Chapter 14, Problem 14.96E

Interpretation Introduction

Interpretation:

Whether the shape of xenon tetrafluoride, ${\text{XeF}}_{\text{4}}$ is tetrahedral or square planar, where IR and Raman spectra each show three vibrations is to be stated.

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,

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

Answer to Problem 14.96E

The shape of xenon tetrafluoride, ${\text{XeF}}_{\text{4}}$ is square planar.

Explanation of Solution

It is assumed that the shape of xenon tetrafluoride is tetrahedral.

The character table for point group $T_{\text{d}}$ is shown below.

operations	$E$	$8C_3$	$3C_2$	$6S_4$	$6{\sigma }_{\text{d}}$
$N_{\text{stationary}}$	$5$	$2$	$1$	$1$	$3$
$\theta (°)$	$0$	$120$	$180$	$90$	$180$
${\chi }_{\text{r}}=\left(1+2\cos \theta \right)$	$3$	$0$	$-1$	$1$	$-1$
${\chi }_{\text{tot}}=\pm N_{\text{stationary}}\left(1+2\cos \theta \right)$	$15$	$0$	$-1$	$-1$	$3$
${\chi }_l=\pm \left(1+2\cos \theta \right)$	$3$	$0$	$-1$	$-1$	$1$
${\chi }_{\text{v}}={\chi }_{\text{tot}}-{\chi }_l-{\chi }_{\text{r}}$	$9$	$0$	$1$	$-1$	$3$

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 orthogonality 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 $A_1$.

$\begin{array}{rll}{a}_{{\text{A}}_1}& =& \frac{1}{24}\left[\left(1\cdot 1\cdot 9\right)+\left(8\cdot 1\cdot 0\right)+\left(3\cdot 1\cdot 1\right)+\left(6\cdot 1\cdot -1\right)+\left(6\cdot 1\cdot 3\right)\right]\\ & =& 1\end{array}$

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 $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{{\text{A}}_2}& =& \frac{1}{24}\left[\left(1\cdot 1\cdot 9\right)+\left(8\cdot 1\cdot 0\right)+\left(3\cdot 1\cdot 1\right)+\left(6\cdot -1\cdot -1\right)+\left(6\cdot -1\cdot 3\right)\right]\\ & =& 0\end{array}$

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.

$\begin{array}{rll}{a}_{\text{E}}& =& \frac{1}{24}\left[\left(1\cdot 2\cdot 9\right)+\left(8\cdot -1\cdot 0\right)+\left(3\cdot 2\cdot 1\right)+\left(6\cdot 0\cdot -1\right)+\left(6\cdot 0\cdot 3\right)\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $E$ 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 $T_{\text{d}}$ point group, character of the class linear combination and number of symmetry operations for $T_1$.

$\begin{array}{rll}{a}_{{\text{T}}_1}& =& \frac{1}{24}\left[\left(1\cdot 3\cdot 9\right)+\left(8\cdot 0\cdot 0\right)+\left(3\cdot -1\cdot 1\right)+\left(6\cdot 1\cdot -1\right)+\left(6\cdot -1\cdot 3\right)\right]\\ & =& 0\end{array}$

The number of times the irreducible representation for $T_1$ appears in a linear combination is $0$.

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.

$\begin{array}{rll}{a}_{{\text{T}}_2}& =& \frac{1}{24}\left[\left(1\cdot 3\cdot 9\right)+\left(8\cdot 0\cdot 0\right)+\left(3\cdot -1\cdot 1\right)+\left(6\cdot -1\cdot -1\right)+\left(6\cdot 1\cdot 3\right)\right]\\ & =& 2\end{array}$

The number of times the irreducible representation for $T_2$ appears in a linear combination is $2$.

The character of $E$ symmetry species is $2$. This shows that the $E$ is doubly degenerate. The character of $T_2$ symmetry species is $3$. This shows that the $T_2$ is triply degenerate.

Therefore, $A_1$ irreducible representation contains $x^2+y^2+z^2$, this indicates that it will give one Raman-active vibration. The $E$ irreducible representation contains $\left(x^2-y^2,2z^2-x^2-y^2\right)$, this indicates that it will give one Raman-active vibration. This will be doubly degenerate. The $T_2$ irreducible representation contains $\left(xy,yz,zx\right)$ and $\left(x,y,z\right)$, this indicates that it will give two Raman-active vibration and IR-active vibrations.

Therefore, there are four Raman-active vibrations and two IR-active vibrations are given by tetrahedral. This is not matched with the given information. Therefore, the shape of xenon tetrafluoride, ${\text{XeF}}_{\text{4}}$ is not tetrahedral.

It is assumed that the shape of xenon tetrafluoride is square planar.

The character table for point group $D_{\text{4h}}$ is shown below.

operations	$E$	$2C_4$	$C_2$	$2C^{\text{'}}{}_2$	$2C^{\text{'}\text{'}}{}_2$	$i$	$2S_4$	${\sigma }_{\text{h}}$	$2{\sigma }_{\text{v}}$	$2{\sigma }_{\text{d}}$
$N_{\text{stationary}}$	$5$	$1$	$1$	$3$	$1$	$3$	$1$	$5$	$3$	$1$
$\theta (°)$	$0$	$90$	$180$	$180$	$180$	$90$	$90$	$180$	$180$	$180$
${\chi }_{\text{r}}=\left(1+2\cos \theta \right)$	$3$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$
${\chi }_{\text{tot}}=\pm N_{\text{stationary}}\left(1+2\cos \theta \right)$	$15$	$1$	$-1$	$-3$	$-1$	$-3$	$-1$	$5$	$3$	$1$
${\chi }_l=\pm \left(1+2\cos \theta \right)$	$3$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
${\chi }_{\text{v}}={\chi }_{\text{tot}}-{\chi }_l-{\chi }_{\text{r}}$	$9$	$-1$	$1$	$-1$	$1$	$-3$	$-1$	$5$	$3$	$1$

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 $16$.

The great orthogonality theorem orthogonality of the irreducible representation of $A_{\text{1g}}$, $B_{\text{1g}}$, $B_{\text{2g}}$, $A_{\text{1u}}$ and $E_{\text{u}}$ is shown below.

Substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_{\text{4h}}$ point group, character of the class linear combination and number of symmetry operations for $A_{\text{1g}}$.

$\begin{array}{rll}{a}_{A_{\text{1g}}}& =& \frac{1}{16}\left[\begin{array}{l}\left(1\cdot 1\cdot 9\right)+\left(2\cdot 1\cdot -1\right)+\left(1\cdot 1\cdot 1\right)+\left(2\cdot 1\cdot -1\right)+\left(2\cdot 1\cdot 1\right)+\left(1\cdot 1\cdot -3\right)\\ +\left(2\cdot 1\cdot -1\right)+\left(1\cdot 1\cdot 5\right)+\left(2\cdot 1\cdot 3\right)+\left(2\cdot 1\cdot 1\right)\end{array}\right]\\ =1\end{array}$

The number of times the irreducible representation for $A_{\text{1g}}$ appears in a linear combination is $1$.

Similarly, for $B_{\text{1g}}$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_{\text{4h}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{B_{\text{1g}}}& =& \frac{1}{16}\left[\begin{array}{l}\left(1\cdot 1\cdot 9\right)+\left(2\cdot -1\cdot -1\right)+\left(1\cdot 1\cdot 1\right)+\left(2\cdot 1\cdot -1\right)+\left(2\cdot -1\cdot 1\right)+\left(1\cdot 1\cdot -3\right)\\ +\left(2\cdot -1\cdot -1\right)+\left(1\cdot 1\cdot 5\right)+\left(2\cdot 1\cdot 3\right)+\left(2\cdot -1\cdot 1\right)\end{array}\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $B_{\text{1g}}$ appears in a linear combination is $1$.

Similarly, for $B_{\text{2g}}$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_{\text{4h}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{B_{\text{2g}}}& =& \frac{1}{16}\left[\begin{array}{l}\left(1\cdot 1\cdot 9\right)+\left(2\cdot -1\cdot -1\right)+\left(1\cdot 1\cdot 1\right)+\left(2\cdot -1\cdot -1\right)+\left(2\cdot 1\cdot 1\right)+\left(1\cdot 1\cdot -3\right)\\ +\left(2\cdot -1\cdot -1\right)+\left(1\cdot 1\cdot 5\right)+\left(2\cdot -1\cdot 3\right)+\left(2\cdot 1\cdot 1\right)\end{array}\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $B_{\text{2g}}$ appears in a linear combination is $1$.

Similarly, for $A_{\text{1u}}$, substitute the value of order of the group, character of the class of the irreducible representation from character table of $D_{\text{4h}}$ point group, character of the class linear combination and number of symmetry operations.

$\begin{array}{rll}{a}_{A_{\text{1u}}}& =& \frac{1}{16}\left[\begin{array}{l}\left(1\cdot 1\cdot 9\right)+\left(2\cdot 1\cdot -1\right)+\left(1\cdot 1\cdot 1\right)+\left(2\cdot -1\cdot -1\right)+\left(2\cdot -1\cdot 1\right)+\left(1\cdot -1\cdot -3\right)\\ +\left(2\cdot -1\cdot -1\right)+\left(1\cdot -1\cdot 5\right)+\left(2\cdot 1\cdot 3\right)+\left(2\cdot 1\cdot 1\right)\end{array}\right]\\ & =& 1\end{array}$

The number of times the irreducible representation for $A_{\text{1u}}$ 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 $D_{\text{4h}}$ point group, character of the class linear combination and number of symmetry operations for $E_{\text{u}}$.

$\begin{array}{rll}{a}_{E_{\text{u}}}& =& \frac{1}{16}\left[\begin{array}{l}\left(1\cdot 2\cdot 9\right)+\left(2\cdot 0\cdot -1\right)+\left(1\cdot -2\cdot 1\right)+\left(2\cdot 0\cdot -1\right)+\left(2\cdot 0\cdot 1\right)+\left(1\cdot -2\cdot -3\right)\\ +\left(2\cdot 0\cdot -1\right)+\left(1\cdot 2\cdot 5\right)+\left(2\cdot 0\cdot 3\right)+\left(2\cdot 0\cdot 1\right)\end{array}\right]\\ & =& 2\end{array}$

The number of times the irreducible representation for $E_{\text{u}}$ appears in a linear combination is $2$.

The character of $E$ symmetry species is $2$. This shows that the $E$ is doubly degenerate. The character of $T_2$ symmetry species is $3$. This shows that the $T_2$ is triply degenerate.

Therefore, $A_{\text{1g}}$ irreducible representation contains $x^2+y^2$ and $z^2$, this indicates that it will give one Raman-active vibration. This will be doubly degenerate. The $B_{\text{1g}}$ irreducible representation contains $\left(x^2-y^2\right)$, this indicates that it will give one Raman-active vibration. The $B_{\text{2g}}$ irreducible representation contains $\left(xy\right)$, this indicates that it will give one Raman-active vibration. The $A_{\text{2u}}$ irreducible representation contains $\left(z\right)$, this indicates that it will give one IR-active vibration. The $E_{\text{u}}$ irreducible representation contains $\left(x,y\right)$, this indicates that it will give two IR-active vibration.

Therefore, there are three Raman-active vibrations and three IR-active vibrations are given by square planar. This is matched with the given information. Therefore, the shape of xenon tetrafluoride, ${\text{XeF}}_{\text{4}}$ is square planar.

Conclusion

The shape of xenon tetrafluoride, ${\text{XeF}}_{\text{4}}$ is square planar.