Matrices, bases and matrix elements for cubic double crystallographic groups

TL;DR

This paper derives matrices for irreducible representations of specific double crystallographic point groups, develops a method for matrix element calculation, and demonstrates the unreliability of an existing alternative approach.

Contribution

It introduces a new method for deriving matrix elements of vector/tensor quantities in double groups and critiques an existing method as unreliable.

Findings

Derived matrices for irreducible representations of key double groups.

Developed a general method for matrix element derivation.

Showed the unreliability of the LS-diagonalization method.

Abstract

Matrices of the irreducible representations of double crystallographic point groups O, Td, Ox{1,I} and Tdx{1,I} are derived. The characteristic polynomials (spinor bases) up to the sixth power are obtained. The method for the derivation of the general form of an arbitrary matrix element of a vector/tensor quantity is developed; as an application, the kp matrix elements are calculated. It is demonstrated that the other known method for obtaining the bases of the irreducible representations of the double groups (LS-diagonalization of a linear combination of spherical harmonics) is unreliable.