Bonding
Molecular Orbital Theory
LCAO — atomic orbitals combine into bonding (lower E) and antibonding (higher E) molecular orbitals; explains O₂ paramagnetism
Molecular orbital (MO) theory describes chemical bonding by combining atomic orbitals from each atom into delocalized molecular orbitals that span the whole molecule. The dominant approximation, linear combination of atomic orbitals (LCAO), takes ψ_MO = c1·φ1 + c2·φ2 + … and solves a Hartree-Fock or Kohn-Sham eigenvalue problem to find orbital energies and coefficients. Two atomic orbitals always combine into one bonding (lower energy) and one antibonding (higher energy) MO. Bond order = (n_bonding − n_antibonding)/2 predicts molecular stability and magnetism. The theory was developed by Friedrich Hund and Robert Mulliken from 1928 to 1932; Mulliken won the 1966 Chemistry Nobel for it. The triumph: predicting O₂'s paramagnetism (two unpaired electrons in degenerate π* orbitals) — a fact valence-bond theory could not explain.
- Bond orderBO = (n_bonding − n_antibonding)/2
- O₂BO 2; 2 unpaired e⁻ in π*; paramagnetic
- N₂BO 3; all paired; diamagnetic
- FoundersHund & Mulliken, 1928–1932
- NobelMulliken, Chemistry 1966
- ApproximationLCAO; Hartree-Fock, then DFT
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Why MO theory matters
- Predicts magnetism correctly. O₂ has two unpaired electrons in π* and is paramagnetic; NO, NO₂, ClO₂ are odd-electron radicals with one unpaired e⁻; B₂ has two unpaired in π. Each is exactly what MO predicts and exactly what valence-bond theory using paired Lewis structures gets wrong.
- Foundation of computational chemistry. Every Gaussian, ORCA, or Q-Chem calculation outputs MOs. Hartree-Fock (1930s) and DFT (since Kohn-Sham 1965, 1998 Nobel) both express electrons in molecular orbitals; geometry optimization, IR/Raman frequencies, and electronic spectra all flow from the MO eigenvectors and eigenvalues.
- Frontier orbital reactivity (HOMO/LUMO). Fukui's frontier-orbital theory and the Woodward-Hoffmann rules (1965, 1981 Nobel to Hoffmann and Fukui) explain pericyclic reactions — Diels-Alder, electrocyclizations, sigmatropic shifts — entirely from HOMO/LUMO symmetry. The 4n+2 thermal/4n photochemical rules require an MO picture.
- UV-Vis spectra map onto MO transitions. π→π* transitions in conjugated π systems, n→π* in carbonyls, d-d in transition-metal complexes, and charge-transfer bands all correspond to specific HOMO→LUMO promotions visible in MO diagrams.
- Aromaticity and antiaromaticity. Hückel's 4n+2 rule for aromatic stability comes directly from filling the closed-shell bonding π MOs of a cyclic conjugated system. Antiaromatic 4n systems (cyclobutadiene) have an open-shell ground state and are kinetically unstable for purely MO-symmetry reasons.
- Spectroscopic predictions. Photoelectron spectroscopy (Koopmans' theorem: IE_n ≈ −ε_n where ε_n is the orbital energy) directly measures occupied MO energies. Modern XPS and UPS calibration of metals, semiconductors, and proteins all rely on this MO interpretation.
- Generalizes to solids. Tight-binding (Bloch's theorem with LCAO basis) extends MO theory to crystals, where MOs become bands. Band structure of graphene (Dirac cones), silicon (indirect gap), and metals (partially filled bands) is the same LCAO mathematics, just with k-space periodicity.
Common misconceptions
- Antibonding orbitals are 'empty' or 'unreal'. Antibonding MOs are real orthogonal eigenfunctions of the Fock operator; they exist whether or not they are occupied. Excited states populate them. He₂⁺ has BO = 0.5 because removing one of the four electrons leaves σ(1s)² σ*(1s)¹ — antibonding occupancy of one electron is non-zero.
- The bond order must be an integer. Half-integer bond orders are common in radicals: O₂⁻ (superoxide) BO = 1.5, O₂²⁻ (peroxide) BO = 1, NO BO = 2.5, He₂⁺ BO = 0.5. Bond lengths and dissociation energies scale predictably with non-integer BO.
- π MOs are weaker than σ. Per overlap, yes — but in N₂ the two π bonds together contribute about half the total bond energy of the triple bond, and ethylene's π bond at ~270 kJ/mol is plenty strong on its own. The right comparison is total bonding energy contribution, not single-orbital strength.
- MO theory abandons hybridization. Strictly, MO theory does not require hybrids — it can use pure s, p, d basis functions. But chemists often pre-mix atomic orbitals into sp³, sp², sp hybrids before LCAO to better match the local geometry. Hybridization and MO theory coexist; Walsh diagrams interpolate between them as a function of geometry.
- One MO = one bond. Bonds in MO theory are not pinned to atom pairs. The single 1a₁ MO of methane is delocalized over all five atoms; localizing to four C-H bonds requires a unitary transformation (Boys, Edmiston-Ruedenberg, Pipek-Mezey). Both pictures are valid; localized orbitals match Lewis intuition, canonical MOs match spectroscopy.
- Heteronuclear MOs split symmetrically. When two AOs have different energies (e.g., H 1s at −13.6 eV and F 2p at −18.7 eV in HF), the bonding MO is mostly the lower-energy AO (F 2p in HF), the antibonding is mostly the higher (H 1s). Polarization of the bond toward F follows from the LCAO coefficients, not from any added physics.
Constructing an MO diagram for a homonuclear diatomic
Consider O₂. Each oxygen contributes a 1s, 2s, and three 2p orbitals; we focus on valence (2s, 2p), 8 AOs total. By symmetry, only AOs of the same symmetry combine. The 2s_A and 2s_B make σ(2s) bonding and σ*(2s) antibonding. The 2p_z (along the bond axis) make σ(2p_z) bonding and σ*(2p_z) antibonding. The 2p_x and 2p_y (perpendicular) make degenerate π(2p_x), π(2p_y) bonding and degenerate π*(2p_x), π*(2p_y) antibonding pairs. Energy ordering for O₂ and F₂ is σ(2s) < σ*(2s) < σ(2p_z) < π(2p_x,y) < π*(2p_x,y) < σ*(2p_z); for B₂, C₂, N₂ the σ(2p_z) and π(2p_x,y) order swaps because of s-p mixing — the 2s and 2p_z orbitals interact when their energy gap is small, pushing σ(2p_z) above the π pair.
Fill 12 valence electrons (6 per oxygen) bottom up: σ(2s)² σ*(2s)² σ(2p_z)² π(2p_x)² π(2p_y)² π*(2p_x)¹ π*(2p_y)¹. The last two electrons go to the degenerate π* pair with parallel spins by Hund's rule, giving total spin S = 1 — paramagnetic. Bond order = (8 bonding − 4 antibonding)/2 = 2. The full MO diagram explains both the experimental bond length (1.21 Å), bond energy (498 kJ/mol), and magnetic susceptibility consistently.
Going from O₂ to O₂⁻ (superoxide) adds one electron to the π* pair — the third electron pairs with one of the existing unpaired electrons. BO drops from 2.0 to 1.5; bond length stretches to 1.33 Å; only one unpaired electron remains. Going to O₂²⁻ (peroxide) adds two electrons, fully filling π* and giving BO = 1, length 1.49 Å, diamagnetic. The progression O₂⁺ (BO 2.5, 1.12 Å) → O₂ (BO 2, 1.21 Å) → O₂⁻ (BO 1.5, 1.33 Å) → O₂²⁻ (BO 1, 1.49 Å) is the classic MO-theory exercise; bond length increases monotonically as antibonding orbitals fill, and the smoothness of the trend is empirical confirmation of the MO picture.
MO vs valence-bond theory — six dimensions of comparison
| Property | Molecular Orbital (MO) | Valence Bond (VB) |
|---|---|---|
| Orbital description | Delocalized over whole molecule from the start | Localized between pairs of atoms; possibly hybridized |
| Bonding picture | Bonding/antibonding pairs from each AO combination | Two singly-occupied AOs overlap to share an electron pair |
| Hybridization vs MO mixing | Optional pre-step before LCAO | Central concept (sp³, sp², sp), explains tetrahedral methane |
| Paramagnetism prediction | Correct for O₂ (2 unpaired in π*); for B₂; for radicals | Wrong for O₂ if drawn as O=O with paired e⁻; needs 'three-electron bonds' fix |
| O₂ / N₂ / F₂ bond orders | 2 / 3 / 1 directly from MO filling and bond-order formula | 2 / 3 / 1 from Lewis structures, but no electronic structure detail |
| Treatment of resonance | Delocalization is built in; no resonance structures needed | Sum of weighted Lewis-style resonance structures (Pauling 1928) |
| UV-Vis / spectroscopy | Direct from HOMO/LUMO and orbital energies | Indirect; needs configuration interaction |
| Computational use | Universal in HF, DFT, post-HF | VB methods (BOVB, GVB) less common; mostly didactic |
Famous MO-theory results
- O₂ paramagnetism — the founding triumph. Faraday placed liquid O₂ in his magnetic series in 1846; Curie measured its susceptibility in 1895. Lewis's 1916 cubical-atom theory and Pauling's 1928 valence-bond resonance both predicted diamagnetic O₂. Hund (1928) and Mulliken (1932) showed in MO terms that the two highest electrons go to degenerate π* with parallel spins, predicting paramagnetism — the result agrees with experiment to four decimal places in the magnetic moment. The liquid-O₂-on-a-magnet demonstration is now a fixture of general-chemistry classrooms.
- Hückel benzene calculation, 1931. Erich Hückel's 1931 paper computed benzene's π MOs analytically. Energies α + 2β, α + β (×2), α − β (×2), α − 2β; six π electrons fill the three bonding orbitals at total energy 6α + 8β, lower than three isolated ethylene π bonds (6α + 6β) by 2β ≈ 150 kJ/mol — the resonance/aromatic stabilization energy. Hückel's 4n+2 rule for aromaticity was the first quantitative prediction from MO theory.
- Walsh diagrams, A.D. Walsh 1953. Plotting MO energies as a function of bond angle for triatomic molecules predicted shapes from electron count alone. AH₂ molecules with 4 valence electrons (BeH₂) are linear; with 5–8 (CH₂, NH₂⁻) are bent. Diagram reading replaces 30 minutes of full computation with a 30-second symmetry argument, still taught in inorganic-chemistry courses.
- Woodward-Hoffmann rules, 1965. R.B. Woodward and Roald Hoffmann showed that pericyclic reactions (Diels-Alder, electrocyclic ring closures, sigmatropic shifts) proceed thermally only when the symmetry of the HOMO is correct for orbital overlap. The 4n+2 rule for thermal disrotatory closure of conjugated polyenes vs 4n for conrotatory was a central result. Hoffmann shared the 1981 Chemistry Nobel with Kenichi Fukui (frontier orbital theory).
- Density functional theory and the Kohn-Sham equations, 1965. Walter Kohn and Lu Jeu Sham reformulated MO theory in terms of electron density rather than wavefunction, but kept Kohn-Sham orbitals as the workhorse representation. DFT now dominates computational chemistry — >90% of all molecular calculations published since 2000 use Kohn-Sham orbitals. Kohn shared the 1998 Chemistry Nobel.
Frequently asked questions
What does LCAO actually mean?
LCAO stands for Linear Combination of Atomic Orbitals. The MO wavefunction is approximated as ψ_MO = Σ c_i φ_i, a weighted sum of atomic orbitals φ_i centered on each nucleus. For H₂ this is ψ_bonding = (φ_1s,A + φ_1s,B)/√(2 + 2S) and ψ_antibonding = (φ_1s,A − φ_1s,B)/√(2 − 2S), where S is the overlap integral. The bonding MO has electron density piled up between the nuclei; the antibonding has a node between them. Coefficients c_i are determined variationally — chosen to minimize the total energy — by solving the Roothaan equations from Hartree-Fock theory. LCAO is a basis-set choice; the truth is in principle exact, but in practice the basis is finite (e.g., 6-31G*, def2-TZVP), which limits accuracy.
Why is O₂ paramagnetic and what does MO theory say?
Liquid O₂ is famously attracted to a magnet, and in 1846 Faraday placed it in his magnetic series — but the explanation came only with MO theory. O₂ has 16 electrons. Filling the MO diagram σ(2s)², σ*(2s)², σ(2p_z)², π(2p_x,y)⁴, π*(2p_x,y)² — the last two electrons go into degenerate π* orbitals, and by Hund's rule occupy them with parallel spins. Result: two unpaired electrons, total spin S = 1, paramagnetic. Bond order = (8 − 4)/2 = 2, so O₂ is double-bonded. Valence-bond theory using a Lewis structure O=O with all electrons paired predicted diamagnetism — a clear failure that MO theory resolved cleanly. The liquid-O₂ levitating-in-a-magnet demo is a standard general-chemistry visualization.
How is bond order calculated and what does it predict?
Bond order BO = (n_bonding − n_antibonding)/2. For homonuclear diatomics: H₂ BO = (2 − 0)/2 = 1, He₂ BO = (2 − 2)/2 = 0 (no bond, He is monatomic), Li₂ BO = 1, Be₂ BO = 0, B₂ BO = 1 (with two unpaired e⁻ in π), C₂ BO = 2 (no σ bond from p_z, both bonds π), N₂ BO = 3 (triple bond, all paired), O₂ BO = 2 (with two unpaired e⁻ in π*), F₂ BO = 1, Ne₂ BO = 0. Bond order correlates with both bond length (higher BO → shorter) and bond dissociation energy (higher BO → stronger). N₂ at BO 3, length 1.10 Å, BDE 945 kJ/mol; O₂ at BO 2, length 1.21 Å, BDE 498 kJ/mol; F₂ at BO 1, length 1.42 Å, BDE 158 kJ/mol.
How does MO theory differ from valence bond theory?
VB theory (Heitler-London 1927; Pauling 1931 development of resonance and hybridization) keeps electrons in localized atomic or hybrid orbitals on each atom and forms bonds by overlap of two singly-occupied orbitals to form a paired-spin σ or π bond. Resonance superposes Lewis structures. MO theory (Hund-Mulliken) makes orbitals delocalized over the whole molecule from the start. For H₂, both reach essentially identical energies after configuration interaction; they are equivalent in the full-CI limit. Differences: MO naturally handles odd-electron species (NO, O₂, NO₂) and predicts ionization energies and electronic spectra directly from orbital energies (Koopmans' theorem). VB more closely matches chemical intuition (a bond between A and B is a localized object) and is preferred for qualitative reactivity arguments and arrow pushing. Modern computational chemistry uses MO/Hartree-Fock and DFT exclusively, while teaching uses VB for introduction and MO for diatomics and conjugated systems.
What does Hückel theory simplify?
Erich Hückel (1931) reduced MO theory for planar conjugated π systems to a 2x2 matrix problem per pair of carbons by treating only the p_z orbitals perpendicular to the molecular plane and lumping all interactions into two parameters: α (Coulomb integral, energy of an electron in an isolated p_z) and β (resonance integral, energy of overlap between adjacent p_z, ~ −75 kJ/mol). The Hückel secular determinant gives orbital energies and coefficients for benzene, naphthalene, polyenes, and any aromatic. Benzene comes out with three bonding π MOs at α + 2β, α + β, α + β and three antibonding at α − β, α − β, α − 2β; six π electrons fill all three bonding orbitals — an exceptionally stable closed shell, which is the modern definition of aromaticity (4n + 2 rule).
Why do we need both σ and π MOs?
Atomic orbitals overlap differently along versus perpendicular to the internuclear axis. Head-on overlap (s-s, s-p_z, p_z-p_z when z is the bond axis) makes σ orbitals — cylindrically symmetric about the axis. Side-on overlap (p_x-p_x, p_y-p_y) makes π orbitals — with a node containing the axis. σ MOs concentrate density between the nuclei and are usually the strongest bond. π MOs are weaker per overlap but allow multiple bonds: in N₂, one σ + two π = triple bond. d orbitals introduce δ overlap (four lobes around the axis). Symmetry of the AO determines the MO label. The HOMO/LUMO ordering and frontier-orbital reactivity (Woodward-Hoffmann rules, 1965) are inherently molecular-orbital concepts impossible to derive from purely localized VB theory.