Historical Context & Motivation
By the 1920s, the early quantum theory of Bohr and Sommerfeld had explained atomic spectra with remarkable precision, but it offered no satisfying account of how atoms combine to form stable molecules. The valence bond (VB) theory of Heitler and London (1927) provided a first quantum-mechanical treatment of the H2 bond, but its localized picture struggled to explain the paramagnetism of O2 and certain spectroscopic details of polyatomic species. A complementary framework was needed—one that treated electrons as delocalized over the entire molecule rather than confined between specific atom pairs.
The central question MO theory addresses is deceptively simple: When atoms approach one another, how do their atomic orbitals combine, and what determines whether the resulting interaction is stabilizing or destabilizing? Answering this question requires a formalism that constructs molecular-wide wave functions from atomic building blocks and assigns electrons to them according to the same quantum-mechanical rules that govern atoms—the aufbau principle, Pauli exclusion, and Hund's rule.
Core Principles & Definitions
Molecular orbital theory rests on the idea that atomic orbitals (AOs) on different atoms can be combined mathematically to produce molecular orbitals (MOs) that extend over the entire molecule. The most common approximation for constructing MOs is the Linear Combination of Atomic Orbitals (LCAO) method: if two atomic orbitals φA and φB are located on atoms A and B, respectively, then the resulting MOs are ψ = cAφA + cBφB. Two AOs produce exactly two MOs—a fundamental conservation rule. The in-phase combination concentrates electron density between the nuclei and lowers the energy, yielding a bonding MO; the out-of-phase combination depletes density between the nuclei and raises the energy, yielding an antibonding MO.
LCAO Approximation
Bonding vs. Antibonding
Bond Order
Symmetry Labels (σ and π)
Electron Filling Rules
MO Energy-Level Diagram for Homonuclear Diatomics
The canonical way to visualize molecular orbital theory is through an MO energy-level diagram (also called a correlation diagram). In this diagram, atomic orbital energies are shown on the left and right sides for atoms A and B, while the resulting molecular orbitals are placed in the center at their respective energies. Dashed lines connect each MO to the AOs from which it was constructed. The diagram below illustrates the general scheme for a second-period homonuclear diatomic like O2 or F2 (without s–p mixing).
Several important features emerge from this diagram. First, for every bonding MO formed below the parent AO energy, an antibonding counterpart appears above—the energy lowering of the bonding MO is roughly matched (and in fact slightly exceeded) by the energy raising of the antibonding MO. Second, the π2p bonding orbitals are doubly degenerate (two orbitals of equal energy), as are the π*2p antibonding orbitals. This degeneracy is critical because it provides slots for unpaired electrons—precisely the feature that explains the paramagnetism of O2. Third, the overall ordering of MO energies depends on whether s–p mixing (also called s–p hybridization of MOs) is significant: for elements earlier in the second period (Li through N), the σ2p bonding level is pushed above the π2p level, while for O and F the standard ordering shown in the diagram is restored.
Mathematical Framework
The quantitative backbone of MO theory is the variational method applied to the LCAO expansion. For a homonuclear diatomic, we write the trial wave function as ψ = cAφA + cBφB and minimize the energy expectation value E = ⟨ψ|Ĥ|ψ⟩ / ⟨ψ|ψ⟩ with respect to the coefficients. This procedure yields a secular determinant whose solutions give the MO energies and the coefficient ratios.
MO Configurations & Properties of Second-Period Diatomics
One of the most powerful applications of MO theory is systematically predicting the electronic configurations, bond orders, bond lengths, bond dissociation energies, and magnetic properties of the homonuclear diatomic molecules of the second period. The table below summarizes these properties, illustrating the s–p mixing crossover that occurs between N2 and O2.
| Molecule | Total e⁻ | MO Configuration (valence) | Bond Order | Magnetism |
|---|---|---|---|---|
| Li2 | 6 | (σ2s)² | 1 | Diamagnetic |
| Be2 | 8 | (σ2s)²(σ*2s)² | 0 | Does not form |
| B2 | 10 | …(π2p)¹(π2p)¹ | 1 | Paramagnetic |
| C2 | 12 | …(π2p)²(π2p)² | 2 | Diamagnetic |
| N2 | 14 | …(π2p)²(π2p)²(σ2p)² | 3 | Diamagnetic |
| O2 | 16 | …(σ2p)²(π2p)²(π2p)²(π*2p)¹(π*2p)¹ | 2 | Paramagnetic |
| F2 | 18 | …(π*2p)²(π*2p)² | 1 | Diamagnetic |
| Ne2 | 20 | …(σ*2p)² | 0 | Does not form |
Notice from the table that the trend in bond order across the second period mirrors the experimental trend in bond dissociation energy and is inversely related to bond length. N2 has the highest bond order (3) and accordingly possesses the shortest bond length (109.8 pm) and the largest dissociation energy (945 kJ mol−1). The prediction that O2 is paramagnetic (two unpaired electrons occupying degenerate π* MOs) was historically one of MO theory's greatest triumphs, since Lewis structures and VB theory both predict O2 to be diamagnetic.
Worked Example: MO Analysis of O₂
Let us apply the full MO procedure to molecular oxygen, O2, to determine its electron configuration, bond order, and magnetic character.
MO Theory vs. Valence Bond Theory
MO theory and VB theory are complementary quantum-mechanical approaches to chemical bonding, each with distinct strengths and limitations. In practice, modern computational chemistry often blends insights from both: VB-like resonance structures provide chemical intuition, while MO-based methods (Hartree–Fock, DFT, post-Hartree–Fock) deliver quantitative accuracy. The following table summarizes the key differences.
| Feature | Valence Bond (VB) Theory | Molecular Orbital (MO) Theory |
|---|---|---|
| Electron treatment | Localized between atom pairs | Delocalized over entire molecule |
| Bond description | Overlap of hybrid or pure AOs on adjacent atoms | Electrons fill MOs spanning all nuclei |
| Magnetic prediction | Incorrectly predicts O₂ as diamagnetic | Correctly predicts O₂ paramagnetism |
| Resonance / delocalization | Handled via superposition of resonance structures | Naturally built in through delocalized MOs |
| Chemical intuition | Strong—maps onto Lewis structures | Less intuitive for large molecules without localization procedures |
| Dissociation limit | Correctly dissociates H₂ → H + H | Simple MO theory gives ionic contamination at dissociation (requires CI correction) |
Connections to Advanced Computational Methods
The simple LCAO-MO picture presented in this lesson is the conceptual foundation for virtually all modern electronic structure methods. In practice, the atomic orbital basis sets are far more extensive than minimal bases, and electron–electron correlation—entirely absent from simple MO theory—must be accounted for to achieve chemical accuracy (≈ 1 kcal mol−1). The table below sketches how the basic MO framework evolves into the tools used in contemporary research.
| Concept in This Lesson | Advanced Extension | What Changes |
|---|---|---|
| LCAO with minimal basis (one AO per atom) | Hartree–Fock with large Gaussian basis sets (cc-pVTZ, etc.) | Many basis functions per atom; self-consistent field iterates MO coefficients until convergence. |
| Independent-electron picture (each e⁻ in one MO) | Post-HF methods: MP2, CCSD(T), CASSCF | Electron correlation included via perturbation theory, coupled cluster, or multi-reference approaches. |
| Bond order from electron counting | Mayer bond order, Wiberg indices, QTAIM | Bond order computed from the density matrix or electron density topology, allowing fractional and multi-center bonds. |
| Delocalized canonical MOs | Localized MOs (NBO, Boys, Pipek–Mezey) | Unitary transformation of canonical MOs recovers localized 'lone pairs' and 'bonds' for chemical interpretation. |
Density functional theory (DFT), the workhorse of modern quantum chemistry, also uses the MO framework: electrons are placed in Kohn–Sham orbitals that resemble MOs but are constructed so that the electron density—not the many-body wave function—is the fundamental variable. Despite this philosophical shift, the practical workflow (build a basis, solve for orbital coefficients, fill orbitals) mirrors the LCAO-MO approach you have learned here. Mastering bond order and the bonding/antibonding dichotomy thus provides the conceptual scaffolding for all subsequent computational chemistry coursework.
Practice Problems
Lesson Summary
Molecular orbital theory constructs delocalized wave functions for molecules by taking linear combinations of atomic orbitals (LCAO). Combining n AOs always yields n MOs. In-phase overlap produces bonding MOs (lower energy, enhanced internuclear electron density), while out-of-phase overlap produces antibonding MOs (higher energy, nodal plane between nuclei). The bond order—defined as ½(Nb − Na)—predicts bond strength, bond length, and whether a molecule is stable at all.
Electrons fill MOs according to the aufbau principle, Pauli exclusion, and Hund's rule. For second-period diatomics, awareness of s–p mixing is essential: it inverts the σ2p/π2p ordering for Li2 through N2. The triumph of MO theory is its correct prediction of O₂ paramagnetism—a result that eludes Lewis structures and simple VB theory. MO theory also provides the conceptual foundation for modern computational methods including Hartree–Fock, DFT, and post-Hartree–Fock correlation methods.