Electron-Deficient Molecules: Boranes and Three-Centre Bonds

Here is a puzzle that stumped chemists for decades. Boron has three valence electrons, so boron hydride “ought” to be BH₃, a flat molecule like BF₃. But BH₃ is not what you find in a bottle. What exists as a stable gas is diborane, B₂H₆, two boron atoms and six hydrogens. Try to draw a Lewis structure for it with ordinary two-electron bonds and you run out of electrons before you run out of bonds.

Molecules like this are called electron-deficient: they don’t have enough valence electrons to connect every pair of neighbouring atoms with a normal two-centre, two-electron (2c-2e) bond. Their solution is elegant, and learning it will stretch your idea of what a chemical bond can be.

Counting the electrons

Start with the obvious attempt: an ethane-like structure, H₃B–BH₃. That needs seven bonds (six B–H plus one B–B), which would require 14 electrons.

Now count what diborane actually has:

• 2 boron atoms × 3 valence electrons = 6
• 6 hydrogen atoms × 1 valence electron = 6
• Total = 12 valence electrons

Twelve electrons can make only six ordinary bonds. Yet each boron is attached to four hydrogens, and there are eight B–H contacts in the real molecule. Something other than 2c-2e bonds must be doing the work.

The actual structure

Electron diffraction and spectroscopy show that diborane is not ethane-shaped. Its structure looks like this:

• The two boron atoms and four terminal hydrogens lie in one plane, two hydrogens on each boron.
• The remaining two hydrogens bridge between the borons, one above that plane and one below.
• Each boron sits at the centre of a distorted tetrahedron of four hydrogens.
• There is no direct B–B bond in the usual sense; the borons are held together through the bridges.

The bridging B–H distances are noticeably longer than the terminal B–H distances, a first hint that the bridge bonds are weaker, “shared” bonds.

Now reallocate the electrons:

• Four terminal B–H bonds are ordinary 2c-2e bonds: 8 electrons.
• That leaves 4 electrons for the two B–H–B bridges.
• So each bridge holds 2 electrons spread over 3 atoms.

This is a three-centre two-electron bond (3c-2e bond), often drawn as a curved “banana” bond passing through the bridging hydrogen.

How can two electrons hold three atoms?

The most natural way to understand this is with molecular orbital theory, because MO theory never assumed bonds had to be between two atoms.

Focus on one bridge. Three atomic orbitals take part: an approximately sp³ hybrid orbital from each boron, pointing towards the bridge, and the 1s orbital of the bridging hydrogen. Three atomic orbitals combine to give three molecular orbitals:

1. A bonding MO: all three orbitals combine in phase. It has electron density spread across B, H and B, and it is lower in energy than the starting orbitals.
2. A nonbonding MO: the two boron hybrids combine with opposite signs, and the hydrogen 1s makes no net contribution by symmetry. Its energy is roughly that of the starting boron orbitals.
3. An antibonding MO: high in energy, with nodes between the atoms.

Two electrons go into the bonding MO, and the nonbonding and antibonding MOs stay empty. One pair of electrons in one orbital that spans three atoms is enough to hold all three together. Each B–H link in the bridge is roughly “half a bond”, which fits the longer bridge distance.

An analogy

Picture two climbers on either side of a narrow gap, with only one rope between them. They can’t each have a rope to a central anchor, so they both clip onto the same rope passing through a single ring in the middle. The rope (the electron pair) is shared three ways: climber, ring, climber. It isn’t as strong as giving everyone their own rope, but it holds.

The simplest example of all

The ion H₃⁺, a triangle of three protons with only two electrons, is held together by exactly the same kind of 3c-2e bonding. It shows that three-centre bonding is not a special trick of boron. It is what happens whenever orbitals are plentiful but electrons are scarce.

Why does boron do this?

Boron is short of electrons but not of orbitals. With four valence orbitals (2s and three 2p) and only three electrons, a BH₃ unit leaves one orbital completely empty. An empty orbital next door to an electron-rich bond is an invitation. In BH₃ the only electron pairs available are the B–H bonds of a second BH₃ unit, so two units pool their resources. Each boron uses one of its B–H bonds to partly fill the empty orbital on the other boron, forming the two bridges.

The energy trade-off is favourable: dimerisation releases energy, so B₂H₆ rather than BH₃ is the stable species at room temperature. Free BH₃ can be captured when a real lone pair is offered, as in adducts with ethers or amines such as H₃B·NH₃, where boron simply accepts an electron pair and reaches an octet. That is ordinary Lewis acid–base behaviour. Its coordinate bond is a 2c-2e bond, so the adducts are not electron-deficient.

For boron’s basic data, see the boron element page.

Beyond diborane: higher boranes

Diborane is only the smallest member of a large family. Boron hydrides such as B₄H₁₀, B₅H₉ and B₁₀H₁₄ form cages and open “baskets” of boron atoms. Their bonding combines several kinds of links:

Bond type	Electrons	Atoms involved
Terminal B–H	2	2
Direct B–B	2	2
Bridging B–H–B	2	3
Closed B–B–B	2	3 (a triangle of borons)

William Lipscomb worked out a systematic way of assigning these bonds to borane structures and received the 1976 Nobel Prize in Chemistry for his work on the structures and bonding of boranes. Later, Wade’s rules linked the number of skeletal electron pairs to the shape of the cage, classing clusters as closo (closed), nido (nest-like) and arachno (web-like). These rules treat the cage bonding as delocalised over the whole cluster, much as the π electrons in benzene are delocalised over the ring.

Other electron-deficient bridges

Similar 3c-2e bridges show up outside boron:

• Trimethylaluminium exists as a dimer, Al₂(CH₃)₆, in which two methyl groups bridge the aluminium atoms. The bridging carbon uses one hybrid orbital for two Al–C links, exactly as the hydrogen does in diborane.
• Beryllium hydride, BeH₂, forms a polymeric solid held together by Be–H–Be bridges.

A look-alike that is not electron-deficient

Aluminium chloride vapour contains Al₂Cl₆ dimers that look very like diborane: two chlorine atoms bridge two aluminium atoms. It is tempting to call these 3c-2e bonds too. They aren’t.

Each bridging chlorine has lone pairs. It forms a normal bond to one aluminium and donates a lone pair to the other, making an ordinary dative bond. Every Al–Cl link in the bridge is a 2c-2e bond, and each aluminium ends up with an octet. Count the electrons and there are plenty; the bridge is electron-precise, not electron-deficient. The shape is similar, but the bonding is different. (The exceptions to the octet rule list also covers AlCl₃.)

A quick test: a bridging atom with lone pairs to spare (Cl, and often O or N) usually makes dative bridges. A bridging atom with no lone pairs (H, or C in a methyl group) must be making a 3c-2e bond.

Where this matters

Electron deficiency makes boranes reactive. Diborane is highly toxic and ignites spontaneously in air, so it is handled only in specialist laboratories. Its tendency to accept electron pairs underlies hydroboration, a widely used reaction in organic synthesis for adding hydrogen and boron across C=C bonds. Borohydride ions (BH₄⁻), where boron has a full octet and four ordinary B–H bonds, are the familiar reducing agents of organic chemistry.

The deeper lesson is conceptual. The “one line = two electrons between two atoms” rule is a useful convention, not a law of nature. Molecular orbitals can spread over any number of atoms, and electron-deficient molecules are where that fact becomes impossible to ignore.

Common mistakes

• Drawing diborane like ethane. There is no H₃B–BH₃ structure with a B–B bond; the borons are linked through two hydrogen bridges.
• Giving the bridging hydrogen two full bonds. Hydrogen has only a 1s orbital and cannot hold four electrons. Each bridge contains just two electrons in total.
• Calling Al₂Cl₆ a 3c-2e compound. Its chlorine bridges are dative bonds using chlorine lone pairs, so it is electron-precise.
• Assuming “electron-deficient” means “unstable or impossible”. Diborane is a real, isolable compound; it simply uses fewer electrons per bond.
• Confusing 3c-2e with 3c-4e bonding. Three-centre four-electron bonds occur in electron-rich species such as I₃⁻ and XeF₂ (see hypervalent molecules). They also fill the nonbonding orbital, which is a different situation.
• Treating borane adducts as electron-deficient. In H₃B·NH₃ boron receives a lone pair and has an ordinary octet.

Key takeaways

• Electron-deficient molecules have too few valence electrons for 2c-2e bonds between every pair of connected atoms.
• Diborane, B₂H₆, has 12 valence electrons: 8 go into four terminal B–H bonds and 4 into two B–H–B bridges.
• Each bridge is a three-centre two-electron bond: one bonding MO spread over B, H and B holds two electrons.
• Boron has an empty orbital and few electrons, so BH₃ units dimerise to share electron pairs.
• Higher boranes combine 2c-2e and 3c-2e bonds into cages; Lipscomb’s work on them earned the 1976 Nobel Prize.
• Look-alike bridges in Al₂Cl₆ use lone pairs and are electron-precise, not electron-deficient.