The irreducible representations of a point group aremostly countable and offundamental importance!!!The character table of a point group lists up allessentialinformationofitsirreduciblerepresentations
The irreducible representations of a point group are mostly countable and of fundamental importance!!! The character table of a point group lists up all essential information of its irreducible representations
3.5.2. CharacterTables of Point GroupsExample-pointgroupC2CharactertableEC2Sh=4ov(xz) oy(yz)AABBx2+Frequently usedbasisRzxye.g., translation.x,RX2rotation,and so onRVCharactersTop line: point groupsymmetryoperationsorderofgroup,h=numberofsymmetryoperationsSymmetry speciesofirreduciblerepresentations
3.5.2. Character Tables of Point Groups Frequently used basis, e.g., translation, rotation, and so on Symmetry species of irreducible representations. Characters Top line: point group symmetry operations order of group, h = number of symmetry operations
Characters &reducing representation!x(A)=Zai(A)Character of a matrix A.(sumofitsdiagonalelements!)EC2OxzOyz210000003-D Rep.000000I xyz000000x(E) =3x(αx) = 1x(gy) = 1x(C) =-1Reducedto1DmatricesEC2Oxz6yB11-1x [1] [-1] [1] [-1]1-1B2y[ 1] [-1] [-1] [1]1-1-17A11111z [1][1][1][1]R311-1Ixyz =A, 甲B, @BxyzIf T=F,④T, ④I, ④.. Xr(R) =ZXr,(R)F---I.R.s
Characters & reducing representation! • Character of a matrix A: (E) =3 (C2 ) = -1 (xz) = 1 (yz) = 1 xyz xyz =A1 B1 B2 C2v 3-D Rep. E C2 xz yz 3 -1 1 1 B1 B2 A1 xyz i ( A ) aii( A) (sum of its diagonal elements!) i Γ Γ Γ Γ Γ Γ χ (R) χ (R) i If 1 2 3 ., i -I.R.s
TranslationsMovements of whole molecule-representbyvectorsEoperationy (after operation)=ye.g.y vectorC2y'= -y (i.e. y' = -1 xy)ov(xz)y'=-yov(yz)y'=yall operations=Zz vector23Eoperationx'=xx vectorCx'=-xHov(xz)x'=xov(yz)x'=-X
TranslationsConsider effectof symmetryoperation on the vectorWrite+1fornochange,-1for reversalEov(xz)o(yz)A+1z vector?B1x+1EC2C2oy(xz) v(yz)Labels A, etc. aresymmetryspecies;A+7they summarise theeffectsof symmetryB1operations ontheB2+1+vectors