MO theory of heteronuclear diatomics: CO, NO, HF and the Walsh correlation diagrams

Intuition Beginner

When two different atoms form a bond, their atomic orbitals sit at different energies. Oxygen's 2p orbitals are lower in energy than carbon's because oxygen is more electronegative — its nucleus pulls electrons harder. This energy mismatch changes the molecular orbitals that form.

In a homonuclear molecule like N2, each MO is shared equally between the two atoms. The bonding orbital has the electron density split 50/50. In a heteronuclear molecule, the bonding MO leans toward the more electronegative atom. The electrons spend more time on one side of the molecule than the other. This is what makes a bond polar — not a full transfer of charge (that would be ionic), but an uneven sharing.

Carbon monoxide, CO, is the textbook example. Carbon has 6 electrons, oxygen has 8, giving 14 total — the same as N2. Because CO and N2 have the same electron count, they are isoelectronic. Their MO diagrams look remarkably similar. CO has a triple bond, just like N2, and a bond dissociation energy of 1072 kJ/mol — even stronger than N2's 945 kJ/mol. The difference is polarity: CO has a small dipole moment because oxygen pulls the bonding electrons toward itself.

Hydrogen fluoride, HF, is a different story. Hydrogen brings only a 1s orbital; fluorine brings 2s and 2p orbitals that sit much lower in energy. The energy gap between H 1s and F 2p is so large that only one strong bonding MO forms. The remaining fluorine orbitals are essentially unchanged — they become nonbonding orbitals, holding electron pairs that belong entirely to fluorine. These nonbonding pairs are the lone pairs you draw in Lewis structures.

Visual Beginner

A heteronuclear MO diagram has atomic orbitals at different heights on the left and right sides. The more electronegative atom (higher ionisation energy, stronger pull on electrons) has its orbitals drawn lower. When the atomic orbitals combine, the bonding MO sits closer in energy and character to the more electronegative atom. The antibonding MO sits closer to the less electronegative atom.

For CO, the MO filling is: $\sigma (2s{)}^2{\sigma }^{\ast }(2s{)}^2\pi (2p{)}^4\sigma (2p_z{)}^2$. The bond order is $(8-2)/2=3$, a triple bond. The two antibonding electrons in ${\sigma }^{\ast }(2s)$ partially cancel two of the ten bonding electrons, netting three bonds. The diagram closely mirrors N2, confirming the isoelectronic relationship.

For HF, the picture is simpler. The H 1s orbital overlaps with one of the F 2p orbitals (the one pointing along the bond axis). They form one bonding $\sigma$ MO and one antibonding ${\sigma }^{\ast }$ MO. The other two F 2p orbitals (perpendicular to the bond) do not overlap with H 1s and remain as nonbonding lone pairs. Filling gives 2 electrons in the bonding $\sigma$, 4 in the nonbonding F 2p orbitals, and 2 in the nonbonding F 2s orbital. Bond order = 1.

Worked example Beginner

Problem. Construct the MO diagram for CO and predict (a) the bond order, (b) whether CO is paramagnetic or diamagnetic, and (c) the direction of the dipole moment.

Step 1: Electron count. C has 6 electrons, O has 8, total = 14. Ignoring the core 1s electrons (2 on each atom, chemically inert), we have 10 valence electrons to place.

Step 2: Energy ordering. Oxygen is more electronegative (3.44 on the Pauling scale) than carbon (2.55), so all oxygen atomic orbitals sit lower in energy. The MO ordering for CO follows the same pattern as N2 because they are isoelectronic: $\sigma <{\sigma }^{\ast }<\pi <\sigma <{\pi }^{\ast }<{\sigma }^{\ast }$.

Step 3: Fill with 10 valence electrons.

MO	Type	Electrons
$\sigma (2s)$	bonding	2
${\sigma }^{\ast }(2s)$	antibonding	2
$\pi (2p_x),\pi (2p_y)$	bonding pair	4
$\sigma (2p_z)$	bonding	2

All 10 electrons are placed. No electrons occupy antibonding ${\pi }^{\ast }$ or ${\sigma }^{\ast }(2p_z)$.

Step 4: Answers.

(a) Bonding = 2 + 4 + 2 = 8. Antibonding = 2. Bond order = (8 − 2) / 2 = 3. A triple bond.

(b) All electrons are paired. CO is diamagnetic.

(c) The bonding MOs are polarised toward oxygen (the more electronegative atom). The small dipole moment points from C to O, with the negative end on oxygen. The experimental dipole moment of CO is 0.11 D — small because the lone pair on carbon partially opposes the oxygen-directed polarity.

Check your understanding Beginner

Formal definition Intermediate+

For a heteronuclear diatomic AB with atomic orbitals ${\varphi }_A$ on atom A and ${\varphi }_B$ on atom B, the LCAO-MO framework from 14.05.01 generalises by allowing unequal Coulomb integrals:

$${\alpha }_A:=⟨{\varphi }_A|\hat{H}|{\varphi }_A⟩\ne {\alpha }_B:=⟨{\varphi }_B|\hat{H}|{\varphi }_B⟩.$$

The difference $\Delta \alpha :={\alpha }_A-{\alpha }_B$ encodes the electronegativity mismatch. The resonance integral $\beta :=⟨{\varphi }_A|\hat{H}|{\varphi }_B⟩$ and overlap $S:=⟨{\varphi }_A|{\varphi }_B⟩$ remain as before.

The secular equation for a two-orbital model is

$$\det \left(\begin{array}{cc}{\alpha }_A-E& \beta -ES\\ \beta -ES& {\alpha }_B-E\end{array}\right)=0,$$

which expands to the quadratic

$$(1-S^2)E^2-({\alpha }_A+{\alpha }_B-2\beta S)E+({\alpha }_A{\alpha }_B-{\beta }^2)=0.$$

In the zero-overlap Hückel limit ($S=0$), the roots simplify to

$$E_{\pm }=\frac{{\alpha }_A+{\alpha }_B}{2}\pm \sqrt{{\left(\frac{{\alpha }_A-{\alpha }_B}{2}\right)}^2+{\beta }^2}.$$

The corresponding eigenvectors satisfy

$$\frac{c_A}{c_B}=\frac{E_{\pm }-{\alpha }_B}{\beta }.$$

For the bonding root $E_{-}$ (the lower energy): since $E_{-}<{\alpha }_B<{\alpha }_A$ (when ${\alpha }_A>{\alpha }_B$), the numerator $E_{-}-{\alpha }_B<0$ while $\beta <0$, giving $c_A/c_B<1$ in magnitude. The bonding MO has a **larger coefficient on the more electronegative atom** (atom B with lower ${\alpha }_B$). For the antibonding root $E_{+}$: $c_A/c_B>1$ in magnitude, so the antibonding MO is weighted toward the less electronegative atom.

Polar-covalent crossover. Define the polarity parameter $\eta :=|\beta |/|\Delta \alpha |$. When $\eta \gg 1$ (strong coupling relative to the energy gap), the MOs approach the homonuclear limit with nearly equal coefficients. When $\eta \ll 1$ (large energy gap, weak coupling), the bonding MO approaches the pure atomic orbital ${\varphi }_B$ and the antibonding approaches ${\varphi }_A$. The bond transitions from covalent ($\eta \gg 1$) to ionic ($\eta \ll 1$) as the electronegativity difference grows.

Mulliken population analysis. The electron density in an MO ${\psi }_k=c_A^{(k)}{\varphi }_A+c_B^{(k)}{\varphi }_B$ is partitioned between the atoms by the gross atomic population:

$$q_A^{(k)}=|c_A^{(k)}{|}^2+c_A^{(k)\ast }{c}_B^{(k)}{S}_{AB},$$

with $q_B^{(k)}$ defined symmetrically and $q_A^{(k)}+q_B^{(k)}=1$ for a normalised MO. Summing over all occupied MOs gives the total electron population on each atom. The difference from the neutral-atom electron count gives the Mulliken partial charge, quantifying the polarity predicted by the MO diagram.

Full heteronuclear diatomic MO construction. For second-row heteronuclear diatomics like CO and NO, the basis includes all valence atomic orbitals (2s, 2p${}_x$, 2p${}_y$, 2p${}_z$) on both atoms — 8 basis functions yielding 8 MOs. The secular equation is an $8\times 8$ generalised eigenvalue problem. By symmetry, the $\sigma$ orbitals (from 2s and 2p${}_z$ on each atom) and the $\pi$ orbitals (from 2p${}_x$ and 2p${}_y$) decouple into separate blocks: a $4\times 4$ $\sigma$ block and two identical $2\times 2$ $\pi$ blocks.

For CO specifically, the 8 MOs in order of increasing energy are:

$${\sigma }_1<{\sigma }_2^{\ast }<{\pi }_x\sim {\pi }_y<{\sigma }_3<{\pi }_x^{\ast }\sim {\pi }_y^{\ast }<{\sigma }_4^{\ast },$$

where $\sim$ denotes degeneracy. The labels omit the $g/u$ subscript used in homonuclear diatomics because the inversion symmetry is broken — heteronuclear molecules lack a centre of symmetry and belong to the $C_{\infty v}$ point group rather than $D_{\infty h}$.

Nonbonding orbitals and the HF limit

When $|\Delta \alpha |\gg |\beta |$, one or more atomic orbitals may have no partner of compatible energy and symmetry on the other atom. These orbitals enter the MO picture with coefficients almost entirely on one atom and energy nearly equal to their atomic value. They are called nonbonding orbitals. In HF, the fluorine 2s orbital and the two perpendicular fluorine 2p orbitals are nonbonding: hydrogen has no 2s or 2p orbitals to pair with them, and H 1s is too far in energy from F 2s for meaningful mixing. These become the lone pairs of Lewis theory.

Counterexamples to common slips

• "The more electronegative atom always has the larger coefficient in every MO." This is true only for bonding MOs. Antibonding MOs have larger coefficients on the less electronegative atom. Each MO has its own coefficient ratio determined by the eigenvalue equation.

• "Heteronuclear diatomics cannot have degenerate MOs." The $\pi$ orbitals (from 2p${}_x$ and 2p${}_y$) remain degenerate in heteronuclear diatomics because rotation around the bond axis is still a symmetry operation of $C_{\infty v}$. The loss of inversion symmetry does not break the cylindrical symmetry.

• "CO is isoelectronic with N2, so they have identical properties." Isoelectronic species share the same electron count and bond order, but properties like dipole moment, bond length, and reactivity differ. CO has a dipole moment (0.11 D); N2 does not. CO binds to haemoglobin; N2 does not.

• "The HOMO of CO sits on carbon because carbon is less electronegative." This is correct and has profound biochemical consequences. The highest occupied MO (HOMO) of CO is the $\sigma (2p_z)$ orbital, which is concentrated on carbon. This carbon-localised HOMO is what donates electron density to the iron in haemoglobin, forming the coordinate bond that causes carbon monoxide poisoning.

Key theorem with proof Intermediate+

Theorem (Heteronuclear MO energies and polarisation). For a heteronuclear diatomic AB with Coulomb integrals ${\alpha }_A>{\alpha }_B$, resonance integral $\beta <0$, and overlap $S=0$ (Hückel approximation), the bonding MO energy is

$$E_{-}=\frac{{\alpha }_A+{\alpha }_B}{2}-\sqrt{{\left(\frac{\Delta \alpha }{2}\right)}^2+{\beta }^2},$$

and the ratio of coefficients in the bonding MO satisfies

$${\left|\frac{c_A}{c_B}\right|}^2=\frac{(\Delta \alpha /2{)}^2+{\beta }^2-|\beta |\sqrt{(\Delta \alpha /2{)}^2+{\beta }^2}}{(\Delta \alpha /2{)}^2+{\beta }^2+|\beta |\sqrt{(\Delta \alpha /2{)}^2+{\beta }^2}}<1,$$

confirming that the bonding MO is polarised toward the more electronegative atom B.

Proof. The secular determinant with $S=0$ is

$$\det \left(\begin{array}{cc}{\alpha }_A-E& \beta \\ \beta & {\alpha }_B-E\end{array}\right)=({\alpha }_A-E)({\alpha }_B-E)-{\beta }^2=0.$$

Expanding: $E^2-({\alpha }_A+{\alpha }_B)E+({\alpha }_A{\alpha }_B-{\beta }^2)=0$. By the quadratic formula:

$$E_{\pm }=\frac{{\alpha }_A+{\alpha }_B}{2}\pm \sqrt{{\left(\frac{{\alpha }_A-{\alpha }_B}{2}\right)}^2+{\beta }^2}.$$

Since ${\beta }^2>0$, the square root exceeds $|\Delta \alpha /2|$, so $E_{-}$ is the lower root (bonding) and $E_{+}$ is the upper root (antibonding).

For the eigenvector at $E_{-}$, the secular equation gives $({\alpha }_A-E_{-})c_A+\beta c_B=0$, hence

$$\frac{c_A}{c_B}=-\frac{\beta }{{\alpha }_A-E_{-}}.$$

Substituting $E_{-}$:

$${\alpha }_A-E_{-}={\alpha }_A-\frac{{\alpha }_A+{\alpha }_B}{2}+\sqrt{{\left(\frac{\Delta \alpha }{2}\right)}^2+{\beta }^2}=\frac{\Delta \alpha }{2}+\sqrt{{\left(\frac{\Delta \alpha }{2}\right)}^2+{\beta }^2}.$$

Since $\Delta \alpha >0$ and the square root is positive, ${\alpha }_A-E_{-}>0$. With $\beta <0$:

$$\frac{c_A}{c_B}=\frac{|\beta |}{\frac{\Delta \alpha }{2}+\sqrt{(\Delta \alpha /2{)}^2+{\beta }^2}}.$$

The denominator exceeds $|\beta |$ (since $\Delta \alpha /2>0$ adds to the square root), so $|c_A/c_B|<1$. The bonding MO has a larger coefficient on atom B (the more electronegative atom). $\square$

Corollary (Perturbative limits).

1. Covalent limit $\Delta \alpha \to 0$: $|c_A/c_B|\to 1$, recovering the homonuclear result.

2. Ionic limit $|\Delta \alpha |\gg |\beta |$: expanding the square root, $E_{-}\approx {\alpha }_B-{\beta }^2/\Delta \alpha$ and $|c_A/c_B|\approx |\beta /\Delta \alpha |\ll 1$. The bonding MO is almost entirely on atom B.

Exercises Intermediate+

Walsh diagrams for triatomic molecules Master

The MO theory of heteronuclear diatomics generalises to polyatomic molecules. For triatomic molecules (H2O, CO2, BeH2, etc.), a central question is: what geometry does the molecule adopt? Walsh diagrams (also called correlation diagrams) answer this by tracking how each MO energy changes as a function of the bond angle.

A.D.E. Walsh, in his landmark 1953 paper, showed that the preferred geometry of a polyatomic molecule is determined by the number of valence electrons. Each MO has a characteristic dependence on bond angle: some MOs decrease in energy as the angle opens from 90 to 180 degrees (favoured by more electrons), while others increase. The total energy, obtained by filling the MOs and summing, determines the equilibrium angle.

H2O Walsh diagram. Water has 8 valence electrons (O provides 6, each H provides 1). The relevant MOs, labelled by $C_{2v}$ symmetry, are:

• $1a_1$: a low-lying bonding orbital combining O 2s with H 1s orbitals. Energy decreases (stabilises) as the H-O-H angle decreases from 180 to 90 degrees.
• $2a_1$: a bonding orbital from O 2p${}_z$ (along the $C_2$ axis) mixed with H 1s. Energy decreases as the angle decreases.
• $1b_2$: a bonding orbital from O 2p${}_y$ (perpendicular to the molecular plane for the linear configuration) mixed with H 1s. Energy increases as the angle decreases from 180 to 90.
• $3a_1$: a predominantly nonbonding orbital on oxygen. Relatively flat as a function of angle.
• $1b_1$: a nonbonding orbital (pure O 2p${}_x$, perpendicular to the molecular plane). Energy is independent of the H-O-H angle because this orbital has no overlap with H 1s at any angle.

For H2O with 8 valence electrons, the filling is $1a_1^{2}1b_2^{2}2a_1^{2}1b_1^2$ (or equivalently $1a_1^{2}2a_1^{2}1b_2^{2}1b_1^2$ depending on the ordering at the equilibrium angle). The angle-dependent orbitals $1a_1$ and $2a_1$ both prefer a bent geometry, and their stabilisation outweighs the destabilisation of $1b_2$. The predicted angle is in the range 90–110 degrees — experiment gives 104.5 degrees.

CO2 Walsh diagram. Carbon dioxide has 16 valence electrons. The Walsh diagram for a symmetric triatomic ABA shows that the first several MOs prefer a linear geometry. With 16 electrons, all the angle-destabilising orbitals are filled for bent geometries, and the linear geometry (D${}_{\infty h}$, angle 180 degrees) is strongly favoured. Experimentally, CO2 is indeed linear.

Walsh's rules. Walsh summarised his analysis into a set of empirical rules:

1. Molecules with 4 or fewer valence electrons around the central atom tend to be linear.
2. Trihydrides (AH2) with 5–8 valence electrons are bent.
3. The critical electron count at which the geometry changes depends on the specific orbital ordering, which in turn depends on the electronegativity of the atoms.

These rules are qualitative predictions of MO theory. Modern computational chemistry (DFT, coupled cluster) computes the exact energy as a function of angle and locates the minimum numerically, but Walsh's insight — that the geometry is determined by how the occupied MOs respond to angle changes — remains the conceptual foundation.

Connections Master

• Homonuclear diatomic MO theory 14.05.02. This unit generalises the homonuclear LCAO framework by relaxing the constraint ${\alpha }_A={\alpha }_B$. The secular equation, variational principle, and Aufbau filling carry over directly. Every result of 14.05.02 is recovered in the limit $\Delta \alpha \to 0$.

• Periodic trends and electronegativity 14.01.02. The Coulomb integral difference $\Delta \alpha$ that drives MO polarisation is a direct consequence of the periodic trends in ionisation energy and electron affinity covered in 14.01.02. Electronegativity quantifies the orbital energy mismatch that MO theory uses to predict polarity.

• Lewis structures and VSEPR 14.02.01. The nonbonding orbitals in HF correspond exactly to the lone pairs drawn in Lewis structures. Walsh diagrams provide the MO-theoretic justification for VSEPR geometry predictions — the same electron-count rules that VSEPR uses are explained by how MO energies depend on bond angle.

• Spectroscopy 14.12.01. The MO energy level diagrams for CO, NO, and HF predict their UV-visible and photoelectron spectra. The HOMO-LUMO gap of CO ($3\sigma \to {\pi }^{\ast }$) corresponds to a well-known UV absorption. Photoelectron spectroscopy measures the ionisation energies of individual MOs, providing direct experimental validation of the orbital ordering.

• Biochemistry 17.04.01. The carbon-localised HOMO of CO is the electronic-structure reason for carbon monoxide's toxicity: it donates electron density to the iron in haemoglobin more effectively than O2, outcompeting oxygen for the binding site. The MO diagram is the mechanistic explanation at the quantum level.

Historical notes Master

The extension of MO theory to heteronuclear molecules followed quickly after the homonuclear treatment. Robert Mulliken's 1932 papers on "The Assignment of Quantum Numbers for Electrons in Molecules" included heteronuclear diatomics and introduced the concept of electronegativity as a determinant of orbital mixing. Mulliken's electronegativity scale, defined as $(I+A)/2$ (the average of ionisation energy and electron affinity), was directly motivated by the role of $\Delta \alpha$ in the MO secular equation.

The isoelectronic principle — that species with the same electron count have similar MO diagrams — was recognised early. The CO/N2 isoelectronic pair was a key test case. Both have bond order 3 and similar bond lengths (113 pm for CO, 110 pm for N2). The small differences (dipole moment, reactivity) are attributed to the polarisation of the MOs.

A.D.E. Walsh's 1953 paper "The Electronic Orbitals of Polyatomic Molecules" in the Journal of the Chemical Society was a landmark in theoretical chemistry. Walsh constructed correlation diagrams showing how each MO energy varies with bond angle for triatomic and tetratomic molecules. His rules for predicting molecular geometry from electron count were remarkably successful and predated the computational tools that would later confirm them. The Walsh diagram remains a standard tool in qualitative MO theory and is taught in every advanced inorganic chemistry course.

Mulliken's 1955 paper on electron population analysis formalised the assignment of partial charges to atoms within the MO framework. The Mulliken population analysis, despite its well-known basis-set dependence, remains the simplest and most widely used method for partitioning electron density in quantum chemistry calculations. More sophisticated schemes (Lowdin, Natural Population Analysis, Bader's AIM) address its limitations but share the same goal: assigning fractional charges to atoms based on the MO wave function.

The development of density functional theory (DFT) in the 1960s and its computational implementation in the 1990s made quantitative MO calculations accessible for heteronuclear molecules. Modern DFT calculations on CO, NO, HF, and triatomic molecules like H2O and CO2 reproduce experimental geometries, dipole moments, and vibrational frequencies to within a few percent — validating the qualitative MO picture that Walsh and Mulliken constructed from first principles.

Bibliography Master

• Walsh, A.D.E., "The electronic orbitals of polyatomic molecules," J. Chem. Soc. (1953), 2260–2266. The founding paper on Walsh correlation diagrams and qualitative geometry prediction from MO theory.

• Mulliken, R.S., "Electron population analysis on LCAO-MO molecular wave functions," J. Chem. Phys. 23 (1955), 1833–1840. The Mulliken population analysis for partitioning electron density between atoms.

• Mulliken, R.S., "The assignment of quantum numbers for electrons in molecules," Phys. Rev. 32 (1928), 186; 41 (1932), 49; 43 (1933), 279. The extension of MO theory to heteronuclear diatomics and the origin of Mulliken electronegativity.

• DeKock, R.L. & Gray, H.B., Chemical Structure and Bonding (Benjamin/Cummings, 1980), Ch. 4. A pedagogical treatment of heteronuclear MO diagrams and Walsh diagrams.

• Zumdahl, S.S. & DeCoste, D.J., Chemical Principles, 8e (Cengage, 2017), Ch. 9.4–9.5. Introductory treatment of heteronuclear diatomic MO theory.

• Atkins, P. & Friedman, R., Molecular Quantum Mechanics, 5e (Oxford University Press, 2010), Ch. 5. Rigorous treatment of heteronuclear LCAO-MO theory.

• McQuarrie, D.A., Quantum Chemistry, 2e (University Science Books, 2008), Ch. 9. MO theory for heteronuclear diatomics and polyatomics.