The characters of the f orbitals in an octahedral environment are to be predicted. 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, a Γ = 1 h ∑ all classes of point group N ⋅ χ Γ ⋅ χ linear combo Where, • a Γ is the number of times the irreducible representation appears in a linear combination. • h is the order of the group. • χ Γ is the character of the class of the irreducible representation. • χ linear combo is the character of the class linear combination. • N is the number of symmetry operations.

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Answer 1

Question

Chapter 13, Problem 13.53E

Interpretation Introduction

Interpretation:

The characters of the f orbitals in an octahedral environment are to be predicted.

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,

aΓ=1h∑all classesof point groupN⋅χΓ⋅χlinear combo

Where,

• aΓ is the number of times the irreducible representation appears in a linear combination.

• h is the order of the group.

• χΓ is the character of the class of the irreducible representation.

• χlinear combo is the character of the class linear combination.

• N is the number of symmetry operations.

Expert Solution & Answer

Answer to Problem 13.53E

The characters of the f orbitals in an octahedral environment are A2u⊕T1u⊕T2u.

Explanation of Solution

The symmetry elements present in octahedral symmetry are, E, C3, C2, C4, C2', i, S4, S6, σh and σd.

From Rh(3) character table the formula to calculate the value of χC2 is,

χC2=1+2cosθ+2cos2θ+2cos3θ …(1)

Substitute the value of θ=180° in equation (1).

χC2=1+2cos180°+2cos2(180°)+2cos3(180°)=1−2+2−2=−1

From Rh(3) character table the formula to calculate the value of χC3 is,

χC3=1+2cosθ+2cos2θ+2cos3θ …(1)

Substitute the value of θ=120° in equation (1).

χC3=1+2cos120°+2cos2(120°)+2cos3(120°)=1−1−1+2=1

From Rh(3) character table the formula to calculate the value of χC4 is,

χC4=1+2cosθ+2cos2θ+2cos3θ …(1)

Substitute the value of θ=90° in equation (1).

χC4=1+2cos90°+2cos2(90°)+2cos3(90°)=1+0−2+0=−1

From Rh(3) character table the formula to calculate the value of χS4 is,

χS4=−1+2cosθ−2cos2θ+2cos3θ …(2)

Substitute the value of θ=90° in equation (2).

χS4=−1+2cos90°−2cos2(90°)+2cos3(90°)=−1+0+2+0=1

From Rh(3) character table the formula to calculate the value of χS6 is,

χS6=−1+2cosθ−2cos2θ+2cos3θ …(2)

Substitute the value of θ=60° in equation (2).

χS6=−1+2cos60°−2cos2(60°)+2cos3(60°)=−1+1+1−2=−1

Therefore, the character table for f orbital in octahedral environment is shown below.

OhE8C33C26C46C2'i8S63σh6S46σd71−1−1−1−7−1111

The great orthogonality theorem for the reducible representation can be represented as,

aΓ=1h∑all classesof point groupN⋅χΓ⋅χlinear combo

Where,

• aΓ is the number of times the irreducible representation appears in a linear combination.

• h is the order of the group.

• χΓ is the character of the class of the irreducible representation.

• χlinear combo is the character of the class linear combination.

• N is the number of symmetry operations.

The order of the group is 48.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations for A1g.

aA1g=148[(1⋅1⋅7)+(8⋅1⋅1)+(3⋅1⋅−1)+(6⋅1⋅−1)+(6⋅1⋅−1)+(1⋅1⋅−7)+(8⋅1⋅−1)+(3⋅1⋅1)+(6⋅1⋅1)+(6⋅1⋅1)]=148[0]=0

The number of times the irreducible representation for A1g appears in a linear combination is 0.

Similarly, for A2g, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aA2g=148[(1⋅1⋅7)+(8⋅1⋅1)+(3⋅1⋅−1)+(6⋅−1⋅−1)+(6⋅−1⋅−1)+(1⋅1⋅−7)+(8⋅1⋅−1)+(3⋅1⋅1)+(6⋅−1⋅1)+(6⋅−1⋅1)]=0

The number of times the irreducible representation for A2g appears in a linear combination is 0.

Similarly, for Eg, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aEg=148[(1⋅2⋅7)+(8⋅−1⋅1)+(3⋅2⋅−1)+(6⋅0⋅−1)+(6⋅0⋅−1)+(1⋅2⋅−7)+(8⋅−1⋅−1)+(3⋅2⋅1)+(6⋅0⋅1)+(6⋅0⋅1)]=0

The number of times the irreducible representation for Eg appears in a linear combination is 0.

Similarly, for T1g, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aT1g=148[(1⋅3⋅7)+(8⋅0⋅1)+(3⋅−1⋅−1)+(6⋅1⋅−1)+(6⋅−1⋅−1)+(1⋅3⋅−7)+(8⋅0⋅−1)+(3⋅−1⋅1)+(6⋅1⋅1)+(6⋅−1⋅1)]=0

The number of times the irreducible representation for T1g appears in a linear combination is 0.

Similarly, for T2g, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aT2g=148[(1⋅3⋅7)+(8⋅0⋅1)+(3⋅−1⋅−1)+(6⋅−1⋅−1)+(6⋅1⋅−1)+(1⋅3⋅−7)+(8⋅0⋅−1)+(3⋅−1⋅1)+(6⋅−1⋅1)+(6⋅1⋅1)]=0

The number of times the irreducible representation for T2g appears in a linear combination is 0.

Similarly, for A1u, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aTA1u=148[(1⋅1⋅7)+(8⋅1⋅1)+(3⋅1⋅−1)+(6⋅1⋅−1)+(6⋅1⋅−1)+(1⋅−1⋅−7)+(8⋅−1⋅−1)+(3⋅−1⋅1)+(6⋅−1⋅1)+(6⋅−1⋅1)]=0

The number of times the irreducible representation for A1u appears in a linear combination is 0.

Similarly, for A2u, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aTA2u=148[(1⋅1⋅7)+(8⋅1⋅1)+(3⋅1⋅−1)+(6⋅−1⋅−1)+(6⋅−1⋅−1)+(1⋅−1⋅−7)+(8⋅−1⋅−1)+(3⋅−1⋅1)+(6⋅1⋅1)+(6⋅1⋅1)]=148[48]=1

The number of times the irreducible representation for A2u appears in a linear combination is 1.

Similarly, for Eu, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aEu=148[(1⋅2⋅7)+(8⋅−1⋅1)+(3⋅2⋅−1)+(6⋅0⋅−1)+(6⋅0⋅−1)+(1⋅−2⋅−7)+(8⋅1⋅−1)+(3⋅−2⋅1)+(6⋅0⋅1)+(6⋅0⋅1)]=0

The number of times the irreducible representation for Eu appears in a linear combination is 0.

Similarly, for T1u, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aT1u=148[(1⋅3⋅7)+(8⋅0⋅1)+(3⋅−1⋅−1)+(6⋅1⋅−1)+(6⋅−1⋅−1)+(1⋅−3⋅−7)+(8⋅0⋅−1)+(3⋅1⋅1)+(6⋅−1⋅1)+(6⋅1⋅1)]=148[48]=1

The number of times the irreducible representation for T1u appears in a linear combination is 1.

Similarly, for T2u, substitute the value of order of the group, character of the class of the irreducible representation from character table of Oh point group, character of the class linear combination and number of symmetry operations.

aT1u=148[(1⋅3⋅7)+(8⋅0⋅1)+(3⋅−1⋅−1)+(6⋅−1⋅−1)+(6⋅1⋅−1)+(1⋅−3⋅−7)+(8⋅0⋅−1)+(3⋅1⋅1)+(6⋅1⋅1)+(6⋅−1⋅1)]=148[48]=1

The number of times the irreducible representation for T2u appears in a linear combination is 1.

Thus, the linear combination is A2u⊕T1u⊕T2u.

Conclusion

The characters of the f orbitals in an octahedral environment are A2u⊕T1u⊕T2u.

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