The theory that can completely/properly explain the nature of bonding in [Ni(Co)₄] is :

Solution

Chemistry ma’am · 3 steps

A metal carbonyl means back-bonding, and only MO theory has room for the empty π^* orbital of CO.

1. Two-way bond. In [Ni(CO)₄], each CO gives its carbon lone pair to an empty orbital on Ni, forming a σ bond. At the same time, filled Ni d orbitals push electron density back into the empty π^* orbitals of CO. This synergic bonding (σ donation plus π back-donation) is the real nature of the Ni–C bond.

   Theory	What it explains	What it misses
   Werner's theory	Primary and secondary valency, coordination number 4	Uses no orbitals at all, so it says nothing about how the bond forms
   Valence bond theory	Ni(0) becomes 3d¹⁰ after pairing, sp³, tetrahedral, diamagnetic	No back-bonding, so it cannot explain the strong Ni–C bond or the weaker C–O bond
   Crystal field theory	Splitting of the d orbitals	Treats ligands as point charges and ignores covalency, yet CO is neutral and bonds covalently
   Molecular orbital theory	σ donation and π back-donation into CO π^*	Nothing important for this complex

2. The evidence. Back-donation partly fills the π^* orbital of CO, so the C–O bond in the complex is weaker and longer than in free CO, and its stretching frequency drops. The Ni–C bond gains partial double-bond character. Only an orbital-overlap picture of the whole molecule accounts for both effects.
3. So. Only molecular orbital theory explains both the σ donation and the π back-bonding in [Ni(CO)₄], so the answer is (d).

The idea to keep. Metal carbonyls bond both ways: σ donation from CO and π back-donation from the metal into CO π^*. VBT gives shape and magnetism, but only MO theory explains the bonding itself.

Answer from the NTA final answer key. Solution by PrepWiser.