CharactersThenumbers+1and-1arecalled characters.The charactertablehasall possiblesymmetryspeciesforthe point group.It isthe same for all molecules belonging tothepointgroup-e.g.C2yforH,O,SiH,Cl2,Fe(CO).Cl2,etc.Note:the charactertableCharactertableliststhesymmetryECov(xz) ov(yz)h = 4福21speciesfortranslationsAandrotationsARzBX, RyA,BshowsymmetrywithB2respect to rotation.y, Rx1,2distinguishsymmetrywithrespecttoreflections
2.2 symmetry species:Mulliken symbolsCrE30,2C311Al1x? + y2, z?Z1R:1A2-12E0-1(x2 y2, xy)(xz, yz)(x, y)(Rx, R,)111IIIIVAll 1-D irreducible reps. are labeled by either A or B, 2-Dirreducible rep. by E, 3-D irreducible rep. by T and so on.A: symmetric with respect to C, rotation, i.e., x(Cr)=1B: asymmetric with respect to C, rotation, i.e., x(Cn)=-1.Subscriptions 1 or 2 designates those symmetric or asymmetricwith respect to a C2l or a v :SubscriptsgoruforuniversalparityordisparitySuperscriptsordesignatesthosesymmetric orasymmetricwithrespect to On
2.2 symmetry species: Mulliken symbols • All 1-D irreducible reps. are labeled by either A or B, 2-D irreducible rep. by E, 3-D irreducible rep. by T and so on. • A: symmetric with respect to Cn rotation, i.e., (Cn )=1. • B: asymmetric with respect to Cn rotation, i.e., (Cn )=-1. • Subscriptions 1 or 2 designates those symmetric or asymmetric with respect to a C2 or a sv . • Subscripts g or u for universal parity or disparity. • Superscripts ‘ or ‘’ designates those symmetric or asymmetric with respect to sh
2.3 Symmetry of molecular propertiesTranslationsand rotations can be assignedtosymmetryspecies-andso canothermolecularpropertiese.g.p,orbitalon OECNatom of H,0ov(xz) o(yz)+1+1+1Unchanged byall+1HHoperationsPy orbitalB2HHsymmetricstretchof O-HB2HbondsThis set of characters is the representation of the symmetric stretch
2.3 Symmetry of molecular properties
Characters for more than one objectoractionWecanmakerepresentationsof several thingse.g.H1sorbitalsinH,Oorbital 1orbital 2Eoperationorbital1"=orbital1orbital2'=orbital2Eachisunchanged1xitself),sothecharacteris2Strictly speakingthe characteristhe trace(sum of diagonal terms)1of the transformationmatrix
Charactersformorethan one objectoractionRepresentations of several thingse.g.H1sorbitalsin H,0orbital1orbital 2orbital1'=orbital2C2operationorbital2=orbital1There is no contribution from the old orbital 1 to the new one (=0 xitself)so thecharacteris0The trace of the transformationmatrix is zero