Orbital Hybridization

Orbital hybridization is the construction of new atomic orbitals by combining orbitals centered on the same atom. These hybrid orbitals have directional properties useful for describing chemical bonds and the spatial arrangement of atoms in a molecule. Most commonly associated with valence bond theory, hybridization provides the familiar sp, sp², and sp³ descriptions of linear, trigonal-planar, and tetrahedral bonding environments. It is a mathematical representation of electronic structure rather than a separate physical process that must precede bond formation. (goldbook.iupac.org)

Mathematical meaning

In quantum mechanics, an orbital is represented by a one-electron wave function. A hybrid orbital is a linear combination of selected atomic-orbital functions:

where $\phi_j$ denotes an original orbital and $c_{ij}$ specifies its contribution. Hybridization normally combines orbitals with different angular-momentum quantum numbers, such as s and p orbitals. The combination changes the shape and orientation of the functions used to describe bonding. (goldbook.iupac.org)

For example, two normalized sp hybrids can be constructed from mutually orthogonal s and p functions:

Their large lobes point in opposite directions. Combining two orbitals produces two hybrids; combining four produces four. In mathematical terms, an invertible transformation supplies a different basis for the same linear span, not additional orbitals. The equations illustrate the linear-combination definition of hybridization. (goldbook.iupac.org)

Hybridization combines orbital functions, not electrons. Orbital occupation is a separate question: hybrid orbitals may participate in bonds or accommodate nonbonding electron pairs. (openstax.org)

Principal hybridization schemes

The three standard s–p schemes differ in the number of p orbitals combined with one s orbital. Their ideal arrangements refer to the directions of the hybrids’ large lobes. The superscript counts contributing p orbitals, not electrons or bonds. (openstax.org)

The s-character fractions follow from distributing one s orbital equally among the equivalent hybrids. In sp hybridization, two p orbitals remain outside the hybrid set; in sp², one remains; in sp³, all three p orbitals enter the set. These remaining p orbitals are important in descriptions of multiple bonding. (chem.purdue.edu)

Carbon compounds and multiple bonds

Carbon provides the best-known examples, especially in organic chemistry. In methane, CH₄, four equivalent carbon sp³ orbitals overlap with the 1s orbitals of four hydrogen atoms. This describes four equivalent covalent bonds directed toward the vertices of a tetrahedron. Linus Pauling presented the mathematical explanation of this tetrahedral orbital construction in 1931. (openstax.org)

A sigma bond involves head-on orbital overlap along the internuclear axis. In ethane, each carbon is described as sp³-hybridized: one hybrid forms the carbon–carbon sigma bond, while three form carbon–hydrogen bonds. Hybridization therefore applies to individual atomic environments, not necessarily to a molecule as a whole. (openstax.org)

In ethylene, H₂C=CH₂, each carbon has three sp² hybrids and one remaining p orbital perpendicular to their plane. The hybrids form the sigma-bond framework. Sideways overlap of the two p orbitals forms a pi bond, giving the carbon–carbon double bond one sigma and one pi component. This overlap helps explain the planar arrangement and restricted rotation about the double bond. (openstax.org)

In acetylene, HC≡CH, each carbon uses two sp hybrids for its sigma bonds. Two mutually perpendicular p orbitals remain on each carbon and form two pi bonds. The carbon–carbon triple bond consequently contains one sigma and two pi components, while the molecular framework is linear. (openstax.org)

Lone pairs and geometric assignment

Hybrid orbitals need not all form bonds. In the elementary sp³ description of ammonia, three nitrogen hybrids participate in N–H bonding and the fourth contains a lone pair. For water, two oxygen hybrids participate in O–H bonding and two accommodate lone pairs. These descriptions distinguish an approximately tetrahedral arrangement of electron pairs from the pyramidal or bent arrangement of the atoms themselves. (openstax.org)

A common assignment method begins with a Lewis structure and counts regions of electron density around the atom. Following valence-shell electron-pair repulsion theory, each single bond, multiple bond, or lone pair counts as one region. Two, three, and four regions suggest sp, sp², and sp³ descriptions respectively. This is a useful correspondence between geometry and a bonding model, rather than an independent measurement of hybridization. (openstax.org)

Scope and limitations

Hybridization labels describe idealized orbital constructions. Actual bonding environments need not contain perfectly equivalent hybrids, and calculated hybrid orbitals can be adjusted to reflect their local surroundings. Modern electronic-structure work also constructs localized, hybrid-like bases from molecular or solid-state electronic states. Thus, hybrid-orbital language is not confined to elementary diagrams. (arxiv.org)

Molecular orbital theory describes electrons using orbitals that can extend across several atoms. It offers an alternative representation to a localized hybrid-orbital picture; directional hybrids alone do not provide a complete description of every bonding situation. (openstax.org)

A particularly important limitation concerns historical sp³d and sp³d² descriptions of main-group compounds such as PCl₅ and SF₆. Their geometries do not establish substantial d-orbital participation in bonding. Calculations support descriptions using delocalized or three-center four-electron bonding without requiring the conventional expanded d-hybrid schemes. The geometric labels should therefore not be interpreted as proof that central-atom d orbitals supply the bonds. (sciencedirect.com)