Parity, and signature or simplex

In the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group, the possibility of having at ones disposal two different quantum numbers simultaneously is very limited. Indeed in D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ one has only three pairs of commuting linear operators, namely, ( $\hat{\cal{R}}_{k}$, $\hat{\cal{P}}$) for k=x, y, or z. For each such pair, the corresponding simplex operator $\hat{\cal{S}}_{k}$ is also conserved, but it does not give any additional quantum number. Only one generic two-generator subgroup, $\{\hat{\cal{R}}_{k},\hat{\cal{P}}\}$, see 2-I_B in Table 1, allows, therefore, for two quantum numbers. Similarly, only three generic three-generator subgroups allow for two quantum numbers, namely, (i) $\{\hat{\cal{R}}_{k},\hat{\cal{T}},\hat{\cal{P}}\}$, which allows only for stationary solutions, (ii) $\{\hat{\cal{R}}_{l},\hat{\cal{R}}_{m},\hat{\cal{P}}\}$, which does not allow for non-zero average values of the angular-momentum, and (iii) $\{\hat{\cal{R}}_{l},\hat{\cal{S}}_{m}^T,\hat{\cal{P}}\}$, which is the only two-quantum-number subgroup which allows for rotating mean-field states. Needless to say, this latter case is most often used in cranking calculations to date, see Sec. 3.3.