SUPEREXCHANGE-INTERACTION-AND-SYMMETRY-PROPERTIES-

J.Phys.Chem.SolidsPergamonPress1959.Vol.10.pp.87-98.SUPEREXCHANGEINTERACTIONANDSYMMETRYPROPERTIESOFELECTRONORBITALSJUNJIROKANAMORI*DepartmentofPhysics,OsakaUniversity,Osaka,Japan(Received31May1958;revised6November1958)Abstract-Therelationbetweenthesymmetryofelectronorbitalsandsuperexchangeinteractionisdiscussed.Itisshownthatthesignofthesuperexchangeinteractioniscloselyconnectedwiththecationorbitalstate,whenthecationissubjecttothecrystallinefieldarisingfromoctahedrallyortetrahedrallysurroundinganions.Insomecases,thesignofthesuperexchangeinteractionisde-finitelydeterminedfromthesymmetryrelations.Thecasesinwhicheachcationissubjecttoanoctahedra1cubicfieldandthelinesconnectingtheinteractingcationstotheinterveninganionmakeanangleofeither180”or90”arediscussedinparticular.Ourdiscussionofthe180”caseisapplicabletocrystalsoftheperovskitetypeandNaCltypeandthatofthe90”casetoanhydrouschlorides.Inthecasewhereeachcationissubjecttoatetrahedralcubicfield,thereisadefiniterelationbetweenthesymmetryofthecationelectronorbitalsandsuperexchangeinteraction,ifonlythes-orbitalofanionparticipatesinthesuperexchangeinteraction.TiHsisanexampleofthiscase.Theinteractionbetweennearest-neighborcationsinthecrystaloftheNaCltypeisalsodiscussed.1.INTRODUCTIONIN1950ANDERSON(~)formulatedKRAMERS’idea(s)ofthesuperexchangeinteractioninmagneticcom-poundsandshowedthatKRAMERS’mechanismgivesrisetoanantiferromagneticinteractionbe-tweencationswithamore-than-half-filledd-shellandaferromagneticinteractionbetweencationswithaless-than-half-filledd-shell.Hisconclusionfortheless-than-half-filledd-shellis,however,notalwayssupportedexperimentally,thoughitprovedtobevalidinafewcompoundssuchasCrTe,CrOs,etc.,whichareferromagnetic.Inthemean-while,variousothermechanismswhichgiverisetoantiferromagneticinteractionevenintheless-than-halfcasehavebeenproposedbySLATERR),GOODENOUGHandLOMB,GOODENOUGH(andANDERSONandHASEGAWA(@.Thepresentpaperwillshowthatmoreattentionhastobepaidtothesymmetryrelationbetweentheoccupiedd-orbitalsofthecationandthep-ors-orbitalsoftheanionbeforeitispossibletoconcludewhetherthesuperexchangemechanismgivesrisetoferro-magneticorantiferromagneticinteraction.In-*NowattheInstitutefortheStudyofMetals,UniversityofChicago,Chicago37,Ill.,onleavefromOsakaUniversity.87certaincasesitcanbeshownthatthesignofthesuperexchangeinteractionissocloselyconnectedwiththeorbitalstateofthecationthatitdoesnotdependonanyspecificmechanismconceivable.Inthecrystaltheorbitalofthecationandthatoftheanionsoverlapeachotherconsiderably,sothatwecannotgenerallyconsiderthemtobeortho-gonaltoeachother,exceptwhentheyhavecertaindifferentsymmetryproperties.Inthecaseinwhicheachcationissurroundedoctahedrallyortetra-hedrallybyanions,thereexistdefiniteorthogonal-ityrelationsbetweentheoccupiedd-orbitalsofthecationandtheoccupiedorbitalsoftheanionthroughtheirdifferentangulardependences.Theseorthogonalitiesdetermine,inthefirstplace,thepairofthed-orbitalandanionorbitalformingapartialcovalentbondand,inthesecondplace,thesignoftheexchangeintegralbetweentheoccupiedd-orbitalandanionorbital.Asanillustrativeex-ampleofthediscussionsinthefollowingsections,weshallherepresentthecaseofCaMnOs.CaMnOsisanantiferromagnethavingthecrystalstructureoftheperovskitetype.(v)EachMn4+issurroundedoctahedrallybyOs-,andeachOs-issituatedatthecenterofeachlineconnectingtwonearest-neighborMn4+.EachMn4fissubject88JUNJIROKANAMORItoacrystallinefieldofcubicsymmetry,inwhichtheloweststateofthefreeMn*+,*F,issplitintothreesublevels.Thelowestofthesesublevelsisnon-degenerate,withsuchanelectronicconfigurationthatthree3d+orbitals,d(q),d(y,z),andd(zx),aresinglyoccupied,whereyz,etc.,representtheangulardependencesofthed-orbitalsandtheco-ordinatesrefertothecubicprincipalaxes.Itisusuallyunderstoodthatintheoctahedralcubicfieldthe&-level,whosewavefunctionsare&cY),4~4and441ieslowerthanthe&-levelwhosewavefunctionsared(xz-~2)andd(2z2---x2-9).Now,lettheanionorbitalwhichpar-takesinthesuperexchangemechanismbethep-orbitalwhoseaxispointstooneoftheMn*+(pa-orbital).Thispa-orbitaldoesnotchangeitssignwhenthecoordinateaxesarerotatedaboutthelineconnectingthesetwoions,whereastheoc-cupiedde-orbitalsofMn*+dochangetheirsign(seeFig.1).Thereforethepo-orbitalisorthogonalFIG.1.Thesymmetryrelationbetweenthedc-andPO-orbitals.tothede-orbitals.Asforthedy-orbitals,whichareunoccupied,wecanconstructanappropriatelinearcombinationoftheminsuchawaythatitdoesnotchangetheirsignunderthesamerotation;forinstance,42.x2--y2-2s)doesnotchangesignbyrotatingthecoordinateaxesaboutthex-axis.Therefore,apartialcovalentbondinvolvingthepa-orbitalcanbeformedwiththe&-orbital,i.e.electrontransferfrompucantakeplacetothe&orbital,butnottothedc-orbitals.Furthermore,theexchangeintegralsbetweenpuandtheoccupiedda’sarepositive(ferromagnetic),astheyaremutuallyorthogonal.ANDERSON’Smechanismisdescribedintheschemeofperturbationtheorybyvirtualprocessesinwhichanelectronisfirsttransferredfromthepa-orbitaltooneoftheneighboringMn*+andthentheunpairedspinleftonOs-coupleswiththespinoftheotherneighbor-ingMn*+throughexchangeinterac