Molecular Orbital Theory

01 / UNDERSTAND & APPLY

1. Oxygen exposes the limit of a single Lewis picture

The usual Lewis drawing of O₂ has a double bond and paired lone pairs. It correctly suggests two units of bonding but does not explain the experimentally observed unpaired electrons of ground-state oxygen. A bonding model must account for more than a bond count. Molecular-orbital theory treats electrons as occupying orbitals that belong to the molecular system rather than assigning every electron pair to one localized bond or atom.

An orbital is a quantum-mechanical state described by a wavefunction; its squared magnitude is related to probability density. It is not a little circular track. In a diatomic molecule, the relevant wavefunctions extend over both nuclei. Occupying different molecular orbitals can strengthen, weaken or leave nearly unchanged the bonding between those nuclei.

Valence-bond and molecular-orbital descriptions are useful complementary models. A localized sigma-plus-pi picture is convenient for explaining geometry and restricted rotation in ethene. A delocalized MO occupancy picture is especially useful for bond order and magnetism in diatomic molecules. Do not assume that a failure of one simplified drawing invalidates every insight obtained from that model.

This lesson uses elementary ground-state MO diagrams for isolated light diatomic species, with the ordering stated explicitly. It does not apply a single universal diagram to every molecule, transition-metal complex or solid. Those systems require their own symmetry, energy and interaction information.

02 / UNDERSTAND & APPLY

2. Combine orbitals without inventing extra states

In a linear-combination picture, two compatible atomic orbitals generate two molecular orbitals: one bonding and one antibonding. Constructive phase overlap increases electron density in the internuclear region in the bonding combination. The antibonding combination has an additional node between the nuclei and is higher in energy within the simple pair interaction. An asterisk identifies antibonding character.

Positive and negative signs on orbital lobes indicate wavefunction phase, not positive and negative electric charge. The sign of an entire wavefunction can be reversed without changing its probability density. What matters for overlap is relative phase. Confusing phase colors with charged lobes leads to an incorrect electrostatic explanation of bonding and antibonding combinations.

Effective interaction requires appropriate symmetry, appreciable overlap and reasonably compatible orbital energies. Two orbitals perpendicular by symmetry may have zero net overlap even if their drawings touch. In a heteronuclear bond, the combining atomic orbitals can have different energies and contribute unequally to the resulting MOs. An orbital largely localized on one atom may be effectively nonbonding with respect to the particular bond.

The number of molecular orbitals equals the number of atomic basis orbitals combined. Two 1s orbitals yield one σ1s and one σ*1s; six 2p orbitals from two atoms yield six p-derived MOs. Electron count is a separate question: an orbital can be empty, singly occupied or doubly occupied, with no more than two electrons of opposite spin in one spatial orbital.

03 / UNDERSTAND & APPLY

3. Sigma and pi describe overlap symmetry

Take the internuclear axis as z. End-on overlap of orbitals with the appropriate symmetry produces σ-type molecular orbitals, whose electron density is cylindrically symmetric about that axis in the elementary diatomic picture. Two s orbitals, two p_z orbitals, or a compatible s and p_z pair can generate sigma interactions. “Sigma” is not synonymous with “made from s orbitals.”

Side-by-side overlap of p_x orbitals forms a π_x bonding and π_x* antibonding pair. The p_y pair produces another such set. Each π orbital has a nodal plane containing the internuclear axis; a π* orbital has an additional antibonding node between nuclei. The two perpendicular π orbitals are degenerate in a linear homonuclear diatomic molecule under the usual field-free model.

In the familiar localized bond description, a single bond is one σ bond; a double bond contains one σ and one π; a triple bond contains one σ and two π. There is only one internuclear line for an atom pair, so adding a second sigma bond is not the usual description of a double bond. Count all other single bonds, such as C–H, separately when a question asks for the whole molecule.

Rotation about an isolated σ bond can preserve end-on overlap, although other interactions create rotational barriers. Rotation about the double-bond axis would misalign the p orbitals and disrupt π overlap, so ordinary double bonds have strongly restricted rotation. In a triple bond, two perpendicular π systems surround the same axis; they do not create two different C–C sigma connections.

An MO labeled π is an orbital with a particular symmetry, while a localized “π bond” is a bonding description. They should not be mixed as if every filled π orbital in a delocalized polyatomic system were automatically one independent bond between a chosen pair of atoms. The simple counting rule is reliable for ordinary localized structural formulae within its stated scope.

04 / UNDERSTAND & APPLY

4. Use the correct second-period energy ladder

Fill the lowest-energy MOs first, place no more than two opposite-spin electrons in one orbital, and apply Hund’s rule to degenerate orbitals: occupy them singly with parallel spins before pairing. Orbital energy order must be specified before filling. Using one memorized sequence for all second-period diatomics can give incorrect magnetic predictions.

For the common elementary B₂, C₂ and N₂ diagrams, the valence sequence is σ2s < σ*2s < π2p_x = π2p_y < σ2p_z < π*2p_x = π*2p_y < σ*2p_z. Mixing between sigma orbitals derived from 2s and 2p shifts the relative energies, placing the π2p set below σ2p_z in this standard teaching model.

For O₂ and F₂, the standard valence sequence is σ2s < σ*2s < σ2p_z < π2p_x = π2p_y < π*2p_x = π*2p_y < σ*2p_z. The σ2p_z and π2p ordering is therefore reversed relative to the earlier sequence. The axis labels are arbitrary as long as one axis is consistently identified as the bond axis.

The core σ1s and σ*1s orbitals are both filled for these species. Their bonding and antibonding contributions cancel in the elementary bond-order count, so a valence-only configuration can omit them if that convention is stated. Do not count total electrons as though all were valence electrons and accidentally place the 1s core electrons into the 2p set.

For molecular ions, remove electrons from the highest occupied MO of the neutral reference or add them to the lowest available level, maintaining the appropriate ground-state filling. This is not the same as removing an electron from the atom with the largest atomic number. The electron belongs to a molecular state.

05 / UNDERSTAND & APPLY

5. Bond order counts the net bonding occupancy

The elementary MO bond order is (N_b−N_a)/2, where N_b is the number of bonding electrons and N_a the number of antibonding electrons. Each bonding electron contributes +1/2 and each antibonding electron −1/2 in this model. Nonbonding electrons do not enter this particular difference. Count electrons, not the number of occupied orbital boxes.

H₂ has two bonding electrons and none antibonding, giving order 1. H₂⁺ has one bonding electron, giving order 1/2. Adding a third electron to form H₂⁻ places it in σ*1s, again giving order 1/2. The same formal bond order does not make two differently charged species identical in energy or stability; the model summarizes one aspect of bonding.

Within a closely related series, larger bond order usually corresponds to a shorter, stronger bond. This is most reliable when comparing ions of the same diatomic skeleton. It is not a universal numerical rule across different elements: a bond of order two between one atom pair cannot be assigned a specific energy merely by comparing it with a different single bond.

Zero bond order predicts no net ordinary covalent stabilization in the elementary diagram, as for He₂ with equal σ1s and σ*1s occupancy. Weak dispersion-bound dimers and excited species are outside that simple conclusion. Therefore “bond order zero” should not be inflated into the absolute claim that no association of those atoms can exist under any circumstances.

06 / UNDERSTAND & APPLY

6. Unpaired electrons reveal magnetic behavior

A species with unpaired electrons is paramagnetic in the ordinary molecular sense: it is attracted into a magnetic field. A closed-shell species with all electrons paired is diamagnetic and is weakly repelled. Count unpaired electrons only after applying Hund’s rule to the correct orbital-energy ladder. An even total electron count does not guarantee that all electrons are paired.

For B₂, six valence electrons fill σ2s², σ*2s² and then one electron in each of the two degenerate π2p orbitals. The two remaining electrons do not pair in one π orbital while the other stays empty. B₂ therefore has two unpaired electrons in this model. C₂ fills both π2p orbitals completely and is diamagnetic in its ground state.

O₂ is the central example. Its twelve valence electrons give σ2s²σ*2s²σ2p_z²π2p_x²π2p_y²π*2p_x¹π*2p_y¹. The last two occupy separate degenerate antibonding orbitals. Bond order is (8−4)/2 = 2, but there are still two unpaired electrons. The model explains both the double-bond-like order and paramagnetism that a simple closed-shell Lewis drawing cannot.

The spin-only approximation μ ≈ √[n(n+2)] μ_B uses n unpaired electrons and neglects orbital contributions and other complications. For n=2 it gives √8≈2.83 μ_B. Treat it as an idealized estimate, not an exact magnetization measurement for every molecule or condensed phase. Paramagnetism is also not the same phenomenon as permanent ferromagnetic order in a bulk solid.

07 / UNDERSTAND & APPLY

7. Ionization can strengthen a bond if it removes antibonding electrons

In O₂, the highest occupied orbitals are π*2p. Removing one electron to make O₂⁺ reduces antibonding occupancy from four to three while leaving bonding occupancy at eight. Bond order rises from 2 to 2.5. Thus positive ion formation does not always weaken a bond; what matters is whether the removed electron was bonding or antibonding.

Adding one electron gives O₂⁻, with three electrons in the π* pair and bond order 1.5. One orbital is paired and the other singly occupied, leaving one unpaired electron. Adding a second gives O₂²⁻, fully occupying both π* orbitals, giving order 1 and no unpaired electrons. These formal species correspond to familiar superoxide and peroxide electronic descriptions, although their actual properties in salts depend on the environment.

The simple bond-length trend is O₂⁺ shortest, then O₂, then O₂⁻, then O₂²⁻ longest, following decreasing bond order. The two middle magnetic conclusions are easy to confuse: neutral oxygen has two unpaired electrons, superoxide one, peroxide zero. A negative charge does not automatically mean all spins have paired.

For F₂, fourteen valence electrons occupy through π*2p fully, giving bond order 1 and diamagnetism. If an electron is removed from its antibonding frontier orbital in the elementary model, order increases. Always establish the neutral configuration before applying an ionization shortcut; nitrogen loses a bonding electron whereas oxygen and fluorine lose antibonding electrons.

08 / UNDERSTAND & APPLY

8. Equal electron counts do not imply identical orbitals

Isoelectronic species have the same number of electrons, but their nuclear charges and orbital energies can differ. CO and N₂ each have ten valence electrons and are often assigned elementary bond order three and closed-shell ground states. Their polarity and frontier-orbital distributions are not identical because CO contains unlike nuclei and unequal atomic orbital energies.

In a heteronuclear MO diagram, the lower-energy atomic orbital usually contributes more to the lower-energy bonding combination, while the higher-energy partner contributes more to the antibonding combination in the simplest two-orbital model. The resulting unequal distribution helps connect MO theory to bond polarity. Exact coefficients and energies require more than electron counting.

The elementary N_b−N_a formula and standard ladders are teaching models. Some bonding cases, including detailed C₂ bonding interpretations, are more subtle in higher-level electronic-structure treatments. Use the supplied model for an examination-style question, while avoiding claims that its bond-order number captures every measurable aspect of the molecule.

Nor does a ground-state filling automatically describe an electronically excited state. Promoting an electron can change bond order and spin. If a problem explicitly specifies an excited configuration, count that configuration rather than refill it into the ground state. State whether your answer concerns the ground state, a specified ion or a supplied excited state.

09 / UNDERSTAND & APPLY

9. Original MO and bond-counting practice

These original exercises use the standard elementary ground-state sequences stated above unless another configuration is supplied. Write the valence electron count first, then the occupied orbitals. A correct final bond order with an impossible occupancy is not a valid solution.

Check three independent outputs: total electrons, net bonding count and unpaired-electron count. They answer different questions. If removing an electron changes bonding occupancy, bond order falls by one-half; if it changes antibonding occupancy, it rises by one-half. This provides a fast check after the full configuration is written.