《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