MOT- Features, Types of MO, Linear Combination of Atomic Orbitals, Shapes of MO, Energy Level Diagrams of MO

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Why do we study molecular orbital theory?
To overcome the limitation of VBT, F. Hund and R.S. Mulliken proposed a new theory and named it Molecular orbital theory (MOT). MOT is used to determine the bond parameters and magnetic nature of molecules.

Table of content:

What is molecular orbital theory
Features of Molecular Orbital Theory
Types of MO
Linear Combination of Atomic Orbitals
Symmetrical and Asymmetrical MO
Energy Level Diagram of Molecular Orbitals
Bond order calculation
Practice problems
FAQs

What is molecular orbital theory?

Molecular orbital theory is a theory in which electrons are treated as wave mechanical objects moving under the influence of the nuclei in the whole molecule.

Features of Molecular Orbital Theory

Electrons are associated with the molecules as a whole and not with individual atoms.
Atomic orbitals (AOs) of individual atoms combine to form molecular orbitals (MOs).
Atomic orbitals are monocentric, whereas molecular orbitals are polycentric.
Energies of the atomic orbitals of the combining atoms must be comparable.
The number of molecular orbitals formed is equal to the number of atomic orbitals combined.
Two atomic orbitals produce two molecular orbitals, one with lower energy and the other with higher energy than the energy of combining atomic orbitals.

Atomic orbital combine to form molecular orbital

Types of MO

Bonding molecular orbitals (BMO): Atom A and atom B are to be bonded. The electron cloud of atom A is attracted towards the nucleus of atom B and vice versa. This leads to the formation of orbitals having lower energy and maximum stability. These orbitals are known as bonding molecular orbitals (BMO).

Antibonding molecular orbitals (ABMO): Some electron clouds around the two nuclei tend to pull the nuclei of the two atoms away from each other, leading to an empty region between the nuclei of the involved atoms. These electron clouds tend to keep the two nuclei apart from each other. Therefore, these orbitals have higher energy and are unstable.

Types of Molecular Orbitals Based on Overlapping:

Sigma (σ) BMO: Electron density is directly shared between the bonding atoms along the bonding axis.

Sigma (σ*) ABMO: Normally, this orbital is empty i.e., out of BMO and ABMO, BMO is filled first and if ABMO is also filled then the wave nature of electron density (when present) will be out of phase and cancelling in nature. There is a node exactly in the middle of the bonding atoms (zero electron density).

Pi (π) BMO: The bonding electron density lies above and below, or in front and the back of the bonding axis with no electron density on the bonding axis.

Pi (π*) ABMO: The combination of two atomic orbitals of 2 px or 2py out of phase with each other results in the formation of π*2 px or π* 2py antibonding molecular orbital.

Linear Combination of Atomic Orbitals:

Electrons are associated with dual nature (particle and wave nature). Therefore, an electron is associated with a wave function Ψ, which is the solution to the Schrodinger wave equation and is said to occupy molecular orbitals.

Bonding molecular orbitals: Constructive interference

Consider two atomic orbitals A and B combining linearly. The wave function of BMO is the sum of the wave functions of atomic orbitals A and B.

Ѱ_BMO = Ѱ_A + Ѱ_B

Antibonding molecular orbitals: Destructive interference
Consider two atomic orbitals A and B combined linearly. The wave function of ABMO is the difference between the wave functions of atomic orbitals A and B.

Ѱ_ABMO = Ѱ_A − Ѱ_B

Symmetrical and Asymmetrical MO

σ symmetry:

A molecular orbital will have σ symmetry if the orbital is symmetric concerning the axis joining the two nuclear centres, i.e., the internuclear axis. This means that the rotation of the molecular orbital about the internuclear axis does not result in a phase change.

Symmetry of σ BMO

A σ* molecular orbital (sigma antibonding orbital) also maintains the symmetry when rotated about the internuclear axis. The σ* molecular orbital has a nodal plane that is between the nuclei and perpendicular to the internuclear axis.

π symmetry:

A molecular orbital with π symmetry results from the interaction of either two px- px atomic orbitals or two py- py atomic orbitals.
A molecular orbital will have π symmetry if the orbital is asymmetric concerning rotation about the internuclear axis.

Asymmetry of π BMO

A π* molecular orbital (pi antibonding orbital) will also produce a phase change when rotated

about the internuclear axis. The π* molecular orbital also has a second nodal plane between the nuclei.

Gerade and Ungerade:

The orbital is designated g (for gerade, even or zero nodes) if the phase is the same. The orbital is designated u (for ungerade, uneven or odd nodes) if the phase changes sign.
In the figure, the direction of the arrow mark shows the operation of inversion, in which the sign of the lobes remains the same. Therefore, the orbital is gerade (g).

In the figure, the direction of the arrow mark shows the operation of inversion, in which the sign of the lobes changes. Therefore, the orbital is ungerade (u).

Conclusion: Table showing the symmetry of molecular orbitals.

MO	Symmetry of MO gerade (g), or ungerade(u)
σ(BMO)	g
σ(ABMO)	u
(BMO)	u
(ABMO)	g

Energy Level Diagram of Molecular Orbitals

Molecular Orbital Diagram for 1–14 electron System:

The order of increasing energy of molecular orbitals as follows:
σ1s < σ*1s < σ2s < σ*2s < π2px= π2py < σ2pz < π*2px = π*2py < σ*2pz

Energies of molecular orbital for 15–20 electron system:

The order of increasing energy of molecular orbitals is as follows:

σ1s < σ*1s < σ2s < σ*2s < σ2pz < π2px = π2py < π*2px = π*2py < σ*2pz

Orbitals are filled as per the Aufbau principle, Hund’s rule, and Pauli’s exclusion principle.

Bond order calculation :

(B.O.) is defined as one half the difference between the number of electrons present in the bonding

and the antibonding orbitals.

Where, N_b= number of bonding electrons, N_a= number of antibonding electrons

Practice problems

Example 1: Draw the molecular orbital diagram for B2 molecule.
Answer: The electronic configuration of B is 1s² 2s² 2p¹ . In the B_² molecule, a total of 10 electrons are present that occupy the molecular orbitals.The molecular electronic configuration of B2 is σ1s² σ*1s² σ2s² σ*2s² (π2p_x¹ = π2p_y¹).

Example 2: What is the bond order of a Ne molecule?

a. 1
b. 2
c. 0
d. 1.5

Answer: The molecular electronic configuration of Ne2 is as follows:

Example 3. Draw the molecular orbital diagram for He₂⁺ molecule.
Solution: The electronic configuration of He is 1s². The molecular electronic configuration of He₂⁺ is σ1s2 σ*1s¹

Example 4. In the molecular orbital diagram for molecular ion, N2+ the number of electrons in σ2p molecular orbital is

a. 0
b. 2
c. 3
d. 1

Answer: The molecular electronic configuration of N2+ is σ1s2 σ*1s2 σ2s2 σ*2s2 (π2px2 = π2py2) σ2pz1.
From electronic configuration, we can easily predict the number of electrons in σ2p which is equal to 1.

FAQs:

Q 1. What is bonding axis?
Answer: Shared electron density is directly between the bonding atoms along with one of the axis and by convention z-axis that is taken to be internuclear axis is considered as bonding axis.

Q 2. Helium molecule has zero bond order. What does this mean?
Answer: Helium molecule has zero bond order, this means that the number of electrons in bonding molecular orbital is equal to the number of electrons in anti- molecular orbital.

Q 3. What is a node?
Answer: Node is a point in the wave where the probability of finding electron density is zero.

Q 4. What is constructive interference and distructive interference?Answer: Constructive interference occurs when the maxima of the two waves add together (the two waves are in the same phase) so that the amplitude of the resulting wave is equal to the sum of the individual amplitudes. Destructive interference occurs when the maxima of two waves are out of phase. A positive displacement of one wave is cancelled by a negative displacement of the other. The amplitude of the resulting wave is zero or less than the amplitude of the combining waves.