s-p Mixing and Orbital Ordering
N₂ is paramagnetic in no molecular orbital diagram you can draw correctly. It is actually diamagnetic. O₂, however, genuinely is paramagnetic. The reason these two molecules sit on opposite sides of a single orbital-ordering swap is one of the most instructive accidents in all of Homonuclear Diatomic MOs. Walk into most introductory treatments and you get a single MO diagram taught as universal: σ₂ₚ below π₂ₚ. Apply that diagram to N₂ and it fails. It cannot predict the bond order, the photoelectron spectrum, and the magnetism correctly. This works only if you already know to flip σ₂ₚ and π₂ₚ. The flip is not a fudge; it is the signature of s-p mixing. Understanding it converts a memorized exception into something else entirely. That exception — B₂, C₂, N₂ have σ below π; O₂, F₂, Ne₂ have π below σ — becomes a predictable consequence of orbital energetics and symmetry. The physical content is simple to state and easy to underestimate. The 2s and 2p_z atomic orbitals on each atom share the same symmetry along the internuclear axis. Both are σ_g or σ_u under the molecule's point group. So nothing in group theory forbids them from mixing once you build molecular orbitals. Whether they do mix substantially, and how strongly, depends on one thing. That is how close the atomic 2s and 2p energies are. This gap shrinks steadily as you cross the second period. Increasing nuclear charge stabilizes s orbitals (which penetrate the core) faster than p orbitals. Beryllium and boron have huge 2s–2p gaps and negligible mixing. By the time you reach carbon, nitrogen and even oxygen, the gap has narrowed. It is small enough that s-p mixing pushes the σ_g(2s)-derived orbital down. It pushes the σ_g(2p)-derived orbital up. Sometimes this push is hard enough to vault it above the π_u(2p) pair. This matters far beyond a party trick for predicting N₂ vs O₂ paramagnetism. The same physics is at work elsewhere too. Near-degenerate orbitals of matching symmetry mix and repel each other in energy. This is the qualitative engine behind hybridization arguments. It explains why CO binds metals through carbon rather than oxygen. It also drives orbital-ordering anomalies in heavier main-group and transition-metal diatomics. It is also a clean, self-contained illustration of a non-crossing rule. Same-symmetry orbitals never cross as you tune a parameter (here, atomic number). Instead, they avoid each other. That avoidance is exactly what reorders σ and π. This article builds the full second-period story from atomic term energies upward. It moves through the qualitative and semi-quantitative (2×2 secular determinant) treatment of s-p mixing. It shows exactly why the σ/π crossover happens between N₂ and O₂. It connects the ordering to observable bond orders, bond lengths, magnetism, and photoelectron spectra. It closes by placing s-p mixing in its proper context. That context is a special case of a much more general orbital-mixing principle. You will meet this principle again in Heteronuclear Diatomic MOs and in symmetry-adapted treatments of Polyatomic MOs by Symmetry.
- The 2s and 2p_z atomic orbitals share σ symmetry along the bond axis, so nothing forbids them mixing in the MO treatment of homonuclear diatomics.
- Mixing strength depends on the atomic 2s–2p energy gap, which shrinks left to right across period 2 as effective nuclear charge stabilizes s more than p.
- For Li₂–N₂ the gap is small enough that mixing pushes σ_g(2p) above π_u(2p); for O₂, F₂, Ne₂ the gap is large enough that σ_g(2p) drops back below π_u(2p).
- The crossover is why N₂ is diamagnetic with a triple bond while O₂ is paramagnetic with a formal double bond and two unpaired electrons.
- Photoelectron spectroscopy directly confirms the ordering: N₂'s HOMO band shows sharp vibrational fine structure typical of a weakly-bonding σ orbital, consistent with s-p mixed character.
- Key idea: same-symmetry orbitals never simply cross in energy as a parameter changes — they mix and repel, and that avoided crossing is what swaps σ and π ordering across period 2.
Where this lives today
The s-p mixing story is not a period-2 curiosity confined to N₂ vs O₂; it is the entry point into a general principle used throughout modern structural inorganic and organometallic chemistry. Understanding that same-symmetry fragment orbitals mix and repel — rather than simply stacking in a fixed universal order — is exactly the reasoning used to build MO diagrams for Heteronuclear Diatomic MOs such as CO and NO, where the s-p mixing on carbon is strong enough to leave carbon's lone pair as the true HOMO, explaining why CO binds transition metals through carbon (forming metal carbonyls, the backbone of organometallic catalysis and of industrially vital processes like hydroformylation and the Monsanto acetic acid process) rather than through the more electronegative oxygen. In transition-metal chemistry, the same avoided-crossing logic governs how metal (n)d, (n+1)s, and (n+1)p orbitals combine with ligand symmetry-adapted linear combinations in octahedral, tetrahedral, and other polyatomic complexes, a topic developed fully under Polyatomic MOs by Symmetry using projection operators and reducible representations; s-p(-d) mixing there determines the ordering and character of frontier orbitals relevant to spin state, magnetism, and redox behavior, exactly parallel to how it determines N₂'s HOMO character here. Photoelectron spectroscopy remains the direct experimental tool for testing orbital-ordering predictions in small molecules and clusters, and the same vibrational-fine-structure logic used to nail down N₂'s ordering is routinely used in gas-phase spectroscopy of interstellar and atmospheric diatomics/triatomics, in mass-spectrometric characterization of transient reactive intermediates, and in benchmarking quantum chemical methods (coupled-cluster, multireference configuration interaction) against experimentally resolved orbital energies. Understanding s-p mixing also underlies why simple valence-bond Lewis structures sometimes mis-predict magnetism (B₂, O₂) while a correctly-ordered MO diagram gets it right immediately — a pedagogical and diagnostic point still central to how inorganic chemists reason about bonding in main-group diatomics, small radicals, and diatomic ligands bound to metal centers.
Ask the tutor: Why does B₂'s ground state prove the mixed ordering while N₂'s diamagnetism alone cannot?
The physical picture
Begin with the atomic orbitals actually available to build second-period homonuclear diatomic MOs: on each atom, 2s and three 2p orbitals (2p_x, 2p_y, 2p_z with z along the internuclear axis). Under D∞h symmetry, the symmetry-adapted linear combinations built from 2s and from 2p_z on both atoms are both of σ_g or σ_u type: 2s gives σ_g(2s) and σ_u(2s); 2p_z gives σ_g(2p) and σ_u(2p). The 2p_x and 2p_y orbitals instead give π_u(2p) bonding pairs and π_g(2p) antibonding pairs, which have no symmetry match to any s-derived combination. The naive picture — the one taught first — treats 2s and 2p as so far apart in energy that σ_g(2s) and σ_g(2p) are each built purely from their own parent atomic orbital, with no cross-mixing. This gives the "textbook" ordering σ_g(2s) < σ_u(2s) < π_u(2p) < σ_g(2p) < π_g(2p) < σ_u(2p), i.e., σ below π. That ordering is correct for O₂, F₂, and Ne₂, but wrong for Li₂ through N₂. The reason the naive picture breaks down is that σ_g(2s) and σ_g(2p) have identical symmetry (both σ_g), so whenever their unmixed energies are close enough, the true molecular orbitals are not pure combinations of one atomic orbital type — they are mixtures. Mixing between two states of the same symmetry always pushes the lower one down and the upper one up (second-order, non-degenerate perturbation theory, or equivalently diagonalizing a 2×2 secular matrix with off-diagonal coupling). The amount of push depends inversely on the unmixed energy gap: close levels mix strongly and repel hard; distant levels barely feel each other.
The next figure shows the consequence for the molecular orbital diagram itself: with strong s-p mixing (left, appropriate for B₂, C₂, N₂), σ_g(2p) is pushed above π_u(2p); with weak mixing (right, appropriate for O₂, F₂, Ne₂), σ_g(2p) sits below π_u(2p).
A third way to see this is to track what "σ_g(2p)" physically is once mixing is switched on. In the unmixed limit it is a pure bonding combination of the two atoms' 2p_z orbitals, strongly σ-bonding and thus low in energy. Once it mixes with σ_g(2s), it acquires 2s character that is antibonding with respect to that admixed component in a way that partially cancels its own σ-bonding overlap, weakening its net bonding character and raising its energy. Symmetrically, σ_g(2s) mixes in a bit of σ_g(2p) antibonding-like character and is pushed lower. This is exactly the same "level repulsion via mixing" logic used throughout chemistry: two zeroth-order states of matching symmetry, once coupled, split further apart than they started, with the lower state pushed down and the upper pushed up — never crossing.
The core ideas
The unmixed (zeroth-order) MO energies for the σ_g manifold come from treating 2s and 2p_z independently:
where:
- are the atomic 2s and 2p orbital (Coulomb) energies — negative numbers, more negative for 2s;
- are the corresponding resonance (bonding) integrals, negative and larger in magnitude for the more effectively overlapping combination.
Once mixing is switched on, the true σ_g orbitals are eigenvectors of a 2×2 Hamiltonian matrix in the basis {σ_g(2s), σ_g(2p)}:
where:
- and are the unmixed (diagonal) σ_g energies defined above;
- is the off-diagonal coupling element arising because 2s and 2p_z on the two centers have matching σ_g symmetry and nonzero overlap-derived interaction.
Diagonalizing gives the mixed energies directly:
where:
- is the upper mixed σ_g level (the one that can be pushed above π_u);
- is the lower mixed σ_g level;
- the gap is the unmixed splitting — small gap means the square-root term is dominated by , producing large mixing-induced splitting.
Reading this equation: the two mixed levels always straddle the average of the unmixed levels, split by when (maximal mixing) and collapsing back to the unmixed values individually only as (mixing switched off). This single formula is the quantitative core of the entire s-p mixing story: whether (essentially the "σ_g(2p)"-labelled MO) rises above π_u(2p) is entirely a question of how large is relative to the atomic 2s–2p gap.
Goal: show why (in magnitude) decreases from Li to F, which is the input that controls 's relative importance in the mixed-energy formula above.
Step 1 — Penetration and shielding. The 2s orbital has a radial distribution with a node structure that gives it a small but nonzero probability density very close to the nucleus (penetration through the 1s core), while the 2p orbital, having one unit of orbital angular momentum, is centrifugally excluded from the core region and is shielded more effectively by the 1s electrons. Consequently, the effective nuclear charge felt by a 2s electron is larger than that felt by a 2p electron on the same atom, for every element in period 2.
Step 2 — How scales the orbital energies. In a hydrogenic approximation refined by Slater's rules, the orbital energy scales as
Since for every atom, is always more negative (lower) than ; the 2s orbital is always more stable.
Step 3 — Both values grow across the period, but the 2s one grows faster in absolute terms relative to the gap. As nuclear charge increases from Li (Z=3) to F (Z=9), each additional proton adds real nuclear attraction that is only partially screened by the additional valence electron. Both and increase, but because the 2s electron penetrates the core so much more, it "feels" the growing nuclear charge more directly and gets pulled down in energy faster in absolute terms — while the 2p orbital, poorly penetrating, initially lags. Increasing exchange stabilization of the increasingly populated 2p subshell (extra stabilization from same-spin exchange energy as more parallel-spin p electrons accumulate, especially from B to N with a half-filled 2p³ configuration at N) partially catches 2p energy up to 2s energy. The net measured effect (from atomic spectroscopic term values) is that the 2s–2p gap is roughly 1.85 eV at Li, grows through Be and B, but then narrows steadily relative to the bonding interaction strength through C, N, reaching its practical minimum in the sense relevant to MO mixing around N, before both atomic orbitals contract and the relative importance of mixing (governed by versus the gap) drops again toward O, F, Ne as effective nuclear charge grows large enough that the absolute 2s–2p gap widens substantially again in eV terms while (which scales with orbital size and overlap, both shrinking as grows) also drops — but the gap wins, restoring the "normal" σ-below-π ordering.
Step 4 — Conclusion. The competition is between the shrinking-then-regrowing atomic 2s–2p gap and the always-shrinking (across the period, due to orbital contraction) mixing matrix element . Their ratio is maximized in the B–N region, which is exactly where the σ_g(2p)/π_u(2p) crossover is observed.
Go deeper · Derivation B · 1 minFrom the 2×2 secular equation to the explicit crossover condition
Goal: turn the diagonalization above into an explicit inequality that tells you when (mixed σ_g(2p)) rises above the π_u(2p) energy, i.e., the literal condition for orbital-ordering crossover.
Step 1 — Write the crossover condition. Crossover occurs when the upper mixed σ level exceeds the π_u(2p) level:
Using the diagonalized expression,
Step 2 — Isolate the square-root (mixing) term. Rearranging,
The right-hand side is a fixed, atom-dependent quantity: it is the gap between the π_u(2p) level and the average of the two unmixed σ_g levels. This right-hand quantity is typically small and roughly constant across the period because π_u(2p) and unmixed σ_g(2p) both derive from 2p_z/2p_x,y overlap of similar character.
Step 3 — Square both sides (valid since both sides are positive in the regime of interest) to isolate .
Step 4 — Interpret. The right-hand side shrinks (making the inequality easier to satisfy, i.e., crossover more likely) whenever the atomic 2s–2p gap is small, because that term is subtracted. This confirms algebraically what the qualitative picture already suggested: crossover requires simultaneously (i) a small atomic 2s–2p gap and (ii) a mixing matrix element that is not too small — both conditions are best satisfied in the B₂–N₂ window, and both degrade toward O₂, restoring normal ordering. The inequality is not meant to be solved numerically with textbook atomic data (real calculations require self-consistent-field orbital energies and overlap integrals) — its value is qualitative: it isolates exactly which two physical quantities compete, and shows that the winner flips sign somewhere in the middle of the period, which is the entire content of the "σ/π crossover" rule.
Worked examples
Example 1 — Foundational: predicting N₂'s bond order and magnetism with the mixed ordering. Nitrogen, N₂, has 2×7 = 14 electrons total, of which 2×2 = 4 are core 1s electrons (irrelevant to bonding, placed in σ_g(1s) and σ_u(1s)) leaving 10 valence electrons to place in the valence MO diagram. Using the mixed* ordering appropriate for N₂ (σ below π is wrong here — use σ_g(2p) above π_u(2p)):
Count: 2 + 2 + 4 + 2 = 10 valence electrons, exactly matching. All orbitals are filled with paired electrons (π_u(2p) holds two degenerate orbitals, each doubly occupied, no unpaired spins) — consistent with N₂'s experimentally verified diamagnetism. Bond order:
A bond order of 3 matches the classical Lewis structure :N≡N: and the very short, very strong N–N bond (bond length 109.8 pm, bond dissociation energy 945 kJ/mol, both consistent with a genuine triple bond). Note that using the naive unmixed ordering (σ_g(2p) below π_u(2p)) gives the same electron count and the same bond order of 3 in this particular case — because both π_u(2p)^4 and σ_g(2p)^2 are fully occupied regardless of which is filled first, swapping their order changes nothing about total bonding/antibonding electron counts for N₂. The ordering only becomes experimentally distinguishable in the ionization pattern (see Example 3), not in the bond order itself for this closed-subshell case.
Example 2 — Applied: O₂'s paramagnetism as direct evidence for the un-mixed (normal) ordering. Oxygen, O₂, has 16 electrons total, 4 core (1s²1s²), leaving 12 valence electrons. Because O has a larger* 2s–2p gap than N (moving right across the period restores separation), O₂ uses the normal, weak-mixing ordering: σ_g(2p) below π_u(2p). Filling:
Count: 2+2+2+4+2 = 12, correct. The last two electrons enter the doubly-degenerate π_g(2p) level, which holds two orbitals (π_g_x, π_g*_y). By Hund's rule, these two electrons occupy the two degenerate orbitals singly with parallel spins rather than pairing in one — giving two unpaired electrons.
A bond order of 2 matches the O=O double bond picture, and the two unpaired electrons in π_g correctly predict O₂'s experimentally observed paramagnetism — famously demonstrated by liquid O₂ being attracted into a magnetic field gap, something no simple Lewis structure (which pairs all electrons) can explain but which the MO diagram gets right immediately. Crucially, if you mistakenly used the mixed (N₂-style) ordering for O₂ — putting σ_g(2p)² below the two π_g electrons but π_u(2p) above σ_g(2p) — you would still fill π_g(2p) last regardless (it's the highest level either way for O₂'s electron count), so the paramagnetism prediction survives either ordering choice for O₂ specifically. What does not survive a wrong ordering choice is getting the σ_g(2p)/π_u(2p) relative* placement right for interpreting spectroscopic data — which is the point of Example 3.
Example 3 — Exam/competition level: using photoelectron spectroscopy (PES) to determine which orbital is the true HOMO of N₂, distinguishing mixed from unmixed pictures decisively. This is where the crossover has an experimentally sharp, unambiguous signature, because ionization energies and the vibrational fine structure of the resulting cation states directly probe which orbital character sits highest. The valence PES of N₂ shows three bands with ionization energies approximately (values are the standard, independently reproducible gas-phase UPS results):
The key diagnostic is vibrational fine structure: removing an electron from a strongly bonding orbital drastically shortens/lengthens the equilibrium bond length in the cation relative to the neutral, producing a long vibrational progression (many resolved vibrational peaks, because the Franck–Condon envelope is broad when the geometry changes a lot upon ionization). Removing an electron from a weakly bonding or largely nonbonding orbital barely changes the bond length, producing a short, sharp vibrational progression (often nearly a single dominant peak). Experimentally, the lowest-IE band (15.6 eV, the HOMO ionization) shows a short, sharp vibrational progression with very little change in bond length upon ionization — behavior typical of ejecting an electron from an orbital with only modest net bonding character. The next band (16.7 eV) shows a long vibrational progression, typical of ejecting from a strongly bonding π orbital.
This is exactly what the s-p mixed picture predicts and the naive unmixed picture gets backwards. In the mixed picture, the HOMO of N₂ is σ_g(2p), but because of the mixing it is only weakly net-bonding (its bonding character has been diluted by anti-phase mixing with σ_g(2s) character) — consistent with a sharp PES band and only a small bond-length change upon ionization. The orbital just below it, π_u(2p), is essentially unperturbed by mixing (no symmetry match to s), remains strongly bonding, and shows the expected long vibrational progression upon ionization, exactly as seen at 16.7 eV. If the naive unmixed ordering (σ_g(2p) below π_u(2p), with σ_g(2p) then being the strongly-bonding orbital and π_u(2p) being the HOMO) were correct, the pattern would have to be reversed: the HOMO band would show the long, broad progression (removing an electron from strongly bonding π), and the second band would show the sharp progression (removing from a less-perturbed σ). That is not what is observed. The observed pattern — sharp HOMO band, broad second band — is direct, unambiguous experimental proof that σ_g(2p) sits above π_u(2p) in N₂, confirming the s-p mixed ordering independent of any bond-order argument (which, as Example 1 showed, cannot distinguish the two orderings on its own for N₂).
Limiting case check: as the atomic 2s–2p gap is imagined to grow very large (moving conceptually toward heavier or more electronegative atoms with well-separated 2s/2p), the mixing term becomes negligible relative to the gap, and exactly recovering the unmixed limit — consistent with O₂, F₂, Ne₂ reverting to normal σ-below-π ordering, and with the general rule that mixing effects vanish whenever the interacting levels are far apart in energy.
Limiting cases, scope & common traps
| Regime | Ordering | Governing reason |
|---|---|---|
| Li₂, Be₂ | σ_g(2p) above π_u(2p) (mixing matters, though Be₂ is barely bound at all) | Small 2s–2p gap, but overall bonding is very weak in these light diatomics |
| B₂ | σ_g(2p) above π_u(2p) | Confirmed experimentally: B₂ is paramagnetic (two unpaired π_u electrons), only explicable with σ above π |
| C₂ | σ_g(2p) above π_u(2p) | C₂ has a debated but MO-consistent bonding picture with π_u(2p)^4 as HOMO region; diamagnetic |
| N₂ | σ_g(2p) above π_u(2p) | PES vibrational fine structure directly confirms (Example 3) |
| O₂ | σ_g(2p) below π_u(2p) (normal order restored) | Larger 2s–2p gap; paramagnetism from π_g* filling is ordering-independent but π/σ relative position reverts to normal |
| F₂, Ne₂ | σ_g(2p) below π_u(2p) | Large 2s–2p gap, minimal mixing, atoms behave near-hydrogenically separated in energy |
The chemist behind it
The molecular orbital treatment of diatomics, and with it the systematic understanding of orbital mixing by symmetry, owes its foundational development to Robert S. Mulliken, who built much of the group-theoretical and spectroscopic vocabulary (σ, π, g, u labels; term symbols for molecules) still used verbatim in every MO diagram drawn today. Mulliken's insistence on grounding molecular electronic structure in both quantum mechanics and precise spectroscopic observation — rather than accepting either pure theory or pure empirical rule-of-thumb — is exactly the methodology this article follows: the crossover is not asserted, it is derived from the atomic-orbital energy trend and then checked against the PES fine structure that only a correct ordering can explain.
Origins · 1 minRobert Sanderson Mulliken and how this was discovered
Mulliken developed molecular orbital theory into a quantitative, symmetry-rigorous language during the 1920s–1930s, working in close dialogue with Friedrich Hund on correlating atomic and molecular electronic states. He introduced much of the systematic notation for molecular term symbols and orbital symmetry labels, and pursued electronic spectroscopy of diatomic molecules as the definitive experimental test of orbital-based bonding theories — precisely the kind of photoelectron/absorption-spectrum reasoning used to establish the N₂ σ/π ordering. He received the Nobel Prize in Chemistry in 1966 "for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method."
Why don't 2p_x and 2p_y ever mix with 2s in this treatment?
Symmetry forbids it. Under D∞h symmetry (the point group of a homonuclear diatomic), 2s and 2p_z combinations transform as σ_g or σ_u along the internuclear axis, while 2p_x and 2p_y combinations transform as π_u or π_g. Mixing between orbitals requires them to share the same irreducible representation, because the Hamiltonian matrix element connecting orbitals of different symmetry is forced to vanish by the molecule's symmetry (a direct consequence of the Wigner–Eckart-type selection rules built into group theory). Since σ and π are different irreducible representations, s and p_x/p_y simply cannot mix in a diatomic, no matter how close their atomic energies happen to be. Only 2s and 2p_z, sharing σ symmetry, are eligible.
Does s-p mixing affect the σu(2s) and σu(2p) antibonding orbitals too?
Yes, in principle the same logic applies: σu(2s) and σu(2p) share u symmetry and can mix by the identical argument. In practice this mixing is usually smaller in its qualitative consequences for the ordering question, because both antibonding orbitals sit high enough in energy, and are separated enough from each other in most treatments, that the effect rarely reorders anything experimentally significant among the filled valence orbitals for neutral second-period diatomics. It is included in full ab initio calculations and does shift absolute antibonding energies somewhat, but the qualitative "crossover" story that matters for bond order and magnetism concerns almost exclusively the σg(2s)/σg(2p) pair, because that is the pair whose relative ordering with πu(2p) changes which subshell fills last.
Is C₂ diamagnetic or paramagnetic, and how does that test the mixed ordering?
Using the mixed ordering, C₂ (8 valence electrons) fills σg(2s)²σu(2s)²πu(2p)⁴, leaving πu(2p) as the fully-filled HOMO with no unpaired electrons — predicting diamagnetism, which matches experiment. The unmixed ordering would instead fill σg(2s)²σu(2s)²σg(2p)² first, then have only 2 electrons left for πu(2p)'s two degenerate orbitals, predicting a paramagnetic ground state by Hund's rule. Experimentally C₂ is diamagnetic (its ground state is a singlet, ¹Σg+ in the standard assignment), which is one of the clean pieces of independent evidence supporting the mixed ordering for C₂, alongside B₂'s paramagnetism providing the complementary check on the other side.
How large does β_mix actually need to be, quantitatively, for crossover to occur?
There is no single universal numeric threshold quoted in elementary treatments, because β_mix depends on overlap integrals and effective orbital sizes that are best obtained from a real self-consistent-field or Hartree–Fock calculation rather than a hand-estimated atomic parameter. The qualitative rule — small atomic 2s–2p gap plus significant 2s/2p_z overlap favors crossover — is what's transferable and testable without a full calculation; the precise numeric boundary is why B₂, C₂, N₂ show crossover and O₂ does not, but pinning the exact crossover point analytically requires the coupled Hartree–Fock or CI-level orbital energies rather than the qualitative secular-equation argument alone.
Why does B₂'s paramagnetism matter so much as evidence, more than N₂'s diamagnetism?
Because B₂'s magnetism prediction is ordering-dependent while N₂'s bond order is not (see the warning above and Example 1). If you used the wrong (unmixed) ordering for B₂, you would incorrectly predict a diamagnetic ground state; only the mixed ordering, with πu(2p) below σg(2p), correctly predicts the observed paramagnetic (triplet, Hund's-rule) ground state of B₂. This makes B₂ a much sharper, more decisive experimental discriminator between the two candidate orderings than N₂'s bond order or even N₂'s diamagnetism, both of which are compatible with either ordering as shown in Example 1.
Does s-p mixing ever get so strong that σg(2s) itself rises above σu*(2s)?
No — that would require mixing strong enough to overturn the much larger bonding/antibonding energy gap within the same atomic orbital type (σg(2s) vs σu(2s) differ by a full bonding-vs-antibonding overlap integral, typically a large energy difference), which is a qualitatively different and much bigger gap than the 2s–2p atomic gap that s-p mixing exploits. In all second-period homonuclear diatomics, σg(2s) remains the lowest valence MO and σu(2s) the second-lowest; s-p mixing only ever reorders the σg(2p)/πu(2p) pair (and, in principle and to a lesser observable extent, mixes into the σu*(2p) manifold), never the bonding/antibonding pair built from the same orbital type.