The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry. Because the fermions change phase with 360 degree rotation, enhanced symmetry groups that describe band degeneracy and topological properties of magnonic systems are needed, which depend not only on geometric rotation, but on the corresponding fermionic phase factor in representations (for the related mathematical concept, see the formal definition). They were introduced for studying complexes of ions like Ti3+, that have a single unpaired electron in the metal ion's valence electron shell, and complexes of ions like Cu2+, which have a single "vacancy" in the valence shell.[1][2]
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.