Double group

The character tables of the binary tetrahedral, octahedral and icosahedral groups following Frobenius (1899)

In mathematics, G. Frobenius derived and listed in 1899 the character tables of the finite binary subgroups of SU(2), the double cover of the rotation group SO(3). In 1875, F. Klein had already classified the subgroups into the cyclic groups, the binary dihedral groups, the binary tetrahedral group, the binary octahedral group and the binary icosahedral group. Alternative derivations of the character tables were given by I. Schur and H. E. Jordan in 1907; further branching rules and tensor product formulas were also determined. In a landmark 1929 article on splitting of atoms in crystals, the physicist H. Bethe coined the term "double group" (Doppelgruppe), applying the theory to relativistic quantum mechanics and crystallographic point groups, where a natural physical restriction to 32 point groups occurs. Subsequently, the non-crystallographic icosahedral case has also been investigated more extensively, resulting most recently in groundbreaking advances on carbon 60 and fullerenes in the 1980s and 90s, for which R. Curl, H. Kroto and R. Smalley were jointly awarded a Nobel Prize in Chemistry in 1996.

In physical chemistry, 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 such as D₄ or O_h.

Background

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

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

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 a rotation through an angle of $2\pi +\alpha$ is equal to $-\chi ^{J}(\alpha )$. 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 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.

Character table: double group D'₄

D'4			C4	C43	C2	2C'2	2C''2
	E	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

Character table: point group D₄

D4	E	2 C4	C2	2 C2'	2 C2
A1	1	1	1	1	1 +
A2	1	1	1	−1	−1
B1	1	−1	1	1	−1
B2	1	−1	1	−1	1
E	2	0	−2	0	0

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).

Applications

Sub-structure at the center of an octahedral complex

Structure of a square-planar complex ion such as [AgF₄]^2-

An atom or ion (red) held in a C₆₀ fullerene cage

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]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_4h). 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⁺@C₆₀^3-}.

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.^{(Issue 12)}

See also

• McKay graph
• ADE classification
• Molecular symmetry
• Point group
• Space group

Further reading

• Lipson, R.H. "Spin-orbit coupling and double groups". (web site)
• Earnshaw, Alan (1968). Introduction to Magnetochemistry. Academic Press.
• Figgis, B.N.; Lewis, J. (1960). "The Magnetochemistry of Complex Compounds". In Lewis. J. and Wilkins. R.G. (ed.). Modern Coordination Chemistry. New York: Wiley.
• Orchard, A.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.