Double group
The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry of complexes of ions like Ti3+, that have a single unpaired electron in the metal ion's valence electron shell and to complexes of ions like Cu2+ which have a single "vacancy" in the valence shell. In the specific instances of complexes of metal ions that have the electronic configurations 3d1, 3d9, 4f1 and 4f13, rotation by 360° must be treated as a symmetry operation R, in a separate class from the identity operation E. This arises from the nature of the wave function for electron spin. A double group is formed by combining a molecular point group with the group {E, R} that has two symmetry operations, identity and rotation by 360°. The double group has twice the number of symmetry operations compared to the molecular point group. (Wikipedia).
