Interpretation:
The characters of the
Concept introduction:
The characters of the irreducible representations of the given point group can be multiplied by each other. The only condition is 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,
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•
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Answer to Problem 13.53E
The characters of the
Explanation of Solution
The symmetry elements present in octahedral symmetry are,
From
Substitute the value of
From
Substitute the value of
From
Substitute the value of
From
Substitute the value of
From
Substitute the value of
Therefore, the character table for
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 characters of the
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Chapter 13 Solutions
Physical Chemistry
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- Identify the symmetry elements present in the following objects. a A ream of blank paper, no holes. b A ream of blank three-holed paper. c A round pencil, unsharpened, with cylindrical eraser. d A round pencil, sharpened, with cylindrical eraser.arrow_forwardIdentify all the symmetry elements present in the tetrahedron, the cube, and the octahedron.arrow_forwardShow that any two of the irreducible representations of the following point groups are orthogonal to each other. a C2 b C2v c D2h d Oh e Tdarrow_forward
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- List the symmetry elements of the following molecules and name the point groups to which they belong: (a) staggered CH3CH3, (b) chair and boat cyclohexane, (c) B2H6, (d) [Co(en)3]3+, where en is 1,2-diaminoethane (ignore its detailed structure), (e) crown-shaped S8. Which of these molecules can be (i) polar, (ii) chiral?arrow_forwardThe algebraic forms of the f orbitals are a radial function multiplied by one of the factors (a) z(5z2 − 3r2), (b) y(5y2 − 3r2), (c) x(5x2 − 3r2), (d) z(x2 − y2), (e) y(x2 − z2), (f) x(z2 − y2), (g) xyz. Identify the irreducible representations spanned by these orbitals in the point group C2v.arrow_forwardUsing the coordinate system provided, derive the irredicuble representations (set of 4 characters) for the the following orbitals in the C2v point group. Give the Muliken symbol for each irreducible representation. a. pz b. py c. dz2 d. dxyarrow_forward
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