1) Chemical bonding: Formation of molecular orbitals

 CHEMICAL BONDING: FORMATION OF MOLECULAR ORBITALS

Introduction

When two half-filled atomic orbitals belonging to two same or different atoms are brought near each other, they overlap and form a new orbital called molecular orbital. It surrounds both the nuclei and contains both electrons. The Schrödinger equation can be solved exactly only for one electron system such as a hydrogen atom. If it could be solved exactly for molecules containing two or more electrons, we would have a precise picture of the shape of the orbitals available to each electron and the energy for each orbital. Since exact solutions are not available, drastic approximations must be made. There are two chief general methods of approximation: The valence-bond method and the molecular orbital method.

Valence-Bond Method

This approach is due mainly to the work of Heitler, London, Slater, and Pauling. In this method, a molecule is regarded to be formed by the interaction between the electron waves of the half-filled
atomic orbitals only. When two atoms are brought closer together, their half-filled atomic orbitals
(A.O.) overlap to form a molecular orbital (M.O.). All other orbitals (except bonding orbital) on the atom remain undisturbed and preserve their individual character. In this approach, a wave equation is written for each of the various possible electronic structures (also known as Lewis structures or canonical forms) for a molecule and the total wave function for the molecule is the sum of all individual wave tasks for the canonical forms each with a weighting factor.
...

For example, a wave function can be written for each of the following canonical forms of a hydrogen molecule, H2.

The weighting factor e, which determines its contribution to the total form, is obtained by solving the equation for various values of each c and choosing the solution of lower energy. In the case of a hydrogen molecule, e will be very high for (1) and low for (II) and (III), suggesting that (1) has a greater contribution to the total structure of the hydrogen molecule.

As mentioned above, a molecule is regarded to be formed by the approach of two initially distinct atomic orbitals, each containing one electron (half-filled orbital) of opposite spin. The most useful concept invoked in understanding chemical bonding is that it results from the overlap of two atomic orbitals thus lowering the potential energy of the system.

The principles of overlap can be enunciated as

(1) Only those orbitals overlap which participate in the bond formation. All other orbitals on the atom remain undisturbed and preserve their individual character.

(2) The greater overlapping of the atomic orbitals leads to a lowering of energy due to the attractive forces between electrons and the nuclei between atomic orbitals. Thus the greater the overlap between atomic orbitals, the stronger the covalent bond. This is known as the principle of maximum overlap, and 

(3) The direction of the electron density in the bonds determines the bond angle.

From the above, it is clear that the bond strength increases with decreasing internuclear distance During bond formation, the two orbitals approach each other and start to overlap. Energy is released as the electron in each atom is attracted to the positively charged nucleus of the other atom as well as to its own nucleus (Fig. 1.1). The more the orbital overlap, the more the energy decreases until the atoms approach each other so closely that their nuclei start to repel each other electrostatically. This causes a large increase in energy. This means that maximum stability (minimum energy) is achieved when the nuclei are at a certain, finite distance apart. This distance corresponds to the bond length of the new covalent bond.