Quantum Mechanical Treatment of Molecular Orbital Theory

Molecular orbital (M.O.) or Hund-Mulliken (H.M.) Theory

Molecules have quantised energy levels corresponding to molecular orbitals which represent the region along which there is maximum probability of finding electrons. During molecule formation, atomic orbitals of same symmetry and same or similar energy combine to form molecular orbitals. The single electron of H₂⁺ could be over the H-atom - A ($\phi_1$) or over the H-atom - B ($\phi_2$). The real wave function is the linear combination of two probable wave functions:

$$\psi_{mo} = C_1 \phi_1 + C_2 \phi_2$$

Co-efficients $C_1$ & $C_2$ should be such that they should result in the lowest energy of molecular orbital wave function i.e.

$$\frac{\partial E}{\partial C_1} = 0 \quad \text{and} \quad \frac{\partial E}{\partial C_2} = 0 \quad \text{where} \quad E = \frac{\int \psi H \psi^* d\tau}{\int \psi \psi^* d\tau}$$

Putting the value of $\psi_{mo}$ and differentiating with respect to $C_1$ & $C_2$, two energy states are obtained—

$$E_b = \frac{H_{11} + H_{12}}{S_{11} + S_{12}} \quad \text{and} \quad E_a = \frac{H_{11} - H_{12}}{S_{11} - S_{12}}$$

Thus two atomic orbitals combine to give one molecular orbital $\psi_b$ of lower energy and the other of higher energy than the atomic orbitals. The composition of molecular wave functions are—

$$\psi_b = C_1 (\phi_1 + \phi_2) \quad \& \quad \psi_a = C_1 (\phi_1 - \phi_2)$$

On normalisation, normalising factors are—

$$N_b = \frac{1}{\sqrt{2(1 + S_{12})}} \quad \& \quad N_a = \frac{1}{\sqrt{2(1 - S_{12})}}$$

Since $S_{12} < 1$, hence it is neglected giving following molecular wave functions:

$$\psi_b = \frac{1}{\sqrt{2}} (\phi_1 + \phi_2); \quad \psi_a = \frac{1}{\sqrt{2}} (\phi_1 - \phi_2)$$

$$\therefore \quad \int \psi_b^2 d\tau = \frac{1}{2} \left[ \int \phi_1^2 d\tau + 2\int \phi_1 \phi_2 d\tau + \int \phi_2^2 d\tau \right]$$

$$\text{and} \quad \int \psi_a^2 d\tau = \frac{1}{2} \left[ \int \phi_1^2 d\tau - 2\int \phi_1 \phi_2 d\tau + \int \phi_2^2 d\tau \right]$$

Thus in $\psi_b$, the probability of electron being found between nuclei is greater than at the extreme ends. This is formed by combination of atomic orbitals in phase and give lower energy state. In case of $\psi_a$, the probability of electrons being found between nuclei is less than the extreme ends. This is formed by atomic orbitals in opposite phases and give higher energy state. After the formation of molecular orbitals, there is rearrangement of electrons in them. Electrons going to the lower energy $\psi_b$ liberate energy which is translated into the forces of attraction between atoms and causes bonding of atoms. This is called bonding molecular orbitals. Electrons going to $\psi_a$ are raised to higher energy. They destabilise the molecule and results in repulsion between atoms. Thus they counteract the bonding force and such molecular orbitals are called antibonding. The separation of bonding and antibonding molecular orbitals depends upon overlap integral ($S_{12}$). They should be equally separated from atomic orbital energy ($E$). However it can be seen that $E_a - E > E - E_b$. In cases where atomic orbitals do not undergo overlap or overlap is orthogonal, they remain as such in molecule. They have neither bonding nor antibonding effect and are called non-bonding molecular orbitals.

Related Topics
Molecular Orbital Theory
Principles of Linear Combination
Quantum Mechanical Treatment of Valence Bond Theory
Quantum Mechanical Treatment of a Rigid Rotor