Bonding

Valence Bond Theory

Heitler-London 1927 → Pauling — bonds form by overlap of half-filled atomic orbitals; resonance is superposition

Valence Bond Theory (VB) describes covalent bonds as forming when two atoms each contribute a half-filled atomic orbital that overlaps in the internuclear region, producing a localized two-electron bond with paired spins. Walter Heitler and Fritz London solved the H2 molecule in 1927 with a wavefunction ψ = φA(1)φB(2) + φA(2)φB(1), capturing about 73% of the binding energy without any adjustable parameters. Linus Pauling extended VB through the 1930s with hybridization (sp, sp2, sp3, sp3d, sp3d2) and the resonance concept — superposition of multiple Lewis structures — culminating in The Nature of the Chemical Bond (1939) and the 1954 Nobel Prize in Chemistry.

  • First quantum bondHeitler & London 1927
  • H2 binding recovered~73% (no fitting)
  • Hybridizationssp, sp2, sp3, sp3d, sp3d2
  • Tetrahedral angle109.47°
  • Benzene resonance E~150 kJ/mol
  • Pauling NobelChemistry 1954

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Why valence bond theory matters

  • First successful quantum description of a chemical bond. Before 1927, chemists used Lewis dot structures with no quantum mechanical justification. Heitler and London applied Schrödinger's equation to two hydrogen atoms and obtained a binding curve with a minimum at about 87 pm and ~3.14 eV — vs the experimental 74 pm and 4.75 eV — using the antisymmetric singlet combination ψ = φA(1)φB(2) + φA(2)φB(1). It was the proof that quantum mechanics could explain chemistry.
  • Five hybridization schemes cover most main-group geometry. sp gives linear (180°, BeCl2, HC≡CH); sp2 gives trigonal planar (120°, BF3, ethylene); sp3 gives tetrahedral (109.47°, methane, ammonia, water with lone pairs); sp3d gives trigonal bipyramidal (PF5); sp3d2 gives octahedral (SF6, [Co(NH3)6]3+). One framework predicts >90% of organic and main-group geometries.
  • Resonance quantifies delocalization. Benzene's experimental C-C bond length of 139 pm — uniformly between single (154 pm) and double (134 pm) — is naturally explained as a 50:50 superposition of two Kekulé structures. Resonance energy ~150 kJ/mol (relative to a hypothetical 1,3,5-cyclohexatriene) explains aromatic stability.
  • Sigma vs pi distinction is a VB construct. Sigma bonds form by head-on (axial) overlap and have cylindrical symmetry around the internuclear axis; pi bonds form by side-on overlap of unhybridized p orbitals and have a node along that axis. A C=C double bond is one σ + one π (controls planarity); C≡C is one σ + two π (controls linearity).
  • Reaction mechanisms are taught in VB language. Every arrow-pushing diagram in undergraduate organic chemistry — SN1, SN2, E1, E2, addition, elimination, pericyclic — assumes localized bonds, lone pairs, and partial charges. None of those concepts survive in pure MO language without reconstruction.
  • Modern VB methods compete with CASSCF. Sason Shaik and Philippe Hiberty's BOVB (Breathing Orbital Valence Bond, 1992) and VBSCF give chemically interpretable wavefunctions for transition states and excited states with accuracy rivaling multi-reference MO methods. Packages like XMVB are actively used in computational chemistry research.
  • Predicts ionicity quantitatively. Pauling's electronegativity scale was derived from VB resonance: bond energy = covalent term + ionic term, with the ionic stabilization growing as Δχ2. The empirical formula gave electronegativity differences directly from thermochemistry and remains the standard chemistry-classroom scale today.

Common misconceptions

  • Hybrid orbitals exist in isolated atoms. Wrong. Hybridization is a mathematical construction — a unitary mixing of degenerate atomic solutions — that becomes useful only when other atoms approach to bond. An isolated carbon atom has 2s and 2p orbitals; sp3 hybrids are a basis transformation that maximizes overlap with four neighbors.
  • Resonance means the molecule oscillates between structures. No. The wavefunction is a fixed quantum superposition; the molecule has one true electron distribution at all times. Picturing benzene as flickering between two Kekulé forms is a beginner's confusion that violates the time-independent nature of stationary states.
  • VB and MO theory contradict each other. They give the same observables for ground-state geometries and energies (when both include enough configurations). They are unitary transformations of each other in the full configuration interaction limit. The differences are pedagogical and computational, not physical.
  • O2 is diamagnetic per VB. Simple VB with the structure :O=O: predicts diamagnetism, but liquid O2 is paramagnetic (sticks to a magnet). MO immediately explains this with two unpaired electrons in degenerate π* orbitals. This was the textbook strike against naive VB; modern VB recovers paramagnetism with three-electron bonds.
  • sp3d hybridization in PF5 uses real d orbitals. Modern computational chemistry shows that the 3d orbitals on phosphorus are too high in energy to participate significantly. The bonding is better described as 3-center-4-electron bonds along the axial F-P-F axis. The sp3d label is still pedagogically useful but not literally accurate.
  • More resonance structures = more stable. Only equivalent or near-equivalent structures contribute meaningfully. Adding high-energy charge-separated structures with bad geometry to the resonance hybrid does not lower the energy. Coefficients fall off rapidly with structural energy.

How orbitals overlap and pair

The Heitler-London calculation for H2 starts with two hydrogen atoms A and B at separation R, each with a 1s electron. The total wavefunction must be antisymmetric under electron exchange (fermion statistics), so the spatial part is taken as the symmetric combination ψ+ = φA(1)φB(2) + φA(2)φB(1) paired with the antisymmetric singlet spin function (αβ - βα)/√2. Plugging into the two-electron Hamiltonian and integrating gives the energy as a function of R: a Coulombic term (electron 1 on A, electron 2 on B), an exchange term (electrons swap atoms), and an overlap S = ∫φAφBdτ. The exchange integral is the key quantum mechanical ingredient; without it there is no bond. Pauling later refined the wavefunction by adding ionic terms φA(1)φA(2) and φB(1)φB(2), capturing the few percent of the time both electrons sit on the same atom.

Hybridization extends this to polyatomic molecules. For methane, build four sp3 hybrids by orthogonal linear combination of (2s, 2px, 2py, 2pz): h1 = ½(s + px + py + pz), h2 = ½(s + px - py - pz), and so on. Each hybrid points to a tetrahedron vertex; each pairs with a hydrogen 1s to form a localized two-electron bond. The four C-H bonds are equivalent by construction, the H-C-H angle comes out to arccos(-1/3) = 109.47°, and the bond energy of ~414 kJ/mol per C-H is recovered without further fitting. The same procedure for ethylene gives sp2 + pπ (planar, 120°), and for acetylene gives sp + 2pπ (linear, 180°).

Resonance is implemented as a linear combination of multiple VB structures: ψ = c1ψKekule-A + c2ψKekule-B + smaller terms (Dewar structures, ionic structures). The coefficients ci are optimized variationally; the energy of the mixture is lower than any single contributor, and that lowering is the resonance stabilization. For benzene the two Kekulé contributors are identical by symmetry so c1 = c2, and the calculation predicts equal C-C bond lengths and a stabilization of ~150 kJ/mol vs hypothetical localized 1,3,5-cyclohexatriene. The theory generalizes to allyl, carbonate, nitrate, peptide bonds, and every aromatic system.

Comparison: VB versus MO theory

PropertyValence Bond (VB)Molecular Orbital (MO)
Electron localizationLocalized bonding pairs between atom pairsDelocalized over entire molecule
Building blocksAtom-centered (hybrid) orbitalsLCAO over all atomic orbitals
Pictorial intuitionLewis structures, arrow-pushingEnergy-level diagrams, nodes
O2 paramagnetismFails at simple levelPredicted directly (degenerate π*)
Bond dissociationCleanly to neutral atomsRHF MO gives 50% ionic limit
Spectroscopy / excited statesClumsy without configuration interactionNatural — TD-DFT, CIS, etc.
Reaction mechanismsNative language of organic chemRequires Walsh / FMO reinterpretation
Modern computational useBOVB, CASVB, VBSCFHF, DFT, CCSD(T), CASSCF

Applications and examples

  • Methane (CH4) tetrahedral geometry. sp3 hybridization, four equivalent C-H bonds at 109.47°, C-H bond length 109 pm, bond energy 414 kJ/mol. The textbook explanation for why diamond is hard, methane is the smallest alkane, and why all sp3 carbons in organic molecules look alike.
  • Benzene aromaticity. Six sp2 carbons in a planar hexagon, each with one unhybridized p orbital perpendicular to the ring forming a 6-electron delocalized π system. Two-Kekulé resonance gives equal C-C bonds at 139 pm and ~150 kJ/mol stabilization, the foundation for all aromatic chemistry from pyridine to porphyrins.
  • Water and ammonia bond angles. sp3 hybridization on O (in H2O) and N (in NH3) with lone pairs occupying hybrid orbitals; lone-pair-bond-pair repulsion compresses the H-O-H angle to 104.5° and H-N-H to 107° from the ideal 109.47°. VSEPR is the qualitative shorthand for this VB analysis.
  • Octahedral coordination complexes. [Co(NH3)6]3+ uses sp3d2 (or in modern terms, d2sp3) hybrids on Co3+ overlapping with the lone pairs of six NH3 ligands. Pauling's analysis of inner-sphere vs outer-sphere complexes (1948) was the first quantum mechanical treatment of transition-metal coordination chemistry.
  • Peptide bond planarity. The C-N bond in proteins has partial double-bond character from resonance with the carbonyl, restricting rotation around φ and ψ backbone angles. The Ramachandran plot, which underlies all of structural biology, depends on this VB resonance argument.

Frequently asked questions

What is the central postulate of valence bond theory?

A covalent bond forms when a half-filled atomic orbital on one atom overlaps a half-filled atomic orbital on a neighboring atom, with the two electrons sharing that overlap region having opposite spins (Pauli pairing). The greater the overlap, the stronger the bond. The Heitler-London 1927 wavefunction for H2 made this quantitative: ψ = φA(1)φB(2) + φA(2)φB(1), an antisymmetric combination of products of 1s orbitals on atoms A and B that recovered roughly 73% of the experimental binding energy of 4.75 eV without any empirical parameters. Pauling later added a small ionic term φA(1)φA(2) + φB(1)φB(2) to capture residual correlation, raising the recovered binding energy to about 80%.

What is hybridization and why is it needed?

Pauling proposed hybridization in 1931 to reconcile valence bond theory with observed molecular geometries. Atomic orbitals as solved for the isolated atom (2s, 2px, 2py, 2pz) cannot directly explain why methane has four equivalent C-H bonds at 109.47°, since carbon's ground state has only two unpaired p electrons in orthogonal directions. Hybridization mixes the 2s and three 2p orbitals into four equivalent sp3 hybrids pointing toward the corners of a tetrahedron. The catalog: sp gives linear (180°, e.g. BeH2, alkynes), sp2 gives trigonal planar (120°, e.g. BF3, alkenes), sp3 gives tetrahedral (109.47°, methane), sp3d gives trigonal bipyramidal (PF5), and sp3d2 gives octahedral (SF6). The hybrids retain s-orbital strength near the nucleus while gaining p-orbital directionality, maximizing overlap with neighboring atoms.

What is resonance in valence bond theory?

When a single Lewis structure cannot adequately describe a molecule, the true wavefunction is taken as a quantum superposition of several contributing structures, each weighted by its own coefficient. Benzene is the canonical example: two equivalent Kekulé structures with alternating single and double bonds contribute equally, producing six C-C bonds of intermediate length (139 pm, between 154 pm single and 134 pm double) and a delocalized π system. Resonance is a mathematical statement about the wavefunction, not a physical oscillation between structures — the molecule never has alternating bonds at any instant. The energy lowering relative to the most stable single contributor is the resonance energy, ~150 kJ/mol for benzene.

How does VB theory differ from molecular orbital theory?

VB localizes electrons into bonding pairs between specific atom pairs, building the molecule from atom-centered orbitals that overlap; MO delocalizes electrons across the entire molecular framework into linear combinations of all atomic orbitals (LCAO). For ground-state H2 the two methods give nearly identical energies and the same equilibrium bond length (~74 pm). They diverge on excited states and bond dissociation: simple MO predicts H2 dissociates partly to ionic H+ + H- because the bonding orbital is half ionic, while VB cleanly gives two H atoms. Modern computational chemistry uses both: VB for chemical intuition, ionicity, and reaction mechanisms; MO for spectroscopy, photochemistry, and ab initio calculations on extended systems.

Why does VB theory predict diamagnetism for O2 incorrectly?

Drawing the Lewis structure O=O pairs all valence electrons into bonds and lone pairs, predicting diamagnetic O2. Experimentally, liquid oxygen sticks to a magnet — it is paramagnetic with two unpaired electrons. MO theory immediately explains this: the bond order is 2, but the highest occupied levels are two degenerate π* orbitals, each singly occupied by Hund's rule. This was historically the most cited failure of simple VB and the strongest early evidence for MO theory. Modern VB recovers the correct paramagnetism through three-electron bonds or symmetry-adapted spin couplings, but the textbook Pauling-era VB picture genuinely missed it.

Is VB theory still used today?

Yes — both pedagogically and in research. Every introductory organic chemistry course teaches arrow-pushing mechanisms in valence bond language: lone pairs, bonds, hybridization, formal charges. In computational chemistry, modern VB methods (CASVB, BOVB, VBSCF) compete with MO-based CASSCF for studying multi-reference problems like transition states, bond breaking, and excited states. They are particularly useful when chemists want to recover interpretable structures (e.g. percent ionic character) that get smeared out in MO orbitals. Sason Shaik and Philippe Hiberty have led a modern VB revival since the 1990s, and packages like XMVB and VB2000 are actively maintained.