In magnetochemistry, the need for a double group arises in a very particular circumstance, namely, in the treatment of the magnetic properties of complexes of a metal ion in whose electronic structure there is a single unpaired electron (or its equivalent, a single vacancy) in a metal ion's d- or f- shell. This occurs, for example, with the elements copper, silver and gold in the +2 oxidation state, where there is a single vacancy in the 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, by an angle α, of a wavefunction for half-integer angular momentum is given by

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

where angular momentum is the vector sum of spin and orbital momentum, ${\displaystyle J=L+S}$. This formula applies with angular momentum in general.

In atoms with a single unpaired electron the character for a rotation through an angle of ${\displaystyle 2\pi +\alpha }$ is equal to ${\displaystyle -\chi ^{J}(\alpha )}$. 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 operation is labelled R in this example. The character table for the point group D₄ is shown for comparison.

In the table for the double group, the symmetry operations such as C₄ and C₄R belong to the same class but the header is shown, for convenience, in two rows, rather than C₄, C₄R in a single row . Character tables for double groups can be found in many books on applications of group theory; for example, the tables for the double groups D'₄ and O' are given in appendix VII of Cotton (1971).

Core structure of an octahedral complex

An example where the need for a double group arises is in the treatment of magnetic properties of some compounds of copper in the +2 oxidation state.

(1) Six-coordinate complexes of the Cu(II) ion, with the generic formula [CuL₆]²⁺, are subject to Jahn-Teller distortion so that the symmetry is reduced from octahedral (point group O_h) to tetragonal (point group D_4h). Since d orbitals are centrosymmetric the related atomic term symbols can be classified in the D₄ subgroup of D_4h.

(2) To a first approximation spin-orbit coupling can be ignored and the magnetic moment can be predicted, using the spin-only approximation, to be 1.73 Bohr magnetons. However, for a more accurate prediction spin-orbit coupling must be taken into consideration. This means that the relevant quantum number is J, where J = L + S.

(3) When J is half-integer, the character for a rotation by an angle of α + 2π radians is equal to minus the character for rotation by an angle α. This cannot be true for an identity operation, so the point group must be extended to include rotations by α + 2π as separate symmetry operations. This group is known as the double group, D₄'.

The use of the double group is more important in the case of silver(II) as the extent of spin-orbit coupling is greater than in copper(II).

A double group is also used for compounds of titanium in the +3 oxidation state. 6-coordinate compounds of titanium(III) with the generic formula [TiL₆]ⁿ⁺ have a single electron in the 3d shell. The magnetic moments have been found in the range 1.63 - 1.81 B.M. at room temperature.[3] When the compounds are octahedral, the double group O', rather than the point group O_h, is used to classify electronic states.