Chapter 10, Problem 10B.2ST

Interpretation Introduction

Interpretation:

The symmetry species for the combination of wavefunction ${\psi }_{\text{A}}-{\psi }_{\text{B}}+{\psi }_{\text{C}}-{\psi }_{\text{D}}$ for ${\text{PtCl}}_4^{2-}$ ion has to be stated.

Concept introduction:

The molecules are classified on the basis of the symmetry elements presents in them.  These symmetry elements form a group and are known as point group.  The point groups have the specific symmetry elements which is characteristic of the molecule.

Answer to Problem 10B.2ST

The symmetry species for the combination of wavefunction ${\psi }_{\text{A}}-{\psi }_{\text{B}}+{\psi }_{\text{C}}-{\psi }_{\text{D}}$ for ${\text{PtCl}}_4^{2-}$ ion is $B_{1\text{g}}$.

Explanation of Solution

The combination of wavefunction given is shown below.

    $\psi ={\psi }_{\text{A}}-{\psi }_{\text{B}}+{\psi }_{\text{C}}-{\psi }_{\text{D}}$

The sign of the wavefunction $\left(\psi \right)$ under the symmetry operations give the character of the corresponding symmetry element.

The geometry of ${\text{PtCl}}_4^{2-}$ ion is shown in figure 1.

Figure 1

The sign of wavefunction $\left(\psi \right)$ remains same on operating identity symmetry element.  Therefore, the character for identity symmetry element is written as shown below.

    $\chi \left(E\right)=1$

The sign of wavefunction $\left(\psi \right)$ remains same on operating $C_2$ symmetry element as on rotating the symmetrical molecule ${\text{PtCl}}_4^{2-}$ by $90°$, same molecule is obtained.  Therefore, the character for $C_2$ symmetry element is written as shown below.

    $\chi \left(C_2\right)=1$

The sign of wavefunction $\left(\psi \right)$ remains same on operating ${\sigma }_{\text{v}}$ symmetry element as on observing the symmetrical molecule ${\text{PtCl}}_4^{2-}$ through the $xz$ or $yz$ planes, same molecule is obtained.  Therefore, the character for ${\sigma }_{\text{v}}$ symmetry element is written as shown below.

    $\chi \left({\sigma }_{\text{v}}\right)=1$

The characters obtained for the symmetry elements match with the symmetry species $B_{1\text{g}}$ of $D_{4\text{h}}$ character table.

Therefore, the symmetry species for the combination of wavefunction ${\psi }_{\text{A}}-{\psi }_{\text{B}}+{\psi }_{\text{C}}-{\psi }_{\text{D}}$ for ${\text{PtCl}}_4^{2-}$ ion is $B_{1\text{g}}$.