RotationsSimilarlyfor rotations of the moleculesEov(xz)ov(yz)A+1+1zvector+1-?y++B1X+1+17Rz?+1+1-11RyB1+17RB2+1+1
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心21speciesfortranslationsAandrotationsARzBX, RyA,BshowsymmetrywithB2respect to rotation.y, Rx1,2distinguishsymmetrywithrespecttoreflections
: Mulliken symbolsSymmetry species:ECr2C,30,111x2 + y2, 22A,z1R:A21-12E0-1(x2 y2, xy)(xz, yz)(x, y)(Rx, R,)All 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 Cn rotation, i.e., x(C,)=1B: asymmetric with respect to C, rotation, i.e., x(C,)=-1 Subscriptions 1 or 2 designates those symmetric orasymmetric with respect to a C2l or a v :Subscripts g or u for universal parity or disparitySuperscripts 'or " designates those symmetric orasymmetric with respect to o
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 v . • Subscripts g or u for universal parity or disparity. • Superscripts or designates those symmetric or asymmetric with respect to h
3.5.3The “Great Orthogonality Theorem" and ItsConsequences广义正交定理Some notations:h - the order of a group; R - operations (elements) of a point groupl,-the dimension of ith representation (i.e., the order of its matrices)F(R)mn - the element in the mth row and nth column of the matrixcorresponding to the operation R in the ith representationhSSZ/T,(R)mnI[,(R)mrnnmm1,1RIt means that in the set of matrices constituting any oneirreducible representation, any set of corresponding matrixelements, one from each matrix, behaves as the components of avectorinah-dimensional spacesuchthatallthesevectorsaremutually orthogonal and each is normalized so that the square its length is h/l
3.5.3 The “Great Orthogonality Theorem” and Its Consequences Some notations: h – the order of a group; R – operations (elements) of a point group. l i – the dimension of ith representation (i.e., the order of its matrices) i (R)mn – the element in the mth row and nth column of the matrix corresponding to the operation R in the ith representation. ' ' * ' ' [ ( ) ][ ( ) ] i j m m n n R i j i m n j m n l l h R R It means that in the set of matrices constituting any one irreducible representation, any set of corresponding matrix elements, one from each matrix, behaves as the components of a vector in a h-dimensional space such that all these vectors are mutually orthogonal and each is normalized so that the square of its length is h/li . 广义正交定理
Five important rulesregarding irreducible representations and their characters:Rule 1 - the sum of the squares of the dimensions of theirreducible representations of a group is egqual to the order of agroup.E30,C.r2C;Zt?=he.g., for C3v,i1A,1A21Zl? =1 +1? +2? =6= hE20-1x(E)=l|=Z[x(E)}? = hRule 2-thesum of the square of the characters in anyirreduciblerepresentationofagroupequalshZ/ x,(R)/? = he.g., A2 forC3v, 1? +2.1? +3.(-1) =6R
E h i i 2 [ ( )] Five important rules regarding irreducible representations and their characters: l h i i 2 Rule 2 – the sum of the square of the characters in any irreducible representation of a group equals h, e.g., for C3v, l h i i 1 1 2 6 2 2 2 2 i i ( E ) l R h R i 2 [ ( )] 1 2 1 3 1 6 2 2 2 e.g.,A2 forC3v , ( ) Rule 1 – the sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of a group