Molecular-Orbital Theory as a Delocalised Model

Introduction

Molecular-orbital theory describes electrons in orbitals of a whole molecule rather than assigning every electron pair to one atom–atom bond. It is especially useful for diatomic bond order and magnetism. The theory does not discard Lewis or valence-bond diagrams; it answers questions those simpler representations may leave open.

Core explanation

When atomic orbitals of suitable energy and symmetry interact, they produce molecular orbitals. A simple two-atomic-orbital combination produces a lower-energy bonding orbital and a higher-energy antibonding orbital. Electrons fill these orbitals according to energy and spin rules. Bonding orbital occupation tends to stabilise the molecule; antibonding occupation opposes that stabilisation. For diatomics, a common bond-order expression is (number of bonding electrons − number of antibonding electrons)/2.

The word delocalised means a molecular orbital may extend over both atoms of a diatomic molecule or across several atoms in a larger one. It does not mean an electron is spread as a featureless cloud with no structure. Orbital wavefunctions have shapes, nodes and energies. A sigma MO has symmetry around the internuclear axis, while pi MOs have different symmetry; stars mark antibonding orbitals in standard notation.

H₂ provides a first calculation. Two electrons occupy the low-energy σ1s bonding MO, and none occupy σ1s . Bond order is (2 − 0)/2 = 1, consistent with a stable H–H bond. If two electrons also filled σ1s , bonding and antibonding contributions would cancel in this simple model, as in the formal He₂ picture with bond order zero. This does not by itself describe all possible excited states or weak intermolecular interactions.

O₂ reveals a special advantage. Its ground-state MO filling places two electrons singly in two degenerate π orbitals. They remain unpaired under Hund's rule, making O₂ paramagnetic. A simple Lewis O=O diagram supports a rough bond order of two but normally pairs all electrons on paper and does not explain paramagnetism directly. MO theory links the observed magnetic response to electron occupancy.

MO theory is an approximation too. Basic diagrams use simplified orbital energies and may change ordering across the second-period diatomics due to s–p mixing. For complex molecules, quantitative calculations are required. A diagram is meaningful only when the species, electron count and orbital ordering are specified.

Step-by-step reasoning

1. Count all electrons to be placed in the chosen MO diagram. 2. Determine bonding and antibonding orbital energy ordering for the species. 3. Fill from low energy using Pauli and Hund rules. 4. Calculate bond order from occupations and count unpaired electrons. 5. Compare predictions with measured bond length or magnetism where available.

Visual explanation

Draw two H 1s atomic levels on the left and right combining into a low σ1s box and a high σ1s box in the centre. Put two opposite-spin electrons in σ1s. Below draw a small O₂ π pair of equal-energy boxes, each with one electron.

Real-world analogy

A whole orchestra score tracks contributions spread across many instruments, while a duet sheet focuses on one pair. MO theory treats electron states of the whole molecule; a local bond drawing focuses on individual connections. The analogy is about scope, not literal sound waves in a bond.

Real-world example

Oxygen's attraction to a magnetic field is evidence of unpaired electrons. MO occupancy predicts this feature, illustrating how an electronic model can explain an observable property that a simple Lewis picture misses.

Why?

Why does antibonding occupancy reduce bond order? An antibonding orbital has a node or reduced stabilising density between nuclei relative to the corresponding bonding orbital. Filling it offsets some bonding stabilisation in the simple MO count.

Common misconception

“MO theory says there are no bonds between atoms.” It uses delocalised orbitals to represent electrons but still predicts net bonding, bond order and molecular stability. It changes the electron description, not the existence of bonded molecules.

Worked example

For H₂⁺, one electron occupies σ1s and none σ1s . Its bond order is (1 − 0)/2 = 0.5. For H₂, two occupy σ1s, giving bond order 1. The model therefore predicts H₂ has greater bonding occupation than H₂⁺, while both can be bonded species. The half-integer result is a model-derived bond order, not half of a literal electron-pair stick.

Quick check

1. What MO occupation causes ground-state O₂ paramagnetism? Answer: Two unpaired electrons in separate degenerate π antibonding orbitals.

Exam focus

Specify electron count and correct orbital ordering before filling. Use bonding-minus-antibonding count for bond order and unpaired electrons for magnetism. Do not infer an MO diagram directly from a Lewis bond line without checking the species.

Advanced insight

Molecular orbitals are one-electron functions used in many-electron approximations; electron correlation can require methods beyond a single simple diagram. The introductory model nevertheless explains key qualitative results such as O₂ paramagnetism.

Summary

MO theory places electrons in molecule-wide bonding and antibonding orbitals. Their occupations predict bond order and magnetism, including the unpaired electrons of O₂. It complements local bond models while carrying its own assumptions about orbital energies.

Practice questions

1. Calculate H₂ bond order from two bonding and zero antibonding electrons. Answer: (2 − 0)/2 = 1. 2. Calculate H₂⁺ bond order from one bonding electron. Answer: 0.5. 3. What does an asterisk usually denote in MO notation? Answer: An antibonding orbital. 4. Why is a Lewis O=O drawing insufficient to explain paramagnetism directly? Answer: It pairs electrons in the simple diagram and does not show the two unpaired π electrons.