Molecular orbital theory

Result

Molecular orbital theory

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Statement

Two atomic orbitals \(\phi_A,\phi_B\) combine linearly into a bonding molecular orbital \(\psi_+\propto\phi_A+\phi_B\) (in-phase, lower energy, electron density concentrated between the nuclei) and an antibonding orbital \(\psi_-\propto\phi_A-\phi_B\) (out-of-phase, higher energy, with a node between the nuclei). Filling these molecular orbitals by the same aufbau and Hund's-rule procedure used for atoms gives a bond order \(\text{BO}=\tfrac{n_{\text{bonding}}-n_{\text{antibonding}}}{2}\), and correctly predicts molecular magnetism — including the paramagnetism of \(\text{O}_2\), which the Lewis/valence-bond picture (this unit's earlier results) cannot explain.

Why it matters

Molecular orbital (MO) theory and the Lewis/hybridisation picture (lewis-structures-octet, orbital-hybridisation) are two different approximations to the same underlying quantum mechanics, and for most simple molecules they agree on bond order. \(\text{O}_2\) is the classic case where they visibly diverge: a Lewis structure for \(\text{O}_2\) (\(\text{O=O}\), all electrons paired in bonds and lone pairs) predicts a diamagnetic molecule, yet liquid oxygen is experimentally, dramatically paramagnetic — it is attracted to a magnet, a textbook demonstration performable with nothing more than liquid \(\text{O}_2\) and a strong magnet. MO theory, by filling two electrons into a pair of degenerate antibonding \(\pi^*\) orbitals according to Hund's rule (aufbau-hund-pauli, applied one level up, at the molecular rather than atomic scale), predicts exactly this outcome, with no special pleading required.

Hypotheses

Molecular orbitals are approximated as linear combinations of atomic orbitals (LCAO).This is an approximation, not exact quantum mechanics: the true molecular orbitals are solutions of the full multi-nuclear Schrödinger equation, generally without a closed form. LCAO is justified because, near each nucleus, the true molecular orbital must resemble that atom's own atomic orbital (the electron still predominantly "belongs" to whichever nucleus it is near), and because LCAO, despite its simplicity, reproduces bond orders, bond lengths, and magnetic properties correctly for the great majority of simple diatomics. For \(\text{Li}_2\) through \(\text{N}_2\), mixing between the 2\(s\) and 2\(p\) atomic orbitals (since their energies are close for these lighter elements) pushes the \(\sigma_{2p}\) molecular orbital above the doubly-degenerate \(\pi_{2p}\) pair; for \(\text{O}_2\), \(\text{F}_2\), \(\text{Ne}_2\) the larger 2\(s\)–2\(p\) energy gap (increasing effective nuclear charge, per periodic-trends, pulls the 2\(s\) level down faster than 2\(p\) as atomic number increases across period 2) suppresses this mixing, restoring the "naive" order with \(\sigma_{2p}\) below \(\pi_{2p}\).This ordering swap is not a minor detail: applying the wrong order for \(\text{B}_2\) (in the \(\text{Li}_2\)–\(\text{N}_2\) group) would predict a diamagnetic molecule, directly contradicted by \(\text{B}_2\)'s well-established experimental paramagnetism (see Fails without and Problems) — a second, independent experimental confirmation of the s-p mixing order, alongside \(\text{O}_2\)'s paramagnetism confirming the other order.

Proof

\psi_{\pm} = c_1\phi_A \pm c_2\phi_B

The two possible linear combinations of two atomic orbitals: in-phase (\(+\)) constructive interference builds up electron density in the region between the nuclei, while out-of-phase (\(-\)) destructive interference produces a node (zero electron density) between the nuclei. A

E(\psi_+) < E(\phi_A),E(\phi_B) < E(\psi_-)

Electron density concentrated between two positively charged nuclei (as in \(\psi_+\)) is attracted to both nuclei simultaneously, lowering the total energy relative to the separate atomic orbitals — a bonding orbital. A node between the nuclei (as in \(\psi_-\)) removes exactly the electron density that would otherwise screen internuclear repulsion, raising the energy above the atomic orbitals — an antibonding orbital, conventionally marked with an asterisk (e.g. \(\sigma^*\)). A

\text{Fill molecular orbitals lowest-energy first (aufbau); split degenerate orbitals singly before pairing (Hund's rule).}

Exactly the same two filling principles established for atomic orbitals (aufbau-hund-pauli) apply unchanged to molecular orbitals, since both are ultimately about minimising total electronic energy subject to the Pauli exclusion principle. For homonuclear diatomics, 1\(s\) orbitals combine into \(\sigma_{1s}\)/\(\sigma_{1s}^*\), 2\(s\) into \(\sigma_{2s}\)/\(\sigma_{2s}^*\), and the three 2\(p\) orbitals combine into \(\sigma_{2p}\)/\(\sigma_{2p}^*\) (from the head-on-overlapping \(p\) orbital) and two degenerate pairs \(\pi_{2p}\)/\(\pi_{2p}^*\) (from the two sideways-overlapping \(p\) orbitals). A

\text{BO} = \frac{n_{\text{bonding}} - n_{\text{antibonding}}}{2}

By the same electron-pair-per-bond convention already used in the Lewis picture, net bonding character is measured by the surplus of bonding over antibonding electrons, divided by two (electrons per orbital/bond). This convention is validated, not proven from energetics alone, by its agreement with the already-known Lewis bond orders for simple cases (Worked examples: \(\text{N}_2\) triple bond, \(\text{O}_2\) double bond) once the correct filling order (Step 3, Hypotheses) is used. A

Result

\text{BO} = \frac{n_{\text{bonding}}-n_{\text{antibonding}}}{2}

Molecule	Valence e⁻	Bond order	Magnetism
Li₂	2	1	diamagnetic
Be₂	4	0 (unbound)	—
N₂	10	3	diamagnetic
O₂	12	2	paramagnetic (2 unpaired)
F₂	14	1	diamagnetic
Ne₂	16	0 (unbound)	—

Reading. A single filling procedure, applied consistently across the whole period-2 diatomic series, reproduces every known bond order (matching the Lewis single/double/triple bond count exactly) and every known magnetic property, including the two cases (\(\text{Be}_2\), \(\text{Ne}_2\)) where bonding and antibonding electrons exactly cancel and no stable covalent molecule forms.

Scope. The table's filling order (\(\sigma_{2p}\) below \(\pi_{2p}\)) applies to \(\text{O}_2\), \(\text{F}_2\), \(\text{Ne}_2\); \(\text{Li}_2\) through \(\text{N}_2\) use the s-p-mixed order (\(\pi_{2p}\) below \(\sigma_{2p}\)) per Hypotheses — the bond orders shown are correct under whichever order applies to that molecule, and (for \(\text{Li}_2\)/\(\text{Be}_2\), with no occupied \(2p\)-derived orbitals at all) the order distinction does not affect the result.

Corollaries & converses

Bond order correlates with both bond length (higher BO, shorter bond) and bond dissociation energy (higher BO, stronger bond) across the period-2 series: \(\text{N}_2\) (\(\text{BO}=3\)) has both the shortest bond length and the largest dissociation energy of the series, while \(\text{F}_2\) (\(\text{BO}=1\)) has a comparatively long, weak bond — consistent with more net bonding electron density holding the nuclei more tightly together.
Removing an electron from an antibonding orbital (forming a cation) increases bond order and shortens the bond, while removing one from a bonding orbital decreases bond order and lengthens it — the opposite of the intuition that removing any electron should always weaken a bond.
Converse: an experimentally observed paramagnetic diatomic (attracted to a magnetic field) implies, via Hund's rule at the MO level, that its highest-occupied molecular orbitals include a degenerate, partially-filled set (typically the \(\pi^*\) pair) — the presence of paramagnetism is itself indirect evidence for a specific electron configuration, without needing to compute it from scratch.

Fails without

Apply the \(\text{O}_2\)/\(\text{F}_2\)/\(\text{Ne}_2\) filling order (\(\sigma_{2p}\) below \(\pi_{2p}\)) to \(\text{B}_2\) instead of the correct s-p-mixed order: this would fill \(\text{B}_2\)'s two \(2p\)-derived valence electrons into a single \(\sigma_{2p}\) orbital, both paired, predicting a diamagnetic molecule. Experimentally, \(\text{B}_2\) is paramagnetic (verified by its attraction to a magnetic field, exactly as for \(\text{O}_2\)) — only the s-p-mixed order (electrons entering the degenerate \(\pi_{2p}\) pair first, one each per Hund's rule, unpaired) correctly predicts this.
Drop Hund's rule at the molecular-orbital level, force electron pairing whenever possible: \(\text{O}_2\)'s final two electrons would be predicted to pair up in a single \(\pi_{2p}^*\) orbital, leaving the other \(\pi_{2p}^*\) orbital empty and the molecule diamagnetic — directly contradicted by \(\text{O}_2\)'s experimentally observed paramagnetism, the central motivating fact for this entire result.

Common errors

Using a single, fixed filling order for every period-2 homonuclear diatomic, rather than switching between the s-p-mixed and unmixed orders at the correct point in the series (between \(\text{N}_2\) and \(\text{O}_2\)).
Forgetting Hund's rule when filling degenerate \(\pi\) or \(\pi^*\) pairs, leading to an incorrect diamagnetic prediction where the true molecule is paramagnetic (directly the \(\text{O}_2\)/\(\text{B}_2\) error above).
Assuming bond order must always be a whole number; odd-electron species and molecular ions (e.g. \(\text{O}_2^+\), Problems) routinely have half-integer bond orders, which are entirely physical, not a sign of an error.
Applying MO filling to core (1\(s\)) electrons when only a qualitative valence-level bond order is needed; core-level bonding and antibonding contributions are, to excellent approximation, equal and cancel (the core electrons remain essentially atomic and do not participate meaningfully in bonding), so most working diagrams for period-2 diatomics omit the 1\(s\)-derived orbitals entirely and count only valence electrons, as done throughout this Result.

Discussion

Molecular orbital theory was developed chiefly by Robert Mulliken and Friedrich Hund from the late 1920s through the 1930s, in direct competition with the valence-bond approach that Linus Pauling was simultaneously developing (the same Pauling responsible for hybridisation theory, orbital-hybridisation). Mulliken received the 1966 Nobel Prize in Chemistry substantially for this work. For decades the two approaches were seen as rival schools; \(\text{O}_2\)'s paramagnetism became the standard textbook example cited in MO theory's favour, since the valence-bond/Lewis picture (predicting a fully-paired, diamagnetic \(\text{O=O}\)) offered no natural way to accommodate it without ad hoc modification.

The modern view treats MO and valence-bond/hybridisation theory as complementary approximations rather than competitors: valence-bond theory (with hybridisation) gives an intuitive, localised picture well suited to predicting shape and reactivity, particularly in organic chemistry, while MO theory gives a delocalised picture better suited to predicting spectroscopic properties, ionisation patterns, and magnetism. Full modern quantum-chemical calculations do not strictly use either simplified picture, but both remain in active pedagogical and even research use because each captures different chemically-useful information cheaply.

Common misconception: that MO theory and the Lewis/VSEPR/hybridisation picture are mutually exclusive, with one being "right" and the other "wrong." Both are approximations to the same underlying quantum mechanics; where they agree (the great majority of ordinary closed-shell molecules), either is a valid, useful description, and where they disagree (as for \(\text{O}_2\) or \(\text{B}_2\)), it is specifically because the Lewis picture's simplifying assumption (electrons always pair up into two-centre bonds and lone pairs) breaks down for open-shell configurations that MO theory, built around delocalised orbitals and explicit degeneracy, handles naturally.

Worked examples

\text{N}_2:\ 10\ \text{valence e}^-,\ \sigma_{2s}^2\sigma_{2s}^{*2}\pi_{2p}^4\sigma_{2p}^2 \ (\text{s-p-mixed order}),\ \text{BO}=\frac{8-2}{2}=3

All electrons paired (no partially-filled degenerate set), so \(\text{N}_2\) is diamagnetic; the bond order of exactly 3 matches the familiar Lewis triple bond \(\text{N}{\equiv}\text{N}\), and correctly identifies \(\text{N}_2\) as having the shortest, strongest bond in the period-2 diatomic series (Corollaries). A

\text{O}_2:\ 12\ \text{valence e}^-,\ \sigma_{2s}^2\sigma_{2s}^{*2}\sigma_{2p}^2\pi_{2p}^4\pi_{2p}^{*2} \ (\text{unmixed order}),\ \text{BO}=\frac{8-4}{2}=2

The final two electrons fill the doubly-degenerate \(\pi_{2p}^*\) pair singly, one in each orbital (Hund's rule), leaving two unpaired electrons and making \(\text{O}_2\) paramagnetic — while the bond order of 2 still matches the familiar Lewis double bond \(\text{O=O}\), so MO and Lewis theory agree on bond order here even as they differ sharply on magnetism. A

\text{N}_2:\ \text{BO}=3,\ \text{diamagnetic}; \qquad \text{O}_2:\ \text{BO}=2,\ \textbf{paramagnetic}

Reading. The identical filling procedure applied to two adjacent period-2 diatomics gives, in one case, full agreement with simple Lewis-structure intuition, and in the other, the same bond order but a qualitatively different (and experimentally verifiable) magnetic prediction.

Scope. Every homonuclear period-2 diatomic in the Result's table is filled by the identical aufbau-plus-Hund procedure, with only the s-p-mixed/unmixed order distinction (Hypotheses) changing between the two halves of the series.

Problems

Determine the valence electron configuration, bond order, and magnetism of \(\text{F}_2\) (14 valence electrons, unmixed order).
Solution
\(\sigma_{2s}^2\sigma_{2s}^{*2}\sigma_{2p}^2\pi_{2p}^4\pi_{2p}^{*4}\) (total \(2+2+2+4+4=14\)). Bonding electrons: \(2+2+4=8\); antibonding: \(2+4=6\). \(\text{BO}=\tfrac{8-6}{2}=1\), matching the familiar single bond \(\text{F}\text{–}\text{F}\). All orbitals are completely filled (both \(\pi_{2p}^*\) orbitals fully occupied, no unpaired electrons), so \(\text{F}_2\) is diamagnetic.
\(\text{Ne}_2\) would have 16 valence electrons. Determine its bond order under the unmixed filling order, and explain why this result is consistent with neon not forming a stable diatomic molecule.
Solution
Filling all valence molecular orbitals: \(\sigma_{2s}^2\sigma_{2s}^{*2}\sigma_{2p}^2\pi_{2p}^4\pi_{2p}^{*4}\sigma_{2p}^{*2}\) (total \(2+2+2+4+4+2=16\)). Bonding electrons: \(2+2+4=8\); antibonding: \(2+4+2=8\). \(\text{BO}=\tfrac{8-8}{2}=0\): every bonding orbital's stabilisation is exactly cancelled by a corresponding antibonding orbital, so there is no net covalent attraction holding two neon atoms together, consistent with neon (like all noble gases) not forming a stable, covalently-bound diatomic molecule.
\(\text{B}_2\) has 6 valence electrons and uses the s-p-mixed order (\(\pi_{2p}\) below \(\sigma_{2p}\)). Determine its electron configuration, bond order, and magnetism, and state which experimental observation this configuration is consistent with.
Solution
\(\sigma_{2s}^2\sigma_{2s}^{*2}\pi_{2p}^2\) (total \(2+2+2=6\)). The final two electrons enter the doubly-degenerate \(\pi_{2p}\) pair; by Hund's rule they occupy the two orbitals singly rather than pairing in one, leaving \(\text{B}_2\) with 2 unpaired electrons and paramagnetic. Bonding electrons: \(2+2=4\); antibonding: \(2\). \(\text{BO}=\tfrac{4-2}{2}=1\), a single bond. This matches \(\text{B}_2\)'s experimentally observed paramagnetism (Fails without) — a result that specifically requires the s-p-mixed order; the unmixed order would have incorrectly predicted a diamagnetic \(\sigma_{2p}^2\) configuration instead.
The dioxygenyl cation \(\text{O}_2^+\) (formed by removing one electron from neutral \(\text{O}_2\)) has 11 valence electrons. Determine its bond order, and predict, using the Corollaries' bond-length trend, whether its O–O bond is longer or shorter than neutral \(\text{O}_2\)'s.
Solution
Removing one electron from \(\text{O}_2\)'s highest-occupied orbital (one of the two degenerate \(\pi_{2p}^*\) orbitals) gives \(\sigma_{2s}^2\sigma_{2s}^{*2}\sigma_{2p}^2\pi_{2p}^4\pi_{2p}^{*1}\) (total \(2+2+2+4+1=11\)). Bonding electrons: \(2+2+4=8\); antibonding: \(2+1=3\). \(\text{BO}=\tfrac{8-3}{2}=2.5\) — a legitimate half-integer bond order, higher than neutral \(\text{O}_2\)'s \(\text{BO}=2\). Since the removed electron came from an antibonding orbital, net bonding character increases; by the Corollaries' bond-order/bond-length correlation, \(\text{O}_2^+\)'s bond is predicted to be shorter (and stronger) than neutral \(\text{O}_2\)'s — consistent with the experimentally measured O–O bond length in \(\text{O}_2^+\) being shorter than in \(\text{O}_2\).