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 \chi, for rotation of a molecular
wavefunction for angular momentum by an angle
α is given by : \chi^J (\alpha) = \frac{\sin (J+{1\over 2})\alpha} {\sin {1\over 2} \alpha } where 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, \chi^J , for a rotation through an angle of 2\pi+\alpha about an axis through that atom is equal to minus the character for a rotation through an angle of \alpha : \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 2\pi, is classified as being distinct from the identity operation, is used. A character table for the double group 4 is as follows. The new symmetry operations are shown in the second row of the table. : : -->The symmetry operations such as
C4 and
C4
R belong to the same
class but the column header is shown, for convenience, in two rows, rather than
C4,
C4
R in a single row. Character tables for the double groups , , , , , , , , , , , , and are given in Salthouse and Ware. == Applications ==