Symmetry Operations

Foundations of Group Theory in Physics

The Hexagonal Mystery

Why do snowflakes must have six-fold symmetry?

Is it a physical law, a geometric coincidence, or an inevitable mathematical consequence of the water molecule's structure?

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Macroscopic Shape

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Molecular Symmetry

What is a Symmetry Operation?

Definition

An action that moves an object into a configuration that is indistinguishable from the original.

The center of mass remains fixed.
Distances and angles are preserved (Isometry).
Points are mapped to points of the same type.

C₄

Example: A 90° rotation leaves a square invariant.

Element vs. Operation

Symmetry Element

The "Geometrical Entity"

A point, line, or plane with respect to which the operation is performed.

Example: Rotation Axis

Symmetry Operation

The "Action"

The actual movement that leaves the object invariant.

Example: Rotation by \( \theta = \frac{2\pi}{n} \)

The Geometric Toolkit

\( E \)

Identity

The operation of "doing nothing." Present in every object. Mathematically essential for group structure.

\( C_n \)

Proper Rotation

Rotation by \( 360/n \) degrees about an axis. \( C_2 \), \( C_3 \), \( C_4 \), etc.

\( \sigma \)

Reflection

Reflection through a plane. Vertical (\( \sigma_v \)), Horizontal (\( \sigma_h \)), or Dihedral (\( \sigma_d \)).

\( i \)

Inversion

Every point \( (x, y, z) \) is mapped to \( (-x, -y, -z) \) through a center of symmetry.

Mathematical Group Criteria

1

Closure

If \( A, B \in G \), then \( A \cdot B \in G \).

2

Associativity

\( (A \cdot B) \cdot C = A \cdot (B \cdot C) \).

3

Identity

There exists \( E \) such that \( A \cdot E = E \cdot A = A \).

4

Inverses

For every \( A \), there is an \( A^{-1} \) where \( A \cdot A^{-1} = E \).

“Symmetry operations of a molecule form a point group because they satisfy all four axioms.”

Improper Rotations \( S_n \)

A composite operation: Rotation by \( 360/n \) followed by reflection through a plane perpendicular to the rotation axis.

\( S_n = \sigma_h \cdot C_n \)

Note: The object may not have \( C_n \) or \( \sigma_h \) individually, but can still have \( S_n \)!

Reflected + Rotated

Symmetry and Conservation Law

Symmetry Basics and Groups

Molecular Point Groups

Matrix Transformations

Noethers Theorem

Symmetry Selection Rules

Symmetry Ops Slides

Polyhedron Analysis Worksheet

Symmetry Fundamentals Teacher Guide

Point Group Slides

Schoenflies Classification Guide

Molecular Sorting Activity

Matrix Representation Slides

Transformation Math Problem Set

Transformation Math Answer Key

Noether Theorem Slides

Lagrangian Symmetry Handout

Selection Rules Slides

Character Table Reference

Spectroscopic Predictor Assessment

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