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In molecular orbital (MO) theory, bond order is half the number of bonding electrons minus the number of antibonding electrons: bond order = (bonding electrons − antibonding electrons) / 2. It measures the net bonding contribution in the MO model. This electron-counting approach also explains why oxygen, O2, has two unpaired electrons and is paramagnetic.
What bond order means in molecular orbital theory
Molecular orbitals form when atomic orbitals combine mathematically. Unlike an atomic orbital, which is associated with an atom, a molecular orbital belongs to the molecule as a whole. Electrons in a bonding MO occupy a distribution that favors bonding between the nuclei. An antibonding MO has a node between the nuclei, and electrons in it oppose bonding. OpenStax explains that bond order is a guide to bond strength: for a given pair of atoms, a higher bond order generally indicates a stronger bond. It is not itself a measured bond energy or a universal scale for comparing every kind of bond.
The calculation gives each bonding electron a positive contribution and each antibonding electron an offsetting contribution. Dividing the difference by two expresses the net contribution in bond-pair units. Counting only bonding electrons would therefore overstate the net bonding.
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How to calculate bond order from an MO diagram
- Count the electrons included in the diagram, then fill the orbitals in the energy order shown, following the usual electron-filling rules.
- Count bonding electrons across the occupied bonding orbitals.
- Count antibonding electrons across the occupied antibonding orbitals.
- Apply the formula: subtract the antibonding count from the bonding count and divide by two.
- Interpret the result: a positive value indicates net bonding in this model. A value of zero means the bonding and antibonding populations cancel; the cited OpenStax treatment says a stable bond does not form at zero bond order.
Use the orbital ordering appropriate to the molecule. Second-row homonuclear diatomic diagrams can differ in the relative energies of 2p-derived orbitals because of s-p mixing; the ordering is part of the calculation, not merely a diagram style choice.
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Worked examples: H2, N2, and O2
| Molecule | MO filling and counts | Bond-order calculation | Magnetic prediction |
|---|---|---|---|
| H2 | The two electrons occupy the lower-energy σ1s bonding MO; σ1s* is empty. Bonding: 2; antibonding: 0. | (2 − 0) / 2 = 1 | No unpaired electrons; diamagnetic. |
| N2 | Using the second-row ordering appropriate to N2, the eight valence electrons fill σ2s, σ2s*, and the 2p-derived bonding orbitals before the 2p-derived antibonding levels. Bonding: 8; antibonding: 2. | (8 − 2) / 2 = 3 | No unpaired electrons in this filling; diamagnetic. |
| O2 | Using the ordering appropriate to O2, eight valence electrons occupy bonding MOs and four occupy antibonding MOs, including two singly occupied 2p-derived antibonding orbitals. | (8 − 4) / 2 = 2 | Two unpaired electrons; paramagnetic. |
For H2, the two hydrogen 1s orbitals form a lower-energy σ1s bonding MO and a higher-energy σ1s* antibonding MO. OpenStax notes that H2 is lower in energy than two isolated hydrogen atoms.
The N2 and O2 counts are valence-electron counts. In the 2p-derived levels, Purdue’s treatment uses one ordering for O2 and F2, and a model incorporating hybridization for B2, C2, and N2. Do not assume a single 2p ordering applies to every second-row diatomic.
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Why O2 is paramagnetic
In the O2 MO filling, the last two electrons occupy separate 2p-derived antibonding orbitals rather than pairing in one orbital. Those two unpaired electrons make oxygen paramagnetic. The conventional Lewis structure showing an O=O double bond does not display these unpaired electrons, which is one of the useful explanatory advantages of MO theory.
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Bond-order values are not always whole numbers. For sulfur dioxide, Purdue describes an average Lewis-structure bond order of 1.5: one resonance structure has an S–O single bond, and another has an S=O double bond, so the average for the S–O bonds is 1.5. This is a resonance-based Lewis-structure average; it is distinct from calculating MO bond order by subtracting antibonding electrons from bonding electrons and dividing by two.
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Sources
- OpenStax, Chemistry: Atoms First 2e, section 5.4, “Molecular Orbital Theory”
- Purdue University Department of Chemistry, molecular orbital theory resource
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