Double group

The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry.

Because the fermions' phase changes 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).

[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.

In magnetochemistry, the need for a double group arises in a very particular circumstance, namely, in the treatment of the paramagnetism of complexes of a metal ion in whose electronic structure there is a single electron (or its equivalent, a single vacancy) in a metal ion's d- or f-shell.

This formula applies with most paramagnetic chemical compounds of transition metals and lanthanides.

[4] The need for a double group occurs, for example, in the treatment of magnetic properties of 6-coordinate complexes of copper(II).

Titanium(III) has a single electron in the 3d shell; the magnetic moments of its complexes have been found to lie in the range 1.63–1.81 B.M.

The magnetic properties of octahedral complexes of this ion are treated using the double group O′.

When a cerium(III) ion is encapsulated in a C60 cage, the formula of the endohedral fullerene is written as {Ce3+@C603−}.

[7] The magnetic properties of the compound are treated using the icosahedral double group I2h.