Transcript Slide 1

Lecture 11
REPRESENTATIONS OF SYMMETRY POINT GROUPS
1) Mulliken labels
A means symmetric with respect to the highest order axis Cn;
B - antisymmetric
One- (A, B), two- (E), three- (T), four (G), five (H) dimensional representation
Symmetric (') or antisymmetric (") with respect to sh
A'1g
Presence (g) or absence (u) of a center of inversion
Symmetric (1) or antisymmetric (2) behavior with respect to a second
symmetry element (C2 or sv)
C2v
E
C2
sxz
syz
G
A11
1
1
1
1
z
G
A22
1
1
-1
-1
Rz
B31
G
1
-1
1
-1
x, Ry
G
B42
1
-1
-1
1
y, Rx
2) Character tables. Simplest cases
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•
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C1 group. Consists of a single operation E; thus its order h=1
and number of classes is 1. There is a single irreducible
representation. Its character with any basis function is 1 (E).
Cs group. Consists of two operations, E and sh; thus its order
h is 2 and the number of classes is 2. There are two
irreducible representations. They can be one-dimensional
only, (1)2 + (1)2 = 2.
Ci group. Consists of two operations, E and i. Both its order
h and number of classes is 2. Similarly to Cs, the group
includes two irreducible one-dimensional representations.
C1
E
A 1
Cs
E
sh
A'
A"
1
1
x,y, Rz
1
-1
z,Rx,Ry
Ci
E
i
Ag
Au
1
1
Rx
1
-1
x,y,z
3) The character table for the C3v point group
sv
•
Symmetry operations constituting C3v point group
are E, 2 C3 and 3sv (h = 6).
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The number of classes is 3. Thus, there are three
irreducible representations. One of them is always with
all characters equal to 1 (E). Note that 112 + 2(1)2 + 3(1)2 = 6.
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The only combination of dimensions of the three representations which
squares sum to 6 is 1, 1 and 2 (12 + 12 + 22 = 6).
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The second one-dimension representation G2 is orthogonal to the G1:
1(1)(1)+2(1)(1)+3(1)(-1)=0
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The third representation G3 is of order 2. Thus, c3(E) = 2.
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Orthogonality of 3rd, 1st and 2nd representations allows us to find c3(C3) = -1
and c3(sv) = 0 and the complete set of characters:
1(1)(2)+2(1)(x)+3(1)(y)=0 (G1 ┴ G3)
1(1)(2)+2(1)(x)+3(-1)(y)=0 (G2 ┴ G3)
give x = -1 and y = 0
C3v
C3
E
2C3
3sv
AG11
1
1
1
AG22
1
1
-1
G
E3
2
-1
0
4) Reducible representations
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Any reducible representation Gr of a group can always be decomposed into
a sum of irreducible ones Gi:
Gr   N i Gi
i
•
Number of times a particular irreducible representation Gi appears in this
decomposition, Ni, can be calculated according to the formula:
1
N i   c r ( X ) c i ( X )  n( X )
h X
here h is the group order, cr(X) is the character of a symmetry operation X in
the reducible representation; ci(X) is the character of X in the irreducible
representation i and n(X) is the number of symmetry operations in the class.
Ni
C2v
E
C2
sxz
syz
3
A1
1
1
1
1
z
1
A2
1
1
-1
-1
Rz
2
B1
1
-1
1
-1
x, Ry
3
B2
Gr
1
9
-1
-1
-1
1
1
3
y, Rx
Gr = 3 A1 + 1 A2 + 2 B1 + 3 B2