Explain $H_{2}$ molecule by molecular orbital theory.

(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 $LCAO$ (Linear Combination of Atomic Orbitals) 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 addition or subtraction of the wave functions of individual atomic orbitals as shown below:
$\psi_{MO} = \psi_{A} + \psi_{B} \quad \text{or} \quad \psi^{*}_{MO} = \psi_{A} - \psi_{B}$
- 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,$\psi_{MO} = \sigma(H_{2}) = \psi_{A} + \psi_{B}$.
- 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,$\psi^{*}_{MO}(H_{2}) = \sigma^{*}_{(H_{2})} = \psi_{A} - \psi_{B}$.
- The energy level diagram shows that the bonding orbital has lower energy than the atomic orbitals,while the antibonding orbital has higher energy.