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Molecular Orbital Theory

Molecular Orbital Theory

The Molecular Orbital (MO) Theory, developed by F. Hund and R. S. Mulliken, provides a more advanced and comprehensive description of chemical bonding compared to Valence Bond Theory. MO theory proposes that when atoms combine to form a molecule, their atomic orbitals merge to form new molecular orbitals (MOs) that are spread across the entire molecule. Electrons in these molecular orbitals belong to the molecule as a whole, not to individual atoms.

MO theory successfully explains phenomena that Valence Bond Theory struggles with, such as the paramagnetism of oxygen ($$O_2$$), the existence of fractional bond orders, and the bonding in molecules with delocalized electrons.

Formation Of Molecular Orbitals Linear Combination Of Atomic Orbitals (LCAO)

Molecular orbitals are formed from the combination of atomic orbitals of the constituent atoms. The Linear Combination of Atomic Orbitals (LCAO) method is a mathematical approximation used to describe this combination. According to LCAO, molecular orbitals ($\psi_{MO}$) can be approximated by the linear combination (sum or difference) of atomic orbitals ($\psi_{AO}$):

$$ \psi_{MO} = \psi_{AO1} \pm \psi_{AO2} $$

When two atomic orbitals combine, they form two molecular orbitals:

The total number of molecular orbitals formed is always equal to the total number of atomic orbitals that combined. This principle is analogous to conservation of orbitals.

Conditions For The Combination Of Atomic Orbitals

For effective combination of atomic orbitals to form stable molecular orbitals, certain conditions must be met:

  1. Symmetry: The combining atomic orbitals must have the same or similar symmetry with respect to the internuclear axis. For example, s orbitals can combine with s orbitals, and p orbitals oriented along the internuclear axis (e.g., $$p_z$$ orbitals if the bond is along the z-axis) can combine with each other. An s orbital generally cannot combine effectively with a $$p_x$$ or $$p_y$$ orbital if the bond is along the z-axis due to symmetry mismatch.
  2. Energy Overlap: The combining atomic orbitals must have comparable or similar energies. Atomic orbitals with very different energies combine poorly. For example, the 1s orbital of Hydrogen combines well with the 1s orbital of another Hydrogen atom but combines poorly with a 3d orbital of another atom.
  3. Effective Overlap: The overlapping atomic orbitals must overlap to a significant extent. Orbitals that are further from the nucleus or are very diffuse may not overlap effectively, leading to very weak MOs.

Types Of Molecular Orbitals

Molecular orbitals are classified based on the type of overlap of atomic orbitals:

Energy Level Diagram For Molecular Orbitals

The energy levels of molecular orbitals are represented in an energy level diagram. For diatomic molecules, the order of filling MOs can vary slightly depending on the specific atoms involved. However, a general order for homonuclear diatomic molecules of the second period (like $$Li_2$$ to $$N_2$$) is:

$$ \sigma_{1s} < \sigma^{*}_{1s} < \sigma_{2s} < \sigma^{*}_{2s} < \pi_{2p} < \sigma_{2p} < \pi^{*}_{2p} < \sigma^{*}_{2p} $$

For diatomic molecules like $$O_2$$ and $$F_2$$ (and heteronuclear molecules where atomic orbital energies differ significantly), the order of $\sigma_{2p}$ and $\pi_{2p}$ MOs is reversed:

$$ \sigma_{1s} < \sigma^{*}_{1s} < \sigma_{2s} < \sigma^{*}_{2s} < \sigma_{2p} < \pi_{2p} < \pi^{*}_{2p} < \sigma^{*}_{2p} $$

Filling Molecular Orbitals:

Electronic Configuration And Molecular Behaviour

The electronic configuration of a molecule in terms of its molecular orbitals allows us to predict its stability, bond order, and magnetic properties.

1. Bond Order (BO):

Bond order is a measure of the number of covalent bonds between two atoms. It is calculated as:

$$ \text{Bond Order (BO)} = \frac{1}{2} (\text{Number of electrons in BMOs} - \text{Number of electrons in ABMOs}) $$

2. Magnetic Properties:

3. Stability:

Example: Oxygen Molecule ($$O_2$$)**

  • Electronic configuration of O atom: $1s^2 2s^2 2p^4$.
  • For $$O_2$$, we combine the 1s and 2s, 2p atomic orbitals.
  • MO configuration (using the order for $$O_2$$ and $$F_2$$):
    • $$ (\sigma_{1s})^2 (\sigma^{*}_{1s})^2 (\sigma_{2s})^2 (\sigma^{*}_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^{*}_{2p})^2 $$
  • Number of electrons in BMOs = 2+2+2+4+2 = 12
  • Number of electrons in ABMOs = 2+2+2 = 6
  • Bond Order = $$(12 - 6) / 2 = 6 / 2 = 3$$. Wait, bond order for $$O_2$$ is 2. Let's recheck the MO diagram.

    • Rechecking the MO configuration for $$O_2$$:

    $$ (\sigma_{1s})^2 (\sigma^{*}_{1s})^2 (\sigma_{2s})^2 (\sigma^{*}_{2s})^2 (\sigma_{2p})^2 (\pi_{2p})^4 (\pi^{*}_{2p})^2 $$

    BMOs: $$( \sigma_{1s})^2, (\sigma_{2s})^2, (\sigma_{2p})^2, (\pi_{2p})^4$$ = 2+2+2+4 = 10 electrons

    ABMOs: $$(\sigma^{*}_{1s})^2, (\sigma^{*}_{2s})^2, (\pi^{*}_{2p})^2$$ = 2+2+2 = 6 electrons

    Bond Order = $$(10 - 6) / 2 = 4 / 2 = 2$$. This matches the double bond in $$O_2$$.

    Magnetic Property: The $$(\pi^{*}_{2p})^2$$ orbitals contain two unpaired electrons (one in each degenerate $\pi^{*}_{2p}$ orbital). Therefore, $$O_2$$ is paramagnetic, which was a key experimental observation that VBT could not explain but MO theory successfully predicts.

MO theory provides a powerful framework for understanding the electronic structure, stability, and properties of molecules.