Double Group

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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.

In the specific instances of complexes of metal ions that have the electronic configurations 3''d''¹, 3''d''⁹, 4''f''¹ and 4''f''¹³, 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 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 3''d'' shell and with cerium(III) which has a single electron in the 4''f'' shell.

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

:$\chi^J (\alpha) = \frac$

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 ''D₄ is as follows. The new symmetry operations are shown in the second row of the table.

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The symmetry operations such as ''C''₄ and ''C''₄''R'' belong to the same ''class'' but the column header is shown, for convenience, in two rows, rather than ''C''₄, ''C''₄''R'' in a single row .

Character tables for the double groups T', O', T_d', D_3h', C_6v', D₆', D_2d', C_4v', D₄', C_3v', D₃', C_2v', D₂' and R(3)' are given in Salthouse and Ware.

Applications

170px, left, Sub-structure at the center of an octahedral complex
170px, Structure of a square-planar complex ion such as gF₄sup>2-

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 r''d''⁹. It can be said that there is a single vacancy, or hole, in the copper 3''d''-electron shell, which can contain up to 10 electrons. The ion u(H₂O)₆sup>2+ is a typical example of a compound with this characteristic.
:(1) Six-coordinate complexes of the Cu(II) ion, with the generic formula uL₆sup>2+, are subject to the Jahn-Teller effect so that the symmetry is reduced from octahedral (point group ''O_h)'' to tetragonal (point group ''D''_4''h''). Since ''d'' orbitals are centrosymmetric the related atomic term symbols can be classified in the subgroup ''D''₄ .
:(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, ''D''₄'.

With species such as the square-planar complex of the silver(II) ion gF₄sup>2- the relevant double group is also ''D''₄'; 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. Titanium(III) has a single electron in the 3''d'' shell; the magnetic moments of its complexes 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 4''f'' 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 . The magnetic properties of the compound are treated using the icosahedral double group ''I''²_''h''.

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.

See also

* Molecular symmetry
* Point group
* Magnetochemistry

References

Further reading

{{cite book, last1=Vulfson , first1=Sergey G. , last2=Arshinova , first2=Rose P. , title=Molecular Magnetochemistry , year=1998, publisher=Taylor & Francis, isbn=90-5699-535-9

Group theory
Molecular physics
Theoretical chemistry
Materials science

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