Explain Back Bonding in B2H6. Is it a 3 center bond?

Yes, diborane, or ##B_2H_6##, has three-center 2-electron bonds, two of them to be exact, each one consisting of the two boron atoms and a hydrogen atom sharing two electrons.

Diborane has more orbitals than it has , which makes it electron deficient. The molecule has a total of 12 - 3 from each boron atom and 1 from each hydrogen atom. However, it only has 10 bonding orbitals - 4 from each boron atom and 6 from each hydrogen atom.

boron is ##"sp"^3## and can thus form four bonds. In the diborane molecule, each of the two boron atoms will be bonded to two hydrogen atoms via .

These bonds will use up 8 of the 12 . The other 4 valence electrons will be shared by the remaining two hydrogen atoms and by the boron atoms , each atom contributing one orbital to the formation of the 3-center 2-electron bond.

This is what a three-center two-electron bond actually is - three atoms sharing two electrons.

As you can see, each boron uses two of its four ##"sp"^3## hybrid orbitals to form the terminal ##"B-H"## bonds. This leaves each boron atom with two ##"sp"^3## orbitals available - one empty and one containing one electron. In this case, the bond between two ##"sp"^3## hybrid orbitals - one coming from each boron atom - and a ##"1s"## orbital is called a hydrogen bridge.

The strong electrostatic repulsion felt by the positively charged nuclei of the two hydrogen atoms that form the hydrogen bridge (due to the proximity of the atoms) will cause the bond to be bent - you'll sometimes see this referred to as a banana bond.

Here's a more exact image of what makes up the 3-c 2-electrron bond

Each of the two hydrogen bridges will contain 1 electron from a boron atom and 1 electron from a hydrogen atom, with three orbitals coming together to form the bridge.