The type of molecular orbitals in the allyl ligand (CH₂ = CH‐CH₂^-) that are used for σ‐donation and π back donation with metal d‐orbitals, respectively are

Concept:

• Allyl ligands can donate  electrons as well as  electrons. They also are good  acceptor due to availability of anti- bonding  orbitals.
• The molecular orbital diagram for Allyl system can be drawn using huckel Molecular Orbital Theory (HMOT) according to which  electron can be treated separately from  electrons.
• Only  electrons are considered in MOT of allyl  conjugated systems to describe their property. 

Explanation:

The MOT of allyl system is:

1. The lowest energy level, that is  is most stable  and thus is used for  donation. 

2.  orbital acts the -donor orbital.

3. The last orbital which is unoccupied can accept  electrons.  is used for -donation.

Conclusion:

Therefore, in the allyl ligand (CH2 = CH‐CH2-), orbitals that are used for σ‐donation and  ‐back donation with metal d‐orbitals, respectively are  and .