Walsh Diagrams

Walsh diagrams are not a historical curiosity confined to textbook AH₂/AH₃ examples; they are the conceptual backbone of how chemists read photoelectron and photodetachment spectra of small radicals and ions relevant to atmospheric and combustion chemistry. The bent-to-linear structural change of NO₂ upon ionization to NO₂⁺, and the associated long bending-mode vibrational progression, is a standard diagnostic used in atmospheric photochemistry studies of NOₓ species, where NO₂ photolysis and ionization pathways matter for tropospheric ozone chemistry. Similarly, the CO₂/CO₂⁺/CO₂⁻ family — central to greenhouse-gas monitoring, Mars-atmosphere spectroscopy, and CO₂-reduction electrocatalysis — is routinely rationalized with exactly this diagram: CO₂ is linear (16 valence electrons with a fully engaged π-bonding frontier), while adding an electron to form CO₂⁻ populates a bending-favoring orbital and bends the anion to roughly 135°, a fact directly relevant to understanding the transition state and intermediate geometries in CO₂ electroreduction catalysis, an active area of energy-storage and carbon-capture materials research. In main-group and organometallic chemistry, the AH₃-type $D_{3h}\leftrightarrow C_{3v}$ correlation underlies the discussion of pyramidal versus planar geometries at nitrogen, phosphorus, and carbanion centers, connecting directly to the study of amine inversion barriers (relevant to chiral amine stability and to the design of bridgehead amines that cannot invert), and to isoelectronic main-group anion/cation series used in modern frustrated Lewis pair and low-valent main-group catalysis, where the frontier-orbital symmetry of a bent versus planar E(II)/E(III) center (E = group 13–15 element) predicts its Lewis basicity and reactivity toward small-molecule activation (H₂, CO₂, N₂O). In spectroscopy instrumentation, Walsh-diagram thinking is built directly into the interpretation software and analysis conventions for ultraviolet photoelectron spectroscopy (UPS) and threshold photoelectron spectroscopy, where identifying which ionized orbital is angle-sensitive (long vibrational progression) versus angle-insensitive (short, sharp progression, near-vertical Franck–Condon envelope) is the first diagnostic step in assigning an experimental spectrum to a particular molecular orbital. The same logic, generalized to Delocalized Pi Systems and to Renner–Teller vibronic coupling in linear/quasi-linear radicals (important in interstellar-medium and combustion-radical spectroscopy, e.g. C₃H, HCCN), extends the two-orbital AH₂ story into a general framework for reading structural information directly out of orbital-symmetry correlation diagrams.

Ask the tutor: How would the Walsh diagram change if I built it for an AH₂ fragment where A is a third-row atom with accessible low-lying d orbitals — would the frontier-orbital bending logic still hold the same way?

Start with an AH₂ fragment — one central atom A with valence $ns$ and $np$ orbitals, two hydrogens each contributing a $1s$ orbital. Build the MOs first in the linear limit, point group $D_{\infty h}$, then again in the bent limit, point group $C_{2v}$, and finally connect the two sets by correlating irreducible representations as the angle $\theta$ (H–A–H) sweeps from 180° down through some hypothetical continuum to 90°. In $D_{\infty h}$, the two H $1s$ orbitals form symmetric ($\sigma_g^+$) and antisymmetric ($\sigma_u^+$) combinations. The central atom's $s$ orbital is $\sigma_g^+$; its $p_z$ (along the molecular axis) is $\sigma_u^+$; its $p_x, p_y$ (perpendicular to the axis) form a degenerate $\pi_u$ pair that cannot interact with the H group orbitals at all in a linear geometry, because $\pi_u$ has no totally-symmetric overlap with the $\sigma$-symmetry H₂ combinations. So in linear AH₂ we get, in order of increasing energy: a strongly bonding $1\sigma_g$ (A $s$ + H₂ symmetric), a bonding $1\sigma_u$ (A $p_z$ + H₂ antisymmetric), a nonbonding degenerate pair $1\pi_u$ (pure A $p_x, p_y$, entirely on the central atom, since no ligand orbital of matching symmetry exists), and above that an antibonding $2\sigma_g^*$. Now bend the molecule. The point group drops to $C_{2v}$, with symmetry elements $E, C_2, \sigma_v(xz), \sigma_v'(yz)$ (convention: molecule in the $yz$ plane, $C_2$ along $z$). Under this lower symmetry, $\sigma_g^+ \to a_1$, $\sigma_u^+ \to b_2$ (or $a_1$, depending on axis convention — here we place the H–H direction along $y$ so the in-plane, symmetric-under-$\sigma_v$ combinations become $a_1$), and crucially the degenerate $\pi_u$ pair splits, because degeneracy is a linear-molecule-only feature. One component of $\pi_u$ (the one lying in the molecular plane, i.e. along $y$, the H–H direction) becomes $a_1$ symmetry and correlates with — and mixes with — the $A_1$ combinations already present; the other component (perpendicular to the plane, along $x$) becomes $b_1$ and has no other $b_1$ orbital to mix with, so its energy is comparatively angle-insensitive. The mixing is the whole story. The in-plane $p$ orbital, once it becomes $a_1$ upon bending, can hybridize with the central $s$ orbital (also $a_1$) — precisely the same $s$–$p$ mixing that builds $sp$-like hybrids pointing at the two hydrogens. This mixing is angle-dependent: at 180° there is a symmetry prohibition on $s$–$p_\perp$ mixing (the relevant $p$ orbital is $\pi_u$, orthogonal by parity to the $\sigma_g$ $s$ orbital), but as the angle closes, the increasing overlap allows more $s$ character into this orbital, and by the noncrossing rule it is pushed down in energy as it moves away from the higher-lying $\sigma$-type $a_1$ combination it correlates with. That is why this orbital's Walsh curve slopes down from 180° to 90°. The out-of-plane $b_1$ orbital, remaining a nearly pure unhybridized $p$ orbital throughout (it is always orthogonal to the H group orbitals in the molecular plane), is comparatively flat, with a slight net destabilization on bending because the H atoms swing toward its nodal cone and reduce a small stabilizing overlap it had at linearity.

Two more small effects sharpen the picture for real molecules. The $1\sigma_g$-derived orbital ($1a_1$) is essentially the deep bonding combination of A $s$ with the symmetric H₂ group orbital; it is nearly always doubly filled and nearly flat across all angles because it is dominated by the strong A–H $\sigma$ bonding interaction, which is not very sensitive to the angle between the two bonds. The truly decisive curves are the frontier pair originating from $\pi_u$: the $a_1$ member that plunges in energy on bending (because it soaks up stabilizing $s$ character), and the $b_1$ member that stays comparatively flat or rises gently. Whichever of these two is the HOMO controls geometry. The extension to AH₃ (planar $D_{3h}$ vs pyramidal $C_{3v}$) runs by exactly the same logic, correlating $D_{3h}\to C_{3v}$. In planar $D_{3h}$ NH₃-like AH₃, the central $p_z$ orbital perpendicular to the H₃ plane is symmetry label $a_2''$; it is strictly nonbonding by symmetry (no matching combination of H $1s$ orbitals has $a_2''$ symmetry) and, being oriented along the $C_3$ axis, it is exactly the orbital that gains bonding/lone-pair character as the molecule pyramidalizes. On going to $C_{3v}$, $a_2'' \to a_1$, and this $a_1$ mixes strongly with the central $s$ orbital (also $a_1$ once pyramidal) to make the familiar rehybridized lone pair pointing away from the three H atoms. This orbital's Walsh curve drops steeply as the molecule folds away from planarity — which is exactly why NH₃ (8 valence electrons, HOMO = this orbital) is pyramidal with a shallow inversion barrier (~24 kJ mol⁻¹, fast umbrella inversion), while a species with one fewer electron in that orbital, or none at all in a nonbonding frontier level, prefers planarity (e.g., BH₃, 6 valence electrons, is rigorously planar $D_{3h}$; the CH₃⁺ cation is likewise planar).

The unifying theme, worth stating once cleanly: degenerate orbitals in the high-symmetry (linear or planar) structure lose their degeneracy on distortion, and one branch is stabilized while the other is not; whichever branch ends up occupied by the last electrons decides whether the distortion is energetically favorable. This is the molecular-orbital face of the Delocalized Pi Systems logic that will reappear when discussing ring puckering and pseudorotation, and it is the rigorous version of second-order Jahn–Teller reasoning.

The central relationship is not a single closed-form equation (Walsh diagrams are built from correlation and perturbation theory, not from one master formula), but the underlying energetic bookkeeping can be written compactly. Define the total valence electronic energy as a sum over occupied MOs:

where:

• $\theta$ is the bond angle (H–A–H for AH₂, or the pyramidalization angle for AH₃),
• $\varepsilon_i(\theta)$ is the energy of MO $i$ as a function of $\theta$, read directly off the Walsh diagram,
• $n_i \in \{0,1,2\}$ is the occupation number of orbital $i$.

The equilibrium angle is the $\theta^*$ that minimizes $E_{\text{tot}}(\theta)$:

where:

• the first condition locates a stationary point,
• the second confirms it is a minimum rather than a maximum (an unstable, non-equilibrium angle).

Because most of the doubly-filled low-lying $\sigma$-bonding orbitals are nearly flat in $\theta$ (they are dominated by strong, angle-insensitive A–H bonding), the sum is, to good approximation, dominated by the frontier orbital(s) straddling the HOMO:

where:

• $n_{\text{HOMO}}$ is the occupation of the frontier orbital (0, 1, or 2 electrons),
• $\frac{d\varepsilon_{\text{HOMO}}}{d\theta}$ is the local slope of that orbital's Walsh curve.

Reading: if the HOMO's Walsh curve slopes downward toward smaller $\theta$ (as the in-plane $a_1$ derived from $\pi_u$ does for AH₂), then populating it makes bending energetically favorable; if the HOMO is a flat or angle-insensitive orbital (like $1\sigma_u/b_2$), added electrons produce little geometric preference, and the molecule remains near-linear or is set by weaker secondary effects. The mixing coefficient that produces the slope itself comes from second-order perturbation theory: the in-plane $\pi_u$-derived orbital mixes with the $\sigma$-symmetric AO combination with a stabilization

where:

• $H_{12}(\theta)$ is the angle-dependent mixing (resonance) integral between the central $p$-orbital component and the $s$-derived $a_1$ combination, vanishing at $\theta = 180^\circ$ by symmetry and growing as $\theta$ decreases,
• $\varepsilon_p - \varepsilon_{s,\text{comb}}$ is the unperturbed energy gap between the interacting basis combinations (an energy denominator, always taken positive here since we stabilize the lower state and destabilize the upper one).

This is the microscopic origin of the downward-sloping Walsh curve: as $\theta$ shrinks from 180°, $H_{12}(\theta)$ grows from zero, and second-order mixing pushes the lower (in-plane, eventually $a_1$) combination down while pushing a higher unoccupied $\sigma$-antibonding combination up by the same mechanism (level repulsion, a two-state avoided crossing).

Example 1 (foundational): predict and rationalize the shape of BeH₂ versus NH₂⁻. BeH₂: Be contributes 2 valence electrons, each H contributes 1, total $N = 2 + 1 + 1 = 4$. Filling the Walsh diagram from the bottom: $1a_1^2$ (2 electrons) then $1b_2^2$ (2 electrons) exactly exhausts the count — both flat, strongly A–H bonding orbitals, nothing in the bending-sensitive $2a_1$ or $1b_1$. By the electron-counting rule derived above, $N - 4 = 0$: no frontier occupation, so no bending driving force. Predicted geometry: linear, $\theta = 180°$. This matches the experimentally/computationally established linear structure of gas-phase BeH₂ (isoelectronic in valence count with CO₂'s central-atom-only comparison is not exact, but BeH₂ is a textbook linear AH₂). NH₂⁻: N contributes 5 valence electrons, each H contributes 1, plus the extra negative charge contributes 1 more electron, total $N = 5 + 1 + 1 + 1 = 8$. This is the same electron count as neutral H₂O (isoelectronic species). Filling: $1a_1^2 1b_2^2 2a_1^2 1b_1^2$ — both frontier orbitals doubly occupied, with $2a_1$ dominating the angle dependence. Predicted geometry: strongly bent, similar to or even more acute than water's angle because the extra electron density (anion) can enhance $s$–$p$ mixing effects modestly. Experimentally, NH₂⁻ has a bond angle near 102–104°, consistent with the isoelectronic H₂O comparison and the Walsh prediction.

Example 2 (applied): use the Walsh diagram to explain why ozone (O₃) is bent while CO₂ is linear, treating the central-atom-and-two-ligands unit with the AH₂-type diagram generalized to AB₂. The AH₂ Walsh diagram generalizes directly to any AB₂ system (A central, B terminal) as long as B contributes orbitals of matching symmetry to the H $1s$ — for main-group terminal atoms this means using B's frontier $\sigma$-type combination (e.g. an $sp$ hybrid pointing at A) in place of H $1s$, with additional $\pi$-symmetry orbitals on B entering a separate manifold; the frontier σ/n-orbital counting logic carries over unchanged for judging linear-vs-bent preference, since it is dictated by the same central-atom nonbonding $p\to (a_1,b_1)$ splitting. CO₂: count valence electrons used in the $\sigma$-framework and nonbonding sets relevant to the bending coordinate. Total valence electrons: C (4) + 2 O (6 each) = 16. Two C=O σ bonds and two sets of C=O π bonds absorb 4 (σ) + 8 (two π bonds, 4 electrons each accounting for both O lone-pair-derived and bonding character in the full MO treatment) electrons in strongly bonding combinations that are essentially angle-insensitive by the same "no other orbital of that symmetry to mix with" argument used for $1\sigma_g,1\sigma_u$ in AH₂ — except here there are extra oxygen lone pairs. The decisive point: the central-atom nonbonding-derived frontier orbitals analogous to $2a_1/1b_1$ in AH₂ are, for CO₂, both empty or both fully bonding-committed because carbon's $p_x, p_y$ (perpendicular set) are fully engaged in $\pi$ bonding with both oxygens (forming bonding $\pi_u$ combinations that are filled, and antibonding $\pi_g^*$ combinations that remain empty). There is no extra pair of electrons left over to populate a bending-favoring $2a_1$-type orbital the way there is in H₂O; the molecule's frontier occupied orbitals are the filled $\pi_u$ bonding MOs, which (analogous to the flat $1\pi_u$ nonbonding pair of BeH₂ before any splitting is favorable) do not gain enough stabilization from bending to overcome the loss of $\pi$-bonding overlap that bending would cause. Predicted and observed: linear, $\theta = 180°$. Ozone: O₃ has $3\times 6 = 18$ valence electrons, two fewer available bonding partners for full triple-bond-like delocalization than in CO₂ relative to its electron count, and critically the central oxygen carries a nonbonding lone pair in an orbital directly analogous to AH₂'s $2a_1$ — populated because ozone's π system is a 3-center-4-electron arrangement (allyl-anion-like) that leaves a central-oxygen in-plane lone pair only partially engaged in π bonding, exactly the electron-count situation ($N-4 \geq 1$ in the frontier-orbital sense) that favors bending. Predicted and observed: bent, experimental angle 116.8°, substantially closer to 120° than water's 104.5° because oxygen's frontier orbital in ozone is less purely nonbonding than in H₂O (partial π delocalization stiffens the bending potential), but still bent rather than linear, exactly as the Walsh-diagram electron-counting logic requires.

Example 3 (competition-level): use the Walsh diagram together with Koopmans' theorem to predict how the bond angle of NO₂ changes upon photoionization to NO₂⁺, and rationalize the vibrational progression seen in the photoelectron spectrum. NO₂ (neutral) is an AB₂-type 17-valence-electron radical (N: 5, two O: 6 each, total 17) with bond angle 134°, intermediate between ozone's 117° and a hypothetical linear limit — consistent with it having one unpaired electron in the frontier $2a_1$-type (bending-favoring) orbital rather than the two paired electrons found in ozone's analogous, more strongly populated frontier level combined with additional antibonding population effects; the qualitative Walsh-diagram point is that NO₂'s unpaired electron sits in the same symmetry-type orbital ($a_1$, derived from the central-atom in-plane nonbonding orbital of the $D_{\infty h}\to C_{2v}$ correlation) that drives bending in AH₂ systems, but with a shallower net stabilization than in ozone or water because of partial electron count and competing π-delocalization onto both oxygens. Apply Koopmans' theorem: the first vertical ionization energy corresponds, to a good approximation, to removing an electron from the HOMO while leaving the geometry fixed at the neutral's equilibrium angle (134°), i.e. $IE_1 \approx -\varepsilon_{\text{HOMO}}(\theta = 134°)$. Removing the single electron from the bending-favoring $2a_1$-type HOMO of NO₂ produces NO₂⁺, which — carrying zero electrons in that orbital rather than one — should relax toward a larger bond angle, because the driving force for bending (populated $2a_1$-type orbital) has been entirely switched off; NO₂⁺ is in fact known to be linear or very close to linear (isoelectronic with CO₂ in valence count: N (5) + 2 O(6) − 1 (cation) = 16, identical valence electron count to CO₂, so the Walsh-diagram prediction directly matching CO₂'s established linear frontier-orbital occupation is that NO₂⁺ should indeed be linear or nearly so, and experimentally NO₂⁺ is linear, $D_{\infty h}$). Because ionization drives a large geometry change (134° neutral → 180° cation), the Franck–Condon principle predicts a long vibrational progression in the bending mode in the photoelectron spectrum of NO₂ — the photoelectron spectrum should show many bending-mode vibronic peaks with significant intensity far from the 0–0 transition, exactly as is observed experimentally, and this vibronic richness is a direct, textbook spectroscopic fingerprint of the underlying Walsh-diagram-predicted angle change on ionization. This is the same reasoning used generally to interpret photoelectron spectra of small polyatomics: whichever orbital is ionized, its Walsh-diagram slope tells you immediately whether to expect a short (angle-insensitive orbital) or long (angle-sensitive orbital) vibrational progression.