《结构化学 Structural Chemistry》课程教学资源（课件讲稿）Chapter 3 Molecular Symmetry and Point Group Part B Representation of Point Group and Applications Download BW version

Chapter 3Molecular symmetry and symmetry point groupPart B(ref. Chemical Application of Group Theory, 3rd ed., F.ACotton, by John Wiley & Sons, 1990.)

Chapter 3 Molecular symmetry and symmetry point group Part B (ref. Chemical Application of Group Theory, 3rd ed., F.A. Cotton, by John Wiley & Sons, 1990.)

S 3.5 Group representation Theory and irreduciblerepresentation of point groups3.5.1Representationsofapointgroup:reducible vs.irreducibleFora point group,> Each element is a unique symmetry operation (operator)Eachoperationcanberepresentedbya squarematrix.These matrices constitute a matrixgroup,i.e.,a matrixrepresentationofthispointgroupExample: C,=(E, i}~ a general point (x,y,z) in spaceXR0a matrix group

For a point group,  Each element is a unique symmetry operation (operator).  Each operation can be represented by a square matrix.  These matrices constitute a matrix group, i.e., a matrix representation of this point group. 3.5.1 Representations of a point group: reducible vs. irreducible §3.5 Group representation Theory and irreducible representation of point groups Example: Ci = {E, i} ，               0 0 1 0 1 0 1 0 0 i ˆ ~ a general point (x,y,z) in space.                                                 z y x z y x z y x i 0 0 1 0 1 0 1 0 0 ˆ            0 0 1 0 1 0 1 0 0 Eˆ a matrix group

Example: C,one unit vectorxE(x)=(1)(x)= (x)i(x)=(-1)(x)=(-x(1),(-1)The corresponding matrix representation of C, isQl:How many representations can be found for a particular group?A large number, limited on our ingenuity in devising ways togenerate themQ2: If we were to assign three small unit vectors directed along the x,y and z axes to each of the atoms in H,O and write down the matricesrepresenting the changes and interchanges of these upon theoperations, what would be obtained?A matrix representation consisting of four 9x9 matrices would beobtained upon operating on a column matrix (xo, Yo, zo, XHi, YHI, ZHI)Xh2 YH2, ZH2)

Q1:How many representations can be found for a particular group? A large number, limited on our ingenuity in devising ways to generate them. Q2: If we were to assign three small unit vectors directed along the x, y and z axes to each of the atoms in H2O and write down the matrices representing the changes and interchanges of these upon the operations, what would be obtained? Example: Ci one unit vector x ix1x  x ˆ Ex1x x ˆ The corresponding matrix representation of Ci is 1，-1 A matrix representation consisting of four 9x9 matrices would be obtained upon operating on a column matrix (xO , yO , zO , xH1, yH1, zH1, xH2, yH2, zH2)

Example:three unit vectors (x,y,z) or a general point[E, C2, Oxz, Oyz]Principal axis: z-axis00OXxX00010-100xxxxX00<60-V一1X0OZamatrix representation ofC2VEC200yz0000000-11-1000000-1-110000000010101

Example: C2v three unit vectors (x,y,z) or a general point {E, C2 , xz, yz} Principal axis: z-axis.                                            0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 E C 2  xz  y z                                           z y x z y x z y x E 0 0 1 0 1 0 1 0 0 ˆ                                             z y x z y x z y x xz 0 0 1 0 1 0 1 0 0 ˆ                                           z y x z y x z y x yz 0 0 1 0 1 0 1 0 0 ˆ                                               z y x z y x z y x C 0 0 1 0 1 0 1 0 0 2 ˆ a matrix representation of C2v

Bases, representations and their dimensionsDimension of a representation =The order of matrices..DifferentbasisDifferentrepresentation.Example: C2yBasis ~ a general point or three unit vectorsEC2aOyz000000A 3-D rep000000000福Simple basis: a translational vector as x, y, or z, or a rotor RzReducedto 1Dmatricesirreduciblerepresentation1-D Reps.XIF11-11711T-11-11F=1-11[1] [-1] [-1] [1]-1-1111[] [[ []1=1-1-1I Rz = 15Z

Example: C2v Bases, representations and their dimensions Basis ~ a general point or three unit vectors. Simple basis: a translational vector as x, y, or z, or a rotor Rz • Different basis  Different representation. 1-D Reps. • Dimension of a representation = The order of matrices.                                            0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 E C 2  xz  y z A 3-D rep

Reducing of representationsSuppose that we have a set of n-dimensional matrices, A, B,C, ... , which form a representation of a group. These n-Dmatrices themselves constitute a matrix group I= (A, B,..].If we make the same similarity transformation on eachmatrix, we obtain a newset ofmatrices, namelyA'= X-IAX, B'= X-'BX,C'= X-'CX, ...I'=(A', B',C',..that forms a new matrix group: F' is also a representation of the group!

Reducing of representations • Suppose that we have a set of n-dimensional matrices, A, B, C, . , which form a representation of a group. These n-D matrices themselves constitute a matrix group  = {A，B,.}. • If we make the same similarity transformation on each matrix, we obtain a new set of matrices，namely,   A' ， B' ，C' ,. •  is also a representation of the group! that forms a new matrix group: , . 1 1 1 C X CX A X AX B X BX       ' ' ， '

It is provable that if any of the matrix (e.g., A') in I' is ablock-factored matrix, then all other matrices (e.g., B,c',...)inrare also blocked-factored000[c]0[B, ]00[4]000[4.]0[B, ] 0[c,] 0000000A':B':[c,] [4, ][B,] 000000000[c.][B ]000000000Ain which A,A2,A3... are n,n2,n3...-order submatrices with n =ng + n2 + n3 +.These n-order matrices can be simply expressed asA'= A,④A2 ④A ④...,B'= B,④B2 ④ B3 ④ ...,C'= C,④C, ④ C 甲...,(Directsumofsubmatrices!

                        ' , ' , ' ，.                                        4 3 2 1 4 3 2 1 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C C C C C B B B B B A A A A A in which A1 ,A2 ,A3. are n1 ,n2 ,n3.-order submatrices with n = n1 + n2 + n3 + . • It is provable that if any of the matrix (e.g., A ) in  is a block-factored matrix, then all other matrices (e.g., B,C ,.) in  are also blocked-factored. A= A1A2  A3  ., B= B1B2  B3  ., C= C1C2  C3  ., . • These n-order matrices can be simply expressed as (Direct sum of submatrices！）

It is also provable that the various sets of submatricesT,={A,B1,C....], T2-{A2,B2,C2..], T3-{A3,B3,C3...], ...,are in themselves representations of the groupWe then call the set of matrices I={A,B,C, ...} a reduciblerepresentation of the group, which breaks up into a directsum of the representations, i.e., I = T, ④ T, ④ T3 @ ...If it is not possibleto find a similarity transformationtoreduce a representation in the above manner, therepresentationissaidtobeirreducibleThe irreducible representations of apoint group are mostlycountableandoffundamentalimportance!

• It is also provable that the various sets of submatrices, T1={A1 ,B1 ,C1.}, T2={A2 ,B2 ,C2.}, T3={A3 ,B3 ,C3.}, ., are in themselves representations of the group. • We then call the set of matrices ={A,B,C, .} a reducible representation of the group, which breaks up into a direct sum of the representations, i.e.,  = T1  T2  T3  . • If it is not possible to find a similarity transformation to reduce a representation in the above manner, the representation is said to be irreducible. • The irreducible representations of a point group are mostly countable and of fundamental importance!

Example: C2vIs this 3-D Rep. reducible?Yes. These matrices are block-factored!EC2OxzOyz00000000一L0000000xyz000000000Y1RedicedoDmatriceirreduciblerepresentation1[1] [-1] [1] [-1]1-1-1X1-11] [-1] [-11 [1]-11S1111=Z[1][1][1][1]Fxz=F ④, ④The 3-D rep. is reduced to 3 1-D rep

Example: C2v Is this 3-D Rep. reducible? xyz xyz =x y z Yes. These matrices are block-factored! The 3-D rep. is reduced to 3 1-D rep

(symm. ops.)Point group RR={RA, Rp, Rc,...]Exerted on any set of bases(e.g., AO's, MO's, vectors, rotations etc.)Amatrix group, I = {A, B, C, ...}(a matrix rep. of group R, dimension = order of the matrix)Similarity transformations (reducing of a representation!A block-factored matrix group, I' ={A, B', C', ...(A'=A,④A,④..., B"=B,B,④..., C' =C,④C,@... ....)and F ={Aj,Bi,C1....] , I, ={A2,B2,C2....] ...&=I④④Direct sum of irreducible representations!

Point group R R={RA，RB，RC，.} A matrix group,  = {A，B，C，.} (a matrix rep. of group R, dimension = order of the matrix) Exerted on any set of bases （e.g., AOs, MOs, vectors, rotations etc.) Similarity transformations (reducing of a representation!) Direct sum of irreducible representations! (symm. ops.) A block-factored matrix group,  = {A ，B ，C ，.} (A = A1 A2 ., B = B1 B2 ., C = C1 C2 . ,.) and 1 ={A1 ,B1 ,C1 ,.} , 2 ={A2 ,B2 ,C2 ,.} . &  = 1  2 