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Irreducible bases and correlations of spin states for double point groups

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 $|j,μ\rangle$ 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.