The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry of complexes 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 has twice the number of symmetry operations compared to the molecular point group.
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 , for rotation of a molecular wavefunction for angular momentum by an angle α is given by
where ; 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, , for a rotation through an angle of about an axis through that atom is equal to minus the character for a rotation through an angle of [3]
The change of sign cannot be true for an identity operation in any point group. Therefore, a double group, in which rotation by , 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.
D'4 | E | C4 | C43 | C2 | 2C'2 | 2C''2 | |
---|---|---|---|---|---|---|---|
R | C4R | C43R | C2R | 2C'2R | 2C''2R | ||
A'1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
A'2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
B'1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 |
B'2 | 1 | 1 | -1 | -1 | 1 | -1 | 1 |
E'1 | 2 | -2 | 0 | 0 | -2 | 0 | 0 |
E'2 | 2 | -2 | √2 | -√2 | 0 | 0 | 0 |
E'3 | 2 | -2 | -√2 | √2 | 0 | 0 | 0 |
The symmetry operations such as C4 and C4R belong to the same class but the column header is shown, for convenience, in two rows, rather than C4, C4R in a single row .
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.[4]
In mathematics, the term "double group" can be applied to any group which is the direct product of two groups. The term "double" may also be taken to mean "double-valued" in this context.
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.
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).[5]
A double group is also used for some compounds of titanium in the +3 oxidation state. 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. at room temperature.[6] 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 of the endohedral fullerene is written as {Ce3+@C603-}. [7] The magnetic properties of the compound are treated using the icosahedral double group I2h. [8]
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.[9]
Earnshaw, Alan (1968). Introduction to Magnetochemistry. Academic Press.
Figgis, Brian N.; Lewis, Jack (1960). "The magnetochemistry of complex compounds". In Lewis, J.; Wilkins, R.G. (eds.). Modern Coordination Chemistry. New York: Interscience. pp. 400–451.
Orchard, Anthony F. (2003). Magnetochemistry. Oxford Chemistry Primers. Oxford University Press. ISBN 0-19-879278-6.
Vulfson, Sergey G.; Arshinova, Rose P. (1998). Molecular Magnetochemistry. Taylor & Francis. ISBN 90-5699-535-9.