Chapter 10, Problem 10B.9P

(a)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $z\left(5z^2-3r^2\right)$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

A representative in the group theory is the mathematical operator used for the representation of the physical symmetry operation.  The set of these mathematical operators representing all the symmetry operations of the group are represented in matrix form in the group theory.  The set of all matrices representing all the symmetry operations of the group are known as a representation.

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $z\left(5z^2-3r^2\right)$ in the point group $C_{2\text{v}}$ is ${\text{A}}_{\text{1}}$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $z\left(5z^2-3r^2\right)$ transforms as $z\left({\text{A}}_{\text{1}}\right)$.  Therefore, the irreducible representation spanned by the function $z\left(5z^2-3r^2\right)$ is ${\text{A}}_{\text{1}}$.

(b)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $y\left(5y^2-3r^2\right)$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

As mentioned in the concept of introduction in part (a).

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $y\left(5y^2-3r^2\right)$ in the point group $C_{2\text{v}}$ is ${\text{B}}_2$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $y\left(5y^2-3r^2\right)$ transforms as $y\left({\text{B}}_{\text{2}}\right)$.  Therefore, the irreducible representation spanned by the function $y\left(5y^2-3r^2\right)$ is ${\text{B}}_2$.

(c)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $x\left(5x^2-3r^2\right)$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

As mentioned in the concept of introduction in part (a).

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $x\left(5x^2-3r^2\right)$ in the point group $C_{2\text{v}}$ is ${\text{B}}_1$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $x\left(5x^2-3r^2\right)$ transforms as $x\left({\text{B}}_1\right)$.  Therefore, the irreducible representation spanned by the function $x\left(5x^2-3r^2\right)$ is ${\text{B}}_1$.

(d)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $z\left(x^2-y^2\right)$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

As mentioned in the concept of introduction in part (a).

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $z\left(x^2-y^2\right)$ in the point group $C_{2\text{v}}$ is ${\text{A}}_1$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $z\left(x^2-y^2\right)$ transforms as $z\left({\text{A}}_1\right)$.  Therefore, the irreducible representation spanned by the function $z\left(x^2-y^2\right)$ is ${\text{A}}_1$.

(e)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $y\left(x^2-z^2\right)$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

As mentioned in the concept of introduction in part (a).

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $y\left(x^2-z^2\right)$ in the point group $C_{2\text{v}}$ is ${\text{B}}_{\text{2}}$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $y\left(x^2-z^2\right)$ transforms as $y\left({\text{B}}_{\text{2}}\right)$.  Therefore, the irreducible representation spanned by the function $y\left(x^2-z^2\right)$ is ${\text{B}}_{\text{2}}$.

(f)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $x\left(z^2-y^2\right)$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

As mentioned in the concept of introduction in part (a).

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $x\left(z^2-y^2\right)$ in the point group $C_{2\text{v}}$ is ${\text{B}}_1$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $x\left(z^2-y^2\right)$ transforms as $x\left({\text{B}}_1\right)$.  Therefore, the irreducible representation spanned by the function $x\left(z^2-y^2\right)$ is ${\text{B}}_1$.

(g)

Interpretation Introduction

Interpretation:

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $xyz$ has to be identified in the point group $C_{2\text{v}}$.

Concept introduction:

As mentioned in the concept of introduction in part (a).

Answer to Problem 10B.9P

The irreducible representation spanned by the f orbital when its radial function is multiplied by a factor $xyz$ in the point group $C_{2\text{v}}$ is ${\text{A}}_{\text{2}}$.

Explanation of Solution

The character table for the $C_{\text{2v}}$ point group is shown below.

  $\begin{array}{ccccccc}{C}_{\text{2v}}& E& C_{\text{2}}& {\sigma }_{\text{v}}& {{\sigma }^{\prime }}_{\text{v}}& & \\ {\text{A}}_{\text{1}}& 1& 1& 1& 1& z,z^2,x^2,y^2& \\ {\text{A}}_2& 1& 1& -1& -1& xy& R_z\\ {\text{B}}_{\text{1}}& 1& -1& 1& -1& x,xz& R_y\\ {\text{B}}_2& 1& -1& -1& 1& y,yz& R_x\end{array}$

The function $x^2$, $y^2$, $z^2$ and $r^{\text{2}}$ are invariant for all symmetry operations.  From the character table of $C_{\text{2v}}$ the function $xyz$ transforms as ${\text{B}}_{\text{1}}\times {\text{B}}_{\text{2}}\times {\text{A}}_{\text{1}}={\text{A}}_2$.  Therefore, the irreducible representation spanned by the function $xyz$ is ${\text{A}}_{\text{2}}$.