelsisemphasizedbydashedlines. Figure9.DifferencesofRHF/6-31G*andDF(BP86)/6-31G*calcu-latedorbitalenergiesplottedvsRHF/6-31G*orbitalenergiesforShamOrbitalsandEigenaluesJ.Am.Chem.Soc.,Vol.121,No.14,1999 0.27,isnicelycloseto0.24(HO)and).Itremainstobeseenifthereismeaninginthis.WhatorbitalswouldoneexpectforPdCl?AsidefromcoreorbitalsofPdandCl(includingtheaCl2sset)wewouldexpectlowerlyingPdClbondingorbitalsofasymmetry,andCllonepairsspanningarepresentations.ThePdlevelsshouldfortheseweexpectacharacteristic4below1splittingofasquare-planarcomplex.TheLUMOshouldbetheorbitalofbForthepalladiumcomplex,theenergyorderingofthePdClbondingorbitalsa,andb(thelowestoccupiedorbitalblockinFigure8)isinexcellentagreementforallthemethods.Someoccupiedorbitalsofhigherenergy(CllonepairsandPddorbitals)areinterchangedinorder(cf.Figure8;forclarity,thesymmetryofonlythehighestsevenoccupiedorbitalsisgiven).ForinstancetheeHcalculationsleadtoaHOMOofsymmetry,whiletheHFandDFcalculationshaveanaHOMO.IncontrasttothepreviouslydiscussedCrHthevirtualorbitalsoftypeIVforthePdClcomplexarenicelyseparatedenergeticallyfromthevirtualtypeIIIorbitals.WithallmethodwefindconsistentlyaLUMOofbsymmetry.Thetrendspreviouslyobtainedforthemain-group-elementmol-eculesthusholdupforPdCl3.SummaryLetusreturntothetitlequestion:WhatmeaningistheretotheKSorbitalsandeigenvalues?IfwewishtouseKSorbitalstorationalizechemicalphenomena,wehavetoidentifytheorder,symmetry,andshapeofKSorbitals.Whenwearedealingwithunoccupiedorbitals,wehavetodistinguishinsomecasesbetweeninterchangedvirtualorbitalsoftypeIIIandtypeIV.Oncethisisdone,thenwethinkonemayapplyKSorbitalsinaqualitativemannerinMOarguments,inthewaywearecomfortabletodoingwitheHorbitals.Theirnumber,symmetryproperties,andshapearejustlikethoseoftheexpectedone-electronorbitals.ThesituationismuchlikeBaerendsdescribestheseseemtobetheorbitalsaqualitative,chemicalanalysisAlso,asareviewerremarks,ªsincetheshapeandsizeoftheKSorbitalsisincaseofa`good'(KSpotential)more`physical'thanthoseofothersingle-particleIfwewanttogoastepbeyondaqualitativeinterpretationandlookatorbitalenergiesasroughionizationpotentials,andiftheDFTcalculationsaredonewithcommonlyusedpotentials,thenitappearswemusttaketheabsoluteconstantandlinearorbitalenergyshiftintoaccountbyapplyingasuitablescaling.Perhapsthesituationwillchangewithnewfunctionalsandnewmethods.WegratefullyacknowledgefinancialsupportfromtheDeutscheForschungsgemeinschaftforagenerousfellowshiptoR.S.andtheCornellUniversityTheoryCenterforprovidingcomputationaltime.WealsowishtothankE.J.Baerends,W.Kohn,W.Glassey,E.Merschrod,andR.Rytzforhelpfuldiscussionsandcomments.AlsowewanttothankProf.Politzerforsendinguspreprintsofhiswork. (53)Cf.thecaseofthelowestexcitation(ttoecalculatedwitheH)inCr(CO)whichwasrecentlyreassignedin:Pollak,C.;Rosa,A.;Baerends,E.J.J.Am.Chem.Soc.,7324.(54)Johnson,J.B.;Klemperer,W.G.J.Am.Chem.Soc.,7132.(55)Zhao,Q.;Morrison,R.C.;Parr,R.G.Phys.Re,2138.J.Am.Chem.Soc.,Vol.121,No.14,1999StowasserandHoffmann