Group Theory with Applications to Electronic Structure

George Tritsaris, Marco Vanin, Federico Calle Vallejo, Ask Hjorth Larsen

Electronic Structure Journal Club E08

October 22, 2008

Plan

Topics

AHL Basics of group theory and representations MV Symmetry in quantum mechanics GT Point groups FCV Lattices and space groups etc Miscellaneous applications

The big spoiler page

Why this is interesting

• Hamiltonian is invariant under symmetry transformations. Symmetries lead to degeneracies, selection rules

• The symmetry operations of a system form a group, for which reason we shall indulge ourselves in group theory.

• Systems can be characterized in terms of point groups and space groups

Groups

Definition

A group is set G with the following properties:

• There exists a binary operation ∗ defined for all A, B in G, such that A ∗ B ∈ G

• The operation ∗ is associative, i.e. for all A, B, C in G,

A ∗ (B ∗ C) = (A ∗ B) ∗ C

• There exists an identity element I ∈ G such that for all A ∈ G,

A ∗ I = I ∗ A = A

• Each element A ∈ G has an inverse A^{−1} ∈ G such that

A ∗ A^{−1} = A^{−1} ∗ A = I

Examples of groups

• The set of regular n × n matrices with matrix product, identity matrix as identity element and matrix inversion

• The set of symmetry operations on a physical system

matrix Reducibility