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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.

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Group representations

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  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

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