Molecular Orbital Theory for Homonuclear Diatomics
Oxygen's usual Lewis structure pairs all its electrons, yet oxygen is attracted to a magnetic field. Molecular orbital theory uses orbitals spanning both atoms to predict bond order and the number of unpaired electrons.
What a learner can do afterwards
- Combines two atomic orbitals into a bonding and an antibonding combination and explains the energy difference
- Fills a diagram for a second-row diatomic, including the s and p mixing that reorders the levels before oxygen
- Calculates bond order and explains its usual relationship to bond length and dissociation energy for comparable bonds
- Uses the diagram to predict that oxygen is paramagnetic and explains why the shared-pair picture cannot
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You have used atomic orbitals and shared pairs. Molecular orbitals describe electrons across a molecule. Two identical atomic orbitals with suitable overlap form bonding and antibonding combinations. Matching wave phases increase electron density between the nuclei in the bonding combination, lowering its energy. The antibonding combination has a node between the nuclei and higher energy. A node is a region where the wavefunction is zero.
Take the bond as the z axis. End-on p overlap forms sigma orbitals; side-on overlap forms two equal-energy pi orbitals. A star marks antibonding. Fill lower energies first. Each orbital holds at most two electrons with opposite spins. Within an equal-energy set, place one electron in each orbital with parallel spins before pairing. This is the same filling rule you used for atomic orbitals.
B, C, N, O and F supply 3, 4, 5, 6 and 7 valence electrons per atom. The valence ladder starts sigma(2s), sigma*(2s). In B2, C2 and N2, s-p mixing puts bonding pi(2p) below sigma(2p). In O2 and F2, sigma(2p) is below pi(2p). The remaining levels are pi*(2p), then sigma*(2p). Each pi set contains two orbitals. The larger atomic 2s-to-2p gap in oxygen and fluorine reduces mixing enough to leave the sigma-first order.
Bond order = (bonding electrons - antibonding electrons) / 2. Subtract first, then halve. Li2 has two electrons in bonding sigma(2s) and none in antibonding levels, giving (2 - 0) / 2 = 1. Equal bonding and antibonding counts give order zero in this elementary model. Filled sigma(1s) and sigma*(1s) core levels contribute two electrons each and cancel in the subtraction. Count unpaired electrons separately to predict magnetism.
For comparable bonds between the same elements, higher bond order usually means a shorter bond and a greater dissociation energy. Dissociation energy is the energy needed to separate the bonded atoms. Different elements also differ in orbital size and overlap, so bond order cannot rank every bond's length or strength. N2, O2 and F2 have orders 3, 2 and 1, and their dissociation energies follow that order.
Ground-state oxygen is paramagnetic: it is attracted to an applied magnetic field. Its MO configuration leaves one electron in each of the two equal-energy pi* orbitals, giving two unpaired electrons. Oxygen is not a permanent magnet. The usual O=O Lewis structure shows all electrons paired and misses this magnetic behaviour. To predict magnetism for another diatomic, fill its complete scheme and count electrons without an opposite-spin partner.
Add both atoms' valence electrons. Order: sigma(2s), sigma*(2s), pi(2p), sigma(2p), pi*(2p), sigma*(2p) for B2 to N2. For O2 and F2, swap bonding pi and sigma(2p). Each sigma holds 2; each pi pair holds 4. Fill low first, singly across equal-energy orbitals before pairing. Bond order is (unstarred count - starred count) / 2. Singles give paramagnetism; all paired gives diamagnetism. Include empty sets when recording the full occupation.
Choose the level order and fill it, spreading electrons across equal-energy orbitals before pairing. Bond order is (bonding - antibonding electrons) / 2. Count unpaired electrons separately.
Lesson, questions and cards written for this dataset. OpenStax chemistry records were consulted for facts and teaching structure.
Readings and cards written for this dataset. Linked video: Bond length and bond energy, credited to Khan Academy.
More to read
From atomic orbitals to molecular orbitals
An orbital describes an electron's wavefunction. In a molecule, atomic orbitals provide starting functions for a molecular description. Two identical orbitals with suitable overlap give a bonding combination below their starting energy and an antibonding combination above it. The bonding combination has increased electron density between the nuclei. The antibonding combination has a node there, where the wavefunction is zero.
Electron filling follows familiar rules. Start at the lowest level. An orbital holds at most two electrons, with opposite spins. If several orbitals share one energy, occupy them singly with parallel spins before making pairs. This last rule matters for oxygen: its last two valence electrons occupy different pi* orbitals. They do not form a pair in one orbital.
Choosing and filling the valence ladder
Use the internuclear line as the z axis. End-on p overlap gives sigma combinations. The transverse p orbitals give two bonding pi orbitals and two antibonding pi* orbitals. The members of each pi pair have equal energy.
Begin the valence ladder with sigma(2s), then sigma*(2s). For B2, C2 and N2, the next levels are bonding pi, bonding sigma(2p), pi*, then sigma*(2p). For O2 and F2, swap the bonding pi and sigma(2p) levels. S-p mixing changes the energies of existing sigma orbitals; it does not create extra orbitals.
Each B, C, N, O or F atom supplies 3, 4, 5, 6 or 7 valence electrons. Double the count for a neutral diatomic. Fill its ladder using the equal-energy filling rule, then check that the total electron count is unchanged.
Two different questions about the same configuration
A filled diagram answers two different questions. For bond order, subtract the antibonding electron count from the bonding count and divide by two. For magnetism, count the electrons that do not have an opposite-spin partner in their orbital.
In the introductory scheme, B2, C2, N2, O2 and F2 have bond orders 1, 2, 3, 2 and 1. Their unpaired-electron counts are 2, 0, 0, 2 and 0. B2 and F2 therefore share a bond order without sharing magnetic behaviour. Bond order alone does not determine magnetism.
B2 and O2 are paramagnetic in their ground states because they have unpaired electrons. C2, N2 and F2 are diamagnetic, with all electrons paired. Attraction to an applied magnetic field does not make a paramagnetic substance a permanent magnet.
What bond order can tell you
Compare bonds between the same elements in similar environments. A higher bond order usually accompanies a shorter bond and a larger dissociation energy. The dissociation energy measures the energy needed to separate bonded atoms. Comparisons across different elements also depend on orbital size and overlap, so a bond-order number cannot rank every bond.
Core electrons can be omitted from the introductory valence calculation when their bonding and antibonding contributions cancel. The filled sigma(1s) and sigma*(1s) levels each contain two electrons, so including them leaves the bond-order difference unchanged.
Equal bonding and antibonding counts give zero bond order in this model. That result is a statement about the model's count, not proof that weak association or bonding effects beyond the elementary model can never occur.
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