2.（CharacterTables ofPointGroupsExample-pointgroupC2vCharactertableECh=4ov(xz) oy(yz)+1x2, y2, z2ARzBasesBBX RLRVTop line:point groupsymmetryoperationsorder ofgroup,h, =number of symmetry operations
2. Character Tables of Point Groups Bases
2.1Constructionof Character TableTotal Representation for C2vIndividually block diagonalized matricesEC2OyzOxz000000000000000000000000Reducedto 1Dmatricesirreducible representation-1x[ 1] [-1] [ 1] [-1]-111T=1y[ 1] [-1] [-1] [1]-1-11=1111.川z [ 1] [1] [1] [ 1]1-1-1=1Z
2.1 Construction of Character Table
TranslationsMovementsof wholemolecule-representbyvectorsEoperationy(afteroperation)=ye.g-y vectorC2y'=-y (i.e. y'=-1 xy)ov(xz)y'=-yov(yz)y'=yall operationsz vector=Z29Eoperationx'=Xx vectorC2X'=-X一ov(xz)X'=Xov(yz)X'=-X
TranslationsConsidereffectof symmetryoperation onthevectorWrite+1fornochange,-1forreversalEov(xz)ov(yz)C2A1+1+1+1z vector+1B2+17yB1-1x+1ECCoy(xz) oy(yz)Labels A, etc.aresymmetryspecies;A+1+1+1+1theysummarisetheAeffects of symmetryBBoperations onthe+1+1工vectors.These translation vectors constitute a set of bases of C2y group
These translation vectors constitute a set of bases of C2v group
RotationsSimilarlyfor rotations of the moleculesEov(xz)ov(yz)A+1+1z vector-?y+7B,X+1+1RzA2+1+1-11RyB1+1店RxB2+1+1