8.4 Molecular Orbital Theory

Molecular Orbital Theory explains how atoms combine to form molecules. It shows how atomic orbitals merge to create molecular orbitals, which determine a molecule's properties and behavior.

This theory helps us understand chemical bonding, molecular stability, and electron configurations. By studying how electrons are distributed in molecules, we can predict their reactivity, magnetic properties, and overall structure.

Molecular Orbital Theory

Derivation of molecular orbitals

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• Molecular orbitals (MOs) formed by linear combination of atomic orbitals (LCAO) method
  • Atomic orbitals (AOs) with similar energy and symmetry combine to create MOs (s + s, p + p)
  • Constructive interference of wave functions produces bonding MOs ($\sigma_{1s}$, $\sigma_{2s}$, $\pi_{2p}$)
  • Destructive interference of wave functions produces antibonding MOs ($\sigma_{1s}^*$, $\sigma_{2s}^*$, $\pi_{2p}^*$)
• Quantum mechanics calculates wave functions and energies of MOs
  • Schrödinger equation solved for molecular system determines MO properties
  • Wave functions represent probability distribution of electrons in MOs (90% within bonds)

Bonding vs antibonding orbitals

• Bonding MOs stabilize molecules by concentrating electron density between nuclei
  • Formed by constructive interference of AOs (in-phase overlap)
  • Lower energy than constituent AOs (more stable)
  • Examples: $\sigma_{1s}$, $\sigma_{2s}$, $\pi_{2p}$ in H₂, N₂, O₂
• Antibonding MOs destabilize molecules by concentrating electron density away from internuclear region
  • Formed by destructive interference of AOs (out-of-phase overlap)
  • Higher energy than constituent AOs (less stable)
  • Examples: $\sigma_{1s}^*$, $\sigma_{2s}^*$, $\pi_{2p}^*$ in He₂, Ne₂, F₂
• Nodes in MOs indicate regions of zero electron probability (node)

Bond order calculations

• Bond order = $\frac{1}{2}$ (number of bonding electrons - number of antibonding electrons)
  • Bonding electrons occupy bonding MOs ($\sigma_{1s}$, $\sigma_{2s}$, $\pi_{2p}$)
  • Antibonding electrons occupy antibonding MOs ($\sigma_{1s}^*$, $\sigma_{2s}^*$, $\pi_{2p}^*$)
• Higher bond order indicates stronger bond and shorter bond length
  • Bond order of 1 = single bond (H₂, 2 bonding e⁻)
  • Bond order of 2 = double bond (O₂, 8 bonding e⁻ - 4 antibonding e⁻)
  • Bond order of 3 = triple bond (N₂, 8 bonding e⁻ - 2 antibonding e⁻)

Electron Configurations and Molecular Properties

Diatomic molecule configurations

• Determine total valence electrons in molecule (H₂ = 2, N₂ = 10, O₂ = 12)
• Fill MOs in order of increasing energy following quantum principles
  1. Aufbau principle: electrons occupy lowest energy MOs first ($\sigma_{1s}$ < $\sigma_{1s}^*$ < $\sigma_{2s}$ < $\sigma_{2s}^*$ < $\pi_{2p}$ < $\pi_{2p}^*$ < $\sigma_{2p}$)
  2. Pauli exclusion principle: each MO holds maximum of 2 electrons with opposite spins ($\uparrow\downarrow$)
  3. Hund's rule: degenerate MOs ($\pi_{2p}$) singly occupied before pairing
• Represent configuration using MO diagrams or notation
  • H₂: $\sigma_{1s}^2$
  • N₂: $\sigma_{1s}^2 \sigma_{1s}^{*2} \sigma_{2s}^2 \sigma_{2s}^{*2} \pi_{2p}^4 \sigma_{2p}^2$
  • O₂: $\sigma_{1s}^2 \sigma_{1s}^{*2} \sigma_{2s}^2 \sigma_{2s}^{*2} \pi_{2p}^4 \pi_{2p}^{*2} \sigma_{2p}^2$

Molecular stability predictions

• Molecules with higher bond orders are more stable (N₂ > O₂ > F₂)
• Fully filled bonding MOs and empty antibonding MOs maximize stability (N₂, CO)
• Paramagnetic molecules have unpaired electrons and are attracted to magnetic fields
  • O₂: 2 unpaired electrons in $\pi_{2p}^*$ ($\uparrow\downarrow\uparrow\downarrow\uparrow\uparrow$)
  • NO: 1 unpaired electron in $\pi_{2p}^*$ ($\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow$)
• Diamagnetic molecules have no unpaired electrons and are repelled by magnetic fields
  • N₂: all electrons paired ($\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow$)
  • CO: all electrons paired ($\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow$)

Molecular Symmetry and Orbital Interactions

• Orbital symmetry determines allowed interactions between MOs
• Hybridization of atomic orbitals affects molecular geometry and bonding
• Electron delocalization occurs in molecules with extended π-systems, stabilizing the structure