Symmetry and Group Theory

Symmetry and Group Theory – Complete Notes, Point Groups, Operations & Tables

1. What is Symmetry?

In chemistry, symmetry refers to the balanced arrangement of atoms in a molecule.

👉 A molecule is said to be symmetric if it remains unchanged after certain operations.

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🔍 Simple Example

• Methane (CH₄) → highly symmetrical
• Water (H₂O) → less symmetrical

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🧠 Why Symmetry is Important?

Symmetry helps in:

✔ Predicting molecular properties
✔ Understanding spectroscopy
✔ Studying bonding and orbitals
✔ Simplifying quantum calculations

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📌 2. Symmetry Elements and Symmetry Operations

🧠 Key Concept

• Symmetry Element → Geometric entity (point, line, plane)
• Symmetry Operation → Action performed on molecule

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🔹 2.1 Identity (E)

Definition

Doing nothing.

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🧠 Important

Every molecule has E

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🔹 2.2 Rotation Axis (Cₙ)

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Definition

Rotation by 360°/n gives same molecule.

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Example

C₂ → rotation by 180°

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🔹 2.3 Mirror Plane (σ)

Types

• σᵥ → vertical
• σₕ → horizontal
• σ_d → diagonal

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🔹 2.4 Inversion Center (i)

Definition

Every atom moves through center to opposite side.

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🔹 2.5 Improper Rotation (Sₙ)

Definition

Rotation + reflection

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📌 3. Types of Symmetry Elements

Element	Symbol	Meaning
Identity	E	No change
Rotation axis	Cₙ	Rotation
Mirror plane	σ	Reflection
Inversion center	i	Inversion
Improper rotation	Sₙ	Rotation + reflection

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📌 4. Symmetry Operations

🧠 Definition

Action that leaves molecule unchanged.

Examples

• Rotation
• Reflection
• Inversion

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📌 5. Point Groups (VERY IMPORTANT)

🧠 What is Point Group?

A set of symmetry operations that describe a molecule.

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🔥 Common Point Groups

1. C₁

No symmetry except identity

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2. Cₛ

Contains mirror plane

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3. C₂

Contains C₂ axis

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4. C₂ᵥ

Contains:

• C₂
• Two σᵥ

Example: H₂O

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5. D₂h

Highly symmetric molecules

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6. T_d

Example: CH₄

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7. O_h

Example: SF₆

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📌 6. How to Determine Point Group (Step-by-Step)

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Step 1

Check linear or non-linear

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Step 2

Find highest Cₙ axis

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Step 3

Check for σ, i, Sₙ

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Step 4

Assign point group

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📌 7. Group Theory (Basic Concept)

🧠 Definition

A group is a set of elements that follow certain rules.

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📌 Conditions of Group

1. Closure
2. Associativity
3. Identity
4. Inverse

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📌 8. Matrix Representation

Symmetry operations can be written as matrices.

Example

Rotation matrix:

$$𝑅=\left[\begin{array}{cc}\cos 𝜃& -\sin 𝜃\\ \sin 𝜃& \cos 𝜃\end{array}\right]$$

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📌 9. Representations

Types

• Reducible representation
• Irreducible representation

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🧠 Importance

Used in:

✔ Vibrational spectroscopy
✔ Orbital symmetry

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📌 10. Character Tables

What is Character Table?

A table that shows symmetry properties of a point group.

Example: C₂ᵥ

Operation	E	C₂	σᵥ	σᵥ
A₁	1	1	1	1
A₂	1	1	-1	-1
B₁	1	-1	1	-1
B₂	1	-1	-1	1

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📌 11. Applications of Group Theory

🧪 1. Spectroscopy

Predict:

✔ IR active vibrations
✔ Raman active vibrations

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⚛️ 2. Molecular Orbitals

Helps in:

✔ Orbital symmetry
✔ Bonding analysis

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🧠 3. Quantum Chemistry

Simplifies calculations

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📌 12. Symmetry in Spectroscopy

🧠 Key Idea

Only those vibrations are IR active which:

👉 Change dipole moment

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Using Group Theory

We can predict:

✔ Number of IR peaks
✔ Number of Raman peaks

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📌 13. Vibrational Modes

Formula

$$3𝑁-6(\text{non-linear})$$
$$3𝑁-5(\text{linear})$$

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📌 14. IR and Raman Activity

Type	Condition
IR active	Change in dipole moment
Raman active	Change in polarizability

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📌 15. Advantages of Group Theory

✔ Simplifies calculations
✔ Predicts spectra
✔ Helps in structure determination

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📌 16. Limitations

❌ Requires practice
❌ Abstract concept
❌ Mathematical complexity

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📌 17. MCQs (Exam Ready)

Q1. Identity operation is:
✔ E

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Q2. Rotation by 180° is:
✔ C₂

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Q3. Water belongs to:
✔ C₂ᵥ

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Q4. CH₄ belongs to:
✔ T_d

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Q5. IR active vibrations require:
✔ Change in dipole moment

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📌 18. FAQ

❓ What is symmetry in chemistry?

Symmetry refers to balanced arrangement of atoms such that molecule remains unchanged after certain operations.

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❓ What is point group?

A set of symmetry operations describing a molecule.

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❓ Why is group theory important?

It helps in understanding molecular structure and spectroscopy.

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📌 19. Quick Revision Table

Concept	Key Point
Symmetry	Balance in molecule
Operation	Action
Element	Geometry
Point group	Classification
Group theory	Mathematical tool

Symmetry and Group Theory Video Presentation | Group theory in chemistry

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