Post on 30-May-2018
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Group theory as applied to chemical problems (bonding,etc) usually involves:
Generating as reducible representation
Reducing it to its component irreducible representations
This info can be used to generate qualitiative MOdiagrams and other useful info!
Reducible Representations
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Reducible Representations
These 4 are the only irreducible representations of the C 2v point group
There are reducible representations for this group (symbolized by )
= 3 3 1 1
Reducible representations can be reduced to a sum of irreducible reps
Can you express as a sum of A 1, A2, B 1 and B 2???
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This was an easy example, but what if its much harder???
There is a formula:
# of times irred. rep. appears = 1/h R I N
h = order of the group (number of operations in the group) R = character of the reducible rep
I= character of the irreducible rep
N = number of equivalent operations in each class
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Reducible Representations
# of times irred. rep. appears = 1/h R I N
h = order of the group (number of operations)
R = character of the reducible rep I = character of the irreducible repN = number of equivalent operations
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= 3 3 1 1
# of times irred. rep. appears = 1/h R I N
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Practice Problem on Reducible Representations
0-12E
-111A2111A1
3 v2C 3 (z)EC 3v
Reduce the following representation of C 3v:
= 4 1 -2
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For more practice on reducible representations, see p. 56-59 in Programme 3, MSGT by Vincent
What about C 4V?How can we represent the effect of sym ops on x, y and z (axis or p orbs)?
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But first before matrices can we say anything more about px
and py
in C4v
?
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Matrices
Read and work probs in Programme 4 in MSGT by Vincent
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Matrix Multiplication
Matrices that can be multiplied are called conformable .In order to multiply 2 matrices (eg. [x] [y]),the columns in [x] must equal the rows in [y]
Note: order matters, [x] [y] [y] [x]
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Matrix Multiplication
Always, product matrix will have-the same number of rows as the 1 st matrix &-the same number of columns as the 2 nd matrix
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Using Matrices To Represent Symmetry Operations
Inversion (i):
(x, y, z) (-x, -y, -z)i
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Using Matrices To Represent Symmetry Operations
Horizontal Mirror Plane ( h):
(x, y, z) (x, y, -z) h
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Using Matrices To Represent Symmetry Operations
Rotation by 180
(C 2(z)):
(x, y, z) (-x, -y, z)C2
In point groups with proper axes of order greater than 2 ( Cn where n>2 ),determining the transformation matrix is more complex, directional properties
mix (they are degenerate) so we need a matrix with off diagonal elements .
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Using Matrices To Represent Symmetry Operations
Rotation by 90
(C 4(z)):
(x, y, z) (x, y, z)C4
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Before We Tackle More Complex Examples, SomeGeneral Rules for Finding the Transformation Matrix
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Using Matrices To Represent Symmetry Operations
Rotation by 120
(C 3(z)):
(x, y, z) (x, y, z)C3
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Character of a Matrix
Character of a Matrix = = Sum of diagonal elements (top left to bottom right)
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Practice Problems