Heteronuclear Diatomic MOs

Carbon monoxide has one of the strongest bonds known in the periodic table. It measures 1072 kJ/mol, stronger even than N≡N. Yet CO is a notorious poison. That same bonding scheme leaves a lone pair pointing the "wrong way," off the carbon. This lone pair is primed to attack a metal center in hemoglobin or in a catalytic converter. Explaining this apparent contradiction requires more than counting electrons. It requires an asymmetric molecular orbital diagram. In this diagram, the atomic orbitals of carbon and oxygen enter at different energies. They mix with different weights and produce orbitals that are polarized. Some point toward oxygen; some, strangely, point back toward carbon. Homonuclear diatomics like N₂ and O₂ are the training wheels of MO theory. Two identical atoms contribute orbitals of identical energy. So every bonding MO is an exactly equal blend of both atoms. Orbital ordering puzzles reduce to a single question of s-p Mixing and Orbital Ordering. Real chemistry, however, is overwhelmingly heteronuclear. CO, NO, HF, HCl, and every polar bond in organic and inorganic chemistry demand a different diagram. In it, the two atomic orbital manifolds sit at different energies on the same vertical axis. The resulting MOs are unequal mixtures. They weight toward whichever atom's atomic orbital lies closer in energy to the MO itself. This asymmetry is not a minor correction. It is the entire origin of bond polarity. It explains the correlation between MO coefficients and Percent Ionic Character. It also links orbital shape to the molecular dipole discussed under Bond Polarity and Dipole Moments. The big picture is a single organizing rule, borrowed from perturbation theory. An atomic orbital contributes most heavily to whichever molecular orbital lies closest to it in energy. The mixing coefficient shrinks as the energy gap between the two atomic orbitals grows. When the gap is small (C 2p vs O 2p, both around −11 to −16 eV), the MOs are strongly mixed and nearly covalent. When the gap is enormous (H 1s at −13.6 eV vs F 2p at −18.7 eV, but more importantly F 2s buried at −40 eV), the "bonding" MO collapses almost entirely onto the electronegative atom. The bond then looks almost, but not quite, ionic. This single principle — energy match controls orbital mixing — lets us build CO, NO, and HF from first principles. From the diagram, we can read off bond order, magnetism, dipole direction, and reactivity directly. This article builds the three canonical heteronuclear cases explicit atomic-orbital energy ladder by explicit atomic-orbital energy ladder. These are CO (isoelectronic with N₂ but polarized), NO (one electron beyond CO, radical, colored), and HF (the maximally asymmetric limit, essentially a lone pair plus an ionic-covalent bond). Along the way we derive why CO binds metals through carbon. We show why NO is a stable radical that dimerizes only weakly. We also explain why HF's bonding MO is 80% fluorine in character. This is true despite looking, on a Lewis structure, perfectly ordinary.

Key takeaways

In a heteronuclear MO diagram, atomic orbitals of the two atoms are placed at different energies on the vertical axis, set by each atom's orbital ionization energy.
Each MO is a weighted combination $\psi = c_A\phi_A + c_B\phi_B$ ; the coefficients are unequal, and the more electronegative atom's orbital dominates the bonding MOs while the electropositive atom's orbital dominates the antibonding MOs.
CO has a triple bond (σ + 2π, bond order 3) like N₂, but the HOMO is a mostly-carbon lone pair — this is why CO binds transition metals through C, not O.
NO has one electron in a π* orbital, making it a stable, colored, paramagnetic radical with bond order 2.5, weaker and longer than the CO/CN⁻/N₂ bond.
HF has no π system at all; F 2s stays nonbonding (too low in energy to mix), and the σ bonding MO is roughly 80% F 2p / 20% H 1s, giving high ionic character without full ionic transfer.
Key idea: the fractional "ownership" of each MO by each atom is fixed by the atomic orbital energy gap, and reading that ownership off the diagram is reading off the bond polarity, the dipole direction, and the reactive lone pair.

Where this lives today

CO's carbon-localized HOMO is the working basis of essentially all transition-metal carbonyl chemistry: Ni(CO)₄, Fe(CO)₅, Mo(CO)₆, and every metal carbonyl catalyst intermediate in hydroformylation and Fischer–Tropsch chemistry rely on CO donating that carbon-centered lone pair into empty metal d-orbitals (σ-donation) while metal-filled d-orbitals back-donate into CO's empty 2π orbital — a synergic bonding scheme (the Dewar–Chatt–Duncanson model) that is only comprehensible once you know the HOMO sits on carbon and the LUMO is the π pair spread over both atoms. The same carbon-donor orbital explains why CO poisoning of hemoglobin and of heme-based enzymes occurs at the iron center via the carbon end, and why industrial CO poisoning of Pd/Pt hydrogenation catalysts and fuel-cell electrodes is such a persistent engineering problem — CO binds transition-metal surface sites about two orders of magnitude more strongly than H₂ or O₂, precisely because of this excellent σ-donor/π-acceptor combination. NO's radical character and its bond-order sensitivity to oxidation state (NO⁺ at bond order 3, NO at 2.5, NO⁻ at 2) underlie the entire chemistry of nitrosyl transition-metal complexes, used both as spectroscopic probes of metal oxidation state (the Enemark–Feltham {MNO}ⁿ formalism explicitly exploits this MO picture) and, biologically, as the basis of nitroglycerin and related nitrate drugs, which work by releasing NO to activate soluble guanylate cyclase and relax vascular smooth muscle — the discovery that earned the 1998 Nobel Prize in Physiology or Medicine. HF's heavily polarized σ bond, and the near-nonbonding fluorine lone pairs, show up directly in the extreme reactivity and hydrogen-bonding behavior of HF (it is a weak acid in water despite its strong bond, because the strength of the H–F σ bond, not electronegativity alone, controls gas-phase acidity) and in why HF etches glass (attacking Si–O bonds via a mechanism that depends on the same fluorine lone-pair nucleophilicity visible in this diagram). More broadly, the entire heteronuclear MO framework — asymmetric orbital energy ladders producing polarized bonding/antibonding pairs — is the direct conceptual bridge to building MO diagrams for polyatomic molecules using full point-group symmetry, where central-atom and ligand-group orbitals of different energies mix in exactly this same asymmetric way, as developed under Polyatomic MOs by Symmetry.

Ask the tutor: How does the same asymmetric-mixing framework extend to central-atom/ligand-group orbital diagrams in polyatomic molecules like H₂O or NH₃?

The physical picture

Building a heteronuclear diagram starts with a step that homonuclear diagrams skip entirely: placing the two atoms' valence orbitals at different heights on the energy axis, using experimental valence orbital ionization energies (VOIEs), which track (with sign flipped) the atomic orbital energies from Hartree–Fock or from photoelectron spectroscopy. For period-2 atoms these energies drop steadily left to right because increasing effective nuclear charge pulls all valence orbitals down: roughly $-10.7$ eV (2p) for B, $-13.4$ eV for C, $-15.4$ eV for N (2p), $-15.9$ eV for O 2p, and $-18.7$ eV for F 2p, with the corresponding 2s orbitals sitting 10–20 eV below their own 2p on the same atom. This single monotonic trend across a period is the whole reason heteronuclear diagrams are asymmetric, and it is worth committing to memory as a mental number line before drawing any diagram. Once the two orbital ladders are drawn, the rule for mixing is: orbitals combine to form MOs only if they (i) share the same irreducible representation along the internuclear (z) axis — σ-type with σ-type, π-type with π-type — and (ii) lie reasonably close in energy. The resulting bonding MO sits below the lower of the two contributing AOs, is weighted more heavily by whichever AO started closer in energy to it, and correspondingly the antibonding partner sits above the higher AO and is weighted more heavily by the other atom. This is the polarization rule: bonding MOs lean toward the more electronegative atom; antibonding MOs lean toward the less electronegative atom. It is the direct MO-theory statement of what a Lewis/VSEPR treatment calls "the bonding pair is pulled toward the more electronegative atom" — see Bond Polarity and Dipole Moments — except now it is quantitative, atom-by-atom, and orbital-by-orbital rather than a single averaged bond dipole. CO is built from carbon (2s ≈ −19.4 eV, 2p ≈ −10.7 eV as free-atom term-averaged values, though VOIE tables differ slightly in convention — the ordering and the ~9 eV 2s–2p gap on carbon is the reliable part) and oxygen (2s ≈ −32.4 eV, 2p ≈ −15.9 eV). Because oxygen is more electronegative, both its 2s and 2p lie below carbon's, and every bonding MO in CO is oxygen-rich while every antibonding MO is carbon-rich — with one crucial exception at the top of the diagram, the HOMO, which despite being an antibonding combination of the 2s manifold ends up carbon-rich and non-bonding in character, projecting outward from carbon. That lone pair is the σ-donor orbital that makes CO such a good ligand.

Figure

Atomic orbital energy ladders for C, N, O, and F valence orbitals (schematic, eV)

NO adds one more electron beyond CO/CN⁻/N₂'s isoelectronic 14-valence-electron count, landing it in the doubly-degenerate π level. That single unpaired electron is the entire personality of the molecule: NO is paramagnetic, it is a radical (yet kinetically fairly stable because the electron sits in an orbital with modest antibonding character spread over both atoms), it dimerizes only weakly to (NO)₂, and it is a genuine biological signaling molecule (endothelium-derived relaxing factor) precisely because a lone diffusible radical with one unpaired π electron reacts fast with heme iron and with O₂ but does not immediately self-destruct. HF is the limiting case in which the electronegativity gap becomes so large that one entire atomic orbital — F 2s — simply refuses to mix with anything on hydrogen, because it sits some 25+ eV below H 1s, an energy mismatch far outside the range where mixing does anything but produce a trivially small perturbative correction. F 2s becomes an almost pure nonbonding lone pair. Only F 2p_z (the one directed along the bond axis) is close enough in energy to H 1s to mix, forming one bonding σ and one antibonding σ, both heavily weighted toward fluorine, while F's two perpendicular 2p orbitals (2p_x, 2p_y) have no partner of the right symmetry on H and remain nonbonding lone pairs on fluorine. The net picture — one bonding pair, three fluorine lone pairs — reproduces the ordinary Lewis structure of HF, but MO theory now tells you how ionic* that one bond is and exactly what fraction of the bonding electron density sits on which atom.

Figure

CO molecular orbital diagram: C 2s/2p (left) and O 2s/2p (right) combine into polarized MOs

The core ideas

The mixing of two atomic orbitals $\phi_A$ (on the electropositive atom) and $\phi_B$ (on the electronegative atom) into bonding and antibonding MOs is governed by a two-level perturbation problem:

\psi_\pm = c_A\phi_A \pm c_B\phi_B, \qquad |c_A|^2+|c_B|^2 = 1

where: $\psi_+$ is the lower-energy (bonding) MO, $\psi_-$ the higher-energy (antibonding) MO, and $c_A, c_B$ are real mixing coefficients (S taken as small, non-orthogonality corrections neglected for the qualitative picture). Reading: equal coefficients ( $c_A = c_B$ ) is the homonuclear special case; the heteronuclear case has $c_A \ne c_B$ , and which one is larger is exactly the physical content of the next equation. The degree of polarization is controlled by the energy gap between the unmixed atomic orbitals relative to the interaction (resonance) integral between them:

\frac{c_B}{c_A} \approx \frac{\beta}{\alpha_A - \alpha_B}, \qquad \text{(bonding MO, } \alpha_B < \alpha_A)

where: $\alpha_A$ and $\alpha_B$ are the atomic orbital energies (Coulomb integrals, both negative, $\alpha_A$ less negative i.e. higher in energy — this is the electropositive atom), $\beta$ is the resonance (interaction) integral (negative, and roughly proportional to orbital overlap $S_{AB}$ ), and the ratio is only valid when $|\beta| \ll |\alpha_A - \alpha_B|$ (weak-mixing / non-degenerate perturbation regime). Reading: as $\alpha_A - \alpha_B \to 0$ (atoms electronically identical, the homonuclear limit) the ratio $c_B/c_A \to \beta/0$ , which diverges — signaling that the weak-mixing formula breaks down and equal mixing ( $|c_A|=|c_B|$ ) must be used instead, exactly as required for N₂ or O₂. As $\alpha_A - \alpha_B$ grows large (HF-like), $c_B/c_A \to 0$ in the bonding MO's complementary antibonding partner, meaning the bonding MO becomes almost pure $\phi_B$ (the electronegative atom) and the antibonding MO becomes almost pure $\phi_A$ . The overall bond order and magnetism follow the same electron-filling rules used for homonuclear diatomics, with bond order defined identically:

\text{Bond order} = \tfrac{1}{2}\left(n_{\text{bonding}} - n_{\text{antibonding}}\right)

where: $n_{\text{bonding}}$ and $n_{\text{antibonding}}$ are the numbers of electrons in bonding and antibonding MOs respectively (nonbonding electrons, such as F 2s in HF or the polarized lone pairs in CO's 1σ/3σ, are excluded from the count). Reading: this formula is diagram-agnostic — it works exactly the same way whether the MOs are 50/50 mixed (homonuclear) or 90/10 polarized (HF), because bond order counts electrons in bonding vs. antibonding levels, not the atomic character of those levels.

Derivation ATwo-orbital mixing and the polarization of CO's frontier orbitals

Goal. Show explicitly why CO's HOMO (3σ) ends up localized on carbon even though carbon is the less electronegative atom, and connect this to the standard "1π below 3σ, no s–p crossover past N₂" ordering used for CO.

Step 1 — set up the atomic energy ladder. Using representative valence orbital ionization energies, C 2s ≈ −19.4 eV, C 2p ≈ −10.7 eV (free-atom-like values; different tabulations shift these by 1–3 eV, but the relative ordering is what matters), O 2s ≈ −32.4 eV, O 2p ≈ −15.9 eV. Oxygen's orbitals sit uniformly below carbon's, consistent with oxygen's larger effective nuclear charge and higher electronegativity.

Step 2 — form the σ-symmetry combinations. Along the C–O axis, C 2s, C 2p_z, O 2s, and O 2p_z all transform as σ and are allowed to mix (this is the same 4-orbital σ manifold that in N₂ produces the famous s–p mixing responsible for π-below-σ ordering; see s-p Mixing and Orbital Ordering). Because O 2s (−32.4 eV) is far below C 2p (−10.7 eV) — a gap of over 20 eV — O 2s mixes only weakly with the higher carbon orbitals and drops out to become the deep, largely non-bonding 1σ MO, almost pure O 2s in character.

Step 3 — the second σ level (2σ). The next combination pulls together C 2s (−19.4 eV) with O 2p (−15.9 eV) — now a much smaller gap of only about 3.5 eV — plus a smaller admixture of O 2s and C 2p. Because O 2p is the higher of this close pair (less negative, i.e. O 2p sits energetically above O 2s and reasonably close to C 2s), this bonding combination is weighted toward oxygen: applying the polarization rule, the bonding MO leans toward whichever atomic orbital is closer in energy to the resulting MO, and here that is predominantly the oxygen contribution. 2σ is therefore O-rich and strongly bonding.

Step 4 — the frontier σ level (3σ) and the two π levels. The remaining σ combination (largely C 2s + C 2p mixing against a smaller O 2p_z contribution) is pushed up in energy by its antibonding interaction with the deep O 2s/2p combinations, exactly the s–p mixing mechanism familiar from N₂. This upward push is large enough that 3σ rises above the two degenerate 1π (bonding, C 2p_x,y + O 2p_x,y, O-rich because O 2p sits lower) orbitals. Crucially, because 3σ is built mostly from the carbon 2s/2p combination (carbon's orbitals being higher in energy and less able to stabilize by bonding with the very deep O 2s), the resulting 3σ MO is polarized onto carbon — it becomes a lobe pointing away from oxygen, essentially a carbon-centered lone pair with slight antibonding character.

Step 5 — fill ten valence electrons. CO has 4 (C) + 6 (O) = 10 valence electrons. Filling from the bottom: $1\sigma^2\,2\sigma^2\,1\pi^4\,3\sigma^2$ — that is 2 + 2 + 4 + 2 = 10 electrons, all levels through 3σ (HOMO) full, 2π and 4σ empty.

\text{Bond order (CO)} = \tfrac12\left[(2+2+4+2) - 0\right]_{\text{bonding count only}} = \tfrac12(2+4+2) = 3

(1σ, being essentially nonbonding O 2s lone pair, is not counted as a bonding pair in the simplified bond-order tally, matching CO's experimental description as a triple bond: one σ, two π.)

Step 6 — interpret. The HOMO (3σ) is carbon-rich and points away from the O atom — precisely the orbital that donates into empty metal d-orbitals in metal carbonyls, explaining why CO binds through carbon in every known simple carbonyl complex, and why this binding mode is retained even though a naive electronegativity argument might have predicted oxygen-end coordination.

Go deeper · Derivation B · 2 minHF: quantifying "80% fluorine character" from the atomic orbital gap

Derivation BHF: quantifying "80% fluorine character" from the atomic orbital gap

Goal. Turn the qualitative claim "HF's bonding MO is mostly on fluorine" into an explicit coefficient calculation using a simple two-orbital (H 1s, F 2p_z) model, and connect the result to Percent Ionic Character.

Step 1 — the relevant orbitals. Only H 1s (α_H ≈ −13.6 eV) and F 2p_z (α_F ≈ −18.7 eV) are close enough in energy and correct in symmetry to combine into a σ bond; F 2s (≈ −40 eV) is too deep to mix appreciably, and F 2p_x, 2p_y have no σ-symmetric H partner, so they stay as fluorine lone pairs.

Step 2 — write the 2×2 secular problem. In a simple Hückel-like treatment, the bonding/antibonding energies solve

E_\pm = \frac{\alpha_H + \alpha_F}{2} \pm \sqrt{\left(\frac{\alpha_H - \alpha_F}{2}\right)^2 + \beta^2}

Taking $\alpha_H = -13.6$ eV, $\alpha_F = -18.7$ eV, and a representative resonance integral $\beta \approx -3.0$ eV (order-of-magnitude, scaled to typical H–F overlap):

\frac{\alpha_H - \alpha_F}{2} = \frac{-13.6 - (-18.7)}{2} = \frac{5.1}{2} = 2.55\ \text{eV}

E_\pm = -16.15 \pm \sqrt{(2.55)^2 + (3.0)^2} = -16.15 \pm \sqrt{6.50 + 9.00} = -16.15 \pm 3.94

giving $E_+ = -20.09$ eV (bonding σ) and $E_- = -12.21$ eV (antibonding σ*).

Step 3 — extract the mixing coefficients. For the bonding root, the secular equations give the coefficient ratio

\frac{c_H}{c_F} = \frac{\beta}{E_+ - \alpha_H} = \frac{-3.0}{-20.09-(-13.6)} = \frac{-3.0}{-6.49} = 0.462

Normalizing $c_H^2 + c_F^2 = 1$ with $c_H = 0.462\,c_F$ :

(0.462)^2 c_F^2 + c_F^2 = 1 \implies c_F^2(1.2135) = 1 \implies c_F^2 = 0.824,\ \ c_H^2 = 0.176

Step 4 — interpret as percent orbital character. The bonding σ MO is therefore

\%\,\text{F character} = 82.4\%, \qquad \%\,\text{H character} = 17.6\%

Step 5 — connect to ionic character. This is not the same number as the "percent ionic character" extracted from the measured dipole moment (HF: μ = 1.82 D, bond length 0.917 Å, giving an ionic-limit-referenced ionic character of roughly 41% via $\mu_{\text{obs}}/\mu_{\text{ionic}}$ , see Percent Ionic Character) — the two percentages answer different questions. The 82%/18% split describes how the shared bonding pair is distributed between the atoms' basis orbitals; the 41% ionic character describes how far the actual charge distribution has moved from perfectly covalent (50/50 charge, μ = 0) toward perfectly ionic (full electron transfer, μ = μ_ionic). A bonding MO can be heavily polarized in orbital coefficient while the resulting dipole moment is a smaller fraction of the full ionic limit, because polarization of the shared pair is a different (and typically larger-looking) number than fractional charge transferred, and because core/lone-pair contributions to μ partly cancel the bonding-pair contribution.

Step 6 — the limiting check. As $\alpha_H \to \alpha_F$ (an artificial homonuclear H–H-like limit), $E_+ - \alpha_H \to \beta$ , so $c_H/c_F \to 1$ and $c_F^2 \to 0.5$ : the formula correctly recovers equal 50/50 mixing in the no-electronegativity-difference limit, as it must.

Worked examples

Example 1 (foundational): Electron count and bond order of CO from the diagram directly. CO contributes 4 valence electrons from C (2s²2p²) and 6 from O (2s²2p⁴), for 10 total. Filling the CO MO diagram built above, in order of increasing energy: 1σ² 2σ² 1π⁴ 3σ² (10 electrons; 2π and 4σ empty). Counting bonding pairs (2σ, 1π×2 — three bonding pairs) against zero antibonding pairs filled gives

\text{Bond order} = \tfrac12(2 + 4 + 2 - 0) = 3

matching the experimental C≡O bond length of 112.8 pm and force constant near 1902 N/m, both close to the C≡C-like values expected for a genuine triple bond, and consistent with CO being isoelectronic with N₂ (bond order 3, 109.8 pm) and CN⁻ (bond order 3). The diagnostic difference from N₂ is entirely in the polarization: N₂'s 3σg is delocalized equally over both N atoms (a genuine lone pair on each end, both good donors, explaining why N₂ can bind metals from either end in principle although weakly), whereas CO's analogous 3σ is skewed onto carbon, making CO an enormously stronger and more universal σ-donor ligand than N₂.

Example 2 (applied): Why NO's bond is weaker and longer than CO's, and why NO is colored and paramagnetic. NO carries 4(from N: 2s²2p³ → 5) — more carefully, N contributes 5 valence electrons and O contributes 6, for 11 total, one more than CO/CN⁻/N₂. Using the same MO ordering (1σ² 2σ² 1π⁴ 3σ² then the extra electron must go into 2π*):

1\sigma^2\,2\sigma^2\,1\pi^4\,3\sigma^2\,2\pi^{*1}

\text{Bond order} = \tfrac12\big[(2+4+2) - 1\big] = \tfrac12(8-1) = 3.5

Wait — check the bonding/antibonding partition carefully: 1σ is essentially nonbonding (mostly O 2s, excluded), 2σ and 1π (six electrons) are bonding, and 2π* (one electron) is antibonding, giving

\text{Bond order} = \tfrac12\big[(2+4)-1\big] = \tfrac12(6-1) = 2.5

which matches the standard textbook value for NO (bond order 2.5, intermediate between the double bond of NO⁺'s isoelectronic-with-CO analogue at bond order 3 minus one antibonding electron's worth of weakening). Losing that one π electron to form NO⁺ (nitrosyl cation, isoelectronic with CO and CN⁻) raises the bond order back to 3 and shortens the bond (NO 115.1 pm → NO⁺ 106 pm), while adding an electron to form NO⁻ lowers it toward bond order 2. The single unpaired electron in neutral NO's 2π level is responsible for (i) paramagnetism (one unpaired spin, confirmed by EPR), (ii) the visible/near-UV absorption bands that give NO its role as a spectroscopic probe, and (iii) NO's characteristic reactivity as a radical — it reacts readily with O₂ (itself a diradical, see O₂'s own π* half-filling) to give NO₂, and with heme Fe(II) centers, the basis of nitric oxide's role as a vasodilating signaling molecule.

Example 3 (exam/competition level): Estimating the dipole moment direction and rough magnitude of CO from orbital polarization, and reconciling it with the observed near-zero (0.11 D) dipole. This is the classic "gotcha" of introductory MO teaching: naive electronegativity reasoning (O more electronegative than C) predicts a dipole with the negative end on oxygen and a moment on the order of 1–2 D, similar to other C–O single/double bonds. The measured dipole moment of CO is only 0.122 D, and — remarkably — the negative end is on the carbon, not the oxygen (established by microwave Stark spectroscopy and confirmed by high-level ab initio calculations of the CO charge distribution). To see why the diagram predicts this, tabulate where each pair of electrons "sits" using the polarization derived above: 1σ (2 electrons, ~90% O character, deep and inert) and 2σ (2 electrons, O-rich bonding pair) both pull electron density toward oxygen, exactly as electronegativity would suggest, and the 1π pairs (4 electrons, moderately O-rich) do the same, all consistent with an oxygen-negative dipole contribution. But the 3σ HOMO (2 electrons) is carbon-rich — essentially a lone pair bulging outward from carbon, away from oxygen — and this orbital's dipole contribution points the opposite way, partially canceling the oxygen-ward polarization of the σ and π bonding pairs.

\mu_{\text{total}} \approx \mu(1\sigma) + \mu(2\sigma) + \mu(1\pi\times2) + \mu(3\sigma)

Each term on the right is individually sizable (on the order of 1 D or more, consistent with the strong O-character of 1σ/2σ/1π), but the sign of $\mu(3\sigma)$ is opposite to the other three, so the sum is a small residual — and detailed calculation (and experiment) shows that residual comes out net carbon-negative. This is the single most instructive numerical fact in heteronuclear MO theory: a near-zero, "wrong-signed" dipole moment is not evidence against orbital polarization — it is evidence of several large, partially-canceling polarized orbitals, exactly analogous to how a molecule can have zero net dipole (CO₂, BF₃) despite every individual bond being highly polar, except here the cancellation is orbital-by-orbital within a single bond rather than bond-by-bond around a symmetric molecule.

Limiting cases, scope & common traps

Regime	AO energy gap	MO behavior	Example
Homonuclear limit	$\Delta\alpha \to 0$	Equal mixing, $c_A = c_B$ ; standard symmetric diagram applies	N₂, O₂, F₂
Weak polarization	$\Delta\alpha$ small (few eV)	Mild lean toward electronegative atom; s–p mixing still possible; π/σ ordering can still invert	CO, CN⁻, NO
Strong polarization	$\Delta\alpha$ large (∼10+ eV)	Bonding MO dominated (≳70–85%) by electronegative atom; antibonding MO dominated by electropositive atom	HF, HCl
Extreme mismatch	$\Delta\alpha \gg \|\beta\|$	Orbitals fail to mix at all; become purely nonbonding lone pairs	F 2s in HF; core orbitals in general
One-electron-beyond-closed-shell	add 1 e⁻ to isoelectronic closed-shell parent	Radical character, half-filled π or σ, paramagnetism, lower bond order	NO relative to CO/CN⁻/N₂

The chemist behind it

The systematic construction of heteronuclear MO diagrams, and the recognition that atomic orbital energy differences (rather than just symmetry) control bond polarity, grew out of the broader molecular orbital program built by Robert Mulliken in the 1920s–1930s, extended through the ligand-field and photoelectron-spectroscopy era by chemists who needed quantitative orbital energy ladders (valence orbital ionization energies) to make heteronuclear diagrams predictive rather than merely qualitative.

Origins · 1 minRobert S. Mulliken and how this was discovered

Robert S. Mulliken

1896–1986 · American physical chemist, pioneer of molecular orbital theory

Mulliken, working from the 1920s onward — first at Harvard and the University of Chicago — developed and named the molecular orbital approach to chemical bonding as an alternative and complement to Heitler–London valence bond theory, introducing the σ/π/g/u orbital nomenclature still used today and championing correlation diagrams that connect atomic orbitals of separated atoms to molecular orbitals of the bound molecule — the same logical structure used to build the CO, NO, and HF diagrams here. He received the 1966 Nobel Prize in Chemistry "for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method." Mulliken also pioneered practical schemes for population analysis (Mulliken charges) and for extracting atomic orbital energies from spectroscopic and ionization data, laying the numerical groundwork (valence orbital ionization energies) that makes it possible to draw a heteronuclear diagram with real, defensible AO energies rather than a cartoon.

History

- 1916 — G. N. Lewis introduces the shared electron-pair bond, the conceptual ancestor of the "bonding pair" language still used to describe MOs.

- 1927 — Heitler and London give the first quantum mechanical treatment of the H₂ bond, founding valence bond theory as a rival framework.

- 1928–1932 — Mulliken and Hund independently develop the molecular orbital approach, with Mulliken introducing σ, π, bonding/antibonding terminology for diatomics.

- 1930s–1950s — Application of MO theory to heteronuclear diatomics (CO, NO, HF, CN) becomes standard, using empirical atomic term values to set relative AO energies.

- 1960s–1970s — Photoelectron spectroscopy (Turner and coworkers) provides direct experimental confirmation of MO energy orderings, including the closely-spaced 3σ/1π ordering in CO and N₂ that motivates the s–p mixing discussion.

Why can't we just use a single averaged energy for both atoms' 2p orbitals in a heteronuclear diagram, the way we do for homonuclear ones?

Because the entire physical content of a heteronuclear diagram — bond polarity, dipole direction, and which atom "owns" each MO — comes from the fact that the two atoms' orbitals sit at different energies. Averaging them away would silently convert every heteronuclear molecule back into an artificial homonuclear-style diagram with 50/50 mixing everywhere, which is exactly wrong: it would predict CO, NO, and HF all have zero dipole moment and perfectly shared bonding pairs, contradicting experiment. The asymmetric ladder is not a drawing convenience; it is the model's core physical claim, directly tied to each atom's Percent Ionic Character and to measured dipole moments.

Does a more electronegative atom always dominate the HOMO of a heteronuclear diatomic?

Not necessarily — CO is the standard counterexample. Oxygen dominates the deep 1σ, 2σ, and 1π orbitals, but the HOMO (3σ) is carbon-rich because it arises from an antibonding-flavored s–p mixing interaction that pushes a largely carbon-based combination upward past the 1π level. The general rule is that bonding MOs favor the more electronegative atom and antibonding MOs favor the less electronegative one; whether the HOMO itself is bonding, antibonding, or a distorted mix depends on the specific electron count and the strength of s–p mixing in that molecule, so each case must be traced through its own diagram.

How do valence orbital ionization energies (VOIEs) get measured or estimated, and how reliable are the numbers used in these diagrams?

VOIEs are typically obtained from atomic term-value spectroscopy (average energies over the relevant spectroscopic terms of the atom or its ions) or from semi-empirical fits calibrated against photoelectron spectroscopy of simple molecules; different tabulations (Hinze–Jaffé, Basch–Viste–Gray, and others) can disagree by 1–3 eV depending on convention (neutral atom vs. valence-state ionization, Slater vs. Hartree–Fock orbitals). For qualitative diagram-building the relative ordering — that F sits below O below N below C in both 2s and 2p, and that each atom's own 2s sits well below its own 2p — is robust across all tabulations and is the only feature actually needed to predict polarization direction and orbital ordering correctly.

Why does NO dimerize only weakly to (NO)₂, unlike, say, two NO₂ radicals combining readily to N₂O₄?

The unpaired electron in NO sits in a doubly-degenerate 2π level, and pairing two NO radicals' π electrons into a single N–N σ bond requires significant orbital reorganization and gives only a very weak, long, easily-broken N–N interaction in the resulting (NO)₂ dimer (stable only at very low temperature, in the gas phase or as a matrix species) — the dimerization enthalpy is only a few kJ/mol. This contrasts with NO₂'s unpaired electron, which sits substantially in an orbital with better direct overlap for N–N σ-bond formation, giving the much more robust N₂O₄ dimer that exists at room temperature. The lesson is that "has one unpaired electron" does not automatically mean "dimerizes easily" — the symmetry and spatial character of the singly-occupied orbital matters as much as the electron count.

Is HF's bond considered ionic or covalent, given how polarized its bonding MO is?

Neither label alone is adequate — HF is best described as a highly polar covalent bond. The bonding MO is a genuine two-center combination (not a discrete transferred electron plus a bare proton), so some covalent (shared, delocalized) character always remains, consistent with H–F's substantial bond dissociation energy (∼565 kJ/mol) being far higher than a purely electrostatic ion-pair interaction would predict at that bond length. The ∼80% fluorine orbital character reflects strong polarization of the shared pair, while the dipole-based ionic character (∼40%) reflects how far the actual charge distribution has shifted toward the full-transfer limit — both numbers describe the same bond as substantially, but not completely, polarized.

How does this framework extend to HCl, HBr, and HI, where the electronegativity gap to hydrogen shrinks down the halogen group?

The same two-orbital (H 1s / halogen np) framework applies, but the AO energy gap $\Delta\alpha$ between H 1s and the halogen's valence p orbital shrinks steadily from F to I (halogen np orbital energies rise, i.e., become less negative, down the group as electronegativity falls), so the bonding MO becomes progressively less polarized and the coefficients approach more equal mixing. This directly parallels the falling ionic character and falling dipole moments down the series (HF 1.82 D → HCl 1.11 D → HBr 0.83 D → HI 0.44 D), and is a clean, testable prediction of the same $c_B/c_A \approx \beta/(\alpha_A - \alpha_B)$ relationship used for HF, now with a shrinking denominator down the group.

Why is CO such a stronger σ-donor than N₂, given that both are isoelectronic with the same bond order and similar orbital diagrams?

In N₂, the analogous HOMO (3σg) is symmetric and shared equally between the two identical nitrogen atoms, so it is a comparatively modest, delocalized donor orbital at both ends. In CO, the polarization derived here concentrates the HOMO almost entirely onto the single carbon atom, making that lone pair far more spatially concentrated, higher in energy, and geometrically well-directed for overlap with an empty metal d-orbital — all factors that make it a dramatically better σ-donor despite the two molecules sharing the same electron count and the same nominal bond order.