Molecular Orbital Diagram: Period 2 Homonuclear Diatomics

What this adds on top of the sibling simulations. The 1s-1s simulation models one exactly-solvable orbital pair with real bond energetics; the s/p-overlap simulation generalises the orbital shapes but stays a single, isolated pair. This one drops both the potential-energy curve and the R slider entirely and instead shows a complete valence MO diagram — all eight 2s/2p-derived molecular orbitals at once, filled with the correct number of electrons for whichever period-2 homonuclear diatomic you pick, from Li₂ to F₂.

Core electrons are omitted on purpose. Only the four MOs built from 2s and 2p are shown; the two 1s-derived core MOs (σ1s, σ*1s) aren't drawn at all. They sit far below the valence set, are essentially non-bonding in character (the core doesn't overlap meaningfully at any real bond length), and their filled bonding/antibonding pair cancels exactly in the bond-order sum — so leaving them out changes nothing about bond order or magnetism, which is the standard convention for these diagrams.

The shape panel now draws a real 2s radial node. A true 2s orbital's radial part changes sign partway out from the nucleus (proportional to (1−ζr)·e^−ζr, versus 1s's plain e^−ζr with no such node) — the contour plot for σ(2s)/σ*(2s) now shows this honestly, as a small nodal circle around each nucleus in addition to the larger bonding/antibonding node between them, for every species except H₂ (whose "2s" really is a 1s and correctly stays nodeless). This is a display-only correction, though: the energy ladder's overlap integral (S, feeding the secular equation and the s–p mixing coupling V) deliberately keeps the older nodeless approximation, because swapping in the nodal version changes S enough — particularly at the longer bond lengths of Li₂/Be₂/B₂ — to break the K_MIX calibration and flip B₂'s magnetism prediction wrong (checked directly). Redoing that calibration properly is a bigger job than fixing the picture, so for now the shape you see and the energy you read are computed from two different (but both individually standard) levels of approximation for the same orbital — a real, acknowledged mismatch rather than a hidden one.

Orbital exponents (ζ), one per element. Each period-2 element's 2s/2p orbitals share a single Slater exponent ζ = Z_eff/2, with Z_eff = Z − [0.85 × 2 for the two 1s core electrons + 0.35 × (v−1) for the other v−1 valence 2s/2p electrons]. This is the same convention (and, for carbon, the same value ζ=1.625) used in the sibling s/p-overlap simulation — Slater's rules don't distinguish 2s from 2p, so one exponent stands in for both, which is a real simplification: accurate SCF calculations give 2p a noticeably larger exponent (more contracted) than 2s for the same atom. The resulting values run ζ = 0.65 (Li) → 0.975 (Be) → 1.3 (B) → 1.625 (C) → 1.95 (N) → 2.275 (O) → 2.6 (F), growing steadily as the nuclear charge increases across the row. H₂ isn't on this ladder at all — Slater's screening constants only apply once there's a core to screen, so its 1s exponent is the exact, unscreened ζ=Z_eff/n=1/1=1.

Real bond lengths. Each molecule's nuclei are drawn at its actual experimental bond length (H₂ 0.741 Å, Li₂ 2.673 Å, Be₂ 2.45 Å, B₂ 1.590 Å, C₂ 1.243 Å, N₂ 1.098 Å, O₂ 1.208 Å, F₂ 1.412 Å) — used both for the orbital-shape panel and (new) to compute the genuine overlap integrals behind the energy ladder below.

Real AO energies, and why the 2s-2p gap widens so sharply across the row. The two atomic reference lines are real valence-orbital ionization energies (eV), not placeholders: 2s and 2p for B through F come from Gray's Electrons and Chemical Bonding (1964) — the standard spectroscopically-derived VOIE table (Li 2s −5.46; Be 2s −9.30; B −14.01/−8.31; C −19.47/−10.66; N −25.54/−13.14; O −32.36/−15.87; F −46.37/−18.72). Li and Be have no occupied 2p electron to measure, so their 2p values are estimated: across B–F, the effective nuclear charge felt by a 2p electron (back-solved from each energy via E=−13.6×(Z_eff/2)²) rises in an almost perfectly straight line with atomic number — extending that line to Li/Be and converting back gives 2p ≈ −4.7 eV (Li) and −6.4 eV (Be). The result is a 2s–2p gap that grows from just 0.7 eV at Li₂ to 27.6 eV at F₂ — the real effect (not a hand-tuned schematic) that the s–p mixing model below depends on.

MO energies are computed, via the same secular equation as the s/p-overlap sibling — now with real numbers. For two identical atomic orbitals with ionization energy H_ii and overlap S, the non-orthogonal 2×2 secular equation gives E_bond=(H_ii+H_ij)/(1+S), E_anti=(H_ii−H_ij)/(1−S). S is a genuine numerically-integrated two-centre overlap (same quadrature as the sibling simulation) at each molecule's real bond length and Slater exponent; H_ij is estimated by the standard Wolfsberg–Helmholz approximation H_ij=K·S·H_ii (K=1.75, the usual extended-Hückel constant) — a well-established way to turn overlap into a resonance energy without a full SCF calculation. This is done independently for all three same-symmetry orbital pairs — σ(2s), σ(2p), π(2p) — giving six real numbers in eV, which is what's actually plotted (see the axis).

s–p mixing, honestly: an avoided crossing, not a relabelling. σ(2s) and σ(2p) share σ-gerade symmetry, so beyond their own independent bonding/antibonding pairs they can interact further — and separately, so can σ*(2s) and σ*(2p), which share σ-ungerade symmetry. Both are modelled the same way: a second, smaller avoided crossing between the relevant pair of already-computed levels, with mixed energies at the midpoint ± √(halfGap²+V²), where halfGap is half their unmixed separation and V is a coupling term built the same Wolfsberg–Helmholz way from the genuine 2s–2p cross overlap integral (the same V both times — the cross overlap integral itself doesn't know which pair it's being applied to). There is no independently-measured value for V in a model this simple, so its overall scale is calibrated once — chosen so that switching mixing on reproduces the well-established textbook pattern (σ(2p) pushed above π(2p) for Li₂–N₂, unswapped for O₂/F₂) without needing a full multi-orbital CI calculation. Applying the same interaction to the antibonding pair doesn't change any bond order or magnetism prediction for any of the seven species — σ*(2p) is never partially occupied in this row, so exactly where it sits relative to π*(2p) never affects filling — it only reshuffles some unoccupied levels (most visibly for Li₂, where σ*(2s) drops below σ(2p); Li₂ has only 2 valence electrons, so nothing filled is affected there either). The orbital shown in the shape panel for σ(2p) is still always the pure head-on 2p+2p combination regardless of the toggle; only its energy (and the resulting fill order) responds to mixing, not its recomputed hybrid character.

Two honest surprises this real-numbers version turns up. First, σ*(2s) — the antibonding partner of the deepest, most-stabilised level — actually sits below (more stable than) the whole 2p-derived bonding set for N₂, exactly matching the ordering seen in N₂'s real photoelectron spectrum (only σ_g(2p), π_u(2p) and σ_u*(2s) appear in a typical He(I) spectrum, with σ_g(2s) too deep to ionize at all) — a feature a hand-drawn schematic ladder would likely get wrong by placing σ*(2s) just above σ(2s) and well below everything else. Second, F₂'s entire 2p-derived manifold (σ(2p) through σ*(2p)) is compressed into a narrow band compared to its own σ(2s)/σ*(2s) split — a direct consequence of F's small 2p–2p overlap at its real, short bond length, and part of the same underlying reason F–F is a surprisingly weak single bond.

Filling: Aufbau + Hund's rule, and why the toggle matters most for B₂ and C₂. Electrons fill the six valence levels lowest-energy-first, two per orbital, with the two doubly-degenerate π levels filled singly (Hund's rule) before any pairing. Bond order is (bonding electrons − antibonding electrons)/2. For most of the row, σ(2p) and π(2p) end up either both empty or both completely full regardless of which one fills first, so the mixing toggle changes nothing observable (try O₂ or F₂ with mixing switched on anyway — same bond order, same magnetism). The exception is exactly when the electron count lands between the two: B₂ (2 electrons split across the doubly-degenerate π pair, one unpaired each way → predicted paramagnetic, matching experiment) and C₂ (4 electrons exactly fill the π pair → predicted diamagnetic, matching experiment). Switch mixing off for either and the ordering flips which level fills first, silently getting the magnetism wrong while leaving the bond order unchanged — a genuinely important, non-cosmetic difference, not just a relabelling.

H₂ has no 2p at all, so it's not really a "period 2" diagram — it's included anyway as the simplest possible reference point. Its σ_g(1s)/σ_u*(1s) pair is pushed through the exact same secular-equation and overlap-integral machinery as everything else, just fed hydrogen's own real numbers: e2s here is actually the 1s ionization energy (−13.6 eV, the exact hydrogen ground-state energy — no VOIE estimate needed), the Slater exponent ζ=1 is exact (a lone 1s electron feels the full nuclear charge, no core to screen), and the bond length is the real 0.741 Å H–H distance. Two valence electrons fill σ(1s) completely, giving bond order 1 and zero unpaired electrons — the textbook H₂ picture, with no 2p, no π levels, and no s–p mixing to speak of (the toggle is hidden for this reason).

Li₂ and Be₂ drop σ(2p)/π(2p) from the diagram entirely, not just leave them unfilled. Neither element has any 2p valence electrons, so those levels are never populated and can't affect bond order or magnetism — same logic as omitting the 1s core (below), just one row up. Showing four empty levels that never matter would only clutter what is, for these two molecules, really just a σ(2s)/σ*(2s) picture; the s–p mixing toggle is hidden for the same reason it's hidden for H₂ — there's no 2p left for 2s to mix with. (Their 2s/2p atomic reference energies and overlap integrals are still computed exactly as before, purely so the σ(2s)/σ*(2s) energies plotted are the genuine full calculation, not a re-derived special case.)

Be₂ is included even though its predicted bond order is zero. σ(2s) and σ*(2s) are both completely filled with nothing left over, so simple MO theory predicts no net bond at all. Be₂ is in fact only very weakly bound in reality (a few kJ/mol, confirmed spectroscopically), via dispersion-type effects this simple valence-bond-order picture doesn't capture — its inclusion here is deliberately honest about that mismatch, not a bug.

"Model" (A₂) is not a real element — it's the idealized textbook diagram, made honest. Every general-chemistry course eventually draws a generic period-2 MO diagram to teach the concept of s–p mixing, independent of any particular atom. That's what this eighth option is: a hypothetical diatomic with illustrative, round-number inputs (ζ = 1.625, the same Slater exponent as carbon; bond length 1.40 Å; atomic 2s/2p reference energies −17.0/−9.5 eV, a 7.5 eV gap chosen to sit safely in the same "mixing matters" regime as B₂/C₂) rather than anything measured. Critically, it is not a separate hand-drawn ladder — it is pushed through the exact same secular-equation and numerically-integrated-overlap machinery as Li₂–F₂ above, so the resulting diagram is a genuine computed consequence of those inputs, not a schematic fake. With 8 valence electrons, toggling mixing here flips the predicted magnetism outright: off gives σ(2p) filling before π(2p), leaving the π(2p) pair half-full and two electrons unpaired (paramagnetic); on gives π(2p) filling first and completing, pairing all four electrons (diamagnetic) — bond order stays 2 either way, so the only thing that changes is magnetism, isolating exactly what mixing does and does not affect. This mirrors real C₂'s behaviour but without tying the demonstration to one specific element.