The coordination number of central metal atom in a complex is determined by

The coordination number (CN) of a central metal atom or ion in a complex is strictly defined as the total number of coordinate covalent bonds (donor-acceptor bonds) formed directly between the ligand donor atoms and the metal center.

Let's look at the nature of these bonds to evaluate the options:

• When a ligand coordinates to a metal ion, the initial Lewis base-acid interaction always occurs via the formation of a $$\sigma$$-bond (sigma bond), where the ligand donates a lone pair of electrons into an empty d, s, or p orbital of the metal.
• Some specific ligands (like $$\text{CO}$$, $$\text{CN}^-$$, or $$\text{PR}_3$$) can also participate in secondary back-bonding or multiple bonding, which introduces $$\pi$$-bonds (pi bonds).
• However, these supplementary $$\pi$$-bonds are formed purely to stabilize the electron density on the metal; they do not alter the spatial arrangement, the number of attached donor atoms, or the spatial geometry of the complex. Therefore, $$\pi$$-bonds are completely ignored when calculating the coordination number.

• In $$[\text{Fe(CN)}_6]^{4-}$$, there are six $$\text{CN}^-$$ monodentate ligands. Each forms one $$\sigma$$-bond with the $$\text{Fe}^{2+}$$ ion, making the coordination number 6 (despite the presence of internal $$\pi$$-bonding character).
• In $$[\text{Co(en)}_3]^{3+}$$, ethylenediamine ($$\text{en}$$) is a bidentate ligand. Even though there are only three ligand molecules, each molecule forms two separate $$\sigma$$-bonds, giving a total of $$3 \times 2 = 6$$ $$\sigma$$-bonds. Thus, the coordination number is 6.

The coordination number is determined solely by counting the number of coordinate $$\sigma$$-bonds directed toward the central metal ion, regardless of whether the ligands are anionic, cationic, neutral, or capable of secondary $$\pi$$-backbonding.

Answer: Option A — the number of ligands around a metal ion bonded by sigma bonds