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1、OCTOBER15,193ZPHYSICALREVIEW'VOLUME4ZAtomicEigenfunctionsandEnergiesByC.K.UFFoRDANDG.H.SHoRTLEYPalmerPhysica/Laboratory,PrincetonUniversity(ReceivedAugust6,1932)In$1,thecalculationofthenondiagonalelementsofelectrostaticinteractionissketched.In$2,attentioniscalledtothefactt
2、hatthematrixelementsofL(thex-componentoforbitalangularmomentum),ascalculatedbetweensphericalharmoniceigenfunctionstakenwithpositivephase,arenegativewhenmfis+andpositivewhenm&is—.ThecalculationofeigenfunctionsinLScouplinghasalwaysbeendoneusingonlypositivematrixelementsofI,.
3、Thisamountseffectivelytousingazero-orderschemeinwhichonetakessphericalharmoniceigenfunctionswithnegativephaseforpositiveoddvaluesofmf,,andpositivephaseforallothervalues.In$3thecalculationoffirstorderenergiesforconfigurationswhichgivemorethanonemultipletofakindisconsidered,
4、andexplicitformulasgivenfortheelectrostaticenergiesandmagneticsplittingofthetwo'D'sofd3.Theresultsareshowntocomparesatisfactorilywiththeobserveddata.Thesecond'Dispredictedingeneraltobeextremelyhighandinverted.'"NTHISpaperwewishtocalculatetheseparateenergiesandintervalsof~-
5、multipletswhichoccurmorethanonceinaconfiguration.Forthispurposeweneedthenondiagonalelementsofthematrixofelectrostaticinteraction,whicharegiveninf1.Sincethismatrixmustbeobtainedalgebraically,wediscussin$2theinterpretation,intermsofalgebraicfunctions,oftheeigen-functionsinI.
6、Scouplingwhichhavebeenfoundbyvariousinvestigators.Thisconsiderationisnecessarybecauseofaquestionofphasetowhichatten-tionhasnothithertobeencalled.Wethenproceedin)3todiscusstheenergylevelsandseparations,calculatingindetailthetwo'D'sofd'.$1.MATRIXELEMENTSOFELECTROSTATICINTERA
7、CTIONWeshallfirstsketchthecalculationofthematrixoftheelectrostaticinteraction(11)i)j=l~iji&j'=linthezero-orderscheme.Inthisschemetheeigenfunctionbelongingtothestate.A=(a',a',a~)willbetheantisymmetriccombinationf(A)=+(X!)&g(—1)"ENI(a')e(a')e~(a~))(12)whereI'representsapermu
8、tationofquantumnumbersrelativetoelectronindices,andPhastheparityofP.Herethesymbolu;(a&)in
