Point group

Summary

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules. Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1). The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems

Official source

https://en.wikipedia.org/wiki/Point_group

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New refinement of the crystal structure of Zn(NH3)(2)Cl-2 at 100 K

Trpimir Ivsic, Arnaud Magrez

The crystal structure of [ZnCl2(NH3)(2)], diamminedichloridozinc, was reinvestigated at low temperature, revealing the positions of the hydrogen atoms and thus a deeper insight into the hydrogen-bonding scheme in the crystal packing. In comparison with previous crystal structure determinations [MacGillavry & Bijvoet (1936). Z. Kristallogr. 94, 249-255; Yamaguchi & Lindqvist (1981). Acta Chem. Scand. 35, 727-728], an improved precision of the structural parameters was achieved. In the crystal, tetrahedral [Zn(NH3)(2)Cl-2] units (point-group symmetry mm2) are linked through N-H center dot center dot center dot Cl hydrogen bonds into a three-dimensional network.

2019

The crystal structure of [ZnCl2(NH3)2], diamminedichloridozinc, was re-investigated at low temperature, revealing the positions of the hydrogen atoms and thus a deeper insight into the hydrogen-bonding scheme in the crystal packing. In comparison with previous crystal structure determinations [MacGillavry & Bijvoet (1936). Z. Kristallogr. 94, 249–255; Yamaguchi & Lindqvist (1981). Acta Chem. Scand. 35, 727–728], an improved precision of the structural parameters was achieved. In the crystal, tetrahedral [Zn(NH3)2Cl2] units (point-group symmetry mm2) are linked through N—H...Cl hydrogen bonds into a three-dimensional network.

2019

Quantum spin liquid phases in the bilinear-biquadratic two-SU(4)-fermion Hamiltonian on the square lattice

Olivier Clément Romain Gauthé

We consider the phase diagram of the most general SU(4)-symmetric two-site Hamiltonian for a system of two fermions per site (i.e., self-conjugate 6 representation) on the square lattice. It is known that this model hosts magnetic phases breaking SU(4) symmetry and quantum disordered dimerlike phases breaking lattice translation symmetry. Motivated by a previous work [O. Gauthe, S. Capponi, and D. Poilblanc, Phys. Rev. B 99, 241112(R) (2019)], we investigate the possibility of the existence of SU(4) quantum spin liquid phases in this model, using SU(4)-symmetric projected entangled pair states (PEPS) of small bond dimensions, which can be classified according to point group and charge (C) symmetries. Among several (disconnected) families of SU(4)-symmetric PEPS, breaking or not C-symmetry, we identify critical or topological spin liquids which may be stable in some regions of the phase diagram. These results are confronted to exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations.

AMER PHYSICAL SOC2020