Magnetism from MO Diagrams

Oxygen's paramagnetism, the flagship example throughout this article, is exploited directly in clinical and industrial oxygen sensors: paramagnetic oxygen analyzers measure the magnetic susceptibility of a gas sample to determine $\mathrm{O_2}$ concentration without any chemical reaction, a technique used in anesthesia machines, industrial combustion monitoring, and life-support systems, precisely because almost no other common atmospheric gas ($\mathrm{N_2}$, $\mathrm{CO_2}$, Ar) shares oxygen's unpaired-spin ground state. The same triplet ground state governs why ground-state $\mathrm{O_2}$ reacts slowly with most organic molecules (spin-forbidden pathways to singlet organic products) while its excited singlet form, $^1\mathrm{O_2}$, is a ferociously reactive oxidant central to photodynamic cancer therapy and to oxidative damage in biological tissue. The reasoning behind $\mathrm{B_2}$'s anomalous magnetism — that orbital ordering depends sensitively on atomic-orbital energy gaps and their element-dependent mixing — is the same reasoning used to rationalize bonding in main-group clusters and in boron hydrides, where boron's small $s$–$p$ gap drives characteristically electron-deficient, multicenter bonding patterns quite different from carbon's. More broadly, the electron-counting method demonstrated here for homonuclear diatomics generalizes directly into heteronuclear diatomics (CO, NO, CN⁻) and into the Isoelectronic Principle, where species sharing a valence electron count are recognized to share MO filling patterns, bond orders, and magnetic behavior — the logic used routinely to predict, for instance, that $\mathrm{NO^+}$ (isoelectronic with $\mathrm{N_2}$, diamagnetic, strong triple bond) behaves utterly differently from neutral $\mathrm{NO}$ despite differing by only one electron. Magnetic susceptibility measurements (Gouy balance, SQUID magnetometry, EPR spectroscopy) remain the direct experimental test of every prediction made from an MO diagram, closing the loop between the drawn diagram and the measurable, physical property it claims to explain.

Ask the tutor: Why does N₂ have such an unusually high bond dissociation energy compared to its paramagnetic neighbors B₂ and O₂, and how does the MO diagram make that quantitative?

Every homonuclear diatomic of period-2 elements is built from the 2s and 2p valence orbitals of two identical atoms. Combining two 2s orbitals gives $\sigma_{2s}$ (bonding) and $\sigma_{2s}^*$ (antibonding). Combining the two $2p_z$ orbitals that point along the internuclear axis gives $\sigma_{2p}$ and $\sigma_{2p}^*$. The remaining $2p_x$ and $2p_y$ orbitals on each atom, perpendicular to the axis, combine in pairs to give two degenerate bonding $\pi_{2p}$ orbitals and two degenerate antibonding $\pi_{2p}^*$ orbitals. This is the standard qualitative MO picture built on Homonuclear Diatomic MOs; what this article adds is the electron-counting rule that turns the diagram into a magnetism prediction, and the crucial detail of where $\sigma_{2p}$ sits relative to the $\pi_{2p}$ pair. The subtlety is this: $\sigma_{2s}$ and $\sigma_{2p}$ have the same symmetry ($\sigma_g$), so they are allowed by group theory to mix. When the 2s and 2p atomic orbitals are close in energy — true for the lighter elements Li through N, where the effective nuclear charge is small and the 2s–2p gap is modest — this $s$–$p$ mixing is strong. It pushes $\sigma_g(2s)$ down and pushes $\sigma_g(2p)$ up, high enough that $\sigma_g(2p)$ rises above the $\pi_u(2p)$ pair. For O, F, and Ne, the 2s–2p energy gap has widened enough (increasing $Z_{\rm eff}$ pulls 2s down much more than 2p) that mixing is weak, and the "intuitive" order $\pi_u(2p) < \sigma_g(2p)$ is restored. This single crossover, occurring between $\mathrm{N_2}$ and $\mathrm{O_2}$, is what makes $\mathrm{B_2}$ paramagnetic and is essential to get right. Once the energy ladder is fixed, filling it is exactly like filling atomic orbitals: lowest energy first, Pauli exclusion (two electrons per orbital, opposite spin), and Hund's rule — maximize total spin among degenerate orbitals before pairing. Only the outermost, partially-filled degenerate set matters for magnetism; fully filled shells (bonding and antibonding both full) always pair up and contribute nothing to net spin. The single deciding question for any diatomic is: does the last set of electrons land in a still-partially-filled degenerate pair?

Notice the crucial structural difference between the two diagrams: in the left-hand (Li₂–N₂) ordering, the degenerate $\pi_u(2p)$ pair sits below $\sigma_g(2p)$, so it fills before $\sigma_g(2p)$ does; in the right-hand (O₂–Ne₂) ordering, $\sigma_g(2p)$ fills first and the degenerate pair that matters for unpaired spins is $\pi_g^*(2p)$, higher up. Both diagrams have a degenerate pair that can trap an odd or half-filled electron pair; which diagram you use just changes which electron count lands in the degenerate slot.

The pattern is unmistakable: bond order rises then falls symmetrically about $\mathrm{N_2}$ (triple bond, bond order 3, the strongest and shortest of the row), and only two species in the whole row — $\mathrm{B_2}$ and $\mathrm{O_2}$ — are paramagnetic, precisely because only they land electrons singly into a still-open degenerate orbital pair.

The bond order from MO theory is:

where: $n_{\text{bonding}}$ is the total number of electrons in bonding MOs ($\sigma_{2s}, \sigma_{2p}, \pi_{2p}\times 2$), and $n_{\text{antibonding}}$ is the total in antibonding MOs ($\sigma_{2s}^*, \sigma_{2p}^*, \pi_{2p}^*\times 2$).

Reading: every electron placed in a bonding orbital adds constructive overlap between the nuclei; every electron in an antibonding orbital subtracts it. The factor of $\tfrac12$ converts a raw electron difference into the same "half-a-bond-per-electron-pair" units used for a Lewis single/double/triple bond, so MO bond order and Lewis bond order agree whenever they can be compared (e.g. $\mathrm{N_2}$: both give 3). The magnetism criterion follows directly from Hund's rule applied to the filled diagram:

where: the sum runs only over MO sets that are degenerate (the two $\pi_u$ or two $\pi_g^*$ orbitals); non-degenerate $\sigma$ orbitals always fill in pairs and contribute zero net spin once occupied by 2 electrons.

Reading: if $n_{\text{unpaired}} > 0$, the molecule is paramagnetic (attracted into a magnetic field, weakly, because unpaired electron spins have permanent magnetic moments); if $n_{\text{unpaired}} = 0$, it is diamagnetic (weakly repelled, all spins paired and all orbital motions individually canceled). The magnetic moment magnitude follows the familiar spin-only formula, useful for connecting bond diagrams to measured susceptibilities:

where: $n$ is the number of unpaired electrons and $\mu_B$ is the Bohr magneton ($9.274\times 10^{-24}\ \mathrm{J\,T^{-1}}$).

Reading: this formula, borrowed from the same spin-counting logic used for transition-metal complexes, converts the MO-diagram electron count directly into an experimentally checkable number; for $n=2$ (both $\mathrm{O_2}$ and $\mathrm{B_2}$) it predicts $\mu = \sqrt{2\times4} = \sqrt8 \approx 2.83\ \mu_B$.

Example 1 (foundational) — $\mathrm{C_2}$, an in-between case. Carbon contributes $2s^2 2p^2$ per atom, so $N_{\rm val} = 2\times4 = 8$. Using the Li₂–N₂ ordering (still applicable, since carbon is early enough for strong $s$–$p$ mixing):

Check: $2+2+4 = 8$. Here the degenerate $\pi_u(2p)$ pair is completely filled (4 electrons, 2 per orbital), so there is no unpaired-electron opportunity: $\mathrm{C_2}$ is diamagnetic. Bond order:

This bond order of 2, built from a filled $\pi$-orbital pair with an empty $\sigma_g(2p)$, is a genuinely unusual bonding picture — a "double bond with no $\sigma$ component" — and is part of why $\mathrm{C_2}$ has attracted decades of debate over whether it should be assigned a bond order as high as 4 in more sophisticated valence-bond treatments; the simple MO count gives 2, and diamagnetism either way, which is the testable magnetic prediction this article is built around.

Example 2 (applied) — Is the superoxide ion $\mathrm{O_2^-}$ paramagnetic? Superoxide adds one electron to neutral $\mathrm{O_2}$'s 12 valence electrons, giving $N_{\rm val}=13$. Using the O₂-type ordering, the extra electron enters the $\pi_g^*(2p)$ set alongside the two already there:

Check: $2+2+2+4+3 = 13$. With three electrons distributed over the two degenerate $\pi_g^*$ orbitals, one orbital must hold a pair (opposite spins) and the other a single electron: exactly one electron remains unpaired.

Bond order: bonding electrons $= 2+2+4=8$; antibonding electrons $=2+3=5$,

correctly reproducing superoxide's known bond order of 1.5, intermediate between $\mathrm{O_2}$'s double bond and peroxide $\mathrm{O_2^{2-}}$'s single bond, and correctly predicting that superoxide, like $\mathrm{O_2}$ itself, is paramagnetic — a fact directly relevant to its role as a reactive oxygen species in biochemistry, where its single unpaired spin underlies both its detectability by EPR and its chemical reactivity.

Example 3 (exam/competition level) — $\mathrm{CN}^-$ vs. $\mathrm{CN}$ (radical) vs. $\mathrm{NO}$: three heteronuclear cousins tested by the same logic. Although this article's MO diagrams were built for homonuclear diatomics, the electron-counting method extends immediately to heteronuclear species built from similar-energy AOs, which is exactly the reasoning that generalizes into the Isoelectronic Principle. Cyanide ion $\mathrm{CN}^-$ has valence electron count $4(\mathrm{C}) + 5(\mathrm{N}) + 1(\text{charge}) = 10$, isoelectronic with $\mathrm{N_2}$:

giving bond order 3 and diamagnetism, matching $\mathrm{CN^-}$'s known behavior as a strong-field, closed-shell ligand. The neutral radical $\mathrm{CN}^\bullet$ has one fewer electron, $N_{\rm val}=9$, isoelectronic with $\mathrm{N_2^+}$: the highest-energy $\sigma_g(2p)$-type orbital (for the N₂-type ordering, this is the HOMO) holds a single electron.

$\mathrm{CN}^\bullet$ is paramagnetic (it is, after all, a radical — the odd-electron count guarantees at least one unpaired spin regardless of orbital ordering details, a useful universal check). Nitric oxide $\mathrm{NO}$, valence count $5+6=11$, isoelectronic with $\mathrm{O_2^+}$, uses the O₂-type ordering:

Check: $2+2+2+4+1=11$. One electron sits alone in the doubly-degenerate $\pi^*$ set:

correctly matching NO's known bond order of 2.5 (between double and triple, consistent with its bond length of 115 pm, shorter than a genuine double bond) and its well-established paramagnetism (NO is one of the very few stable, isolable free-radical gases, detectable directly by EPR — a property exploited in NO's biological role as a vasodilator signaling molecule, where its odd-electron nature is mechanistically central).

The crossover point sits between $\mathrm{N_2}$ and $\mathrm{O_2}$, not at some arbitrary spot; the physical driver is the growing 2s–2p atomic orbital gap with increasing nuclear charge across the row, which progressively suppresses $s$–$p$ mixing. This table also shows the correct limiting behavior at both ends of the row: $\mathrm{Be_2}$ and $\mathrm{Ne_2}$ both have bond order zero (equal bonding and antibonding electron counts) and are not expected to exist as stable bound molecules under ordinary conditions — a useful "as $N_{\rm val}\to$ closed shell, BO$\to 0$" limiting check on the whole framework.