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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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...
1
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
2
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
3
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'
'
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 00-1 0 -1 0 0 = 1 0 00 0 1 0 0 1 0 0 1
v C4 d'
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 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
'
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 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
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 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
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 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
3
9
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
10
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)
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 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 '
'
12
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)
13
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
14
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
15
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
16
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
17
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
18
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
' “
19
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
20
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
21
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