For mathematical details, see Finite subgroups of SU(2).

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 study omplexes 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.^[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.

Background

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 occurs, for example, with the elements copper and silver in the +2 oxidation state, where there is a single vacancy in a d-electron shell, with titanium(III) which has a single electron in the 3d shell and with cerium(III) which has a single electron in the 4f shell.

In group theory, the character ${\displaystyle \chi }$, for rotation of a molecular wavefunction for angular momentum by an angle α is given by

${\displaystyle \chi ^{J}(\alpha )={\frac {\sin(J+{1 \over 2})\alpha }{\sin {1 \over 2}\alpha }}}$

where ${\displaystyle J=L+S}$; angular momentum is the vector sum of orbital and spin angular momentum. This formula applies with most paramagnetic chemical compounds of transition metals and lanthanides. However, in a complex containing an atom with a single electron in the valence shell, the character, ${\displaystyle \chi ^{J}}$, for a rotation through an angle of ${\displaystyle 2\pi +\alpha }$ about an axis through that atom is equal to minus the character for a rotation through an angle of ${\displaystyle \alpha }$^[3]

${\displaystyle \chi ^{J}(2\pi +\alpha )=-\chi ^{J}(\alpha )}$

The change of sign cannot be true for an identity operation in any point group. Therefore, a double group, in which rotation by ${\displaystyle 2\pi }$, is classified as being distinct from the identity operation, is used. A character table for the double group D'₄ is as follows. The new symmetry operations are shown in the second row of the table.