1 / 21

Group representations

Group representations. Example molecule: SF 5 Cl. z. Consider the group C 4v Element Matrix E 1 0 0 0 1 0 0 0 1 C 4 0 1 0 -1 0 0 0 0 1 C 2 -1 0 0 0 -1 0 0 0 1 C 4 0 -1 0 1 0 0 0 0 1. F. F. F. F. y.

harperd
Download Presentation

Group representations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Group representations Example molecule: SF5Cl z Consider the group C4v Element Matrix E 1 0 0 0 1 0 0 0 1 C4 0 1 0 -1 0 0 0 0 1 C2 -1 0 0 0 -1 0 0 0 1 C4 0 -1 0 1 0 0 0 0 1 F F F F y S F x Cl 3

  2. Group representations Example molecule: SF5Cl z Consider the group C4v Element Matrix E 1 0 0 0 1 0 0 0 1 C4 0 1 0 -1 0 0 0 0 1 C2 -1 0 0 0 -1 0 0 0 1 C4 0 -1 0 1 0 0 0 0 1 F F F F y S F x Cl (yxz) 3 (xyz)

  3. Group representations Example molecule: SF5Cl z Consider the group C4v Element Matrix E 1 0 0 0 1 0 0 0 1 C4 0 1 0 -1 0 0 0 0 1 C2 -1 0 0 sv 1 0 0 sv -1 0 0 0 -1 0 0 -1 0 0 1 0 0 0 1 0 0 1 0 0 1 C4 0 -1 0 sd 0 -1 0 sd 0 1 0 1 0 0 -1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 F F F F y S F x Cl ' 3 '

  4. Group representations These matrices obey all rules for a group when combination rule is matrix multiplication: Identity exists - E 1 0 0 0 1 0 0 0 1 Products in group 1 0 0 0 1 0 0 1 0 0-1 0 -1 0 0 = 1 0 0 0 0 1 0 0 1 0 0 1 sv C4 sd '

  5. Group representations These matrices obey all rules for a group when combination rule is matrix multiplication: Identity exists - E 1 0 0 0 1 0 0 0 1 Products in group 1 0 0 0 1 0 0 1 0 0-1 0 -1 0 0 = 1 0 0 0 0 1 0 0 1 0 0 1 sv C4 sd Inverses in group Transpose matrix; determine co-factor matrix of transposed matrix; divide by determinant of original matrix '

  6. Group representations These matrices obey all rules for a group when combination rule is matrix multiplication: Inverses in group Transpose matrix; determine co-factor matrix of transposed matrix ; divide by determinant of original matrix 0-1 0 0 1 0 0 1 0 1 0 0 -1 0 0 -1 0 0 0 0 1 0 0 1 0 0 1 C4 transpose co-factor matrix det = 1 3

  7. Group representations These matrices obey all rules for a group when combination rule is matrix multiplication: Inverses in group Transpose matrix; determine co-factor matrix of transposed matrix ; divide by determinant of original matrix 0-1 0 0 1 0 0 1 0 1 0 0 -1 0 0 -1 0 0 0 0 1 0 0 1 0 0 1 C4 transpose inverse = C4 All matrices listed show these properties 3

  8. Group representations These matrices obey all rules for a group when combination rule is matrix multiplication: Inverses in group Transpose matrix; determine co-factor matrix of transposed matrix ; divide by determinant of original matrix 0-1 0 0 1 0 0 1 0 1 0 0 -1 0 0 -1 0 0 0 0 1 0 0 1 0 0 1 C4 transpose inverse = C4 The matrices represent the group Each individual matrix represents an operation 3

  9. Group representations Set of representation matrices that can be block diagonalized termed a reducible representation Ex: 1 0 0 1 0 trace = 0 0-1 0 0-1 0 0 1 1 trace = 1

  10. Group representations Set of representation matrices that can be block diagonalized termed a reducible representation Ex: 1 0 0 1 0 trace = 0 0-1 0 0-1 0 0 1 1 trace = 1 Character c of matrix is its trace (sum of diagonal elements)

  11. Group representations Consider the group C4v Element Matrix E 1 0 0 all matrices can be block diagonalized - all 0 1 0 are reducible 0 0 1 C4 0 1 0 -1 0 0 0 0 1 C2 -1 0 0 sv 1 0 0 sv -1 0 0 0 -1 0 0 -1 0 0 1 0 0 0 1 0 0 1 0 0 1 C4 0 -1 0 sd 0 -1 0 sd 0 1 0 1 0 0 -1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 ' 3 '

  12. Irreducible Representations Sum of squares of dimensions di of the irreducible representations of a group = order of group Sum of squares of characters ci in any irreducible representation = order of group Any two irreducible representations are orthogonal (sum of products of characters representing each operation = 0) No. of irreducible representations of group = no. of classes in group (class = set of conjugate elements)

  13. Irreducible Representations Ex: C2h (E, C2, i, sh) Each operation constitutes a class C2 – E-1 C2 E = C2 (C2)-1 C2 C2 = C2 i-1 C2 i = C2 (sh)-1 C2 sh = C2 Other elements behave similarly C2h

  14. Irreducible Representations Ex: C2h (E, C2, i, sh) Each operation constitutes a class Must be 4 irreducible representations Order of group = 4: d12 + d22 + d32 + d42 = 4 All di = ±1 All ci = ±1

  15. Irreducible Representations Ex: C2h (E, C2, i, sh) Each operation constitutes a class Thus, must be 4 irreducible representations Order of group = 4: d12 + d22 + d32 + d42 = 4 All di = ±1 All ci = ±1 Let G1 = 1 1 1 1 Array G1 of matrices represents the group – thus exhibits all group props. & has same mult. table E = 1 E-1 = 1 1 1 = 1 1-1 = 1

  16. Irreducible Representations Ex: C2h (E, C2, i, sh) Thus, must be 4 irreducible representations Order of group = 4: d12 + d22 + d32 + d42 = 4 All di = ±1 All ci = ±1 4 representations: E C2 i sh G1 1 1 1 1 G2 1 1 –1 –1 G3 1 –1 –1 1 G4 1 –1 1 –1

  17. Irreducible Representations E 1 0 0 0 1 0 0 0 1 C2 -1 0 0 0-1 0 0 0 1 i -1 0 0 0-1 0 0 0-1 sh 1 0 0 0 1 0 0 0-1 Ex: C2h (E, C2, i, sh) 4 representations: E C2 i sh G1 1 1 1 1 G2 1 1 –1 –1 G3 1 –1 –1 1 G4 1 –1 1 –1 These irreducible representations are orthogonal Ex: 1 1 + 1 1 + 1 (-1) + 1 (-1) = 0

  18. Irreducible Representations Ex: C3v ([E], [C3, C3 ], [sv, sv, sv,]) 3 classes, 3 representations: Order of group = 6 Dimensions given by d12 + d22 + d32 = 6 ––> 1 1 2 E 2C3 3sv G1 1 1 1 G2 1 1 –1 G3 2 –1 0 ' “

  19. Irreducible Representations Ex: C3v ([E], [C3, C3 ], [sv, sv, sv,]) 3 classes, 3 representations: Order of group = 6 Dimensions given by d12 + d22 + d32 = 6 ––> 1 1 2 E 2C3 3sv G1 1 1 1 G2 1 1 –1 G3 2 –1 0 ' “ 1 0 0 1

  20. Irreducible Representations Ex: C3v ([E], [C3, C3 ], [sv, sv, sv,]) 3 classes, 3 representations: Order of group = 6 Dimensions given by d12 + d22 + d32 = 6 ––> 1 1 2 E 2C3 3sv G1 1 1 1 G2 1 1 –1 G3 2 –1 0 ' “ 1 0 0 1 -1/2 3/2 - 3/2 -1/2

  21. Irreducible Representations Ex: C2h (E, C2, i, sh) C2h E C2 i sh Ag 1 1 1 1 Rz Bg 1 –1 1 –1 Rx Ry Au 1 1 –1 –1 z Bu 1 –1 –1 1 x y 1-D representations called A (+), B(–) 2-D representations called E 2-D representations called T Subscript 1 - symmetric wrt C2 perpend to rotation axis g, u – character wrt i

More Related