Three-generator subgroups

Apart from the unique case of the whole D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group being conserved, which amounts to conserving its four generators, we also have 15 different three-generator subgroups (Table 1), which when conserved, may lead to physically different mean-field solutions. Conserved three-generator subgroups are exceptional in that they do not lead to further simplifications of the matrix elements of mean-field Hamiltonians.

This is so, because cases enumerated in Sec. 3.1.1-3.1.8 exhaust different possibilities of using conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators to simplify the structure of operators by suitable choices of the single-particle bases. Indeed, the type ``III'' subgroups of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$, Table 1, which involve operators for three different Cartesian axes, do not induce any new simplifications. The signature or simplex operators for different axes (the $\hat{\cal{X}}$ operators of Sec. 3.1.3) do not commute, and hence cannot give independent quantum numbers of single-particle states. Similarly, T-signature or T-simplex operators for different axes (the $\hat{\cal{Z}}$ operators of Sec. 3.1.2) do not commute either, and hence cannot simultaneously define phases of single-particle states.

One should stress, however, that even if a given conserved symmetry does not allow for any further simplification of the matrix elements of a mean-field Hamiltonian (like each third generator of a three-generator subgroup), its conservation or its non-conservation may induce entirely different solutions of the mean-field problem.