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Juggling with representations Matrix representation of Symmetry - PowerPoint PPT Presentation

Sep 13, 2022 •441 likes •860 views

Juggling with representations Matrix representation of Symmetry Point Groups C2v Irreducible Representation E C2 ( z ) v ( zx ) v ( yz ) Basis O 1 1 1 1 a 1 0 0 1 1 0 0 1 b 0 1 1 0 0

  1. Juggling with representations

  2. Matrix representation of Symmetry Point Groups C2v Irreducible Representation E C2 ( z ) σ v ( zx ) σ v ’( yz ) Basis O 1 1 1 1 a 1 0 0 1 1 0 0 1 b 0 1 1 0 0 1 1 0 σ v’ Two dimensional representation O Reducible? How many such representations? Ha Hb σ v C2

  3. Function space Z ( e3 ) Collection of functions f1 , f2 , …, fi, …, f n p1 • fi + fj = fk p2 • n fm = fn Y ( e2 ) p3 Σ ai fi = fq • X ( e1 ) ( fi , fj )= ∫ fi* fi d τ • p4 • If n of the functions are linearly Linear independence: independent, then any of the other functions can be represented as a Σ αι φι = 0 linear combination of these n functions. The space is n -dimensional if and only if • Orthonormal basis functions ai = 0 for all values of i 3

  4. Transformation operators, OR for a function space made up of n linearly independent basis functions, fj 4

  5. Transformation operators leave the scalar product of two functions unchanged d’ d d d 5

  6. Transformation operators are linear (a) If f and g are functions, a is a number and g = a f (b) If f , g , h are functions, h = f + g = 6

  7. Transformation operators produce a unitary representation if orthonormal basis functions are used ⇒ ∫ * δ ij ∫ διϕ * * διϕ * kl * διϕ D( R )†D( R ) = E 8

  8. Transformation operators produce a unitary representation if orthonormal basis functions are used ⇒ How does one switch to an orthonormal basis?? Similarity Transformation διϕ * D( R )†D( R ) = E 9

  9. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = 7 3 3 e g a P 10

  10. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = A 11

  11. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = B 12

  12. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = B = A -1 B 13

  13. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = B = A -1 B Let A = A ‘ and B = B ‘ 14

  14. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = B = A -1 B Let A = A ‘ and B = B ‘ ⇒ B = A -1 15

  15. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = B = A -1 B Let A = A ‘ and B = B ‘ ⇒ B = A -1 Σ n fk = gl Alk l = 1 16

  16. Switching bases Let f 1, f 2, …., fn and g 1, g 2, …., gn be two sets of linearly independent basis functions for the same space e.g. px, py, pz and p1 , p-1 , p0 orbitals = B = A -1 B Let A = A ‘ and B = B ‘ ⇒ B = A -1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 17

  17. Now what? Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 18

  18. Transformation operators Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 19

  19. Transformation operators Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 20

  20. Transformation operators Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 21

  21. Transformation operators Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 22

  22. Transformation operators Σ n fi Bij i = 1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 23

  23. Transformation operators Σ Σ n n = fi fi Bij Bij since OR is a linear operator i = 1 i = 1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 24

  24. Transformation operators Σ Σ n n = fi Bij fi Bij i = 1 i = 1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 25

  25. Transformation operators Σ Σ n n = fi Bij i Bij i = 1 i = 1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 26

  26. Transformation operators Σ Σ n n = fi Bij i Bij i = 1 i = 1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 27

  27. Transformation operators Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 Σ Σ n n fk = gl Alk gj = fi Bij l = 1 i = 1 28

  28. Transformation operators Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ n gl l = 1 29

  29. Transformation operators Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ Σ n n gl Bij Alk l = 1 i = 1 30

  30. Transformation operators Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ Σ n n gl Bij Alk l = 1 i = 1 31

  31. Transformation operators Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ Σ n n gl Bij Alk l = 1 i = 1 32

  32. A relationship between Df(R) and Dg(R) Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ Σ n n gl Bij Alk l = 1 i = 1 Dg ( R ) = A Df ( R ) B 33

  33. A relationship between Df(R) and Dg(R) Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ Σ n n gl Bij Alk But B = A -1 l = 1 i = 1 Dg ( R ) = A Df ( R ) B 34

  34. Similarity transformation switches the bases Σ Σ Σ n n n = fi Bij i Bij gl Alk i = 1 i = 1 l = 1 = Σ Σ n n gl Bij Alk l = 1 i = 1 Dg ( R ) = A Df ( R ) A -1 35

  35. Similarity transformation switches the bases • Transformation matrices for two linearly idependent bases are conjugate to each other Dg ( R ) = A Df ( R ) A -1 36

  36. Similarity transformation switches the bases • Transformation matrices for two linearly idependent bases are conjugate to each other • The similarity transformation is achieved by the matrices that relate the bases linearly Dg ( R ) = A Df ( R ) A -1 37

  37. Similarity transformation switches the bases • Transformation matrices for two linearly idependent bases are conjugate to each other • The similarity transformation is achieved by the matrices that relate the bases linearly • Change of basis does not affect the multiplication rules Dg ( R ) = A Df ( R ) A -1 38

  38. Similarity transformation switches the bases • Transformation matrices for two linearly idependent bases are conjugate to each other • The similarity transformation is achieved by the matrices that relate the bases linearly • Change of basis does not affect the multiplication rules If Df ( SR ) = Df ( S ) Df ( R ), then Dg ( SR ) = Dg ( S ) Dg ( R ) Dg ( R ) = A Df ( R ) A -1 39

  39. Equivalent representations • Transformation matrices for two linearly idependent bases are conjugate to each other • The similarity transformation is achieved by the matrices that relate the bases linearly • Change of basis does not affect the multiplication rules If Df ( SR ) = Df ( S ) Df ( R ), then Dg ( SR ) = Dg ( S ) Dg ( R ) Two representations of a point group are EQUIVALENT if, for every symmetry operation R , Dg ( R ) = A Df ( R ) A -1 using the same pair of matrices A and A -1 Dg ( R ) = A Df ( R ) A -1 40

  40. Homework Problem 7 3 3 e g a P • Consider the two sets of p orbitals: px , py , pz and p +1, p -1, p 0 • Work out the transformation matrices for both the sets, for all the symmetry elements of point group C3v • With the help of the matrices that correlate px , py , pz with p +1, p -1, p 0, prove the equivalence of the two representations thus generated 41

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