Chapter 28, Problem 28.2P

Interpretation Introduction

(a)

Interpretation:

In conjugated triene, $1,3,5-\text{hexatriene}$ the number of $\text{π}$ MO is to be stated.

Concept introduction:

The molecular orbital is a combination of two atomic orbitals. It is used to represent the regions in a molecule where the electron is likely to be present in an orbital. It represents the wave-like nature of an electron in a molecule. It may be symmetric or antisymmetric. It may be bonding, antibonding or non-bonding. It may be HOMO or LUMO.

Answer to Problem 28.2P

In $1,3,5-\text{hexatriene}$ molecule a total of six carbon atoms are present which as a result form six $\text{π}$ molecular orbitals (MO).

Explanation of Solution

The structure of $1,3,5-\text{hexatriene}$ is shown below.

${\text{CH}}_{\text{2}}=\text{CH}-\text{CH}=\text{CH}-\text{CH}={\text{CH}}_{\text{2}}$

The electron configuration of carbon atom is ${\text{1s}}^{\text{2}}{\text{2s}}^{\text{2}}{\text{2p}}^{\text{2}}$ where p orbital of carbon atom overlapped together to form molecular orbital. In $1,3,5-\text{hexatriene}$ molecule a total of six carbon atoms are involved whose atomic p orbital overlapped together to give the same number of molecular orbitals. Therefore, a total of six $\text{π}$ molecular orbitals (MO) are present in $1,3,5-\text{hexatriene}$ molecule. The $\text{π}$ molecular orbital (MO) with increasing energy is shown below.

Figure 1

Conclusion

The atomic p orbital of carbon atom overlapped together to form molecular orbital. A total of six molecular orbitals are formed due to six carbon atomic orbital overlapping in $1,3,5-\text{hexatriene}$ molecule.

Interpretation Introduction

(b)

Interpretation:

Each molecular orbital is to be classified as symmetric and antisymmetric.

Concept introduction:

The molecular orbital is a combination of two atomic orbitals. It is used to represent the regions in a molecule where the electron is likely to be present in an orbital. It represents the wave-like nature of an electron in a molecule. It may be symmetric or antisymmetric. It may be bonding, antibonding or non-bonding. It may be HOMO or LUMO.

Answer to Problem 28.2P

The molecular orbitals ${\text{π}}_1{\text{, π}}_3{\text{, π}}_5^{\ast }$ are symmetric and the molecular orbitals ${\text{π}}_2{\text{, π}}_4^{\ast }{\text{, π}}_6^{\ast }$ are antisymmetric.

Explanation of Solution

In molecular orbital theory, the MO is said to be symmetric or anti-symmetric is depend on the relative phase of the two terminal carbons. In symmetric MO, the peaks reflect across the reference plane into the peaks and troughs reflect into troughs. On the other hand, in antisymmetric MO, the peaks reflect into troughs and vice versa. According to the general principle, the even number molecular orbitals are antisymmetric and odd number molecular orbitals are symmetric. Therefore, the molecular orbitals ${\text{π}}_1{\text{, π}}_3{\text{, π}}_5^{\ast }$ are symmetric and the molecular orbitals ${\text{π}}_2{\text{, π}}_4^{\ast }{\text{, π}}_6^{\ast }$ are antisymmetric.

Conclusion

The odd number of molecular orbitals is symmetric molecular orbitals that are ${\text{π}}_1{\text{, π}}_3{\text{, π}}_5^{\ast }$ symmetric molecular orbitals. The even number of molecular orbitals is antisymmetric molecular orbitals that are ${\text{π}}_2{\text{, π}}_4^{\ast }{\text{, π}}_6^{\ast }$ antisymmetric.

Interpretation Introduction

(c)

Interpretation:

Bonding and antibonding molecular orbitals are to be identified.

Concept introduction:

The molecular orbital is a combination of two atomic orbitals. It is used to represent the regions in a molecule where the electron is likely to be present in an orbital. It represents the wave-like nature of an electron in a molecule. It may be symmetric or antisymmetric. It may be bonding, antibonding or non-bonding. It may be HOMO or LUMO.

Answer to Problem 28.2P

The molecular orbital ${\text{π}}_1{\text{, π}}_2{\text{, π}}_3$ are bonding molecular orbitals and ${\text{π}}_4^{\ast }{\text{, π}}_5^{\ast }{\text{, π}}_6^{\ast }$ are antibonding molecular orbitals.

Explanation of Solution

In $1,3,5-\text{hexatriene}$ molecule, a total of six $\text{π}$ molecular orbitals are present. Out of them, half are bonding molecular orbital which has lower energy and other half are antibonding molecular orbital which has higher energy. Therefore, among six molecular orbitals ${\text{π}}_1{\text{, π}}_2{\text{, π}}_3$ are bonding molecular orbitals and ${\text{π}}_4^{\ast }{\text{, π}}_5^{\ast }{\text{, π}}_6^{\ast }$ are antibonding molecular orbitals. The type of MO bond with energy is shown below.

Figure 2

Conclusion

The molecular orbital with lower energy are bonding MO ${\text{π}}_1{\text{, π}}_2{\text{, π}}_3$ and molecular orbital with higher energy are antibonding MO ${\text{π}}_4^{\ast }{\text{, π}}_5^{\ast }{\text{, π}}_6^{\ast }$.

Interpretation Introduction

(d)

Interpretation:

Among the molecular orbitals, the frontier molecular orbital is to be identified.

Concept introduction:

The molecular orbital is a combination of two atomic orbitals. It is used to represent the regions in a molecule where the electron is likely to be present in an orbital. It represents the wave-like nature of an electron in a molecule. It may be symmetric or antisymmetric. It may be bonding, antibonding or non-bonding. It may be HOMO or LUMO.

Answer to Problem 28.2P

The molecular orbital ${\text{π}}_3$ is a HOMO orbital and ${\text{π}}_4^{\ast }$ is a LUMO orbital. They both constitute frontier molecular orbitals.

Explanation of Solution

The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is known as frontier molecular orbitals. In $1,3,5-\text{hexatriene}$ molecule, a total of six $\text{π}$ electrons are present which is occupied by the first three MO of $1,3,5-\text{hexatriene}$ molecule. The orbital with the highest energy that occupies the electrons is known as the highest occupied molecular orbital (HOMO) and the orbitals with the lowest energy that does not occupy the electrons are known as the lowest unoccupied molecular orbitals (LUMO). Therefore, ${\text{π}}_3$ is a HOMO orbital and ${\text{π}}_4^{\ast }$ is a LUMO orbital. The filling of the molecular orbital is also shown in Figure 2.

Conclusion

The molecular orbital ${\text{π}}_3$ and ${\text{π}}_4^{\ast }$ constitute frontier molecular orbitals. The molecular orbital ${\text{π}}_3$ is a HOMO orbital and ${\text{π}}_4^{\ast }$ is a LUMO orbital.

Interpretation Introduction

(e)

Interpretation:

Whether the phase at terminal carbons in HOMO orbitals is the same or different is to be stated.

Concept introduction:

The molecular orbital is a combination of two atomic orbitals. It is used to represent the regions in a molecule where the electron is likely to be present in an orbital. It represents the wave-like nature of an electron in a molecule. It may be symmetric or antisymmetric. It may be bonding, antibonding or non-bonding. It may be HOMO or LUMO.

Answer to Problem 28.2P

The HOMO orbital that is ${\text{π}}_3$ is symmetric molecular orbital or its phase terminal carbons are same.

Explanation of Solution

According to the general principle, the even number molecular orbitals are antisymmetric or its phase terminal carbons are different and odd number molecular orbitals are symmetric or its phase terminal carbons are same. The HOMO orbital is ${\text{π}}_{\text{3}}$ molecular orbital which confirms that it is symmetric molecular orbital with same phase terminal carbons.

Conclusion

The HOMO orbital is ${\text{π}}_{\text{3}}$ molecular orbital which confirms that it is symmetric molecular orbital or its phase terminal carbons are same.

Interpretation Introduction

(f)

Interpretation:

Whether the phase at terminal carbons in LUMO orbitals is the same or different is to be stated.

Concept introduction:

The molecular orbital is a combination of two atomic orbitals. It is used to represent the regions in a molecule where the electron is likely to be present in an orbital. It represents the wave-like nature of an electron in a molecule. It may be symmetric or antisymmetric. It may be bonding, antibonding or non-bonding. It may be HOMO or LUMO.

Answer to Problem 28.2P

The LUMO orbital that is ${\text{π}}_4^{\ast }$ is an antisymmetric molecular orbital or its phase terminal carbons are different.

Explanation of Solution

According to the general principle, the even number molecular orbitals are antisymmetric or its phase terminal carbons are different and odd number molecular orbitals are symmetric or its phase terminal carbons are same. The LUMO orbital is ${\text{π}}_4^{\ast }$ molecular orbital which confirms that it is an antisymmetric molecular orbital.

Conclusion

The LUMO orbital is ${\text{π}}_4^{\ast }$ molecular orbital which confirms that it is an antisymmetric molecular orbital or its phase terminal carbons are different.