Chapter 13, Problem 13.64E

Interpretation Introduction

(a)

Interpretation:

The given integral which is exactly zero due to symmetry considerations is to be identified.

Concept introduction:

The integral is represented as shown below.

${\displaystyle \int {\Psi }_{\Gamma 1}^{*}}{\Psi }_{\Gamma 2}^{*}d\tau$

Where,

• $\Gamma$ represents the symmetry species of wavefunction.

If the above integral has nonzero numeric value then the above integral can be written as given below.${\left(\Gamma 1\right)}^{*}\otimes \Gamma 2=A_1$

Where,

• $A_1$ is the totally symmetric irreducible representation.

If the product of irreducible representation will not be equal to $A_1$ then the integral must be exactly zero.

Interpretation Introduction

(b)

Interpretation:

The given integral which is exactly zero due to symmetry considerations is to be identified.

Concept introduction:

The integral is represented as shown below.

${\displaystyle \int {\Psi }_{\Gamma 1}^{*}}{\Psi }_{\Gamma 2}^{*}d\tau$

Where,

• $\Gamma$ represents the symmetry species of wavefunction.

If the above integral has nonzero numeric value then the above integral can be written as given below.${\left(\Gamma 1\right)}^{*}\otimes \Gamma 2=A_1$

Where,

• $A_1$ is the totally symmetric irreducible representation.

If the product of irreducible representation will not be equal to $A_1$ then the integral must be exactly zero.

Interpretation Introduction

(c)

Interpretation:

The given integral which is exactly zero due to symmetry considerations is to be identified.

Concept introduction:

The integral is represented as shown below.

${\displaystyle \int {\Psi }_{\Gamma 1}^{*}}{\Psi }_{\Gamma 2}^{*}d\tau$

Where,

• $\Gamma$ represents the symmetry species of wavefunction.

If the above integral has nonzero numeric value then the above integral can be written as given below.${\left(\Gamma 1\right)}^{*}\otimes \Gamma 2=A_1$

Where,

• $A_1$ is the totally symmetric irreducible representation.

If the product of irreducible representation will not be equal to $A_1$ then the integral must be exactly zero.

Interpretation Introduction

(d)

Interpretation:

The given integral which is exactly zero due to symmetry considerations is to be identified.

Concept introduction:

The integral is represented as shown below.

${\displaystyle \int {\Psi }_{\Gamma 1}^{*}}{\Psi }_{\Gamma 2}^{*}d\tau$

Where,

• $\Gamma$ represents the symmetry species of wavefunction.

If the above integral has nonzero numeric value then the above integral can be written as given below.${\left(\Gamma 1\right)}^{*}\otimes \Gamma 2=A_1$

Where,

• $A_1$ is the totally symmetric irreducible representation.

If the product of irreducible representation will not be equal to $A_1$ then the integral must be exactly zero.