Chapter 13, Problem 13.43E

Interpretation Introduction

Interpretation:

The $12\times 12$ matrices for all three planes of symmetry $\sigma$, $\sigma \text{'}$, and $\sigma "$ are to be stated. Also the similarities and differences between the three matrices are to be stated.

Concept introduction:

The group of symmetry operations of which atleast one point is kept fixed is called point group. The symmetry operations can be identity, rotation, reflection, inversion and improper rotation.

Answer to Problem 13.43E

The $12\times 12$ matrices that specify the change in atomic coordinates of ${\text{NH}}_3$ upon operation of $\sigma$, $\sigma \text{'}$, and $\sigma "$ symmetry operations are shown below.

$\sigma =\left[\begin{array}{cccccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

$\sigma \text{'}=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

$\sigma "=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

The coordinates for hydrogen in ${\text{NH}}_3$ for $\sigma \text{'}$, and $\sigma "$ are present off diagonal while in case of $\sigma$, the coordinates are present on the diagonal. In all the matrices, there are four submatrices which follow the same pattern.

Explanation of Solution

The structure of ammonia molecule is,

Figure 1

The total coordinates in the given system is $3N$, where $N$ is the number of atoms. Therefore, total coordinates are $12$.

The ammonia molecule has non-linear structure and $C_3$ as principal axis. In this molecule $3{\sigma }_{\text{v}}$ is present, each passes through one of the hydrogen atom. Therefore, the point group is $C_{\text{3v}}$.

On applying the $\sigma \left(yz\right)$ or $\sigma$ symmetry operation on the total coordinates of ammonia molecule, it gives the negative coordinates of $x$ as shown below.

Figure 2

The $12\times 12$ matrix that specifies the change in atomic coordinates of ${\text{NH}}_3$ upon operation of the $\sigma \left(yz\right)$ symmetry operation is shown below.

$\sigma =\left[\begin{array}{cccccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

Similarly on applying the $\sigma \left(xz\right)$ or $\sigma \text{'}$ symmetry operation on the total coordinates of ammonia molecule, it gives the negative coordinates of $y$ as shown below. The $y$ coordinate of ${\text{H}}_1$ moves to the $y$ coordinate of ${\text{H}}_2$ which implies $0$ on the diagonal position and $-1$ on the off diagonal position.

The $12\times 12$ matrix that specifies the change in atomic coordinates of ${\text{NH}}_3$ upon operation of the $\sigma \left(xz\right)$ symmetry operation is shown below.

$\sigma \text{'}=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

Similarly on applying the $\sigma \left(xy\right)$ or $\sigma "$ symmetry operation on the total coordinates of ammonia molecule, it gives the negative coordinates of $z$ as shown below. The $z$ coordinate of ${\text{H}}_1$ moves to the $z$ coordinate of ${\text{H}}_2$ which implies $0$ on the diagonal position and $-1$ on the off diagonal position.

The $12\times 12$ matrix that specifies the change in atomic coordinates of ${\text{NH}}_3$ upon operation of the $\sigma \left(xy\right)$ symmetry operation is shown below.

$\sigma "=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

The coordinates for hydrogen for $\sigma \text{'}$, and $\sigma "$ are on off diagonal while in case of $\sigma$, the coordinates are on the diagonal. In all the matrices there is submatrix which follows the same pattern.

Conclusion

The $12\times 12$ matrices that specify the change in atomic coordinates of ${\text{NH}}_3$ upon operation of $\sigma$, $\sigma \text{'}$, and $\sigma "$ symmetry operations are shown below.

$\sigma =\left[\begin{array}{cccccccccccc}-1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

$\sigma \text{'}=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

$\sigma "=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{{\text{H}}_1}\\ y_{{\text{H}}_1}\\ z_{{\text{H}}_1}\\ x_{\text{N}}\\ y_{\text{N}}\\ z_{\text{N}}\\ x_{{\text{H}}_2}\\ y_{{\text{H}}_2}\\ z_{{\text{H}}_2}\\ x_{{\text{H}}_3}\\ y_{{\text{H}}_3}\\ z_{{\text{H}}_3}\end{array}\right]$

The coordinates for hydrogen in ${\text{NH}}_3$ for $\sigma \text{'}$, and $\sigma "$ are present on the off diagonal while in case of $\sigma$, the coordinates are present on the diagonal. In all the matrices, there are four submatrices which follow the same pattern.