Chapter 9, Problem 9.2PR

(a)

Interpretation Introduction

Interpretation:

The expression of molecular orbital as $\psi ={\psi }_A\cos \theta +{\psi }_B\sin \theta$ obtained as a linear combination of atomic orbitals has to be explained.

Concept Introduction:

Molecular orbitals can be expressed as the linear combination of atomic orbitals (LCAO).  Certain conditions should be satisfied for the formation of molecular orbitals from atomic orbitals.  They are, combining atomic orbitals should have comparable energy, combining orbitals should have similar symmetry and should have proper overlap. 

Bonding molecular orbital are formed by the constructive interference of two atomic orbitals (additive effect) and the amplitude of new wave can be represented as $\psi ={\psi }_1+{\psi }_2$, where ${\psi }_{1}and{\psi }_2$ are amplitude of wave function of combining atomic orbitals.

Anti-bonding molecular orbital are formed by the destructive interference of two atomic orbitals (subtractive effect) and the amplitude of new wave can be represented as $\psi ={\psi }_1-{\psi }_2$, where ${\psi }_{1}and{\psi }_2$ are amplitude of wave function of combining atomic orbitals.

(b)

Interpretation Introduction

Interpretation:

$\psi$ is normalized to 1 if ${\psi }_{A}and{\psi }_B$ are orthogonal and each are normalized to 1has to be proved.

Concept Introduction:

Molecular orbitals can be expressed as the linear combination of atomic orbitals (LCAO).  Certain conditions should be satisfied for the formation of molecular orbitals from atomic orbitals.  They are, combining atomic orbitals should have comparable energy, combining orbitals should have similar symmetry and should have proper overlap. 

Bonding molecular orbital are formed by the constructive interference of two atomic orbitals (additive effect) and the amplitude of new wave can be represented as $\psi ={\psi }_1+{\psi }_2$, where ${\psi }_{1}and{\psi }_2$ are amplitude of wave function of combining atomic orbitals.

Anti-bonding molecular orbital are formed by the destructive interference of two atomic orbitals (subtractive effect) and the amplitude of new wave can be represented as $\psi ={\psi }_1-{\psi }_2$, where ${\psi }_{1}and{\psi }_2$ are amplitude of wave function of combining atomic orbitals.

Two functions are said to be orthogonal if the integral of their product is zero.

Consider two functions X and Y. If these functions are orthogonal then ${\displaystyle \int X.Y}dz=0.$

A function is said to be normalized if the integral of its product to itself is 1.

Consider a function X, if this function is normalized then ${\displaystyle \int X.X}dz=1.$

(c)

Interpretation Introduction

Interpretation:

The value of $\theta$ corresponds to the bonding and anti-bonding orbitals has to be mentioned.

Concept Introduction:

Molecular orbitals can be expressed as the linear combination of atomic orbitals (LCAO).  Certain conditions should be satisfied for the formation of molecular orbitals from atomic orbitals.  They are, combining atomic orbitals should have comparable energy, combining orbitals should have similar symmetry and should have proper overlap. 

Bonding molecular orbital are formed by the constructive interference of two atomic orbitals (additive effect) and the amplitude of new wave can be represented as $\psi ={\psi }_1+{\psi }_2$, where ${\psi }_{1}and{\psi }_2$ are amplitude of wave function of combining atomic orbitals.

Anti-bonding molecular orbital are formed by the destructive interference of two atomic orbitals (subtractive effect) and the amplitude of new wave can be represented as $\psi ={\psi }_1-{\psi }_2$, where ${\psi }_{1}and{\psi }_2$ are amplitude of wave function of combining atomic orbitals.