Molecular Orbital Theory

Molecular Orbital (MO) theory—also known as Hund–Mulliken theory—is a quantum-mechanical framework in physical, inorganic, and organic chemistry that describes the electronic structure of molecules using delocalized one-electron wavefunctions (molecular orbitals, ψ_i) that extend across all constituent atomic nuclei rather than being confined to localized two-center bonds between pairs of atoms. Formulated between 1927 and 1932 by Friedrich Hund, Robert S. Mulliken, John E. Lennard-Jones, and Erich Hückel, MO theory complements localized Valence Bond (VB) theory, orbital hybridization, and VSEPR theory by directly explaining phenomena that localized Lewis structures cannot easily capture: the triplet paramagnetism of ground-state dioxygen (O₂), fractional bond orders in one- and three-electron species (H₂⁺, He₂⁺), hypervalent three-center four-electron bonds, Hückel 4*n* + 2 aromaticity, ultraviolet-visible electronic excitation spectra, and Frontier Molecular Orbital (HOMO–LUMO) chemical reactivity.

🧪 Interactive Curriculum: Compare Valence Bond hybridization and Molecular Orbital models in the VSEPR & Orbital Bonding Mechanics learning module and VSEPR Lab.

Constructive Bonding (ψ₊) versus Destructive Antibonding (ψ₋)

Within the Born–Oppenheimer approximation (which holds the heavier atomic nuclei at fixed coordinates R_A while solving for the rapid motion of the electrons), each molecular orbital ψ_i is approximated as a Linear Combination of Atomic Orbitals (LCAO) centered on the M nuclei of the molecule:

ψ_i = ∑_*μ*=1^K c_μi φ_μ

For a homonuclear or heteronuclear two-orbital interaction between atomic orbital φ_A on atom A (with Coulomb integral α_A = ⟨*φ*_A|*Ĥ*|*φ*_A⟩) and atomic orbital φ_B on atom B (with α_B = ⟨*φ*_B|*Ĥ*|*φ*_B⟩), applying the Rayleigh–Ritz variational principle (∂⟨*ψ*|*Ĥ*|*ψ*⟩/∂*c*_μ = 0) yields the 2 × 2 secular determinant:

| α_A − E , β_AB − E S_AB | = 0 (and symmetrically for row 2)

where S_AB = ⟨*φ*_A|*φ*_B⟩ is the dimensionless overlap integral (0 ≤ S_AB < 1) and β_AB = ⟨*φ*_A|*Ĥ*|*φ*_B⟩ < 0 is the resonance (exchange) integral, which scales approximately with the overlap integral via the Mulliken–Wolfsberg–Helmholz relation β_AB ≈ ½ K (α_A + α_B) S_AB.

Solving the secular equation for a homonuclear system (α_A = α_B = α) yields two normalized molecular orbitals and their corresponding energy eigenvalues:

1. The Bonding Molecular Orbital (ψ₊, σ or π):

• ψ₊ = (φ_A + φ_B) / √[ 2(1 + S_AB) ]
• E₊ = (α + β_AB) / (1 + S_AB) ≈ α − |*β*_AB| + S_AB|*β*_AB|
• In-phase constructive interference increases electron probability density in the internuclear midpoint (ρ_mid = (φ_A + φ_B)²/[2(1 + S_AB)] > ½(φ_A² + φ_B²)), electrostatic shielding the two positively charged nuclei and stabilizing the molecule below α.

1. The Antibonding Molecular Orbital (ψ₋, σ\* or π\*):

• ψ₋ = (φ_A − φ_B) / √[ 2(1 − S_AB) ]
• E₋ = (α − β_AB) / (1 − S_AB) ≈ α + |*β*_AB| + S_AB|*β*_AB|
• Out-of-phase destructive interference creates a nodal plane perpendicular to the internuclear axis where electron density drops to zero (ψ₋ = 0), pushing electron density onto the outer flanks of the nuclei and destabilizing the orbital above α. Because the denominator (1 − S_AB) is smaller than (1 + S_AB), an antibonding orbital is always destabilized more than its partner bonding orbital is stabilized (Δ*E*_antibonding > |Δ*E*_bonding|). Consequently, filling both the bonding and antibonding orbitals with four electrons (as in hypothetical He₂, 1*σ*_g² 1*σ*_u*²) yields a net four-electron Pauli repulsion (closed-shell steric repulsion) rather than a covalent bond.

1. Non-Bonding Molecular Orbitals (n): If two atomic orbitals have orthogonal symmetries (S_AB = 0, such as a p_x orbital meeting an s orbital along the z-axis) or carry a large energy mismatch (|*α*_A − α_B| ≫ |*β*_AB|), no mixing occurs, leaving a non-bonding molecular orbital at its original atomic energy E ≈ α (corresponding to a localized lone pair).

Formal Bond Order and Second-Period Homonuclear Diatomics

In MO theory, the covalent bond order (BO) is defined as half the difference between the number of electrons occupying bonding orbitals (N_b) and the number occupying antibonding orbitals (N_a*):

Bond Order (BO) = ½ ( N_bonding − N_antibonding )

Across the second-period homonuclear diatomic series (Li₂ through Ne₂), the valence 2*s* and 2*p* subshells combine into eight molecular orbitals: two σ orbitals from 2*s* (2*σ*_g and 2*σ*_u*), two coaxial σ orbitals from 2*p*_z (3*σ*_g and 3*σ*_u*), and two degenerate pairs of lateral π orbitals from 2*p*_x and 2*p*_y (1*π*_u and 1*π*_g*). Here the subscripts g (gerade, even) and u (ungerade, odd) denote whether the wavefunction retains (+1) or inverts (−1) its sign upon inversion through the molecular center of symmetry (i): σ bonding orbitals are g (σ_g) and *σ*\* antibonding orbitals are u (σ_u*), whereas π bonding orbitals are u (π_u) and *π*\* antibonding orbitals are g (π_g*).

In the lighter diatomics (Li₂, Be₂, B₂, C₂, and N₂), the atomic 2*s*–2*p* energy separation is relatively small (from 1.85 eV in Li to 10.9 eV in N). Because the 2*σ*_g (from 2*s*) and 3*σ*_g (from 2*p*_z) molecular orbitals share identical σ_g symmetry, they undergo second-order s–p mixing: 2*σ*_g is pushed lower in energy while 3*σ*_g is pushed upward above the degenerate 1*π*_u(2p_x,*y*) pair, producing the "inverted" (π-below-σ) ordering (1*π*_u < 3*σ*_g) for B₂, C₂, and N₂. In O₂ and F₂, the large effective nuclear charge (Z_eff) widens the atomic 2*s*–2*p* energy gap (>15–20 eV), suppressing s–p mixing and restoring the "normal" (σ-below-π) ordering (3*σ*_g < 1*π*_u).

Perturbation Theory and HOMO–LUMO Donor-Acceptor Reactivity

In heteronuclear bonds (A–B where χ_B > χ_A, so α_B < α_A), second-order perturbation theory gives the stabilization energy Δ*E* ≈ −|*β*_AB|² / |*α*_A − α_B|:

• The lower-energy bonding MO (ψ₊) has a larger coefficient on the more electronegative atom B (|*c*_B| > |*c*_A|), polarizing bonding electrons toward B.
• The higher-energy antibonding MO (ψ₋, σ\* or π\*) has a larger coefficient on the less electronegative atom A (|*c*_A| > |*c*_B|), explaining why nucleophilic attack on a polar C=O or C–Br bond always occurs at the electropositive carbon atom where the LUMO lobe is largest.

As formulated by Kenichi Fukui (1952) and Roald Hoffmann and Robert B. Woodward (1965, awarded the 1981 Nobel Prize in Chemistry), most bimolecular chemical reactions are governed by the interaction between the Highest Occupied Molecular Orbital (HOMO) of the electron-donor nucleophile and the Lowest Unoccupied Molecular Orbital (LUMO) of the electron-acceptor electrophile:

1. Nucleophilic Substitution (S_N2): The nucleophile's HOMO (lone pair n) attacks the backside lobe of the substrate's C–X *σ*\* LUMO at 180°, populating *σ*\* to cleave the C–X bond while forming the new C–Nu σ bond with stereochemical Walden inversion.
2. Woodward–Hoffmann Conservation of Orbital Symmetry: Concerted pericyclic reactions (such as the [4+2] Diels–Alder cycloaddition between a 4π diene HOMO and a 2π dienophile LUMO, or electrocyclic ring closures of vitamin D precursors in human skin under ultraviolet light) proceed with low activation energy only when the terminal lobes of the interacting frontier orbitals have matching algebraic phase signs (suprafacial/antarafacial selection rules).
3. Hückel's 4*n* + 2 Aromaticity Rule: In planar cyclic conjugated polyenes (C_mH_m), solving the Hückel secular determinant yields a Frost circle of π molecular orbitals with a unique lowest orbital (a_2u in benzene) followed by degenerate pairs (e_1g, e_2u, …). Planar rings containing 4*n* + 2 π electrons (n = 0, 1, 2, …; e.g., 6 π electrons in benzene, adenine, and tryptophan, or 18 π electrons in the porphyrin macrocycle of heme and chlorophyll) completely fill all bonding shells to achieve closed-shell aromatic stabilization, whereas 4*n* rings (such as 4π cyclobutadiene) leave half-filled non-bonding degenerate orbitals that undergo Jahn–Teller distortion (antiaromaticity).

Why does ionizing an electron from nitric oxide (NO → NO⁺) or dioxygen (O₂ → O₂⁺) shorten and strengthen the bond instead of weakening it?

In most molecules (such as N₂ → N₂⁺, where bond order drops from 3.0 to 2.5 and bond length stretches from 1.098 Å to 1.116 Å), removing a valence electron ejects it from a bonding molecular orbital. However, in nitric oxide (NO, 11 valence electrons) and dioxygen (O₂, 12 valence electrons), the Highest Occupied Molecular Orbital (HOMO) is the antibonding *π*\*_2*p* orbital (occupied by one electron in NO and two unpaired electrons in O₂). Ionizing NO to the nitrosonium cation (NO⁺, isoelectronic with N₂ and CO) or O₂ to the dioxygenyl cation (O₂⁺) removes an antibonding *π*\* electron, increasing the formal bond order from 2.5 to 3.0 in NO⁺ (shortening the N–O bond from 1.15 Å to 1.06 Å and raising the IR stretching frequency from 1,876 cm⁻¹ to 2,377 cm⁻¹) and from 2.0 to 2.5 in O₂⁺ (r₀ drops from 1.208 Å to 1.116 Å; *D*° rises from 498 to 643 kJ mol⁻¹).

How does ground-state triplet oxygen (³O₂) differ chemically from reactive singlet oxygen (¹O₂)?

In its lowest-energy ground state (³Σ_g⁻ triplet dioxygen, ³O₂), the two degenerate antibonding 1*π*_g* orbitals each contain one electron with parallel spins (↑ and ↑) in accordance with Hund's rule. Because nearly all organic biomolecules (lipids, proteins, DNA) have closed-shell singlet ground states (all electrons spin-paired, ↑↓), a direct concerted two-electron reaction between triplet ³O₂ and a singlet organic molecule to give singlet products is spin-forbidden by conservation of angular momentum, creating a kinetic barrier that prevents spontaneous combustion of living tissue in air. When a photosensitizer (such as excited triplet chlorophyll or porphyrin) transfers energy (94.3 kJ mol⁻¹ / 0.98 eV) to ³O₂, it flips one spin to generate singlet oxygen (¹Δ_g, ¹O₂), in which both electrons are paired in one π_g* orbital while the second π_g* orbital is completely vacant—turning ¹O₂ into a spin-allowed electrophile (reactive oxygen species) that rapidly attacks alkene double bonds in polyunsaturated membrane lipids.

What is the relationship between canonical Molecular Orbitals and the localized bonds of Valence Bond / VSEPR theory?

For any closed-shell single-determinant wavefunction (such as Hartree–Fock or Kohn–Sham Density Functional Theory), the total many-electron Slater determinant Ψ(1, 2, …, N), the total 3D electron density ρ(r), the molecular energy E, and the equilibrium nuclear geometry are invariant under any unitary linear transformation among the occupied molecular orbitals. Canonical Molecular Orbitals diagonalize the Fock energy matrix to yield delocalized symmetry-adapted orbitals (such as the a₁ and t₂ orbitals of CH₄) that directly predict vertical ionization energies (Koopmans' theorem) and electronic absorption spectra. Applying a unitary localization transformation (such as Boys, Edmiston–Ruedenberg, or Natural Bond Orbital localization) to those exact same occupied canonical MOs transforms them into localized bonding and lone-pair orbitals that match the two-center σ/π bonds and lone pairs of VSEPR theory and orbital hybridization.

1. Mulliken, R. S. (1967). "Spectroscopy, molecular orbitals, and chemical bonding." Science, 157(3784), 13–24. https://doi.org/10.1126/science.157.3784.13
2. Hoffmann, R., & Woodward, R. B. (1968). "The conservation of orbital symmetry." Accounts of Chemical Research, 1(1), 17–22. https://doi.org/10.1021/ar50001a003
3. Fukui, K. (1982). "Role of frontier orbitals in chemical reactions." Science, 218(4574), 747–754. https://doi.org/10.1126/science.218.4574.747
4. Albright, T. A., Burdett, J. K., & Whangbo, M.-H. (2013). Orbital Interactions in Chemistry (2nd ed.). John Wiley & Sons. ISBN 978-0471080398