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 \(\sigma\) electrons as well as \(\pi\) electrons. They also are good \(\pi\) acceptor due to availability of anti- bonding \(\pi^*\) orbitals.
• The molecular orbital diagram for Allyl system can be drawn using huckel Molecular Orbital Theory (HMOT) according to which \(\pi\) electron can be treated separately from \(\sigma\) electrons.
• Only \(\pi\) 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 \(\pi_1\) is most stable  and thus is used for \(\sigma\) donation. 

2. \(\pi_2 \) orbital acts the \(\pi\)-donor orbital.

3. The last orbital which is unoccupied can accept \(\pi\) electrons. \(\pi_3\) is used for \(\pi\)-donation.

Conclusion:

Therefore, in the allyl ligand (CH2 = CH‐CH2-), orbitals that are used for σ‐donation and \(\pi\) ‐back donation with metal d‐orbitals, respectively are \(\pi_1\) and \(\pi_3\).