Chemistry: Atoms First 2e · Advanced Theories of Bonding

Molecular Orbital Theory

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In 30 seconds
Why this matters
The college version
Eli explains
Worked example
Key takeaway
Check yourself
Study tools
Sources & references

In 30 seconds

Valence bond theory (earlier in this chapter) pictures a bond as two atomic orbitals overlapping, with electrons localized between specific pairs of atoms. theory takes a different view: when atoms combine, their atomic orbitals merge into molecular orbitals — new orbitals that belong to the whole molecule, not to any single atom. Electrons are assigned to these molecule-wide orbitals, filling them by the same rules used for atoms (Aufbau, Pauli exclusion, Hund's rule).

The theory is built on the linear combination of atomic orbitals (): combining N atomic orbitals produces N molecular orbitals. Roughly half are bonding orbitals (lower energy, electron density concentrated between nuclei, stabilizing the molecule) and half are antibonding orbitals (higher energy, with a node between nuclei, destabilizing). The key quantitative result is :

$Bond order = (number of bonding electrons) - (number of antibonding electrons) 2$

MO theory explains observations that valence bond theory cannot — most famously the paramagnetism of O₂ — and provides a unified picture of bonding, magnetism, and bond strength.

Why this matters

MO theory is the most general bonding model in general chemistry and the gateway to much of modern chemistry and physics. It correctly predicts that O₂ is (attracted into a magnetic field) because it has unpaired electrons in antibonding π orbitals — a fact Lewis structures and simple valence bond theory get wrong. It explains why He₂ does not exist while H₂ does, why N₂ is exceptionally stable, and why removing one electron from O₂ (O₂⁺) actually strengthens the bond. The same orbital picture underlies color in transition-metal complexes, the conductivity of solids (band theory is MO theory on a huge scale), and the frontier-orbital reasoning used to design catalysts.

The college version

Core Concepts

Combining atomic orbitals: LCAO

When two atoms approach, their orbitals interfere like waves. The constructive combination adds amplitudes between the nuclei, producing a bonding molecular orbital (σ or π) that is lower in energy than either starting atomic orbital. The destructive combination cancels amplitude between the nuclei, producing an *antibonding molecular orbital (σ\ or π*)** — written with an asterisk — that is higher in energy and has a node between the nuclei.

For s orbitals on two atoms, the results are σ_1s (bonding) and σ_1s^* (antibonding). For p orbitals oriented along the bond axis, you get σ_2p and σ_2p^*; for p orbitals perpendicular to the axis, you get π_2p and π_2p^* orbitals (two of each, from the two perpendicular directions), with the antibonding versions again starred. Each molecular orbital holds up to two electrons (Pauli), and electrons fill lowest-energy orbitals first, one at a time per orbital before pairing (Hund's rule).

Bond order and bond stability

The bond order formula above is the theory's central tool. A bond order of 1 corresponds to a stable single bond; a bond order of 0 means no net bonding (the molecule is not stabilized). Fractional bond orders (0.5, 1.5, 2.5) occur for ions and radicals and correspond to weaker-than-integer bonds. Higher bond order → shorter, stronger, more stable bond.

Diatomic molecules of the first and second periods

H₂ has two electrons in σ_1s: bond order = (2 − 0)/2 = 1 — a stable single bond. He₂ would need four electrons: two in σ_1s and two in σ_1s^*: bond order = (2 − 2)/2 = 0 — no net bond, which is why helium exists only as isolated atoms.

For second-row diatomics (Li₂ through F₂), the molecular orbitals in order of increasing energy are:

$σ 2s < σ 2s * < π 2p y, π 2p z < σ 2p x < π 2p y *, π 2p z * < σ 2p x *$

(for B₂, C₂, and N₂ the π_2p orbitals lie below σ_2p; for O₂ and F₂ the order of these two switches, with σ_2p lower). Filling this diagram with the valence electrons of each molecule and applying the bond-order formula reproduces the observed bond orders: Li₂ = 1, Be₂ = 0, B₂ = 1, C₂ = 2, N₂ = 3, O₂ = 2, F₂ = 1, Ne₂ = 0 — an excellent match to experiment.

Why O₂ is paramagnetic

Oxygen has 12 valence electrons. Filling the diagram: σ_2s², σ_2s^*2, σ_2p², π_2p⁴, then two electrons go into the degenerate π_2p^* orbitals one at a time (Hund's rule). So O₂ has two unpaired electrons in antibonding orbitals:

$Bond order (O 2) = 10 - 6 2 = 2$

The unpaired electrons make O₂ paramagnetic — attracted into a magnetic field, exactly as measured. Lewis structures cannot explain this; MO theory can.

Common Confusions

Do Not Confuse	With	Difference
Bonding vs antibonding	"Bonding = filled, antibonding = empty"	Bonding/antibonding is about energy and nodes, not occupancy; antibonding orbitals can hold electrons (e.g., in He₂, O₂).
σ vs π molecular orbitals	Atomic orbital labels	MOs are molecule-wide; σ MOs have cylindrical symmetry about the axis, π MOs have a nodal plane containing the axis.
Orbital order for O₂ vs N₂	One universal diagram	For B₂–N₂, π_2p fills before σ_2p; for O₂–F₂ the order flips. Using the wrong order gives wrong configurations.
"Bond order = number of bonds drawn"	(bonding − antibonding)/2	For neutral diatomics the numbers often match, but for ions/radicals (O₂⁺, He₂⁺) you must use the formula.
Paramagnetic vs diamagnetic	One is "magnetic"	Paramagnetic = unpaired electrons (attracted); diamagnetic = all paired (weakly repelled). O₂ is paramagnetic despite having a "double bond."

Eli explains

The same idea, in plain words

Explain it like I’m 10

Think of two flashlights shining toward each other. Where their beams overlap brightly, you get a “bonding” light that holds them together; where the beams cancel out, you get an “antibonding” light that pushes them apart. Electrons fill the bright spots first. If bright and dark spots balance exactly, the two flashlights don’t stick at all — that’s why two helium atoms never hold hands, but two hydrogen atoms do.

Worked example

Example 1: Does He₂⁺ exist? Bond order of a cation

He₂⁺ has 3 electrons total (2 + 2 − 1). Filling: σ_1s², σ_1s^*1.

$Bond order (He 2 +) = 2 - 1 2 = 0.5$

A bond order of 0.5 means a real but weak bond — He₂⁺ does exist as a transient species in mass spectrometers, while neutral He₂ (bond order 0) does not. This shows how MO theory handles ions and radicals that Lewis structures cannot even draw.

Example 2: O₂, O₂⁺, and O₂⁻ — removing or adding an electron

O₂ has 16 total electrons: bond order (10 − 6)/2 = 2. Removing an electron from the highest occupied orbital (a π_2p^* antibonding orbital) gives O₂⁺ with 15 electrons: (10 − 5)/2 = 2.5. Adding an electron gives O₂⁻ (superoxide) with 17 electrons: (10 − 7)/2 = 1.5.

Predictions: O₂⁺ has the strongest, shortest bond (highest bond order) and O₂⁻ the weakest — confirmed by experiment. This pattern is why removing an electron from O₂ strengthens the bond: you are removing an antibonding electron.

Example 3: B₂ and C₂ — the "swapped" orbital order

Boron has 3 valence electrons each, so B₂ has 6. Using the B₂/C₂/N₂ ordering (π_2p below σ_2p): σ_2s², σ_2s^*2, then two electrons enter the degenerate π_2p orbitals singly (Hund's rule), giving π_2p¹, π_2p¹.

$Bond order (B 2) = 4 - 2 2 = 1$

with two unpaired electrons → B₂ is predicted (and found) to be paramagnetic with a single, relatively weak bond. C₂ (8 valence electrons) fills π_2p², π_2p²: bond order (6 − 2)/2 = 2, diamagnetic. The magnetic predictions are confirmed experimentally and are impossible to obtain from simple Lewis structures.

Key takeaways

Combining N atomic orbitals gives N molecular orbitals: about half bonding (lower energy), half antibonding (higher, starred, with a node between nuclei).
Bond order = (bonding electrons − antibonding electrons)/2. Zero → no bond; higher → stronger, shorter bond.
H₂ has bond order 1; He₂ has bond order 0 (does not exist); N₂ has bond order 3; O₂ has bond order 2.
O₂ has two unpaired electrons in π_2p^* orbitals → paramagnetic. This is MO theory's signature success.
For B₂, C₂, N₂ the π_2p orbitals fill before σ_2p; for O₂ and F₂ the order flips. Know which diagram to use for which molecule.
Bonding orbitals concentrate electron density between nuclei; antibonding orbitals have a node there and raise energy.

Check yourself

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

What is the bond order of He₂, and what does that value predict?
Show answer
Bond order = (2 − 2)/2 = 0; it predicts no net bond, so He₂ does not exist as a stable molecule.
Why is O₂ paramagnetic according to MO theory, and why is that a famous result?
Show answer
O₂'s two highest electrons occupy the degenerate π_2p^* orbitals singly (Hund's rule), leaving two unpaired electrons; experiment confirms O₂ is attracted into a magnetic field, which Lewis/valence bond pictures could not explain.
Which has the stronger bond: O₂ or O₂⁺? Explain using bond order.
Show answer
O₂⁺: bond order 2.5 vs 2 for O₂, because removing an electron removes an antibonding electron — a higher bond order means a stronger, shorter bond.
How many molecular orbitals result from combining six atomic orbitals?
Show answer
Six: N atomic orbitals combine to give exactly N molecular orbitals (three bonding, three antibonding).
What is the bond order of N₂, and how does it connect to nitrogen's very high bond energy?
Show answer
N₂ has bond order (10 − 4)/2 = 3, a very strong triple bond, matching its high bond energy (~941 kJ/mol) and low reactivity.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

molecular orbital (MO): An orbital spread over the whole molecule, formed by combining atomic orbitals.
LCAO: Linear combination of atomic orbitals: adding/subtracting wavefunctions to build MOs.
bonding orbital (σ, π): Lower-energy orbital with electron density between nuclei.
antibonding orbital (σ*, π*): Higher-energy orbital with a node between nuclei.
bond order: (bonding e⁻ − antibonding e⁻)/2.
paramagnetic: Attracted into a magnetic field because unpaired electrons are present.
antibonding orbital (σ π): Higher-energy orbital with a node between nuclei.

Sources & references

openstax.org — Chemistry Atoms First 2e

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

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