Irreducible bases and correlations of spin states for double point groups

TL;DR

This paper develops a method to construct symmetrical adapted bases for spin states using irreducible bases of double groups, with explicit calculations for the tetrahedral double group, enhancing understanding of spin correlations in these symmetries.

Contribution

It introduces an analytic formula for combining spin states into symmetrical bases for double groups, applicable to all such groups and demonstrated explicitly for the tetrahedral double group.

Findings

Analytic formula for symmetrical basis construction

Explicit spin state correlations for T' group

Method applicable to all double point groups

Abstract

In terms of the irreducible bases of the group space of the octahedral double group {\bf O'}, an analytic formula is obtained to combine the spin states $|j,\mu \rangle$ into the symmetrical adapted bases, belonging to a given row of a given irreducible representation of {\bf O'}. This method is effective for all double point groups. However, for the subgroups of {\bf O'}, there is another way to obtain those combinations. As an example, the correlations of spin states for the tetrahedral double group {\bf T'} are calculated explicitly.