1. 1 Group representations Consider the group C 4v ElementMatrix E1 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 Example molecule: SF 5 Cl S F F F F Cl F x y z 3

2. 2 Group representations Consider the group C 4v ElementMatrix E1 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 Example molecule: SF 5 Cl S F F F F Cl F x y z (xyz) (yxz) 3

3. 3 Group representations Consider the group C 4v Element Matrix E1 0 0 0 1 0 0 0 1 C 4 0 1 0 -1 0 0 0 0 1 C 2 -1 0 0  v 1 0 0  v -1 0 0 0 -1 00 -1 00 1 0 0 0 10 0 10 0 1 C 4 0 -1 0  d 0 -1 0  d 0 1 0 1 0 0 -1 0 01 0 0 0 0 10 0 10 0 1 Example molecule: SF 5 Cl S F F F F Cl F x y z 3 ' '

4. 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  v C 4  d '

5. 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  v C 4  d Inverses in group Transpose matrix; determine co-factor matrix of transposed matrix; divide by determinant of original matrix '

6. 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 10 0 1 C 4 transpose co-factor matrix det = 1 3

7. 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 10 0 1 C 4 transpose inverse = C 4 All matrices listed show these properties 3

8. 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 10 0 1 C 4 transpose inverse = C 4 The matrices represent the group Each individual matrix represents an operation 3

9. 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 1trace = 1

10. 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 1trace = 1 Character  of matrix is its trace (sum of diagonal elements)

11. 11 Group representations Consider the group C 4v Element Matrix E1 0 0all matrices can be block diagonalized - all 0 1 0 are reducible 0 0 1 C 4 0 1 0 -1 0 0 0 0 1 C 2 -1 0 0  v 1 0 0  v -1 0 0 0 -1 00 -1 00 1 0 0 0 10 0 10 0 1 C 4 0 -1 0  d 0 -1 0  d 0 1 0 1 0 0 -1 0 01 0 0 0 0 10 0 10 0 1 3 ' '

12. 12 Irreducible Representations 1. Sum of squares of dimensions d i of the irreducible representations of a group = order of group 2. Sum of squares of characters  i in any irreducible representation = order of group 3. Any two irreducible representations are orthogonal (sum of products of characters representing each operation = 0) 4. No. of irreducible representations of group = no. of classes in group (class = set of conjugate elements)

13. 13 Irreducible Representations Ex: C 2h (E, C 2, i,  h ) Each operation constitutes a class C 2 – E -1 C 2 E = C 2 (C 2 ) -1 C 2 C 2 = C 2 i -1 C 2 i = C 2 (  h ) -1 C 2  h = C 2 Other elements behave similarly C 2h

14. 14 Irreducible Representations Ex: C 2h (E, C 2, i,  h ) Each operation constitutes a class Must be 4 irreducible representations Order of group = 4: d 1 2 + d 2 2 + d 3 2 + d 4 2 = 4 All d i = ±1 All  i = ±1

15. 15 Irreducible Representations Ex: C 2h (E, C 2, i,  h ) Each operation constitutes a class Thus, must be 4 irreducible representations Order of group = 4: d 1 2 + d 2 2 + d 3 2 + d 4 2 = 4 All d i = ±1 All  i = ±1 Let  1 = 1 1 1 1 Array  1 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. 16 Irreducible Representations Ex: C 2h (E, C 2, i,  h ) Thus, must be 4 irreducible representations Order of group = 4: d 1 2 + d 2 2 + d 3 2 + d 4 2 = 4 All d i = ±1 All  i = ±1 4 representations: E C 2 i  h  1 1 1 1 1  2 1 1 –1 –1  3 1 –1 –1 1  4 1 –1 1 –1

17. 17 Irreducible Representations Ex: C 2h (E, C 2, i,  h ) 4 representations: E C 2 i  h  1 1 1 1 1  2 1 1 –1 –1  3 1 –1 –1 1  4 1 –1 1 –1 These irreducible representations are orthogonal Ex:1 1 + 1 1 + 1 (-1) + 1 (-1) = 0 E 1 0 0 0 1 0 0 0 1 C 2 -1 0 0 0-1 0 0 0 1 i -1 0 0 0-1 0 0 0-1  h 1 0 0 0 1 0 0 0-1

18. 18 Irreducible Representations Ex: C 3v ([E], [C 3, C 3 ], [  v,  v,  v,]) 3 classes, 3 representations: Order of group = 6 Dimensions given by d 1 2 + d 2 2 + d 3 2 = 6 ––> 1 1 2 E 2C 3 3  v  1 1 1 1  2 1 1 –1  3 2 –1 0 '“

19. 19 Irreducible Representations Ex: C 3v ([E], [C 3, C 3 ], [  v,  v,  v,]) 3 classes, 3 representations: Order of group = 6 Dimensions given by d 1 2 + d 2 2 + d 3 2 = 6 ––> 1 1 2 E 2C 3 3  v  1 1 1 1  2 1 1 –1  3 2 –1 0 '“ 1 0 0 1

20. 20 Irreducible Representations Ex: C 3v ([E], [C 3, C 3 ], [  v,  v,  v,]) 3 classes, 3 representations: Order of group = 6 Dimensions given by d 1 2 + d 2 2 + d 3 2 = 6 ––> 1 1 2 E 2C 3 3  v  1 1 1 1  2 1 1 –1  3 2 –1 0 '“ 1 0 0 1 -1/2 3/2 - 3/2 -1/2

21. 21 Irreducible Representations Ex: C 2h (E, C 2, i,  h ) C 2h E C 2 i  h A g 1 1 1 1 R z B g 1 –1 1 –1 R x R y A u 1 1 –1 –1 z B u 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 C 2 perpend to rotation axis g, u – character wrt i