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Double groups are used in the treatment of the magnetochemistry of complexes of metal ions that have a single unpaired electron in the d-shell or f-shell. Instances when a double group is commonly used include 6-coordinate complexes of copper(II), titanium(III) and cerium(III). In these double groups rotation by 360° is treated as a symmetry operation, R, separate from the identity operation, E; the double group is formed by combining the symmetry operations the group {E,R} with the symmetry operations of a point group.

Background
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 and silver 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.

With group theory, the character $$\chi$$, for rotation, by an angle α, of a wavefunction for half-integer angular momentum is given by


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

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

In atoms with a single, unpaired, electron the character for spin angular momentum corresponding to a rotation through an angle of $$ 2\pi+\alpha$$ is equal to minus the character corresponding to a rotation through an angle of $$\alpha$$.
 * $$ \chi(2\pi+\alpha) = -\chi (\alpha)$$

The change of sign cannot be true for an identity operation ($$ \alpha = 0$$) in any point group, for which eq. 1 applies. Therefore, a double group, in which rotation by $$ 2\pi$$ 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.
 * {| class="wikitable" style=text-align:center

!D'4 || || ||C4||C43||C2||2C'2||2C''2 ! ||E ||R ||C4R||C43R||C2R||2C'2R||2C''2R !A'1 !A'2 !B'1 !B'2 !E'1 !E'2 !E'3 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. The theory for double groups can be found in many books on applications of group theory. .
 * + Character table: double group D'4
 * 1||1||1||1||1||1||1|1
 * 1||1||1||1||1||1|-1||-1
 * 1||1||-1||-1||1||1||-1
 * 1||1||-1||-1||1||-1||1
 * 2||-2||0||0||-2||0||0
 * 2||-2||√2||-√2||0||0||0
 * 2 ||-2 ||-√2||√2 ||0 ||0 ||0
 * }

Character tables for the double groups T', O', Td', D3h', C6v', D6', D2d', C4v', D4', C3v', D3', C2v', D2' and R(3)' are given in Salthouse and Ware.

Applications
The need for a double group occurs, for example, in the treatment of magnetic properties of 6-coordinate complexes of copper(II). The electronic configuration of the central Cu2+ ion can be written as [Ar]3d9. It can be said that there is a single vacancy, or hole, in the copper 3d-electron shell, which can contain up to 10 electrons. The ion [Cu(H2O)6]2+ is a typical example of a compound with this characteristic.
 * (1) Six-coordinate complexes of the Cu(II) ion, with the generic formula [CuL6]n+, are subject to the Jahn-Teller effect so that the symmetry is reduced from octahedral (point group Oh) to tetragonal (point group D4h). Since d orbitals are centrosymmetric the related atomic term symbols can be classified in the subgroup D4.
 * (2) To a first approximation spin-orbit coupling can be ignored and the magnetic moment is then predicted to be 1.73 Bohr magnetons, the so-called spin-only value. 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 in a point group. Consequently, a group must be used in which rotations by α + 2π are classed as symmetry operations distinct from rotations by an angle α. This group is known as the double group, D4'.

With species such as the square-planar complex of the silver(II) ion [AgF4]2- the relevant double group is also D4'; deviations from the spin-only value are greater as the magnitude of spin-orbit coupling is greater for silver(II) than for copper(II).

A double group is also used for some compounds of titanium in the +3 oxidation state. Compounds of titanium(III) have a single electron in the 3d shell. The magnetic moments of octahedral complexes with the generic formula [TiL6]n+ have been found to lie in the range 1.63 - 1.81 B.M. at room temperature. The double group O' is used to classify their electronic states.

The cerium(III) ion, Ce3+, has a single electron in the 4f shell. 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-}. The magnetic properties of this ion are treated using the icosahedral double group I'.

Free radicals
Double groups may be used in connection with free radicals. This has been illustrated for the species CH3F+ and CH3BF2+ which both contain a single unpaired electron.

History
Bethe developed the theory of double groups, and applied it in magnetochemistry.

How double groups may be used in the study of quasicrystals has been outlined.

In mathematics, the term "double group" has been used for any group that is the "product" of one group with a group that contains two elements, one of which is the identity operation. The term "double" signifies that the number of symmetry operations is double the number of operations in the first originating group.