Bond Order Calculator

Bond order counts the net number of electron pairs holding two atoms together. In molecular orbital theory the atomic orbitals of the two atoms combine into an equal number of molecular orbitals, half bonding, with density concentrated between the nuclei, and half antibonding, with a node there. Bonding electrons pull the nuclei together while antibonding electrons push them apart slightly harder, so net bonding is the difference divided by two. Dinitrogen with 10 bonding and 4 antibonding electrons gives 3, matching its 945 kJ/mol dissociation energy and 1.10 angstrom bond length. Dioxygen gives 2, and its final two electrons occupy degenerate pi antibonding orbitals singly with parallel spins, making O2 paramagnetic, which Lewis structures cannot explain. Dihelium gives zero, which is why He2 does not exist. Bond order is inversely related to bond length and directly related to bond strength. Carbon-carbon bonds run about 1.54 angstroms at order 1, 1.34 at order 2 and 1.20 at order 3, with dissociation energies near 346, 614 and 839 kJ/mol. Fractional values arise from delocalisation: each carbon-carbon bond in benzene is 1.5, each nitrogen-oxygen bond in nitrate 1.33, and both show up as bond lengths intermediate between single and double. Two limitations are worth noting. The counting formula is a heuristic tied to a particular orbital diagram and does not reproduce the continuous bond orders of Mayer, Wiberg or natural bond orbital analysis. And inferring magnetism from whether electron counts are odd or even is unreliable: a species can have an even total yet carry two unpaired electrons, exactly as O2 does.

Molecular orbital theory grew out of spectroscopy rather than structural chemistry. In the late 1920s Robert Mulliken at the University of Chicago and Friedrich Hund in Germany independently developed the idea that electrons in a molecule occupy orbitals belonging to the molecule as a whole, delocalised over all nuclei, rather than localised in the shared pairs of the Lewis picture of 1916. Mulliken coined the term orbital in 1932 and received the Nobel Prize in Chemistry in 1966 for this body of work. John Lennard-Jones supplied a decisive early success in 1929, using a linear combination of atomic orbitals to show that the ground state of O2 has two unpaired electrons in antibonding pi orbitals, explaining the long-standing puzzle of liquid oxygen being attracted to a magnet. The bond order expression became a teaching staple once the diatomic correlation diagrams were disseminated through mid-century textbooks, particularly Charles Coulson Valence of 1952. Fractional bond order came from the parallel effort to describe benzene, where Linus Pauling resonance treatment of the 1930s and the Huckel molecular orbital method published by Erich Huckel in 1931 both concluded that the six carbon-carbon bonds are equivalent and intermediate in character, a result confirmed by X-ray and electron diffraction measurements showing a uniform 1.39 angstrom bond length. Later quantum chemists, including Istvan Mayer in 1983, defined bond order as a continuous quantity computable directly from the electron density matrix.