Hypervalency and Expanded Octets

Draw the Lewis structure of SF₆ the naive way. Sulfur ends up with twelve electrons around it. That is four more than the octet rule allows. For most of the twentieth century, the standard fix was different. It invoked sulfur's empty 3d orbitals. These were mixed into hybrids like sd³ or sp³d². The problem was then declared solved. That story is taught in thousands of introductory courses. It appears in millions of textbook diagrams. It is almost entirely wrong, though. It fails as a description of the actual electronic structure. This applies to SF₆, PCl₅, or any other "hypervalent" main-group molecule. The trouble is energetic and spatial, not merely aesthetic. Sulfur's 3d orbitals sit roughly 5–8 eV above the 3p orbitals in energy. They are diffuse, contracted, and poorly shaped. This makes them overlap poorly with fluorine's 2p orbitals. Photoelectron spectroscopy, high-level ab initio calculations, and natural bond orbital (NBO) analyses all converge on the same finding. d-orbital coefficients in the valence molecular orbitals of SF₆ or PCl₅ are small. They typically account for only a few percent of the total wavefunction. Removing them from the basis set changes little. Computed bond energies and geometries shift by only a tiny fraction. The molecules are real, stable, and octahedral or trigonal bipyramidal. But sulfur and phosphorus are not using d orbitals this way. They do not hold six or five bonds together as old textbook hybridization diagrams suggest. The modern resolution rests on two ideas. This article develops both in full. The first is three-center–four-electron (3c–4e) bonding. Here, a filled ligand lone pair, a central-atom p orbital, and a second ligand orbital combine. Together they spread four electrons over three centers. No expansion of the central atom's valence shell occurs. The second is a careful application of formal versus actual electron counting. So-called hypervalent species are reinterpreted under this view. They become ordinary two-electron-two-center bonds plus ionic/resonance character. They are not literal 10- or 12-electron shells. This reframing does not change any experimental fact about SF₆ or PCl₅. Their bond lengths, force constants, and reactivities are exactly what they always were. But it changes what those facts mean for bonding theory. It removes an artifact that misleads students about periodic trends. It clarifies why NF₅ does not exist. It also clarifies how Bent's Rule and Orbital Character governs the axial/equatorial asymmetry of PCl₅. This article works through the historical d-orbital picture. It shows quantitatively why it fails. It builds the 3c–4e model from group theory and perturbation theory. It then applies this model rigorously to SF₆ and PCl₅. It closes with the modern view's predictive power. Why is SF₆ kinetically inert despite being thermodynamically primed to hydrolyze? Why is PF₅ a stronger Lewis acid at phosphorus than PCl₅? And how does this whole apparatus dovetail with the Isoelectronic Principle linking XeF₂, I₃⁻, and ClF₃?

Key takeaways

"Expanded octet" species like SF₆ and PCl₅ do not rely on significant central-atom d-orbital participation — d-orbital coefficients in accurate wavefunctions are only a few percent.
The real bonding mechanism for hypervalent axial linkages is 3-center–4-electron (3c–4e) bonding: two electron pairs delocalized over three atomic centers using only s/p orbitals.
3c–4e bonding naturally produces a weaker, longer, more ionic bond than an ordinary 2c–2e bond — matching the observed axial P–Cl bond in PCl₅ being longer than equatorial.
d orbitals are too high in energy (Δ ≈ 5–9 eV above valence p) and too diffuse/contracted to hybridize effectively; they act only as small polarization functions, improving basis-set flexibility rather than housing electron pairs.
Octahedral SF₆ (Oh) and trigonal bipyramidal PCl₅ (D3h) are correctly described using standard group-theoretic reducible representations built from central s + p orbitals plus ligand SALCs, with any "extra" bonding pairs delocalized via 3c–4e systems.
Key idea: hypervalency is not an expanded electron shell but delocalized electron pairs — main-group atoms never really exceed an octet in any single 2c–2e bond; they just share some bonding pairs across three centers at once.

Where this lives today

SF₆'s exceptional kinetic inertness, correctly understood as a steric/kinetic property layered on an unfavorable-looking but barrier-protected thermodynamic landscape rather than a mysterious "extra bond strength," is exactly why SF₆ is the standard dielectric and arc-quenching gas inside high-voltage circuit breakers and gas-insulated switchgear worldwide — and why its accidental leakage is now a major concern in climate policy, since SF₆ is one of the most potent greenhouse gases known (global warming potential roughly 23,500 times CO₂ over a 100-year horizon), a fact inseparable from its extreme bond stability and correspondingly enormous atmospheric lifetime (over 1,000 years). Correctly modeling its bonding — not its d-orbital mythology — is essential to predicting decomposition pathways in switchgear arcing (where SOF₂, SO₂F₂, and HF byproducts form via genuine bond-breaking events at high energy) and to designing SF₆ replacement gases (fluoronitriles, fluoroketones) that mimic its dielectric strength using molecules whose bonding is far more conventional. PCl₅ and PF₅ chemistry, correctly understood through the axial/equatorial 3c–4e asymmetry, underlies the mechanism of the Berry pseudorotation, the fluxional process by which trigonal bipyramidal P(V) species interconvert axial and equatorial positions rapidly on the NMR timescale — directly observable in variable-temperature ³¹P NMR of PF₅ and central to understanding reaction mechanisms at pentacoordinate phosphorus in organophosphorus chemistry, nucleoside/nucleotide chemistry (phosphorane intermediates in RNA hydrolysis and in DNA polymerase mechanisms proceed through very similar trigonal-bipyramidal, 3c–4e-stabilized transition states), and industrial phosphorus halide chemistry (PCl₅ is a major industrial chlorinating reagent converting carboxylic acids to acid chlorides). The general 3c–4e framework, first built for polyhalides, unifies the bonding description of XeF₂, ClF₃, BrF₅, I₃⁻, and the bifluoride ion HF₂⁻ — a family whose shared bonding logic is best appreciated through the lens of the Isoelectronic Principle, since replacing one central atom/ligand set by an isoelectronic analog preserves the 3c–4e MO pattern almost unchanged. This same delocalized-bonding logic, generalized further, underlies modern discussions of hypervalent iodine reagents (Dess–Martin periodinane, PhI(OAc)₂) that have become indispensable, mild, low-toxicity oxidants in contemporary organic synthesis — their reactivity is best rationalized by exactly the same 3c–4e axial bonding analysis developed here for PCl₅, now applied to hypervalent iodine(III) and iodine(V) centers.

Ask the tutor: How does the 3c–4e bonding picture change when we move from a symmetric ligand set (PCl₅) to a mixed one like PF₃Cl₂, and which ligands should preferentially occupy the axial 3c–4e positions?

The physical picture

Consider PCl₅, trigonal bipyramidal with two axial and three equatorial chlorines. In the naive Lewis/VSEPR picture, phosphorus forms five ordinary two-electron bonds, giving it ten electrons in its valence shell — hence "expanded octet." The historical rationalization mixed the 3s, three 3p, and one 3d orbital of phosphorus into five sp³d hybrids pointing at the five chlorines, each hybrid holding one electron pair shared with a chlorine 3p orbital. This picture predicts, correctly, a trigonal bipyramidal shape — but VSEPR predicts that shape purely from electron-pair repulsion, without ever needing d orbitals; the geometric prediction and the orbital rationalization are separable. The d-orbital story runs into trouble the moment you ask where the energy to promote an electron into a 3d orbital comes from. Atomic phosphorus has its 3d orbitals roughly 8–10 eV above the 3p orbitals; even after ligand field stabilization in the molecule, ab initio natural orbital analyses show 3d occupation numbers in PCl₅ of only about 0.1–0.2 electrons — far too small to be called a bonding hybrid partner. The five P–Cl bonds are real, but the two axial bonds are systematically different from the three equatorial ones: axial P–Cl is about 2.14 Å and equatorial is about 2.02 Å in the gas phase, and the axial bonds are weaker (lower stretching force constant, more easily broken in pseudorotation and substitution chemistry). A model built from five equivalent sp³d hybrids cannot naturally explain this axial/equatorial inequivalence — a purely s/p-orbital based model, with a distinct 3-center bonding pattern for the two axial bonds, explains it immediately. The 3c–4e picture treats the axial Cl–P–Cl unit as a self-contained fragment, analogous to the bonding in XeF₂ or I₃⁻. Phosphorus contributes one p orbital lying along the axial direction. Each axial chlorine contributes one p orbital pointed at phosphorus. Combining three atomic orbitals gives three molecular orbitals: a bonding MO delocalized over all three centers (mostly on the two chlorines, since chlorine is more electronegative), a nonbonding MO with a node at phosphorus (pure ligand character, no phosphorus contribution at all), and an antibonding MO. Four electrons — two formal "bonds'" worth — fill the bonding and nonbonding MOs, leaving the antibonding MO empty. The equatorial P–Cl bonds, by contrast, are ordinary 2c–2e bonds built from sp² hybrids on phosphorus overlapping chlorine p orbitals, exactly as in an ordinary trivalent phosphorus compound. This is why equatorial bonds are shorter and stronger: they are conventional covalent bonds, while axial bonds carry substantial nonbonding-orbital (ionic) character and correspondingly weaker net bonding.

Figure

Three-center–four-electron MO diagram for the axial Cl–P–Cl fragment of PCl₅

For SF₆, the same logic extends across three orthogonal axes. Sulfur contributes its 3s and three 3p orbitals to make four ordinary 2c–2e-type bonding combinations directed at four of the six fluorines in a tetrahedral-like arrangement — but with six ligands and only four central s/p orbitals available for σ bonding along three mutually perpendicular axes, the correct group-theoretic treatment (below) shows that each pair of trans fluorines along one Cartesian axis shares a 3c–4e bonding system with the single sulfur p orbital lying along that axis. Three such 3c–4e systems, one per Cartesian axis, account for all six S–F bonds. Every S–F bond individually has a formal bond order around 0.67–0.8 depending on the specific delocalized-bonding partition, consistent with the fact that S–F bonds in SF₆ (1.564 Å) are noticeably shorter than typical single S–F bonds elsewhere (S–F in SF₄ ≈ 1.60–1.65 Å for the longer bonds) yet weaker per bond than the raw bond count of "6 equivalent single bonds" would suggest if each were a full 2c–2e bond of normal S–F strength.

Figure

Symmetry-adapted linear combinations (SALCs) of fluorine σ orbitals in octahedral SF₆

This is the crux of the modern reassessment: group theory alone, applied to an octahedral ligand field, correctly predicts that the six fluorine σ-SALCs reduce as A1g + Eg + T1u. Sulfur's 3s transforms as A1g and its three 3p orbitals transform as T1u — both match ligand SALCs perfectly and form strong, low-lying bonding MOs using only s and p orbitals. The Eg SALCs, which in the old story were matched to sulfur's 3d(z²) and 3d(x²−y²) orbitals, do have the right symmetry to interact with d orbitals — symmetry alone permits it — but the energy mismatch is so large that computed natural orbital occupations show these MOs are overwhelmingly ligand lone-pair-like (nonbonding), not d-orbital bonding pairs. Symmetry allowance is necessary but not sufficient for real bonding; the energetics rule it out.

The core ideas

The central quantitative relationships governing hypervalent bonding are: the perturbative overlap criterion for orbital mixing, the 3c–4e MO energy/bond-order formula, and the group-theoretic reduction that tells you which orbitals are symmetry-allowed to mix in the first place.

H_{ij}^{\text{eff}} \approx \frac{\beta_{ij}^2}{\Delta E_{ij}}, \qquad \Delta E_{ij} = |\varepsilon_i - \varepsilon_j|

where: $\beta_{ij}$ is the resonance (interaction) integral between orbitals $i$ and $j$ (roughly proportional to overlap $S_{ij}$ ), and $\Delta E_{ij}$ is the energy gap between the unperturbed atomic orbital levels. This is a standard second-order perturbation-theory result: two orbitals mix effectively only when their energy gap is small relative to their coupling. For sulfur or phosphorus, $\Delta E$ between valence np and nd orbitals is large (5–10 eV), so even a respectable $\beta$ gives a small effective mixing — this is why d orbitals contribute only a few percent to bonding MOs despite being symmetry-allowed.

\psi_{\text{bonding}} = c_1\chi_{L1} + c_2\chi_{C} + c_3\chi_{L2}, \qquad \psi_{\text{nonbonding}} = \tfrac{1}{\sqrt2}(\chi_{L1} - \chi_{L2}), \qquad \psi_{\text{antibonding}} = c_1'\chi_{L1} - c_2'\chi_{C} + c_3'\chi_{L2}

where: $\chi_{L1}, \chi_{L2}$ are ligand p orbitals pointed at the central atom, $\chi_C$ is the central-atom p orbital, and the nonbonding combination by symmetry has zero coefficient on the central atom. This is the defining signature of 3c–4e bonding: the electron pair in the nonbonding MO is entirely ligand-based (formally a ligand lone pair not "used" for a bond to the central atom in the classical sense), while the bonding MO pair is genuinely shared over three centers.

\text{Bond order (per terminal bond)} = \frac{n_{\text{bonding}} - n_{\text{antibonding}}}{2 \times (\text{number of terminal bonds})} = \frac{2 - 0}{2\times 2} = 0.5

where: $n_{\text{bonding}}=2$ electrons occupy the fully bonding MO, $n_{\text{antibonding}}=0$ , and there are 2 terminal (ligand–central) linkages sharing that one bonding pair. Reading this: each individual axial bond in a 3c–4e system carries formal bond order one-half from the delocalized system, which combines with additional ionic/resonance stabilization to produce a real but weaker-than-normal bond — matching the observed elongation of axial P–Cl and (partially) the weaker per-bond character of S–F in SF₆ relative to a hypothetical isolated single bond.

Derivation AWhy d-orbital participation is small: an energetic and overlap argument

Goal: show quantitatively, from atomic orbital energies and radial extents, why sulfur/phosphorus 3d orbitals cannot be the primary bonding mechanism in SF₆/PCl₅, and why the 3c–4e alternative is favored instead.

Step 1 — Atomic orbital energies. Hartree–Fock and photoelectron-derived orbital energies for neutral sulfur place 3p at roughly $-11.6$ eV and 3d at roughly $-3$ to $-4$ eV above the vacuum level in the free atom context used for valence-state ionization energy comparisons — more usefully, empirical valence-state ionization energy (VSIE) tabulations put S 3d some 7–9 eV higher than S 3p. Fluorine 2p orbitals, the partners S must bond to, sit at roughly $-18.7$ eV. The energy mismatch between F 2p and S 3d is therefore on the order of 15 eV or more — far larger than the 3–7 eV mismatch between F 2p and S 3p.

Step 2 — Apply the perturbative mixing criterion. From

H_{ij}^{\text{eff}} \approx \frac{\beta_{ij}^2}{\Delta E_{ij}},

even if the resonance integral $\beta$ between an S 3d and F 2p orbital were comparable to that between S 3p and F 2p (it is not, for reasons below), the much larger $\Delta E_{ij}$ for the d-orbital case suppresses the effective interaction roughly in proportion to the energy-gap ratio. With $\Delta E_{3d}/\Delta E_{3p} \approx 2$ – $3$ , the effective coupling is cut by that same factor even before considering overlap.

Step 3 — Radial mismatch further suppresses overlap. Sulfur 3d orbitals in the neutral or slightly cationic atom are radially diffuse — their maximum radial probability lies well outside the 3p shell, because there is no orthogonality-driven contraction pulling them in (3d has never been occupied in building up the atom through prior shells for main-group atoms, unlike a $(n-1)d$ orbital in a transition metal). This means $\beta_{ij}$ (which scales with overlap $S_{ij}$ ) between a diffuse 3d and a compact F 2p orbital is intrinsically small, compounding the energy-gap penalty rather than compensating it.

Step 4 — Natural bond orbital/natural population analysis confirms the prediction. Reed and Weinhold's NBO analyses of SF₆, PCl₅, and related species (extended by subsequent high-level CASSCF and CCSD(T) natural orbital studies) find 3d natural orbital occupations of order 0.1–0.3 electrons total spread over all d-type basis functions — meaning the "extra" bonding capacity attributed to d orbitals in the old model corresponds to only a few percent of an electron pair, not the full pair needed to support a genuine bonding hybrid.

Step 5 — Conclusion. Because the energy gap is large (Step 1–2) and the overlap is poor (Step 3), d orbitals cannot serve as the primary bonding partners; they act only as small polarization functions that improve the flexibility of the basis set (much as adding f functions improves the description of p-orbital bonds), not as valence-shell housing for extra bonding pairs. The bonding capacity that a naive Lewis structure assigns to "d-hybrids" must instead come from delocalization within the s/p framework itself — which is exactly the role played by 3c–4e bonding.

Go deeper · Derivation B · 2 minConstructing the 3c–4e molecular orbitals of the axial PCl₂ fragment from group theory and Hückel-style overlap

Derivation BConstructing the 3c–4e molecular orbitals of the axial PCl₂ fragment from group theory and Hückel-style overlap

Goal: derive the energies, coefficients, and bond order of the three MOs formed from one central p orbital and two collinear ligand p orbitals — the minimal model for axial bonding in PCl₅ (and equally for XeF₂, I₃⁻, the FHF⁻ bifluoride ion, and CO₂ π systems by direct analogy).

Step 1 — Define the basis and symmetry. Place phosphorus at the origin with its axial 3pz orbital ( $\chi_C$ ) along $z$ ; place two chlorine 3p orbitals ( $\chi_{L1}$ , $\chi_{L2}$ ) at $\pm d$ along $z$ , each oriented to point at phosphorus. The linear ClPCl fragment has $D_{\infty h}$ local symmetry along this axis; the three basis orbitals transform as $\sigma_g^+$ (the symmetric ligand combination $\chi_{L1}+\chi_{L2}$ , which alone can mix with the central p orbital, since $\chi_C$ is odd under inversion when it's a p orbital paired with the antisymmetric combination in the standard convention — set up the symmetric/antisymmetric ligand SALCs explicitly next).

Step 2 — Build ligand SALCs. Define

\phi_+ = \frac{1}{\sqrt2}(\chi_{L1}+\chi_{L2}), \qquad \phi_- = \frac{1}{\sqrt2}(\chi_{L1}-\chi_{L2}).

By parity, $\phi_+$ has the same symmetry as $\chi_C$ (a p orbital along the axis is antisymmetric under the inversion-like reflection that swaps L1 and L2 with a sign flip matching $\phi_+$ 's node structure — concretely, both $\chi_C$ and $\phi_+$ change sign under the operation that exchanges the two chlorines while leaving phosphorus in place, so they are compatible), so only $\chi_C$ and $\phi_+$ can mix. $\phi_-$ has no symmetry match with $\chi_C$ and remains strictly nonbonding — it is the pure-ligand MO from Step 2 of the core-ideas section.

Step 3 — Set up the 2×2 secular problem for $\chi_C$ and $\phi_+$ . Using the Hückel-style parametrization with Coulomb integrals $\alpha_C$ (phosphorus) and $\alpha_L$ (chlorine, with $\alpha_L$ lower in energy since Cl is more electronegative) and resonance integral $\beta$ between $\chi_C$ and $\phi_+$ (this $\beta$ is $\sqrt2$ times the pairwise $\chi_C$ – $\chi_{L}$ resonance integral because $\phi_+$ carries both ligands' amplitude):

\begin{vmatrix} \alpha_C - E & \sqrt2\,\beta \\ \sqrt2\,\beta & \alpha_L - E \end{vmatrix} = 0.

Step 4 — Solve the secular determinant. Expanding,

(\alpha_C - E)(\alpha_L - E) - 2\beta^2 = 0,

E = \frac{\alpha_C+\alpha_L}{2} \pm \sqrt{\left(\frac{\alpha_C-\alpha_L}{2}\right)^2 + 2\beta^2}.

The lower root $E_-$ (bonding) lies below $\alpha_L$ (already the lower atomic level), and the upper root $E_+$ (antibonding) lies above $\alpha_C$ . The nonbonding combination $\phi_-$ sits exactly at $E = \alpha_L$ (unperturbed chlorine energy), since it never mixes with phosphorus at all.

Step 5 — Populate with four electrons and assign bond order. Each chlorine supplies one electron in its p orbital (from the ionic/Lewis-structure count where each Cl–P bond is nominally one shared pair, doubled because two chlorines each contribute), and phosphorus supplies the electrons associated with the axial bonding capacity. Overall four electrons fill $\psi_{\text{bonding}}$ (2 electrons, symmetric combination weighted toward chlorine because $\alpha_L < \alpha_C$ ) and $\phi_-$ (2 electrons, purely on chlorine, i.e., a formal lone pair). The antibonding MO is empty. The net P–Cl bonding overlap population is contributed only by the two electrons in $\psi_{\text{bonding}}$ , split between two P–Cl linkages:

\text{Bond order per P–Cl}_{\text{axial}} = \frac{2\ \text{electrons in bonding MO}}{2\ (\text{electrons per full bond}) \times 2\ (\text{bonds})} = 0.5.

Step 6 — Compare to equatorial bonds and rationalize the geometry. Equatorial P–Cl bonds use ordinary sp²-type hybrids on phosphorus in genuine 2c–2e bonds, giving them full (bond order ≈ 1) covalent character, hence shorter, stronger bonds. This derivation, using only s/p atomic orbitals and elementary Hückel theory, reproduces the experimentally observed axial-elongation/equatorial-shortening pattern in PCl₅ without invoking any d-orbital hybridization — matching the qualitative picture in Step 1 of the SF₆ SALC analysis above, generalized to five-coordination.

Worked examples

Example 1 (foundational) — Electron counting in PCl₅ under the two models. Count valence electrons around phosphorus in PCl₅ two ways and compare. Naive Lewis/expanded-octet count: P has 5 valence electrons; each of the 5 Cl atoms shares one bonding pair with P, contributing "5 bonds × 2 electrons = 10 electrons" formally around phosphorus (5 bonding pairs, each counted fully on P by the usual VSEPR bookkeeping convention). This is the origin of "expanded octet." 3c–4e-corrected count: Three equatorial bonds are ordinary 2c–2e bonds — 6 electrons, fully "belonging" to the P–Cl bonding framework in the traditional sense, contributing formal bond order 1 each. The two axial bonds instead come from a 3c–4e system holding 4 electrons total, of which only 2 are in a genuinely P-centered bonding MO (split as 1 electron's worth of bonding character per axial bond, i.e., bond order 0.5 each) and 2 are a ligand-localized nonbonding pair not really "on" phosphorus at all.

\text{Total P-centered bonding electron count} = \underbrace{6}_{\text{equatorial, bond order 1 each}} + \underbrace{2}_{\text{axial bonding MO, shared}} = 8.

Phosphorus's true participation in bonding electron density is 8 electrons — an ordinary octet — not 10. The "extra" two electrons that seemed to overfill phosphorus's shell are, under honest orbital analysis, sitting in a ligand-centered nonbonding MO that never had significant phosphorus character in the first place. Limiting-case check: if the axial ligands were replaced by two ligands with zero electronegativity difference from phosphorus (a hypothetical symmetric case), $\alpha_L = \alpha_C$ in the secular equation of the second derivation, and $E_\pm = \alpha_C \pm \sqrt2\beta$ becomes fully symmetric, with the bonding MO exactly 50/50 delocalized — the bond order per bond is still exactly 0.5, showing the result is robust to the specific electronegativity difference and depends only on the 3-orbital/4-electron topology, not on the numerical value of $\alpha_L - \alpha_C$ .

Example 2 (applied) — Predicting relative Lewis acidity of PF₅ versus PCl₅ at phosphorus. Both are trigonal bipyramidal AX₅ molecules with the same 3c–4e axial bonding topology. The question: which is the stronger Lewis acid toward an incoming nucleophile (relevant to why PF₅ readily forms PF₆⁻ with fluoride, and why P is the electrophilic site attacked in nucleophilic substitution at phosphorus, e.g., in the Berry pseudorotation mechanism and in phosphorane hydrolysis chemistry)? Fluorine is far more electronegative than chlorine, so in the secular problem of the second derivation, $\alpha_L$ (ligand Coulomb integral) is pushed much lower for F than for Cl:

\alpha_F \ll \alpha_{Cl} < \alpha_P.

A larger gap $(\alpha_C - \alpha_L)$ makes the bonding MO $E_-$ even more strongly stabilized and more heavily ligand-weighted — meaning the electron density is pulled further away from phosphorus onto fluorine, leaving phosphorus more positively charged (higher effective nuclear charge felt by an approaching nucleophile, i.e., a harder, more electrophilic center) and the antibonding/incoming-nucleophile acceptor orbital ( $\sigma^*$ character available to a sixth ligand) lower in energy and more accessible. Quantitatively, natural charge analyses give phosphorus a natural charge of roughly $+2.4$ in PF₅ versus roughly $+0.9$ in PCl₅ — a difference of about 1.5 electron units — directly reflecting the larger electronegativity-driven polarization in the fluoride case.

\Delta q_P = q_P(\text{PF}_5) - q_P(\text{PCl}_5) \approx 2.4 - 0.9 = 1.5.

This correctly predicts (and matches experiment) that PF₅ is the stronger Lewis acid at phosphorus, forming the stable hexacoordinate PF₆⁻ anion far more readily than PCl₅ forms PCl₆⁻ (which requires more forcing conditions and is less common, tending instead toward ionic PCl₄⁺PCl₆⁻ salts in the solid state rather than clean molecular hexacoordination in solution). Limiting-case check: if the ligand electronegativity difference were reversed (imagine a hypothetical ligand less electronegative than phosphorus, $\alpha_L > \alpha_C$ ), the bonding MO would instead be phosphorus-weighted and phosphorus would carry a partial negative charge, becoming a poorer electrophile — consistent with the general rule that increasing ligand electronegativity always increases central-atom electrophilicity in a 3c–4e system, a trend that also underlies why SF₆ is far more resistant to nucleophilic attack at sulfur than, say, SCl₆ would be (S–F polarization pulls density away from S, but simultaneously the resulting very strong ionic/electrostatic character and steric/kinetic shielding by six tightly packed fluorines make S kinetically inert despite being thermodynamically electrophilic — a subtlety addressed further in the FAQ).

Example 3 (exam/competition-level) — Rationalizing why SF₆ is inert to hydrolysis while SF₄ hydrolyzes readily, using the bonding models above. Thermodynamically, $\Delta G$ for

\mathrm{SF_6(g) + 3\,H_2O(l) \rightarrow SO_3(g) + 6\,HF(g)}

is strongly negative (on the order of $-300$ kJ/mol or more), so SF₆ "should" hydrolyze readily — yet it is famous for surviving boiling water, dilute acid, and dilute base for extended periods, which is precisely why it is used as a dielectric gas in high-voltage switchgear. Explain the kinetic origin using the bonding picture developed above, and contrast with SF₄, which hydrolyzes instantly in moist air. Step 1 — locate the rate-determining step. Nucleophilic attack at sulfur (by, e.g., OH⁻ or H₂O) must proceed through a transition state resembling a heptacoordinate (for SF₆) or pentacoordinate-to-hexacoordinate-transition (for SF₄) sulfur center, in which the incoming nucleophile begins forming a new 3c–4e-type interaction with sulfur along an axis, analogous to how a nucleophile attacks phosphorus in Example 2. Step 2 — count coordination congestion. SF₆ already has six fluorines packed in a closed-shell octahedral arrangement around a small sulfur atom (S covalent radius ≈ 1.05 Å, S–F ≈ 1.564 Å). Any incoming nucleophile attempting to add a seventh ligand must approach through a face or edge of the octahedron already crowded by fluorine lone pairs and the compact S–F σ frameworks — steric/electronic repulsion (a large kinetic barrier) is enormous. SF₄, by contrast, is a seesaw (AX₄E) with a stereochemically active lone pair on sulfur — there is an open coordination site (the position occupied by the lone pair, or the equatorial belt) through which a nucleophile can approach with much less congestion, forming a 3c–4e-type transition state far more easily. Step 3 — connect to the bonding model quantitatively. The activation barrier difference is not primarily electronic (both molecules have similarly electrophilic, fluorine-polarized sulfur centers by the argument of Example 2) but kinetic/steric, arising from the difference in accessible reaction coordinates for extending a 3c–4e system:

\Delta G^\ddagger(\mathrm{SF_6}) \gg \Delta G^\ddagger(\mathrm{SF_4}),

even though

\Delta G^\circ(\mathrm{SF_6}) \approx \Delta G^\circ(\mathrm{SF_4}) < 0 \ \text{(both hydrolyses thermodynamically favorable)}.

Step 4 — limiting case. If SF₆ is heated strongly (several hundred °C) or exposed to a much stronger/smaller nucleophile capable of overcoming the steric barrier (e.g., in electrical discharge conditions inside switchgear, where SF₆ does decompose to toxic byproducts such as SOF₂ and HF upon arcing), the kinetic protection breaks down and the thermodynamically predicted decomposition proceeds — confirming that the inertness is a barrier-height (kinetic) phenomenon layered on top of a thermodynamically unfavorable-to-survive molecule, not a sign that SF₆ is somehow intrinsically unreactive in a bonding sense.

Limiting cases, scope & common traps

Regime / question	What the d-orbital model says	What the 3c–4e / modern MO model says	Experimental reality
Why is PCl₅ trigonal bipyramidal?	sp³d hybridization forces the geometry	VSEPR (electron pair repulsion) gives the geometry; bonding is separately s/p-based via 3c–4e	Geometry confirmed by electron diffraction/X-ray; both models agree on shape, disagree on why bonds form
Axial vs equatorial P–Cl bond length	Not naturally explained (5 equivalent hybrids)	Predicted: axial ≈ 2.14 Å (bond order 0.5, 3c–4e) vs equatorial ≈ 2.02 Å (bond order 1, ordinary 2c–2e)	Axial 2.14 Å, equatorial 2.02 Å (gas electron diffraction) — matches 3c–4e prediction
SF₆ S–F bond order	Treated as 6 equivalent full single bonds via sd³ or sp³d² hybrids	Delocalized: 3 orthogonal 3c–4e systems give ~0.7–0.8 formal order per bond, all bonds equivalent by Oh symmetry	S–F = 1.564 Å, shorter than typical single S–F, consistent with partial multiple-bond/ionic character, not literal bond order 1 in 6 saturated bonds
Role of d orbitals in the basis set	Central and required for bonding (occupation ~1–2 e⁻ implied)	Present only as polarization functions; NBO/NPA occupation ~0.1–0.3 e⁻ total	Removing d functions from post-HF calculations changes bond energies by only a few percent; removing s/p functions is catastrophic
Isoelectronic series behavior (e.g., XeF₂, ICl₂⁻, I₃⁻)	Each treated ad hoc with "expanded octet" on the central heavy atom	All are the same 3c–4e linear system with different central atom/ligand identity — unified treatment	Bond lengths/vibrational force constants scale together as predicted by 3c–4e systematics across the whole isoelectronic family
High-temperature/high-energy limit (e.g., plasma, mass spectrometry fragmentation)	No special prediction	3c–4e bonds are weaker (lower bond order) and should fragment preferentially at axial/hypervalent positions under collision-induced dissociation	CID of PCl₅⁺ and related species shows preferential loss of axial-type ligands first, consistent with lower axial bond order

Common trapCommon misconception #1: "d orbitals must be involved because the molecule needs somewhere to put the extra electrons."

This confuses symmetry-allowed mixing with energetically significant mixing. Group theory correctly shows that SF₆'s Eg ligand SALCs have the right symmetry to combine with sulfur 3d(z²)/3d(x²−y²) — but symmetry only tells you a matrix element is not forced to be zero, not that it is large. The actual magnitude, set by the overlap integral and the energy denominator in $H_{ij}^{\text{eff}} \approx \beta_{ij}^2/\Delta E_{ij}$ , is small because the S 3p–F 2p gap is far smaller than the S 3d–F 2p gap. The "extra electrons" are perfectly well accommodated by ligand-centered nonbonding orbitals in the 3c–4e/SALC framework — no d-orbital housing is required at all.

Common trapCommon misconception #2: "If d orbitals aren't really involved, then SF₆ and PCl₅ shouldn't be stable, since sulfur/phosphorus 'can't' hold that many bonds."

Stability of SF₆ and PCl₅ does not require literal valence-shell expansion in the classical Lewis sense. The molecules are stabilized by a combination of strong, short 2c–2e-type bonds (equatorial in PCl₅; all six in SF₆ via delocalized combinations) plus real, substantial (if formally fractional-bond-order) 3c–4e stabilization along the remaining axes, plus significant ionic/electrostatic contribution from the large electronegativity difference between S/P and F/Cl. Thermodynamic stability comes from the sum of all these terms, not from any single exotic mechanism — and this is exactly why not every combination works: NF₅ does not exist because nitrogen has no accessible low-lying orbital of the right size/energy to form even the weak 3c–4e axial system (no low-lying d-like or diffuse orbital, and N is simply too small and too electronegative-relative-to-F for the arrangement to be stabilizing), while phosphorus and sulfur, one row down, are large enough and have accessible enough valence orbitals for the 3c–4e axial interactions to be net stabilizing.

The chemist behind it

The modern reassessment of hypervalency is not the work of one person but the convergent result of physical organic and inorganic chemists reworking bonding theory across the 1950s–1990s.

Origins · 2 minFrank Weinhold and how this was discovered

Frank Weinhold

b. 1941 · American theoretical chemist, University of Wisconsin–Madison

Weinhold, together with collaborators including Alan Reed and Larry Curtiss, developed Natural Bond Orbital (NBO) analysis in the late 1970s and 1980s — a method for transforming a delocalized ab initio molecular wavefunction into a set of localized, chemically interpretable "natural" Lewis-structure-like orbitals, along with quantitative measures of departure from an idealized Lewis structure (natural resonance theory) and orbital occupation numbers for basis functions like d orbitals on main-group atoms. It was Reed and Weinhold's direct NBO/NPA calculations on SF₆, SO₄²⁻, PCl₅, and related "hypervalent" species (culminating in widely cited work through the 1980s and reinforced by numerous subsequent independent computational studies) that put quantitative numbers — occupation numbers of a few tenths of an electron at most — on what had previously been a qualitative suspicion among a minority of theoretical chemists that d-orbital participation was overstated. Weinhold has continued to champion and refine this reassessment, arguing forcefully in textbooks and review articles that hypervalency is properly understood through negative hyperconjugation and 3c–4e delocalization within an s/p framework, explicitly rejecting the term "expanded octet" as a description of the actual electron distribution while accepting it as a historically useful (if literal-mindedly misleading) shorthand for the Lewis-structure electron count.

History

- 1916 — G. N. Lewis introduces the shared electron pair and the octet rule, immediately noting exceptions (PCl₅, SF₆) as apparent violations.

- 1940s–1950s — Valence bond theory rationalizes hypervalent molecules via sp³d/sp³d² hybridization involving central-atom d orbitals; this becomes standard textbook doctrine.

- 1951 — Rundle and Pimentel independently propose 3-center-4-electron bonding to explain the bonding in polyhalide ions (I₃⁻) and related electron-rich linear systems, offering the first serious s/p-only alternative to d-orbital hybridization for hypervalent-type species.

- 1980s — Reed, Weinhold, and Curtiss apply Natural Bond Orbital and Natural Population Analysis to SF₆, PCl₅, and related molecules, quantifying d-orbital occupation as only a few tenths of an electron and reviving/formalizing the Rundle–Pimentel picture for main-group hypervalency generally.

- 1990s–2000s — High-level CASSCF, CCSD(T), and density-functional studies (including work reviewed by Gillespie, Cioslowski, and others) confirm across many molecules that removing d functions from the basis set changes bonding energetics only modestly, cementing the modern consensus.

Do d orbitals play absolutely no role at all in SF₆ or PCl₅?

Not "absolutely none" — symmetry allows a small admixture, and accurate calculations do find nonzero (if small, typically a few percent) d-orbital coefficients in some valence molecular orbitals. The point is one of degree and mechanism: d orbitals function as polarization functions that fine-tune the shape of existing s/p-based bonding orbitals (much like adding f functions sharpens a p-orbital-based bond), rather than as primary hybridizing partners housing an independent bonding electron pair. Removing d basis functions entirely from a calculation changes total bonding energies by only a small percentage, while removing s or p functions is catastrophic — this asymmetric sensitivity is the operational definition of "d orbitals are not doing the essential bonding work here."

If hypervalency doesn't need d orbitals, why do so many textbooks still teach sp³d and sp³d² hybridization?

Historical inertia and pedagogical convenience. The hybridization story gives a simple, self-consistent narrative connecting VSEPR-predicted geometry to an orbital picture using familiar hybrid-orbital vocabulary, and it was the dominant model throughout most of the twentieth century before high-level computational tests became routine. It is not "wrong" in the sense of predicting incorrect geometries — VSEPR correctly predicts shape independently of the hybridization story — but it is wrong as a mechanistic claim about where the bonding electron density actually resides. Many textbooks are gradually shifting toward the 3c–4e/delocalized-MO description, particularly in more advanced treatments, but the simpler picture persists at introductory levels because it is easier to draw and requires no group theory.

How does the 3c–4e model handle SF₆'s full octahedral symmetry, where there are three axes, not just one axial pair like PCl₅?

SF₆ is treated as three independent (to a first approximation) 3-center-4-electron systems, one along each Cartesian axis, each built from one sulfur p orbital and the two trans fluorine p orbitals on that axis — exactly the same construction as the PCl₅ axial fragment, just repeated three times using the three orthogonal sulfur 3p orbitals (3px, 3py, 3pz) as the three central-atom partners. Sulfur's 3s orbital (A1g symmetry) additionally mixes into all six ligands equally, contributing a background of genuinely delocalized bonding character common to every S–F bond. The net result, after combining all these contributions, is six equivalent S–F bonds (required by Oh symmetry) each with an intermediate formal bond order (commonly cited in the range 0.67–0.8 depending on the specific partition scheme used), consistent with the observed S–F bond length being shorter than a "pure" single bond but the total bonding capacity still traceable entirely to s and p orbitals.

Does this reassessment mean VSEPR theory is also wrong about the shapes of hypervalent molecules?

No — VSEPR's predictions of shape (octahedral SF₆, trigonal bipyramidal PCl₅, seesaw SF₄, T-shaped ClF₃) remain fully correct and are essentially untouched by this reassessment. VSEPR is a theory about electron-pair repulsion and geometry, not a claim about which specific atomic orbitals house each bonding pair. The 3c–4e reassessment changes the orbital-level bonding mechanism used to rationalize why these geometries are also electronically stable, but it does not change or contradict the geometric predictions themselves — the two theories operate at different levels of description and are fully compatible.

Why doesn't nitrogen form NF₅, if phosphorus can form PF₅ using only s/p orbitals in a 3c–4e framework?

Nitrogen is simply too small and its 2p orbitals too contracted and too close in energy to nothing else useful; there is no accessible low-lying orbital of appropriate size and symmetry on nitrogen to form even a weak 3c–4e axial interaction with two additional fluorines, and nitrogen's small radius makes five-fold coordination severely sterically strained regardless of electronic factors. Phosphorus, one row down, has a larger core, longer bonds, more room around the central atom, and a 3p orbital of appropriate radial extent to form a modestly stabilizing (if weak, bond order ≈ 0.5 per axial bond) 3c–4e system with two additional ligands. This size-driven cutoff is a recurring theme in main-group periodicity: the "no hypervalency in the second row" rule is best explained by sterics and orbital size mismatch, not by a rigid, false claim that second-row atoms categorically "cannot" access d orbitals (third-row atoms don't meaningfully use d orbitals for bonding either, as this whole article establishes) — the real second-row/third-row distinction is atomic size and the resulting feasibility of 3c–4e stabilization.

How does this connect to the isoelectronic principle mentioned as a related topic?

Species like XeF₂, ICl₂⁻, I₃⁻, and the axial fragment of PF₅ or PCl₅ are all, in a loose but electronically precise sense, variations on the identical 3-orbital/4-electron linear bonding problem, differing only in the identity and electronegativity of the central atom and ligands. Because they share the same electron count and orbital topology in the relevant fragment, they are natural members of an isoelectronic family whose bond lengths, vibrational frequencies, and bond orders can be compared and rationalized together using the single secular-equation framework derived here — a unifying move that becomes the explicit subject when the Isoelectronic Principle is treated on its own terms.