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WHAT IS LCAO THEORY

Linear Combination of Atomic Orbitals [L.C.A.O].

The molecular orbitals are formed by the linear combination of the wave functions of the participating atomic orbitals. They may combine either by addition or by subtraction. Let  ψ_A and ψ_B represent the wave functions (or amplitude) of the two combining atomic orbitals A and B taking part in chemical combination.

(a) Combination by addition. When the two electron waves are in phase i.e. they have the same sign*, they will add up to give a new wave function expressed as ψ (or ψ) = ψ_A + ψ_B.

(b) Combination by subtraction. When the two electron waves are out of phase i.e. they have opposite sign of the wave functions, then they will combine by subtraction and the resulting wave function ψ* (or ψ)*) = ψ_A - ψ_B as shown below :

We know that the probable electron density is given by ψ² and not by ψ. On squaring we get :

Here ψ_A² and ψ_B² represent the probable electron density in the two atomic orbitals whereas ψ² and ψ^*2 indicate the electron density in the two molecular orbitals. These are called Bonding Molecular orbital (ψ²) and Anti bonding Molecular orbital (ψ^*2).

Bonding Molecular Orbital (ψ²) is formed by the linear combination of wave functions of the combining atomic orbitals (L.C.A.O) by addition.

Anti bonding Molecular Orbital (ψ*2) is formed by the linear combination of wave functions of the combining atomic orbitals (L. C .A.O) by subtraction. Let us illustrate both these orbitals by the combination of the atomic orbitals of hydrogen atoms.

The formation of bonding molecular orbital when two atomic orbitals of hydrogen atoms ( Is orbital) combine by the addition of their wave functions is shown below :

Similarly, the anti bonding molecular orbital arising from the subtraction of the wave functions of the two participating atomic orbitals is shown as follows:

Thus we conclude that whenever two atomic orbitals combine, two molecular orbitals are formed. One of them is bonding molecular orbital (ψ²M.O) while the other is antibonding molecular orbital (ψ^*2M.O).

Bonding Molecular Orbital has less energy than the Antibonding Molecular orbital.

The bonding molecular orbital has less energy as compared to the antibonding molecular orbital. In case of bonding molecular orbital, most of the electron density is located between the nuclei of the bonded atoms. As a result, the nuclei are shielded from each other and the repulsion in them is small. On the other hand, in an antibonding molecular orbital, most of the electron density is located away from the space between the nuclei. As a result, the nuclei are less shielded and rather exposed to each other more than in bonding molecular orbitals. We can also say that the electrons in the bonding molecular orbital hold the nuclei together and stabilise the molecule whereas in the antibonding molecular orbital, the effect is the reverse and the electrons tend to destabilise it. The decrease in the energy of the bonding molecular orbital as compared to the atomic orbitals is called stabilising energy while the increase in antibonding molecular orbital is known as destabilising energy.