Identification of Coordination Complexes Exhibiting Geometrical Isomerism

To determine which of the given coordination complexes exhibit geometrical isomerism, we analyze each complex based on its geometry and the arrangement of its ligands:

1. Option (A): $[\text{Pt}(\text{en})\text{Cl}_2]$

   • The central metal atom $\text{Pt}^{2+}$ is a $d^8$ transition metal ion with a coordination number of 4, forming a square planar complex.
   • The ligand $\text{en}$ (ethylenediamine) is a symmetrical bidentate chelating ligand. Because of the bite-angle restriction of the ethylenediamine ring, the donor nitrogen atoms can only coordinate at adjacent positions ($\text{cis}$ to each other).
   • Consequently, the two chloride ligands are restricted to the remaining two adjacent ($\text{cis}$) positions. A $\text{trans}$ arrangement is not geometrically possible.
   • Therefore, this complex does not exhibit geometrical isomerism.

2. Option (B): $[\text{Pt}(\text{en})_2]\text{Cl}_2$

   • The complex cation is $[\text{Pt}(\text{en})_2]^{2+}$, which has a square planar geometry of the type $[\text{M}(\text{AA})_2]^{2+}$, where $\text{AA}$ is a symmetrical bidentate ligand.
   • All four coordination sites are occupied by identical donor atoms ($-\text{NH}_2$ groups), and swapping the ligands does not generate any distinct stereoisomer.
   • Therefore, this complex does not exhibit geometrical isomerism.

3. Option (C): $[\text{Pt}(\text{en})_2\text{Cl}_2]\text{Cl}_2$

   • Here, platinum is in the $+4$ oxidation state ($d^6$), with a coordination number of 6, adopting an octahedral geometry.
   • The complex ion has the general formula $[\text{M}(\text{AA})_2\text{X}_2]^{2+}$, where $\text{AA} = \text{en}$ and $\text{X} = \text{Cl}^-$.
   • It exhibits two distinct geometrical isomers:
     • $\text{cis}$-isomer: The two $\text{Cl}^-$ ligands occupy positions at an angle of $90^\circ$ to each other.
     • $\text{trans}$-isomer: The two $\text{Cl}^-$ ligands occupy positions at an angle of $180^\circ$ to each other (opposite axial or equatorial positions).
   • Therefore, this complex exhibits geometrical isomerism.

4. Option (D): $[\text{Pt}(\text{NH}_3)_2\text{Cl}_2]$

   • The central metal atom $\text{Pt}^{2+}$ is in a square planar geometry of the type $[\text{MA}_2\text{B}_2]$, where $\text{A} = \text{NH}_3$ and $\text{B} = \text{Cl}^-$.
   • It exists in two distinct geometrical isomers:
     • $\text{cis}$-isomer (cisplatin): The identical ligands are adjacent ($90^\circ$ apart).
     • $\text{trans}$-isomer (transplatin): The identical ligands are opposite ($180^\circ$ apart).
   • Therefore, this complex exhibits geometrical isomerism.

Thus, the complexes exhibiting geometrical isomerism are (C) and (D).