s-p Mixing in Second-Row Diatomics

The N₂/O₂ crossover is not a historical curiosity confined to introductory courses — it is a live diagnostic and design principle. In atmospheric and combustion chemistry, the unusually strong, mixing-diluted N≡N triple bond (and the correspondingly high N₂ dissociation energy near 945 kJ/mol) explains why N₂ is such a poor ligand and such a thermodynamically stubborn feedstock, motivating the entire field of dinitrogen activation catalysis (industrial Haber–Bosch chemistry and biological nitrogenase enzymes both must overcome the same mixing-stabilized bond). In materials and magnetism, the paramagnetism of O₂ arising directly from the weak-mixing πg SOMO pair underlies everything from the design of oxygen sensors (paramagnetic susceptibility measurement is a real, standard industrial technique for measuring O₂ concentration in gas streams) to understanding triplet oxygen's distinctive photochemistry and its role as a biradical in autoxidation and combustion initiation. In <strong>spectroscopy and instrumentation</strong>, photoelectron spectroscopy (UPS/XPS) routinely uses exactly the band-ordering logic of Example 3 to assign orbital characters in new molecules, and s–p mixing corrections are now standard in interpreting core-to-valence and valence photoemission spectra of diatomics and small polyatomics alike. In <strong>quantitative computational chemistry</strong>, modern Hartree–Fock and density functional calculations reproduce the crossover automatically (it requires no special treatment once the full secular problem is solved self-consistently), but understanding why* it happens — via the widening atomic 2s–2p gap — remains essential for interpreting orbital energy diagrams, predicting trends in isoelectronic series (CO, CN⁻, NO⁺ all inherit qualitatively similar mixing physics), and building chemical intuition before reaching for a computer. The same symmetry-driven mixing logic extends directly into heteronuclear systems, where unequal atomic orbital energies on the two centers introduce additional asymmetric mixing — the natural next step covered under Heteronuclear Diatomics: HF and CO — and into the broader question of predicting magnetic ground states, covered under Magnetic Properties and Ground-State Configurations.

:::quiz[In the two-level mixing formula E± = (H11+H22)/2 ± √[((H11−H22)/2)² + H12²], what happens as H11 → H22 (degenerate zeroth-order levels)?]

• The splitting vanishes completely
• The splitting approaches 2|H12|, the maximum possible value ✓
• The mixing angle θ approaches 0°
• E+ and E− become equal to H12

Ask the tutor: How would the s–p mixing picture change if I tried to apply the same two-level analysis to a third-row diatomic like P₂ or S₂, where 3s–3p gaps and d-orbital involvement complicate the story?

Start from the familiar picture of homonuclear diatomic MOs built from valence 2s and 2p atomic orbitals, as developed for Homonuclear Diatomics: MO Diagrams for Li2 to F2. Placing the internuclear axis along $z$, the atomic orbitals combine according to their symmetry: 2s + 2s and 2p$_z$ + 2p$_z$ combinations are both cylindrically symmetric about the axis (σ symmetry), while 2p$_x$ + 2p$_x$ and 2p$_y$ + 2p$_y$ combinations have a nodal plane containing the axis (π symmetry), as classified in Sigma and Pi Molecular Orbitals and Symmetry Labels. In a naive treatment, one assumes the 2s atomic orbitals combine only with each other and the 2p$_z$ orbitals combine only with each other, producing four σ orbitals (1σg, 1σu, 2σg, 2σu — bonding/antibonding pairs from 2s and from 2p$_z$) that are then simply stacked in energy according to the parent atomic orbital energies. But 1σg, 2σg (often relabeled 3σg once 2p is included) share the same symmetry label, σg. Group theory guarantees that any two orbitals of matching symmetry in a molecule can mix without violating any selection rule; nothing prevents the true molecular eigenstates from being linear combinations of the "naive" 2s-based σg and the "naive" 2p$_z$-based σg orbitals. The true situation is a 2×2 (or larger) secular problem, and its solution pushes the two mixed levels apart — the lower one drops further, the upper one rises further, exactly as in any two-level quantum mechanical interaction. Because the upper σg (mostly 2p$_z$ bonding character) is pushed higher, it can rise above the πu (2p$_x$,2p$_y$ bonding) level if the mixing is strong enough. This is precisely what happens for B₂ through N₂.

The physical origin of the widening gap is orbital penetration and shielding. The 2s orbital has a large-amplitude region very close to the nucleus (better penetration through the 1s core), so it feels a larger effective nuclear charge than the more diffuse, angularly-excluded-from-the-core 2p orbital. As $Z$ increases across the row, this differential penetration effect compounds, and the 2s orbital energy drops faster than the 2p orbital energy. The atomic 2s–2p gap is roughly 5 eV for boron, climbs past 10 eV by nitrogen, and exceeds 15 eV for oxygen and fluorine. Since the strength of configuration mixing between two states scales inversely with their unperturbed energy separation, this widening gap is the single variable controlling the σ/π crossover.

Notice the essential difference: in N₂ the 3σg orbital (the mixed, mostly-2p$_z$-bonding combination) sits above the 1πu pair and is the HOMO; all ten valence electrons pair up, giving a closed-shell, diamagnetic, triple-bonded molecule. In O₂ the 3σg orbital sits below the 1πu pair (which is now filled) and the two extra electrons (twelve valence electrons total) go into the degenerate 1πg* antibonding pair, one each by Hund's rule, giving the two unpaired spins responsible for O₂'s paramagnetism. The ordering flip is not decorative — it is exactly why O₂ sticks to a magnet and N₂ does not.

The starting point is the observation that the "naive" MOs $2\sigma_g$ (bonding combination of 2s) and $3\sigma_g$ (bonding combination of $2p_z$) both belong to the $\sigma_g$ irreducible representation of the molecular point group ($D_{\infty h}$). Because they share a symmetry label, the true Hamiltonian has nonzero off-diagonal matrix elements between them:

where:

• $2\sigma_g^{(0)}$, $3\sigma_g^{(0)}$ — the unmixed zeroth-order LCAO-MOs built purely from 2s and purely from 2p$_z$ atomic orbitals respectively
• $\hat H$ — the one-electron effective molecular Hamiltonian (in a Hückel-like or extended-Hückel sense)
• $H_{12}$ — the coupling matrix element responsible for mixing

Reading: symmetry does not forbid this matrix element from being nonzero; only if the two orbitals belonged to different irreducible representations would $H_{12}$ vanish identically by the great orthogonality theorem. Since both are $\sigma_g$, mixing is symmetry-allowed and its magnitude is a quantitative, not qualitative, question. Treating the problem as a 2×2 secular equation with diagonal (zeroth-order) energies $H_{11}$ and $H_{22}$ and off-diagonal element $H_{12}$, and (for simplicity) neglecting the overlap correction beyond zeroth order, the two mixed eigenvalues are

where:

• $E_+$ — energy of the upper mixed $\sigma_g$ level (pushed up relative to $H_{22}$)
• $E_-$ — energy of the lower mixed $\sigma_g$ level (pushed down relative to $H_{11}$)
• $H_{11}, H_{22}$ — zeroth-order (unmixed) energies of the 2s-based and 2p$_z$-based $\sigma_g$ MOs
• $H_{12}$ — the coupling element from above

Reading: this is exactly the two-level "avoided crossing" formula familiar from perturbation theory and from Hückel theory. The splitting between $E_+$ and $E_-$ always exceeds the unmixed gap $|H_{11}-H_{22}|$, and the excess splitting grows with $H_{12}^2$ divided by the unmixed gap when the mixing is weak, or dominates entirely when the unmixed gap is small.

where:

• $\Delta E_{\text{splitting}}$ — total energy gap between the two mixed $\sigma_g$ levels
• all other symbols as above

Reading: as $H_{12}\to 0$ (symmetry mixing switched off artificially), $\Delta E_{\text{splitting}} \to |H_{11}-H_{22}|$, recovering the naive unmixed picture; as $|H_{11}-H_{22}| \to 0$ (degenerate zeroth-order levels — the extreme limit), $\Delta E_{\text{splitting}} \to 2|H_{12}|$, the maximum possible mixing-induced splitting. The remaining physical question is what sets $H_{12}$ and $(H_{11}-H_{22})$ as functions of atomic number. Because $2\sigma_g^{(0)}$ is built from 2s atomic orbitals and $3\sigma_g^{(0)}$ from 2p$_z$ atomic orbitals, both $H_{11}$ and $H_{22}$ track the atomic 2s and 2p orbital energies respectively (via a Coulomb-integral-like relationship), so

where:

• $\varepsilon_{2s}, \varepsilon_{2p}$ — atomic orbital energies (ionization energies with a sign flip) of the 2s and 2p atomic orbitals of the atom in question
• the approximation holds because the bonding combinations largely inherit the parent atomic orbital energy shifted by a roughly constant bonding stabilization

Reading: as $Z$ increases from boron to fluorine, $\varepsilon_{2s}$ drops (becomes more negative/stable) faster than $\varepsilon_{2p}$, so $|\varepsilon_{2s}-\varepsilon_{2p}|$ grows monotonically. Since $H_{12}$ (an overlap-and-resonance-type integral between 2s-based and 2p$_z$-based combinations) changes comparatively little across the row, the ratio $H_{12}/(H_{11}-H_{22})$ shrinks steadily left to right, and with it the fractional mixing.

Example 1 — Foundational: constructing the correct qualitative MO diagram for C₂. Carbon has atomic number 6, well inside the "small 2s–2p gap" regime (the atomic 2s–2p separation for carbon is about 9 eV, smaller than for nitrogen and much smaller than for oxygen). By the physical trend established above, C₂ should show strong s–p mixing, placing the mixed 3σg orbital above the 1πu pair — the same ordering as N₂, not the "naive" ordering. Count valence electrons: each carbon atom contributes 4 valence electrons (2s²2p²), so C₂ has $4\times 2 = 8$ valence electrons. Fill the mixed-order MO ladder,

Filling in order: 1σg (2 electrons, 2 total), 1σu (2 electrons, 4 total), 2σg (2 electrons, 6 total), then the two degenerate 1πu orbitals receive the remaining 2 electrons. With only 2 electrons to place among the doubly-degenerate 1πu pair (capacity 4), Hund's rule says they go in singly with parallel spins unless* additional stabilization from pairing overrides it — and indeed experimentally and computationally, C₂'s ground state is found to be a singlet with the configuration $(1\pi_u)^4$ description debated historically, but the standard teaching-level configuration is

placing all 8 valence electrons with the 1πu pair fully filled and the 3σg (mixed, 2p-based) orbital left empty — because with the crossover in effect, 1πu lies below 3σg and fills first. This gives C₂ a formal bond order of

Wait — check the antibonding count carefully: only 1σu* is antibonding among the filled orbitals (2 electrons), while 1σg, 2σg, and 1πu (2+2+4 = 8 electrons) are bonding. So

This matches the now widely accepted view (supported by modern ab initio and experimental bond-length/force-constant studies) that C₂ has substantial quadruple/near-triple bond character debates aside; at the introductory level the crossover correctly predicts a closed-shell, diamagnetic C₂ with no unpaired electrons — exactly as observed — which the naive (unmixed) ordering would get wrong, since naive ordering would fill 3σg before completing 1πu and would leave two electrons unpaired in the degenerate 1πu set instead, wrongly predicting paramagnetism. Limiting-case check: if we pretended carbon behaved like the heavy-atom (weak-mixing) limit, we would place 3σg below 1πu, fill 2σg² then 3σg², leaving only 2 electrons for the degenerate 1πu pair — predicting a paramagnetic diradical. Experiment (C₂ is a singlet, closed-shell to a good approximation, no permanent magnetic moment observed in the ground state) sides decisively with the strong-mixing ordering, confirming that carbon sits on the small-gap side of the crossover as the theory predicts.

Example 2 — Applied: a numerical two-level model of the mixing-induced shift for N₂. Use representative extended-Hückel-type parameters (illustrative, chemically reasonable numbers, deliberately asymmetric) for the unmixed $2\sigma_g^{(0)}$ and $3\sigma_g^{(0)}$ levels of N₂:

with a coupling element estimated from overlap-based resonance integrals,

Apply the two-level formula:

Average of the diagonal elements:

So the two mixed eigenvalues are

Compare to the unmixed values $H_{11}=-33.5$ eV and $H_{22}=-17.2$ eV: mixing has pushed the lower level down by $34.71-33.5 = 1.21$ eV and pushed the upper level up by $17.2-15.99 = 1.21$ eV (the shifts are equal in magnitude, as they must be for a traceless perturbation to a 2×2 block — the trace $H_{11}+H_{22}$ is conserved). The upper (true 3σg, HOMO) level at $-15.99$ eV is now comfortably above a representative unmixed 1πu level (typically estimated near $-16.5$ eV to $-17$ eV for N₂ using similar parameter sets), confirming the σg-above-πu ordering quantitatively rather than just qualitatively. Compute the mixing angle for completeness:

A mixing angle of nearly 15° (rather than 0°) confirms substantial — though not maximal (45° would be complete degeneracy-driven mixing) — s–p character exchange between the two σg orbitals, consistent with N₂ sitting close to, but not exactly at, the point of strongest possible mixing. Limiting-case check: if the atomic 2s–2p gap were artificially forced to zero ($H_{11}=H_{22}$), the formula gives $E_\pm = H_{11} \pm |H_{12}| = -33.5\pm 4.6$, i.e., $\theta = 45^\circ$ exactly and maximal splitting of $2|H_{12}| = 9.2$ eV split symmetrically about the (now degenerate) diagonal energy — reproducing the textbook maximal-mixing limit exactly as the general formula predicts.

Example 3 — Exam/competition level: using photoelectron spectroscopy band ordering to test the crossover. Photoelectron spectroscopy (PES) measures ionization energies by ejecting electrons from successive occupied MOs; by Koopmans' theorem the vertical ionization energy from orbital $i$ approximately equals $-\varepsilon_i$. The PES spectrum of N₂ shows three well-separated bands (ignoring core 1s and the deepest 2s-derived bands) in the valence region, with vertical ionization energies (in eV) approximately:

The task: does this ordering support the s–p mixing picture, or contradict it?

Reasoning. The lowest ionization energy corresponds to removing an electron from the highest-lying occupied orbital (since less energy is needed to remove a more weakly bound electron). Observing $IE(3\sigma_g) < IE(1\pi_u)$ means the 3σg orbital sits above 1πu in energy — precisely the s–p-mixed ordering, not the naive one. If the naive (unmixed) ordering held, we would expect $IE(1\pi_u) < IE(3\sigma_g)$ instead, i.e., the π-band should appear at lower ionization energy than the σg band. The observed order is the opposite, directly and independently confirming the crossover derived from the two-level mixing model — this is, historically, one of the key experimental proofs that forced the mixed-orbital ordering onto textbook diagrams rather than the naive one.

Extension. The narrowness of the $3\sigma_g$ band (little vibrational structure) is itself a clue about orbital character: because the true 3σg orbital contains substantial 2s antibonding admixture diluting its "pure" 2p bonding character, removing an electron from it perturbs the N–N bond length comparatively little — consistent with a mixed orbital that is only weakly bonding rather than strongly bonding. Compare this with a hypothetical unmixed, purely-2p-bonding 3σg orbital, which would be expected to show a much broader, more structured band upon ionization (large bond-length change) — the opposite of what is observed. The narrow bandwidth is therefore independent, spectroscopic evidence for substantial s–p mixing diluting the bonding character of the 3σg MO, beyond just the energy-ordering argument. Limiting-case check: for O₂, where mixing is weak, the analogous σg-derived band should show more vibrational structure (more purely 2p-bonding, larger bond-length sensitivity) and should appear at higher ionization energy than the π band — both features are indeed observed experimentally in the O₂ PES spectrum, consistent with the weak-mixing regime.

The molecular orbital framework that makes s–p mixing calculable traces to the independent 1927–1932 work of Friedrich Hund and Robert Mulliken, who built the LCAO-MO method and systematically assigned symmetry labels to molecular orbitals — the same labeling scheme (σg, σu, πg, πu) used throughout this analysis.