Isomorphism and matrix representation of point groups

In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

Three-dimensional hydrogen-bonded frameworks in organic crystals: a topological study

1551 homomolecular single hydrogen-bonded frameworks in organic crystals have been classified into 148 topological types of three-periodic nets. Different representations of hydrogen-bonded frameworks as nets of molecular centroids, edge or ring nets are discussed. To study the influence of hydrogen bonds on the topology of molecular packings, 42 270 molecular crystals without hydrogen bonds have been considered. The topologies of molecular packings are found to be independent of hydrogen bonding. Analysis of 231 homomolecular frameworks composed of crystallographically different molecules shows that molecules not related by symmetry tend to form the same hydrogen-bond pattern. The relations between net topological types, space-group symmetry of crystals, site symmetry and point-group symmetry of molecules are discussed. As a result, a set of rules for the crystal design of molecular frameworks is proposed.

New computer program to calculate the symmetry of molecules

In this paper we, present some MATLAB and GAP programs and use them to find the automorphism group of the Euclidean graph of the C80 fullerence with connectivity and geometry of Ih symmetry point group. It is proved that this group has order 120 and is isomorphic to Ih≊Z2×A5, where Z2 is, a cyclic group of order 2 and A5 is the alternating group on five symbols.

A database survey of molecular and crystallographic symmetry

The point of contact between molecular and crystallographic symmetries is that of the Wyckoff position, the position at which a molecule resides in a crystal structure. These Wyckoff positions may have the same symmetry as the molecules, some symmetry in common with the molecules or no symmetry at all. Using CSDSymmetry [Yao et al. (2002). Acta Cryst. B58, 640–646], a relational database containing information pertaining to the symmetry of molecules and the crystal structures that play host to them, the distribution of molecules over Wyckoff positions and the occupancy of Wyckoff positions in crystal structures is presented. Analysis of these data has led to the characterization of some relationships between molecular and crystallographic symmetry.