Optical isomerism occurs in coordination compounds when a complex is chiral (non-superimposable on its mirror image) and lacks a plane of symmetry. This is common with bidentate ligands like ethylenediamine (en) which can create chiral geometries.

Let's analyze each complex:

1. [Co(en)₃]³⁺: This is an octahedral complex with three bidentate en ligands. It has no plane of symmetry and exists as a pair of enantiomers (Δ and Λ forms). Exhibits optical isomerism.

2. [Co(NH₃)₃Cl₃]: This is a neutral, octahedral complex. The three Cl^- and three NH₃ ligands can be arranged in two ways: the facial (fac) isomer (C_3v symmetry, has a plane of symmetry) and the meridional (mer) isomer (C_2v symmetry, has planes of symmetry). Neither isomer is chiral. Does not exhibit optical isomerism.

3. [Co(en)₂Cl₂]⁺: This complex has two bidentate en ligands and two monodentate Cl^- ligands. The cis isomer (C₂ symmetry, no plane of symmetry) is chiral and exhibits optical isomerism. The trans isomer (D_4h symmetry, has planes of symmetry) is achiral. Since the question refers to the species in general, the cis form is possible. Can exhibit optical isomerism.

4. [Co(en)(NH₃)₂Cl₂]⁺: This complex has one bidentate en ligand, two NH₃, and two Cl^- ligands. The cis isomer where the two Cl^- ligands are adjacent (C₁ symmetry, no plane of symmetry) is chiral and exhibits optical isomerism. The trans isomer is achiral. Can exhibit optical isomerism.

Final Answer: The complex species [Co(NH₃)₃Cl₃] is not expected to exhibit optical isomerism because all of its possible geometric isomers possess planes of symmetry.

Related Topics & Theory

Isomerism in Coordination Compounds: This is a fundamental concept where compounds with the same molecular formula have different arrangements of atoms. Optical isomerism is a type of stereoisomerism.

Chirality: A molecule is chiral if it is not superimposable on its mirror image. These mirror images are called enantiomers.

Role of Symmetry: The presence of a plane of symmetry (an imaginary plane that divides the molecule into two mirror-image halves) makes a molecule achiral (not chiral) and therefore incapable of optical activity.

Key Formulae & Concepts

While there is no single formula, the key is to draw the possible geometric structures and look for a plane of symmetry. $\text{If Plane of Symmetry exists }\Rightarrow \text{ Achiral }\Rightarrow \text{ No Optical Isomerism}$ $\text{If No Plane of Symmetry exists }\Rightarrow \text{ Chiral }\Rightarrow \text{ Exhibits Optical Isomerism}$