Interpretation:
The
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is that the characters of the same symmetry operations are multiplied together. The multiplication of the characters is commutative.
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
Answer to Problem 13.84E
The
Explanation of Solution
The symmetry elements present in octahedral symmetry are,
Figure 1
For the symmetry element
Figure 2
All the
For
Figure 3
The
The
The
On inversion of the given set of orbitals, all the hybrid orbitals exchange their position and their contribution is
For
For
For
For
This plane can be represented as,
Figure 4
Thus, the reducible representations from the given contribution is,
The great orthogonality theorem for the reducible representation can be represented as,
Where,
•
•
•
•
•
The order of the group is
Substitute the value of order of the group, character of the class of the irreducible representation from character table of
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Similarly, for
The number of times the irreducible representation for
Thus, the linear combination is
The
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Chapter 13 Solutions
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