For mathematical details see Direct product of groups

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 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 that have the electronic configurations 3d¹, 3d⁹, 4f¹ and 4f¹³, rotation by 360° must be treated as a symmetry operation R, separate from the identity operation. This arises from the nature of the wave function for electron spin. The double group is formed by combining the the molecular point group with a group that has the two operations, identity and R; it has double 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 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 $J=L+S$ and angular momentum is the vector sum of spin and orbital momentum. This formula applies with angular momentum in general.

In a chemical compound 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 $\alpha$ ^[3]

\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 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	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

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 character table for the double group, 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 this and other double groups can be found in many publications on applications of group theory.

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.

Applications

Sub-structure at the center of an octahedral complex

Structure of a square-planar complex ion such as [AgF₄]^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 [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 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).^[4]

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 the generic formula [TiL₆]ⁿ⁺ have been found to lie in the range 1.63 - 1.81 B.M. at room temperature.^[5] 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⁺@C₆₀^3-}. ^[6] The magnetic properties of the compound are treated using the icosahedral double group I²_h.^[7]

Complexes of zirconium (III), with 13 electrons in the 4f valence shell may also be treated using a double group.^[5]

References

^ Cotton, F. Albert (1971). Chemical Applications of Group Theory. New York: Wiley. pp. 289–294, 376. ISBN 0 471 17570 6.
^ Tsukerblat, Boris S. (2006). Group Theory in Chemistry and Spectroscopy. Mineola, New York: Dover Publications Inc. pp. 245–253. ISBN 0-486-45035-X.
^ Lipson, R.H. "Spin-orbit coupling and double groups" (PDF).
^ Foëx, D.; Gorter, C. J.; Smits, L.J. (1957). Constantes Sélectionées Diamagnetism et Paramagnetism. Paris: Masson et Cie.
^ ^a ^b Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 971. doi:10.1016/C2009-0-30414-6. ISBN 978-0-08-037941-8.
^ Chai, Yan; Guo, Ting; Jin, Changming; Haufler,, Robert E.; Chibante, L. P. Felipe; Fure, Jan; Wang, Lihong; Alford, J. Michael; Smalley, Richard E. (1991). "Fullerenes with metals inside". The Journal of Physical Chemistry. 95 (20): 7564–7568. doi:10.1021/j100173a002.{{cite journal}}: CS1 maint: extra punctuation (link)
^ Balasubramanian, K. (1996). "Double group of the icosahedral group (I_h) and its application to fullerenes". Chemical Physics Letters. 260: 476–484.

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Background

Applications

References

Further reading