Irreducible-Tensor Theory for the Group O. I.VandWCoefficients

The irreducible-tensor theory is extended to the double group O. A complex basis is used because the representations $E^{\prime }$, $E^{\prime \prime }$, and $U^{\prime }$ are necessarily complex, and a set of phase factors are presented. The presence of repeated representations in the reduction of direct products of representations implies that more than one set of $V$ coefficients may be associated with a given set of representations $\mathrm{abc}$. For the group O, representations are repeated a maximum of two times in any direct product and the corresponding sets of $V$ coefficients are labeled $V_1$ and $V_2$. This requires a modification in the standard definition of $W$ so that the subscripts on the $V$ coefficients are included in the definition. This in turn calls for a rederivation of the useful matrix elements of double-tensor operators and several of the more important formulas are developed with particular emphasis on matrix elements in a spin-orbit basis. Complete tables of all $V$ and $W$ coefficients for O are also given.