Chemical Bonding and Molecular Structure · Lesson 9 of 12

Building Molecular Orbitals

“Construct bonding and antibonding orbitals and learn the energy patterns needed to describe a molecule’s electrons.”

Learning Objectives

• Distinguish molecular orbitals from atomic orbitals and explain why molecular orbitals extend over nuclei. • Describe bonding and antibonding combinations using wavefunction phase and electron density. • State the energy, symmetry, and overlap conditions for atomic-orbital combination. • Recognise sigma and pi molecular orbitals and the two relevant second-period energy orders. • Apply Aufbau, Pauli, and Hund rules when preparing to fill molecular orbitals.

An orbital can belong to the whole molecule

Valence bond theory gives a useful localised picture of shared pairs and overlapping orbitals. Molecular orbital theory uses a different starting description: electrons occupy orbitals extending over the molecular framework. This makes it possible to discuss the molecule’s electron distribution and magnetic properties as a whole.

Definition

Molecular orbital

A wavefunction describing an electron in the field of two or more nuclei in a molecule, rather than one isolated nucleus.

An atomic orbital is centred on one atom, whereas a molecular orbital is associated with more than one nucleus. Its squared wavefunction is related to the probability density for the electron. The nuclei are not barriers confining an electron to its original parent atom after a molecular orbital has formed.

The number of molecular orbitals equals the number of atomic orbitals used to construct them. Two input orbitals give two molecular orbitals; eight input orbitals give eight. Electrons fill the resulting orbitals according to their energies and the same occupancy principles used for atoms.

Adding and subtracting wavefunctions

A detailed molecular wavefunction is difficult to obtain directly. The linear combination of atomic orbitals, abbreviated LCAO, provides an approximate construction. For two suitable orbitals, one combination adds their wavefunctions and the other subtracts them. Both possible combinations must be retained.

Two qualitative LCAO combinationsLaTeX

ψ_{bonding} \propto ψ_{A} + ψ_{B} ψ_{antibonding} \propto ψ_{A} - ψ_{B}

The symbols psi_A and psi_B represent atomic wavefunctions. The proportionality sign leaves out normalisation factors; these expressions show the phase relationship rather than a complete numerical calculation.

When matching phases reinforce between the nuclei, constructive interference increases the electron density in that region. The electrons are attracted to both nuclei, and the distribution helps stabilise the system. The resulting bonding molecular orbital is lower in energy than the parent atomic orbitals in the simple symmetric case.

Destructive interference produces a node between the nuclei, where the orbital’s electron probability density is zero. Density is displaced away from the central region, and occupying this antibonding orbital opposes the stabilising effect of a bonding orbital. An asterisk identifies an antibonding orbital.

Bonding and antibonding combinations— The black dots mark nuclei. The dashed line indicates the additional node in the antibonding orbital.

Definition

Bonding and antibonding molecular orbitals

Bonding orbitals contribute to stabilisation when occupied; antibonding orbitals reduce the net bonding effect when occupied. The star in sigma-star or pi-star identifies an antibonding orbital.

An empty antibonding orbital still exists

Constructing two MOs does not mean both must contain electrons. Orbital existence and orbital occupancy are separate questions. Also, an asterisk is an orbital label, not a statement that the whole molecule carries an electric charge.

Example — Counting outputs

Problem
Two atoms each contribute one 2s and three 2p orbitals. How many molecular orbitals arise from this selected set?

1.Each atom contributes four orbitals, so the input total is $4 + 4 = 8$ .
2.The number of output MOs must also be eight. In the simple diatomic construction, they include four bonding and four antibonding combinations.
3.The electron total is counted separately. Eight orbitals could hold up to sixteen electrons, but they are not automatically all occupied.

Which atomic orbitals can combine effectively?

Orbital combination is restricted by energy, symmetry, and spatial overlap. Drawing two orbitals close together does not guarantee a useful interaction. Their energies must be sufficiently similar and their shapes must permit a nonzero overlap with compatible symmetry about the molecular axis.

Condition	Meaning	Illustration
Comparable energies	Very different-energy orbitals interact weakly in the simple construction.	Two equivalent 1s orbitals on identical atoms combine readily.
Compatible symmetry	The overlap must not cancel through equal opposite-phase contributions.	An axial p orbital can combine with an equivalent axial p orbital.
Effective spatial overlap	The orbitals must meet appropriately in space.	Better overlap generally produces a greater bonding–antibonding energy separation.

For a homonuclear diatomic molecule, the two atoms are identical. Their corresponding orbitals have matching energies, making the construction especially simple. By convention we can choose z as the internuclear axis. The p_z orbitals then point along that axis, while p_x and p_y point perpendicular to it.

Example — Testing symmetry

Problem
With z as the molecular axis, compare 2p_z–2p_z and 2p_z–2p_x combinations.

1.The two 2p_z orbitals have compatible symmetry and can overlap head-on along the axis.
2.The 2p_x orbital points perpendicular to the axis. Overlap with 2p_z gives opposite-phase contributions that cancel by symmetry.
3.The fact that the orbitals are both called “2p” does not override the directional symmetry requirement.

Sigma and pi molecular orbitals

A sigma molecular orbital is symmetric on rotation around the internuclear axis. Combining two s orbitals gives sigma and sigma-star MOs. Combining axial p_z orbitals also gives a sigma pair. The suffix identifies the parent orbital type, such as sigma-1s or sigma-2p-z.

Sideways combinations of p_x orbitals produce pi-2p-x and pi-star-2p-x orbitals. The corresponding p_y combination produces a second pi pair with the same energy in a free linear diatomic molecule. Such equal-energy orbitals are called degenerate orbitals.

A bonding pi orbital has density on two sides of the axis and a nodal plane containing the axis. Its antibonding partner has an additional node separating the nuclei. “Pi” alone describes orbital symmetry; the star is what distinguishes the antibonding member. The sigma/pi labels therefore do not replace the bonding/antibonding distinction.

Parent orbitals, z as the axis	Bonding MO	Antibonding MO
1s and 1s	$σ_{1 s}$	$σ_{1 s}^{*}$
2s and 2s	$σ_{2 s}$	$σ_{2 s}^{*}$
2p_z and 2p_z	$σ_{2 p_{z}}$	$σ_{2 p_{z}}^{*}$
2p_x and 2p_x	$π_{2 p_{x}}$	$π_{2 p_{x}}^{*}$
2p_y and 2p_y	$π_{2 p_{y}}$	$π_{2 p_{y}}^{*}$

The two energy orders must be kept distinct

Orbital occupancy depends on energy order, so one diagram cannot be applied indiscriminately to all second-period diatomic molecules. For O₂ and F₂, the sigma-2p-z orbital lies below the two bonding pi-2p orbitals. For B₂, C₂, and N₂, the bonding pi pair lies below sigma-2p-z.

Energy order for oxygen and fluorine moleculesLaTeX

σ_{1 s} < σ_{1 s}^{*} < σ_{2 s} < σ_{2 s}^{*} < σ_{2 p_{z}} < π_{2 p_{x}} = π_{2 p_{y}} < π_{2 p_{x}}^{*} = π_{2 p_{y}}^{*} < σ_{2 p_{z}}^{*}

The less-than sign means lower energy. Equal signs identify the degenerate pi orbitals.

Energy order for boron through nitrogen moleculesLaTeX

σ_{1 s} < σ_{1 s}^{*} < σ_{2 s} < σ_{2 s}^{*} < π_{2 p_{x}} = π_{2 p_{y}} < σ_{2 p_{z}} < π_{2 p_{x}}^{*} = π_{2 p_{y}}^{*} < σ_{2 p_{z}}^{*}

The important change is the relative order of bonding sigma-2p-z and bonding pi-2p orbitals.

Comparing the two second-period MO orders— Focus on the exchange in position of the bonding sigma-2p level and bonding pi-2p pair.

The source diagram associates this difference with 2s–2p mixing in the lighter molecules. At this level the important skill is choosing the correct order before filling electrons. This changes the predicted occupancy of B₂ and C₂, and therefore their magnetic properties.

Rules for filling the diagram

The Aufbau principle places electrons in lower-energy orbitals first. The Pauli exclusion principle allows no more than two electrons in one orbital, with opposite spins. Hund’s rule places electrons singly with parallel spins in degenerate orbitals before pairing them.

Example — Filling two equal-energy orbitals

Problem
How should two electrons be placed in an empty degenerate pi pair?

1.The two pi orbitals have equal energy, so either may receive the first electron.
2.Place the second electron in the other pi orbital with parallel spin, following Hund’s rule.
3.There is then one electron in each orbital and two unpaired electrons. Placing both in the first orbital would miss the required ground-state occupancy pattern.

Reading the energy diagram correctly

A line or box represents one orbital, not one electron. Two adjacent lines at the same height often represent a degenerate pi pair. Electron arrows show occupancy and spin; their direction on the page is not the direction of electron travel through the molecule.

Check your understanding

Before predicting a molecule’s properties, check which orbitals exist, how they are ordered, and how many electrons must fill them. Confusing these stages creates incorrect bond-order and magnetic predictions.

Quick check

Two atomic orbitals used in an LCAO construction produce how many molecular orbitals?

Quick check

Which feature identifies the antibonding combination of two s orbitals?

Quick check

With z as the axis, which pair can form an axial sigma MO?

Quick check

For B₂, C₂, and N₂, the bonding pi-2p pair lies:

Quick check

Two electrons entering an empty degenerate pi pair normally occupy it as:

Practice Problems

Explain how an atomic orbital differs from a molecular orbital in the nuclei associated with it.
Sketch the constructive and destructive combinations of two s orbitals, indicating density and the antibonding node.
List the three conditions for effective atomic-orbital combination and explain one example of failed symmetry overlap.
With z as the axis, identify which combinations of 2p orbitals generate sigma and pi orbitals.
Write the two MO energy sequences for B₂–N₂ and O₂–F₂, highlighting only the part that changes.
Show how one, two, three, and four electrons fill a degenerate pi pair, and count unpaired electrons in each case.
Explain why counting eight available MOs does not tell you how many electrons a particular molecule contains.

Key Takeaways

• Molecular orbitals describe electrons over a molecular framework rather than one isolated atom. • Two input orbitals produce both bonding and antibonding combinations. • Compatible energy, symmetry, and overlap are required for an effective combination. • Sigma/pi symmetry and bonding/antibonding character are separate labels. • Choose the correct energy order, then use Aufbau, Pauli, and Hund rules to fill electrons.

Previous · Lesson 8

Hybridisation and the Shapes of Covalent Molecules

Next · Lesson 10

Molecular Orbital Configurations and Predictions

An orbital can belong to the whole molecule

Adding and subtracting wavefunctions

Which atomic orbitals can combine effectively?

Sigma and pi molecular orbitals

The two energy orders must be kept distinct

Rules for filling the diagram

Check your understanding

Quiz

Practice Problems

Key Takeaways