📌 Why Group Theory is a High-Scoring Unit

In CSIR UGC NET Chemical Sciences and GATE Chemistry, Group Theory consistently contributes 8 to 14 marks. Unlike subjective topics, Group Theory is governed by strict mathematical symmetry theorems, meaning questions have definitive, unambiguous answers.

• Predictable Scoring: Direct formulas for irreducible representation reduction (Reducible to Irreducible via the Great Orthogonality Theorem).
• Spectroscopic Applications: Deciding active vibrational modes in IR and Raman spectroscopy using character tables.
• Molecular Orbital & Hybridization: Deriving symmetry-adapted linear combinations (SALCs) and predicting allowed electronic transitions.
• Companion Download: You can also download our popular Name Reagents Handwritten Notes PDF for organic conversions.

🔬 Core Topics Covered in These Notes

1. Symmetry Elements & Operations

• Identity (E): The “do-nothing” operation present in every molecule.
• Proper Rotation Axis (C_n): Rotation by an angle θ = 360° / n around an axis. (Principal axis has highest n).
• Plane of Symmetry (σ):
  • Vertical Plane (σ_v): Contains the principal symmetry axis (e.g., in H₂O, NH₃).
  • Horizontal Plane (σ_h): Perpendicular to the principal symmetry axis (e.g., in BF₃, C₆H₆).
  • Dihedral Plane (σ_d): Bisects the angle between two C₂ axes perpendicular to the principal axis (e.g., in allene, ferrocene staggered).
• Center of Inversion (i): Inversion through a center point (x, y, z → −x, −y, −z). Found in trans-dichloroethylene, SF₆, C₆H₆.
• Improper Rotation Axis / Alternating Axis (S_n): Rotation by 360° / n followed by reflection through a plane perpendicular to the rotation axis (S_n = C_n × σ_h). Notice that S₁ = σ and S₂ = i.

2. Systematic Point Group Determination Flowchart

• Linear Molecules:
  • With center of inversion i (e.g., H₂, CO₂, C₂H₂) → D_∞h
  • Without center of inversion i (e.g., HCl, CO, HCN) → C_∞v
• High Symmetry Cubic Groups: Tetrahedral (T_d, e.g., CH₄), Octahedral (O_h, e.g., SF₆), Icosahedral (I_h, e.g., C₆₀ fullerene).
• Molecules with n C₂ axes ⊥ Principal C_n (Dihedral Groups):
  • With σ_h → D_nh (e.g., BF₃ is D_3h, PtCl₄²⁻ is D_4h, Benzene is D_6h)
  • With σ_d (no σ_h) → D_nd (e.g., Allene is D_2d, Staggered ethane is D_3d)
  • No planes → D_n
• Molecules without C₂ ⊥ Principal C_n:
  • With σ_h → C_nh (e.g., trans-N₂F₂ is C_2h, Boric acid is C_3h)
  • With σ_v → C_nv (e.g., H₂O is C_2v, NH₃ is C_3v, PCl₃ is C_3v)
  • No planes → C_n (e.g., H₂O₂ non-planar is C₂)
• Low Symmetry Groups: C₁ (no symmetry), C_s (only σ plane, e.g., SOCl₂), C_i (only center of inversion, e.g., anti-1,2-dibromo-1,2-dichloroethane).

3. Great Orthogonality Theorem (GOT) & Character Tables

The Great Orthogonality Theorem is the foundation of group representation theory. The five key rules derived from GOT are:

1. Sum of Squares of Dimensions: ∑ (l_i)² = h, where h is the order of the group and l_i is the dimension of the i-th irreducible representation (IRR).
2. Sum of Squares of Characters: For any irreducible representation, ∑ [ χ_i(R) ]² = h.
3. Orthogonality of Different IRRs: ∑ χ_i(R) χ_j(R) = 0 (when i ≠ j).
4. Invariance in Conjugate Classes: The characters of all symmetry operations belonging to the same class are identical.
5. Number of IRRs: The total number of irreducible representations equals the number of conjugate classes of operations in the group.

4. Mulliken Symbols for Irreducible Representations

• A: Singly degenerate, symmetric with respect to principal rotation axis [χ(C_n) = +1].
• B: Singly degenerate, antisymmetric with respect to principal axis [χ(C_n) = −1].
• E: Doubly degenerate representation (dimension = 2).
• T: Triply degenerate representation (dimension = 3).
• Subscripts 1 and 2: Symmetric (+1) or antisymmetric (−1) with respect to a C₂ perpendicular to C_n (or σ_v in groups lacking C₂ ⊥ C_n).
• Subscripts g and u: Gerade (symmetric, +1) or ungerade (antisymmetric, −1) with respect to center of inversion i.
• Primes ‘ and ”: Symmetric (+1) or antisymmetric (−1) with respect to horizontal reflection plane σ_h.

5. Reduction Formula & Vibrational Spectroscopy (IR / Raman)

• Total Degrees of Freedom: 3N (where N is the number of atoms).
• Vibrational Modes (Γ_vib): Γ_vib = Γ_3N − Γ_trans − Γ_rot.
  For non-linear molecules: 3N − 6. For linear molecules: 3N − 5.
• IR Activity: An irreducible representation is IR active if it transforms like translational coordinates (x, y, or z) in the character table.
• Raman Activity: An irreducible representation is Raman active if it transforms like quadratic products (x², y², z², xy, xz, yz).
• Rule of Mutual Exclusion: For centrosymmetric molecules possessing an inversion center i (e.g., CO₂, C₂H₄, SF₆), transitions that are IR active are Raman inactive, and vice versa. No mode can be both IR and Raman active simultaneously.

📊 Exam Weightage: Group Theory

Mark distribution across competitive chemistry examinations:

❓ Frequently Asked Questions (FAQ)