symmetrry operations 2_9

8/14/2019 Symmetrry Operations 2_9

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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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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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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



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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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Horizontal Mirror Plane ( h):

(x, y, z) (x, y, -z) h


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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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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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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

symmetrry operations 2_9

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