Molecular Orbital Theory

Overview

Molecular Orbital (MO) theory describes bonding as the combination of atomic orbitals to form molecular orbitals that extend over the entire molecule. Unlike Lewis structures, MO theory explains magnetic properties and bond orders.

Key Concepts

Atomic Orbitals Combine to Form Molecular Orbitals

• Bonding orbitals ($\sigma$, $\pi$): Lower energy, electron density between nuclei
• Antibonding orbitals ($\sigma^*$, $\pi^*$): Higher energy, node between nuclei

Number of MOs = Number of AOs Combined

Types of Molecular Orbitals

Sigma ($\sigma$) Orbitals

• Head-on overlap of s or p orbitals
• Electron density along the bond axis
• $\sigma$ (bonding) and $\sigma^*$ (antibonding)

Pi ($\pi$) Orbitals

• Side-by-side overlap of p orbitals
• Electron density above and below bond axis
• $\pi$ (bonding) and $\pi^*$ (antibonding)

MO Diagram for Homonuclear Diatomics

For O₂, F₂ (Z > 7)

Energy
  ↑
       σ*₂p  ___
       
       π*₂p  ___ ___
       
       σ₂p   ___
       
       π₂p   ___ ___
       
       σ*₂s  ___
       
       σ₂s   ___

For Li₂ to N₂ (Z ≤ 7)

Energy
  ↑
       σ*₂p  ___
       
       π*₂p  ___ ___
       
       π₂p   ___ ___
       
       σ₂p   ___
       
       σ*₂s  ___
       
       σ₂s   ___

Note: $\sigma_{2p}$ and $\pi_{2p}$ swap order!

Bond Order

$$\text{Bond Order} = \frac{\text{Bonding electrons} - \text{Antibonding electrons}}{2}$$

Significance

• Bond Order = 0 → No bond (molecule doesn't exist)
• Higher bond order → Stronger, shorter bond
• Fractional bond orders are possible

Examples

H₂ (2 electrons)

σ*₁s  ___
σ₁s   ↑↓

$$\text{Bond Order} = \frac{2 - 0}{2} = 1$$

Single bond, diamagnetic

He₂ (4 electrons)

σ*₁s  ↑↓
σ₁s   ↑↓

$$\text{Bond Order} = \frac{2 - 2}{2} = 0$$

Does not exist as a stable molecule

O₂ (16 electrons)

σ*₂p  ___
π*₂p  ↑   ↑
σ₂p   ↑↓
π₂p   ↑↓  ↑↓
σ*₂s  ↑↓
σ₂s   ↑↓

$$\text{Bond Order} = \frac{10 - 6}{2} = 2$$

Double bond, paramagnetic (2 unpaired electrons)

N₂ (14 electrons)

σ*₂p  ___
π*₂p  ___  ___
π₂p   ↑↓  ↑↓
σ₂p   ↑↓
σ*₂s  ↑↓
σ₂s   ↑↓

$$\text{Bond Order} = \frac{10 - 4}{2} = 3$$

Triple bond, diamagnetic

Magnetic Properties

Paramagnetic

• Has unpaired electrons
• Attracted to magnetic field
• Examples: O₂, B₂

Diamagnetic

• All electrons paired
• Slightly repelled by magnetic field
• Examples: N₂, F₂

Bond Order Comparisons

Species	Electrons	Bond Order	Magnetic
H₂	2	1	Diamagnetic
H₂⁺	1	0.5	Paramagnetic
He₂	4	0	N/A
O₂	16	2	Paramagnetic
O₂⁺	15	2.5	Paramagnetic
O₂⁻	17	1.5	Paramagnetic
O₂²⁻	18	1	Diamagnetic
N₂	14	3	Diamagnetic
F₂	18	1	Diamagnetic

Heteronuclear Diatomics

For molecules like CO, NO:

• More electronegative atom's orbitals are lower in energy
• MOs are not equally shared between atoms
• Similar diagrams but asymmetric

Carbon Monoxide (CO)

• Isoelectronic with N₂ (14 electrons)
• Bond order = 3
• Diamagnetic

Nitric Oxide (NO)

• 15 electrons
• Bond order = 2.5
• Paramagnetic (1 unpaired electron)

Comparison: MO vs VB Theory

Feature	MO Theory	VB Theory
Electrons	Delocalized	Localized
Explains	Magnetic properties	Molecular shape
Bond order	Calculated exactly	Assumed from Lewis
Complexity	More complex	Simpler

Advantages of MO Theory

1. Explains paramagnetism of O₂
2. Predicts bond orders accurately
3. Describes delocalized bonding
4. Explains stability of ions (H₂⁺, He₂⁺)