Physical Chemistry/Molecular Orbital Theory

Introduction
The valence electrons are considered to be associated with all the nuclei in the molecule. Thus, the atomic orbitals from different atoms must be combined to product molecular orbitals. Developed by Friedrich Hund and Robert S. Millikan in 1932, molecular orbital theory takes the view that a molecule is similar to an atom in one important respect. Both have energy levels that correspond to various orbitals that can be populated by electrons. In atoms, these orbitals are known as atomic orbitals; in molecules however, they are called molecular orbitals (MOs).

Assumptions to be considered

 * 1) When two atoms approach each other, their atomic orbitals lose their identity and mutually overlap to form new orbitals called molecular orbitals (MOs).
 * 2) The MOs are polycentric and are associated with the nuclei of all the atoms constituting the molecule. The electron probability distribution around the group of nuclei constituting the molecule is given by the molecular orbitals just as atomic orbitals give probability of finding an electron around the nucleus in an atom.
 * 3) Only atomic orbitals of about the same energy and same symmetry interact significantly.
 * 4) The total number of MOs produced is always equal to the total number of atomic orbitals contributed by the atoms that have combined.
 * 5) When two atomic orbital overlap in-phase, it leads to an increase in the intensity of negative charge in region of overlap; the MO formed has lower potential energy than the separate atomic orbitals and is called a bonding molecular orbital.
 * 6) When two atomic orbital overlap out-of-phase, it leads to an decrease in the intensity of negative charge between the nuclei; the MO formed has higher potential energy than the separate atomic orbitals and is called an anti-bonding molecular orbital. Electrons in this type of MO destabilize the bond between atoms.
 * 7) Amount of stabilization of bonding orbitals equals the amount of destabilization of the anti-bonding orbitals.
 * 8) Each MO can accommodate electrons according to Pauli's exclusion principle and Hund's rule of maximum multiplicity.

Formation of Molecular Orbitals by Linear Combination of Atomic Orbitals (LCAO)
Consider two atoms $$A$$ and $$B$$ which have atomic orbitals described by the wave functions $$\Psi_a$$ and $$\Psi_b $$. When the electron clouds of these two atoms overlap as the atoms approach, the wave function for the molecule can be obtained by linear combination of the atomic orbitals $$\Psi_a$$ and $$\Psi_b$$.

The atomic orbitals $$\Psi_a$$ and $$\Psi_b$$ combine to give rise to a pair of molecular orbitals $$\Psi_g$$ and $$\Psi_u$$. The function $$\Psi_g$$ corresponds to the increased electron density in between the nuclei due to in-phase overlap and is, therefore, a bonding molecular orbital $$[\Psi_g = \Psi_a + \Psi_b]$$. It is thus lower in energy than the original atomic orbitals. Similarly, the function $$\Psi_u$$ corresponds to the minimized electron density in between the nuclei. It is called the anti-bonding orbital which is higher in energy $$[\Psi_g = \Psi_a - \Psi_b]$$.