Central symmetry inversion group

The particular cases Cj, C = Cj = C v and C, = S2 correspond to the trivial group, the plane symmetry group, and the central symmetry inversion group, respectively. 

The point groups Td,Oh, and / are the respective symmetry group of Tetrahedron, Cube, and Icosahedron the point groups T, O, and I are their respective normal subgroup of rotations. The point group 7 is generated by T and the central symmetry inversion of the centre of the Isobarycenter of the Tetrahedron. 

Thus, the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Q, C, h, (m even), Dmh (m even), Dmd (m odd), 7, Oh, and 7. A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list C,-, C2h, Dm, Da, D3d, Du, 7, and /. Clearly, a fullerene with v vertices can be centrally symmetric only if v is divisible by four as p6 must be even. After the minimal case v = 20, the first centrally symmetric fullerenes are at v = 32 (Dm) and v = 36 (Dm). 

Since our i-basis has even inversion symmetry, the matrix elements connected with the perturbation from a given water molecule are independent of whether this molecule is situated on one or on the other side of the central ion. This means that if we want to discuss the perturbation from our six water molecules with octahedrally positioned ligators (point group symmetry Tn), we can as well take into account only three of them, nos. 1, 2, and 3, say, and eventually multiply all perturbation matrix elements by two. One may say that the holohedrized symmetry (9, 21, 22, 23) of the three water molecules around the central ion is T1. ... 

Consider a complex in which the metallic atom is surrounded by four ligands that are placed at the corners of a square (6-23). The symmetry elements of this system are characteristic of the Dau point group. The axes are shown in 6-24. The planes of symmetry are xy (oj,), xz and yz (cfdb), respectively, together with the planes that bisect xz and yz and each contain two M—L bonds (Oya and respectively). The inversion centre is of course at the origin, coincident with the central atom. 

The symmetry of the 3D-slab model is given by one of the 3D space groups G (see Table 11.1) and may depend on the slab thickness, i.e. number of layers in the slab and its termination. As is seen from Fig. 11.1, for MgO crystal (001) surface slabs of an odd number of atomic planes have inversion symmetry (relative to the central atomic plane) but slabs of an even number of atomic planes have no inversion symmetry. For the cubic perovskite ABO3 (001) surface the stoichiometric slabs (AO-BO2-AO-BO2-) consist of an even number of atomic planes and have no inversion symmetry. But the nonstoichiometric AO- or B02-terminated slabs have inversion symmetry relative to the central AO or BO2 planes, respectively. 

In another example, at temperatures >393 K, barium titanate has the perovskite structure, which is simple cubic with all of the symmetry elements of the cubic lattice, so its point group is Oh or m3m. As the temperature is reduced to its Curie temperature, the lattice contracts and the oxygen ions on the faces of the cube squeeze the titanium ion in the center of the cube so that it is displaced in one direction while the oxygen ions are displaced in the opposite direction, destroying the inversion symmetry as well as the mirror symmetry about the central plane and the rotational symmetry about several of... 

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