# Molecular Orbital Theory

Molecular Orbital Theory (MOT)

[Based on Linear Combination of Atomic Orbitals (L.C.A.O)]

Hund and Mulliken have developed an approach to covalent bond formation which is based upon the effects of the various electron fields upon each other and which employs molecular orbitals rather than atomic orbitals. Each such orbital characterizing the molecule as a whole is described by a definite combination of quantum numbers and possesses relative energy value.

Comparison between atomic and molecular orbitals

 Atomic Orbitals Molecular Orbitals They belongs to a particular atom and are influenced by one nucleus, i.e., monocentric. They are denoted by s, p, d, f etc. They have definite shapes. They can accommodate a maximum of two electrons of opposite spins. They belong to the whole molecular (or ion) and are influenced by more than one nucleus, i.e., polycentric. They are denoted by $\sigma, \sigma^*, \pi, \pi^*, \delta, \delta^*$etc. They do not have definite shapes. They can also accommodate a maximum of two electrons of opposite spins.

According ot this theory the atomic orbitals combine and form a resultant orbital known as the molecular orbital in which the identity of both the atomic orbitals is lost. All the electrons pertaining to both the atoms are considered to be moving along the entire molecule under the influence of all the nuclei.

When tow atoms are brought very close together as required by a chemical bond the interaction of the system is such that the quantum number of the atoms has no meaning. Atomic orbitals are monocentric but molecular orbitals are polycentric. Each electron belongs to the entire molecule.

Let $\Psi _A \text{and} \Psi_B$ be the two atomic orbitals (AOs)of the atoms A and B. These AOs combine to form two molecular orbitals (MOs)known as bonding $(\Psi_b)$ and anti-bonding $(\Psi_a)$ MOs. These orbitals may be represented as:

$\Psi_b = \Psi_A + \Psi_B$ …………….(I)

And

$\Psi_a = \Psi_A- \Psi_B$ ………………(II)

It is to be noted that in the case of bonding orbital electron density is concentrated between the two nuclei of the two atoms. While in the case of anti-bonding orbital nuclei of the two atoms come close to each other. Due to same charge and absence of the electron density the nuclei repel each other. We known that square of the wave function $(\Psi)$ is known as probability of finding the electrons hence on squaring the eq. I and II, we get

$\Psi^b_b = \Psi^2_A + \Psi_B^2 + 2 \Psi_A \Psi_B$ …………..(III)

And

$\Psi_a^2 = \Psi^2_A + \Psi_B^2- 2 \Psi_A \Psi_B$ ……………(IV)

It is clear from eq. III that the value of [late size=”2”]\Psi_B^2[/latex] is greater than the sum of $\Psi_A^2 \text{and} \Psi^2_B$. It means that the probability of finding the electrons in the MOs obtained by the linear combination in accordance with eq. I is greater than that is either of the AOs $\Psi_A \text{and} \Psi_B$. In other word the energy of $\Psi_b$ is lower than either of $\Psi_A \text{and} \Psi_B$. Hence this orbital forms stable chemical bond and is named as bonding molecular orbital (B.M.O).

In the same way, we can say, form eq. IV, that the value $\Psi_a$ is less than $\Psi_A + \Psi_B$. It means that the probability of finding the electron in the MOs obtained by the linear combination in accordance with eq. II is less than that in either of the AOS $\Psi_A \text{and} \Psi_B$. Hence this orbital cannot form a stable chemical bond and is named antibonding molecular orbital (A.B.M.O.).

The shapes of different bonding and anti-bonding molecular orbitals are as follows:

$\sigma 1s, \sigma 2s \to \text{Has no Nodal plane}$ $*\sigma 1 p_z *\sigma 2 p_z \pi 2p_x or \pi 2p_y \to \text{has one Nodal plane each}$ $*\pi 2p_x or * \pi 2 p_y \to \text{Has two Nodal planes each}$

Comparison between B.M.O. and A.B.M.O.

 B.M.O. A.B.M.O. 1.These are formed by the addition of orbitals of same phase. I.e., $\Psi = \Psi_a + \Psi_b$ 2.they have lower energy than atomic orbitals involved. 3.electron density increases between the nuclei involved thus stability of the molecule increases. 4.may or may not have nodal planes. They are formed by the addition of the orbitals of different phase, i.e. $\Psi = \Psi_a- \Psi_b$ They have higher energy than atomic orbitals involved. Electron density decreases between the nuclei involved thus stability of the molecule decreases. Always have nodal planes.

The energy level of these molecular orbitals is given here and shown in the figure

$\sigma (1s) < \sigma^* (1s) < \sigma (2s) < \sigma^* (2s) < \pi (2p_x) \\ = \pi (2p_y) < \sigma (2p_z < \pi^* (2p_x) = \pi^* (2p_y) < \sigma^* (2p_z)$

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