In molecular orbital (MO) theory, bond order is (bonding electrons − antibonding electrons) ÷ 2. It measures the net bonding contribution in bond-pair units: bonding electrons contribute to a bond, while antibonding electrons offset that contribution. For example, O2 has bond order 2 and two unpaired electrons, which explains why oxygen is paramagnetic.
What bond order means in molecular orbital theory
Molecular orbitals form when atomic orbitals combine mathematically. Unlike an atomic orbital, an MO belongs to the molecule as a whole. Electrons in a bonding MO occupy a distribution that stabilizes bonding; an antibonding MO has a node between the nuclei, and electrons in it oppose bonding.
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The formula is:
Bond order = (number of bonding electrons − number of antibonding electrons) ÷ 2
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A positive bond order indicates a net bonding contribution. If the bonding and antibonding populations cancel, the value is zero; in this introductory model, a stable bond does not form. Bond order is an index, not a direct measurement of bond energy. OpenStax describes it as “a guide to its strength” and says a bond between the same two atoms becomes stronger as bond order increases.
How to calculate bond order from an MO diagram
- Count the relevant electrons for the molecule or ion.
- Fill the molecular orbitals in the energy order shown by the appropriate diagram, following the electron-filling rules.
- Count the electrons in bonding orbitals and, separately, those in antibonding orbitals.
- Subtract the antibonding count from the bonding count, then divide by two.
- Use the result as a net bonding index. When comparing the same pair of atoms, a higher bond order is a guide to a stronger bond, not a universal strength scale or a measured energy.
Worked examples: H2, N2 and O2
The examples below compare electron filling, bonding and antibonding counts, bond order, and magnetic behavior. For N2 and O2, the chosen 2p orbital ordering matters to how the electrons are filled.
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| Molecule | Electron filling | Bonding electrons | Antibonding electrons | Bond order | Unpaired electrons and magnetism |
|---|---|---|---|---|---|
| H2 | Both electrons occupy σ1s; σ1s* is empty. | 2 | 0 | (2 − 0) ÷ 2 = 1 | No unpaired electrons; diamagnetic. |
| N2 | For this second-row molecule, the 2p-derived levels are ordered with π2p below σ2p. The valence configuration is σ2s2 σ2s*2 π2p4 σ2p2. | 8 | 2 | (8 − 2) ÷ 2 = 3 | No unpaired electrons; diamagnetic. |
| O2 | For this second-row molecule, σ2p lies below π2p. After filling the lower levels, the two highest electrons occupy separate π2p* orbitals. | 8 | 4 | (8 − 4) ÷ 2 = 2 | Two unpaired electrons; paramagnetic. |
H2: bond order 1
Two hydrogen 1s atomic orbitals combine to form a lower-energy σ1s bonding MO and a higher-energy σ1s* antibonding MO. Both electrons occupy σ1s, so the calculation is (2 − 0) ÷ 2 = 1. OpenStax notes that H2 is lower in energy than two isolated hydrogen atoms.
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N2 has ten valence electrons. Using the ordering in which π2p lies below σ2p, fill σ2s2, σ2s*2, π2p4, then σ2p2. That gives eight bonding electrons and two antibonding electrons, so the bond order is 3. All occupied orbitals contain paired electrons, making N2 diamagnetic.
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O2: bond order 2 and paramagnetism
Purdue’s MO treatment counts eight valence electrons in bonding orbitals and four in antibonding orbitals for O2. The result is (8 − 4) ÷ 2 = 2. In the O2 diagram, the last two electrons occupy separate degenerate π2p* antibonding orbitals, leaving them unpaired. Substances with unpaired electrons are attracted to a magnetic field, so oxygen is paramagnetic.
A conventional Lewis structure shows an O=O double bond but does not account for those two unpaired electrons. The MO picture explains both the bond order and the magnetic behavior.
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Why the 2p MO ordering changes across second-row diatomics
There is not one universal ordering for every 2p-derived molecular orbital in second-row homonuclear diatomics. The relative energies are affected by s-p mixing. In the convention described by Purdue, B2, C2 and N2 are better represented by an ordering with π2p below σ2p; O2 and F2 are better represented with σ2p below π2p.
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This distinction is part of the electron-filling calculation, not just a drawing preference. Use the ordering appropriate to the molecule rather than applying one second-row diagram to every example.
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Can bond order be fractional?
Yes. Fractional values can arise in contexts such as resonance, where equivalent structures represent bonding shared across multiple positions. Purdue gives sulfur dioxide as an example: averaging an S–O single bond in one Lewis structure and an S=O double bond in another gives a Lewis-structure average bond order of 1.5.
That resonance average is not the same calculation as counting bonding and antibonding electrons in an MO diagram. The number 1.5 is useful context for how bond-order values can be fractional, but it should not be presented as an MO electron-count result for SO2.
Quick Recap
Sources
- OpenStax, Chemistry: Atoms First, section 5.4, “Molecular Orbital Theory”.
- Purdue University Department of Chemistry, molecular orbital theory resource.
- OpenStax, Chemistry: Atoms First 2e, instructional material on molecular orbital theory and N2.
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