Bonding
Three-Center Two-Electron Bonds: How Diborane Glues Two Borons With One Pair
Diborane (B2H6) has only 12 valence electrons but needs at least 14 to draw every bond as a normal shared pair — it is two electrons short of a conventional Lewis structure. Chemistry solves this deficit not by breaking a rule but by spreading one electron pair across three atoms at once. The result is the three-center two-electron (3c-2e) bond: a single pair of electrons that simultaneously binds a bridging hydrogen to two boron atoms in a curved "banana" of electron density.
A 3c-2e bond is a chemical bond in which two electrons are shared among three atomic centers, occupying one bonding molecular orbital built from three atomic orbitals. It is the archetypal electron-deficient bond, and it is what holds together the boron hydrides, carbocation bridges, boranes, and many organometallic and hydride-bridged species.
- TypeElectron-deficient (delocalized) covalent bond
- Proposed byLonguet-Higgins & R. P. Bell, 1943
- Developed byWilliam Lipscomb (Nobel Prize, 1976)
- Canonical exampleB-H-B bridge in diborane, B2H6
- Bridge B-H length1.33 Å (vs 1.19 Å terminal)
- Measured byElectron diffraction, X-ray/neutron, IR, 11B NMR
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What a 3c-2e bond is and where it shows up
A three-center two-electron bond arises whenever a molecule has fewer valence electrons than it has orbitals and adjacencies to fill with ordinary two-center pairs. Rather than leaving atoms unbonded, one electron pair occupies a single bonding molecular orbital delocalized over three atomic centers.
The defining example is diborane, B2H6. Each boron uses two of its electrons and two terminal hydrogens to form ordinary 2c-2e B-H bonds. The two remaining hydrogens sit in bridge positions above and below the B-B axis, each held by a 3c-2e B-H-B bond. Two borons plus six hydrogens supply only 12 valence electrons: 8 go into the four terminal B-H bonds, leaving just 4 electrons (2 pairs) for the two bridges.
- Boron hydrides: B2H6, B4H10, B5H9, and the closo-boranes.
- Carbocations: the non-classical bridged 2-norbornyl cation and protonated methane (CH5+).
- Organometallics: bridging hydrides and Al2(CH3)6 (bridging methyls).
The mechanism: building the bond from three orbitals
Consider the B-H-B bridge. Each boron is roughly sp3-hybridized. One sp3 hybrid from the left boron, the 1s orbital of the bridging hydrogen, and one sp3 hybrid from the right boron combine. Three atomic orbitals always yield three molecular orbitals: one bonding (ψ1), one nonbonding/weakly bonding (ψ2), and one antibonding (ψ3).
- ψ1 (bonding): all three orbitals in phase — a single lobe of electron density curving over the B, H, and B. Energy is lowest.
- ψ2 (nonbonding): a node on the central H; the two boron lobes are out of phase.
- ψ3 (antibonding): alternating phases, highest energy, empty.
Only two electrons are available per bridge, and they drop into ψ1. That one filled bonding MO is the 3c-2e bond. Because the pair is spread over three nuclei rather than two, each B-H link has a formal bond order near 0.5, which is exactly why the bridge bonds are longer and weaker than the terminal ones. The curved shape of ψ1 is the origin of the nickname banana bond.
Key quantities and a worked electron count
Electron bookkeeping for B2H6. Valence electrons = 2(3) from B + 6(1) from H = 12. A conventional structure with 8 bonds would need 16 electrons; even the minimal connected framework needs 14. The 2-electron shortfall is absorbed by the two 3c-2e bridges.
- Terminal B-H bond length: 1.19 Å
- Bridging B-H bond length: 1.33 Å (longer, weaker)
- B-B internuclear distance: 1.77 Å
- Terminal H-B-H angle: ≈ 121-122°
- Bridge B-H-B angle: ≈ 97°
- Molecular symmetry: D2h
Wade's rules connection. For cluster boranes the electron count is generalized by Wade's rules: a closo-borane [BnHn]2- has n+1 skeletal electron pairs. Diborane is treated in Lipscomb's styx topological scheme, where s = B-H-B bridges, t = 3c-2e B-B-B bonds, y = 2c-2e B-B bonds, x = extra terminal B-H2. For B2H6 the styx code is 2002: two B-H-B bridges (s=2) and two BH2 groups (x=2).
How the 3c-2e bond is measured and used
Structure. The bridged D2h geometry was pinned down by gas-phase electron diffraction (Hedberg & Schomaker, 1951), which resolved the ethane-like ‘ethylenic' controversy in favor of the hydrogen-bridged structure. Neutron and X-ray diffraction on related boranes locate the bridging H atoms directly.
- Infrared: the bridge B-H-B stretch appears near 1600-2100 cm-1, distinct from sharp terminal B-H stretches around 2500-2600 cm-1; the bridge deformation sits lower.
- 11B NMR: boron in B2H6 resonates near δ +16 ppm as a triplet from coupling to two terminal H (1J(B-H) ≈ 130 Hz); bridging H couple more weakly.
- 1H NMR: bridging protons appear upfield/broadened relative to terminal protons.
Use. Diborane is a workhorse reagent for hydroboration (Brown, Nobel 1979): B-H adds across C=C with anti-Markovnikov, syn regiochemistry, feeding oxidation to alcohols. The same electron-deficient bonding underlies boron cluster chemistry, boron neutron capture therapy precursors, and CVD boron films.
How it differs from its close cousins
The 3c-2e bond is easy to confuse with several neighbors; the distinctions are sharp.
- vs 2c-2e (ordinary covalent) bond: a normal bond localizes 2 electrons between 2 nuclei. A 3c-2e bond spreads the same 2 electrons over 3 nuclei — same electron pair, one extra center, lower bond order per link.
- vs 3c-4e bond: in a three-center four-electron bond (e.g. the F-Xe-F axis of XeF2, or the bifluoride ion FHF-), the bonding and the nonbonding MOs are both filled. That is hypervalent, electron-rich bonding — the opposite of electron-deficient 3c-2e.
- vs hydrogen bonding: a hydrogen bond (X-H···Y) is a largely electrostatic, longer, weaker interaction (5-40 kJ/mol). A 3c-2e B-H-B is a genuine covalent bond with shared electrons and comparable strength to a normal covalent bond, just split two ways.
- vs agostic interaction: an agostic M···H-C donates a C-H bonding pair into a metal — also 3-center, but the pair originates in an existing bond.
History, exceptions, and famous cases
Origin. The bridged structure and the 3c-2e concept were proposed by H. Christopher Longuet-Higgins with R. P. Bell in 1943 — Longuet-Higgins was still an Oxford undergraduate. William N. Lipscomb then built the general topological (styx) and MO framework for the entire borane family, work recognized by the 1976 Nobel Prize in Chemistry.
Famous cases and controversies.
- The 2-norbornyl cation: decades of debate (H. C. Brown vs S. Winstein) over whether it is a rapidly equilibrating classical pair or a single non-classical 3c-2e bridged ion. Low-temperature X-ray crystallography (Scholz, Meyer et al., 2013) confirmed the symmetric bridged structure — a triumph for 3c-2e bonding.
- CH5+ (methanium): a fluxional protonated methane whose 3c-2e H-C-H bridge has no fixed structure at all.
- Al2(CH3)6: bridging methyls form B-H-B-analogous 3c-2e Al-C-Al bonds.
Limit: 3c-2e descriptions are one valid resonance/MO picture; a full symmetry-adapted MO treatment of the whole molecule reproduces the same electron distribution and is required for spectroscopy.
| Feature | Terminal B-H (2c-2e) | Bridge B-H-B (3c-2e) |
|---|---|---|
| Electrons per bond | 2 | 2 (shared over 3 atoms) |
| Atomic orbitals in the MO | 2 | 3 |
| Bond length | 1.19 Å | 1.33 Å (B-H) |
| Characteristic angle | H-B-H ≈ 120° | B-H-B ≈ 97° |
| Bond order per B-H link | 1.0 | ~0.5 |
| Electron density | Localized between 2 nuclei | Delocalized, banana-shaped over 3 |
Frequently asked questions
Why does diborane form 3c-2e bonds instead of a normal B-B single bond like ethane?
Diborane has only 12 valence electrons, two short of what a fully 2c-2e (ethane-like) structure needs. Boron is electron-deficient, so rather than leave bonds unfilled, two hydrogens bridge the borons and each shared pair spreads over three atoms. This uses the available electrons maximally and is lower in energy than any 12-electron localized alternative.
What does the 'banana bond' shape actually mean?
It refers to the shape of the filled bonding molecular orbital (ψ1) in the B-H-B bridge. Because the orbital is built from a boron hybrid, the hydrogen 1s, and a second boron hybrid all in phase, its electron density curves in an arc over the three nuclei rather than lying straight along a single axis — like a banana. It signals delocalized, multicenter bonding.
How is a 3c-2e bond different from a 3c-4e bond?
Both share electrons over three centers, but the electron count and character are opposite. A 3c-2e bond fills only the bonding MO with 2 electrons and is electron-deficient (boranes, carbocations). A 3c-4e bond fills both the bonding and the nonbonding MOs with 4 electrons and is electron-rich or hypervalent, as in XeF2 or the bifluoride ion FHF-.
What are the actual bond lengths and angles in diborane?
Terminal B-H bonds are about 1.19 Å with H-B-H angles near 120°. The bridging B-H bonds are longer at about 1.33 Å because their bond order is roughly 0.5, and the B-H-B bridge angle is about 97°. The B-B distance is about 1.77 Å. The molecule has D2h symmetry.
How do you count electrons to predict a 3c-2e bond?
Add all valence electrons and compare to the electrons needed for a fully localized 2c-2e structure. For B2H6: 2(3)+6(1) = 12 electrons. Eight go to four terminal B-H bonds, leaving 4 electrons — exactly two pairs — for two B-H-B bridges. For cluster boranes, Wade's rules (n+1 skeletal pairs for closo) or Lipscomb's styx notation generalize the counting.
Where do 3c-2e bonds matter outside of boron chemistry?
They appear in non-classical carbocations like the bridged 2-norbornyl cation and in protonated methane (CH5+), in bridging metal hydrides, and in trimethylaluminum Al2(CH3)6 where methyl groups bridge two aluminums. They are the conceptual basis for hydroboration chemistry and for understanding electron-deficient clusters throughout inorganic and organometallic chemistry.