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C4 Symmetry in Group Theory – Simple Explanation

Introduction to Symmetry and Group Theory
In chemistry, symmetry helps us understand the structure and properties of molecules.
Group theory is the mathematical language used to describe symmetry.
Every molecule has some symmetry elements (like rotation axes, mirror planes, inversion centers).
When we apply these symmetry operations, the molecule may look the same as before.
These operations together form a group in mathematics.
One very important rotation axis is the C4 axis.

What is a C4 Symmetry Axis?
The letter C stands for proper rotation (from Latin “cyclic”).
The number 4 means rotation by 360° ÷ 4 = 90°.
So, a C4 axis allows the molecule to be rotated by 90°, 180°, 270°, or 360° and look the same.

Example of C4 Symmetry:
A square or square planar molecule like [PtCl₄]²⁻ or XeF₄ has a C4 axis perpendicular to the plane.
If you rotate the square by 90°, 180°, or 270°, it still looks the same.

Operations in C4 Symmetry
The C4 axis generates four rotation operations:
C4 (90° rotation) → The molecule looks the same after rotating by 90°.
C4² (180° rotation) → Two 90° rotations; molecule looks the same.
C4³ (270° rotation) → Three 90° rotations; molecule looks the same.
C4⁴ (360° rotation = identity, E) → Full rotation, molecule is unchanged.
This group is called a cyclic group (C4 group).

Importance of C4 Symmetry in Chemistry
🔹 1. Molecular Shapes
Many molecules like XeF₄, PtCl₄²⁻, and square rings (cyclobutane) show C4 symmetry.
Helps classify molecules into point groups in group theory.
🔹 2. Spectroscopy
Symmetry decides which molecular vibrations are IR active or Raman active.
For C4 molecules, vibrations repeat after 90° rotation.
🔹 3. Molecular Orbital Theory
Symmetry helps simplify orbital combinations in molecules.
C4 symmetry restricts how orbitals overlap.
🔹 4. Exams (NEET, JEE, IIT-JAM, CSIR-NET)
You must recognize C4 in square planar complexes.
C4 symmetry = rotation by 90°.
Belongs to C4 group (cyclic group of order 4).

✅ Summary
C4 axis = 90° rotation symmetry.
Example: square, XeF₄, PtCl₄²⁻.
Group elements: {E, C4, C4², C4³}.
Important for point groups, spectroscopy, and molecular orbital theory.