The translational and inversion pseudosymmetry of the atomic crystal structures of organic and organometallic compounds
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TALLOGRAPHIC SYMMETRY Dedicated to the 100th birthday of G.B. Bokiі
The Translational and Inversion Pseudosymmetry of the Atomic Crystal Structures of Organic and Organometallic Compounds N. V. Somov and E. V. Chuprunov Nizhni Novgorod State University, pr. Gagarina 23, Nizhni Novgorod, 603950 Russia email: [email protected] Received December 18, 2008
Abstract—The translational and inversion pseudosymmetry of 211162 atomic crystal structures from the Cambridge Structural Database have been investigated. PACS numbers: 61.50.Ah DOI: 10.1134/S1063774509050022
The main geometric feature characteristic of all crystals is symmetry, i.e., the invariance of atomic structure with respect to the transformations of one of the 230 symmetry space groups. However, some crys tals are also characterized by the approximate symme try of their atomic structure; in this case, a sufficiently large atomic fragment is invariant with respect to a certain supergroup of the symmetry space group of the crystal. Such crystals are generally referred to as pseu dosymmetric. Pseudosymmetric features of atomic crystal structures are responsible for the special types of diffraction patterns and features of pyroelectric (and other properties) of crystals [1–3]. It is well known that some atomic structures of lowsymmetry crystalline phases that are characterized by second order phase transitions have pseudosymmetry proper ties [4, 5]. The additional symmetry of such an atomic frag ment can be noncrystallographic; in this case, super symmetry or hypersymmetry are generally mentioned. Different types of hypersymmetry of molecular crys tals were investigated in detail by Zorkiі and colleagues [6, 7]. They described hypersymmetry groups (non Fedorov space groups), which, along with conven tional symmetric operations, include hypersymmetric ones. Let us consider the case where a large part of a crys tal structure is invariant with respect to the lattice compatible symmetry operations. Such a pseudosym metry will be referred to as Fedorov to indicate the specifically crystallographic (Fedorov) character of the increase in the symmetry of the crystal structure fragments [1]. It is of interest to perform a systematic quantitative analysis of the pseudosymmetric features of atomic crystal structures with Fedorov pseudosymmetry for which reliable structural information exists. In this
paper we report the results of an analysis of the atomic structures included in one of the versions of Cam bridge Structural Database [8]. The atomic structure of a pseudosymmetric crystal can be described by the symmetry space group G (with respect to which the entire atomic structure is invari ant (the total function of electron density and nucleus of atoms)) and its supergroup T (to which the struc tural fragment is invariant). In the case of Fedorov pseudosymmetry, the group T is one of 230 space groups. The group T can be represented as a combination of adjacent classes with respect to its subgroup G. Let us expand group T in lefthanded adjacent classes
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