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Chemistry 2e · Advanced Theories of Covalent Bonding

Molecular Orbital Theory

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On this page 8 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Key takeaway
  6. Check yourself
  7. Study tools
  8. Sources & references

In 30 seconds

Valence bond theory treats a molecule as atoms with localized bonds. theory takes the opposite view: the molecule is one system, and its electrons occupy molecular orbitals spread over the whole molecule. Each MO is built by combining atomic orbitals from all the atoms — the linear combination of atomic orbitals () method — and electrons fill MOs from lowest energy up, following the same rules (Pauli, Hund) used for atoms.

Combining N atomic orbitals produces N molecular orbitals: roughly half are bonding (lower energy, electron density concentrated between nuclei) and half are antibonding (higher energy, with a node between the nuclei, often marked with a star, e.g., σ*). The distribution of electrons between bonding and antibonding orbitals gives the :

bond order = electrons in bonding MOs - electrons in antibonding MOs2

A bond order of zero means no bond (the molecule does not form); 1, 2, or 3 correspond to single, double, and triple bonds. MO theory succeeds where VB theory stumbles: it predicts that O2 is (has unpaired electrons), explains why He2 does not exist, and handles molecules with delocalized or fractional bonding naturally.

Why this matters

  • Magnetic and optical properties: MO theory explains why liquid oxygen is attracted to a magnet (two unpaired electrons), and why many molecules absorb visible light (transitions between MOs) — the basis of color in dyes, and of spectroscopy used in labs and astronomy.
  • Stability and existence: Bond order from MO theory tells you immediately whether a molecule can exist at all (He2, Be2, Ne2 → bond order 0), and how its bond length/energy compare with related species.
  • Ions and radicals: Species like O2+, O2−, NO, and NO2 have non-integer bond orders and unpaired electrons; MO theory handles them cleanly, while Lewis structures require awkward resonance.
  • Semiconductors and materials: Band theory — the extension of MO theory to millions of orbitals — explains why metals conduct, insulators don't, and semiconductors sit between. Every electronic device traces back to this idea.

The college version

Core Concepts

Building molecular orbitals from atomic orbitals

Two atomic orbitals on different atoms combine in two ways:

  • In phase (same sign): the wavefunctions add → electron density piles up between the nuclei → , lower energy than the separate atomic orbitals.
  • Out of phase (opposite signs): the wavefunctions cancel between the nuclei, creating a node → , higher energy.

For s orbitals you get σ and σ* MOs; for p orbitals, σp/σp* (head-on) and degenerate πp/πp* pairs (sideways).

Filling order for second-row diatomic molecules

For O2, F2, and Ne2, the MO energy order is:

σ2s < σ2s* < σ2pz < π2px = π2py < π2px* = π2py* < σ2pz*

For Li2 through N2, the σ2pz and the π2p pair swap order (the π pair sits lower) because of s–p mixing — a detail to check in the textbook's diagrams. In both cases, fill electrons from the bottom up, two per orbital, opposite spins first, then Hund's rule (one electron per degenerate orbital before pairing).

Bond order from electron counts

After filling, subtract antibonding electrons from bonding electrons and divide by 2 (formula above). Examples:

  • H2: 2 electrons in σ1s → bond order 1.
  • He2: 2 in σ1s, 2 in σ1s* → bond order 0 → does not exist.
  • N2: 10 valence electrons → bond order 3 (matches the very strong triple bond).
  • O2: 12 valence electrons → bond order 2.

Paramagnetism: the O2 prediction

Hund's rule forces O2's last two electrons into the two degenerate π* orbitals unpaired. Unpaired electrons make a molecule paramagnetic — weakly attracted into a magnetic field. Liquid O2 is indeed paramagnetic, and MO theory predicted this while localized VB/Lewis pictures could not explain it. Paramagnetism is direct evidence that molecular orbitals are real, not just a bookkeeping device.

Heteronuclear diatomics: CO, NO, HF

When the atoms differ, the atomic orbitals have different energies; the more electronegative atom contributes lower-energy orbitals, so its character dominates the bonding MOs (the bonding pair sits closer to it — polarity emerges naturally). CO has 10 valence electrons → bond order 3 (isoelectronic with N2); NO has 11 → bond order 2.5 (an odd electron → paramagnetic radical). Bond order need not be an integer.

MO vs VB: choosing the right tool

QuestionVB theoryMO theory
Where is each bond?Localized between two atomsElectrons spread over the whole molecule
O2 magnetismCannot explainExplains (unpaired π* electrons)
Existence of He2Predicts a bondBond order 0 → no bond
Geometry/hybridizationExcellent (topics 1–3)Less intuitive
Fractional/odd-electron bondsNeeds resonanceNatural (e.g., NO, 2.5)

Both are models; chemists use whichever fits the question.

How It Works / Step-by-Step Process

Worked example 1: H2, He2, and H2+

Problem. Use MO theory to find bond orders for H2, He2, and H2+, and state which exist.

Solution.

  1. H2: 2 valence electrons. Fill σ1s (2 e⁻) → bond order = (2 − 0)/2 = 1. Exists as a stable single bond.
  2. He2: 4 valence electrons. Fill σ1s (2) then σ1s* (2) → bond order = (2 − 2)/2 = 0. No net bond → He2 does not exist.
  3. H2+: 1 electron. σ1s (1 e⁻) → bond order = (1 − 0)/2 = 0.5. A weak bond; the ion exists (bond energy 255 kJ/mol, less than H2's 436 kJ/mol). Fractional bond orders are fine in MO theory.

Worked example 2: bond order and magnetism of O2

Problem. Calculate the bond order of O2 and predict its magnetic behavior.

Solution.

  1. Count valence electrons: each O contributes 6 → 12 total.
  2. Fill in order for the O2/F2 family: σ2s (2), σ2s* (2), σ2pz (2), π2p pair (4), then the last 2 electrons go singly into the two degenerate π2p* orbitals (Hund's rule) → π2p* has 2 electrons.
  3. Bond order = (8 bonding − 4 antibonding)/2 = 2 — a double bond, matching the 121 pm bond length and 495 kJ/mol bond energy.
  4. Because the two π* electrons are unpaired, O2 is paramagnetic — confirmed by experiment (liquid oxygen is drawn into a magnetic field).

Worked example 3: why CO binds so strongly, and isoelectronic species

Problem. CO and N2 both have 14 total electrons (10 valence). Predict CO's bond order and explain its toxicity.

Solution.

  1. CO valence electrons: C (4) + O (6) = 10 — same count as N2 (2 × 5). These are isoelectronic.
  2. Same filling → bond order 3 → a very strong triple bond.
  3. Polarity: O is more electronegative, so the bonding electrons (especially the σ pair) sit closer to O, but the lone pair on carbon remains available — CO binds to the iron in hemoglobin more tightly than O2 does, blocking oxygen transport. The strength and directionality of CO's bonding MOs explain both its stability and its danger.

Common Confusions

Do Not ConfuseWithDifference
Bonding MOLower energy = always filled firstBonding MOs are lower energy and fill first — but antibonding MOs still get electrons in larger molecules (that's why bond orders drop)
Antibonding MO"No bond" zoneAntibonding MOs are real orbitals that electrons occupy; they subtract from bonding, they don't just disappear
VB "bond pair"MO "bond order"VB pairs are localized between two atoms; MO bond order is a molecule-wide count that can be fractional (2.5 in NO)
O2 paramagnetismN2 paramagnetismO2 has unpaired π* electrons (paramagnetic); N2's electrons are all paired (diamagnetic)
Number of atomic orbitalsNumber of molecular orbitalsCombining N atomic orbitals yields N MOs — never more, never fewer
σ2pz below π2pσ2pz above π2pOrdering differs between Li2–N2 and O2–Ne2 due to s–p mixing; check the diagram for the molecule at hand
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

Think of atomic orbitals as two drums. If you hit them at the same time and the drumheads push together, the sound is stronger between them — that's a bonding orbital. If they push apart, there's a quiet spot (a node) between them — that's an antibonding orbital. Fill the "strong sound" spots first; if as many electrons end up in the quiet spots as in the loud spots, the molecule falls apart. Some molecules, like oxygen, end up with leftover single electrons that act like tiny magnets.

Key takeaways

  • MO theory: electrons occupy molecular orbitals spanning the whole molecule; combining N atomic orbitals gives N MOs (bonding + antibonding).
  • Bond order = (bonding electrons − antibonding electrons)/2; 0 → no bond; 1, 2, 3 → single, double, triple.
  • H2 bond order 1; He2 and Ne2 bond order 0 (don't exist); N2 order 3; O2 order 2; CO order 3 (isoelectronic with N2); NO order 2.5.
  • O2 has two unpaired electrons in π* MOs → paramagnetic; this is MO theory's signature success over VB theory.
  • For O2/F2, MO order: σ2s < σ2s* < σ2pz < π2p < π2p* < σ2pz*; for Li2–N2, π2p sits below σ2pz.
  • Antibonding MOs have a node between nuclei; bonding MOs concentrate density there.
  • In heteronuclear molecules, bonding electrons sit closer to the more electronegative atom → polarity.

Check yourself

5 review questions from the chapter. Try each one, then open the answer.

  1. State the bond order formula and the meaning of a bond order of 0.

    Show answer

    Bond order = (bonding electrons − antibonding electrons)/2; a bond order of 0 means bonding and antibonding effects cancel, so no net bond (molecule doesn't form).

  2. Why does He2 not exist, according to MO theory?

    Show answer

    He2 fills both σ1s and σ1s* with 2 electrons each → bond order 0.

  3. What is the bond order of N2? Of NO?

    Show answer

    N2: 10 valence electrons → bond order 3. NO: 11 valence electrons → bond order (8 − 3)/2 = 2.5.

  4. How does MO theory explain O2's paramagnetism?

    Show answer

    The last two electrons enter the degenerate π2p* orbitals singly (Hund's rule) → two unpaired electrons → paramagnetic.

  5. What does isoelectronic mean, and which famous pair of diatomics is isoelectronic?

    Show answer

    Isoelectronic = same number of valence (total) electrons; CO and N2 (both 10 valence electrons) are the classic pair.

Keep learning

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

molecular orbital (MO)
Orbital spread over the whole molecule, built from atomic orbitals
LCAO
Combining atomic orbitals to make molecular orbitals
bonding MO
Lower-energy MO with density between the nuclei
antibonding MO
Higher-energy MO with a node between the nuclei
bond order
(bonding e⁻ − antibonding e⁻)/2
degenerate orbitals
MOs with the same energy (e.g., the two π2p MOs)
paramagnetic
Attracted into a magnetic field due to unpaired electrons

Sources & references

  1. openstax.org — Chemistry 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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