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Summary

Bond order, as introduced by Linus Pauling, is defined as the difference between the number of bonds and anti-bonds.

The bond number itself is the number of electron pairs (bonds) between a pair of atoms. For example, in diatomic nitrogen N≡N the bond number is 3, in ethyne H−C≡C−H the bond number between the two carbon atoms is also 3, and the C−H bond order is 1. Bond number gives an indication of the stability of a bond. Isoelectronic species have the same bond number.

In molecules which have resonance or nonclassical bonding, bond number may not be an integer. In benzene, the delocalized molecular orbitals contain 6 pi electrons over six carbons essentially yielding half a pi bond together with the sigma bond for each pair of carbon atoms, giving a calculated bond number of 1.5. Furthermore, bond numbers of 1.1, for example , can arise under complex scenarios and essentially refer to bond strength relative to bonds with order 1.

Bond order in molecular orbital theory

In molecular orbital theory, bond order is defined as half the difference between the number of bonding electrons and the number of antibonding electrons as per the equation below. This often but not always yields similar results for bonds near their equilibrium lengths, but it does not work for stretched bonds. Bond order is also an index of bond strength and is also used extensively in valence bond theory.

${\text{B.O.}}={\frac {{\text{number of bonding electrons}}-{\text{number of antibonding electrons}}}{2}}\$ Generally, the higher the bond order, the stronger the bond. Bond orders of one-half may be stable, as shown by the stability of H+
2
(bond length 106 pm, bond energy 269 kJ/mol) and He+
2
(bond length 108 pm, bond energy 251 kJ/mol).

Hückel MO theory offers another approach for defining bond orders based on MO coefficients, for planar molecules with delocalized π bonding. The theory divides bonding into a sigma framework and a pi system. The π-bond order between atoms r and s derived from Hückel theory was defined by Charles Coulson by using the orbital coefficients of the Hückel MOs:

$p_{rs}=\sum _{i}n_{i}c_{ri}c_{si}$ ,

Here the sum extends over π molecular orbitals only, and ni is the number of electrons occupying orbital i with coefficients cri and csi on atoms r and s respectively. Assuming a bond order contribution of 1 from the sigma component this gives a total bond order (σ + π) of 5/3 = 1.67 for benzene rather than the commonly cited 1.5, showing some degree of ambiguity in how the concept of bond order is defined.

For more elaborate forms of MO theory involving larger basis sets, still other definitions have been proposed. A standard quantum mechanical definition for bond order has been debated for a long time. A comprehensive method to compute bond orders from quantum chemistry calculations was published in 2017.

Other definitions

The bond order concept is used in molecular dynamics and bond order potentials. The magnitude of the bond order is associated with the bond length. According to Linus Pauling in 1947, the bond order between atoms i and j is experimentally described as

$s_{ij}=\exp {\left[{\frac {d_{1}-d_{ij}}{b}}\right]}$ where $d_{1}$ is the single bond length, $d_{ij}$ is the bond length experimentally measured, and b is a constant, depending on the atoms. Pauling suggested a value of 0.353 Å for b, for carbon-carbon bonds in the original equation:

$d_{1}-d_{ij}=0.353~{\text{ln}}s_{ij}$ The value of the constant b depends on the atoms. This definition of bond order is somewhat ad hoc and only easy to apply for diatomic molecules.