Natural bond orbital

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Natural bond orbital (NBO) is a way to describe bonding in quantum chemistry by focusing on the electron density that most clearly represents a Lewis-type picture. NBOs are part of a family of natural localized orbitals that sit between atomic orbitals and molecular orbitals: natural atomic orbitals (NAO), natural hybrid orbitals (NHO), natural bond orbitals (NBO), and natural (semi-)localized molecular orbitals (NLMO). They help show how electrons are distributed in atoms and in bonds.

NBOs are derived from the N-electron wavefunction and aim to give an optimal, localized description of bonding. Bonding NBOs (sigma_AB) typically have high occupancy (near 2 electrons) and reflect the strongest, most localized bonding between two atoms A and B. Antibonding NBOs (sigma*_AB) are the complementary, non-Lewis orbitals with occupations near zero. Small occupancies in antibonding or other non-Lewis orbitals indicate some delocalization away from a perfect Lewis picture.

Origin and idea: The concept of natural orbitals was introduced by Per-Olov Löwdin in 1955 to describe orthonormal one-electron functions intrinsic to the N-electron wavefunction.

Theory in simple terms: A bonding NBO between atoms A and B, sigma_AB, can be written as a mix of two valence hybrids on A and B: sigma_AB = c_A h_A + c_B h_B, where h_A and h_B are the valence hybrids (NHOs) on A and B, and c_A and c_B are polarization coefficients. The strength and character of the bond change with these coefficients: if c_A ≈ c_B, the bond is more covalent; if c_A greatly exceeds c_B, the bond has more ionic character. Each bonding NBO has a corresponding antibonding NBO, sigma*_AB = c_A h_A − c_B h_B, which completes the valence space.

Lewis structures and resonance: NBO analysis seeks the Lewis structure with the maximum amount of electron density in Lewis-type orbitals (the Lewis charge). Real molecules can exhibit resonance, but NBOs incorporate bond polarity and other effects directly, so separate covalent-ionic resonance is often not needed to explain the bonding in many cases. For example, in amides, NBO calculations typically show the carbonyl double bond as the dominant Lewis structure.

See also: Molecular orbital theory; Basis set (chemistry).

References and further reading: Foundational work by Weinhold and Landis includes Natural Bond Orbitals and Extensions of Localized Bonding Concepts (2001) and Discovering Chemistry With Natural Bond Orbitals (2012). These sources explain how NBOs provide a practical, localized view of electronic structure and bonding.

External resources: The NBO program page and related literature offer practical tools and definitions for working with NBO calculations.