Homonuclear Diatomics: MO Diagrams for Li2 to F2

The O₂ paramagnetism story is not a historical curiosity — it is the everyday basis of respiratory and combustion chemistry, and the same MO reasoning explains why O₂ reacts sluggishly with most closed-shell organic molecules (spin conservation strongly disfavors direct reaction of a triplet with a singlet, which is exactly why combustion needs an initiation step or a catalyst, and why singlet oxygen, an excited electronic state with paired π electrons, is so much more reactive and central to photodynamic therapy and photooxidation chemistry). Superoxide (O₂⁻) and peroxide (O₂²⁻) species, whose bond orders you can read directly off the O₂ MO diagram by adding electrons to π_g, are central to oxidative stress biochemistry, antioxidant enzymology (superoxide dismutase), and battery chemistry in metal–air cells. The N₂ triple bond and its associated inertness underlie the entire logic of industrial ammonia synthesis (Haber–Bosch) and biological nitrogen fixation by nitrogenase enzymes, both of which must find a way to populate N₂'s antibonding π_g* orbitals to weaken the bond enough for reduction — precisely the mechanism illustrated by the N₂/N₂⁻ bond order calculation above. Photoelectron spectroscopy, used to experimentally verify orbital orderings (as in the exam example above), remains a standard tool for probing valence electronic structure in materials science and atmospheric chemistry, where N₂ and O₂ photoionization cross sections matter for modeling the upper atmosphere and auroral emission. More broadly, the entire diatomic MO framework is the direct conceptual seed for band theory in solids (where an infinite chain of atomic orbitals broadens each discrete MO level into a continuous band) and for s-p Mixing in Second-Row Diatomics, which refines exactly the crossover behavior sketched qualitatively here into a quantitative treatment connected to Magnetic Properties and Ground-State Configurations across the whole periodic table.

Ask the tutor: How does adding s–p mixing quantitatively (not just as an energy-order swap) change the computed bond energies for B₂, C₂, and N₂?

Each atom in the second period brings a 1s core (chemically inert, tucked away as a nonbonding pair on each side) plus a valence shell of one 2s and three 2p atomic orbitals. When two identical atoms approach along an internuclear axis — call it $z$ — their atomic orbitals combine in pairs of matching symmetry. The 2s orbitals combine to give a bonding $\sigma_g(2s)$ and antibonding $\sigma_u^*(2s)$. The $2p_z$ orbitals, pointing along the bond axis, combine head-on to give a strongly bonding $\sigma_g(2p)$ and strongly antibonding $\sigma_u^*(2p)$. The $2p_x$ and $2p_y$ orbitals, perpendicular to the axis, combine side-on to give a degenerate pair of bonding $\pi_u(2p)$ orbitals and a degenerate pair of antibonding $\pi_g^*(2p)$ orbitals. That gives ten valence molecular orbitals in total (counting each π pair as two), built from eight valence atomic orbitals (four 2s + four 2p across the two atoms) — wait, more precisely: 2s(A), 2s(B), 2p_z(A), 2p_z(B), 2p_x(A), 2p_x(B), 2p_y(A), 2p_y(B) — eight atomic orbitals combining into eight molecular orbitals: $\sigma_g(2s)$, $\sigma_u^*(2s)$, $\sigma_g(2p)$, $\sigma_u^*(2p)$, and the two degenerate pairs $\pi_u(2p)$, $\pi_g^*(2p)$. The two 1s orbitals on each atom give a further nonbonding-in-practice $\sigma_g(1s)$/$\sigma_u^*(1s)$ pair, which is so tightly bound and so poorly overlapping that it is simply omitted from valence bonding discussions — but it does count for total electron bookkeeping if you're not working with a "valence-only" diagram. The subtlety that makes this topic interesting rather than merely mechanical is that the energy ordering of these valence MOs is not fixed. Naively, you would expect head-on (σ) overlap to always be stronger than side-on (π) overlap, placing $\sigma_g(2p)$ below the $\pi_u(2p)$ pair. This is true for O₂ and F₂. But for Li₂ through N₂, the 2s and 2p atomic orbitals are close enough in energy that the $\sigma_g(2s)$ and $\sigma_g(2p)$ orbitals — both of $\sigma_g$ symmetry — mix with each other (s–p mixing). This mixing pushes $\sigma_g(2p)$ up in energy and $\sigma_g(2s)$ down, far enough that $\sigma_g(2p)$ rises above the $\pi_u(2p)$ pair. The crossover happens because the 2s–2p energy gap shrinks moving right across the period (increasing effective nuclear charge stabilizes 2s less than it destabilizes the mixing channel), so by O and F the σ/π mixing is too weak to cause a swap.

Look carefully at the crossover: in the left-hand (Li₂–N₂) ladder, $\sigma_g(2p)$ sits above the $\pi_u(2p)$ pair; in the right-hand (O₂, F₂) ladder it sits below. Everything else — the σu(2s), the πg pair, the σu*(2p) — stays in the same relative order. That single swap of two adjacent levels is the entire content of "the s–p mixing effect," and it is the only place where the two ladders differ.

Ten valence electrons (five from each nitrogen atom) fill, in order: $\sigma_g(2s)^2$, $\sigma_u^*(2s)^2$, then the early-period order places the $\pi_u(2p)$ pair below $\sigma_g(2p)$, so $\pi_u(2p_x)^2\pi_u(2p_y)^2$, then $\sigma_g(2p)^2$. Every orbital is completely full — no unpaired electrons are possible — which is exactly why N₂ is diamagnetic and chemically so unreactive: eight bonding electrons, two antibonding electrons, giving a bond order of three, the strongest and shortest bond of the whole second-period diatomic series.

The molecular orbitals themselves are linear combinations of atomic orbitals (LCAO), each pair combining constructively (bonding) or destructively (antibonding):

where:

• $2s_A, 2s_B$ — 2s atomic orbitals on the two identical nuclei A and B,
• $c_1, c_2$ — normalization constants (not equal, since bonding/antibonding overlap integrals differ),
• the subscript $g$ (gerade) or $u$ (ungerade) records the parity of the MO under inversion through the bond midpoint, and the asterisk marks an antibonding orbital.

Reading it: adding the two atomic orbitals in phase builds up electron density between the nuclei (bonding, lower energy); subtracting them creates a node in the internuclear region (antibonding, higher energy). The analogous combinations for $2p_z$ (along the bond axis) and for $2p_x, 2p_y$ (perpendicular, giving π symmetry) follow the same logic, just with different overlap geometry. The bond order — the single number that predicts bond length and bond strength trends across the whole series — is:

where:

• $n_{\text{bonding}}$ — number of electrons in bonding MOs (σ_g and π_u types),
• $n_{\text{antibonding}}$ — number of electrons in antibonding MOs (σ_u and π_g types, marked with asterisks),
• the division by 2 reflects that it takes two electrons (a filled bonding MO) to constitute "one full bond's worth" of stabilization.

Reading it: every electron placed in a bonding orbital adds half a bond order unit of stability; every electron placed in an antibonding orbital cancels half a unit. A fully bonding, fully antibonding pair (both filled) contributes zero net bond order — exactly why the core 1s electrons, whose $\sigma_g(1s)^2\sigma_u^*(1s)^2$ combination is entirely filled, never affect the bond order and can be dropped from valence-only diagrams. The energy ordering that decides which configuration you actually get, for the early-period molecules, comes from second-order mixing between same-symmetry orbitals:

where:

• $E_1, E_2$ — the unperturbed energies of the two same-symmetry basis MOs ($\sigma_g(2s)$-like and $\sigma_g(2p)$-like) before mixing,
• $H_{12}$ — the off-diagonal Hamiltonian matrix element coupling them (nonzero only because both have $\sigma_g$ symmetry),
• $E_\pm$ — the resulting mixed-in energies, pushed apart ("repelled") relative to the unperturbed values.

Reading it: this is the two-level, avoided-crossing formula familiar from any two-state mixing problem. When $E_1$ and $E_2$ (the unmixed 2s-based and 2p-based $\sigma_g$ energies) are close — as they are for Li through N, where the 2s–2p atomic energy gap is small — the mixing term dominates, pushing $\sigma_g(2p)$ up enough to cross above $\pi_u(2p)$. Once the atomic 2s–2p gap widens (O, F), $|E_1-E_2|$ dominates over $H_{12}$, the square root barely shifts the levels, and $\sigma_g(2p)$ stays comfortably below $\pi_u(2p)$, restoring the "naive" order.

Example 1 — Foundational: building and checking Li₂. Lithium has the configuration $1s^2 2s^1$, so each Li atom contributes one valence electron (the 1s core is omitted from the valence diagram), for a total of two valence electrons in Li₂. Using either the early- or late-period ordering makes no difference here, because with only two electrons you never reach the π or higher σ levels: they simply fill the lowest available orbital, $\sigma_g(2s)$.

Bonding electrons $n_{\text{bonding}} = 2$, antibonding electrons $n_{\text{antibonding}} = 0$, so

a single bond, consistent with Li₂ being a genuine (if weak) diatomic molecule observable in lithium vapor, with bond length 267 pm and dissociation energy about 105 kJ/mol — much weaker than a typical single covalent bond precisely because the bonding electrons come from diffuse, poorly-overlapping 2s orbitals with a large internuclear distance. All electrons are paired ($\sigma_g(2s)^2$, one filled orbital), so Li₂ is diamagnetic.

Limiting-case check: if you (incorrectly) tried to also fill $\sigma_u^*(2s)$ for Li₂ you would get bond order zero and predict no bond at all — but that only happens once you reach Be₂ (four valence electrons, filling $\sigma_g(2s)^2\sigma_u^*(2s)^2$), which is why Be₂ is famously either unbound or only very weakly bound by van der Waals forces, not a normal covalent diatomic. This correctly separates the case "one electron short of filling both σ(2s) levels" (Li₂, bond order 1) from "exactly filling both" (Be₂, bond order 0).

Example 2 — Applied: comparing bond orders and reactivity of N₂ and O₂ toward reduction. Take the industrially crucial contrast between N₂ (famously inert, requiring the Haber–Bosch catalyst and high pressure/temperature to reduce) and O₂ (readily reduced in respiration and combustion). N₂ has ten valence electrons; using the early-period ordering ($\pi_u(2p)$ below $\sigma_g(2p)$):

Bonding electrons: $2 (\sigma_g 2s) + 4(\pi_u\text{ pair}) + 2(\sigma_g 2p) = 8$. Antibonding electrons: $2(\sigma_u^* 2s)$. So

Now consider one-electron reduction, $\text{N}_2 + e^- \rightarrow \text{N}_2^-$. The extra electron must go into the lowest unoccupied MO, which for N₂ is the doubly-degenerate $\pi_g^*(2p)$ pair (empty in neutral N₂). Adding one electron there gives bonding electrons still 8, antibonding electrons $2 + 1 = 3$, so

a real, measurable weakening (bond order drops from 3 to 2.5) — this is exactly the difficulty of activating N₂: any process that adds electron density to it (as a nitrogenase enzyme active site, or an electride reagent, must attempt) necessarily populates an antibonding π* orbital and pays an energy penalty, which is a large part of why N₂ fixation is so hard and so enzymatically/catalytically demanding. Contrast O₂, which already has bond order 2 with two electrons sitting in $\pi_g^*$. Reducing it by one electron, $\text{O}_2 + e^- \rightarrow \text{O}_2^-$ (superoxide), adds a third electron to the $\pi_g^*$ pair: Bonding electrons unchanged at 8; antibonding electrons $2(\sigma_u^*2s) + 3(\pi_g^*) = 5$, so

O₂ drops readily from bond order 2 to 1.5 upon one-electron reduction (superoxide formation), and further to bond order 1 upon two-electron reduction to peroxide, $\text{O}_2^{2-}$ (antibonding electrons $2+4=6$, bond order $(8-6)/2=1$) — reflecting the real, much greater ease of reducing O₂ stepwise compared with the sharp all-or-nothing difficulty of reducing N₂. This progressive weakening (bond order 2 → 1.5 → 1 for O₂, O₂⁻, O₂²⁻) matches measured O–O bond lengths that lengthen correspondingly (about 121, 128, and 149 pm).

Example 3 — Exam/competition level: photoelectron spectroscopy assignment and s–p mixing detection. Photoelectron spectroscopy (PES) measures the ionization energy required to remove an electron from each occupied MO — by Koopmans' theorem, the ionization energy from orbital $i$ approximately equals $-\varepsilon_i$, the negative of that orbital's energy. Suppose an unknown second-period homonuclear diatomic $X_2$ shows, in order of increasing binding energy (i.e., in the order the outermost electron is removed first), the following pattern of PES bands: a band from a doubly-degenerate orbital appears at lower binding energy than the band immediately below it from a non-degenerate σ orbital, and there is a large gap before the next σ* band, with total valence electron count 10.

Step 1 — count electrons and infer identity. Ten valence electrons with two atoms of the same element means five valence electrons per atom, i.e., a $2s^2 2p^3$ configuration — nitrogen. So $X_2 = \text{N}_2$.

Step 2 — use the PES ordering to confirm level order. The problem states the highest-occupied (lowest binding energy, removed first) band comes from a doubly-degenerate orbital, and the next one down (higher binding energy) is a single, non-degenerate σ orbital. A doubly-degenerate valence orbital can only be one of the π pairs; the σ orbital directly below it in energy must be $\sigma_g(2p)$. This is only consistent with the early-period ordering, $\pi_u(2p)$ above $\sigma_g(2p)$ — precisely the ordering established above for N₂ via s–p mixing, and it is in fact the real experimental PES signature historically used to prove that the naive "σ always below π" picture fails for N₂: real N₂ photoelectron spectra show the HOMO band (from $\sigma_g(2p)$... Careful — let's re-examine: for N₂ using the early-period order, the actual HOMO (last filled, lowest ionization energy) is $\sigma_g(2p)$ (highest energy occupied level, since it sits above $\pi_u(2p)$ in that ordering), not the π pair. So a spectrum showing the lowest-binding-energy band coming from a non-degenerate orbital, with the next band down being doubly degenerate, is the actual experimental signature for N₂: HOMO $= \sigma_g(2p)$ (nondegenerate, removed first, lowest IE ≈ 15.6 eV), next $= \pi_u(2p)$ pair (degenerate, IE ≈ 16.7 eV), then $\sigma_u^*(2s)$ deeper still (IE ≈ 18.8 eV). This ordering — σ HOMO above a doubly-degenerate π band — is the diagnostic fingerprint of s–p mixing, and it is precisely because this ordering was seen for N₂ (not the "naive" reverse) that spectroscopists confirmed the mixed early-period MO diagram over the naive one.

Step 3 — bond order from the same ten electrons. Regardless of whether $\sigma_g(2p)$ or $\pi_u(2p)$ is drawn on top, the filled set of orbitals for N₂ is the same (both are fully occupied either way, since there are enough electrons to fill both) — the ordering swap changes which orbital is the literal HOMO but not the total count of bonding vs. antibonding electrons. Bonding electrons 8, antibonding 2, bond order 3, exactly as computed above — a useful robustness check: PES can distinguish orbital ordering even when it cannot change the bond order, because bond order depends only on total bonding/antibonding populations, not on the fine energy ordering within the bonding manifold.

Friedrich Hund, working in Göttingen in the late 1920s, established the empirical and theoretical rules for filling degenerate orbitals — the same Hund's rule that forces O₂'s two extra electrons into separate π_g* orbitals with parallel spins rather than pairing up in one. Hund arrived at his rules first from spectroscopic term analysis of atoms, then extended the reasoning, alongside Robert Mulliken, to molecules — Mulliken in particular is the person most responsible for developing molecular orbital theory itself as a systematic alternative to the valence-bond picture that dominated organic chemistry reasoning at the time. Mulliken's insistence on building molecular wavefunctions from delocalized orbitals with definite symmetry labels, filled according to the same Aufbau and Pauli principles used for atoms, is the direct intellectual ancestor of every MO diagram drawn today, including the diatomic diagrams in this article.