Homonuclear Diatomic MOs

The O₂ paramagnetism result is not a museum piece — it is central to why O₂ activation chemistry in biology and catalysis is hard. Because ground-state O₂ is a triplet (two unpaired, parallel-spin electrons) while almost all organic molecules are closed-shell singlets, direct reaction of O₂ with organic substrates is spin-forbidden and kinetically slow at room temperature — this is precisely why life needs metalloenzymes (cytochrome P450s, hemoglobin's partner oxidases, and related metal centers) to first bind and activate O₂, converting it to a reactive singlet-compatible or metal-bound form before it reacts efficiently with organic carbon frameworks. The MO diagram derived above is the entire reason this spin barrier exists. The bond-order/bond-length correlation demonstrated for N₂/N₂⁺/N₂⁻ underlies interpretation of gas-phase photoelectron spectroscopy and mass spectrometry of diatomic radical cations and anions used in atmospheric and combustion chemistry — N₂⁺ is a genuine, spectroscopically characterized species in auroral emission and in plasma physics, and its bond length and vibrational frequency are used as textbook calibration points for exactly this MO bond-order logic. Superoxide (O₂⁻) and peroxide (O₂²⁻) chemistry, predicted correctly above from bond order and magnetism, are central to oxidative stress biochemistry, battery chemistry (metal–air batteries generate and consume superoxide/peroxide intermediates), and industrial bleaching and oxidation processes. More broadly, this same LCAO/aufbau/Hund machinery, generalized to unlike atoms, is the direct route into Heteronuclear Diatomic MOs (needed for CO, NO, and HF, all of industrial or biological importance — CO as a ligand in hemoglobin poisoning and in industrial carbonylation catalysis, NO as a biological signaling radical), and generalized to quantitative spin counting it becomes Magnetism from MO Diagrams, used routinely to predict and rationalize the magnetic behavior of molecular radicals and small inorganic clusters in materials chemistry.

Ask the tutor: How exactly does the shrinking 2s–2p energy gap across Period 2 cause σ(2p) to drop below π(2p) at O₂, in terms of orbital mixing coefficients?

Start from the rule that governs every diatomic in this row: atomic orbitals combine only with partners of the same symmetry along the bond axis, and mixing is strongest when the two atomic orbitals are close in energy. Along the internuclear (z) axis, an s orbital and a $p_z$ orbital both have cylindrical symmetry with no nodal plane containing the axis — both are labeled σ. The $p_x$ and $p_y$ orbitals on each atom have one nodal plane containing the axis and are labeled π; they combine only with π-symmetry partners on the other atom, never directly with σ orbitals, because a σ- and π-symmetry orbital product integrates to zero by symmetry (their overlap integral vanishes exactly, independent of the energies involved). For H₂ there is only one valence orbital per atom (1s), so the whole molecule reduces to the textbook two-orbital problem: in-phase combination gives σ₁ₛ (bonding, lower in energy, node-free between nuclei), out-of-phase combination gives σ₁ₛ (antibonding, higher in energy, with a nodal plane bisecting the bond). Both electrons of H₂ occupy σ₁ₛ, giving bond order 1 — a single covalent bond, matching the Lewis picture exactly. For Period 2, each atom contributes four valence atomic orbitals: 2s, $2p_x$, $2p_y$, $2p_z$ (taking z as the internuclear axis). Four orbitals per atom, eight atoms' worth total, means eight molecular orbitals must appear. The 2s orbitals combine to give σ(2s) and σ(2s). The two $2p_z$ orbitals, pointing directly along the bond, combine head-on to give σ(2p) and σ(2p). The $2p_x$ orbitals combine side-on to give one bonding π(2p) and one antibonding π(2p); the $2p_y$ orbitals do the same, giving a second, energy-degenerate π(2p)/π(2p) pair. So the π(2p) bonding level is doubly degenerate, and so is π(2p) — each holds up to four electrons total (two orbitals × two electrons), which is exactly why Hund's rule becomes decisive for O₂. The one subtlety that changes the whole story partway across the row is s–p mixing: the σ(2s), σ(2s) orbitals are not perfectly "pure" 2s combinations, and the σ(2p) orbital is not pure $2p_z$. Because σ-symmetry MOs built from 2s and from $2p_z$ have the same* symmetry label ($\sigma_g$ or $\sigma_u$ depending on convention), they are allowed to mix with each other, and they do so more strongly when the 2s–2p energy gap on the atom is small. Early in the period (Li, Be, B, C, N) the 2s–2p gap is small, mixing is large, and it pushes σ(2p) up above π(2p). Late in the period (O, F, Ne) the 2s–2p gap has grown large enough that mixing is weak, and σ(2p) drops back below π(2p), which is where a naive "no-mixing" MO diagram would have placed it all along. This crossover is the single most commonly mis-drawn feature of Period 2 MO diagrams, and it is developed in full in s-p Mixing and Orbital Ordering; here it is simply built in as an empirical, well-verified fact for each element.

The bond order relationship already stated is the single quantitative handle on everything else:

where:

• $n_{\text{bonding}}$ = number of electrons in bonding MOs (σ and π combined),
• $n_{\text{antibonding}}$ = number of electrons in antibonding MOs,
• non-bonding electrons (rare in this row, but relevant once core orbitals are included loosely) do not enter the count.

Reading: each bonding electron pair contributes a full bond; each antibonding electron pair cancels one. A bond order of zero, as for He₂ or (hypothetically) Ne₂, means no net stabilization — the molecule is not expected to exist as a stable species relative to separated atoms. Bond order in turn correlates with two measurable quantities. Bond length shortens as bond order rises across an isoelectronic-type series because more bonding character pulls the nuclei closer, and — treating the bond loosely as a stretched spring for comparison purposes — bond dissociation energy rises with bond order because more net bonding electron density must be removed to separate the atoms:

where:

• $r_e$ = equilibrium bond length,
• $D_e$ = bond dissociation energy (or bond enthalpy, in the thermochemical convention),
• the comparison is only valid for the same pair of atoms — comparing bond order across different elements (e.g., N₂ vs O₂) still tracks length/energy well empirically in Period 2, but the correlation is not a rigorous law across arbitrary atom pairs.

A second core relationship is simply electron bookkeeping — the orbital count conservation that makes the whole method predictive rather than descriptive:

where:

• $N_{\text{MO}}$ = total number of molecular orbitals produced,
• $N_{\text{AO,A}}, N_{\text{AO,B}}$ = number of valence atomic orbitals contributed by atom A and atom B respectively.

Reading: for Period 2 homonuclear diatomics, $4+4=8$ molecular orbitals always appear (σ2s, σ2s, σ2p, π2p ×2, π2p ×2, σ*2p), regardless of which element or how many electrons are actually available to fill them. This is what makes the diagram a fixed scaffold that you simply pour electrons into, element by element, exactly as with atomic orbital diagrams and the aufbau principle.

Example 1 (foundational): Li₂, the simplest Period-2 case. Each Li atom is $1s^2 2s^1$; only the single valence 2s electron per atom matters (1s is core, dropped as a canceling nonbonding pair as before). Total valence electrons for Li₂: $2\times 1 = 2$. These two electrons fill the lowest available molecular orbital, σ(2s), completely: configuration σ(2s)². No electrons reach σ*(2s) or anything higher. Bonding electrons $n_b = 2$; antibonding electrons $n_a = 0$. Then

Li₂ is predicted to have a single bond, and indeed it is a known (weakly bound, gas-phase) diatomic molecule with bond length 267 pm and bond energy about 105 kJ mol⁻¹ — much longer and weaker than N₂'s triple bond, exactly as expected for a bond order of only 1 built from large, diffuse 2s orbitals with poor overlap. As a limiting check: Be₂ (four valence electrons, filling σ(2s)²σ*(2s)²) gives $BO = \tfrac12(2-2) = 0$; consistent with this, Be₂ is only very weakly bound (a van der Waals-scale complex in its ground state) rather than a normal covalent diatomic, confirming that the "cancel to zero" prediction is not just a bookkeeping curiosity but matches the real absence of a robust Be–Be bond.

Example 2 (applied): C₂ and the surprising bond order 2 vs the "expected" triple bond. Carbon is $1s^2 2s^2 2p^2$; valence electrons per atom = 4, total for C₂ = 8. Carbon lies in the strong-mixing region of the period (small 2s–2p gap), so the ordering is the same "mixed" ladder used for N₂: σ(2s) < σ(2s) < π(2p) < σ(2p) < π(2p) < σ*(2p). Filling 8 electrons from the bottom:

That is $2+2+2+2 = 8$, exactly using up the supply — note that with the mixed ordering, π(2p) fills before σ(2p), so σ(2p) remains completely empty. Bonding electrons: σ(2s) contributes 2, π(2p) pair contributes 4, subtotal $n_b = 6$. Antibonding electrons: σ*(2s) contributes 2, subtotal $n_a = 2$. Then

This is a genuine, experimentally confirmed prediction of MO theory that a simple bonding-pairs argument gets wrong if you assume σ(2p) fills before π(2p): C₂ is a double bond with both bonding pairs being π bonds and no σ(2p) bonding contribution, and — because both π(2p_x) and π(2p_y) are exactly filled (2 electrons each) — the molecule is diamagnetic, with no unpaired electrons, despite naive expectations (based on the O₂ pattern) that partially filled degenerate orbitals might be involved. As a limiting check, compare the isoelectronic CN⁻ or the cation C₂⁺: removing one electron from C₂ takes it out of π(2p) (the HOMO under mixed ordering), giving π(2p_x)²π(2p_y)¹, one unpaired electron, and $BO = \tfrac12(6-1-2) = 1.5$ — a weaker, longer bond than neutral C₂, consistent with photoelectron spectroscopy assignments for this ordering.

Example 3 (exam/competition level): Rank N₂, N₂⁺, and N₂⁻ by bond order and bond length, and identify which ionization removes a bonding vs an antibonding electron. Starting point: neutral N₂ has 10 valence electrons and configuration σ(2s)²σ(2s)²π(2p)⁴σ(2p)² (mixed ordering, as derived above), $BO=3$, bond length 109.8 pm. For N₂⁺ (9 valence electrons), the electron removed on ionization comes from the highest-filled orbital, which under the mixed ordering is σ(2p) — a bonding orbital. New configuration: σ(2s)²σ(2s)²π(2p)⁴σ(2p)¹. Bonding electrons: $2 + 4 + 1 = 7$; antibonding electrons: $2$. Then

Removing a bonding electron weakens the bond, so N₂⁺ should be longer and weaker-bonded than N₂ — and indeed N₂⁺ has a bond length of about 111.6 pm, longer than N₂'s 109.8 pm, exactly tracking the drop from BO 3 to BO 2.5. For N₂⁻ (11 valence electrons), the added electron must go into the lowest unoccupied orbital, which is π(2p) — an antibonding orbital. New configuration: σ(2s)²σ(2s)²π(2p)⁴σ(2p)²π*(2p)¹. Bonding electrons: $2+4+2 = 8$; antibonding electrons: $2 + 1 = 3$. Then

So both ions land at the same bond order of 2.5 despite opposite processes (one removes a bonding electron, the other adds an antibonding one) — a good self-check, since removing a bonding electron and adding an antibonding electron both reduce net bond order by exactly $\tfrac12$ relative to the neutral. The predicted ranking of bond length is

with N₂ shortest (highest BO = 3) and both ions longer (BO = 2.5) — the diagram distinguishes mechanism (bonding electron lost vs antibonding electron gained) while correctly predicting that the magnitude of the bond-order change, and hence the length change, is comparable either way.

The MO picture of diatomics was not the work of one person alone; it emerged through the 1920s–30s from the convergence of quantum mechanics with spectroscopic data on diatomic molecules, and its two most influential early architects were Robert Mulliken and Friedrich Hund. Robert S. Mulliken spent much of his career turning band spectra of diatomic molecules into a systematic orbital-based language — the very σ/π, bonding/antibonding notation used throughout this article is largely his. He insisted on building molecular states from delocalized orbitals spanning the whole molecule, rather than from pre-formed atomic bonding pairs, which is precisely the leap that gets O₂'s magnetism right where a localized-pair Lewis picture cannot. Mulliken also pushed the LCAO approximation from a qualitative sketch toward a genuinely predictive, spectroscopically calibrated tool, cross-checking orbital orderings against measured ionization energies from early photoelectron and absorption spectroscopy — the same kind of check used above to justify the σ(2p)/π(2p) crossover.