Heteronuclear Diatomics: HF and CO

The asymmetric-mixing picture of heteronuclear diatomics is not a museum piece — it is load-bearing machinery in modern chemistry and biology. Carbon monoxide's carbon-rich $5\sigma$ HOMO is the direct explanation for why CO is such a potent ligand in organometallic chemistry and why it poisons heme proteins: CO donates that carbon lone pair into empty metal d-orbitals (σ-donation) while accepting metal d-electron density back into its empty $2\pi^*$ orbital (π-backbonding), a synergistic bonding mode that makes the Fe–CO bond in carboxyhemoglobin far stronger than the physiological Fe–O₂ bond in oxyhemoglobin — the molecular-orbital root cause of carbon monoxide toxicity. The same $\sigma$-donor/$\pi$-acceptor duality underlies industrial hydroformylation and Fischer–Tropsch catalysis, where metal carbonyl complexes are stabilized by exactly this orbital interaction, and it underlies infrared spectroscopic diagnostics of metal oxidation states, since the CO stretching frequency shifts systematically with how much electron density backfills into the $2\pi^*$ orbital. Hydrogen fluoride's strongly polarized σ bond, quantified above as roughly 85–90% fluorine character in the bonding MO, explains both HF's exceptionally strong hydrogen-bonding ability (the highly polarized bond leaves the hydrogen unusually electron-poor and thus an excellent hydrogen-bond donor) and its behavior as a weak acid in water despite the enormous bond dissociation energy of the H–F bond itself — the same asymmetric orbital picture that stabilizes the bond also localizes the bonding electrons onto fluorine so effectively that heterolytic cleavage to $\text{H}^+ + \text{F}^-$ is comparatively favorable in a polar solvent. Anhydrous HF is used industrially in petroleum alkylation and in etching processes for semiconductor manufacturing, both applications that trace back to this same electronic asymmetry. More broadly, the two-orbital asymmetric-mixing framework developed here is the direct ancestor of the electronegativity concept treated quantitatively in MO Polarity and Electronegativity, and it is the starting point for understanding chemical reactivity through Frontier Orbitals: HOMO and LUMO, since it is precisely the asymmetric polarization of frontier orbitals — carbon-rich HOMOs, oxygen-rich lower orbitals — that dictates which atom in a heteronuclear molecule acts as nucleophile or electrophile in a given reaction.

Ask the tutor: Why does a smaller 2s–2p energy gap on carbon cause stronger s–p mixing there than on oxygen, and how does that specifically flip the polarization of CO's 5σ orbital?

Start from the two-orbital problem that underlies all diatomic MO theory: two atomic orbitals, $\chi_A$ on atom A and $\chi_B$ on atom B, combine to form molecular orbitals $\psi = c_A\chi_A + c_B\chi_B$. In a homonuclear molecule such as N₂, $\chi_A$ and $\chi_B$ have identical orbital energies (Coulomb integrals $\alpha_A=\alpha_B=\alpha$), so by symmetry the bonding and antibonding combinations must have $|c_A|=|c_B|$ — equal-weight in-phase and out-of-phase combinations. The only question left is which AOs mix with which, governed by the energy gaps discussed in s-p Mixing in Second-Row Diatomics. Once $\alpha_A \ne \alpha_B$, that symmetry argument is gone. The variational problem must be solved honestly, and the algebra shows that the lower atomic orbital energy anchors the bonding MO more strongly, while the higher one anchors the antibonding MO more strongly. Physically: an electron in the bonding orbital "prefers" to sit closer to the atom where it would have been more stable even as a lone atomic orbital — the more electronegative atom. This is exactly the intuition chemists use qualitatively (electronegativity pulls electron density) made quantitative through $\alpha$, $\beta$, and overlap $S$. Two atoms illustrate two different regimes of this asymmetry. Fluorine's 2p orbital ($\alpha_F \approx -18.7$ eV) sits far below hydrogen's 1s orbital ($\alpha_H \approx -13.6$ eV), a gap of about 5 eV — large compared to the resonance integral. The bonding σ MO is therefore heavily F-2p in character (roughly 80% F, 20% H in a common parametrization), and the antibonding σ* is heavily H-1s. Oxygen's 2p orbital ($\alpha_O \approx -15.9$ eV) sits below carbon's 2p orbital ($\alpha_C \approx -10.7$ eV) by a smaller but still substantial ~5.2 eV, while the 2s orbitals are separated similarly. The resulting CO diagram looks superficially like N₂'s but with every level tilted toward oxygen — except that the frontier orbital, the $5\sigma$/HOMO, ends up polarized toward carbon due to how s–p mixing interacts with the asymmetric Coulomb integrals, a subtlety that explains CO's chemistry at transition metal centers.

The starting point is the same variational two-orbital secular problem used throughout MO theory, now with unequal diagonal (Coulomb) integrals. Writing $H_{AA}=\alpha_A$, $H_{BB}=\alpha_B$, $H_{AB}=\beta$, and overlap $S_{AB}=S$, the secular determinant is

where:

• $\alpha_A, \alpha_B$ — Coulomb integrals (approximate orbital energies of $\chi_A$, $\chi_B$ in the molecular environment), both negative and $\alpha_A \ne \alpha_B$
• $\beta = H_{AB}$ — resonance (exchange) integral, negative and roughly proportional to overlap
• $S = S_{AB}$ — overlap integral, $0<S<1$
• $E$ — molecular orbital energy (eigenvalue) being solved for

Neglecting overlap ($S=0$, the Hückel-style simplification used for orientation, though real calculations keep $S$), the quadratic reduces to a clean closed form:

where:

• $E_+$ — antibonding MO energy (raised above the higher atomic level)
• $E_-$ — bonding MO energy (lowered below the lower atomic level)
• the square-root term is the avoided-crossing gap: half the energy splitting between bonding and antibonding levels

Reading: when $\alpha_A=\alpha_B$ this collapses to the familiar homonuclear result $E_\pm=\alpha\pm\beta$. As the gap $|\alpha_A-\alpha_B|$ grows relative to $|\beta|$, the square root approaches $|\alpha_A-\alpha_B|/2$, so $E_-\to\alpha_B$ (the lower atomic level, barely perturbed) and $E_+\to\alpha_A$ (the higher atomic level, barely perturbed). Large electronegativity differences produce weak bonding in this idealized sense — the MOs become nearly pure atomic orbitals — which is why bonds between very dissimilar atoms are often better described as ionic than covalent, a limiting case worth remembering. The mixing coefficients (unnormalized ratio form, $S=0$) follow from substituting $E_-$ back into the secular equations:

where:

• for the bonding MO ($E_-$ close to $\alpha_B$), this ratio is large, meaning $c_B \gg c_A$: the bonding orbital is dominated by the lower-energy AO
• for the antibonding MO ($E_+$ close to $\alpha_A$), the analogous ratio is small, meaning the antibonding orbital is dominated by the higher-energy AO

Reading: electron density in the filled bonding MO sits preferentially on the atom whose orbital was already lower in energy — the more electronegative atom. This is the orbital-level definition of bond polarity, and it is exactly what feeds into a Pauling-type electronegativity difference in MO Polarity and Electronegativity.

Example 1 — Foundational: numeric two-orbital mixing for a generic A–B σ bond. Take $\alpha_A = -11.0$ eV, $\alpha_B = -16.0$ eV, $\beta = -2.5$ eV (chosen asymmetric and non-round to avoid masking sign or algebra errors), $S=0$. Compute $\Delta = \alpha_A-\alpha_B = -11.0-(-16.0) = 5.0$ eV, and the average $\Sigma/2 = (-11.0-16.0)/2 = -13.5$ eV.

So

Check the limiting-case sanity: since here $\Delta$ and $2|\beta|$ happen to be comparable ($5.0$ vs $5.0$), this is not deep in the perturbative regime, so both formulas should be checked. Using the perturbative estimate from the derivation, $E_-\approx \alpha_B - \beta^2/\Delta = -16.0 - (2.5)^2/5.0 = -16.0-1.25=-17.25$ eV — close to but visibly different from the exact $-17.036$ eV, exactly as expected since $\beta$ is not really small here; the exact quadratic must be trusted, and it is what was used. Now the coefficients: from $c_B/c_A = (\alpha_A-E_-)/\beta$,

Normalizing $c_A^2+c_B^2=1$ with $c_B = -2.414\,c_A$: $c_A^2(1+5.828)=1 \Rightarrow c_A = \pm0.383$, $c_B=\mp0.925$. Taking the physical (same-sign, bonding) choice, $c_A=0.383$, $c_B=0.925$: the bonding MO is about $(0.925)^2=85.5\%$ atom B and $14.5\%$ atom A — strongly B-polarized, consistent with B being the more electronegative partner.

Example 2 — Applied: the HF σ bond, quantitative polarization and dipole implication. Use literature-typical valence orbital ionization energies (VOIEs, a standard proxy for $\alpha$ in semi-empirical MO theory): $\alpha_H = -13.6$ eV (H 1s), $\alpha_F = -18.7$ eV (F 2p$_z$, the orbital pointing along the bond axis). Take a representative resonance integral $\beta \approx -2.35$ eV (a typical Wolfsberg–Helmholz-derived value for a first-row–H σ bond at bonding distance).

Both electrons of the bonding pair occupy $E_\sigma = -19.62$ eV, below both starting atomic levels — a genuine, if modest, bond stabilization (about 0.9 eV below the F 2p level alone, since $-19.62-(-18.7)=-0.92$ eV lower than fluorine's own orbital, despite fluorine already anchoring most of the density). Coefficient ratio: $c_F/c_H = (\alpha_H-E_\sigma)/\beta = (-13.6-(-19.62))/(-2.35) = 6.02/(-2.35) = -2.562$. Normalizing, $c_H^2(1+6.564)=1 \Rightarrow c_H = 0.364$, $c_F = 0.932$ (opposite-sign convention chosen so both AOs are in-phase in the bonding combination). Electron density fractions: F carries $(0.932)^2 = 86.9\%$, H carries $13.1\%$. This directly rationalizes the large dipole moment of HF (1.82 D, experimental) as an orbital effect: two bonding electrons sit ~87% on fluorine, transferring roughly $2\times0.869 - 1 \approx 0.74$ of an electron's worth of charge asymmetry toward F relative to a nonpolar reference bond — consistent (order of magnitude, not an exact charge count) with the qualitative picture of HF as $\text{H}^{\delta+}\text{–F}^{\delta-}$. As a limiting-case check: if $\alpha_H$ were made equal to $\alpha_F$ (imagine a hypothetical "isoelectronegative" H–F), the formula would collapse to $c_H=c_B=1/\sqrt2$, zero dipole from this bond — correctly recovering the homonuclear-style symmetric limit even though the atoms remain chemically different species.

Example 3 — Exam/competition level: locating the CO frontier orbital polarization including s–p mixing. This is the subtle case that trips up students who apply the naive two-orbital picture to CO without accounting for the near-degenerate 2s/2p manifold on each atom (the physics of s-p Mixing in Second-Row Diatomics). Take typical VOIEs: $\alpha_{C,2s}=-19.4$ eV, $\alpha_{C,2p}=-10.7$ eV, $\alpha_{O,2s}=-32.3$ eV, $\alpha_{O,2p}=-15.9$ eV (values consistent with standard extended-Hückel/semi-empirical tabulations, chosen deliberately non-round). First note the 2s–2p gap within each atom: for carbon it is $19.4-10.7=8.7$ eV; for oxygen it is $32.3-15.9=16.4$ eV. Oxygen's 2s orbital is so much lower than its own 2p (and far below carbon's 2p) that it mixes only weakly into the frontier σ system and behaves almost as a core-like lone pair — this matters because if oxygen's 2s freely mixed with carbon's 2p as strongly as in a homonuclear treatment, the whole level ordering would shift. Build the $3\sigma$ orbital primarily from carbon's 2s/2p hybrid character mixing with oxygen's 2p$_z$. In the s–p mixing regime (same physical mechanism as in N₂, but now with unequal atomic partners), the $4\sigma$ and $5\sigma$ orbitals repel each other. Because carbon's 2s and 2p are much closer together in energy (gap 8.7 eV) than oxygen's, carbon undergoes much stronger internal s–p hybridization than oxygen does. The upshot, confirmed by both semi-empirical and ab initio calculations, is that the $5\sigma$ orbital (HOMO) of CO is built predominantly from a carbon $sp$-hybrid lone pair pointing away from oxygen, with only a small admixture of oxygen character — even though oxygen is the more electronegative atom and anchors essentially every lower bonding MO. Quantitatively, population analyses (e.g., Mulliken partitioning from standard ab initio calculations on CO at its equilibrium bond length 1.128 Å) give roughly 70–75% carbon character in the $5\sigma$ HOMO versus only 25–30% oxygen character — the reverse polarization compared to every other filled valence MO in the molecule, which are all 60–75% oxygen. This single fact, that the HOMO of CO is anomalously carbon-rich due to the interplay of s–p mixing and unequal Coulomb integrals, is the orbital-level explanation for why CO binds transition metals through carbon (donating the carbon-rich $5\sigma$ lone pair into empty metal d-orbitals) rather than through the "obviously more electronegative and more lone-pair-looking" oxygen end. As a limiting-case sanity check: if carbon and oxygen had identical 2s–2p gaps (an artificial symmetric limit), s–p mixing would affect both atoms equally and the naive expectation — HOMO polarized toward the more electronegative atom, like every other filled MO — would hold; it is specifically the unequal degree of s–p mixing on the two centers, layered on top of the unequal Coulomb integrals, that flips the HOMO's polarization.