Molecular Orbital Theory

When two atomic orbitals combine in-phase (same sign of the wavefunction), constructive interference occurs and electron density builds up between the two nuclei, producing a bonding MO that is lower in energy than the original atomic orbitals. Out-of-phase combination (opposite signs) leads to destructive interference, creating a node between the nuclei — this is an antibonding MO, higher in energy than the original AOs. The number of molecular orbitals always equals the number of atomic orbitals combined.

Bond order = (bonding e⁻ − antibonding e⁻) / 2. When BO = 0, the stabilizing effect of bonding electrons is exactly cancelled by the destabilizing effect of antibonding electrons (in fact, antibonding MOs are slightly more destabilizing than bonding MOs are stabilizing). The molecule has no net bond and does not exist as a stable species. A classic example is He₂: (σ_1s)²(σ*_1s)², giving BO = (2 − 2)/2 = 0.

In head-on (σ) overlap, the lobes of two p orbitals point directly at each other along the internuclear axis. This concentrates electron density between the nuclei and maximizes the overlap integral S, producing a large stabilization energy. In lateral (π) overlap, the p lobes are parallel and overlap above and below the axis — the overlap region is more diffuse and less effective. As a result, π bonds are individually weaker than σ bonds, though a triple bond (one σ + two π) is collectively very strong.

O₂ has 16 electrons. After filling the bonding MOs, two electrons remain to occupy the two degenerate π*_2p antibonding orbitals. By Hund's rule, these two electrons enter the two π* orbitals with parallel spins rather than pairing in one orbital. This gives O₂ two unpaired electrons, making it paramagnetic — a fact that Lewis structures (which predict all electrons paired) fail to explain. The bond order is (8 − 4)/2 = 2, consistent with a double bond.

Photoelectron spectroscopy ionizes electrons from occupied MOs. Each peak in the spectrum corresponds to a distinct MO energy level — the ionization energy equals the negative of the orbital energy (Koopmans' theorem). For N₂, the PES shows that the σ_2p MO is higher in energy than the π_2p MOs, confirming the effect of s-p mixing. The vibrational fine structure of each peak also reveals whether the electron was in a bonding (many vibrational states, changed bond length upon ionization) or weakly bonding/non-bonding orbital. This direct experimental evidence validates the MO energy ordering predicted by theory.

s-p mixing arises because the σ_2s and σ_2p MOs have the same symmetry (σ_g). When the 2s and 2p atomic orbital energies are close together (as in Li through N), these σ MOs interact and repel each other in energy: the σ_2s is pushed down and the σ_2p is pushed up. This reverses the expected ordering so that σ_2p lies above π_2p. In O₂ and F₂, the 2s–2p gap is large enough that s-p mixing is negligible, and the "normal" ordering (σ_2p below π_2p) is restored.

For N₂ (10 valence electrons): (σ_2s)²(σ*_2s)²(π_2p)⁴(σ_2p)². Bonding = 8, antibonding = 2, so BO = (8 − 2)/2 = 3. For N₂⁺ (9 valence electrons), one electron is removed from the highest occupied MO (σ_2p): bonding = 7, antibonding = 2, so BO = (7 − 2)/2 = 2.5. Higher bond order correlates with shorter, stronger bonds. N₂ bond length is 109.8 pm; N₂⁺ is 111.6 pm — confirming that the triple bond in N₂ is shorter.

He₂ has 4 electrons: (σ_1s)²(σ*_1s)². BO = (2 − 2)/2 = 0 — the antibonding electrons completely cancel the bonding, and in fact antibonding MOs are slightly more destabilizing than bonding MOs are stabilizing, so the molecule is energetically unfavorable. He₂⁺ has 3 electrons: (σ_1s)²(σ*_1s)¹. BO = (2 − 1)/2 = 0.5 — there is a net bonding interaction, producing a weak but detectable bond. He₂⁺ has been observed with a bond dissociation energy of about 230 kJ/mol.

In heteronuclear diatomics, the bonding MO is weighted toward the more electronegative atom. Fluorine (χ = 4.0) is far more electronegative than hydrogen (χ = 2.1), so the F 2p orbitals lie at lower energy. The bonding MO, being lower in energy, has a larger coefficient on F — meaning the bonding electron density is polarized toward fluorine. Conversely, the antibonding MO is weighted toward H. This asymmetry is the MO explanation for the polar nature of the H–F bond (partial negative charge on F).

Although CO and N₂ are isoelectronic, CO is heteronuclear: the oxygen AOs are lower in energy than the carbon AOs because oxygen is more electronegative. This makes the MO diagram asymmetric — unlike the symmetric N₂ diagram. The bonding MOs have larger coefficients on O (polarized toward the more electronegative atom), while the antibonding MOs and the HOMO lone pair have larger coefficients on C. Despite this asymmetry, CO retains a bond order of 3 and is in fact the strongest diatomic bond known (1076 kJ/mol), even stronger than N₂ (945 kJ/mol).

The HOMO of CO (sometimes labeled 3σ or 5σ) is a lone pair that is largely localized on carbon. This may seem counterintuitive since oxygen is more electronegative, but the HOMO has antibonding character with respect to the 2s interaction, and antibonding MOs are weighted toward the less electronegative atom. This carbon-centered lone pair is the reason CO acts as a strong σ-donor ligand in coordination chemistry: it donates the C lone pair into empty metal d orbitals. CO always binds to metals M–C≡O, never M–O≡C (except in rare isocarbonyl cases).

NO has 15 electrons — an odd number, guaranteeing at least one unpaired electron. The electronic configuration places 8 electrons in bonding MOs and 3 in antibonding MOs (the extra electron relative to N₂ enters a π* orbital). Bond order = (8 − 3)/2 = 2.5. The single unpaired electron in π* makes NO a radical (a species with an unpaired electron). This radical character underlies NO's biological importance as a signaling molecule. Removing the antibonding electron to form NO⁺ increases the bond order to 3 and makes the ion isoelectronic with N₂ and CO.

Frontier molecular orbital (FMO) theory, developed by Kenichi Fukui (Nobel Prize 1981), identifies the HOMO and LUMO as the key orbitals controlling reactivity. A nucleophile donates electrons from its HOMO into the LUMO of the electrophile. The energy gap between HOMO and LUMO determines the kinetic accessibility of a reaction — a smaller gap generally means easier reaction. In coordination chemistry, frontier orbitals explain ligand binding: a ligand's HOMO donates into a metal's LUMO (σ donation), and the metal's filled d orbitals (acting as HOMO) can donate into the ligand's LUMO (π back-bonding).

CO binds to metals through a synergistic two-way interaction. First, the carbon lone pair (the HOMO of CO, localized on C) donates into an empty metal d orbital — this is σ donation, which increases electron density on the metal. Second, filled metal d orbitals with appropriate symmetry donate electron density back into the empty π* antibonding orbitals of CO — this is π back-bonding (or back-donation). The two processes reinforce each other: σ donation increases electron density on the metal, which enhances π back-bonding, which in turn makes CO a better σ donor. A consequence of π back-bonding is that electron density enters CO's antibonding orbitals, weakening and lengthening the C≡O bond (the CO stretching frequency in IR decreases from 2143 cm⁻¹ in free CO).

ICl is a heteronuclear diatomic between two halogens with different electronegativities. Chlorine (χ = 3.2) is more electronegative than iodine (χ = 2.7), so the Cl atomic orbitals lie at lower energy. In the MO diagram, the bonding MOs have larger coefficients on Cl, meaning bonding electron density is polarized toward chlorine. This gives Cl a partial negative charge (δ−) and I a partial positive charge (δ+). The bond is polar covalent — not purely ionic (the electronegativity difference of 0.5 is moderate) and not purely covalent (the atoms are not identical). The MO approach naturally explains the polarity: no additional assumptions about "percent ionic character" are needed, as the unequal orbital coefficients directly produce the charge asymmetry.