Bond Order · aiwiki.page

Bond order describes the multiplicity or degree of a chemical bond between two atoms. In a conventional structural formula, single, double, and triple bonds have formal bond orders of 1, 2, and 3. More general definitions describe bonding through orbital occupations or the distribution of electron density, allowing noninteger values. Bond order is therefore a model-dependent descriptor rather than a uniquely defined property with one universal numerical value. (goldbook.iupac.org)

Formal bond order

In a Lewis structure, bond order is the number of shared electron pairs assigned to a particular connection. A single covalent bond represents one shared pair, a double bond two pairs, and a triple bond three pairs. For example, the H–H bond in molecular hydrogen has formal bond order 1. (openstax.org)

For ordinary localized multiple bonds, these connections can also be described in terms of orbital symmetry: a single bond is generally a sigma bond, a double bond comprises one sigma and one pi bond, and a triple bond comprises one sigma and two pi bonds. This description distinguishes the components of a multiple bond rather than treating them as identical copies of a single bond. (openstax.org)

Resonance and fractional values

When a single Lewis structure cannot adequately represent a molecule or ion, resonance provides a description using several contributing structures. In a valence-bond treatment, bond order may be expressed as a weighted average of the formal orders assigned by those structures. Equivalent contributors receive equal weight in the simplest formal treatment; inequivalent contributors need not. (goldbook.iupac.org)

Benzene illustrates the familiar fractional assignment. Its two equivalent Kekulé structures place alternating single and double bonds around the ring. Averaging their assignments gives each carbon–carbon connection a formal bond order of

\frac{1+2}{2}=1.5.

All six connections are equivalent, with lengths intermediate between typical carbon–carbon single and double bonds. The value 1.5 is a resonance-based formal description, not a requirement that every quantum-chemical bond index produce exactly that number. (openstax.org)

Likewise, the two equivalent resonance forms of the acetate ion assign one single and one double carbon–oxygen bond in opposite positions, giving a formal average of 1.5 for each connection. Resonance does not mean that the molecule switches between these drawings: they represent one electronic structure with equivalent carbon–oxygen bonds. (openstax.org)

Molecular-orbital definition

In the elementary molecular-orbital model, especially for diatomic species, bond order is calculated as

\mathrm{BO}=\frac{N_{\mathrm b}-N_{\mathrm a}}{2},

where $N_{\mathrm b}$ and $N_{\mathrm a}$ are the numbers of electrons occupying bonding and antibonding molecular orbitals, respectively. A bonding electron contributes $+\tfrac12$ , and an antibonding electron contributes $-\tfrac12$ . (openstax.org)

Bonding and antibonding orbitals arise from combinations of atomic orbitals. In the simplest hydrogen example, the bonding combination has lower energy and electron density between the nuclei; the antibonding combination has higher energy and a node between them. Occupying the latter offsets the bonding contribution of the former. (openstax.org)

Hydrogen therefore has bond order $(2-0)/2=1$ . The same elementary treatment assigns neutral He₂ bond order $(2-2)/2=0$ , indicating cancellation of the bonding and antibonding contributions in that orbital configuration. Half-integer orders arise when their occupation difference is odd. These results belong to the specified orbital model; they are not obtained simply by counting lines in a structural formula. (openstax.org)

Quantum-chemical bond indices

For more general electronic structures, bond indices quantify how density is distributed between atomic centers. Different partitioning schemes yield different definitions. Three important examples are:

Mulliken overlap population: a measure involving the density matrix and the overlap between basis functions on different atoms. Positive and negative overlap populations can indicate bonding and antibonding contributions, respectively. (goldbook.iupac.org)
Wiberg bond index: a sum of squared interatomic density-matrix elements in an orthonormal atomic basis. Natural-bond-orbital implementations commonly evaluate it in a natural atomic orbital basis. Because it is nonnegative, it does not distinguish net bonding from antibonding in the same manner as a signed overlap population. (nbo.chem.wisc.edu)
Mayer bond order: an index constructed from density and overlap matrices, with a spin-density contribution in its general open-shell formulation. (orca-manual.mpi-muelheim.mpg.de)

These quantities provide numerical descriptions beyond integer structural assignments. A calculated value should therefore be identified by its definition: a Wiberg index, Mayer order, and formal resonance order are not interchangeable merely because all describe an atomic connection. (goldbook.iupac.org)

Relationship to bond length and strength

For comparable bonds between the same elements, increasing bond order generally corresponds to a shorter bond and greater bond dissociation energy. Carbon–carbon triple bonds, for example, are generally shorter and stronger than double bonds, which are shorter and stronger than single bonds. This makes bond order useful for interpreting structural and energetic trends. (openstax.org)

The relationship is not a universal conversion rule. Bond energies vary with molecular environment, and tabulated average bond energies are not exact dissociation energies for every molecule containing that bond. Nor does a double bond necessarily have twice the dissociation energy of a single bond. Bond order describes bonding multiplicity or electronic sharing; bond length and dissociation energy describe different physical properties. (openstax.org)

Scope and limitations

The principal limitation is dependence on the chosen description. Formal bond orders count assigned pairs, elementary molecular-orbital orders count net bonding occupations, and quantum-chemical indices partition electronic density. Their values can agree in simple cases without being equivalent definitions. Fractional values likewise require context: a resonance average and a density-matrix index need not represent the same numerical construction. (goldbook.iupac.org)

Bond order is most informative when its definition is explicit and comparisons use a consistent framework. It should not be interpreted as an exact count of independently measurable “bonds,” or as sufficient by itself to determine bond strength across unrelated chemical species. (goldbook.iupac.org)

References

4 Molecular Orbital Theory — Chemistry: Atoms First 2eopenstax.org
11 Describing Chemical Bonds: Molecular Orbital Theory — Organic Chemistryopenstax.org
3 Multiple Bonds — Chemistry 2eopenstax.org
2 Structure and Stability of Benzene — Organic Chemistryopenstax.org
4 Resonance — Organic Chemistryopenstax.org
NBO 7.0 Program Manualnbo.chem.wisc.edu
1 Population Analysis — ORCA 6.1.1 Manualorca-manual.mpi-muelheim.mpg.de
5 Strengths of Ionic and Covalent Bonds — Chemistryopenstax.org