Lewis structures and valence bond ideas are extremely useful, but they do not explain every molecular property naturally. Oxygen provides a famous example. A conventional Lewis structure for O₂ shows all electrons paired, yet gaseous O₂ is paramagnetic and is attracted by a magnetic field. Molecular Orbital Theory explains why.
MO Theory becomes even more useful when electrons are delocalised over several atoms. In conjugated molecules such as butadiene and benzene, several parallel p atomic orbitals interact to produce a set of delocalised π molecular orbitals. Once these energy levels are understood, ideas such as bond order, magnetism, HOMO, LUMO and electronic absorption become different applications of the same underlying model.
Try these five questions before reading the chapter. They test the ideas that will become important when you construct and interpret molecular orbital diagrams.
- 5 diagnostic questions
- MO concepts and electron configurations
- Instant explanation for every answer option
What is Molecular Orbital Theory?
An atomic orbital (AO) describes an allowed one-electron wavefunction associated with an atom. When atoms interact, suitable atomic orbitals can be combined mathematically to form molecular orbitals (MOs).
A molecular orbital belongs to the molecule rather than being assigned permanently to one particular atom. In a diatomic molecule it may extend over both atoms, while in a conjugated molecule a π molecular orbital may extend over several atoms.
Molecular orbitals represent discrete electronic energy levels in a molecule. Each MO can contain a maximum of two electrons with opposite spins.
Linear Combination of Atomic Orbitals
The method used to construct molecular orbitals from atomic orbitals is called the Linear Combination of Atomic Orbitals, or LCAO.
The number of orbitals is conserved during this process.
Thus two atomic orbitals produce two molecular orbitals, four interacting p orbitals produce four π molecular orbitals, and six interacting p orbitals produce six π molecular orbitals.
Bonding and antibonding molecular orbitals
Atomic orbitals behave as wavefunctions and therefore have phase. Phase is often represented by + and − signs in orbital diagrams. These signs do not represent positive and negative electrical charges.
When regions of compatible atomic orbitals having the same phase reinforce one another in the internuclear region, the combination is constructive and a bonding molecular orbital is formed.
The corresponding out-of-phase combination is destructive. Electron density is reduced in the internuclear region and an additional node is introduced, producing an antibonding molecular orbital.
Bonding MO: lower in energy than the parent atomic orbitals and stabilised by constructive interaction.
Antibonding MO: higher in energy than the parent atomic orbitals and destabilised by destructive interaction. It is identified by an asterisk, such as σ* or π*.
Why is a bonding MO stabilised?
In a bonding MO, there is increased probability of finding an electron in the internuclear region. An electron in this region is attracted to both positively charged nuclei, which lowers the energy of the system relative to the separated atoms.
Occupying an antibonding MO has the opposite effect because the additional nodal character reduces electron density in the region that would otherwise help hold the nuclei together.
The existence of an antibonding orbital does not mean that the molecule cannot exist. What matters is how many electrons occupy bonding and antibonding orbitals. A molecule can contain antibonding electrons and still have a positive net bond order.
Non-bonding molecular orbitals
Not every atomic orbital must undergo a strong bonding or antibonding interaction. If an orbital has no suitable partner, incompatible symmetry or only a very weak interaction with other orbitals, the resulting MO may remain approximately similar in energy to the original atomic orbital.
Such an orbital is described as a non-bonding molecular orbital, commonly represented by n.
| Type of molecular orbital | Relative effect on energy | Effect of electrons on bonding |
|---|---|---|
| Bonding | Stabilised | Increases bond order |
| Non-bonding, n | Approximately unchanged by the interaction | No direct contribution to bond order |
| Antibonding, * | Destabilised | Decreases bond order |
σ and π symmetry
The labels σ and π describe the symmetry of a molecular orbital with respect to the internuclear axis.
σ molecular orbitals
A σ molecular orbital has electron density centred around the internuclear axis and has cylindrical symmetry about that axis.
Suitable head-on combinations such as s–s, s–p and p–p combinations along the internuclear axis can produce σ symmetry.
π molecular orbitals
A π molecular orbital has a nodal plane that contains the internuclear axis. Electron density occurs on opposite sides of this nodal plane.
Side-on combinations of parallel p orbitals commonly produce molecular orbitals with π symmetry.
σ and π are symmetry labels. They do not by themselves tell you whether an orbital is bonding or antibonding. Both σ and σ* orbitals, and both π and π* orbitals, can occur.
How electrons are placed into molecular orbitals
Once the relative energies of the molecular orbitals are known, electrons are placed into them using the same principles used for atomic orbitals.
| Principle | Meaning | Application to MO diagrams |
|---|---|---|
| Aufbau principle | Lower-energy orbitals are occupied first. | Fill the MO diagram from the bottom upwards. |
| Pauli exclusion principle | One orbital holds at most two electrons with opposite spins. | A filled MO contains ↑↓. |
| Hund's rule | Degenerate orbitals are singly occupied before pairing. | Essential when filling the two degenerate π or π* orbitals. |
Bond order
The overall balance between bonding and antibonding electrons is expressed by the bond order.
Each electron in a bonding orbital contributes +½ to bond order, while each electron in an antibonding orbital contributes −½.
Higher bond order: generally corresponds to a stronger and shorter bond when closely related species are compared.
Lower bond order: means that more of the bonding stabilisation has been cancelled by occupation of antibonding orbitals.
MO diagram for H₂
Each hydrogen atom has the electronic configuration 1s¹. The two 1s atomic orbitals combine to produce two molecular orbitals: a lower-energy σ1s bonding MO and a higher-energy σ*1s antibonding MO.
The two electrons occupy the lower-energy σ1s orbital.
H₂ therefore contains two bonding electrons and no antibonding electrons.
Both electrons are paired, so H₂ is diamagnetic.
Valence MO diagrams for O₂ and F₂
O₂ and F₂ involve the valence 2s and 2p atomic orbitals. Their filled 1s core orbitals can be omitted from a valence MO diagram because the resulting σ1s and σ*1s orbitals are both filled. Their contributions to bond order therefore cancel, and they do not affect the conclusion about unpaired valence electrons.
For the simple O₂ and F₂ diagrams required here, the relevant valence MO order is:
The two π2p orbitals are degenerate, and the two π*2p orbitals are also degenerate. Degenerate orbitals have the same energy.
The diagrams below show one valid electron arrangement in the degenerate atomic 2p orbitals. Since the three isolated-atom 2p orbitals have the same energy, which particular 2p orbital contains a paired electron before bond formation is arbitrary. What matters is the correct total electron count and Hund's rule.
O₂: two unpaired electrons
Each oxygen atom has the valence configuration 2s²2p⁴, so O₂ contains 12 valence electrons.
The final two electrons enter the two degenerate π*2p orbitals. Because these orbitals have the same energy, Hund's rule requires the electrons to occupy them separately with parallel spins before pairing.
O₂ therefore contains two unpaired electrons and is paramagnetic.
The experimentally observed paramagnetism of O₂ is an important success of Molecular Orbital Theory. A simple Lewis structure does not reveal these two unpaired electrons.
F₂: all valence electrons are paired
Each fluorine atom has the valence configuration 2s²2p⁵, giving 14 valence electrons in F₂.
F₂ has two more valence electrons than O₂. These electrons complete the two degenerate π*2p orbitals.
All electrons in F₂ are therefore paired, so F₂ is diamagnetic.
| Molecule | Bond order | Unpaired electrons | Magnetic behaviour |
|---|---|---|---|
| H₂ | 1 | 0 | Diamagnetic |
| O₂ | 2 | 2 | Paramagnetic |
| F₂ | 1 | 0 | Diamagnetic |
HOMO and LUMO
Two molecular orbitals are particularly useful when interpreting an MO diagram.
HOMO — Highest Occupied Molecular Orbital: the occupied molecular orbital, or degenerate set of occupied molecular orbitals, at the highest occupied energy level.
LUMO — Lowest Unoccupied Molecular Orbital: the empty molecular orbital, or degenerate set of empty molecular orbitals, at the lowest unoccupied energy level.
HOMO and LUMO are sometimes referred to collectively as frontier molecular orbitals.
When a highest occupied energy level is degenerate, more than one orbital can share the same HOMO energy. Similarly, more than one degenerate orbital can share the LUMO energy.
In O₂, for example, the two degenerate π*2p orbitals form the highest occupied energy level and are each singly occupied.
Molecular orbitals are discrete electronic energy levels
Molecular orbitals do not form a continuous range of electronic energies. They correspond to discrete, quantised electronic energy levels.
If an electron is promoted from an occupied molecular orbital to a higher unoccupied molecular orbital, energy must be absorbed.
When that energy is supplied by electromagnetic radiation:
Therefore, a smaller electronic energy gap corresponds to a lower-energy photon, lower frequency and longer wavelength.
Smaller ΔE → lower ν → longer λ.
Delocalised π molecular orbitals in linear polyenes
A conjugated polyene contains a continuous sequence of atoms with parallel p orbitals. These p atomic orbitals can interact across the conjugated region rather than forming completely isolated two-atom π bonds.
N interacting p atomic orbitals produce N delocalised π molecular orbitals.
Butadiene
Buta-1,3-diene contains four sp² carbon atoms. Each contributes one p atomic orbital to the conjugated π system.
Butadiene has four π electrons. They occupy the two lowest-energy π molecular orbitals.
The lowest MO, ψ₁, has no node between adjacent carbon centres. Each successive MO contains one additional node, so the amount of destructive interference increases as we move upwards in energy.
Within the π-MO set of a linear polyene, more nodes correspond to higher orbital energy.
For ground-state butadiene, ψ₂ is the HOMO and ψ₃* is the LUMO.
The π molecular orbitals of benzene
Benzene contains six sp² carbon atoms, each contributing one p atomic orbital to the π system.
The six π molecular orbitals consist of one lowest-energy MO, two degenerate MOs at the next energy, two degenerate MOs at the next higher energy and one highest-energy MO.
Benzene contains six π electrons, so the three lowest-energy π molecular orbitals are occupied.
The two occupied degenerate orbitals immediately below the gap are the HOMOs of benzene, while the two empty degenerate orbitals immediately above them are the LUMOs.
It is important not to describe benzene as possessing only one “delocalised π orbital”. Its six p atomic orbitals form six distinct π molecular orbitals. The six π electrons occupy the three lowest-energy members of this set.
Why increasing conjugation generally reduces the HOMO–LUMO gap
As the length of a linear conjugated system increases, more p atomic orbitals interact. The number of π molecular orbitals therefore increases.
For a related series of linear polyenes, the π energy levels become more closely spaced as conjugation extends. The energy difference between the HOMO and LUMO therefore generally decreases.
This connects Molecular Orbital Theory directly to electronic spectroscopy. The important relationship is:
A smaller energy gap therefore corresponds to a larger wavelength.
Do not simply memorise “more conjugation means longer wavelength”. The molecular explanation is that increasing conjugation changes the π-MO energy levels and generally reduces the relevant occupied-to-empty energy separation.
What level of LCAO is required?
At H3 Chemistry level, you are expected to apply LCAO qualitatively. You should be able to use the phases and symmetry of atomic orbitals to understand the shapes and relative energies of the resulting molecular orbitals.
You should also be able to construct and interpret MO diagrams for simple homonuclear diatomic molecules such as H₂, O₂ and F₂, and for the π systems of benzene and linear polyenes.
Quantitative calculation of LCAO coefficients is not required. Detailed knowledge of orbital mixing between orbitals of the same symmetry is also not required.
A reliable method for solving MO questions
| Step | Question to ask | What to do |
|---|---|---|
| 1 | Which atomic orbitals are involved? | Identify their number, symmetry and relative energies. |
| 2 | How many molecular orbitals must result? | Use N atomic orbitals → N molecular orbitals. |
| 3 | Which combinations are bonding and antibonding? | Consider constructive and destructive combinations. |
| 4 | What is the relative energy order? | Place more stabilised orbitals lower and more antibonding orbitals higher. |
| 5 | How many electrons are present? | Count the required electrons carefully. |
| 6 | How should they be filled? | Apply Aufbau, Pauli and Hund's rule. |
| 7 | What does the completed diagram tell you? | Determine bond order, unpaired electrons, HOMO and LUMO. |
Common Mistakes
| Common mistake | Why it is wrong | Better approach |
|---|---|---|
| Treating + and − orbital phases as positive and negative charges. | They indicate the mathematical phase of the wavefunction, not electrical charge. | Use phase only to determine constructive or destructive orbital combination. |
| Assuming two atomic orbitals form only one molecular orbital. | The number of orbitals must be conserved. | Remember N AOs → N MOs. |
| Calling every higher-energy MO an antibonding orbital without checking how it was formed. | Antibonding character results from destructive orbital combination, not simply from being high in a particular diagram. | Identify bonding character from phase, nodes and the orbital interaction. |
| Pairing the final two electrons of O₂ in one π* orbital. | The π*2pₓ and π*2pᵧ orbitals are degenerate. | Apply Hund's rule and singly occupy both before pairing. |
| Concluding that O₂ is diamagnetic from its Lewis structure. | The Lewis structure does not reveal the actual occupancy of the degenerate π* molecular orbitals. | Use the MO diagram and identify the two unpaired electrons. |
| Forgetting antibonding electrons when calculating bond order. | Antibonding electrons oppose the stabilisation produced by bonding electrons. | Use ½(Nbonding − Nantibonding). |
| Assuming a π MO must be antibonding. | π describes symmetry, not bonding character. | Distinguish π from π*. |
| Saying benzene contains only one delocalised π molecular orbital. | Six p atomic orbitals form six π molecular orbitals. | Describe the complete delocalised π system as a set of six MOs. |
| Saying greater conjugation directly makes light “stronger”. | The relevant change is in the spacing between electronic energy levels. | Link conjugation → smaller energy gap → longer absorption wavelength. |
What you should be able to do
Explain LCAO
Explain qualitatively how atomic orbitals combine to produce molecular orbitals.
Distinguish bonding types
Identify bonding, antibonding and non-bonding molecular orbitals.
Recognise σ and π symmetry
Explain the distinguishing symmetry of σ and π molecular orbitals.
Construct diatomic MO diagrams
Construct and interpret appropriate MO diagrams for H₂, O₂ and F₂.
Calculate bond order
Use bonding and antibonding electron counts to calculate bond order.
Predict magnetism
Determine whether a molecule is paramagnetic or diamagnetic from its MO electron configuration.
Identify HOMO and LUMO
Locate the highest occupied and lowest unoccupied MO energy levels, including degenerate cases.
Interpret conjugated π systems
Relate the number of interacting p atomic orbitals to the number and relative energies of π molecular orbitals.
Connect MO Theory to spectroscopy
Relate electronic energy gaps to photon energy, frequency and wavelength.
These ten questions test whether you can apply MO Theory to orbital counting, electron configurations, bond order, magnetic behaviour, conjugated systems and electronic energy gaps.
- 10 mastery questions
- Application rather than simple recall
- Instant explanation for every answer option
Frequently asked questions
Molecular Orbital Theory questions students often ask
Molecular Orbital Theory can feel abstract because an orbital is a wavefunction rather than a small physical object. These questions clear up the distinctions that most often cause confusion.
Why do two atomic orbitals produce two molecular orbitals?
The number of independent orbital wavefunctions is conserved when one orbital basis is transformed into another. Therefore two atomic orbitals produce two molecular orbitals. More generally, N atomic orbitals produce N molecular orbitals.
Are the + and − signs used for atomic orbital phases electrical charges?
No. They represent opposite phases of the orbital wavefunction. Same-phase regions can combine constructively, while opposite-phase regions can combine destructively. They do not indicate positive or negative electron density.
Does occupying an antibonding orbital mean the molecule immediately becomes unstable?
No. The important quantity is the net balance between bonding and antibonding electrons. O₂ and F₂ both contain electrons in antibonding orbitals, but both still have positive bond orders.
Why are the 1s orbitals omitted from the O₂ and F₂ diagrams?
The 1s orbitals form filled σ1s and σ*1s molecular orbitals. Because each contains two electrons, their contributions to bond order cancel. They also contain no unpaired electrons. A valence-only diagram therefore gives the same conclusions about bond order and magnetic behaviour while being easier to read.
Why can the paired electron be shown in different atomic 2p orbitals before O₂ or F₂ forms?
The three 2p atomic orbitals of an isolated atom are degenerate. Therefore, before the molecular axis is introduced, there is no physically preferred choice of which particular p orbital contains the paired electrons. The diagram shows one valid representation. The important features are the correct electron count and compliance with Hund's rule.
Why is O₂ paramagnetic but F₂ diamagnetic?
O₂ has two singly occupied degenerate π*2p orbitals, giving two unpaired electrons. F₂ has two additional electrons, which pair with these electrons in the π* orbitals. F₂ therefore has no unpaired electrons.
Is every π molecular orbital antibonding?
No. π describes symmetry. A bonding π molecular orbital and an antibonding π* molecular orbital have the same general π symmetry classification but different bonding character.
Can more than one orbital be the HOMO or LUMO?
Yes. If several orbitals are degenerate at the highest occupied energy, they share the HOMO energy. Similarly, degenerate empty orbitals can share the LUMO energy. Benzene provides an important example because both its HOMO level and LUMO level are two-fold degenerate.
Does benzene have one delocalised π molecular orbital or six?
Benzene has six π molecular orbitals because six p atomic orbitals are combined. The entire π electron system is delocalised over the ring, but that does not mean there is only one MO. Six π electrons occupy the three lowest-energy π molecular orbitals.
Why does increasing conjugation generally shift absorption to longer wavelength?
In a related series of conjugated polyenes, increasing the number of interacting p orbitals produces more closely spaced π molecular energy levels. The HOMO–LUMO gap therefore generally decreases. Since ΔE = hc/λ, a smaller transition energy corresponds to a longer wavelength.
Do I need to calculate LCAO coefficients mathematically?
No. The syllabus requires qualitative application of LCAO to obtain and interpret molecular orbital shapes and relative energies. Quantitative treatment of LCAO is not required.
Do I need to learn detailed orbital mixing between orbitals of the same symmetry?
No. Knowledge of orbital mixing between orbitals of the same symmetry is explicitly not required for this syllabus. Focus on the required qualitative MO diagrams and their interpretation.
What should I check after drawing an MO diagram?
Check that the number of molecular orbitals matches the number of atomic orbitals combined, that the total number of electrons is correct, and that Aufbau, Pauli and Hund's rule have been obeyed. Then identify bond order, unpaired electrons, HOMO and LUMO.
Molecular Orbital Theory is much easier to understand when the diagram is treated as an energy story rather than something to memorise. Atomic orbitals combine to form molecular orbitals, electrons fill those orbitals according to their energies, and the resulting occupancy determines bond order, magnetic behaviour and the electronic energy gaps that connect naturally to spectroscopy.