Molecular orbital theory: Postulates, LCAO method, and molecular orbital diagram

Molecular orbital theory was developed by F.Hund and R.S. Mulliken in 1932. According to molecular orbitals theory, all the atomic orbitals of combining atoms undergo mixing to form a new set of orbitals termed molecular orbitals.

Features of molecular orbital theory

The main features of molecular orbitals theory are:

1. According to Molecular orbital theory, all the atomic orbitals of combining atoms first mix to form a new set of orbitals called molecular orbitals. Once the molecular orbitals are formed, the combing atoms lose their identity of atomic orbitals.
2. The number of molecular orbitals formed is equal to number of atomic orbitals mixed.
3. The molecular orbitals are formed by the linear combination of atomic orbitals.
4. Molecular orbitals are formed by addition and subtraction overlap of atomic orbitals. The molecular orbitals formed by addition overlap are called bonding molecular orbitals and the molecular orbitals formed by substraction overlap are called anit-bonding molecular orbitals.
5. Addition of electrons in bonding molecular orbitals tends to stabilize the molecule while addition of electrons in anti-bonding molecular orbital tends to destabilize the molecules.
6. The shape of molecular orbitals formed depends on the shape of atomic orbital mixed.
7. The atomic orbitals participating in the formation of molecular orbitals should have same symmetry and comparable energies.
8. Molecular orbitals have different energy, shapes and sizes.
9. Each electrons in the molecular orbitals can be described by its wave function Ψ and Ψ² gives the probability of electrons density in the molecular orbitals.
10. The bonding molecular orbital has lower energy and greater stability than anti-bonding molecular orbitals.
11. The electrons in a molecule are present in various molecular orbital as the the electrons of atom are present in different various shells.
12. In molecular orbitals, electrons are in influence of two or more nuclei.
13. The molecular orbitals are filled in accordance to Aufbau principle, Pauli’s principle and Hund’s rule.

Molecular orbital theory bonding and antibonding

Bonding molecular orbitals

Bonding molecular orbital is formed by the positive overlap of atomic orbitals of combining atoms. The electron density is high between two nuclei. The formation of bonding molecular orbitals is shown below:

Anti bonding molecular orbitals

Anti-bonding molecular orbital is formed by the negative overlap of atomic orbitals of combining atoms. The electron density is very low between the two nuclei. The formation of antibonding molecular orbitals is shown below:

Non-bonding molecular orbitals

A molecular orbital in which the in-phase bonding atomic orbitals overlap equals the number of out-phase antibonding atomic orbitals. Electrons in a nonbonding molecular orbital are higher in energy than in a bonding molecular orbital, but lower in energy than electrons in an antibonding molecular orbital. The formation of non-bonding molecular orbitals is illustrated below;

Difference between Bonding and anti-bonding molecular orbitals

The main difference between bonding and anti-bonding molecular orbitals are shown below:

LCAO method molecular orbital theory

According to LCAO method, molecular orbitals are formed by a linear combination of atomic orbitals which are brought by addition overlap and subtraction overlap. When two atomic orbitals combine to form molecular orbitals, the wave functions are combined both in-phase and out of phase to create one bonding molecular orbitals and one anti-bonding molecular orbitals respectively.

Let two atoms A and B combine to form a molecule AB.

A+B→AB

i.e. Ψ_mo= Ψ_A±Ψ_B

where Ψ_A=wave function of atomic orbital of atom A,

Ψ_B=wave function of atomic orbital of atom B

Ψ_mo =wave function of molecular orbital of AB

Here, Ψ^b=Ψ_A+Ψ_B and Ψ^*=Ψ_A-Ψ_B can be separated from Ψ_mo= Ψ_A±Ψ_B.

Equation Ψ^b=Ψ_A+Ψ_B represent the formation of bonding molecular orbital formed by the addition overlap of atomic orbitals.

Equation Ψ^*=Ψ_A-Ψ_B represents the formation of anti-bonding molecular orbital formed by subtraction overlap of atomic orbitals.

Conditions of LCAO method

The main conditions of the LCAO method are:

1. The atomic orbitals must have comparable energy.
2. The atomic orbitals must overlap appreciably
3. The atomic orbitals must have same symmetry with respect to the molecular axis.

Bond order molecular orbital theory

The bond order is defined as half of the difference between the number of bonding electrons(N_b) and antibonding electrons(N_a).

Mathematically, Bond order= (N_b-N_a)/2

1. Bond order of a diatomic molecule is equal to number of bonds present between two atoms. For examples: bond oder of O₂ molecule= 2
2. Zero bond order of a moelcule represent that it is unstable and does not exit.
3. Bond order is directly proportional to stability of molecules, bond energy but inversly proportional to bond length of molecules.

How to find bond order in molecular orbital theory

For O₂ molecule:

The electronic configuration of O-atom= 1s^2, 2s², 2p⁴

Total electrons in O₂ molecules= 16

Molecular electronic configuration of O₂ molecules= σ1s², σ*1s², σ2s², σ*2s², σ2p_x², π2p_y²=π2p_z², π*2p_y¹=π*2p_z¹

Bond order= (N_b-N_a)/2=(10-6)/2= 2

Bond order 2 means there is a double bond in O₂ molecules.

Valence bond theory and molecular orbital theory differences

Atomic orbitals and molecular orbitals similarities

Some similarities between molecular orbitals and atomic orbitals are given below:

1. Electron whether it is present in atomic orbitals or molecular orbitals, can be denoted by a particular wave function, Ψ.
2. Like atomic orbitals, molecular orbitals also accommodate maximum two electrons with oppostie spin.
3. Like atomic orbitals, the molecular orbitals also have different shapes and energy.
4. Electrons should follow the aufbau principle,pauli’s exclusion principle and hund’s rule when it enters in Atomic orbitals or molecular orbitals.

Atomic orbitals and molecular orbitals differences

The main differences between atomic orbitals and molecular orbitals are:

How to write electronic configuration in molecular orbital theory

The electrons are filled in molecular orbitals in the case of a molecule. The electrons may occupy either bonding molecular orbital or antibonding molecular orbitals. In some molecules, electrons may occupy non-bonding orbitals as well. The bonding orbitals are represented by σ and π while antibonding orbitals are represented by σ* and π*. The electronic configuration in molecular orbitals are written as:

For lighter elements(from H₂ to N₂)

The relative energy of molecular orbitals is

σ1s<σ*1s<σ2s<σ*2s<π2p_y=π2p_z<σ2p_x<π*2p_y=π*2p_z<σ*2p_x

For heavier elements(O₂,F₂ and Ne₂)

The relative energy of molecular orbitals is

σ1s<σ*1s<σ2s<σ*2s<σ2p_x<π2p_y=π2p_z<π*2p_y=π*2p_z<σ*2p_x

Molecular orbital theory diagram

The molecular orbital diagram of flourine as example is shown below:

formation of N₂ molecule by molecular orbital theory

Each nitrogen atom has 7 electrons. Thus, the N2 molecule contains a total of 14 electrons. These electrons are arranged in molecular orbitals as:

σ1s², σ*1s², σ2s², σ*2s², π2p_y²=π2p_z², σ2p_x², π*2p_y⁰=π*2p_z⁰

In such molecules, the energies of π2p_y=π2p_z orbitals are lower than σ2p_x orbitals.

molecular orbital theory of O₂

Each oxygen atom has 8 electrons. Thus, the O2 molecule contains a total of 16 electrons. These electrons are arranged in molecular orbitals as:

σ1s², σ*1s², σ2s², σ*2s², σ2p_x², π2p_y²=π2p_z², π*2p_y¹=π*2p_z¹

The antibonding orbitals π*2py, π*2pz are singly occupied in accordance with Hund’s rule.

Molecular orbital theory of CO

The molecular orbital theory diagram of the CO molecule is shown below:

molecular orbital theory video