Double group

The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry. Because the fermions' phase changes with 360-degree rotation, enhanced symmetry groups that describe band degeneracy and topological properties of magnonic systems are needed, which depend not only on geometric rotation, but on the corresponding fermionic phase factor in representations (for the related mathematical concept, see the formal definition). They were introduced for studying complexes of ions that have a single unpaired electron in the metal ion's valence electron shell, like Ti 3+ , and complexes of ions that have a single "vacancy" in the valence shell, like Cu 2+ . In the specific instances of complexes of metal ions that have the electronic configurations 3d 1 , 3d 9 , 4f 1 and 4f 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.

The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry. Because the fermions' phase changes with 360-degree rotation, enhanced symmetry groups that describe band degeneracy and topological properties of magnonic systems are needed, which depend not only on geometric rotation, but on the corresponding fermionic phase factor in representations (for the related mathematical concept, see the formal definition). They were introduced for studying complexes of ions that have a single unpaired electron in the metal ion's valence electron shell, like Ti 3+ , and complexes of ions that have a single "vacancy" in the valence shell, like Cu 2+ . In the specific instances of complexes of metal ions that have the electronic configurations 3d 1 , 3d 9 , 4f 1 and 4f 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. \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 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 , , , , , , , , , , , , and are given in Salthouse and Ware.

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