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Subject Chemistry

Paper No and Title 2 and Physical Chemistry-I

Module No and Title 27 and Valence Bond Theory I

Module Tag CHE_P2_M27

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

______TABLE OF CONTENTS

1. Learning Outcomes 2. Introduction 3. Valence Bond Theory (VBT) 3.1 Postulates of VBT 3.2 VBT of Hydrogen molecule 4. Summary

1.

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

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1. Learning Outcomes

After studying this module, you shall be able to:

Learn about electronic structure of molecules using VBT Understand the postulates of VBT The VBT of Hydrogen molecule

2. Introduction

Now, at this point we know that the Schrodinger equation cannot be solved exactly for multi-electron system even if the inter-nuclear distances are held constant. This is due to the presence of electron-electron repulsion terms in the Hamiltonian operator. Consequently, various approaches have been developed for the approximate solution of Schrodinger equation. Of these, Molecular Orbital Theory (MOT) and Valence Bond Theory (VBT) have been widely used. These two approaches differ in the choice of trial/approximate wave-function which is optimized via variation method for the system under consideration. Of these, we have already discussed MOT in earlier modules.

In this module, we will take up the formalism of VBT in detail.

3. Valence Bond Theory

Valence Bond Theory was the first quantum mechanical treatment to account for chemical bonding. This theory was first introduced by Heitler and London in 1927 and subsequently by Slater and Pauling in 1930s. It is sometimes referred to as Heitler-London-Slater- Pauling (HLSP) theory. VBT draws on the Lewis concept of a covalent bond as a shared pair of electrons (electron pair bond) i.e., valence bond approach is similar to the concept of valence bond between

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

______two atoms In this method, a molecule is considered as a collection of atoms and the interactions between different atoms are considered. With this, the question is how VBT chooses the trial/approximate wave-function. To elaborate this, we will consider the simplest case of chemical bond viz., hydrogen molecule in this module.

3.1 Postulates of VBT

The main postulates of this theory are as follows:

Overlapping of two half - filled valence atomic orbitals of two different atoms results in a covalent bond.

The electrons in the overlapping orbitals get paired, localized and concentrated between the nuclei of two atoms.

Only two electrons with their spins paired may be shared by one set of overlapping orbitals.

The electron density between two bonded atoms increases due to overlapping. This confers stability to the molecule.

The strength of the bond is directly proportional to the extent of overlap. Greater the extent of overlapping, stronger is the bond formed.

The direction of the covalent bond is along the region of overlapping of the atomic orbitals i.e., covalent bond is directional.

3.2 VBT of Hydrogen molecule Before discussing the valence bond treatment for hydrogen molecule, it is important to note that VBT cannot solve Schrödinger equation for hydrogen molecule ion as VBT approximation considers overlapping of atomic orbitals each with one electron SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

______respectively. And hydrogen molecule ion contains only one electron. For this reason, hydrogen molecule ion cannot be explained by VBT.

Let us now consider the simplest case of chemical bond, i.e., the hydrogen molecule H2 which has two nuclei and two electrons.

So, the total number of terms in Hamiltonian are (2 + 2)(2 + 2 + 1) 푇표푡푎푙 푛푢푚푏푒푟 표푓 푡푒푟푚푠 𝑖푛 퐻푎푚𝑖푙푡표푛𝑖푎푛 = = 10 2

The Hamiltonian for H2 molecule is as follows:

ℎ2 1 ℎ2 1 ℎ2 1 ℎ2 1 푒2 푒2 푒2 푒2 ̂ 2 2 2 2 퐻 = − 2 ∇퐴 − 2 ∇퐵 − 2 ∇1 − 2 ∇2 + − − − 8휋 푚푝 8휋 푚푝 8휋 푚푒 8휋 푚푒 4휋휀표푅 4휋휀표푟1퐴 4휋휀표푟1퐵 4휋휀표푟2퐴 푒2 푒2 − + 4휋휀표푟2퐵 4휋휀표푟12

Applying Born-Oppenheimer approximation (considering the nuclei fixed, the nuclear

kinetic energy terms are neglected, 푚푝 > 푚푒), we get ℎ2 1 ℎ2 1 푒2 푒2 푒2 푒2 푒2 푒2 ̂ 2 2 퐻푒 = − 2 ∇1 − 2 ∇2 − − − − + + 8휋 푚푒 8휋 푚푒 4휋휀표푟1퐴 4휋휀표푟1퐵 4휋휀표푟2퐴 4휋휀표푟2퐵 4휋휀표푟12 4휋휀표푅

which can be rewritten as,

′ 퐻̂푒 = 퐻̂퐴1 + 퐻̂퐵2 + 퐻̂

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

______where 퐻̂퐴1푎푛푑 퐻̂퐵2 are the Hamiltonians of electron 1 in atom HA and electron 2 in atom ′ HB respectively and 퐻̂ is the potential energy operator.

Wave-function for hydrogen molecule Since hydrogen molecule is a two nuclei and two electron system, there are two possibilities.

Case I: If the atoms are widely spaced (with no interaction with each other), the electronic wave-function for the system can be written as,

훙푽푩 = 흍푯푨ퟏ흍푯푩ퟐ

Where 휓퐻퐴1 represents wave-function of electron 1 on atom HA and 휓퐻퐵2 represents

wave-function of electron 2 on atom HB. And the total energy of the system is equal to the sum of energies of the separate atoms.

퐸푡표푡푎푙 = 퐸퐻퐴1 + 퐸퐻퐵2

Case II: If the atoms are close, it is not possible to distinguish whether electron 1 or electron 2 is on atom A (or atom B). When the two outcomes are equally probable, quantum mechanics instructs to describe the state of the system by linear combination of each possibility. So, the wave-function for the molecule is constructed by taking linear combination of both the possibilities as:

훙푽푩 = 흍푯푨ퟏ흍푯푩ퟐ + 흍푯푨ퟐ흍푯푩ퟏ = 풄ퟏ훙ퟏ + 퐜ퟐ훙ퟐ

where 흍푯푨 흍푯푩 represents 1s orbitals (Atomic orbitals) of the two hydrogen atoms A and B respectively (the spins of the electrons are neglected for the sake of simplicity) and

풄ퟏ & 풄ퟐ represent the coefficients. The above wave-function represents covalent structures as each bonded atom contains one electron.

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

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Another possibility of electron distribution that can be taken into account is when both the electrons belong either to atom A or atom. So, the wave-function representing ionic structures is given by the equation,

훙푰풐풏풊풄 = 흍푯푨ퟏ흍푯푨ퟐ + 흍푯푩ퟏ흍푯푩ퟐ With this total VB wave-function becomes,

훙푽푩 = 훙푪풐풗풂풍풆풏풕 + 훙푰풐풏풊풄 = 흍푯푨ퟏ흍푯푩ퟐ + 흍푯푨ퟐ흍푯푩ퟏ + 흍푯푨ퟏ흍푯푨ퟐ + 흍푯푩ퟏ흍푯푩ퟐ

Energy expression for hydrogen molecule Taking into consideration the covalent structure, the energy value of hydrogen molecule is given by,

∫ 훙푽푩퐻̂푒 훙푽푩풅흉 퐸푎 = ∫ 훙푽푩 훙푽푩풅흉

The detailed mathematical of the above equation using VBT for hydrogen molecule will taken up in next module.

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I

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5. Summary

Valence Bond Treatment was the first quantum mechanical treatment to account for chemical bonding. VBT was first introduced by Heitler and London in 1927 and subsequently by Slater and Pauling in 1930s. VBT draws on the Lewis concept of a covalent bond as a shared pair of electrons. VBT treatment for hydrogen molecule considers the wave-function for the molecule as

훙푽푩 = 흍푯푨ퟏ흍푯푩ퟐ + 흍푯푨ퟐ흍푯푩ퟏ = 풄ퟏ훙ퟏ + 퐜ퟐ훙ퟐ

∫ 훙푽푩퐻̂푒 훙푽푩풅흉 퐸푎 = ∫ 훙푽푩 훙푽푩풅흉

SUBJECT PAPER : 2, Physical Chemistry I MODULE : 27, Valence Bond Theory I