MO Diagrams (Diatomic)
Molecular Orbital Theory (MOT) provides a quantum-mechanical description of chemical bonding by combining atomic orbitals (AOs) mathematically to form molecular orbitals (MOs) that belong to the entire molecule, not to individual atoms. Unlike valence bond theory, MOT correctly predicts properties such as the paramagnetism of O₂ and the bond orders of species like He₂⁺. The key principle is that N atomic orbitals always combine to produce exactly N molecular orbitals, each described by a wave function called a molecular orbital wave function. For homonuclear diatomic molecules (both atoms identical), the MO diagram is constructed by placing the valence atomic orbitals of the two atoms on either side and drawing the resulting bonding and antibonding MOs in the centre. Bonding MOs (σ, π) are lower in energy than the original AOs because constructive interference increases electron density between the nuclei. Antibonding MOs (σ, π) are higher in energy due to destructive interference, which creates a node between nuclei. The bond order — a measure of net bonding — is calculated as ½(bonding electrons − antibonding electrons). A critical subtlety arises for elements with atomic number Z < 8 (Li, Be, B, C, N): the 2s and 2p atomic orbitals are close enough in energy that they mix, causing the σ(2p) MO to be pushed above the π(2p) MOs. This s–p mixing inverts the usual σ/π ordering and is essential for correctly drawing MO diagrams for N₂, B₂, and C₂. For O₂ and F₂ (Z ≥ 8), the 2s–2p energy gap is large enough that mixing is negligible and σ(2p) sits below π(2p).
- Count valence atomic orbitals
- Apply s–p mixing rule
- Fill electrons using Aufbau, Pauli, Hund
- Calculate bond order
- Predict magnetic properties
Count valence atomic orbitals
For second-row diatomics, each atom contributes one 2s and three 2p orbitals (2pₓ, 2pᵧ, 2p_z). That gives 4 AOs per atom → 8 MOs total per molecule: σ(2s), σ(2s), σ(2p), π(2p), π(2p), π(2p), π(2p), σ(2p). The two degenerate π orbitals arise from the two p orbitals perpendicular to the bond axis.
Apply s–p mixing rule
For Z < 8 (B₂, C₂, N₂), s–p mixing raises the σ(2p) above both π(2p) orbitals. The correct low-to-high order is: σ(2s) < σ(2s) < σ(2p) < π(2p) = π(2p) < [gap] for N₂. Wait — with s–p mixing the order becomes: σ(2s) < σ(2s) < π(2p) = π(2p) < σ(2p) < π(2p) = π(2p) < σ(2p). For Z ≥ 8 (O₂, F₂), σ(2p) drops below the π(2p) pair: σ(2s) < σ(2s) < σ(2p) < π(2p) = π(2p) < π(2p) = π(2p) < σ*(2p).
Fill electrons using Aufbau, Pauli, Hund
Place all valence electrons into MOs from lowest to highest energy. Apply the Pauli exclusion principle (max 2e⁻ per orbital, opposite spins) and Hund's rule (half-fill degenerate orbitals before pairing). N₂ has 10 valence electrons; O₂ has 12; F₂ has 14. For O₂, the last two electrons half-fill the two degenerate π*(2p) orbitals — one electron each, parallel spins — explaining its paramagnetism, a triumph of MOT over Lewis structures.
Calculate bond order
Bond Order (BO) = ½(Nᵦ − Nₐ), where Nᵦ = electrons in bonding MOs and Nₐ = electrons in antibonding MOs. For N₂: BO = ½(8−2) = 3 (triple bond, consistent with Lewis structure). For O₂: BO = ½(8−4) = 2 (double bond). For F₂: BO = ½(8−6) = 1 (single bond). Higher bond order means shorter bond length and larger bond dissociation energy.
Predict magnetic properties
If all MOs are fully paired, the molecule is diamagnetic (repelled weakly by magnets). If any MO contains unpaired electrons, the molecule is paramagnetic (attracted by magnets). N₂ and F₂ are diamagnetic; O₂ is paramagnetic with two unpaired electrons in its degenerate π*(2p) MOs. This paramagnetism of liquid O₂ can be demonstrated by suspending it between magnet poles.
The Key Equation
- N_b — Bonding electrons (dimensionless): Total number of electrons occupying bonding molecular orbitals (σ and π).
- N_a — Antibonding electrons (dimensionless): Total number of electrons occupying antibonding molecular orbitals (σ and π).
- \text{Bond Order} — Bond order (dimensionless): Net number of bonds; 1 = single, 2 = double, 3 = triple. Higher values correlate with shorter, stronger bonds.
Worked Example
Problem. Draw the MO electron configuration of O₂, calculate its bond order, and state whether it is paramagnetic or diamagnetic.
Answer. Bond order = 2; O₂ is paramagnetic (2 unpaired electrons).
Origins · 1 minRobert S. Mulliken and how this was discovered
Robert Mulliken developed Molecular Orbital Theory in the 1920s–1930s, introducing the concept of molecular orbitals formed by the linear combination of atomic orbitals (LCAO-MO method). He coined the term 'molecular orbital' and created the framework used today to draw MO diagrams for diatomic and polyatomic molecules. His work earned him the Nobel Prize in Chemistry in 1966. Mulliken was known for his extremely dense, equation-heavy papers — his colleagues joked that his writing was 'Mulliken-ese,' almost a language of its own!
Related Concepts
- LCAO Approximation — MO diagrams are built using the Linear Combination of Atomic Orbitals method; understanding LCAO explains why bonding and antibonding MOs form.
- Bond Order and Bond Properties — Bond order derived from MO diagrams directly predicts bond length, bond energy, and vibrational frequency.
- Paramagnetism and Diamagnetism — MO theory uniquely explains the paramagnetism of O₂ through half-filled degenerate π* orbitals, a failure of Lewis structures.
- Valence Bond Theory — Comparing VBT and MOT highlights the strengths and limitations of each model, especially for O₂ and other open-shell molecules.
- Atomic Orbital Shapes (s, p, d) — MO formation depends on the symmetry and shape of the combining AOs; σ MOs form from head-on overlap, π MOs from lateral overlap of p orbitals.