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...

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1 Group representations 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 Example molecule: SF 5 Cl S F F F F Cl F x y z 3

Transcript of 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...

Page 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.

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Group representations

Consider the group C4v

Element Matrix

E 1 0 00 1 00 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 01 0 00 0 1

Example molecule: SF5Cl

S

F

F

F

F

Cl

F

x

y

z

3

Page 2: 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.

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Group representations

Consider the group C4v

Element Matrix

E 1 0 00 1 00 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 01 0 00 0 1

Example molecule: SF5Cl

S

F

F

F

F

Cl

F

x

y

z

(xyz)

(yxz)3

Page 3: 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.

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Group representations

Consider the group C4v

Element Matrix

E 1 0 00 1 00 0 1

C4 0 1 0 -1 0 0

0 0 1

C2 -1 0 0 v 1 0 0 v -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 d 0 -1 0 d 0 1 0

1 0 0 -1 0 01 0 00 0 1 0 0 1 0 0

1

Example molecule: SF5Cl

S

F

F

F

F

Cl

F

x

y

z

3'

'

Page 4: 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.

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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 00-1 0 -1 0 0 = 1 0 00 0 1 0 0 1 0 0 1

v C4 d'

Page 5: 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.

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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 00-1 0 -1 0 0 = 1 0 00 0 1 0 0 1 0 0 1

v C4 d

Inverses in group

Transpose matrix; determine co-factor matrix of transposed

matrix; divide by determinant of original matrix

'

Page 6: 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.

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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 01 0 0 -1 0 0 -1 0 00 0 1 0 0 1 0 0 1

C4 transpose co-factor matrix

det = 1

3

Page 7: 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.

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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 01 0 0 -1 0 0 -1 0 00 0 1 0 0 1 0 0 1

C4 transpose inverse = C4

All matrices listed show these properties

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Page 8: 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.

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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 01 0 0 -1 0 0 -1 0 00 0 1 0 0 1 0 0 1

C4 transpose inverse = C4

The matrices represent the group

Each individual matrix represents an operation

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Page 9: 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.

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Group representations

Set of representation matrices that can be block diagonalized termed a reducible representation

Ex:

1 0 0 1 0 trace = 00-1 0 0-1 0 0 1 1 trace

= 1

Page 10: 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.

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Group representations

Set of representation matrices that can be block diagonalized termed a reducible representation

Ex:

1 0 0 1 0 trace = 00-1 0 0-1 0 0 1 1 trace

= 1

Character of matrix is its trace (sum of diagonal elements)

Page 11: 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.

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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 v 1 0 0 v -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 d 0 -1 0 d 0 1 0

1 0 0 -1 0 01 0 00 0 1 0 0 1 0 0

1

3 '

'

Page 12: 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.

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Irreducible Representations

1. Sum of squares of dimensions di 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)

Page 13: 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.

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Irreducible Representations

Ex: C2h (E, C2, i, h)

Each operation constitutes a class

C2 –E-1 C2 E = C2

(C2)-1 C2 C2 = C2

i-1 C2 i = C2

(h)-1 C2 h = C2

Other elements behave similarly

C2h

Page 14: 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.

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Irreducible Representations

Ex: C2h (E, C2, i, h)

Each operation constitutes a classMust be 4 irreducible representations

Order of group = 4:

d12 + d2

2 + d32 + d4

2 = 4

All di = ±1All i = ±1

Page 15: 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.

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Irreducible Representations

Ex: C2h (E, C2, i, h)

Each operation constitutes a classThus, must be 4 irreducible representations

Order of group = 4:

d12 + d2

2 + d32 + d4

2 = 4

All di = ±1All 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

Page 16: 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.

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Irreducible Representations

Ex: C2h (E, C2, i, h)

Thus, must be 4 irreducible representations

Order of group = 4:

d12 + d2

2 + d32 + d4

2 = 4

All di = ±1All i = ±1

4 representations: E C2 i h

1 1 1 1 1

2 1 1 –1 –1

3 1 –1 –1 1

4 1 –1 1 –1

Page 17: 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.

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Irreducible Representations

Ex: C2h (E, C2, i, h)

4 representations: E C2 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

C2 -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

Page 18: 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.

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Irreducible Representations

Ex: C3v ([E], [C3, C3 ], [v, v, v,])

3 classes, 3 representations:Order of group = 6Dimensions given by d1

2 + d22 + d3

2 = 6 ––> 1 1 2

E 2C3 3v

1 1 1 1

2 1 1 –1

3 2 –1 0

' “

Page 19: 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.

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Irreducible Representations

Ex: C3v ([E], [C3, C3 ], [v, v, v,])

3 classes, 3 representations:Order of group = 6Dimensions given by d1

2 + d22 + d3

2 = 6 ––> 1 1 2

E 2C3 3v

1 1 1 1

2 1 1 –1

3 2 –1 0

' “

1 00 1

Page 20: 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.

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Irreducible Representations

Ex: C3v ([E], [C3, C3 ], [v, v, v,])

3 classes, 3 representations:Order of group = 6Dimensions given by d1

2 + d22 + d3

2 = 6 ––> 1 1 2

E 2C3 3v

1 1 1 1

2 1 1 –1

3 2 –1 0

' “

1 00 1

-1/2 3/2- 3/2 -1/2

Page 21: 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.

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Irreducible Representations

Ex: C2h (E, C2, i, h)

C2h E C2 i h

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 axisg, u – character wrt i