Molecular Orbital Theory
Table of Contents
 Introduction to Molecular Orbital Theory
 Linear Combination of Atomic Orbital(LCAO)
 Difference between Atomic Orbitals and Molecular Orbitals
 Relative Energies of Molecular Orbitals
 Energy Level Diagram
 Rules for Filling of Molecular Orbitals
 Order of Energies of Various Molecular Orbitals
 Bond Order
 Analysis done by Bond Order
Introduction to Molecular Orbital Theory
Valence Bond Theory fails to answer certain questions like Why He_{2} molecule does not exist and why O_{2} is paramagnetic? Therefore in 1932 F. Hood and RS. Mulliken came up with theory known as Molecular Orbital Theory to explain questions like above. According to Molecular Orbital Theory individual atoms combine to form molecular orbitals, as the electrons of an atom are present in various atomic orbitals and are associated with several nuclei.
Fig. No. 1 Molecular Orbital Theory
Electrons may be considered either of particle or of wave nature. Therefore, an electron in an atom may be described as occupying an atomic orbital, or by a wave function Ψ, which are solution to the Schrodinger wave equation. Electrons in a molecule are said to occupy molecular orbitals. The wave function of a molecular orbital may be obtained by one of two method:
1. Linear Combination of Atomic Orbitals (LCAO). 2. United Atom Method.
Linear Combination of Atomic Orbitals (LCAO)
As per this method the formation of orbitals is because of Linear Combination (addition or subtraction) of atomic orbitals which combine to form molecule. Consider two atoms A and B which have atomic orbitals described by the wave functions Ψ_{A} and Ψ_{B} .If electron cloud of these two atoms overlap, then the wave function for the molecule can be obtained by a linear combination of the atomic orbitals Ψ_{A} and Ψ_{B} i.e. by subtraction or addition of wave functions of atomic orbitals
Ψ_{MO}= Ψ_{A} + Ψ_{B}
The above equation forms two molecular orbitals
Bonding Molecular Orbitals
When addition of wave function takes place, the type of molecular orbitals formed are called Bonding Molecular orbitals and is represented by Ψ_{MO }= Ψ_{A} + Ψ_{B}_{.}
They have lower energy than atomic orbitals involved. It is similar to constructive interference occurring in phase because of which electron probability density increases resulting in formation of bonding orbital. Molecular orbital formed by addition of overlapping of two s orbitals shown in figure no. 2. It is represented by s.
AntiBonding Molecular Orbitals
When molecular orbital is formed by subtraction of wave function, the type of molecular orbitals formed are called Antibonding Molecular Orbitals and is represented byΨ_{MO }= Ψ_{A}  Ψ_{B}_{.}
They have higher energy than atomic orbitals. It is similar to destructive interference occurring out of phase resulting in formation of antibonding orbitals. Molecular Orbital formed by subtraction of overlapping of two s orbitals are shown in figure no. 2. It is represented by s* (*) is used to represent antibonding molecular orbital) called Sigma Antibonding.
Fig. No. 2 Formation of Bonding and AntiBonding Orbital
Therefore, Combination of two atomic orbitals results in formation of two molecular orbitals, bonding molecular orbital (BMO) whereas other is antibonding molecular orbital (ABMO).
BMO has lower energy and hence greater stability than ABMO. First BMO are filled then ABMO starts filling because BMO has lower energy than that of ABMO.
Formation of molecular orbitals occurs by the combination of atomic orbitals of proportional symmetry and comparable energy. Therefore, a molecular orbital is polycentric and atomic orbital is monocentric. Number of molecular orbitals formed is equal to the number of atomic orbitals.
Differences between Molecular Orbital and Atomic Orbital
Molecular Orbital  Atomic Orbital 


Relative Energies of Molecular Orbitals
Bonding Molecular Orbitals (BMO)  Energy of Bonding Molecular Orbitals is less than that of Anti Bonding Molecular Orbitals because the attraction of both the nuclei for both the electron (of the combining atom) is increased.
AntiBonding Molecular Orbitals (ABMO)  Energy of Anti Bonding Molecular Orbitals is higher than Bonding Molecular Orbitals because the electron try to move away from the nuclei and are in repulsive state.
The Energies of Bonding Molecular Orbitals and AntiBonding Molecular Orbitals are shown in figure below:
Fig. No.3 Energies of BMO and ABMO
Energy Level Diagram
The factors upon which relative energies of molecular orbitals depend are:
(i) Energies of the Atomic orbitals combining to form Molecular Orbitals.
(ii) The extent of overlapping between the atomic orbitals. The greater the overlap, the more the bonding orbital is lowered and the antibonding orbital is raised in energy relative to AOs
1s Atomic Orbitals (AOs) of two atoms form two Molecular Orbitals (MOs) designated as s1s and s *1s.The 2s and 2p orbitals (eight AOs of two atoms) form four bonding MOs and four antibonding MOs as:
Bonding MOs: σ 2s, σ 2pz, π 2p_{x}, π 2p_{y}
Anti – Bonding MOσ: σ *2s, σ *2p_{z}, π *2p_{x}, π *2p_{y}
Using Spectroscopy, the energy levels of these molecular orbitals are determined experimentally. The order of increasing energy of molecular orbitals obtained by combination of 1s, 2s and 2p orbitals of two atoms is →
σ1s, σ *1s, σ 2s, σ *2s, σ 2p_{z}, π 2p_{x} = π 2p_{y}, π *2p_{x}= π *2p_{y}, σ *2p_{z}
(Energy Increases from left to right)
The molecular orbital diagram representing this order of energy levels is shown in fig.
But experimental evidence for some diatomic molecules have shown that the above sequence of energy levels of MOs is not correct for all the molecules. For example, homonuclear diatomic molecules of second row elements like Li_{2}, Be_{2}, B_{2 }, C_{2}, N_{2} , the σ 2p_{z }MOs is higher in energy than π 2p_{x} and π 2p_{y }MOs
For these atoms, the order is: →
σ1s, σ *1s, σ 2s, σ *2s, [π 2p_{x} = π 2p_{y}], σ 2p_{z} [π *2p_{x}= π *2p_{y}], σ*2p_{z}
The molecular orbital diagram representing this order of energy levels is shown in fig.
Fig. No. 5 Order of Energy Levels for Boron, Carbon, Nitrogen etc.
This kind of energy reversal is due to mixing of 2s and 2p orbitals where the energy difference is very close, that is, for B, C, and N atoms. According to the symmetry interactions, the two orbitals of the same symmetry repel each other and the lower energy orbital lowers down more while the higher energy orbital is energized more. Accordingly, σ 2s and σ 2p_{x} have same symmetry and similarly for σ *2s and σ *2p_{x} the energy of σ 2s is lowered and that of the σ 2p_{x} becomes higher. Similarly, the energy of σ *2s lowered while that of σ *2p_{x} becomes higher. Finally, the energy of the σ *2p_{x} becomes higher than π2p_{y} and π2p_{z} which remain unchanged in the symmetry interaction. This kind of mixing of orbitals or symmetry interaction is not applicable for O_{2} and F_{2} molecule formation because of larger energy gap between 2s and 2p orbitals for these atoms.
Rules for Filling of Molecular Orbitals
Certain rules are to be followed while filling up molecular orbitals with electrons in order to write correct molecular configurations:
 Aufbau Principle – This principle states that those molecular orbital which have the lowest energy are filled first.
 Pauli’s Exclusion Principle – According to this principle each molecular orbital can accommodate maximum of two electrons having opposite spins.
 Hund’s Rule – This rule states that in two molecular orbitals of the same energy, the pairing of electrons will occur when each orbital of same energy consist one electron.
Order of energy of various Molecular Orbitals
Order of energy of various molecular orbitals is as follows:
For O_{2} and higher molecules →
σ1s, σ *1s, σ 2s, σ *2s, σ 2p_{z}, [π2p_{x} = π2p_{y}], [π*2p_{x}= π*2p_{y}], σ *2p_{z}
Fig. no. 6 Order of Energy for O_{2 }and Higher molecules
For N_{2} and lower molecules →
σ 1s, σ *1s, σ 2s, σ *2s, [π 2p_{x} = π 2p_{y}], σ 2px [π *2p_{x}= π *2p_{y}], σ*2p_{z}
Fig. No. 8 Order of Energy for N_{2} and lower molecules
Bond Order
It may be defined as the half of difference between the number of electrons present in the bonding orbitals and the antibonding orbitals that is,
Bond order (B.O.) = (No. of electrons in BMO  No. of electrons in ABMO)/ 2
Those with positive bonding order are considered stable molecule while those with negative bond order or zero bond order are unstable molecule.
Magnetic Behavior: If all the molecular orbitals in species are spin paired, the substance is diamagneti. But if one or more molecular orbitals are singly occupied it is paramagnetic. For Example, if we look at CO Molecule, it is diamagnetic as all the electron in CO are paired as in the figure below:
Fig. No. 9 Molecular Orbital Diagram for CO
Analysis done by Bond Order
 If value of bond order is positive, it indicates a stable molecule and if the value is negative or zero, it means that the molecule is unstable.
 The stability of a molecule is measured by its bond dissociation energy. But the bond dissociation energy is directly proportional to the bond order. Greater the bond order, greater is the bond dissociation energy.
 Bond order is inversely proportional to the bond length. The higher the bond order value, smaller is the bond length. For Example the bond length in nitrogen molecule is shorter than in oxygen molecule
 Magnetic character  If all the electrons in the molecule of a substance are paired, the substance is diamagnetic (repelled by the magnetic field). On the other hand, if there are unpaired electrons in the molecule, the substance is paramagnetic (attracted by the magnetic field).
Let’s take a question to understand it more clearly.
Q1: Arrange the species O_{2}, O_{2}^{}, O_{2}^{+} the decreasing order of bond order and stability and also indicate their magnetic properties.
Sol. The molecular orbital configuration of
O_{2}, O_{2}^{}, O_{2}^{2}, O_{2}^{+} are as follows:
O_{2 }= σ1s^{2}, σ*1s^{2}, σ2s^{2}, σ*2s^{2}, σ2p_{z}^{2}, π2p_{x}^{2 }= π2p_{y}^{2}, π*2p_{x}^{1 }= π*2p_{y}^{1}
Bond order = (106)/2 = 2, Number of unpaired electrons = 2, Therefore paramagnetic
O_{2}^{ }= σ 1s^{2}, σ *1s^{2}, σ 2s^{2}, σ *2s^{2}, σ 2p_{z}^{2}, π2p_{x}^{2 }= π2p_{y}^{2}, π*2p_{x}^{2 }= π*2p_{y}^{1}
Bond order = (107)/2 = 1.5, Number of unπaired electrons = 1, Therefore paramagnetic
O_{2}^{2} = σ1s^{2}, σ*1s^{2}, σ2s^{2}, σ*2s^{2}, σ2p_{z}^{2}, π2p_{x}^{2 }= π2p_{y}^{2}, π*2p_{x}^{2 }= π*2p_{y}^{2}
Bond order = (108)/2 = 1, Number of unpaired electrons = 0, Therefore diamagnetic
O_{2}^{+ }= σ1s^{2}, σ*1s^{2}, σ2s^{2}, σ*2s^{2}, σ2p_{z}^{2}, π2p_{x}^{2 }= π2p_{y}^{2}, π*2p_{x}^{1}= π*2p_{y}^{0}
Bond order = (105)/2 = 2.5, Number of unpaired electrons = 1, Therefore paramagnetic
The bond order decreases in the order is
O_{2}^{+} >O_{2}>O_{2}^{}>O_{2}^{2}
so, we conclude stability is directly proportional to bond order.
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