Chapter 13, Problem 13.59E

Reduce the following reducible representations using the great orthogonality theorem.

(a) In the $C_2$ point group:

$\begin{array}{ccc}& E& C_2\\ \Gamma & 5& 1\end{array}$

(b) In the $C_{\text{3v}}$ point group:

$\begin{array}{cccc}& E& 2C_2& 3{\sigma }_{\text{v}}\\ \Gamma & 6& 0& 0\end{array}$

(c) In the $D_4$ point group:

$\begin{array}{cccccc}& E& 2C_4& C_2& 2C_2^{\text{'}}& 2C_2^{\text{'}\text{'}}\\ \Gamma & 6& -2& 2& 2& -4\end{array}$

(d) In the $T_{\text{d}}$ point group:

$\begin{array}{cccccc}& E& 8C_3& 3C_2& 6S_4& 6{\sigma }_{\text{d}}\\ \Gamma & 7& -2& 3& 1& -1\end{array}$

Interpretation Introduction

(a)

Interpretation:

The given reducible representations are to be reduced by using the great orthogonality theorem.

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,

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

Answer to Problem 13.59E

The linear combination for given irreducible representation of $C_2$ is $3A\oplus 2B$.

Explanation of Solution

The given reducible representation of $C_2$ point group is shown below.

$\begin{array}{ccc}& E& C_2\\ \Gamma & 5& 1\end{array}$

This reducible representation is reduced using great orthogonality theorem as shown below.

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

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

The order of the group is $2$.

The great orthogonality theorem of the irreducible representation of $A$ and $B$ is shown below.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $C_2$ point group, character of the class linear combination and number of symmetry operations for $A$.

$\begin{array}{rll}{a}_{\text{A}}& =& \frac{1}{2}\left[\left(1\cdot 1\cdot 5\right)+\left(1\cdot 1\cdot 1\right)\right]\\ & =& 3\end{array}$

The number of times the irreducible representation for $A$ appears in a linear combination is $3$.

Similarly, for $B$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $C_2$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{\text{B}}& =& \frac{1}{2}\left[\left(1\cdot 1\cdot 5\right)+\left(1\cdot -1\cdot 1\right)\right]\\ & =& 2\end{array}$

The number of times the irreducible representation for $B$ appears in a linear combination is $2$.

Thus, the linear combination is $3A\oplus 2B$.

Conclusion

The linear combination for given irreducible representation of $C_2$ is $3A\oplus 2B$.

Interpretation Introduction

(b)

Interpretation:

The given reducible representations are to be reduced by using the great orthogonality theorem.

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,

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$ is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

Answer to Problem 13.59E

The linear combination for given irreducible representation of $C_{\text{3v}}$ is $A_1\oplus A_2\oplus 2E$.

Explanation of Solution

The given reducible representation of $C_{\text{3v}}$ point group is shown below.

$\begin{array}{cccc}& E& 2C_2& 3{\sigma }_{\text{v}}\\ \Gamma & 6& 0& 0\end{array}$

This reducible representation is reduced using great orthogonality theorem as shown below.

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

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$ is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

The order of the group is $6$.

The great orthogonality theorem of the irreducible representation of $A_1$, $A_2$

and $E$ is shown below.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $C_{\text{3v}}$ point group, character of the class linear combination and number of symmetry operations for $A_1$.

$\begin{array}{rll}{a}_{{\text{A}}_1}& =& \frac{1}{6}\left[\left(1\cdot 1\cdot 6\right)+\left(2\cdot 1\cdot 0\right)+\left(3\cdot 1\cdot 0\right)\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $A_1$ appears in a linear combination is $1$.

Similarly, for $A_2$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $C_{\text{3v}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{{\text{A}}_2}& =& \frac{1}{6}\left[\left(1\cdot 1\cdot 6\right)+\left(2\cdot 1\cdot 0\right)+\left(3\cdot -1\cdot 0\right)\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $A_2$ appears in a linear combination is $1$.

Similarly, for $E$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $C_{\text{3v}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{\text{E}}& =& \frac{1}{6}\left[\left(1\cdot 2\cdot 6\right)+\left(2\cdot -1\cdot 0\right)+\left(3\cdot 0\cdot 0\right)\right]\\ & =& 2\end{array}$

The number of times the irreducible representation for $E$ appears in a linear combination is $2$.

Thus, the linear combination is $A_1\oplus A_2\oplus 2E$.

Conclusion

The linear combination for given irreducible representation of $C_{\text{3v}}$ is $A_1\oplus A_2\oplus 2E$.

Interpretation Introduction

(c)

Interpretation:

The given reducible representations are to be reduced by using the great orthogonality theorem.

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,

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$ is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

Answer to Problem 13.59E

The linear combination for given irreducible representation of $D_4$ is $A_2\oplus 3B_1\oplus E$.

Explanation of Solution

The given reducible representation of $D_4$ point group is shown below.

$\begin{array}{cccccc}& E& 2C_4& C_2& 2C_2^{\text{'}}& 2C_2^{\text{'}\text{'}}\\ \Gamma & 6& -2& 2& 2& -4\end{array}$

This reducible representation is reduced using great orthogonality theorem as shown below.

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

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$ is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

The order of the group is $8$.

The great orthogonality theorem of the irreducible representation of $A_1$, $A_2$, $B_1$, $B_2$ and $E$ is shown below.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_4$ point group, character of the class linear combination and number of symmetry operations for $A_1$.

$\begin{array}{rll}{a}_{{\text{A}}_1}& =& \frac{1}{8}\left[\left(1\cdot 1\cdot 6\right)+\left(2\cdot 1\cdot -2\right)+\left(1\cdot 1\cdot 2\right)+\left(2\cdot 1\cdot 2\right)+\left(2\cdot 1\cdot -4\right)\right]\\ & =& 0\end{array}$

The number of times the irreducible representation for $A_1$ appears in a linear combination is $0$.

Similarly, for $A_2$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_4$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{{\text{A}}_2}& =& \frac{1}{8}\left[\left(1\cdot 1\cdot 6\right)+\left(2\cdot 1\cdot -2\right)+\left(1\cdot 1\cdot 2\right)+\left(2\cdot -1\cdot 2\right)+\left(2\cdot -1\cdot -4\right)\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $A_2$ appears in a linear combination is $1$.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_4$ point group, character of the class linear combination and number of symmetry operations for $B_1$.

$\begin{array}{rll}{a}_{{\text{B}}_1}& =& \frac{1}{8}\left[\left(1\cdot 1\cdot 6\right)+\left(2\cdot -1\cdot -2\right)+\left(1\cdot 1\cdot 2\right)+\left(2\cdot 1\cdot 2\right)+\left(2\cdot -1\cdot -4\right)\right]\\ & =& 3\end{array}$

The number of times the irreducible representation for $B_1$ appears in a linear combination is $3$.

Similarly, for $B_2$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_4$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{{\text{B}}_2}& =& \frac{1}{8}\left[\left(1\cdot 1\cdot 6\right)+\left(2\cdot -1\cdot -2\right)+\left(1\cdot 1\cdot 2\right)+\left(2\cdot -1\cdot 2\right)+\left(2\cdot 1\cdot -4\right)\right]\\ & =& 0\end{array}$

The number of times the irreducible representation for $B_2$ appears in a linear combination is $0$.

Similarly, for $E$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_4$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{\text{E}}& =& \frac{1}{8}\left[\left(1\cdot 2\cdot 6\right)+\left(2\cdot 0\cdot -2\right)+\left(1\cdot -2\cdot 2\right)+\left(2\cdot 0\cdot 2\right)+\left(2\cdot 0\cdot -4\right)\right]\\ & =& 1\end{array}$

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

Thus, the linear combination is $A_2\oplus 3B_1\oplus E$.

Conclusion

The linear combination for given irreducible representation of $D_4$ is $A_2\oplus 3B_1\oplus E$.

Interpretation Introduction

(d)

Interpretation:

The given reducible representations are to be reduced by using the great orthogonality theorem.

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,

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$ is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

Answer to Problem 13.59E

The linear combination for given irreducible representation of $T_{\text{d}}$ is $2E\oplus T_1$.

Explanation of Solution

The given reducible representation of $T_{\text{d}}$ point group is shown below.

$\begin{array}{cccccc}& E& 8C_3& 3C_2& 6S_4& 6{\sigma }_{\text{d}}\\ \Gamma & 7& -2& 3& 1& -1\end{array}$

This reducible representation reduced using great orthogonality theorem as shown below.

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

$a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$

Where,

• $a_{\Gamma }$ is the number of times the irreducible representation appears in a linear combination.

• $h$ is the order of the group.

• ${\chi }_{\Gamma }$ is the character of the class of the irreducible representation.

• ${\chi }_{\text{linear}\text{combo}}$ is the character of the class linear combination.

• $N$ is the number of symmetry operations.

The order of the group is $24$.

The great orthogonality theorem of the irreducible representation of $A_1$, $A_2$$E$, $T_1$ and $T_2$ is shown below.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations for $A_1$.

$\begin{array}{rll}{a}_{{\text{A}}_1}& =& \frac{1}{24}\left[\left(1\cdot 1\cdot 7\right)+\left(8\cdot 1\cdot -2\right)+\left(3\cdot 1\cdot 3\right)+\left(6\cdot 1\cdot 1\right)+\left(6\cdot 1\cdot -1\right)\right]\\ & =& 0\end{array}$

The number of times the irreducible representation for $A_1$ appears in a linear combination is $0$.

Similarly, for $A_2$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{{\text{A}}_2}& =& \frac{1}{24}\left[\left(1\cdot 1\cdot 7\right)+\left(8\cdot 1\cdot -2\right)+\left(3\cdot 1\cdot 3\right)+\left(6\cdot -1\cdot 1\right)+\left(6\cdot -1\cdot -1\right)\right]\\ & =& 0\end{array}$

The number of times the irreducible representation for $A_2$ appears in a linear combination is $0$.

Similarly, for $E$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{\text{E}}& =& \frac{1}{24}\left[\left(1\cdot 2\cdot 7\right)+\left(8\cdot -1\cdot -2\right)+\left(3\cdot 2\cdot 3\right)+\left(6\cdot 0\cdot 1\right)+\left(6\cdot 0\cdot -1\right)\right]\\ & =& 2\end{array}$

The number of times the irreducible representation for $E$ appears in a linear combination is $2$.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations for $T_1$.

$\begin{array}{rll}{a}_{{\text{T}}_1}& =& \frac{1}{24}\left[\left(1\cdot 3\cdot 7\right)+\left(8\cdot 0\cdot -2\right)+\left(3\cdot -1\cdot 3\right)+\left(6\cdot 1\cdot 1\right)+\left(6\cdot -1\cdot -1\right)\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $T_1$ appears in a linear combination is $1$.

Similarly, for $T_2$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{{\text{T}}_2}& =& \frac{1}{24}\left[\left(1\cdot 3\cdot 7\right)+\left(8\cdot 0\cdot -2\right)+\left(3\cdot -1\cdot 3\right)+\left(6\cdot -1\cdot 1\right)+\left(6\cdot 1\cdot -1\right)\right]\\ & =& 0\\ & =& 0\end{array}$

The number of times the irreducible representation for $T_2$ appears in a linear combination is $0$.

Thus, the linear combination is $2E\oplus T_1$.

Conclusion

The linear combination for given irreducible representation of $T_{\text{d}}$ is $2E\oplus T_1$.