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__CROBUTTONMTEditEquationSection2EquationChapter1Section9CHEM614MODULE9__NYELECTRONATOMSWiththeex__ptionofhydrogentheelementsofthePeriodicTableconsistofatomsthath__e2ormoreelectronsintheir__keup.ThisfactmeansthatinourSchrödingerequationfortheenergyweh__emultiplekineticenergyoperatorsandmultiplepotentialenergyoperatorstocontendwith.The______st__nyelectronatomisthatofheliumwithaZ=2i.e.twoelectronsandonenucleonh__ing+2charge.Figure
9.1isasche__ticoftheHeatom.AswiththeHatomwecanneglectthemotionoftheatomasawholethroughspa__ascontributingtotheinternalenergy.Thusweneedtoaccountforthekineticenergyofthetwoelectronsaboutthe__nterof__ssandtherearePEtermsforthevariousCoulombicattractionsandrepulsionspresent.Thehamiltonianoperatorinatomicunitsis__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.1where
12...nidentifiesthespatialcoordinatesofeachofthenelectrons.OfcourseforHeatomn=2andthehamiltonianbecomes__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.2whichcanbewritteninanabbreviatedform__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.3wheretheh1andh2termsareoneelectronhamiltonians.Thethirdtermisatwoelectronoperatorduetotheinter-repulsionofthetwoelectronsanditisthepresen__ofthistermthatpreventsusfromseparatingtheSchrödingerequationintoapairofsinglevariableequations.Thusthe3-particleHeatomisimpossibletosolveexactlyandweneedtoresorttoapproxi__tionsmethodssuchasvariationtheorythatweintrodu__dinModule
8.Averyobviousbutdrasticapproxi__tionistheoneweusedinourvariationtreatmentviz.tosimplyignoretheterm.Thisisthe“independentelectronapproxi__tion”anditisclearthattheh1andh2termsinequation
9.3arehamiltoniansforapairofhydrogenicatomscontainingaZ=2nucleus.Thecorrespondingw__efunctionsarethe1s2s2petcatomicorbitals.Ignoringthe1/r12terminthefullhamiltoniangeneratesanapproxi__teoperatorwhereItisimportanttonotethatthe1s2setcatomicorbitalsareeigenfunctionsofbutnotofthefullhamiltonian.Inthisapproxi__tionzeroorderthegroundstateoftheHeatomcanbewrittenas1s11s2wherethe1and2refertotheindividualelectronsintheatom.AsweshallseelatertheHeatomquestioncanalsobeapproachedusingPerturbationTheory.AreElectronsIndividuallyRecognizableInthe__croscopicworldwecaneasilyi__gineabilliardtableonwhicharepla__dasetofidenticalbilliardballssamecolorandsamesize.Thewaythatwecandistinguishbetweenthemisbytheirpositionsrelativetoafixedpointonthetablesu_____oriftheyareinmotionbytheirindividualvelocityvectors.Theirdifferenttrajectories__kethemdistinguishable.Electronsarealsoallthesamesamerest__sssameintrinsicangularmomentumandsamecharge.Therearenodifferentcolorsetcthatenableustotagindividualelectrons.Inthesub-microscopicworldoftheelectronhoweverthemomentumandpositionvectorsfailtocommuteandthereforethesetwoobservablescannotbesimultaneouslyspecifiedwitharbitraryprecision.Ifthemomentumofanelectronissharplydefineditspositionisveryun__rtainandvi__versa.Thusthereisnopropertyofanelectroninanatomthatwecanusetodistinguishitfromanyotherelectron.Inasituationwherethereareseveralelectronsoccupyingatomicorbitalsweareunabletostatewhichoftheelectronsoccupieswhichorbital.Indeedthequestionitselfiswithoutanymeaning.TheearlierdescriptionoftheelectronicconfigurationofHegroundstateas1s11s2issatisfactorybecauseit____susthatbothoftheelectronsareoccupyingthesamespatialorbitaldesignatedas1sandthereisnoattempttodistinguishbetweentheelectrons.AsweshallseelaterwecanspecifyanexcitedstateofHeas1s12s1indicatingoneelectronisoccupyingthe1sorbitalandtheotherisinthe2sorbital.Thisdescriptionisallowablesin__itdoesnotrequireustostatewhichelectronisin1sandwhichisin2s.Howevertowritethezeroth-orderw__efunctionoftheexcitedstateas1s12s2isnotallowedsin__nowwearespecifyingthatelectron1isinthe1sorbitalandelectron2isin2s.Thisimpliesthatwecansomehowdistinguishelectron1fromelectron2whichwecannot.Insteadwemustusesuperpositionfunctionssuchas__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.4whichdonotspecifythataparticularelectronisinaparticularorbital.Thew__efunctionofasystemofparticlesisafunctionofallthevariablesthatpertainbothspa__andspin.Thusforelectron1inasetofnelectronsthespa__variablesarex1y1andz1andthespinvariableis
1.Forthesakeofbrevitywedenoteallofthesevariablesforelectron1byq1andforelectron2byq2andsoonuptoqn.Thusthew__efunctionofasystemofnidenticalparticlescanbewrittenas.Nowwedefinethepermutationorexchangeoperatorastheoperatorthatexchangesallthespa__andspincoordinatesoftheparticlesjandkinthen-particlesystem.Atthispointitisconvenienttosimplifyoursystemtotwoelectronse.g.theHeatomandwritethefunctionasandtheexchangeoperatoras.Thus__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.5andweseethatapplicationoftheoperatorhasbeentoexchangethecoordinatesofthetwoelectrons.Thiswillnotaffectthestateofthesystemsin__theelectronsarenotdistinguishableanelectronisanelectronisanelectron…sin__ineffectitservesonlytochangethelabelswepla__dontheelectrons.Clearlyapplicationoftheexchangeoperatorasecondtimewillbringthesystembacktotheoriginaldescriptionsowecansay__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.6whereistheunitoperator.Itcanbeshownthattheeigenvaluesofanyoperatorwho’ssquareistheunitoperatorare+1and–
1.Thuswhenourfunctionisaneigenfunctionofweseethat__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.7Whentheresultoftheoperationretainsthesignoftheoriginalfunctionwesaythatthefunctionissymmetricwithrespecttointerchangeofelectrons1and2;whentheresultoftheoperationchangesthesignofthefunctionwesaythatthefunctionisantisymmetricwithrespecttotheinterchange.Theconclusionappliesinthegeneralcase:aw__efunctionforasystemofnidenticalparticlesmustbesymmetricorantisymmetricwithrespecttoeverypossibleinterchangeofanytwooftheparticles.Sin__theparticlesareidenticalitisnotcon__ivablethatsomeinterchangeswouldturnouttobesymmetricandothersantisymmetric.Theperiodictableoftheelementsprovidesabundanteviden__seelaterthatforsystemsofelectronsatomicstatesonlytheantisymmetricresultisfoundandwecanformulateanotherquantummechanicalpostulatethat“Thew__efunctionofasystemofelectronsmustbeantisymmetricwithrespecttotheinterchangeofanytwoelectrons.”ThisisaveryimportantpostulateinQuantumtheoryanditisduetoWolfgangPauliwhoemployedrelativisticquantumfieldtheorytoshowthatfermionsparticleswithhalf-integralspin;andsoonthefamilytowhichelectronsbelongrequireantisymmetricw__efunctionswhereasbosonss=012andsoonrequiresymmetricw__efunctions.TodemonstratetheeffectofthePauliprincipleforidenticalfermionswestartwiththerecognitionfromthepostulatethattheantisymmetryrequirementis__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.8Nowsupposeweassignthesamefourcoordinatestotwoidenticalelectrons.Thusx1=x2y1=y2z1=z2and1=2thusq2=q
1.Substitutingthisinequation
9.8leadsto__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.9or2andtheprobabilityoffindingthetwoelectronsintheorbitaldefinedbyiszero.Thustwoelectronsh__ingthesamespinfunctionh__ezeroprobabilityofbeingfoundh__ingthesamevaluesofthespa__coordinatesinotherwordscannotco-existinthesameregionofspa__.This“Paulirepulsion”for__selectronsofthesamespintokeepapartfromoneanothere.g.indifferentorbitals.TheHeliumAtomWesawabovethattheelectronicconfigurationofthegroundstateofHeinthezerothorderspatialw__efunctioncouldbewrittenas1s11s2i.e.bothelectronsoccupyingthesameorbital.Fromtheforegoingdiscussionthisdescriptionrequiresustoconsiderthespinstateofthetwoelectrons.AtasuperficialleveltherearefourpossiblespincombinationsofthetwoelectronsthusTheprincipleofindistiguishabilityofidenticalparticlesisclearlyupheldinthefirstpairofcombinationssin__bothelectronsareassignedthesamespinfunction.Howeverthisisnotthecaseforthesecondtwocombinationssin__nowweshowaparticularelectronwithaparticularspinfunctionandthisviolatestheprinciple.Applyingtothecombinationsshowsthatthefirstandsecondarebothsymmetricandthethirdandfourthareneithersymmetricnorantisymmetricwithrespecttointerchange.Theselasttwothenareunac__ptable.Moreovereventhoughthefirstandsecondspincombinationsareac__ptableinthemselveswhentheyarecoupledwiththe1s11s2spatialfunctionwhichitselfissymmetrictoparticleinterchangewefindthePauliprincipleforbidsthecombinationsandsin__theyareoverallsymmetrictoelectroninterchangetherebyviolatingthePaulirule.Thewaytopro__edistoseekaspinfunctionthatisanantisymmetricsuperposition.ThetwocandidatesforthisaregivenbyThesetwofunctionsarenor__lizedlinearcombinationsof.Theyarethesymmetric+andantisymmetric-eigenfunctionsof.Sin__wearecombiningoneofthesewiththesymmetricspatialfunction1s11s2to__keanoverallantisymmetrictotalfunctionweneedtheantisymmetricsuperpositioninwhichcasetheHegroundstateisgivenbyThisisthetotalw__efunctionforthegroundstateofHeinzeroth-orderincludingspatialandspincontributions.Notethatitisasinglestate;therearenodegeneracies.Atthispointitisworthnotingthatthehamiltonianoperatorcontainsnospintermssothattheenergyofastateisnotaffectedbythespintermsinthew__efunctionatleasttoareasonableapproxi__tion.NowwecantakeapreliminarylookatHeinanexcitedelectronicstatei.e.astateinwhichoneofthe1selectronsispromotedtothen=2level.Inprinciplethisnewstatecouldbedescribedas1s12s1or1s12px.Howeverinnon-hydrogenicsystemsmorethanasingleelectronthe2sorbitalisataslightlylowerenergythananyofthethree2porbitals.Atfirstsightthetwo-electroncombinationmightbewrittenas1s12s2or1s22s1butthesedescriptionsimplythatwecanidentifyelectrons1and2asbeinginthedesignatedorbitals.Thisclearlyviolatesthecon__ptofindistiguishability.Moreoverthetwoproductfunctionsareneithersymmetricnorantisymmetricwhenoperateduponbytheexchangeoperatorandthuscannotmeaningfullycontributetoatotalw__efunctionthatmustbeoverallantisymmetricwithrespecttoelectronexchange.Insuchasituationweresorttotheuseofasuperpositionwhichcansatisfythesymmetryrequirements.Oneoftheseneww__efunctionsisgivenby__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.10whichissymmetriconelectroninterchangei.e.onoperatingwiththeexchangeoperator.Theantisymmetricsuperpositionis__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.11ThesymmetricspatialcombinationcanbeanallowedstateoftheHe1s2sconfigurationifitisassociatedwithanantisymmetricspintermandsimilarlytheantisymmetricspatialcombinationcanconformtothePauliprincipleifassociatedwithasymmetricspinterm.Thusfourtotalw__efunctionssatisfytherequirementsforthe1s2sexcitedstateofHe.Theseare__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.12__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.13__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.14__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.15Thusfourdistinctstatescanbegeneratedundertheconfiguration1s2s.Inordertoobtainenergytermsforthesefourpossiblestatesweneedtoseehowthew__efunctionstransformundertheapplicationofthefullhamiltonianequation
9.
2.Ourfunctionsh__ebeensetupusingtheindependentelectronapproxi__tioninwhichweignoredtheinter-electronrepulsionterminequation
9.
2.Thereforethefunctionswillnotbeeigenfunctionsofandthereforewecannotsimplycomparetheeigenvalues.Howeverwecancalculatethe__erageenergiesbyevaluatingtheexpectationvaluesofwiththefourfunctionslisted.Thus__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.16Notethattheusualdenominatorismissingsin__bothspa__andspinpartsarenor__lizedandthusthedenominatorisunity.Nothingintheoperatorweareusingwillinteractwithanyofthespincontributionsinthefourfunctionslistedandsin__thespintermsarenor__lizedtheirintegralscomeouttobeunity.Thereforeweseethatthe__erageenergywillbedeterminedbythespatialcontributionandasthreeofthefunctionscontainthesamespatialparttheyformathree-folddegenerateset.[Itisimportanttonotethatthefactorizationintospatialandspinfunctionsisonlypossiblewithtwo-electroncases.]Onlyhasaspatialpartthatisdifferentandthereforewecananticipatethisfunctiontoh__eadifferentenergythanthethreefunctions.FromthisweexpectthattheexcitedstateofHethatweh__edesignated1s2swillh__etwoenergystatesoneofwhichistriplydegenerate.Inordertofindwhattheenergyvaluesareweneedtogrindthroughsomeintegrationsandthepro__dureforthisislaidoutinAppendixI.Therewefindtheconclusionthatthetwoenergiesaregivenby__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.17wherethefirsttwotermsontheRHSaretheenergiesofthe1sand2seigenvaluesforHe+Z=2respectively.TheintegralsJtheCoulombintegralandKtheExchangeintegralaregivenby__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.18__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.19JisalwayspositivebecauseitdealswiththeCoulombicinteractionbetweenthetwoelectrondistributions1s*1sand2s*2s.Thesedistributionsareeverywherenegativeandsotheinteractionisrepulsiveandthustheenergyofthecombinationisraisedlessnegative.TheintegralKhastheproductfunctionsintheintegranddifferingbyexchangeofelectronshen__itsname.AsAppendixIexplainsKhasnegativeandpositivecontributionsbutispositiveoverallandnotaslargeasJ.Thepro__dureintheAppendixshowsthatthetriplydegeneratelevelislowerinenergythanthenon-degeneratelevelandtheseparationis2K.WecanthinkoftheJtermasresultingfromhomogeneous/time-__eragedelectrondensitiesinthecharge-cloudsoftheorbitalsandtheKtermasbeingacorrectionarisingfromtheinhomogeneitiesthatoccurasaconsequen__ofelectronswantingto__oideachother.Forexampleifweapproxi__tethetwolinearcombinationspatialtermsinequations
9.12and
9.13asbeingoftheformandwhereaisafunctionoftheradiusvectorr1andbisafunctionoftheradiusvectorr2thenasr1approachesr2then-tendstovanishi.e.theelectronstendto__oideachotherinthedifferen__combination.ThisleadstotheFermihole.Ina__gneticfieldthetriplydegeneratelevelissplitintothreediscreteenergystates-itisatripletofstatesoratripletstate.SlaterDeterminantsInourabovediscussionoftheHeatomgroundandexcitedstatesweh__efounditeasytowritedowntheantisymmetrictotalw__efunctionsasrelatively______multiplesofspatialandspinparts.Itwaseasybecausewehadtodealwithjusttwoelectrons.Itisanother__ttertoconstructanasymmetricw__efunctionforNelectronsbyinspection.In1930’sSlaterintrodu__dthecon__ptofusingdeterminantstoconstructtherequiredw__efunctions.AsanexampleletustaketheHegroundstatew__efunctionthataboveweh__eshowntobeThisw__efunctioniscomposedofasymmetricspa__functionandanantisymmetricspinfunctionsoitisoverallantisymmetricwithrespecttointerchangeofthepairofelectrons-asPaulirequires.Asadeterminantthisbecomes__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.20Expandingthedeterminantandrearrangingleadstotheantisymmetricw__efunctionshownabove.Theindividualtermsinthedeterminantarespinorbitalsthatdescribethespatialandspinstateoftheappropriateelectron.Thedeterminantisformedbyplacingtheindividualspinorbitalsalongtherowsandtheassociatedelectronsinthecolumnssomeauthorsreversethispro__durebutthishasnoeffectonthedeterminantsin__interchangingcolumnsandrowsle__esthedeterminantunchanged.Determinantssuchasthatinequation
9.20arecalledSlaterdeterminantsandthew__efunctioninequation
9.20iscalledadeterminantalw__efunction.Ifthelabels1and2areinterchangedinthedeterminantthispla__selectron1incolumn2andelectron2incolumn1i.e.twocolumnsofthedeterminantareinterchangedwhichhastheeffectofchangingthesignofthedeterminantseeBarrantechapter9andthereforeofthew__efunctionhen__thedeterminantrepresentsanantisymmetricfunction.Furthermoresupposethatourtwoelectronsinthe1sorbitalh__ethesamespinsay.Thischangesthedeterminantasfollows__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.21Nowtworowsthefirstandsecondareidenticalwhichcausesthedeterminanttoh__ethevalueof0irrespectiveoftheorderofthedeterminant.Thusthew__efunctionforastateinwhichbothelectronsh__eidenticalsetsofspa__andspinvariablesiszeroandthatstatethereforedoesnotexist.ThereareshorthandwaysofwritingSlaterdeterminants.Oneistoindicate-spinbyabaroverthespatialorbitaldesignator;ifthebarisomitted-spinsindicated.Thusthew__efunctioninequation
9.20wouldbecome__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.22Wecanuseafour-electronexampletoillustratethegeneralcasewhereUirepresentstheithspinorbital.Thenthefour-electronconfigurationinwhicheachspinorbitalissinglyoccupiedisU1U2U3U
4.Thecorrespondingdeterminantwillbe4x4withanor__lizationfactorof.Thenthespinorbitalsoccupytherowsandtheoccupyingelectronsareinthecolumnsthus__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.23Theprincipaldiagonalofthedeterminantisidenticaltotheconfigurationofthestateasaboveanditiscommonpracti__torepresenttheSlaterdeterminantbyitsprincipaldiagonalinthefollowingwaywiththenor__lizingfactorunderstood.Letusexaminethew__efunctionfortheLiatomZ=3usingtheSlaterdeterminantmethod.Firstletusseewhetherthe1s3configurationispossible.Ifso__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.24Twooftherowsfirstandthirdareidenticalwhichmeansthatthedeterminantiszeroandthusthe1s3configurationisnotpossible.Sothenexttotryisandwewritethedeterminant__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.25Noti__weh__eassignedthe2selectronash__ingspinbutthisispurelyarbitrarysin__eitherareclearlypossible.ThusthegroundstateofLiexistsasapairofdegeneratestatesitisadoubletofstatesoradoubletstate.Nownotworowsareidenticalandifweinterchangetheelectronslabels1and2weswitchthefirstandsecondcolumnsandthedeterminanttherebychangessignindicatingthew__efunctionisantisymmetricasisne__ssaryonthePauliprinciple.SlaterOrbitalsInModule8weappliedVariationTheorytotheHeatomicgroundstateinordertoevaluateitsenergy.InthecalculationweusedtheatomicnumberZasthevariationalparameterandthetrialfunctionwastakenas__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.26MinimizationoftheenergywithrespecttovariationofZledustowhereastheexperimentalresultis–
2.9033au.Thusthevariationmethodwithonlyonevariationalparametercomeswithin2%oftheactualvalue.Thisappearstobequitegoodagreementbutletusexaminethismoreclosely.TheionizationenergyIEofHeisgivenbytheenergydifferen__betweenHeandHe+__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.27ThefirsttermcanbecalculatedexactlybecauseHe+isahydrogenicionwithZ=2andwecanusethevariationtheoryvalueofEHe1sasthesecondtermthusTheexperimentalvalueofIEis2372kJmol-
1.Thusourcalculatedvalueisca150kJmol-1inerrorwhichisaboutthestrengthofsomechemicalbonds.ThusthesingleparametervariationalcomputationleadstoanenergyforHe1sthatisnotsignificantonachemicalscaleandweneedtodobetter.Onewaytoimproveistochangeourtrialfunction.Intheabovewechosethehydrogenic1sw__efunctionbutthisisunne__ssarilyrestrictivealmostanyfunctionorcombinationoffunctionscouldbeused.In1930Slaterintrodu__danewsetoforbitalsnowcalledSlaterorbitalswhichareoftheform__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.28whereisaconstantfornor__lizationandtheYfunctionistheappropriatesphericalharmonic.ThetermisequaltoZ/nforhydrogenicorbitalsbutintheSlater-typeorbitalsSTOsitbecomesanarbitraryparameter.TheradialpartsofSTOsh__enonodesunlikethehydrogeniccounterpartsforn
1.InthecalculationofHe1sthatwedidinModule8weineffectusedtheSTOS100thusourtrialw__efunctionfollowingthepro__dureinModule8andrememberingthatl=0is__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.29__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.30whichleadstoourearliervaluesof=
1.6875andEmin=-
2.8477auwhichwesawabovetoleadtoavalueofionizationenergythatisabout150kJmol-1inerror.Animprovementaslightonecanbegainedbyalsolettingnvaryinthenor__lizingfactorandintheexponentoftherterminequation
9.
28.Stayingwithinthecon__ptoforbitalssatisfactoryforchemistsespeciallywhenwedealwithbondingwecan__keadditionalimprovementstoEminbyusingtrialfunctionsofevenmoreflexibilityforexamplebydefiningthespatialfunctionasaproductofoneelectronorbitalsthatarecompletelygeneral__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.31ThisapproachleadsustoalimitingvalueforEminthatiscalledtheHartree-Focklimit.Thisisthebestthatcanbedonewithretentionoftheorbitalcon__pt.ForHe1stheHartree-FocklimitisEmin=-
2.8617au.Byremovingtherestrictionthatthetrialfunctionbeaproductofone-electronorbitalsandsimplyexpressthetrialfunctionasaseriesofvariationalparametersthentheexactenergycanbeobtained.ThiswasdonebyPekerisin1959whoused1078parameterstoobtainEmin=-
2.903724375auinexactagreementwiththetruevalue.Inthisapproachtheemphasisisonobtainingtheexactenergybycrunchingnumbersandtheideaofatomicorbitalshasbeenabandoned.Inmoderntimesthepro__dureisusuallytofindtheH-ForbitalsthatgeneratetheH-FlimitandthencorrecttheenergybyapplyingaPerturbationTheorytreatmentseealaterModule.TheSelf-ConsistentFieldMethodItisinstructiveforustoseehowtheH-Fpro__dureworksina______caseviz.theHeatom.Anypro__durethatisgoingtobesuc__ssfulclearlymustaccountfortheinteractionbetweenelectronsinsomeway.IntheHFapproachtheelectron-electronrepulsionistreatedinan__erageway;eachelectronisconsideredtobemovingwithinthefieldofthenucleonandthe__eragefieldofthere__inderoftheelectrons.Asweshallseethisisnotdetailedenoughbutitwillgetusstarted.Westartbywritingthetwo-electronw__efunctionastheproductofone-electronorbitalsasoutlinedabove
9.31Thefunctionsrjareidenticalsin__weallowbothelectronsinHe1s2tobeinthesameorbitalunderthePaulirestriction.Theprobabilitydistributionofelectron2its“charge-cloud”isgivenbyandthinkingclassicallywecaninterpretthisasachargedensity.The__eragepotentialenergythatisexperien__dbyelectron1atsomevalueofr1becauseofthepresen__ofthecharge-cloudofelectron2isinaugivenby__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.32Sowecandefineaone-electronhamiltonianthatincludesthis__eragedpotentialenergytermCoulomboperator__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.33andthentheSchrödingerequationforelectron1becomes__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.34Thereisasimilarequationfortheotherelectronbutsin__thew__efunctionsareidenticalweneedonlyconsideroneequation.Equation
9.34istheH-FequationforHe1s2andsolvingthisequationwillgeneratetheoptimumw__efunctionforthisatomwithintheorbitalcon__pt.Atthispointweseethatthereappearstobeaproblem;theoperatorintheSchrödingerequationdependsonV__1whichdependsinturnonr
2.ThusweneedtoknowthesolutiontotheSchrödingerequationbeforewecanwritedownitsoperator.Thewayaroundthisistheself-consistentfieldmethodofDouglasHartree.Firstweguessaformforr2anduseittoevaluateV__1r1throughequation
9.33thenwesolvetheSchrödingerequation
9.34forr
1.Usuallyafterthefirstcycletheoutputrwilldifferfromtheinputrunlesswehadguessedthecorrectfunctioninthefirstpla__.Iftheyaredifferentweusethecalculatedrtoesti__teanewpotentialenergytermandthen__anotherhamiltonianandsubsequentlyanotherr
1.Recallthatthetwoorbitalfunctionsareidentical.Thenthecycleisrepeateduntiltheoutputandinputw__efunctionsaresufficientlycloseorareself-consistent.Atthispointweh__etheHartree-Fockorbitals.Asimpliedabovesinglefunctionalorbitalsdonotyieldthebestenergyandinpracti__linearcombinationsofSTO’sareemployed.TheoptimumresultfiveSTO’swithdifferentcoefficientsfortheHegroundstateasstatedearlierwasfoundtogiveanenergythatwas109kJmol-1toohighbutitisthebestthatcanbeachievedwithintheorbitalapproxi__tion.ThewaytheCoulomboperatorwascalculatedintheabovepro__duretreatseachelectronasthoughitweremovinginthetime-__eragedfieldoftheothersinthesystem.Howeverthisisonlypartlytruebecauseelectronsaremutuallyrepulsiveandtheytendto__oideachother.Inotherwordstheirmotionsarecorrelated.TheHFpro__dureisnotconstructedtotakethisintoaccountanditgeneratesEHFEexpt.Thedifferen__betweentheHFlimitandthetrueenergyisthecorrelationenergy.IntheHecasethiswas109kJmol-1onlyca
1.5%ofthetotalelectronicenergybutalargeabsoluteerrornonetheless.MorethanTwoElectronsWeh__ebeenabletouseequation
9.31inthisdevelopmentbecauseforthetwo-electroncaseofHetheSlaterdeterminantfactorsintospatialandspinparts.Formorethantwoelectronsthisseparationofspa__andspinbecomesun__ailablesowemustworkwiththefullSlaterdeterminantequation
9.25forexample.Asweh__estated__nytimesthecomplicatingfeatureofallcalculationsistheelectron-electronrepulsionterm1/rijinthehamiltonian.Wesawabovethattheeffectofelectroncorrelationisdifficulttoesti__te.Sowestartbysupposingthatthetrueelectronicw__efunctionthatweseekissimilarinformtoanidealizedw__efunctionwithoutelectron-electronrepulsionTheSchrödingerequationfortheidealizedw__efunctionisgivenby__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.35where
15.8Thetermsarecorehamiltoniansfortheithelectronstotaln.Equation
9.35isann-electronequationthatisseparableinton-oneelectronequationsand0istheproductofoneelectronw__efunctionsorbitalseachofwhichisasolutiontotheSchrödingerequation__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.36whereadenotesorbitalindexandidenoteselectronindex.Theeigenvalueistheenergyofanelectroninorbitala.Thisisanindependentelectronmodel.Thisleadstotherelationship__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.37Thew__efunctionsdependonthespatialcoordinatesofalltheelectrons.Sofarweh__econfinedourattentiontospatialw__efunctionswithoutconsideringspinexplicitly.Thiscanbetakenintoconsiderationbyredefiningequation
9.37intermsofspinorbitals.Rememberthataspinorbitalistheproductofspatialandspinfunctionsandwewillwriteithereaswhereqirepresentsthejointspa__andspincoordinatesofelectroni.FinallywewriteoutourorbitalasaSlaterdeterminant__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.38wheretheprincipaldiagonalonlyhasbeenwritten.Theufunctionswithu=abc...zarespinorbitalsandaretakentobeorthonor__l.Inequation
9.38weh__earrivedatouridealizedw__efunctionwhichistobeourmodelandthestartingpointofavariationalcalculation.Weneedtorememberhoweverthatitisidealizedandelectron-electroninteractionsh__ebeenignoredindependentelectronapproxi__tion.TheHartree-Fockapproachwillgopartwaytoaccountingforthe1/rijpotentials.H-Ftreatselectronrepulsionsinan__erageway.Eachelectronistakentobemovingintheelectrostaticfieldofthenucleiandinthetime-__eragedfieldoftheotherelectrons.TheHartree-Fockequationisdefinedas__CROBUTTONMTPla__Ref\*MERGEFOR__T
9.39forelectron1inorbitalaofenergyaandistheso-calledFockoperator.ThisoperatorwillbeconsideredinmoredetailinalaterModulebutfornowwecanthinkofitasbeingequivalenttothatwemetinthecaseofHe.Toarriveattheexpressionforthespinorbitalswesolveequationslike
9.39afterfirstevaluatingtheFockoperatorwhichdependsonthespinorbitalsofthen-1otherelectronsandasbeforeitseemswemustknowthesolutionsbeforewecanfindthesolutions.Theself-consistentfieldmethoddetailedaboveallowsthistobeperformed.Thusthenoccupiedspinorbitalsinthedeterminantalw__efunction
9.38allcontributetotheFockoperator.On__arrivedattheFockoperatorislikeanyotherHermitianoperatorina__uchasithasaninfinitenumberofeigenfunctionsi.e.thereareaninfinitenumberofspinorbitalsuh__ingenergyu.Aninfinitenumberisanimpracticablenumbersoitisusualtofindmsolutionswheremnthenumberofelectrons.ThemoptimizedH-FSCForbitalsarearrangedinorderofincreasingenergy;thenlowestaretermed“occupied”andthem-nre__iningaretermed“virtual”orbitals.Koop__nsTheoremAlthoughthesumofalltheone-electronenergiescalculatedbytheH-FSCFmethodisnotyetequaltothetotalelectronicenergyofourmoleculeitispossibleforrelationshipstophysicalpropertiestobe__de.ForexampleifwewishtoknowtheenergyofamoleculeafteroneelectronhasbeenremovedtheradicalcationweuseKoop__n’stheorem.Thisstatesthattheionizationenergyofanelectroninanorbitalristhenegativeofthecalculatedone-electronenergyofthatorbitalrThisidentityassumesthatthedistributionofthere__iningelectronsdonotrelaxaftertheremoval.Theionizationenergiesofmostatomsintheperiodictableh__ebeencalculatedusingH-FSCFwithKoop__n’stheorem.Figure
9.1showsacomparisonofthecalculatedxandexperimentalvalues.Theplotclearlyshowstheshellandsub-shellstructureweh__ecometobelieveinfromgeneralchemistry.Noadjustableparametersh__ebeenusedintheseabinitiocalculationsandtheagreementisnothingshortofre__rkable.Thecalculationsalsodemonstratethatunlikethehydrogenicentitiesthe__nyelectronatomsh__el-dependentenergies.Inmulti-electronatomstheinter-electronrepulsionterminthehamiltoniancausesliftingofthedegeneracyofthe2sand2porbitalsandsimilarlythatinthe3s__and3dshell.Thehigherislthetendencyisfortheorbitalenergytobehigherthuswefindtheseries1s2s2p3s__4s3d4p5s4d5p6s5d4f...andweseethatforn=3andgreatertheorderingislost.Theenergy-orderingschemeaboveisthebasisforthe“aufbau”building-upprinciplefortheelectronicconfigurationofatomsinthePeriodicTableoftheelements.InbuildinguptheconfigurationswealsoneedtobecognizantofthePauliexclusionprincipleintheform“notwoelectronscanpossessthesamefourquantumnumbers”andtheHundRuleinadegeneratesetoforbitalsthegreateststabilityisassociatedwiththehighestmultiplicity.InalaterModuleweshallconsidertheangularmomentumsituation.AppendixIN12r1r2r12FIGURE
9.1FIGURE
9.1117。