Valence bond theory is a quantum-mechanical approach to chemical bonding that constructs molecular electronic states from orbitals associated with particular atoms and from specified arrangements of their electrons. Its simplest picture represents a covalent bond as two electrons shared between overlapping atomic orbitals. More general formulations combine several bonding arrangements, allowing both localized bonds and electron delocalization to be described. Together with molecular orbital theory, it provides a fundamental framework for interpreting molecular electronic structure. (goldbook.iupac.org)
Origins and development
The theory developed from the shared-electron-pair interpretation of bonding associated with Gilbert N. Lewis. Walter Heitler and Fritz London supplied its foundational quantum-mechanical treatment in 1927 by analyzing the hydrogen molecule. Their approach combined alternative assignments of two electrons to two hydrogen atoms rather than treating the electrons as permanently distinguishable particles. Linus Pauling subsequently extended the approach to chemical structure, connecting quantum mechanics with directional bonds and the language of electron-pair bonding. His early paper The shared-electron chemical bond emphasized the Pauli exclusion principle and quantum-mechanical resonance as central factors in valence. (arxiv.org)
Modern valence bond methods retain this connection to recognizable bonding patterns while allowing orbitals and structure coefficients to be optimized computationally. The resulting theory is broader than the elementary orbital-overlap model commonly introduced in chemistry courses. (arxiv.org)
Electron pairing and the hydrogen molecule
An atomic orbital specifies a one-electron spatial function, not a classical trajectory. In the simplest valence bond description of the hydrogen molecule, two 1s functions, and , are centered on different nuclei. The covalent spatial wave function has the form
where normalizes the function and the labels 1 and 2 refer to electron coordinates. Both electron assignments occur in the same quantum state. This symmetric spatial function is combined with an antisymmetric singlet spin function, producing an overall electronic wave function that changes sign when the electrons are exchanged. This connects bond pairing with electron spin, rather than merely with two opposite arrows in an orbital diagram. (arxiv.org)
As the nuclei approach, their orbital overlap and electronic interactions change. Bond formation corresponds to a reduction in total energy relative to the separated atoms; excessive compression raises the energy. The equilibrium separation lies at the minimum of the energy curve. Orbital overlap is therefore part of the explanation, but overlapping regions alone do not establish that a stable bond exists. (openstax.org)
Directional bonds and hybridization
Valence bond theory relates bond direction to the spatial orientation of orbitals. A sigma bond involves overlap directed along the internuclear axis. A pi bond can arise from side-by-side overlap of parallel p orbitals, with a nodal plane containing that axis. In the conventional localized description, a carbon–carbon double bond contains one sigma and one pi component, whereas a triple bond contains one sigma and two pi components. These distinctions explain why multiple bonds impose different geometrical constraints from single bonds. (openstax.org)
Orbital hybridization uses mathematical combinations of atomic orbitals on the same atom to construct directional bonding functions. Ideal , , and sets correspond respectively to linear, trigonal-planar, and tetrahedral arrangements. In methane, four equivalent carbon hybrids provide a localized description of the four carbon–hydrogen bonds. Hybridization is a representation of molecular bonding, not a separately observed physical event that must occur before atoms combine. (openstax.org)
Resonance and ionic contributions
A single localized bonding arrangement is often insufficient. A general valence bond state is expressed as a linear combination:
where each represents a spin-adapted electronic structure and the coefficients determine their contributions. Such structures often correspond to Lewis structures, but are mathematical components of a wave function rather than separate molecules. They may describe covalent electron sharing, charge separation, or unpaired electrons. (pubs.rsc.org)
Resonance is the mixing of these contributions. It does not mean that a molecule oscillates between drawings. For benzene, combining alternative bonding patterns permits a valence bond account of delocalization. Even hydrogen can be described more flexibly by adding ionic contributions, with both electrons associated with one nucleus or the other, to its covalent contribution. (goldbook.iupac.org)
Because contributing structures can overlap mathematically, their weights require care in interpretation. They are not automatically ordinary probabilities, and assigning independent observable properties to each structure can produce inconsistencies. (pubs.rsc.org)
Computational formulations and applications
In valence bond self-consistent-field calculations, orbitals and mixing coefficients are optimized together. Breathing-orbital valence bond methods allow different structures to use different orbital sets, providing additional flexibility to describe electron correlation. Research has also combined valence bond wave functions with quantum Monte Carlo methods to calculate molecular binding energies. These developments make valence bond theory a quantitative computational approach, not only a qualitative bonding vocabulary. (arxiv.org)
Valence bond representations are useful for analyzing chemical reactions because reactant-like and product-like structures can be followed as bonds break and form. Empirical valence bond models use coupled, parameterized potential functions representing different bonding arrangements; they are distinct from fully electronic, first-principles valence bond calculations. Such models have been used to construct reactive potentials for molecular simulations. (pubmed.ncbi.nlm.nih.gov)
Relationship to molecular orbital theory
Valence bond and molecular orbital descriptions emphasize different representations of the same electronic problem. The former foregrounds spin coupling and chemically recognizable structures; the latter commonly starts from orbitals extending across a molecule. Neither localization nor delocalization is exclusive to one framework. With sufficiently complete expansions, the approaches become equivalent. Limitations of a single Lewis-like picture therefore should not be mistaken for fundamental limitations of valence bond theory itself. (goldbook.iupac.org)