Characters for more than one objectoractionRepresentationsof several thingse.g.H1sorbitalsin H,Oorbital 1orbital 2Eoperationcharacteris+20C20ov(xz)E+2C2ov(yz)C2ov(xz) ov(yz)so overall:00+2+2This the reducible representation of the set of 2orbitals
ReduciblerepresentationsThis set of characters does not appear in the character table-but itcanalwaysbeexpressed asa sumof linesCharactertableEMustbeanAandaBh=4oy(xz) ov(yz)to make the second+1+1number=0RzAMustthenbeA,+B,toB,X, Rymakefinal number=2B2y, Rx+1.A,isthesymmetriccombinationB,istheasymmetriccombinationA,+B,is the irreduciblerepresentationof thetwo orbitals
ReducingrepresentationsThehardway-solve a setofsimultaneous equationsTheeasyway-usetheformulaprovidedZ8R.X(R).X,(R)aiFormula ishRaisthenumberof‘things(orbitalsetc.)of symmetryspecieslhistheorderof thegroupgRis the order of class R (the number of operations of that type)(R)is the characterforoperation Rin the reducible representationX,(R)is the character foroperation Rin the character table for symmetryspeciesiThis formula was derived from the “Great orthorgonality theorem
• This formula was derived from the “Great orthorgonality theorem
Reducing representationse.g.s orbitalsonFatomsof XeOFXeF1F3一Pointgroup?
Point group algorithm-XeOFStart hereyesDyeslinear?infinite groupsCnonolyesyesh6 C, axes ?icosahedralgroupsnonolyesyesO3 C, axes ?i?octahedralgroupsOnonolyesi?4 C, axes ?tetrahedral groupsyesno3Snonoyeslow-symmetryrotationaxisQ?Gnoorder,n,≥2?groupsyesCnoi?yesCnoyesyesDgroupsDntn C,axes Ito2a.the C, axis ?yesDndnoO=On?noDnoyesCnhCand SgroupsOn?Osnono.XeF3mSnoyes4F4Cno