Explain the linear combination of atomic orbitals $(LCAO)$ with a suitable example.

(N/A) $LCAO$ Method of molecular orbitals: Molecular orbitals are not obtained directly by the $Schrodinger$ wave equation,but they can be obtained by the $LCAO$ method.
$LCAO$ method for Hydrogen Molecule $(H_2)$:
- Hydrogen is a homonuclear diatomic molecule. Consider the hydrogen molecule $(H_2)$ consisting of two atoms $H_A$ and $H_B$.
- Mathematically,the formation of molecular orbitals may be described by the linear combination of atomic orbitals,which can take place by the addition and subtraction of wave functions of individual atomic orbitals as shown below:
$\psi_{MO} = \psi_A + \psi_B$ (Bonding)
$\psi^*_{MO} = \psi_A - \psi_B$ (Antibonding)
Bonding molecular orbital $(\psi_{MO})$ e.g.,$\sigma$: The molecular orbital $\sigma$ formed by the addition of atomic orbitals is called the bonding molecular orbital. Here,for $\sigma$ type molecular orbital,$\psi_{MO} = \sigma(H_2) = \psi_A + \psi_B$.
Antibonding molecular orbital $(\psi^*_{MO})$ e.g.,$\sigma^*$: The molecular orbital $\sigma^*$ formed by the subtraction of atomic orbitals $(\psi_A$ and $\psi_B)$ is called the antibonding molecular orbital. Here,for $\sigma^*$ type antibonding molecular orbital,$\psi^*_{MO}(H_2) = \sigma^*(H_2) = \psi_A - \psi_B$.