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1、CHEM614MODULE9MANYELECTRONATOMSN12r1r2r12FIGURE9.1WiththeexceptionofhydrogentheelementsofthePeriodicTableconsistofatomsthathave2ormoreelectronsintheirmakeup.ThisfactmeansthatinourSchrödingerequationfortheenergywehavemultiplekineticenergyoperatorsandmultiplepotentialen
2、ergyoperatorstocontendwith.ThesimplestmanyelectronatomisthatofheliumwithaZ=2,i.e.,twoelectronsandonenucleonhaving+2charge.Figure9.1isaschematicoftheHeatom.AswiththeHatomwecanneglectthemotionoftheatomasawholethroughspaceascontributingtotheinternalenergy.Thusweneedtoacc
3、ountforthekineticenergyofthetwoelectronsaboutthecenterofmass,andtherearePEtermsforthevariousCoulombicattractionsandrepulsionspresent.Thehamiltonianoperatorinatomicunitsis(9.1)where1,2,...,nidentifiesthespatialcoordinatesofeachofthenelectrons.OfcourseforHeatomn=2,andth
4、ehamiltonianbecomes(9.2)whichcanbewritteninanabbreviatedform(9.3)19wheretheh(1)andh(2)termsareoneelectronhamiltonians.Thethirdtermisatwoelectronoperator(duetotheinter-repulsionofthetwoelectrons)anditisthepresenceofthistermthatpreventsusfromseparatingtheSchrödingerequa
5、tionintoapairofsinglevariableequations.Thusthe3-particleHeatomisimpossibletosolveexactlyandweneedtoresorttoapproximationsmethods,suchasvariationtheorythatweintroducedinModule8.Averyobvious(butdrastic)approximationistheoneweusedinourvariationtreatment,viz.,tosimplyigno
6、retheterm.Thisisthe“independentelectronapproximation”anditisclearthattheh(1)andh(2)termsinequation(9.3)arehamiltoniansforapairofhydrogenicatomscontainingaZ=2nucleus.Thecorrespondingwavefunctionsarethe1s,2s,2p,etcatomicorbitals.Ignoringthe1/r12terminthefullhamiltoniang
7、eneratesanapproximateoperatorwhereItisimportanttonotethatthe1s,2s,etcatomicorbitalsareeigenfunctionsof,butnotofthefullhamiltonian.Inthisapproximation(zeroorder)thegroundstateoftheHeatomcanbewrittenas1s(1)1s(2),wherethe(1)and(2)refertotheindividualelectronsintheatom.As
8、weshallseelater,theHeatomquestioncanalsobeapproachedusingPerturbationTheory.AreElectronsIndividuallyRecognizable?Inthemacros
